diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpbdm" "b/data_all_eng_slimpj/shuffled/split2/finalzzpbdm" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpbdm" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \n\nGalaxies continue to surprise us, both in their observed structure and\nby their inferred evolution. Some of these surprises come from moving\nto new wavelengths and re-observing old friends; while other surprises\ncome staying at familiar wavelengths but looking at the galaxies from\na somewhat new perspective or in a slightly revised context.\n\nWith the launch of the GALEX satellite into low-Earth orbit on April\n28, 2003 it has become possible to image a large number and a wide\nrange of different types of galaxies at largely unexplored\nwavelengths: in the near and far ultraviolet. Given the high\nsensitivity of the detectors and the very low sky background\n(especially in the far ultraviolet channel) it is possible to see\nfeatures in the ultraviolet out to surface brightness levels\nunequalled by ground-based optical observations. Moreover, the wide\n(one-degree) field of view of GALEX also maximizes the possibility of\nserendipitous discovers, of which there have been many.\n\n\\section{Radially Extended Star Formation}\n\nAll of the above aspects of GALEX contributed to the ``discovery'' of\n{\\it extended ultraviolet} (XUV) disks around a number of nearby\nspirals. Prime examples, found early in the mission are M83 (Thilker\n{\\it et al.} 2005) and NGC~4625 (Gil de Paz {\\it et al.} 2005). A\ncomparison of the optical image of the compact, one-armed spiral NGC~4625\nwith its significantly more extended, and multi-armed UV counterpart\nimage is shown in Figure 1. \n\n\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[scale = 0.6] {N4625B.ps}\n\\includegraphics[scale = 1.1] {N4625UV.ps}\n\\end{center}\n\\caption{NGC 4625. Right panel shows the optical image of the galaxy,\nwhile the left panel shows the NUV (GALEX) of this same galaxy where\nthe extended UV disk of star formation is clearly\nhighlighted}\\label{fig1}\n\\end{figure}\n\n\n\n\n\\subsection{Pre-Discoveries and Thresholds}\n\nWhile these features are striking, they should not have come as a\nsurprise (althogh they did) as they are not unheralded. For instance,\nbased UV imaging data obtained from a balloon-borne experiment (SCAP\n2000) Donas et al. (1981) asked the question ``How Far Does M101\nExtend?'' Clearly they were seeing extended UV features in the outer\ndisk of a well known nearby galaxy. And later Ferguson et al. (1998),\nannounced ``The Discovery of Recent Star Formation in the Extreme\nOuter Regions of Disk Galaxies'' based on deep H$\\alpha$ surveys of\nthree nearby late-type galaxies: NGC~0628, NGC~1058 and NGC~6946. And\nall of this might have been seen to be a natural consequence of the\ndiscovery of extended HI disks around many galaxies (e.g., NGC~2915,\nMuere et al. 1996) except perhaps for the interesting interpretive\npaper by Martin \\& Kennicutt (2001) which seemed to put an end to star\nformation in the outer disks of galaxies by titling their paper ``Star\nFormation Thresholds in Galactic Disks.'' With hindsight, which we all\nnow possess, it might have been more robust to have added ``as Traced\nby HII Regions'' to the title.\n\n\\subsection{Ultraviolet Data}\n\nRecently, using GALEX ultraviolet imaging data, Boissier et al. (2007)\nhave re-examined the correlation of gas density with star formation\n({\\it as traced by FUV\/NUV light}) and have come to very different\nconclusions regarding the existence of any threshold to star formation\nat low gas densities. As can be seen in Figure 2 the left panel shows\nthe run of star formation rates as a function of total gas density for\na combined sample of 43 galaxies. The star formation rate is based on\nextinction-corrected UV surface brightness obtained by the GALEX\nsatellite and the total gas density combines neutral and molecular\nhydrogen contributions. The right panel shows a selection of\nindividual galaxies where the run of UV and the H$\\alpha$ star\nformation rates with surface density of gas are intercompared on an\nindividual basis. Clearly there is a difference. No truncation and no\nthreshold is apparent in the UV data. The surface density of star\nformation as traced by hot, blue stars (which may or may not) include\nO stars (which power the brightest HII regions) is continuous with the\ngas surface density, to the limits of both surveys.\n\nThe hard cut-off in H$\\alpha$ has been challenged, of course, by\nFerguson et al. (1988), but it may also be that other factors are in\nplay. In the outer parts of galaxies we may be seeing small number\nstatistics force the mass function of the molecular clouds down to a\nmass level that while they can support star formation they are not\nindividually large enough to produce even a single O star capable of\nionizing the surrounding medium. The plane thickness may have grown so\nmuch, or the intercloud separation may be so large that the star\nformation regions are density-bounded and that large fraction of the\nUV radiation is leaking out before it can produce a detectable HII\nregion or that the radiation that is intercepted is only reradiated at\na very low emission measure and perhaps and widely distributed. Very\ncompact HII regions have been found in the XUV disk of M83 and they\nare consistent with single-star ionization (Gil de Paz et\nal. 2005). Studies are underway to probe deeper and to lower surface\nbrightness levels in search of any stray radiation.\n\n\n\\begin{figure}[!ht]\n\\includegraphics[scale = 0.360] {f6.eps}\n\\includegraphics[scale = 0.340, bb = 45 110 65 145] {f7.eps}\n\\caption{The left panel shows the radial drop-off of star formation\nrates as a function of gas density for ** galaxies as reproduced from\nBoissier et al. (2007). The vertical grey zone shows the gas-density\nwhere the star-formation threshold of Martin \\& Kennicutt (2001)is\nexpected. For the UV data no threshold is observed. The right panels\nshows an intercomparison of UV star formation rates (dotted lines) and\nH$\\alpha$ star formation rates (solid lines) as a function of radius\n(scaled to the Martin-Kennicutt threshold.) Again, the UV star\nformation shows no sign of any thresholding at any gas density\nplotted.}\\label{fig3}\n\\end{figure}\n\n\n\\section{Gravitating Non-Luminous (GNL) Galaxies}\n\nMost contemporary models of structure formation within the framework\nof a $Lambda$-CDM cosmology predict a steep power-law increase of\nlower mass galaxies fainter than $L^*$ (e.g., Kravtsov, Gnedin \\&\nKlypin 2004). Most of the satellites are deemed ``missing'' because\nthey not seen in optical surveys. This has led to the speculation that\nthey are currently somehow devoid of baryons, and thus invisible by\nconstruction. Invisible but not undetectable. The presence of {\\it\nGravitating Non-luminous} (GNL) galaxies can in principle (and in\npractice) be inferred (and seen) by their gravitational interactions\nwith other nearby (visible) galaxies. Below we explore a few such\ntests for their presence ... or absence.\n\n\\subsection{Ring Galaxies}\n\nTwenty years ago Arp \\& Madore (1987) published a catalog and\nphotographic atlas of peculiar galaxies based on a visual inspection\nof over 94,000 optical images of southern-hemisphere galaxies. More\nrecently Madore, Nelson \\& Petrillo (2007 in prep) extracted from that\na pure sample of about 100 ring galaxies (see Figure 3 for a\nsampling). According to models by Theys \\& Spiegel (1976) and by Lynds\n\\& Toomre (1976) they can be explained by galaxy-galaxy interactions,\nfine-tuned to be head-on collisions between a disk galaxy and a\nlower-mass intruder. The ring galaxies culled from the Arp-Madore\nCatalogue were listed by those original discoverers because of their\nring morphology not because they did or did not have\ncompanions. However, it is gratifyingly strong confirmation of the\ntheory that virtually all of the rings also have adjacent (line of\nsight) companions (many of which already have redshifts confirming\ntheir physical as well as apparent association with the rings.) These\ncompanions are plausible colliders, based on their apparent proximity\nand being only a diameter or two from the ring itself.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[scale = 0.55] {ring_poster.ps}\n\\end{center}\n\\caption{Examples of ring galaxies and their adjacent companions from\nthe soon to be published atlas of Madore, Nelson \\& Petrillo\n(2007)}\\label{fig2}\n\\end{figure}\n\nWhile this is all good news for the theory of ring formation in\nspecific, it is not so good news for the theory of galaxy formation in\ngeneral. Without a single convincing example of an isolated ring, one\nobvious conclusion is that GNL galaxies do not exist in the numbers\npredicted by theory. While it may be counter-argued that ring galaxies\nrequire such peculiar circumstances for their formation (mass ratios,\norbital parameters and galaxy types, etc.) and that they can only\narise from a collision involving concentrated {\\it baryonic} satellite\nintruders for their formation, the same cannot be said for\ncollision-induced peculiarities in general. The CDM simulations\npredict the observed number of optical galaxy only at one mass. Above\nand below this there is a divergence at all masses bewteen theory and\nobservation, reach more than a factor of 10 descrepancy at the fainest\nend. Wehether is is high-mass or low-mass companions that are being\nlooked for in GNL-galaxy interactions they are predicted to be there\nat the level of factors more than their optical counterparts rather\nthan occasionally occurring at very low levels of incidence, as\nappears to be the case.\n\n\\subsection{Arp Peculiar Galaxies}\n\nTurning back to the time at which the original {\\it Atlas of Peculiar\nGalaxies} was published (Arp 1966) it is fair to say that the topic of\npeculiar galaxies was still in its formative stages and that the\nsample illustrated was not premised upon their being or not being\nnearby galaxies to qualify them to be include in the Atlas. Indeed by\nmoder standards many of the galaxies in the {\\it Atlas} are not now\nconsidered to be particularly peculiar at all ({\\it e.g.,}\nlow-surface-brightness galaxies [ARP 001-004], dwarf irregular\ngalaxies [ARP 005-006], and certain alignments in small groups and\nclusters [ARP 311-332]). Rare does not necessarily mean peculiar, but\nthe {\\it Atlas of Peculiar Galaxies} did include many rare types of\nobjects. Our point here is however that of the 338 objects that are\nincluded in the {\\it Atlas} because the galaxy in question is bodily\ndistorted, is considerably asymmetric or has extended tails, the vast\nmajority of those have very nearby companions that can easily be\nimplicated as the source of the interaction and distortion. It is the\nabsence of {\\it isolated} peculiar galaxies that is in itself\nnoteworthy. Clearly the vast majority of peculiarities seen as bodily\ndeformations of the galaxy in question can be explained by\ninteractions with nearby optical companions. And this simple fact\nleaves no room (and no evidence for) bodily deformed galaxies\nresulting from their interaction with gravitating non-luminous (GNL)\ngalaxies (i.e., pure dark-matter halos).\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[scale = 0.5] {arp172.ps}\n\\includegraphics[scale = 0.5] {arp107.ps}\n\\includegraphics[scale = 0.5] {arp173.ps}\n\\end{center}\n\\caption{Arp~172 (left), Arp~107 (center) and Arp ~173 (right) typify\nthe types of bodily distorted galaxies in the Arp Atlas of Peculiar\nGalaxies that almost without exception have obvious interactions\non-going between two optically visible galaxies}\\label{fig1}\n\\end{figure}\n\n\n\n\\subsection{Karachentsev Isolated Galaxies}\n\n\nThere is an alternative path to follow here. Karachentsev (1988)\npublished a list of 1,000 galaxies that are optically isolated from\nother comparable-sized (optically visible) galaxies. This, of course,\nis not to say that apparently isolated cannot and do not have GNL\ngalaxies of comparable (or even larger) sizes orbiting and interacting\nwith them. However this catalog would suggest that this is not\nthe case. With complete certainty we can say that out of the entire\nsample of optically isolated galaxies there are no examples of\nArp-like bodily-distorted systems. \n\nThere are a handful of isolated galaxies that are peculiar to some\nlesser degree. But even this is to be expected without having to\ninvoke GNL galaxies; mergers will deplete the apparent population of\nvisible interactors while still leaving evidence of the collision in\nthe form of tidal debris or lingering asymmetries. For example, even a\ncursory examination of the 2MASS near-infrared image of KIG 0022 (an\nobject that has large `tidal' arms in the optical) shows that it has a\ndouble nucleus; presumably the result of a recent merger of two\n(previously) visible galaxies. Details of these samples and their\nanalysis will be given in a forthcoming paper (Madore. Petrillo \\&\nNelson 2007).\n\\section{Conclusions} \n\nUsing UV light as a tracer for star formation, FUV and NUV imaging\nobservations of nearby galaxies using the GALEX satellite show a\nsmooth and monotonic decline of star formation with total gas surface\ndensity. No thresholding of star formation is visible in this sample,\nat these projected surface densities.\n\nThe summary observations of peculiar galaxies viewed in the context of\n$Lamda$-CDM simulations are as follows: (1) All cataloged ring\ngalaxies have plausible colliders that are optically visible. (2) All\npeculiar galaxies (in the Arp Atlas) that are bodily deformed have\nvisible nearby companions that are plausibly responsible for the\ninteraction-induced deformities. (3) Virtually all isolated galaxies\nare not peculiar, distorted or interacting to any noticeable degree.\n\nThe peculiar galaxy study leads to the following general conclusions:\n(1) No ring galaxy is being produced from a head-on collision between\na spiral and pure dark-matter GNL galaxy (2) No bodily deformed galaxies\nare the result of collisions and\/or near encounters between optical\nand pure dark-matter GNL galaxies. (3) Optically isolated galaxies\nshow no signs of bodily interactions with pure dark matter GNL\ngalaxies.\n\n\n\n\n\n\\vfill\\eject\n\\acknowledgements\nWe sincerely thank all of our colleagues on the GALEX Mission for their\nsupport in enabling the science being conducted by NASA's Galaxy\nEvolution Explorer. Major portions of this research would not have\nbeen possible without the NASA\/IPAC Extragalactic database (NED) which\nis operated by the Jet Propulsion Laboratory, California Institute of\nTechnology, under contract with the National Aeronautics and Space\nAdministration. Significant support for this research was also\nprovided by the Observatories of the Carnegie Institution of\nWashington.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nLight can be utilized as a tool to manipulate and engineer novel phases in quantum materials~\\cite{basov2017towards}. In particular, excitation via intense light pulses has been used to create nonequilibrium states of matter nonexistent at thermal equilibrium, such as transient superconductivity in underdoped cuprates~\\cite{fausti2011light}, metastable ferroelectricity in SrTiO$_3$~\\cite{Nova2019,Li2019}, and unconventional charge-density wave order in LaTe$_3$ \\cite{Kogar2020}. The light pulses excite a transient population of quasiparticles or collective excitations, which acts as a dynamic parameter\nto alter the material's free-energy landscape. \nFor sufficiently strong excitation densities, a nonequilibrium phase transition can eventually occur~\\cite{Kogar2020,teitelbaum2019,dolgirev2019universal}.\nBy the same token, intense pulsed light holds\npromise to manipulate the spin state of frustrated magnets~\\cite{balents2010spin,knolle2019field}. These materials, in fact, can host exotic and elusive phases, such as spin liquids (SL). Whereas SLs harbors rich many-body phenomena resulting from spin frustration and possible spin fractionalization~\\cite{kitaev2006anyons,broholm2020quantum}, these phases often compete with a magnetically ordered ground state, which is typically energetically favoured.\nPulsed light excitation can then provide a mechanism to tip the energetic balance away from the magnetically ordered ground state towards a nonequilibrium proximate spin liquid phase.\n\nWe explore this concept for the Kitaev-Heisenberg frustrated magnet, a type of Mott insulator with a layered honeycomb structure and strong spin-orbit coupling~\\cite{jackeli2009,chaloupka2013,takagi2019concept}. For these materials the large spin-orbit interaction leads to a sizeable bond directional spin exchange, whereas the symmetric Heisenberg exchange cancels out by virtue of the edge-sharing octahedra geometry, making them promising candidates for Kitaev physics~\\cite{jackeli2009,chaloupka2013,takagi2019concept}. Still, the remaining Heisenberg interaction, present due to small structural distortions away from the ideal honeycomb structure, \\cite{johnson2015} is an adversary to spin liquid formation, and generally favors a spin-ordered ground state~\\cite{chaloupka2013,alpichshev2015,nembrini2016}. %\nBy modulating spin entropy through finite temperature effects ~\\cite{do2017majorana,sandilands2015raman} or by adding external magnetic fields ~\\cite{kasahara2018majorana}, one can however stabilize proximate or field-induced \nspin liquid phases at thermal equilibrium.\nThese spin liquid realizations show emergent behavior expected for the pure Kitaev spin liquid,\\cite{kitaev2006anyons} most notably, fractionalized particle statistics \\cite{sandilands2015raman} and quantized conduction phenomena.\\cite{kasahara2018majorana}\n\nA case in point is $\\alpha$-RuCl$_{3}$\\@\\xspace. This honeycomb Mott insulator has nearly-ideal $j_{\\rm eff}$\\,$=$\\,$\\tfrac{1}{2}$ isospins in highly symmetric octahedra,~\\cite{plumb2014,agrestini2017} making it possibly the most promising Kitaev spin liquid host studied to date~\\cite{do2017majorana,sandilands2015raman,kasahara2018majorana}.\nThe ($B$,$T$)-plane in Fig.\\,\\ref{fig:phasediagram} provides the \\textit{equilibrium} phase diagram as a function of magnetic field and temperature. Below $T_{\\rm N}$\\,$\\approx$\\,$7$K, the isospins couple in a zigzag fashion, consistent with the types of magnetic order captured by the Kitaev-Heisenberg model.\\cite{chaloupka2013}\nStrong short-range spin correlations persist between $T_{\\rm N}$ and the crossover temperature $T_{\\rm H}$\\,$\\approx$\\,$100$\\,K, hinting at the formation of a proximate spin liquid (pSL) phase within this intermediate temperature regime.\n\\cite{do2017majorana,banerjee2016proximate,winter2018}\nAbove $100$\\,K thermal fluctuations bring the system into a conventional paramagnetic phase.\nAn additional tuning parameter is provided by an in-plane magnetic field. A field of $B_{\\rm c}$\\,$\\approx$\\,$7$\\,T is sufficient to destabilize the zigzag order.