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Girardi} \\footnote{on leave of absence of LAPTH, UMR\n5108 du CNRS associ\\'ee au L.A.P.P.},\n{\\large R. Grimm} \\\\\n\n{\\em CPT \\footnote{UMR 6207 du CNRS}, C.N.R.S. Luminy, Case 907,\nF-13288 Marseille Cedex 09, France} \\\\[1.2cm]\n\n\n{\\large B. Labonne and J. Orloff} \\\\\n{\\em LPC, Universit\\'{e} Blaise Pascal, F-63177, Aubi\\`{e}re Cedex, France}\\\\\n\\end{center}\n\\vs{15}\n\n\n\\centerline{ {\\bf Abstract}}\n\n\\indent In analogy to the chiral-linear multiplet correspondence we establish a relationship between the 3-form (or gaugino condensate) multiplet and a coupled non-minimal $(0,1\/2)$ multiplet, illustrated by a simple explicit example.\n\n\\vfill\n\\rightline{CPT-P92-2007}\n\\rightline{PCCF-RI-0705}\n\\vskip .2cm \\rightline{\\today}\n\\end{titlepage}\n\n\\newpage\n\\pagenumbering{roman}\n\\pagenumbering{arabic}\n\\section{Introduction}\n \\hspace{1cm} The scalar multiplet \\cite{WZ74c}, commonly termed chiral\nmultiplet, is the most popular realization of the $(0,1\/2)$\nrepresentation (massive or massless) of supersymmetry in terms of local quantum\nfields. It contains as components a complex scalar, a Majorana\nspinor and a complex scalar auxiliary field. Another realization of the\nsame representation (generally massless in this case) is provided\nby the linear multiplet \\cite{FZW74}, given in terms of\na real scalar, a 2-index antisymmetric tensor gauge field, a\nMajorana spinor and no auxiliary field. Contrary to the previous\none the linear multiplet is a gauge multiplet.\n In classical Lagrangian field theory one can establish\n\\cite{Sie79} a certain correspondence between the chiral multiplet\nand the linear multiplet, sometimes referred to as {\\em{chiral-linear\nmultiplet duality}}, in particular in applications where the\nlinear multiplet incarnates a\n{\\em{dilaton-axion multiplet}}. In this note we would like to draw\nattention to yet another couple of realizations of the $(0, 1\/2)$\nrepresentation, the\n{\\em{3-form multiplet}} \\cite{Gat81} and a {\\em{non-minimal $(0,\n1\/2)$ multiplet}} \\cite{GatesSieg}, including simple chiral\nmultiplet couplings.\n\nThe 3-form multiplet made of a 3-index antisymmetric tensor gauge field,\n a complex scalar, a Majorana spinor and a real auxiliary field may be\n understood as a {\\em{further constrained chiral multiplet}}.\n It is the basic ingredient in the context of gaugino condensation,\n but is also relevant in the theory of supersymmetric gauge anomalies\n and in the description of curvature squared terms and Chern-Simons forms\n in supersymmetry. The non-minimal $(0, 1\/2)$ multiplet, on the other\n hand, is less well known. In this note, we would like to outline\n a relation with the 3-form multiplet in very much the same vein as the above\n mentioned chiral-linear correspondence.\nTo be definite, we shall exhibit here a very simple toy model,\n coupling the gaugino-condensate multiplet to a single generic\n chiral multiplet and suggest a corresponding coupling of the non-minimal\n $(0, 1\/2)$ multiplet.\n\n It may be worthwhile to comment briefly on the notion of {\\em{gaugino condensate multiplet}}. In a supersymmetric gauge theory the gauge field-strength tensor is promoted to a multiplet containing as superpartners the gaugino and a real bosonic auxiliary field. The corresponding gaugino superfield, denoted $W_{\\alpha}, {\\bar{W}}^{\\dot{\\alpha}}$ is chiral ($ {\\bar{D}}^{\\dot{\\alpha}} W_{\\alpha}=0, D_{\\alpha} {\\bar{W}}^{\\dot{\\alpha}}=0$) and subject to the additional constraint $D^{\\alpha} W_{\\alpha}= {\\bar{D}}_{\\dot{\\alpha}} {\\bar{W}}^{\\dot{\\alpha}}$.\nIrrespectively of the mechanisms underlying {\\em{gaugino\ncondensation}}, the constraints on $W_{\\alpha},\n{\\bar{W}}^{\\dot{\\alpha}}$ imply that the {\\em{condensate}} superfields $tr(W^{\\alpha} W_{\\alpha}) ,\ntr({\\bar{W}}_{\\dot{\\alpha}} {\\bar{W}}^{\\dot{\\alpha}})$ are not only chiral\nbut fulfill the additional condition $$D^2\\,tr(W^{\\alpha} W_{\\alpha}) -\n{\\bar{D}}^2\\,\\, tr({\\bar{W}}_{\\dot{\\alpha}} {\\bar{W}}^{\\dot{\\alpha}})= i{\\varepsilon}_{klmn} tr \\left( F^{kl}\nF^{mn}\\right).$$ Interestingly enough, this supermultiplet can be\nviewed as a particular realization of a generic 3-form gauge\ntheory in superspace with the 3-form gauge potential related to\nthe Chern-Simons form of the Yang-Mills theory.\n\n\n\n\\sect{ The 3-Form Multiplet \\label{3F2}}\n\n\\hspace{1cm} In multiplets of supersymmetry different components may be\nassigned different $R$-weights, in relation to their supersymmetry\ntransformations and the chiral properties of their generators\n\\cite{FF77}. As it seems reasonable to assign vanishing $R$-weight\nto gauge potential components, the $R$-weights of their\nsupersymmetry partners are then determined correspondingly.\nPrecisely in the case of $C_{klm}(x)$, the 3-form gauge potential\nof the gaugino condensate multiplet with vanishing $R$-weight, the weights of\nthe other components are dictated by supersymmetry: in\nunits where $r(\\theta)=r({\\bar{D}})=+1,\\, r(\\bar{\\theta})=r(D)=-1$, the\ncomplex scalar $Y(x), {\\overline{Y}}(x)$ has $r(Y)=+2,\\, r({\\overline{Y}})=-2$. The fermionic components $\\eta_{\\alpha}(x), \\bar{\\eta}^{\\dot{\\alpha}}(x)$ acquire $r(\\eta)=+1, r(\\bar{\\eta})=-1$, whereas\n$H(x)$, the real auxiliary field has $r(H)=0$. Therefore, $H(x)$ may constitute by itself an R-inert supersymmetric Lagrangian in\nanalogy with the Fayet-Iliopoulos $D$-term familiar in\nsupersymmetric gauge theory.\n\nIn superfield language, the 3-form multiplet is characterized by\nthe superfields $Y, {\\overline{Y}}$ subject to the chirality conditions\n\\begin{equation}\\label{ChiY} {\\bar{D}}^{\\dot{\\alpha}} Y \\,=\\,0 , \\hspace{5mm} D_{\\alpha} {\\overline{Y}} \\,=\\,0 ,\n\\end{equation}\nand the additional constraint\n \\begin{equation}\\label{conY} D^2 Y\n- {\\bar{D}}^2 {\\overline{Y}} \\,=\\, -8i\\, \\partial C (x),\n\\end{equation}\nwith $\\partial C=-\\frac{4}{3} {\\epsilon}^{klmn}\\partial_k C_{lmn}$. These superfield relations\nhave an interpretation as Bianchi identities in superspace geometry \\cite{Gat81}, \\cite{BPGG96}.\nComponent fields are identified as usual by projection to lowest superfield components\n\\begin{equation}\n{\\overline{Y}}{\\loco} \\,=\\,{\\overline{Y}}(x), \\hspace{3mm} {\\bar{D}}^{\\dot{\\alpha}} {\\overline{Y}}{\\loco} \\,=\\,\\sqrt{2} \\,\n\\bar{\\eta}^{\\dot{\\alpha}} (x), \\hspace{3mm} Y{\\loco} \\,=\\,Y (x), \\hspace{3mm} D_{\\alpha} Y{\\loco}\n\\,=\\,\\sqrt{2} \\, \\eta_{\\alpha}(x),\n\\end{equation}\n\\begin{equation}\nD^2 Y{\\loco} + {\\bar{D}}^2 {\\overline{Y}}{\\loco} \\,=\\,-8 \\, H(x), \\hspace{1cm} C_{lmn} \\loco =C_{lmn}(x).\n\\end{equation}\n\n\nSupersymmetry transformations for these components read\n\n\\begin{equation}\n \\delta_{\\xi} C_{mlk} \\,=\\,\\frac{\\sqrt{2}}{16} \\left(\n{\\bar{\\xi}} {\\bar{\\sigma}}^n \\eta - \\xi {\\sigma}^n \\bar{\\eta} \\right) {\\varepsilon}_{nmlk},\n\\end{equation}\n\\begin{equation}\n {\\delta}_{\\xi} Y \\, = \\, \\sqrt{2} \\, \\xi^{\\alpha} \\eta_{\\alpha} , \\hspace{1cm}\n {\\delta}_{\\xi} {\\overline{Y}} \\, = \\, \\sqrt{2} \\, {\\bar{\\xi}}_{\\dot{\\alpha}} \\bar{\\eta}^{\\dot{\\alpha}} ,\n\\end{equation}\n\\begin{equation}\n {\\delta}_{\\xi} \\eta_{\\alpha} = \\sqrt{2} \\xi_{\\alpha} \\left( H +i \\partial C \\right)\n +i\\sqrt{2} ({\\bar{\\xi}}{\\bar{\\sigma}}^m {\\epsilon})_{\\alpha} \\partial_m Y,\n\\end{equation}\n\\begin{equation}\n {\\delta}_{\\xi} \\bar{\\eta}^{\\dot{\\alpha}} = \\sqrt{2} {\\bar{\\xi}}^{\\dot{\\alpha}} \\left( H -i \\partial C \\right) +i \\sqrt{2}\n(\\xi {\\sigma}^m {\\epsilon})^{\\dot{\\alpha}} \\partial_m {\\overline{Y}},\n\\end{equation}\n\\begin{equation}\n {\\delta}_{\\xi} H = \\f{i}{\\sqrt{2}} ({\\bar{\\xi}}{\\bar{\\sigma}}^m\\partial_m \\eta)\n+\\frac{i}{\\sqrt{2}} (\\xi {\\sigma}^m \\partial_m \\bar{\\eta}) . \\label{delH}\n\\end{equation}\n\n\nTaking care of the overall $R$-weights of $Y$ and ${\\overline{Y}}$,\ninvariant component field Lagrangians may be obtained from\n(\"$D$-term integration\")\n\\begin{equation}\\label{LYY}\n {\\cal L}_{Y {\\overline{Y}}} = \\int d^2 \\th d^2 \\bar{\\theta} \\,(Y {\\overline{Y}}) =\n -\\partial_m Y \\partial^m {\\overline{Y}} +\\frac{i}{2}(\\partial_m \\eta \\sigma^m\n \\bar{\\eta} - \\eta \\sigma^m \\partial_m \\bar{\\eta}) +H^2 +{\\partial C}^2,\n\\end{equation}\n the kinetic Lagrangian density and (\"$F$-term integration\")\n\\begin{equation}\\label{Hterm}\n\\int d^2 \\th \\, Y +\\int d^2 \\bar{\\theta}\\, {\\overline{Y}}= 2H(x),\n\\end{equation}\ngiving rise to the $H$-term referred to above.\n\nLet us consider, as a very simple example, the coupling to a single\nchiral superfield\\footnote{Component fields for $ \\phi$ are\n $\\phi \\loco =A(x), D_{\\alpha} \\phi \\loco =\\sqrt{2} \\chi_{\\alpha} (x), D^2\n \\phi \\loco =-4F(x)$, likewise for ${\\bar{\\phi}}$. }, $\\phi$, of vanishing $R$-weight, {\\em {i.e.\\ }} adding a\nkinetic density $$ \\int d^2 \\th d^2 \\bar{\\theta} \\, \\phi {\\bar{\\phi}},$$ and\ngeneralizing (\\ref{Hterm}) to\n$$ \\int d^2 \\th \\, Y \\, U\n(\\phi) +\\int d^2 \\bar{\\theta} \\, {\\overline{Y}} \\, \\bar{U} ({\\bar{\\phi}}), $$ with $U(\\phi),\n\\bar{U}({\\bar{\\phi}})$ at most quadratic in the renormalizable case.\n\nIn fact, using the explicit solutions of\nthe constraints (\\ref{ChiY}, \\ref{conY})\n\\begin{equation}\\label{omega}\n Y= -4 {\\bar{D}} ^2\\ \\Omega, \\hspace{3mm} {\\overline{Y}}= -4 D ^2\\ \\Omega,\n\\end{equation}\nwith $\\Omega$ the real unconstrained pre-potential (undetermined up\nto a linear superfield pre-gauge transformation) and employing\nintegration by parts in superspace, the complete action density\nmay be written as a pure $D$-term integration\n\n\\begin{equation}\\label{complact}\n \\int d^2 \\th d^2 \\bar{\\theta} \\ \\left[ \\frac{}{} Y {\\overline{Y}} + \\phi {\\bar{\\phi}} + 16 \\Omega \\left( U(\\phi)\n +{\\bar{U}} ({\\bar{\\phi}}) \\right) \\right],\n\\end{equation}\nwith suitable superfield equations of motion.\n\n\n\\sect{A simple model }\n\n\\hspace{1cm} In this section, we consider a particular combination of the\n3-form and a chiral multiplet in choosing $U(\\phi)={\\alpha} +\\mu\n\\phi, \\, \\mu \\in \\mathbb{R}$, giving rise to the superfield action\ndensity:\n\\begin{equation}\\label{Laggen}\n \\int d^2 \\th d^2 \\bar{\\theta} \\left( \\phi {\\bar{\\phi}} +Y {\\overline{Y}} \\right)\n +\\int d^2 \\th \\left( {\\alpha} +\\mu \\phi \\right) Y\n + \\int d^2 \\bar{\\theta} \\left( {\\bar{{\\alpha}}}+\\mu {\\bar{\\phi}} \\right) {\\overline{Y}}.\n\\end{equation}\nAt the component field level, this action contains the kinetic\nterms for $A, {\\bar{A}}, \\chi, {\\bar{\\chi}}$ (chiral multiplet) and $Y, {\\overline{Y}},\n\\eta, \\bar{\\eta}$ (3-form multiplet), mixing terms of these with $F,\n{\\bar{F}}$ (chiral multiplet), $H$ (3-form multiplet) and, last but not\nleast, the terms containing $\\partial C$, the field-strength of the\n3-index antisymmetric gauge potential.\n\nIn many cases, in supersymmetric field theories, {\\em{elimination\nof auxiliary fields}} means rather diagonalization in terms\n of non propagating fields (no derivative terms in the action\n density) with trivial algebraic equations of motion. In the case\n at hand, this can be done easily for the part of the action\n density containing $F, {\\bar{F}}$ and $H$, yielding\n\\begin{eqnarray}\\label{LagY}\n{\\cal L}&=&-\\partial^m \\! A \\, \\partial_m {\\bar{A}} - \\f{i}{2} \\left( \\chi {\\sigma}^m \\partial_m\n{\\bar{\\chi}} + {\\bar{\\chi}} {\\bar{\\sigma}}^m \\partial_m \\chi \\right) -\\partial_m Y \\partial^m {\\overline{Y}}\n-\\frac{i}{2}( \\eta\\sigma^m \\partial_m \\bar{\\eta} + \\bar{\\eta}{\\bar{\\sigma}}^m \\partial_m\n\\eta) \\nonumber \\\\&&\n - \\mu\\,\\left( \\chi \\eta + {\\bar{\\chi}} \\bar{\\eta} \\right)-|\\,\\mu\\,Y|{}^2\n -|{\\alpha}+\\mu \\,A|{}^2+\\left[ \\partial C +\\frac{i}{2}\\left( {\\alpha} -\\bar{{\\alpha}}+\\mu\\,(A -{\\bar{A}})\\right) \\right]^2 \\nonumber\n \\\\&& +\\, \\cal{F}\\, \\overline{\\cal{F}}+ {\\cal{H}}{\\cal{H}},\n\\end{eqnarray}\nwith diagonalized auxiliary fields\n \\begin{equation} \\label{auxeq}{\\cal{F}}=F+\\mu {\\overline{Y}}, \\hspace{3mm} \\overline{\\cal{F}}={\\bar{F}}+\\mu\nY,\\hspace{3mm}{\\cal{H}}=H+\\f12 ({\\alpha}+\\bar{{\\alpha}})+ \\frac{\\mu}{2}(A + {\\bar{A}}).\n\\end{equation}\n\nThe complex scalar $Y, {\\overline{Y}}$ satisfies a Klein-Gordon equation\nwith mass $\\mu$, the Weyl spinors $ \\eta,{\\bar{\\chi}},$\ncombine into a Dirac spinor of the same mass. The equations of\nmotion for the fields $A, {\\bar{A}}, C_{klm}$ are most\nconveniently written using\n $A =A_1 +i A_2,\\, {\\alpha}={\\alpha}_1 +i {\\alpha}_2$, so that\n\\begin{eqnarray} \\label{aplus}\n \\Box A_1 - \\mu^2\\,\\left( A_1 +\\frac{{\\alpha}_1}{ \\mu} \\right)&=& 0, \\\\ \\label{amoins}\n \\Box A_2 - \\mu \\partial C \\,&=&0, \\\\\n \\label{dc}\n\\partial_m \\, ( \\partial C - \\mu \\, A_2 ) &=&0.\n \\end{eqnarray}\nThe last equation is compatible with a constant $ K_2 =\\partial C - \\mu \\, A_2 $,\ngiving rise to a shifted Klein-Gordon equation for $A_2$\n\\begin{eqnarray} \\label{KGA}\n&& (\\Box- \\mu^2 )\\,\\left( A_2 +\\frac{ {\\alpha}_2+K_2}{\\mu} \\right) =0.\n \\end{eqnarray}\nWe would like to stress that these features arise necessarily in\nthe context\n of models dealing with gaugino condensation.\n\n \\sect{ The $X-Y$ Correspondence}\n\n\\hspace{1cm} Independently of supersymmetry, the 3-index antisymmetric\ngauge potential $C_{klm}$ has been employed in the context of the\ncosmological constant problem \\cite{Haw84}, \\cite{Duf89}. The\nderivative quadratic action density is proportional to $ (\\partial C)^2$.\nThis density can be related to a constant considering the density\n$$ X^2 +X \\partial C$$ with $X(x)$ a real field. Varying with respect to\n$X$ and substituting back reproduces $(\\partial C)^2$. On the other hand,\nvarying with respect to $C_{klm}$ implies $\\partial_m X=0$, {\\em {i.e.\\ }} $X$ a\nconstant.\n\n This mechanism can be extended to the supersymmetric\ncase, \\textit{e.g.} the 3-form multiplet. Here we consider the combination\n\\begin{equation}\n\\int d^2\\theta d^2\\bar{\\theta} \\left[ \\frac{}{} -X {\\overline{X}} -X Y -{\\overline{Y}} {\\overline{X}}\n+ 16\\Omega \\left( U(\\phi)+{\\bar{U}}({\\bar{\\phi}}) \\right) +\\phi {\\bar{\\phi}} \\ \\right].\n\\end{equation}\n$X, {\\overline{X}}$ is a complex unconstrained superfield, $Y,{\\overline{Y}}$, the\n3-form superfield introduced above and $\\Omega$ its unconstrained\nreal pre-potential. $\\phi, {\\bar{\\phi}}$ are considered as spectator\nsuperfields. Varying with respect to $X, {\\overline{X}}$ just implies $\nX={\\overline{Y}},\\, {\\overline{X}}=Y$ and one recovers (\\ref{complact}) upon\nsubstitution. As to variation with respect to the 3-form multiplet\nwe shall use the solution (\\ref{omega}) of the constraints and\nintegration by parts in superspace to arrive at\n\\begin{equation}\n\\int d^2\\theta d^2\\bar{\\theta} \\left[ \\frac{}{}- X {\\overline{X}} +4 \\Omega \\left( {\\bar{D}}^2 X +D^2\n{\\overline{X}} + 4 U(\\phi)+ 4{\\bar{U}}({\\bar{\\phi}}) \\right) +\\phi {\\bar{\\phi}} \\ \\right],\n\\end{equation}\nwhere $\\Omega$ may be considered as a Lagrange multiplier superfield giving rise to a constraint\n\\begin{equation}\\label{Xcon}\n {\\bar{D}}^2 X +D^2 {\\overline{X}} + 4U(\\phi)+4{\\bar{U}}({\\bar{\\phi}})=0,\n\\end{equation}\n that can be separated into two constraints\n\\begin{equation}\\label{conX}\n {{\\bar{D}}}^2 X = -4U(\\phi)-4 K , \\hspace{1cm} {D}^2 {\\overline{X}}=-4 {\\bar{U}} ({\\bar{\\phi}})-4 \\bar{K},\n\\end{equation}\n related by a constant $K=- \\bar{K} =iK_2$, which might be\nabsorbed in a redefinition of $U(\\phi), {\\bar{U}}({\\bar{\\phi}})$.\n In other words, in supersymmetry, the analogue of the constant\n mentioned above (in the non supersymmetric case) is given by a complex\n superfield, $X,{\\overline{X}}$.\n\nThe component\nfield action is then obtained from\n\\begin{equation}\\label{lXphi}\n\\int d^2\\theta d^2\\bar{\\theta} \\left[ \\frac{}{} - X {\\overline{X}} +\\phi {\\bar{\\phi}} \\ \\right].\n\\end{equation}\n In the\ncase $U=0$, this multiplet has been presented in \\cite{GatesSieg}.\nWe shall call it {\\em{non-minimal}} in what follows and use the\nterm {\\em{coupled non-minimal}} in the case of non vanishing $U$,\nto be discussed in the next section.