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+{"text":"\\section*{Introduction}\nSmall particles, such as laser-cooled atoms or dielectric nanospheres, are nowadays routinely trapped at submicron distances from solids. Structures currently investigated include photonic crystal waveguides~\\cite{thompson_coupling_2013,goban_superradiance_2015,magrini_near-field_2018}, optical nanofibers~\\cite{vetsch_optical_2010,goban_demonstration_2012,beguin_generation_2014,kato_strong_2015,lee_inhomogeneous_2015,corzo_large_2016}, single carbon-nanotubes~\\cite{gierling_cold-atom_2011,schneeweiss_dispersion_2012}, dielectric membranes~\\cite{diehl_optical_2018}, and even macroscopic prisms~\\cite{hammes_cold-atom_2002,bender_probing_2014}. The opportunities in research and application for systems combining atoms and solids are numerous, including the search for novel fundamental forces~\\cite{geraci_improved_2008,geraci_short-range_2010,arkanihamed_hierarchy_1998,dalvit_casimir_2011,klimchitskaya_casimir_2009}, the implementation of quantum metrology and sensing using collective atomic state entanglement~\\cite{beguin_observation_2018}, and integrated quantum memories for photons guided in nanoscale waveguides~\\cite{sayrin_storage_2015,gouraud_demonstration_2015,corzo_waveguide-coupled_2019}. A rich toolbox is already available for the cooling, trapping, positioning, and probing of atoms and nanoparticles. However, not all techniques commonly used in free-space traps for manipulating trapped particles are compatible with the presence of solid structures in their immediate proximity: Control laser beams, for instance, may be reflected or scattered in undesired ways. Moreover, additional effects such as van der Waals forces or coupling of the atoms or particles to thermal excitations in the solid have to be considered.\n\nFull control at the quantum level over the internal as well as external degrees of freedom of individual atoms coupled to a nanophotonic structure was achieved only recently~\\cite{meng_near-ground-state_2018}. A key challenge in this context is the  heating of the atomic motion observed in these systems~\\cite{reitz_coherence_2013,albrecht_fictitious_2016} which can reach rates of several hundred motional quanta per second -- about three orders of magnitude larger than in comparable free-space optical traps. Large cooling rates realized, for example, by ultrastrong spin-motion coupling~\\cite{schneeweiss_cold-atom-based_2018,dareau_observation_2018}, are required to overcome the heating and prepare atoms close to their motional ground state. In essence, the observed storage times of atoms in nanophotonic traps have fallen short of expectations, both for trapped cesium~\\cite{goban_demonstration_2012,beguin_generation_2014,kato_strong_2015,corzo_large_2016,goban_superradiance_2015} and rubidium~\\cite{lee_inhomogeneous_2015} atoms, ever since the first implementation of a nanofiber-based trap for laser-cooled atoms~\\cite{vetsch_optical_2010}. The origin of the strong heating and the corresponding low lifetimes has so far remained elusive. There is a range of conceivable causes, such as Raman scattering of the trapping light fields in the waveguide material \\cite{engelbrecht_nichtlineare_2015}, Brillouin scattering~\\cite{beugnot_brillouin_2014,florez_brillouin_2016}, or Johnson-Nyquist noise~\\cite{henkel_loss_1999}. However, estimates of their effect, provided as supplemental material %\n\\footnote{See supplemental material at the end of this article for estimates of the contribution of other mechanisms to the heating of nanofiber-trapped cold atoms}, %\ndemonstrate that these mechanisms fail to explain heating rates observed in experiments. Additionally, tapered optical fibers, as used for realizing nanofiber-based cold-atom traps, exhibit thermally driven high-$Q$ torsional mechanical resonances which have been considered as a likely candidate for explaining the large heating in these systems~\\cite{wuttke_optically_2013}. In contrast, optical traps that are based on the evanescent field above a prism surface seem to feature small heating rates which are compatible, for instance, with Bose-Einstein condensation of cesium atoms~\\cite{rychtarik_two-dimensional_2004}. Indeed, even at room temperature, one does not expect thermally excited phonon modes of the macroscopic prism to contribute to the heating of the trapped atoms~\\cite{henkel_heating_1999}.\n\nHere, we identify thermally populated flexural phononic modes of the nanoscopic waveguide as the dominant contributor to the large heating rates observed in nanofiber-based cold-atom traps. We give a concise description of the effect of mechanical modes on light guided in optical waveguides and provide a general theory of the resulting atom-phonon interaction in nanophotonic traps. Based on this formalism, we perform a case study for the cesium two-color nanofiber-based trap described in refs.~\\cite{vetsch_optical_2010,reitz_coherence_2013,albrecht_fictitious_2016}. Relying on independently measured system properties, we predict heating rates in excellent quantitative agreement with experimental observations. Surprisingly, the effect of the high-$Q$ torsional mechanical resonances that have previously been observed in this system~\\cite{wuttke_optically_2013} can be neglected, even if they coincide with the trap frequencies. We then use our model to numerically and analytically infer the scaling of the heating rates with system parameters such as the mechanical properties of the fiber, its temperature, or the trap frequencies. This systematic analysis allows us to outline strategies for minimizing the heating, thereby suggesting a solution to a long-standing problem of nanofiber-based cold-atom systems. While we formulate our theory in terms of atoms near nanofibers, it is indeed applicable to any kind of polarizable object trapped by conservative forces due to the light field surrounding a photonic structure. Building on the agreement obtained in the case study, our quantitative formalism might therefore be used for the faithful description of other nanophotonic cold-atom systems and, more generally, optomechanical systems with small particles, such as dielectric nanospheres \\cite{magrini_near-field_2018,chang_cavity_2010,romero-isart_toward_2010,li_millikelvin_2011,gieseler_subkelvin_2012,kiesel_cavity_2013,fonseca_nonlinear_2016,jain_direct_2016}, trapped in close vicinity to hot solid bodies.\n\nThis article is structured as follows: In \\cref{sec: framework}, we provide a general quantum theory describing atoms trapped in the optical near field of a vibrating photonic structure. In particular, we derive the general form of the atom-phonon interaction and discuss the resulting heating rates of the atomic motion. \\Cref{sec: case study} is dedicated to a case study of heating rates expected in a nanofiber-based two-color trap for laser-cooled atoms. In \\cref{sec: photon appendix}, we review the concept of photonic eigenmodes and summarize the modes of a nanofiber, while \\cref{sec: atom appendix} recapitulates the resulting forces acting on trapped atoms. In \\cref{sec: phonon appendix}, we review quantized linear elastodynamics and summarize the phononic eigenmodes of a nanofiber. In \\Cref{sec: interaction appendix}, we supply details on how to calculate the atom-phonon coupling constants based on the framework presented in \\cref{sec: photon appendix,sec: atom appendix,sec: phonon appendix}. The parameters of the experimental setup considered in the case study are listed in \\cref{sec: case study appendix}.\n\n\n\\section{Atoms Trapped near Vibrating Photonic Structures}\n\\label{sec: framework}\nMicro- and nanophotonic traps rely on the optical near fields surrounding a photonic structure to spatially confine laser-cooled atoms in high vacuum. The optical fields are detuned from resonances of the atom such that they do not drive transitions between its internal (electronic) states. Confinement is achieved through gradients in the electric field that result in optical forces acting on the atom, analogous to free-space optical dipole traps \\cite{grimm_optical_2000}. In contrast to free-space setups, a dielectric photonic structure is used to pattern laser light in a way that creates local minima suitable for trapping atoms in the optical potential \\cite{chang_colloquium_2018}.\nThe light can either be guided by the structure such that atoms interact with the evanescent fields surrounding it \\cite{mabuchi_atom_1994,dowling_evanescent_1996,vernooy_quantum_1997,le_kien_atom_2004,vetsch_optical_2010,christensen_trapping_2008,goban_demonstration_2012,hung_trapped_2013,goban_atom-light_2014}, or scattered by the structure \\cite{ovchinnikov_atomic_1991,le_kien_microtraps_2009,thompson_coupling_2013,goban_superradiance_2015}; see \\cref{sec: photon appendix}. In either case a fraction of the light is absorbed, which can lead to a bulk temperature of the dielectric of several hundred kelvins due to the weak thermal coupling to its environment \\cite{wuttke_thermalization_2013}. In consequence, mechanical modes of the photonic structure are thermally excited. These mechanical modes (phonons) are in turn coupled to the external (motional) state of trapped atoms through the optical forces and other forces acting between the atoms and the structure.\n\n\\medskip{}\n\nAn individual atom trapped in the optical near field surrounding a mechanically vibrating photonic structure suspended in high vacuum can be modeled by the Hamiltonian\n\\begin{equation}\\label{eqn: total Hamiltonian}\n  \\op{\\Hamilfunc} = \\op{\\Hamilfunc}_\\text{at}  + \\op{\\Hamilfunc}_\\text{phn} + \\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational}~.\n\\end{equation}\nThe first term describes the dynamics of the trapped atom in the absence of phonons. Atoms are trapped at a distance of a few hundred nanometers from the surface of the structure because the near fields decay on a scale given by the optical wavelength. At such distances, corrections $\\op\\pot_\\text{ad}$ to the optical potential $\\op\\pot_\\text{opt}$ due to surface effects like dispersion forces become relevant \\cite{le_kien_atom_2004,buhmann_dispersion_2012}. Optical forces and dispersion forces are additive to first order \\cite{fuchs_nonadditivity_2018}; hence, the total potential experienced by the atom is $\\op\\pot_0 \\equiv \\op\\pot_\\text{opt} + \\op\\pot_\\text{ad}$. While the potential in general couples all atomic degrees of freedom \\cite{dareau_observation_2018,meng_near-ground-state_2018}, we focus on scenarios without coupling of electronic and motional states and assume that the atom does not change its internal state. In this case $\\op\\pot_0 = V_0(\\op\\atpos)$; that is, the center of mass of the atom is subject to a potential $V_0$ which depends on the internal state of the atom (see \\cref{sec: atom appendix}). Approximating the potential as harmonic for an atom close to its trapped motional ground state yields the atom Hamiltonian\n\\begin{equation}\\label{eqn: atom Hamiltonian harmonic approximation}\n  \\op{\\Hamilfunc}_\\text{at} \\equiv \\sum_i \\hbar \\trapfreq_{ i} \\hconj\\hat{a}_i\\hat{a}_i ~,\n\\end{equation}\nwhere $i$ labels the three orthogonal symmetry axes of the potential in harmonic approximation, $\\trapfreq_{ i} $ are the trap frequencies, $\\hbar$ is the reduced Planck constant, and $\\hat{a}_i$ and $\\hconj\\hat{a}_i$ are ladder operators for the harmonic motion of the trapped atom.\n\nThe second term $\\op{\\Hamilfunc}_\\text{phn}$ in \\cref{eqn: total Hamiltonian} describes the free evolution of the phonon field of the photonic structure. Vibrations at frequencies relevant to atom traps can be modeled by linear elasticity theory because the corresponding phonon wavelengths are sufficiently large not to resolve the microscopic structure of the solid. Linear elasticity theory describes the dynamics of elastic deformations of a continuous body around its equilibrium configuration \\cite{achenbach_wave_1973,auld_acoustic_1973-1,gurtin_linear_1984}. The deformations are described by the \\emph{displacement field} $\\vec{\\ufieldcomp}$, a real-valued vector field which indicates magnitude and direction of the displacement of each point of the body from equilibrium at a given time. A quantum formulation of linear elasticity theory can be obtained through canonical quantization based on phononic eigenmodes; see \\cref{sec: phonon appendix}. The eigenmodes can be labeled by a suitable multi-index $\\gamma$ which may contain both discrete and continuous indices. In terms of ladder operators $\\op{b}_\\gamma$ and $\\hconj\\op{b}_\\gamma$ of the phonon field, the resulting phonon Hamiltonian is\n\\begin{equation}\\label{eqn: Hamiltonian phonons}\n  \\op{\\Hamilfunc}_\\text{phn} \\equiv \\sum_\\gamma \\hbar\\omega_\\gamma \\hconj\\op{b}_\\gamma\\op{b}_\\gamma~,\n\\end{equation}\nwhere the sum symbolizes an integral in the case of the continuous index components.\n\nThe last term $\\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational}$ in the Hamiltonian \\cref{eqn: total Hamiltonian} describes the coupling between the atomic motion and the phonon field. In order to obtain explicit expressions for the atom-phonon coupling, it is necessary to know how the potential experienced by the atom is changed by vibrations. Here, we give an overview of how this dependence can be modeled, while further details as well as explicit expressions for the resulting coupling constants in the case of a nanofiber-based atom trap are provided in \\cref{sec: interaction appendix}. The coupling arises both because vibrations displace the photonic structure relative to the atom and because they change the electromagnetic properties of the structure in two ways \\cite{zoubi_optomechanical_2016}: First, vibrations deform the surface of the structure, as determined by the displacement field $\\vec{\\ufieldcomp}$. Second, they locally change the refractive index and introduce birefringence (photoelastic effect), as determined by the \\emph{strain tensor} $\\tens{\\straintenscomp}$. The strain tensor describes deformations of the solid and has components $S^{ij} \\equiv \\pare{\\partial_i u^j + \\partial_j u^i}\/2$, where $\\partial_i$ indicates a spatial derivative. Both effects modify the photonic eigenmodes and hence the optical trapping fields. The optical fields and surface forces adapt to changes caused by vibrations on a timescale that is fast compared to the motion of the trapped atom. We can therefore treat the total potential as a functional $V[\\vec{\\ufieldcomp},\\tens{\\straintenscomp}](\\vec{\\atpossymbol})$ which, in the absence of vibrations, reduces to the potential $V[\\boldsymbol{0},\\boldsymbol{0}](\\vec{\\atpossymbol}) \\equiv V_0(\\vec{\\atpossymbol})$ included in $\\op{\\Hamilfunc}_\\text{at}$.\n\nThermal vibrations only weakly modify the atom trap. In consequence, it is justified to expand the potential to linear order around \\mbox{$\\vec{\\ufieldcomp} = \\boldsymbol{0}$} and \\mbox{$\\tens{\\straintenscomp} = \\boldsymbol{0}$}, and approximate $V[\\vec{\\ufieldcomp},\\tens{\\straintenscomp}] \\simeq V_0 + \\frechetDV_{(\\boldsymbol{0},\\boldsymbol{0})}[\\vec{\\ufieldcomp},\\tens{\\straintenscomp}]$. The first-order term is the functional derivative of $V[\\vec{\\ufieldcomp}',\\tens{\\straintenscomp}']$, evaluated at $(\\vec{\\ufieldcomp}', \\tens{\\straintenscomp}') = (\\boldsymbol{0},\\boldsymbol{0})$ and in direction $(\\vec{\\ufieldcomp}$, $\\tens{\\straintenscomp})$; see %\n\\footnote{%\n\\label{note: functional derivative}\nThe Fr\u00e9chet derivative $D F$ of a functional $F[\\vec{x}]$ evaluated at $\\vec{x} = \\vec{a}$ and in direction $\\vec{n}$ is defined as \\cite{yamamuro_differential_1974,werner_funktionalanalysis_2011} %\n$$ D F_{\\vec{a}}[\\vec{n}] \\equiv \\lim_{h \\to 0} \\cpare{ F[\\vec{a} + h\\vec{n}] - F[\\vec{a}] }\/h~.$$\nThe derivative is linear in $\\vec{n}$, and can be used in a Taylor \\mbox{expansion \\cite{werner_funktionalanalysis_2011}}. In particular, it is suitable for the linear-order approximation $F[\\vec{x}] \\simeq F[\\vec{a}] + D F_{\\vec{a}}[\\vec{x}]$. The partial Fr\u00e9chet derivative $\\delta_{\\vec{x}}G$ of a multivariate functional $G[\\vec{x},\\vec{y}]$ with respect to $\\vec{x}$ evaluated at $(\\vec{x},\\vec{y}) = (\\vec{a},\\vec{b})$ and in direction $\\vec{n}$ is defined as \\cite{yamamuro_differential_1974}\n$$\\delta_\\vec{x} G_{(\\vec{a},\\vec{b})}[\\vec{n}] \\equiv \\lim_{h \\to 0} \\cpare{ G[\\vec{a} + h\\vec{n},\\vec{b}] - G[\\vec{a},\\vec{b}] }\/h~.$$\nPartial derivatives can be used to express the total derivative $D$ of a multivariate functional \\cite{yamamuro_differential_1974}, for instance\n$$D G_{(\\vec{a},\\vec{b})}[\\vec{n},\\vec{m}] = \\delta_\\vec{x} G_{(\\vec{a},\\vec{b})}[\\vec{n}] + \\delta_\\vec{y} G_{(\\vec{a},\\vec{b})}[\\vec{m}]$$ in the case of a bivariate functional.}.\nThis term approximates phonon-induced variations of the potential and acts as the atom-phonon interaction Hamiltonian\n\\begin{equation}\\label{eqn: definition atom-phonon interaction Hamiltonian}\n  \\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational} \\equiv \\frechetDV_{(\\boldsymbol{0},\\boldsymbol{0})}[\\op{\\ufield},\\op{\\straintens}](\\op\\atpos)~.\n\\end{equation}\nTruncating the expansion at linear order corresponds to assuming that the atom interacts only with single phonons at a time. Since the potential depends on both displacement and strain, there are two contributions to the interaction Hamiltonian, a \\emph{displacement coupling} (\\text{dp}) due to the direct dependence of the potential on $\\vec{\\ufieldcomp}$, and a \\emph{strain coupling} (\\text{st}) due to the dependence on $\\tens{\\straintenscomp}$:\n\\begin{equation}\\label{eqn: atom-phonon interaction Hamiltonian contributions}\n  \\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational} = \\delta_{\\vec{\\ufieldcomp}}V_{(\\boldsymbol{0},\\boldsymbol{0})}[\\op{\\ufield}] + \\delta_{\\tens{\\straintenscomp}}V_{(\\boldsymbol{0},\\boldsymbol{0})}[\\op{\\straintens}]~,\n\\end{equation}\nHere, $\\delta$ is the partial functional derivative \\cite{Note1}. The interaction Hamiltonian is linear in $\\op{\\ufield}$ and $\\op{\\straintens}$ because the functional derivative is linear. By expanding displacement and strain in terms of phononic eigenmodes, the Hamiltonian can thus be expressed in terms of a position-dependent, complex-valued \\emph{coupling function} $g_{\\gamma}(\\vec{\\atpossymbol})$ for each phonon mode $\\gamma$,\n\\begin{equation}\\label{eqn: general form atom-phonon interaction Hamiltonian}\n    \\op{\\Hamilfunc}_{\\atomic\\text{-}\\vibrational} = \\sum_\\gamma\\spare{g_{\\gamma}(\\op\\atpos) \\op{b}_\\gamma + \\text{H.c.}}~,\n\\end{equation}\nwhere $g_{\\gamma}(\\vec{\\atpossymbol}) = g^\\text{dp}_{\\gamma}(\\vec{\\atpossymbol}) + g^\\text{st}_{\\gamma}(\\vec{\\atpossymbol})$.\nThe coupling function $g^\\text{dp}_{\\gamma}(\\vec{\\atpossymbol})$ derives from displacement coupling and $g^\\text{st}_{\\gamma}(\\vec{\\atpossymbol})$ from strain coupling.\n\nFurthermore, we approximate the phonon-induced forces acting on a trapped atom as linear in the atom position by  expanding \\cref{eqn: definition atom-phonon interaction Hamiltonian} to first order around the trap minimum $\\vec{\\atpossymbol}_0$. The interaction Hamiltonian then takes the form %\n\\footnote{%\nThe term $\\frechetDV_{(\\boldsymbol{0},\\boldsymbol{0})}[\\op{\\ufield},\\op{\\straintens}](\\vec{\\atpossymbol}_0)$ at order zero in the expansion describes a light-induced change in the mechanical equilibrium configuration of the photonic structure. We may safely neglect this constant shift, because it is small compared to the dimensions of a nanoscale structure and therefore only weakly modifies its photonic and phononic spectrum.%\n}\n\\begin{equation}\\label{eqn: linear force interaction Hamiltonians}\n  \\begin{split}\n    \\op{\\Hamilfunc}_{{\\atomic\\text{-}\\vibrational}} &\\simeq \\sum_{i\\gamma}  \\hbar (\\hat{a}_i + \\hconj\\hat{a}_i) (g_{\\gamma i}\\op{b}_\\gamma + \\cconjg_{\\gamma i} \\hconj\\op{b}_\\gamma)~,\n  \\end{split}\n\\end{equation}\nwhere the coupling constants are\n\\begin{equation}\\label{eqn: definition coupling constants}\n    g_{\\gamma i} \\equiv \\frac{\\Delta \\atiposb}{\\hbar} \\partial_i g_{\\gamma}(\\vec{\\atpossymbol}_0)~.\n\\end{equation}\nThe length $\\Delta \\atiposb \\equiv \\sqrt{\\hbar\/(2 M \\trapfreq_{ i})}$ is the zero-point motion of the atom of mass $M$ in the trap. The coupling constants quantify the interaction of each phonon mode $\\gamma$ with the motion of the atom in direction $i$. Analogous to the coupling function, there are contributions from both displacement and strain coupling, $g_{\\gamma i} = g^\\text{dp}_{\\gamma i} + g^\\text{st}_{\\gamma i}$.\n\nThe variation of the optical potential caused by displacement can in general be modeled by perturbatively calculating the new photonic eigenmodes in the presence of shifted boundaries of the nanostructure \\cite{johnson_perturbation_2002}. The displacement has two effects: First, it shifts the photonic structure, together with the electromagnetic fields surrounding it, relative to the trapped atom. Second, it deforms the surface of the structure, leading to new photonic eigenmodes and thereby also deforming the electromagnetic fields. The first effect scales with the ratio between the displacement of the surface and the size of the atom trap (the extent of the wave function of the atom). The second effect, on the other hand, scales with the ratio between the displacement and the dimensions of the structure. Since the trap is typically at least one order of magnitude smaller than the photonic structure (see \\cref{sec: case study}), we neglect the second effect and assume that both optical and surface potential are displaced as a whole together with the fiber surface \\cite{le_kien_phonon-mediated_2007}. This model is particularly useful for structures such as nanofibers which have a simple geometrical shape and highly symmetric mechanical modes. The resulting displacement coupling functions $g_{\\gamma}^\\text{dp}(\\vec{\\atpossymbol})$ for a nanofiber-based atom trap in particular are given in \\cref{sec: interaction appendix}.\n\nStrain leads to changes in the optical potential through the photoelastic effect, which can be modeled by a strain-dependent permittivity tensor $\\prtrbd\\tens{\\relpermitttenscomp}[\\tens{\\straintenscomp}]$ \\cite{nelson_theory_1971,narasimhamurty_photoelastic_2012,wuttke_optically_2013}. The modified permittivity is then in general neither homogeneous nor isotropic, and results in modified electric fields $\\prtrbd\\vec{\\Efieldcomp}$ surrounding the fiber and thus in a modified optical potential $V_\\text{opt}[\\prtrbd\\vec{\\Efieldcomp}]$. In consequence, the total potential $V[\\vec{\\ufieldcomp},\\tens{\\straintenscomp}]$ depends on strain. We neglect the influence of strain on the surface forces because they arise from the interaction of the atom with charges in a thin slice at the surface of the fiber and are largely independent of changes in the interior of the fiber \\cite{buhmann_dispersion_2012}. The strain coupling function $g_{\\gamma}^\\text{st}(\\vec{\\atpossymbol})$ can then be obtained by perturbatively calculating the new photonic eigenmodes in the presence of a modified permittivity; see \\cref{sec: interaction appendix}.\n\n\n\\medskip{}\n\nHaving obtained the Hamiltonian of the coupled atom-phonon system, we can now describe the resulting evolution of the atomic motion. The cold atom can absorb kinetic energy from the thermally excited phonon field of the photonic structure (\\emph{heating} of the atomic motion). Provided that the atom-phonon coupling is weak compared to the trap frequencies and the coherence time of phonon excitations, the phonon field can be adiabatically eliminated. The effective evolution of the density matrix $\\densopc(t)$ describing the motional state of the atom is then governed by a master equation \\cite{cohen-tannoudji_atom-photon_1998,breuer_theory_2002}; see \\cref{sec: interaction appendix}. Heating of the atom is reflected in the increase of the expected number of motional quanta $n_i(t) \\equiv \\textrm{tr} [\\densopc(t) \\hconj\\hat{a}_i \\hat{a}_i]$ along a spatial direction $i$. The population grows linearly with heating rate $\\Gamma_i^\\text{th}$ for sufficiently short times,\n\\begin{equation}\\label{eqn: short time population evolution}\n n_i(t) \\simeq \\Gamma_i^\\text{th} t~,\n\\end{equation}\nassuming that the atom is in the motional ground state at $t=0$.\n\nThe phononic eigenmodes supported by the photonic structure can feature both discrete and continuous frequency spectra. Discrete spectra are observed for phonon modes with a spacing in frequency that is larger than their damping rates. In contrast, if a set of modes has frequency spacings much smaller than their damping rates (e.g., because the mechanical excitation is efficiently transmitted from the structure to its suspension), the discrete mechanical resonances are no longer discernible, and the spectrum is effectively continuous. Hence, we distinguish the contribution $\\Gamma^{\\text{d}}_{i}$ of discrete mechanical resonances from the contribution $\\Gamma^{\\text{c}}_{i}$ of a continuum of phonon modes:\n\\begin{equation}\\label{eqn: heating rate contributions}\n \\Gamma^{\\text{th}}_{i} = \\Gamma^{\\text{c}}_{i} + \\Gamma^{\\text{d}}_{i}~.\n\\end{equation}\n\nFor continuous phonon modes, Fermi's golden rule can be employed to calculate the heating rate $\\Gamma^{\\text{c}}_{i}$ \\cite{cohen-tannoudji_atom-photon_1998}:\n\\begin{equation}\\label{eqn: Fermis golden rule}\n  \\Gamma^{\\text{c}}_{i} = 2\\pi \\bar{n}_i \\sum_{\\gamma_i} \\rho_{\\gamma_i} |g_{\\gamma_i i}|^2~.\n\\end{equation}\nThe sum runs over the discrete set of continuous phonon modes $\\gamma_i$ that are resonant with the trap, $\\omega_{\\gamma_i} = \\trapfreq_{ i}$. The thermal occupation of the resonant phonon modes is $\\bar{n}_i \\equiv 1\/\\spare{\\exp\\pare{\\hbar \\trapfreq_{ i}\/k_BT}-1}$, where $T$ is the temperature of the photonic structure and $k_B$ is the Boltzmann constant \\cite{gerry_introductory_2005}. The phonon density of states is given by the inverse slope of the phonon dispersion relation (band structure), $\\rho_\\gamma \\equiv | d\\omega_\\gamma \/ dp |^{-1}$, where $p$ is the propagation constant along the fiber; see \\cref{sec: phonon appendix}.\n\nThe discrete resonances have finite lifetimes corresponding to decay rates $\\kappa_\\gamma$ due to internal losses and nonzero coupling to the suspension. Adiabatic elimination of these discrete mechanical modes in general leads to the heating rate $\\Gamma^{\\text{d}}_{i}$ given in \\cref{eqn: torsional heating general} in \\cref{sec: interaction appendix} \\cite{cirac_laser_1992,wilson-rae_cavity-assisted_2008}. There are two limiting cases that are of interest in \\cref{sec: case study}: In the case where the atom-trap frequency is smaller than the lowest-frequency phonon mode $\\gamma_1$, $\\trapfreq_{ i} < \\omega_{\\gamma_1}$, and detuned from resonance, $\\kappa_{\\gamma_1} \\ll |\\trapfreq_{ i} - \\omega_{\\gamma_1}|$, the ground-state  heating rate of the atom is\n\\begin{equation}\\label{eqn: heating rate discrete phonon detuned}\n  \\Gamma_{i}^\\text{d} \\simeq 2 \\bar{n} \\kappa_{\\gamma_1} |g_{\\gamma_1 i}|^2 \\frac{\\trapfreq_{ i}^2 + \\omega_{\\gamma_1}^2}{(\\trapfreq_{ i}^2 - \\omega_{\\gamma_1}^2)^2}~.\n\\end{equation}\nIn the case where the atom trap is resonant with a single phonon mode $\\gamma$, $\\kappa_{\\gamma} \\gg |\\trapfreq_{ i} - \\omega_{\\gamma}|$, the rate is\n\\begin{equation}\\label{eqn: heating rate discrete phonon}\n  \\Gamma_{i}^\\text{d} \\simeq \\frac{4 \\bar{n}|g_{\\gamma i}|^2}{\\kappa_{\\gamma}}~,\n\\end{equation}\nwhere we assume $\\bar{n} \\gg 1$.\n\n\\medskip{}\n\nThe theory of atom-phonon interaction outlined in this section applies to any optical atom trap that relies on a photonic structure to shape light fields. The explicit calculation of atom-phonon coupling constants requires modeling of the dependence of the potential that the atom experiences on the displacement and the strain caused by the mechanical eigenmodes of the structure. Once the mechanical modes and corresponding atom-phonon coupling constants of a particular structure are known, \\cref{eqn: Fermis golden rule,eqn: heating rate discrete phonon detuned,eqn: heating rate discrete phonon}, or more generally \\cref{eqn: torsional heating general}, can be used to predict the phonon-induced heating of the atomic motion. In the next section, we apply this theory to explain heating rates observed in nanofiber-based atom traps.\n\n\n\\section{Case Study of a Nanofiber-based Trap}\n\\label{sec: case study}\n\\begin{table*}\n  \\begin{tabularx}{\\textwidth}{c @{\\qquad} l @{\\quad}  l @{\\qquad} l @{\\quad} l @{\\qquad} l @{\\quad} l X}\n    \\toprule\n    Trap & \\multicolumn{2}{c}{$\\text{T}_{01}$} & \\multicolumn{2}{c}{$\\text{L}_{01}$} & \\multicolumn{2}{c}{$\\text{F}_{11}$} & ~ \\\\\n    \\cmidrule{2-8}\n    & $|g^\\text{dp}_{\\gamma i}|\/2\\pi$ (\\si{\\hertz}) & $|g^\\text{st}_{\\gamma i}|\/2\\pi$ (\\si{\\hertz})\n    & $|g^\\text{dp}_{\\gamma i}|\/2\\pi$ ($\\si{\\hertz}\\sqrt{\\si{\\meter}}$) & $|g^\\text{st}_{\\gamma i}|\/2\\pi$ ($\\si{\\hertz}\\sqrt{\\si{\\meter}}$)\n    & $|g^\\text{dp}_{\\gamma i}|\/2\\pi$ ($\\si{\\hertz}\\sqrt{\\si{\\meter}}$) & $|g^\\text{st}_{\\gamma i}|\/2\\pi$ ($\\si{\\hertz}\\sqrt{\\si{\\meter}}$) \\\\\n    \\midrule\n    $r$ & \\num{0} & \\num{5.47e-08} & \\num{3.08e-09} & \\num{1.56e-08} & \\num{3.93e-4} & \\num{2.18e-08} \\\\\n    $\\varphi$ & \\num{0} & \\num{7.81e-4} & \\num{0} & \\num{7.76e-11} & \\num{2.28e-4} & \\num{2.99e-10}\\\\\n    $z$ & \\num{0} & \\num{2.19e-12} & \\num{0} & \\num{1.05e-4} & \\num{0} & \\num{1.13e-10} \\\\\n    \\bottomrule\n  \\end{tabularx}\n    \\caption{Atom-phonon coupling constants. Listed are the contributions of displacement (\\text{dp}) and strain (\\text{st}) coupling to the coupling constants. The displacement coupling constants $g^\\text{dp}_{\\gamma i}$ are calculated according to \\cref{eqn: 3d trap radiation pressure coupling constants}. The strain coupling constants  $g^\\text{st}_{\\gamma i}$ are obtained from \\cref{eqn: definition coupling constants} with the coupling functions listed in \\cref{tab: strain coupling functions} in \\cref{sec: interaction appendix}. Coupling to modes on the continuous $\\text{L}_{01}$ and $\\text{F}_{11}$ bands is independent of the position of the trap site along the fiber axis. In contrast, the strain coupling constants to the discrete $\\text{T}_{01}$ modes depend on the position since the torsional modes form standing waves; see \\cref{sec: phonon appendix}. Listed here are the maximal coupling constants; for radial motion, the coupling is maximal at the end of the nanofiber ($z = 0,L$), while it is maximal at the center of the nanofiber ($z = L\/2$) for the azimuthal and axial motion.}\n    \\label{tab: coupling constants}\n\\end{table*}\n\n\\begin{table}\n\\newcolumntype{C}[1]{>{\\centering\\arraybackslash}p{#1}}\n\\newcommand{$\\ll$}{$\\ll$}\n  \\begin{center}\n    \\begin{tabular}{c C{2.1cm} C{2.1cm} C{2.1cm} }\n      \\toprule\n      Trap & $\\text{T}_{01}$ \\qquad & $\\text{L}_{01}$ & $\\text{F}_{11}$ \\\\\n       \\midrule\n      $r$ & $\\ll$ & $\\ll$ & \\SI{446}{\\hertz} \\\\\n      $\\varphi$ & $\\ll$ & $\\ll$ & \\SI{340}{\\hertz} \\\\\n      $z$ & $\\ll$ & \\SI{8.36e-2}{\\hertz} & $\\ll$ \\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption{Atom heating rates. Listed are the contributions of the relevant phonon modes $\\text{T}_{01}$, $\\text{L}_{01}$, and $\\text{F}_{11}$ to the heating rate $\\Gamma_i^\\text{th}$ of a trapped atom in direction $i \\in \\{r,\\varphi,z\\}$ calculated according to \\cref{eqn: Fermis golden rule,eqn: heating rate discrete phonon detuned}. Contributions below $10^{-4}\\,\\si{\\hertz}$ are indicated by \\lq{}$\\ll$.\\rq{} The  rates are independent of the position of the trap site along the fiber. The fiber temperature is assumed to be $T = \\SI{805}{\\kelvin}$, and the remaining parameters are specified in \\cref{sec: case study appendix}.}\n    \\label{tab: heating rates}\n  \\end{center}\n\\end{table}\n\nLet us now use the framework sketched in \\cref{sec: framework} to study the phonon-induced heating rates of the atomic motion in a nanofiber-based two-color atom trap. In particular, we consider a cesium atom trapped in the evanescent optical field surrounding a silica nanofiber \\cite{dowling_evanescent_1996,le_kien_atom_2004}. The nanofiber is formed by the waist of an optical fiber which has been heated and pulled \\cite{ward_optical_2014}. There have been several experimental realizations of this nanophotonic atom trap configuration \\cite{vetsch_optical_2010,goban_demonstration_2012,kato_strong_2015,lee_inhomogeneous_2015,corzo_large_2016,ostfeldt_dipole_2017,meng_near-ground-state_2018,albrecht_fictitious_2016}. We calculate atom heating rates for the setup described in \\cite{albrecht_fictitious_2016}, where a measured heating rate of $\\Gamma_\\varphi^\\text{th} = \\SI{340(10)}{\\hertz}$ in the azimuthal direction was reported. In order to explicitly calculate the phonon-induced heating rates, it is necessary to know the mechanical eigenmodes of the nanofiber close to resonance with the trap frequencies and to obtain the atom-phonon coupling constants. The latter calculation requires knowledge of the trap potential as well as the photonic eigenmodes of the nanofiber. \\Cref{sec: photon appendix} summarizes the photonic eigenmodes of a nanofiber, and \\cref{sec: atom appendix} provides details on the resulting trapping potential. \\Cref{sec: phonon appendix} summarizes the phononic eigenmodes. In \\cref{sec: interaction appendix}, we derive the resulting atom-phonon coupling constants for a nanofiber-based trap. The parameters of the particular setup considered in this \\namecref{sec: case study} are listed in \\cref{sec: case study appendix}.\n\n\nTrapping of atoms is achieved by means of two lasers, one red and the other blue detuned with respect to the $D$ lines of cesium. The lasers are guided as photonic $\\text{HE}_{11}$ spatial modes in the nanofiber region; see \\cref{sec: photon appendix}. \\Cref{fig: trap} in \\cref{sec: interaction appendix} shows the resulting trapping potential. The red-detuned laser is coupled into the fiber at both ends, leading to a standing wave that confines the atoms in the axial direction and creates a one-dimensional optical lattice. The laser beams are linearly polarized when coupled into the fiber, which leads to \\emph{quasilinearly} polarized fields with intensity maxima at opposite poles of the fiber cross section in the nanofiber region \\cite{le_kien_state-dependent_2013}. The corresponding electric field profiles are listed in \\cref{sec: case study appendix}. The red- and blue-detuned field have orthogonal polarizations to obtain stronger azimuthal confinement \\cite{vetsch_optical_2010}. There is an offset magnetic field oriented perpendicular to the fiber axis ($z$ axis) along $\\vec{z}_B = \\cos(\\phi) \\unitvec_\\xpos + \\sin(\\phi) \\unitvec_\\ypos$, with $\\phi = \\SI{66}{\\degree}$.  Atoms are initially prepared in the Zeeman substate $F=4$, $M_\\atF = -4$ of the hyperfine structure, where the offset magnetic field provides the quantization axis. The magnetic field causes a slight azimuthal shift of the trap sites. Nonetheless, the symmetry axes of the potential at the trap minimum are to a good approximation aligned with the radial, azimuthal, and axial unit vectors of a cylindrical coordinate system whose $z$ axis coincides with the nanofiber axis. We can therefore use $i\\in \\{r,\\varphi,z\\}$ for the atom trap directions in the atom Hamiltonian \\cref{eqn: atom Hamiltonian harmonic approximation}. The resulting frequencies of the atom trap are $(\\trapfreq_{\\rpos}, \\,\\trapfreq_{\\phipos}, \\,\\trapfreq_{\\zpos}) = 2\\pi \\times(\\num{123}, \\,\\num{71.8}, \\,\\num{193})\\, \\si{\\kilo\\hertz}$.\n\nAn infinitely long nanofiber supports three phonon bands which do not have a cutoff at low frequencies: the torsional $\\text{T}_{01}$ band, longitudinal $\\text{L}_{01}$ band, and flexural $\\text{F}_{11}$ band; see \\cref{sec: phonon appendix}.\n\\Cref{fig: phonon modes} shows the displacement of the nanofiber caused by phonon modes on each of these bands. The torsional band is linear and the longitudinal band asymptotically linear for low frequencies, with speeds of sound $c_\\trans$ and $c_h$ introduced in \\cref{sec: phonon appendix}, respectively. The flexural band has a quadratic asymptote. The dispersion relations describing these bands as functions of the propagation constant $p$ are\n\\begin{align}\\label{eqn: fundamental phonon bands}\n  \\omega_\\text{T} &= c_\\trans |p| & \\omega_\\text{L} &\\simeq c_h |p| & \\omega_\\text{F} &\\simeq \\frac{c_h R}{2} p^2~.\n\\end{align}\nHere, $R$ is the radius of the nanofiber. These three fundamental bands are the only candidates for phonon-induced heating of the atomic motion since all other bands have frequencies much larger than the trap frequencies.\n\nIn experiments, the optical nanofibers used for atom trapping are typically realized as the waist of a tapered optical fiber \\cite{vetsch_optical_2010}. The mechanical eigenmodes of this system -- including the nanofiber, the tapers, and the surrounding macroscopic fiber -- can be calculated either analytically or using finite-element methods \\cite{wuttke_thermal_2013}. Since the fiber is finite in length, the eigenmodes are standing waves and the spectrum consists of discrete mechanical resonances. The system can in general support the same kinds of excitations as an infinite cylinder: torsional, longitudinal, and flexural. For some modes, the tapers act as reflectors and strongly localize them in the nanofiber region. Others are transmitted through the tapers and are delocalized over the entire fiber \\cite{wuttke_thermal_2013}. In practice, all modes are damped. Dissipation occurs, among others, due to clamping losses \\cite{pennetta_tapered_2016}, friction with the background gas \\cite{wuttke_optically_2013}, material losses \\cite{wiedersich_spectral_2000}, and surface losses \\cite{penn_frequency_2006}. Depending on the magnitude of the damping $\\kappa$ of each mode compared to the free spectral range (FSR), the actual spectrum ranges from  discrete ($\\text{FSR} \\gg \\kappa$) to continuous ($\\text{FSR} \\ll \\kappa$). In the case of a discrete spectrum, standing waves of finite lifetime $1\/\\kappa$ are a useful description of the mechanical dynamics of the fiber. In the limit of a continuous spectrum, the idealized eigenmodes of the system are no longer faithful representations, since the phonons interact too strongly with other degrees of freedom and are dissipated before they can form standing waves. Instead, it is more useful to represent the phonons as propagating modes of an infinite structure which interact with the atom once and then never return (analogous to an atom interacting with fiber-guided or free-space photons). Some of the damping mechanisms can be modeled theoretically \\cite{penn_frequency_2006,wiedersich_spectral_2000}. However, more reliable results are obtained by measuring damping rates for the particular fiber in use. We perform measurements of the mechanical modes of the particular nanofiber setup considered here \\cite{albrecht_fictitious_2016}, similar to \\cite{wuttke_thermal_2013,fenton_spin-optomechanical_2018}. While torsional resonances are clearly visible, there is no indication of resonantly enhanced longitudinal or flexural nanofiber modes. The mode of lowest frequency is at $\\omega_\\text{T} = 2\\pi \\times \\SI{258}{\\kilo\\hertz}$ with a wavelength of $\\SI{14.6}{\\milli\\meter}$ and a decay rate of $\\kappa = 2\\pi \\times \\SI{48(1)}{\\hertz}$. The torsional modes can be modeled faithfully by imposing hard boundary conditions on an elastic cylinder; see \\cite{wuttke_thermal_2013} and \\cref{sec: phonon appendix}. The resulting spectrum is a discrete subset of the $\\text{T}_{01}$ band of an infinite cylinder. In keeping with the absence of discrete resonances corresponding to longitudinal and flexural modes, we model these modes as the propagating modes of an infinite cylinder, with a continuous dispersion relation given by the longitudinal and flexural bands \\cref{eqn: fundamental phonon bands}. The form of the longitudinal and flexural mechanical bands and the corresponding eigenmodes are then determined by the elastic mechanical properties of silica and the fiber radius alone. The wavelengths of the modes resonant with the azimuthal trap frequency, for instance, are $\\SI{80.0}{\\milli\\meter}$ for the $\\text{L}_{01}$ mode and $\\SI{0.251}{\\milli\\meter}$ for the $\\text{F}_{11}$ mode.\n\nThe theory derived in \\cref{sec: framework} allows us to calculate atom heating rates based on these physical parameters. The only parameter not provided by ref.~\\cite{albrecht_fictitious_2016} is the fiber temperature $T$. We choose the temperature such that the azimuthal heating rate $\\Gamma_\\varphi^\\text{th}$ observed in \\cite{albrecht_fictitious_2016} is reproduced. Agreement with the measurement in ref.~\\cite{albrecht_fictitious_2016} is achieved for $T = \\SI{805}{\\kelvin}$, which agrees well with the temperature of $T = \\SI{850\\pm150}{\\kelvin}$ measured independently in \\cite{wuttke_thermalization_2013} for a similar nanofiber at the given transmitted laser power. Heating in the azimuthal direction is dominantly caused by resonant flexural $\\text{F}_{11}$ modes. To our knowledge, this is the first time that a theoretical prediction of the atom heating rate based on measured parameters and in quantitative agreement with measured heating rates has been obtained. We are then able to calculate the phonon-induced heating rates of the atomic motion in the radial, azimuthal, and axial direction, accounting for both displacement and strain coupling. The predicted atom-phonon coupling constants are listed in \\cref{tab: coupling constants} and the resulting heating rates in \\cref{tab: heating rates}.\n\nThe predicted heating rate for the radial degree of freedom is of a magnitude similar to the rate for the azimuthal degree of freedom. The calculated radial heating rate is $\\Gamma_r^\\text{th} = \\SI{446}{\\hertz}$, which agrees with the heating rate assumed in \\cite{reitz_coherence_2013} to explain measured $T_2^\\prime$ decoherence rates for nanofiber-trapped atoms. Heating along the radial axis, like heating in the azimuthal direction, is dominated by coupling to the resonant flexural $\\text{F}_{11}$ modes. The coupling constants in \\cref{tab: coupling constants} reveal that the coupling is due to displacement of the fiber surface, while coupling due to strain is lower by several orders of magnitude. \\latin{A priori}, both longitudinal $\\text{L}_{01}$ and flexural $\\text{F}_{11}$ modes couple to the radial motion by displacement. However, the flexural modes lead to much higher heating rates for two reasons: First, flexural modes displace the fiber surface by a factor of $|w^r_\\text{F}\/w^r_\\text{L}| \\simeq \\sqrt{E\/(2 \\rho)}\/(\\trapfreq_{\\rpos} \\PoissonnuR) \\simeq 10^5$ more than the longitudinal modes which leads to larger displacement coupling constants. Here, $w^r_\\text{F}$ and $w^r_\\text{L}$ are the radial components of the displacement eigenmode for the flexural and longitudinal modes, respectively. The quantity $E$ is Young's modulus and $\\nu$ is the Poisson ratio; together, they describe the elastic properties of the nanofiber. The quantity $\\rho$ is the mass density of the nanofiber and $\\trapfreq_{\\rpos}$ the radial trap frequency. The second reason is that the density of states of the flexural modes is larger than the one of longitudinal modes by a factor of $\\rho_{\\text{F} r}\/\\rho_\\text{L} \\simeq \\sqrt{c_h\/(2 \\trapfreq_{\\rpos} R)} \\simeq 100$, and the heating rates are enhanced accordingly; see \\cref{eqn: Fermis golden rule}.\n\nHeating in the axial direction is predicted to be predominantly due to strain coupling to the resonant longitudinal $\\text{L}_{01}$ mode, with a rate much smaller than the heating rates in the radial and azimuthal direction. To the best of our knowledge, the heating rate in the axial direction has not been measured so far.\n\nOne might expect heating by near-resonant torsional modes to be dominant because they are tightly confined to the nanofiber region, leading to Purcell enhancement of the coupling strength \\cite{gerry_introductory_2005}. The strain induced by torsional modes causes a tilt of the quasilinear polarization of the light fields, see \\cref{fig: discrete T modes blue and red scalar coupling r phi} in \\cref{sec: interaction appendix}, which leads to coupling to the azimuthal motion of the atom in particular. In the present case, the contribution of torsional modes to the heating is negligible due to the large detuning between the torsional mode and trap frequencies compared to the phonon decay rate. However, we can use \\cref{eqn: heating rate discrete phonon} with the coupling constants given in \\cref{tab: coupling constants}  to obtain an estimate of the heating rates expected in the case when the torsional modes are resonant (e.g., in the case the nanofiber is longer). In this worst-case scenario, the predicted contribution to the heating rate in the azimuthal direction is $\\Gamma^\\text{d}_\\varphi = \\SI{17.8}{\\hertz}$, while heating in the other trap directions is still below $10^{-4}\\,\\si{\\hertz}$, despite the Purcell enhancement. For the hypothetical case in which the torsional modes are not reflected at the ends of the nanofiber, our model predicts even lower heating rates. Hence, torsional modes are not a relevant source of heating in \\cite{albrecht_fictitious_2016}, even if they are resonant with the trap frequencies.\n\nIn summary, the atom heating in the radial and azimuthal direction observed in experiments is well explained by the displacement coupling to the continuous $\\text{F}_{11}$ band alone. In this case, \\cref{eqn: Fermis golden rule} simplifies to the single equation\n\\begin{align}\\label{eqn: final heating formula}\n  \\Gamma^\\text{th}_i &\\simeq \\frac{1}{2\\sqrt{2}\\pi}\\frac{k_B}{\\hbar} T M\\sqrt{\\frac{\\trapfreq_{ i}}{R^5 \\sqrt{E \\rho^3}}} & i \\in \\{ r,\\varphi\\}~,\n\\end{align}\nwhere we use that $\\hbar \\trapfreq_{ i} \\ll k_B T$, such that the thermal occupation of the phonon modes is $\\bar{n}_i \\simeq k_B T\/\\hbar \\trapfreq_{ i}$. This simple formula agrees exceedingly well with calculations considering all phonon modes and both displacement and strain coupling.