\nFor fields between $7$\\,-\\,$8$\\,T a much-debated field-induced SL is then stabilized, \\cite{kasahara2018majorana} whereas for higher fields a quantum disordered state with partial field alignment of the effective moments forms.\\cite{johnson2015,sears2017phase,sahasrabudhe2019high} \n\n\\begin{figure}[h!]\n\\center\n\\includegraphics[width=2.75in]{Fig1_phasediagramrcl.png} \n\\caption{A nonequilibrium dimension to $\\alpha$-RuCl$_{3}$\\@\\xspace's magnetic phase diagram. \nThe ($B$,$T$)-plane sketches the \\textit{equilibrium} magnetic phase diagram. Photoexcited holon-doublon pairs $n_{\\gamma}$ form a new \\textit{nonequilibrium} parameter.\nFor small (red) to intermediate (magenta) quenches the system stays inside the zigzag ordered phase. Above a critical density $n_{\\gamma,\\rm crit}$ a nonequilibrium proximate spin liquid state may be induced (light blue arrow).}\n\\label{fig:phasediagram}\n\\end{figure}\n\nIn this work, we report on the observation of a transient long-lived spin disordered state in the Kitaev-Heisenberg magnet $\\alpha$-RuCl$_{3}$\\@\\xspace induced by pulsed light excitation. Holon-doublon pairs are created by photoexcitation above the Mott gap, and provide a new \\textit{nonequilibrium} dimension to $\\alpha$-RuCl$_{3}$\\@\\xspace's magnetic free energy landscape and resulting phase diagram, as illustrated in Fig.\\,\\ref{fig:phasediagram}. The subsequent holon-doublon pair recombination through multimagnon emission leads to a decrease of the zigzag magnetic order. This is tracked through the magnetic linear dichroism (MLD) response of the system. For a sufficiently large holon-doublon density the MLD rotation vanishes, implying that the zigzag ground state is fully suppressed and that a long-lived transient spin-disordered phase is induced. The disordering dynamics of the zigzag order parameter is captured by a time-dependent Ginzburg-Landau model, corroborating the nonequilibrium quasistationary nature of the transient phase. Our work provides insight into the coupling between high-energy electronic and low-energy magnetic degrees of freedom in $\\alpha$-RuCl$_{3}$\\@\\xspace and suggests a new route to reach a proximate spin-liquid phase in honeycomb Mott insulators with residual interactions beyond the bond-directional Kitaev exchange.\n\n\\section*{Results and discussion}\n\nThe photoinduced change in reflected polarization rotation from $\\alpha$-RuCl$_{3}$\\@\\xspace was measured as a function of temperature and photoexcitation density. The sample is excited above the $\\Delta_{\\rm MH}$\\,$\\sim$\\,$1.0$\\,eV Mott-Hubbard gap~\\cite{sandilands2016} with a photon energy of $\\hbar\\omega$\\,$\\approx$\\,$1.55$\\,eV. \nThe probe light has $2.42$\\,eV photon energy. Under zero-field conditions, two contributions to the total optical polarization rotation $\\theta_{\\rm tot}$ can be distinguished: \n\n\\begin{equation}\n\\theta_{\\rm tot}= \\theta_{\\rm LD} + \\theta_{\\rm MLD}(\\vec{L}^2).\n\\label{eq:totalrotation}\n\\end{equation}\n\n\\noindent The first term $\\theta_{\\rm LD}$, linear dichroism, originates from the monoclinic distortion of RuCl$_{3}$\\@\\xspace, \\cite{johnson2015} and will only show a negligible temperature dependence over the relevant temperature range.\\cite{glamazda2017}\nThe second term $\\theta_{\\rm MLD}$, magnetic linear dichroism (MLD), \\cite{smolenskiui1975birefringence,pisarev1991optical} is proportional to the square of the zigzag antiferromagnetic order parameter $\\vec{L}=\\vec{M_{\\uparrow}}-\\vec{M_{\\downarrow}}$, where $\\vec{M_{\\uparrow}}$ and $\\vec{M_{\\downarrow}}$ give the sublattice magnetizations.\nAs such, the MLD rotation provides an optical probe of the zigzag spin order in $\\alpha$-RuCl$_{3}$\\@\\xspace.\n\n\\begin{figure}[h!]\n \\center\n\\includegraphics[scale=1]{Fig2_temperature.png} \n\\caption{Temperature dependent transient polarization rotation and critical slowing down. a) Photoinduced change in polarization rotation $-\\Delta\\theta_{\\rm MLD}(t)$ for various temperatures below and above $T_{\\rm N}$\\,$\\approx$\\,$7$\\,K. The signal below $T_{\\rm N}$ is dominated by the proper stacking phase. Above $T_{\\rm N}$ a small signal with opposite rotational sign is observed, originating from the stacking-fault phase. b) Integrated change in rotation (black spheres) and $\\tau_{\\rm decay}$[ps] (red circles) as a function of temperature. A critical slowing down of the disordering is observed upon approaching the phase transition.}\n\\label{fig:tempdep}\n\\end{figure}\n\n\nFigure \\ref{fig:tempdep}a displays the photoinduced change in polarization rotation $-\\Delta\\theta_{\\rm MLD}(t)$ for various bath temperatures. A low-excitation fluence $F$\\,$\\sim$\\,$1.7$\\,$\\mu$J\/cm$^{2}$ was used, corresponding to a photoexcitation density of $n_{\\gamma}$\\,$\\approx$\\,$0.8$\\,$\\cdot$\\,$10^{17}$\\,cm$^{-3}$ (Ref.\\,\\citenum{footnoteS1}). For temperatures below $T_{\\rm N}$\\,$\\approx$\\,$7$\\,K an initial fast demagnetization on the \ntens of ps timescale is observed, after which the signal recovers on the ns-timescale. Above $T_{\\rm N}$ a small amplitude response is observed with an opposite rotational sense, originating\nfrom a fraction of unavoidable stacking-fault-phase contributions at the sample surface.\\cite{sandilands2016,cao2016}\nIn Fig.\\,\\ref{fig:tempdep}b the integrated change in rotation $\\Delta\\theta_{\\rm max}$ is plotted versus temperature. The integrated rotation change shows a pronounced increase, followed by a rapid reduction upon approaching $T_{\\rm N}$\\,$\\approx$\\,$7$\\,K. This behavior is qualitatively rationalized by considering that the photoexcitation will have the largest transient effect where the derivative of the zigzag order parameter with respect to temperature is the largest~\\cite{banerjee2017neutron}. Concomitantly, we observe a critical slowing down of the disordering upon approaching the phase transition~\\cite{hohenberghalperin1977,zong2019}. This behavior is well captured by a $\\tau_{\\rm decay}$\\,$\\propto$\\,$\\vert 1-T\/T_{\\rm N}\\vert^{-\\nu z}$ power law with critical exponent $\\nu z$\\,$=$\\,$-2.1$, compatible with the universality class of the 2D Ising model-A dynamics, applicable to $\\alpha$-RuCl$_{3}$\\@\\xspace~\\cite{banerjee2017neutron,hohenberghalperin1977,tauber2014critical}.\n\n\nFigure \\,\\ref{fig:densitydep}a shows the transient rotation traces $\\theta_{\\rm MLD}(t)$ for various initial photoexcitation densities $n_{\\gamma}$ (sphere symbols). \nThe photoexcitation dependence of the maximum MLD change, $\\Delta\\theta_{\\rm MLD,max}$, is depicted in Fig.\\,\\ref{fig:densitydep}b. \nQualitatively, two excitation regimes can be distinguished. For lower excitation densities ($n_{\\gamma}$\\,$<$\\,$n_{\\rm \\gamma,crit}$\\,$\\approx$\\,$3$\\,$\\cdot$\\,$10^{17}$\\,cm$^{-3}$), the spin system partially disorders, followed by a subsequent recovery. In this regime the disordering time slows down with increasing photoexcitation density. For the high excitation densities ($n_{\\gamma}$\\,$>$\\,$n_{\\rm\\gamma, crit}$\\,$\\approx$\\,$3$\\,$\\cdot$\\,$10^{17}$\\,cm$^{-3}$) a faster disordering time is observed and\nthe change $\\Delta\\theta_{\\rm MLD,max}$ saturates (Fig.~\\ref{fig:densitydep}b), implying that the photoexcited system resides in a $L$\\,=\\,$0$ state for multiple $100$s of ps.\nReferring to the magnetic phase diagram (Fig.~\\ref{fig:phasediagram} and Refs.~\\citenum{johnson2015,kasahara2018majorana}), this means that for quench strengths above $n_{\\rm \\gamma,crit}$ the zigzag \norder can be fully suppressed, leaving the system in a spin disordered state. \nThe disordering mechanism and the nature of the long-lived transient state is corroborated below.\n\n\n\\begin{figure}[h!]\n \\center\n\\includegraphics[scale=1]{Fig3_density.png} \n\\caption{Nonequilibrium magnetic phase transition and holon-doublon pair recombination by multimagnon emission. a) Density-dependent $\\theta_{\\rm MLD}(t)$ for different excitation densities $n_{\\gamma}$, as indicated with spheres. The modelled rotation $\\theta(t)$ is indicated with thick lines. b) Maximum change in the magnetic linear dichroism (MLD) rotation $\\Delta\\theta_{\\rm MLD}(t)$ as a function of photon density $n_{\\gamma}$. Above the critical density $n_{\\rm \\gamma,crit}$\\,$\\approx$\\,$3$\\,$\\cdot$\\,$10^{17}$\\,cm$^{-3}$ the maximum change in MLD-rotation saturates.\nc) The honeycomb lattice, consisting of Ru-sites (dark-blue sites) and chloride ligand ions (red sites). The lower process\nshows the photogeneration of a holon-doublon pair. The upper process shows the subsequent multimagnon emission by holon-doublon recombination.} \n\\label{fig:densitydep}\n\\end{figure}\n\nThe inherently strong charge-spin coupling of Mott insulators leads to an efficient nonlinear demagnetization mechanism upon photoexcitation above the Mott-Hubbard gap.~\\cite{lenarcic2013,afanasiev2019} \nIn order to illustrate this mechanism, first consider the photoexcitation process corresponding to the lowest $t_{2g}^{5}$\\,$t_{2g}^{5}$\\,$\\rightarrow$\\,$t_{2g}^{4}$\\,$t_{2g}^{6}$ hopping-type excitation across the Mott-Hubbard gap, as illustrated by the lower hopping process in Fig.~\\ref{fig:densitydep}c. Within a quasiparticle picture, this intermediate excited state corresponds to a spinless \\textit{holon} ($t_{2g}^{4}$) and \\textit{doublon} ($t_{2g}^{6}$), by which effectively two magnetic moments are removed from the zigzag lattice. The mere \\textit{creation} of these quasiparticles at the used low densities of $4$\\,-\\,$85$\\,ppm photons\/Ru$^{3+}$-site however does not suffice to explain the magnitude and timescale of the zigzag disordering.\\cite{footnoteS1} Instead, once created, the dominant decay mechanism of the holon-doublon pairs is \\textit{recombination} through multimagnon emission (upper hopping process Fig.\\,\\ref{fig:densitydep}c).\\cite{lenarcic2013}\nAn order of magnitude estimate for the released amount of magnons per decayed $hd$-pair is provided by $\\Delta_{\\rm MH}\/W$\\,$\\sim$\\,$25$ (Refs.\\,\\citenum{lenarcic2013}), with $W$\\,$\\approx$\\,$4.0$\\,meV being\nthe bandwidth of the low-energy spin wave branch in the zigzag phase.\\cite{banerjee2018excitations} As such, this quasiparticle recombination provides an efficient electronic demagnetization mechanism. \n\nIn order to further delineate the excitation mechanism and resulting magnetization dynamics, we model the time-domain data within a dynamic Ginzburg-Landau (GL) model.\\cite{hohenberghalperin1977,tauber2014critical} The holon-doublon density, representing the nonequilibrium dimension in Fig.\\,\\ref{fig:phasediagram}, comes in as a new dynamical variable here. We first consider the modified free energy for the antiferromagnetic order parameter $L$ and the holon-doublon-\\textit{pair} density $n$: \n\\begin{equation}\n\\mathcal{F}(n,L) = \\frac{a_1}{2}(n-n_{\\rm c,eq}) L^2 + \\frac{a_2}{4}L^4 + \\tilde{\\mathcal{F}}(n),\n\\end{equation}\n\\noindent with\n\\begin{equation}\n\\tilde{\\mathcal{F}}(n) = a_3 n + \\frac{a_4}{2} n^2 +\\frac{a_5}{3}n^3 ,\n\\end{equation}\n\n\\noindent where $a_i$, $i=1, \\dots, 5$ are phenomenological parameters.\nThe terms with even powers in the zigzag order parameter $L$ are the standard symmetry-allowed terms in the Landau free energy expansion for an antiferromagnet.\\cite{tauber2014critical,khomskii2010basic} Notice that odd powers of $L$ are ruled out by the inversion symmetry of RuCl$_{3}$\\@\\xspace.\\cite{johnson2015} \nThe initial value of the holon-doublon pair density $n(0)$, or quench strength, is taken proportional to the experimental photoexcitation densities $n_{\\gamma}$, i.e., $n(0)$\\,$\\propto$\\,$n_{\\gamma}$, where each photon creates one $hd$-pair. \nThe first term in $\\mathcal{F}(n,L)$, coupling the $hd$-pair density $n$ to the order parameter $L$, leads to a destabilization of the magnetic order for a sufficiently strong excitation of $hd$-pairs, thus reproducing the process of annihilation of $hd$-pairs into magnons.\\cite{lenarcic2013,footnoteS2}\nThe parameter $n_{\\rm c,eq}$ is introduced as the critical $hd$-pair density at equilibrium. \nThe functional $\\tilde{\\mathcal{F}}(n)$, independent of the order parameter $L$, describes the excess energy of the $hd$-density and its relaxation in the absence of magnetization. It therefore accounts for decay mechanisms other than the nonradiative multimagnon emission discussed above, such as nonradiative phonon emission, spontaneous decay under radiative emission,\\cite{mitrano2014}, and possible $hd$-pair diffusion out of the probe volume.\nThe form of $\\tilde{\\mathcal{F}}(n)$ is chosen as a third-order polynomial, although its exact form is not crucial for the analysis.\n\nThe time evolution of the $hd$-pair density $n$ and magnetic order parameter $L$ is described by the coupled equations of motion:\n\\begin{equation}\n\\frac{d L}{dt}=-\\frac{\\delta \\mathcal{F}}{\\delta L}, \\qquad \\frac{d n}{dt}=-\\frac{\\delta \\mathcal{F}}{\\delta n}.\n\\label{eq:Onsager}\n\\end{equation}\nIn order to relate Eqs.\\,\\eqref{eq:Onsager} to the experimentally measured rotation $\\theta_{\\rm MLD}$, we rewrite the equations in terms of the polarization rotation $\\theta$\\,$=$\\,$L^2\/2$, to finally obtain:\n\\begin{subequations}\n\\label{eq:NEQ-Onsager}\n\\begin{align}\n\\frac{d \\theta}{d t} &=-2a_1 (n-n_{\\rm c,eq})\\theta -4a_2\\theta^2, \\\\\n\\frac{d n}{d t} & = -a_1\\theta -\\frac{\\delta}{\\delta n}\\tilde{\\mathcal{F}}(n)\n\\end{align}\n\\end{subequations}\nBy using Eq. \\eqref{eq:NEQ-Onsager}a, the trajectories $n(t),\\theta(t)$ can be modelled for different initial quench strengths $n(0)$, taken proportional to the experimental $n_{\\gamma}$ densities. The curves for $\\theta(t)$ are superimposed on the experimental $\\theta_{\\rm MLD}(t)$ in Fig.\\,\\ref{fig:densitydep}a. The model captures the dependence of the demagnetization time on the excitation density and the position of $t_{\\rm max}$, i.e., the time at which $\\Delta\\theta_{\\rm max}$ is reached. For the higher excitation densities the magnetic order vanishes, reproducing the long-lived transient $L$\\,$=$\\,$0$ state. The inclusion of the $n^3$-term in $\\tilde{\\mathcal{F}}(n)$ ensures that the GL-description does not overestimate the lifetime of the $L$\\,$=$\\,$0$ state.\\cite{footnoteS3} The density-dependent rotation transients are well captured considering the minimal amount of parameters needed in the \nnonequilibrium GL-description.\n\nThe $hd$-pair density-dependent free-energy landscape $\\mathcal{F}(n,L)$ is shown in Fig.~\\ref{fig:freenenergy}. For low densities, the free energy retains its double-well profile, whereas for higher densities a single well forms.\\cite{zong2019}\nRepresentative trajectories $n(t),L(t)$ for different excitation densities are drawn into the free-energy landscape, with colors corresponding to the conceptual trajectories of Fig.~\\ref{fig:phasediagram}. \nThe quench $n(0)$ brings the system into a high-energy state, after which $n$ and $L$ relax along the minimal energy trajectory. For a small quench (red trajectory)\nthe zigzag order parameter $L(t)$ stays finite and eventually recovers. For the intermediate densities (magenta trajectory)\nthe $n(t),L(t)$ coordinates approach the $L$\\,$=$\\,$0$ line. For the higher excitation densities (light blue trajectory)\nthe $hd$-pairs have sufficient excess energy to let $n(t),L(t)$ follow a trajectory along the $L$\\,$=$\\,$0$ line, i.e., full spin disordering is reached. We emphasize that, for strong quenches, the excitation density $n(t)$ still varies in time, even though $L(t)$ takes the quasistationary value $L(t)$\\,$=$\\,$0$.\nA sufficiently strong photoexcitation quench thus provides a mechanism to dynamically stabilize a \\textit{nonequilibrium quasistationary} spin disorded state in $\\alpha$-RuCl$_{3}$\\@\\xspace.\n\n\\begin{figure}[h!]\n \\center\n\\includegraphics[width=3.375in]{Fig4_freeenergy.png} \n\\caption{Free energy landscape and nonequilibrium quasistationary spin disordered state. Free energy landscape $F(n,L)$ as a function of the zigzag order parameter $L$ and $hd$-pair density $n$ (cf. the phase diagram in Fig.\\,\\ref{fig:phasediagram}). The initial quench $n(0)$ brings the system to a high energy state, after which the system relaxes. For small quenches (red trajectory), the order parameter $L$ stays finite under the relaxation of the density $n$. For intermediate quenches (magenta trajectory) the system approaches the $L$\\,=$0$ line. For strong quenches (light blue trajectory, highest energies not shown) the system relaxes along the $L$\\,=$0$ line, implying that the system is described as a nonequilibrium quasistationary spin disordered state.}\n\\label{fig:freenenergy}\n\\end{figure}\n\n\nThe maximum lifetime of the nonequilibrium quasistationary spin disordered state is dictated by the recombination rate of the $hd$-pairs. The time evolution of $n(t)$ obtained from the nonequilibrium GL model provides us with an estimate of a few nanoseconds for the recombination timescale of the $hd$-pairs.\\cite{footnoteS3}\nThis timescale\nis expected to grow\nexponentially with the number of magnons needed to traverse the Mott gap, i.e., $\\tau$\\,$\\sim$\\,$e^{\\Delta_{\\rm MH}\/W}$. \\cite{lenarcic2013} \nConsidering the weak exchange-interaction scale $W$ in $\\alpha$-RuCl$_{3}$\\@\\xspace, one may indeed expect significantly longer recombination times compared to materials with an order of magnitude stronger exchange, such as Nd$_2$CuO$_4$ and Sr$_2$IrO$_4$, where $hd$-pair lifetimes on the order of $0.1$\\,ps have been reported.\\cite{afanasiev2019,okamoto2011} A large ratio between the Mott gap and the exchange interaction energy thus is the key element to ensure a long lifetime of the nonequilibrium quasistationary state.\n\nThe microscopic nature of the transient long-lived spin disordered state currently remains elusive. The used low excitation densities by far do not provide sufficient energy to drive the material into a conventional paramagnetic state, nor to change the dominant interactions in the system. Considering the phase diagram, it therefore seems plausible that the system is driven into a transient proximate spin liquid phase, reminiscent of the thermodynamic state just above $T_{\\rm N}$. Energy-resolved ultrafast techniques may provide more insight into the microscopic properties of the induced phase. \\cite{nembrini2016,sandilands2015raman,halasz2016} Furthermore, exact diagonalization \\cite{okamoto2013} and nonequilibrium dynamical mean-field theory \\cite{aoki2014} methods may elucidate the role of $hd$-excitations in the Kitaev-Heisenberg model and the resulting phase diagram.\n\n\\section*{Conclusions}\n\nWe have unveiled a pulsed light excitation driven mechanism allowing to trap a Kitaev-Heisenberg magnet into a quasistationary spin disordered state. Photoexcitation above the Mott-gap generates a transient density of holon-doublon quasiparticle pairs. The subsequent recombination of these quasiparticles through efficient multimagnon emission provides a way to dynamically destabilize the competing zigzag ordered ground state and thereby keeps the system in an out-of-equilibrium spin disordered state, up until the transient electronic quasiparticle density gets depleted. Our work provides insight into the coupling between electronic and magnetic degrees of freedom in $\\alpha$-RuCl$_{3}$\\@\\xspace, and suggest a new way to reach a proximate spin liquid phase in Kitaev- Heisenberg magnets. \n\n\\section*{Materials and methods}\n\\paragraph*{Sample growth and characterization} High-quality $\\alpha$-RuCl$_{3}$\\@\\xspace crystals were prepared by vacuum sublimation.