\n\\section{The coupled non-minimal multiplet}\n\n\\hspace{1cm} The superfield constraints (\\ref{conX}) determine a multiplet\nof 12 bosonic and 12 fermionic component field degrees of freedom,\nidentified as usual by successive applications of covariant spinor\nderivatives. We define the component fields contained in $X$ as\n\\begin{eqnarray}\\label{locoX}\n X \\loco &=& X, \\hspace{1cm} D_{\\alpha} X\\loco = \\sqrt{2} \\,{\\psi}_{\\alpha}, \\hspace{5mm} \\ \\\n{{\\bar{D}}}^{\\dot{\\alpha}} X\\loco = -\\sqrt{2} \\, {\\bar{\\omega}}^{\\dot{\\alpha}},\\hspace{5mm}\n \\nonumber \\\\\n {{\\bar{D}}}^{\\dot{\\alpha}} D_{\\alpha} X \\loco &=& {V_{\\alpha}}^{\\dot{\\alpha}},\\hspace{5mm} {{\\bar{D}}}^{\\dot{\\alpha}} D^2 X \\loco\n =-4\\bar{\\rho}^{\\dot{\\alpha}},\\hspace{5mm} D^2 X \\loco = -4E.\n\\end{eqnarray}\nObserve that the $\\bar{\\theta}^2$ component is given in terms of $A$, Cf.(\\ref{conX}). For\n${\\overline{X}}$ we define similarly\n\\begin{eqnarray}\\label{locoXb}\n {\\overline{X}} \\loco &=& {\\overline{X}}, \\hspace{1cm} D_{\\alpha} {\\overline{X}}\\loco = -\\sqrt{2} \\,{\\omega}_{\\alpha}, \\hspace{5mm} \\ \\\n{{\\bar{D}}}^{\\dot{\\alpha}} {\\overline{X}}\\loco = \\sqrt{2} \\, {\\bar{\\psi}}^{\\dot{\\alpha}},\\hspace{5mm}\n \\nonumber \\\\\n D_{\\alpha} {{\\bar{D}}}^{\\dot{\\alpha}} {\\overline{X}} \\loco &=& {{{\\bar{V}}}_{\\alpha}}{}^{\\dot{\\alpha}},\\hspace{5mm} {D}_{\\alpha} {\\bar{D}}^2 {\\overline{X}} \\loco\n =-4{\\rho}_{\\alpha},\\hspace{5mm} {\\bar{D}}^2 {\\overline{X}} \\loco = -4 \\bar{E}.\n\\end{eqnarray}\nProjecting $\\frac{1}{16}D {\\bar{D}}^2 D ( -X {\\overline{X}})$ to lowest\ncomponents gives the canonical component field action density\\footnote{Primes indicate derivatives\nwith respect to $A$ or ${\\bar{A}}$, as the case may be.}\n\\begin{eqnarray}\\label{XX}\n{\\cal L}_X\n&=&- \\partial_m X \\partial^m {\\overline{X}} -\\f{i}{2} \\left( {\\omega} {\\si^m \\prt_m\\ } \\bar{{\\omega}} + \\bar{{\\omega}} {\\sib^m \\prt_m\\ }{\\omega} \\right) \\\\\n &&- |\\,U|^2\n- U' \\left( {\\overline{X}} F +{\\omega} \\chi \\right)- {{\\bar{U}}}' \\left( X\n{\\bar{F}} +\\bar{{\\omega}} {\\bar{\\chi}} \\right) +\\frac{1}{2}\n\\,{\\overline{X}} U'' (\\chi \\chi) +\\frac{1}{2}\\,X {{\\bar{U}}}''({\\bar{\\chi}} {\\bar{\\chi}}) \\nonumber \\\\\n&&- \\frac{1}{2} V_m {\\bar{V}}^m - E \\bar{E} - \\f{i}{2} \\psi\\left( {\\si^m \\prt_m\\ }\n{\\bar{\\psi}} +i \\sqrt{2} \\rho \\right)\n- \\f{i}{2}{\\bar{\\psi}} \\left({\\sib^m \\prt_m\\ }\\psi+i \\sqrt{2} \\bar{\\rho}\\ \\right) ,\\nonumber\n\\end{eqnarray}\ndescribing a complex scalar $X$ and a Majorana spinor ${\\omega}$ as\nphysical fields. Auxiliary fields consist of a complex scalar $E$,\na complex vector $V_m$, and 2 Majorana spinors $\\psi, \\rho$. This\naction density is invariant under supersymmetry transformations:\n\\begin{eqnarray}\\label{suztf}\n&&\\delta_{\\xi} X = \\sqrt{2} \\ (\\xi \\psi - {\\bar{\\xi}}\n{\\bar{\\omega}}),\\hspace{0.8cm}\n \\delta_{\\xi} {\\overline{X}} = \\sqrt{2}\\ ( {\\bar{\\xi}} {\\bar{\\psi}}-\\xi \\omega),\\nonumber \\\\\n&&\\delta_{\\xi} {\\psi}_{\\alpha}= \\sqrt{2} E\\ \\xi_{\\alpha} -\\frac{1}{ \\sqrt{2}}V_m \\\n({\\bar{\\xi}} {\\bar{\\sigma}}^m {\\epsilon})_{{\\alpha}} , \\hspace{0.8cm} \\delta_{\\xi} {{\\bar{\\psi}}}^{\\dot{\\alpha}} =\n\\sqrt{2} \\bar{E}\\ {\\bar{\\xi}}^{\\dot{\\alpha}} -\\frac{1}{ \\sqrt{2}}\\ {\\bar{V}}_m (\\xi\n{\\sigma}^m {\\epsilon})^{\\dot{\\alpha}}, \\nonumber \\\\\n&&\\delta_{\\xi} {\\bar{\\omega}}^{\\dot{\\alpha}} = -\\sqrt{2} U {\\bar{\\xi}}^{\\dot{\\alpha}} -\\frac{1}{\n\\sqrt{2}} (V_m +2i \\partial_m X)(\\xi {\\sigma}^m {\\epsilon})^{\\dot{\\alpha}} ,\\nonumber \\\\ &&\\delta_{\\xi}\n{\\omega}_{\\alpha} = -\\sqrt{2} {\\bar{U}} \\xi_{\\alpha}\n-\\frac{1}{ \\sqrt{2}}\\ ({\\bar{V}}_m +2i \\partial_m {\\overline{X}})({\\bar{\\xi}} {\\bar{\\sigma}}^m {\\epsilon})_{{\\alpha}},\\nonumber \\\\\n&&\\delta_{\\xi} \\ V_m = (\\xi {\\sigma}_{m}\\bar{\\rho})+\\sqrt{2} i \\ (\\xi\n{\\sigma}_{n} {\\bar{\\sigma}}_m\\partial^n \\psi -{\\bar{\\xi}} {\\bar{\\sigma}}_{m}{\\sigma}_n \\partial^n \\bar{\\omega})\n-({\\bar{\\xi}} {\\bar{\\sigma}}_m \\chi)U', \\nonumber \\\\\n&&\\delta_{\\xi} \\ {\\bar{V}}_m = ({\\bar{\\xi}} {\\bar{\\sigma}}_{m}{\\rho})+\\sqrt{2} i \\ ({\\bar{\\xi}} {\\bar{\\sigma}}_{n}{\\sigma}_m \\partial^n \\bar{\\psi}-\\xi\n{\\sigma}_{m}{\\bar{\\sigma}}_n \\partial^n \\omega)\n-(\\xi {\\sigma}_m {\\bar{\\chi}}){\\bar{U}}',\\nonumber \\\\\n&&\\delta_{\\xi} \\bar{\\rho}^{\\dot{\\alpha}} = 2i\\ \\ \\partial_m E \\ (\\xi {\\sigma}^m\n{\\epsilon})^{\\dot{\\alpha}} + (2i\\partial_m V^m\n-2 \\Box X -U'' \\chi \\chi +2 U' F)\\ {\\bar{\\xi}}^{\\dot{\\alpha}} , \\nonumber \\\\\n &&\\delta_{\\xi} {\\rho}_{\\alpha} = 2i\\ \\partial_m \\bar{E}\\ ({\\bar{\\xi}} {\\bar{\\sigma}}^m\n{\\epsilon})_{\\alpha} \\\n + (2i\\partial_m {\\bar{V}}^m -2 \\Box {\\overline{X}} -{\\bar{U}}'' {\\bar{\\chi}} {\\bar{\\chi}} +2 {\\bar{U}}' {\\bar{F}})\\\n \\xi_{\\alpha}\\ ,\\nonumber \\\\\n&&\\delta_{\\xi} E = {\\bar{\\xi}}\\, \\bar{\\rho}, \\hspace{0.8cm}\\delta_{\\xi}\n\\bar{E} = \\xi\\, {\\rho},\\nonumber \\\\ &&\\delta_{\\xi} {\\bar{D}}^2 X =-4 \\sqrt{2} U'\n(\\xi \\chi ),\\hspace{0.8cm} \\delta_{\\xi} D^2 {\\overline{X}} = -4 \\sqrt{2} {\\bar{U}}' ({\\bar{\\xi}}\n{\\bar{\\chi}} ).\n\\end{eqnarray}\n\nAdding the kinetic Lagrangian for $\\phi$\n\\begin{eqnarray}\\label{lska}\n{\\cal{L}_S} &=&-\\partial^m \\! A \\, \\partial_m {\\bar{A}} - \\f{i}{2} \\left( \\chi\n{\\si^m \\prt_m\\ } {\\bar{\\chi}} + {\\bar{\\chi}} {\\sib^m \\prt_m\\ }\\chi \\right) +F {\\bar{F}},\n\\end{eqnarray}\nthe complete Lagrangian is\n\\begin{eqnarray}\\label{LL}\n{\\cal L} &=&-\\partial^m \\! A \\, \\partial_m {\\bar{A}} - \\frac{i}{2}\\left( \\, \\chi {\\si^m \\prt_m\\ } {\\bar{\\chi}}\n+{\\bar{\\chi}} {\\sib^m \\prt_m\\ } \\chi \\right)- \\partial_m X \\partial^m {\\overline{X}} -\\frac{i}{2}\\left( \\, {\\omega} {\\si^m \\prt_m\\ }\n\\bar{{\\omega}}+ \\bar{{\\omega}} {\\sib^m \\prt_m\\ } {\\omega} \\right)\n\\ \\nonumber \\\\\n&&- U'(A)\\, {\\omega} \\chi - {{\\bar{U}}}'({\\bar{A}}) \\, \\bar{{\\omega}} {\\bar{\\chi}} - |U'(A)|^2 X {\\overline{X}} -|U(A)|^2 \\nonumber \\\\&& +\\frac{1}{2}\n\\,{\\overline{X}} U''(A)(\\chi \\chi )+\\frac{1}{2}\\,X {{\\bar{U}}}''({\\bar{A}}) ({\\bar{\\chi}}\n{\\bar{\\chi}}) + {\\cal{F}}\\,\n{\\overline{\\cal{F}}}-\\frac{1}{2} V_m {\\bar{V}}^m - E \\bar{E}\\nonumber \\\\ &&-\n\\f{i}{2} \\psi\\left( {\\si^m \\prt_m\\ } {\\bar{\\psi}} +i \\sqrt{2} \\rho \\right) - \\f{i}{2}{\\bar{\\psi}}\n\\left({\\sib^m \\prt_m\\ }\\psi+i \\sqrt{2} \\bar{\\rho}\\ \\right),\n\\end{eqnarray}\nwith\n \\begin{equation} \\label{auxeqX}{\\cal{F}}=F- X {\\bar{U}} ', \\hspace{3mm} \\overline{\\cal{F}}={\\bar{F}}- {\\overline{X}} U', \\end{equation}\nand exhibiting the general scalar potential\n\\begin{equation}\n{\\cal{V}} =|U'(A)|^2 X {\\overline{X}} +|U(A)|^2.\n\\end{equation}\nIn order to make contact with the simple model of section 3, we\nset $U(A)= {\\alpha} +\\mu A,\\, {\\bar{U}}({\\bar{A}})= \\bar{{\\alpha}}+\\mu {\\bar{A}}$. Then (\\ref{LL}) describes two complex scalar fields and a Dirac field with common mass $\\mu$, just like the Lagrangian\n(\\ref{LagY}).\nThe difference between the two Lagrangians appears in the auxiliary field sector and, correspondingly, in the component field supersymmetry transformations. Moreover, it should be stressed that $Y, {\\overline{Y}}$ represents a gauge multiplet, whereas $X, {\\overline{X}}$ does not; this is also the case for the linear-chiral multiplet correspondence.\n\n\\section{Conclusions}\n\n\\hspace{1cm} The main purpose of this short communication was to establish\na correspondence between the 3-form multiplet and a non-minimal\nmultiplet, in analogy to the well-known relation between the\n2-form ({\\em {i.e.\\ }} linear) multiplet and the chiral multiplet. Observe\nthat in both cases the correspondence can only be established\nunder certain restrictive assumptions.\n\nAlthough the 3-form multiplet and the non-minimal multiplet might be considered as {\\em{exotic multiplets}}, they are not. As indicated in the introduction, the 3-form multiplet describes naturally the gaugino squared chiral superfield $ tr \\left( W^{\\alpha} W_{\\alpha} \\right)$ and its complex conjugate.\nOn the other hand, the non-minimal multiplet appears naturally in the context of the solution of the chiral superfield constraints, {\\em {i.e.\\ }} $\\phi\\, =\\, {\\bar{D}}^2 \\varphi, \\, {\\bar{\\phi}} \\, =\\, D^2 {\\bar{\\vp}} $, in terms of unconstrained potentials $\\varphi, {\\bar{\\vp}}$, defined up to pre-gauge transformations $\\varphi \\rightarrow \\varphi+\\xi, \\, {\\bar{\\vp}} \\rightarrow {\\bar{\\vp}}+{\\bar{\\xi}}$. These superfields are themselves subject to the pre-constraints ${\\bar{D}}^2 \\xi =0,\\, D^2 {\\bar{\\xi}}=0$, leaving $\\phi,{\\bar{\\phi}}$ invariant.\n\nLet us mention as well that the above-mentioned 3-form constraints appear in an intriguing way in supergravity, in the framework of $U(1)$ superspace. The chiral supergravity superfields $R, R^\\dagger$ are intertwined with the vector superfield $G_a$ through the relation ${\\cal D}^2 R -{\\bar{\\cd}}^2 R^\\dagger =4i {\\cal D}^a G_a$.\nRemarkably enough, here, the {\\em{$H$-term }} of $R, R^\\dagger$ corresponds to a {\\em{$D$-term}} of the $U(1)$ supergravity sector.\n\nThe emphasis of the present note was to draw attention to the basic features of the correspondence between the 3-form multiplet and the non-minimal multiplet restricting ourselves to quite elementary considerations. More involved structures as well as the corresponding supergravity couplings will be the subject of forthcoming publications.\n\n\n\\addcontentsline{toc}{section}{\\bf REFERENCES}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:introduction}\n\nAlthough, the elementary particles seem to be exhausted by the Pauli classification into bosons and Fermions, topological considerations allow us to generalize the standard statistics in 1D (exchange\/anyon statistics~\\cite{leinas}) and in 2D (fractional\/exclusion statistics~\\cite{poly}). In arbitrary dimensions, Haldane~\\cite{Hald}, motivated by the properties of quasi-particles in the fractional quantum Hall effect and in one dimensional inverse-square exchange spin chains, introduced the fractional exclusion statistics (FES)~\\cite{Hald}. His proposal is to consider systems with a generalized Pauli blocking interpolating between ``no exclusion'' and ``perfect exclusion''. More explicitly, consider $N$ particles on a lattice of dimension $K$. If we fix the positions of $N-1$ particles, then the remaining single particle can occupy $d_N=K-g(N-1)$ positions. The constant $g$ is a ``statistical interaction'' parameter given by $g=-\\Delta d \/ \\Delta N$ where $\\Delta d$ is the change in the dimension of a single particle space and $\\Delta N$ is the change in the number of particles. The conventional Bose-Einstein (BES) and Fermi-Dirac (FDS) statistics correspond, respectively, to $g = 0$ and $g = 1$. Now, clear confirmations for FES have been found as well as applications in numerous models of interacting particles; see, e.g.,~\\cite{ref2,ref3,ref4,ref5,ref6,anghel,anghel2,CS,Hu} and references therein. In these notes, we designate FES with parameter $g$ by FES$_g$, and term particles obeying FES$_g$ $g$-ons. \n\nIn his seminal paper, Haldane postulated that the full Hilbert space for $g$-ons systems has the size:~\\cite{Hald, ref3} \\begin{equation} \\label{HW} W_{g}(K,N)= {d_{N}+N-1 \\choose N},\\end{equation} where the generic binomial coefficient $\\binom{a}{b}$ equals $a!\/(b!(a-b)!)$ if $0 \\leq b \\leq a$, and vanishes otherwise. However, no concrete counting procedure is behind the interpolating formula \\eqref{HW} as is the case for conventional statistics. Indeed, BES and FDS are examples of ball-in-box models, i.e., models for random allocations of unlabeled balls\/particles in labeled boxes\/states, subject to global exclusion constraints. The combinatorial weight --the number of micro-states-- coincides here with the number of ways to distribute the balls among the boxes. In this letter, we want to see if~\\eqref{HW} is the combinatorial weight of a ball-in-box model.\n\nFirst, we must have $N\\equiv 1 \\pmod r$ so that the dimension $d_N$, and accordingly $W_g(K,N)$, is a whole number. Thus, if $N = r P +1$ for some integer $P$ and $g=q\/r$ ($q (K-1)\/q$. \n\nCertainly, dealing with FES using standard techniques of statistical mechanics, allows to describe the thermodynamic properties of $g$-ons~\\cite{ref3,ref4,Nayak}, but leads to some inconsistencies when we attempt to find the probabilities for various occupation numbers of a single state~\\cite{Nayak}. For instance, Polychronakos~\\cite{Poly} proposed a \\textit{multiplicative} model which accurately gives back the statistical mechanics of FES in the thermodynamic limit. Multiplicativity here means that, at least for large $K$, the grand partition function is the $K$-th power of a $K$-independent function~\\cite{Poly}. However, the price paid for this microscopic realization is the occurrence of negative probabilities. Now, we understand that this problem occurs because Haldane statistics is not multiplicative and, unlike the Pauli principle, the exclusion operates on more than one level. Chaturvedi and Srinivasan~\\cite{chat}, and subsequently Murthy and Shankar~\\cite{neg}, showed with a remarkable tour de force how negative weights may be avoided for $g=1\/2$ (semions) and for $g=1\/3$. The more general case of FES$_{1\/m}$ was worked out subsequently and explicitly realized in models in one dimension, see~\\cite[Chapters 3 \\& 4]{ref6}. \n\nIn this letter, we revisit and solve this problem in a closed form when the parameter $g$ is generally any irreducible fraction: $g=q\/r$, where $q$ and $r$ are coprime and $0 < q \\leq r$. Our approach is purely combinatorial. It consists of regarding FES as a ball-in-box model, i.e. as an ideal quantum gas, with appropriate weighting on the set of occupancy configurations. To do so, we write the dimension $W_{g}(K,N)$ in the generic form~\\eqref{Wbis} below. While calculating the corresponding weights, we deduce the following exclusion rules: An occupancy configuration is allowed if (1) the maximal number of particles that each state can accommodate is $r-q+1$, and not to $g^{-1}$ whenever $q\\neq 1$, and (2) the configurations in which the number of states with two or more particles is greater than $(N-1)\/r$ are forbidden. This will allow us to distinguish infinitely many families of FES$_g$ systems depending on $g$ and $N$.\n\n\n\\section{Multiplicative vs. non-multiplicative statistics}\\label{s2}\nLet us first set the notation for our combinatorial analysis. The terminology comes mainly from the theory of integer partitions. \n\nA \\emph{partition} of a non-negative integer $N$ is a non-increasing sequence of positive integers whose sum is $N$. To indicate that ${\\lambda}$ is a partition of $N$, we write ${\\lambda} \\vdash N$ and denote \\mbox{${\\lambda}=(1^{k_1} 2^{k_2} \\ldots N^{k_N})$}, where $\\sum_{i=1}^N i k_i = N$ and $k_i$ designates the multiplicity of the part $i$; the sum $\\ell({\\lambda})=\\sum_{i=1}^N k_i$ is called the \\emph{length} of ${\\lambda}$. The \\emph{Ferrers diagram} of ${\\lambda}$ is a pattern of dots, with the $j$th row having the same number of dots as the $j$th term in ${\\lambda}$.\n\nSuppose we have $N$ ideal particles to be randomly distributed into $K$ states. In this work, this quantum system will be referred to as a ball-in-box model (indistinguishable balls and labeled boxes). An \\emph{occupancy configuration} of the system is said to be of shape ${\\lambda}=(1^{k_1} 2^{k_2} \\ldots N^{k_N}) \\vdash N$ if $k_i$ states are occupied by $i$ particles ($i=1,\\ldots, N$) and the number of non-vacant states $\\ell({\\lambda})$ is less than or equal to $K$. Moreover, if no parts of ${\\lambda}$ exceed a fixed integer $m$, the corresponding configuration is additionally characterized by $\\ell({\\lambda}^*) \\leq m$, where ${\\lambda}^*$ stands for the \\emph{conjugate} partition of ${\\lambda}$, that is, the partition whose Ferrers diagram is obtained from ${\\lambda}$ by reflection with respect to the diagonal so that rows become columns and columns become rows. For instance, the fermionic configuration is of shape $(1^N)$, characterized by $\\ell({\\lambda}^*) \\leq 1$.\n\n\\subsection{A basic formula}\nWe begin with a simple observation. \n\\begin{proposition} \\label{prop1} For bosons and fermions, the combinatorial weight can be uniquely written as weighted sums over partitions: \\begin{equation} \\label{Wbis} W(K,N)= \\sum_{{{\\lambda}=(1^{k_1} 2^{k_2} \\ldots N^{k_N})\\; \\vdash N}}w({\\lambda}) \\frac{\\ell({\\lambda})!}{k_1! \\, k_2! \\cdots k_N!} \\binom{K}{\\ell({\\lambda})}, \\end{equation} where the function $w$ is given by \\[w({\\lambda})= \\left\\{\n\t\\begin{array}{ll}\n\t1, & \\hbox{\\emph{for bosons};} \\\\\n\t\\delta_{{\\lambda},(1^N)}, & \\hbox{\\emph{for fermions}.}\n\t\\end{array}\n\t\\right.