\n\\begin{figure*}\n  \\centering\n  \\setlength{\\widthFigA}{180.9393pt}\n  \\setlength{\\marginLeftFigA}{25.9393pt}\n  \\setlength{\\marginRightFigA}{7pt}\n  %\n  \\setlength{\\widthFigB}{162pt}\n  \\setlength{\\marginLeftFigB}{7pt}\n  \\setlength{\\marginRightFigB}{7pt}\n  %\n  \\setlength{\\widthFigC}{160.6pt}\n  \\setlength{\\marginLeftFigC}{5.6pt}\n  \\setlength{\\marginRightFigC}{7pt}\n  %\n  \\parbox[t]{\\widthFigA}{\\vspace{0pt}\\includegraphics[width = \\widthFigA]{figure1a.pdf}}\n  %\n  \\parbox[t]{\\widthFigB}{\\vspace{0pt}\\includegraphics[width = \\widthFigB]{figure1b.pdf}}\n  %\n  \\parbox[t]{\\widthFigC}{\\vspace{0pt}\\includegraphics[width = \\widthFigC]{figure1c.pdf}}\\\\\n  %\n  %\n  \\caption{Atom heating rate in the radial and azimuthal direction calculated using \\cref{eqn: final heating formula} as function of (a) the nanofiber radius, (b) the temperature of the nanofiber, and (c) the power of the blue-detuned trapping laser. The difference between \\cref{eqn: final heating formula} and the full theory \\cref{eqn: heating rate contributions} is not discernible at the given scales. In \\subcrefandb{fig: heating vs laser power}{a}{b}, all other parameters, in particular the trap frequencies, are unchanged. In \\subcrefb{fig: heating vs laser power}{c}, the ratio between the power of the red- and blue-detuned laser is kept constant, $P_\\text{b} \/ P_\\text{r} = 14.24$. The relation between the total laser power and temperature is modeled as $T(P) = m_0 + m_1P + m_2P^2$, with $m_0 = \\SI{400}{\\kelvin}$, $m_1 = \\SI{24}{\\kelvin\/\\milli\\watt}$, $m_2 = \\SI{-0.062}{\\kelvin\/\\milli\\watt^2}$ based on the measurements in \\cite{wuttke_thermalization_2013} for a nanofiber of radius $R = \\SI{250}{\\nano\\meter}$ and length $L = \\SI{5}{\\milli\\meter}$. The temperature then varies from $T = \\num{427}$ to $\\SI{2298}{\\kelvin}$ over the shown range of laser power. The trap frequencies simultaneously increase from $(\\trapfreq_{\\rpos},\\,\\trapfreq_{\\phipos}) = 2\\pi \\times (\\num{29.1},\\,\\num{23.9})\\,\\si{\\kilo\\hertz}$ to $2\\pi\\times (\\num{291},\\,\\num{168})\\,\\si{\\kilo\\hertz}$. The remaining parameters are specified in \\cref{sec: case study appendix}.}\n  \\label{fig: heating vs laser power}\n\\end{figure*}\n\\Cref{fig: heating vs laser power} shows the dependence of the predicted heating rates in the radial and azimuthal direction on individual parameters, keeping the remaining parameters unchanged. Most pronounced is the scaling with the nanofiber radius as $\\Gamma^\\text{th}_i \\propto R^{-5\/2}$, see \\subcref{fig: heating vs laser power}{a}. The strong dependence on the radius is mostly due to the increased mechanical stability of larger nanofibers which leads to smaller vibrational amplitudes, see \\cref{eqn: displacement F modes low-frequency limit}, in addition to a lower density of states. In contrast, the dependence on the fiber temperature is linear, see \\subcref{fig: heating vs laser power}{b}, since the thermal occupation of the resonant phonon modes increases linearly with the temperature. Comparison of \\subcref{fig: heating vs laser power}{a} and \\subcref{fig: heating vs laser power}{b} shows that increasing the nanofiber radius by $\\SI{150}{\\nano\\meter}$ to $R = \\SI{400}{\\nano\\meter}$ at constant temperature has an effect comparable to cooling the fiber down to room temperature if all other parameters of the setup could be kept unchanged. \\subCref{fig: heating vs laser power}{c} shows the dependence on the power of the blue-detuned laser, where the ratio of the power of the red- and blue-detuned lasers is kept constant. The temperature of the nanofiber increases with increased laser power since there is more absorption in the fiber \\cite{wuttke_thermalization_2013}; see caption for details. Moreover, higher intensities lead to a tighter confinement of the atoms. The observed increase of the heating rate when raising the laser power is therefore caused by an increase of both the fiber temperature and the trap frequencies. While Young's modulus $E$ also slightly changes with $T$ \\cite{spinner_elastic_1956}, the influence of this effect on the heating rate is negligible due to the weak dependence, $\\Gamma^\\text{th}_i \\propto E^{-1\/4}$.\n\n\\medskip{}\n\nLet us now discuss ways to reduce the atom heating caused by coupling to the continuous $\\text{F}_{11}$ band. Lowering the overall fiber temperature in order to reduce the heating rates is difficult even in cryogenic environments because thermal coupling of the fiber to its surroundings is very weak \\cite{wuttke_thermalization_2013}. However, based on the above analysis, different strategies to minimize the heating rates are conceivable. First of all, the fiber radius should be chosen as large as possible while maintaining the optical properties required for atom trapping. A second approach is to design the nanofiber such that it supports discrete, well-resolved resonances of flexural modes. While precise predictions of phonon linewidths are difficult, it may be possible to optimize the taper at both ends of the nanofiber and ensure that flexural modes are reflected and confined to $z\\in[0,\\len]$ with narrow linewidths, while the transmission of light is not reduced \\cite{pennetta_tapered_2016}. Such a resonator of length $\\len$ for the flexural modes would effectively break the $\\text{F}_{11}$ band into a discrete set of frequencies $\\omega_m$, and allow us to detune the atom trap from resonance with these mechanical modes.\n\\begin{figure*}[t]\n  \\raggedright\n  \\setlength{\\widthFigA}{256.0448pt}\n  \\setlength{\\marginLeftFigA}{29.0448pt}\n  \\setlength{\\marginRightFigA}{7pt}\n  %\n  \\setlength{\\widthFigB}{233pt}\n  \\setlength{\\marginLeftFigB}{7pt}\n  \\setlength{\\marginRightFigB}{6pt} %\n  %\n  \\parbox[t]{\\widthFigA}{\\vspace{0pt}\\includegraphics[width = \\widthFigA]{figure2a.pdf}}\n  %\n  \\parbox[t]{\\widthFigB}{\\vspace{0pt}\\includegraphics[width = \\widthFigB]{figure2b.pdf}}\\\\\n  %\n  %\n  \\caption{Atom heating rate in the radial direction due to flexural resonator modes as a function of (a) the resonator length and (b) the trap frequency. In both cases, an exemplary decay rate of $\\kappa = 2\\pi\\times \\SI{1.2}{\\hertz}$ is assumed for all resonator modes. The bold yellow line corresponds to the heating rate experienced by an atom trapped at the center of the resonator, $\\zpos_0 = \\len\/2$, calculated according to \\cref{eqn: general heating discrete flexural modes}. The thin blue line represents a position-independent upper bound obtained by pretending that the atom sits at an antinode of each phonon mode simultaneously: In consequence, no resonance between the atom and resonator is masked by a vanishing position-dependent coupling rate. This approach is useful, since in experiments an entire ensemble of atoms is trapped at various positions along the fiber. The dashed red lines show the approximations \\cref{eqn: heating off-resonant 1,eqn: heating off-resonant 2,eqn: resonant heating}. \\subCrefb{fig: reduce heating}{a} assumes a trap frequency of $\\trapfreq_{\\rpos} = 2\\pi\\times \\SI{123}{\\kilo\\hertz}$, and \\subcrefb{fig: reduce heating}{b} assumes a resonator length of $\\len = \\SI{600}{\\micro\\meter}$.}\n  \\label{fig: reduce heating}\n\\end{figure*}\nThe flexural eigenmodes are then standing waves (see \\cref{sec: phonon appendix}), with frequency spectrum\n\\begin{align}\\label{eqn: flexural resonator spectrum}\n  \\omega_m &\\equiv m^2 \\frac{\\pi^2 R}{2\\len^2}\\sqrt{\\frac{E}{\\rho}}~, & m &\\in \\mathds{N}~.\n\\end{align}\nThe heating rate in the radial and azimuthal direction due to these flexural resonator modes then depends on the position $z_0$ of the atom along the fiber axis; see \\cref{sec: interaction appendix}. \\Cref{fig: reduce heating} shows the dependence of the heating rate on the resonator length and trap frequency. Three regimes are clearly distinguishable: First, the trap is resonant with a flexural phonon mode. Second, the trap is off resonant and lies below the fundamental resonator frequency. Third, the trap is off resonant and lies above the fundamental resonator frequency. Assuming high thermal occupation of the phonon modes, $\\bar{n}_m \\gg 1$, simplified expressions for the heating rate can be obtained for each regime. If the trap frequency is below the fundamental phonon frequency but still much larger than the corresponding decay rate, $\\kappa_1 \\ll \\trapfreq_{ i} < \\omega_1$, as well as far detuned, $|\\trapfreq_{ i}-\\omega_1|\\gg \\kappa_1$, heating is dominated by off-resonant interaction with the fundamental phonon mode alone. In this case, the heating rate can be approximated as\n\\begin{equation}\\label{eqn: heating off-resonant 1}\n  \\begin{split}\n    \\Gamma_i^\\text{th} &\\simeq \\Gamma_{i<}^\\text{nres} \\sin^2(\\pi \\zpos_0\/\\len)\\\\\n    \\Gamma_{i<}^\\text{nres} &\\equiv \\frac{16}{\\pi^9}\\frac{k_B}{\\hbar}\\frac{T M \\rho \\kappa_1 \\trapfreq_{ i}^3 \\len^7}{E^2 R^6} ~.\n  \\end{split}\n\\end{equation}\n\nIf the trap has a frequency larger than the fundamental resonator frequency, $\\trapfreq_{ i}\\gg \\omega_1$, while still being off resonant, $|\\trapfreq_{ i}-\\omega_m|\\gg \\kappa_m$, heating is mainly due to the low-frequency phonon modes below the trap frequency. Assuming in addition that the phonon decay rate is the same for all relevant modes, $\\kappa_m \\simeq \\kappa$, an upper bound for the heating rate can be obtained:\n\\begin{equation}\\label{eqn: heating off-resonant 2}\n \\Gamma_i^\\text{th} \\lesssim \\Gamma_{i>}^\\text{nres} \\equiv \\frac{2}{45 \\pi} \\frac{k_B}{\\hbar}\\frac{TM \\kappa \\trapfreq_{ i} \\len^3}{E R^4}.\n\\end{equation}\nHere, we replace the sine in the coupling constant with $1$ for all modes, pretending the atom is located at an antinode of all modes simultaneously as a worst-case estimate. This approximation is useful because in experiments many atoms at different sites along the fiber axis are trapped at the same time.\n\nIf the trapped atom is resonant with a flexural phonon mode $m$, $|\\trapfreq_{ i}-\\omega_m|\\ll \\kappa_m$, and the contributions of the off-resonant modes can be neglected, the heating rate is\n\\begin{equation}\\label{eqn: resonant heating}\n  \\begin{split}\n  \\Gamma_i^\\text{th} &\\simeq \\Gamma_{i}^\\text{res} \\sin^2(p_m \\zpos_0)\\\\\n  \\Gamma_{i}^\\text{res} &\\equiv \\frac{2}{\\pi}\\frac{k_B}{\\hbar} \\frac{T M \\trapfreq_{ i}}{\\len \\rho \\kappa_m R^2}~.\n  \\end{split}\n\\end{equation}\nThe limiting expressions \\cref{eqn: heating off-resonant 1,eqn: heating off-resonant 2,eqn: resonant heating} are shown as dashed black lines in \\cref{fig: reduce heating}. Note that the dependence on decay rate and resonator length is inverted for off-resonant heating, \\cref{eqn: heating off-resonant 1,eqn: heating off-resonant 2}, compared to resonant heating, \\cref{eqn: resonant heating}. This inversion is expected, since large phonon linewidths $\\kappa_m$ assist off-resonant coupling, while small linewidths lead to a larger resonant enhancement. Small resonator lengths $\\len$ lead to higher coupling constants (Purcell enhancement), which increases resonant heating due to a single mode. In contrast, large resonator lengths result in a higher number of low-frequency modes and hence overcompensate the decrease in coupling strength and increase the heating due to off-resonant interaction.\n\n\nIn \\cref{fig: reduce heating}, we exemplarily assume a decay rate of $\\kappa_m = 2\\pi \\times \\SI{1.2}{\\hertz}$ for all relevant flexural modes. This corresponds to a quality factor of $\\omega_r\/\\kappa_m = 10^5$ at the frequency of the radial trap. Quality factors of this magnitude have been achieved for silica microspikes by optimization of the shape of the taper \\cite{pennetta_tapered_2016}. \\subCref{fig: reduce heating}{a} shows that a decrease of the radial heating rate below the value expected without a resonator for flexural modes (see \\cref{tab: heating rates}) is predicted for resonator lengths $\\len \\lesssim \\SI{3}{\\milli\\meter}$. A length of $\\len = \\SI{50}{\\micro\\meter}$ to the very left of \\subcref{fig: reduce heating}{a} can still be achieved for nanofibers, the calculated heating rate due to flexural phonon modes with the given decay rate is then as low as \\SI{0.1}{\\milli\\hertz}. \\subCref{fig: reduce heating}{b} assumes a resonator length of $\\len = \\SI{600}{\\micro\\meter}$, achieving heating rates of around $\\SI{1}{\\hertz}$ and shows the dependence on the trap frequency. The spacing between resonances is on the order of $2\\pi \\times \\SI{50}{\\kilo\\hertz}$, which would indeed render it possible to detune the radial and azimuthal trap from resonance.\n\nThese findings suggest that it may be possible to significantly reduce the heating rate of atomic motion in nanofiber-based traps by two orders of magnitude or more through optimization of the phononic properties of the fiber. Moreover, the scaling of the heating rate with the mass of the trapped particles as $\\Gamma^\\text{th}_i \\propto M$ is highly relevant for optomechanical experiments. Setups with levitated nanoparticles, for instance, may feature comparable trap frequencies for particles that are orders of magnitude heavier than a single atom \\cite{magrini_near-field_2018,diehl_optical_2018}. In order to stably trap heavier particles using nanophotonic structures and successfully cool their motion, it is imperative to carefully manage vibrations of the structure, for instance by improving the mechanical stability or by tuning mechanical modes out of resonance with the particle motion.\n\n\\section*{Conclusion}\nIn this article, we formulate a general theoretical framework for calculating the effect of phonons on guided optical modes and the resulting heating of atoms in nanophotonic traps. Our results are applicable to nanophotonic cold-atom systems~\\cite{chang_colloquium_2018} and can readily be extended to the heating of dielectric nanoparticles trapped close to surfaces~\\cite{magrini_near-field_2018,diehl_optical_2018}. In a case study for the example of cold cesium atoms in a two-color nanofiber-based optical trap, we predict heating rates of the atomic center-of-mass motion which are in excellent agreement with independently measured values~\\cite{albrecht_fictitious_2016,reitz_coherence_2013}. In this system, the dominant contribution to heating stems from thermally occupied flexural modes of the nanofiber. We find that the heating rate scales with the fiber radius as $R^{-5\/2}$. As a general design rule, this implies that structures of larger lateral dimensions are preferable regarding heating, albeit at the expense of smaller mode confinement and, hence, potentially lower atom-photon coupling strength. Given the fact that the heating rate is directly proportional to the temperature of the nanophotonic structure, reducing the absorption losses of the guided trapping light fields is advisable~\\cite{ravets_intermodal_2013}. Moreover, heating is expected to decrease for smaller trap frequencies, $\\Gamma \\propto \\sqrt{\\omega}$. In general, our case study shows that careful design of the phononic properties of the nanophotonic system and, in particular, of its mechanical resonances is an effective strategy for reducing the heating. Finally, by providing a coherent theoretical framework in a single source, our work is instrumental in calculating, understanding, and managing heating in a plethora of nanophotonic traps.\n\n\\begin{acknowledgments}\nWe thank Y.~Meng for the experimental characterization of the torsional mode resonances of the tapered optical fiber in the nanofiber-based two-color trap setup. Financial support by the European Research Council (CoG NanoQuaNt) and the Austrian Academy of Sciences (\u00d6AW, ESQ Discovery Grant QuantSurf) is gratefully acknowledged. We acknowledge support by the Austrian Federal Ministry of Science, Research, and Economy (BMWFW).\n\\end{acknowledgments}\n\n\n\n\\section*{Power fluctuations of the trapping laser fields}\nThe two-color optical dipole trap used in \\cite{albrecht_fictitious_2016} is formed by a blue-detuned running-wave light field and a red-detuned field which is in a standing-wave configuration. Beyond polarization fluctuations, which are already treated in the main text of our manuscript, only power fluctuations are relevant for the blue-detuned field and would lead to resonant heating (via a shift of the trap center) of the trap's radial DOF, and to parametric heating (via a change of the trap frequency) of the azimuthal DOF, while no coupling occurs for the axial DOF. For the red-detuned light field, power fluctuations would lead to resonant heating along the radial direction, as well as parametric heating along azimuthal and axial directions. We characterized the power fluctuations of the trap lasers used in experiment~\\cite{albrecht_fictitious_2016}. The measured power noise characteristics cannot explain the large heating rates we observe. We estimate that heating due this technical noise amounts to a heating rate of about~$\\SI{0.02}{quanta\/\\milli\\second}$.\n\n\n\\section*{Off-resonant scattering}\nIn the two-color dipole trapping scheme, the blue-detuned light field leads to an increase of the potential energy of the electronic atomic ground state while the red-detuned light field lowers this energy. The trap is formed as the sum of both potentials, and the trap depth is smaller than the magnitudes of the individual shifts. For this reason, the nanofiber-based two-color trap shows a larger off-resonant scattering rate for a given trap depth compared to, e.g., single-color free-space dipole traps. The expected heating rate due off-resonant scattering amounts to roughly $\\SI{2}{quanta\/s}$ (calculated for the z-axis), much less than what is experimentally observed.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{figureSupplement.png}\n\t\\caption{Measured Stokes Raman scattering spectrum of three pieces of Liekki Passive-6-125 optical fiber. The wavelength of the fiber-guided pump light field is $\\lambda=\\SI{780}{nm}$. The black arrow indicates the Cesium $D_2$ resonance at $\\SI{852}{nm}$.}\n\t\\label{fig:Raman}\n\\end{figure}\n\n\\section*{Raman scattering}\nThe trapping light fields propagating in the tapered optical fiber give rise to Raman scattering in the fiber material, which is fused silica. The contribution of the blue-detuned light field to the relevant wavelength-range of the Raman-scattered light is much larger than that of the red-detuned light field and, therefore, we focus the discussion on the effect of that laser only. We measured the Raman scattering induced by a fiber-guided laser field of $\\SI{780}{\\nano\\meter}$ wavelength and recorded the Stokes-scattered light using a spectrometer. The test fiber was a Liekki Passive-6-125, of the same type as the one used in~\\cite{albrecht_fictitious_2016}. Figure~\\ref{fig:Raman} shows the measured power spectral density of the Raman signal over the frequency difference from the pump laser frequency for three different pieces of fiber. The signal below $\\SI{600}{\\centi\\meter^{-1}}$ is blocked by two edge filters that are required to sufficiently extinguish the pump light. The $D_2$ line of Cesium corresponds to a Stokes shift of about $\\SI{1100}{\\centi\\meter^{-1}}$ where there is still a sizable Raman signal. However, while the Raman-scattered light is many THz wide, the absorption linewidth of cold Cesium atoms is only a few MHz. In order to get a precise estimate, we measure the Raman-scattered signal induced by the blue-detuned trap light field using a SPCM. The fiber including the tapered section as used in the experiment~\\cite{albrecht_fictitious_2016} has a length of about $\\SI{5}{\\meter}$. All the light exiting the fiber passes a filter that removes the trapping light field. It then passes a narrow (FWHM $\\SI{0.12}{\\nano\\meter}$) bandpass filtering stage centered around the Cesium's $D_2$ line, which isolated the fraction of the Raman-scattered light which is (near-)resonant with the atoms' dominant optical transition. From that and further system parameters, we estimate that a single nanofiber-trapped Cesium atom absorbs about $\\SI{50}{photons\/\\second}$ of Raman light, which yields a heating rate even lower than that arising from off-resonant scattering of trap light fields described above.\n\n\\section*{Brillouin scattering}\nGuided light fields such as these that generate the optical dipole trap in experiment~\\cite{albrecht_fictitious_2016} can experience Brillouin scattering. For this, a phase matching condition between the optical mode and an acoustic mode of the nanofiber has to be fulfilled. For a nanofiber of $\\SI{250}{nm}$ radius and a guided light field of $\\SI{780}{nm}$ wavelength, the first matching for Brillouin backscattering occurs for an acoustic wave with a frequency of about $\\SI{11}{GHz}$ (calculations analogous to \\cite{florez_brillouin_2016}. The next resonances occur at higher acoustic frequencies. The spectral width of the Brillouin-scattered light amounts to a few $\\SI{10}{MHz}$, i.e., much narrower than for Raman scattering. The Brillouin-scattered light together with the pump light can, in principle, drive stimulated Raman transitions between different internal states of the nanofiber-trapped cold atoms. When the trapping-potential has a dependence on the internal atomic state (as it can be the case for nanofiber-based traps, see \\cite{le_kien_state-dependent_2013}, this could lead to heating. The only internal atomic states with a comparable energy separation are the two hyperfine ground-state manifolds of the Cesium atom (HFS splitting: about $\\SI{9}{GHz}$). However, the two-photon detuning for this hyperfine state-changing Raman transition is still about $\\SI{2}{GHz}$. Such a large detuning, combined with the low power of the Brillouin-scattered light that we estimate to be at the sub-nW level, and given the narrow width of the Brillouin signal, suggests that Brillouin scattering in the nanofiber is irrelevant for the observations in the manuscript. A similar estimation can be made for the standard fiber part. Here, the Brillouin shift amounts to about $\\SI{22.3}{GHz}$, i.e., an even larger two-photon detuning. Again, a scattered power in the sub-nW level is found. In the view of these numbers, we consider also Brillouin scattering in the bulk fiber to have a negligible effect on heating.\n\nFor the discussion of another mechanism related to Brillouin scattering, we consider the following scenario: Each of the two red-detuned trapping light fields gives rise to a Brillouin scattered (BS) light field that also propagates in the fiber. These two BS fields form an additional standing wave (SW) that has a randomly fluctuating phase with respect to the original trapping SW. This SW formed from BS results in a stochastic modulation a) of the axial trapping frequency and b) of the position of the minima of the trapping potential along the fiber. The former gives rise to parametric heating while the latter leads to resonant heating.\n\nWe first consider parametric heating resulting from BS-induced fluctuations of the axial trap frequency, $\\omega_z$. When a potential minimum of the trapping SW coincides with a potential minimum of the SW formed from BS, the resulting trap frequency is increased. When the relative phase between the two SWs is such that a potential minimum of the trapping SW falls onto a potential maximum of the SW formed from BS, the resulting trap frequency is decreased. Due to the relatively small detuning of the BS light (about 10~GHz vs. about 300~THz), we can neglect the mismatch between the periodicity of the SW formed from BS and the trapping SW. The rate of parametric heating then depends on the power spectral density (PSD) of the relative phase fluctuations of the two SWs evaluated at twice the axial trap frequency, $2\\omega_z$. The fluctuations of the relative phase between the two BS light fields is determined by the individual spectral widths of the two BS fields. The spectrum of BS light for a situation comparable to ours has been published in \\cite{beugnot_brillouin_2014}. An approximately Gaussian spectrum with a FWHM of about $\\SI{25}{MHz}$ was found. In order to apply the formalism by~\\cite{savard_laser-noise-induced_1997} to the calculation of heating rates, we normalize the frequency-integrated PSD of the individual BS light fields to the root-mean square value of the relative intensity noise. For the latter, the ratio of the powers of the trapping and the BS light fields serves as an upper estimate. The PSD of the SW formed from BS then follows from the (appropriately normalized) convolution of the two PSDs of the two BS fields. Using this approach, we compute an exponential increase of the temperature with a time constant $\\tau \\approx 7 \\times 10^7\\,\\mathrm{s}$. This corresponds to a heating rate for the axial degree of freedom on the order of $\\SI{e-8}{quanta\/\\second}$, i.e., ten orders of magnitude smaller than the heating rate measured for the azimuthal degree of freedom.\n\nWe now consider resonant heating that originates from BS-induced position fluctuations of the trapping potential minimum. The heating rate depends on the PSD of the position fluctuations evaluated at the trap frequency. The displacement reaches a maximum value,  $\\Delta z_{\\rm max}$, when the SW formed from BS is shifted by a quarter of the wavelength of the trapping light with respect to the trapping SW. Again, we assume the PSD of the position fluctuations to have the above-mentioned Gaussian spectrum. We use $\\Delta z_{\\rm max}$ as a worst-case estimate for the RMS position fluctuation. For our setting, we estimate a maximal displacement of $\\Delta z_\\mathrm{max} \\approx 10^{-4}\\,\\mathrm{nm}$. This yields a heating rate in the axial direction on the order of $4 \\times 10^{-5}\\,\\mathrm{quanta\/s}$. Thus, we exclude BS-induced fluctuations of the axial trap frequency, or of the axial position of the trapping potential minimum, as mechanisms that currently limit the lifetime of atoms in our nanofiber-based trap.\n\n\\section*{Blackbody radiation and Johnson--Nyquist noise}\nBlackbody radiation and Johnson--Nyquist noise are fundamental processes which have been studied extensively in the context of atom chips. For example, the authors of \\cite{henkel_loss_1999} quantitatively estimate the expected heating rate for a spin confined in a harmonic potential with $\\SI{100}{kHz}$ trap frequency in close proximity to a material half space. While the surface-induced heating rates can be comparably large for atoms close to a conductive material such as copper (on the order of $\\SI{10}{quanta\/s}$), the rates are only on the order of $\\SI{e-14}{quanta\/s}$ for an atom $\\SI{200}{nm}$ away from a glass surface, mainly thanks to the low electrical conductivity of glass. Thus, these heating mechanisms are negligible for our experimental conditions.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nLet us consider variational problems, consisting in the minimization of\n$$J \\,:\\, u \\mapsto \\int_\\Omega L(x,u(x),\\nabla u(x)) \\, \\text{\\normalfont d}\\mu(x) $$\nwhere the usual Lebesgue measure is replaced by a generic Borel measure~$\\mu$, under possible $\\mu$-a.e.\\@ or boundary constraints. In calculus of variations, the direct method consists in extracting a converging subsequence (in a suitable sense) from a minimizing sequence, thanks to a compactness result on the set of admissible functions, and to conclude by semi-continuity of the functional $J$. For our functional, two problems appear:\n\\begin{itemize}\n\\item which functional space should we consider in order to give a sense to the gradient $\\nabla u$? More precisely, if $\\mu$ is the Lebesgue measure or has a density $f$ bounded from above and below, we can work in the classical Sobolev space $H^1(\\Omega)$ (which is exactly the set of functions $u \\in L^2_\\mu$ having weak derivatives in $L^2_\\mu$), but it is not so clear if this assumption on $f$ does not hold or if $\\mu$ has a singular part.\n\\item does there exist a compactness result which allows to extract from a minimizing sequence a subsequence converging, in a suitable sense, to an admissible function? For instance, in the classical Sobolev space $H^1(\\Omega)$, the Rellich theorem allows to extract from any bounded sequence a strongly-convergent subsequence in $L^2(\\Omega)$ which is a.e.\\@ convergent on $\\Omega$.\n\\end{itemize}\n\nLet us fix more precisely the notations. Let $\\Omega$ be a bounded open set of $\\mathbb{R}^d$ and $f$ a measurable and a.e.\\@~positive function on $\\Omega$. If we assume $f$ to be bounded from below and above, it is obvious that the set\n$$ \\{u \\in L^2_\\mu(\\Omega) : \\nabla u \\text{ exists in the weak sense and belongs to } L^2_\\mu(\\Omega)^d \\} $$\nis exactly the classical Sobolev space $H^1(\\Omega)$, since the $L^2_\\mu$-norm is equivalent to the usual $L^2$-norm on $\\Omega$. If $f$ is only assumed to be positive, for $u \\in L^2(\\Omega)$, the Cauchy-Schwarz inequality gives\n$$ \\int_\\Omega |u(x)| \\, \\text{\\normalfont d} x = \\int_\\Omega \\left(|u(x)| \\sqrt{f(x)}\\right) \\frac{\\, \\text{\\normalfont d} x}{\\sqrt{f(x)}} \\leq \\left( \\int_\\Omega |u(x)|^2 f(x) \\, \\text{\\normalfont d} x \\right)^{1\/2} \\, \\left( \\int_\\Omega \\frac{\\, \\text{\\normalfont d} x}{f(x)} \\right)^{1\/2} $$\nthus, under the assumption\n\\begin{equation} 1\/f \\in L^1(\\Omega) \\label{f} \\end{equation}\nwe have the continuous embedding\n$$ L^2_f(\\Omega) := \\left\\{u : \\int_{\\Omega} |u(x)|^2 f(x) \\, \\text{\\normalfont d} x < +\\infty \\right\\} \\hookrightarrow L^1(\\Omega). $$\nIn this case, any function $f \\in L^2_f(\\Omega)$ has a gradient $\\nabla u$ in the weak sense, since it is locally integrable on~$\\Omega$, and we can define the weighted Sobolev space with respect to $f$\n$$H^1_f(\\Omega) = \\left\\{ u \\in L^2_f(\\Omega) \\; : \\; \\nabla u \\in L^2_f(\\Omega) \\right\\}. $$ \nMore generally, if $p \\in \\, ]1,+\\infty[ \\, $, the assumption\n$$ (1\/f)^{1\/(p-1)} \\in L^1_{loc}(\\Omega) $$\nis a well-known sufficient condition to define the weighted Sobolev space $W^{1,p}(\\Omega)$ with respect to $f$ (see~\\cite{kufner} for more details). For our problem, if the Lagrangian functional is quadratic with respect to the gradient, for instance\n$$J(u) = \\int_\\Omega |\\nabla u|^2 \\, \\text{\\normalfont d}\\mu, $$\nit means that the set of admissible functions is well-defined if $\\mu$ has a density $f$ such that $1\/f$ is integrable: it is the set of the elements of the weighted Sobolev space $H^1_f$ satisfying the constraints.\n\nIf $\\mu$ is absolutely continuous with density $f$, the weighted Lebesgue space with respect to $f$ is exactly the space $L^p_\\mu$, so that the space $H^1_f$ can be seen as a Sobolev space with respect to the measure $\\mu$. A natural generalization consists in defining the Sobolev space with respect to the measure $\\mu$, without condition on its density or when $\\mu$ is not anymore assumed to be absolutely continuous with respect to~$\\mathcal{L}^d$. There exists some general definitions of the Sobolev space in a generic metric measure space $(X,d,\\mu)$ (see \\cite{haj2}), but we will not enter to the details of this notions in this paper and we prefer to focus on the case of an open set of $\\mathbb{R}^d$.\n\nWe begin this paper by an overview of the definitions and already known results about this Sobolev spaces \\cite{bbs, fra, f-m, preiss, zhikov2, zhikov}, and present several new results: in particular, we give a precise description of the tangent space to any measure $\\mu$ on the real line. As a corollary of this result, we show a compactness result in $H^1_\\mu$, which states precisely that any bounded sequence admits a pointwise $\\mu$-a.e.\\@ convergent subsequence on the set of points where the tangent space is not null (this result is already known in any dimension under strong conditions on the measure $\\mu$, when the compact embedding of the Sobolev space $W^{1,p}_\\mu$ with respect to $\\mu$ into the Lebesgue space $L^p_\\mu$ still holds; see \\cite{bf, h-k}).\n\nThis is applied to a variational problem coming from optimal transportation: we consider the minimization of the functional\n$$ J(T) = \\int_\\Omega L(x,T(x),D_\\mu T(x)) \\, \\text{\\normalfont d}\\mu(x) $$\namong all the maps $T:\\Omega \\mapsto \\mathbb{R}^d$ which admit a Jacobian matrix $D_\\mu T$ with respect to $\\mu$ and under a constraint on the image measure $T_\\# \\mu$ (it corresponds to the classical Monge-Kantorovich optimal transportation problem \\cite{vil} if $L$ does not depend on its third variable, and is linked to minimization problems under volume-preservation or area-preservation constraints \\cite{tann}). In the one-dimensional case, we get the existence of a solution for any measure $\\mu$ (the optimal map is known if $\\mu$ is assumed to be the uniform measure on the interval, see \\cite{ls} for details). However, we are not able to give a precise description of the tangent space and to obtain the existence of solution to this transport problem in the most general case in any dimension.\n\n\\section{Sobolev spaces with respect to a measure}\n\nThis section is devoted to an overview of the definitions and already known results about tangent spaces to a generic Borel measure $\\mu$ and Sobolev spaces associated to this measure. First, let us recall that there exist some notions of Sobolev spaces in arbitrary metric measure spaces $(X,d,\\mu)$, for instance in the papers by Shanmugalingam \\cite{sha}, Haj\\l asz \\cite{haj} or Haj\\l asz and Koskela \\cite{h-k} (see \\cite{haj2} for a global summary of this notions). In our case, a usual method consists in defining the tangent space to $\\mu$ (which is a function defined $\\mu$-a.e.\\@ on $\\mathbb{R}^d$ and taking values in the set of linear subspaces of $\\mathbb{R}^d$), and the gradient with respect to $\\mu$ for a regular function $u$ through\n$$ \\nabla_\\mu u (x) = p_{T_\\mu(x)} (\\nabla u(x)) \\quad \\text{for }\\mu\\text{-a.e.\\@ } x \\in \\mathbb{R}^d, $$\nwhere $p_{T_\\mu(x)}$ is the orthogonal projection on $T_\\mu(x)$ in $\\mathbb{R}^d$. Then we consider for the Sobolev space $H^1_\\mu$ the closure of $C^\\infty\\left(\\overline{\\Omega}\\right)$ for the norm\n$$u \\in C^\\infty\\left(\\overline{\\Omega}\\right) \\mapsto ||u||_{L^p_\\mu} + ||\\nabla_\\mu u||_{L^p_\\mu}. $$\nThere exist several ways to define the tangent space of a generic measure $\\mu$. Preiss \\cite{preiss} gives a method based on the idea of blow-up: a $k$-dimensional subspace $P_\\mu$ is said to be an approximate tangent space of $\\mu$ at $x$ if we have, for some $\\theta > 0$, the following convergence in the vague topology of measure when $\\rho$~goes to $0$:\n$$ \\mu(x+ \\rho\\, \\cdot \\,) \\rightharpoonup \\theta \\mathcal{H}^k|_{P_\\mu}. $$\nIn order to examine variational problems, Bouchitt\\'e {\\it et al}.\\@ \\cite{bbs} have introduced a dual-formulation of the tangent space: it is the $\\mu$-ess.\\@ union (see \\cite{c-v} or later) $x \\mapsto Q_\\mu(x)$ of the tangent fields, {\\it i.e.} the vector fields belonging to\n$$ X^{p'}_\\mu = \\{ \\phi \\in (L^{p'}_\\mu)^d : \\div(\\mu \\phi) \\in L^{p'}_\\mu \\},  $$ \nwhere the operator $\\div(\\mu v)$ is defined in the distributional sense. Fragal\\`a and Mantegazza \\cite{f-m} have noticed that, with this notation, we have the inclusion $Q_\\mu(x) \\subseteq P_\\mu(x)$ for $\\mu$-a.e.\\@ of $\\mathbb{R}^d$ (see the PhD.\\@ thesis \\cite{fra} for a complete overview and more details about these definitions). \n\nWe are interested in another way to define tangent and Sobolev spaces, introduced by Zhikov \\cite{zhikov2, zhikov}. Let $\\Omega$ be a bounded open set of $\\mathbb{R}^d$ and $\\mu$ a finite positive measure on $\\Omega$ We will say that $u \\in L^2_\\mu$ belongs to the space $H^1_\\mu$ if it can be approximated by a sequence of regular functions whose gradients have a limit in the space $L^2_\\mu$:\n$$u \\in H^1_\\mu \\; \\Longleftrightarrow \\; \\exists (u_n)_n \\in C^\\infty(\\overline{\\Omega}), \\, v \\in (L^2_\\mu)^d : \\, \n\\left\\{\n\\begin{array}{l}\nu_n \\to u \\\\\nv_n \\to v\n\\end{array}\n\\right.\n\\; \\text{for the }L^2_\\mu\\text{-norm}.  $$\nThe set of these limits $v$ is denoted by $\\Gamma(u)$, and its elements are called gradients of $u$. In general, $u$ can have many gradients (see below the example of a measure supported on a segment of $\\mathbb{R}^2$), and it is obvious that $\\Gamma(u)$ is a closed affine subspace of $(L^2_\\mu)^d$ with direction $\\Gamma(0)$. The projection of $0$ onto this subspace (in the Hilbert space $(L^2_\\mu)^d$) is thus the unique element of $\\Gamma(u)$ with minimal $L^2_\\mu$-norm: we call it {\\it tangential gradient} of $u$ with respect to $\\mu$.\n\n\\noindent {\\bf Pointwise description of $\\nabla_\\mu u$ and tangent space to $\\mu$.} We define the tangent space to $\\mu$ as follows: the space $\\Gamma(0)$ can be seen as the set of vector-valued functions which are pointwise orthogonal to the measure $\\mu$. Let us denote by $(e_1, \\dots, e_d)$ the canonical basis of $\\mathbb{R}^d$, and set\n$$\\xi_i = p_{\\Gamma(0)}(e_i)$$\nwhere the projection is taken in the Hilbert space $L^2_\\mu$ (here $e_i$ is seen as a constant function on $\\Omega$). For $x \\in \\Omega$, we denote by\n$$ T_\\mu(x) = \\left(\\operatorname{Vect} (\\xi_1(x), \\dots, \\xi_d(x))\\right)^\\perp $$\nand call $T_\\mu(x)$ (which is defined for $\\mu$-a.e.\\@ $x \\in \\Omega$) the tangent space to $\\mu$ at $x$. Then, the following equivalence holds:\n$$ v \\in \\Gamma(0) \\quad \\Longleftrightarrow \\quad \\text{for $\\mu$-a.e.\\@ } x \\in \\Omega, \\; v(x) \\perp T_\\mu(x). $$\nThis result, combined to the orthogonality property of $\\nabla_\\mu u$ in $L^2_\\mu$, implies a pointwise description of the tangential gradient:\n\\begin{prop} Let $u \\in H^1_\\mu$. Then, for $v \\in \\Gamma(0)$, the function\n$$x \\in \\Omega \\mapsto p_{T_\\mu(x)}(v(x)) $$\nis independent of the function $v$ and only depends on $u$, and we have\n$$\\nabla_\\mu u(x) = p_{T_\\mu(x)} \\qquad \\text{for } \\mu\\text{-a.e.\\@ } x \\in \\Omega. $$\n\\end{prop}\n\n\\noindent {\\bf Some natural examples.} We can see that the words ``{\\it tangential} gradients'' are quite natural in the following cases:\n\\begin{itemize}\n\\item if $\\mu$ is the Lebesgue measure $\\mathcal{L}^1$ concentrated on the segment $I=[0,1]\\times\\{0\\} \\times \\dots \\times \\{0\\}$, then $T_\\mu$ is the line $\\mathbb{R} \\times \\{0\\} \\times \\dots \\times \\{0\\}$ a.e.\\@ on $I$ and\n$$H^1_\\mu = \\left\\{u \\in L^2_\\mu : \\frac{\\partial u}{\\partial x_1} \\in L^2_\\mu\\right\\} \\quad \\text{and} \\quad \\nabla_\\mu u = \\left(\\frac{\\partial u}{\\partial x_1},0,\\dots,0\\right); $$\n\\item more generally, if $\\mu$ is the uniform Hausdorff measure supported on a $k$-dimensional manifold $M$, then $T_\\mu$ is the tangent space to $M$ in the sense of the differential geometry.\n\\end{itemize}\nLet us remark that, if $v$ is a tangent field as defined above, {\\it i.e.}\\@ the operator $\\div(\\phi \\mu)$ is continuous for the $L^2_\\mu$-norm on $\\mathcal{D}(\\Omega)$, we have for any sequence $(u_n)_n$ of smooth functions having $0$ for limit in $L^2_\\mu$:\n$$ \\left|\\int_\\Omega \\nabla u_n \\cdot \\phi \\, \\text{\\normalfont d}\\mu \\right| \\leq C \\, ||u_n||_{L^2_\\mu} \\to 0.  $$\nThen, if $v \\in \\Gamma(0)$, we have $v \\cdot \\phi$ in $L^2_\\mu$ for any element $\\phi \\in X^2_\\mu$. We deduce that, with the above notations, the space $Q_\\mu(x)$ is pointwise orthogonal to $F_\\mu(x)$: it means that, up to a $\\mu$-negligible set, we have the inclusion between tangent spaces\n$$Q_\\mu(x) \\subseteq T_\\mu(x). $$\nWe are not able for the moment to prove the inverse inclusion, but the equality between this linear spaces holds for all the examples that we have~studied.\n \n\\section{Precise description and compactness result in one dimension}\n\n\\subsection{The main results}\n\nLet us now give a precise pointwise description of the tangent space $T_\\mu(x)$ when $d=1$ and $\\Omega$ is a bounded interval of $\\mathbb{R}$ (which we denote by $I$). In this case, there are only two options for $T_\\mu(x)$ which are of course $\\{0\\}$ and $\\mathbb{R}$, and the definitions of the tangent space give the following characterizations:\n\n\\noindent {\\bf Fact.} Let $B \\subseteq I$ be a Borel set with $\\mu(B) > 0$. We have the following implications:\n\\begin{enumerate}\n\\item if any $v \\in \\Gamma(0)$ is $\\mu$-a.e.\\@ null on $B$, then $T_\\mu = \\mathbb{R}$ $\\mu$-a.e.\\@ on $B$;\n\\item if, for any $u \\in H^1_\\mu$, there exists a gradient of $u$ which is $\\mu$-a.e.\\@ null on $B$, then $T_\\mu = 0$ $\\mu$-a.e.\\@ on $B$;\n\\item if there exists a gradient of $0$ which is positive $\\mu$-a.e.\\@ on $B$, then $T_\\mu$ = $0$ $\\mu$-a.e.\\@ on $B$.\n\\end{enumerate}\n\n\\noindent {\\bf Notations.} We denote by:\n\\begin{itemize}\n\\item $\\mu = \\mu_a + \\mu_s$ , where $\\mu_a$ and $\\mu_s$ are respectively the absolutely continuous and the singular part of $\\mu$ with respect to the Lebesgue measure;\n\\item $A$ a Lebesgue-negligible set on which is concentrated $\\mu_s$;\n\\item $f$ the density of $\\mu_a$, and\n$$ M = \\left\\{x \\in I : \\; \\forall \\varepsilon > 0, \\; \\int_{I \\cap B(x,\\varepsilon)} \\frac{\\, \\text{\\normalfont d} t}{f(t)} = +\\infty \\right\\}$$\nwhich is a closed set of $I$ verifying $1\/f \\in L^1_{loc}(I\\setminus M)$.\n\\end{itemize}\nNotice that if $\\mu$ is absolutely continuous with respect to $\\mathcal{L}^1$, the Sobolev space with respect to $\\mu$ (thus, to $f$) is well-defined exactly ``outside of the set $M$''. In our case, we find an analogous result:\n\n\\begin{theo}\nFor $\\mu$-a.e.\\@ $x \\in I$, the tangent space is given by\n$$ T_\\mu(x) =\n\\left\\{\n\\begin{array}{ll}\n\\{0\\} & \\text{if } x \\in M\\cup A \\\\\n\\mathbb{R} & otherwise.\n\\end{array}\n\\right.\n $$\n\\end{theo}\n\nLet us give a short comment of this result. Saying that the tangent space is $\\mathbb{R}$ on a set $B$ means exactly that, if $u \\in H^1_\\mu$ is given, all the gradients of $u$ are equals on $B$. In our case, let us denote by $V = I \\setminus (M \\cup A)$ and $U = I \\setminus M$. Notice that $U$ is an open subset of $I$ coinciding with $V$ up to the $\\mathcal{L}^1$-negligible set $A$. Let us fix $u \\in H^1_\\mu$. We will prove that the distributional derivative of $u|_U$ is well-defined, belongs to $L^2_f$ and that, if $v \\in \\Gamma(u)$, $u'=v$ $\\mu$-a.e.\\@ on $V$; therefore, $u'|_V$ is the only gradient of $u$ on the set $V$.\n\nFirst, let us recall that if $u$ is an element of $L^2_\\mu$, its restriction to $U$ belongs to $L^2_f(U)$, which is included into $L^1_{loc}(U)$ by definition of $M$. The weak derivative of $u|_U$ is thus well-defined. If $\\phi$ is a test function with support in $U$ and $(u_n)_n$ a sequence of regular functions such that $(u_n,u'_n) \\to (u,v)$ in $L^2_\\mu$, testing $v-u'$ against $\\phi$ gives\n$$|< v-u',\\phi>_{\\mathcal{D}'(U),\\mathcal{D}(U)}| = \\lim\\limits_{n \\to +\\infty} \\left| \\int_I (u_n-u) \\phi'\\right| \\leq ||u_n-u||_{L^2_f} \\left(\\int_I \\frac{(\\phi')^2}{f} \\right)^{1\/2} $$\nwhere the last inequality comes from the H\\\"o lder inequality, and the last term is finite since $\\phi'$ is bounded and $1\/f$ integrable on the support of $\\phi$. This proves that $u'_n \\to v$ in the sense of distributions on $U$. Then $v|_U$ is the weak derivative of $u$ on this set, and we know that $v|_U \\in L^2_f(U)$.\n\nFinally, any element $u \\in H^1_\\mu$ gives by restriction an element of the weighted Sobolev space $H^1_f(U)$ and, on $U$, $\\nabla_\\mu u$ and $u'$ are coinciding a.e.\\@ for the regular part $f \\, \\mathcal{L}^d$ of $\\mu$. To summarize, we have just proved the following:\n\n\\begin{prop} We denote by $V = I \\setminus (M\\cup A)$. Let us recall that $A$ is Lebesgue-negligible and that $V \\cup A$ is open; we still denote by $H^1_f(V)$ the weighted Sobolev space $H^1_f(V\\cup A)$. Then, a measurable function $u$ belongs to the Sobolev space $H^1_\\mu(I)$ if and only if the two following conditions are satisfied:\n$$u \\in L^2_\\mu(I) \\quad and \\quad u|_V \\in H^1_f(V) $$\nand in this case, its $\\mu$-Sobolev norm is given by\n$$||u||_{H^1_\\mu(I)}^2 = ||u||_{L^2_\\mu(I)}^2 + ||u'||_{L^2_f(V)}^2 $$\nwhere $u'$ is the weak derivative of $u|_V$.\n\\end{prop}\n\n\\noindent {\\bf Compactness result in $H^1_\\mu(I)$.} In order to examine variational problems in this Sobolev spaces, the following compactness result is useful (it is already known in the case of the Lebesgue measure, as a consequence of the Rellich theorem):\n\n\n\\begin{prop} Let $(u_n)_n$ be a bounded sequence of $H^1_\\mu(I)$. Then there exists a subsequence $(u_{n_k})_k$ which admits a pointwise limit $u$ on $\\mu$-a.e.\\@ every point on which $T_\\mu$ is $\\mathbb{R}$.\n\\end{prop}\n\n\\begin{proof} We know that $V$ is exactly (up to a $\\mu$-negligible set) the set of points where $T_\\mu$ is $\\mathbb{R}$. We still denote by $U = I \\setminus M$. $U$ is an open set and we have $U = V \\cup A$. We will show that $(u_n)_n$ admits a subsequence which is pointwise convergent on $\\mathcal{L}^1$-a.e.\\@ any point of $U$: it will be enough to conclude that this subsequence is $\\mu$-a.e.\\@ convergent on $V$, since $\\mu|_V$ is absolutely continuous with respect to the Lebesgue measure.\n\nThe sequence $(u_n)$ is bounded in $H^1_\\mu(I)$, thus the sequence $(u_n|_U)_n$ is bounded in the weighted Sobolev space $H^1_f$. But since $U$ is exactly the set of points around which $1\/f$ is integrable, we know that $L^2_f(U) \\hookrightarrow L^1_{loc}(U)$; this implies that the sequence of the weak derivatives of $u_n$ (which are functions of $L^2_f(U)$) is bounded in $L^1_{loc}(U)$. Then $(u_n)_n$ is bounded in the Sobolev space $W^{1,1}_{loc}(U)$, and admits a subsequence which is strongly convergent in $L^1(K)$, for any compact subset $K$ of $I$. We can again extract a subsequence which is pointwise convergent on $\\mu$-a.e.