\\cite{sahasrabudhe2019high} Different samples of the batch were characterized by SQUID magnetometry, showing a sharp phase transition at $T_{\\rm N}$\\,$\\approx$\\,$7$K. This bulk technique can only provide a first indication of sample quality for an optics study. Cleaving or polishing of RuCl$_{3}$\\@\\xspace samples introduces strain, which leads to stacking faults. For the optics study, we therefore refrained from any sample treatment, and used an as-grown RuCl$_{3}$\\@\\xspace sample with a shiny $\\sim$\\,$1.5$\\,$\\times$\\,$1.5$\\,mm$^2$ surface area. The temperature dependence shown in Fig.\\,\\ref{fig:tempdep}b shows a clear phase transition at $T_{\\rm N}$\\,$\\approx$\\,$7$K. \n\n\\paragraph*{Time-resolved magneto-optical experiment}\nThe $\\alpha$-RuCl$_{3}$\\@\\xspace sample is mounted in a bath cryostat. The time-resolved magneto-optical experiment was performed using $800$\\,nm pump pulses with a temporal with of $40$\\,fs, and probe pulses of $512$\\,nm with a temporal width of $250$\\,fs. The pump and probe beam were focused down to a radius of $r_{\\rm pump}$\\,$\\approx$\\,$39$\\,$\\mu$m and $r_{\\rm probe}$\\,$\\approx$\\,$25$\\,$\\mu$m, respectively. The repetition rate of the amplified laser system was set to $f$\\,$=$\\,$30$\\,kHz in order to ensure that the system can relax back to the ground state between consecutive pulses. The change in polarization rotation of the reflected probe pulse is measured via a standard polarization bridge scheme. \nThe optical conductivity reported in Ref.\\,\\citenum{sandilands2016} and the structural properties reported in Ref.\\,\\citenum{johnson2015} allows us to calculate the photoexcitation densities, as outlined in more detail in the Supplementary Material.\\cite{footnoteS1}\n\n\\section*{Acknowledgements}\nThe authors thank A.~Rosch (Cologne, DE) and Z.~Lenar\\v{c}i\\v{c} (Berkeley, USA) for fruitful discussions.\nThis project was partially financed by the Deutsche\nForschungsgemeinschaft (DFG) through Project No.\n277146847 - Collaborative Research Center 1238: Control\nand Dynamics of Quantum Materials (Subprojects No. B05\nand No. C04) and through project INST 216\/783-1 FUGG.\nS.D. acknowledges support by the European\nResearch Council (ERC) under the Horizon 2020 research and innovation program, Grant Agreement No. 647434 (DOQS).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLow energy effective actions are important in physical situations in\nwhich one is interested in phenomena at an energy scale which is\nsmall compared to the masses of some of the fundamental degrees of\nfreedom in the theory. Although these heavy degrees of freedom can only\noccur as virtual states at the energy scale of interest, they still\nproduce observable effects, which are summarized in a low energy\neffective action for the ``light\" degrees of freedom. The low energy\neffective action is determined by ``integrating out\" the heavy degrees\nof freedom.\n\n Recently there has been renewed interest in perturbative computations of\n effective actions in supersymmetric Yang-Mills theories\n\\cite{henningson,dewit,pickering,grisaru}. This interest was\nstimulated by the work of Seiberg and Witten \\cite{SeibergWitten}, who\ndeduced the full\nnonperturbative low energy effective action for an N=2 supersymmetric\nYang-Mills theory. The perturbative computations have focused on the\ncorrections to the low energy K\\\"ahler potential for the scalar\nsupermultiplets in the N=1 superfield formulation of N=2\nsupersymmetric Yang-Mills theory. The effective K\\\"ahler potential\nwas first considered in superfield form in the work of\nBuchbinder et al \\cite{buchbinder1,buchbinder2}.\n\nIn this paper, we introduce a new technique for the perturbative\ncalculation of\nlow energy effective actions, and then illustrate it in superspace by\ndetermining the low energy effective action obtained by\nintegrating out massive N=1 scalar multiplets in the presence of an N=1\nsupersymmetric\nYang-Mills background. This is a case not treated fully in the earlier\ncalculations quoted above. The approach is from the point of view of\nheat kernels and zeta functional regularization, as opposed to the\nsupergraph calculations in \\cite{dewit,pickering,grisaru}.\nBuchbinder et al \\cite{buchbinder1,buchbinder2} have also developed functional\ntechniques for computing low energy effective actions in superspace,\nbut\nour approach differs significantly from theirs. Also, although\nzeta function\nregularization is only really useful in the computation of one-loop effective\nactions, there are\nnonrenormalization theorems in N=2 supersymmetric Yang-Mills theories\n which ensure the absence of higher loop corrections\nto the effective action, so this is not a disadvantage if one is\nultimately interested in these theories.\n\nThe plan of the paper is as follows. We begin with a brief summary of an\napproach to the computation of nonsupersymmetric low-energy effective actions\nwhich relies on the similarity of the functional trace of the heat kernel\nto a\nGaussian integral. The technique is then applied in superspace to determine\nthe low energy effective action obtained by\nintegrating out massive N=1 scalar multiplets coupled to an N=1\nsupersymmetric\nYang-Mills background. The paper ends with a short discussion.\\\\\n\n\n\\section{ The Nonsupersymmetric Case}\nAn efficient way to generate the one-loop effective action is via zeta\nfunction\nregularization. We consider for definiteness a massive scalar field in the\npresence of a Yang-Mills background with Euclidean action $$S[A, \\phi] =\n\\frac12\\int d^4x \\, \\phi^{\\dagger} (-D_aD_a + m^2) \\phi,$$ where $D_a$ is the\ncovariant derivative in the representation $R$ of the gauge group to\nwhich the\nscalar fields belong, with $[D_a,D_b] = F_{ab}.$ The one loop effective\naction\nfor the Yang-Mills fields obtained by ``integrating out'' the scalar\nfields is\n$$\\Gamma[A] =- \\ln \\int [d\\phi] e^{-S[A,\\phi]} = \\ln \\det (-D_aD_a + m^2). $$\n Using zeta function regularization, this is just $ - \\zeta'(0), $ where the\nzeta function is defined by\n$$ \\zeta(s) = \\frac{1}{\\Gamma(s)} \\int_0^{\\infty} ds \\, t^{s-1} e^{-m^2\nt\/\\mu^2}\\, Tr {\\tilde K}(t\/\\mu^2). $$\nIn this expression, $Tr$ is a trace over gauge indices and ${\\tilde K}(t)$ is\nthe functional trace of the heat kernel for the operator $D_aD_a,$\n$$ {\\tilde K}(t) = \\int d^4x \\, \\lim_{x \\rightarrow x'} e^{tD_aD_a}\\,\n\\delta^{(4)}(x,x') \\equiv \\int d^4x K(t). $$\nSince $t$ in $K(t)$ has dimensions of inverse mass squared, the usual\nparameter\n$\\mu$ with dimensions of mass (representing the renormalization point) has\nbeen\nintroduced into the zeta function to make $t$ dimesionless.\n\nIf we introduce a plane wave basis for the delta function, $K(t)$ takes the\nform\n \\begin{eqnarray*}\nK(t) & =& \\lim_{x \\rightarrow x'} \\int \\frac{d^4k}{(2\\pi)^4} \\, e^{i\nk.(x-x')}\n\\, \\left( e^{-ik.(x-x')} \\, e^{t D_a D_a} \\, e^{ik.(x-x')} \\right) \\\\\n&=& \\int \\frac{d^4k}{(2\\pi)^4} \\, e^{t X_a X_a }\n\\end{eqnarray*}\nwhere $X_a = D_a + i k_a.$ It then follows that $K(t)$ satisfies the\ndifferential equation\n\\begin{equation}\n\\frac{dK(t)}{dt} = K_{aa}(t),\n\\label{deq}\n\\end{equation}\n where the tensors $K_{a_1 \\, \\cdots \\, a_n}(t)$ are defined by\n$$K_{a_1 \\, \\cdots \\, a_n}(t) = \\int \\frac{d^4k}{(2\\pi)^4} \\, X_{a_1} \\,\n\\cdots\n\\, X_{a_n} \\, e^{t X.X}. $$\nTo solve the differential equation, it is necessary to obtain an expression\nfor\n$K_{ab}(t)$ in terms of $K(t).$ The approach we take is to use the identity\n$$ 0 = \\int \\frac{d^4k}{(2\\pi)^4} \\, \\frac{\\partial}{\\partial k^{a_m}} \\,\n\\left(\nX_{a_1} \\, \\cdots \\, X_{a_{m-1}} \\, e^{tX.X} \\right) $$\nto study the properties of these tensors (the boundary term in the integral\nvanishes because of the $e^{-k^2}$ factor in the integrand). This is the same\nmethod that can be used to determine the moments $\\int\n\\frac{d^4k}{(2\\pi)^4} \\,\nk_{a_1} \\cdots k_{a_n} e^{-k^2}$ of an ordinary Gaussian in terms of the\nGaussian itself.\n\nIn particular, applying it in the case $m=2,$\n\\begin{eqnarray}\n 0 &= & i \\delta_{ab } K(t) + \\int \\frac{d^4k}{(2\\pi)^4} \\, X_{a} \\,\n\\frac{\\partial}{\\partial k^{b}} \\, e^{tX.X} \\nonumber \\\\\n &= & i \\delta_{ab} K(t) + 2 i t \\int \\frac{d^4k}{(2\\pi)^4} \\, X_{a} \\left(\n\\int_0^1 ds \\, e^{-st X.X} X_{b} \\, e^{stX.X}\\right) e^{tX.X}.\n\\label{Kuv}\n\\end{eqnarray}\nBecause $\\int_0^1 ds \\, e^{-st X.X} X_{b}\\, e^{stX.X} = X_b + \\,\n\\cdots, $\nthis identity yields an expression for $K_{ab}(t) $ in terms of $K(t)$ which\ncan then be used to solve the linear differential equation (\\ref{deq}).\n\nThe quantity\n\\begin{equation}\n\\int_0^1 ds \\, e^{-st X.X} \\, X_{b} \\, e^{stX.X} = \\sum_{n=0}^{\\infty}\n\\frac{t^n}{(n+1)!} \\, ad^{(n)}(X.X)(X_b)\n\\label{ints}\n\\end{equation}\ncannot be evaluated exactly and so must be approximated in some way. In the\ncase\nof a computation of the low energy effective action, one is interested in the\npiece of the effective action which contains no covariant derivatives of the\nfield strength of the Yang-Mills background. This is the covariant\ngeneralization of the low energy effective action for the $U(1)$ case, where a\nconstant field strength is equivalent to the long wavelength limit for the\nbackground electromagnetic field. A covariantly constant background\ncorresponds to setting $(D_{a} F_{bc}) = [X_{a}, [X_{b}, X_{c}]] $ to\nzero, so\nthat only the commutators $[X_a,X_b ] = F_{ab}$ need be retained in\nevaluating\n(\\ref{ints}). In this case, it is easy to show that $ ad^{(n)}(X.X)(X_{b}) =\n(-2)^n (F^n)_{b c} X_{c}, $ so that\n$$ \\int_0^1 ds \\, e^{-st X.X} X_{b} \\, e^{stX.X} = B_{bc}(t) \\, X_{c} $$\nwhere\n$$ B_{bc}(t) = \\left[ \\frac{e^{-2tF}-{\\bf 1}}{-2tF} \\right]_{bc}. $$\nNote that the power series expansion for the matrix $B_{bc}(t) $ begins with\n$\\delta_{bc}$ and only involves positive powers of $F_{ab}.$ As a result, the\ninverse matrix $B^{-1}_{bc}(t) = \\left[ \\frac{-2tF}{e^{-2tF}-{\\bf\n1}}\\right]_{bc}$ exists and does not require the inverse of $F_{ab}$ to be\ndetermined.\nInserting this result into (\\ref{Kuv}), and using the fact that $B_{bc}(t)$\ncommutes with $X_a$ to the order that we are working,\none finds\n$$ 0 = \\delta_{ab} \\, K(t) + 2 t B_{bc}(t) K_{ac}(t).$$\nIt follows that\n$$ K_{ab} = \\left[ \\frac{F}{e^{-2tF}-{\\bf 1}} \\right]_{ba}. $$\nThus (\\ref{deq}) becomes\n$$\n\\frac{d K(t)}{dt} = tr \\left[ \\frac{F}{e^{-2 t F}-{\\bf 1}}\\right] K(t),\n$$\nwhere the trace $tr$ is over spacetime indices and {\\em not} gauge indices;\nthe\nkernel is still a matrix with respect to its gauge indices. Noting that $ tr\n\\left[ \\frac{F}{e^{-2 t F}-{\\bf 1}}\\right]$ $ = tr \\left[\n\\frac{Fe^{2tF}}{{\\bf\n1} -e^{2tF}}\\right] = -\\frac12 tr \\left[C^{-1} ({\\bf 1}-e^{2tF})^{-1}\n\\frac{d}{dt} ({\\bf 1}-e^{2tF}) C\\right],$ with $C$ a matrix independent of\n$t$,\nand using the boundary condition that $K(t)$ reduces to the ordinary Gaussian\n$\\int \\frac{d^4k}{(2\\pi)^4} e^{-k^2}$ in the limit $F_{ab} \\rightarrow 0,$\none finds the standard result \\cite{schwinger,brown,reuter,schubert} for the\nfunctional trace of the heat kernel:\n\\begin{equation}\nK(t) = \\frac{1}{4 \\pi^2} \\det \\left[\\frac{ {\\bf 1}-e^{2tF}\n}{F}\\right]^{-\\frac12} = \\frac{1}{16 \\pi^2 t^2} \\, \\det \\left[ \\frac{t\nF}{\\sinh\ntF} \\right]^{\\frac12}.\n\\label{le}\n\\end{equation}\n\nThis technique is readily generalises to quantum fields of different spin,\nor to\nthe inclusion of a potential for the scalar fields. For later use, we note\nthat\nfor a Dirac spinor in a Yang-Mills background, the kernel for the Laplace-type\noperator given by the square of the Dirac operator is \\cite{schwinger,brown}\n\\begin{equation}\nK_{\\frac12}(t) = tr \\left(e^{- t \\Sigma_{ab}F_{ab}} \\right) \\, K(t),\n\\label{spin12}\n\\end{equation}\nwhere $\\Sigma_{ab} = \\frac{1}{4}[\\gamma_a,\\gamma_b],$ the trace $tr$ is over\nthe spinor indices on the gamma matrices and $K(t)$ is the spin zero kernel\n(\\ref{le}).\n\nThe above considerations were motivated to a certain extent by the work of\nAvramidi \\cite{avramidi}, who computed\n$$\\int \\frac{d^4k}{(2\\pi)^4} \\, \\sqrt{\\gamma} \\, e^{-t( \\gamma_{ab}k_ak_b -\n2\nk_a D_a)},$$\nwhere $\\gamma_{ab}$ is a metric in $k$ space, and $\\gamma = \\det \\gamma_{ab}.$\nThe result is of the form $f(t,\\gamma,F) \\, e^{t g_{ab}(t)D_aD_b},$ where\n$g_{ab}$ is a functional of $\\gamma_{ab}$ and $F_{ab}.$ By choosing\n$\\gamma_{ab}$ so that $g_{ab} = \\delta_{ab},$ it is possible to obtain an\nexpression for $e^{tD_aD_a}$ which can be used to compute the heat kernel and\nhence its functional trace. It is not immediately clear how this could be\napplied to superspace calculations.\n\n\nAnother common approach to computing effective actions is to determine the\nGreen's function for the quantum fields in the presence of the background;\nthis is then used to compute the functional trace of the heat kernel. In the\nabove approach, we compute the functional trace of the heat kernel directly\nwithout the need for this intermediate step. This is an advantage when it\ncomes\nto computing effective actions in supersymmetric theories, as there are many\ndifferent Green's functions. In \\cite{buchbinder1}, where the effective\nK\\\"ahler\npotential for the Wess-Zumino model is computed, Buchbinder et al\nexpressed all\nthe Green's functions in terms of a single one, thereby providing some\nsimplification. In the next section, we show that it is possible to extend the\ntechniques developed above to calculations of effective actions for\nsupersymmetric theories in superspace. To illustrate this, we compute the\nsupersymmetric analogue of the results (\\ref{le}) and (\\ref{spin12}).\n\n\n\n\n\n\\section{ Massive Scalar Multiplet in Gauge Superfield Background}\nHere, we will be concerned with the computation of the one-loop low energy\neffective action for a gauge supermultiplet which results from integrating\nout\nmassive scalar multiplets coupled to the gauge background.\n This is the superspace analogue of the nonsupersymmetric theory treated in\n\\S2. The superspace action for the scalar supermultiplet $\\Phi(x,\\theta)$\ntransforming in some (real) representation $R$ of the gauge group $G$ is\n$$S = \\int d^4x \\, d^2\\theta \\, d^2\\bar{\\theta} \\, \\, \\bar{\\Phi} \\, e^{-V}\n\\Phi\n+ \\int d^4x \\,d^2\\theta \\, \\frac{m}{2} \\Phi^2 + \\int d^4x \\,d^2\\bar{\\theta} \\,\n\\frac{m}{2} \\bar{ \\Phi}^2. $$\nGauge indices are suppressed, and the superspace conventions of of Wess and\nBagger \\cite{Wess} have been adopted, except that the metric has been Wick\nrotated to be Euclidean. Because the quantum superfields $\\Phi$ are subject to\nthe chirality constraint $\\bar{D}_{\\dot{\\alpha}} \\Phi = 0,$ the effective\naction\ncannot simply be formed as the logarithm of the superdeterminant of the\noperator\nappearing in the quadratic part of the action. Rather, it is first\nnecessary to\nwrite the action in terms of unconstrained superfields. To this purpose we\nintroduce complex scalar superfields $\\Psi$ and solve the constraint by\nexpressing $\\Phi = \\bar{D}^2 \\Psi;$ the superfields transform under the\naction\nof the gauge group in the same way as $\\Phi$. In terms of the unconstrained\nfields, the action can be expressed in the full superspace as\n$$ S = 8 \\int d^4x \\, d^2\\theta \\, d^2\\bar{\\theta} \\left( \\begin{array}{ll}\n\\Psi, & \\Psi^{\\dagger} \\end{array} \\right) \\left(\n\\begin{array}{ll} \\frac{1}{16} \\bar{D}^2e^V D^2 e^{-V} & -\\frac{m}{4}\n\\bar{D}^2e^V \\\\ -\\frac{m}{4} D^2e^{-V} & \\frac{1}{16} D^2e^{-V} \\bar{D}^2 e^V\n\\end{array} \\right) \\left( \\begin{array}{c} e^{V} \\Psi^{\\dagger} \\\\\ne^{-V}\n\\Psi \\end{array} \\right) .$$\nSo the effective action is $\\Gamma_{eff}[V] = - \\frac12 \\ln {\\rm sdet}\n\\Delta, $\nwhere $\\Delta $ is the superspace operator in the unconstrained action above\n\\cite{buchbinder1}. The appropriate zeta function is thus\n$ \\zeta(s) = \\frac{1}{\\Gamma(s)} \\int_0^{\\infty} dt \\, t^{s-1} \\int d^8Z\n\\, Tr\n K(t\/\\mu^2), $\nwhere $Tr$ denotes the trace over gauge indices, $\\mu$ is the renormalization\npoint and\n$$\nK(t) = tr \\lim_{Z \\rightarrow Z'} \\exp t \\left( \\begin{array}{ll}\n\\frac{1}{16} \\bar{D}^2e^V D^2e^{-V} & -\\frac{m}{4} \\bar{D}^2e^V \\\\\n-\\frac{m}{4}\nD^2e^{-V} & \\frac{1}{16} D^2e^{-V} \\bar{D}^2 e^V \\end{array} \\right) \\,\n\\delta^{(8)}(Z,Z'). $$\nHere, the trace $tr$ is over the $ 2\\times 2 $ matrices, and $\\int d^8Z$\nand\n$ \\delta^{(8)}(Z,Z') $ denote the integration measure and the delta\nfunction on\nthe full superspace.\nApplying the Baker-Campbell-Hausdorff formula, the terms involving the\nmass $m$\n can be placed in a separate exponential; performing the two dimensional trace\nthen projects out even powers of $m$, giving\n\\begin{eqnarray*}\n K(t) & = & \\sum_{n=0}^{\\infty} \\frac{(mt)^{2n}}{(2n)!}\\, \\lim_{Z\n\\rightarrow\nZ'}\\,\n\\biggl( \\left( \\frac{1}{16} \\bar{D}^2e^VD^2e^{-V} \\right)^n \\,\ne^{\\frac{t}{16} \\bar{D}^2e^VD^2e^{-V}}\\delta^{(8)}(Z,Z') \\\\\n& + &\n\\left(\\frac{1}{16} D^2e^{-V}\\bar{D}^2e^{V} \\right)^n \\, e^{\\frac{t}{16}\nD^2e^{-V}\\bar{D}^2e^{V}} \\delta^{(8)}(Z,Z') \\biggr) \\\\\n& = & \\sum_{n=0}^{\\infty} \\frac{(mt)^{2n}}{(2n)!}\\, \\frac{d^n}{dt^n} \\,\n\\lim_{Z \\rightarrow Z'} \\biggl( e^{\\frac{t}{16}\n\\bar{D}^2e^VD^2e^{-V}}\\delta^{(8)}(Z,Z') \\\\\n& + & e^{\\frac{t}{16} D^2e^{-V}\\bar{D}^2e^{V}} \\delta^{(8)}(Z,Z') \\biggr).\n\\end{eqnarray*}\n Using $\\int d^8Z = \\int d^4x \\int d^2 \\theta ( - \\frac14 ) \\bar{D}^2$ and\n$\\int d^8Z = \\int d^4x \\int d^2 \\bar{ \\theta} ( - \\frac14 ) D^2,$\nfactors of\n$( - \\frac14 ) \\bar{D}^2$ and $ (- \\frac14 ) D^2$ can be extracted from the\nexponentials to act on the full superspace delta function to convert the zeta\nfunction to the ``chiral'' form\n\\begin{eqnarray}\n\\zeta(s) &=& \\frac{1}{\\Gamma(s)} \\int_0^{\\infty} dt \\, t^{s-1} \\,\n\\sum_{n=0}^{\\infty} \\frac{1}{(2n)!} \\left( \\frac{mt}{\\mu^2}\\right)^{2n}\\,\n\\mu^{2n} \\frac{d^n}{dt^n}\\, Tr \\,\\biggl( \\int d^4x \\int d^2 \\theta \\,\nK_L(t\/\\mu^2) \\nonumber \\\\\n& + & \\int d^4x \\int d^2 \\bar{\\theta} \\, K_R(t\/\\mu^2) \\biggr) ,\n\\label{zetasum}\n\\end{eqnarray} where the chiral kernels are\n\\begin{eqnarray}\nK_L(t) &=& \\lim_{Z \\rightarrow Z'} e^{\\frac{t}{16} \\bar{D}^2e^VD^2e^{-V}}\n\\delta^{(4)}(x,x') \\delta^{(2)}(\\theta, \\theta'), \\nonumber \\\\\nK_R(t) &=& \\lim_{Z \\rightarrow Z'} e^{\\frac{t}{16} D^2e^{-V}\\bar{D}^2e^{V}}\n\\delta^{(4)}(x,x') \\delta^{(2)}(\\bar{\\theta}, \\bar{\\theta}').