\\] ($\\delta$ is the Kronecker symbol.) \n\\end{proposition} \n\\begin{proof} We shall apply the generating function method dear to combinatorialists~\\cite{wilf}. For bosons, it is well-known that the grand partition function (or the generating function of $W(K,N)$ with respect to $N$) is \\[\\sum_{N=0}^{\\infty} \\binom{K+N-1}{N} z^N= (1-z)^{-K}= \\left(\\sum_{i=0}^\\infty z^i\\right)^K.\\] where $z$ is the fugacity. On the other hand, by the expansion of power series raised to integral powers~\\cite[page 823]{stegun}:\n\t\\begin{equation} \\left(\\sum_{i=0}^{\\infty}a_{i}z^{i} \\right)^{K} =\n\t\\sum_{n=0}^{\\infty} \\left(\\sum \\frac{K!}{k_{0}! k_{1}! \\cdots\n\t\tk_{n}!} a_{0}^{k_{0}}a_{1}^{k_{1}} \\cdots a_{n}^{k_{n}}\\right)\n\tz^n ,\\end{equation} where the inner sum is over the set $\\{k_{i} |\n\t\\sum_{i=0}^{n} k_{i}=K \\, \\mbox{and} \\sum_{i=1}^{n}i k_{i}=n \\}$,\nwe write the partition function as\t\n\t\\[\\sum_{N=0}^{\\infty} \\binom{K+N-1}{N} z^N = \\sum_{N=0}^{\\infty} \\left(\\sum_{\\{k_i\\}} \\frac{K!}{(K-k_1- \\ldots -k_N)! \\, k_1! \\cdots k_N!}\\right) z^N,\\] where the inner sum runs over all $N$-tuples $(k_1, \\ldots , k_N)$ subject to $k_1+2k_2 + \\ldots + N k_N =N $, i.e., over all partitions of $N$. Equating the coefficients on both sides, the bosonic combinatorial weight takes the form \\[\\sum_{\\{k_i\\}}\\binom{K}{k_1+ \\ldots + k_N} \\frac{(k_1+ \\ldots + k_N)!}{k_1! k_2! \\cdots k_N!}=\\sum_{{{\\lambda}\\; \\vdash N}} \\frac{\\ell({\\lambda})!}{k_1! \\, k_2! \\cdots k_N!} \\binom{K}{\\ell({\\lambda})}.\\] Thus, for bosons, $w({\\lambda})=1$. As for fermions, since the partition function is $(1+z)^K$, we find similarly that $w({\\lambda})=1$ if $k_1=N$ and $w({\\lambda})=0$ otherwise. \t\n\\end{proof}\nThe expression~\\eqref{Wbis} is generic for multiplicative models where the grand partition function is the $K$-th power of an analytic function: $\\sum_i a_n z^n$ with positive and $K$-independent Taylor coefficients ($a_0=1$). Reproducing the proof of Proposition~\\ref{prop1} yields the weight $w(\\lambda) = \\prod_{n=1}^N a_n^{k_n}$~\\cite{fahssi}. A particularly well-studied example of multiplicative models is the intermediate Gentile statistics of order $G \\geq 1$~\\cite{gent} for which $a_n=1$ if $n \\leq G$ and $a_n=0$ otherwise. So, the one-configuration weight reads here\n\\begin{equation} \\label{wG} w_G({\\lambda})= \\left\\{\n\\begin{array}{ll}\n1, & \\hbox{if} \\quad \\ell({\\lambda}^*) \\leq G; \\\\\n0, & \\hbox{otherwise.}\n\\end{array}\n\\right. \\end{equation} \n\nThe summands of Eq.~\\eqref{Wbis} have the following interpretation: for a fixed configuration of shape ${\\lambda}$, the factor $\\frac{\\ell({\\lambda})!}{k_1! \\, k_2! \\cdots k_N!} \\binom{K}{\\ell({\\lambda})}$ is the number of ways to choose $\\ell({\\lambda})$ non-vacant states out of $K$ ones and arrange $k_i$ states with $i$ particles among them; $i=1, \\ldots, N$. The result is then weighted by a non-negative function $w({\\lambda})$. Of course, a configuration ${\\lambda}$ with $w({\\lambda})=0$ does not contribute to the counting and therefore is not permitted. Thus, the function $w(\\lambda)$ encodes the counting rules of the statistics : for bosons, all the configurations contribute equally in the counting, while, for fermions, the Kronecker's delta $\\delta_{{\\lambda},(1^N)}$ is a manifestation of the Pauli principle.\n\nThe FES is not multiplicative in the sense of the definition above. For instance, it is easy to verify that the grand partition function for semions is given by \\begin{equation} \\sum_{n\\geq0}W_{1\/2}(K,N)z^N=\\frac{2}{\\sqrt{z^2+4}}\\left(\\frac{z}{2}+\\sqrt{1+\\frac{z^2}{4}}\\right)^{2K}.\\end{equation}\nAs discussed in the introduction, Polychronakos, through a slight modification of $W_g(K,N)$~\\cite[Eq.5]{Poly} which leads to the same statistical mechanics, proposed a microscopic realization of FES based on multiplicativity. It can be shown that the Polychronakos modified grand partition function is effectively the $K$-th power of \\begin{equation} {1 \\over 4}\\left(z+\\sqrt{4+z^2}\\right)^2= 1+z+\\frac{z^2}{2}+\\frac{z^3}{8}-\\frac{z^5}{128}+\\frac{z^7}{1024}-\\frac{5\nz^9}{32768}+\\cdots \\end{equation} The Taylor coefficients of this function are not always positive. Therefore, their interpretation as probabilities is problematic~\\cite{Nayak}.\n\nInterestingly, the dimension~\\eqref{HWbis} can be cast in the generic form~\\eqref{Wbis} with a well-defined and positive weighting function $w({\\lambda})$. It is the latter that must be perceived as probability of (global) occupation. More precisely, we show\n\n\\begin{theorem} \\label{HWweight} For $g=q\/r$ and $N \\equiv 1 \\pmod{r}$ , the number of micro-states \\emph{\\eqref{HWbis}} can uniquely be written in the form~\\eqref{Wbis}, where\n \\begin{equation} w_g({\\lambda}) = \\label{HWw} \\binom{(N-1)\/r}{\\ell({\\lambda})-k_1} \\binom{\\ell({\\lambda})}{k_1}^{\\!\\!-1} \\,\\prod_{j=0}^{r-q} \\binom{r-q}{j}^{k_{j+1}}. \\end{equation} In particular, $w_g({\\lambda})=0$ if $\\ell({\\lambda}^*) > r-q+1$. \n\\end{theorem} \\noindent We postpone the proof of our main result to Sub-section.~\\ref{proof}.\n \nIn the case with $g=1\/2$, the weight~\\eqref{HWw} reads\n\\[w_{1\/2}({\\lambda})=\\binom{(N-1)\/2}{k_2}\\binom{k_1+k_2}{k_1}^{-1}, \\qquad {\\lambda}=(1^{k_1}2^{k_2}) \\vdash N\\] which is exactly the formula derived by Chaturvedi and Srinivasan in their microscopic interpretation of semion statistics~\\cite{chat}. For $g=1\/3$, \\[w_{1\/3}({\\lambda})= \\binom{(N-1)\/3}{k_2+k_3}\\binom{k_1+k_2+k_3}{k_1}^{-1} 2^{k_2}, \\qquad {\\lambda}=(1^{k_1}2^{k_2}3^{k_3}) \\vdash N, \\] formula obtained by Murthy and Shankar using a different approach~\\cite{neg}.\n\nFrom the expression~\\eqref{HWw}, we underline the following features: \\begin{enumerate}\n\t\\item[(1)] the weights $w_g ({\\lambda})$ are fractional and non-negative definite,\n\t\\item[(2)] the weights $w_g({\\lambda})$ depend only upon $P \\coloneqq (N-1)\/r$ and the difference $r-q$,\n\t\\item[(3)] the allowed occupation number for a single-state does not exceed $r-q+1$ and \\emph{not} $1\/g$ whenever $q\\ne 1$. We recall, however, that the average occupation number is less than $1\/g \\leq r-q+1$~\\cite{ref3},\n\t\\item[(4)] Since the binomial coefficient $\\binom{P}{\\ell({\\lambda})-k_1}$ in~\\eqref{HWw} vanishes if $P < \\sum_{i=2}^m k_i$ , the corresponding configuration does not contribute to the counting.\n\\end{enumerate} The last observation is crucial. It stipulates that a necessary condition for permissible configurations is that the number of states occupied by two particles or more is less than or equal to $P$. In other words, the Ferrers diagram of the partition $(2^{k_{2}} \\ldots (r-q+1)^{k_{r-q+1}})$, extracted from ${\\lambda}$, must fit inside the rectangle $[P \\times (r-q+1)]$.\n\nLet us incorporate the above-formulated rules as follows: \n\\medskip\n\n\\noindent \\textbf{Generalized Exclusion Principle}.\n\t\\emph{A configuration of shape $ {\\lambda} \\vdash n$ is admissible if and only if the following constraints are fulfilled:}\n\\begin{enumerate}[leftmargin=5cm]\n\t\\item[$C_1$\\; \\emph{:}] \\quad $\\ell({\\lambda}) \\leq K$ \\; \\quad (\\emph{by definition}),\n\t\\item[$C_2$\\; \\emph{:}] \\quad $\\ell({\\lambda}^*) \\leq r-q+1$ \\quad (\\emph{at most $r-q+1$ particles per state}),\n\t\\item[$C_3$\\; \\emph{:}] \\quad $\\displaystyle \\sum_{i=2}^m k_i \\leq \\frac{N-1}{r} \\leq \\frac{K-1}{q}$ \\quad ($w_g({\\lambda}) \\neq 0$ \\emph{and} $d_N \\geq 1$). \\end{enumerate}\n\nTherefore, the exclusion operates not only on the ``microscopic'' level (condition $C_2$), but also on the ``macroscopic'' level (condition $C_3$). To illustrate, we implement this in two specific examples. First, let, say, $g=1\/3$ and $N=10$. Here the maximal allowed occupancy of a state is $3$. By the constraint $C_2$, 14 configurations may contribute (depending on $K \\geq 4$), among which the configurations $(1^2 2^4)$, $(2^5)$, $(1 2^3 3)$ and $(2^2 3^2)$ are forbidden by the constraint $C_3$:\n\\begin{center}\\qquad \\quad \\includegraphics[width=6cm]{ferrers.pdf}\\end{center}\nAs a second example, take $g=3\/5$ and $N=16$. Here the maximal allowed occupancy is again $3$. Among the 231 partitions of 16, only 10 may contribute to the total weight: \n\\begin{center}\n{\\footnotesize\t\\begin{tabular}\n\t\t\t{c||c|c|c|c|c|c|c|c|c|c}\n\t\t\n\t\t\t$\\lambda$ & $(1^{16})$ & $(1^{14} 2)$ & $(1^{12} 2^2)$ & $(1^{10} 2^3)$ & $(1^{13} 3)$ & $(1^{11} 2\\,3)$ & $(1^9 2^2 3)$ & $(1^{10} 3^2)$ & $(1^7 3^3)$&$(1^8 2\\,3^2)$\\\\\n\t\t\n\t\t\t\\hline\n\t\t\n\t\t\t$w(\\lambda)$ & $1$& $2\/5$ & $12\/91$ & $4\/143$ & $3\/14$ & $1\/13$ & $1\/55$ & $1\/22$ & $1\/120$& $2\/165$ \\\\\n\t\\end{tabular},}\n\\end{center}\neach of which contributes only if its length is less than or equal to $K \\geq 10$. We check readily that summing the contributions of all allowed configurations yields $\\binom{K+6}{16}=W_{3\/5}(k,16)$.\n\nIt is worth noting that, in view of the constraints $C_2$ and $C_3$, we can distinguish infinitely many families of $g$-ons systems according to $P=(N-1)\/r$ and the difference $ r-q $. Indeed, representing an $N$-particle system fulfilling FES$_g$ by the pair $(N,g=q\/r)$, two systems $(N,g=q\/r)$ and $(N',g')$ are subject to the same exclusion rules if there exist an integer $j >0$ not a multiple of $r-q$ such that\n\\begin{equation} g'=\\frac{j}{r-q+j}, \\quad \\hbox{and} \\quad\n\\frac{N'-1}{r-q+j}=\\frac{N-1}{r}.\\end{equation}\nThe semions, for example, belong to the family with $g=j\/(j+1)$, the \\emph{semionic family}. Clearly, the Bose and Fermi statistics are recovered in the limits $j=0$ and $j \\to \\infty$ respectively.\n\n\\begin{proposition} \\label{prop3}\n\tFor $N \\leq K$, the number of permissible configurations is \\begin{equation} \\label{conf} \\binom{(N-1)\/r +r-q}{r-q}. \\end{equation}\n\\end{proposition} \n\\begin{proof} Clearly, when $N \\leq K$ the condition $C_1$ and the inequality in the right of the constraint $C_3$ are satisfied. Thus, a configuration ${\\lambda}$ is likely if and only if the inequality in the left of the condition $C_3$ holds true. Therefore, the number of allowed configurations is the number of solutions of $k_2 + k_3 + \\cdots + k_m \\leq (N-1)\/r$ in nonnegative integers. The result follows from the known fact that the number of solutions of $x_1 + x_2 + \\cdots + x_k \\leq p$ is given by $\\binom{p+k}{k}$ (cf.~\\cite[p.103]{vanLint}). \\end{proof}\nBy way of comparison, the exact exclusion rules for the Gentile statistics are, in addition to $C_1$, $\\ell({\\lambda}^*) \\leq G$ and $N \\leq G K$. Thus, the number of permitted configurations is simply that of the partitions of $N$ with no more than $K$ parts; no part exceeding $G$. This number is the coefficient of $q^N$ in the Gaussian polynomial $\\left[ \\!{K+G \\atop K} \\!\\right]_q$~\\cite[Chap.3]{Andr}. When $N\\leq K$, this reduces to the number of partitions with largest part not exceeding $G$. We also emphasize that if $G=r-q+1$, then $W_G(K,N)$ majorizes $W_g(K,N)$ since the exclusion principle of FES is more restrictive.\n\n\\subsubsection{Proof of Theorem~\\ref{HWweight}\\label{s4}}\n\\label{proof}\t\nTo prove Theorem~\\ref{HWweight}, we need the following identity:\n\\begin{lemma} Let $P$, $n$ and $k$ be positive integers. Then\n\t\\begin{equation} \\label{HF} \\binom{k P}{n}= \\sum_{\\{l_i\\}} \\frac{P!}{l_1! \\cdots l_k!(P-l_1- \\cdots -l_k)!}\n\t\\prod_{i=1}^k \\binom{k}{i}^{l_i}, \\end{equation} where the sum runs over all $k$-tuples $(l_1, \\ldots , l_k)$ subject to the constraint $l_1+2l_2 + \\ldots +k l_k =n $. \\end{lemma}\n\n\\begin{proof} We shall again use the technique of generating function. Let $z$ be an indeterminate. On one hand, we have by application of the binomial theorem \\begin{equation} \\label{ex1} (1+z)^{kP}=\\sum_{n=0}^{kP}\\binom{kP}{n} z^n,\\end{equation} and, on the other hand, by the well-known multinomial theorem: \\begin{eqnarray} \\nonumber (1+z)^{kP}&=&\\left((1+z)^k\\right)^P = \\left(\\sum_{i=0}^k \\binom{k}{i}z^i\\right)^P = \\sum_{l_0 + l_1 + \\cdots l_k=P} \\frac{P!}{l_0! \\, l_1! \\, \\cdots l_k!}\\prod_{i=0}^k \\left( \\binom{k}{i}z^i \\right)^{l_i} \\\\\n\t\\label{ex2} &=& \\sum_{(l_1 , \\ldots , l_k)} \\left(\\frac{P!}{ l_1! \\, \\cdots l_k! \\, (P-l_1-l_2-\\cdots l_k)! }\\prod_{i=0}^k \\binom{k}{i}^{l_i}\\right) \\; z^{l_1+2l_2+\\cdots +k l_k}. \\end{eqnarray} The identity~\\eqref{HF} follows by equating the coefficients of $z^n$ in the two expansions~\\eqref{ex1} and~\\eqref{ex2}.\\end{proof} \\begin{proof}[Proof of Theorem~\\ref{HWweight}] Inserting the weight $w_g({\\lambda})$, the RHS of \\eqref{Wbis} can be displayed as \\[ \\sum_{{{\\lambda} \\vdash N \\atop \\ell({\\lambda}^*) \\leq r-q+1}} \\left(\n\t\\frac{P!}{k_2! \\cdots k_{r-q+1}!(P-k_2-\\cdots-k_{r-q+1})!} \\prod_{i=1}^{r-q}\n\t{{r-q}\\choose i}^{k_{i+1}} \\right) {{K}\\choose \\ell({\\lambda})}, \\] where $P=(N-1)\/r$. Taking into account that $\\ell({\\lambda})=\\sum_{i=1}^{r-q}k_{i}=N-\\sum_{i=1}^{r-q}ik_{i+1}$\n\tand putting $s=\\sum_{i=1}^{r-q}ik_{i+1}$ (the integer $s$ ranges\n\tfrom 0 to $(r-q)P$ since $k_{r-q+1} \\leq P$), we\n\tre-express the last formula as a double sum:\n\t\\begin{equation} \\label{A2} \\sum_{s=0}^{(r-q)P} \\left( \\sum_{\\sum_{i=1}^{r-q}ik_{i+1}=s}\\frac{P!}{k_2! \\cdots k_{r-q+1}!(P-k_2-\\cdots-k_{r-q+1})!} \\prod_{i=1}^{r-q} {{r-q}\\choose i}^{k_{i+1}}\\right) {{K}\\choose N-s}.\\end{equation} Now we make the change of summation indices $l_i=k_{i+1}$ to write the inner sum as the RHS of formula~\\eqref{HF}:\n\t\\begin{equation} \\label{A3} \\sum_{\\sum_{i=1}^{r-q}i l_{i}=s}\\frac{P!}{l_1! \\cdots\n\t\tl_{r-q}!(P-l_1-\\cdots-l_{r-q})!} \\prod_{i=1}^{r-q} {{r-q}\\choose i}^{l_{i}} = {{(r-q)P}\\choose s}.\n\t\\end{equation}\n\tWe deduce finally that the RHS of Eq.~\\eqref{Wbis} reads \\begin{equation} \\label{Vander} \\sum_{s=0}^{(r-q)P}{{(r-q)P}\\choose s}{{K}\\choose N-s}={{K+(r-q)P}\\choose N}=W_g(K,N), \\end{equation} where, to obtain the last equality, we employed the well-known Vandermonde's formula for binomial coefficients~\\cite{vander}. \\end{proof} \n\nWe stress, finally, that for $g>1$ ($r0\\}, \\quad P_i, Q_j\\in\\mathbb{R}[x_,\\dots,x_\\ell],\n\\end{equation}\nsuch that the sum of the degrees of all $P$ and $Q$ over all these basic sets defining $X$ is $D$. The filtration $\\Omega'$ is not \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal. However, the filtration $\\Omega$ sharply generated by $\\Omega'$ is \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal. This is a non-trivial fact that follows from effective cell decomposition in the semialgebraic category, see \\cite{Algos_real_algebrometry}.\n\\subsubsection{The analytic structure ${\\mathbb R}_\\textnormal{an}$}\nNot surprisingly, $\\ranalytic$ is not sharply o-minimal with respect to\nany FD-filtration. In fact, it is not even presharp, but by \\Cref{thm:for_show} it is sufficient to show that it is not sharp. Assume the contrary. Let $\\omega_1=1$ and\n$\\omega_{n+1}=2^{\\omega_n}$, and let $\\Gamma=\\{y=f(z)\\}\\subset{\\mathbb C}^2$\ndenote the graph of the holomorphic function\n$f(z)=\\sum_{j=1}^\\infty z^{\\omega_j}$ restricted to the disc of radius\n$1\/2$ (this function is definable in ${\\mathbb R}_\\textnormal{an}$). By axioms of sharpness, the\nnumber of points in\n\\begin{equation}\n \\Gamma\\cap\\big\\{y=\\epsilon_{n}+\\sum_{j=1}^n z^{\\omega_j}\\big\\},\n\\end{equation}\nshould be bounded by $P(\\omega_n)$, where $P$ is some polynomial \ndefined by the format and degree of $\\Gamma$. However, by basic complex analysis, if $\\epsilon_{n}$ is small enough, then the number of points in this set is at least \n$\\omega_{n+1}=2^{\\omega_n}$, giving a contradiction for large $n$.\n\\subsubsection{Pfaffian structures}\n\\label{subsec:Pffafian_example}\n\\label{marker}\nLet $B\\subset{\\mathbb R}^\\ell$ be an open box. A sequence\n$f_1,\\ldots,f_m:B\\to{\\mathbb R}$ of real-analytic functions is called a\n\\emph{Pfaffian chain} if they satisfy a triangular system of algebraic\ndifferential equations of the form\n\\begin{equation}\n \\pderivative{f_i}{x_j} = P_{ij}(x_1,\\ldots,x_\\ell,f_1,\\ldots,f_i), \\qquad \\forall i,j,\n\\end{equation}\nwhere $P_{ij}$ are polynomials. \nThe Pfaffian chain is called \\emph{restricted} if $B$ is bounded and\n$f_1,\\ldots,f_m$ extend as real analytic functions to a neighborhood of $\\bar B$. A\nPfaffian function $f$ is a polynomial\n$Q(x_1,\\ldots,x_\\ell,f_1,\\ldots,f_m)$. The degree of $f$ is defined to be the degree of $Q$. We denote the structure\ngenerated by the Pfaffian functions by $\\pfaff$, and its restricted\nanalog by $\\rpfaff$.