\\@ point of $I$; the proof is complete.\\end{proof}\n\n\n\\subsection{First part of the proof: the regular part, outside of the critical set}\n\nFirst, let us prove that $T_\\mu = \\mathbb{R}$ outside of $M \\cup A$. Using the first characterization of the tangent space, we take an element $g$ of $\\Gamma(0)$ and we want to show that $g = 0$ $\\mu$-a.e.\\@ outside of $M \\cup A$; by definition of~$A$, it is enough to show that $g = 0$ $\\mathcal{L}^1$-a.e.\\@ on $U$. As in the above remark, taking a sequence of regular functions $u_n \\to 0$ with $u'_n \\to g$ and a test function $\\varphi$ such that $1\/f$ is integrable on the support of $\\varphi$, we~obtain\n$$ \\left|\\int_U u_n' \\varphi\\right| = \\left|\\int_U u_n \\varphi'\\right| \\leq \\int_U \\left|u_n \\sqrt{f}\\right| \\left|\\frac{\\varphi'}{\\sqrt{f}}\\right| \\leq \\left( \\int_U u_n^2 f \\right)^{\\frac{1}{2}} \\left( \\int_U \\frac{\\varphi'^2}{f} \\right)^{\\frac{1}{2}} $$\nwhich goes to $0$ as $n \\to +\\infty$. The same computation gives $\\int_U u'_n \\varphi \\to \\int_U g\\varphi$. We deduce that $g = 0$~$\\mathcal{L}^1$-a.e.\n\n\\subsection{Second part: the singular part of the measure}\n\nSecond, we prove that $T_\\mu = \\{0\\}$ for the singular part of $\\mu$. We use the third characterization of the tangent space and build a sequence of $C^1$ functions $(u_n)_n$ such that\n$$u_n \\to 0 \\quad \\text{and} \\quad u'_n \\to \\mathds{1}_A \\quad \\text{in } L^2_\\mu.$$\nwhere $\\mathds{1}_A$ is the characteristic function of the set $A$; this will prove that $\\mathds{1}_A \\in \\Gamma(0)$ and imply the result.\n\nFor $n \\in \\mathbb{N}$, let $\\Omega_n$ be an open set such that $A \\subseteq \\Omega_n$ and $\\mu(\\Omega_n \\setminus A)+\\mathcal{L}^1(\\Omega_n) \\leq 1\/n$. By Lusin theorem, there exists a continuous function $v_n$ with $0 \\leq v_n \\leq 1$ on $I$ and\n$$ (\\mu+\\mathcal{L}^1)(\\{x \\in I : v_n(x) \\neq \\mathds{1}_{\\Omega_n}(x) \\}) \\leq 1\/n $$\nLet us consider $u(x) = \\int_a^x v_n(x) \\, \\text{\\normalfont d} x$, where $a$ is the lower bound of $I$. Then we have:\n\\begin{itemize}\n\\item for any $x \\in I$, \n$$|u_n(x)| \\leq \\int_I(|v_n-\\mathds{1}_{\\Omega_n}|(t)+\\mathds{1}_{\\Omega_n}(t)) \\, \\text{\\normalfont d} t \\leq \\mathcal{L}^1(\\{v_n \\neq \\mathds{1}_{\\Omega_n}\\}) + \\mathcal{L}^1(\\Omega_n) \\leq 2\/n $$\nthus $(u_n)_n$ goes to $0$ uniformly, and also in the space $L^2_\\mu$;\n\\item on the other hand, since $u'_n = v_n$ coincides with $\\mathds{1}_A$ outside of a set $E_n$ such that $\\mu(E_n) \\leq 1\/n$, we~have\n$$ \\int_I |u'_n(x)-\\mathds{1}_A(x)|^2 \\, \\text{\\normalfont d}\\mu(x) \\leq ||v_n-\\mathds{1}_A||_\\infty^2 \\, \\mu(E_n) \\leq 4\/n $$\nthus $u'_n \\to \\mathds{1}_A$ in $L^2_\\mu(I)$.\n\\end{itemize}\nWe obtain that $\\mathds{1}_A \\in \\Gamma(0)$, which guarantees that $T_\\mu = 0$ on $A$.\n\n\\subsection{Third part: the critical set}\n\nThis part is more difficult. Given a function $u \\in C^1\\left(\\overline{I}\\right)$, we build a sequence $(u_n)_n$ of regular functions (say, $C^1$) such that $u_n \\to u$ and $u'_n \\to v$ for the $L^2_\\mu$-norm, with $v=0$ on $M$. The strategy is the following:\n\\begin{itemize}\n\\item given a set $\\Omega_n$ which is ``almost'' $M$, we start from a function $u_n$ which coincides with $u$ outside of $\\Omega_n$ and is piecewise constant on $\\Omega_n$ (so that its derivative is null on $M$);\n\\item then, using the fact that the discontinuity points of $u_n$ belong to $M$, we regularize $u_n$ around this points so that its derivative stays small for the $L^2_\\mu$-norm.\n\\end{itemize}\n\nFirst, we build our set $\\Omega_n$:\n\n\\begin{lem} Let us denote by $(x_n)_n$ a sequence containing all the atoms of $\\mu$. For $n \\in \\mathbb{N}$, there exists a set $\\Omega_n$ such that:\n\\begin{itemize}\n\\item $\\Omega_n = \\bigcup\\limits_{i=1}^{p_n} \\, ]a_i,b_i[ \\, $, with $b_i < a_{i+1}$ for each $i$, and $\\, ]a_i,b_i[ \\,  \\cap M \\neq \\emptyset$;\n\\item $\\Omega_n \\supseteq M \\setminus \\{x_1,\\dots,x_n\\}$;\n\\item $\\mu(\\Omega_n \\setminus ( M \\setminus \\{x_1,\\dots,x_n\\})) \\leq 1\/n.$\n\\end{itemize}\n\\end{lem}\n\n\n\\begin{proof} Let $U_n$ be an open set such that $M \\subseteq \\Omega_n$ and $\\mu(U_n \\setminus M) \\leq 1\/n$ (such a set exists since $\\mu$ is regular from above); $U_n$ is a union of open intervals, and since $M$ is compact we can assume this union to be finite. We denote by $\\Omega_n = U_n \\setminus \\{ x_1,\\dots,x_n\\}$. It is still a finite union of open intervals, containing $M \\setminus \\{ x_1, \\dots, x_n\\}$ and with $\\mu(\\Omega_n \\setminus ( M \\setminus \\{x_1,\\dots,x_n\\})) \\leq 1\/n$. Moreover, we may assume that all these intervals contain an element of $M$: it is enough to remove from $\\Omega_n$ the intervals which do not contain any element of $M$ (if after that we obtain $\\Omega_n = \\emptyset$, it means that $M \\subseteq A$ and we already know that $T_\\mu = \\{0\\}$ on $A$, so there is nothing to prove). \\end{proof}\n\n\\par Let us thus take a sequence $(g_n)_n$ of piecewise constant functions such that $g_n \\to u$ in $L^2_\\mu$ (it is possible since $u$ is continuous, thus can be approximated uniformly on $I$ by a sequence of piecewise functions) and $||g_n||_\\infty \\leq C$, where $C$ only depends on $u$; we replace $g_n$ by $u$ outside of the set $\\Omega_n$ (the new function will still be called $g_n$), so that we have now\n\\begin{itemize}\n\\item $g_n \\to u$ in $L^2_\\mu$;\n\\item $g_n$ coincides with $u$ outside of $\\Omega_n$;\n\\item $g_n$ coincides on $\\Omega_n$ with a piecewise constant function.\n\\end{itemize}\n\n\\par We begin by regularizing $g_n$ around the endpoints of the intervals forming $\\Omega_n$. Let $\\varepsilon_n > 0$ be small enough so that:\n\\begin{itemize}\n\\item $a_i+\\varepsilon_n < b_i-\\varepsilon_n$, for each $i$ (we will set $a'_i = a_i+\\varepsilon_n$ and $b'_i = b_i-\\varepsilon_n$);\n\\item $]a'_i,b'_i[ \\, $ contains at least an element of $M$, for each $i$;\n\\item on $\\, ]a_i,b_i[ \\, $, $g_n$ has not any discontinuity point outside $\\, ]a'_i,b'_i[ \\, $;\n\\item if we denote by $\\Omega'_n$ the union of the intervals $\\, ]a'_i,b'_i[ \\, $, we have $\\mu(\\Omega_n \\setminus \\Omega'_n) \\leq 1\/n$.\n\\end{itemize}\n\n\\begin{lem} There exists a function $w_n$ coinciding with $g_n$ outside of $\\Omega_n \\setminus \\Omega'_n$, and such that, on each interval $\\, ]a_i,a'_i[ \\, $ and $\\, ]b'_i,b_i[ \\, $,\n\\begin{itemize}\n\\item $w_n$ and $w'_n$ are bounded by constants depending only on $u$ and $u'$;\n\\item $w_n(a_i) = u(a_i)$, $w'_n(a_i) = u'(a_i)$ and $w'_n=0$ on a (small) open interval having $a'_i$ for upper bound;\n\\item $w_n(b_i) = u(b_i)$, $w'_n(b_i) = u'(b_i)$ and $w'_n=0$ on a (small) open interval having $b'_i$ for lower bound.\n\\end{itemize}\n\\end{lem}\n\n\\begin{proof} It is enough to replace $g_n$ on the interval $\\, ]a_i,a_i+\\varepsilon_n[ \\, $ by the function $x \\mapsto Q(a_i+x)$ where\n$$Q(t) = -\\frac{u'(a_i)}{2\\varepsilon_n} t^2 + u'(a_i) t + u(a_i) $$\n(so that $Q(0)=u(a_i)$, $Q'(0)=Q'(a_i)$ and $Q'(\\varepsilon_n)=0$), to scale the new function on the interval $\\, ]a_i,a'_i[ \\, $ by replacing it by\n$$ x \\mapsto \\left\\{ \\begin{array}{ll}\nw_n(a_i+2(x-a_i)) & \\text{if } a_i \\leq x \\leq a_i+\\varepsilon_n\/2 \\\\\nw_n({a'_i}^-) & \\text{otherwise}\n\\end{array} \\right.$$\nand to make a similar construction on the interval $\\, ]b_i-\\varepsilon_n,b_i[$. \\end{proof}\n\nSince $g_n$ and $w_n$ are bounded uniformly in $n$ and coincide outside of the set $\\Omega_n \\setminus \\Omega'_n$, whose measure is at most $1\/n$, the sequence $(w_n)_n$ still converges to $u$ in $L^2_\\mu$; moreover, we have\n$$ ||w'_n||_{L^2_\\mu(\\Omega_n \\setminus \\Omega'_n)} \\leq (2\/n) ||u'||_\\infty $$\nand for any discontinuity point $y$ of $w_n$, $w_n$ is piecewise constant on a (small) neighborhood of $y$. We now have to regularize $w_n$ around its discontinuity points, which belong to $\\Omega'_n$; this is possible with a small cost only if these points belong to the set $M$. For this reason we are interested by the following ``displacement'' procedure of the discontinuity points:\n\n\\begin{lem}\nFor any $n$, there exists a function $v_n$ such that\n\\begin{itemize}\n\\item $v_n = w_n$ outside of $\\Omega'_n$;\n\\item $v_n$ is still piecewise constant on $\\Omega'_n$;\n\\item any discontinuity point of $v_n$ belongs to $M$;\n\\item $v_n \\to u$ in $L^2_\\mu$.\n\\end{itemize}\n\\end{lem}\n\n\\begin{proof} We have to modify $w_n$ only on each interval $\\, ]a'_i,b'_i[ \\, $. On this interval, the number of discontinuity points of $w_n$ is finite; we denote these points by $a'_i \\leq x_1 < \\dots < x_n = b'_i$. We make the following construction:\n\\begin{itemize}\n\\item Let $m = \\inf([a'_i,b'_i] \\cap M)$. We define $v_n$ on the interval $[a'_i,m[ \\, $ (if it is nonempty) by setting $v_n = w_n({a'_i}^+)$.\n\\item Then we reiterate the construction starting from $m$:\n\\begin{itemize}\n\\item if $\\, ]m,b'_i[ \\,  \\cap M = \\emptyset$, we set $v_n = w_n({b'_i}^-)$ on this interval, and we are done;\n\\item otherwise, let $m'=\\inf(\\, ]m,b'_i[ \\,  \\cap M)$. We have naturally $m' \\geq m$. If $m \\geq x_n$, then we set $v_n = w_n({b'_i}^-)$ on $\\, ]m',b'_i[ \\, $,$w_n$ on $[m,m'[ \\, $ and we are done;\n\\item if $m = m' < x_n$, then we denote by $j$ the smallest index such that $x_j > m$, we set $v_n = w_n$ on $[m,x_j[ \\, $ and we reiterate this construction starting from $x_j$;\n\\item finally, if $m < m' < x_n$, we set $v_n = w_n$ on $[m,m'[ \\, $ and we reiterate this construction starting from $m'$.\n\\end{itemize}\n\\end{itemize}\n\nWith this construction, $w_n-v_n \\neq 0$ only on $\\Omega'_n$ and outside of the set $M$. Since $\\mu(\\Omega'_n \\setminus M) \\leq 1\/n $ and $w_n, v_n$ are uniformly bounded, we get $||v_n-w_n||_{L^2_\\mu} \\leq C\/n$, and we thus still have $v_n \\to u$. Moreover, by construction, $v_n$ is still piecewise constant on the set $\\Omega'_n$ and all its discontinuity points belong to $M$. \\end{proof}\n\nTo finish, we have to modify $v_n$ around each discontinuity point, so that the new function $u_n$ is regular and admits a derivative which is small for the $L^2_\\mu$-norm. This is possible thanks to the following result about embeddings between functional spaces:\n\n\\begin{lem} Let $J$ a bounded interval of $\\mathbb{R}$, and $\\mu$ a finite measure on $J$ with density $f > 0$. The following assertions are equivalent:\n\\begin{enumerate}\n\\item The function $1\/f$ belongs to $L^1(J)$\n\\item The space $L^2_\\mu(J)$ is continuously embedded into $L^1(J)$\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof} The direct implication is obvious and comes directly from the Cauchy-Schwarz inequality. For the converse one, let us assume that $\\int_J 1\/f = +\\infty$ and set\n$$E_n = \\left\\{t \\in J : \\frac{1}{n+1} \\leq f(t) < \\frac{1}{n} \\right\\} \\quad \\text{and} \\quad l_n = \\mathcal{L}^1(E_n). $$\nWe know that $\\sum\\limits_n l_n < +\\infty$ (it is the length of $J$) and\n$$ \\sum\\limits_n nl_n = \\sum\\limits_n \\int_J n \\, \\mathds{1}_{\\{n-1 \\leq 1\/f \\leq n\\}} \\geq \\sum\\limits_n \\int_J \\frac{1}{f} \\, \\mathds{1}_{\\{n-1 \\leq 1\/f \\leq n\\}} \\geq \\int_J \\frac{1}{f} = +\\infty $$\nthus $\\sum\\limits_n nl_n = +\\infty$. We will build a function $U$ which is constant on each set $E_n$, belongs to $L^2_\\mu$ and does not belong to $L^1$. If we denote by $u_n$ the value of $U$ on $E_n$, it is equivalent to find a sequence $(u_n)_n$ verifying\n$$ \\sum\\limits_n u_n^2 \\, (nl_n) < +\\infty \\quad \\text{and} \\quad \\sum\\limits_n |u_n| \\, l_n = +\\infty $$\nTo summarize, we want to prove the following statement: for any sequence $(l_n)_n$ of positive numbers such that $\\sum\\limits_n n l_n = +\\infty$ and $\\sum\\limits_n l_n < +\\infty$, there exists a sequence $(u_n)_n$ of positive numbers such that $\\sum\\limits_n u_n^2 (nl_n) < +\\infty$ and $\\sum\\limits_n nu_n = +\\infty$. By contraposition, it is equivalent to the following: for any sequence $(l_n)_n$ of positive numbers such that $\\sum\\limits_n l_n < +\\infty$, if the following implication holds:\n$$ \\left( \\sum\\limits_n u_n^2 \\,(n l_n) < + \\infty \\right) \\Rightarrow \\left(\\sum\\limits_n l_n |u_n| < + \\infty \\right)\n $$\nthen we have $\\sum\\limits_n n l_n < +\\infty$. This result can be seen as a corollary of the Banach-Steinhaus theorem. Denoting by $\\ell^2_{nl_n}$ the space of sequences $(u_n)_n$ such that $\\sum\\limits_n u_n^2 \\,(n l_n) < + \\infty$, the operator\n$$ T_N : u \\in \\ell^2_{nl_n} \\longmapsto \\sum\\limits_{n=0}^N l_n u_n  $$\nis linear continuous with norm $\\left(\\sum\\limits_{n=0}^N nl_n\\right)^{1\/2}$ and the assumption about $(l_n)_n$ is equivalent to\n$$\\forall u \\in \\ell^2_{nl_n} \\quad \\sup\\limits_{N \\in \\mathbb{N}} |T_N(u)| < + \\infty.$$\nBy Banach-Steinhaus theorem, we get $\\sup\\limits_{N \\in \\mathbb{N}} ||T_N|| < +\\infty$ and $\\sum\\limits_{n \\in \\mathbb{N}} nl_n < +\\infty$; the proof is complete.\n\\end{proof}\n\n\\noindent {\\bf End of the proof of Theorem 2.1.} Thanks to the two last lemmas, we are now able to transform the function $w_n$ into a $C^1$ function $u_n$, which will provide us our approximation of $u$. Let us recall that $v_n$ coincides with $u$ outside $\\Omega_n$, is piecewise constant on $\\Omega'_n$, all its discontinuity points are located in $M$ and each of this points admits a neighborhood where $v_n'$ is null. Denoting by $y_1 < \\dots < y_p$ the discontinuity points of $v_n$, we find $\\varepsilon_n$ such that, for each $j$, $v_n$ is constant on $\\, ]y_j-\\varepsilon_n,y_j[ \\, $ and $\\, ]y_j,y_j+\\varepsilon_n[ \\, $. Moreover, we have:\n$$ \\sum\\limits_{j=1}^p \\mu(\\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\, ) \\leq 2p\\varepsilon_n + \\sum\\limits_{j=1}^p \\mu(\\{y_j\\}). $$\nWe take $\\varepsilon_n$ small enough so that $2p \\varepsilon_n \\leq 1\/n$. On the other hand, since each $y_j$ does not belong to the set $\\{x_1,\\dots,x_n\\}$ of the ``big atoms'' of $\\mu$, we have\n$$ \\sum\\limits_{j=1}^p \\mu(\\{y_j\\}) \\leq \\sum\\limits_{k \\geq n} \\mu(\\{x_k\\}). $$\nTherefore,\n$$ \\mu \\left( \\bigcup\\limits_{j=1}^p \\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\,  \\right) \\xrightarrow[n \\to +\\infty]{} 0. $$\nOn the interval $\\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\, $, thanks to Lemma 2.4, $L^2_\\mu$ is not embedded into $L^1$, thus we can find a regular function $g_j$ such that\n$$ \\int_{y_j-\\varepsilon_n}^{y_j+\\varepsilon_n} g_j = v_n(y_j^+)-v_n(y_j^-) \\quad \\text{and} \\quad \\int_{y_j-\\varepsilon_n}^{y_j+\\varepsilon_n} g_j^2 \\, \\text{\\normalfont d}\\mu \\leq \\frac{1}{nq}. $$\nThen, we set\n$$ u_n(x) = \\left \\{ \\begin{array}{ll}\n\\tilde{v}_n(y_j-\\varepsilon_n) + \\int_{y_j-\\varepsilon_n}^x g_j & \\text{ if } y_j-\\varepsilon_n \\leq x \\leq y_j+\\varepsilon_n \\\\\nv_n(x) & \\text{ otherwise.}\n\\end{array}\n\\right. $$\nThis functions $u_n$ form our desired approximation of $u$:\n\n\\begin{prop} This sequence $(u_n)_n$ satisfies $u_n \\to u$ and $u'_n \\to v$ in the space $L^2_\\mu$, where\n\n$$v(x) =  \\left\\{ \\begin{array}{ll} u'(x) & \\text{if } x \\notin M \\text{ or is an atom of } \\mu  \\\\ 0 & \\text{otherwise.} \\end{array} \\right.$$\nConsequently, $T_\\mu = \\{0\\}$ $\\mu$-a.e.\\@ on $M$. \\end{prop}\n\n\\begin{proof} We know that $v_n \\to u$, thus $u_n \\to u$ in the space $L^2_\\mu$ outside of the intervals $\\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\, $. But since the total mass of these intervals goes to $0$ and $(u_n)_n$ is uniformly bounded, we get $u_n \\to u$. For the derivative, since $u_n=u$ outside of $\\Omega_n$, we have\n$$ ||u'_n-v||_{L^2_\\mu}^2 = ||u'_n-v||^2_{L^2_\\mu(\\Omega_n)}  = ||u'_n-v||^2_{L^2_\\mu(\\Omega_n \\setminus M)} + ||u'_n-v||^2_{L^2_\\mu(M \\setminus \\{x_1,...,x_n\\})} $$\nwhere the first term goes to 0 (since $(u_n)_n$ is uniformly bounded and $\\mu(\\Omega_n \\setminus M)$ goes to 0); for the second one, we have $v=0$ on $M$, thus it is enough to prove that $u'_n$ goes to $0$ for the $L^2_\\mu$-norm on $M \\setminus \\{x_1,\\dots,x_n\\}$; this term is bounded by\n$$ ||u'_n||_{L^2_\\mu(\\Omega'_n \\setminus \\{y_1,\\dots,y_p\\})}^2 + \\sum\\limits_{j=1}^p u'_n(y_j) \\mu(\\{y_j\\}).  $$\nSince $(u'_n)_n$ is uniformly bounded, we know that the second term goes to $0$, and since $u_n$ is constant outside of the intervals $\\, ]y_j-\\varepsilon_n,y_j+\\varepsilon_n[ \\, $ the first one is equal to\n$$\\sum\\limits_{j=1}^p \\int_{y_j-\\varepsilon_n}^{y_j+\\varepsilon_n} g_j^2 d\\mu$$\nwhich, by definition of $g_j$, is smaller than $1\/n$. This completes the proof. \\end{proof}\n\n\\section{Application to a transport problem with gradient penalization}\n\n\\subsection{Problem statement, and the easiest case}\n\nWe investigate the following problem, which is somehow intermediate between optimal transportation and elasticity theory:\n\n$$ \\inf\\left\\{ \\int_\\Omega (|T(x)-x|^2 + |\\nabla T(x)|^2) \\, \\text{\\normalfont d} \\mu(x) \\right\\} \\, , $$\nwhere the infimum is taken among all maps $T:\\Omega \\to \\mathbb{R}^d$ with prescribed image measure $T_\\# \\mu = \\nu$ and admitting a Jacobian matrix $\\nabla T$ in a suitable sense. Contrary to the Monge-Kantorovich optimal transport problem, if $\\mu$ has a density $f$ bounded from above and below, then the existence of a solution is obvious and comes from the direct method of the calculus of variations; more precisely:\n\n\\begin{prop} Let $f:\\Omega \\to \\mathbb{R}^d$ a measurable function such that $0 < c < f < C < +\\infty$ for some constants $c,C>0$. Let $\\nu \\in \\mathcal{P}(\\mathbb{R}^d)$. We assume that there exists at least one Sobolev transport map between $\\, \\text{\\normalfont d}\\mu = f \\cdot \\, \\text{\\normalfont d}\\mathcal{L}^d$ and $\\nu$. Then the problem\n$$ \\inf \\left\\{ \\int_\\Omega (|T(x)-x|^2 + |\\nabla T(x)|^2) f(x) \\, \\text{\\normalfont d} x : T \\in H^1(\\Omega), \\, T_\\#\\mu = \\nu \\right \\} $$\nadmits at least one solution.\n\\end{prop}\n\n\\begin{proof} Let $(T_n)_n$ be a minimizing sequence. We can extract from $(T_n)_n$ a sequence having, thanks to the Rellich theorem, a strong limit $T$ in $L^2$, and we also can assume that $T_n \\to T$ $\\mathcal{L}^1$-a.e.\\@ on $\\Omega$, thus $\\mu$-a.e.\\@ on $\\Omega$. Then for any function $\\phi \\in C_b(\\mathbb{R}^d)$ we have\n$$ \\forall n \\in \\mathbb{N} \\quad \\int_\\Omega \\phi(T_n(x)) \\, \\text{\\normalfont d}\\mu(x) = \\int_{\\mathbb{R}^d} \\phi(y) \\, \\text{\\normalfont d}\\nu(y). $$\nThanks to the pointwise $\\mu$-a.e.\\@ convergence of $(T_n)_n$, we can pass to the limit in the left-hand-side of this equality, which gives\n$$ \\forall \\phi \\in C_b(\\mathbb{R}^d) \\quad \\int_\\Omega \\phi(T(x)) \\, \\text{\\normalfont d}\\mu(x) = \\int_{\\mathbb{R}^d} \\phi(y) \\, \\text{\\normalfont d}\\nu(y) $$\nand $T$ satisfies the constraint on the image measure. Moreover, the functional that we consider is of course lower semicontinuous with respect to the weak convergence in $H^1(\\Omega)$, and $T$ minimizes our problem. \\end{proof}\n\n\\subsection{The general formulation, and the one-dimensional case}\n\nIf $\\mu$ is a generic Borel measure, we replace the term with the jacobian matrix of $T$ by $\\nabla_\\mu T$, so that our problem is now written\n\\begin{equation} \\inf \\left\\{ \\int_\\Omega (|T(x)-x|^2+|\\nabla_\\mu T(x)|^2) \\, \\text{\\normalfont d}\\mu(x) \\; : \\; T \\in H^1_\\mu(\\Omega) \\right\\}. \\label{1} \\end{equation}\n\nThe existence of solutions is not clear in general. In the case of the classical Sobolev space $H^1(\\Omega)$, we have seen that the key point to prove the existence is the following: from any minimizing sequence $(T_n)_n$ we can extract a sequence which converges $\\mathcal{L}^d$-a.e.\\@ on $\\Omega$, and this is enough to obtain that the limit is admissible. This is not possible in general, since we don't have any equivalent of Rellich compactness theorem for the Sobolev spaces with respect to a generic measure $\\mu$.\n\nIn the one-dimensional case, if $\\mu$ is the Lebesgue measure, it is known that the monotone transport map between $\\mathcal{L}^1$ and $\\nu$ is optimal for the problem \\eqref{1} (see \\cite{ls}). This result does not hold if we do not make any assumption of $\\mu$, but we can get an existence result thanks to the $\\mu$-a.e.\\@ compactness result of the second section:\n\n\\begin{theo} In dimension 1, the problem \\eqref{1} admits at least one solution. \\end{theo}\n\n\\begin{proof} Let us begin by rewriting precisely the functional that we consider in this case: we know that $T_\\mu = \\{0\\}$ on $M \\cup A$ and $\\mathbb{R}$ on $V$, so that we are now minimizing\n$$ J : U \\in H^1_\\mu(I) \\longmapsto \\int_{V} ((U(x)-x)^2+U'(x)^2) f(x) \\, \\text{\\normalfont d} x + \\int_{M \\cup A} (U(x)-x)^2 \\, \\text{\\normalfont d}\\mu(x). $$\nLet $(U_n)_n$ be a minimizing sequence. On the set $V$, which is exactly the set where $T_\\mu$ is~$\\mathbb{R}$, we can extract from $(U_n)_n$ a $\\mu$-a.e.\\@ (which means $\\mathcal{L}^1$-a.e.\\@ wherever $f \\neq 0$) pointwise convergent subsequence, whose limit is denoted by $U$; let us remark that $U$ is the weak limit of $(U_n)_n$ (up to a subsequence) in the space $H^1_f$, and by semicontinuity, we have\n$$ \\int_V ((U(x)-x)^2+U'(x)^2) f(x)\\, \\text{\\normalfont d} x \\leq \\liminf \\left(\\int_V ((U_n(x)-x)^2+U_n'(x)^2) f(x)\\, \\text{\\normalfont d} x \\right). $$\nMoreover, let us set, for $n \\in \\mathbb{N}$, $\\nu_n = (U_n)_\\#(\\mu|_{M \\cup A})$ and $\\tilde{U}_n$ the optimal transport map for the Monge-Kantorovich quadratic cost between the measures $\\mu|_{M \\cup A}$ and $\\nu_n$. It is well-known that $\\tilde{U}_n$ is the unique nondecreasing transport map between $\\mu|_{M \\cup A}$ and $\\nu_n$; because of compactness properties of nondecreasing maps, we can assume that $(\\tilde{U}_n)_n$ admits, for the $\\mu$-a.e.\\@ convergence, a limit $\\tilde{U}$. For any $n$, thanks to the optimality of $\\tilde{U}_n$, we have\n$$ \\int_{M \\cup A} (\\tilde{U}_n(x)-x)^2 \\, \\text{\\normalfont d}\\mu(x) \\leq \\int_{M \\cup A} (U_n(x)-x)^2 \\, \\text{\\normalfont d} \\mu(x) $$\nand by semicontinuity\n$$ \\int_{M \\cup A} (\\tilde{U}(x)-x)^2 \\, \\text{\\normalfont d} \\mu(x) \\leq \\liminf \\left(\\int_{M \\cup A} (U_n(x)-x)^2 \\, \\text{\\normalfont d} \\mu(x) \\right). $$\nThus, if we denote by\n$$ T_n(x) = \\left\\{ \\begin{array}{ll}\nU_n(x) & \\text{if } x \\in V \\\\\n\\tilde{U}_n(x) & \\text{if } x \\in M \\cup A\n\\end{array} \\right.\n\\qquad \\text{and} \\qquad\nT(x) = \\left\\{ \\begin{array}{ll}\nU_(x) & \\text{if } x \\in V \\\\\n\\tilde{U}(x) & \\text{if } x \\in M \\cup A\n\\end{array} \\right. $$\nwe have $T_n \\to T$ $\\mu$-a.e.\\@ on $I$, and\n$$ J(T) \\leq \\liminf \\left(\\int_V ((U_n(x)-x)^2+U_n'(x)^2) f(x)\\, \\text{\\normalfont d} \\right) +  \\liminf \\left(\\int_{M \\cup A} (U_n(x)-x)^2 \\, \\text{\\normalfont d} \\mu(x) \\right) = \\liminf J(U_n) $$\nwhere $(U_n)_n$ is a minimizing sequence for $J$ on the set of $H^1_\\mu$ transport maps between $\\mu$ and $\\nu$. Thus, it is enough to prove that $T$ satisfies the constraint on image measure to conclude. But for each $n$, by construction, $(T_n)_\\# \\mu = \\nu$ and the $\\mu$-a.e.\\@ convergence allows to obtain the same for the limit~$T$; the proof is complete. \\end{proof}\n\n\\noindent {\\bf Remark.} This result can be generalized to any functional $J: U \\mapsto \\int_\\Omega (L_1(x,U(x)) + L_2(\\nabla_\\mu U(x))) \\, \\text{\\normalfont d}\\mu(x)$, where $L_1$ and $L_2$ have one of the following forms:\n\\begin{itemize}\n\\item $L_1$ is a transport cost such that the nondecreasing map is optimal for the Monge-Kantorovich problem: it is the case if $L_1(x,u) = h(|x-u|)$, where $h$ is a convex function. Let us notice that in particular the statement holds if we study the problem of minimization of the norm of the gradient among all Sobolev transport maps (this corresponds to $L_1 = 0$). Of course we need to assume that the class\n$$ \\left \\{U \\in H^1_\\mu : U_\\# \\mu = \\nu \\text{ and } \\int_\\Omega L_1(x,U(x)) \\, \\text{\\normalfont d}\\mu(x) < +\\infty \\right \\} $$\nis nonempty (to guarantee that $J \\not\\equiv +\\infty$ on the set of admissible functions). Thanks to the quadratic structure of $H^1_\\mu$, this is automatically the case if $L_1$ is the quadratic cost and there exists a Sobolev transport map.\n\\item $L_2$ is ``quadratic'', so that the space where we study the problem is actually the Sobolev space $H^1_\\mu$. The natural cases are $L_2(\\nabla_\\mu U) = |\\nabla_\\mu U|^2$ or $|\\nabla_\\mu U - I_d|^2$, where $I_d$ is the identity matrix (in this last case, we can consider the functional $U \\mapsto ||U-\\text{id}||_{H^1_\\mu}$, which is a Sobolev version of the quadratic transport problem where we minimize $||U-\\text{id}||_{L^2_\\mu}$).\n\\end{itemize}\n\n\\subsection{Difficulties and partial results in any dimension}\n\nAs we said in the second section of this paper, we don't have a precise pointwise description of the $\\mu$-Sobolev space $H^1_\\mu(\\Omega)$ if $\\Omega$ is an open set of $\\mathbb{R}^d$, which was the key point for the compactness result. More precisely, the following results still hold in any dimension:\n\n\\begin{itemize}\n\\item Outside of the set\n$$ M = \\left\\{ x \\in \\Omega : \\forall \\varepsilon > 0, \\, \\int_{\\Omega \\cap B(x,\\varepsilon)} \\frac{\\, \\text{\\normalfont d} x}{f(x)} = +\\infty \\right\\} $$\nwe have $T_\\mu = \\mathbb{R}^d$, a.e.\\@ for the regular part of $\\mu$. The proof is identical to the one-dimensional case, based on the Cauchy-Schwarz inequality and the embedding $L^2_f(\\Omega \\setminus M) \\hookrightarrow L^1_{loc}$.\n\\item Of course, we don't have anymore $T_\\mu = 0$ for the singular part of $\\mu$: for instance, if $\\mu$ is uniform and supported on a segment, then $\\mu$ is singular, and we know that $\\operatorname{dim} T_\\mu = 1$ on any point. However, the tangent space on the atoms of $\\mu$ is known:\n\n\\begin{prop} If $x_0$ is an atom of $\\mu$, then $T_\\mu(x_0) = \\{0 \\}$. \\end{prop}\n\n\\begin{proof} Let us prove it if $x_0 = 0$. We want to build a sequence of functions $(u_n)_n$ such that\n$$ u_n \\to 0 \\quad \\text{and} \\quad \\nabla u_n \\to e \\, \\mathds{1}_{\\{0\\}} $$\nwhere $e$ is an arbitrary unit vector (this shows that any unit vector belongs to the space $T_\\mu(0)^\\perp$, thus $T_\\mu(0) = \\{0\\}$). For this goal, let us consider a smooth cutoff function $\\chi$ such that $0 \\leq \\chi \\leq 1$ and\n$$ \\chi_n(x) = 1 \\text{ if } 0 \\leq |x| \\leq 1 \\text{ and } \\chi_n(x) = 0 \\text{ if } |x| \\geq 2 $$\nand we denote by $\\chi_n(x) = \\chi(nx)$. We then set $u_n(x) = \\langle x,e \\rangle \\, \\chi_n(x)$ and show that $(u_n)_n$ is the function that we are looking for. First, noting that $u_n(0) = 0$, that $u_n$ is null outside of $B(0,2\/n)$ and that $0 \\leq u_n \\leq 1$ for any $n$, we have\n$$ ||u_n||_{L^2_\\mu} = ||u_n||_{L^2\\mu(\\Omega \\setminus x_0)} \\leq \\mu(B(0,2\/n))-\\mu(\\{0\\}) $$\nwhich goes to $0$ as $n \\to +\\infty$; this gives us $u_n \\to 0$ in $L^2_\\mu$. Second, for any~$n$, we have\n$$ \\nabla u_n(x) = \\chi_n(x) \\, e + \\langle e,x \\rangle \\, \\nabla \\chi_n(x) = \\chi_n(x) \\, e + \\langle e,x \\rangle \\, n \\nabla\\chi(nx). $$\nLet us notice that $\\nabla u_n(0) = e$ for any $n$, thus it is enough to prove that $||\\nabla u_n||_{L^2_\\mu (\\Omega \\setminus \\{0\\})} \\to 0 $. But $\\chi_n$ and $\\nabla \\chi_n$ are null outside of $B(0,2\/n)$ and if $0 < |x| \\leq 2\/n$ we have\n$$ |\\nabla u_n(x)| \\leq |e| |\\chi(nx)| + |\\langle e,x \\rangle| n |\\nabla \\chi(x)| \\leq C(1+n|x|) \\leq 3C $$\nwhere $C$ is an upper bound of $\\chi$ and $\\nabla \\chi$. Thus, $(\\nabla u_n)_n$ is uniformly bounded by a positive constant, and $\\nabla u_n - e\\, \\mathds{1}_{\\{0\\}}$ is supported on the set $B(0,2\/n) \\setminus \\{0\\}$ whose measure $\\mu$ goes to $0$. This completes the proof. \\end{proof}\n\\item Finally, we can prove that there exists absolutely continuous measure $\\mu$ such that $T_\\mu$ is neither $\\{0\\}$ nor $\\mathbb{R}^n$ on any point of $\\Omega$. We provide an explicit example:\n\\begin{prop} Let be $g : \\,]0,1[ \\,  \\to \\, ]0,+\\infty[ \\, $ such that $\\int_Jg = +\\infty$ for any open interval $J \\subseteq \\, ]0,1[ \\, $. Let $\\Omega = \\, ]0,1[ \\, ^2$, $f:(x,y) \\in \\Omega \\mapsto g(x)$, and $\\mu$ the measure with density $f$. Then the tangent space is the vertical line $\\mathbb{R} \\cdot e_2$ on $\\mu$-a.e.\\@ point of $\\Omega$.\\end{prop}\n\\begin{proof} We first show that $T_\\mu(x)$ is at most one-dimensional on $\\mu$-a.e.\\@ $x \\in \\Omega$. Let be $u(x,y)=x$. Since the tangent space of the measure $g(x) \\cdot \\mathcal{L}^1$ on $\\, ]0,1[ \\, $ is $\\{0\\}$, we can find a sequence of functions $(w_n)_n$ such that\n$$ \\int_{0}^1 |w_n(x)-x|^2 g(x) \\, \\text{\\normalfont d} x \\to 0 \\quad \\text{and} \\quad \\int_0^1 |w'_n(x)|^2 g(x) \\, \\text{\\normalfont d} x \\to 0 $$\nand we denote by $u_n(x,y)=w_n(x)$. It is clear that $\\nabla u_n \\to 0$ and $u_n \\to u$ in $L^2_\\mu$. Then, $\\nabla u = (1,0)$ but $\\nabla_\\mu u = (0,0)$ $\\mu$-a.e.\\@ on $\\Omega$. This would impossible if $T_\\mu$ was $\\mathbb{R}^2$ on a non-negligible set of $\\Omega$.\n\nWe now have to show that $\\mathbb{R} \\cdot e_2$ is included to $T_\\mu(x)$ for $\\mu$-a.e.\\@ $x$. For this, we prove that any element $v=(v_1,v_2)$ of $\\Gamma(0)$ satisfies $v_2 = 0$. Indeed, we have $v_2 = \\lim \\partial_2 u_n$ with $u_n \\to 0$. For any test function $\\phi$, integrating by parts with respect to $y$ (since the density of $\\mu$ depends only on $x$)~gives\n$$ \\int_\\Omega \\partial_2 u_n \\phi \\, \\text{\\normalfont d}\\mu = -\\int_\\Omega u_n \\partial_2\\phi \\, \\text{\\normalfont d}\\mu $$\nwhich goes to $0$ since $u_n \\to 0$ in $L^2_\\mu$. This gives $\\int_\\Omega v_2 \\phi \\, \\text{\\normalfont d}\\mu = 0$ for any $\\phi \\in \\mathcal{D}(\\Omega)$, thus $v_2 = 0$, and the proof is complete. \\end{proof}\n\nThis shows that we cannot hope to obtain a compactness result analogous to the one-dimensional case, where any bounded sequence in $H^1_\\mu$ has a subsequence which converges on $\\mu$-a.e.\\@ $x$ such that $T_\\mu(x) \\neq \\{0\\}$: it is enough to take a sequence of functions $(u_n)_n$ depending only on~$x$ and non-compact for the a.e.\\@ convergence.\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Intro}\n\nThe primary utilitarian motivation for supersymmetry being accessible\nto experiments at the electroweak scale, e.g., at the LHC, depends on\nits ability to alleviate the problem of fine-tuning of electroweak\nsymmetry breaking present in the Standard Model\n\\cite{EENZ,hep-ph\/0312378}.  A supplementary phenomenological\nmotivation for weak-scale supersymmetry is its ability to provide the\ncold dark matter required by astrophysics and cosmology \\cite{EHNOS,\nhep-ph\/9506380}. This is a natural feature of supersymmetric models\nthat conserve R parity, with the lightest neutralino $\\tilde{\\chi}^0_1$ being\nparticularly well-suited to provide the preferred amount of cold dark\nmatter if it is the lightest supersymmetric particle (LSP) and weighs\nless than about 1~TeV \\cite{EHNOS,etcEllis:1985jn}. Within the general\nsupersymmetric framework, one may find more plausible regions of the\nsupersymmetric parameter space that are less fine-tuned, in the sense\nthat the values of the model parameters chosen at some high input\nscale require less delicate adjustment in order to obtain the correct\nvalue of the electroweak scale \\cite{EENZ,Barbieri:1987fn}, as\nmeasured by $M_Z$, or the correct value of the cold dark matter\ndensity $\\Omega_{CDM}h^2$\n\\cite{Ellis:2001zk,hep-ph\/0603095,hep-ph\/0608135,King:2007vh}.\n\nIt is hard to make this type of plausibility argument at all rigorous:\nit is notoriously difficult to make probabilistic statements about the\nunique (by definition) Universe in which we live, it is largely a\nmatter of personal choice which derived quantity one should consider\nand which input parameters one wishes to avoid fine-tuning, it is\ndifficult to argue conclusively for the superiority of one measure of\nfine-tuning over any other, and even less easy to agree on a `pain\nthreshold' in the amount of fine-tuning one is prepared to tolerate\n\\cite{Barbieri:1987fn}. Nevertheless, within a given model framework\nwith its specific input parameters, it is legitimate to consider some\nimportant derived quantity such as $\\Omega_{CDM}h^2$, and compare the\namounts of fine-tuning required in different regions of its parameter\nspace, which frequently do not depend very sensitively on the specific\nsensitivity measure employed.\n\nMoreover, even if one does not accept that the less sensitive\nparameter regions are more plausible, measuring the dark matter\nfine-tuning may have other uses. For example, one hopes (expects) some\nday to discover supersymmetry and start to measure the values of its\nparameters. Unavoidably, these will have non-negligible measurement\nerrors, and these uncertainties propagate via the dark matter\nfine-tuning parameters into the calculation, e.g., of\n$\\Omega_{CDM}h^2$. One of the key features of supersymmetry is its\nability to provide a calculable amount of cold dark matter, and it is\ninteresting to know how accurately which of its parameters must be\nmeasured in order to calculate $\\Omega_{CDM}h^2$ with an accuracy\ncomparable to that quoted by astrophysicists and cosmologists\n\\cite{Ellis:2001zk,Battaglia:2003ab}. An accurate calculation of\n$\\Omega_{CDM}h^2$ might also reveal some deficiency of the\nsupersymmetric explanation of the cold dark matter, and possibly the\nneed for some other new physics in addition.\n\nIt is important to note that the parameters we refer to here are the\nGUT-scale soft supersymmetry-breaking masses and couplings, whereas\nexperiments would measure directly physical masses and mixings at much\nlower energies. Ideally, one would calculate the relic density\ndirectly from the low-energy measurements of MSSM parameters. However\nit will be difficult, if not impossible, to pin down all the key\nparameters using LHC data alone, except by making supplementary\nassumptions about the pattern of supersymmetry breaking at the GUT\nscale, as we do here. Assuming a structure of GUT-scale unification,\none may use experimental measurements to constrain these fewer\nhigh-energy parameters. The strength of the constraints will depend on\nthe magnitudes of these parameters and the experimental tools\navailable. Very likely some accelerator beyond the LHC will be needed,\nbut we do not yet know what will be available.  The fine-tuning\nmeasures we calculate here show clearly which of the high-energy\nparameters are most important for a precise calculation of the relic\ndensity, and hence contribute to the `wish list' for such an\naccelerator.\n\nFor these reasons, we make no further apologies for considering the\nfine-tuning of $\\Omega_{CDM}h^2$ in this paper, which we shall refer\nto as ``dark matter fine-tuning''\n\\cite{Ellis:2001zk,hep-ph\/0603095,hep-ph\/0608135,King:2007vh} to\ndistinguish it from the more commonly studied ``electroweak\nfine-tuning'' \\cite{EENZ,Barbieri:1987fn}, which we also consider for\ncompleteness. The issue of dark matter fine-tuning has been considered\npreviously in the context of several different models including the\nconstrained minimal supersymmetric extension of the Standard model\n(CMSSM) \\cite{Ellis:2001zk}, in which the soft supersymmetry-breaking\nscalar masses $m_0$, gaugino masses $m_{1\/2}$ and trilinear parameters\n$A_0$ are each assumed to be universal, a more general MSSM with\nnon-universal third family scalars and gaugino masses\n\\cite{hep-ph\/0603095}, a string-inspired non-universal model\n\\cite{hep-ph\/0608135} and SUSY GUTs with non-universal gaugino masses\n\\cite{King:2007vh}. Here we extend such considerations to models with\nnon-universal soft supersymmetry-breaking contributions to the Higgs\nmasses (NUHM).  Within this NUHM framework, the independent input\nparameters may be taken as \\cite{oldnuhm,Ellis:2002wv,hep-ph\/0210205}\n\\begin{equation}\n  a_{NUHM}=\\left\\{m_0,~m_{H_1},~m_{H_2},~m_{1\/2},~A_0,~\\tan\\beta,\n  ~\\text{sign}(\\mu)\\right\\},\n\\end{equation}\nand we take as our measure of dark matter fine-tuning the quantity\n\\begin{equation}\n  \\Delta_\\Omega \\; \\equiv \\; {\\rm Max}_i \\left| \\frac{a_i}{\\Omega_\\chi}\n  \\frac{\\partial \\Omega_\\chi}{\\partial a_i} \\right| .\n\\end{equation}\nOur objective will be three-fold: to compare the amount of dark matter\nfine-tuning required within the NUHM to that required within the\nCMSSM, to identify the regions of the NUHM parameter space that\nrequire relatively less (or more) dark matter fine-tuning, and thereby\nto quantify the accuracy in the determination of the GUT-scale NUHM\nparameters that would be needed in order to calculate $\\Omega_\\chi\nh^2$ with any desired accuracy.\n\nThe regions of the NUHM parameter space where $\\Omega_\\chi h^2$ falls\nwithin the range favoured by WMAP and other experiments has been\nstudied quite extensively, for example in \\cite{hep-ph\/0210205}.  It\nshares several features in common with the more restrictive CMSSM\nframework proposed in \\cite{Kane:1993td} and extensively studied in\n\\cite{Ellis:1999mm}.  For example, there are regions where $\\tilde{\\chi}^0_1$ -\nstau coannihilation is important, and others where $\\tilde{\\chi}^0_1$ pairs\nannihilate rapidly via direct-channel $H, A$ poles. However, other\npossibilities also occur. For example, there are regions where $\\tilde{\\chi}^0_1$\n- sneutrino coannihilation is dominant. Also there are regions where\nrapid-annihilation and bulk regions, which are normally separated by a\ncoannihilation strip, approach each other and may even merge. As we\ndiscuss below in more detail, the sneutrino coannihilation regions\nexhibit relatively high dark matter fine-tuning, whereas the `merger'\nregions may require significantly less dark matter fine-tuning.\n\nIn this work we provide a first calculation of the dark matter\nfine-tuning for the regions of the NUHM that are favoured by dark\nmatter measurements. In addition, we present a first calculation of\nthe electroweak fine-tuning within this model and update the parameter\nscans for the current measurement of the top mass.\n\nThe rest of the paper is laid out as follows. In\nSection~\\ref{sec:methods} we summarise the methods used in our\nnumerical studies. Next, in Section~\\ref{CMSSM} we review the familiar\ncase of the CMSSM, which serves as a baseline for later\ncomparison. Then, in Section~\\ref{NUHM} we study dark matter within\nthe NUHM model in which universality between the soft\nsupersymmetry-breaking masses of the sfermions (squarks and sleptons)\nand Higgs multiplets is broken. Finally, in Section~\\ref{Conc} we\npresent our conclusions.\n\n\\section{Methodology}\n\\label{sec:methods}\n\n\\subsection{Codes}\n\nIn order to study the low-energy phenomenology of the NUHM, we need a\ntool to run the mass spectrum from the GUT scale down to the\nelectroweak scale using the renormalisation group equations\n(RGEs)\\cite{Martin:1993zk}. For this purpose we use the RGE code {\\tt\nSoftSusy}~\\cite{hep-ph\/0104145}. This interfaces with the MSSM package\nwithin {\\tt micrOMEGAs}~\\cite{hep-ph\/0112278}, which we use to\ncalculate the dark matter relic density $\\Omega_{CDM}h^2$, $BR(b\\rightarrow s \\gamma)$ and $\\delta a_{\\mu}$. We\ntake $m_t=170.9$~GeV throughout.\n\n\\subsection{Theoretical, Experimental and Cosmological Bounds}\n\nAfter running the mass spectrum of any chosen model parameter set from\nthe GUT scale down to the electroweak scale, we perform a number of\nchecks on the phenomenological acceptability of the point chosen. A\npoint is ruled out if:\n\\begin{enumerate}\n\\item It does not provide radiative electroweak symmetry breaking\n  (REWSB). Such regions are displayed in light red in the subsequent\n  figures.\n\\item It violates the bounds on particle masses provided by the\n  Tevatron and LEP~2. Such regions are displayed in light\n  blue~\\footnote{The current LEP~2 bound on the lightest MSSM Higgs\n  stands at $114.4$~GeV. However, there is a theoretical uncertainty\n  of some $3$~GeV in the determination of the mass of the light\n  Higgs~\\cite{Allanach:2004rh}. Rather than placing a hard cut on the\n  parameter space for the Higgs mass, instead we plot a line at\n  $m_h=111$~GeV and colour the region in which $m_h <111$~GeV in very\n  light grey-blue.}.\n\\item It results in a lightest supersymmetric particle (LSP) that is\n  not the lightest neutralino. We colour these regions light green.\n\\end{enumerate}\n\nIn the remaining parameter space we display the 1- and 2-$\\sigma$\nregions for $\\delta a_{\\mu}$ and $BR(b\\rightarrow s \\gamma)$, as well as plotting the 2-$\\sigma$\nregion for the relic density allowed by WMAP and other observations.\n\n\\subsubsection{$\\delta a_{\\mu}$}\n\nPresent measurements of the anomalous magnetic moment of the muon\n$a_\\mu$ deviate from theoretical calculations of the SM contribution\nbased on low-energy $e^+ e^-$ data~\\footnote{There is a long-running\ndebate whether the calculation of the hadronic vacuum polarisation in\nthe Standard Model should be done with $e^+e^-$ data, or with $\\tau$\ndecay data. The weight of evidence indicates the $e^+e^-$ estimate is\nmore reliable so we use the $e^+ e^-$ value in our work.}. Taking the\ncurrent experimental world average and the state-of-the-art SM value\nfrom~\\cite{hep-ph\/0703049}, there is a discrepancy:\n\\begin{equation}\n  (a_\\mu)_{exp}-(a_\\mu)_{SM}=\\delta a_\\mu = (2.95\\pm\n  0.88)\\times 10^{-9},\n\\end{equation}\nwhich amounts to a 3.4-$\\sigma$ deviation from the SM value.  As\nalready mentioned, we use {\\tt micrOMEGAs} to calculate the SUSY\ncontribution to $(g-2)_\\mu$. The dominant theoretical errors in this\ncalculation are in the SM contribution, so in our analysis we neglect\nthe theoretical error in the calculation of the SUSY contribution.\n\n\\subsubsection{$BR(b\\rightarrow s \\gamma)$}\n\nThe variation of $BR(b\\rightarrow s \\gamma)$ from the value predicted by the Standard Model\nis highly sensitive to SUSY contributions arising from charged\nHiggs-top loops and chargino-stop loops. To date no deviation from the\nStandard Model has been detected. We take the current world average\nfrom~\\cite{hfag}, based on the BELLE~\\cite{hep-ex\/0103042},\nCLEO~\\cite{hep-ex\/0108033} and BaBar~\\cite{Aubert:2005cu}\nmeasurements:\n\\begin{equation}\n  BR(b\\rightarrow s \\gamma) = (3.55 \\pm 0.26) \\times 10^{-4}.\n\\end{equation}\nAgain, we use {\\tt micrOMEGAs} to calculate both the SM value of\n$BR(b\\rightarrow s \\gamma)$ and the SUSY contributions. It is hard to estimate the\ntheoretical uncertainty in the calculation of the SUSY contributions,\nbut note that there is an uncertainty of $10\\%$ in the NLO SM\nprediction of $BR(b\\rightarrow s \\gamma)$~\\cite{bsgNLO}~\\footnote{We recall that {\\tt\nmicrOMEGAs} calculates the SM contribution to $BR(b\\rightarrow s \\gamma)$ to NLO. A first\nestimate of the SM prediction of $BR(b\\rightarrow s \\gamma)$ to NNLO was presented\nin~\\cite{Misiak:2006zs}. This showed a decrease of around $0.4\\times\n10^{-4}$ in the central value of the SM prediction. The implementation\nof the NNLO contributions in the calculation is non-trivial and its\nimplementation in {\\tt micrOMEGAs} is currently underway. As a result\nwe do not include this decrease in the results we present, but instead\nnote that \\textit{positive} SUSY contributions to $BR(b\\rightarrow s \\gamma)$ look likely\nto be favoured in future. This would favour a negative sign of $\\mu$\nand thus cause tension with $(g-2)_\\mu$.}. As with $\\delta a_{\\mu}$, we plot the\n1-$\\sigma$ and 2-$\\sigma$ experimental ranges, and do not include a\ntheoretical error in the calculation.\n\n\\subsubsection{$\\Omega_{CDM}h^2$}\nEvidence from the cosmic microwave background, the rotation curves of\ngalaxies and other astrophysical data point to a large amount of cold\nnon-baryonic dark matter in the universe. The present\nmeasurements~\\cite{astro-ph\/0603449} indicate the following value for\nthe current cold dark matter density:\n\\begin{equation}\n  \\Omega_{CDM}h^2 = 0.106 \\pm 0.008.\n\\end{equation}\nWe calculate the relic dark matter density with {\\tt micrOMEGAs} using\nthe \\textit{fast} approximation. Given a low-energy mass spectrum,\n{\\tt micrOMEGAs} gives an estimated precision of $1\\%$ in the\ntheoretical prediction of the relic density. This is negligible\ncompared to the present observational error, so the 2-$\\sigma$ band\nplotted takes into account only the experimental error~\\footnote{We\nemphasize that the quoted $1\\%$ accuracy is for a given low-energy\nspectrum, which is obtained using {\\tt softsusy}.  However, there are\ndifferences in the details of the mass spectrum between\ncodes~\\cite{Allanach:2003jw}, for given high-energy inputs, and\ndifferent dark matter regions have different levels of sensitivity to\nthese variations: see~\\cite{Belanger:2005jk} for a detailed study. The\nresult of the discrepancies between codes is to move the dark matter\nregions slightly in the GUT scale parameter space. As we are\ninterested in broad features of these regions, rather than their\nprecise locations, these uncertainties are not important for our\npurposes.}.\n\nIn the following Sections, we calculate the dark-matter fine-tuning\nfor any point that lies within the $2\\sigma$ allowed region, and\nindicate the amount using colour coding. We also display electroweak\nfine-tuning contours over the different regions.\n\n\\section{The Constrained Minimal Supersymmetric Standard Model}\n\\label{CMSSM}\n\nWe first review the familiar Constrained Minimal Supersymmetric\nStandard Model (CMSSM) \\cite{Kane:1993td,Ellis:1999mm}. \nwhich serves as a standard to which we compare\nthe parameter space of the NUHM.