\n\\label{chiralkernels}\n\\end{eqnarray}\n\nThe mass dependence in (\\ref{zetasum}) involves derivatives of the chiral\nkernels. It is convenient to remove these derivatives by repeated\nintegration by\nparts. The boundary terms at $t= \\infty $ vanish as the kernels vanish in this\nlimit (this can be checked explicitly using the result (18) for the kernel);\nthe boundary term at $t=0$ involves a factor $t^{n+s+1}$ and also vanishes as\nwe are only interested in $\\zeta(s)$ for $s$ in a small neighbourhood of\n$s=0.$\nThe result can be expressed in the form\n\\begin{eqnarray*}\n\\zeta(s) &=& \\frac{1}{\\Gamma(s)} \\int_0^{\\infty} dt \\, t^{s-1} \\, \\left[\n\\sum_{n=0}^{\\infty} \\frac{1}{(2n)!} \\left(-\\frac{m^2t}{\\mu^2}\\right)^n\\,\n\\frac{\\Gamma (2n+s)}{\\Gamma (n+s) }\\right ]\\\\ & & Tr \\,\\biggl( \\int d^4x \\int\nd^2 \\theta \\, K_L(t\/\\mu^2)\n + \\int d^4x \\int d^2 \\bar{\\theta} \\, K_R(t\/\\mu^2) \\biggr) .\n\\end{eqnarray*}\nThe sum in square brackets is the generalized hypergeometric function\n$$ {}_2F_2\\left[ \\frac{s}{2} + \\frac12,\\frac{s}{2};\n\\frac12,s;-\\frac{m^2t}{\\mu^2} \\right]. $$\n It will be seen later that the chiral kernels $ \\int d^4x \\int d^2 \\theta \\,\nK_L(t) + \\int d^4x \\int d^2 \\bar{\\theta} \\, K_R(t)$ have a power series\nexpansion in $t$ of the form $\\sum_{n=0}^{\\infty} V_n t^n,$ where $V_n$\nrepresents an effective vertex for the interaction of $n+2$ particles. So,\nwith\na rescaling of $t$,\n$$ \\zeta(s) = \\sum_{n=0}^{\\infty} \\frac{V_n}{ m^2}\n\\left(\\frac{m^2}{\\mu^2}\\right)^{-s}\\, \\frac{1}{\\Gamma(s)} \\int_{0}^{\\infty} dt\n\\, t^{n+s-1} {}_2F_2\\left[ \\frac{s}{2}+\\frac12 ,\\frac{s}{2}; \\frac12,s ;-t\n\\right]. $$\nAlthough we have not been able to do the integral (which is the Mellin\ntransform\nof a generalized hypergeometric function) explicitly, it is possible to\ndeduce\nthe form of the zeta function for $s $ near zero using the fact that $\n{}_2F_2\\left[ \\frac{s}{2}+\\frac12 ,\\frac{s}{2}; \\frac12,s;-t \\right]\n\\rightarrow\n\\frac12 e^{-t}$ as $s \\rightarrow 0.$ Thus for\n$ n\\neq 0, $ the integral is regular in the limit $s \\rightarrow 0,$ giving\n$\\frac12 \\Gamma(n). $ This means that in calculating $\\zeta'(0), $ the\nderivative must\nact on $\\frac{1}{\\Gamma (s)} = s + O(s^2)$ to eliminate the zero at $s=0.$ For\n$n=0,$ the integral behaves like $\\frac12 \\Gamma(s) $ for small $s,$ and so\nthe\nderivative acts on $ (m^2\/\\mu^2)^{-s}. $ The result is\n\\begin{equation}\\zeta'(0) = \\sum_{n=1}^{\\infty}\\frac12\\, \\Gamma(n)\n\\frac{V_n}{m^{2n}} -\n\\frac12 \\,V_0 \\ln \\frac{ m^2}{\\mu^2}. \\label{seriesexp}\n\\end{equation}\nAlternatively, if Schwinger proper time regularization is used, it is only\nnecessary to know the hypergeometric at $s=0;$ in this case there is a\ndivergence in the $n=0$ contribution to the effective action which must be\nremoved by hand, but the $n\\geq 1$ contributions are as above.\n\nIt therefore remains to evaluate the chiral kernels; we do this for\n$K_L(t),$ as\nthe calculation for $K_R(t)$ is identical except that left chiral quantities\nare replaced by right chiral quantities. For the rest of the paper, $K_L(t)$\nwill simply be denoted $K(t).$ Acting on left chiral superfields, the\noperator $\n\\frac{1}{16} \\bar{D}^2e^VD^2e^{-V}$ is equivalent to the Laplace-type\noperator\n${\\cal D}_a {\\cal D}_a + W^{\\alpha} {\\cal D}_{\\alpha} + \\frac12 ({\\cal\nD}^{\\alpha}W_{\\alpha}),$ where the ${\\cal D}$ denote gauge covariant\nsuperspace\nderivatives in the left chiral basis: ${\\cal D}_{\\alpha} = e^V D_{\\alpha}\ne^{-V},$ $\\bar{{\\cal D}}_{\\dot{\\alpha}} = \\bar{ D}_{\\dot{\\alpha}},$ $ \\{ {\\cal\nD}_{\\alpha}, \\bar{{\\cal D}}_{\\dot{\\alpha}} \\} = -2 i (\\sigma_a)_{\\alpha\n\\dot{\\alpha} } \\,{\\cal D}_a, $ and $W_{\\alpha} = -\\frac14 [\\bar{{\\cal\nD}}_{\\dot{\\alpha}}, \\{ \\bar{{\\cal D}}^{\\dot{\\alpha}}, {\\cal D}_{\\alpha} \\} ].$\nThus the left chiral kernel contains a Laplace-type operator, as required\nfor a\nwell-defined heat kernel:\n$$ K(t) = \\lim_{Z \\rightarrow Z'} e^{t({\\cal D}_a {\\cal D}_a + W^{\\alpha}\n{\\cal\nD}_{\\alpha} + \\frac12 ({\\cal D}^{\\alpha}W_{\\alpha}))} \\, \\delta^{(4)}(x,x')\n\\delta^{(2)}(\\theta, \\theta'). $$\nThe left chiral delta function has the representation\n$$\\frac14 \\delta^{(4)}(x,x') \\delta^{(2)}(\\theta, \\theta') = \\int\n\\frac{d^4k}{(2\\pi)^4}\\, e^{ik_a (x_a - x_a' - i \\theta \\sigma_a\n\\bar{\\theta}' +\ni \\theta' \\sigma_a \\bar{\\theta})} \\int d^2 \\epsilon\\, e^{i\n\\epsilon^{\\alpha}(\\theta - \\theta')_{\\alpha}}, $$\nwhere $\\epsilon_{\\alpha}$ is a Grassmann parameter which is the supersymmetric\npartner of $k_a.$ The delta function in $x $ is a function of the\nsupertranslation invariant interval $x_a - x_a' - i \\theta \\sigma_a\n\\bar{\\theta}' + i \\theta' \\sigma_a \\bar{\\theta}$ on superspace\\footnote{ Note\nthat $ \\bar{{\\cal D}}_{\\dot{\\alpha }} (x_a - x_a' - i \\theta \\sigma_a\n\\bar{\\theta}' + i \\theta' \\sigma_a \\bar{\\theta}) = -i (\\theta -\n\\theta')^{\\alpha} (\\sigma_a)_{\\alpha \\dot{\\alpha}},$ so that although the\nsuperspace invariant interval is not itself left chiral, the fact that\n$(\\theta\n- \\theta')^3 = 0$ means that the full delta function on left chiral superspace\n{\\em is} annihilated by $ \\bar{{\\cal D}}_{\\dot{\\alpha}},$ as required.}.\nMoving\nthe exponential to the left through the differential operators, the\ncoincidence\nlimit of the left chiral kernel has the expression\n\\begin{equation}\n K(t) =4 \\int \\frac{d^4k}{(2\\pi)^4} \\int d^2 \\epsilon\\,\\, e^{t(X_aX_a +\nW^{\\alpha}X_{\\alpha} + \\frac12 ({\\cal D}^{\\alpha}W_{\\alpha}))},\n\\label{chiralK}\n\\end{equation}\nwhere\n\\begin{equation}\n X_a = {\\cal D}_a + i k_a , \\, \\, \\, \\, \\, X_{\\alpha} = {\\cal D}_{\\alpha} + i\n\\epsilon_{\\alpha} .\n\\label{X}\n\\end{equation}\nNote that there is also a shift $ - k_a (\\sigma_a)_{\\alpha \\dot{\\alpha}}\n(\\bar{\\theta } - \\bar{\\theta}')^{\\dot{\\alpha}}$ in ${\\cal D}_{\\alpha}$;\nhowever,\nthis vanishes in the coincidence limit as there are no $\\bar{{\\cal\nD}}_{\\dot{\\alpha}}$ operators present to annihilate the $\\bar{\\theta} -\n\\bar{\\theta}'.$\nAlso note that the integrand in (\\ref{chiralK}) contains an explicit factor\n$e^{-k^2}$ necessary for the convergence of the $k$ integral.\nTo compute the kernel, we will solve the differential equation\n\\begin{equation}\n \\frac{dK(t)}{dt} = K_{aa}(t) + W^{\\alpha} K_{\\alpha}(t) + \\frac12({\\cal\nD}^{\\alpha}W_{\\alpha}) K(t) ,\n\\label{dess}\n\\end{equation}\nwhere\n$$K_{A_1 A_2 \\, \\cdots A_n}(t) = 4 \\int \\frac{d^4k}{(2\\pi)^4} \\int d^2\n\\epsilon\\,\\,X_{A_1} X_{A_2} \\, \\cdots \\, X_{A_n} \\, e^{t\\Delta}, $$\nand $X_{A} $ can represent either a bosonic operator $X_a $ or a fermionic\noperator $X_{\\alpha}; $ the abbreviation $\\Delta = X_aX_a +\nW^{\\alpha}X_{\\alpha} + \\frac12 ({\\cal D}^{\\alpha}W_{\\alpha}) $ has also been\nintroduced. The aim is to use identities similar to those used in \\S2 to\nexpress\n$ K_{aa}(t) $ and $ W^{\\alpha} K_{\\alpha}(t) $ in terms of $K(t).$ In\nsuperspace, there are two kinds of identities involving the vanishing the\nintegral of a total derivative which can be employed, one involving\nderivatives\nwith respect to the bosonic variables $k_a$ and the other involving\nderivatives\nwith respect to the fermionic variables $\\epsilon_{\\alpha}$:\n\\begin{eqnarray}\n 0 &=& \\int \\frac{d^4k}{(2\\pi)^4} \\int d^2\n\\epsilon\\,\\,\\frac{\\partial}{\\partial\nk^b} \\left(X_{A_1} X_{A_2} \\, \\cdots \\, X_{A_n} \\, e^{t\\Delta}\n\\right),\\nonumber\n\\\\\n 0 &=& \\int \\frac{d^4k}{(2\\pi)^4} \\int d^2\n\\epsilon\\,\\,\\frac{\\partial}{\\partial\n\\epsilon^{\\beta}} \\left(X_{A_1} X_{A_2} \\, \\cdots \\, X_{A_n} \\, e^{t\\Delta}\n\\right).\n\\label{idents}\n\\end{eqnarray}\nThe expression for $K_{ab}(t)$ will arise from the use of the first identity\nwith $X_{A_1} X_{A_2} \\, \\cdots \\, X_{A_n}$ replaced by $X_a,$ as to the\naction of the derivative on the exponential pulls down an operator which to\nleading order is $2 i t X_a.$ Similarly, an expression for $W_{\\beta}\nK_{\\alpha}(t) $ can be obtained from the second identity with $X_{A_1}\nX_{A_2}\n\\, \\cdots \\, X_{A_n}$ replaced by $X_{\\alpha},$ as the action of the\nderivative\non the exponential pulls down a factor which to leading order is $-i t\nW_{\\beta}.$\n\nIn both cases, the factors $2 i t X_a $ and $-i t W_{\\beta}$ will be\naccompanied by additional terms containing commutators of $\\Delta$ with these\nfactors. As was the case in \\S2, these will form an infinite series which\ncannot\nbe summed in general. It is necessary to truncate to a given order in\ncommutators and anticommutators of $X_{a}$ and $X_{\\alpha},$ corresponding\nto a\nparticular order in the (super)derivative expansion of the effective\naction. To\nlowest order, the effective action contains the superfields $W_{\\alpha},$\nbut\nno derivatives of them. Since $W_{\\alpha}$ can be expressed as a double\n(anti)commutator of spinor derivatives ${\\cal D}_{\\alpha}$ and $\\bar{{\\cal\nD}}_{\\dot{\\alpha}},$ this corresponds to truncation to at most two\n(anti)commutators of spinor derivatives (with ${\\cal D}_{a}$ counting as a\nfirst order anticommutator via $ \\{ {\\cal D}_{\\alpha}, \\bar{{\\cal\nD}}_{\\dot{\\alpha}} \\} = -2 i (\\sigma_a)_{\\alpha \\dot{\\alpha} } \\,{\\cal\nD}_a $).\nThe left chiral kernel is trivial to compute in this approximation, because\n$[X_a,X_b],$ $[X_a, X_{\\alpha}], $\n $[X_a, W_{\\alpha}] $ and $\\{X_{\\alpha}, W_{\\beta}\\} $ all involve at least\nthree (anti)commutators of spinor derivatives and therefore must be set to\nzero. The only potential anticommutator at this order is\n$\\{X_{\\alpha},X_{\\beta}\\},$ but this vanishes due to the torsion and curvature\nconstraints imposed in supersymmetric Yang-Mills theory \\cite{Wess}. The\nkernel\nthus reduces in this lowest order approximation to\n\\begin{equation}\nK(t) = 4 \\int \\frac{d^4k}{(2\\pi)^4} \\int d^2 \\epsilon\\,\\, e^{-tk^2}\ne^{itW^{\\alpha} \\epsilon_{\\alpha}} = \\frac{1}{16 \\pi^2} W^{\\alpha}W_{\\alpha}.\n\\label{K0}\n\\end{equation}\nNote that this does not even reproduce the the nonsupersymmetric low energy\neffective actions considered in \\S2; it contains only the leading term,\nquadratic in the Yang-Mills field strength. To reproduce the supersymmetric\nanalogue of these results, it is necessary to go to the next order in the\nsuperspace derivative expansion, namely to consider up to three\n(anti)commutators of spinor derivatives. This is the truncation which will be\nmade here.\n\nUsing the curvature and torsion constraints for supersymmetric Yang-Mills\ntheory\n\\cite{Wess}, and letting $\\bar{M}_{ab} = (\\bar{{\\cal D}} \\bar{\\sigma}_{ab}\n\\bar{W}), $ $M_{ab} = ({\\cal D}\\sigma_{ab}W), $ and $N_{\\alpha \\beta} = ({\\cal\nD}_{\\alpha}W_{\\beta}),$ the nonvanishing commutators in this truncation are:\n$$ [X_a,X_b] = -\\frac12(\\bar{M}_{ab} - M_{ab}), \\, \\, [X_a, X_{\\alpha}] = i\n(\\sigma_a)_{\\alpha \\dot{\\alpha}} \\bar{W}^{\\dot{\\alpha}},\\, \\, \\{X_{\\alpha},\nW_{\\beta}\\} = N_{\\alpha \\beta}.$$ In particular, note that\n$[X_a, W_{\\alpha}]$ must be set to zero at this order.\n\nWe first consider the calculation of $W^{\\alpha}K_{\\alpha}(t)$ in terms of\n$K(t).$ The relevant identity is\n\\begin{eqnarray*}\n 0 & =& 4 \\int \\frac{d^4k}{(2\\pi)^4} \\int d^2\n\\epsilon\\,\\,\\frac{\\partial}{\\partial \\epsilon_{\\beta}} \\left(X_{\\alpha} \\,\ne^{t\\Delta} \\right) \\\\\n&=& i \\delta_{\\alpha}{}^{\\beta} K(t) - 4 \\int \\frac{d^4k}{(2\\pi)^4} \\int d^2\n\\epsilon\\,\\, X_{\\alpha} \\,\\frac{\\partial}{\\partial \\epsilon_{\\beta}}\ne^{t\\Delta}.\n\\end{eqnarray*}\nThe derivative of the exponential is computed using\n$$\\frac{\\partial}{\\partial \\epsilon_{\\beta}} e^{t\\Delta} =\\left( \\int_0^1\nds \\,\ne^{st\\Delta}\\, (-i t W^{\\beta}) \\, e^{-st \\Delta}\\right) e^{t \\Delta} = -i t\n\\sum_{n=0}^{\\infty} \\frac{t^n}{(n+1)!} ad^{(n)}(\\Delta)(W^{\\beta})\ne^{t\\Delta}.$$\nThe commutators in the series are easily evaluated to the required order, and\nwe obtain\n\\begin{equation}\n \\frac{\\partial}{\\partial \\epsilon_{\\beta}} e^{t\\Delta} = -i t\n\\sum_{n=0}^{\\infty} \\frac{t^n}{(n+1)!} W^{\\gamma} (N^n)_{\\gamma }{}^{\n\\beta} = -\ni W^{\\gamma}\\left( \\frac{e^{tN} - {\\bf 1}}{N}\\right)_{\\gamma}{}^{\\beta}.\n\\label{epsilonderiv}\n\\end{equation} Note that the power series expansion begins at order $N^0$ and\ndoes not involve any negative powers of the matrix $N;$ thus the inverse\nmatrix\n$ \\left( \\frac{N}{e^{tN} - 1} \\right)_{\\gamma}{}^{\\beta} $ exists and does not\nrequire the inversion of $N.$ Substituting this result and being careful to\ninclude the anticommutator which arises from moving $W^{\\gamma}$ through\n$X_{\\alpha},$\nwe obtain\n$$ 0 = \\delta_{\\alpha}{}^{\\beta} K(t) + N_{\\alpha}{}^{\\gamma}\\left(\n\\frac{e^{tN}\n- {\\bf 1}}{N}\\right)_{\\gamma}{}^{\\beta} K(t) - W^{\\gamma}\\left(\n\\frac{e^{tN} -\n{\\bf 1}}{N}\\right)_{\\gamma}{}^{\\beta} K_{\\alpha}(t). $$\nMultiplying by $ \\left( \\frac{N}{e^{tN} - {\\bf 1}} \\right)_{\\beta}{}^{\\rho} $\nand contracting indices appropriately yields a result of the desired form:\n\\begin{equation}\nW^{\\alpha}K_{\\alpha}(t) = tr \\left( \\frac{N}{e^{tN} - \\bf{1}} \\right)\\,\nK(t) +\ntr(N) \\, K(t).\n\\label{WK}\n\\end{equation}\nNote that the trace here is over the spinor indices, $tr(N) =\nN_{\\alpha}{}^{\\alpha};$ there is {\\em not} a trace over gauge indices, as the\nexpression is still a matrix with respect to gauge indices.\n\n\nThe first identity in (\\ref{idents}) is used to compute $K_{aa}(t)$ in the\nsame\nmanner as in the bosonic case in \\S2.\nTo the order in which we are working, one finds\n$$ \\frac{\\partial}{\\partial k^b} e^{t \\Delta} =\n2 i t B_{bc}(t) X_c + 2 i t A_b(t),$$\nwhere\n\\begin{eqnarray*}\n B_{bc}(t) & = & \\left( \\frac{e^{-t(M-\\bar{M})} - {\\bf 1}}{-t(M-\\bar{M})}\n\\right)_{bc}\\\\\nA_b(t) & = & \\frac{i}{t} \\biggl[\\frac{(e^{-t(M-\\bar{M})} - {\\bf 1})}{(M-\n\\bar{M}) (\\bar{M} - \\frac12 tr(N) {\\bf 1}) }\\\\\n& - & \\frac{(e^{-t(M-\\frac12 tr(N) {\\bf 1})}-{\\bf 1})}{ (M - \\frac12 tr(N){\n\\bf 1}) (\\bar{M} - \\frac12 tr(N) {\\bf 1})} \\biggr]_{bc}\\, (W^{\\alpha}\\sigma_{c\n\\alpha \\dot{\\alpha}} \\bar{W}^{\\dot{\\alpha}}).\n\\end{eqnarray*}\nThus the first identity in (\\ref{idents}) with $X_{A_1} X_{A_2} \\, \\cdots \\,\nX_{A_n}$ replaced by $X_{a}$ yields\n$$ 0 = i \\delta_{ab} K(t)+ 2 i t B_{bc}(t) K_{ac}(t) + 2 i t A_b(t) K_{a}(t).$$\nThe matrix $B_{bc}(t)$ has a power series expansion in positive powers of $M$\nand $\\bar{M}$ which begins at order ${\\bf1},$ and so it is invertible, yielding\n$$ K_{ab}(t) = -\\frac{1}{2t} (B^{-1})_{ba}(t) K(t) - (B^{-1})_{bc}(t) A_c(t)\nK_a(t).$$\nThe right hand side involves $K_{a}(t); $ this is evaluated by using the\nfirst\nidentity in identity (\\ref{idents}) with $ X_{A_1} X_{A_2} \\, \\cdots \\,\nX_{A_n}\n$ replaced by $1.$ The result is\n\\begin{equation}\n K_a(t) = - (B^{-1})_{ab}(t) A_b(t) K(t).\n\\label{Ka}\n\\end{equation}\n\nAt this point an important simplification can be achieved by noting that the\nkernel in the order that we currently working must reduce to the ``zero'th\norder'' result $K(t) = \\frac{1}{16\\pi^2}\\, W^2$ as $M,\\bar{M}$ and $ N\n\\rightarrow 0$ i.e. when the higher order commutators we have allowed at this\norder vanish. So the kernel must be of the form\n$K(t) = F[M,\\bar{M},N] \\, \\frac{1}{16\\pi^2} W^2$ with $ F[M,\\bar{M},N]\n\\rightarrow 1$ as $M,\\bar{M}, N \\rightarrow 0.$ On the other hand, to the\norder\nthat we are working, $\\{W_{\\alpha}, W_{\\beta}\\}$ can be taken to be zero,\nsince\nit involves a five commutators of spinor derivatives\\footnote{In a nonabelian\ngauge theory, it is not true in general that $\\{W_{\\alpha}, W_{\\beta}\\}= 0;$\nrather, $\\{W_{\\alpha}, W_{\\beta}\\} = W_{\\alpha}^a W_{\\beta}^b f_{ab}{}^c\nT_c,$\nwhere $T_a$ are generators of the gauge group satisfying $[T_a,T_b] =\nf_{ab}{}^cT_c.$ Also note that using the general result\n$W_{\\alpha} W_{\\beta} = \\frac12 \\epsilon_{\\alpha \\beta} W^2 + \\frac12 \\{\nW_{\\alpha}, W_{\\beta} \\},$\nit follows that to the order we are working, $W_{\\alpha} W_{\\beta} = \\frac12\n\\epsilon_{\\alpha \\beta} W^2.$ }. As a result, expressions involving more than\ntwo $W's$ vanish at the order we are working because one index must be\nrepeated\nand $(W_{\\alpha})^2 = 0.$ Therefore, using (\\ref{Ka}), $K_a(t)$ vanishes\nbecause $A_b(t)$ involves one factor of $W$ and $K(t)$ involves two. Thus the\nexpression for $K_{ab}(t)$ becomes simply:\n\\begin{equation}\n K_{ab}(t) = -\\frac{1}{2t} (B^{-1})_{ba}(t) K(t). \\label{Kabss}\n\\end{equation}\n\nSubstituting (\\ref{WK}) and (\\ref{Kabss}) into (\\ref{dess}), the differential\nequation for the left chiral kernel is\n$$ \\frac{dK(t)}{dt} =\\frac12 tr\\left( \\frac{(M-\n\\bar{M})}{e^{-t(M-\\bar{M})}-{\\bf 1}} \\right)\\, K(t) + tr\\left(\n\\frac{N}{e^{tN}-{\\bf 1}}\\right) \\, K(t) + \\frac12 tr(N) \\, K(t).$$\nNoting the similarity of the first and second terms on the right-hand side of\nthe equation with the differential equation in \\S2, the solution is easily\nseen\nto be\n$$K(t) = c_1 \\,e^{\\frac{t}{2} tr(N)} \\, \\det \\left(\\frac{{\\bf 1} - e^{-t\nN}}{A_1} \\right) \\, \\det \\left(\\frac{{\\bf 1} - e^{t(M-\n\\bar{M})}}{A_2}\\right)^{-\\frac12},$$\nwhere $c_1, A_1$ and $A_2$ are constants (independent of $t$). The latter are\ndetermined by the requirement that in the limit $M, \\bar{M}, N \\rightarrow 0,$\nthe kernel must be of the form (\\ref{K0}), from which it follows that\n\\begin{equation} K(t) = \\frac{W^2}{16 \\pi^2} \\,\\, e^{\\frac{t}{2} tr(N)} \\,\n\\det\n\\left(\\frac{{\\bf 1} - e^{-t N}}{N} \\right) \\, \\det \\left(\\frac{{\\bf 1} -\ne^{t(M-\n\\bar{M})}}{(M- \\bar{M}) }\\right)^{-\\frac12}.\n\\label{kernelresult}\n\\end{equation}\nThe power series expansion of the chiral kernel begins at order $t^0,$ and\nso is\nof the form $\\sum_{n=0}^{\\infty} V_n t^n.$ This yields the\n vertices in the effective action for the Yang Mills superfield background via\n(\\ref{seriesexp}). As mentioned in the introduction, this case is\nnot considered in the recent computations of effective actions for\nsupersymmetric Yang-Mills theories by graphical techniques\n\\cite{dewit,grisaru}, or in earlier results of\nBuchbinder et al \\cite{buchbinder1,buchbinder2} using functional\nmethods. Pickering and West \\cite{pickering} have computed the piece\nof the effective action corresponding to the lowest order\napproximation (\\ref{K0}) to the heat kernel, but have not computed the\ncorrections involving $ \\cal{D}_{\\alpha}W_{\\beta}.$\n\n\n\nWe can perform two checks on the result (\\ref{kernelresult}). The first is\nto go back to the\nexpression (\\ref{chiralK}) and perform the $\\epsilon$ integral explicitly.\nThis\nis done using $ 4\\int d^2 \\epsilon = \\frac{\\partial}{\\partial\n\\epsilon_{\\alpha}} \\frac{\\partial}{\\partial \\epsilon^{\\alpha}}|_{\\epsilon =\n0}$\nand (\\ref{epsilonderiv}). One finds\n\\begin{eqnarray*}\nK(t) &=& \\frac12 \\, W^2\\, tr \\left[ \\left( \\frac{e^{tN}-{\\bf 1}}{N}\n\\right) \\,\n\\left( \\frac{e^{-t(N - tr(N) {\\bf 1})}-{\\bf 1}}{N- tr(N) {\\bf 1}}\\right)\n\\right]\n\\\\\n& & \\int \\frac{d^4k}{(2\\pi)^4} \\, \\exp \\left\\{ t(X_a X_a + W^{\\alpha}{\\cal\nD}_{\\alpha} + \\frac12 (D^{\\alpha}W_{\\alpha}))\\right\\}.