\n\nGabrielov and Vorobjov \\cite{Pfaffaian_complexity} defined an FD-filtration $\\Omega$ on $\\rpfaff$, which is not knwon to be sharp (or even presharp). Roughly speaking, in $\\Omega$, the format of a semipfaffian set $X\\subset{\\mathbb R}^{k}$ is the maximum among $k$ and the length of the Pfaffian chains defining the Pfaffian functions appearing in a representation of $X$ as a finite union of basic sets, and the degree of $X$ is the sum of the degrees of all Pfaffian functions and the polynomials $P_{ij}$ in all Pfaffian chains appearing in the same representation. If $Y$ is a projection of $X$, the format and degree of the subpfaffian set $Y$ were defined to be those of $X$. While Gabrielov and Vorobjov were able to obtain bounds on the sum of the Betti numbers of a semipfaffian set $X$ which are polynomial in the degree of $X$, they were not able to obtain the same bounds for subpffafian sets in full generality. The main, and crucial, difficulty is that if $A\\in\\Omega_{{\\mathcal F},D}$ then it is only known that $A^{c}\\in\\Omega_{{\\poly_{\\cF}(D)},{\\poly_{\\cF}(D)}}$, i.e. the format of $A^c$ depends also on the degree of $A$.\\\\ \n\nIn \\cite{Pfaffian_cells}, Binyamini and Vorobjov introduce a new notion of degree for subpfaffian sets, with which they do achieve polynomial bounds on the Betti numbers of subpfaffian sets. Essentially, they introduce an FD-filtration $\\Omega^{*}$, based on $\\Omega$, that makes $\\rpfaff$ into a W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal structure with \\raisebox{.5ex}{\\scalebox{0.6}{\\#}} CD, and conjecturally makes $\\pfaff$ into a W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal structure with \\raisebox{.5ex}{\\scalebox{0.6}{\\#}} CD as well. As $\\Omega\\le\\Omega^{*}$, they obtain bounds on the sum of the Betti numbers of subpfaffian sets which are polynomial in the degree in the sense of \\cite{Pfaffaian_complexity}. As mentioned above, our construction generalizes the construction of $\\Omega^{*}$ from \\cite{Pfaffian_cells} to the settings of presharp o-minimal structures. \n\\subsubsection{A remark on notation} The symbol $O_{a}(1)$ denotes a specific universally fixed function (possibly different at each occurrence) $a\\mapsto C(a)$, where $C(a)$ is positive. We will make the non-restrictive (for our purposes) assumption that if $a=(a_1,\\dots,a_{t})$ ranges over ${\\mathbb N}^{t}$, then $C(a)\\geq\\max\\{a_1,\\dots,a_{t}\\}$, and moreover that $C(a)$ is (weakly) monotone increasing with respect to every variable $a_{i}$. The symbol $\\poly_{a}(b)$ denotes a polynomial $P_{a}(b)$ in $b$ with positive coefficients, where $a\\mapsto P_{a}$ is a specific universally fixed function. Thus, rather then representing classes of functions like ordinary asymptotic notations, these symbols are simply stand-ins for specific constants and polynomials we do not keep track of. Additionally, we will write ${O_{\\cF}(1)},\\;{\\poly_{\\cF}(D)}$ (perhaps with extra variables) as short hand for $O_{{\\mathcal F},\\Sigma}(1),\\;\\poly_{{\\mathcal F},\\Sigma}(D)$ (perhaps with extra variables) respectively. In some of our results, such as \\Cref{prop:fordeg_of_formula_set}, this is redundant as there is no dependence on $\\Sigma$.\n\\subsection{Format and degree of first order formulae}\nLet $\\Sigma=({\\mathcal S},\\Omega)$ be a \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal structure, and let ${\\mathcal L}$ be the language with atomic predicates of the form $(x\\in X)$ for every definable set $X$, and with neither constants nor function symbols. We will assume that the variables of ${\\mathcal L}$ are linearly ordered, so that if a formula $\\psi$ has $n$ free variables, then it defines a set in ${\\mathbb R}^{n}$. The goal of this subsection is to filter the formulae in ${\\mathcal L}$ by format and degree, such that if a formula $\\psi$ has format ${\\mathcal F}$ and degree $D$, then the set it defines is in $\\Omega_{{O_{\\cF}(1)},{\\poly_{\\cF}(D)}}$. \n\\begin{defi}[Format and degree of formulae]\n\\label{defi:fordeg_of_sformula}\nLet $\\psi$ be a formula. Suppose that there are $n$ different variables appearing in $\\psi$ (either free or quantified), and let $X_{j}\\in\\Omega_{{\\mathcal F}_{j},D_{j}}$ be the sets appearing in the atomic predicates of $\\psi$. Denote ${\\mathcal F}=\\max{{\\mathcal F}_{j}}$ and $D:=\\sum D_{j}$, then we say that $\\psi$ has format $\\max\\{{\\mathcal F},n\\}$ and degree $D$. \n\\end{defi}\nIn this text we will also need a notion of P-format, and it is natural to consider a notion of W-format as well. Since in the presharp or weakly-sharp case, geometric operations such as intersections and projections may increase the format, unlike the format of a formula $\\psi$, the P-format and W-format of $\\psi$ can't be defined just in terms of its atoms with \\Cref{prop:fordeg_of_formula_set} in mind. Rather, the P-format and W-format will depend on the parse-tree of $\\psi$. \n\\begin{defi}[P-format]\n\\label{defi:fordeg_of_formula}\nLet $\\psi$ be a formula, and $d$ be the depth of the parse-tree of $\\psi$. Then the P-format of $\\psi$ is defined to be $\\max\\{{\\mathcal F},d\\}$, where ${\\mathcal F}$ is the format of $\\psi$.\n\\end{defi}\nFor the definition of W-format, one needs to consider a different kind of parse-tree (one which is not necessarily binary), and moreover disregard vertices of the tree that are associated with disjunction. Since we won't actually need the W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal case of the following proposition in this paper, we omit a formal definition of W-format. The following proposition is clear from the definition. In fact, axiom (\\s1) is not even needed. \n\\begin{prop}\n\\label{prop:fordeg_of_formula_set}\nLet $({\\mathcal S},\\Omega)$ be a \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal structure (resp. P\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal, W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal), and let $\\psi$ be a formula of format (resp. P-format, W-format) ${\\mathcal F}$ and degree $D$, then the set that $\\psi$ defines is in $\\Omega_{{O_{\\cF}(1)},{\\poly_{\\cF}(D)}}$.\n\\end{prop}\n\\subsection{Sharp cellular decomposition, $\\Omega^{*}$}\n\\label{sec:Sharp_cellular_decomposition}\nWe recall the notion of a \\emph{cell}. A\ncell $C\\subset{\\mathbb R}$ is either a point or an open interval (possibly\ninfinite). A cell $C\\subset{\\mathbb R}^{\\ell+1}$ is either the graph of a\ndefinable continuous function $f:C'\\to{\\mathbb R}$ where $C'\\subset{\\mathbb R}^\\ell$ is\na cell, or the set\n\\begin{equation}\n\\{(x,y)\\in C'\\times{\\mathbb R}|f(x)0$ then there exists a functional $f$ such that $\\dim X_{f}<\\dim X$.\n\\end{lem}\n\\begin{proof}[Proof of \\Cref{prop:bound_on_H_0}]\nSuppose first that $X\\subset D(R)$, where $D(R)$ is an open disk of radius $R$ around the origin. Let $X=X_{1}\\cup\\dots\\cup X_{N}$ be the decomposition of $X$ into its connected components. The proof is by induction on $\\dim X$. The idea is to use \\Cref{lem:dim_of_local_maxima} and the induction hypothesis, but the problem is that the $X_{i}$ may have intersecting closures. \\\\ \n\nIf $\\dim X=0$, then there exists a linear functional $f:{\\mathbb R}^{\\ell}\\to{\\mathbb R}$ such that $f|_{X}$ is injective. Thus $\\dim H_0(X;{\\mathbb R})=\\dim H_0(f(X);{\\mathbb R})$, but $f(X)$ has format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$, so according to (P1) we have $\\dim H_0(f(X);{\\mathbb R})\\le{\\poly_{\\cF}(D)}$. \\\\ \n\nNow suppose $\\dim X>0$. Let $\\epsilon>0$ and consider $X_{\\epsilon}:=\\{x\\in X|\\;d(x,\\partial X)<\\epsilon\\}$ and $Y:=X\\backslash X_{\\epsilon}$. Clearly $Y$ has format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$. We claim that for $\\epsilon$ small enough, $Y$ is a union of $N$ sets with disjoint closures. Define $Y_{i}=X_{i}\\backslash X_{\\epsilon}$, and to ensure that the $Y_{i}$ are nonempty, choose $\\epsilon<\\underset{i}{\\min}\\underset{x\\in X_{i}}{\\sup}{d(x,\\partial X)}$. \\\\ \\\\ \nLet us prove that the $Y_{i}$ have disjoint closures. Say $x\\in\\overline{Y_{i}}\\cap\\overline{Y_{j}}$, and so $x\\in\\overline{X}$. If $x\\notin X$, then $x\\in\\partial X$, but $d(\\overline{Y_{i}},\\partial X)\\geq\\epsilon$, contradicting $x\\in\\overline{Y_{i}}$. So $x\\in X$, say $x\\in X_k$, but since $x\\in\\overline{X_{i}}$ we conclude that ${X_{i}}\\cup{X_{k}}$ is connected. This forces $k=i$, and by repeating the argument that $k=j$. Of course this shows that $Y$ has at least $N$ components. \\\\ \\\\ \nSince $\\dim{\\overline{Y}}>0$, by \\Cref{lem:dim_of_local_maxima} there exists a functional $f$ such that $\\dim\\left(\\overline{Y}\\right)_{f}<\\dim Y$. Certainly, $\\left(\\overline{Y}\\right)_{f}$ has format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$. Moreover, since $Y$ is bounded, $\\left(\\overline{Y}\\right)_{f}$ meets every $\\overline{Y}_{i}$, and since the $\\overline{Y}_{i}$ are disjoint we conclude $N\\leq\\dim H_{0}\\left(\\left(\\overline{Y}\\right)_{f};{\\mathbb R}\\right)$. As $\\dim\\left(\\overline{Y}\\right)_{f}<\\dim Y\\leq \\dim X$, by induction $N\\leq\\dim H_{0}(\\left(\\overline{Y}\\right)_{f};{\\mathbb R})\\leq{\\poly_{\\cF}(D)}$. \\\\ \n\nTo end the proof, note that while we assume that $X\\subset D(R)$, our bounds on $\\dim H_0(X;{\\mathbb R})$ does not depend on $R$. Assume now that $X$ is not bounded. Since homology commutes with direct limit, we see that $H_0(X;{\\mathbb R})=\\underset{\\longleftarrow}{\\lim}H_0(X\\cap D(R);{\\mathbb R})$, and so $\\dim H_0(X;{\\mathbb R})\\leq\\limsup\\dim H_0(X\\cap D(R);{\\mathbb R})\\le{\\poly_{\\cF}(D)}$.\n\\end{proof}\n\\section{Stratification}\n\\label{sec:sharp_stratification}\nLet $r$ be a positive integer, and fix a presharp structure $\\left({\\mathcal S},\\Omega\\right)$, then we have the following.\n\\begin{prop}\n\\label{prop:effectivity_of_derivatives}\nLet $f:{\\mathbb R}^{\\ell}\\to{\\mathbb R}^{k}$ be a definable map of format ${\\mathcal F}$ and degree $D$. Then $f$ is $C^{r}$ outside a definable set $V$ of codimension $\\geq 1$ of format ${O_{\\cF,r}(1)}$ and degree $\\poly_{{\\mathcal F}}(D,k,r)$.\n\\end{prop}\n \\begin{proof}\n We prove it for $r=1$ and leave the general case for the reader. Fix $1\\leq i\\leq k$, then the set $A_i=\\{x\\in{\\mathbb R}^{\\ell}|f_{i}$ is differentiable at $x\\}$ can be given by the following formula:\n \\begin{equation}\n \\exists L_{i}\\;\\forall\\epsilon>0\\;\\exists\\delta>0\\;\\forall y\\; \\left(|y-x|<\\delta\\to\\left|f_i(y)-f_i(x)-L_i\\cdot (y-x)\\right|<\\epsilon|y-x|\\right),\n \\end{equation}\n where $L_{i}$ should be understood as a tuple of $\\ell$ variables.\n Thus, by \\Cref{prop:fordeg_of_formula_set} $A_i$ has format $O_{{\\mathcal F}}(1)$ and degree ${\\poly_{\\cF}(D)}$. Therefore $\\cap_{i}A_{i}$ has format $O_{{\\mathcal F},k}(1)\\le O_{{\\mathcal F}}(1)$ and degree $\\poly_{{\\mathcal F}}(D,k)$. By o-minimality, ${\\mathbb R}^{\\ell}\\setminus A_{i}$ has codimension $\\geq 1$, and thus $V:={\\mathbb R}^{\\ell}\\backslash\\cap_{i}A_{i}$ has codimension $\\geq 1$.\n\\end{proof}\nGiven a positive integer $r$ and a set $X\\subset{\\mathbb R}^{\\ell}$, it is always possible to stratify $X$ in the following sense. \n\\begin{prop}\n\\label{prop:stratification}\nLet $X\\subset{\\mathbb R}^{\\ell}$ be of format ${\\mathcal F}$ and degree $D$. Then there exists a stratification of $X=X_{1}\\cup\\dots\\cup X_{s}$ where $s=\\dim X$ and each $X_{i}\\in\\Omega_{{O_{\\cF,r}(1)},{\\poly_{\\cF}(D,r)}}$ is a $C^{r}$ smooth embedded submanifold (possibly disconnected) of ${\\mathbb R}^{\\ell}$. \n\\end{prop}\n\\begin{proof}\nWhile a proof of this proposition appears in \\cite{Wilkie_conj_Proof}, it is instructive to repeat it here.\nThe proof is by induction on dimension. Let $X_{\\textnormal{reg}}\\subset X$ be the set of points near which $X$ is a $C^{r}$ manifold. By an argument similar to that of \\Cref{prop:effectivity_of_derivatives}, $X_{\\textnormal{reg}}$ has format ${O_{\\cF,r}(1)}$ and degree ${\\poly_{\\cF}(D,r)}$, and in the case of sharp derivatives, format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D,r)}$. By o-minimality $\\dim X\\backslash X_{\\textnormal{reg}}<\\dim X$, so we can apply the induction hypothesis on $X\\backslash X_{\\textnormal{reg}}$, and we are done.\n\\end{proof}\n\\begin{rem}\nIn \\cite{Forts} a similar result more compatible with cellular decomposition is proved. In particular, we may assume that the $X_{i}$ form a cellular decomposition of $X$, albeit then $s={\\poly_{\\cF}(D)}$ instead of $s=\\dim X$.\n\\end{rem}\nSimilarly it follows that the derivatives of $f$, where defined, have format ${O_{\\cF,r}(1)}$ and degree $\\poly_{{\\mathcal F}}(D,r)$. The fact that the format of $f^{(r)}$ depends on $r$ is very restrictive in general applications, and so far it seems generally unavoidable, even in \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal structures. In structures like ${\\mathbb R}_{\\textnormal{alg}}$ and ${\\mathbb R}_{\\rpf}$ however, the format of the derivatives is independent of $r$. We therefore recall the notion of \\emph{sharp derivatives}, first introduced in \\cite{Wilkie_conj_Proof}.\n\\begin{defi}\n\\label{defi:sharp_derivatives}\nLet ${\\mathcal S}$ be an o-minimal structure and $\\Omega$ be an FD filtration. We say that $({\\mathcal S},\\Omega)$ has \\emph{sharp derivatives} if for every\n ${\\mathcal F}\\in{\\mathbb N}$ there are\n \\begin{align}\n a_{\\mathcal F}&\\in{\\mathbb N}, & b_{\\mathcal F}&\\in{\\mathbb N}[D,k]\n \\end{align}\n such that the following holds. Given a definable $f:{\\mathbb R}^n\\to{\\mathbb R}$ with\n $f\\in\\Omega_{{\\mathcal F},D}$, we have for every $\\alpha\\in{\\mathbb Z}_{\\ge0}^n$\n \\begin{equation}\n f^{(\\alpha)}\\in\\Omega_{a_{\\mathcal F},b_{\\mathcal F}(D,|\\alpha|)}.\n \\end{equation}\n Where by $f^{(\\alpha)}$ we mean that $f$ is restricted to the locus where it is in $C^{|\\alpha|}$.\n\\end{defi}\nIn \\Cref{prop:stratification} above, if the structure has sharp derivatives, then the strata $X_{i}$ can be taken to have format ${O_{\\cF}(1)}$ and degree $\\poly_{{\\mathcal F}}(D,r)$.\n\\section{Sharp definable choice}\n\\label{sec:sharp_definable_choice}\n Fix a presharp structure. We prove the following form of sharp definable choice. Our proof is completely inspired by the construction in \\cite{van_den_dries_book}.\n\\begin{prop}\n \\label{prop:sharp_definable_choice}\n Let $\\{X_{\\lambda}\\subset{\\mathbb R}^{\\ell}\\}_{\\lambda\\in\\Lambda}$ be a definable family whose elements are nonempty, such that the format of the total space $X_{\\Lambda}:=\\{(\\lambda,x)|\\;x\\in X_{\\lambda},\\;\\lambda\\in\\Lambda\\}$ is ${\\mathcal F}$ and its degree is $D$. Then there exists a definable map $g:\\Lambda\\to{\\mathbb R}^{\\ell}$ of format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$ such that $g(\\lambda)\\in X_{\\lambda}$ for all $\\lambda\\in\\Lambda$.\n \\end{prop}\n \\begin{proof}\n We prove this by induction on $\\ell$. First we show the induction step. We use the induction hypothesis on the family $\\pi_{\\ell-1}(X_{\\lambda})\\subset {\\mathbb R}^{\\ell-1}$ to obtain a map $g_1:\\Lambda\\to{\\mathbb R}^{\\ell-1}$ with $g_1(\\lambda)\\in X_{\\lambda}$ for every $\\lambda\\in\\Lambda$. Since $g_1$ has format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$, the total space of the family $\\{x\\in{\\mathbb R}: (g_{1}(\\lambda),x)\\in X_{\\Lambda}\\}_{\\lambda\\in \\Lambda}$ has format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$. Finally, we use the induction hypothesis on this family, producing a map $g_2:\\Lambda\\to{\\mathbb R}$ of format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$. Then the map $(g_1,g_2)$ satisfies the requirements of the proposition. \\\\ \n\nFinally, we show the statement for $\\ell=1$. For $X\\subset{\\mathbb R}$, define $a(X):=\\inf X$ and $b(X):=\\sup\\{x:(a(X),x)\\subset X\\}$. We decompose $\\Lambda$ into the following sets, and define $g$ on them:\n\\begin{equation*}\n\\begin{aligned}\n A=&\\{\\lambda:a(X_{\\lambda})=-\\infty,\\;b(X_{\\lambda})=\\infty\\},\\;g(\\lambda)=0, \\\\\n B=&\\{\\lambda:a(X_{\\lambda})=-\\infty,\\; b(X_{\\lambda})\\in{\\mathbb R}\\},\\;g(\\lambda)=b(X_{\\lambda})-1, \\\\\n C=&\\{\\lambda:a(X_{\\lambda})\\in{\\mathbb R},\\;b(X_{\\lambda})=\\infty\\},\\;g(\\lambda)=a(X_{\\lambda})+1, \\\\\n D=&\\{\\lambda:a(X_{\\lambda}),b(X_{\\lambda})\\in{\\mathbb R}\\},\\;g(\\lambda)=\\frac{a(X_{\\lambda})+b(X_{\\lambda})}{2}\n\\end{aligned}\n\\end{equation*}\nIt remains to check that $A,B,C,D$, as well as the functions $a(X_{\\lambda}),b(X_{\\lambda})$ are of format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$. We will only check this for $B,a(X_{\\lambda})$, and leave the rest for the reader. $B$ can be described by the formula \\begin{equation}\n \\left(\\forall M\\;\\exists x\\in X_{\\lambda}\\;x0\\exists z\\in X_{\\lambda}\\;a+\\epsilon>z\\right)\\}\n\\end{equation}\nThus, just as before, we can bound the format and degree of $a(X_{\\lambda})$.