\n\nThe CMSSM has a much simpler spectrum of soft masses than the full\nMSSM. First, all of the soft squark and slepton (mass)$^2$ matrices\nare chosen to be diagonal and universal at the GUT scale with the\ndiagonal entries equal to $m_0^2$. Secondly both the soft Higgs\n(mass)$^2$ are also set equal to $m_0^2$. Additionally, all the\ngaugino masses are assumed to be unified with a value $m_{1\/2}$ at the\nGUT scale. Finally, we take the trilinear coupling matrices to have\nonly one non-zero entry (the third-family dominance approximation) and\nassume that all these entries are equal to a common value\n$A_0$. Requiring that electroweak symmetry be broken radiatively to\ngive the observed electroweak boson masses, we trade the soft\nparameters $\\mu$ and $B$ for $\\tan\\beta$, the ratio of the Higgs vevs, and\nthe sign of $\\mu$. This results in a model with four free parameters\nand a sign:\n\\begin{equation}\n  a_{CMSSM} \\in \\left\\{\n  m_0,~m_{1\/2},~A_0,~\\tan\\beta,~\\text{sign}(\\mu)\\right\\}.\n\\end{equation}\nAlthough our main focus is the dark-matter fine-tuning, we also report\nthe required amounts of electroweak fine-tuning for\nspecific cases of interest.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/CMSSM,t,10.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small The $(m_{1\/2},m_0)$ plane of the CMSSM\n     with $A_0=0,~\\tan\\beta=10$ and sign$(\\mu)$\n    +ve.\\label{f:CMSSM,t10}}\n\\end{figure}\n\nIn Fig.~\\ref{f:CMSSM,t10} we show the $(m_0,m_{1\/2})$ plane of the\nCMSSM for $A_0=0,~\\tan\\beta=10$ and sign$(\\mu)$ positive. At low $m_0$ the\nparameter space is ruled out because $m_{\\tilde{\\tau}}<m_{\\tilde{\\chi}^0_1}$, resulting\nin a stau LSP (light green). Regions at low $m_{1\/2}$ are ruled out by\nLEP~2 bounds on the masses of the charginos and sleptons (light\nblue). A larger range of $m_0$ is ruled out by the absence of a light\nHiggs boson, and we shade the region with $m_h<111$~GeV (light grey\nwith a black boundary). Finally, the model fits the current $\\delta a_{\\mu}$\nmeasurement for low $m_0$ and $m_{1\/2}$. The $1\\sigma$ and $2\\sigma$\nbounds are shown as a short and long-dashed green lines\nrespectively. At larger $m_0$ and $m_{1\/2}$, the SUSY contribution\nbecomes small as the sparticles that contribute in loops become\nheavy. Therefore $\\delta a_{\\mu}\\neq 0$ favours relatively light soft masses.\n\nIn the remainder of the parameter space we find the regions that fit\nthe WMAP strip at $2\\sigma$, and for each such point we calculate the\nfine-tuning of the dark matter density. Each point is then plotted\nwith a colour that corresponds to the value of the tuning via the\nlog-scale on the right hand side.\n\n\\begin{table}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{A1} &\n      \\multicolumn{2}{|c|}{A2}\\\\\n      \\cline{2-5}\n                & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$     & 60    & 0.62 & 100 & 5.7\\\\\n      $m_{1\/2}$ & 200   & 0.99 & 500 & 5.8\\\\\n      $\\tan\\beta$   & 10    & 0.13 & 10  & 1.5\\\\\n      \\hline\n      $\\Delta_\\Omega$ & & 0.99 &     & 5.8\\\\\n      \\hline\n      $\\Delta_{EW}$   & & 37   &     & 190\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small A summary of the properties of points A1 and A2,\n    shown in Fig.~\\ref{f:CMSSM,t10}, chosen as representatives of the\n    bulk region (A1) and stau-coannihilation region (A2). We also\n    present a breakdown of the dark-matter and electroweak\n    fine-tunings with respect to each parameter of the CMSSM, except\n    for $A_0$, which we fix: $A_0=0$ here and elsewhere in this\n    paper.\\label{t:CMSSM,t10}}\n\\end{table}\n\nThere are two distinct regions that fit the WMAP measurement of the\nrelic density. The first region lies at $m_{1\/2}\\approx 200$~GeV,\n$m_0\\approx 70$~GeV and contains the point A1. This is the bulk\nregion, which is adjacent to the light blue region that is ruled out\nby the LEP~2 slepton mass constraint. The lightness of the sleptons\nenhances neutralino annihilation via $t$-channel slepton exchange to\nthe extent that it allows bino dark matter to fit the WMAP relic\ndensity measurement. This process is relatively insensitive to the\nprecise masses of the neutralino and the sleptons. This is reflected\nin the fact that much of the bulk region is plotted in yellow,\nsignifying $\\Delta^{\\Omega}<1$. From the breakdown of the tunings of point A1\nin Table~\\ref{t:CMSSM,t10} it is clear that the tuning is mainly in\nthe parameters $m_{1\/2}$ and $m_0$. The sensitivity to $m_0$ is to be\nexpected, as $t$-channel slepton exchange is clearly dependent on the\nmass of the exchanged particle. This also explains the sensitivity to\n$m_{1\/2}$ since, although the masses of the sleptons are determined by\n$m_0$ at the GUT scale, the running to low energies is dominated by\n$m_{1\/2}$. Thus, in this region the masses of the light sleptons are\nsensitive to both $m_0$ and $m_{1\/2}$. The relic density is also\ndependent upon the mass of the LSP, in this case (mainly) a bino with\na mass determined primarily by $m_{1\/2}$. It is therefore apparent why\nthe sensitivity of the bulk regions lies primarily with $m_0$ and\n$m_{1\/2}$. The fact that no precise cancellations or balancing of\nparameters is required explains why the tuning is low. However, the\nbulk region lies inside a light grey region signifying a Higgs with a\nmass less than $111$~GeV. Therefore we do not consider this region\nfurther here.\n\nThe second region that fits the observed relic density is the band\nthat contains point A2. This band lies alongside the area that is\nruled out by a stau LSP (light green). Along the edge of the light\ngreen region, the mass of the stau is close to that of the lightest\nneutralino, resulting in comparable number densities of both particles\naround the time of freeze-out. Thus many more annihilation channels\nmust be considered, such as the annihilations of $\\tilde{\\tau}-\\tilde{\\tau}$ and\n$\\tilde{\\tau}-\\tilde{\\chi}^0_1$ in addition to $\\tilde{\\chi}^0_1-\\tilde{\\chi}^0_1$. This suppresses the number\ndensity of SUSY particles and thus reduces the resultant dark matter\nrelic density. The effect of coannihilation depends strongly upon the\nnumber density of NLSPs at freeze-out. This is in turn very sensitive\nto the mass difference between the NLSP and the LSP. Thus, it may at\nfirst sight seem surprising that the $\\tilde{\\tau}$ coannihilation band is\nplotted here in green and red, corresponding to relatively low dark\nmatter fine-tuning $\\Delta^{\\Omega}=3-11$. The reason, as discussed\nin~\\cite{hep-ph\/0603095}, is that in this region the RGE running\nresults in the mass of the right handed $\\tilde{\\tau}$ and the mass of the\nlightest neutralino both being dominantly dependent on $m_{1\/2}$, with\nthe stau having a smaller secondary dependence on $m_0$. As a result,\ntheir masses vary together as the soft masses are varied, and $\\Delta\nm$ is remarkably insensitive to $m_0$ or $m_{1\/2}$. This mitigates the\nnormal sensitivity of the coannihilation region. We take the point A2\nas a representative point in this band and provide a breakdown of the\nindividual tunings in Table~\\ref{t:CMSSM,t10}. The tuning is equally\ndependent upon $m_0$ and $m_{1\/2}$, as expected.\n\nAs well as the dark-matter fine-tuning, for reference we also\ncalculate the fine-tuning required to fit the electroweak boson\nmasses. This calculation is performed across the parameter space using\nthe same measure as we use for the dark matter fine-tuning. We plot\nthe tuning using blue dot-dashed contours in Fig.~\\ref{f:CMSSM,t10},\nand label each contour with their respective values of\n$\\Delta^{EW}=\\text{max}(\\Delta^{EW}_a)$. We also list the value of\n$\\Delta^{EW}$ in the last row of Table~\\ref{t:CMSSM,t10} for both points\nwe consider.  The $W$ boson mass is in general the result of a careful\nbalancing act between the soft masses: the larger the soft masses, the\nmore precisely they must cancel to give the $W$ boson mass. Therefore\nit is unsurprising that the amount of fine-tuning required to obtain\nthe correct electroweak symmetry breaking increases smoothly with the\nincreasing soft masses. The electroweak and dark matter fine-tunings\nare largely independent.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/CMSSM,t,50.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small The $(m_0,~m_{1\/2})$ plane of the\n    CMSSM parameter space for $A_0=0$, $\\tan\\beta=50$ and sign($\\mu$)\n    +ve.\\label{f:CMSSM}}\n\\end{figure}\n\nOther regions in which the CMSSM fits the observed relic density are\nseen when we consider larger values of $\\tan\\beta$. In Fig.~\\ref{f:CMSSM}\nwe consider a wider range in the $(m_0,m_{1\/2})$ plane and take\n$\\tan\\beta=50$.\n\nLow $m_0$ is once again ruled out because the stau is the LSP (light\ngreen). Equally low $m_{1\/2}$ and $m_0$ are ruled out by searches for\nthe lightest Higgs (light grey with a black boundary) or LEP~2 bounds\non the chargino and light sleptons (solid light blue). Finally large\n$m_0$ and low $m_{1\/2}$ are ruled out as they do not give REWSB (light\nred).  We plot the different regions that fit the relic density using\nthe same colour coding for dark-matter fine-tuning as was used in\nFig.~\\ref{f:CMSSM,t10}.\n\n\\begin{table}\n  \\begin{center}\n    \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{A3} &\n      \\multicolumn{2}{|c|}{A4} &\n      \\multicolumn{2}{|c|}{A5} &\n      \\multicolumn{2}{|c|}{A6}\\\\\n      \\cline{2-9}\n                & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value\n& $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$     & 2030  & 27   & 540 & 5.0  & 277   & 19   & 1400& 13   \\\\\n      $m_{1\/2}$ & 500   & 18   & 600 & 8.0  & 350   & 12   & 250 & 22   \\\\\n      $\\tan\\beta$   & 50    & 2.0  & 50  & 76   & 50    & 48   & 50  & 8.2  \\\\\n      \\hline\n      $\\Delta_\\Omega$ & & 27   &     & 76   &       & 48   &     & 22   \\\\\n      \\hline\n      $\\Delta_{EW}$  &  & 150  &     & 230  &       & 92   &     & 48   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Here we summarize properties of the points A3-6,\n    shown in Fig.~\\ref{f:CMSSM}, which represent the higgsino\/bino\n    region (A3,6), the pseudoscalar Higgs funnel (A4) and the\n    stau-coannihilation region at large $\\tan\\beta$ (A5). We also present\n    a breakdown of the dark matter fine-tuning with respect to each\n    parameter, and also list the electroweak\n    fine-tuning.\\label{t:CMSSM}}\n\\end{table}\n\nWe now distinguish four regions in which the CMSSM can account for the\nobserved relic density of dark matter. The least interesting of these\nregions is the bulk region, which once again appears at low $m_0$ and\nlow $m_{1\/2}$, but requires a Higgs lighter than current search limits\nallow.  Moving on from the inaccessible bulk region, we again find the\ncoannihilation strip along the side of the region ruled out because\nthe stau is the LSP. In comparison to Fig.\\ref{f:CMSSM,t10}, it is\nthin and broken. This is a sampling artefact, arising because we have\nextended the range of the $m_0$ scan, and the resolution is not fine\nenough to resolve the full strip. In contrast to\nFig.\\ref{f:CMSSM,t10}, here the strip is plotted in purple,\ndesignating a dark matter fine-tuning in the range $\\Delta^{\\Omega}=30-100$.\n\nConsider now the representative point A5 and the corresponding\nfine-tuning. At large $\\tan\\beta$, the mass of the $\\tilde{\\tau}_R$ runs down\nmore than for low $\\tan\\beta$, and therefore a larger soft mass is\nnecessary to avoid a $\\tilde{\\tau}$ LSP. For low $\\tan\\beta$ the $\\tilde{\\tau}$\ncoannihilation region occurred when $m_0\\ll m_{1\/2}$, whereas here\n$m_0$ and $m_{1\/2}$ are much closer in magnitude. This reduces the\ndominance of $m_{1\/2}$ in the running and restores the need for a\nprecise balance of $m_0$ and $m_{1\/2}$ to keep the $\\tilde{\\tau}-\\tilde{\\chi}^0_1$ mass\ndifference small. However, it is the influence of $\\tan\\beta$ over the\nrunning that dominates the sensitivity.\n\nThe second region of interest shows up as two diagonal bands starting\nbeside the light green region ruled out because the stau is the\nLSP. These lines lie to either side of the line along which\n$2m_{\\tilde{\\chi}^0_1}=m_A$, and neutralino annihilation proceeds via the\nresonant production of a pseudoscalar Higgs boson. This region is\nknown as the pseudoscalar Higgs funnel region. The WMAP lines lie\nalong either side of this resonance, where there is just enough of an\nenhancement in the annihilation cross section to allow a bino LSP to\naccount for the observed relic density. As a result, the dark matter\nfine-tuning price of such a region is large. This is reflected in the\npurple and blue shading of the funnel region, corresponding to a dark\nmatter fine-tuning in the range $\\Delta^{\\Omega}=30-300$.\n\nWe consider the representative point A4 and break down the\nfine-tuning. The mass of the pseudoscalar Higgs is dependent upon the\ndetails of the running of the soft Higgs mass-squared terms, which also\ndetermine the Higgs vevs and thus $\\tan\\beta$. As we require REWSB and set\n$\\tan\\beta$ at the start, we can run this chain of logic back the other\nway. A fixed value of $\\tan\\beta$ requires a specific value of the soft\nHiggs mass-squared terms at the low-energy scale. Thus the value of\n$\\tan\\beta$ has a significant impact on the pseudoscalar Higgs boson\nmass. This explains the dominant sensitivity of the pseudoscalar Higgs\nfunnel region to $\\tan\\beta$. This is to be contrasted with the NUHM,\nwhich will allow access to the pseudoscalar Higgs funnel region\nwithout large $\\tan\\beta$, so that naively one might expect a more natural\nfunnel region.\n\nThe final WMAP strip lies at large $m_0$ in all panels. The corner of\nthe parameter space at large $m_0$ and low $m_{1\/2}$ (shown in light\nred) is ruled out as $\\mu^2<0$, signifying a failure of REWSB. Along\nthe perimeter of this region $\\mu\\approx 0$, and $\\mu$ decreases\nsteadily as one approaches the boundary. As $\\mu$ decreases, the\nhiggsino component of the LSP increases. Higgsino dark matter\nannihilates very efficiently resulting in $\\Omega_{CDM}h^2\\ll \\Omega^{WMAP}_{CDM}h^2$, whereas bino\ndark matter generally gives $\\Omega_{CDM}h^2\\gg\\Omega^{WMAP}_{CDM}h^2$. Along the WMAP strip, the\nhiggsino component of the $\\tilde{\\chi}^0_1$ is large enough that the\nannihilation cross section is enhanced sufficiently for the relic\ndensity to fit the WMAP data. This region is known as the focus-point\nregion.  It is sensitive to the composition of the lightest\nneutralino, and so depends upon the difference between $\\mu$ and $M_1$\nat the electroweak scale. The region is plotted in red and purple,\nwith a couple of green regions at low $m_{1\/2}$. This corresponds to a\ndark matter fine-tuning of $\\Delta^{\\Omega} \\approx 10$ at low $m_{1\/2}$,\nrising to a dark matter fine-tuning of $\\Delta^{\\Omega}\\approx 60$ at large\n$m_0$ and $m_{1\/2}$. The kink at $m_{1\/2}\\approx450~GeV$, where the\ntuning drops (signified by a change from purple to red) corresponds to\nthe top quark mass threshold where processes of the form $\\tilde{\\chi}^0_1\\neut\n\\rightarrow t\\overline{t}$ become kinematically allowed.\n\nWe take points A3 and A6 as representative points in this region and\nbreak the dark matter fine-tuning down into its individual\nelements. At point A3 the LSP is predominantly bino with a small, but\nsignificant, higgsino component. The higgsino component is determined\nby the relative size of $\\mu(EW)$ to $M_1(EW)$. As we set $\\mu(EW)$ by\nrequiring REWSB, $\\mu(EW)$ is determined by the running of the soft\nHiggs masses. Thus $\\mu(EW)$ is sensitive to the soft Higgs mass-squared\nterms at the GUT scale (set to $m_0^2$ in the CMSSM). It is also\nsensitive to strongly-interacting sparticle masses through the\nRGEs. This once again brings in a sensitivity to $m_0$, but also to\n$M_3$ through its strong influence on the stop quark mass. Therefore\n$\\mu(EW)$ is sensitive to both $m_0$ and $m_{1\/2}$. To achieve the\ncorrect balance of bino and higgsino components one needs to balance\n$\\mu(EW)$ and $M_1(EW)$. These both depend strongly upon $m_{1\/2}$,\nwith $\\mu$ also dependent upon $m_0$. This common dependence on\n$m_{1\/2}$ reduces the dark matter fine-tuning below what one would expect.\n\nAt point A6 we consider the kink at the bottom of the higgsino\/bino\nline, where the higgsino and bino components are almost equal. This\nwould normally result in extremely efficient annihilation of\nneutralinos in the early universe and give $\\Omega_{CDM}h^2\\ll\\Omega^{WMAP}_{CDM}h^2$. However at\nthis point the neutralino mass is $m_{\\tilde{\\chi}^0_1}=79.6$~GeV, disallowing\nthe annihilation of neutralinos to Higgs, $Z$ bosons and\n$t\\overline{t}$. It is also on the $W$ boson threshold, suppressing\nannihilation to $W$s. With these annihilation channels ruled out, the\nnormally efficient annihilation of an LSP with a substantial higgsino\nfraction is suppressed sufficiently to fit the WMAP relic density. As\nwith point A3, the determining factors for the annihilation cross\nsection are the relative sizes of $\\mu(EW)$ and $M_1(EW)$, resulting\nin a similar pattern and magnitude of dark matter fine-tuning measures.\n\nOnce again we also calculate the electroweak fine-tuning across the\nplane and plot the corresponding contours of $\\Delta^{EW}$. As in the\ncase of Fig.~\\ref{f:CMSSM,t10}, the degree of electroweak tuning\nincreases steadily with increasing $m_{1\/2}$. This is because the\ndominant soft term is $M_3$, the gluino mass. As $m_{1\/2}$ increases,\nthe gluino mass and squark masses increase, requiring more precise\ncancellations to reproduce the $W$ mass, and thus greater\nelectroweak fine-tuning.\n\n\\begin{table}[ht!]\n  \\begin{center}\n    \\begin{tabular}{|l|c|}\n      \\hline\n      {\\bf Region} & {\\bf Tuning Range} \\\\\n      \\hline\n      Pseudoscalar Higgs Funnel     & 60-1200+ \\\\\n      $\\tilde{\\tau}-\\tilde{\\chi}^0_1$ coannihilation (large $m_0,~m_{1\/2}$ or $\\tan\\beta$)  & 30-60\\\\\n      $\\tilde{\\tau}-\\tilde{\\chi}^0_1$ coannihilation (low $m_0,~m_{1\/2}$ and $\\tan\\beta$)  & 3-10\\\\\n      bino\/higgsino region          & 10-60\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Here we summarize the DM annihilation\n    channels present within the CMSSM and their associated\n    tunings. This provides a reference to which we will compare the\n    regions accessible within the NUHM.\\label{t:CMSSMTunings}}\n\\end{table}\n\nNow that we have considered the different dark matter regions present\nwithin the CMSSM, we summarize the typical tunings in each case. We\ndistinguish in Table \\ref{t:CMSSMTunings} the principal dark matter\nregions present within the CMSSM and the corresponding amounts of\ndark matter fine-tuning.\nWith this as our starting point, we now consider the case\nof the NUHM model in which the universality between the soft Higgs\nmasses and the soft sfermion masses is broken.\n\n\\section{The NUHM}\n\\label{NUHM}\n\nWe now consider the MSSM with non-universal Higgs soft masses\n(NUHM) \\cite{oldnuhm,Ellis:2002wv,hep-ph\/0210205}.\nAfter breaking the universality between the soft Higgs and\nsfermion masses of the CMSSM we have the following independent inputs:\n\\begin{equation}\n  a_{NUHM}=\\left\\{m_0,~m_{H_1},~m_{H_2},~m_{1\/2},~A_0,~\\tan\\beta,\n  ~\\text{sign}(\\mu)\\right\\},\n\\label{NUHMPar}\n\\end{equation}\nwhere $m_{H_1}$ and $m_{H_2}$ characterize the independent soft Higgs\nmasses.  These are subject to constraints arising from vacuum\nstability and cosmological considerations, and may be negative.  As\nlong as $m_{H_{1,2}}^2 + \\mu^2 > 0$ at the GUT scale, there is no\ndangerous high-scale vacuum state, but specifying the precise\nboundaries of the NUHM parameter space lies beyond the scope of this\nwork.\n\nAs with the CMSSM, the NUHM contains a finite number of distinct\nregions in which it can provide the observed dark matter relic\ndensity, which were catalogued in~\\cite{hep-ph\/0210205}. Here we\nfollow the approach of this previous work and reproduce the same\nregions of the parameter space. The plots we present here show the\nupdated parameter space for the current world average for the top\nmass, $m_t=170.9$~GeV, and include the current dark matter and $\\delta a_{\\mu}$\nconstraints. However, the primary goal of this work is rather to\nanalyse the fine-tuning of the dark-matter regions of the NUHM. To\nthis end we calculate and plot the dark-matter fine-tuning in the\nallowed parameter space, and also make some comments on the amount of\nelectroweak fine-tuning.\n\nAs the NUHM contains the CMSSM as a limiting case, all the dark-matter\nregions present in the CMSSM are present in the NUHM. In addition,\nthere are four new regions that are not present in the CMSSM:\n\n\\begin{itemize}\n\\item A pseudoscalar Higgs funnel at low $\\tan\\beta$.\n\\item A bulk region where $\\tilde{\\chi}^0_1$ annihilation is dominantly mediated\n  via $t$-channel $\\tilde{\\tau}$ exchange which does not violate Higgs mass\n  bounds.\n\\item A $\\tilde{\\nu}-\\tilde{\\chi}^0_1$ coannihilation region.\n\\item A mixed bino\/higgsino region at low $m_0$.\n\\end{itemize}\n\nWe shall be particularly interested in understanding how finely tuned\nthe NUHM parameters must be in each of these new regions.\n\n\\subsection{Comparison with the CMSSM}\n\nThe NUHM contains all the points in the CMSSM parameter\nspace. Therefore, we start by studying the tuning of the dark matter\npoints A1-6, presented in Tables~\\ref{t:CMSSM,t10},\\ref{t:CMSSM}, with\nrespect to the parameters of the NUHM.\n\n\\begin{table}[ht!]\n\\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{A1} &\n      \\multicolumn{2}{|c|}{A2} &\n      \\multicolumn{2}{|c|}{A3} &\n      \\multicolumn{2}{|c|}{A4}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 60    & 0.62 & 100    & 5.7 & 2030   & 200  & 540    & 8.1  \\\\\n      $m_{H_1}$   & 60    & 0.017& 100    & 0.26& 2030   & 14   & 540    & 28   \\\\\n      $m_{H_2}$   & 60    & 0.014& 100    & 0.26& 2030   & 230  & 540    & 30   \\\\\n      $m_{1\/2}$   & 200   & 0.99 & 500    & 5.8 & 500    & 18   & 600    & 8.0  \\\\\n      $\\tan\\beta$     & 10    & 0.13 & 10     & 1.5 & 50     & 2.0  & 50     & 76   \\\\\n      \\hline\n      $\\Delta_\\Omega$  &  & 0.99 &        & 5.8 &        & 230  &        & 76   \\\\\n      \\hline\n      $\\Delta_{EW}$    &  & 37   &        & 190 &        & 1300 &        & 230  \\\\\n      \\hline\n    \\end{tabular}\n    \\begin{tabular}{|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{A5} &\n      \\multicolumn{2}{|c|}{A6}\\\\\n      \\cline{2-5}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$  \\\\\n      \\hline\n      $m_0$       & 277    & 23  & 1400   & 230  \\\\\n      $m_{H_1}$   & 277    & 1.5 & 1400   & 10  \\\\\n      $m_{H_2}$   & 277    & 2.5 & 1400   & 73  \\\\\n      $m_{1\/2}$   & 350    & 12  & 250    & 22  \\\\\n      $\\tan\\beta$     & 50     & 48  & 50     & 8.2 \\\\\n      \\hline\n      $\\Delta_\\Omega$ &    & 48  &        & 230 \\\\\n      \\hline\n      $\\Delta_{EW}$   &    & 92  &        & 600 \\\\\n      \\hline\n    \\end{tabular}\n    \\end{center}\\vskip -0.5cm\n  \\caption{\\small A re-analysis of the representative points A1-6 from\n    Figs.~\\ref{f:CMSSM,t10},\\ref{f:CMSSM}, calculating their tunings\n    with respect the NUHM rather than the CMSSM.\\label{t:CMSSMComp}}\n\\end{table}\n\nWe show the dark matter\nfine-tuning of these points with respect to the parameters\n$a_{NUHM}$ in Table~\\ref{t:CMSSMComp}. Point A1 represents the bulk\nregion of the CMSSM, which is inaccessible because the Higgs is\nlight. The primary annihilation channel is $t$-channel slepton\nexchange, and the sensitivity in the CMSSM is primarily due to $m_0$\nand $m_{1\/2}$ as they determine the neutralino and slepton\nmasses. This is also true in the NUHM, and the sensitivities to the\nHiggs soft masses are negligible.\n\nPoint A2 represents the low-$\\tan\\beta$ coannihilation region of the\nCMSSM, in which the primary sensitivities were to $m_{1\/2}$ and $m_0$,\nas these determine the stau mass and the neutralino mass. Once again,\nthis picture changes very little in the NUHM, with the sensitivity to\nthe soft Higgs masses being negligible.\n\nPoints A3-6 have large $\\tan\\beta$. We recall that A3 and A6 lie in the\nhiggsino-bino focus-point region. In the CMSSM the primary\nsensitivities were to $m_0$ and $m_{1\/2}$, as $m_0$ (and to a lesser\nextent $M_3$) determine the size of $\\mu$, and $m_{1\/2}$ determines\n$M_1(EW)$. Therefore these two parameters determine the mass and\ncomposition of the lightest neutralino, and the total CMSSM dark matter\nfine-tuning of the point in the CMSSM was $\\Delta^{\\Omega}=27$. In the NUHM we\nhave a very different picture. Here the total dark matter fine-tuning is\n$\\Delta^{\\Omega}=230$, and the primary sensitivities are to $m_0$ and\n$m_{H_2}$. This can be explained by the process of radiative\nelectroweak symmetry breaking. For electroweak symmetry breaking to\noccur, the Higgs (mass)$^2$ must become negative. By requiring this\nprocess to give the correct electroweak boson masses we set the size\nof $\\mu$, and thus the magnitude of the higgsino component of the\nlightest neutralino. Therefore to understand the sensitivity of a\nhiggsino-bino dark matter region, we must look for the terms that\ncontribute to the running of the Higgs mass-squared. First there is the\nsoft Higgs mass at the GUT scale, and then there are the running\neffects, primarily the contribution from the stop mass. In the CMSSM,\nthese two terms are coupled, reducing the dependence on either one\nindividually. Therefore even though the scalar masses are large, the\nsensitivity of $\\mu$ to $m_0$ remains small. In the NUHM there is no\nconnection between the soft sfermion masses and the soft Higgs masses,\ntherefore the sensitivity returns. Therefore one should not expect\nnatural bino-higgsino dark matter at large $m_0$ in the NUHM. The\nsignificant increase in the electroweak fine-tuning for these points\nis due to exactly the same physics.\n\nPoints A4 and A5 represent the pseudoscalar Higgs funnel and the\nstau-coannihilation band. At this value of $\\tan\\beta$, the primary\nsensitivity is to $\\tan\\beta$, a feature not altered by breaking the\nuniversality amongst the scalars.\n\n\\subsection{Detour: RGE behaviour with negative masses-squared}\n\nTo understand the dependence of the dark matter phenomenology on the\nNUHM GUT scale parameters we need to understand how the soft Higgs\nmasses affect the RGEs, and through them the low-energy\nparameters. Four low-energy parameters in particular are useful to\nconsider when we talk about dark matter: $\\mu$, $m_A$, and\n$\\tilde{\\tau}_{L,R}$. The higgsino component of the LSP is determined by\n$\\mu$, $m_A$ determines the position of the pseudoscalar Higgs funnel,\nand the lightest stau (a mixture of $\\tilde{\\tau}_{L,R}$) mediates the\nprevalent t-channel slepton exchange annihilation diagrams and\ndetermines the efficiency of $\\tilde{\\tau}$ coannihilation channels.\n\nAfter EW symmetry breaking we can write $\\mu$ as:\n\\begin{equation}\n  \\mu^2=\\frac{m_{H_1}^2-m_{H_2}^2\\tan^2\\beta}{\\tan^2\\beta-1} -\n  \\frac{1}{2}m_Z^2.\n\\label{e:musq}\n\\end{equation}\nClearly $\\mu$ depends on the soft Higgs mass-squared terms and $\\tan\\beta$, as\nwell as other soft parameters through the RGEs. It is also useful to\nconsider the limit of large $\\tan\\beta$ where we can approximate\n(\\ref{e:musq}) as:\n\\begin{equation}\n\\mu^2=-m_{H_2}^2+\\frac{m_{H_1}^2}{\\tan^2\\beta},\n\\end{equation}\nassuming $|m_{H_{1,2}}^2|\\gg m_Z^2$. Therefore for large $\\tan\\beta$, to\nachieve REWSB and have $\\mu^2>0$ we require either negative\n$m_{H_2}^2$, or very large positive $m_{H_1}^2$.\n\nThe pseudoscalar Higgs mass is determined after EWSB by the relation:\n\\begin{equation}\n  m_A^2=m_{H_1}^2+m_{H_2}^2+2\\mu^2.\n\\label{mAsq}\n\\end{equation}\nClearly $m_A^2$ is strongly dependent upon the soft Higgs mass-squared\nterms, $\\tan\\beta$ through its effect on $\\mu$, and other soft terms\nthrough their influence on the Higgs RGEs.\n\nWe now consider the explicit form of the soft Higgs mass-squared RGEs:\n\\begin{eqnarray}\n\\nonumber \\frac{d(m_{H_1}^2)}{dt}&=&\\frac{1}{8\\pi^2}\n\\left(-3g_2^2M_2^2-g_1^2M_1^2+h_\\tau^2 (m_{\\tilde{\\tau}_L}^2 +\nm_{\\tilde{\\tau}_R}^2 + m_{H_1}^2 + A_\\tau^2) \\right.\\\\\n&&\\left.+3h_b^2(m_{\\tilde{b}_L}^2 +\nm_{\\tilde{b}_R}^2+m_{H_1}^2+A_b^2) - 2S\\right),\\\\\n\\frac{d(m_{H_2}^2)}{dt}&=&\\frac{1}{8\\pi^2} \\left(-3g_2^2M_2^2-g_1^2M_1^2\n+3h_t^2(m_{\\tilde{t}_L}^2 + m_{\\tilde{t}_R}^2 + m_{H_2}^2 +\nA_t^2) + 2S\\right),\n\\end{eqnarray}\nwhere $S$ is definedly:\n\\begin{eqnarray}\n\\nonumber S&\\equiv&\\frac{g_1^2}{4}\\left(m_{H_2}^2-m_{H_1}^2 +\n2\\left(m_{\\tilde{Q}_L}^2-m_{\\tilde{L}_L}^2-2m_{\\tilde{u}_R}^2 +\nm_{\\tilde{d}_R}^2+m_{\\tilde{e}_R}^2\\right)\\right.\\\\\n&&+\\left.\\left(m_{\\tilde{Q}_{3L}}^2-m_{\\tilde{L}_{3L}}^2 -\n2m_{\\tilde{t}_R}^2 + m_{\\tilde{b}_R}^2+m_{\\tilde{\\tau}_R}^2\\right)\\right).\n\\end{eqnarray}\n\nThe only parameters in these RGEs that we are not free to set at the\nGUT scale are the Yukawa couplings $h_i$. These are set by the\nrequirement that the Higgs vevs should give the correct SM particle\nmasses:\n\\begin{equation}\nm_{\\tau,b}=\\frac{1}{\\sqrt{2}}h_{\\tau,b} v_1,\n~m_{t}=\\frac{1}{\\sqrt{2}}h_{t} v_2.\n\\end{equation}\nTherefore $\\tan\\beta$ influences the RGEs indirectly through its\ndetermination of the size of the Yukawa couplings. The Yukawa\ncouplings multiply the contribution to the RGEs from the soft squark\nand slepton mass-squared terms and the soft Higgs terms. Therefore varying\nthe Yukawa couplings has a large impact on the running. As we increase\n$\\tan\\beta$, we increase $v_2$ with respect to $v_1$, and so we must\ndecrease $h_t$ and increase $h_{\\tau,b}$. Therefore we reduce the\nYukawa contribution to the running of $m_{H_2}^2$, while increasing\nthe contribution to the running of $m_{H_1}^2$.\n\nNow consider the RGEs for the right and left handed stau masses:\n\\begin{eqnarray}\n\\frac{d(m_{\\tilde{L}_{3L}}^2)}{dt}&=&\\frac{1}{8\\pi^2}\n\\left(-3g_2^2M_2^2-g_1^2M_1^2+h_\\tau^2 \\left(m_{\\tilde{L}_{3L}}^2 +\nm_{\\tilde{\\tau}_R}^2+m_{H_1}^2+A_\\tau^2\\right) -2S\\right)\\\\\n\\frac{d(m_{\\tilde{\\tau}_R}^2)}{dt}&=&\\frac{1}{8\\pi^2} \\left(-4g_1^2M_1^2 +\n2h_\\tau^2 \\left(m_{\\tilde{L}_{3L}}^2 +\nm_{\\tilde{\\tau}_R}^2+m_{H_1}^2+A_\\tau^2\\right)+4S\\right)\n\\end{eqnarray}\nIn both cases $m_{H_1}^2$ provides a substantial contribution to the\nrunning, with a coefficient of $h_\\tau$. As we have seen, increasing\n$\\tan\\beta$ increases $h_\\tau$ and thus increases the impact of the Higgs\nmasses on the running of the staus. Therefore we expect any effects of\nnon-universal soft Higgs masses on the stau running to be amplified\nfor large $\\tan\\beta$. In the CMSSM, $m_{H_1}^2$ will remain positive from\nthe GUT scale to the EW scale. Indeed, it is harder to achieve REWSB\nif $m_{H_1}^2$ runs negative. Therefore generally this term provides a\npositive contribution to both the left and right handed stau RGE and\nacts to suppress the stau masses.\n\nIn the CMSSM this poses a problem. As we increase $\\tan\\beta$ we must\nincrease the soft stau mass to avoid it becoming the LSP. However as\nwe increase $m_0$ we are also increasing $m_{H_1}^2$, and thus\nincrease the effect on the running. This can be avoided in the\nNUHM. We can set $m_{H_1}^2$ small and so avoid a very light\n$\\tilde{\\tau}$.\n\nHowever, there is another more subtle effect. The interaction of the\nneutralinos with the stau also depends upon the composition of the\nlightest stau which is determined by the mixing between\n$\\tilde{\\tau}_{L,R}$. This mixing is increased if the two states are close in\nmass. In the CMSSM $S$ is negligible and so $d(m_{\\tilde{\\tau}_R}^2)\/dt \\gg\nd(m_{\\tilde{L}_{3L}}^2)\/dt$, resulting in the right handed stau always\nbeing considerably lighter than the left-handed stau. In the NUHM we\ncan avoid this by having a large negative $S$. This acts to suppress\nthe left handed stau mass while increasing the right handed stau\nmass. As we increase the component of the left-handed stau, we\nincrease the annihilation rate of neutralinos via t-channel stau\nexchange.\n\n\\subsection{Sample $(m_0,m_{1\/2})$ planes in the NUHM}\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/m0,m12,t,10.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small The $(m_0,~m_{1\/2})$ plane of the NUHM\n  parameter space with $A_0=0$, $\\tan\\beta=10$ and sign$(\\mu)$ +ve. The\n  values of $\\mu$ and $m_A$ vary between the panels: (a)\n  $\\mu=400$~GeV, $m_A=400$~GeV, (b) $\\mu=400$~GeV, $m_A=700$~GeV, (c)\n  $\\mu=700$~GeV, $m_A=400$~GeV, (d) $\\mu=700$~GeV, $m_A=700$~GeV. This\n  figure can be compared directly to Fig.~2\n  in~\\protect\\cite{hep-ph\/0210205}. The Roman cross in panel (b)\n  indicates the single point where the parameter space makes contact\n  with the CMSSM.\\label{f:m0,m12,t,10}}\n\\end{figure}\n\nHaving analysed the CMSSM points from the perspective of the NUHM, we\nnow turn to a sampling of the full NUHM parameter space. In\nFig.~\\ref{f:m0,m12,t,10}, we show $(m_{1\/2}, m_0)$ planes for\n$\\tan\\beta=10$, $A_0=0$ and sign$(\\mu)$ positive. We set the\nelectroweak scale parameters $\\mu$ and $m_A$ to different discrete\nvalues in each panel as explained in the figure caption.\n\nAs we saw in the previous section, $\\mu$ and $m_A$ are not high-scale\ninputs into the theory, rather they are the low-energy numbers derived\nfrom a given set of the true input parameters. However, displaying\nresults as functions of these parameters can be more informative. As\nboth have a strong dependence on $m_{H_{1,2}}^2$, we can fit a\nparticular value of $\\mu$, $m_A$ with the correct choice of\n$m_{H_{1,2}}^2$ at the GUT scale. Therefore we use a code that varies\n$m_{H_{1,2}}^2$ across the parameter space to fit the designated\nlow-energy values of $\\mu$ and $m_A$. All fine-tunings are calculated\nin terms of the inputs of the NUHM as listed in (\\ref{NUHMPar}).\n\nBy starting with $(m_0, m_{1\/2})$ planes, we make contact with the\nparameter space of the CMSSM as displayed in\nFigs.~\\ref{f:CMSSM,t10},~\\ref{f:CMSSM}~\\footnote{ We note in panel (b)\nof Fig.~\\ref{f:m0,m12,t,10} the appearance of a CMSSM point, the only\npoint in any of these planes where full GUT-scale universality is\nrecovered.}.  As before, low $m_0$ is ruled out by a $\\tilde{\\tau}$ LSP\n(light green), and low $m_{1\/2}$ results in a Higgs with $m_h<\n111$~GeV (light grey with black boundary). As before, $\\delta a_{\\mu}$ favours\nlight sleptons, and thus low $m_0$ and $m_{1\/2}$.\n\n\\begin{table}[ht!]\n\\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{B1} &\n      \\multicolumn{2}{|c|}{B2} &\n      \\multicolumn{2}{|c|}{B3} &\n      \\multicolumn{2}{|c|}{B4}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 500   & 32   & 500    & 8.6 & 500    & 4.6  & 500    & 12   \\\\\n      $m_{H_1}^2$ & -80249& 16   & -126930& 12  & -248480& 0.61 & 90625  & 2.3  \\\\\n      $m_{H_2}^2$ & 461380& 62   & 675760 & 25  & 1202900& 24   & 1194100& 60   \\\\\n      $m_{1\/2}$   & 435   & 39   & 540    & 19  & 750    & 18   & 750    & 38   \\\\\n      $\\tan\\beta$     & 10    & 5    & 10     & 3.2 & 10     & 1.1  & 10     & 2.9  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &  & 62   &        & 25  &        & 24   &        & 60   \\\\\n      \\hline\n      $\\Delta_{EW}$    &  & 150  &        & 220 &        & 390  &        & 390  \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 400   & -    & 400 & -   & 400 & -   & 400 & -   \\\\\n      $m_A$       & 400   & -    & 400 & -   & 400 & -   & 700 & -   \\\\\n      \\hline\n    \\end{tabular}\n    \\\\\n    \\begin{tabular}{|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{B5} &\n      \\multicolumn{2}{|c|}{B6} &\n      \\multicolumn{2}{|c|}{B7}\\\\\n      \\cline{2-7}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 500    & 40  & 100    & 6.3 & 0      & 0   \\\\\n      $m_{H_1}^2$ & -416350& 110 & -400510& 12  & -79656 & 2.2 \\\\\n      $m_{H_2}^2$ & -24320 & 4.1 & -332200& 10  & -266010& 7.4 \\\\\n      $m_{1\/2}$   & 442    & 52  & 400    & 3.5 & 445    & 4.3 \\\\\n      $\\tan\\beta$     & 10     & 5.8 & 10     & 0.55& 10     & 1.9 \\\\\n      \\hline\n      $\\Delta_\\Omega$ &    & 110 &        & 12  &        & 7.4 \\\\\n      \\hline\n      $\\Delta_{EW}$   &    & 250 &        & 250 &        & 250 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 700 & -   & 700 & -   & 700 & -   \\\\\n      $m_A$       & 400 & -   & 400 & -   & 700 & -   \\\\\n      \\hline\n    \\end{tabular}\n    \\end{center}\\vskip -0.5cm\n  \\caption{\\small Analysis of the points B1-7, shown in\n    Fig.~\\ref{f:m0,m12,t,10}, which are representative of the\n    pseudoscalar Higgs funnel (B1,2,4,5), mixed bino-higgsino dark\n    matter (B3) and $\\tilde{\\tau}$ coannihilation regions (B6,7). We present a\n    breakdown of the dark matter fine-tuning with respect to each parameter\n    of the NUHM. We give the value of $m_{H_{1,2}}^2$, but the tunings\n    are calculated with respect to $m_{H_{1,2}}$.\\label{t:m0,m12,t,10}}\n\\end{table}\n\nThe dark matter phenomenology shows some similarities to and some\nmarked differences from the CMSSM. First, we see a familiar $\\tilde{\\tau}$\ncoannihilation region alongside the region with a $\\tilde{\\tau}$ LSP. As in\nthe CMSSM, this region is plotted in red and green, designating a\ntuning of $\\Delta^{\\Omega}=3-30$. The only new feature of the coannihilation\nregion here is that effects of the non-universal Higgs soft masses\nalter the running of the stau mass, which allows access to regions\nwith $m_0=0$. We can access $m_0=0$ with small negative $m_{H_1}^2$\nand larger negative $m_{H_2}^2$. The combination of a small Yukawa\ncontribution (due to low $\\tan\\beta$ along with small $|m_{H_1}^2|$) along\nwith negative $S$ results in the stau mass that increases as we run\ndown from the GUT scale, allowing an acceptable stau mass even with\n$m_0=0$.\n\nThe points B6 and B7 are representative of the $\\tilde{\\tau}$ coannihilation\nregion, and the breakdowns of their tunings are also shown in\nTable~\\ref{t:m0,m12,t,10}. The dependences on $m_0$ and $m_{1\/2}$ are\nsimilar to what was observed in the CMSSM. However, the dominant\nsensitivities are now to $m_{H_{1,2}}$. For both these points the soft\nHiggs mass-squared terms are large and negative at the GUT scale. As we\nhave seen, these soft parameters have a significant effect on the stau\nRGE. Therefore the coannihilation strip exhibits tuning with respect to\nthese parameters. The total sensitivity remains low suggesting that,\neven though the soft Higgs masses play a role in the running, the\ndominant contribution to the stau mass is still from $M_1$.\n\nMore distinctive deviations from the familiar CMSSM phenomenology\narise in the forms of the strong vertical dark matter regions at\nparticular values of $m_{1\/2}$. In panel (a) three vertical strips are\npresent. To understand these lines we need to consider the mass and\ncomposition of the lightest neutralino. The bino component of the\nlightest neutralino is determined by $M_1(EW)\\approx 0.4\nm_{1\/2}(GUT)$, whereas the wino component is determined by\n$M_2(EW)\\approx 0.8 m_{1\/2}(GUT)$. Hence, $M_2(EW)>M_1(EW)$ throughout\nthe NUHM parameter space, and we never have a large wino component in\nthe LSP. Of more importance is the higgsino component, determined by\n$\\mu(EW)$. When $\\mu(EW)\\approx M_1(EW)$, there will be a sizeable\nhiggsino component in the LSP. In panel (a) we have set $\\mu=400$~GeV\nand $m_A=400$~GeV. Therefore, when $m_{1\/2}\\approx 1000$~GeV,\n$M_1(EW)\\approx \\mu$ and the lightest neutralino will be a\nbino\/higgsino mixture.  However, for $m_{1\/2}\\gg 1000$~GeV the\nlightest neutralino is mainly a higgsino, with a mass\n$m_{\\tilde{\\chi}^0_1}\\approx 400$~GeV, whereas for $m_{1\/2}\\ll 1000$~GeV the\n$\\tilde{\\chi}^0_1$ is predominantly a bino and has a mass determined by\n$M_1(EW)$.\n\nWith this in mind, we can understand the vertical lines in panel (a)\nat particular values of $m_{1\/2}$. At $m_{1\/2}=500$~GeV, the lightest\nneutralino is a bino with a mass $m_{\\tilde{\\chi}^0_1}\\approx 200$~GeV. As the\npseudoscalar Higgs mass is $m_A=400$~GeV throughout, this results in\nresonant neutralino annihilation through the pseudoscalar Higgs. As a\nresult, the relic density is below the WMAP value across the region\n$450<m_{1\/2}<530$. On the edges of the resonance the relic density may\nfall within the narrow range favoured by astrophysics.  The lines are\nmostly plotted in purple and blue, corresponding to a large degree of\ndark matter fine-tuning $\\Delta^{\\Omega}=30-300$.\nHowever, it is interesting that the edge\nof the resonance at larger $m_{1\/2}$ is plotted in red. This is the\nfirst instance of an acceptable pseudoscalar Higgs resonance region\nwith relatively low dark-matter fine-tuning, thanks to the larger\nhiggsino fraction in the LSP at larger $m_{1\/2}$. Both the LSP mass\nand the mass of $m_A$ are sensitive to $\\mu$, so the mass of the LSP\nand $m_A$ are coupled. This mitigates the dark matter\ntuning of the pseudoscalar\nHiggs funnel to an extent.\n\nPoints B1 and B2 lie on either side of the pseudoscalar Higgs funnel,\nwhere it is interesting to break the dark matter\nfine-tuning measure down into its\ncomponent parts. Unsurprisingly, the dark matter\nfine-tuning is due to a balancing\nact between the mass of the pseudoscalar Higgs, as shown by the large\nsensitivity to $m_{H_2}$, and the neutralino mass, as shown by the\nsensitivity to $m_{1\/2}$.\n\nThe other vertical strip is a wide region around $m_{1\/2}\\approx\n750$~GeV, where the predominantly bino LSP acquires a sufficient\nhiggsino admixture to suppress the relic density to the observed\nvalue. This wide band is plotted in red and purple, representing a\ntuning of $10-30$, similar to that of the low-$m_0$ end of the\nbino\/higgsino region within the CMSSM. There is also a thin line of\ngreen at the base of this band where it meets the coannihilation\nstrip. This shows that the interaction of a stau coannihilation region\nwith a bino\/higgsino LSP reduces the overall tuning of either region\nalone.\n\nPoint B3 is representative of this region. The primary dark matter\nfine-tunings are clearly with respect to $m_{H_2}$ and $m_{1\/2}$. In\nthis case, the dark matter fine-tuning is related to the balance\nbetween the roles of these terms in determining $\\mu$ and $M_1$ at the\nelectroweak scale.  This is the first instance of a mixed\nbino-higgsino region at low $m_0$ that we find in the NUHM. It is\ntherefore interesting to see that, even with a TeV-scale value of\n$m_{H_2}$, the tuning remains relatively small. This is in contrast to\nthe point A3, at which a TeV-scale soft Higgs mass gave large\nfine-tuning. To understand the origin of both electroweak and dark\nmatter fine-tunings, it is useful to consider the analytic form of the\ndependence of the low-energy parameter $\\mu(EW)$ on the GUT-scale soft\ninputs. For $\\tan\\beta=10$, we have:\n\\begin{eqnarray}\n  \\nonumber \\frac{m_Z^2}{2}=&&-0.94\\mu^2+0.010m_{H_1}^2-0.19M_2^2\n  -0.0017M_1^2-0.63m_{H_2}^2+0.38m_{Q_3}^2\\\\\n  \\nonumber &&+0.38m_{U_3}^2\n  +0.093A_t^2-0.011A_tM_1-0.070A_tM_2-0.30A_tM_3\\\\\n  &&+2.51M_3^2+0.0059M_1M_2+0.028M_1M_3+0.195M_2M_3 ,\n  \\label{musq}\n\\end{eqnarray}\nfrom which we can see that, when $m_{Q_3}^2=m_{H_{1,2}}^2$, the terms\nfrom the scalars approximately cancel. This explains the jump in\nsensitivity when this universality is broken. On the other hand, in\norder to obtain a small value of $\\mu$, these soft scalar terms need\nto provide a large contribution to balance out the contribution from\n$M_3^2$. This explains why we find a higgsino\/bino only at large $m_0$\nin the CMSSM. In the NUHM, it is unnecessary to go to large $m_0$,\njust large $m_{H_2}^2$. By keeping $m_0$, and thus $m_{Q_3}^2$, small\none keeps the dark matter and electroweak fine-tunings associated with\nthese parameters under control. The remaining large electroweak\nfine-tuning associated with point B3 is due to $M_{1\/2}$ being quite\nsizeable.\n\nIn panel (b) we have $\\mu=400$~GeV and $m_A=700$~GeV. Here we see only\nthe lower edge of the pseudoscalar resonance. This band lies at\n$750$~GeV and is plotted in purple, once again showing the large dark\nmatter fine-tuning we expect of such resonances. The upper edge of the\nresonance never appears, because at larger $m_{1\/2}$ the lightest\nneutralino is dominantly higgsino. Therefore, at values of $m_{1\/2}$\nabove the resonance, the higgsino nature of the LSP suppresses the\nrelic density so that the relic density never rises enough to fit the\nWMAP measurement.\n\nPoint B4 illustrates the component dark matter fine-tunings at the\nresonance. As with point B1, the dark matter fine-tuning is large, and\nprimarily due to sensitivities to $m_{H_2}^2$ and $m_{1\/2}$. These are\ndue to their effects on the pseudoscalar Higgs mass and the mass of\nthe LSP, respectively.\n\nIn panel (c) we set $\\mu=700$~GeV and $m_A=400$~GeV. As the\npseudoscalar mass is the same, the pseudoscalar Higgs funnel is\ncentred in the same place as in panel (a). However, the higgsino\nfraction of the LSP has dropped, increasing the overall relic\ndensity. Therefore one must go closer to the resonance before the\nenhancement to the annihilation cross section is sufficient to fit the\nobserved relic density. Therefore the WMAP lines are closer to the\npeak of the resonance, and require more precise dark matter\nfine-tuning than for the lower value of $\\mu$. This is shown by\nexamining the component dark matter fine-tunings for point B5. The\ndark matter fine-tuning is large, and primarily due to $m_{H_2}$\nthrough its influence on the pseudoscalar Higgs mass. In panel (c)\nthere is no region in which the LSP is higgsino, due to the larger\nvalue of $\\mu$. One would have to go to $m_{1\/2}\\approx 1700$~GeV\nbefore the LSP acquires a significant higgsino fraction as\n$\\mu=700$~GeV.\n\nFinally, in panel (d) we take $\\mu=700$~GeV and $m_A=700$~GeV, and we\nsee the pseudoscalar Higgs resonance at $m_{1\/2}=800$~GeV as expected,\nand it remains finely tuned, as before. As in panel (c), the value of\n$\\mu$ is too large to find a region of the parameter space in which\nthe neutralino is higgsino.\n\nOverall, Fig.