\n\\end{eqnarray*}\nNote that since $\\epsilon$ has been set to zero, the integral involves the\noperator $W^{\\alpha}{\\cal D}_{\\alpha} $ rather than $W^{\\alpha} X_{\\alpha}.$\nThe remaining integral can be evaluated by the trick of differentiating with\nrespect to $t$ and using total $k$ derivative identities. The result is\n\\begin{eqnarray*}\nK(t) & = & -\\frac{1}{32\\pi^2} \\, W^2 \\, tr \\left[ \\left( \\frac{e^{tN}-{\\bf\n1}}{N} \\right) \\, \\left( \\frac{e^{-t(N - tr(N) {\\bf 1})}-{\\bf 1}}{N- tr(N)\n{\\bf\n1}}\\right) \\right]\\, e^{-\\frac{t}{2} tr N}\\\\\n& & \\det \\left(\\frac{{\\bf 1}-e^{t(M-\\bar{M})}}{(M-\\bar{M})}\n\\right)^{-\\frac12}.\n\\end{eqnarray*}\nAlthough this seems to differ from (\\ref{kernelresult}), in fact the two\nexpressions can be shown to be equivalent. This relies on the fact that $N$\n is a $2\\times 2$ matrix, so that $\\det N = \\frac12 (trN)^2 - \\frac12\ntr(N^2).\n$ Also, $tr \\left(e^{-tN}\\right) $ $= e^{-\\frac{t}{2} tr(N)} \\, tr \\,\n\\left[e^{-t(N-\\frac12 tr(N) {\\bf 1})}\\right],$ and since $N-\\frac12 tr(N) {\\bf\n1}$ is a traceless $2\\times 2$ matrix, only traces of even powers are\nnonvanishing, with $tr[(N-\\frac12 tr(N) {\\bf 1})^{2n}] = 2 [\\frac12\ntr(N-\\frac12\ntr(N) {\\bf 1})^2]^n.$\n\nThe second check is to compare the result (\\ref{kernelresult}) with component\nresults. In the case where the fermionic components of the supersymmetric\nYang-Mills background are set to zero, we expect that the supersymmetric\nkernel\nshould reduce to the difference of the bosonic kernels in \\S2 for spin zero\nand\nspin half quantum fields in a Yang-Mills background. To see that this is so,\nconsider a supersymmetric theory in the presence of a supersymmetric\nbackground. The one loop effective action for the background fields is\nminus the\n logarithm of the superdeterminant of the operator appearing in the part\nof the\naction quadratic in the quantum fields. If the fermionic components of the\nbackground fields are set to zero, then this superdeterminant factorises\ninto a\nratio of ordinary determinants, so\n$$ \\Gamma_{eff} = \\ln \\det \\Delta_B - \\ln \\det \\Delta_{F}, $$\nwhere $ \\Delta_B $ and $ \\Delta_{F}$ denote the operators in the pieces\nof the\naction quadratic in the bosonic and fermionic quantum fields respectively.\nThis\nis equivalent to $-\\zeta^{\\prime}_{B}(0) + \\zeta^{\\prime}_{F}(0),$ where\n$\\zeta_{B}(s) $ and $ \\zeta_{F}(s) $are the usual zeta functions associated\nwith\nthe operators $ \\Delta_B $ and $ \\Delta_{F}$ respectively. The\ndifference $\n\\zeta_{B}(s) - \\zeta_{F}(s)$ of the zeta functions is of the form\n$$\\frac{1}{\\Gamma(s)} \\int_0^{\\infty} ds \\, t^{s-1} \\int d^4x \\, K(t\/\\mu^2), $$\nwhere\n$$ K(t) = \\lim_{x \\rightarrow x'} \\, ( e^{t\\Delta_B} - e^{t \\Delta_F}) \\,\n\\delta^{(4)}(x,x') \\equiv K_B(t) - K_F(t). $$\nAs the latter is difference of functional traces, it is a functional {\\em\nsupertrace}.\n\nIn the case at hand, if the fermionic components of the background Yang-Mills\nsuperfields are set to zero, then, with the convention $[D_a,D_b] = F_{ab},$\n$ N_{\\alpha}{}^{\\beta} \\rightarrow F_{ab}\n(\\sigma_{ab})_{\\alpha}{}^{\\beta},\n$ $ Tr(N) \\rightarrow 0, $ $ (M - \\bar{M})_{ab} \\rightarrow -2 F_{ab}, $\nand $ W_{\\alpha} \\rightarrow - F_{ab} (\\sigma_{ab})_{\\alpha}{}^{\\beta}\n\\theta_{\\beta}. $\nUsing the antisymmetry of $F_{ab},$ the supersymmetric kernel\n(\\ref{kernelresult}) becomes\n$$K(t) = - \\theta^2 \\det \\left({\\bf 1}-e^{-tF}\\right) \\frac{1}{4\\pi^2} \\det\n\\left(\\frac{{\\bf 1}- e^{2tF}}{F}\\right)^{-\\frac12}$$\nwhere the first determinant is over $2\\times 2$ undotted spinor indices with\n$F \\equiv F_{ab} (\\sigma_{ab})_{\\alpha}{}^{\\beta}$, and the second determinant\nis over vector indices with $F \\equiv F_{ab}$. On the other hand, using\n(\\ref{le}) and (\\ref{spin12}), the difference of the kernel for a scalar and\nfor a left-handed spinor\\footnote{The normalization of the left handed spinor\nkernel is changed by a factor of $\\frac12$ relative to that of the scalar\nkernel because the kernel (\\ref{spin12}) is computed for the {\\em square}\nof the\nDirac operator, whereas the effective action contains the determinant of the\nDirac operator and not its square.}\n$$ \\frac12 tr \\left( {\\bf 1}- e^{-tF}\\right) \\frac{1}{4\\pi^2} \\det\n\\left(\\frac{{\\bf 1}- e^{2tF}}{F}\\right)^{-\\frac12},$$\nwhere again the trace is over $2\\times 2$ spinor indices and the\ndeterminant is\nover vector indices. Using the fact that $ F_{ab}\n(\\sigma_{ab})_{\\alpha}{}^{\\beta}$ is a traceless $2\\times 2$ matrix, it is\nrelatively easy to show the equivalence of $- \\det \\left({\\bf\n1}-e^{-tF}\\right)$ and $tr\\left( {\\bf 1}- e^{-tF}\\right),$ thus ensuring\nthat\nthe supersymmetric kernel (\\ref{kernelresult}) for a purely bosonic\nbackground\ndoes indeed reduce to a difference of kernels for bosonic and fermionic\nquantum\nfields.\n\n\n\n\n\n\n\n\n\n\n\n\\section{ Discussion}\nIn this paper, we have illustrated a new approach to computing effective\nactions\nin supersymmetric theories by applying it to the determination of\n the low energy effective action obtained by\nintegrating out massive scalar supermultiplets in the presence of a\nsupersymmetric Yang-Mills background. The approach relies on computing the\nfunctional trace of an appropriate heat kernel; however, it is not\nnecessary to\ncompute Green's functions for the quantum fields in the presence of the\nbackground, as advocated by Buchbinder et al \\cite{buchbinder1,buchbinder2}.\nInstead, we make use of the similarity of the functional trace of the heat\nkernel expressed in momentum space to a Gaussian integral. It is expected\nthat\nthe method will also be applicable in the case of scalar superfield\nbackgrounds.\n\nThe mass dependence of the effective action involved the Mellin transform of a\ngeneralized hypergeometric function. Although we were unable to evaluate the\nMellin transform explicitly, the fact that the hypergeometric function reduced\nto an exponential in the limit $s \\rightarrow 0$ was enough to determine the\nform of the mass dependence of the effective action. We note in passing\nthat in\nthe calculation of Buchbinder et al \\cite{buchbinder1} of the low energy\neffective K\\\"ahler potential for the N=1 Wess-Zumino model, the kernel also\ninvolves a generalized hypergeometric function.\n The functional supertrace of the appropriate heat kernel is \\cite{buchbinder1}\n\\begin{equation}\nK(t) = \\frac{\\eta \\bar{\\eta}}{16 \\pi^2} \\sum_{n=1}^{\\infty}\n\\frac{n!}{(2n)!} (-t\n\\eta \\bar{\\eta})^{n-1}.\n\\label{hyp}\n\\end{equation}\nHere, $\\eta$ is the left chiral superfield $ \\frac{\\partial^2 W}{\\partial \\Phi\n\\partial \\Phi},$ where $W(\\Phi)$ is the superpotential for the chiral scalar\nsuperfields $\\Phi$ in the Wess-Zumino model.\nThe sum on the right hand side is actually a generalized hypergeometric\nfunction,\n$$K(t) = \\frac{\\eta \\bar{\\eta}}{32\\pi^2}\\, {}_1F_1[1,\\frac32 ; -\\frac{t \\eta\n\\bar{\\eta}}{4}].$$\nThis simplifies considerably the computation of the zeta function\n$$ \\zeta(s) = \\frac{1}{\\Gamma(s)} \\int_0^{\\infty} dt \\, t^{s-1}\\, \\int d^8Z\n\\,\\, K(t\/\\mu^2).$$\nUsing the integral representation\n$$ {}_1F_1[\\alpha, \\gamma; z] = \\frac{\\Gamma(\\gamma)}{\\Gamma(\\alpha)\n\\Gamma(\\gamma -\\alpha)} \\int_0^1 dx \\, e^{zx} \\, x^{\\alpha -1} (1-x)^{\\gamma -\n\\alpha -1} $$\nfor the confluent hypergeometric functions,\n\\begin{eqnarray*}\n \\zeta(s) &=& \\int d^8Z \\, \\frac{\\eta \\bar{\\eta}}{64\\pi^2 \\Gamma(s)} \\,\n\\, \\,\n\\int_0^1 dx \\, (1-x)^{-\\frac12} \\, \\int_0^{\\infty} dt \\, t^{s-1} \\,\ne^{-\\frac{t\n\\eta \\bar{\\eta x }}{4 \\mu^2}} \\\\\n&=&\\int d^8Z \\, \\frac{\\eta \\bar{\\eta}}{64\\pi^2 } \\, \\left (\\frac{4\n\\mu^2}{\\eta\n\\bar{\\eta}} \\right)^s \\int_0^1 dx \\, \\frac{(1-x)^{-\\frac12}}{x^s}.\n\\end{eqnarray*}\nThe integral over $x $ is just the beta function $B(\\frac12, 1-s) =\n\\frac{\\Gamma(\\frac12)\\Gamma(1-s)}{\\Gamma(\\frac32 -s)}.$\n>From this it is easy to show\n$$\\zeta^{\\prime}(0) =\\int d^8Z \\, \\frac{\\eta \\bar{\\eta}}{32\\pi^2 }\\, (2 -\n\\ln\n\\frac{\\eta \\bar{\\eta}}{\\mu^2}),$$\nwhich is the result obtained by Pickering and West in \\cite{pickering} using\nsupergraph techniques. The Schwinger proper time regularization employed by\nBuchbinder et al \\cite{buchbinder1} yields an expression for the effective\nK\\\"ahler potential which contains a finite constant in the form of an\ninfinite series, which is not evaluated explicitly. Thus recognizing that the\nkernel is essentially a hypergeometric function in this case yields\nconsiderable\nsimplification in the computation of the effective action if zeta function\nregularization is employed.\n\nFinally, we make a comment of the form of the effective action\n(\\ref{kernelresult}). The effective action computed using the lowest\norder approximation (\\ref{K0}) to the heat kernel is holomorphic in\nthe sense of Seiberg \\cite{Seiberg}, in that it contains only terms\nwith chiral or antichiral gauge superfields. However, when\ncorrections incorporating $\\cal{D}_{\\alpha}W_{\\beta}$ are included as\nin (\\ref{kernelresult}), this is no longer true, and the result\ncontains both $\\bar{M}_{ab} = (\\bar{{\\cal D}} \\bar{\\sigma}_{ab}\n\\bar{W}) $ and $M_{ab} = ({\\cal D}\\sigma_{ab}W)$ in the one term.\n In calculating the\neffective action for N=2 supersymmetric Yang-Mills theory in N=1\nsuperfield formulation, the result (\\ref{kernelresult}) will thus represents\na nonholomorphic correction to the effective action which should be\nincluded with the nonholomorphic corrections involving scalar\nsuperfields computed in \\cite{dewit,pickering,grisaru}.\n\n\\ack{ IMcA is grateful to the Alexander von Humboldt-Stiftung for support.\nBoth\nauthors wish to thank Professor J. Wess for his generous hospitality at the\nUniversity of M\\\"unchen during the period in which this this work was done.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe copious production of heavy quarkonium states at high energy colliders has inaugurated a new era of precision studies of such states~\\cite{Brambilla:2010cs}. In proton-proton collisions (p+p), next-to-leading order (NLO) perturbative studies are available \\cite{Butenschoen:2012px,Chao:2012iv,Gong:2012ug,Shao:2014yta} within the Non-Relativistic QCD (NRQCD) factorization framework~\\cite{Bodwin:1994jh}. These computations can be further improved by employing QCD factorization \\cite{Kang:2014tta,Kang:2014pya} to resum large logarithms $\\ln(p_\\perp\/M)$ in the ratio of the transverse momentum $p_\\perp$ to the quark mass $M$.\nA comparison of these studies with collider data therefore provides key insight into the formation and hadronization of heavy quark-antiquark pair ($Q\\bar{Q}$-pair) states in QCD.\n\nIn proton-nucleus (p+A) collisions, additional features of $Q\\bar{Q}$-pair production and hadronization can be tested. These include many-body QCD effects such as multiple scattering and shadowing of gluon distributions in nuclei, as well as the radiative energy loss induced in the scattering of the $Q\\bar{Q}$-pair off the colored medium. Besides these insights into many-body QCD dynamics, p+A collisions also provide an important benchmark for understanding the interactions of heavy quarks in the hot and dense medium created in heavy ion collisions.\n\nFor small gluon momentum fractions $x$, their distributions saturate with a dynamically generated saturation scale $Q_S(x)$ \\cite{Gribov:1984tu,Mueller:1985wy,McLerran:1993ni,McLerran:1993ka}. This regime is accessed when $p_\\perp \\lesssim Q_S$. The Color Glass Condensate (CGC) effective theory \\cite{Iancu:2003xm,Gelis:2010nm} provides a quantitative framework to study many-body QCD effects in high energy scattering processes when $Q_S(x) >> \\Lambda_{\\rm QCD}$, where $\\Lambda_{\\rm QCD}$ is the fundamental QCD scale. In this limit, multiple scattering contributions can be absorbed into light like Wilson line correlators, which govern the shadowing and $p_\\perp$ broadening of $Q\\bar{Q}$-pair distributions at small $x$. Energy evolution of these correlations at small $x$ is described by the Balitsky-JIMWLK hierarchy of renormalization group equations~\\cite{Balitsky:1995ub,Iancu:2000hn,JalilianMarian:1997dw}. Energy loss contributions, included in some models in the literature~\\cite{Arleo:2012rs}, are formally NLO in the CGC framework~ \\cite{Liou:2014rha}.\n\nExpressions for $Q\\bar{Q}$-pair production in p+A collisions in the CGC framework were derived previously in~\\cite{Blaizot:2004wv,Fujii:2005vj,Fujii:2006ab,Kharzeev:2005zr,Kharzeev:2008nw,Dominguez:2011cy,Akcakaya:2012si}\nas well as in related dipole approaches~\\cite{Kopeliovich:2002yv,Motyka:2015kta}. For $p_\\perp >> Q_S$, the results can be matched to those derived in perturbative QCD frameworks~\\cite{Gelis:2003vh}.\nIn \\cite{Fujii:2013gxa}, the matrix elements in the CGC framework were combined with the Color Evaporation hadronization Model (CEM) to compute $J\/\\psi$ production in proton-proton and proton (deuteron)-nucleus collisions at the LHC (RHIC)\\footnote{For simplicity, we will generically call both sorts of collisions p+A collisions in the rest of the paper.}. The quantity\n\\begin{align}\\label{eq:RpA}\nR_{pA}=\\frac{d\\sigma_{pA}}{A\\times d\\sigma_{pp}}\\,,\n\\end{align}\nthe ratio of the cross-sections in p+A collisions to p+p collisions, normalized by the atomic number $A$, was found to be suppressed relative to the data~\\cite{Abelev:2013yxa,Aaij:2013zxa}. Very recently, the authors of \\cite{Ducloue:2015gfa} argued that better agreement of the CGC+CEM model with the $R_{pA}$ data is obtained if nuclear effects were treated differently. Here we shall apply NRQCD to describe the hadronization of $Q\\bar{Q}$-pair and compute $J\/\\psi$ production in a CGC+NRQCD framework~\\cite{Kang:2013hta}. In addition to providing a more systematic power counting, NRQCD allows one to smoothly match the CGC computations to successful NLO NRQCD computations for $p_\\perp >> Q_S$. This strategy was previously applied to successfully describe p+p collisions at RHIC and LHC~\\cite{Ma:2014mri}.\n\nFor completeness, we outline the CGC+NRQCD formalism~\\cite{Kang:2013hta,Ma:2014mri}. In NRQCD factorization, the production cross section of a quarkonium $H$ in the forward region of a p+A collision is expressed as \\cite{Bodwin:1994jh}\n\\begin{equation} \\label{eq:NRQCD}\nd\\sigma^H_{pA}=\\sum_\\kappa d\\hat{\\sigma}_{pA}^\\kappa\\langle{\\cal O}^H_\\kappa\\rangle\\,,\n\\end{equation}\nwhere $\\kappa=\\state{2S+1}{L}{J}{c}$ denotes the quantum numbers of the intermediate $Q\\bar{Q}$-pair in the standard spectroscopic notation for angular momentum. The superscript $c$ denotes the color state of the pair, which can be either color singlet (CS) with $c=1$ or color octet\n(CO) with $c=8$. For $J\/\\psi$ production that will be studied here, the most important intermediate states are $\\CScSa$, $\\COaSz$, $\\COcSa$ and $\\COcPj$. In Eq.~\\eqref{eq:NRQCD}, $\\langle{\\cal O}^H_\\kappa\\rangle$ are non-perturbative universal long distance matrix elements (LDMEs), which can be extracted from data, and $d\\hat{\\sigma}^\\kappa$ are short-distance coefficients (SDCs) for the production of a $Q\\bar{Q}$-pair, computed in perturbative QCD.\n\nTo calculate the SDCs in Eq.~\\eqref{eq:NRQCD}, we apply the CGC effective field theory \\cite{Gelis:2010nm,Blaizot:2004wv}, which results in the expressions~\\cite{Kang:2013hta,Ma:2014mri},\n\\begin{align}\\label{eq:dsktCS}\n\\begin{split}\n\\frac{d \\hat{\\sigma}_{pA}^\\kappa}{d^2{\\vt{p}} d\ny}\\overset{\\text{CS}}=&\\frac{\\alpha_s (\\pi \\bar{R}_A^2)}{(2\\pi)^{9}\n(N_c^2-1)} \\underset{{\\vtn{k}{1}},{\\vt{k}},{\\vtp{k}}}{\\int}\n\\frac{\\varphi_{p,y_p}({\\vtn{k}{1}})}{k_{1\\perp}^2}\\\\\n&\\hspace{-1.5cm}\\times \\mathcal{N}_{Y}({\\vt{k}})\\mathcal{N}_{Y}({\\vtp{k}})\\mathcal{N}_{Y}({\\vt{p}}-{\\vtn{k}{1}}-{\\vt{k}}-{\\vtp{k}})\\,\n{\\cal G}^\\kappa_1,\n\\end{split}\n\\end{align}\nfor the color-singlet $\\CScSa$ channel, and\n\\begin{align}\\label{eq:dsktCO}\n\\begin{split}\n\\frac{d \\hat{\\sigma}_{pA}^\\kappa}{d^2{\\vt{p}} d\ny}\\overset{\\text{CO}}=&\\frac{\\alpha_s (\\pi \\bar{R}_A^2)}{(2\\pi)^{7}\n(N_c^2-1)} \\underset{{\\vtn{k}{1}},{\\vt{k}}}{\\int}\n\\frac{\\varphi_{p,y_p}({\\vtn{k}{1}})}{k_{1\\perp}^2}\\\\\n&\\times \\mathcal{N}_Y({\\vt{k}})\\mathcal{N}_Y({\\vt{p}}-{\\vtn{k}{1}}-{\\vt{k}})\n\\,\\Gamma^\\kappa_8,\n\\end{split}\n\\end{align}\nfor the color-octet channels. Here $\\varphi_{p,y_p}$ is the unintegrated gluon distribution inside the proton, which can be expressed as\n\\begin{align}\\label{eq:unintegrated}\n\\varphi_{p,y_p}({\\vtn{k}{1}})=\\pi \\bar{R}_p^2 \\frac{N_c k_{1\\perp}^2}{4\\alpha_s} \\widetilde{\\mathcal{N}}^A_{y_p}({\\vtn{k}{1}})\\,.\n\\end{align}\nThe functions ${\\cal G}^\\kappa_1$ ($\\Gamma^\\kappa_8$) are calculated perturbatively--the expressions are available in \\cite{Ma:2014mri} (\\cite{Kang:2013hta}).\n${\\cal N}$ ($\\widetilde{\\mathcal{N}}^A$) are the momentum-space dipole forward scattering amplitudes with Wilson lines in the fundamental (adjoint) representation, and $\\pi \\bar{R}_p^2$ ($\\pi \\bar{R}_A^2$) is the effective transverse area of the dilute proton (dense nucleus). These formulas can be used to compute quarkonium production in p+A collisions. By replacing ``$A$'s by $p$'s\", they can also be used to compute quarkonium production in p+p collisions~\\cite{Ma:2014mri}.\nFor deuteron-gold (d+Au) collisions at RHIC, since gluon shadowing effects are weak for deuteron side, we assume $\\varphi_{d,y_d}({\\vtn{k}{1}})= 2\\, \\varphi_{p,y_p}({\\vtn{k}{1}})$.\n\nBefore we confront our framework to data on p+A collisions, there are a number of parameters that have to be fixed. Nearly all the parameters are identical to those previously determined in our study~ \\cite{Ma:2014mri} of p+p collisions. {The charm quark mass is set to be $m=1.5\\mathrm{~GeV}$, approximately one half the $J\/\\psi$ mass.\nThe CO LDMEs were extracted in the NLO collinear factorized NRQCD formalism~\\cite{Chao:2012iv} by fitting Tevatron high $p_\\perp$ prompt ${J\/\\psi}$ production data; one obtains $\\mops=1.16\/(2N_c) \\mathrm{~GeV}^3$, $\\mopa=0.089\\pm0.0098 \\mathrm{~GeV}^3$, $\\mopb=0.0030\\pm0.0012 \\mathrm{~GeV}^3$ and $\\mopc=0.0056\\pm0.0021\\mathrm{~GeV}^3$.\nWe emphasize, as previously, that the high sensitivity of short distance cross-sections to quark mass\ncan be mitigated by the mass dependence of the LDMEs. Note that the uncertainties of these CO LDMEs include only uncorrelated statistic errors, but not correlated errors \\cite{Chao:2012iv}.}\nFurther, as in \\cite{Ma:2014mri}, ${\\cal N}$ and $\\widetilde{\\mathcal{N}}^A$ are obtained by solving the running coupling Balitsky-Kovchegov\n(rcBK) equation~\\cite{Balitsky:1995ub,Kovchegov:1999yj} in momentum space with McLerran-Venugopalan (MV) initial conditions~\\cite{McLerran:1993ni,McLerran:1993ka} for the dipole amplitude at the initial rapidity scale $Y_0\\equiv\\ln(1\/x_0)$ (with $x_0=0.01$) for small $x$ evolution. In the case of p+p collisions, all the parameters in the rcBK evolution are fixed from fits to the HERA DIS data~\\cite{Albacete:2012xq}. In \\cite{Ma:2014mri}, we devised a matching scheme that allowed us to interpolate between the proton's collinearly factorized gluon distribution at large $x$ with the unintegrated distribution in Eq.