\n \\end{proof}\n \\section{Proof of the main results}\n\\label{sec:main_proof}\n\\subsection{Sketch of the proof of \\Cref{thm:*omega_effective_cellular_decomposition}}\nWe will explain the main ideas and the main steps of the proof, while trying to avoid as many technical details as possible. The interested reader should consult \\cite{Pfaffian_cells}. We may replace ${\\mathbb R}$ by $I$, using the definable homeomorphism $\\frac{x-1\/2}{x-x^2}:I\\to{\\mathbb R}$. The following proposition is the key ingredient in the proof. \n\\begin{prop}\n\\label{prop:weird_prop}\n Let $\\{X_{\\alpha}\\}$ be a collection of $N$ definable sets of format ${\\mathcal F}$ and degree $D$ in $I^{\\ell}$. Let $n\\leq\\ell$ be a positive integer, then there exists a cylindrical decomposition of $I^{n}$ of size $\\poly_{{\\mathcal F}}(D,N)$, compatible with the collection $\\{\\pi_{n}(X_{\\alpha})\\}$, and whose cells have *-format ${O_{\\cF}(1)}$ and *-degree ${\\poly_{\\cF}(D)}$. \n\\end{prop}\nLet us show how \\Cref{thm:*omega_effective_cellular_decomposition} follows from \\Cref{prop:weird_prop}. Given $X_{1},\\dots,X_{k}\\subset I^{n}$ in $\\Omega^{*}_{{\\mathcal F},D}$, each $X_{i}$ is union of projections of connected components of sets $X_{\\alpha,i}$. We may suppose without loss of generality that the $X_{\\alpha,i}$ are contained in the same ambient space $I^{\\ell}$. Now we can use \\Cref{prop:weird_prop} on all the sets $X_{\\alpha,i}$, producing a cellular decomposition compatible with them. Then it must also be compatible with their connected components, and the projections of the cells to $I^{n}$ will be compatible with the sets $X_{i}$.\n\nWe next turn to prove \\Cref{prop:weird_prop}. The proof is by lexicographic induction on $(n,k)$ where $k:=\\underset{\\alpha}{\\max}\\dim\\left(\\pi_{n-1}(X_{\\alpha})\\right)$. For a collection $\\Pi$ of subsets of $I^{\\ell}$, let $\\bigcup\\Pi$ be the union of the sets in $\\Pi$, and for a positive integer $n\\leq\\ell$ denote $\\pi_{n}(\\Pi):=\\{\\pi_n(X)|X\\in\\Pi\\}$. For a positive integer $t$, we denote by $\\Pi_{s_{\\beta}$ holds globally over $C$, then there exists a point $x$ such that $s_{\\alpha}(x)=s_{\\beta}(x)$, but $s_{\\alpha},s_{\\beta}$ are not equal in a neighborhood of $x$. Then $(x,s_{\\alpha}(x))$ is in a strata of $Z_{\\alpha,\\beta}$ of dimension $1$, then $T_{v}$ is either the union of, or the intersection of, the sets $T_{v_{1}},\\dots,T_{v_{k}}$.\n \\item If $k=1$, then $T_{v}$ is one of the following: $\\pi_{\\ell-1}(T_{v_{1}}),\\left(T_{v_{1}}\\right)^{c},T_{v_{1}}\\times{\\mathbb R}$.\n\\end{enumerate}\nA slanted structure tree is defined similarly to a structure tree, the only difference is that in item (3) above the operation ${\\mathbb R}\\times T_{v_{1}}$ is also allowed.\n\\end{defi}\n\\begin{defi}[$\\Omega$-format of structure trees]\nLet $\\Omega$ be any FD-filtration. We define $\\Omega$-format of structure trees by induction on the depth. If $T$ has a single vertex $r$ and the associated set $T_{r}$ is in $\\Omega_{{\\mathcal F},D}$, then the $\\Omega$-format of $T$ is defined to be ${\\mathcal F}$.\n\nLet $(T,r)$ be a structure tree, let $v_{1},\\dots,v_{k}$ be the children of $r$, and denote by $T_{1},\\dots,T_{k}$ the subtrees defined by them. Suppose that $T_{i}$ has $\\Omega$-format ${\\mathcal F}_{i}$, then: \n\\begin{enumerate}\n \\item If $k=1$ and $T_{r}=T_{v_{1}}\\times{\\mathbb R}$ (or ${\\mathbb R}\\times T_{v_{1}}$ in the slanted case), then the $\\Omega$-format of $T$ is $\\max\\{{\\mathcal F}_{i}\\}+1$.\n \\item In any other case, the $\\Omega$-format of $T$ is $\\max\\{{\\mathcal F}_{i}\\}$.\n\\end{enumerate} \nIf $A_{j}\\in\\Omega_{{\\mathcal F}_{j},D_{j}}$ are the sets associated to the leaves of $T$, then the degree of $T$ is defined to be $\\sum D_{j}$.\n\\end{defi} \nThe following proposition is the key ingredient in the proof of \\Cref{prop:Wsharp_SCD_isSharp}.\n\\begin{prop}\n \\label{prop:ST_defines_good_set}\n Let $({\\mathcal S},\\Omega)$ be a W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal structure with \\raisebox{.5ex}{\\scalebox{0.6}{\\#}} CD, and let $(T,r)$ be a structure tree of $\\Omega$-format ${\\mathcal F}$ and degree $D$. Then $T_{r}\\in\\Omega_{O_{{\\mathcal F}}(1),{\\poly_{\\cF}(D)}}$. \n\\end{prop}\n\\begin{proof}\nLet $A_{i}\\in{\\mathbb R}^{\\ell_i}$ be the sets associated to the leaves of $T$. Denote $m:=\\max\\{\\ell_i\\}$, and for a set $X\\subset{\\mathbb R}^{m}$ and an integer $\\ell$ we denote $P_{\\ell}(X):=\\pi_{\\min\\{\\ell,m\\}}(X)\\times{\\mathbb R}^{\\max\\{\\ell-m,0\\}}$. Let ${\\mathcal C}_1,\\dots,{\\mathcal C}_{N}$ be a cellular decomposition of ${\\mathbb R}^{m}$ compatible with the sets $A_{i}\\times{\\mathbb R}^{m-\\ell_{i}}$. We claim that for every vertex $v$ of $T$, if the associated set $T_{v}$ is a subset of ${\\mathbb R}^{\\ell}$, then $P_{\\ell}({\\mathcal C}_1),\\dots,P_{\\ell}({\\mathcal C}_{N})$ is a cellular decomposition of ${\\mathbb R}^{\\ell}$ compatible with $T_{v}$. \n\nWe prove this by descending induction on the distance from $v$ to $r$. If $v$ is a leaf then the claim is clear by definition. Now let $v$ be any vertex, and let $v_1,\\dots,v_k$ be its children. Then by the definition of structure trees, one of the following holds.\n\\begin{enumerate}\n \\item If $k>1$, then $T_{v}$ is either the union of, or the intersection of, the sets $T_{v_{1}},\\dots,T_{v_{k}}$, but by the inductive hypothesis $P_{\\ell}({\\mathcal C}_{1}),\\dots,P_{\\ell}({\\mathcal C}_{N})$ are compatible with $T_{v_{1}},\\dots,T_{v_{k}}$, and thus they are compatible with $T_{v}$. \n \\item If $k=1$, then a straightforward check shows the same conclusion.\n\\end{enumerate}\nSince we assumed $({\\mathcal S},\\Omega)$ has \\raisebox{.5ex}{\\scalebox{0.6}{\\#}} CD, the cells ${\\mathcal C}_{i}$ can be chosen to have format ${O_{\\cF}(1)}$, degree ${\\poly_{\\cF}(D)}$ and their number $N$ can be chosen to be ${\\poly_{\\cF}(D)}$. Assume that $T_{r}\\subset{\\mathbb R}^{n}$, then ${\\mathcal F}\\geq n$, and the cells $P_{n}({\\mathcal C}_{i})$ have format $O_{{\\mathcal F},n}(1)\\le O_{{\\mathcal F}}(1)$ and degree ${\\poly_{\\cF}(D)}$, so $T_{r}$ has format $O_{{\\mathcal F}}(1)$ and degree ${\\poly_{\\cF}(D)}$ as well. \n\\end{proof}\n\\subsection{Proof of \\Cref{prop:Wsharp_SCD_isSharp}}\nLet $({\\mathcal S},\\Omega)$ be W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal with \\raisebox{.5ex}{\\scalebox{0.6}{\\#}} CD, we define an FD-filtration $\\Omega'$ in the following way. The set $X\\subset{\\mathbb R}^{n}$ is in $\\Omega'_{{\\mathcal F},D}$ if there exists a structure tree $(T,r)$ of $\\Omega$-format ${\\mathcal F}$ and degree $D$ such that $T_{r}=X$. It is clear that $({\\mathcal S},\\Omega')$ satisfies the axioms of \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal structures, except for (\\raisebox{.5ex}{\\scalebox{0.6}{\\#}} 1). Perhaps the only other unclear point is the second part of axiom (\\s2), i.e. that if $A\\in\\Omega'_{{\\mathcal F},D}$ then ${\\mathbb R}\\times A\\in\\Omega'_{{\\mathcal F}+1,D}$, but this follows from the simple observation that if $(T,r)$ is any structure tree, one can replace all the associated sets $T_{v}$ by ${\\mathbb R}\\times T_{v}$ and obtain a new structure tree of $\\Omega$-format greater by $1$ than the $\\Omega$-format of $T$.\n\nIt is also clear that $\\Omega\\subset\\Omega'$, since if $X$ is in $\\Omega_{{\\mathcal F},D}$, then $T$ can be taken to be a single vertex with associated set $X$. By \\Cref{prop:ST_defines_good_set}, we have $\\Omega'\\le\\Omega$, thus we simultaneously obtain the following. \n\\begin{enumerate}\n \\item The FD-filtration $\\Omega'$ satisfies (\\s1), or in other words, $({\\mathcal S},\\Omega')$ is \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal.\n \\item The filtrations $\\Omega,\\Omega'$ are equivalent, therefore $({\\mathcal S},\\Omega')$ has \\raisebox{.5ex}{\\scalebox{0.6}{\\#}} CD. \n\\end{enumerate}\n\\qedsymbol\\\\\nBefore turning to the proof of \\Cref{prop:main_result}, we need the following lemma on the normalization of format.\n\\begin{lem}\n\\label{lem:format_normalized}\nLet ${\\mathcal S}$ be an o-minimal expansion of ${\\mathbb R}$, and let $\\Omega$ be an FD-filtration such that $({\\mathcal S},\\Omega)$ satisfies the axioms \\textnormal{(\\raisebox{.5ex}{\\scalebox{0.6}{\\#}} j)} (resp. \\textnormal{(Wj),(Pj)}) for $j=1,\\dots,6$, where every occurrence of ${\\mathcal F}+1$ is replaced by ${O_{\\cF}(1)}$. Then $\\Omega$ is equivalent to a filtration $\\Omega'$ such that $({\\mathcal S},\\Omega')$ is \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal (resp. W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal, P\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal).\n\\end{lem}\n\\begin{proof}\n We will prove this in the \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal case, the proof being similar (but somewhat more complicated) in the W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal and P\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal case. We will denote by $C({\\mathcal F})$ the appearence of ${O_{\\cF}(1)}$ in axiom (2). Let $\\Omega'$ be the FD-filtration sharply generated by $\\Omega$. It is sufficient to show that $\\Omega'\\le\\Omega$. Let $X\\subset{\\mathbb R}^{n}$ be in $\\Omega'_{{\\mathcal F},D}$, then there exists a slanted structure tree $(T,r)$ of $\\Omega$-format ${\\mathcal F}$ and degree $D$, such that $T_{r}=X$. In particular it follows that $T_{r}\\in\\Omega_{C^{{\\mathcal F}}({\\mathcal F}),D}$, where by $C^{{\\mathcal F}}$ we mean the composition of $C$ with itself ${\\mathcal F}$ times, which is sufficient. \n\\end{proof} \nAs mentioned above, the proof in the W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal and P\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal case is somewhat more involved. It requires a notion of \\say{weak} and \\say{pre} $\\Omega$-format for structure trees. The same conclusion holds - the weak$\\backslash$ pre $\\Omega$-format of sets in $\\Omega'_{{\\mathcal F},D}$ is bounded by\n\\begin{equation}\n \\underset{[{\\mathcal F}]\\to\\{C\\}}{\\max}\\{C_{{\\mathcal F}}(\\dots(C_{1}({\\mathcal F}))\\dots)\\},\n\\end{equation}\nwhere the set $\\{C\\}$ is the set all functions $C({\\mathcal F})$ replacing appearances of ${\\mathcal F}+1$ in the axioms that $\\Omega$ satisfies, and the maximum is taken over all sequences of length ${\\mathcal F}$ in this set.\n\\subsection{Proof of \\Cref{prop:main_result}}\nLet $({\\mathcal S},\\Omega)$ be P\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal. By definition of $\\Omega^{*}$, the pair $({\\mathcal S},\\Omega^{*})$ satisfies axiom (W4). Moreover, according to \\Cref{thm:*omega_effective_cellular_decomposition}, the pair $({\\mathcal S},\\Omega^{*})$ satisfies the axioms (Wj) for $j\\in\\{1,2,3,5,6\\}$, where every appearance of ${\\mathcal F}+1$ is replaced by ${O_{\\cF}(1)}$. By \\Cref{lem:format_normalized} we can define an FD-filtration $\\Omega'$ which is equivalent to $\\Omega^{*}$, and such that $({\\mathcal S},\\Omega')$ is W\\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal with \\raisebox{.5ex}{\\scalebox{0.6}{\\#}} CD. We finish by applying \\Cref{prop:Wsharp_SCD_isSharp}. \\qedsymbol\n\\section{Sharp Triangulation}\n\\label{sec:Sharp_Triangulation}\nIn \\cite{Pfaffian_cells}, a sharp triangulation theorem for $\\rpfaff$ is deduced from \\raisebox{.5ex}{\\scalebox{0.6}{\\#}} CD and the ordinary proof of triangulation in o-minimality as it appears in \\cite{Intro_o_min}. The deduction is simple; one need only verify that the formulas describing the operation in \\cite{Intro_o_min} have format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$. Thus we have the following. Fix a \\raisebox{.5ex}{\\scalebox{0.6}{\\#}}\\kern-.02em{}o-minimal structure with sharp cellular decomposition. \n\\begin{thm}\n\\label{thm:sharp_triangulation}\nLet $X\\subset I^{\\ell}$ be closed and definable, $X_{1},\\dots,X_{k}\\subset X$ definable subsets such that all these sets have format ${\\mathcal F}$ and degree $D$. Then there exists a simplicial complex $K$ of size ${\\poly_{\\cF}(D)}$ with vertices in ${\\mathbb Q}^{\\ell}$ and a definable homeomorphism $\\Phi:|K|\\to X$ of format ${O_{\\cF}(1)}$ and degree ${\\poly_{\\cF}(D)}$, such that each $X_{i}$ is a union of images of simplices. \n\\end{thm}\nWe immediately conclude the following.\n\\begin{thm}[Bound on sum of Betti numbers]\nLet $\\Sigma$ be a presharp structure. If $X\\in\\Omega_{{\\mathcal F},D}$ is compact, then the sum of the Betti numbers of $X$ is bounded by ${\\poly_{\\cF}(D)}$.\n\\end{thm}\n\\begin{proof}\nThe theorem follows from \\Cref{thm:sharp_triangulation} and \\Cref{rem:star_counting}.\n\\end{proof}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIt is now an established fact that neutrinos are massive and leptonic\nflavors are not symmetries of Nature~\\cite{Pontecorvo:1967fh,\n Gribov:1968kq}. This picture has now become fully proved thanks to\nthe upcoming of a set of precise experiments which have confirmed the\nresults obtained with solar and atmospheric neutrinos using\nterrestrial beams of neutrinos produced in nuclear reactors and\naccelerators facilities~\\cite{GonzalezGarcia:2007ib}.\nThe minimum joint description of the neutrino data requires mixing\namong all the three known neutrinos ($\\nu_e$, $\\nu_\\mu$, $\\nu_\\tau$),\nwhich can be expressed as quantum superposition of three massive\nstates $\\nu_i$ ($i=1,2,3$) with masses $m_i$. Consequently when\nwritten in terms of mass eigenstates, the weak charged current\ninteractions of leptons~\\cite{Maki:1962mu, Kobayashi:1973fv} contain a\nleptonic mixing matrix which can be parametrized as:\n\\begin{equation}\n \\label{eq:matrix}\n U_\\text{vac} =\n \\begin{pmatrix}\n c_{12} c_{13}\n & s_{12} c_{13}\n & s_{13} e^{-i\\delta_\\text{CP}}\n \\\\\n - s_{12} c_{23} - c_{12} s_{13} s_{23} e^{i\\delta_\\text{CP}}\n & \\hphantom{+} c_{12} c_{23} - s_{12} s_{13} s_{23} e^{i\\delta_\\text{CP}}\n & c_{13} s_{23} \\hspace*{5.5mm}\n \\\\\n \\hphantom{+} s_{12} s_{23} - c_{12} s_{13} c_{23} e^{i\\delta_\\text{CP}}\n & - c_{12} s_{23} - s_{12} s_{13} c_{23} e^{i\\delta_\\text{CP}}\n & c_{13} c_{23} \\hspace*{5.5mm}\n \\end{pmatrix},\n\\end{equation}\nwhere $c_{ij} \\equiv \\cos\\theta_{ij}$ and $s_{ij} \\equiv\n\\sin\\theta_{ij}$. In addition to the Dirac-type phase $\\delta_\\text{CP}$,\nanalogous to that of the quark sector, there are two physical phases\nassociated to the Majorana character of neutrinos, which are not\nrelevant for neutrino oscillations~\\cite{Bilenky:1980cx,\n Langacker:1986jv} and which are therefore omitted in the following.\n\nIn the simplest quantum-mechanical picture, flavor oscillations are\ngenerated by the kinematical Hamiltonian for this ensemble, \n$H_\\text{vac}$, which in the flavor basis $(\\nu_e, \\nu_\\mu, \\nu_\\tau)$\nreads\n\\begin{equation}\n \\label{eq:hvac}\n H_\\text{vac} = U_\\text{vac} D_\\text{vac} U_\\text{vac}^\\dagger\n \\quad\\text{with}\\quad\n D_\\text{vac} = \\frac{1}{2E_\\nu} \\mathop{\\mathrm{diag}}(0, \\Delta m^2_{21}, \\Delta m^2_{31})\n\\end{equation}\nThe quantities $\\Delta m^2_{21}$, $|\\Delta m^2_{31}|$, $\\theta_{12}$,\n$\\theta_{23}$, and $\\theta_{13}$ are relatively well determined by the\nanalysis of solar, atmospheric, reactor and accelerator experiments,\nwhile barely nothing is known on the CP phase $\\delta_\\text{CP}$ and on the sign\nof $\\Delta m^2_{31}$~\\cite{nufit-1.1, GonzalezGarcia:2012sz, Fogli:2012ua,\n Tortola:2012te}.\nGiven the observed hierarchy between the solar and atmospheric\nmass-squared splittings there are two possible non-equivalent\norderings for the mass eigenvalues, which are conventionally chosen as\n\\begin{align}\n \\label{eq:normal}\n \\Delta m^2_{21} &\\ll (\\Delta m^2_{32} \\simeq \\Delta m^2_{31})\n \\text{ with } (\\Delta m^2_{31} > 0) \\,;\n \\\\\n \\label{eq:inverted}\n \\Delta m^2_{21} &\\ll |\\Delta m^2_{31} \\simeq \\Delta m^2_{32}|\n \\text{ with } (\\Delta m^2_{31} < 0) \\,.\n\\end{align}\nAs it is customary we refer to the first option,\nEq.~\\eqref{eq:normal}, as the \\emph{normal} ordering, and to the\nsecond one, Eq.~\\eqref{eq:inverted}, as the \\emph{inverted}\nordering. Clearly they correspond to the two possible choices of the\nsign of $\\Delta m^2_{31}$.