~\\ref{f:m0,m12,t,10} shows some similarities and some\nmarked deviations from the phenomenology of the CMSSM. A $\\tilde{\\tau}$\ncoannihilation band appears in roughly the same region of the\nparameter space and exhibits slightly larger tuning. This increase is\ndue to the effect of the soft Higgs masses squared on the $\\tilde{\\tau}$\nrunning. A further deviation from the CMSSM comes in the form of a\npseudoscalar Higgs funnel at low $\\tan\\beta$. In most cases this exhibits\ndark matter fine-tuning similar to the CMSSM, supporting the\nobservation that resonances require significant dark matter\nfine-tuning wherever they appear. The exception is when the LSP has a\nsignificant higgsino fraction. The sensitivity of both $m_A$ and\n$m_{\\tilde{\\chi}^0_1}$ to $\\mu$ lowers the required dark matter fine-tuning\nsignificantly.  There also appears a band of higgsino\/bino dark matter\nand we find that the tuning is of the same order as in the CMSSM,\nwhich is surprising. In the CMSSM the focus-point region has\nrelatively low dark matter fine-tuning because of the cancellation\namong the scalar masses. By breaking this universality, this\ncancellation is broken. However, it also allows us to access mixed\nbino\/higgsino regions at low $m_0$. This enhances other annihilation\nchannels, such as that via $t$-channel slepton exchange, and we\nrequire a smaller higgsino admixture to obtain a suitable dark matter\nrelic density. These regions recover the relatively low dark matter\nfine-tuning of the focus-point region of the CMSSM.\n\nFinally, we note that we have also calculated the electroweak\nfine-tuning across these planes in the NUHM parameter space, and find\nit to be very similar to that of the CMSSM planes studied previously.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/mu,400,mA,700.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample $(m_{1\/2}, m_0)$ planes of the\n  NUHM parameter space with $A_0=0$, $\\mu=400$~GeV, $m_A=700$~GeV and\n  sign$(\\mu)$ and the following values of $\\tan\\beta$: (a) $\\tan\\beta=10$, (b)\n  $\\tan\\beta=20$, (c) $\\tan\\beta=35$, (d) $\\tan\\beta=50$. The Roman cross in panel\n  (a) shows the single point where the NUHM makes contact with the\n  MSSM. \\label{f:mu,400,mA,700}}\n\\end{figure}\n\nWe now consider the behaviours of these regions as $\\tan\\beta$\nincreases. In Fig.~\\ref{f:mu,400,mA,700} we take $\\mu=400$~GeV,\n$m_A=700$~GeV, $A_0=0$ and sign($\\mu$) positive, and in panels (a)-(d)\nwe take successively larger values of $\\tan\\beta$~\\footnote{ Note the\nCMSSM point in panel (a) of Fig.~\\ref{f:mu,400,mA,700}.}.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{B8} &\n      \\multicolumn{2}{|c|}{B9}\\\\\n      \\cline{2-5}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 800     & 21   & 500    & 150 \\\\\n      $m_{H_1}^2$ & 1609600 & 30   & 892230 & 98  \\\\\n      $m_{H_2}^2$ & 1357100 & 75   & 379620 & 1.3 \\\\\n      $m_{1\/2}$   & 730     & 14   & 432    & 35  \\\\\n      $\\tan\\beta$     & 50      & 50   & 50     & 290 \\\\\n      \\hline\n      $\\Delta_\\Omega$  &  & 75   &        & 290 \\\\\n      \\hline\n      $\\Delta_{EW}$    &  & 420  &        & 130 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 400   & -    & 400 & -   \\\\\n      $m_A$       & 700   & -    & 700 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points B8 and B9, shown in\n    Fig.~\\ref{f:mu,400,mA,700} exemplify the behaviour of the\n    pseudoscalar Higgs funnel (B8) and the stau-coannihilation region\n    (B9) at large $\\tan\\beta$ within the NUHM. We present a breakdown of\n    the dark matter fine-tuning with respect to each parameter of the\n    NUHM. We give the value of $m_{H_{1,2}}^2$, although the tunings\n    are calculated with respect to\n    $m_{H_{1,2}}$.\\label{t:mu,400,mA,700}}\n\\end{table}\n\nAs the value of $\\tan\\beta$ increases, the mass and composition of the\nneutralino across the $(m_{1\/2}, m_0)$ plane remains essentially\nunaltered. Therefore we find the lower edge of the pseudoscalar funnel\nin the same place in all panels. Studying the breakdown of the dark matter\nfine-tunings of the pseudoscalar Higgs funnel for large $\\tan\\beta$ at\npoint B8, we find that in this case all parameters show a significant\ntuning. The primary tunings come from $m_{H_2}$ and $\\tan\\beta$. From\n(\\ref{mAsq}),(\\ref{e:musq}) we can see that this comes from the\ndominant sensitivity of $m_A^2$, through $\\mu$, on both $\\tan\\beta$ and\n$m_{H_2}^2$.\n\nThe pseudoscalar Higgs funnel remains in the same place as $\\tan\\beta$\nincreases, whereas the $\\tilde{\\tau}$-coannihilation strip moves\nconsiderably. For larger $\\tan\\beta$ the Yukawa contribution to the stau\nrunning is enhanced. This suppresses the stau mass, requiring larger\n$m_0$ to avoid a stau LSP. For very low $m_0$ there is a brick red\nregion. This is not due to a failure of REWSB, but rather it is due to\na tachyonic stau mass.\n\nBy increasing $\\tan\\beta$ we reduce the dominance of $M_1$ in the running\nof the stau mass-squared. This breaks the link between the masses of the\n$\\tilde{\\tau}$ and the $\\tilde{\\chi}^0_1$, resulting in an increase of the tuning\nrequired for the coannihilation region at larger values of\n$\\tan\\beta$. Point B9 is representative of the $\\tilde{\\tau}$ coannihilation\nregion at large $\\tan\\beta$. Though there is large sensitivity to the soft\ngaugino mass $m_{1\/2}$, this is eclipsed by the sensitivities to\n$\\tan\\beta$ and $m_0$.\n\n\\subsection{Sample $(\\mu,m_A)$ planes}\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/mu,mA,t,10.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_A)$ planes with\n$A_0=0$, $\\tan\\beta=10$ and sign$(\\mu)$ positive, and different values of\n$m_0$ and $m_{1\/2}$: (a) $m_0=100$~GeV, $m_{1\/2}=300$~GeV, (b)\n$m_0=100$~GeV, $m_{1\/2}=500$~GeV, (c) $m_0=300$~GeV,\n$m_{1\/2}=300$~GeV, (d) $m_0=300$~GeV, $m_{1\/2}=500$~GeV. The Roman\ncrosses in each panel show where the NUHM meets the\nCMSSM.\\label{f:mu,mA,t,10}}\n\\end{figure}\n\nWe now discuss other planes so as to explore more features of the NUHM\nmodel. In Fig.~\\ref{f:mu,mA,t,10} we consider $(\\mu,~m_A)$ planes with\n$A_0=0$, $\\tan\\beta=10$ and sign$(\\mu)$ positive, taking different values of\n$m_0$ and $m_{1\/2}$ in the various panels. As before, though we plot\nresults in terms of $\\mu$ and $m_A$, the primary variables are\n$m_{H_{1,2}}^2$.\n\nWe first consider some overall features. Low $\\mu$ is ruled out in all\ncases by the appearance of a light chargino (light blue). In panels (a)\nand (c), the low value of $m_{1\/2}=300~\\text{GeV}$ results in a Higgs\nboson with $m_h<111$~GeV across much of the parameter space. This bound is\nvery sensitive to variations in the top mass, so regions that are excluded\nhere could be allowed with a higher top mass. On the other hand, in panels\n(b) and (d), only very low values of $m_A$ result in a problematically\nlight Higgs. In panels (a) and (b), the low value of $m_0=100~\\text{GeV}$\nresults in a light $\\tilde{\\tau}$. This gives a region at low $\\mu$ and $m_A$ in\nwhich the $\\tilde{\\tau}$ is the LSP and as such is ruled out (light green).\nFinally, in panel (a) $m_{1\/2}$ and $m_0$ are light enough to give light\nsneutrinos. For large $\\mu$ and $m_A$, one of the sneutrinos becomes the\nLSP, ruling out this corner of the parameter space~\\footnote{Sneutrinos\nare massive, neutral and weakly-interacting, and so could in principle\naccount for the dark matter. However, they generally give $\\Omega_{CDM}h^2\\ll\\Omega^{WMAP}_{CDM}h^2$.\nThis can be avoided in some models with right-handed neutrinos, but this\npossibility lies beyond the scope of the MSSM, so we do not consider it\nfurther here.}.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C1}&\n      \\multicolumn{2}{|c|}{C2}&\n      \\multicolumn{2}{|c|}{C3}&\n      \\multicolumn{2}{|c|}{C4}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 0.99 & 100    & 4.7  & 100     & 13   & 100      & 2.7 \\\\\n      $m_{H_1}^2$ & 279530  & 0.38 & -27130 & 0.99 & -2289300& 79   & -966510  & 390 \\\\\n      $m_{H_2}^2$ & 188070  & 17   & 111220 & 3.5  & -3637400& 110  & -1248800 & 330 \\\\\n      $m_{1\/2}$   & 300     & 16   & 300    & 0.20 & 300     & 30   & 300      & 46  \\\\\n      $\\tan\\beta$     & 10      & 0.8  & 10     & 0.51 & 10      & 0.57 & 10       & 1.4 \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 17   &        & 4.7  &         & 110  &          & 390 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 74   &        & 73   &         & 1300 &          & 500 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 190   & -    & 300 & -   & 1620  & -    & 1000& -   \\\\\n      $m_A$       & 600   & -    & 315 & -   & 600   & -    & 268 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C5}&\n      \\multicolumn{2}{|c|}{C6}&\n      \\multicolumn{2}{|c|}{C7}&\n      \\multicolumn{2}{|c|}{C8}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 3.3  & 100    & 5.6  & 100     & 4.7  & 300      & 0.64\\\\\n      $m_{H_1}^2$ & 319380  & 2.7  & -22696 & 0.59 & -1445500& 35   & -70487   & 6.7 \\\\\n      $m_{H_2}^2$ & 222330  & 76   & -25802 & 0.68 & -1440300& 34   & 246160   & 2.4 \\\\\n      $m_{1\/2}$   & 300     & 85   & 500    & 5.6  & 500     & 3.4  & 300      & 3.6 \\\\\n      $\\tan\\beta$     & 10      & 2.7  & 10     & 1.3  & 10      & 0.068& 10       & 1.2 \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 85   &        & 5.6  &         & 35   &          & 6.7 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 74   &        & 210  &         & 590  &          & 79  \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 120   & -    & 640 & -   & 1170  & -    & 235 & -   \\\\\n      $m_A$       & 600   & -    & 700 & -   & 200   & -    & 150 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C9}&\n      \\multicolumn{2}{|c|}{C10}&\n      \\multicolumn{2}{|c|}{C11}\\\\\n      \\cline{2-7}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 300     & 6.1  & 300    & 0.55 & 300     & 1.3  \\\\\n      $m_{H_1}^2$ & 401250  & 0.62 & -106330& 5.1  & -211970 & 0.067\\\\\n      $m_{H_2}^2$ & 568170  & 40   & 525900 & 2.5  & 497010  & 6.8  \\\\\n      $m_{1\/2}$   & 500     & 31   & 500    & 0.023& 500     & 6.5  \\\\\n      $\\tan\\beta$     & 10      & 1.5  & 10     & 1.0  & 10      & 0.30 \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 40   &        & 5.1  &         & 6.8  \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 180  &        & 180  &         & 180  \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 260   & -    & 320 & -   & 350   & -    \\\\\n      $m_A$       & 750   & -    & 325 & -   & 150   & -    \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points C1-11, shown in Fig.~\\ref{f:mu,mA,t,10},\n    illustrate the behaviour of mixed bino-higgsino dark matter\n    (C1,5,9), sneutrino coannihilation (C3), the pseudoscalar Higgs\n    funnel (C4), stau-coannihilation (C2,6,7,11), and the\n    pseudoscalar funnel region with a mixed bino-higgsino LSP\n    (C8,10). We present breakdowns of the dark matter fine-tuning with\n    respect to each parameter of the NUHM. We give the value of\n    $m_{H_{1,2}}^2$, but the fine-tunings are calculated with respect\n    to $m_{H_{1,2}}$.\\label{t:mu,mA,t,10}}\n\\end{table}\n\nBefore considering the dark matter regions, we first note the\ncomposition of the lightest neutralino in different regions of the\nparameter space. In panels (a) and (c) $M_1(EW)\\approx 120$~GeV,\nwhereas in panels (b) and (d) $M_1(EW)\\approx 200$~GeV. Therefore for\n$\\mu\\gg 120(200)$~GeV the LSP is predominantly a bino with a mass\n$m_{\\tilde{\\chi}^0_1}\\approx 120(200)$~GeV. Below these values of $\\mu$, the\nlightest neutralino acquires a significant higgsino fraction.\n\nAs before, the regions that fit the relic density favoured by WMAP are\ndisplayed with the corresponding dark matter fine-tuning colours.  The\nvertical dark matter band at low $\\mu$ in all panels features a mixed\nbino\/higgsino dark matter particle, and is plotted in red and purple,\ncorresponding to a dark matter fine-tuning $\\Delta^{\\Omega}\\approx 20-40$.\nPoints C1 and C9 are representative of the higgsino\/bino band, and\ntheir dark matter fine-tunings are broken down in\nTable~\\ref{t:mu,mA,t,10}. As one would expect, the fine-tuning is due\nto a balancing act between $m_{H_2}$ and $m_{1\/2}$, as these determine\nthe composition of the lightest neutralino, and thus its annihilation\nrate.\n\nPoint C5 also represents a bino\/higgsino region. It is unusual in\nthat, normally, the larger the higgsino component, the more efficient\nthe annihilation. However, in this case, as $\\mu$ is lowered from\npoint C1, the higgsino fraction increases and yet the dark matter\nrelic density rises again. Indeed, C5 is almost evenly split between\nhiggsino and bino components and yet it fits the WMAP relic density\nmeasurement. This is due to the fact that, as $\\mu$ drops, the mass of\nthe neutralino is lowered. At point C5 the LSP becomes lighter than\nthe $Z$, shutting off the annihilation channel $\\tilde{\\chi}^0_1 \\tilde{\\chi}^0_1\n\\rightarrow ZZ$. This reduces the annihilation cross section enough\nthat the relic density is acceptable once more.\n\nAnother region that is easy to pick out is the pseudoscalar Higgs\nfunnel along $m_A=240(400)$~GeV for $m_{1\/2}=300(500)$~GeV\nrespectively. As before, these funnels require significant dark matter\nfine-tuning and as such are predominantly plotted in blue and black,\nshowing a tuning $\\Delta^{\\Omega}>100$. Point C4 is a representative point\nwhose dark matter fine-tuning breakdown we display in\nTable~\\ref{t:mu,mA,t,10}. In previous pseudoscalar Higgs regions the\ndark matter fine-tuning was due primarily to $m_{H_{1,2}}$ and\n$m_{1\/2}$. Here we find that the sensitivity to the Higgs masses has\nincreased significantly. From (\\ref{mAsq}) this is easy to\nunderstand. If we increase $\\mu$ while keeping $m_A$ the same we must\ncarefully balance the large $m_{H_{1,2}}^2$ contributions to give the\nrequired $m_A$. This careful balancing manifests as a steadily\nincreasing sensitivity of $m_A$ to the Higgs soft masses as we\nincrease $\\mu$. This translates to a large sensitivity of the\npseudoscalar Higgs funnel.\n\nAt the other end of the spectrum, there is a region of the\npseudoscalar Higgs funnel at low $\\mu$ with remarkably low\ntuning. This occurs when there is a significant higgsino fraction in\nthe LSP, such as at points C8 and C10. In this region, both $m_A$ and\nthe neutralino mass are sensitive to $\\mu$. This results in the mass\nof the neutralino and the pseudoscalar being coupled, and reduces the\nsensitivity of the mass difference $\\Delta_m=m_A-2m_{\\tilde{\\chi}^0_1}$. At\npoints C8 and C10 the dominant annihilation channels are to heavy\nquarks via an $s$-channel pseudoscalar Higgs. Remarkably the total\ndark matter fine-tunings of the points are only 6.7 and 5.1\nrespectively.\n\nAs the $\\tilde{\\nu}_{e,\\mu}$ become the LSPs in the large $\\mu$, large\n$m_A$ region of panel (a), there is a corresponding sneutrino\ncoannihilation region lying parallel to its boundary, which is plotted\nin purple and blue indicating a dark matter fine-tuning\n$\\Delta^{\\Omega}>80$. Point C3 is a representative of this region, whose dark\nmatter fine-tuning breakdown is also displayed in Table\n\\ref{t:mu,mA,t,10}. The dark matter fine-tuning is large, and comes\nprimarily from the Higgs sector. It is the existence of large negative\n$m_{H_1}^2$ that allows for light sneutrinos. Thus the sneutrino\nmasses are very sensitive to the Higgs soft mass-squared parameters, and\nthis is reflected in the dark matter fine-tuning. There is also some\ndark matter fine-tuning with respect to $m_{1\/2}$ that is typical of\nthe need to balance the bino mass against that of a coannihilation\npartner with an uncorrelated mass.\n\nFinally, the light $\\tilde{\\tau}$ at low $\\mu$ and $m_0$ has an effect on the\ndark matter relic density. As the mass of the $\\tilde{\\tau}$ is reduced, the\nannihilation cross section is increased via $t$-channel slepton\nexchange. Also, as one approaches the region in which the stau is the\nLSP, there are additional contributions from $\\tilde{\\tau}-\\tilde{\\chi}^0_1$\ncoannihilation processes. These two effects combine to give dark\nmatter bands along the edges of the stau LSP region in panels (a) and\n(b). Points C2 in panel (a) and C6,7 in panel (b) are representative\npoints. At point C2 the annihilation proceeds through equal parts of\n$t$-channel $\\tilde{e}_R,~\\tilde{\\mu}_R,~\\tilde{\\tau}$ annihilation (15-20\\% each),\nannihilation to $b,\\overline{b}$ via off-shell pseudoscalar Higgs\nbosons (18\\%) and $\\tilde{\\tau}$ coannihilation (15\\%). Only the\ncoannihilation processes would be expected to exhibit a high\nsensitivity to the soft parameters, as $t$-channel processes are\nfairly insensitive and the point is far from the pseudoscalar\nresonance, reducing significantly the sensitivity of the $s$-channel\npseudoscalar process. As a result, we have a region that arises from a\nmixture of channels and exhibits low tuning. The subdominant role of\ncoannihilation explains why there is so little dark matter fine-tuning\nwith respect to $m_{1\/2}$. The role of the stau in both the\ncoannihilation and $t$-channel processes explains the dominant dark\nmatter fine-tuning with respect to $m_0$, and the dependence on\n$m_{H_2}$ appears from running effects.\n\nUnfortunately, point C2 results in a light Higgs with $m_{h}=110$~GeV,\nwhich is probably unacceptably low, even allowing for the theoretical\nuncertainty in the calculation of its mass. On the other hand, panel\n(b) has a larger value of $m_{1\/2}$ and hence Higgs mass.  However,\nthe masses of the LSP and the sleptons are also increased. This\ndecreases the slepton $t$-channel annihilation cross sections,\nrequiring larger contributions from processes that are finely tuned in\norder to fit the WMAP relic density, which is apparent at points C6\nand C7~\\footnote{ We note that there is a CMSSM point very close\nto C6}. At point C6, $t$-channel slepton annihilation only accounts\nfor 3\\% of the annihilation rate via each channel (9\\% overall). The\nremaining 91\\% is made up entirely of coannihilation processes,\ndominantly with $\\tilde{\\tau}$, but also $\\tilde{e}_R,\\tilde{\\mu}_R$. As this plane has low\n$m_0$, the slepton masses are predominantly determined by\n$m_{1\/2}$. Once again, there is the familiar pattern of dark matter\nfine-tunings for a low-$\\tan\\beta$, low-$m_0$ slepton coannihilation\nregion. The overall dark matter fine-tuning is low, and what\nfine-tuning does exists is due to $m_0$ and $m_{1\/2}$. Point C7 tells\na slightly different story. The pattern of annihilation channels is\nalmost identical to C6, and we see the typical dark matter\nfine-tunings of a coannihilation region in the sensitivity to $m_0$\nand $m_{1\/2}$. However, the dark matter fine-tuning with respect to\n$m_{H_{1,2}}$ has increased dramatically, due to the massive increase\nin $m_{H_{1,2}}^2$ between points C6 and C7. Now the stau running is\ndominated by the Higgs mass-squared terms rather than the gaugino\nmass, and the coannihilation region becomes fine-tuned once again.\n\nThere is one further interesting region. In panel (d) at low $m_A$\nthere is a kink in the higgsino\/bino region. The band moves to larger\n$\\mu$ and the dark matter fine-tuning drops dramatically. The band is\nplotted in green rather than purple, indicating a dark matter\nfine-tuning of less than 10. The kink in the band appears at\n$m_A=280~$GeV. Around this region the LSP is predominantly a bino with\na small but significant higgsino component, and the LSP has a mass of\naround 200~GeV. As the pseudoscalar mass drops, the masses of the\nheavy Higgs, $H$, and the charged Higgses, $H^\\pm$, also\ndecrease. Around $m_A=280$~GeV, the annihilation channels\n$\\tilde{\\chi}^0_1\\neut\\rightarrow hA,W^\\pm H^\\mp,ZH$ open up, which are\nkinematically forbidden at larger $m_A$. These can progress through\neither $t$-channel neutralino (chargino) exchange or $s$-channel Higgs\nand Z processes. They require a small higgsino component, but\nsignificantly less than the higgsino\/bino region represented by point\nC9. This balance of the higgsino and bino components of the LSP\nappears in the sensitivity of point C11 on $m_{1\/2}$ and\n$m_{H_2}$. Thus C11 represents a higgsino\/bino region with low dark\nmatter fine-tuning - something that does not exist in the CMSSM. This\nis because a large negative $m_{H_1}^2$ is needed to achieve low\n$m_{A,H,H^\\pm}$.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/m0,100,m12,300.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_A)$ planes with\n  $A_0=0$, $m_0=100$~GeV, $m_{1\/2}=300$~GeV and sign$(\\mu)$ positive,\n  and the following values of $\\tan\\beta$: (a) $\\tan\\beta=10$, (b) $\\tan\\beta=20$,\n  (c) $\\tan\\beta=35$. We do not show a plot for $\\tan\\beta=50$ as the\n  parameter space is entirely excluded. The Roman crosses in each\n  panel show where the NUHM meets the CMSSM.\\label{f:m0,100,m12,300}}\n\\end{figure}\n\nWe now consider in Fig.~\\ref{f:m0,100,m12,300} the behaviours of these\nregions as $\\tan\\beta$ increases. We have set $m_0=100$~GeV,\n$m_{1\/2}=300$~GeV, $A_0=0$ and increase $\\tan\\beta$ in steps in each\npanel.\n\nWe note first the bulk features of the plane. As noted previously,\nincreasing $\\tan\\beta$ decreases the mass of the lightest stau. Thus plots\nat larger $\\tan\\beta$ have larger regions ruled out because the LSP is a\n$\\tilde{\\tau}$, and we do not show very large $\\tan\\beta$ because at $\\tan\\beta=50$\nthe stau mass becomes tachyonic across the entire plane. By $\\tan\\beta=35$\nthe light stau rules out all the parameter space below\n$\\mu=200$~GeV. The mass of the light Higgs is also sensitive to\n$\\tan\\beta$, and is in all cases very close to $m_h=111$~GeV, so it only\ntakes a small shift to cause a significant change in the area plotted\nin light grey.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C12}&\n      \\multicolumn{2}{|c|}{C13}&\n      \\multicolumn{2}{|c|}{C14}&\n      \\multicolumn{2}{|c|}{C15}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 1.0  & 100    & 15   & 100     & 7.2  & 100    & 8.7 \\\\\n      $m_{H_1}^2$ & 477840  & 1.4  & 858150 & 48   & -532000 & 6.5  & -14857 & 0.28\\\\\n      $m_{H_2}^2$ & 175680  & 17   & 96420  & 6.3  & -1263800& 12   & -2379  & 0.041\\\\\n      $m_{1\/2}$   & 300     & 16   & 300    & 32   & 300     & 10   & 300    & 4.6  \\\\\n      $\\tan\\beta$     & 20      & 0.56 & 20     & 21   & 20      & 8.7  & 20     & 6.5  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 17   &        & 48   &         & 12   &        & 8.7  \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 71   &        & 71   &         & 480  &        & 78   \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 185   & -    & 275 & -   & 1000  & -    & 400 & -   \\\\\n      $m_A$       & 700   & -    & 940 & -   & 700   & -    & 400 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points C12-15, shown in Fig.~\\ref{f:m0,100,m12,300},\n    are representative of the higgsino\/bino region (C12), the\n    sneutrino coannihilation region (C13) and the\n    stau-coannihilation\/bulk region (C15,14) with increasing $\\tan\\beta$\n    within the NUHM. We present a breakdown of the dark matter\n    fine-tuning with respect to each parameter of the NUHM. We give\n    the value of $m_{H_{1,2}}^2$ but the tunings are calculated with\n    respect to $m_{H_{1,2}}$.\\label{t:m0,100,m12,300}}\n\\end{table}\n\nThe most significant change in the dark matter phenomenology is due to\nthe varying $\\tilde{\\tau}$ mass. Between panels (a) and (b) the stau\nbulk\/coannihilation region moves to larger $\\mu$ and $m_A$. We also\nfind a significant stau region at large $m_A$. These features are\nrepresented by points C15 and C14 respectively~\\footnote{ We note that\nthere is a CMSSM point very close to C15.}. Comparing C15 directly\nto C2, we see from Table~\\ref{t:m0,100,m12,300} that the dark matter\nfine-tuning is due primarily to $m_0$ and $\\tan\\beta$. This is because\nthese parameters determine the mass of the lighter stau and this is\nthe primary source of sensitivity for bulk regions. There is also a\ndegree of sensitivity to $m_{1\/2}$, as there is a significant\ncoannihilation contribution that requires the LSP and stau mass to be\nbalanced. At point C14 one has similar degrees of dark matter\nfine-tuning with respect to $\\tan\\beta,~m_{1\/2}$ and $m_0$. However, there\nis now also large dark matter fine-tuning with respect to\n$m_{H_{1,2}}$, due to the larger magnitude of the soft higgsino\nmass-squared terms. The stau mass in this region becomes highly\nsensitive to $m_{H_2}^2$.\n\nThe other regions are little changed from before. Point C12\nexemplifies the mixed bino\/higgsino region at increasing $\\tan\\beta$. It\ncan be compared directly to point C1, and we see that the component\ndark matter fine-tunings are virtually identical.  Point C13 is\nrepresentative of the sneutrino coannihilation region and can be\ncompared to point C3. Once again the dark matter fine-tuning is due\nprimarily to the soft Higgs masses through their impacts on the\nrunning of the sneutrino masses.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/m0,300,m12,500.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_A)$ planes with\n  $A_0=0$, $m_0=300$~GeV, $m_{1\/2}=500$~GeV, sign$(\\mu)$ positive and\n  different values of $\\tan\\beta$: (a) $\\tan\\beta=10$, (b) $\\tan\\beta=20$, (c)\n  $\\tan\\beta=35$, (d) $\\tan\\beta=50$. The Roman crosses in each panel show\n  where the NUHM meets the CMSSM.\\label{f:m0,300,m12,500}}\n\\end{figure}\n\nMuch of the low-$m_0$ parameter space is forbidden by a light Higgs\nand\/or a light stau. We now consider the effect of increasing $\\tan\\beta$\nin a more open part of the parameter space. We take\nFig.~\\ref{f:mu,mA,t,10}(d) with $m_0=300$~GeV, $m_{1\/2}=500$~GeV as a\nstarting point and increase $\\tan\\beta$ steadily, as seen in\nFig.~\\ref{f:m0,300,m12,500}.  In contrast to\nFig.~\\ref{f:m0,100,m12,300}, the bulk features remain fairly stable\nfor moderate values of $\\tan\\beta$. The first hint of a change appears in\npanel (c) at $\\tan\\beta=35$, where we see a small region at large $m_A$ in\nwhich the stau is the LSP. This expands to cut off low $\\mu$ for\n$\\tan\\beta=50$.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{C16}&\n      \\multicolumn{2}{|c|}{C17}&\n      \\multicolumn{2}{|c|}{C18}&\n      \\multicolumn{2}{|c|}{C19}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 300     & 1.0  & 300    & 0.94 & 300     & 52   & 300    & 37  \\\\\n      $m_{H_1}^2$ & -47935  & 2.0  & -170090& 0.86 & 1021100 & 22   & -52957 & 4.4 \\\\\n      $m_{H_2}^2$ & 518240  & 3.8  & 475340 & 4.2  & 390800  & 0.58 & -281880& 2.3 \\\\\n      $m_{1\/2}$   & 500     & 3.1  & 500    & 3.7  & 500     & 25   & 500    & 20  \\\\\n      $\\tan\\beta$     & 20      & 3.0  & 20     & 0.21 & 35      & 86   & 50     & 49  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 3.8  &        & 4.2  &         & 86   &        & 49  \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 170  &        & 170  &         & 170  &        & 290 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 315   & -    & 360 & -   & 380   & -    & 780 & -   \\\\\n      $m_A$       & 325   & -    & 150 & -   & 950   & -    & 600 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points C16-19, shown in\n    Fig.~\\ref{f:m0,300,m12,500}, illustrate the behaviours of\n    the mixed bino-higgsino, the pseudoscalar Higgs funnel (C16) and\n    the stau-coannihilation\/bulk regions (C17,18,19) at increasing\n    values of $\\tan\\beta$ within the NUHM. We present a breakdown of the\n    dark matter fine-tuning with respect to each parameter of the NUHM. We\n    give the value of $m_{H_{1,2}}^2$, but the tunings are calculated\n    with respect to $m_{H_{1,2}}$.\\label{t:m0,300,m12,500}}\n\\end{table}\n\nThere are few dark matter surprises at larger $\\tan\\beta$. The\npseudoscalar Higgs funnel and mixed higgsino\/bino regions remain\nrelatively unaltered throughout. The interaction of the pseudoscalar\nHiggs funnel with the higgsino\/bino LSP continues to provide a\nfavourable degree of tuning in panels (b) and (c). We take point C16\nas a representative point, and break the tuning down in\nTable~\\ref{t:m0,300,m12,500}. As for point C10, the tuning is small\nand the annihilation is primarily due to annihilation to heavy quarks\nvia an $s$-channel pseudoscalar Higgs.\n\nPoint C17 exemplifies the behaviour of a predominantly bino LSP with a\nsmall higgsino admixture that can annihilate to $hA,~ZH$ and $W^\\pm\nH\\mp$. As with point C11, the dark matter fine-tuning is small and\nmostly due to the composition of the LSP, through $m_{1\/2}$ and\n$m_{H_2}$.\n\nPoint C18 is in the new stau coannihilation region that appears at\nlarge $\\tan\\beta$. For $m_0=300$~GeV, $m_{1\/2}=500$~GeV the staus are too\nheavy to contribute significantly to $t$-channel slepton exchange, so\nthis region is pure coannihilation. The stau mass is mainly determined\nby $m_0$ and $\\tan\\beta$, and must be balanced against a predominantly\nbino LSP. Therefore, the tuning is dominated by $\\tan\\beta$ and $m_0$ with\na secondary dependence on $m_{1\/2}$.  The coannihilation grows\nsignificantly by $\\tan\\beta=50$ and point C19 represents this trend.  As\nwith point C18, we find the tuning to be due to $m_0$ and $\\tan\\beta$,\nwith a secondary dependence on $m_{1\/2}$.\n\nThroughout all of these parameter scans we have also calculated the\nelectroweak fine-tuning and found it to be of the same order as that\nfound in the CMSSM for typical scales of soft masses considered.\n\n\\subsection{Sample $(\\mu,m_{1\/2})$ planes}\n\nFinally,we consider sample $(\\mu,~m_{1\/2})$ planes in the NUHM. These\nare interesting, e.g., because $\\mu$ and $m_{1\/2}$ are the parameters\nthat determine the mass and composition of the lightest\nneutralino~\\footnote{Note that in the following plots $m_{1\/2}$ is the\nGUT-scale soft mass, whereas $\\mu$ is the electroweak-scale Higgs\nterm. This is in contrast to the plots of~\\cite{hep-ph\/0210205} where\nthe plots were in terms of $M_2(EW)$ and $\\mu(EW)$.}.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/mu,m12,t,10.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_{1\/2})$ planes\n  with $A_0=0$, $\\tan\\beta=10$, sign$(\\mu)$ positive and varying $m_0$\n  and $m_A$: (a) $m_0=100$~GeV, $m_A=500$~GeV, (b)\n  $m_0=100$~GeV, $m_A=700$~GeV, (c) $m_0=300$~GeV, $m_A=500$~GeV, (d)\n  $m_0=300$~GeV, $m_A=700$~GeV. The Roman crosses in each\n  panel show where the NUHM meets the CMSSM.\\label{f:mu,m12,t,10}}\n\\end{figure}\n\nIn Fig.~\\ref{f:mu,m12,t,10} we set $A_0=0$, $\\tan\\beta=10$ and take\ndiscrete values of $m_0$ and $m_A$. We see that either low $\\mu$ or\nlow $m_{1\/2}$ results in a light chargino that violates particle\nsearches (light blue). Low $m_{1\/2}$ also results in problems with a\nlight Higgs (light grey with a black boundary). On the other hand,\nlarge $m_{1\/2}$ results in a neutralino with a mass above that of the\nstau (light green). The exception is low $\\mu$ where the neutralino is\na higgsino and $m_{\\tilde{\\chi}^0_1}$ is insensitive to $m_{1\/2}$. In panels (a)\nand (b) we have $m_0=100$~GeV. This, combined with low $m_{1\/2}$ and\nlarge $\\mu$ results in a region in which the LSP is a sneutrino (light\ngreen).\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{D1}&\n      \\multicolumn{2}{|c|}{D2}&\n      \\multicolumn{2}{|c|}{D3}&\n      \\multicolumn{2}{|c|}{D4}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 1.0  & 100    & 7.2  & 100     & 3.6  & 100     & 1.1 \\\\\n      $m_{H_1}^2$ & 160370  & 0.30 & -97736 & 3.3  & -2206600& 39   & -2147800& 360 \\\\\n      $m_{H_2}^2$ & 255000  & 18   & -20502 & 0.70 & -2345100& 41   & -2588200& 300 \\\\\n      $m_{1\/2}$   & 350     & 17   & 400    & 4.7  & 670     & 5.9  & 570     & 59  \\\\\n      $\\tan\\beta$     & 10      & 0.51 & 10     & 1.1  & 10      & 0.027& 10      & 0.070\\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 18   &        & 7.2  &         & 41   &         & 360 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 96   &        & 140  &         & 1100 &         & 1100\\\\\n      \\hline\n      \\hline\n      $\\mu$       & 210   & -    & 530 & -   & 1500  & -    & 1500& -   \\\\\n      $m_A$       & 500   & -    & 500 & -   & 500   & -    & 500 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{D5}&\n      \\multicolumn{2}{|c|}{D6}&\n      \\multicolumn{2}{|c|}{D7}&\n      \\multicolumn{2}{|c|}{D8}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 13   & 300    & 7.1  & 300     & 0.38 & 300     & 7.4 \\\\\n      $m_{H_1}^2$ & -2963700& 110  & 109920 & 0.42 & -120550 & 5.7  & -216760 & 0.91\\\\\n      $m_{H_2}^2$ & -4438200& 140  & 490790 & 39   & 845460  & 5.1  & 1153100 & 9.0 \\\\\n      $m_{1\/2}$   & 310     & 32   & 450    & 30   & 680     & 4.7  & 800     & 2.6 \\\\\n      $\\tan\\beta$     & 10      & 0.52 & 10     & 1.1  & 10      & 1.9  & 10      & 0.70\\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 140  &        & 39   &         & 5.7  &         & 9.0 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 1600 &        & 160  &         & 300  &         & 400 \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 1785  & -    & 240 & -   & 410   & -    & 430 & -   \\\\\n      $m_A$       & 500   & -    & 500 & -   & 500   & -    & 500 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\begin{tabular}{|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{D9}\\\\\n      \\cline{2-3}\n                  & value & $\\Delta^{\\Omega}$\\\\\n      \\hline\n      $m_0$       & 300     & 5.6  \\\\\n      $m_{H_1}^2$ & 291010  & 0.38 \\\\\n      $m_{H_2}^2$ & 661620  & 42   \\\\\n      $m_{1\/2}$   & 550     & 34   \\\\\n      $\\tan\\beta$     & 10      & 1.4  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 42   \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 210  \\\\\n      \\hline\n      \\hline\n      $\\mu$       & 280   & -    \\\\\n      $m_A$       & 700   & -    \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Points D1-9, shown in\n    Fig.~\\ref{f:mu,m12,t,10}, are representative of bino-higgsino dark\n    matter (D1,6,9), stau-coannihilation (D2,3,8), the pseudoscalar\n    Higgs funnel (D4) and its interaction with mixed bino-higgsino\n    dark matter (D7), and sneutrino coannihilation (D5). We present\n    breakdowns of the dark matter fine-tuning with respect to each parameter\n    of the NUHM. We give the value of $m_{H_{1,2}}^2$ but the tunings\n    are calculated with respect to $m_{H_{1,2}}$.\\label{t:mu,m12,t,10}}\n\\end{table}\n\nWe see once again the familiar dark matter regions of the previous\nplots. The pseudoscalar Higgs funnel appears as a pair of horizontal\nlines and exhibits large dark matter fine-tuning, and is characterized\nby the point D4 in Table~\\ref{t:mu,m12,t,10}. Here we see that the\nlarge dark matter fine-tuning is due to the soft Higgs masses through\ntheir influence on $m_A$, and to $m_{1\/2}$ through its influence on\nthe neutralino mass.\n\nThe exception to this large dark matter fine-tuning is where the\npseudoscalar funnel interacts with a higgsino\/bino LSP and there is a\nsmall corner with low fine-tuning, as characterized by point D7. The\nannihilation here is mainly to heavy quarks via an $s$-channel\npseudoscalar Higgs, and yet the total tuning is only 5.7. As noted\npreviously, this relatively small dark matter fine-tuning comes from\nthe common sensitivity of $m_A$ and $m_{\\tilde{\\chi}^0_1}$ on $\\mu$.\n\nThere is also a $\\tilde{\\tau}$ coannihilation region in all four plots, which\nlies alongside the region ruled out due to a stau LSP. It exhibits\nsimilar tuning to the CMSSM. We break down the dark matter\nfine-tunings of this region at points D2 and D3, finding that at both\npoints the tuning with respect to $m_0$ and $m_{1\/2}$ is standard for\na stau coannihilation strip at low $m_0$~\\footnote{This is also true\nfor the CMSSM point seen in panel (b).}. Point D3 has larger tuning\nbecause this region of parameter space requires large negative soft\nHiggs masses, which now dominate the determination of the mass of the\nlight stau.\n\nThe sneutrino coannihilation region shows up alongside the sneutrino\nLSP region. Once again we find it to require significant dark matter\nfine-tuning, although this decreases steadily as one moves to lower\n$\\mu$. Point D5 is a representative point with, as before, large dark\nmatter fine-tuning that depends on the soft Higgs masses.\n\nEach plot also has a dark matter region at low $\\mu$ that lies along a\ndiagonal in the $(\\mu, m_{1\/2})$ plane, incorporating points\nD1,6,9. These regions are mixed bino\/higgsino regions. In all cases\nthe pseudoscalar Higgs and heavy Higgs bosons are sufficiently massive\nthat annihilation of the mixed LSP proceeds mainly through the\nchannels $\\tilde{\\chi}^0_1 \\tilde{\\chi}^0_1 \\rightarrow W^+W^-(ZZ)$, via $t$-channel\nchargino (neutralino) exchange. This process is very sensitive to the\ncomposition of the LSP and the masses of the exchanged\nparticles. Therefore there is significant dark matter fine-tuning with\nrespect to $m_{H_2}$ and $m_{1\/2}$ at all these points.\n\nFinally, we consider the point D8 where the coannihilation strip and\nthe mixed bino\/higgsino strips meet. The combination of annihilation\nchannels has a beneficial effect, with the overall dark matter fine-tuning\ndropping to 9.\n\n\\begin{figure}[ht!]\n  \\begin{center}\n    \\scalebox{1.0}{\\includegraphics{.\/Figs\/m0,100,mA,500.eps}}\n  \\end{center}\n  \\vskip -0.5cm \\caption{\\small Sample NUHM $(\\mu,~m_{1\/2})$ planes\n  with $A_0=0$, $m_0=100$~GeV, $m_A=500$~GeV, sign$(\\mu)$ positive and\n  $\\tan\\beta$ varying: (a) $\\tan\\beta=10$, (b) $\\tan\\beta=20$, (c) $\\tan\\beta=35$. We\n  do not show a plane for $\\tan\\beta=50$, as this part of the parameter\n  space is entirely excluded. The Roman crosses in each panel show\n  where the NUHM meets the CMSSM.\\label{f:m0,100,mA,500}}\n\\end{figure}\n\nOnce again it is interesting to go beyond $\\tan\\beta=10$, to understand\nhow the phenomenology changes with $\\tan\\beta$. In\nFig.~\\ref{f:m0,100,mA,500} we consider $(\\mu, m_{1\/2})$ planes with\n$m_0=100$~GeV, $m_A=500$~GeV and steadily increasing values of\n$\\tan\\beta$. As we saw before, increasing $\\tan\\beta$ decreases the $\\tilde{\\tau}$\nmass, causing the stau LSP regions to encroach on the parameter\nspace. By $\\tan\\beta=35$ the light stau rules out all values of low $\\mu$.\nAs noted earlier, at such a low value of $m_0$, $\\tan\\beta=50$ has a\ntachyonic stau and so is not shown here.\n\n\\begin{table}[ht!]\n  \\begin{center}\n  \\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n      \\hline\n      \\multicolumn{1}{|c|}{Parameter} &\n      \\multicolumn{2}{|c|}{D10} &\n      \\multicolumn{2}{|c|}{D11} &\n      \\multicolumn{2}{|c|}{D12} &\n      \\multicolumn{2}{|c|}{D13}\\\\\n      \\cline{2-9}\n                  & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ & value & $\\Delta^{\\Omega}$ &\nvalue & $\\Delta^{\\Omega}$ \\\\\n      \\hline\n      $m_0$       & 100     & 8.8  & 100    & 2.6  & 100     & 5.0  & 100     & 0.69\\\\\n      $m_{H_1}^2$ & -21261  & 0.37 & -377   & 0.0  & -658190 & 15   & -2200200& 320 \\\\\n      $m_{H_2}^2$ & -73998  & 1.5  & -243830& 0.91 & -772520 & 0.27 & -2597000& 280 \\\\\n      $m_{1\/2}$   & 345     & 4.7  & 220    & 2.1  & 470     & 4.1  & 567     & 32  \\\\\n      $\\tan\\beta$     & 20      & 8.5  & 20     & 3.1  & 35      & 0.99 & 35      & 11  \\\\\n      \\hline\n      $\\Delta_\\Omega$  &    & 8.8  &        & 3.1  &         & 15   &        & 320 \\\\\n      \\hline\n      $\\Delta_{EW}$    &    & 120  &        & 120  &         & 420  &        & 1100\\\\\n      \\hline\n      \\hline\n      $\\mu$       & 500   & -    & 500 & -   & 930   & -    & 1500& -   \\\\\n      $m_A$       & 500   & -    & 500 & -   & 500   & -    & 500 & -   \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\\vskip -0.5cm\n  \\caption{\\small Properties of points D10-13, shown in\n    Fig.~\\ref{f:m0,100,mA,500} which are representative of the\n    pseudoscalar Higgs funnel (D13) and the stau-coannihilation\/bulk\n    region (D10,11,12) at increasing $\\tan\\beta$ within the NUHM. We\n    present a breakdown of the dark matter fine-tuning with respect to\n    each parameter of the NUHM. We give the value of $m_{H_{1,2}}^2$\n    but the tunings are calculated with respect to\n    $m_{H_{1,2}}$.\\label{t:m0,100,mA,500}}\n\\end{table}\n\nThe change in the stau mass is the dominant factor that changes the\ndark matter phenomenology. With the lighter stau, the contribution to\nneutralino annihilation from $t$-channel stau exchange increases. We\nconsider two points D10 and D11 in panel (b). At point D10 the\nannihilation is still dominated by coannihilation effects, but the\ngrowing contribution from $t$-channel stau exchange helps to lower the\ndark matter tuning. The dark matter fine-tuning is predominantly due\nto $m_0$ and $\\tan\\beta$ through their influence on the mass of the\nlighter stau, with a subsidiary fine-tuning with respect to\n$m_{1\/2}$. In contrast, point D11 lies in a dark matter band where the\nannihilation of neutralinos is dominantly through $t$-channel slepton\nexchange. As a result the dark matter fine-tuning is small, and due\nprimarily to $m_0$ and $\\tan\\beta$ through their influence on the slepton\nmasses.\n\nAs we move to larger $\\tan\\beta$, the coannihilation and bulk regions\nmeet. In panel (c) we take point D12 as a representative of the\nmeeting of these two regions. However, by this stage one needs large\nsoft Higgs mass-squared parameters and the stau mass is sensitive to\nthese, rather than to $m_0$ and $m_{1\/2}$. Therefore there is large\nfine-tuning with respect to $m_{H_1}$.  Finally, point D13 is\nrepresentative point of the pseudoscalar Higgs funnel for large\n$\\tan\\beta$. As before, we find the dark matter fine-tuning to be large and\npredominantly due to the soft Higgs masses. This chimes with the\ngeneral behaviour of the pseudoscalar Higgs funnel throughout our\nstudy.\n\n\\section{Conclusions}\n\\label{Conc}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|l|c|}\n\\hline\n{\\bf Region} & {\\bf Typical $\\Delta^{\\Omega}$} \\\\\n\\hline\n$\\tilde{\\tau}$ bulk region                                          & 1-5\\\\\n$\\tilde{\\tau}-\\tilde{\\chi}^0_1$ coannihilation                                 & 4-80\\\\\nBino annihilation via pseudoscalar Higgs Funnel              & 30-1200+\\\\\nBino\/higgsino annihilation via pseudoscalar Higgs Funnel     & 3-10\\\\\nBino\/higgsino region, $m_{\\tilde{\\chi}^0_1}>m_{H,A}$                    & 30-300\\\\\nBino\/higgsino region, $m_{\\tilde{\\chi}^0_1}<m_{H,A}$                    & 2-10\\\\\n$\\tilde{\\nu}-\\tilde{\\chi}^0_1$ coannihilation                                & 15-200\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\vskip -0.5cm \\caption{\\small A summary of the different dark matter\n  regions present within the NUHM and typical values of the\n  corresponding dark matter fine-tunings. We also note that\n  combinations from many channels decrease the overall\n  tuning. \\label{t:NUHMTunings}}\n\\end{table}\n\nWe summarize in Table~\\ref{t:NUHMTunings} the ranges of dark matter\nfine-tunings found in the different dark matter regions appearing in\nthe NUHM, which may be compared to those found previously in the\nCMSSM, as shown in Tables~\\ref{t:CMSSM,t10},\\ref{t:CMSSM} and\nsummarised in Table~\\ref{t:CMSSMTunings}. Comparing first the bulk\nregions, which require the smallest amounts of dark matter fine-tuning\nin both the CMSSM and the NUHM, we see that CMSSM point A2 has a low\namount of dark matter fine-tuning that is at the end of the range\nfound in the NUHM. However, point A2 has too small a value of $m_h$,\nand hence is not acceptable, whereas the NUHM can circumvent this\nrestriction. Thus, {\\it the NUHM provides access to dark matter which\nis less fine-tuned than in the CMSSM}. Turning then to the $\\tilde{\\tau}$\ncoannihilation regions, we see that the NUHM offers an option of lower\nfine-tuning than that found in the CMSSM at point A5. For reasons\nexplained earlier in the text, there are very large variations in the\namounts of dark matter fine-tuning required in the pseudoscalar Higgs\nfunnel region of the NUHM, and the amount found at the CMSSM point A4\nlies within this range. In addition, the NUHM contains a region in\nwhich a bino\/higgsino LSP annihilates via a pseudoscalar Higgs boson\nthat requires dramatically less fine-tuning than the pseudoscalar\nfunnel in the CMSSM. Likewise, NUHM analogues of the focus points A3\nand A6 shown in Table~\\ref{t:CMSSM} require dark matter fine-tunings\nthat are substantially less than in the CMSSM. One new region appears\nin the NUHM that has no CMSSM analogue, namely the $\\tilde{\\nu}-\\tilde{\\chi}^0_1$\ncoannihilation region.  At least in the examples studied, this\nrequires rather more dark matter fine-tuning than the $\\tilde{\\tau}-\\tilde{\\chi}^0_1$\ncoannihilation region.\n\nAlthough it was not the primary focus of this paper, we have also\ncalculated the electroweak fine-tuning across the NUHM parameter\nspace, and found it to be of the same order of magnitude as in other\nMSSM studies.\n\nGenerally speaking, the fact that the NUHM has more parameters offers\nmore possibilities to find regions with particularly small (or large)\ndark matter fine-tunings. The smaller amounts of dark matter\nfine-tuning generally occur in regions where several different\n(co)annihilation processes contribute to the final dark matter\ndensity, e.g., where a bino\/higgsino band meets a stau coannihilation\nregion. Conversely, there are instances where a tight correlation is\nnecessary between two {\\it a priori} independent MSSM parameters, such\nas the stau and neutralino masses along a coannihilation strip, which is\nimposed automatically in the lower-dimensional parameter space of the\nCMSSM, resulting in smaller amount of dark matter fine-tuning than\nmight otherwise have been expected.