~(\\ref{eq:unintegrated}). This allowed us to fix the remaining free parameter, the effective gluon radius of the proton $\\bar{R}_p=0.48$ fm.\n\nTurning to p+A collisions, there are two additional parameters in our framework, the initial saturation scale $Q_{s0,A}$ in the nucleus and the effective transverse radius $\\bar{R}_A$. The latter is not the charge radius of the nucleus, but parametrizes the overall non-perturbative cross-section of relevance to quarkonium production. A more detailed treatment would take into account the impact parameter dependence of the unintegrated distributions, and model the inelastic proton-nucleus cross-section as in \\cite{Miller:2007ri,d'Enterria:2003qs}. We will return to this point shortly.\nIn general, we can express the initial saturation scale in the nucleus as $Q_{s0,A}^2=N\\times Q_{s0, p}^2$, where $N$ is a number to be determined and $Q_{s0, p}^2$ is the initial saturation scale in proton, fixed by the fit to HERA DIS data~\\cite{Albacete:2012xq}. Good fits to extant electron-nucleus (e+A) DIS data were obtained in \\cite{Dusling:2009ni} for rcBK evolution with the following initial conditions:\ni) MV model with anomalous dimension $\\gamma=1.13$, ii) MV model with anomalous dimension $\\gamma=1$. For the initial conditions i), one obtains a good fit to e+A data for $N\\approx 3$,\nwhile for initial conditions ii), $N\\approx 1.5$. In this paper, rcBK evolution for nuclei was performed for initial conditions ii). To avoid fine tuning, we will choose $N=2$ for the results presented in this paper\\footnote{The quality of fit for $N=2$ is marginally better than that for $N=3$ but significantly better than those for $N=1$ or higher values of integer $N$. For the IP-sat model, for median impact parameters in e+A DIS, $N=6$ in contrast to $N\\approx 2$ for $b=0$~\\cite{Kowalski:2007rw}.}.\n\n\nSimilar to $\\bar{R}_p$ for the proton, the effective radius $\\bar{R}_A$ providing the non-perturbative normalization of the cross-section here can be different from the transverse charge radius of the nucleus because we have a specific heavy particle produced in the final state. Fortunately, there is a physical condition which we can use to constrain it. When $p_\\perp$ is much larger than the saturation scale involved, the gluon distribution becomes dilute and the nuclear suppression effect should be negligible. Thus $R_{pA}$ must approach 1 for high $p_\\perp>> Q_{s0,A}$. Using Eqs.~(\\ref{eq:RpA})-(\\ref{eq:unintegrated}) one can derive (the argument is presented in Appendix), the expression,\n\\begin{align}\\label{eq:RA}\n\\frac{\\bar{R}_A^2}{A \\bar{R}_p^2}\\frac{ Q_{s0,A}^{2\\gamma}}{Q_{s0,p}^{2\\gamma}}\\approx1.\n\\end{align}\nWe will see later that Eq.~\\eqref{eq:RA} indeed guarantees $R_{pA}\\to 1$ at high $p_\\perp$ limit, within a few percent. By choosing $\\gamma=1$ and $N=2$, we obtain $\\bar{R}_A=\\sqrt{A\/2} \\bar{R}_p$, which equals to $4.9\\mathrm{~fm}$ for Pb and $4.8\\mathrm{~fm}$ for Au\\footnote{Interestingly, the ratio $\\bar{R}_A\/\\bar{R}_p\\sim 10$ here is close to the ratio of radii extracted from estimates of the inelastic p+A and p+p cross-sections at both LHC and RHIC.}.\n\nBecause Eqs.~\\eqref{eq:dsktCS}-\\eqref{eq:unintegrated} are computed only at LO in the CGC power counting, the CGC+NRQCD framework cannot be extended to describe high $p_\\perp$ p+p and p+A data, one might challenge that using Eq.~(\\ref{eq:RA}) to determine $\\bar{R}_A$ is not especially meaningful. We emphasize however that this condition must be satisfied for the CGC+NRQCD framework to be self-consistent at each order in the perturbative expansion. The $p_\\perp$ at which Eq.~(\\ref{eq:RA}) is saturated may differ. At NLO, the above procedure should be redone to determine a new self-consistent condition.\n\n\n\\begin{figure}[!htbp]\n\\center{\n\\includegraphics*[scale=0.4]{pt\n\\caption{\\label{fig:pt-pa} $p_\\perp$ spectrum of $J\/\\psi$ production in p+Pb collisions at 5.02 TeV and d+Au collisions at 0.2 TeV. NLO NRQCD results are taken from Ref. \\cite{hfzhang2015}. The experimental data are taken from Refs.~\\cite{Aaij:2013zxa,Adare:2012qf}.}\n}\n\\end{figure}\n\nTo better present the p+A results, we define a cross section per nucleon-nucleon collision, $d\\sigma_{NN}=\\frac{d\\sigma_{AB}}{AB}$. Fig.~\\ref{fig:pt-pa} displays the $p_\\perp$ spectrum of $J\/\\psi$ production in p+Pb collisions at 5.02 TeV and d+Au collisions at 0.2 TeV. The bands of our CGC+NRQCD results estimate uncorrelated errors of LDMEs and an additional global $30\\%$ uncertainty to account for correlated errors of LDMEs, errors from treatment of feed down, velocity corrections and radiative corrections. We find that the contribution of the CS channel is about $15-20\\%$ at small $p_\\perp$ and decreases as $p_\\perp$ becomes larger. The NLO NRQCD predictions are taken from \\cite{hfzhang2015}, where the PDF shadowing model EPS09~\\cite{Eskola:2009uj} was employed to estimate the (small) nuclear shadowing effects at large $p_\\perp$. For all rapidity bins available,\nthe CGC+NRQCD curves match on to the NLO NRQCD ones smoothly, providing a good description of all experimental data. Interestingly, one finds that the CGC+NRQCD curves overshoot the data at smaller values of $p_\\perp$ at RHIC relative to the LHC data. This may be anticipated because, for a given $p_\\perp$, small $x$ logs are less important at lower energies. However, a full NLO computation in this framework is needed to understand better the matching in $p_\\perp$ of the two formalisms.\n\n\\begin{figure}\n\\center{\n\\includegraphics*[scale=0.4]{y\n\\caption{\\label{fig:y-pa} Rapidity distribution of $J\/\\psi$ production in p+Pb collisions at 5.02 TeV and d+Au collisions at 0.2 TeV.\nThe experimental data are taken from Refs.~\\cite{Aaij:2013zxa,Adare:2010fn}.}\n}\n\\end{figure}\n\nThe rapidity distribution of $J\/\\psi$ production in p+Pb collisions at 5.02 TeV and d+Au collisions at 0.2 TeV is shown in Fig.~\\ref{fig:y-pa},\nwhere the the bands are generated similarly to those in Fig.~\\ref{fig:pt-pa}. Since these data are integrated over $p_\\perp$, the low $p_\\perp$ region dominates and the CGC+NRQCD formalism at LO should apply. Both LHC data and forward RHIC data are well covered by our uncertainty band; the central value for mid-rapidity RHIC data however is slightly below the band. For this data point, our theory curves should have a larger systematic uncertainty because our framework is most reliable for dilute-dense collisions corresponding to high energies and forward rapidities. The key observation though is that both the relative shapes as well as the absolute magnitudes of the curves are well captured in the CGC+NRQCD formalism.\nThe quality of the fits to the $p_\\perp$ and rapidity spectra in Figs.~\\ref{fig:pt-pa} and \\ref{fig:y-pa} are similar to those in p+p collisions~\\cite{Ma:2014mri}. Thus we should be able to describe the $R_{pA}$ ratio, which we shall now discuss.\n\n\nA key point is that the the large uncertainties for LDMEs, feed down contributions and velocity corrections, largely cancel in the ratio of each NRQCD channel contributing to $J\/\\Psi$ production. The band spanned by different channels should be able to bracket the $R_{pA}$ value for ${J\/\\psi}$ production. With this method, the bounded value of $R_{pA}$ extracted for ${J\/\\psi}$ production is independent of the LDMEs and their statistical uncertainties. This is especially noteworthy since independent extractions of the LDMEs from present data are not feasible; their magnitudes, especially between the various CO channels, can vary significantly. Finally, since the CEM is a special case of NRQCD with the choice of certain LDMEs \\cite{Bodwin:2005hm}, our calculation of $R_{pA}$ will also cover the range of CEM predictions. In this sense, the range of theoretical estimates of $R_{pA}$ for $J\/\\psi$ production are independent of the ${J\/\\psi}$ hadronization model and are directly sensitive to the short distance physics.\n\nWe will employ here the principal channels for ${J\/\\psi}$ production given by NRQCD power counting--these correspond to the $\\CScSa$, $\\COaSz$, $\\COcSa$ and $\\COcPj$ channels. \\begin{figure}\n\\center{\n\\includegraphics*[scale=0.4]{rpa-pt-alice}\\\\% Here is how to import EPS art\n\\includegraphics*[scale=0.4]{rpa-y-alice\n\\caption{\\label{fig:rpa-alice} $R_{pA}$ as a function of $p_\\perp$ (upper) and rapidity (lower) at LHC.\nThe experimental data are taken from Refs.~\\cite{Adam:2015iga, Abelev:2013yxa,Aaij:2013zxa}.}\n}\n\\end{figure}\n\\begin{figure}\n\\center{\n\\includegraphics*[scale=0.4]{rpa-pt-rhic}\\\\% Here is how to import EPS art\n\\includegraphics*[scale=0.4]{rpa-y-rhic\n\\caption{\\label{fig:rpa-rhic} $R_{pA}$ as a function of $p_\\perp$ (upper) and rapidity (lower) at RHIC.\nThe experimental data are taken from Refs.~\\cite{Adare:2012qf, Adare:2010fn}.}\n}\n\\end{figure}\nOur results for $R_{pA}$ as a function of $p_\\perp$ and rapidity, compared to data from the LHC and RHIC, respectively, are presented in Figs.~\\ref{fig:rpa-alice} and \\ref{fig:rpa-rhic}, where a 5\\% systematical error is assumed for each channel to account for the approximation in Eq.~\\eqref{eq:RA}. The $R_{pA}$ of all CO channels approaches 1 at high $p_\\perp$, confirming that condition Eq.~\\eqref{eq:RA} indeed is satisfied by the full theoretical calculation. On the contrary, $R_{pA}$ of the CS channel $\\CScSa$ increases to be larger than 1 at high $p_\\perp$. Since forming a color singlet requires two gluons from the target, the additional gluon exchange from the nucleus, at high $p_\\perp$, is enhanced relative to that from a proton (by an amount that is proportional asymptotically to the ratio of their saturation scales at the rapidity of interest). Nevertheless, as we find the contribution of the CS channel is small relative to the CO terms in both p+p and p+A collisions, it does not affect our estimate of $R_{pA}$. Thus the band representing the $R_{pA}$ spanned by the CO channels corresponds to our result for $R_{pA}$ of $J\/\\psi$ production.\n\nThe $p_\\perp$ and rapidity $R_{pA}$ data from both RHIC and LHC lie within our uncertainty bands.\nAt the LHC, the $\\COcSa$ state lies closest to the central values of the data, while at RHIC, the $\\COaSz$ and $\\COcPj$ channels are closest to the data. Our results suggest that the $R_{pA}$ data, in a future global analysis within the CGC\/NLO+NRQCD framework, can help constrain the LDMEs more stringently, thereby providing a further test of NRQCD.\n\nTo summarize, we have shown here that $J\/\\psi$ spectra in p+A collisions both at RHIC and the LHC are well described by our CGC+NRQCD computations. The two free non-perturbative parameters are related by Eq.~\\eqref{eq:RA}; further, the value of the initial nuclear saturation scale $Q_{s0,A}$ is consistent with the values that best describe fixed target e+A DIS data. The fact that the $R_{pA}$ $p_\\perp$ data lie within the bands spanned by our computations for the different color octet channels is a strong evidence for the robustness of our framework since these curves are insensitive to details of how heavy quark pairs hadronize to form the $J\/\\psi$. The results in this paper, when combined with those in \\cite{Ma:2014mri}, provide the first comprehensive description of $J\/\\psi$ production in both p+p and p+A collisions at collider energies.\n\nSeveral outstanding questions remain. Firstly, the NLO CGC computation needs to be performed to confirm that the framework established is solid. Secondly, other quarkonium states remain to be studied; these come with unique challenges. For instance, for $\\Upsilon$ production, Sudakov type double logs in $M\/P_\\perp$ are important and need to be resummed~\\cite{Berger:2004cc,Sun:2012vc,Qiu:2013qka}. A systematic computation of $\\psi(2S)$ production in p+A collisions, may require that we include relativistic contributions in the computation of the heavy quark matrix elements. All these questions can be explored in the framework discussed here.\n\nWe thank Roberta Arnaldi and Prithwish Tribedy for helpful communications. This work was supported in part by the U.S. Department of Energy Office of Science under Award Number DE-FG02-93ER-40762, U. S. Department of Energy under Contract No. de-sc0012704, and the National Natural Science Foundation of China No. 11405268.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\n\nWhile classical planning focuses on determining a plan that accomplishes a fixed goal, a goal\nreasoning agent reasons about which goals to pursue and continuously refines its goals during\nexecution. It may decide to suspend a goal, re-prioritize its goals, or abandon a goal completely.\nGoals are represented explicitly, including goal constraints, relations between multiple goals, and\ntasks for achieving a goal.\nGoal reasoning is particularly interesting on mobile robots, as a mobile\nrobot necessarily acts in a dynamic environment and thus needs to be able to react to unforeseen\nchanges. Computing a single plan to accomplish an overall objective is often unrealistic, as\nthe plan becomes unrealizable during execution. While planning techniques such as contingent planning\n\\cite{hoffmannContingentPlanningHeuristic2005}, conformant planning\n\\cite{hoffmannConformantPlanningHeuristic2006}, hierarchical planning\n\\cite{kaelblingHierarchicalTaskMotion2011}, and continual planning\n\\cite{brennerContinualPlanningActing2009,hofmannContinualPlanningGolog2016} allow to deal with some\ntypes of execution uncertainty, planning for the overall objective often simply is too time consuming. Goal\nreasoning solves these problems by splitting the objective into smaller goals that can be\neasily planned for, and allows to react to changes dynamically by refining the agent's goals.\nThis becomes even more relevant in multi-agent settings, as not only the environment is dynamic, but\nthe other agents may also act in a way that affects the agent's goals. While some work on\nmulti-agent goal reasoning exists\n\\cite{robertsCoordinatingRobotTeams2015,robertsGoalLifecycleNetworks2021,hofmannMultiagentGoalReasoning2021},\nprevious approaches focus on conflict avoidance, rather than facilitating active cooperation\nbetween multiple agents.\n\nIn this paper, we propose an extension to multi-agent goal reasoning that allows effective\ncollaboration between agents. To do this, we attach a set of \\emph{promises} to a goal, which intuitively describe\na set of facts that will become true after the goal has been achieved. In order to make use of\npromises, we extend the goal lifecycle \\cite{robertsIterativeGoalRefinement2014} with \\emph{goal\noperators}, which, similar to action operators in planning,\nprovide a formal framework that defines when a goal can be formulated and which objective it pursues.\nWe use promises to evaluate the goal precondition not only\nin the current state, but also in a time window of future states (with some fixed time horizon). This allows the\ngoal reasoning agent to formulate goals that are currently not yet achievable, but will be in the\nfuture. To expand the goal into a plan, we then translate promises into \\emph{timed initial\nliterals} from PDDL2.2 \\cite{edelkampPDDL2LanguageClassical2004}, thereby allowing a PDDL planner to\nmake use of the promised facts to expand the goal.\nWith this mechanism, promises enable active collaborative behavior between multiple, distributed goal reasoning\nagents.\n\nThe paper is organized as follows: In Section~\\ref{sec:background}, we summarize goal reasoning and\ndiscuss related work. As basis of our implementation, we summarize the main concepts of the \\acf{CX}\nin Section~\\ref{sec:cx}. In Section~\\ref{sec:goal-preconditions}, we introduce a formal notion of a\ngoal operator, which we will use in Section~\\ref{sec:promises} to define promises. In\nSection~\\ref{sec:evaluation}, we evaluate our approach in a distributed multi-agent scenario, before\nwe conclude in Section~\\ref{sec:conclusion}.\n\n\n\\section{Background and Related Work}\\label{sec:background}\n\\paragraph{Goal Reasoning.}\nIn goal reasoning, agents can ``deliberate on and self-select their objectives''\n\\cite{ahaGoalReasoningFoundations2018}. Several kinds of goal reasoning have been proposed\n\\cite{vattamBreadthApproachesGoal2013}:\nThe \\acfiu{GDA}\nframework~\\cite{munoz-avilaGoaldrivenAutonomyCasebased2010,coxModelPlanningAction2016} is based on\nfinite state-transition systems known from planning \\cite{ghallabAutomatedPlanningActing2016} and\ndefine a goal as conjunction of first-order literals, similar to goals in classical planning. They\ndefine a goal transformation function $\\beta$, which, given the current state $s$ and goal $g$,\nformulates a new goal $g'$. In the \\ac{GDA} framework, the goal reasoner also produces a set of\n\\textit{expectations}, which are constraints that are predicted to be fulfilled during the\nexecution of a plan associated with a goal. In contrast to promises, which we also intend as\na model of constraints to be fulfilled, those\nexpectations are used for discrepancy detection rather than multi-agent coordination.\nThe \\acfiu{T-REX} architecture \\cite{mcgannDeliberativeArchitectureAUV2008} is a goal-oriented system that employs multiple\nlevels of reasoning abstracted in reactors, each of which operates in its own functional and\ntemporal scope (from the entire mission duration to second-level operations). Reactors on lower levels\nmanage the execution of subgoals generated at higher levels, working in synchronised timelines\nwhich capture the evolution of state-variables over time. While through promises we also\nattempt to encode a timeline of (partial) objective achievement, our approach has no hierarchy\nof timelines and executives. Timing is not used to coordinated between levels of abstraction\non one agent, but instead is used to indicate future world-states to other, independent agents\nreasoning in the same temporal scope.\nA \\acfiu{HGN} \\cite{shivashankarHierarchicalGoalbasedFormalism2012} is a partial order of goals,\nwhere a goal is selected for execution if it has no predecessor. \\acp{HGN} are used in the\n\\textsc{GoDeL} planning system \\cite{shivashankarGoDeLPlanningSystem2013} to decompose a planning\nproblem, similar to HTN planning. \\citet{robertsGoalLifecycleNetworks2021} extend \\acp{HGN} to\n\\acfiup{GLN} to integrate them with the goal lifecycle.\nThe goal lifecycle~\\cite{robertsIterativeGoalRefinement2014} models goal reasoning as an iterative\nrefinement process and describes how a goal progresses over time.\nAs shown in Figure~\\ref{fig:cx-goal-lifecycle}, a goal is first \\emph{formulated}, merely stating\nthat it may be relevant. Next, the agent \\emph{selects} a goal that it deems the goal to be useful.