\n\nThe flavor evolution of this neutrino ensemble is also affected by the\ndifference in the matter potential induced by neutrino-matter\ninteractions through the so-called Mikheev-Smirnov-Wolfenstein (MSW)\nmechanism~\\cite{Wolfenstein:1977ue, Mikheev:1986gs}. Within the\ncontext of the Standard Model (SM) of particle interactions, this\neffect is fully determined and leads to a matter potential which, for\nneutral matter, is proportional to the number density of electrons in\nthe background $N_e(r)$, $V=\\sqrt{2} G_F N_e(r)$, and which only\naffects electron neutrinos. The evolution of the ensemble is then\ndetermined by the Hamiltonian $ H^\\nu = H_\\text{vac} +\nH^\\text{SM}_\\text{mat}$, with $H^\\text{SM}_\\text{mat} = \\sqrt{2} G_F\nN_e(r) \\mathop{\\mathrm{diag}}(1, 0, 0)$. The magnitude and the presence of\nnon-standard forms of the matter potential can be tested in solar\nneutrino experiments (and in combination with\nKamLAND)~\\cite{Roulet:1991sm, Guzzo:1991hi, Barger:1991ae,\n Fogli:1993xv, Bergmann:1997mr, Bergmann:2000gp, Guzzo:2000kx,\n Fogli:2002hb, Friedland:2004pp, Escrihuela:2009up, Bolanos:2008km,\n Minakata:2010be, Palazzo:2011vg, Bonventre:2013loa}, as well and in\nthe propagation of atmospheric and long-baseline\nneutrinos~\\cite{Grossman:1995wx, GonzalezGarcia:2001mp, Gago:2001xg,\n Fornengo:2001pm, Huber:2001zw, Ota:2001pw, Huber:2002bi,\n Campanelli:2002cc, Ota:2002na, GonzalezGarcia:2004wg,\n Friedland:2004ah, Friedland:2005vy, Blennow:2005qj, Kitazawa:2006iq,\n Friedland:2006pi, Blennow:2007pu, Kopp:2007mi, Kopp:2007ne,\n Ribeiro:2007ud, Bandyopadhyay:2007kx, Ribeiro:2007jq,\n EstebanPretel:2008qi, Blennow:2008ym, Kopp:2008ds, Ohlsson:2008gx,\n Palazzo:2009rb, GonzalezGarcia:2011my}.\n\nIn this article we address our current knowledge of the size and\nflavor structure of the matter background effects in the evolution of\nsolar, atmospheric, reactor and long-baseline (LBL) accelerator\nneutrinos based on the global analysis of oscillation data. To this\naim, in Sec.~\\ref{sec:formalism} we briefly present the most general\nparametrization of the matter potential and its connection with\nnon-standard neutrino interactions (NSI) in matter, which provide a\nwell-known theoretical framework for this kind of phenomenological\nstudies. We also discuss the simplifications used in the analysis of\nthe solar+KamLAND sector and the atmospheric+LBL sector\nrespectively. In Sec.~\\ref{sec:solar} we present the results from the\nupdated analysis of solar+KamLAND data and quantify the impact of the\nmodified matter potential on the data description, as well as the\nstatus of the well-known ``dark-side'' solution which appears in\npresence of NSI. In Ref.~\\cite{GonzalezGarcia:2011my} an analysis of\natmospheric and LBL neutrino data was performed in the framework of a\ngeneralized matter potential, which extended the standard one by\nallowing for an arbitrary rescaling of the potential strength, a\ngeneral rotation from the $ee$ sector, and a rephasing with respect to\n$H_\\text{vac}$. It was concluded that the strength of the potential\ncannot be determined solely by these data, whereas its flavor\ncomposition is very much constrained. In Sec.~\\ref{sec:global} we\nupdate this analysis and revisit its conclusions after combining the\nresults from atmospheric, LBL and reactor experiments with those from\nsolar+KamLAND data. We show to what degree the determination of\nneutrino masses and mixing is robust even in the presence of this\ngeneral form of the matter potential and we derive the most up-to-date\nallowed ranges on NSI parameters. Finally in Sec.~\\ref{sec:summary} we\nsummarize our results.\n\n\\section{Formalism}\n\\label{sec:formalism}\n\nIn the three-flavor oscillation picture, the neutrino evolution\nequation reads:\n\\begin{equation}\n i\\frac{d}{dx}\n \\begin{pmatrix}\n \\nu_e\\\\\n \\nu_\\mu\\\\\n \\nu_\\tau\n \\end{pmatrix}\n = H^\\nu\n \\begin{pmatrix}\n \\nu_e\\\\\n \\nu_\\mu\\\\\n \\nu_\\tau\n \\end{pmatrix}\n\\end{equation}\nwhere $x$ is the coordinate along the neutrino trajectory and the\nHamiltonian for neutrinos and antineutrinos is:\n\\begin{equation}\n H^\\nu = H_\\text{vac} + H_\\text{mat}\n \\quad\\text{and}\\quad\n H^{\\bar\\nu} = ( H_\\text{vac} - H_\\text{mat} )^* \\,,\n\\end{equation}\nwith $ H_\\text{vac}$ given in Eq.~\\eqref{eq:hvac}. Thus the vacuum\nterm has $6$ parameters: $\\Delta m^2_{21}$, $\\Delta m^2_{31}$, $\\theta_{12}$,\n$\\theta_{13}$, $\\theta_{23}$, $\\delta_\\text{CP}$.\nIn the Standard Model $H_\\text{mat}$ is fully determined both in its\nstrength and flavor structure to be $H^\\text{SM}_\\text{mat} =\\sqrt{2}\nG_F N_e(r) \\mathop{\\mathrm{diag}}(1, 0, 0)$ for ordinary matter. Generically ordinary\nmatter is composed by electrons ($e$), up-quarks ($u$) and down-quarks\n($d$), thus in the most general case a non-standard matter potential\ncan be parametrized as:\n\\begin{equation}\n \\label{eq:hmatNSI}\n H_\\text{mat} = \\sqrt{2} G_F N_e(r)\n \\begin{pmatrix}\n 1 & 0 & 0 \\\\\n 0 & 0 & 0 \\\\\n 0 & 0 & 0\n \\end{pmatrix}\n + \\sqrt{2} G_F \\sum_{f=e,u,d} N_f(r)\n \\begin{pmatrix}\n \\varepsilon_{ee}^f & \\varepsilon_{e\\mu}^f & \\varepsilon_{e\\tau}^f\n \\\\\n \\varepsilon_{e\\mu}^{f*} & \\varepsilon_{\\mu\\mu}^f & \\varepsilon_{\\mu\\tau}^f\n \\\\\n \\varepsilon_{e\\tau}^{f*} & \\varepsilon_{\\mu\\tau}^{f*} & \\varepsilon_{\\tau\\tau}^f\n \\end{pmatrix} .\n\\end{equation}\nSince this matter term can be determined by oscillation experiments\nonly up to an overall multiple of the identity, without loss of\ngenerality one can assume $\\varepsilon_{\\mu\\mu}^f = 0$. With this, we have 8\nparameters (for each $f$) since $\\varepsilon_{ee}^f$ and $\\varepsilon_{\\tau\\tau}^f$\nmust be real whereas $\\varepsilon_{e\\mu}^f$, $\\varepsilon_{e\\tau}^f$ and\n$\\varepsilon_{\\mu\\tau}^f$ can be complex.\n\nIn order to determine the relevant ranges for the parameters in the\nproblem we must study which transformations leave the probabilities\ninvariant. In particular we notice that any rephasing $H^\\nu \\to Q H\nQ^*$ where $Q = \\mathop{\\mathrm{diag}}\\left( e^{ia}, e^{ib}, e^{ic} \\right)$ leads to a\nrephasing of the scattering matrix $\\exp(-iH^\\nu L) \\to Q \\exp(-iH^\\nu\nL) Q^*$, which does not affect the probabilities. In the standard\noscillation scenario these symmetries are used to reduce the range of\nthe mixing parameters, most commonly to $0 \\leq \\theta_{ij} \\leq\n\\pi\/2$ and $0 \\leq \\delta_\\text{CP} \\leq 2\\pi$. In the presence of the\nnon-standard matter potential they can be used just in the same way,\nthus reducing the range of the mixing parameters while keeping the\nphases of all the off-diagonal $\\varepsilon_{\\alpha\\neq\\beta}^f$.\nAlternatively, one could instead reduce the range for some of the\n$\\varepsilon_{\\alpha\\neq\\beta}^f$, at the price of retaining a wider range of\nthe vacuum mixing angles. Furthermore, in the particular case of a\nunique $f$ and in the absence of the vacuum term it would be possible\nto use these symmetries to reduce the matter potential parameters from\neight to six: two real flavor diagonal parameters, the absolute value\nof the flavor off-diagonal parameters, $|\\varepsilon_{\\alpha\\neq\\beta}^f|$,\nand one combination of their three complex phases, while the two\nadditional phases would become unphysical. Only when both the vacuum\nterm and the matter general potential are present the two additional\nphases become observable. Hence it is clear from this discussion that\nit is a matter of convention to include them in the matter potential\nor in the vacuum term. For real matter potential this means that only\nan overall sign of the three off-diagonal $\\varepsilon_{\\alpha\\neq\\beta}^f$\ncan be considered a generic feature of the matter potential, while the\nother two signs of $\\varepsilon_{\\alpha\\neq\\beta}^f$ can be traded off by\nenlarging the vacuum mixing parameters to $-\\pi\/2 \\leq \\theta_{ij}\n\\leq \\pi\/2$. We will go back to this issue in the next section.\n \nThe standard theoretical framework for our proposed parametrization of\nthe matter potential is provided by NSI affecting neutrino\ninteractions in matter. They can be described by effective\nfour-fermion operators of the form\n\\begin{equation}\n \\label{eq:def}\n \\mathcal{L}_\\text{NSI} =\n - 2\\sqrt{2} G_F \\varepsilon_{\\alpha\\beta}^{fP}\n (\\bar\\nu_{\\alpha} \\gamma^\\mu \\nu_{\\beta})\n (\\bar{f} \\gamma_\\mu P f) \\,,\n\\end{equation}\nwhere $f$ is a charged fermion, $P=(L,R)$ and\n$\\varepsilon_{\\alpha\\beta}^{fP}$ are dimensionless parameters encoding the\ndeviation from standard interactions. NSI enter in neutrino\npropagation only through the vector couplings so the induced matter\nHamiltonian takes the form~\\eqref{eq:hmatNSI} with\n$\\varepsilon_{\\alpha\\beta}^f = \\varepsilon_{\\alpha\\beta}^{fL} +\n\\varepsilon_{\\alpha\\beta}^{fR}$.\n\n\\subsection{Earth matter potential for atmospheric and LBL neutrinos}\n\\label{sec:form-atmos}\n\nAs seen above, in principle a generalized potential involves different\nparameters for the different charged fermions $f=e,u,d$ in the matter.\nIn practice, however, for the propagation of atmospheric and LBL\nneutrinos the neutron\/electron ratio $Y_n$ is reasonably constant all\nover the Earth. This implies that neutrino atmospheric and LBL\noscillations are only sensitive to the \\emph{sum} of these\ninteractions, weighted with the relative abundance of each\nparticle. We can therefore define:\n\\begin{equation}\n \\label{eq:compoconst}\n \\varepsilon_{\\alpha\\beta} \\equiv\n \\sum_{f=e,u,d} \\left< \\frac{Y_f}{Y_e} \\right> \\varepsilon_{\\alpha\\beta}^f\n = \\varepsilon_{\\alpha\\beta}^e + Y_u\\, \\varepsilon_{\\alpha\\beta}^u + Y_d\\, \\varepsilon_{\\alpha\\beta}^d\n\\end{equation}\nThe PREM model~\\cite{Dziewonski:1981xy} fixes $Y_n = 1.012$ in the\nMantle and $Y_n = 1.137$ in the Core, with an average value $Y_n =\n1.051$ all over the Earth. Since a proton has 2 up-quarks and 1\ndown-quark, a neutron has 1 up-quark and 2 down-quarks, and neutral\nmatter obviously has the same number of protons and electrons ($Y_p =\n1$), we get $Y_u = 2 + Y_n = 3.051$ and $Y_d = 1 + 2 Y_n = 3.102$ in\nthe Earth. With this in mind, the matter part of the Hamiltonian can\nbe written as:\n\\begin{equation}\n \\label{eq:hmatNSIatm}\n H_\\text{mat} = \\sqrt{2} G_F N_e(r)\n \\begin{pmatrix}\n 1 + \\varepsilon_{ee} & \\varepsilon_{e\\mu} & \\varepsilon_{e\\tau}\n \\\\\n \\varepsilon_{e\\mu}^* & \\varepsilon_{\\mu\\mu} & \\varepsilon_{\\mu\\tau}\n \\\\\n \\varepsilon_{e\\tau}^* & \\varepsilon_{\\mu\\tau}^* & \\varepsilon_{\\tau\\tau}\n \\end{pmatrix}\n\\end{equation}\nwhere the standard interactions are accounted by the ``$1\\,+$'' term\nin the $ee$ entry, and the non-standard interactions are accounted by\nthe $\\varepsilon_{\\alpha\\beta}$ terms.\nSince $H_\\text{mat}$ is Hermitian and its trace is irrelevant for\noscillations, we have 8 parameters.\n\nIn Ref.~\\cite{GonzalezGarcia:2011my} an alternative parametrization\nfor $H_\\text{mat}$, mimicking the structure of the vacuum term in\nEq.~\\eqref{eq:hvac}, was introduced as\n\\begin{equation}\n \\label{eq:hmatgen}\n H_\\text{mat} = Q_\\text{rel} U_\\text{mat} D_\\text{mat}\n U_\\text{mat}^\\dagger Q_\\text{rel}^\\dagger\n \\text{~~with~~}\n \\left\\lbrace\n \\begin{aligned}\n Q_\\text{rel} &= \\mathop{\\mathrm{diag}}\\left(\n e^{i\\alpha_1}, e^{i\\alpha_2}, e^{-i\\alpha_1 -i\\alpha_2} \\right) ,\n \\\\\n U_\\text{mat} &= R_{12}(\\varphi_{12})\n \\tilde{R}_{13}(\\varphi_{13}, \\delta_\\text{NS})\n R_{23}(\\varphi_{23}) \\,,\n \\\\\n D_\\text{mat} &= \\sqrt{2} G_F N_e(r) \\mathop{\\mathrm{diag}}(\\varepsilon, \\varepsilon', 0)\n \\end{aligned}\\right.\n\\end{equation}\nwhere we denote by $R_{ij}(\\varphi_{ij})$ a rotation of angle\n$\\varphi_{ij}$ in the $ij$ plane and\n$\\tilde{R}_{13}(\\varphi_{13},\\delta_\\text{NS})$ is a \\emph{complex}\nrotation by angle $\\psi_{13}$ and phase $\\delta_\\text{NS}$. Just as\nEq.~\\eqref{eq:hmatgen} this parametrization also contains 8 real\nparameters: 2 eigenvalues, 3 angles and 3 phases. The two phases\n$\\alpha_1$ and $\\alpha_2$ included in $Q_\\text{rel}$ are not a feature\nof neutrino-matter interactions, but rather a relative feature of the\nvacuum and matter term: they would become unphysical if any of the two\nterms were not there. Reinterpreted in the notation of\nEq.~\\eqref{eq:hmatNSIatm}, this means that only one particular\ncombination of the three complex phases of $\\varepsilon_{e\\mu}$,\n$\\varepsilon_{e\\tau}$, $\\varepsilon_{\\mu\\tau}$ is a genuine property of NSI. In\nother words, the relation in Eq.~\\eqref{eq:compoconst} implies that\nthe matter potential behaves as composed of a unique effective\nfermion, and in this case, as discussed in the previous section, it is\na matter of convention to define the off-diagonal elements of the\nmatter potential as three complex parameters, or as three positive\nreal parameters plus a matter CP phase, and the two additional phases\nbeing assigned to either vacuum or matter part.\n\nFurther simplification follows from neglecting $\\Delta m^2_{21}$ in the\nanalysis of atmospheric, LBL and all reactor experiments but KamLAND,\nand by imposing that two eigenvalues of the $H_\\text{mat}$ are equal\n($\\varepsilon'=0$). In the limit $\\Delta m^2_{21} \\to 0$ the $\\theta_{12}$ angle\nand the $\\delta_\\text{CP}$ phase become unphysical, even in the presence of the\ngeneralized $H_\\text{mat}$ in Eq.~\\eqref{eq:hmatgen}.\nSimilarly, for $\\varepsilon' \\to 0$ the $\\varphi_{23}$ angle and the\n$\\delta_\\text{NS}$ phase become unphysical and the general\n$H_\\text{mat}$ contains 5 real parameters: $\\varepsilon$ which represents a\nrescaling of the matter potential strength, $\\varphi_{12}$ and\n$\\varphi_{13}$ which allows for projection of the potential into the\n$\\nu_\\mu$ and $\\nu_\\tau$ flavors, and the 2 vacuum-matter relative\nphases $\\alpha_1$ and $\\alpha_2$.\nIn Ref.~\\cite{Friedland:2004ah} it was shown that strong cancellations\nin the oscillation of atmospheric neutrinos occur when two eigenvalues\nof $H_\\text{mat}$ are equal, so that although the limit $\\varepsilon' = 0$\nconsidered here is only a subspace of the most general case on\nnon-standard interactions, it is precisely in this subspace where the\nweakest constraints can be placed.\nUnder these assumptions the relations between the original\n$\\varepsilon_{\\alpha\\beta}$ in Eq.~\\eqref{eq:hmatNSIatm} and the parameters\nin Eq.~\\eqref{eq:hmatgen} read:\n\\begin{equation}\n \\label{eq:eps_atm}\n \\begin{aligned}\n \\varepsilon_{ee} - \\varepsilon_{\\mu\\mu}\n &= \\hphantom{-} \\varepsilon \\, (\\cos^2\\varphi_{12} - \\sin^2\\varphi_{12})\n \\cos^2\\varphi_{13} - 1\\,,\n \\\\\n \\varepsilon_{\\tau\\tau} - \\varepsilon_{\\mu\\mu}\n &= \\hphantom{-} \\varepsilon \\, (\\sin^2\\varphi_{13}\n - \\sin^2\\varphi_{12} \\, \\cos^2\\varphi_{13}) \\,,\n \\\\\n \\varepsilon_{e\\mu}\n &= -\\varepsilon \\, \\cos\\varphi_{12} \\, \\sin\\varphi_{12} \\,\n \\cos^2\\varphi_{13} \\, e^{i(\\alpha_1 - \\alpha_2)} \\,,\n \\\\\n \\varepsilon_{e\\tau}\n &= -\\varepsilon \\, \\cos\\varphi_{12} \\, \\cos\\varphi_{13} \\,\n \\sin\\varphi_{13} \\, e^{i(2\\alpha_1 + \\alpha_2)} \\,,\n \\\\\n \\varepsilon_{\\mu\\tau}\n &= \\hphantom{-} \\varepsilon \\, \\sin\\varphi_{12} \\, \\cos\\varphi_{13} \\,\n \\sin\\varphi_{13} \\, e^{i(\\alpha_1 + 2\\alpha_2)} \\,,\n \\end{aligned}\n\\end{equation}\nwhich makes explicit that the diagonal terms ($\\varepsilon_{ee}$,\n$\\varepsilon_{\\mu\\mu}$, $\\varepsilon_{\\tau\\tau}$) can only be determined up to an\noverall additive constant. The term ``$-\\,1$'' at the end of\n$\\varepsilon_{ee} - \\varepsilon_{\\mu\\mu}$ arises from the standard matter term. The\nfermion-specific coefficients $\\varepsilon_{\\alpha\\beta}^f$ are obtained from\nthe effective ones $\\varepsilon_{\\alpha\\beta}$ just by rescaling:\n\\begin{equation}\n \\label{eq:eps_true}\n \\varepsilon_{\\alpha\\beta}^e = \\varepsilon_{\\alpha\\beta} \\,,\n \\qquad\n \\varepsilon_{\\alpha\\beta}^u = \\varepsilon_{\\alpha\\beta} \\big\/ Y_u \\,,\n \\qquad\n \\varepsilon_{\\alpha\\beta}^d = \\varepsilon_{\\alpha\\beta} \\big\/ Y_d \\,.\n\\end{equation}\nThus, in summary, the relevant flavor transition probabilities for\natmospheric and LBL experiments depend on eight parameters:\n($\\Delta m^2_{31}$, $\\theta_{13}$, $\\theta_{23}$) for the vacuum part,\n($\\varepsilon$, $\\varphi_{12}$, $\\varphi_{13}$) for the matter part, and\n($\\alpha_1$, $\\alpha_2$) as relative phases. As for reactor\nexperiments other than KamLAND, matter effects are completely\nirrelevant due to the very small amount of matter crossed, so the\ncorresponding $P_{ee}$ survival probability only depends on the two\nparameters ($\\Delta m^2_{31}$, $\\theta_{13}$).\n\nAs shown in Appendix B of Ref.~\\cite{GonzalezGarcia:2011my}, only the\nrelative sign of $\\Delta m^2_{31}$ and $\\varepsilon$ is relevant for atmospheric\nand LBL neutrino oscillations. Concerning the angles, in the general\ncase of unconstrained $\\alpha_i$ it is enough to consider $0 <\n\\theta_{ij} < \\pi\/2$ and $0 < \\varphi_{ij} < \\pi\/2$, whereas for the\ncase of \\emph{real} NSI (corresponding to $\\alpha_i \\in \\lbrace 0,\\pi\n\\rbrace$) we can set $\\alpha_1 = \\alpha_2 = 0$ and extend the\n$\\varphi_{ij}$ range to $-\\pi\/2 < \\varphi_{ij} < \\pi\/2$.