\n\nThere have been several studies of the implications of prospective LHC\nand\/or ILC measurements for the accuracy with which the astrophysical\ndark matter density could be calculated on the basis of collider\nmeasurements\\cite{Battaglia:2003ab}. These studies have emphasized\nrelatively favourable points in the CMSSM coannihilation region at low\n$m_{1\/2}$, where the dark matter fine-tuning is relatively low and the\nprospective collider measurements relatively accurate. Our analysis\noffers some prognosis as to which NUHM regions might be favourable for\nextensions of these analyses. Clearly, the presence of NUHM points\nwith low dark matter fine-tuning and relatively light sparticles is\nencouraging {\\it a priori}.  However, we note that in general the NUHM\nrelic density is sensitive to the separate and independent values of\nthe soft Higgs masses $m_{H_{1,2}}$. Detailed consideration of their\ndeterminations using collider data has not yet been given, as far as\nwe know, and we recall that the prospects for measuring directly the\nmasses of the heavier Higgs bosons at the LHC and ILC are limited,\nthough the prospects of the latter would be improved with the $\\gamma\n- \\gamma$ option, or by going to higher energies as at CLIC.\n\nThe extension of these collider dark matter studies to the NUHM is a\nlarge task that lies beyond the scope of this paper. However, this\nexploratory study has revealed some of the prospective opportunities\nand pitfalls.\n\n\\section*{Acknowledgements}\n\nJPR would like to thank Ben Allanach for useful advice. The work of JPR\nwas funded under the FP6 Marie Curie contract MTKD-CT-2005-029466.\nWe also acknowledge partial support from the following grants:\nPPARC Rolling Grant PPA\/G\/S\/2003\/00096;\nEU Network MRTN-CT-2004-503369;\nEU ILIAS RII3-CT-2004-506222\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:Intro}\nConsider the following first order system of differential equations\n\\begin{equation} \\label{eq:system}\n \\mathcal{L} y = -i B^{-1} y' + Q(x) y=\\l y, \\qquad y = \\col(y_1,y_2),\n \\qquad x \\in [0,1],\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:BQ}\n B = \\begin{pmatrix} b_1 & 0 \\\\ 0 & b_2 \\end{pmatrix},\n \\quad b_1 < 0 < b_2 \\quad \\text{and}\\quad\n Q = \\begin{pmatrix} 0 & Q_{12} \\\\ Q_{21} & 0 \\end{pmatrix} \\in \\LL{2}.\n\\end{equation}\n\nIf $B = \\begin{psmallmatrix} -1 & 0 \\\\ 0 & 1 \\end{psmallmatrix}$\nsystem~\\eqref{eq:system} is equivalent to the Dirac system (see the classical\nmonographs~\\cite{LevSar88},~\\cite{Mar77}).\n\nLet us associate linearly independent boundary conditions\n\\begin{equation} \\label{eq:cond}\n U_j(y) := a_{j 1}y_1(0) + a_{j 2}y_2(0) + a_{j 3}y_1(1) + a_{j 4}y_2(1)= 0,\n \\quad j \\in \\{1,2\\},\n\\end{equation}\nwith system~\\eqref{eq:system}, and denote as $L_U(Q) := L_U(Q)$ an operator, associated in $\\mathfrak{H} := \\LLV{2}$ with the boundary value problem (BVP)~\\eqref{eq:system}--\\eqref{eq:cond}. It is defined by differential expression $\\mathcal{L}$ on the domain\n\\begin{equation}\n \\dom(L_U(Q)) = \\{f \\in {\\rm AC}([0,1]; \\mathbb{C}^2) :\\ \\mathcal{L} f \\in \\mathfrak{H},\n \\ U_1(f) = U_2(f) = 0\\}.\n\\end{equation}\n\nTo the best of our knowledge, the spectral properties of the general $n \\times n$ system of the form~\\eqref{eq:system} with a nonsingular diagonal $n\\times n$ matrix $B$ with complex entries and a potential matrix $Q(\\cdot)$ of the form\n\\begin{equation} \\label{1.2}\n B = \\diag(b_1, \\ldots, b_n) \\in \\mathbb{C}^{n\\times n} \\quad \\text{and}\\quad\n Q(\\cdot) =: (q_{jk}(\\cdot))_{j,k=1}^n \\in C^1([0,1]; \\mathbb{C}^{n\\times n}).\n\\end{equation}\nhas first been investigated by G.D.~Birkhoff and R.E.~Langer~\\cite{BirLan23}. Namely, they introduced the concepts of \\emph{regular and strictly regular boundary conditions}~\\eqref{eq:cond} and investigated the asymptotic behavior of eigenvalues and eigenfunctions of the corresponding operator $L_{U}(Q)$. Moreover, they proved \\emph{a pointwise convergence result} on spectral decompositions of the operator $L_{U}(Q)$ corresponding to the BVP~\\eqref{eq:system}--\\eqref{eq:cond} with regular boundary conditions.\n\nThe completeness property in $L^2([0,1]; \\mathbb{C}^n)$ of the system of root vectors of BVP for general $n \\times n$ system of the form~\\eqref{eq:system} with matrices $B = \\diag(b_1, \\ldots, b_n)$ and $Q \\in L^1([0,1]; \\mathbb{C}^{n\\times n})$ was established for the first time by M.M.~Malamud and L.L.~Oridoriga in~\\cite{MalOri12} for a wide class of BVPs, although for $2 \\times 2$ Dirac system with $Q\\in C([0,1]; \\mathbb{C}^{2 \\times 2})$ it was proved earlier by V.A.~Marchenko in~\\cite[Chapter 1.3]{Mar77}. As a development of~\\cite{MalOri12},\nin~\\cite{AgiMalOri12,LunMal14IEOT,LunMal15JST,ALMO19}\ncompleteness conditions for non-regular and even degenerate boundary conditions were found with applications to dissipative and normal operators.\nIn the joint paper~\\cite{LunMal15JST} the author and M.M.~Malamud\nalso established the Riesz basis property with parentheses of the system of root vectors for different classes of BVPs for $n \\times n$ system with arbitrary $B$ of the form~\\eqref{1.2} and $Q \\in L^{\\infty}([0,1]; \\mathbb{C}^{n\\times n})$.\nNote also that BVP for $2m \\times 2m$ Dirac equation ($B=\\diag(-I_m, I_m)$) were investigated in~\\cite{MykPuy13} (Bari-Markus property for Dirichlet BVP with $Q \\in L^2([0,1]; \\mathbb{C}^{2m \\times 2m})$ and in~\\cite{KurAbd18,KurGad20} (Bessel and Riesz basis properties on abstract level).\n\nThe Riesz basis property in $\\LLV{2}$ of BVP~\\eqref{eq:system}--\\eqref{eq:cond}\nfor $2 \\times 2$ Dirac system ($b_2 = -b_1 = 1$) with various assumptions on the potential matrix $Q$ was investigated in numerous papers (see~\\cite{TroYam01,TroYam02,HasOri09,DjaMit10,Bask11,DjaMit12UncDir,DjaMit12Crit,LunMal14Dokl,SavShk14,LunMal16JMAA}\nand references therein). The case of separated boundary conditions and $Q \\in C^1([0,1]; \\mathbb{C}^{2 \\times 2})$ was treated in~\\cite{TroYam01,TroYam02} and later in~\\cite{HasOri09} for Dirac-type system. For Dirichlet and periodic boundary conditions the case $Q \\in \\LL{2}$ was first treated by P.~Djakov and B.~Mityagin~\\cite{DjaMit10} and later by A.~Baskakov, A.~Derbushev, A.~Shcherbakov~\\cite{Bask11}. Shortly after, P.~Djakov and B.~Mityagin~\\cite{DjaMit12UncDir} extended these results to general regular boundary conditions.\n\nThe most complete result on the Riesz basis property for $2\\times 2$ Dirac and Dirac-type systems, respectively, with $Q \\in \\LL{1}$ and strictly regular boundary conditions was obtained independently by different methods and at the same time by A.M.~Savchuk and A.A.~Shkalikov~\\cite{SavShk14}\nand by the author and M.M.~Malamud~\\cite{LunMal14Dokl,LunMal16JMAA} (in~\\cite{LunMal14Dokl} the sketches of the proof are given). The case of regular boundary conditions is treated in~\\cite{SavShk14} for the first time. Other proofs were obtained later on in~\\cite{SavSad15DAN,SavSad15,LunMal16JMAA} (see also recent survey~\\cite{SavSad20} and references therein).\n\nIn~\\cite{LunMal16JMAA} results for Dirac operator were also applied to Timoshenko beam model. In general one can show that dynamic generators of many physical models governed by systems of linear PDE are similar to certain first order differential operators. Hence spectral properties of such operators are of significant importance in the study of stability of solutions and corresponding $C_0$-semigroups of many physical models. In particular, in Section~\\ref{sec:damped.string} we establish explicit connection between $2 \\times 2$ Dirac type operators (with $b_1 \\ne -b_2$) on the one hand and certain \\textbf{non-canonical initial-boundary value problem for a damped string} on the other hand. This allows us to apply results on Bari and Riesz basis property (see Definition~\\ref{def:bases}) for Dirac type operators obtained here and in~\\cite{LunMal16JMAA} to the damped string equation.\n\nRecall, that boundary conditions~\\eqref{eq:cond} are called \\textbf{regular}, if and only if they are \\emph{equivalent} to the following conditions\n\\begin{equation} \\label{eq:cond.canon.intro}\n \\widehat{U}_{1}(y) = y_1(0) + b y_2(0) + a y_1(1) = 0,\\quad\n \\widehat{U}_{2}(y) = d y_2(0) + c y_1(1) + y_2(1) = 0,\n\\end{equation}\nwith certain $a,b,c,d \\in \\mathbb{C}$ satisfying $ad-bc \\ne 0$. Recall also that regular boundary conditions~\\eqref{eq:cond} are called \\textbf{strictly regular}, if the sequence $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the eigenvalues of the unperturbed ($Q=0$) BVP~\\eqref{eq:system}--\\eqref{eq:cond} (of the operator $L_U(0)$), is asymptotically separated. In particular, the eigenvalues $\\{\\l_n^0\\}_{|n| > n_0}$ are geometrically and algebraically simple.\n\nIt is well known that \\emph{non-degenerate separated} boundary conditions are always \\emph{strictly regular}. Moreover, conditions~\\eqref{eq:cond.canon.intro} \\emph{are strictly regular for Dirac operator if and only if} $(a-d)^2 \\ne -4bc$. In particular, antiperiodic ($a=d=0$, $b=c=1$) boundary conditions \\emph{are regular but not strictly regular} for Dirac system, while they \\emph{become strictly regular for Dirac-type system} if $-b_1, b_2 \\in \\mathbb{N}$ and $b_2 - b_1$ is odd.\n\nNote in this connection that \\emph{periodic and antiperiodic (necessarily non-strictly regular) BVP} for $2 \\times 2$ Dirac and Sturm-Liouville equations have also attracted certain attention during the last decade. For\\ instance, a criterion for the system of root vectors of the \\emph{periodic} BVP for $2 \\times 2$ Dirac equation to contain a Riesz basis (without parentheses!) was obtained by P. Djakov and B. Mityagin in~\\cite{DjaMit12Crit} (see also\nrecent papers~\\cite{Mak19},~\\cite{Mak20} by A.S.~Makin and the references therein). It is also worth mentioning that F.~Gesztesy and V.~Tkachenko~\\cite{GesTka09,GesTka12} for $q \\in L^2[0,\\pi]$ and P.~Djakov and B.~Mityagin~\\cite{DjaMit12Crit} for $q \\in W^{-1,2}[0,\\pi]$ established by different methods a \\emph{criterion} for the system of root vectors to contain a Riesz basis for Sturm-Liouville operator $-\\frac{d^2}{dx^2} + q(x)$ on $[0,\\pi]$ (see also survey~\\cite{Mak12}).\n\nLet us emphasize that the proof of the Riesz basis property in~\\cite{DjaMit10,Bask11,DjaMit12UncDir,MykPuy13} substantially relies on the Bari-Markus property: the quadratic closeness in $\\LLV{2}$ of the spectral projectors of the operators $L_U(Q)$ and $L_U(0)$.\nAssuming boundary conditions to be strictly regular, let $\\{f_n\\}_{n \\in \\mathbb{Z}}$ and $\\{f_n^0\\}_{n \\in \\mathbb{Z}}$ be the systems of root vectors of the operators $L_U(Q)$ and $L_U(0)$, respectively. Then Bari-Markus property states the implication: $Q \\in L^2 \\Rightarrow \\sum_{n \\in \\mathbb{Z}} \\|f_n - f_n^0\\|_2^2 < \\infty$. Later, this property was generalized to the case $Q \\in \\LL{p}$, $p \\in [1,2]$, in~\\cite{SavShk14,Sad16,LunMal22JDE}. The most complete results in this direction were established\nin the joint paper~\\cite{LunMal22JDE} by the author and M.M.~Malamud. One of these results reads as follows.\n\\begin{theorem}[Theorem 7.15 in \\cite{LunMal22JDE}] \\label{th:ellp-close}\nLet $\\mathcal{K} \\in \\LL{p}$ be a compact set for some $p \\in [1,2]$, let $Q, \\widetilde{Q} \\in \\mathcal{K}$ and boundary conditions~\\eqref{eq:cond} be strictly regular. Then for some normalized systems of root vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ and $\\{\\widetilde{f}_n\\}_{n \\in \\mathbb{Z}}$ of the operators $L_U(Q)$ and $L_U(\\widetilde{Q})$ the following uniform relations hold for $Q, \\widetilde{Q} \\in \\mathcal{K}$:\n\\begin{align}\n\\label{eq:sum.fn-fn0}\n & \\sum_{|n| > N} \\|f_n - \\widetilde{f}_n\\|_{\\infty}^{p'} \\le C\n \\|Q - \\widetilde{Q}\\|_p^{p'}, \\qquad p \\in (1,2], \\quad 1\/p'+1\/p=1, \\\\\n\\label{eq:sum.fn-fn0.hardy}\n & \\sum_{|n| > N} (1+|n|)^{p-2} \\|f_n - \\widetilde{f}_n\\|_{\\infty}^{p} \\le\n C \\|Q - \\widetilde{Q}\\|_p^p, \\qquad p \\in (1,2], \\\\\n\\label{eq:lim.fn-fn0.c0}\n & \\lim_{n \\to \\infty} \\sup_{Q, \\widetilde{Q} \\in \\mathcal{K}}\n \\|f_n - \\widetilde{f}_n\\|_{\\infty} = 0, \\qquad p = 1.\n\\end{align}\n\\end{theorem}\nHere and throughout the paper we denote by $\\|f\\|_s$ the $L^s$-norm of the element $f$ of a scalar, vector or matrix $L^s$-space.\n\nEmphasize, that the proof of the estimates~\\eqref{eq:sum.fn-fn0}--\\eqref{eq:sum.fn-fn0.hardy} is based on the deep Carleson-Hunt theorem. Note, however, that these estimates with $\\|\\cdot\\|_{p'}$-norm instead of $\\|\\cdot\\|_{\\infty}$-norm can be proved in a more direct way, which is elementary in character. Note also that these results substantially rely on transformation operators method that goes back to~\\cite{Mal94,Mal99,LunMal16JMAA}.\n\nRecall that the concepts of Riesz bases and bases quadratically close to the orthonormal bases were introduced by N.K.~Bari in~\\cite{Bari51}. Results of this fundamental paper can also be found in the classical monograph~\\cite{GohKre65} where a basis quadratically close to the orthonormal basis is called a Bari basis. Let us recall the definition of Riesz and Bari bases following~\\cite[Section IV]{GohKre65}.\n\\begin{definition} \\label{def:bases}\n\\textbf{(i)} A sequence of vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ in a separable Hilbert space $\\mathfrak{H}$ is called a \\textbf{Riesz basis} if it admits a representation $f_n = T e_n$, $n \\in \\mathbb{N}$, where $\\{e_n\\}_{n \\in \\mathbb{Z}}$ is an orthonormal basis in $\\mathfrak{H}$ and $T : \\mathfrak{H} \\to \\mathfrak{H}$ is a bounded operator with bounded inverse.\n\n\\textbf{(ii)} A sequence of vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ in a separable Hilbert space $\\mathfrak{H}$ is called a \\textbf{Bari basis} if it is quadratically close to some orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$ in $\\mathfrak{H}$, i.e.\n\\begin{equation} \\label{eq:sum.fn-en}\n \\sum_{n \\in \\mathbb{Z}} \\|f_n - e_n\\|_{\\mathfrak{H}}^2 < \\infty.\n\\end{equation}\n\\end{definition}\nA.S.~Markus in~\\cite{Markus69} studied in detail bases of subspaces with the property similar to~\\eqref{eq:sum.fn-en}. Bari basis property for different classes of differential operators was studied in~\\cite{BDL00,Zhidkov02,Allah14}.\nNote, however, that to the best of our knowledge the question of whether system of root vectors of the operator $L_U(Q)$ forms \\emph{a Bari basis  has not been studied before}. Namely, results of papers~\\cite{DjaMit10,Bask11,DjaMit12UncDir,MykPuy13,SavShk14,LunMal22JDE} in the case of $Q \\in L^2$ and strictly regular boundary conditions establish quadratic closeness of systems of root vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ and $\\{f_n^0\\}_{n \\in \\mathbb{Z}}$, but whether $\\{f_n\\}_{n \\in \\mathbb{Z}}$ is quadratically close to some orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$ remained an open question. The goal if this paper is to close this gap.\nOne of our main results establishes the criterion for the system of root vectors of the operator $L_U(Q)$ to form a Bari basis and reads as follows.\n\\begin{theorem} \\label{th:crit.bari}\nLet boundary conditions~\\eqref{eq:cond.canon.intro} be strictly regular and let $Q \\in \\LL{2}$. Then some normalized system of root vectors of the operator $L_U(Q)$ is a Bari basis in $\\LLV{2}$ if and only if the operator $L_U(0)$ is self-adjoint. The latter holds if and only if the coefficients $a,b,c,d$ in boundary conditions~\\eqref{eq:cond.canon.intro} satisfy the following relations:\n\\begin{equation} \\label{eq:abcd.sa.intro}\n |a|^2 + \\beta |b|^2 = 1, \\qquad\n |c|^2 + \\beta |d|^2 = \\beta, \\qquad\n a \\overline{c} + \\beta b \\overline{d} = 0, \\qquad \\beta := -b_2\/b_1 > 0.\n\\end{equation}\nIn this case every normalized system of root vectors of the operator $L_U(Q)$ is a Bari basis in $\\LLV{2}$.\n\\end{theorem}\nCombining Theorem~\\ref{th:crit.bari} with the results of the previous papers\n\\cite{DjaMit10,Bask11,DjaMit12UncDir,MykPuy13,LunMal14Dokl,LunMal16JMAA,SavShk14}\nconcerning the Riesz basis property we get the following surprising result.\n\\begin{corollary} \\label{cor:not.bari}\nLet $Q \\in \\LL{2}$ and let boundary conditions~\\eqref{eq:cond.canon.intro} be strictly regular but not self-adjoint, i.e. the operator $L_U(0)$ is not self-adjoint. Then every normalized system of root vectors of the operator $L_U(Q)$ is \\textbf{a Riesz basis but not a Bari basis} in $\\LLV{2}$.\n\\end{corollary}\n\\section{Definitions and formulations of the main results}\nLet us recall the following abstract criterion for Bari basis property.\n\\begin{proposition}~\\cite[Theorem VI.3.2]{GohKre65} \\label{prop:crit.bari}\nA complete system $\\mathfrak{F} = \\{f_n\\}_{n \\in \\mathbb{Z}}$ of unit vectors in a separable Hilbert space $\\mathfrak{H}$ forms a Bari basis if and only if there exists a sequence $\\{g_n\\}_{n \\in \\mathbb{Z}}$ biorthogonal to $\\mathfrak{F}$ that is quadratically close to $\\mathfrak{F}$:\n\\begin{equation} \\label{eq:sum.fn-gn.2}\n\\sum_{n \\in \\mathbb{Z}} \\|f_n - g_n\\|_{\\mathfrak{H}}^2 < \\infty, \\qquad (f_n, g_m)_{\\mathfrak{H}} = \\delta_{nm}, \\quad n,m \\in \\mathbb{Z}.\n\\end{equation}\n\\end{proposition}\nBased on this abstract criterion we will introduce a generalization of Bari basis concept. Let $p \\in [1,2]$ and $p' = p\/(p-1) \\in [2,\\infty]$. It is well-known that for the dual space of $\\ell^p := \\ell^p(\\mathbb{Z})$ we have,\n\\begin{equation} \\label{eq:ellp*}\n(\\ell^p(\\mathbb{Z}))^* \\cong \\ell^{p'}(\\mathbb{Z}), \\quad p \\in (1,2],\n\\qquad\\text{and}\\qquad\n(\\ell^1(\\mathbb{Z}))^* \\cong c_0(\\mathbb{Z}).\n\\end{equation}\nFor simplicity we identify $(\\ell^p(\\mathbb{Z}))^*$ with $\\ell^{p'}(\\mathbb{Z})$ for $p \\in (1,2]$ and with $c_0(\\mathbb{Z})$ for $p=1$, respectively. E.g. $\\{a_n\\}_{n \\in \\mathbb{Z}} \\in (\\ell^p(\\mathbb{Z}))^*$ for $p>1$ means that $\\sum_{n \\in \\mathbb{Z}} |a_n|^{p'} < \\infty$.\nWith this in mind, we can extend Definition~\\ref{def:bases}(ii) using equivalence from Proposition~\\ref{prop:crit.bari} to more general concept of closeness of sequences $\\{f_n\\}_{n \\in \\mathbb{Z}}$ and $\\{g_n\\}_{n \\in \\mathbb{Z}}$.\n\\begin{definition} \\label{def:bari.c0}\nLet $p \\in [1,2]$, let $\\mathfrak{F} := \\{f_n\\}_{n \\in \\mathbb{Z}}$ be a complete minimal sequence of unit vectors in a separable Hilbert space $\\mathfrak{H}$ and let $\\mathfrak{G} := \\{g_n\\}_{n \\in \\mathbb{Z}}$ be its (unique) biorthogonal sequence: $(f_n, g_m)_{\\mathfrak{H}} = \\delta_{nm}$, $n, m \\in \\mathbb{Z}$. A sequence $\\mathfrak{F}$ is called a \\textbf{Bari $(\\ell^p)^*$-sequence} if it is ``($\\ell^p)^*$-close'' to its biorthogonal sequence $\\mathfrak{G}$, i.e. $\\curl{\\|f_n - g_n\\|_{\\mathfrak{H}}}_{n \\in \\mathbb{Z}} \\in (\\ell^p)^*$. In view of~\\eqref{eq:ellp*} it means that\n\\begin{equation} \\label{eq:sum.fn-gn}\n\\sum_{n \\in \\mathbb{Z}} \\|f_n - g_n\\|_{\\mathfrak{H}}^{p'} < \\infty \\quad\\text{if}\\quad\np \\in (1,2],\n\\quad\\text{and}\\quad\n \\lim_{n \\to \\infty} \\|f_n - g_n\\|_{\\mathfrak{H}} = 0\n \\quad\\text{if}\\quad p = 1.\n\\end{equation}\nFor brevity we will call \\emph{Bari $(\\ell^1)^*$-sequence} as \\textbf{Bari $c_0$-sequence} and \\emph{Bari $(\\ell^p)^*$-sequence} as \\textbf{Bari $\\ell^{p'}$-sequence} for $p \\in (1,2]$.\n\n\\end{definition}\nProposition~\\ref{prop:crit.bari} implies that the notion of Bari $\\ell^2$-sequence coincides with the notion of Bari basis. Note also that every Bari $(\\ell^p)^*$-sequence is Bari $c_0$-sequence. We specifically chose the word ``sequence'' because it is not clear if Bari $c_0$-sequence is a Riesz basis or even a regular basis in general case.\n\\begin{remark} \\label{rem:c0.bari.diff}\nNote that Bari $c_0$-property from definition~\\ref{def:bari.c0} is not equivalent to more conventional formulation of $c_0$-closeness of $\\{f_n\\}_{n \\in \\mathbb{Z}}$ to a certain orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$ even if $\\{f_n\\}_{n \\in \\mathbb{Z}}$ is already a Riesz basis. Indeed, in this case $f_n = e_n + K e_n$, where $K$ and $(I+K)^{-1}$ are bounded operators in $\\mathfrak{H}$. Hence $\\|f_n - e_n\\|_{\\mathfrak{H}} = \\|K e_n\\|_{\\mathfrak{H}}$. It is easily seen that $g_n = \\((I+K)^{-1}\\)^* e_n = e_n - \\((I+K)^{-1}\\)^* K^* e_n$, and hence $\\|g_n - e_n\\| \\to 0$ as $n \\to \\infty$ is equivalent to $\\|K^* e_n\\| \\to 0$ as $n \\to \\infty$. If $K$ is not compact then $\\lim_{n \\to \\infty}\\|K e_n\\| = 0$ is generally not equivalent to $\\lim_{n \\to \\infty}\\|K^* e_n\\| = 0$ for a given orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$.\n\\end{remark}\nLet us also recall the notion of the system of root vectors of an operator with compact resolvent. Firt, we recall a few basic facts regarding the eigenvalues of a\ncompact, linear operator  $T \\in \\mathcal{B}_{\\infty}(\\mathfrak{H})$ in a\nseparable complex Hilbert space $\\mathfrak{H}$. The {\\it geometric\nmultiplicity}, $m_g(\\lambda_0,T)$, of an eigenvalue $\\lambda_0\n\\in \\sigma_p (T)$ of $T$ is given by\n$\nm_g(\\lambda_0,T) := \\dim(\\ker(T - \\lambda_0)).\n$\n\nThe {\\it root subspace} of $T$ corresponding to $\\lambda_0 \\in\n\\sigma_p(T)$ is given by\n\\begin{equation}\\label{root.subspace}\n\\mathcal{R}_{\\lambda_0}(T) = \\big\\{f\\in\\mathfrak{H}\\,:\\, (T - \\lambda_0)^k f = 0 \\\n\\ \\text{for some}\\ \\ k\\in\\mathbb N \\big\\}.\n\\end{equation}\nElements of $\\mathcal{R}_{\\l_0}(T)$ are called {\\it root vectors}.\nFor $\\lambda_0 \\in \\sigma_p (T) \\backslash \\{0\\}$, the set\n$\\mathcal{R}_{\\lambda_0}(T)$ is a closed linear subspace of\n$\\mathfrak{H}$ whose dimension equals to the {\\it algebraic\nmultiplicity}, $m_a(\\lambda_0,T)$, of $\\lambda_0$,\n$\nm_a(\\lambda_0,T) := \\dim\\big(\\mathcal R_{\\lambda_0}(T)\\big)<\\infty.\n$\n\nDenote by $\\{\\l_j\\}_{j=1}^{\\infty}$ the sequence of non-zero\neigenvalues of $T$ and let $n_j$ be the algebraic multiplicity\nof $\\l_j$. By the {\\it system of root vectors} of the operator\n$T$ we mean any sequence of the form\n$\n\\cup_{j=1}^{\\infty}\\{e_{jk}\\}_{k=1}^{n_j},\n$\nwhere $\\{e_{jk}\\}_{k=1}^{n_j}$ is a basis in $\\mathcal{R}_{\\l_j}(T)$,\n$n_j = m_a(\\lambda_j,T) < \\infty$. The system or root vectors of the operator $T$ is called \\emph{normalized} if $\\|e_{jk}\\|_{\\mathfrak{H}} = 1$, $j \\in \\mathbb{N}$, $k \\in \\{1, \\ldots, n_j\\}$.\n\nWe are particularly interested in the case where $A$ is a\ndensely defined, closed, linear operator in $\\mathfrak{H}$ whose\nresolvent is compact, that is,\n$\nR_A(\\l):=(A - \\l)^{-1} \\in \\mathcal{B}_{\\infty}(\\mathfrak{H}), \\ \\l \\in \\rho (A).\n$\nVia the spectral mapping theorem all eigenvalues of $A$\ncorrespond to eigenvalues of its resolvent $R_A(\\l)$, $\\l \\in\n\\rho (A)$, and vice versa. Hence, we use the same notions of\nroot vectors, root subspaces, geometric and algebraic\nmultiplicities associated with the eigenvalues of $A$, and the\nsystem of root vectors of $A$.\n\nNow we are ready to formulate the main result of this paper, which involve notions of Bari $(\\ell^p)^*$-sequences and $c_0$-sequences from Definition~\\ref{def:bari.c0} above.\n\\begin{theorem} \\label{th:crit.lp.bari}\nLet boundary conditions~\\eqref{eq:cond.canon.intro} be strictly regular and let $Q \\in \\LL{p}$ for some $p \\in [1,2]$. Then some normalized system of root vectors of the operator $L_U(Q)$ is a Bari $(\\ell^p)^*$-sequence in $\\LLV{2}$ if and only if the operator $L_U(0)$ is self-adjoint, i.e. when relations~\\eqref{eq:abcd.sa.intro} hold for the coefficients $a,b,c,d$ in boundary conditions~\\eqref{eq:cond.canon.intro}. In this case every normalized system of root vectors of the operator $L_U(Q)$ is a Bari $(\\ell^p)^*$-sequence in $\\LLV{2}$.\n\\end{theorem}\nAs an immediate consequence of Theorem~\\ref{th:crit.lp.bari} we get Theorem~\\ref{th:crit.bari}: the criterion of Bari basis property for Dirac-type operator $L_U(Q)$ with $L^2$-potential and strictly regular boundary conditions.\n\nLet us briefly comment on the proof of our main result, Theorem~\\ref{th:crit.lp.bari}.\nFirst, we apply Theorem~\\ref{th:ellp-close} to reduce the Bari $(\\ell^p)^*$-property of the system of root vectors of operator $L_U(Q)$ with strictly regular boundary conditions to a certain explicit condition in terms of the eigenvalues $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$, which reads as follows for the case $p=1$.\n\\begin{proposition} \\label{prop:c0.close.cond}\nLet $Q \\in \\LL{1}$ and boundary conditions~\\eqref{eq:cond} be strictly regular. Then some normalized systems of root vectors $\\{f_n\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(Q)$ is a Bari $c_0$-sequence in $\\LLV{2}$ if and only if:\n\\begin{equation} \\label{eq:lim1.lim2.intro}\n b_1 |c| + b_2 |b| = 0, \\qquad \\lim_{n \\to \\infty} \\Im \\l_n^0 = 0\n \\quad\\text{and}\\quad \\lim_{n \\to \\infty} z_n = |bc|,\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:zn.def.intro}\n z_n := \\(1 + d \\exp(- i b_2 \\l_n^0)\\)\\overline{\\(1 + a \\exp(i b_1 \\l_n^0)\\)},\n\\end{equation}\nand $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ is the sequence of the eigenvalues of the operator $L_U(0)$, counting multiplicity.\n\\end{proposition}\nWith condition~\\eqref{eq:lim1.lim2.intro} established, the main difficulty arises in reducing this condition to the desired explicit condition~\\eqref{eq:abcd.sa.intro}.\nIn this connection, recall that the sequence $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the eigenvalues of the operator $L_U(0)$ coincides with the sequence of zeros of characteristic determinant\n\\begin{equation} \\label{eq:Delta0.intro}\n \\Delta_0(\\l) = d + a e^{i (b_1+b_2) \\l} + (ad-bc) e^{i b_1 \\l}\n + e^{i b_2 \\l}.\n\\end{equation}\nIf $b_2 \/ b_1 \\in \\mathbb{Q}$ then the sequence $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ has a simple explicit form: it is the union of arithmetic progression that lie on the lines parallel to the real axis, which simplifies the problem a lot.\n\nThe case $b_2 \/ b_1 \\notin \\mathbb{Q}$ is much more complicated.\nNamely, if $|a|+|d|>0$ and $bc \\ne 0$ there is no explicit description of the spectrum of the operator $L_U(0)$. Nevertheless, we were able to establish equivalence of~\\eqref{eq:lim1.lim2.intro} and~\\eqref{eq:abcd.sa.intro} using Weyl's equidistribution theorem (see~\\cite[Theorem 4.2.2.1]{SteinShak03}). It implies the following crucial property of zeros of $\\Delta_0(\\cdot)$.\n\\begin{proposition} \\label{prop:nlim.inf.intro}\nLet $b_2\/b_1 \\notin \\mathbb{Q}$ and boundary conditions~\\eqref{eq:cond.canon.intro} be regular, i.e. $ad-bc \\ne 0$. Let $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of zeros of the characteristic determinant $\\Delta_0(\\cdot)$ counting multiplicity. Then each of the sequences $\\{\\exp(i b_1 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ and $\\{\\exp(i b_2 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ has infinite set of limit points.\n\\end{proposition}\nThis result was key for proving equivalence of~\\eqref{eq:lim1.lim2.intro} and~\\eqref{eq:abcd.sa.intro}, which in turn implies our main result, Theorem~\\ref{th:crit.lp.bari}, and its main corollary, Theorem~\\ref{th:crit.bari}.\n\\section{Regular and strictly regular boundary conditions}\n\\label{subsec:regular}\nIn this section we recall known properties of BVP~\\eqref{eq:system}--\\eqref{eq:cond} subject to regular or strictly regular boundary conditions from~\\cite{LunMal16JMAA}.\nLet us set\n\\begin{align}\n A := \\begin{pmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\\\\n a_{21} & a_{22} & a_{23} & a_{24} \\end{pmatrix}, \\qquad\n\\label{eq:Ajk.Jjk}\n A_{jk} := \\begin{pmatrix} a_{1j} & a_{1k} \\\\ a_{2j} & a_{2k} \\end{pmatrix},\n \\quad J_{jk} := \\det (A_{jk}), \\quad j,k\\in\\{1,\\ldots,4\\}.\n\\end{align}\nLet\n\\begin{equation} \\label{eq:Phi.def}\n \\Phi(\\cdot, \\l) =\n \\begin{pmatrix} \\varphi_{11}(\\cdot, \\l) & \\varphi_{12}(\\cdot, \\l)\\\\\n \\varphi_{21}(\\cdot,\\l) & \\varphi_{22}(\\cdot,\\l)\n \\end{pmatrix} =: \\begin{pmatrix} \\Phi_1(\\cdot, \\l) & \\Phi_2(\\cdot, \\l)\n \\end{pmatrix}, \\qquad \\Phi(0, \\l) = I_2,\n\\end{equation}\nbe a fundamental matrix solution of the\nsystem~\\eqref{eq:system}, where $I_2 = \\begin{psmallmatrix} 1 & 0 \\\\ 0 & 1 \\end{psmallmatrix}$. Here $\\Phi_k(\\cdot, \\l)$ is the\n$k$th column of $\\Phi(\\cdot, \\l)$.\n\nThe eigenvalues of the problem~\\eqref{eq:system}--\\eqref{eq:cond} counting multiplicity\nare the zeros (counting multiplicity) of the characteristic determinant\n\\begin{equation} \\label{eq:Delta.def}\n \\Delta_Q(\\l) := \\det\n \\begin{pmatrix}\n U_1(\\Phi_1(\\cdot,\\l)) & U_1(\\Phi_2(\\cdot,\\l)) \\\\\n U_2(\\Phi_1(\\cdot,\\l)) & U_2(\\Phi_2(\\cdot,\\l))\n \\end{pmatrix}.\n\\end{equation}\nInserting~\\eqref{eq:Phi.def} and~\\eqref{eq:cond} into~\\eqref{eq:Delta.def}, setting $\\varphi_{jk}(\\l) := \\varphi_{jk}(1,\\l)$, and taking notations~\\eqref{eq:Ajk.Jjk} into account we arrive at the following expression for the characteristic determinant\n\\begin{equation} \\label{eq:Delta}\n \\Delta_Q(\\l) = J_{12} + J_{34}e^{i(b_1+b_2)\\l}\n + J_{32}\\varphi_{11}(\\l) + J_{13}\\varphi_{12}(\\l)\n + J_{42}\\varphi_{21}(\\l) + J_{14}\\varphi_{22}(\\l).\n\\end{equation}\nIf $Q=0$ we denote a fundamental matrix solution as $\\Phi^0(\\cdot, \\l)$. Clearly\n\\begin{equation} \\label{eq:Phi0.def}\n \\Phi^0(x, \\l)\n = \\begin{pmatrix} e^{i b_1 x \\l} & 0 \\\\ 0 & e^{i b_2 x \\l} \\end{pmatrix}\n =: \\begin{pmatrix}\n \\varphi_{11}^0(x, \\l) & \\varphi_{12}^0(x, \\l)\\\\\n \\varphi_{21}^0(x,\\l) & \\varphi_{22}^0(x,\\l)\n \\end{pmatrix}\n =: \\begin{pmatrix} \\Phi_1^0(x, \\l) & \\Phi_2^0(x, \\l) \\end{pmatrix},\n\\end{equation}\nfor $x \\in [0,1]$ and $\\l \\in \\mathbb{C}$. Here $\\Phi_k^0(\\cdot, \\l)$ is the $k$th column of $\\Phi^0(\\cdot, \\l)$. In\nparticular, the characteristic determinant $\\Delta_0(\\cdot)$ becomes\n\\begin{equation} \\label{eq:Delta0}\n \\Delta_0(\\l) = J_{12} + J_{34}e^{i(b_1+b_2)\\l}\n + J_{32}e^{ib_1\\l} + J_{14}e^{ib_2\\l}.\n\\end{equation}\nIn the case of Dirac system $(B =\\diag (-1,1))$ this formula is\nsimplified to\n\\begin{equation} \\label{eq:Delta0_Dirac}\n \\Delta_0(\\l) = J_{12} + J_{34} + J_{32}e^{-i\\l} + J_{14}e^{i\\l}.\n\\end{equation}\nLet us recall the definition of regular boundary conditions.\n\\begin{definition} \\label{def:regular}\nBoundary conditions~\\eqref{eq:cond} are called \\textbf{regular} if\n\\begin{equation} \\label{eq:J32J14ne0}\n J_{14} J_{32} \\ne 0.\n\\end{equation}\n\\end{definition}\nLet us recall one more definition (cf.~\\cite{Katsn71}).\n\\begin{definition} \\label{def:incompressible}\nLet $\\L := \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ be a sequence of complex numbers. It is\ncalled \\textbf{incompressible} if for some $d \\in \\mathbb{N}$ every rectangle\n$[t-1,t+1] \\times \\mathbb{R} \\subset \\mathbb{C}$ contains at most $d$ entries of the sequence,\ni.e.\n\\begin{equation} \\label{eq:card.incomp}\n \\card\\{n \\in \\mathbb{Z} : |\\Re \\l_n - t| \\le 1 \\} \\le d, \\quad t \\in \\mathbb{R}.\n\\end{equation}\n\\end{definition}\nRecall that $\\mathbb{D}_r(z) \\subset \\mathbb{C}$ denotes the disc of radius $r$ with a\ncenter $z$.\n\nLet us recall certain important properties from~\\cite{LunMal16JMAA} of the characteristic determinant $\\Delta(\\cdot)$ in the case of regular boundary conditions.\n\\begin{proposition}~\\cite[Proposition 4.6]{LunMal16JMAA} \\label{prop:sine.type}\nLet the boundary conditions~\\eqref{eq:cond} be regular. Then the characteristic determinant $\\Delta_Q(\\cdot)$ of the problem~\\eqref{eq:system}--\\eqref{eq:cond} given by~\\eqref{eq:Delta}\nhas infinitely many zeros $\\L := \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ counting multiplicities and\n\\begin{equation} \\label{eq:ln.in.Pih}\n|\\Im \\l_n| \\le h, \\quad n \\in \\mathbb{Z}, \\qquad\\text{for some}\\ \\ h \\ge 0.\n\\end{equation}\nMoreover, the sequence $\\L$ is incompressible\nand can be ordered in such a way that the following asymptotical formula holds\n\\begin{equation} \\label{eq:lam.n=an+o1}\n \\Re \\l_n = \\frac{2 \\pi n}{b_2 - b_1} (1 + o(1)) \\quad\\text{as}\\quad n \\to\\infty.\n\\end{equation}\n\\end{proposition}\nClearly, the conclusions of Proposition~\\ref{prop:sine.type} are valid for the characteristic determinant $\\Delta_0(\\cdot)$ given by~\\eqref{eq:Delta0}. Let $\\L_0 = \\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of its zeros counting multiplicity. Let us order the sequence $\\L_0$ in a (possibly non-unique) way such that $\\Re \\l_n^0 \\le \\Re \\l_{n+1}^0$, $n \\in \\mathbb{Z}$.\nLet us recall an important result from~\\cite{LunMal14Dokl,LunMal16JMAA}\nand~\\cite{SavShk14} concerning asymptotic behavior of the eigenvalues.\n\\begin{proposition}[Proposition 4.7 in~\\cite{LunMal16JMAA}]\n\\label{prop:Delta.regular.basic}\nLet $Q \\in \\LL{1}$ and let boundary conditions~\\eqref{eq:cond} be regular. Then the sequence $\\L = \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ of zeros of $\\Delta_Q(\\cdot)$ can be ordered in such a way that the following asymptotic formula holds\n\\begin{equation} \\label{eq:l.n=l.n0+o(1)}\n \\l_n = \\l_n^0 + o(1), \\quad\\text{as}\\quad n \\to \\infty, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\n\\end{proposition}\nTo define strictly regular boundary conditions we need the following definition.\n\\begin{definition} \\label{def:sequences}\n\\textbf{(i)} A sequence $\\L := \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ of complex numbers is\nsaid to be \\textbf{separated} if for some positive $\\tau > 0,$\n\\begin{equation} \\label{separ_cond}\n |\\l_j - \\l_k| > 2 \\tau \\quad \\text{whenever}\\quad j \\ne k.\n\\end{equation}\nIn particular, all entries of a separated sequence are distinct.\n\n\\textbf{(ii)} The sequence $\\L$ is said to be \\textbf{asymptotically\nseparated} if for some $N \\in \\mathbb{N}$ the subsequence $\\{\\l_n\\}_{|n| > N}$ is\nseparated.\n\\end{definition}\nLet us recall a notion of strictly regular boundary conditions.\n\\begin{definition} \\label{def:strictly.regular}\nBoundary conditions~\\eqref{eq:cond} are called \\textbf{strictly regular}, if they\nare regular, i.e. $J_{14} J_{32} \\ne 0$, and the sequence of zeros $\\l_0 =\n\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the characteristic determinant $\\Delta_0(\\cdot)$ is\nasymptotically separated. In particular, there exists $n_0$ such that zeros\n$\\{\\l_n^0\\}_{|n| > n_0}$ are geometrically and algebraically simple.\n\\end{definition}\nIt follows from Proposition~\\ref{prop:Delta.regular.basic} that the sequence $\\L = \\{\\l_n\\}_{n \\in \\mathbb{Z}}$ of zeros of $\\Delta_Q(\\cdot)$ is asymptotically separated if the boundary conditions are strictly regular.\n\nAssuming boundary conditions~\\eqref{eq:cond} to be regular, let us rewrite them in\na more convenient form. Since $J_{14} \\ne 0$, the inverse matrix $A_{14}^{-1}$\nexists. Therefore writing down boundary conditions~\\eqref{eq:cond} as the vector\nequation $\\binom{U_1(y)}{U_2(y)} = 0$ and multiplying it by the matrix\n$A_{14}^{-1}$ we transform these conditions as follows\n\\begin{equation} \\label{eq:cond.canon}\n\\begin{cases}\n \\widehat{U}_{1}(y) = y_1(0) + b y_2(0) + a y_1(1) = 0, \\\\\n \\widehat{U}_{2}(y) = d y_2(0) + c y_1(1) + y_2(1) = 0,\n\\end{cases}\n\\end{equation}\nwith some $a,b,c,d \\in \\mathbb{C}$. Now $J_{14} = 1$ and the boundary conditions\n~\\eqref{eq:cond.canon} are regular if and only if $J_{32} = ad-bc \\ne 0$. Thus, the\ncharacteristic determinants $\\Delta_0(\\cdot)$ and $\\Delta(\\cdot)$ take the form\n\\begin{align}\n\\label{eq:Delta0.new}\n \\Delta_0(\\l) &= d + a e^{i (b_1+b_2) \\l} + (ad-bc) e^{i b_1 \\l}\n + e^{i b_2 \\l}, \\\\\n\\label{eq:Delta.new}\n \\Delta(\\l) &= d + a e^{i (b_1+b_2) \\l} + (ad-bc) \\varphi_{11}(\\l)\n + \\varphi_{22}(\\l) + c \\varphi_{12}(\\l) + b \\varphi_{21}(\\l).\n\\end{align}\n\\begin{remark} \\label{rem:cond.examples}\nLet us list some types of \\emph{strictly regular} boundary\nconditions~\\eqref{eq:cond.canon}. In all of these cases except 4b the set of zeros\nof $\\Delta_0$ is a union of finite number of arithmetic progressions.\n\n\\begin{enumerate}\n\n\\item Regular boundary conditions~\\eqref{eq:cond.canon} for Dirac operator ($-b_1 = b_2 = 1$) are\nstrictly regular if and only if $(a-d)^2 \\ne -4bc$.\n\n\\item Separated boundary conditions ($a=d=0$, $bc \\ne 0$) are always strictly regular.\n\n\\item Let $b_2 \/ b_1 \\in \\mathbb{Q}$, i.e. $b_1 = -n_1 b_0$, $b_2 = n_2 b_0$, $n_1, n_2 \\in \\mathbb{N}$, $b_0 > 0$ and $\\gcd(n_1,n_2)=1$. Since $ad \\ne bc$, $\\Delta_0(\\cdot) e^{-i b_1 \\l}$ is a polynomial in $e^{i b_0 \\l}$ of degree $n_1 + n_2$ with non-zero roots. Hence, boundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if this polynomial does not have multiple roots. Let us list some cases with explicit conditions.\n\n\\begin{enumerate}\n\n\\item~\\cite[Lemma 5.3]{LunMal16JMAA} Let $ad \\ne 0$ and $bc=0$. Then\nboundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if\n\\begin{equation} \\label{eq:bc=0.crit.rat}\n b_1 \\ln |d| + b_2 \\ln |a| \\ne 0 \\quad\\text{or}\\quad\n n_1 \\arg(-d) - n_2 \\arg(-a) \\notin 2 \\pi \\mathbb{Z}.\n\\end{equation}\n\n\\item In particular, antiperiodic boundary conditions ($a=d=1$, $b=c=0$) are strictly regular if\nand only if $n_1 - n_2$ is odd. Note that these boundary conditions are not strictly regular in\nthe case of a Dirac system.\n\n\\item~\\cite[Proposition 5.6]{LunMal16JMAA} Let $a=0$, $bc \\ne 0$. Then\nboundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if\n\\begin{equation} \\label{eq:a=0.crit.rat}\n n_1^{n_1} n_2^{n_2} (-d)^{n_1 + n_2} \\ne (n_1 + n_2)^{n_1 + n_2} (-b c)^{n_2}.\n\\end{equation}\n\n\\end{enumerate}\n\n\\item Let $\\alpha := -b_1 \/ b_2 \\notin \\mathbb{Q}$. Then the problem of strict regularity of boundary conditions is generally much more complicated. Let us list some known cases:\n\n\\begin{enumerate}\n\n\\item~\\cite[Lemma 5.3]{LunMal16JMAA} Let $ad \\ne 0$ and $bc=0$. Then\nboundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if\n\\begin{equation} \\label{eq:bc=0.crit.irrat}\n b_1 \\ln |d| + b_2 \\ln |a| \\ne 0.\n\\end{equation}\n\n\\item~\\cite[Proposition 5.6]{LunMal16JMAA} Let $a=0$ and $bc, d \\in \\mathbb{R}\n\\setminus \\{0\\}$. Then boundary conditions~\\eqref{eq:cond.canon} are strictly regular if and only if\n\\begin{equation} \\label{eq:a=0.crit}\n d \\ne -(\\alpha+1)\\(|bc| \\alpha^{-\\alpha}\\)^{\\frac{1}{\\alpha+1}}.\n\\end{equation}\n\n\\end{enumerate}\n\n\\end{enumerate}\n\\end{remark}\nIt is well-known that the biorthogonal system to the system of root vectors of the operator $L_U(Q)$ coincides with the system of root vectors of the adjoint operator $L_U^*(Q) := (L_U(Q))^*$ after proper normalization. In this connection we give the explicit form of the operator $L_U(Q)^*$ in the case of boundary conditions~\\eqref{eq:cond.canon}.\n\\begin{lemma}\n\\label{lem:adjoint}\nLet $L_{U}(Q)$ be an operator corresponding to the problem~\\eqref{eq:system}, \\eqref{eq:cond.canon}. Then the adjoint operator $L_U^*(Q)$ is given by the differential expression~\\eqref{eq:system} with $Q^*(x) = \\begin{pmatrix} 0 & \\overline{Q_{21}(x)} \\\\ \\overline{Q_{12}(x)} & 0 \\end{pmatrix}$ instead of $Q$ and the boundary conditions\n\\begin{equation} \\label{eq:cond*}\n\\begin{cases}\n U_{*1}(y) = \\overline{a} y_1(0) + y_1(1) + \\beta^{-1} \\overline{c} y_2(1) &= 0, \\\\\n U_{*2}(y) = \\beta \\overline{b} y_1(0) + y_2(0) + \\overline{d} y_2(1) &= 0,\n\\end{cases}\n\\end{equation}\nwhere as before $\\beta = - b_2\/b_1 > 0$. I.e. $L_U^*(Q) = L_{U*}(Q^*)$. Moreover, boundary conditions~\\eqref{eq:cond*} are regular (strictly regular) simultaneously with boundary conditions~\\eqref{eq:cond.canon}.\n\\end{lemma}\n\\begin{corollary} \\label{cor:sa.crit}\nThe operator $L_U(0)$ corresponding to the problem~\\eqref{eq:system},~\\eqref{eq:cond.canon} with $Q=0$ is selfadjoint if and only if\n\\begin{equation} \\label{eq:abcd.sa2}\na = \\overline{d} u, \\quad d = \\overline{a} u, \\quad b = -\\beta^{-1} \\overline{c} u,\n\\quad c = -\\beta \\overline{b} u, \\qquad u := ad-bc \\ne 0,\n\\end{equation}\nwhich in turn is equivalent to~\\eqref{eq:abcd.sa.intro}.\n\\end{corollary}\n\\begin{proof}\nBoundary conditions~\\eqref{eq:cond.canon} and~\\eqref{eq:cond*} can be rewriten in a matrix form as\n\\begin{equation}\n\\binom{y_1(0)}{y_2(1)} + \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}\n\\binom{y_1(1)}{y_2(0)} = 0 \\quad \\text{and} \\quad\n\\begin{pmatrix} \\overline{a} & \\beta^{-1} \\overline{c} \\\\ \\beta \\overline{b} & \\overline{d} \\end{pmatrix} \\binom{y_1(0)}{y_2(1)} +\n\\binom{y_1(1)}{y_2(0)} = 0,\n\\end{equation}\nrespectively. Hence boundary conditions~\\eqref{eq:cond.canon} and~\\eqref{eq:cond*} are equivalent if and only if\n\\begin{equation}\n\\begin{pmatrix} \\overline{a} & \\beta^{-1} \\overline{c} \\\\ \\beta \\overline{b} & \\overline{d} \\end{pmatrix}  = \\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}^{-1} =\n\\frac{1}{ad-bc}\\begin{pmatrix} d & -b \\\\ -c & a \\end{pmatrix} =\n\\frac{1}{u}\\begin{pmatrix} d & -b \\\\ -c & a \\end{pmatrix},\n\\end{equation}\nwhich is equivalent to~\\eqref{eq:abcd.sa2}.\n\nOn the other hand we can rewrite condtions~\\eqref{eq:cond.canon} as\n\\begin{equation} \\label{eq:Cy0+Dy1}\nC y(0) + D y(1) = 0, \\qquad\nC = \\begin{pmatrix} 1 & b \\\\ 0 & d \\end{pmatrix}, \\qquad\nD = \\begin{pmatrix} a & 0 \\\\ c & 1 \\end{pmatrix}.\n\\end{equation}\nAccording to~\\cite[Lemma 5.1]{LunMal14IEOT}\noperator $L_U(0)$ with boundary conditions rewritten as~\\eqref{eq:Cy0+Dy1} is selfadjoint if and only if $C B C^* = D B D^*$. Straightforward calculations show that\n\\begin{align}\n\\label{eq:CBC*}\n b_1^{-1} C B C^* = b_1^{-1} \\begin{pmatrix} b_1 + b_2 |b|^2 &\n b_2 b  \\overline{d} \\\\ b_2 \\overline{b} d & b_2 |d|^2 \\end{pmatrix} &=\n \\begin{pmatrix} 1 - \\beta |b|^2 & -\\beta b \\overline{d} \\\\ -\\beta \\overline{b} d &\n -\\beta |d|^2 \\end{pmatrix}, \\\\\n\\label{eq:DBD*}\n b_1^{-1} D B D^* = b_1^{-1} \\begin{pmatrix} b_1 |a|^2 & b_1 a \\overline{c} \\\\\n b_1 \\overline{a} c & b_1 |c|^2 + b_2 \\end{pmatrix} &= \\begin{pmatrix}\n |a|^2 & a \\overline{c} \\\\ \\overline{a} c & |c|^2 - \\beta \\end{pmatrix}.\n\\end{align}\nHence $C B C^* = D B D^*$ is equivalent\nto the condition~\\eqref{eq:abcd.sa.intro}. It is interesting to note that establishing equivalence of~\\eqref{eq:abcd.sa.intro} and~\\eqref{eq:abcd.sa2} directly is somewhat tedious.\n\\end{proof}\n\\section{Properties of the spectrum of the unperturbed operator} \\label{sec:unperturb}\nIn this section we obtain some properties of the sequence $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of the characteristic determinant $\\Delta_0(\\cdot)$ in the case of regular boundary conditions~\\eqref{eq:cond.canon} that will be needed in Section~\\ref{sec:bari.c0} to study Bari $c_0$-property of the system of root vectors of the operator $L_U(0)$ (see Definition~\\ref{def:bari.c0}).\nRecall that $x_n \\asymp y_n$, $n \\in \\mathbb{Z}$, means that there exists $C_2 > C_1 > 0$ such that $C_1 |y_n| \\le |x_n| \\le C_2 |y_n|$, $n \\in \\mathbb{Z}$. We start the following simple property of zeros of $\\Delta_0(\\cdot)$.\n\\begin{lemma} \\label{lem:ln0.exp.asymp}\nLet boundary conditions~\\eqref{eq:cond.canon} be regular and $\\L_0 := \\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of zeros of $\\Delta_0(\\cdot)$ counting multiplicity. Set\n\\begin{equation} \\label{eq:ekn.def}\n e_{1n} := e_{1,n} := e^{i b_1 \\l_n^0}, \\qquad\n e_{2n} := e_{2,n} := e^{-i b_2 \\l_n^0},\n \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\n\n\\textbf{(i)} Let $bc \\ne 0$. Then\n\\begin{equation} \\label{eq:1+ae1.1+de2}\n 1 + a e_{1n} \\asymp 1,\n \\qquad 1 + d e_{2n} \\asymp 1, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\n\n\\textbf{(ii)} Let boundary conditions~\\eqref{eq:cond.canon} be strictly regular. Then\n\\begin{equation} \\label{eq:|1+de|+|1+ae|.asymp.1}\n \\abs{1 + a e_{1n}}^2 + \\abs{1 + d e_{2n}}^2\n \\asymp 1, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nNote that\n\\begin{equation}\n \\Delta_0(\\l)\n = \\(1 + a e^{i b_1 \\l}\\) \\(d + e^{i b_2 \\l}\\) - b c \\cdot e^{i b_1 \\l}\n = e^{i b_2\\l} \\(1 + a e^{i b_1 \\l}\\) \\(1 + d e^{-i b_2\\l}\\)\n - b c \\cdot e^{i b_1 \\l}, \\quad \\l \\in \\mathbb{C}.\n\\end{equation}\nSince $\\Delta(\\l_n^0) = 0$, $n \\in \\mathbb{Z}$, then with account of notation~\\eqref{eq:ekn.def} we have\n\\begin{equation} \\label{eq:Delta_0_in_roots}\n \\(1 + a e_{1n}\\) \\(1 + d e_{2n}\\)\n = b c e_{1n} e_{2n}, \\qquad n \\in \\mathbb{Z},\n\\end{equation}\nAccording to Proposition~\\ref{prop:sine.type} the relation~\\eqref{eq:ln.in.Pih} holds.\nHence\n\\begin{equation} \\label{eq:e.bj.ln.asymp.1}\n e_{jn} \\asymp 1, \\qquad n \\in \\mathbb{Z}, \\quad j \\in \\{1, 2\\}.\n\\end{equation}\n\n\\textbf{(i)} Since $bc \\ne 0$, then combining~\\eqref{eq:Delta_0_in_roots} with~\\eqref{eq:e.bj.ln.asymp.1} yields the following estimate with some $C_3 > C_2 > C_1 > 0$,\n\\begin{equation} \\label{eq:||+||>=bc.e}\n C_3 > C_2 \\abs{1 + a e_{1n}}\n \\ge \\abs{(1 + a e_{1n})(1 + d e_{2n})}\n = 2 |bc| \\cdot \\abs{e_{1n} e_{2n}} > C_1, \\quad |n| \\in \\mathbb{Z},\n\\end{equation}\nwhich proves the first relation in~\\eqref{eq:1+ae1.1+de2}. The second relation is proved similarly.\n\n\\textbf{(ii)} If $bc \\ne 0$ then~\\eqref{eq:|1+de|+|1+ae|.asymp.1} is implied by~\\eqref{eq:1+ae1.1+de2}. Let $b c = 0$. In this case $a d \\ne 0$ and $\\Delta_0(\\l) = e^{i b_2\\l} \\(1 + a e^{i b_1 \\l}\\) \\(1 + d e^{-i b_2\\l}\\)$. It is clear that $\\L_0 = \\L_0^1 \\cup \\L_0^2$, where $\\L_0^1 = \\{\\l_{1,n}^{0}\\}_{n \\in \\mathbb{Z}}$ and $\\L_0^2 = \\{\\l_{2,n}^{0}\\}_{n \\in \\mathbb{Z}}$ are the sequences of zeros of the first and second factor, respectively. Clearly, these sequences constitute arithmetic progressions lying on the lines, parallel to the real axis. More precisely,\n\\begin{equation} \\label{eq:l1n.l2n.bc=0}\n \\l_{1,n}^{0} =\n \\frac{\\arg(-a^{-1}) + 2 \\pi n}{b_1} + i\\frac{\\ln|a|}{b_1},\n \\qquad\n \\l_{2,n}^{0} = \\frac{\\arg(-d) + 2 \\pi n}{b_2} - i\\frac{\\ln|d|}{b_2},\n\\end{equation}\nfor $n \\in \\mathbb{Z}$. Since boundary conditions~\\eqref{eq:cond.canon} are strictly regular, then the union of these arithmetic progressions $\\L_0 = \\L_0^1 \\cup \\L_0^2$ is asymptotically separated. It is easily seen that, in fact, $\\L_0$ is separated: if $b_2\/b_1 \\in \\mathbb{Q}$ then $\\L_0$ is periodic and if $b_2\/b_1 \\notin \\mathbb{Q}$ then arithmetic progressions $\\L_0^1$ and $\\L_0^2$ necessarily lie on different parallel lines.