\nThe goal is then \\emph{expanded}, e.g., by querying a PDDL planner for a plan that accomplishes the\ngoal. A goal may also be expanded into multiple plans, the agent then \\emph{commits} to one\nparticular plan. Finally, it \\emph{dispatches} a goal by starting to execute the plan.\nIt has been implemented in \\emph{ActorSim} \\cite{robertsActorSimToolkitStudying2016} and in the\n\\ac{CX} \\cite{niemuellerGoalReasoningCLIPS2019}, which we extend in this work with promises.\nMost goal reasoning approaches focus on scenarios with single agents or scenarios where a central\ncoordinating instance is available. In contrast, our approach focuses on a distinctly decentralized\nmulti-agent scenario with no central coordinator.\n\n\\paragraph{Multi-Agent Coordination.}\nIn scenarios where multiple agents interact in the same environment, two or more agents might\nrequire access to the same limited resources at the same time. In such cases, multi-agent\ncoordination is necessary to avoid potential conflicts, create robust solutions and enable\ncooperative behavior between the agents \\cite{diasMarketBasedMultirobotCoordination2006}.\nAgents may coordinate implicitly by recognizing or learning a model of another agent's behavior\n\\cite{sukthankarPlanActivityIntent2014,albrechtAutonomousAgentsModelling2018}.\nParticularly relevant is \\emph{plan recognition}~\\cite{carberryTechniquesPlanRecognition2001}, where\nthe agent tries to recognize another agent's goals and actions.\n\\textsc{Alliance}~\\cite{parkerALLIANCEArchitectureFault1998} also uses an implicit coordination\nmethod based on \\emph{impatience} and \\emph{acquiescence}, where an agent eventually takes over a\ngoal if no other agent has accomplished the goal (impatience), and eventually abandons a goal if it\nrealizes that it is making little progress (acquiescence).\nIn contrast to such approaches, we propose that the agents explicitly communicate (parts of) their\ngoals, so other agents may directly reason about them.\nTo obtain a conflict-free plan, each agent may start with its own individual plan and iteratively\nresolve any flaws~\\cite{coxEfficientAlgorithmMultiagent2005}, which may also be modeled as a\ndistributed constraint optimization problem~\\cite{coxDistributedFrameworkSolving2005}.\nInstead of starting with individual plans, \\citet{jonssonScalingMultiagentPlanning2011} propose to\nstart with some initial (sub-optimal) shared plan and then let each agent improve its own actions.\nAlternatively, agents may also \\emph{negotiate} their goals\n\\cite{davisNegotiationMetaphorDistributed1983,krausAutomatedNegotiationDecision2001}, e.g., using\n\\emph{argumentation}~\\cite{krausReachingAgreementsArgumentation1998}, which typically requires an\nexplicit model of the mental state.\n\\citet{vermaMultiRobotCoordinationAnalysis2021}\nclassify coordination mechanisms based on properties such as static vs dynamic, weak vs strong,\nimplicit vs explicit, and centralized vs decentralized.\nHere, we focus on dynamic, decentralized and distributed multi-robot coordination.\nAlternatively, coordination approaches can be classified based on organization structures\n\\cite{horlingSurveyMultiagentOrganizational2004}, e.g., coalitions, where agents cooperate but each\nagent attempts to maximize its own utility, or teams, where\nthe agents work towards a common goal.\nSuch teams may also form dynamically, e.g., by dialogues~\\cite{dignumAgentTheoryTeam2001}.\nHere, we are mainly interested in fixed teams, where a fixed group of robots have a common goal.\nRole assignment approaches attempt to assign fixed and distinct roles to each of the\nparticipating agents and thereby fixing the agent's plan.\nThis can be achieved either by relying on a central instance or through distributed approaches~\n\\cite{iocchiDistributedCoordinationHeterogeneous2003,vailDynamicMultiRobotCoordination2003,jinDynamicTaskAllocation2019}.\nIntention sharing approaches allow agents to incorporate the objectives of other agents\ninto their own reasoning in an effort to preemptively avoid conflicting actions~\\cite{holvoetBeliefsDesiresIntentions2006,grantLogicBasedModelIntentions2002,sarrattPolicyCommunicationCoordination2016}.\n\\emph{SharedPlans}~\\cite{groszCollaborativePlansComplex1996} use complex actions that involve\nmultiple agents. They use an intention sharing mechanism, formalize collaborative\nplans, and explicitly model the agents' mental states, forming mutual beliefs.\nSimilar to promises, an agent may share its intention with \\emph{intending-that} messages, which\nallow an agent to reason about another agent's actions.\nDepending on the context, agents must have some form of\n\\emph{trust}~\\cite{yuSurveyMultiAgentTrust2013,pinyolComputationalTrustReputation2013} before they\ncooperate with other agents.\nMarket-based approaches as surveyed by \\citet{diasMarketBasedMultirobotCoordination2006}\nuse a bidding process to allocate different conflict-free tasks between the agents of a multi-robot system.\nThis can be extended to auction of planning problems in an effort to optimize temporal multi-agent\nplanning~\\cite{hertleEfficientAuctionBased2018}.\nIn the context of multi-agent goal reasoning, \\citet{wilsonGoalReasoningModel2021} perform\nthe goal lifecycle for an entire multi-robot system on each agent separately and then\nuse optimization methods to prune the results.\n\n\n\\section{The CLIPS Executive}\\label{sec:cx}\n\nThe \\acf{CX}~\\cite{niemuellerGoalReasoningCLIPS2019} is a goal reasoning system implemented in the\nCLIPS rule-based production system \\cite{wygantCLIPSPowerfulDevelopment1989}. Goals follow the\ngoal lifecycle \\cite{robertsIterativeGoalRefinement2014}, with domain specific rules guiding\ntheir progress. It uses PDDL to define\na domain model, which is parsed into the CLIPS environment to enable continuous reasoning on the\nstate of the world, using a representation that is compatible with a planner. Therefore, the notions\nof plans and actions are the ones known from planning with PDDL. Multiple instances\nof the \\ac{CX} (e.g., one per robot), can be coordinated through locking and\ndomain sharing mechanisms, as demonstrated in the \\acf{RCLL} \\cite{hofmannMultiagentGoalReasoning2021}.\nHowever, these mechansims only serve to avoid conflicts as each agent still reasons independently,\nwithout considering the other agents intents and goals.\n\n\n\\paragraph{Goals.}\nGoals are the basic data structure which model certain objectives to be achieved. \nEach goal comes with a set of preconditions which \nmust be satisfied by the current world state before being formulated. In the \\ac{CX}, these\npreconditions are modelled through domain-specific CLIPS rules.\n\nEach instance of a goal belongs to a class, which represents a certain objective\nto be achieved by the agent. Additionally, the goal can hold parameters that might be required\nfor planning and execution. A concrete \\emph{goal} is obtained by grounding all parameters of a goal class.\nDepending on the current state of the world, each robot may formulate multiple goals and multiple\ninstances of the same goal class.\n\nEach goal follows a \\emph{goal lifecycle}, as shown in Figure~\\ref{fig:cx-goal-lifecycle}.\nThe agent formulates a set of goals based on the current state of the world. These can\nbe of different classes, or multiple differently grounded versions of the same class. After\nformulation, one specific goal is selected, e.g., by prioritizing goals\nthat accomplish important objectives. The selection mechanism may be modelled in\na domain-specific way, which allows adapting the behavior according to the domain-specific requirements.\nIn the next step, the selected goal is \\emph{expanded} by\ncomputing a plan with an off-the-shelf PDDL planner. Alternatively,\na precomputed plan can be fetched from a plan library. Once a plan\nfor a goal is determined, the executive attempts to acquire its required\nresources. Finally, the plan is executed on a per-action basis.\n\n\n\\paragraph{Goals Example. }\nTo illustrate goals and goal reasoning in the CX, let us consider a simple resource mining scenario which we will\nevolve throughout this paper and finally use as the basis for our conceptual evaluation. Different case examples \nbased on the given setting are presented in Figure~\\ref{fig:timeline-promises}.\nUp to two robots (\\textsc{Wall-E}\\xspace and \\textsc{R2D2}\\xspace) produce the material \\emph{Xenonite} by first mining \\emph{Regolith} in a mine, \nrefining the Regolith to \\emph{Processite} in a refinery machine, and\nthen using Processite to produce Xenonite in a production machine. Let us assume the following setting as the\nbasis for our example:\none machine is filled with Regolith, one machine is empty, the robots currently pursue no goals. \nIn this scenario, each robot may pursue five classes of goals:\n\\begin{enumerate*}[label=(\\alph*)]\n \\item \\textsc{FillContainer} to fill a container with Regolith at the mine or with Processite,\n \\item \\textsc{CleanMachine} to fill a container with Processite or Xenonite at a machine,\n \\item \\textsc{Deliver} a container to a machine,\n \\item \\textsc{StartMachine} to use a refinery or production machine to process the material,\n \\item \\textsc{DeliverXenonite} to deliver the finished Xenonite to the storage.\n\\end{enumerate*}\nBecause one machine is filled, a goal of class \\textsc{StartMachine} can be formulated. The parameters of the\ngoal class are grounded accordingly, i.e., the target machine is set to the filled machine. Should both machines \nbe filled at the same time, two goals of class \\textsc{StartMachine} could be formulated, one for each of the machines.\nThe agent now selects one of the formulated goals for expansion. Since in our case only one goal is formulated,\nit will be selected. Should there be \\textsc{StartMachine} goals for each machine, then, e.g., the one operating on \nsecond machine in the production chain might be prioritized. \n\n\\paragraph{Execution Model.}\nAfter a goal has been expanded (i.e., a plan has been generated),\nthe \\ac{CX} commits to a certain plan and then dispatches the goal\nby executing the plan.\nFor sequential plans, whenever no action of the plan is executed, the \\ac{CX} selects the next\naction of the plan.\nIt then checks whether the action is executable by evaluating the action's precondition, and if so,\nsends the action to a lower level execution engine.\nIf the action's precondition is not satisfied, it remains in the pending state until either the\nprecondition is satisfied or a timeout occurs. In a multi-agent scenario, each \\ac{CX} instance\nperforms its own planning and plan execution independently of the other agents.\n\nThe execution of two goals with one robot in our example scenario is visualized in with one robot is visualized in Figure \\ref{fig:timeline-promises} \nas \\textsc{Scenario 1}. After finishing the goal of class \\textsc{StartMachine}, a goal of class \\textsc{CleanMachine} \nis formulated and dispatched, clearing the machine's output. \n\n\\paragraph{Multi-Agent Coordination.}\nThe \\ac{CX} supports two methods for multi-agent coordination: \\emph{locking actions} acquire a lock\nfor exclusive access, e.g., a location so no two robots try to drive to the same location.\nThey are part of a plan and executed similar to regular actions.\n\\emph{Goal resources} allow coordination on the goal rather than the action level.\nEach goal may have a set of \\emph{required resources} that it needs to hold exclusively while the\ngoal is dispatched. If a second goal requires a resource that is already acquired by another goal,\nthe goal is rejected. This allows for coordination of conflicting goals, e.g., two robots picking up the\nsame container. Both methods are intended to avoid conflicts and overlapping goals between the agents,\nrather than fostering active cooperation.\n\nTo illustrate coordination by means of goal resources, let us extend the previous example \\textsc{Scenario 1} \nby adding the second robot. In \\textsc{Scenario 2}, illustrated in Figure \\ref{fig:timeline-promises}, both \n\\textsc{Wall-E}\\xspace and \\textsc{R2D2}\\xspace formulate and select \\textsc{StartMachine} for \\texttt{M1}, \nwhich requires the machine as a resource.\n\\textsc{Wall-E}\\xspace starts formulating first and manages to acquire the resources. Therefore it is able to dispatch its goal.\n\\textsc{R2D2}\\xspace on the other hand must reject the goal as it is not able to acquire the machine resource.\nAfter its goal has been rejected, it may select a different goal. Since now the machine is occupied,\nit has to wait until the preconditions of some class are met and a new goal can be formulated.\nThis is the case once the effects of \\textsc{StartMachine} are\napplied, which causes \\textsc{R2D2}\\xspace to formulate and dispatch \\textsc{CleanMachine} to clear the machine.\n\n\\paragraph{Execution Monitoring.}\nThe \\ac{CX} includes execution monitoring functionality to handle exogenous events and action failure\n\\cite{niemuellerCLIPSbasedExecutionPDDL2018}. Since most actions interact with other agents or\nthe physical world, it is critical to monitor their execution for failure or delays. \nAction may either be retried or failed, depending on domain-specific evaluations of the execution behavior. \nAdditionally, timeouts are used to detect potentially stuck actions. The failure of an action might lead to reformulation\nof a goal. This means that the plan of the goal might be adapted w.r.t. the partial\nchange in the world state since original formulation. Alternatively, a completely new objective might be followed by the agent.\n\n\n\\section{Goal Reasoning with Goal Operators}\\label{sec:goal-preconditions}\n\\begin{figure}[htb]\n \\centering\n \\includestandalone[width=1.0\\linewidth]{figures\/goal-lifecycle}\n \\caption{\n The \\ac{CX} goal lifecycle \\cite{niemuellerGoalReasoningCLIPS2019} of two goals and their\n interaction by means of promises.\n When the first goal is dispatched, it makes a set of promises, which can be used by the second\n goal for its precondition check.\n If the goal precondition is satisfied based on the promises, the goal can already be formulated\n (and subsequently selected, expanded, etc.) even if the precondition is not satisfied yet.\n Additionally, promises are also used as timed initial literals (TILs) for planning in order to\n expand a goal.\n \n \n \n}\n \\label{fig:cx-goal-lifecycle}\n\\end{figure}\n\nTo define promises and how promises affect goal formulation in the context of goal reasoning,\nwe first need to formalize goals.\nSimilar to \\citet{coxModelPlanningAction2016}, we base our definition of a goal on classical\nplanning problems.\nHowever, as we need to refer to time later, we assume that each state is timed.\nFormally, given a finite set of logical atoms $A$, a state is a pair $\\left(s, t\\right) \\in 2^A\n\\times \\mathbb{Q}$, where $s$ is the set of logical atoms that are currently true, and $t$ is the\ncurrent global time.\nUsing the closed world assumption, logical atoms not mentioned in $s$ are implicitly considered to\nbe false.\nA literal $l$ is either a positive atom (denoted with $a$) or a negative atom (denoted with\n$\\overline{a}$).\nFor a negative literal $l$, we also denote the corresponding positive literal with $\\overline{l}$\n(i.e., $\\overline{\\overline{l}} = l$).\nA set of atoms $s$ satisfies a literal $l$, denoted with $s \\models l$ if\n\\begin{enumerate*}[label=(\\arabic*)]\n \\item $l$ is a positive literal $a$ and $a \\in s$, or\n \\item $l$ is a negative literal $\\overline{a}$ and $a \\not\\in s$.\n\\end{enumerate*}\nGiven a set of literals $L = \\{ l_1, \\ldots, l_n \\}$, the state $s$ satisfies $L$, denoted with $s\n\\models L$, if $s \\models l_i$ for each $l_i \\in L$.\n\nWe define a \\emph{goal operator} similar to a planning operator in classical planning~\n\\cite{ghallabAutomatedPlanningActing2016}:\n\n\\begin{definition}[Goal Operator]\n A \\emph{goal operator} is a tuple\n $\\gamma = \\left(\\mathrm{head}(\\gamma), \\mathrm{pre}(\\gamma), \\mathrm{post}(\\gamma)\\right)$, where\n \\begin{itemize}\n \\item $\\mathrm{head}(\\gamma)$ is an expression of the form $\\mathit{goal}(z_1, \\ldots, z_k)$, where\n $\\mathit{goal}$ is the \\emph{goal name} and $z_1, \\ldots, z_k$ are the \\emph{goal parameters}, which\n include all of the variables that appear in $\\mathrm{pre}(\\gamma)$ and\n $\\mathrm{post}(\\gamma)$,\n \\item $\\mathrm{pre}(\\gamma)$ is a set of literals describing the condition when a\n goal may be formulated,\n \\item $\\mathrm{post}(\\gamma)$ is a set of literals describing the objective that the\n goal pursues, akin to a PDDL goal.\n \\end{itemize}\n\\end{definition}\n\n\nA goal is a ground instance of a goal operator.\nA goal $g$ can be \\emph{formulated} if $s \\models \\mathrm{pre}(g)$. When a goal is\nfinished, its post condition holds, i.e., $s \\models \\mathrm{post}(g)$.\nNote that in contrast to an action operator (or a macro operator), the effects of a goal are not completely determined; any\nsequence of actions that satisfies the objective $\\mathrm{post}(g)$ is deemed feasible and\nadditional effects are possible.\nThus, the action sequence that accomplishes a goal is not pre-determined but computed by a planner\nand may depend on the current state $s$.\nConsider the goal \\textsc{CleanMachine} from Listing \\ref{lst:goal-operator}. Its objective defines\nthat the robot carries a container and that the container is filled. However, to achieve this\nobjective given the set preconditions, the robot also moves away from its current location,\nwhich will be an additional effect of any plan that achieves the goal.\n\n\n\n\\begin{lstlisting}[style=ReallySmallCLIPSFloat,label={lst:goal-operator},\ncaption={%\nThe definition of the goal operator \\textsc{Clean-Machine}. The precondition defined when a goal may be\nformulated; the objective defines the PDDL goal to plan for once the goal has been selected.}]\n(goal-operator (class CleanMachine)\n (param-names r side machine c mat)\n (param-types robot loc machine cont material)\n (param-quantified)\n (lookahead-time 20)\n (preconditions \"\n (and\n (robot-carries ?r ?c)\n (container-can-be-filled ?c)\n (location-is-free ?side)\n (location-is-machine-output ?side)\n (location-part-of-machine ?side ?machine)\n (machine-in-state ?machine READY)\n (machine-makes-material ?machine ?mat)\n (not (storage-is-full))\n )\n \")\n (objective \n \"(and (robot-carries ?r ?c)\n (container-filled ?c ?mat)\n )\")\n)\n\\end{lstlisting}\n\n\nWe have extended the \\ac{CX} with goal operators (shown in\nListing~\\ref{lst:goal-operator}). For each goal operator and every possible grounding of the\noperator, the goal reasoner instantiates the corresponding goal precondition and tracks its\nsatisfaction while the system is evolving. Once the goal precondition is satisfied, the goal is\nformulated. Afterwards, the goal follows the goal lifecycle as before\n(Figure~\\ref{fig:cx-goal-lifecycle}).\n\n\n\n\\section{Promises}\\label{sec:promises}\n\n\\begin{figure*}[p]\n \n \\centering\n \\includestandalone{figures\/scenarios}\n\n \\caption{\n Multiple scenarios of interaction between two goals and robots in the \\ac{CX} with and without promises\n as discussed in the running example.