\n\n\\subsection{Earth matter potential for solar and KamLAND neutrinos}\n\\label{sec:form-solar}\n\nFor the study of propagation of solar and KamLAND neutrinos one can\nwork in the one mass dominance approximation, $\\Delta m^2_{31} \\to \\infty$\n(which effectively means that generically $G_F \\sum_f N_f(r)\n\\varepsilon_{\\alpha\\beta}^f \\ll \\Delta m^2_{31} \/ E_\\nu$). In this approximation\nthe survival probability $P_{ee}$ can be written as~\\cite{Kuo:1986sk,\n Guzzo:2000kx}\n\\begin{equation}\n \\label{eq:peesun}\n P_{ee} = c_{13}^4 P_\\text{eff} + s_{13}^4\n\\end{equation}\nwhere $c_{ij} \\equiv \\cos\\theta_{ij}$ and $s_{ij} \\equiv\n\\sin\\theta_{ij}$. The probability $P_\\text{eff}$ can be calculated in\nan effective $2\\times 2$ model with the Hamiltonian $H_\\text{eff} =\nH_\\text{vac}^\\text{eff} + H_\\text{mat}^\\text{eff}$, where:\n\\begin{align}\n \\label{eq:hvacsol}\n H_\\text{vac}^\\text{eff}\n &= \\frac{\\Delta m^2_{21}}{4 E_\\nu}\n \\begin{pmatrix}\n -\\cos 2\\theta_{12} & \\sin 2\\theta_{12} \\\\\n \\hphantom{+} \\sin 2\\theta_{12} & \\cos 2\\theta_{12}\n \\end{pmatrix} ,\n \\\\\n \\label{eq:hmatsol}\n H_\\text{mat}^\\text{eff}\n &= \\sqrt{2} G_F N_e(r)\n \\begin{pmatrix}\n c_{13}^2 & 0 \\\\\n 0 & 0\n \\end{pmatrix}\n + \\sqrt{2} G_F \\sum_f N_f(r)\n \\begin{pmatrix}\n -\\varepsilon_D^{f\\hphantom{*}} & \\varepsilon_N^f \\\\\n \\hphantom{+} \\varepsilon_N^{f*} & \\varepsilon_D^f\n \\end{pmatrix} .\n\\end{align}\nThe coefficients $\\varepsilon_D^f$ and $\\varepsilon_N^f$ are related to the original\nparameters $\\varepsilon_{\\alpha\\beta}^f$ by the following relations:\n\\begin{align}\n \\label{eq:eps_D}\n \\begin{split}\n \\varepsilon_D^f &=\n c_{13} s_{13} \\Re\\left[ e^{i\\delta_\\text{CP}} \\big( s_{23} \\, \\varepsilon_{e\\mu}^f\n + c_{23} \\, \\varepsilon_{e\\tau}^f \\big) \\right]\n - \\big( 1 + s_{13}^2 \\big) c_{23} s_{23} \\Re\\!\\big( \\varepsilon_{\\mu\\tau}^f \\big)\n \\\\\n & \\hphantom{={}}\n -\\frac{c_{13}^2}{2} \\big( \\varepsilon_{ee}^f - \\varepsilon_{\\mu\\mu}^f \\big)\n + \\frac{s_{23}^2 - s_{13}^2 c_{23}^2}{2}\n \\big( \\varepsilon_{\\tau\\tau}^f - \\varepsilon_{\\mu\\mu}^f \\big) \\,,\n \\end{split}\n \\\\[2mm]\n \\label{eq:eps_N}\n \\varepsilon_N^f &=\n c_{13} \\big( c_{23} \\, \\varepsilon_{e\\mu}^f - s_{23} \\, \\varepsilon_{e\\tau}^f \\big)\n + s_{13} e^{-i\\delta_\\text{CP}} \\left[\n s_{23}^2 \\, \\varepsilon_{\\mu\\tau}^f - c_{23}^2 \\, \\varepsilon_{\\mu\\tau}^{f*}\n + c_{23} s_{23} \\big( \\varepsilon_{\\tau\\tau}^f - \\varepsilon_{\\mu\\mu}^f \\big)\n \\right] ,\n\\end{align}\nso effectively the relevant probabilities for solar and KamLAND\nneutrinos depend on the 3 real oscillation parameters $\\Delta m^2_{21}$,\n$\\theta_{12}$, and $\\theta_{13}$ as well as one real $\\varepsilon_D^f$ and\none complex $\\varepsilon_N^f$ matter parameter for each $f$. Notice also\nthat the matter chemical composition of the Sun varies substantially\nalong the neutrino production region, with $Y_n$ dropping from about\n$1\/2$ in the center to about $1\/6$ at the border of the solar\ncore. Therefore, unlike the case of Eq.~\\eqref{eq:compoconst} for the\nEarth it is not possible to introduce a common set of parameters\naccounting simultaneously for all the different $f$. Consequently in\nthe analysis of solar data we will consider only one particular choice\nof $f=e$, $f=u$ or $f=d$ at a time.\n\nConcerning the parameter ranges, the situation is very similar to the\nstandard case without NSI. The angle $\\theta_{13}$ only enters through\nEq.~\\eqref{eq:peesun}, so it is sufficient to consider $0 \\le\n\\theta_{13} \\le \\pi\/2$. The Hamiltonian~\\eqref{eq:hmatsol} is\ninvariant under the transformation $\\Delta m^2_{21} \\to -\\Delta m^2_{21}$ $\\wedge$\n$\\theta_{12} \\to \\theta_{12} + \\pi\/2$, so without loss of generality\nwe can assume $\\Delta m^2_{21} > 0$. In Eq.~\\eqref{eq:hvacsol} $\\theta_{12}$\nappears multiplied by $2$, so we can restrict its range to $-\\pi\/2 \\le\n\\theta_{12} \\le +\\pi\/2$. Finally, the probabilities are insensitive to\nthe overall sign of the non-diagonal entry of \\eqref{eq:hmatsol},\nresulting in a symmetry $\\theta_{12} \\to -\\theta_{12}$ $\\wedge$\n$\\varepsilon_N^f \\to -\\varepsilon_N^f$, which can be used to further restrict the\n$\\theta_{12}$ range to $0 \\le \\theta_{12} \\le \\pi\/2$. Thus in the most\ngeneral case we have $\\Delta m^2_{21} > 0$, $0 \\le \\theta_{ij} \\le \\pi\/2$,\n$\\varepsilon_D^f$ real, and $\\varepsilon_N^f$ complex. Notice however that, as\ndiscussed before, from the point of view of neutrino oscillations the\nphase of $\\varepsilon_N^f$ is not a genuine NSI property but rather a\nrelative feature of the vacuum and matter parts.\n\nIn the specific case of non-standard interactions with\n\\emph{electrons} ($f=e$) there is another exact symmetry. Both the\nstandard and the non-standard terms in Eq.~\\eqref{eq:hmatsol} scale\nwith the same matter density profile $N_e(r)$, so they can be merged\ninto a single term and $H_\\text{mat}^\\text{eff}$ takes the form:\n\\begin{equation}\n \\label{eq:hmat_elec}\n H_\\text{mat}^\\text{eff}\n = \\sqrt{2} G_F N_e(r)\n \\begin{pmatrix}\n -\\varepsilon_D^e + c_{13}^2\/2 & \\varepsilon_N^e \\\\\n \\varepsilon_N^{e*} & \\varepsilon_D^e - c_{13}^2\/2\n \\end{pmatrix} .\n\\end{equation}\nThe probabilities are invariant under $H \\to -H^*$, which is realized\nfor $\\Delta m^2_{21} \\to -\\Delta m^2_{21}$ $\\wedge$ $\\left( \\varepsilon_D^e - c_{13}^2\/2\n\\right) \\to -\\left( \\varepsilon_D^e - c_{13}^2\/2 \\right)$ $\\wedge$ $\\varepsilon_N^e\n\\to -\\varepsilon_N^{e*}$. Combining this with the general symmetries\ndiscussed above we can reabsorb the sign flip of both $\\Delta m^2_{21}$ and\n$\\varepsilon_N^e$ into $\\theta_{12}$, resulting in the transformation\n$\\theta_{12} \\to \\pi\/2 - \\theta_{12}$ $\\wedge$ $\\varepsilon_D^e \\to c_{13}^2\n- \\varepsilon_D^e$ $\\wedge$ $\\varepsilon_N^e \\to \\varepsilon_N^{e*}$. This invariance\nimplies that for each point in the so-called ``light-side'' of the\nparameter space (\\textit{i.e.}, the region with $\\theta_{12} <\n45^\\circ$) there is a point in the ``dark-side'' (the region with\n$\\theta_{12} > 45^\\circ$) which cannot be distinguished experimentally\nby oscillations alone. In the case of NSI with $f=u$ or $f=d$ such a\nsymmetry is no longer exact, however as we will see in\nSec.~\\ref{sec:solar} it is still realized with considerable accuracy.\n\nAs mentioned in Sec.~\\ref{sec:form-atmos} the transition probabilities\nin the atmospheric+LBL sector are invariant under a simultaneous sign\nflip of $\\Delta m^2_{31}$, $\\varepsilon$ and $\\alpha_i$. If this transformation is\nextended to the solar+KamLAND sector through Eqs.~\\eqref{eq:eps_atm},\n\\eqref{eq:eps_true} and then~\\eqref{eq:eps_D}, \\eqref{eq:eps_N} (with\n$\\delta_\\text{CP} = 0$ as in the atmospheric approximation) it leads to\n$\\varepsilon_D^f \\to c_{13}^2 \/ Y_f - \\varepsilon_D^f$ $\\wedge$ $\\varepsilon_N^f \\to\n-\\varepsilon_N^{f*}$. For $f=e$ this transformation becomes an exact symmetry\nif combined with a sign flip of $\\Delta m^2_{21}$, as we have just seen.\nHowever, for $f=u$ or $f=d$ such symmetry is only approximate, so that\nthe inclusion of solar data can (at least in principle) lift the sign\ndegeneracy between $\\Delta m^2_{31}$ and $\\varepsilon$.\n\n\\section{Analysis of solar and KamLAND data}\n\\label{sec:solar}\n\nLet us start by presenting the results of the updated analysis of\nsolar and KamLAND experiments in the context of oscillations with the\ngeneralized matter potential in Eq.~\\eqref{eq:hmatsol}. For KamLAND\nwe include the observed energy spectrum in the DS-1 and DS-2 data\nsets~\\cite{Gando:2010aa} with a total exposure of $3.49\\times 10^{32}$\ntarget-proton-year (2135 days). In the analysis of solar neutrino\nexperiments we include the total rates from the radiochemical\nexperiments Chlorine~\\cite{Cleveland:1998nv},\nGallex\/GNO~\\cite{Kaether:2010ag} and\nSAGE~\\cite{Abdurashitov:2009tn}. For real-time experiments we include\nthe 44 data points of Super-Kamiokande phase I (SK1) energy-zenith\nspectrum~\\cite{Hosaka:2005um}, the 33 data points of\nSK2~\\cite{Cravens:2008aa} and 42 data points of SK3~\\cite{Abe:2010hy}\nenergy and day\/night spectra, and the 24 data points of the 1097-day\nenergy spectrum and day-night asymmetry of SK4~\\cite{Smy2012}. We\nalso include the main set of the 740.7 days of Borexino\ndata~\\cite{Bellini:2011rx} as well as their high-energy spectrum from\n246 live days~\\cite{Bellini:2008mr}.\n\nThe results of the three phases of SNO are included in two different\nforms. First, we perform our own combined analysis of the 34 data\npoints of the day-night spectrum data of SNO-I~\\cite{Aharmim:2007nv},\nthe 38 data points of the day-night spectrum of\nSNO-II~\\cite{Aharmim:2005gt} and the three total rates of\nSNO-III~\\cite{Aharmim:2008kc}. We label this analysis as\n\\textsc{SNO-data}. Second, we use the results of their the low energy\nthreshold analysis of the combined SNO phases\nI--III~\\cite{Aharmim:2011vm} which is given in the form of an\neffective \\emph{MSW-like} polynomial parametrization for the day and\nnight survival probabilities --~under the assumption of unitarity of\nthe oscillation probabilities~-- in terms of 7 parameters for which\nthe collaboration give the best fit values and covariant matrix. We\nlabel this analysis as \\textsc{SNO-poly}. Strictly the results of this\neffective parametrization cannot be used for study of exotic scenarios\nin which either unitarity in the active neutrino sector does not hold\n(like for scenarios with sterile neutrinos) or the energy dependence\nof the oscillation probability cannot be well represented by a simple\nquadratic function. Thus in order to verify the robustness of our\nconclusions on the matter potential we present our results for both\nvariants of the SNO analysis. In both cases we have used the solar\nfluxes from the Standard Solar Model GS98~\\cite{Bahcall:2004pz,\n PenaGaray:2008qe}.\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\textwidth]{fig-region-sun}\n \\caption{Two-dimensional projections of the 90\\%, 95\\%, 99\\% and\n $3\\sigma$ CL (2~dof) allowed regions from the analysis of solar\n and KamLAND data in the presence of non-standard matter\n potential. The results are shown for a fix value of\n $\\sin^2\\theta_{13}=0.023$ and after marginalizing over the two\n undisplayed parameters. The left (right) panels corresponds to\n $f=u$ ($f=d$). The colored filled (black-contour void) regions in\n each panel correspond to the \\textsc{SNO-poly} (\\textsc{SNO-data})\n variants of the solar analysis, see text for details. The best\n fit point is marked with a star (triangle). For comparison we\n show also in the lower panels the two green dotted areas\n correspond to the 90\\% and $3\\sigma$ CL allowed regions from the\n analysis of the atmospheric and LBL data.}\n \\label{fig:region-sun}\n\\end{figure}\n\nWe present the results of the analysis of solar and KamLAND data in\nFigs.~\\ref{fig:region-sun} and~\\ref{fig:chisq-sun}. The presence of\nNSI with electrons, $f=e$, would affect not only neutrino propagation\nin matter as described in Eq.~\\eqref{eq:hmatsol}, but also the\nneutrino-electron cross-section in experiments such as SK and\nBorexino. Since here we are only interested in studying the bounds to\npropagation effects we will consider only the cases $f=u$ and $f=d$.\nAlso for simplicity the results are shown for real\n$\\varepsilon_N^f$. Strictly speaking, as discussed in\nSec.~\\ref{sec:form-solar}, the sign of $\\varepsilon_N^f$ is not physically\nobservable in oscillation experiments, as it can be reabsorbed into a\nredefinition of the sign of $\\theta_{12}$. However, for definiteness\nwe have chosen to present our results in the convention $\\theta_{12}\n\\geq 0$, and therefore we consider both positive and negative values\nof $\\varepsilon_N^f$.\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\textwidth]{fig-chisq-sun}\n \\caption{Dependence of the $\\Delta\\chi^2$ function for the analysis\n of the solar and KamLAND data on the relevant oscillation and\n matter potential parameters for $f=u$ (upper panels) and $f=d$\n (lower panels), for both LMA and LMA-D regions and the two\n variants of the SNO analysis, as labeled in the figure.}\n \\label{fig:chisq-sun}\n\\end{figure}\n\nFig.~\\ref{fig:region-sun} shows the two-dimensional projections on the\noscillation parameters ($\\Delta m^2_{21}$, $\\sin^2\\theta_{12}$) and the\nmatter potential parameters ($\\varepsilon_N^f$, $\\varepsilon_D^f$) with $f=u,d$\nafter marginalizing on the undisplayed parameters, for a fix value of\n$\\sin^2\\theta_{13}=0.023$ which is the best fit for the global\nanalysis for $3\\nu$ oscillations~\\cite{GonzalezGarcia:2012sz,\n nufit-1.1}.\nThe first thing to notice is that for both \\textsc{SNO-data} and\n\\textsc{SNO-poly} variants there are two disconnected regions in the\nparameter space. The leftmost region in each panel, whose projection\non the oscillation parameters lies in the first octant of\n$\\theta_{12}$ ($0 \\leq \\theta_{12} \\leq 45^\\circ$) and whose\nprojection on the matter potential parameters contains SM case\n(\\textit{i.e.}, the point $\\varepsilon_N^f = \\varepsilon_D^f = 0$), corresponds to\nthe variation of the ``standard'' LMA solution in the presence of NSI,\nso we will refer to it simply as LMA. The rightmost region in each\npanel, whose projection on the oscillation parameters lies the second\noctant of $\\theta_{12}$ ($45^\\circ \\leq \\theta_{12} \\leq 90^\\circ$)\nand whose projection on the matter potential does not contain the SM\npoint $\\varepsilon_N^f = \\varepsilon_D^f = 0$, corresponds to the ``dark-side''\nsolution found in Ref.~\\cite{Miranda:2004nb} where it was labeled as\nLMA-D. The existence of this new solution, almost degenerate with the\nusual one, is consequence of the quasi-symmetry of the matter\npotential discussed below Eq.~\\eqref{eq:hmat_elec}. We find that at\npresent the best fit point is in most of the cases in the LMA region,\nbut LMA-D lies only at a $\\Delta\\chi^2 = -0.06$ ($f=u$) and $0.4$\n($f=d$) in the \\textsc{SNO-data} variant, increasing to $\\Delta\\chi^2\n= 0.3$ ($f=u$) and $1.4$ ($f=d$) in the \\textsc{SNO-poly} variant. As\nseen in the lower panels the LMA-D solution requires a non-standard\nmatter potential with quite sizable values of $\\varepsilon_D^f$. An obvious\nquestion is whether such large values are in contradiction with other\nneutrino oscillation data, in particular with atmospheric neutrinos.\nWe will return quantitatively to this point in the next section but\nfor illustration we show also in the lower panels as dotted green\nregions the corresponding 90\\% and $3\\sigma$ CL (2~dof) from the\nanalysis of atmospheric and LBL experiments. We see from the figure\nthat still a sizable fraction of the required non-standard matter\npotential parameters for the LMA-D solution is compatible with all the\noscillation data.\n\nIn what respects the dependence on $f$ the figure shows that\nalthough there are small quantitative differences, qualitatively the\nresults are rather similar for non-standard potential for $u$ or $d$\nquarks. Also both variants of the SNO analysis yield similar results.\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\textwidth]{fig-energy-sun}\n \\caption{Survival probabilities in the Sun for different sets of\n oscillation and matter potential parameters as labeled in the\n figure. In all cases we set $\\sin^2\\theta_{13} = 0.023$; the\n quoted value of $\\Delta m^2_{21}$ is given in units of\n $10^{-5}~\\ensuremath{\\text{eV}^2}$. For illustration we also show the extracted\n average survival probabilities from different experiments. See\n text for details.}\n \\label{fig:energy-sun}\n\\end{figure}\n\nFig.~\\ref{fig:chisq-sun} contains the dependence of $\\Delta\\chi^2$ on\neach of the four parameters $\\Delta m^2_{21}$, $\\theta_{12}$, $\\varepsilon_D^f$,\n$\\varepsilon_N^f$, again for $\\sin^2\\theta_{13} = 0.023$ after marginalizing\nover the other three. In each panel the four curves correspond to the\nLMA and LMA-D solutions for both variants of the SNO analysis. The\nmain feature to notice is that in all cases the fit prefers some\nnon-standard value of the matter potential parameters, while for any\n$f$ the SM potential lies at a $\\Delta\\chi^2 = 5.3$ and $\\Delta\\chi^2\n= 7.9$ for \\textsc{SNO-poly} and \\textsc{SNO-data}, respectively. This\narises from the well-known fact that neither the SNO nor SK4 low\nenergy threshold analysis nor the \\Nuc{8}{B} measurement in Borexino\nseem to show evidence of the low energy turn-up of the spectrum\npredicted in the standard LMA MSW solution. This behavior can be\nbetter described in the presence of a non-standard matter potential.