\nThis implies the following asymptotic relations:\n\\begin{equation} \\label{eq:1+de.1+ae.asymp.1}\n 1 + d e^{-i b_2 \\l_{2,n}^0} \\asymp 1, \\quad\n 1 + a e^{i b_1 \\l_{1,n}^0} \\asymp 1, \\qquad n \\in \\mathbb{Z},\n\\end{equation}\nSince $\\L_0 = \\L_0^1 \\cup \\L_0^2$, relations~\\eqref{eq:1+de.1+ae.asymp.1} trivially imply~\\eqref{eq:|1+de|+|1+ae|.asymp.1}.\n\\end{proof}\nThroughout the rest of the section we will denote by $\\fr{x} := x - \\floor{x}$ the fractional part of $x \\in \\mathbb{R}$. To treat the tricky case of $\\beta = -b_2\/b_1 \\notin \\mathbb{Q}$, we need Weyl's equidistribution theorem (see~\\cite[Theorem 4.2.2.1]{SteinShak03}). More precisely, we need the following its consequence.\n\\begin{lemma} \\label{lem:weyl}\nLet $\\beta \\in \\mathbb{R} \\setminus \\mathbb{Q}$ and $0 \\le a < b \\le 1$. Then for any $\\varepsilon>0$ there exists $M_{a,b,\\varepsilon} > 0$ such that for $M \\in \\mathbb{N}$ we have:\n\\begin{equation}\n \\card\\{m \\in \\{-M, \\ldots, M\\} : \\fr{\\beta m} \\in [a,b]\\}\n \\le 2 (b - a + \\varepsilon) M, \\qquad M \\ge M_{a,b,\\varepsilon}.\n\\end{equation}\n\\end{lemma}\n\nFirst, let us recall some simple properties of the sequences that have a finite set of limit points. For brevity we denote the cardinality of the limit points set of a bounded sequence $\\{z_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{C}$ as $\\nlim\\{z_n\\}_{n \\in \\mathbb{Z}}$,\n\\begin{multline}\n \\nlim\\curl{z_n}_{n \\in \\mathbb{Z}} := \\card\\left\\{z \\in \\mathbb{C} :\n \\lim_{k \\to \\infty} z_{n_k} = z \\right. \\\\\n \\left. \\text{for some} \\ \\ \\{n_k\\}_{k \\in \\mathbb{N}} \\subset \\mathbb{Z} \\ \\ \\text{such that} \\ \\ n_j \\ne n_k\n \\ \\ \\text{for} \\ \\ j \\ne k \\right\\}.\n\\end{multline}\nIf the set of limit points is infinite we set $\\nlim\\curl{z_n}_{n \\in \\mathbb{Z}} := \\infty$.\n\\begin{lemma} \\label{lem:limit}\nThe following statements hold:\n\n\\begin{enumerate}\n\\item[(i)] Let $\\{a_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{C}$ be bounded, $f$ be continuous on $\\cup_{|n| > N} \\overline{\\mathbb{D}_{\\varepsilon}(a_n)}$ for some $\\varepsilon>0$ and $N > 0$, and $\\nlim \\{a_n\\}_{n \\in \\mathbb{Z}} = m \\in \\mathbb{N}$. Then $\\nlim \\{f(a_n)\\}_{n \\in \\mathbb{Z}} \\le m$.\n\\item[(ii)] Let $\\{a_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{C}$ and $\\{b_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{C}$ be bounded sequences and let $\\nlim \\{a_n\\}_{n \\in \\mathbb{Z}} = m_a$ and $\\nlim \\{b_n\\}_{n \\in \\mathbb{Z}} = m_b \\in \\mathbb{N}$. Then $\\nlim \\{a_n + b_n\\}_{n \\in \\mathbb{Z}} \\le m_a m_b$ and $\\nlim \\{a_n b_n\\}_{n \\in \\mathbb{Z}} \\le m_a m_b$.\n\\item[(iii)] Let $y_n \\in [0, 1)$, $n \\in \\mathbb{Z}$, and let $\\nlim \\{\\sin (2 \\pi y_n)\\}_{n \\in \\mathbb{Z}} = m \\in \\mathbb{N}$. Then $\\nlim \\{y_n\\}_{n \\in \\mathbb{Z}} \\le 2m+1$.\n\\item[(iv)] Let $a, b \\in \\mathbb{R}$, $\\{x_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{R}$ be bounded and $\\nlim \\{x_n\\}_{n \\in \\mathbb{Z}} = m \\in \\mathbb{N}$. Then $$\\nlim\\{\\fr{a x_n + b}\\}_{n \\in \\mathbb{Z}} \\le m+1.$$\n\\end{enumerate}\n\\end{lemma}\nThe following result of Diophantine approximation nature plays crucial role in treating the tricky case of $b_2\/b_1 \\notin \\mathbb{Q}$.\n\\begin{lemma} \\label{lem:sin.weyl}\nLet $b_1,b_2 \\in \\mathbb{R} \\setminus \\{0\\}$ and $b_2\/b_1 \\notin \\mathbb{Q}$. Further, let $\\{\\alpha_n\\}_{n \\in \\mathbb{Z}} \\subset \\mathbb{R}$ be an incompressible sequence such that\n\\begin{equation} \\label{eq:card.alpn>}\n\\card\\{n \\in \\mathbb{Z} : |\\alpha_n| \\le M\\} \\ge \\gamma M,\n\\qquad M \\ge M_0,\n\\end{equation}\nfor some $\\gamma, M_0 > 0$.\nThen one of the sequences $\\{\\sin(b_1 \\alpha_n)\\}_{n \\in \\mathbb{Z}}$ and $\\{\\sin(b_2 \\alpha_n)\\}_{n \\in \\mathbb{Z}}$\nhas an infinite set of limit points.\n\\end{lemma}\n\\begin{proof}\nAssume the contrary. Namely, let\n$$\n\\nlim \\curl{\\sin(b_1 \\alpha_n)}_{n \\in \\mathbb{Z}} = m_1 \\in \\mathbb{N}\n\\qquad\\text{and}\\qquad\n\\nlim \\curl{\\sin(b_2 \\alpha_n)}_{n \\in \\mathbb{Z}} = m_2 \\in \\mathbb{N}.\n$$\nLet us set\n\\begin{equation} \\label{eq:psi1}\n b_1 \\alpha_n = 2 \\pi (k_n + \\delta_n), \\qquad\n k_n := \\floor{\\frac{b_1 \\alpha_n}{2 \\pi}} \\in \\mathbb{Z}, \\quad\n \\delta_n = \\fr{\\frac{b_1 \\alpha_n}{2 \\pi}} \\in [0, 1).\n\\end{equation}\nIt is clear that $\\sin (2 \\pi \\delta_n) = \\sin(b_1 \\alpha_n)$. Hence by Lemma~\\ref{lem:limit}(iii)\n\\begin{equation} \\label{eq:nlim.deltan}\n \\nlim \\{\\delta_n\\}_{n \\in \\mathbb{Z}} \\le 2m_1+1.\n\\end{equation}\nIt is clear from~\\eqref{eq:psi1} that\n$$\nb_2 \\alpha_n = 2 \\pi \\( \\beta k_n + \\beta \\delta_n\\),\n\\quad n \\in \\mathbb{Z}, \\qquad \\beta := b_2\/b_1 \\notin \\mathbb{Q}.\n$$\nThe same reasoning as above shows that\n$$\n\\nlim \\curl{u_n}_{n \\in \\mathbb{Z}} \\le 2m_2+1, \\qquad\nu_n := \\fr{\\beta k_n + \\beta \\delta_n}.\n$$\nFurther, combining~\\eqref{eq:nlim.deltan} with by Lemma~\\ref{lem:limit}(iv) implies that\n\\begin{equation}\n\\nlim\\curl{v_n}_{n \\in \\mathbb{Z}} \\le 2m_1+2, \\quad\\text{where}\\quad\nv_n := \\fr{\\beta \\delta_n}, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\nFinally, note that $\\fr{\\beta k_n} = \\fr{u_n - v_n}$, $n \\in \\mathbb{Z}$. Hence by parts (ii) and (iv) of Lemma~\\ref{lem:limit}\nthe sequence $\\curl{\\fr{\\beta k_n}}_{n \\in \\mathbb{Z}}$ has exactly $p \\le (2m_2+1)(2m_1+2)+1$ limit points $0 \\le x_1 < \\ldots < x_p \\le 1$.\n\nLet $\\varepsilon > 0$ be fixed. Then there exists $N_{\\varepsilon} \\in \\mathbb{N}$ such that\n\\begin{equation} \\label{eq:beta.kn.in.Ieps}\n \\fr{\\beta k_n} \\in \\mathcal{I}_{\\varepsilon} := [0,1) \\cap\n \\bigcup_{j=1}^p (x_j-\\varepsilon, x_j+\\varepsilon), \\qquad |n| \\ge N_{\\varepsilon}.\n\\end{equation}\nSince $\\beta \\notin \\mathbb{Q}$, Lemma~\\ref{lem:weyl} implies that\n\\begin{align} \\label{eq:card.JepsM}\n \\card(\\mathcal{J}_{\\varepsilon,M}) & \\le 6 p \\varepsilon M,\n \\qquad M \\ge M_{\\varepsilon}, \\quad M \\in \\mathbb{N}, \\quad\\text{where} \\\\\n\\label{eq:JepsM.def}\n \\mathcal{J}_{\\varepsilon,M} & :=\n \\{m \\in \\{-M, \\ldots, M\\} : \\fr{\\beta m} \\in \\mathcal{I}_{\\varepsilon}\\},\n \\quad M \\in \\mathbb{N},\n\\end{align}\nFor $M_{\\varepsilon} := \\max\\bigl\\{M_{x_j-\\varepsilon,x_j+\\varepsilon,\\varepsilon} : j \\in \\{1,\\ldots, p\\}\\bigr\\}$.\n\nLet $M \\in \\mathbb{N}$ and consider the set\n$$\n\\mathcal{K}_{\\varepsilon,M} := \\curl{|n| \\ge N_{\\varepsilon} : |k_n| \\le M} \\subset \\mathbb{Z},\n$$\nIt is clear from~\\eqref{eq:psi1} and inequality $|[x]| < |x|+1$ that\n$$\n\\mathcal{K}_{\\varepsilon,M} \\supset \\curl{|n| \\ge N_{\\varepsilon} : |\\alpha_n| \\le \\widetilde{M}},\n\\qquad \\widetilde{M} := \\frac{2 \\pi (M-1)}{|b_1|}.\n$$\nHence if $\\widetilde{M} \\ge M_0$ condition~\\eqref{eq:card.alpn>} implies that\n\\begin{equation} \\label{eq:card.KepsM>}\n \\card(\\mathcal{K}_{\\varepsilon,M}) \\ge \\gamma \\widetilde{M} - 2 N_{\\varepsilon} + 1 \\ge \\gamma_1 M, \\qquad M \\ge \\widetilde{M}_{\\varepsilon},\n\\end{equation}\nwith $\\gamma_1 := \\pi \\gamma |b_1^{-1}| > 0$ and some $\\widetilde{M}_{\\varepsilon} \\ge M_{\\varepsilon}$.\nCondition~\\eqref{eq:beta.kn.in.Ieps} and definition~\\eqref{eq:JepsM.def} of $\\mathcal{J}_{\\varepsilon,M}$ imply that for $n \\in \\mathcal{K}_{\\varepsilon,M}$ we have $k_n \\in \\mathcal{J}_{\\varepsilon,M}$. Since $\\curl{\\alpha_n}_{n \\in \\mathbb{Z}}$ is incompressible then so is $\\curl{k_n}_{n \\in \\mathbb{Z}}$. Hence multiplicities $d_m := \\card\\curl{n \\in \\mathbb{Z} : k_n = m}$ are bounded, $d_m \\le d$, $m \\in \\mathbb{Z}$, for some $d \\in \\mathbb{N}$. Hence for every $m \\in \\mathcal{J}_{\\varepsilon,M}$ there are at most $d$ values of $n \\in \\mathcal{K}_{\\varepsilon,M}$ for which $k_n = m$. Combining this observation with the estimate~\\eqref{eq:card.JepsM} we arrive at\n\\begin{equation} \\label{eq:card.KepsM<}\n \\card\\(\\mathcal{K}_{\\varepsilon,M}\\) \\le d \\card\\(\\mathcal{J}_{\\varepsilon,M}\\) \\le 6 d p \\varepsilon M.\n\\end{equation}\nNow picking $\\varepsilon > 0$ such that that $6 d p \\varepsilon < \\gamma_1$ and $M > \\widetilde{M}_{\\varepsilon}$ we see that cardinality estimates~\\eqref{eq:card.KepsM>} and~\\eqref{eq:card.KepsM<} contradict to each other, which finishes the proof.\n\\end{proof}\n\\begin{remark}\nIt is clear from the proof of Lemma~\\ref{lem:sin.weyl} that the statement remains valid if we relax condition~\\eqref{eq:card.alpn>} to only hold for $M \\in \\mathcal{M} \\subset \\mathbb{N}$, where $\\mathcal{M}$ is some fixed unbounded subset of $\\mathbb{N}$.\n\\end{remark}\nTo apply Lemma~\\ref{lem:sin.weyl} we first need to establish property~\\ref{lem:sin.weyl} for the sequence $\\{\\Re \\l_n^0\\}_{n \\in \\mathbb{Z}}$. It easily follows from the asymptotic formula~\\eqref{eq:lam.n=an+o1}.\n\\begin{lemma} \\label{lem:density}\nLet the boundary conditions~\\eqref{eq:cond} be regular. Then for every $\\varepsilon > 0$ there exists $N_{\\varepsilon} > 0$ such that\n\\begin{equation} \\label{eq:card.Re.ln0}\n\\card \\left\\{ n \\in \\mathbb{Z} : |\\Re \\l_n^0| \\le N \\right\\} \\ge \\frac{N}{\\sigma + \\varepsilon}, \\quad N \\ge N_{\\varepsilon},\n\\qquad \\sigma := \\frac{\\pi}{b_2-b_1} > 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nAsymptotic formula~\\eqref{eq:lam.n=an+o1} for $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ implies that $|\\Re \\l_n^0| \\le (2 \\sigma + \\varepsilon) |n|$, $|n| \\ge n_{\\varepsilon}$, for some $n_{\\varepsilon} \\in \\mathbb{N}$. Hence\n$$\n\\mathbb{Z} \\cap \\(\\[-\\frac{N}{2\\sigma + \\varepsilon}, -n_{\\varepsilon}\\] \\cup \\[n_{\\varepsilon}, \\frac{N}{2\\sigma + \\varepsilon}\\]\\) \\subset \\mathcal{I}_N := \\left\\{ n \\in \\mathbb{Z} : |\\Re \\l_n^0| \\le N \\right\\},\n$$\nfor $N \\ge (2\\sigma+\\varepsilon) n_{\\varepsilon}$. Taking cardinalities in this inclusion implies\n\\begin{equation}\n\\card \\mathcal{I}_N \\ge 2 \\(\\floor{\\frac{N}{2\\sigma + \\varepsilon}} - n_{\\varepsilon} + 1\\)\n\\ge \\frac{N}{\\sigma + \\varepsilon\/2} - 2 n_{\\varepsilon} \\ge \\frac{N}{\\sigma + \\varepsilon},\n\\qquad N \\ge N_{\\varepsilon},\n\\end{equation}\nwith $N_{\\varepsilon} := 2(\\sigma\/\\varepsilon+1)(2\\sigma+\\varepsilon) n_{\\varepsilon}$.\n\\end{proof}\nCombining two previous results leads to the following important property of zeros of characteristic determinant $\\Delta_0(\\cdot)$, which coincides with Proposition~\\ref{prop:nlim.inf.intro} and is formulated again for reader's convenient.\n\\begin{proposition} \\label{prop:nlim.inf}\nLet $b_2\/b_1 \\notin \\mathbb{Q}$ and boundary conditions~\\eqref{eq:cond.canon.intro} be regular, i.e. $u := ad-bc \\ne 0$. Let $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of zeros of the characteristic determinant $\\Delta_0(\\cdot)$ counting multiplicity. Then each of the sequences $\\{\\exp(i b_1 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ and $\\{\\exp(i b_2 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ has infinite set of limit points.\n\\end{proposition}\n\\begin{proof}\n\\textbf{(i)} First, let $bc=0$. Then according to the proof of Lemma~\\ref{lem:ln0.exp.asymp}, zeros of the characteristic determinant $\\Delta_0(\\cdot)$ are simple and split into two separated arithmetic progressions $\\L_0^1 = \\{\\l_{1,n}^{0}\\}_{n \\in \\mathbb{Z}}$ and $\\L_0^2 = \\{\\l_{2,n}^{0}\\}_{n \\in \\mathbb{Z}}$ given by~\\eqref{eq:l1n.l2n.bc=0}. Let $k \\in \\{1,2\\}$ and $j=2\/k$. Since $E(z) = e^{2 \\pi i z}$ is periodic with period 1, we have for $n \\in \\mathbb{Z}$,\n\\begin{equation}\n\\exp(i b_k \\l_{j,n}^0) = \\exp\\(2 \\pi i n b_k\/b_j + \\omega_{k,j,a,d}\\)\n= \\exp\\(2 \\pi i \\fr{n b_k\/b_j} + \\omega_{k,j,a,d}\\),\n\\end{equation}\nwhere $\\omega_{k,j,a,d}$ is an explicit constant that can be derived from~\\eqref{eq:l1n.l2n.bc=0}. Since $b_k\/b_j \\notin \\mathbb{Q}$, then by the classical Kronecker theorem, the sequence $\\curl{\\fr{n b_k\/b_j}}_{n \\in \\mathbb{Z}}$ is everywhere dense on $[0,1]$. This implies that the sequence $\\{\\exp(i b_k \\l_{j,n}^0)\\}_{n \\in \\mathbb{Z}}$ has infinite set of limit points, which finishes the proof in this case.\n\n\\textbf{(ii)} Now, let $bc \\ne 0$ and assume the contrary: one of the sequences $\\{\\exp(i b_1 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ and $\\{\\exp(i b_2 \\l_n^0)\\}_{n \\in \\mathbb{Z}}$ has a finite set of limit points. For definiteness assume that\n\\begin{equation} \\label{eq:nlim.e1n}\n \\nlim \\{\\exp(i b_1 \\l_n^0)\\}_{n \\in \\mathbb{Z}} = m_1 \\in \\mathbb{N}.\n\\end{equation}\nRecall that $e_{1n} := \\exp(i b_1 \\l_n^0)$ and $e_{2n} := \\exp(-i b_2 \\l_n^0)$, $n \\in \\mathbb{Z}$, and also set\n\\begin{equation} \\label{eq:ln0=an+ibn}\n \\l_n^0 = \\alpha_n + i \\beta_n, \\qquad \\alpha_n := \\Re \\l_n^0,\n \\quad \\beta_n := \\Im \\l_n^0, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nIt is clear that\n\\begin{equation} \\label{eq:abs.e1n}\n |e_{1n}| = |\\exp(i b_1 \\l_n^0)| = \\exp(-b_1 \\Im \\l_n^0)\n = \\exp(-b_1 \\beta_n), \\qquad n \\in \\mathbb{Z}\n\\end{equation}\nIt follows from~\\eqref{eq:abs.e1n}, \\eqref{eq:nlim.e1n}, \\eqref{eq:e.bj.ln.asymp.1} and Lemma~\\ref{lem:ln0.exp.asymp}(i), applied with $f_1(z) = -b_1^{-1} \\log |z|$, that\n\\begin{equation} \\label{eq:nlim.Imln0}\n \\nlim \\{\\beta_n\\}_{n \\in \\mathbb{Z}} =\n \\nlim \\{-b_1^{-1} \\log |e_{1n}| \\}_{n \\in \\mathbb{Z}} \\le m_1.\n\\end{equation}\nIn turn, since $e_{1n} = |e_{1n}| e^{i b_1 \\alpha_n} \\asymp 1$, $n \\in \\mathbb{Z}$,\nthen by Lemma~\\ref{lem:limit}(i) applied with $f_2(z) = \\Im z \/ |z|$ we have\n\\begin{equation} \\label{eq:nlim.sinb1}\n \\nlim \\curl{\\sin\\(b_1 \\alpha_n\\)}_{n \\in \\mathbb{Z}} =\n \\nlim \\curl{\\Im e_{1n} \/ |e_{1n}|}_{n \\in \\mathbb{Z}} \\le m_1.\n\\end{equation}\nSince $bc \\ne 0$ and boundary conditions~\\eqref{eq:cond.canon} are regular, then Lemma~\\ref{lem:ln0.exp.asymp}(i) implies~\\eqref{eq:1+ae1.1+de2}. Recall that $u := ad-bc \\ne 0$. Since $\\Delta(\\l_n^0)=0$, $1 + a e_{1n} \\ne 0$ and $1 + d e_{2n} \\ne 0$, $n \\in \\mathbb{Z}$, it follows from~\\eqref{eq:Delta0.new} that for $n \\in \\mathbb{Z}$:\n\\begin{equation} \\label{eq:e1.via.e2}\n 1 + d e_{2n} + a e_{1n} + u e_{1n} e_{2n} = 0, \\qquad\n e_{2n} = - \\frac{1 + a e_{1n}}{d + u e_{1n}}, \\qquad\n e_{1n} = - \\frac{1 + d e_{2n}}{a + u e_{2n}}.\n\\end{equation}\nSince $e_{1n} \\asymp 1$, $e_{2n} \\asymp 1$, $n \\in \\mathbb{Z}$, relations~\\eqref{eq:1+ae1.1+de2} and~\\eqref{eq:e1.via.e2} imply that\n\\begin{equation} \\label{eq:a+ue2.d+ue1}\n d + u e_{1n} \\asymp 1, \\qquad a + u e_{2n} \\asymp 1, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nHence $f_3(z) := - \\frac{1 + a z}{d + u z}$ is continuous in the neighborhood of $\\{e_{1n}\\}_{n \\in \\mathbb{Z}}$. Combining this with Lemma~\\ref{lem:limit}(i), the second identity in~\\eqref{eq:e1.via.e2} and relation~\\eqref{eq:nlim.e1n} we arrive at\n\\begin{equation} \\label{eq:nlim.e2n}\n \\nlim \\{e_{2n}\\}_{n \\in \\mathbb{Z}} = \\nlim \\{f_3(e_{1n})\\}_{n \\in \\mathbb{Z}} \\le m_1.\n\\end{equation}\nSimilarly to~\\eqref{eq:nlim.sinb1} we get\n\\begin{equation} \\label{eq:nlim.sinb2}\n \\nlim \\curl{\\sin\\(b_2 \\alpha_n\\)}_{n \\in \\mathbb{Z}} =\n \\nlim \\curl{\\Im e_{2n} \/ |e_{2n}|}_{n \\in \\mathbb{Z}} \\le m_1.\n\\end{equation}\nSince boundary conditions~\\eqref{eq:cond.canon} are regular then Proposition~\\ref{prop:sine.type} implies that the sequence $\\{\\alpha_n\\}_{n \\in \\mathbb{Z}}$ is incompressible and Lemma~\\ref{lem:density} implies the estimate~\\eqref{eq:card.Re.ln0}, which in turn yields the estimate~\\eqref{eq:card.alpn>} for $\\{\\alpha_n\\}_{n \\in \\mathbb{Z}}$ with $\\gamma = \\frac{1}{2\\sigma} = \\frac{b_2-b_1}{2\\pi}$. Since $b_2\/b_1 \\notin \\mathbb{Q}$ then by Lemma~\\ref{lem:sin.weyl} one of sequences $\\curl{\\sin(b_1 \\alpha_n)}_{n \\in \\mathbb{Z}}$ and $\\curl{\\sin(b_2 \\alpha_n)}_{n \\in \\mathbb{Z}}$ has infinite set of limit points. This contradicts relations~\\eqref{eq:nlim.sinb1} and~\\eqref{eq:nlim.sinb2} and finishes the proof.\n\\end{proof}\n\\section{Bari $c_0$-property of the system of root vectors of the unperturbed operator} \\label{sec:bari.c0}\n\nIn this section assuming boundary conditions~\\eqref{eq:cond.canon} to be strictly regular, we show that the system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$ if and only if the operator $L_U(0)$ is selfadjoint. Since eigenfunctions of $L_U(0)$ in their ``natural form'' are not normalized we need the following simple practical criterion of Bari $c_0$-property.\n\\begin{lemma} \\label{lem:crit.bari.not.norm}\nLet $\\mathfrak{F} = \\{f_n\\}_{n \\in \\mathbb{Z}}$ be a complete minimal system of vectors in a Hilbert space $\\mathfrak{H}$. Let also $\\{g_n\\}_{n \\in \\mathbb{Z}}$ be ``almost biorthogonal'' to $\\mathfrak{F}$. Namely, $(f_n, g_m) = 0$, $n \\ne m$, $(f_n, g_n) \\ne 0$, $n, m \\in \\mathbb{Z}$. Then the normalized system\n$$\n\\mathfrak{F}' := \\curl{f_n'}_{n \\in \\mathbb{Z}}, \\quad f_n' := \\frac{1}{\\|f_n\\|} f_n,\n\\quad n \\in \\mathbb{Z},\n$$\nis a Bari $c_0$-sequence in $\\mathfrak{H}$ (see Definition~\\ref{def:bari.c0}) if and only if\n\\begin{equation} \\label{eq:sum.fn.gn-1}\n \\frac{\\|f_n\\|_{\\mathfrak{H}} \\cdot \\|g_n\\|_{\\mathfrak{H}}}{|(f_n, g_n)_{\\mathfrak{H}}|} \\to 1\n \\quad\\text{as}\\quad n \\to \\infty.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nFor brevity we set $\\|\\cdot\\| := \\|\\cdot\\|_{\\mathfrak{H}}$ and $(\\cdot,\\cdot) := (\\cdot,\\cdot)_{\\mathfrak{H}}$.\nIt is clear that for the system $\\mathfrak{G}' := \\{g_n'\\}_{n \\in \\mathbb{Z}}$ that is biorthogonal to $\\mathfrak{F}'$ we have,\n\\begin{equation} \\label{eq:fn'gm'}\n (f_n', g_m') = \\delta_{nm}, \\quad n,m \\in \\mathbb{Z}; \\qquad\n g_n' = \\frac{\\|f_n\\|}{(f_n, g_n)} \\cdot g_n, \\quad n \\in \\mathbb{Z}.\n\\end{equation}\nRelations~\\eqref{eq:fn'gm'} imply that\n\\begin{equation} \\label{eq:|fn'-gn'|}\n \\|f_n' - g_n'\\|^2 = \\|f_n'\\|^2 - (f_n', g_n') - \\overline{(f_n', g_n')}\n + \\|g_n'\\|^2 = \\|g_n'\\|^2 - 1\n = \\frac{\\|f_n\\|^2 \\cdot \\|g_n\\|^2}{|(f_n, g_n)|^2} - 1.\n\\end{equation}\nHence, systems $\\mathfrak{F}'$ and $\\mathfrak{G}'$ are $c_0$-close if and only if condition~\\eqref{eq:sum.fn.gn-1} holds.\n\\end{proof}\nThe following simple property of compact operators with asymptotically simple spectrum will be also useful in the next section.\n\\begin{lemma} \\label{lem:some.every}\nLet $T$ be an operator with compact resolvent in a separable Hilbert space $\\mathfrak{H}$ and let $\\{\\l_n\\}_{n \\in \\mathbb{Z}}$ be a sequence of its eigenvalues counting multiplicities. Let also $p \\in [1,2]$.\nAssume that for some $N \\in \\mathbb{N}$ eigenvalues $\\l_n$, $|n| \\ge N$, are algebraically simple.\nThen if some normalized system of root vectors of the operator $T$ is a Bari $(\\ell^p)^*$-sequence in $\\mathfrak{H}$ then every normalized system of root vectors of the operator $T$ is a Bari $(\\ell^p)^*$-sequence in $\\mathfrak{H}$.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathfrak{F} = \\{f_n\\}_{n \\in \\mathbb{Z}}$\nbe a normalized system of root vectors of the operator $T$, which is a Bari $(\\ell^p)^*$-sequence in $\\mathfrak{H}$. By definition,\nthe system $\\mathfrak{F}$ is complete and minimal in $\\mathfrak{H}$. Let $\\mathfrak{G} = \\{g_n\\}_{n \\in \\mathbb{Z}}$ be its (unique) biorthogonal system. Further, let $\\mathfrak{F}' = \\{f_n'\\}_{n \\in \\mathbb{Z}}$ be any other normalized system of root vectors of the operator $T$. Since eigenvalue $\\l_n$, $|n| \\ge N$, is algebraically simple then $\\dim(\\mathcal{R}_{\\l_n}(T)) = 1$, $|n| \\ge N$. Hence $f_n' = \\alpha_n f_n$, $|n| \\ge N$, for some $\\alpha_n \\in \\mathbb{T} := \\{z \\in \\mathbb{C} : |z|=1\\}$. It is clear that $\\mathfrak{F}'$ is also complete and minimal and for its biorthogonal sysyem $\\mathfrak{G}' = \\{g_n'\\}_{n \\in \\mathbb{Z}}$ we have that $g_n' = \\overline{\\alpha_n^{-1}} g_n = \\alpha_n g_n$, $|n| \\ge N$, since $|\\alpha_n|=1$. Hence $\\|f_n'-g_n'\\|_{\\mathfrak{H}} = \\|\\alpha_n \\cdot (f_n-g_n)\\|_{\\mathfrak{H}} = \\|f_n-g_n\\|_{\\mathfrak{H}}$, $|n| \\ge N$. This implies that $\\curl{\\|f_n'-g_n'\\|_{\\mathfrak{H}}}_{n \\in \\mathbb{Z}} = \\curl{\\|f_n-g_n\\|_{\\mathfrak{H}}}_{n \\in \\mathbb{Z}} \\in (\\ell^p)^*$ and finishes the proof.\n\\end{proof}\n\\begin{remark} \\label{rem:sa.c0.bari}\nLet $A$ be a selfadjoint operator with compact resolvent. Then every its normalized system of root vectors is an orthonormal basis in $\\LLV{2}$ and coincides with its biorthogonal sequence. This implies that every normalized system of root vectors of the operator $A$ is a Bari $c_0$-sequence.\n\\end{remark}\nTo study norms $\\|f_n^0\\|_2$ and $\\|g_n^0\\|_2$ of the eigenvectors of the operators $L_U(0)$ and $L_U^*(0)$, we first need to obtain some properties of simple integrals $\\int_0^1 |e^{\\pm 2 i b_j \\l x}| dx$, $j \\in \\{1,2\\}$, $\\l \\in \\mathbb{C}$.\n\\begin{lemma} \\label{lem:ejx.ej.Ej}\nDenote for $j \\in \\{1,2\\}$ and $\\l \\in \\mathbb{C}$:\n\\begin{equation} \\label{eq:Ejpm.def}\n E_{j}^{\\pm}(\\l) := \\int_0^1 \\abs{e^{\\pm 2 i b_j \\l x}} dx\n = \\int_0^1 e^{\\mp 2 b_j \\Im \\l x} dx\n = \\frac{e^{\\mp 2 b_j \\Im \\l} - 1}{\\mp 2 b_j \\Im \\l}.\n\\end{equation}\nThen the following estimate holds:\n\\begin{align}\n\\label{eq:Ej+Ej->1}\n & E_j^+(\\l) E_j^-(\\l) - 1 \\ge \\frac{(b_j \\Im \\l)^2}{3},\n \\qquad j \\in \\{1,2\\}, \\quad \\l \\in \\mathbb{C}.\n\\end{align}\nIn particular, $E_j^+(\\l) E_j^-(\\l) - 1 > 0$ if $\\Im \\l \\ne 0$.\n\\end{lemma}\n\\begin{proof}\nLet $h \\ge 0$. It is clear that\n\\begin{equation} \\label{eq:Ej=f}\n E_j^{\\pm}(\\l) = f(\\mp 2 b_j \\Im \\l), \\quad\\text{where}\\quad\n f(x) := \\frac{e^x - 1}{x} = 1 + \\frac{x}{2} + O(x^2), \\quad |x| < h.\n\\end{equation}\nIt follows from Taylor expansion of $e^x$ that for $x \\in \\mathbb{R}$:\n\\begin{equation} \\label{eq:fx.f-x>1}\n f(x)f(-x) = \\frac{e^x - 1}{x} \\cdot \\frac{e^{-x} - 1}{-x}\n = \\frac{e^x + e^{-x} - 2}{x^2}\n = 2 \\sum_{k=1}^{\\infty} \\frac{x^{2k-2}}{(2k)!}\n \\ge 1 + \\frac{x^2}{12}.\n\\end{equation}\nEstimate~\\eqref{eq:Ej+Ej->1} now immediately follows from~\\eqref{eq:Ej=f} and~\\eqref{eq:fx.f-x>1}.\n\n\\end{proof}\nFirst we establish the Bari $c_0$-property criterion in a special case $b=c=0$.\n\\begin{proposition} \\label{prop:crit.bari.period}\nLet boundary conditions~\\eqref{eq:cond.canon} be strictly regular with $b=c=0$, i.e. they are of the form\n\\begin{equation} \\label{eq:quasi.per.bc}\n y_1(0) + a y_1(1) = d y_2(0) + y_2(1) = 0, \\qquad a d \\ne 0.\n\\end{equation}\nThen some normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$ (see Definition~\\ref{def:bari.c0}) if and only if $|a| = |d| = 1$.\n\\end{proposition}\n\\begin{proof}\n\\textbf{(i)} If $|a|=|d|=1$ (and $b=c=0$) then by Corollary~\\ref{cor:sa.crit} the operator $L_U(0)$ with boundary conditions~\\eqref{eq:quasi.per.bc} is self-adjoint. Remark~\\ref{rem:sa.c0.bari} now finishes the proof.\n\n\\textbf{(ii)} Now assume that some normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$. Since boundary conditions~\\eqref{eq:quasi.per.bc} are strictly regular then by definition, eigenvalues of the operator $L_U(0)$ are asymptotically simple.\nHence by Lemma~\\ref{lem:some.every} every normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$.\n\nAccording to the proof of Lemma~\\ref{lem:ln0.exp.asymp} the eigenvalues of the operator $L_U(0)$ are simple and split into two separated arithmetic progressions $\\L_0^1 = \\{\\l_{1,n}^{0}\\}_{n \\in \\mathbb{Z}}$ and $\\L_0^2 = \\{\\l_{2,n}^{0}\\}_{n \\in \\mathbb{Z}}$ given by~\\eqref{eq:l1n.l2n.bc=0}.\nIt is easy to verify that the vectors\n\\begin{equation}\n f_{1,n}^0(x) = \\binom{e^{i b_1 \\l_{1,n}^0 x}}{0}, \\qquad\n g_{1,n}^0(x) = \\binom{e^{i b_1 \\overline{\\l_{1,n}^0} x}}{0},\n \\qquad n \\in \\mathbb{Z},\n\\end{equation}\nare the eigenvectors of the operators $L_U(0)$ and $L_U^*(0)$ corresponding to the eigenvalues $\\l_{1,n}^0$ and $\\overline{\\l_{1,n}^0}$, and the vectors\n\\begin{equation}\n f_{2,n}^0(x) = \\binom{0}{e^{i b_2 \\l_{2,n}^0 x}}, \\qquad\n g_{2,n}^0(x) = \\binom{0}{e^{i b_2 \\overline{\\l_{2,n}^0} x}}, \\qquad n \\in \\mathbb{Z},\n\\end{equation}\nare the eigenvectors of the operators $L_U(0)$ and $L_U^*(0)$  corresponding to the eigenvalues $\\l_{2,n}^0$ and $\\overline{\\l_{2,n}^0}$ respectively. It is clear that\n\\begin{equation} \\label{eq:fjn.gkm=delta}\n \\(f_{j,n}^0, g_{k,m}^0\\)_2 = \\delta_{j,n}^{k,m}, \\qquad j,k \\in \\{1,2\\},\n \\quad n,m \\in \\mathbb{Z}.\n\\end{equation}\nThus the union system $\\mathfrak{F} := \\{f_{1,n}^0\\}_{n \\in \\mathbb{Z}} \\cup \\{f_{2,n}^0\\}_{n\n\\in \\mathbb{Z}}$ is the system of root vectors of the operator $L_U(0)$ and $\\mathfrak{G} :=\n\\{g_{1,n}^0\\}_{n \\in \\mathbb{Z}} \\cup \\{g_{2,n}^0\\}_{n \\in \\mathbb{Z}}$ is biorthogonal to it. Hence normalization of the system $\\mathfrak{F}$ is a Bari $c_0$-sequence in $L^2([0,1]; \\mathbb{C}^2)$. According to Lemma~\\ref{lem:crit.bari.not.norm} we have\n\\begin{equation} \\label{eq:sumj.sumn.alp}\n \\alpha_{j,n} := \\frac{\\bigl\\|f_{j,n}^0\\bigr\\|_2 \\cdot\n \\bigl\\|g_{j,n}^0\\bigr\\|_2}{\\abs{\\bigl(f_{j,n}^0, g_{j,n}^0\\bigr)_2}}\n \\to 1 \\quad\\text{as}\\quad n \\to \\infty, \\qquad j \\in \\{1,2\\}.\n\\end{equation}\nLet $j=1$. Then taking into account Lemma~\\ref{lem:ejx.ej.Ej} and formula~\\eqref{eq:fjn.gkm=delta} we have\n\\begin{equation} \\label{eq:alp1n>Im}\n \\alpha_{1,n}^2 = \\bigl\\|f_{1,n}^0\\bigr\\|_2^2 \\cdot\n \\bigl\\|g_{1,n}^0\\bigr\\|_2^2 = E_1^+(\\l_{1,n}^0) E_1^-(\\l_{1,n}^0)\n \\ge 1 + \\frac13 \\bigabs{b_2 \\Im \\l_{1,n}^0}^2, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nIt follows from~\\eqref{eq:l1n.l2n.bc=0} that $b_1 \\Im \\l_{1,n}^0 = \\ln|a|$. Since $\\alpha_{1,n} \\to 1$ as $n \\to \\infty$, formula~\\eqref{eq:alp1n>Im} implies that $\\ln|a|=0$, or $|a|=1$.\n\nSimilarly considering the case $j=2$ we conclude that $|d|=1$, which finishes the proof.\n\\end{proof}\nIn the following intermediate result we reduce condition~\\eqref{eq:sum.fn.gn-1} of Bari $c_0$-property of the system of root vectors of the operator $L_U(0)$ to explicit condition in terms of eigenvalues $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$.\nRecall that $\\beta = -b_2\/b_1 > 0$.\n\\begin{proposition} \\label{prop:crit.bari.b.ne.0}\nLet boundary conditions~\\eqref{eq:cond.canon} be strictly regular and let one of the parameters $b$ or $c$ in them be non-zero, $|b|+|c|>0$. Let $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be the sequence of the eigenvalues of the operator $L_U(0)$ counting multiplicities. Then some normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$ (see Definition~\\ref{def:bari.c0}) if and only if the following conditions hold\n\\begin{equation} \\label{eq:lim1.lim2}\n |c| = \\beta |b|, \\qquad \\lim_{n \\to \\infty} \\Im \\l_n^0 = 0\n \\quad\\text{and}\\quad \\lim_{n \\to \\infty} z_n = |bc|,\n\\end{equation}\nwhere\n\\begin{equation} \\label{eq:zn.def}\n z_n := \\(1 + d e^{- i b_2 \\l_n^0}\\)\\overline{\\(1 + a e^{i b_1 \\l_n^0}\\)}.\n\\end{equation}\n\n\\end{proposition}\n\\begin{proof}\nWithout loss of generality we can assume that $b \\ne 0$. By definition of strictly regular boundary conditions there exists $n_0 \\in \\mathbb{N}$ such that eigenvalues $\\l_n^0$ of $L_U(0)$ for $|n| > n_0$ are algebraically simple and separated from each other. According to the proof of Theorem~1.1 in~\\cite{LunMal16JMAA} vector-functions $f_n^0(\\cdot)$ and $g_n^0(\\cdot)$, $|n| > n_0$, of the following form:\n\\begin{equation} \\label{eq:fn0x.gn0x}\n f_n^0(x) := \\binom{b e^{i b_1 \\l_n^0 x}}{\n - (1 + a e^{i b_1 \\l_n^0}) e^{i b_2 \\l_n^0 x}}, \\qquad\n g_n^0(x) := \\overline{\\binom{(1 + d e^{-i b_2 \\l_n^0})\n e^{-i b_1 \\l_n^0 x}}{ - \\beta b e^{-i b_2 \\l_n^0 x}}},\n\\end{equation}\nare non-zero eigenvectors of the operators $L_U(0)$ and $L_U^*(0)$ corresponding to the eigenvalues $\\l_n^0$ and $\\overline{\\l_n^0}$ for $|n| > n_0$, respectively. Let $f_n^0(\\cdot)$ and $g_n^0(\\cdot)$ be some root vectors of operators $L_U(0)$ and $L_U^*(0)$ corresponding to the eigenvalues $\\l_n^0$ and $\\overline{\\l_n^0}$ for $|n| \\le n_0$. Clearly $\\mathfrak{F} := \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ is a system of root vectors of the operator $L_U(0)$ and $\\mathfrak{G} := \\{g_n^0\\}_{n \\in \\mathbb{Z}}$ is the corresponding system for the adjoint operator $L_U^*(0)$. Let us show that normalization of $\\mathfrak{F}$ is a Bari $c_0$-sequence in $\\LLV{2}$ if and only condition~\\eqref{eq:lim1.lim2} holds. Since eigenvalues of the operator $L_U(0)$ are asymptotically simple Lemma~\\ref{lem:some.every} will imply the statement of the proposition.\nClearly, $\\mathfrak{G}$ is almost biorthogonal to $\\mathfrak{F}$. Hence Lemma~\\ref{lem:crit.bari.not.norm} implies that normalization of $\\mathfrak{F}$ is a Bari $c_0$-sequence in $\\LLV{2}$ if and only if condition~\\eqref{eq:sum.fn.gn-1} holds.\n\nSet for brevity $E_{jn}^{\\pm} := E_j^{\\pm}(\\l_n^0)$, $j \\in \\{1,2\\}$, $n \\in \\mathbb{Z}$, where $E_j^{\\pm}(\\l)$ is defined in~\\eqref{eq:Ejpm.def}. With account of this notation and notation~\\eqref{eq:ekn.def} we get after performing straightforward calculations:\n\\begin{align}\n\\label{eq:|fn|2}\n \\|f_n^0\\|_2^2 &= |b|^2 E_{1n}^+ + |1 + a e_{1n}|^2 E_{2n}^+, \\\\\n\\label{eq:|gn|2}\n \\|g_n^0\\|_2^2 &= \\abs{1 + d e_{2n}}^2 E_{1n}^- + \\beta^2 |b|^2 E_{2n}^-, \\\\\n\\label{eq:fn.gn}\n (f_n^0, g_n^0)_2 &= b\\((1 + d e_{2n}) + \\beta (1 + a e_{1n})\\).\n\\end{align}\nSince boundary conditions~\\eqref{eq:cond.canon} are strictly regular, it follows from the proof of Theorem~1.1 in~\\cite{LunMal16JMAA} that the following estimate holds\n\\begin{equation}\n (f_n^0, g_n^0)_2 \\asymp \\Delta'(\\l_n^0) \\asymp 1, \\quad |n| > n_0.\n\\end{equation}\nHence condition~\\eqref{eq:sum.fn.gn-1} is equivalent to\n\\begin{equation} \\label{eq:fn.gn-fngn.to0}\n \\|f_n^0\\|_2^2 \\cdot \\|g_n^0\\|_2^2 - |(f_n^0, g_n^0)_2|^2\n \\to 0 \\quad\\text{as}\\quad n \\to \\infty.\n\\end{equation}\nWith account of~\\eqref{eq:|fn|2}--\\eqref{eq:fn.gn} we get\n\\begin{multline} \\label{eq:fn2.gn2-fn.gn2=tau.sum}\n \\|f_n^0\\|_2^2 \\cdot \\|g_n^0\\|_2^2 - |(f_n^0, g_n^0)_2|^2\n = \\(|b|^2 \\cdot E_{1n}^+ + |1 + a e_{1n}|^2 \\cdot E_{2n}^+\\) \\cdot\n \\(\\abs{1 + d e_{2n}}^2 E_{1n}^- + \\beta^2 |b|^2 E_{2n}^-\\) \\\\\n - |b|^2 \\abs{(1 + d e_{2n}) + \\beta (1 + a e_{1n})}^2\n = \\tau_{1,n} + \\tau_{2,n} + \\tau_{3,n}, \\qquad |n| > n_0,\n\\end{multline}\nwhere\n\\begin{align}\n\\label{eq:tau1}\n \\tau_{1,n} &:= |b|^2 \\cdot \\abs{1 + d e_{2n}}^2 \\cdot\n (E_{1n}^+ E_{1n}^- - 1), \\\\\n\\label{eq:tau2}\n \\tau_{2,n} &:= \\beta^2 |b|^2 \\cdot |1 + a e_{1n}|^2 \\cdot (E_{2n}^+ E_{2n}^- - 1), \\\\\n\\label{eq:tau3}\n \\tau_{3,n} &:= \\beta^2 |b|^4 E_{1n}^+ E_{2n}^- +\n |z_n|^2 \\cdot E_{2n}^+ E_{1n}^- -\n 2 \\beta |b|^2 \\cdot \\Re z_n,\n\\end{align}\nwhere $z_n = (1 + d e_{2n}) \\overline{(1 + a e_{1n})}$ is defined in~\\eqref{eq:zn.def}. According to Proposition~\\ref{prop:sine.type}, $|\\Im \\l_n^0| \\le h$, $n \\in \\mathbb{Z}$, for some $h \\ge 0$. Hence terms $|1 + d e_{2n}|$, $|1 + a e_{1n}|$, $|z_n|$, $E_{1n}^{\\pm}$ and $E_{2n}^{\\pm}$ are all bounded for $n \\in \\mathbb{Z}$.\n\nFirst assume that $\\Im \\l_n^0 \\to 0$ as $n \\to \\infty$. Then it is clear from~\\eqref{eq:Ej=f} that\n\\begin{equation} \\label{eq:ejnto1}\n |e_{jn}| \\to 1 \\quad\\text{and}\\quad\n E_{jn}^{\\pm} \\to 1  \\quad\\text{as}\\quad n \\to \\infty,\n \\qquad j \\in \\{1,2\\}.\n\\end{equation}\nHence $\\tau_{1,n} + \\tau_{2,n} \\to 0$ as $n \\to \\infty$, while\n$\\tau_{3,n} \\to 0$ as $n \\to \\infty$ if and only if\n\\begin{equation} \\label{eq:tau4n.def}\n \\tau_{4,n} :=\n \\beta^2 |b|^4 + |z_n|^2\n - 2 \\beta |b|^2 \\cdot \\Re z_n = \\abs{z_n - \\beta |b|^2}^2 \\to 0 \\quad\\text{as}\\quad n \\to \\infty.\n\\end{equation}\nIt follows from~\\eqref{eq:Delta_0_in_roots} and~\\eqref{eq:ejnto1} that $|z_n| = |bc|\\cdot|e_{1n} e_{2n}| \\to |bc|$ as $n \\to \\infty$. Hence, since $b \\ne 0$, then\n\\begin{equation} \\label{eq:tau4n.to0}\n \\Bigl(\\tau_{4,n} \\to 0 \\ \\ \\text{as}\\ \\ n \\to \\infty\\Bigr)\n \\ \\ \\Leftrightarrow \\ \\\n \\Bigl(|c| = \\beta |b| \\ \\ \\text{and}\\ \\ z_n \\to |bc|\n \\ \\ \\text{as}\\ \\ n \\to \\infty\\Bigr).\n\\end{equation}\nNow if condition~\\eqref{eq:lim1.lim2} holds then~\\eqref{eq:tau4n.to0} and previous observations on $\\tau_{1,n}$, $\\tau_{2,n}$, $\\tau_{3,n}$, $\\tau_{4,n}$ imply the desired condition~\\eqref{eq:fn.gn-fngn.to0}.\n\nNow assume that condition~\\eqref{eq:fn.gn-fngn.to0} holds.\nIt follows from~\\eqref{eq:Ej+Ej->1} and~\\eqref{eq:Ej=f} that\n\\begin{equation} \\label{eq:Ej.Ej-1.asymp}\n 0 \\le E_{jn}^+ E_{jn}^- - 1 \\asymp |\\Im \\l_n^0|^2, \\quad n \\in \\mathbb{Z},\n \\qquad j \\in \\{1,2\\}.\n\\end{equation}\nSince $b \\ne 0$ and $\\beta > 0$ relations~\\eqref{eq:tau1}--\\eqref{eq:tau2} and~\\eqref{eq:Ej.Ej-1.asymp}\ncombined with Lemma~\\ref{lem:ln0.exp.asymp} imply that\n\\begin{equation} \\label{eq:tau1+tau2.asymp}\n \\tau_{1,n} \\ge 0, \\quad \\tau_{2,n} \\ge 0, \\quad \\tau_{1,n} + \\tau_{2,n} \\asymp |\\Im \\l_n^0|^2, \\quad |n| > n_0.\n\\end{equation}\nWith account of~\\eqref{eq:Ej.Ej-1.asymp} we get for $n \\in \\mathbb{Z}$:\n\\begin{equation} \\label{eq:tau3.estim}\n \\beta^2 |b|^4 E_{1n}^+ E_{2n}^- + |z_n|^2\n \\cdot E_{2n}^+ E_{1n}^-\n \\ge 2 \\beta |b|^2 |z_n|\n \\sqrt{E_{1n}^+ E_{1n}^- \\cdot E_{2n}^- E_{2n}^+} \\nonumber \\\\\n \\ge 2 \\beta |b|^2 \\cdot \\Re z_n.\n\\end{equation}\nHence $\\tau_{3,n} \\ge 0$, $n \\in \\mathbb{Z}$. Since $\\tau_{1,n} + \\tau_{2,n} + \\tau_{3,n} = \\|f_n^0\\|_2^2 \\cdot \\|g_n^0\\|_2^2 - |(f_n^0, g_n^0)|_2^2 \\to 0$ as $n \\to \\infty$, then $\\tau_{j,n} \\to 0$ as $n \\to \\infty$, $j \\in \\{1,2,3\\}$. Then condition~\\eqref{eq:tau1+tau2.asymp} implies that $\\Im \\l_n^0 \\to 0$ as $n \\to \\infty$. Combining this with the fact that $\\tau_{3,n} \\to 0$ as $n \\to \\infty$, implies that $\\tau_{4,n} \\to 0$ as $n \\to \\infty$, where $\\tau_{4,n}$ is defined in~\\eqref{eq:tau4n.def}. Now, equivalence~\\eqref{eq:tau4n.to0} finishes the proof.\n\\end{proof}\nIn the next result we reduce part of the condition~\\eqref{eq:lim1.lim2} to an explicit condition on the coefficients $a, b, c, d$ in the boundary conditions~\\eqref{eq:cond.canon} in the difficult case $b_2\/b_1 \\notin \\mathbb{Q}$.\n\\begin{lemma} \\label{lem:lim.Imln}\nLet boundary conditions~\\eqref{eq:cond.canon} be regular, i.e. $u := ad-bc \\ne 0 $. Let also $\\beta = -b_2\/b_1 \\notin \\mathbb{Q}$. Let $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ be a sequence of zeros of the characteristic determinant $\\Delta_0(\\cdot)$ counting multiplicity. Let\n\\begin{equation} \\label{eq:lim.Imln.zn}\n bc \\ne 0, \\qquad |a|+|d|>0, \\qquad \\Im \\l_n^0 \\to 0\n \\quad\\text{and}\\quad\n z_n \\to |bc|  \\quad\\text{as}\\quad n \\to \\infty,\n\\end{equation}\nwhere $z_n$ is defined in~\\eqref{eq:zn.def}.\nThen\n\\begin{equation} \\label{eq.ad-bc=d\/a}\n |a|=|d|>0, \\qquad u = ad - bc = d\/\\overline{a}\n \\qquad\\text{and}\\qquad ad\\overline{bc} < 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nSince $\\Im \\l_n^0 \\to 0$ as $n \\to \\infty$ then $|e_{1n}| \\to 1$ as $n \\to \\infty$.\nFurther, since boundary conditions are regular, $b_2\/b_1 \\notin \\mathbb{Q}$ and $bc \\ne 0$ then all considerations in the proof of Proposition~\\ref{prop:nlim.inf} are valid. Since $u-ad=-bc$, then the second relation in~\\eqref{eq:e1.via.e2} implies:\n\\begin{multline} \\label{eq:zn=frac.e1}\n z_n = (1 + d e_{2n}) \\cdot \\overline{(1+a e_{1n})}\n = \\(1 - d \\frac{1 + a e_{1n}}{d + u e_{1n}}\\) \\cdot\n (1 + \\overline{a} \\overline{e_{1n}}) \\\\\n = \\frac{-bc e_{1n} (1 + \\overline{a} \\overline{e_{1n}})}{d + u e_{1n}}\n = \\frac{-bc (e_{1n} + \\overline{a} |e_{1n}|^2)}{d + u e_{1n}},\n \\qquad n \\in \\mathbb{Z}.\n\\end{multline}\nRecall that $d + u e_{1n} \\asymp 1$, $n \\in \\mathbb{Z}$, as established in~\\eqref{eq:a+ue2.d+ue1}. Since $z_n \\to |bc|$ and $|e_{1n}| \\to 1$ as $n \\to \\infty$, then~\\eqref{eq:zn=frac.e1} implies that\n\\begin{equation}\n |bc| (d + u e_{1n}) + bc (e_{1n} + \\overline{a}) \\to 0\n \\quad\\text{as}\\quad n \\to \\infty,\n\\end{equation}\nor\n\\begin{equation} \\label{eq:f.e1.lim}\n (|bc| u + bc) e_{1n} + |bc| d + bc \\overline{a} \\to 0\n \\quad\\text{as}\\quad n \\to \\infty,\n\\end{equation}\nBut by Proposition~\\ref{prop:nlim.inf} the sequence $\\{e_{1n}\\}_{n \\in \\mathbb{Z}}$ has infinite set of limit points. Hence relation~\\eqref{eq:f.e1.lim} is possible only if\n\\begin{equation} \\label{eq:ubc.ua=d}\n |bc| u = -bc \\quad\\text{and}\\quad |bc| d = -bc \\overline{a}.\n\\end{equation}\nSince $|a|+|d|>0$ and $bc\\ne 0$, then the second relation in~\\eqref{eq:ubc.ua=d} implies that $|a| = |d| > 0$ and that $bc \\overline{ad} = - |bc| |d|^2 < 0$. This implies the first and the third relations in~\\eqref{eq.ad-bc=d\/a}. Further, combining both relations in~\\eqref{eq:ubc.ua=d} implies the second relation in~\\eqref{eq.ad-bc=d\/a}: $u = d\/\\overline{a} = -bc\/|bc|$, which finishes the proof.\n\\end{proof}\nNow we are ready to state the main result of this section.\n\\begin{theorem} \\label{th:crit.c0.bari}\nLet boundary conditions~\\eqref{eq:cond.canon} be strictly regular. Then some normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$ (see Definition~\\ref{def:bari.c0}) if and only the operator $L_U(0)$ is self-adjoint. The latter holds if and only if coefficients $a,b,c,d$ from boundary conditions~\\eqref{eq:cond.canon} satisfy the following relations:\n\\begin{equation} \\label{eq:abcd.sa}\n |a|^2 + \\beta |b|^2 = 1, \\qquad\n |c|^2 + \\beta |d|^2 = \\beta, \\qquad\n a \\overline{c} + \\beta b \\overline{d} = 0, \\qquad \\beta := -b_2\/b_1 > 0.\n\\end{equation}\nIn this case every normalized system of root vectors of the operator $L_U(0)$ is a Bari $c_0$-sequence in $\\LLV{2}$.\n\\end{theorem}\n\\begin{proof}\n\n\\textbf{(i)} If conditions~\\eqref{eq:abcd.sa} hold then by Corollary~\\ref{cor:sa.crit} the operator $L_U(0)$ with boundary conditions~\\eqref{eq:quasi.per.bc} is self-adjoint. Remark~\\ref{rem:sa.c0.bari} now finishes the proof.\n\n\\textbf{(ii)} Now assume that some normalized system of root vectors of the operator $L_U(0)$ forms a Bari basis in $\\LLV{2}$.\n\nIf $b=c=0$ then Proposition~\\ref{prop:crit.bari.period} yields that $|a|=|d|=1$, in which case operator $L_U(0)$ is self-adjoint. This finishes the proof in this case.\n\nNow let $|b|+|c| \\ne 0$. Proposition~\\ref{prop:crit.bari.b.ne.0} implies that relations~\\eqref{eq:lim1.lim2} take place. In particular, $|c| = \\beta |b|$. Consider three cases.\n\n\\textbf{Case A.} Let $b_1 \/ b_2 \\in \\mathbb{Q}$. In this case $b_1 = -m_1 b_0$, $b_2 = m_2 b_0$, where $b_0 > 0$, $m_1, m_2 \\in \\mathbb{N}$. Set $m = m_1 + m_2$. Since $ad \\ne bc$, then $\\Delta_0(\\l) e^{-i b_1 \\l}$ is a polynomial in $e^{i b_0 \\l}$ of degree $m$ with non-zero roots $e^{i\\mu_k}$, $\\mu_k \\in \\mathbb{C}$, $k \\in \\{1, \\ldots, m\\}$, counting multiplicities. Hence, the sequence of zeros $\\{\\l_n^0\\}_{n \\in \\mathbb{Z}}$ of $\\Delta_0(\\cdot)$ is a union of arithmetic progressions $\\left\\{\\frac{\\mu_k + 2 \\pi n}{b_0}\\right\\}_{n \\in \\mathbb{Z}}$, $k \\in \\{1, \\ldots, m\\}$. Clearly $\\Im \\l_n^0 = \\Im \\mu_{k_n} \/ b_0$ for some $k_n \\in \\{1, \\ldots, m\\}$. It is clear, that if $\\Im \\mu_k \\ne 0$, for some $k \\in \\{1, \\ldots, m\\}$, then $\\Im \\l_n^0$ does not tend to $0$ as $n \\to \\infty$.\nHence $\\Im \\l_n^0 = 0$, $n \\in \\mathbb{Z}$. This implies that\n\\begin{equation} \\label{eq:Ejn=1}\n E_{jn}^{\\pm} = \\int_0^1 \\abs{e^{\\pm 2 i b_j \\l_n^0 x}} dx = 1,\n \\qquad n \\in \\mathbb{Z}, \\qquad j \\in \\{1,2\\}.\n\\end{equation}\nIt is clear that $e^{-i b_2 \\l_n^0} = (e^{-i \\mu_{k_n}})^{m_2}$ for some $k_n \\in \\{1, \\ldots, m\\}$, $n \\in \\mathbb{Z}$, and $\\{k_n\\}_{n \\in \\mathbb{Z}}$ is a periodic sequence. Hence the sequence $\\{e^{-i b_2 \\l_n^0}\\}_{n \\in \\mathbb{Z}}$ is periodic. Similarly the sequence $\\{e^{i b_1 \\l_n^0}\\}_{n \\in \\mathbb{Z}}$ is periodic. Hence, the sequence\n\\begin{equation}\n\\{z_n\\}_{n \\in \\mathbb{Z}}, \\qquad z_n = \\(1 + d e^{- i b_2 \\l_n^0}\\)\\overline{\\(1 + a e^{i b_1 \\l_n^0}\\)} = \\(1 + d e_{2n}\\)\\overline{\\(1 + a e_{1n}\\)},\n\\end{equation}\nis periodic. Since $z_n \\to |bc|$ as $n \\to \\infty$ and $|c| = \\beta|b|$, it implies that $z_n = |bc| = \\beta|b|^2$, $n \\in \\mathbb{Z}$. It now follows from~\\eqref{eq:fn2.gn2-fn.gn2=tau.sum}--\\eqref{eq:tau4n.def} and~\\eqref{eq:Ejn=1} that\n\\begin{equation}\n\\|f_n^0\\|_2^2 \\cdot \\|g_n^0\\|_2^2 - |(f_n^0, g_n^0)|_2^2 = \\tau_{4,n} =\n\\abs{z_n - \\beta |b|^2}^2 = 0, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nTaking into account formula~\\eqref{eq:|fn'-gn'|} we see that the normalized eigenvectors $f_n'$ and $g_n'$ of the operators $L_U(0)$ and $L_U^*(0)$ corresponding to the common eigenvalue $\\l_n^0 = \\overline{\\l_n^0}$ are equal for all $n \\in \\mathbb{Z}$, which implies that $L_U(0) = L_U^*(0)$.\n\n\\textbf{Case B.} Let $a=d=0$.\nThen $z_n = 1$, $n \\in \\mathbb{Z}$. Since $z_n \\to |bc|$ as $n \\to \\infty$, then $|bc| = 1$. Combined with $|c| = \\beta|b|$, this implies the desired condition~\\eqref{eq:abcd.sa}, and finishes the proof in this case.\n\n\\textbf{Case C.} Finally, let $b_1 \/ b_2 \\notin \\mathbb{Q}$, $|a|+|d|>0$ and $bc \\ne 0$. Since $\\Im \\l_n^0 \\to 0$ and $z_n \\to |bc|$ as $n \\to \\infty$, then Lemma~\\ref{lem:lim.Imln} implies condition~\\eqref{eq.ad-bc=d\/a}. In particular, $|a|=|d|>0$ and $ad\\overline{bc} < 0$. Since, in addition, $|c| = \\beta |b| > 0$, then\n\\begin{equation} \\label{eq:adbc=b2d2}\n -ad\\overline{bc} = |ad \\cdot bc| = |d|^2 \\cdot \\beta |b|^2 = \\beta b \\overline{d} \\cdot d \\overline{b}.\n\\end{equation}\nSince $d \\overline{b} \\ne 0$, this implies that $a \\overline{c} + \\beta b \\overline{d} = 0$ and coincides with the third condition in~\\eqref{eq:abcd.sa}. Further, the second relation in~\\eqref{eq.ad-bc=d\/a}, combined with relations~\\eqref{eq:adbc=b2d2}, $|a|=|d|$ and $|c| = \\beta |b|$, implies that\n\\begin{multline} \\label{eq:0=a+b=c+d}\n 0 = \\(-ad + bc + d \/ \\overline{a}\\)\\overline{bc} =\n -ad\\overline{bc} + |bc|^2 + \\frac{ad\\overline{bc}}{|a|^2}\n = |ad|\\cdot|bc| + |bc|^2 - \\frac{|ad|\\cdot|bc|}{|a|^2} \\\\\n = |bc|(|ad| + |bc|-1)\n = |bc|(|a|^2 + \\beta|b|^2-1) = |bc|(|d|^2 + \\beta^{-1}|c|^2-1).\n\\end{multline}\nSince $bc \\ne 0$, relation~\\eqref{eq:0=a+b=c+d} implies the first and second relations in~\\eqref{eq:abcd.sa}, which finishes the proof.\n\\end{proof}\n\\section{The proof of the main result}\nThis section is devoted to the proof of the main result of the paper, Theorem~\\ref{th:crit.lp.bari}. Throughout the section we use the following notations:\n\\begin{equation}\n\\mathfrak{H} := \\LLV{2}, \\qquad\n\\|\\cdot\\| := \\|\\cdot\\|_2 = \\|\\cdot\\|_{\\mathfrak{H}} \\quad\\text{and}\\quad\n(\\cdot,\\cdot) := (\\cdot,\\cdot)_{2} = (\\cdot,\\cdot)_{\\mathfrak{H}}.\n\\end{equation}\n\nFirst we need the following trivial corollary from Theorem~\\ref{th:ellp-close}.\n\\begin{corollary} \\label{cor:every.SRV}\nLet $Q \\in \\LL{p}$ for some $p \\in [1,2]$ and boundary conditions~\\eqref{eq:cond} be strictly regular. Let $\\mathfrak{F} := \\{f_n\\}_{n \\in \\mathbb{Z}}$ be a system of root vectors of the operator $L_U(Q)$ such that $\\|f_n\\| \\asymp 1$, $n \\in \\mathbb{Z}$. Then there exists a system of root vectors $\\mathfrak{F}_0 := \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$ such that $\\curl{\\|f_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$ and $\\|f_n\\| = \\|f_n^0\\|$, $n \\in \\mathbb{Z}$.\n\\end{corollary}\n\\begin{proof}\nCombining relations~\\eqref{eq:sum.fn-fn0} and~\\eqref{eq:lim.fn-fn0.c0} from Theorem~\\ref{th:ellp-close} applied with $\\widetilde{Q}=0$, implies existence of normalized systems of root vectors $\\widetilde{\\mathfrak{F}} := \\{\\widetilde{f}_n\\}_{n \\in \\mathbb{Z}}$ and $\\widetilde{\\mathfrak{F}}_0 := \\{\\widetilde{f}_n^0\\}_{n \\in \\mathbb{Z}}$ of the operators $L_U(Q)$ and $L_U(0)$, respectively, such that $\\curl{\\|\\widetilde{f}_n - \\widetilde{f}_n^0\\|_{\\infty}}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Hence\n$\\curl{\\|\\widetilde{f}_n - \\widetilde{f}_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. By Proposition~\\ref{prop:Delta.regular.basic} eigenvalues of $L_U(Q)$ are asymptotically simple. Hence vectors $f_n$ and $\\widetilde{f}_n$, $|n| \\ge N$, are proportional for some $N \\in \\mathbb{N}$, i.e. $f_n = \\alpha_n \\widetilde{f}_n$, $|n| \\ge N$, for some $\\alpha_n \\in \\mathbb{C}$.\nLet us set\n\\begin{equation}\n \\mathfrak{F}_0 := \\{f_n^0\\}_{n \\in \\mathbb{Z}}, \\qquad\n f_n^0 := \\begin{cases}\n  \\alpha_n \\widetilde{f}_n^0, & |n| \\ge N, \\\\\n  \\|f_n\\| \\widetilde{f}_n^0, & |n| < N. \\\\\n \\end{cases}\n\\end{equation}\nIt is clear that $\\mathfrak{F}_0$ is a system of root vectors of the operator $L_U(0)$ and $\\|f_n^0\\| = \\|f_n\\|$, $n \\in \\mathbb{Z}$. Moreover, $\\|f_n - f_n^0\\| = |\\alpha_n| \\cdot \\|\\widetilde{f}_n - \\widetilde{f}_n^0\\|$, $|n| \\ge N$. Since $|\\alpha_n| = \\|f_n\\| \\asymp 1$, $|n| \\ge N$, and $\\curl{\\|\\widetilde{f}_n - \\widetilde{f}_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, then $\\curl{\\|f_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, which finishes the proof.\n\\end{proof}\nNow we are ready to prove our main result on Bari $(\\ell^p)^*$-property.\n\\begin{proof}[Proof of Theorem~\\ref{th:crit.lp.bari}]\nRecall that $Q \\in \\LL{p}$ for some $p \\in[1,2]$. Also note that if $L_U(0)$ is selfadjoint then Theorem~\\ref{th:crit.c0.bari} implies conditions~\\eqref{eq:abcd.sa.intro} on the coefficients from boundary conditions~\\eqref{eq:cond.canon}.