\n \\textsc{Scenario 1}: one robot \\textsc{Wall-E}\\xspace and no promises. It successively formulates and executes\n the applicable goals.\n \\textsc{Scenario 2}: two robots (\\textsc{Wall-E}\\xspace and \\textsc{R2D2}\\xspace) and no promises. Both attempt to \n dispatch \\textsc{StartMachine}, illustrating interaction between multiple agents and goal resources. \n \\textsc{Scenario 3}: two robots, promises. \\textsc{R2D2}\\xspace formulates \\textsc{CleanMachine}\n earlier from the promises, but the action \\texttt{collect-processite} is pending until\n the effects from \\textsc{Wall-E}\\xspace are applied. \n \\textsc{Scenario 4}: two robots, promises, and failed resource handover. \\texttt{collect-processite} is pending\n until the resource \\texttt{M1} is handed over. A timeout occurs and the goal fails.\n }\n \\label{fig:timeline-promises}\n\\end{figure*}\n\n\n\\newcommand*{\\ensuremath{\\mathtt{From}}}{\\ensuremath{\\mathtt{From}}}\n\\newcommand*{\\ensuremath{\\mathtt{Until}}}{\\ensuremath{\\mathtt{Until}}}\n\nA goal reasoning agent relying on goal operators has a model of the conditions that have to be met\nbefore it executes a goal $g$ ($\\mathrm{pre}(g)$) and a partial model of the world\nafter the successful execution of its plan (its objective $\\mathrm{post}(g)$). Thus, when the agent\ndispatches $g$, it assumes that it will accomplish the objective at some point\nin the future (i.e. $s \\models \\mathrm{post}(g)$). However, the other agents are oblivious\nto $g$'s objectives and the corresponding changes to the world and therefore need to wait until the\nfirst agent has finished its goal. To avoid this unnecessary delay, we introduce \\emph{promises},\nwhich intuitively give some guarantee for certain literals to be true in a future state:\n\\begin{definition}[Promise]\nA \\emph{promise} is a pair $\\left(l, t\\right)$, stating that the literal $l$ will be satisfied at\n(global) time $t$.\n\\end{definition}\n\nPromises are made to the other agents by a \\emph{promising} agent when it dispatches a goal.\nThe \\emph{receiving} agent may now use those promises to evaluate whether a goal can be formulated, even\nif its precondition is not satisfied by the world.\nMore precisely, given a set of promises, we can determine the point in time when a goal precondition\nwill be satisfied:\n\\begin{definition}[Promised Time]\nGiven a state $\\left(s, t\\right)$ and a set of promises $P = \\{ \\left(l_i, t_i\\right) \\}_i$, we define\n$\\ensuremath{\\mathtt{From}}(l, s, t, P)$ as the timepoint when a literal $l$ is satisfied and $\\ensuremath{\\mathtt{Until}}(l, s, t,\nP)$ as the timepoint when $l$ is no longer satisfied:\n\\begin{align*}\n \\ensuremath{\\mathtt{From}} (l, s, t, P) &=\n \\begin{cases}\n t & \\text{ if } s \\models l\n \\\\\n \\min_{\\left(l, t_i\\right) \\in P} t_i & \\text{ if } \\exists t_i: \\left(l, t_i\\right) \\in P\n \\\\\n \\infty & \\text{ else }\n \\end{cases}\n \\\\\n \\ensuremath{\\mathtt{Until}} (l, s, t, P) &=\n \\begin{cases}\n t & \\text{ if } s \\models \\overline{l}\n \\\\\n \\min_{\\left(\\overline{l}, t_i\\right) \\in P} t_i & \\text{ if } \\exists t_i: \\left(\\overline{l}, t_i\\right) \\in P\n \\\\\n \\infty & \\text{ else }\n \\end{cases}\n\\end{align*}\n\nWe extend the definition to a set of literals $L$:\n\\begin{align*}\n \\ensuremath{\\mathtt{From}}(L, s, t, P) &= \\max_{ l_i \\in L} \\ensuremath{\\mathtt{From}}(l_i, s, t, P)\n \\\\\n \\ensuremath{\\mathtt{Until}}(L, s, t, P) &= \\min_{l_i \\in L} \\ensuremath{\\mathtt{Until}}(l_i, s, t, P)\n\\end{align*}\n\\end{definition}\n\nPromises are tied to goals and can either be based on the effects of the concrete plan actions\nof an expanded goal, or simply be based on a goal's postcondition. They can be extracted automatically\nfrom these definitions, or hand-crafted in simple domains. Though the concept can be applied more\nflexibly (e.g. dynamic issuing of promises based on active plans and world states), in our\nimplementation promises are static and created during goal expansion.\n\n\\subsection{Goal Formulation with Promises}\nWe continue by describing how promises and promise times for literals can be used for goal\nformulation.\nAs shown in Listing~\\ref{lst:goal-operator}, we\naugment the goal operator with a \\emph{lookahead time}, which is used to evaluate the goal\nprecondition for future points in time. Given a set of promises $P$, a lookahead time $t_l$ and a goal precondition\n$\\mathrm{pre}(\\gamma)$, the goal reasoner formulates a goal in state $\\left(s, t\\right)$ iff the\nprecondition\nwill be satisfied within the next $t_l$ time units, i.e., iff\n\\[\n \n \\ensuremath{\\mathtt{From}}(\\mathrm{pre}(\\gamma),s, t, P) \\leq t + t_l\n\\]\nNote that if $s \\models \\mathrm{pre}(\\gamma)$, then $\\ensuremath{\\mathtt{From}}(\\mathrm{pre}(\\gamma), s, t, P) = t$ and thus\nthe condition is trivially satisfied.\nAlso, if the lookahead time is set to $t_l = 0$, then the goal is formulated iff its precondition is\nsatisfied in the current state, thus effectively disabling the promise mechanism.\n\nFurthermore, this condition results in an optimistic goal formulation, as the\ngoal will still be formulated even if the goal's precondition is promised to be no longer satisfied\nat a future point in time.\nTo ensure that a goal is only formulated if its precondition is satisfied for the whole lookahead\ntime, we can additionally require:\n\\[\n \\ensuremath{\\mathtt{Until}}(\\mathrm{pre}(\\gamma), s, t, P) \\geq t + t_l\n\\]\nIn our implementation, $\\ensuremath{\\mathtt{Until}}$\nis currently not considered, as we are mainly interested in optimistic goal formulation.\n\nIncorporating promises into goal formulation leads to more cooperative behavior\nbetween the agents as goals that build on other goals can be considered by an agent\nduring selection, thereby enabling parallelism.\nThus, it is a natural extension of the goal lifecycle with an intention sharing mechanism.\n\n\\paragraph{Promises in the CX.}\nIn the \\ac{CX}, promises are continously evaluated to compute $\\ensuremath{\\mathtt{Until}}$ and $\\ensuremath{\\mathtt{From}}$ for each\navailable grounding of a goal precondition formula. The results are stored for each\nformula and computed in parallel to the normal formula satisfaction evaluation.\nTherefore, promises do not directly interfere with the world model, but rather are\nintegrated into the goal formulation process by extending the precondition check to\nalso consider $\\ensuremath{\\mathtt{From}}$ for a certain lookahead time $t_l$.\n\nThe lookahead time is manually chosen based on the expected time to accomplish the objective and are\nspecific for each goal. In scenarios with constant execution times for actions, those can be\nused directly as the estimate. If execution times for actions are not constant\n(e.g. driving actions with variable\nstart and end positions), average recorded execution times or more complex estimators might be used,\nat the loss of promise accuracy. By choosing a lookahead time of $0$, promises can be ignored.\n\nPromises are shared between the agents through the shared world model of the \\ac{CX}. For now,\nwe hand-craft promises for each goal operator $\\gamma$ based on $\\mathrm{post}(g)$. However,\nextracting promises automatically from postconditions and plans may be considered in future work.\nIf a goal is completed or fails, the promises are removed from the world\nmodel.\n\n\n\\subsection{Using Promises for Planning}\nWith promises, an agent is able to formulate a goal in expectation of another agent's\nactions. However, promises also need to be considered when expanding a goal with a PDDL\nplanner. To do so, a promise is translated into a\n\\acfiu{TIL}~\\cite{edelkampPDDL2LanguageClassical2004}. Similar to a promise, a \\ac{TIL} states that\nsome literal will become true at some point in the future, e.g.,\n\\lstinline|(at 5 (robot-at M1)| states that\n\\lstinline|(robot-at M1)| will become true in 5 time steps.\nWe extended the PDDL plan expansion of the \\ac{CX} to\ntranslate promises into \\acp{TIL} and to use\n\\textsc{POPF}\\xspace~\\cite{colesForwardchainingPartialorderPlanning2010}, a temporal PDDL planner that\nsupports \\acp{TIL}.\n\n\n\\begin{figure*}[t]\n \\centering\n \\begin{subfigure}{\\textwidth}\n \n \\scalebox{0.5}{\\input{figures\/report-baseline.pgf}}\n \\caption{A run without promises.\n Each goal formulation must wait until the previous goal has finished.\n }\n \\label{fig:xenonite-baseline}\n \\end{subfigure}\n \\begin{subfigure}{\\textwidth}\n \n \\scalebox{0.5}{\\input{figures\/report-promises.pgf}}\n \\caption{\n A run with promises.\n The striped goals have been formulated based on promises.\n Compared to the run without promises, the promise-dependent goals (e.g., the first\n \\textsc{CleanMachine} goal by \\textsc{Eve}\\xspace) are dispatched earlier, because a future effect of another goal \n (i.e. the second \\textsc{StartMachine} goal by \\textsc{Wall-E}\\xspace)\n is considered for goal formulation and expansion, thus leading to an overall better\n performance.\n }\n \\label{fig:xenonite-promises}\n \\end{subfigure}\n \\caption{A comparison of two exemplary runs in the Xenonite domain.}\n \\label{fig:xenonite}\n\\end{figure*}\n\n\\subsection{Goal Execution with Promises}\n\nWhen executing a goal that depends on promises (i.e., it has been formulated based on promises), it is necessary to check whether the\npromise has actually been realized. To do so, we build on top of the execution engine of the\n\\ac{CX} \\cite{niemuellerCLIPSbasedExecutionPDDL2018}. For each plan action, the \\ac{CX} continuously\nchecks the action's precondition and only starts executing the action once the precondition\nis satisfied. While the goal's preconditions can be satisfied (in parts or fully) by\nrelying on promises, they are not applied to action preconditions.\nThus, the action execution blocks until the precondition is satisfied. This\nensures that actual interactions in the world only operate under the assumption of correct\nand current world information. Otherwise, actions might fail or lead to unexpected behavior. \n\nLet us revise our examples from \\textsc{Scenario 1} and \\textsc{Secenario 2} by introducing\npromises, but ignoring the role of goal resources for now. Similarly to the previous scenarios, \n\\textsc{Wall-E}\\xspace formulates, selects and\nexecutes a goal of class \\textsc{StartMachine} on machine \\texttt{M1}. With promises, \nthe expected outcome of that goal (\\lstinline|machine-in-state(M1, READY)|) will \nbe announced to the other agents when the goal gets dispatched ($t_1$). \\textsc{R2D2}\\xspace now uses\nthe provided information to formulate the goal \\textsc{CleanMachine} ahead of time, even though\nthe effects of \\textsc{StartMachine} have not been applied to the world yet. This behavior is shown in \n\\textsc{Scenario 3} of Figure~\\ref{fig:timeline-promises}. The first action of the goal's plan (\\lstinline|move|) \ncan be executed directly. However, the precondition of \naction \\lstinline|collect-processite| is not yet satisifed, since the action \n\\lstinline|start-machine| is still active on \\textsc{Wall-E}\\xspace until $t_2$. The executive of \\textsc{R2D2}\\xspace notices this\nand \\lstinline|collect-processite| is pending until the effects have been applied. Once\nthe effects are applied, \\textsc{R2D2}\\xspace can start executing its action.\nAn example of this behavior occuring in simulation is visualized in Figure~\\ref{fig:xenonite}.\n\n\n\\subsection{Execution Monitoring with Promises}\nGoals that issue promises might take longer than expected, get stuck, or fail completely, possibly\nleading to promises that are never realized. Therefore,\ngoals that have been formulated based on promises may have actions whose preconditions will never\nbe satisfied. To deal with this situation, we again rely on execution monitoring.\nIf a pending action is not executable for a certain amount of time, a timeout occurs and the action\nand its corresponding goal are aborted. However, the timeout spans are increased\nsuch that the potentially longer wait times from promise-based execution can be accounted for.\n\nPromises may not cause deadlocks, as a deadlock may only appear if two\ngoals depend on each other w.r.t. promises. However, this is not possible,\nas promises are used to formulate a goal, but only are issued when\nthe goal is dispatched. Thus, cycles of promises are not possible.\n\nRecall our previous example \\textsc{Scenario 3}. Action \\lstinline|collect-processite| \nremains in the state pending until \\textsc{Wall-E}\\xspace finishes the execution of action\n\\lstinline|start-machine| and its effects are applied. Should \\textsc{Wall-E}\\xspace not complete the action\n(correctly), execution monitoring will detect the timeout on the side of \\textsc{R2D2}\\xspace and will trigger\nnew goal formulation. If the promised-from time has elapsed, the promises\n will not be considered by \\textsc{R2D2}\\xspace anymore when formulating new goals, leading to a different objective\n to be chosen by the agent. Should \\textsc{Wall-E}\\xspace manage to fulfill its goal, then the effects will be applied, and\n\\textsc{R2D2}\\xspace's original goal's preconditions are satisfied through the actual world state.\n\n\n\\subsection{Resource Locks with Promises}\nIn the \\ac{CX}, resource locks at the goal level are used to coordinate conflicting goals.\nNaturally, goals that are formulated on promises often rely on some of the same resources that are\nrequired by the goal that issued the promise. A goal that fills a machine requires the machine as a\nresource. Another goal that is promise-dependent and operates on the same machine might require it\nas a resoure as well. In this scenario, the second goal may be formulated but would immediately be\nrejected, as the required resource is still held by the promising goal. To resolve this issue,\npromise-dependent goals will delay the resource acquisition for any resource that is currently\nheld by the promising goal. As soon as the promising goal is finished and therefore\nthe resource is released, the resource is acquired by the promise-dependent goal. To make sure that\nthere is no conflict between two different promise-dependent goals, for each such delayed resource,\na \\emph{promise resource} is acquired, which is simply the main resource prefixed with\n\\texttt{promised-}. Effectively, a promise-dependent goal first\nacquires the promise resource, then, as soon as the promising goal releases the resource, the\npromise-dependent goal acquires the main resource, and then releases the promise resource.\n\nTo illustrate the mechanism, let us once again consider the example from \\textsc{Scenario 3}.\nWhen dispatched, the goal \\textsc{StartMachine}\nholds the resource \\texttt{M1} and \\textsc{R2D2}\\xspace formulates the goal \\textsc{CleanMachine} based on \\textsc{Wall-E}\\xspace's\npromise. Since the goal operates on the same machine, it needs to acquire the same goal resource. As\nthe goal is promise-dependent, it first acquires the resource \\texttt{promised-M1}. This resource is\ncurrently held by no other agent, thus it can be acquired and the goal dispatched. \\textsc{R2D2}\\xspace first\nexecutes the action \\texttt{move(BASE, M1)}. As before, the next action\n\\texttt{collect-processite(M1)} remains pending until \\textsc{Wall-E}\\xspace has completed its goal. At this point,\n\\textsc{Wall-E}\\xspace releases the resource \\texttt{M1}, which is then acquired by \\textsc{R2D2}\\xspace. Should this resource\nhandover fail, e.g., because the goal \\textsc{StartMachine} by \\textsc{Wall-E}\\xspace never releases its resources,\nthe action eventually times out and \\textsc{R2D2}\\xspace's goal is aborted, as shown in \\textsc{Scenario 4} of\nFigure~\\ref{fig:timeline-promises}.\n\n\n\\section{Evaluation}\\label{sec:evaluation}\nWe evaluate our prototypical implementation\\footnote{\\url{https:\/\/doi.org\/10.5281\/zenodo.6610426}} of promises for multi-agent cooperation as an extension\nof the \\ac{CX} in the same simplified production logistics scenario\nthat we used as a running example throughout this paper. In our evaluation scenario, three robots were\ntasked with filling $n=5$ containers with the materials Xenonite and to deliver it to a\nstorage area. The number of containers was intentionally chosen to be larger than the\nnumber of robots to simulate an oversubscription problem and highlight the effects of\nbetter cooperation on task efficiency. The three robots first collect raw materials and then \nrefine them stepwise by passing them through a series of two machines in sequence.\nOnce all containers are filled and stored, the objective is completed.\n\n\nWe compare the performance of the three robots using \\textsc{POPF}\\xspace to expand goals.\nFigure \\ref{fig:timeline-promises} shows how promises can lead to faster execution of goals by starting\na goal (which always starts with a move action) while the goal's precondition (e.g., that the machine is in\na certain state) is not fulfilled yet.\nFigure~\\ref{fig:xenonite} shows exemplary runs of three\nrobots producing and delivering 5 containers of Xenonite. The expected behavior as highlighted\nin Figure~\\ref{fig:timeline-promises} can indeed be observed in the simulation runs.\n\nTo compare the performance, we simulated each scenario five times.\nAll experiments were carried out on an Intel Core i7 1165G7 machine with \\SI{32}{\\giga\\byte} of RAM.\nWithout promises, the three robots needed \\SI{303}{\\sec} to fill and store all containers,\ncompared to \\SI{284.4 \\pm 0.55}{\\sec} with promises.\nAt each call, \\textsc{POPF}\\xspace needed less than \\SI{1}{\\sec} to generate a plan for the given goal.\nThis shows, at least in this simplified scenario, that promises lead to more effective\ncollaboration, and that the planner can make use of promises when expanding a goal.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn multi-agent goal reasoning scenarios, effective collaboration is often difficult, as one agent is\nunaware of the goals pursued by the other agents. We have introduced \\emph{promises}, a method for\nintention sharing in a goal reasoning framework. When dispatching a goal, each agent promises a set of \nliterals that will be true at some future timepoint, which can be used by another agent to formulate and\ndispatch goals early. We have described how promises can be defined based on \\emph{goal operators},\nwhich describe goals similar to how action operators describe action instances. By translating\npromises to timed initial literals, we allow the PDDL planner to make use of promises\nduring goal expansion. The evaluation showed that using promises with our prototypical\nimplementation improved the performance of a team of three robots in a simplified logistics\nscenario. For future work, we plan to use and evaluate promises in a real-world scenario from the\n\\acf{RCLL}~\\cite{niemuellerPlanningCompetitionLogistics2016} and compare it against our distributed\nmulti-agent goal reasoning system~\\cite{hofmannMultiagentGoalReasoning2021}.\n\n\n\\FloatBarrier\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}