\n\n\\textbf{(i)} First assume that the operator $L_U(0)$ is selfadjoint and let $\\mathfrak{F} := \\{f_n\\}_{n \\in \\mathbb{Z}}$ be some normalized system of root vector of the operators $L_U(Q)$. By Corollary~\\ref{cor:every.SRV} there exists normalized system of root vectors $\\mathfrak{F}_0 := \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$ such that $\\curl{\\|f_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Since $L_U(0)$ is selfadjoint, then $\\{f_n^0\\}_{n \\in \\mathbb{Z}}$ is an orthonormal basis in $\\mathfrak{H}$. If $p=2$ then the proof would be already finished since $\\mathfrak{F}$ is $\\ell^2$-close to the orthonormal basis $\\mathfrak{F}_0$. But as Remark~\\ref{rem:c0.bari.diff} shows for $p \\in [1,2)$, the $(\\ell^p)^*$-closeness to the orthonormal basis is not equivalent to the Bari $(\\ell^p)^*$-property.\n\nTo this end, let $\\mathfrak{G} := \\{g_n\\}_{n \\in \\mathbb{Z}}$ be the system of vectors in $\\mathfrak{H}$ that is biorthogonal to the system $\\mathfrak{F}$. We need to prove that $\\curl{\\|f_n - g_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Clearly, $\\mathfrak{G}$ is (not normalized) system of root vectors of the adjoint operator $L_U^*(Q)$. Since $L_U(0)$ is self-adjoint then by Lemma~\\ref{lem:adjoint} we have $L_U^*(Q) = L_U(Q^*)$. Using Corollary~\\ref{cor:every.SRV} in the ``opposite'' direction we can find a normalized system of root vectors $\\widetilde{\\mathfrak{G}} := \\{\\widetilde{g}_n\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(Q^*)$ such that $\\curl{\\|\\widetilde{g}_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Therefore, $\\curl{\\|f_n - \\widetilde{g}_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$.\nSince both systems $\\mathfrak{G}$ and $\\widetilde{\\mathfrak{G}}$ are root vector systems of the operator $L_U^*(Q) = L_U(Q^*)$ and eigenvalues of $L_U(Q^*)$ are asymptotically simple due to Proposition~\\ref{prop:Delta.regular.basic}, then vectors $g_n$ and $\\widetilde{g}_n$, $|n| \\ge N$, are proportional for some $N \\in \\mathbb{N}$. Since $(f_n, g_n) = 1$, $n \\in \\mathbb{Z}$, it follows that $\\widetilde{g}_n = (f_n, \\widetilde{g}_n) g_n$, $|n| \\ge N$.\nNote that if $f, g \\in \\mathfrak{H}$ and $\\|f\\| = 1$, then\n\\begin{multline} \\label{eq:f.g-1}\n |(f, g) - 1|^2 = |(f, g)|^2 + 1 - 2 \\Re (f, g) \\\\\n \\le \\|f\\|^2 \\|g\\|^2 + 1 - 2 \\Re (f, g)\n = \\|f\\|^2 + \\|g\\|^2 - 2 \\Re (f, g) = \\|f - g\\|^2.\n\\end{multline}\nSince $\\|f_n\\| = 1$, $n \\in \\mathbb{Z}$, then~\\eqref{eq:f.g-1} implies that $|(f_n, \\widetilde{g}_n) - 1| \\le \\|f_n - \\widetilde{g}_n\\|$, $n \\in \\mathbb{Z}$. Hence for $|n| \\ge N$ we have,\n\\begin{equation} \\label{eq:gn-wcgn}\n\\|\\widetilde{g}_n - g_n\\| = \\|(f_n, \\widetilde{g}_n) g_n - g_n\\| =\n|(f_n, \\widetilde{g}_n) - 1| \\cdot \\|g_n\\| \\le \\|f_n - \\widetilde{g}_n\\| \\cdot \\|g_n\\|.\n\\end{equation}\nBy the main result of\n\\cite{LunMal14Dokl,LunMal16JMAA,SavShk14}\nthe system $\\mathfrak{F} = \\{f_n\\}_{n \\in \\mathbb{Z}}$ is a Riesz basis in $\\mathfrak{H}$. Hence so is its biorthogonal system $\\mathfrak{G} = \\{g_n\\}_{n \\in \\mathbb{Z}}$. This in particular implies that $\\|g_n\\| \\asymp 1$, $n \\in \\mathbb{Z}$.\nSince $\\curl{\\|f_n - \\widetilde{g}_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$ and $\\|g_n\\| \\asymp 1$, $n \\in \\mathbb{Z}$, then inequality~\\eqref{eq:gn-wcgn} implies that $\\curl{\\|\\widetilde{g}_n - g_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, which in turn implies the desired inclusion $\\curl{\\|f_n - g_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$.\n\n\\textbf{(ii)} Now assume that some normalized system of root vectors $\\mathfrak{F} := \\{f_n\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(Q)$ is a Bari $(\\ell^p)^*$-sequence in $\\mathfrak{H}$. By definition $\\curl{\\|f_n - g_n\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, where $\\mathfrak{G} := \\{g_n\\}_{n \\in \\mathbb{Z}}$ is a system biorthogonal to $\\mathfrak{F}$ in $\\mathfrak{H}$. Clearly, $\\mathfrak{G}$ is a system of root vectors of the adjoint operator $L_U^*(Q)$.\nBy Corollary~\\ref{cor:every.SRV} there exists normalized system of root vectors $\\mathfrak{F}_0 := \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$ such that $\\curl{\\|f_n - f_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. Similarly there exists (possibly not normalized) system of root vectors $\\mathfrak{G}_0 := \\{g_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U^*(0) = L_{U*}(0)$ such that $\\curl{\\|g_n - g_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$. It is clear, now that $\\curl{\\|f_n^0 - g_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$.\n\nLet $\\widetilde{\\mathfrak{G}}_0 := \\{\\widetilde{g}_n^0\\}_{n \\in \\mathbb{Z}}$ be a system biorthogonal to $\\mathfrak{F}_0$. As in part (i), $\\widetilde{\\mathfrak{G}}_0$ is a Riesz basis in $\\mathfrak{H}$ and $g_n^0 = (f_n^0, g_n^0) \\widetilde{g}_n^0$, $|n| \\ge N$. Since $\\|f_n^0\\| = 1$, $n \\in \\mathbb{Z}$, then~\\eqref{eq:f.g-1} implies that $|(f_n^0, g_n^0) - 1| \\le \\|f_n^0 -g_n^0\\|$, $n \\in \\mathbb{Z}$. Hence\n\\begin{equation} \\label{eq:gn0-wtgn0}\n\\|g_n^0 - \\widetilde{g}_n^0\\| = |(f_n^0, g_n^0) - 1| \\cdot\n\\|\\widetilde{g}_n^0\\| \\le \\|f_n^0 - g_n^0\\| \\cdot \\|\\widetilde{g}_n^0\\|, \\qquad n \\in \\mathbb{Z}.\n\\end{equation}\nSince\n$\\widetilde{\\mathfrak{G}}_0$ is a Riesz basis, then $\\|\\widetilde{g}_n^0\\| \\asymp 1$, $n \\in \\mathbb{Z}$.\nThus, inequality~\\eqref{eq:gn0-wtgn0} and inclusion $\\curl{\\|f_n^0 - g_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$ imply that $\\curl{\\|f_n^0 - \\widetilde{g}_n^0\\|}_{n \\in \\mathbb{Z}} \\in (\\ell^{p})^*$, which means that the normalized root vectors system $\\mathfrak{F}_0 = \\{f_n^0\\}_{n \\in \\mathbb{Z}}$ of the operator $L_U(0)$ is a Bari $(\\ell^p)^*$-sequence and, in particular, is a Bari $c_0$-sequence. Theorem~\\ref{th:crit.c0.bari} now implies that the operator $L_U(0)$ is selfadjoint and finishes the proof.\n\\end{proof}\n\\section{Application to a non-canonical string equation}\n\\label{sec:damped.string}\nIn this section we show the connection of $2 \\times 2$ Dirac type operators with a non-canonical string equation with $u_{xt}$ term, and apply our results on Riesz and Bari basis property.\n\nConsider the following non-canonical hyperbolic equation on a complex-valued function $u(x,t)$ defined for $x \\in [0, 1]$ and $t \\in [0, \\infty)$:\n\\begin{equation} \\label{eq:string}\n u_{tt} - (\\beta_1 + \\beta_2) u_{xt} + \\beta_1 \\beta_2 u_{xx} + a_1(x) u_x + a_2(x) u_t = 0,\n\\end{equation}\nwith the boundary conditions\n\\begin{equation} \\label{eq:string.cond}\n u(0,t)=0, \\qquad h_0 u_x(0,t) + h_1 u_x(1,t) + h_2 u_t(1,t)=0, \\qquad t \\in [0, \\infty),\n\\end{equation}\nand initial conditions\n\\begin{equation} \\label{eq:string.init}\n u(x, 0) = u_0(x), \\qquad u_t(x, 0) = u_1(x), \\qquad x \\in [0,1].\n\\end{equation}\nHere $\\beta_1, \\beta_2$ are constants and\n\\begin{equation} \\label{eq:b1.b2.a1.a2.h}\n \\beta_1 < 0 < \\beta_2, \\quad a_1, a_2 \\in L^1[0,1],\n \\quad h_0, h_1, h_2 \\in \\mathbb{C}, \\quad |h_1| + |h_2| > 0.\n\\end{equation}\n\nIf $-\\beta_1 = \\beta_2 = \\rho^{-1} > 0$ and $h_0 = 0$, the initial-boundary value problem~\\eqref{eq:string}--\\eqref{eq:string.init} governs the small vibrations of a string of length $1$\nand density $\\rho$ with the presence of a damping coefficient $a_2(x)$;\nthe string is fixed at the left end ($x=0$), while the right end ($x=1$) is damped with the coefficient $h_2\/h_1 \\in \\mathbb{C} \\cup \\{\\infty\\}$. Functions $u_0$ and $u_1$ represent the initial position and velocity of the string, respectively.\n\nIf $-\\beta_1 \\ne \\beta_2$ one can use linear transform of the variables $x$ and $t$ to reduce it to a classical string equation, but with damping that depends on $t$ and non-classical initial and boundary conditions: initial condition will be on a segment non-parallel to the $x$-axis ($t=0$), while boundary conditions will be on the rays non-parallel to the $t$-axis ($x=0$).\n\nRecall that $W^{1,p}[0, 1]$, $p \\ge 1$, denotes the Sobolev space of absolutely continuous functions\nwith the finite norm\n\\begin{equation}\n \\|f\\|_{W^{1,p}[0,1]}^p\n := \\int_0^{1} \\bigl(|f(x)|^p + |f'(x)|^p\\bigr) dx < \\infty.\n\\end{equation}\nFor convenience, we introduce the following notations:\n\\begin{equation} \\label{eq:W1p0}\n\\widetilde{W}^{1,p}[0,1] := \\{f \\in W^{1,p}[0,1] :\n f(0) = 0\\}, \\qquad \\widetilde{H}^1_0[0,1] := \\widetilde{W}^{1,2}[0,1],\n\\end{equation}\nwhere $p \\in [1, \\infty]$.\n\nThe non-canonical initial-boundary value problem~\\eqref{eq:string}--\\eqref{eq:string.init} of a damped string can be transformed into an abstract Cauchy problem in a Hilbert space $\\mathfrak{H}$ of the form\n\\begin{equation} \\label{eq:string.fH.norm}\n \\mathfrak{H} :=\n \\widetilde{H}^1_0[0, 1] \\times L^2[0, 1], \\qquad\n\\end{equation}\nwith the inner product\n\\begin{equation} \\label{eq:scal.fH}\n \\scal{f,g}_{\\mathfrak{H}} := \\int_0^1\n \\(f_1'(x) \\cdot \\overline{g_1'(x)}\n + f_2(x) \\cdot \\overline{g_2(x)} \\)\\,dx,\n\\end{equation}\nwhere $f = \\col(f_1, f_2)$, $g = \\col(g_1, g_2) \\in \\mathfrak{H}$.\n\nNow the new representation of the problem~\\eqref{eq:string}--\\eqref{eq:string.cond} reads as follows:\n\\begin{equation} \\label{eq:Y'=LhY}\n Y'(t) = i \\mathcal{L} Y(t), \\quad Y(t) := \\binom{u(\\cdot,t)}{u_t(\\cdot,t)},\n \\quad t \\ge 0, \\qquad Y(0) = \\binom{u_0}{u_1},\n\\end{equation}\nwhere the linear operator $\\mathcal{L} : \\dom(\\mathcal{L}) \\to \\mathfrak{H}$ is defined by\n\\begin{equation} \\label{eq:Lh.def}\n \\mathcal{L} y = \\mathcal{L} \\binom{y_1}{y_2} = -i \\, \\binom{y_2}{-\\beta_1\n \\beta_2 y_1'' + (\\beta_1 + \\beta_2) y_2' - a_1 y_1' - a_2 y_2},\n\\end{equation}\nwhere $y = \\col(y_1, y_2) \\in \\dom(\\mathcal{L})$, with\n\\begin{equation} \\label{eq:dom.cL}\n \\dom(\\mathcal{L}) = \\{y = \\col(y_1, y_2) \\in \\mathfrak{H}:\n y_1' \\in W^{1,1}[0,1], \\quad\n \\mathcal{L} y \\in \\mathfrak{H}, \\quad\n h_0 y_1'(0) + h_1 y_1'(1) + h_2 y_2(1)=0\\}.\n\\end{equation}\nIt is clear from the definition of $\\mathcal{L}$ and $\\dom(\\mathcal{L})$ that for $y = \\col(y_1, y_2) \\in \\dom(\\mathcal{L})$ we have: $y_1 \\in \\widetilde{W}^{1,1}_0[0,1]$ and $y_2 \\in \\widetilde{H}^1_0[0,1]$. In particular, $y_1(0)=y_2(0)=0$.\n\nSpectral properties of the operator $\\mathcal{L}$ play important role in the study of stability of solutions of the corresponding string equation. For example, Riesz basis property of the root vectors system of $\\mathcal{L}$ guarantees the exponential stability of the corresponding $C_0$-semigroup. The Riesz basis property and behavior of the spectrum of the operator $\\mathcal{L}$ have been studied in numerous papers (see~\\cite{CoxZua94,CoxZua95,Shubov96IEOT,Shubov97,BenRao00,GesHol11,GomRze15,Rzep17} and references therein).\n\nLet us show that the operator $\\mathcal{L}$ is similar to a certain $2 \\times 2$ Dirac type operator $L_U(Q)$. Since many spectral properties are preserved under similarity transform, known spectral properties for $2 \\times 2$ Dirac type operators will translate to corresponding properties of the dynamic generator $\\mathcal{L}$.\n\nTo this end, we need to introduce some notations. Set\n\\begin{equation} \\label{eq:string.B.def}\n B := \\diag(b_1,b_2), \\qquad b_1 := \\beta_1^{-1},\n \\quad b_2 := \\beta_2^{-1},\n\\end{equation}\n\\begin{equation} \\label{eq:string.Q.def}\n Q(x) := \\frac{i}{b_2-b_1} \\begin{pmatrix}\n 0 & w(x) \\cdot \\(b_2^2 a_1(x) + b_2 a_2(x)\\) \\\\\n \\frac{-1}{w(x)} \\cdot \\(b_1^2 a_1(x) + b_1 a_2(x) \\) & 0 \\end{pmatrix}, \\end{equation}\nwhere\n\\begin{equation} \\label{eq:wx.def}\n w(x) := w_1(x) w_2(x),\n\\end{equation}\n\\begin{equation} \\label{eq:wj.def}\n w_j(x) := \\exp\\(\\frac{b_1 b_2}{b_2-b_1} \\int_0^x (b_j a_1(t) + a_2(t)) dt\\),\n \\qquad x \\in [0,1], \\quad j \\in \\{1,2\\}.\n\\end{equation}\nNote, that $w_1(\\cdot)$, $w_2(\\cdot)$ are well defined and $Q \\in L^1([0,1], \\mathbb{C}^{2 \\times 2})$ in view of condition~\\eqref{eq:b1.b2.a1.a2.h}. Finally let\n\\begin{align}\n\\label{eq:string.U1}\n U_1(y) &:= y_1(0) + y_2(0) = 0, \\\\\n\\label{eq:string.U2}\n U_2(y) &:= b_1 h_0 y_1(0) + b_2 h_0 y_2(0) + (b_1 h_1 + h_2) w^{-1}_1(1)y_1(1)\n + (b_2 h_1 + h_2) w_2(1) y_2(1) = 0,\n\\end{align}\nbe boundary conditions for a Dirac operator $L_U(Q)$. Here $w_1(\\cdot)$, $w_2(\\cdot)$ are given by~\\eqref{eq:wj.def}.\n\\begin{proposition} \\label{prop:Lh.simil}\nOperator $\\mathcal{L}$ is similar to the $2 \\times 2$ Dirac type operator $L_U(Q)$ with the matrix $B$ given by~\\eqref{eq:string.B.def}, the potential matrix $Q(\\cdot)$ given by~\\eqref{eq:string.Q.def} and boundary conditions $Uy=\\{U_1,U_2\\}y=0$ given by~\\eqref{eq:string.U1}--\\eqref{eq:string.U2}.\n\\end{proposition}\n\\begin{proof}\nWe will transform the operator $\\mathcal{L}$ into the desired operator $L_U(Q)$ via series of similarity transformations.\n\n\\textbf{Step 1.} Define\n\\begin{equation}\n\\mathcal{V}_0 : \\mathfrak{H} \\to \\LLV{2} \\quad\\text{as}\\quad\n\\mathcal{V}_0 y := \\binom{y_1'}{y_2}, \\quad y = \\binom{y_1}{y_2} \\in \\mathfrak{H}.\n\\end{equation}\nSince $\\frac{d}{dx}$ isometrically maps $\\widetilde{H}_0^1[0,1] = \\{f \\in W^{1,2}[0,1] : f(0)=0\\}$ onto $L^2[0,1]$,\nthen the operator $\\mathcal{V}_0$ is bounded with bounded inverse. It is easy to verify that\n\\begin{equation} \\label{eq:L1def}\n L_1 y := \\mathcal{V}_0 \\mathcal{L} \\mathcal{V}_0^{-1} y\n = -i \\,\\binom{y_2'}{-\\beta_1\n \\beta_2 y_1' + (\\beta_1 + \\beta_2) y_2' - a_1 y_1 - a_2 y_2},\n\\end{equation}\nwhere\n\\begin{multline} \\label{eq:string.dom.wtL}\n y = \\binom{y_1}{y_2} \\in \\dom(L_1) := \\mathcal{V}_0 \\dom(\\mathcal{L}) = \\{y \\in W^{1,1}([0,1]; \\mathbb{C}^2) : \\\\\n L_1y \\in L^2([0,1]; \\mathbb{C}^2), \\quad y_2(0) = 0, \\ \\\n h_0 y_1(0) + h_1 y_1(1) + h_2 y_2(1)=0\\},\n\\end{multline}\nin view of~\\eqref{eq:dom.cL} and definition of $\\widetilde{H}_0^1[0,1]$. Thus, the operator $\\mathcal{L}$ is similar to the operator $L_1$,\n\\begin{equation*}\n L_1y = -i B_1 y' + Q_1(x)y,\n\\end{equation*}\nwith the domain $\\dom(L_1)$ given by~\\eqref{eq:string.dom.wtL}, and the matrices $B_1$, $Q_1(\\cdot)$, given by\n\\begin{equation}\n B_1 := \\begin{pmatrix} 0 & 1 \\\\ -\\beta_1 \\beta_2 & \\beta_1 + \\beta_2 \\\\\n \\end{pmatrix}, \\qquad\n Q_1(x) := \\begin{pmatrix} 0 & 0 \\\\ i a_1(x) & i a_2(x) \\end{pmatrix}.\n\\end{equation}\nNote, that $Q_1 \\in L^1([0,1], \\mathbb{C}^{2 \\times 2})$ in view of condition~\\eqref{eq:b1.b2.a1.a2.h}.\n\n\\textbf{Step 2.} Next we diagonalize the matrix $B_1$. To this end let\n\\begin{equation} \\label{eq:string.V1def}\n V_1 := \\begin{pmatrix} 1\/\\beta_1 & 1\/\\beta_2 \\\\ 1 & 1 \\end{pmatrix}\n = \\begin{pmatrix} b_1 & b_2 \\\\ 1 & 1 \\end{pmatrix},\n\\quad \\text{and so} \\quad\n V_1^{-1}\n = \\frac{1}{b_2-b_1} \\begin{pmatrix}\n -1 & b_2 \\\\ 1 & -b_1 \\end{pmatrix},\n\\end{equation}\nwhere $b_1$ and $b_2$ are defined in~\\eqref{eq:string.B.def}. We easily get after straightforward calculations that\n\\begin{equation} \\label{eq:string.V1B1V}\n V_1^{-1} B_1 V_1 = \\diag(\\beta_1, \\beta_2) =\n \\diag(b_1^{-1}, b_2^{-1}) = B^{-1},\n\\end{equation}\n\\begin{equation} \\label{eq:string.V1Q1V}\n V_1^{-1} Q_1(x) V_1 = \\frac{i}{b_2-b_1} \\begin{pmatrix}\n b_1 b_2 a_1(x) + b_2 a_2(x) &\n b_2^2 a_1(x) + b_2 a_2(x) \\\\\n - b_1^2 a_1(x) - b_1 a_2(x) &\n - b_1 b_2 a_1(x) - b_1 a_2(x)\n \\end{pmatrix} =: Q_2(x), \\qquad x \\in [0,1].\n\\end{equation}\nNote, that $Q_2 \\in L^1([0,1], \\mathbb{C}^{2 \\times 2})$ in view of condition~\\eqref{eq:b1.b2.a1.a2.h}. Introducing bounded operator $\\mathcal{V}_1 : y \\to V_1 y$ in $L^2([0,1]; \\mathbb{C}^2)$, noting that it has a bounded inverse, and taking into account~\\eqref{eq:string.V1B1V} and~\\eqref{eq:string.V1Q1V}, we obtain\n\\begin{align}\n\\nonumber\n L_2 y & := \\mathcal{V}_1^{-1} L_2 \\mathcal{V}_1 y\n = -i V_1^{-1} B_1 V_1 y' + V_1^{-1} Q_1(x) V_1 y \\\\\n\\label{eq:L2.def}\n & = -i B^{-1} y' + Q_2(x) y,\n \\qquad y \\in \\mathcal{V}_1^{-1} \\dom(L_1) =: \\dom(L_2),\n\\end{align}\nwhere\n\\begin{multline} \\label{eq:string.dom.L2}\n \\dom(L_2) = \\{y \\in W^{1,1}([0,1]; \\mathbb{C}^2) : \\\n L_2 y \\in L^2([0,1];\\mathbb{C}^2), \\\n y_1(0) + y_2(0) = 0, \\\\\n b_1 h_0 y_1(0) + b_2 h_0 y_2(0) + (b_1 h_1 + h_2) y_1(1)\n + (b_2 h_1 + h_2) y_2(1) = 0\\},\n\\end{multline}\nwith account of formula~\\eqref{eq:string.dom.wtL} for the domain $\\dom(L_1)$ and the formula~\\eqref{eq:string.V1def} for the matrix $V_1$.\n\n\\textbf{Step 3.} On this step we make potential matrix $Q_2$ to be off-diagonal. To this end, Let $\\widetilde{Q}_2$ be a diagonal of $Q_2$, i.e.\n$$\n \\widetilde{Q}_2(x) := \\frac{i}{b_2-b_1} \\begin{pmatrix}\n b_1 b_2 a_1(x) + b_2 a_2(x) & 0 \\\\\n 0 & - b_1 b_2 a_1(x) - b_1 a_2(x)\n \\end{pmatrix}.\n$$\nLet $V_2(\\cdot)$ be a solution of the initial value problem\n\\begin{equation} \\label{eq:V2.equ}\n -i B^{-1} V_2'(x) + \\widetilde{Q}_2(x) V_2(x) = 0,\n \\qquad V_2(0) = I_2.\n\\end{equation}\nIt is easily seen that\n\\begin{equation} \\label{eq:V2.def}\n V_2(x) := \\begin{pmatrix} w_1(x) & 0 \\\\\n 0 & w_2^{-1}(x) \\end{pmatrix}, \\qquad x \\in [0,1],\n\\end{equation}\nwhere $w_1(\\cdot)$, $w_2(\\cdot)$ are defined in~\\eqref{eq:wj.def}. Let us introduce operator $\\mathcal{V}_2 : y \\to V_2(x) y$ in $L^2([0,1]; \\mathbb{C}^2)$. Since $a_1, a_2 \\in L^1[0,1]$, the operator $\\mathcal{V}_2$ is bounded and has a bounded inverse. Combining relation~\\eqref{eq:V2.equ}, definition~\\eqref{eq:string.Q.def} of $Q$ and definition~\\eqref{eq:wx.def} of $w$, we get\n\\begin{align}\n\\nonumber\n L_3 y & := \\mathcal{V}_2^{-1} L_2 \\mathcal{V}_2 y \\\\\n\\nonumber\n & = -i [V_2(x)]^{-1} B^{-1} V_2(x) y'\n + [V_2(x)]^{-1} (-i B^{-1} V_2'(x) + Q_2(x) V_2(x)) y \\\\\n\\nonumber\n & = -i B^{-1} y'\n + [V_2(x)]^{-1} (Q_2(x) - \\widetilde{Q}_2(x)) V_2(x)) y \\\\\n\\label{eq:L3.def}\n & = -i B^{-1} y' + Q(x) y,\n \\qquad y \\in \\mathcal{V}_2^{-1} \\dom(L_2) =: \\dom(L_3).\n\\end{align}\nIt is clear from the definition of $\\mathcal{V}_2$ that $\\dom(L_3)$ coincides with $\\dom(L_U(Q))$ defined via~\\eqref{eq:string.U1}--\\eqref{eq:string.U2}. Hence $L_3 = L_U(Q)$. Combining all the steps of the proof one concludes that $\\mathcal{L}$ is similar to $L_U(Q)$.\n\\end{proof}\nCombining Proposition~\\ref{prop:Lh.simil} with our previous results for $2 \\times 2$ Dirac type operators we obtain the Riesz basis property and analogous of Bari basis property for the dynamic generator $\\mathcal{L}$ of the non-canonical initial-boundary value problem~\\eqref{eq:string}--\\eqref{eq:string.init} for a damped string equation. The part (i) of the following result improves known results in the literature on the Riesz basis property for the operator $\\mathcal{L}$ in the case $-\\beta_1=\\beta_2$, $a_1 \\equiv 0$, $h_0=0$ (see~\\cite{CoxZua94,CoxZua95,Shubov96IEOT,Shubov97,BenRao00,GesHol11,GomRze15,Rzep17} and references therein). The part (ii) shows the application of one of our main results Theorem~\\ref{th:crit.bari}.\n\\begin{theorem} \\label{th:string.riesz}\n\\textbf{(i)} Let parameters of the damped string equation satisfy relaxed conditions~\\eqref{eq:b1.b2.a1.a2.h},\n\\begin{equation} \\label{eq:h2neh1}\n\\beta_2 h_2 + h_1 \\ne 0, \\qquad \\beta_2 h_2 + h_1 \\ne 0,\n\\end{equation}\nand in addition boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2} are strictly regular. Then the system of root vectors of the operator $\\mathcal{L}$ \\ \\textbf{forms a Riesz basis} in $\\mathfrak{H} = \\widetilde{H}^1_0[0,1] \\times L^2[0,1]$.\n\n\\textbf{(ii)} Let in addition $a_1, a_2 \\in L^2[0,1]$. Let also $\\mathcal{V}_0$, $\\mathcal{V}_1$, $\\mathcal{V}_2$ be the operators defined in the steps of the proof of Proposition~\\ref{prop:Lh.simil}. Then the system of root vectors of the operator $\\mathcal{L}$ is quadratically close in $\\mathfrak{H}$ to a system of the form $\\{\\mathcal{V}_0^{-1} \\mathcal{V}_1 \\mathcal{V}_2 e_n\\}_{n \\in \\mathbb{Z}}$, where $\\{e_n\\}_{n \\in \\mathbb{Z}}$ is an orthonormal basis in $\\LL{2}$, if and only if boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2} are self-adjoint, which is equivalent to the condition\n\\begin{equation} \\label{eq:h0=0.b1=-b2}\n h_0 = 0, \\quad \\beta_1 = -\\beta_2, \\quad \\int_0^{1} \\Im a_2(t) dt = \\beta_2 \\log \\abs{\\frac{\\beta_2 h_2 + h_1}{\\beta_2 h_2 - h_1}}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nFirst, let us transform boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2} to a canonical form~\\eqref{eq:cond.canon.intro} assuming condition~\\eqref{eq:h2neh1}. For this we multiply the first condition $U_1$ by $b_1 h_0$ and subtract from $U_2$ and then multiple the second condition $U_2$ by $(b_2 h_1 + h_2)^{-1} w_2^{-1}(1)$. Boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2} will take the form\n\\begin{equation} \\label{eq:cond.canon.string}\n\\begin{cases}\n \\widehat{U}_{1}(y) = y_1(0) + y_2(0) = 0, \\\\\n \\widehat{U}_{2}(y) = d y_2(0) + c y_1(1) + y_2(1) = 0,\n\\end{cases}\n\\end{equation}\nwhere\n\\begin{equation}\nd = \\frac{(b_2-b_1) h_0}{(b_2 h_1 + h_2) w_2 (1)}, \\qquad\nc = \\frac{b_1 h_1 + h_2}{(b_2 h_1 + h_2) w(1)}.\n\\end{equation}\nHere $w, w_1, w_2$ are given by~\\eqref{eq:wx.def}--\\eqref{eq:wj.def}. In particular\n\\begin{equation} \\label{eq:wlen}\n w(1) := \\exp\\(\\frac{b_1 b_2}{b_2-b_1} \\int_0^{1} ((b_1 + b_2) a_1(t) + 2a_2(t)) dt\\).\n\\end{equation}\n\n\\textbf{(i)} Proposition~\\ref{prop:Lh.simil} implies that the operator $\\mathcal{L}$ is similar to the operator $L_U(Q)$ with the matrix $B$ given by~\\eqref{eq:string.B.def}, the potential matrix $Q(\\cdot)$ given by~\\eqref{eq:string.Q.def} and boundary conditions $Uy=\\{U_1,U_2\\}y=0$ given by~\\eqref{eq:string.U1}--\\eqref{eq:string.U2}. Note that condition~\\eqref{eq:h2neh1} implies regularity of boundary conditions~\\eqref{eq:string.U1}--\\eqref{eq:string.U2}. In addition they are strictly regular by the assumption. Hence operator $L_U(Q)$ has compact resolvent and by Proposition~\\ref{prop:Delta.regular.basic} its eigenvalues are asymptotically simple and separated. Moreover, Theorem 1.1 from~\\cite{LunMal16JMAA} implies that the system of root vectors of the operator $L_U(Q)$ forms a Riesz basis in $\\LLV{2}$. Similarity of $\\mathcal{L}$ and $L_U(Q)$ implies the same properties for $\\mathcal{L}$ in the space $\\mathfrak{H}$, which finishes the proof of part (i).\n\n\\textbf{(ii)} Since $a_1, a_2 \\in L^2[0,1]$ it follows that $Q \\in \\LL{2}$. Since boundary conditions~\\eqref{eq:cond.canon.string} are strictly regular then by Theorem~\\ref{th:crit.bari} (any and every) system of root vectors of the operator $L_U(Q)$ forms a Bari basis in $\\LLV{2}$ if only if boundary conditions~\\eqref{eq:cond.canon.string} are self-adjoint, which in turn is equivalent to conditions~\\eqref{eq:abcd.sa.intro}. Since $a=0$ and $b=1$ then~\\eqref{eq:abcd.sa.intro} is equivalent to\n\\begin{equation}\nd=0, \\quad b_1 = -b_2, \\quad |c| = 1.\n\\end{equation}\nSince $\\beta_1 = b_1^{-1}$ and $\\beta_2 = b_2^{-1}$, this in turn is equivalent to~\\eqref{eq:h0=0.b1=-b2}.\n\nLet us set $\\mathcal{V} := \\mathcal{V}_0^{-1} \\mathcal{V}_1 \\mathcal{V}_2$ and let $\\{f_n\\}_{n \\in \\mathbb{Z}}$ be some system of root vectors of the operator $L_U(Q)$. It follows from the proof of Proposition~\\ref{prop:Lh.simil} that $\\{\\mathcal{V} f_n\\}_{n \\in \\mathbb{Z}}$ is a system of root vectors of the operator $\\mathcal{L}$. Hence $\\{f_n\\}_{n \\in \\mathbb{Z}}$ is quadratically close to an orthonormal basis $\\{e_n\\}_{n \\in \\mathbb{Z}}$ in $\\LLV{2}$ if and only if $\\{\\mathcal{V} f_n\\}_{n \\in \\mathbb{Z}}$ is quadratically close to $\\{\\mathcal{V} e_n\\}_{n \\in \\mathbb{Z}}$ in $\\mathfrak{H}$. This completes the proof.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWith the discovery of the Higgs boson at the CERN Large Hadron Collider (LHC) \\cite{atlas:2012gk,cms:2012gu} particle physics has entered a new era. Both LHC collaborations, ATLAS and CMS, have confirmed the existence of a boson with a mass of about 126~GeV and properties consistent with those of the scalar CP-even particle predicted by the Standard Model \\cite{Aad:2013xqa,Chatrchyan:2012jja}. In order to fully establish the nature of the Higgs boson, a precise determination of its couplings to fermions and gauge bosons is essential \\cite{Zeppenfeld:2000td,Duhrssen:2004cv,LHCHiggsCrossSectionWorkingGroup:2012nn}.  \n\nA rather clean environment for such coupling measurements is provided by the vector-boson fusion (VBF) production mode \\cite{Rainwater:1998kj,Plehn:1999xi,Rainwater:1999sd,Kauer:2000hi,Rainwater:1997dg,Eboli:2000ze}, where the Higgs boson is produced via quark-scattering mediated by weak gauge boson exchange in the $t$-channel, $qq'\\to qq'H$. Because of the low virtuality of the exchanged weak bosons, the tagging jets emerging from the scattered quarks are typically located in the forward and backward regions of the detector, while the central-rapidity region exhibits little jet activity due to the color-singlet nature of the $t$-channel exchange. These features can be exploited to efficiently suppress QCD backgrounds with a priori large cross sections at the LHC. \nIn the context of central-jet veto (CJV) techniques, events are discarded if they exhibit one or more jets in between the two tagging jets. To quantitatively employ such selection strategies, a precise knowledge of the VBF cross section with an additional jet, i.e. the reaction $pp\\to H jjj$, is crucial. \n\nNext-to-leading order (NLO) QCD corrections to VBF-induced $Hjjj$ production have first been computed in \\cite{Figy:2007kv}, yielding results with only small residual scale uncertainties of order 10\\% or less. In particular, in that approach the survival probability for the Higgs signal has been estimated to exhibit a perturbative accuracy of about 1\\%. The calculation of \\cite{Figy:2007kv} is implemented in the {\\tt VBFNLO} package~\\cite{Arnold:2008rz,Arnold:2011wj,Arnold:2012xn} in the form of a flexible parton-level Monte-Carlo program.   \nMore recently, an NLO-QCD calculation for electroweak $Hjjj$ production has been presented~\\cite{Campanario:2013fsa}, where several approximations of Ref.~\\cite{Figy:2007kv} have been dropped. \n\nIn this work, we merge the parton-level calculation of \\cite{Figy:2007kv} with a parton-shower Monte-Carlo in the framework of the \\POWHEG{} formalism~\\cite{Nason:2004rx,Frixione:2007vw}, a method for the matching of an NLO-QCD calculation with a transverse-momentum ordered parton-shower program. For our implementation we are making use of version~2 of the  \\POWHEGBOX{}~\\cite{Alioli:2010xd,Nason:2013ydw}, a repository that provides the process-independent ingredients of the \\POWHEG{} method. The code we develop yields precise, yet realistic predictions for VBF-induced $Hjjj$ production at the LHC in a public framework that can easily be used by the reader for further phenomenological studies. \n\nThis article is organized as follows: In Sec.~\\ref{sec:tech} we describe some technical details of our implementation. Phenomenological results are presented in Sec.~\\ref{sec:pheno}. We conclude in Sec.~\\ref{sec:concl}.\n\\section{Technical details of the implementation}\n\\label{sec:tech}\nThe implementation of $Hjjj$ production via VBF in the context of the \\POWHEGBOX{} requires, as major building blocks, the matrix elements for all relevant partonic scattering processes at Born level and at next-to-leading order. These have first been calculated in \\cite{Figy:2007kv} and are publicly available in the {\\tt VBFNLO} package~\\cite{Arnold:2008rz}. We extracted the matrix elements from {\\tt VBFNLO} and adapted them to the format required by the \\POWHEGBOX. \n\nAt leading order (LO), processes of the type $qq'\\to qq'gH$ and all crossing-related channels are taken into account, if they include the exchange of a weak boson in the $t$-channel . Some representative Feynman diagrams are depicted in Fig.~\\ref{fig:lo-graphs}. \n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\textwidth]{.\/figs\/Higgs3j_LO-crop.ps}\n \\caption{Representative tree-level diagrams for VBF $Hjjj$ production. }\n \\label{fig:lo-graphs}\n\\end{figure}\nThe gauge-invariant class of diagrams involving weak-boson exchange in the $s$-channel is considered as part of the Higgs-strahlung process, and disregarded in the context of our work on VBF-induced Higgs production. The interference of $t$-channel with $u$-channel diagrams in flavor channels with quarks of the same type is neglected. Once VBF-specific selection cuts are imposed, these approximations are well justified~\\cite{Ciccolini:2007ec}.\nThroughout, we assume a diagonal CKM matrix. \nWe refer to the electroweak $Hjjj$ production process at order $\\mathcal{O}(\\alpha_s\\alpha^3)$ within these approximations as ``VBF $Hjjj$ production''.  \n\nThe virtual corrections to this reaction comprise the interference of the Born amplitudes with one-loop diagrams where a virtual gluon is attached to a single fermion line [c.f.~Fig.~\\ref{fig:virt-graphs}~(a)--(g)], and diagrams where a virtual gluon is exchanged between the two different fermion lines, see~Fig.~\\ref{fig:virt-graphs}~(h),(i). \n\\begin{figure}[tp]\n\\vspace*{8cm}\n \\includegraphics[width=\\textwidth,bb=0 0 447 0]{.\/figs\/Higgs3j_virt-crop.ps}\n \\\\\n \\includegraphics[width=\\textwidth]{.\/figs\/Higgs3j_penta-crop.ps}\n \\caption{\n Representative one-loop diagrams for $qq'\\to qq'gH$, with the virtual gluon being attached to one single fermion line [graphs ($a$)-($g$)], or to the two different fermion lines  [graphs ($h$) and ($i$)]. }\n \\label{fig:virt-graphs}\n\\end{figure}\nAs discussed in some detail in \\cite{Figy:2007kv}, the \nlatter contributions are strongly suppressed by color factors and due to the VBF dynamics. They can be neglected, if the respective color structures of the real-emission contributions are disregarded as well, as these would serve to cancel the  infrared singularities of the pentagon and hexagon contributions that we drop. \n\nThe real-emission contributions involve subprocesses with four external (anti-)quarks and two gluons such as $qq'\\to qq'ggH $, as well as pure quark scattering processes of the type $qq'\\to qq'Q\\bar QH $, and all crossing-related channels with $t$-channel weak boson exchange. \nBecause of the approximations we have employed in the virtual contributions, where we dropped color-suppressed contributions giving rise to pentagon and hexagon integrals, the respective color structures have to be disregarded in the real-emission contributions as well. In practice this means to neglect interference terms between diagrams where a given gluon is emitted once from the upper and once from the lower quark line in the Feynman graphs. For example, interference terms like $2\\,\\mathcal{R}e\\left( {\\cal B}^3_4 {\\cal B^*}^4_3\\right)$, with ${\\cal B}^3_4$ as depicted in Fig.~\\ref{fig:real-gg-supp}, are dropped, \n\\begin{figure}[tp]\n \\includegraphics[width=1\\textwidth]{.\/figs\/Higgs3j_real3-crop.ps}\n \\caption{\nRepresentative diagrams of the color structure $\\mathcal{B}_4^3$ as introduced in Ref.~\\cite{Figy:2007kv} for the subprocess $qq' \\rightarrow  qq'\\, gg \\, H$. \n}\n \\label{fig:real-gg-supp}\n\\end{figure}\nwhile their squares or the squares of the topologies sketched in Fig.~\\ref{fig:real-gg} are fully considered. \n\\begin{figure}[tp]\n\\begin{center}\n \\includegraphics[width=1\\textwidth]{.\/figs\/Higgs3j_real1-crop.ps} \\\\\n \\vspace{0.5cm} \n \\hspace{1cm} \\includegraphics[width=0.66\\textwidth]{.\/figs\/Higgs3j_real2-crop.ps}  \n \\caption{Representative diagrams  of the color structure $\\mathcal{A}_{43}^{1a}$ as introduced in Ref.~\\cite{Figy:2007kv} for the subprocess $qq' \\rightarrow  qq'\\, gg \\, H$. }\n \\label{fig:real-gg}\n \\end{center}\n\\end{figure}\nRepresentative diagrams for a pure quark subprocess are depicted in Fig.~\\ref{fig:real-QQ}.  For this class of subprocesses we require that the $Q\\bar Q$ pair stems from a gluon. Contributions involving the hadronic decay of a weak boson, $V\\to Q\\bar Q$, such as graph~\\ref{fig:real-QQ}~(c),  are disregarded within our VBF~setup. \n\\begin{figure}[tp]\n \\centering\n \\includegraphics[width=1\\textwidth]{.\/figs\/Higgs3j_realQQ-crop.ps}\n \\caption{\n Representative diagrams for the subprocess $qq' \\to qq' \\, Q\\bar{Q}\\, H$. }\n \\label{fig:real-QQ}\n\\end{figure}\n\nWhile in \\cite{Figy:2007kv} soft and collinear singularities have been taken care of by a dipole subtraction procedure, the \\POWHEGBOX{} makes use of the so-called FKS subtraction scheme \\cite{Frixione:1995ms}. From the color- and spin-correlated amplitudes provided by the user, the \\POWHEGBOX{} internally constructs the counterterms that are needed to cancel soft and collinear singularities in the real-emission contributions. Because we are disregarding certain color-suppressed contributions in the virtual and real-emission amplitudes, we have to make sure that only the counterterms relevant for our setup are constructed. This is achieved by passing only those  color- and spin-correlated Born amplitudes to the \\POWHEGBOX{} that correspond to the color structures we consider within our approximations. \n\nWe have carefully tested that the counter terms constructed in this way approach the real-emission amplitudes in the soft and collinear limits. Additionally, we have compared the tree-level and real emission amplitudes for selected phase space points with code generated by {\\tt MadGraph} \\cite{Stelzer:1994ta} that has been adapted to match the approximations of our calculation. We found agreement at the level of more than ten digits. The virtual amplitudes have been compared to {\\tt VBFNLO}, again showing full agreement for single phase space points. We note that some care has been necessary in this latter check, as finite parts of the subtraction terms are included in the virtual amplitudes in the default setup of {\\tt VBFNLO}.  \nTo verify the entire setup of our code, we have compared cross sections and distributions for various sets of selection cuts at LO and NLO-QCD accuracy as obtained with the \\POWHEGBOX{} with respective results of {\\tt VBFNLO}. We found full agreement for all considered scenarios. \n\nWe note that special care is needed when performing the phase-space integration of VBF $Hjjj$\\;~production in the framework of the \\POWHEGBOX{}. In contrast to the VBF-induced $Hjj$ production cross section that is entirely finite at leading order, the inclusive VBF $Hjjj$ cross section diverges already at leading order when a pair of partons becomes collinear  or a soft gluon is encountered in the final state. While such divergent contributions disappear after phenomenologically sensible selection cuts are imposed, their presence considerably reduces the efficiency of the numerical phase space integration. This effect can be avoided by appropriate phase-space cuts at generation level, or by a so-called Born-suppression factor $F(\\Phi_n)$ that dampens the integrand whenever singular configurations in phase-space are approached. In order to \nensure that our results are independent of technical cuts in the phase-space integration, we recommend the use of a Born-suppression factor. In our  \\POWHEGBOX{} implementation we provide two alternative versions of Born-suppression factors: \n\\begin{itemize}\n\\item\nIn our first, multiplicative, approach, the factor is of the form \n\\begin{equation}\n\\label{eq:bsupp1}\nF(\\Phi_n) = \n\\prod_{i=1}^3\n\\left(\n\\frac{p_{T,i}^2}{p_{T,i}^2+\\Lambda_p^2}\n\\right)^2\n\\prod_{i,j=1;\\\\\nj\\neq i}^3\n\\left(\n\\frac{m_{ij}^2}{m_{ij}^2+\\Lambda_m^2}\n\\right)^2\\,,\n\\end{equation}\nwhere the $p_{T,i}$ and $m_{ij}=\\sqrt{(p_i+p_j)^2}$ respectively denote the transverse momenta and invariant masses of the three final-state partons of the underlying Born configuration. The $\\Lambda_p$ and $\\Lambda_m$ are cutoff parameters that are typically set to values of a few GeV.   %\n\\item\nFollowing the procedure suggested for the related case of trijet production in the framework of the \\POWHEGBOX{}~\\cite{Kardos:2014dua}, we use an exponential suppression factor of the form\n\\begin{equation}\nS_1 = \\exp \\left[ \n- \\Lambda_1^4 \\cdot \\left(\\sum_{i=1}^3 \\frac{1}{p^4_{T,i}} + \\sum_{i,j=1;\\\\\nj\\neq i}^3 \\frac{1}{q^2_{ij}}\\right) \n\\right]\\,,\n\\end{equation}\nwith \n\\begin{equation}\nq_{ij}=p_i \\cdot p_j \\, \\frac{E_i \\, E_j}{E_i^2 + E_j^2}\\,,\n\\end{equation}\nfor the suppression of infrared divergent configurations in the underlying Born kinematics, \naccompanied by a factor \n\\begin{equation}\n S_2 = \\left(\\frac{H_T^2}{H_T^2 + \\Lambda_2^2}\\right)^2,\n\\end{equation}\nwhere\n\\begin{equation}\nH_T = p_{T,1} + p_{T,2} + p_{T,3}.\n\\end{equation}\nThe factor $S_2$ serves to suppress configurations where all partons are having small transverse momenta, and at the same time increase the fraction of events generated with large transverse momenta. \nCombining $S_1$ with $S_2$, we construct\n\\begin{equation}\n\\label{eq:bsupp2}\nF(\\Phi_n) = S_1 \\cdot S_2\\,.\n\\end{equation}\n\\end{itemize}\nFor the generation of the phenomenological results shown below we are using a Born suppression factor of the form given in Eq.~(\\ref{eq:bsupp2}) with $\\Lambda_1 = 10 $~GeV and  $\\Lambda_2 = 30$~GeV, supplemented by a small generation cut on the transverse momenta of the three outgoing partons of the underlying Born configuration, $p_{T,i}^\\mathrm{gen} > 1$~GeV.  \n\nTo make sure our results do not depend on these technical parameters, in addition to our default setup we ran our code using the Born suppression factor of Eq.~(\\ref{eq:bsupp1}) with $\\Lambda=20$~GeV and, again, $p_{T,i}^\\mathrm{gen} > 1$~GeV. The results in the two setups are in full agreement with each other and, at fixed order, also with respective results obtained with {\\tt VBFNLO} that is using an entirely different phase-space generator. \n\\section{Phenomenological results}\n\\label{sec:pheno}\nOur implementation of VBF $Hjjj$\\; production at the LHC is made publicly available in version~2 of  the \\POWHEGBOX{}, and can be obtained as explained at the project webpage,  {\\tt http:\/\/powhegbox.mib.infn.it\/}. \n\nHere, we are providing phenomenological results for a representative setup at the LHC with a center-of-mass energy of $\\sqrt{s}=8$~TeV. \nWe are using the CT10 fixed-four-flavor set \\cite{Lai:2010vv} for the parton distribution functions of the proton as implemented in the {\\tt LHAPDF} library \\cite{Whalley:2005nh} and the accompanying value of the strong coupling, $\\alpha_s(m_Z)=0.1127$.  Jets are reconstructed via the anti-$k_T$ algorithm with a resolution parameter of $R=0.5$, with the help of the {\\tt FASTJET}~package~\\cite{Cacciari:2005hq,Cacciari:2008gp,Cacciari:2011ma}. As electroweak input parameters we are using the masses of the weak gauge bosons, $m_W=80.398$~GeV and $m_Z=91.188$~GeV, and the Fermi constant, $G_F=1.16639\\times 10^{-5}$~GeV$^{-1}$. Other electroweak parameters are computed thereof via tree-level relations. \nThe widths of the massive gauge bosons are set to $\\Gamma_W =  2.095$~GeV and $\\Gamma_Z=2.51$~GeV, respectively. For the Higgs boson, we are using $m_H = 126$~GeV and $\\Gamma_H=4.095$~MeV. \nThe renormalization and factorization scales are identified as $\\mu_\\mr{R}=\\mu_\\mr{F}=m_H\/2$. \nIn order to assess uncertainties that remain after matching the NLO calculation with a parton shower program, we consider three different tools:  \\PYTHIA{}~{\\tt 6.4.25} with the Perugia~0 tune~\\cite{Sjostrand:2006za}, \\HERWIGPP{}~{\\tt 2.7.0}~\\cite{Bahr:2008pv,Bellm:2013lba} with its default angular-ordered shower, and with a transverse-momentum ordered dipole shower~\\cite{Platzer:2011bc} which we tag as {\\tt PYT}, {\\tt HER}, and {\\tt DS++}, respectively. We note that wide-angle, soft radiation that is in principle needed when matching an NLO calculation with a parton-shower program using the \\POWHEG{} method, is missing in the default angular-ordered \\HERWIGPP{} shower. The impact of this missing piece on observables can only be estimated by a comparison with predictions obtained with transverse momentum ordered showers, such as the \\DSPP{} version of \\HERWIGPP{}. We do not consider hadronization, QED radiation, multiple parton interactions, and underlying event effects in this work. \n\nIn order to define a $Hjjj$ event, we demand at least three well-observable jets with \n\\begin{equation} \np_{T,j}>20 \\text{ GeV}\\,,\n\\quad\n|y_j| < 4.5\\,.\n\\end{equation}\nIn addition, we impose VBF-specific selection cuts.  The two hardest jets, referred to as ``tagging jets'', are required to fulfill \n\\begin{equation}\np_{T,j}^\\mathrm{tag}>30~\\mathrm{GeV}\\,,\\quad\n|y_j^\\mathrm{tag}|<4.5\\,,\n\\end{equation}\nand be well-separated from each other, \n\\begin{equation}\n|y_{j_1}^\\mathrm{tag}-y_{j_2}^\\mathrm{tag}|>4.0\\,, \n\\quad\ny_{j_1}^\\mathrm{tag}\\times y_{j_2}^\\mathrm{tag}<0\\,, \n\\quad\nm_{jj}^\\mathrm{tag}>500~\\mathrm{GeV}\\,.\n\\end{equation}\nThe kinematics of the Higgs boson is not restricted. \n\nWith these cuts, we obtain a cross section of $\\sigma^\\mathrm{NLO}=71.5 \\pm  0.4$~fb at fixed order, where the error is the statistical error of the Monte Carlo calculation. After matching the NLO result with a parton shower, some of the events fail to pass the cuts, resulting in slightly smaller cross sections of $\\sigma^{\\tt PYT}=65.8 \\pm  0.3$~fb, $\\sigma^{\\tt HER}=68.3 \\pm 0.3$~fb,  and $\\sigma^{\\tt DS++}=69.8\\pm 0.5$~fb, respectively. \nApart from this change in normalization the impact of the parton shower on observables related to the tagging jets is very mild, as  illustrated in Fig.~\\ref{fig:tag-jet}\n\\begin{figure}[t]\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/ptj1.eps}\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/mjj.eps}\n\\caption{\nTransverse momentum distribution of the hardest tagging jet (left) and invariant mass distribution of the two tagging jets (right) at NLO (black), and at NLO+PS level: \\PYT{}~(red), \\HERPP{}~(blue), \\DSPP{}~(cyan). \nThe lower panels show the NLO+PS results normalized to the pure NLO prediction together with its statistical uncertainty (yellow band). \n}\n\\label{fig:tag-jet}\n\\end{figure}\nfor the transverse momentum distribution of the hardest tagging jet and the invariant mass of the tagging jet pair. \n\nIn contrast to NLO calculations for VBF $Hjj$ production, where the third jet can be described only with LO accuracy, our calculation is NLO accurate in distributions related to the third jet. In Fig.~\\ref{fig:jet3}, NLO+PS results for the transverse momentum and the rapidity distribution of the third jet are shown for different parton shower programs together with the fixed-order NLO result. \n\\begin{figure}[t]\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/ptj3.eps}\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/etaj3.eps}\n\\caption{\nTransverse momentum and rapidity distributions of the third jet at NLO, and at NLO+PS level (line styles as in Fig.~\\ref{fig:tag-jet}). \n}\n\\label{fig:jet3}\n\\end{figure}\nFor all considered parton showers, the difference between the NLO and the NLO+PS results is small. However, \\PYTHIA{} tends to produce slightly more jets in the central-rapidity region of the detector, while \\HERWIGPP{} preferentially radiates in the collinear region between the two tagging jets and the beam axis. We will see below that this effect is more pronounced in the case of sub-leading jets. \n\nLarger differences between the fixed-order and the various matched predictions occur in distributions related to the fourth jet. \nIn the parton-level NLO calculation a fourth jet can only stem from the real-emission contributions, and can thus be described only at tree-level accuracy. Larger theoretical uncertainties are therefore expected for observables related to the fourth jet. Fig.~\\ref{fig:jet4} \n\\begin{figure}[t]\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/ptj4.eps}\n\t\\includegraphics[angle=-90,width=0.5\\textwidth]{.\/plots\/y4star.eps}\n\\caption{\nTransverse momentum distribution of a fourth jet for our default setup with an extra cut of $p_{T,j_4}>1$~GeV~(left) and rapidity distribution of a fourth hard jet with $p_{T,j_4}>20$~GeV relative to the two tagging jets (right) at NLO, and at NLO+PS level (line styles as in Fig.~\\ref{fig:tag-jet}). \n}\n\\label{fig:jet4}\n\\end{figure}\nillustrates the effect of the \\POWHEG{}-Sudakov factor on the transverse momentum of the fourth jet and clarifies how extra radiation in the VBF setup is distributed by the different parton shower programs via the $y_4^\\star$ variable. This quantity\t is defined as\n\\begin{equation} \ny_4^\\star=y_{j_4} - \\frac{y_{j_1} + y_{j_2}}{2}\\,,\n\\label{eq:y4star}\n\\end{equation}\nin order to parameterize the rapidity of the fourth jet relative to the two hard tagging jets. The respective distribution shows, more pronouncedly than in the case of the third jet, that \\PYTHIA{} and \\HERWIGPP{} tend to produce radiation in different regions of phase space. The differences between the various NLO+PS curves can thus be considered as inherent uncertainty of the matched prediction. \n\\section{Conclusions}\n\\label{sec:concl}\nIn this work, we have presented an implementation of VBF $Hjjj$\\; production in version~2 of the \\POWHEGBOX{} repository. We have performed the matching of an existing NLO-QCD calculation with parton-shower programs using the \\POWHEG{} formalism and presented phenomenological results for a representative setup at the LHC. The code we developed is publicly available and can be adapted to the user's need in a straightforward manner. \n\nWe have shown that theoretical uncertainties associated with the description of the third jet by  genuinely different parton-shower programs are mild at NLO+PS level, contrary to what is observed in studies based on matrix elements for VBF $Hjj$ production that are only LO accurate in the third jet. Our implementation thus provides an important improvement in the theoretical assessment of central-jet veto observables that are crucial for VBF analyses at the LHC.  \n\\section*{Acknowledgments} \nWe are grateful to Carlo Oleari for help with implementing the code in the \\POWHEGBOX{} repository. \nThe work of\nB.~J.\\ is supported by the Institutional Strategy of the University of T\\\"ubingen (DFG, ZUK~63). \nF.~S.\\ is supported by the ``Karlsruher Schule f\\\"ur Elementar\\-teilchen- und Astroteilchenphysik: Wissenschaft und Technologie (KSETA)'' and D.~Z.\\ by the BMBF under ``Verbundprojekt 05H2012 -- Theorie''.  \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}