\nThis is illustrated in Fig.~\\ref{fig:energy-sun} where we show the\nsurvival probability of solar neutrinos as a function of the neutrino\nenergy, for the best fit of oscillations only (black line) as well as\nthe best fits for $f=u$ and $f=d$ in the presence of NSI from the\nanalysis of solar+KamLAND data (red lines) and from the global\nanalysis discussed in the next section (green lines). In order to take\ninto account the dependence on the neutrino production point, which is\nof particular relevance in the presence of non-standard matter\npotential, we define the average survival probability $\\langle P_{ee}\n\\rangle$ as\n\\begin{equation}\n \\langle P_{ee} (E_\\nu) \\rangle\n = \\dfrac{\\sum_i \\Phi_i(E_\\nu) \\int \\rho_i(r) \\, P_{ee}(E_\\nu,r) \\, dr}\n {\\sum_i \\Phi_i(E_\\nu)}\n\\end{equation} \nwhere $i = \\text{pp}$, pep, \\Nuc{7}{Be}, \\Nuc{13}{N}, \\Nuc{15}{O},\n\\Nuc{17}{F}, \\Nuc{8}{B} and hep labels the neutrino production\nreaction and $\\rho_i(r)$ is the distribution of production points for\nthe reaction $i$ normalized to 1.\n\n\\section{Results of global analysis}\n\\label{sec:global}\n\nWe now present the results of the global analysis including also\natmospheric, LBL and all other reactor data. The data samples\nincluded here are the same as in the NuFIT~1.1 analysis described in\nRef.~\\cite{nufit-1.1}. For atmospheric data we use the\nSuper-Kamiokande results from phases 1--4~\\cite{skatm1-4}, adding the\n1097 days of phase 4 to their published data from phases\n1--3~\\cite{Wendell:2010md}. For what concerns long-baseline\naccelerator experiments, we combine the energy distribution obtained\nby MINOS in both $\\nu_\\mu$ ($\\bar\\nu_\\nu$)\ndisappearance~\\cite{Adamson:2013whj} and $\\nu_e$ ($\\bar\\nu_e$)\nappearance with $10.7~(3.36) \\times 10^{20}$ protons on\ntarget~\\cite{Adamson:2013ue}, and T2K $\\nu_e$ appearance and $\\nu_\\mu$\ndisappearance data for phases 1--3 corresponding to $3.01\\times\n10^{20}$ pot~\\cite{t2k:moriond13}.\nFor oscillation signals at reactor experiments, besides KamLAND, we\ninclude data from the finalized experiments\nCHOOZ~\\cite{Apollonio:1999ae} (energy spectrum data) and Palo\nVerde~\\cite{Piepke:2002ju} (total rate) together with the recent\nspectrum from Double Chooz with 227.9 days live\ntime~\\cite{Abe:2012tg}, and the total even rates in the near and far\ndetectors in Daya Bay~\\cite{An:2012bu} and Reno with 402 days of\ndata-taking~\\cite{reno:venice13}. For the reactor fluxes we follow\nhere the approach of Ref.~\\cite{Schwetz:2011qt}, \\textit{i.e.}, we\nintroduce an overall flux normalization which is then fitted to the\ndata together with the oscillation and matter potential parameters.\nTo better constrain such reactor flux normalization we also include in\nthe analysis the results of the reactor experiments\nBugey4~\\cite{Declais:1994ma}, ROVNO4~\\cite{Kuvshinnikov:1990ry},\nBugey3~\\cite{Declais:1994su}, Krasnoyarsk~\\cite{Vidyakin:1987ue,\n Vidyakin:1994ut}, ILL~\\cite{Kwon:1981ua},\nG\\\"osgen~\\cite{Zacek:1986cu}, SRP~\\cite{Greenwood:1996pb}, and\nROVNO88~\\cite{Afonin:1988gx}, which due to their short baselines ($L\n\\lesssim 100$~m) are insensitive to the neutrino oscillation effects\ndiscussed here.\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\textwidth]{fig-region-eps}\n \\caption{Two-dimensional projections on the matter potential\n parameters ($\\varepsilon$, $\\varphi_{12}$, $\\varphi_{13}$) of the 90\\%,\n 95\\%, 99\\% and $3\\sigma$ CL (2~dof) allowed regions from the\n global analysis of solar, atmospheric, reactor and LBL data after\n marginalization with respect to the undisplayed parameters. The\n colored filled (black-contour void) regions in each panel\n correspond to $f=u$ and the \\textsc{SNO-poly} (\\textsc{SNO-data})\n variants of the solar analysis. The best fit point is marked with\n a star (triangle). For comparison we also show as green dotted\n areas the 90\\% and $3\\sigma$ CL regions from the analysis of\n atmospheric, LBL and reactor neutrinos (without solar nor\n KamLAND).}\n \\label{fig:region-eps}\n\\end{figure}\n\nWe present the results of the global analysis in\nFigs.~\\ref{fig:region-eps}, \\ref{fig:region-osc}, and\n\\ref{fig:chisq-eps}.\nIn Fig.~\\ref{fig:region-eps} we display the two-dimensional\nprojections of the allowed regions in the matter potential parameters\n$\\varepsilon$, $\\varphi_{12}$ and $\\varphi_{13}$ (in the parametrization of\nEq.~\\eqref{eq:hmatgen} with the additional constraint of equal matter\neigenvalues $\\varepsilon'=0$) after marginalizing over the oscillation\nparameters $\\Delta m^2_{21}$, $\\Delta m^2_{31}$, $\\theta_{12}$, $\\theta_{23}$, and\n$\\theta_{13}$. Since the $\\alpha_i$ phases have little impact on our\nresults, we set for simplicity $\\alpha_1 = \\alpha_2 = 0$. Also, for\nthe sake of concreteness we focus here on $f=u$.\nThe filled colored (black-contour void) regions correspond to the\nglobal analysis with the \\textsc{SNO-poly} (\\textsc{SNO-data}) variant\nof the solar data. For comparison we show also the dotted green\nregions which correspond to the analysis of atmospheric, LBL and\nreactor neutrinos (without solar nor KamLAND) and therefore update our\nprevious results of Ref.~\\cite{GonzalezGarcia:2011my}. As discussed\nin Refs.~\\cite{Friedland:2004ah, Friedland:2005vy,\n GonzalezGarcia:2011my} no bound on the magnitude of the matter\neffects, $\\varepsilon$, can be derived from the analysis of atmospheric and\nLBL experiments in this general scenario. Specific bounds on $\\varepsilon$\ncan be derived if a certain flavor structure of the matter potential\nis assumed \\emph{a priori} (for example, if we assume that no matter\neffects are present in the $e\\mu$ and $e\\tau$ projections, which\ncorresponds to $\\varphi_{12} = \\pi\/2$), implying that $\\varphi_{12}$\nand\/or $\\varphi_{13}$ are larger than some given value. Conversely\nwhen marginalizing over $\\varepsilon$ the full flavor projection\n($\\varphi_{12}$, $\\varphi_{13}$) plane is allowed. However, as seen\nfrom the figure, once the results of solar and KamLAND experiments\n(\\textit{i.e.}, the samples involving $\\nu_e$ or $\\bar\\nu_e$ and long\nenough distances to see both oscillations and NSI effects) are\nincluded in the analysis, a bound on the magnitude of the matter\neffects $\\varepsilon$ is obtained. Furthermore the flavor structure of the\npotential is dramatically constrained as seen in upper-left panel.\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\textwidth]{fig-region-osc}\n \\caption{Two-dimensional projections of the 90\\%, 95\\%, 99\\% and\n $3\\sigma$ CL (2~dof) allowed regions of the oscillation parameters\n for $f=u$ and the \\textsc{SNO-poly} variant of the solar analysis,\n after marginalizing over the matter potential parameters and the\n undisplayed oscillation parameters. The full regions and the star\n correspond to the global analysis including NSI, while the\n black-contour void regions and the triangle correspond to the\n analysis with the usual SM potential. The green and red dotted\n areas show the 90\\% and $3\\sigma$ CL allowed regions from partial\n analyses where the effects of the non-standard matter potential\n have been neglected either in the solar+KamLAND (green) or in the\n atmospheric+LBL (red) sectors.}\n \\label{fig:region-osc}\n\\end{figure}\n\nFig.~\\ref{fig:region-osc} shows the two-dimensional projections of the\nallowed regions from our global analysis in different combinations of\nthe oscillation parameters, again for $f=u$. The regions are obtained\nafter marginalizing over the undisplayed oscillation and matter\npotential parameters. For comparison we also show as black-contour\nvoid regions the corresponding results with the usual SM matter\npotential.\\footnote{Notice that in this analysis we are neglecting\n $\\Delta m^2_{21}$ effects in the atmospheric and LBL oscillations, hence\n the standard oscillation results have no sensitivity to CP violation\n and only very marginal sensitivity to the mass ordering and the\n $\\theta_{23}$ octant. For fully updated results and a complete\n treatment of neutrino oscillations in the standard case we address\n the reader to Refs.~\\cite{GonzalezGarcia:2012sz, nufit-1.1}.}\nThe figure clearly shows the robustness of the determination of the\noscillation parameters even in the presence of a generalized matter\npotential, with the exception of the octant of $\\theta_{12}$. In this\nrespect, we find that the LMA-D solution is still allowed in the\nglobal analysis at $\\Delta \\chi^2 = 0.1$ ($0.2$) for $f=u$ and the\n\\textsc{SNO-data} (\\textsc{SNO-poly}) variants, and at $\\Delta\\chi^2 =\n1.1$ ($1.9$) for $f=d$ and the \\textsc{SNO-data} (\\textsc{SNO-poly})\nanalysis. In the figure we also show as green or red dotted regions\nthe results of the analysis when the effects of the non-standard\nmatter parameters are neglected in either Solar+KamLAND (green, upper\npanels) or in atmospheric+LBL (red, lower panels). The comparison of\nthe global analysis with these partial analyses illustrates the\ncomplementarity of the solar+KamLAND and the atmospheric+LBL data in\nthe robustness of the global fit. We also notice how in the upper\npanels the green regions are perfectly symmetric under a sign flip of\n$\\Delta m^2_{31}$, as explained at the end of\nSec.~\\ref{sec:form-atmos}. However, for NSI with quarks ($f=u,d$)\nthis degeneracy is lifted once the solar data are also included in the\nanalysis, as discussed in Sec.~\\ref{sec:form-solar}. Thus the colored\nregions are not exactly identical for both orderings, although with\npresent data the asymmetry is still minimal.\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\textwidth]{fig-chisq-eps}\n \\caption{Dependence of the $\\Delta\\chi^2$ function for the global\n analysis of solar, atmospheric, reactor and LBL data on the NSI\n parameters $\\varepsilon_{\\alpha\\beta}^f$ for $f=u$ (upper panels) and\n $f=d$ (lower panels), for both LMA and LMA-D regions and the two\n variants of the SNO analysis, as labeled in the figure.}\n \\label{fig:chisq-eps}\n\\end{figure}\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\textwidth]{fig-mixed-eps}\n \\caption{Constraints on the effective matter potential parameters\n $\\varepsilon_D^f$ and $\\varepsilon_N^f$ relevant in solar neutrino propagation\n for $f=u$ (upper panels) and $f=d$ (lower panels). In the left\n panels we show as colored filled (black-contour void) areas the\n two-dimensional projections of the 90\\%, 95\\%, 99\\% and $3\\sigma$\n CL (2~dof) allowed regions from the global analysis, for the\n \\textsc{SNO-poly} (\\textsc{SNO-data}) variants of the solar\n analysis. The best fit point is marked with a star (triangle).\n The green dotted areas correspond to the 90\\% and $3\\sigma$ CL\n allowed regions from the analysis of atmospheric, LBL and reactor\n data (without solar and KamLAND). The central and right panels\n show the dependence of $\\Delta\\chi^2$ from the global analysis on\n $\\varepsilon_D^f$ and $\\varepsilon_N^f$, as labeled in the figure.}\n \\label{fig:mixed-eps}\n\\end{figure}\n\nIn Fig.~\\ref{fig:chisq-eps} we plot the dependence of the\n$\\Delta\\chi^2$ function for the global analysis on the NSI parameters\n$\\varepsilon_{\\alpha\\beta}^f$, after marginalizing over the undisplayed\noscillation and matter potential parameters. Similarly, in\nFig.~\\ref{fig:mixed-eps} we show the present determination on the\neffective matter potential parameters $\\varepsilon_D^f$ and $\\varepsilon_N^f$\nrelevant in the propagation of solar and KamLAND neutrinos. In both\nfigures we display separately the results of the marginalization in\nthe LMA and the LMA-D regions of the parameter space, as well as both\nthe \\textsc{SNO-data} and \\textsc{SNO-poly} variants of the solar\nanalysis.\nFrom these figures we derive the 90\\% and $3\\sigma$ allowed ranges for\nthe NSI parameters implied by the global analysis, which we summarize\nin Table~\\ref{tab:final}. The results in this table correspond to the\n\\textsc{SNO-poly} analysis and have been obtained for real matter\npotential parameters. As discussed in Sec.~\\ref{sec:formalism}, in\nsuch a case only the \\emph{relative} sign of the various\n$\\varepsilon_{\\alpha\\neq\\beta}^f$ and the vacuum mixing angles can be\ndetermined by oscillations. Thus strictly speaking once the results\nare marginalized with respect to all other parameters in the most\ngeneral parameter space, the oscillation analysis can only provide\nbounds on $|\\varepsilon_{\\alpha\\neq\\beta}^f|$. Still, for the sake of\ncompleteness we have decided to retain in Table~\\ref{tab:final} the\nsigns of the non-diagonal $\\varepsilon_{\\alpha\\neq\\beta}^f$, which is correct\nas long as such signs are understood to be relative vacuum-matter\nquantities and not intrinsic NSI features.\n\nNeutrino scattering experiments such as CHARM~\\cite{Dorenbosch:1986tb,\n Allaby:1987vr}, CDHSW~\\cite{Blondel:1989ev} and\nNuTeV~\\cite{Zeller:2001hh} are sensitive to NSI with $u$ and $d$, and\ncan therefore yield information on\n$\\varepsilon_{\\alpha\\beta}^f$~\\cite{Davidson:2003ha}. In\nRef.~\\cite{Miranda:2004nb} it was found that the combination with\nCHARM scattering results~\\cite{Dorenbosch:1986tb, Allaby:1987vr} for\n$f=d$ substantially lifts the statistical difference between LMA and\nLMA-D. Although a rigorous combined analysis of the oscillation\nresults presented here with those from scattering experiments is\nbeyond the scope of this paper,\\footnote{Notice that neutrino\n scattering results also depend on the axial NSI interactions and a\n rigorous global study of neutrino oscillation and scattering data\n will contain a larger number of parameters which makes it\n technically challenging.} in Table~\\ref{tab:final} we present\nseparate ranges for marginalization over $0 \\le \\theta_{12} \\le\n45^\\circ$ (denoted ``LMA'') and over the complete parameter space $0\n\\le \\theta_{12} \\le 90^\\circ$ (denoted ``$\\text{LMA} \\oplus\n\\text{LMA-D}$''), so to give at least an idea of what could be gained\nfrom scattering experiments. In most of the cases the $\\text{LMA}\n\\oplus \\text{LMA-D}$ marginalization yield just a slightly wider\ninterval than the marginalization within the LMA region. However, for\n$\\varepsilon_{ee}^f - \\varepsilon_{\\mu\\mu}^f$ and $\\varepsilon_D^f$ the general allowed\nrange is composed by two separated intervals, one arising from the LMA\nregion and the other from the LMA-D region, so the full $\\text{LMA}\n\\oplus \\text{LMA-D}$ range has to be intended as the direct sum of the\nbound provided in the LMA case and the extra interval quoted in the\n$\\text{LMA} \\oplus \\text{LMA-D}$ column.\n\n\\begin{table}\\centering\n \\definecolor{grey}{gray}{0.75}\n \\newcommand{~\\color{grey}\\vrule\\!\\!}{~\\color{grey}\\vrule\\!\\!}\n \\begin{tabular}{|l|r|r@{~\\color{grey}\\vrule\\!\\!}r|r@{~\\color{grey}\\vrule\\!\\!}r|}\n \\hline\n \\multicolumn{2}{|c|}{}\n & \\multicolumn{2}{c|}{90\\% CL}\n & \\multicolumn{2}{c|}{$3\\sigma$}\n \\\\\n \\hline\n Param. & best-fit\n & \\multicolumn{1}{c@{~\\color{grey}\\vrule\\!\\!}}{LMA}\n & \\multicolumn{1}{c|}{$\\text{LMA} \\oplus \\text{LMA-D}$}\n & \\multicolumn{1}{c@{~\\color{grey}\\vrule\\!\\!}}{LMA}\n & \\multicolumn{1}{c|}{$\\text{LMA} \\oplus \\text{LMA-D}$}\n \\\\\n \\hline\n \\hline\n $\\varepsilon_{ee}^u - \\varepsilon_{\\mu\\mu}^u$ & $+0.298$\n & $[+0.00, +0.51]$ & ${}\\oplus [-1.19, -0.81]$\n & $[-0.09, +0.71]$ & ${}\\oplus [-1.40, -0.68]$\n \\\\\n $\\varepsilon_{\\tau\\tau}^u - \\varepsilon_{\\mu\\mu}^u$ & $+0.001$\n & $[-0.01, +0.03]$ & $[-0.03, +0.03]$\n & $[-0.03, +0.20]$ & $[-0.19, +0.20]$\n \\\\\n $\\varepsilon_{e\\mu}^u$ & $-0.021$\n & $[-0.09, +0.04]$ & $[-0.09, +0.10]$\n & $[-0.16, +0.11]$ & $[-0.16, +0.17]$\n \\\\\n $\\varepsilon_{e\\tau}^u$ & $+0.021$\n & $[-0.14, +0.14]$ & $[-0.15, +0.14]$\n & $[-0.40, +0.30]$ & $[-0.40, +0.40]$\n \\\\\n $\\varepsilon_{\\mu\\tau}^u$ & $-0.001$\n & $[-0.01, +0.01]$ & $[-0.01, +0.01]$\n & $[-0.03, +0.03]$ & $[-0.03, +0.03]$\n \\\\\n \\hline\n $\\varepsilon_D^u$ & $-0.140$\n & $[-0.24, -0.01]$ & ${}\\oplus [+0.40, +0.58]$\n & $[-0.34, +0.04]$ & ${}\\oplus [+0.34, +0.67]$\n \\\\\n $\\varepsilon_N^u$ & $-0.030$\n & $[-0.14, +0.13]$ & $[-0.15, +0.13]$\n & $[-0.29, +0.21]$ & $[-0.29, +0.21]$\n \\\\\n \\hline\n \\hline\n $\\varepsilon_{ee}^d - \\varepsilon_{\\mu\\mu}^d$ & $+0.310$\n & $[+0.02, +0.51]$ & ${}\\oplus [-1.17, -1.03]$\n & $[-0.10, +0.71]$ & ${}\\oplus [-1.44, -0.87]$\n \\\\\n $\\varepsilon_{\\tau\\tau}^d - \\varepsilon_{\\mu\\mu}^d$ & $+0.001$\n & $[-0.01, +0.03]$ & $[-0.01, +0.03 ]$\n & $[-0.03, +0.19]$ & $[-0.16, +0.19]$\n \\\\\n $\\varepsilon_{e\\mu}^d$ & $-0.023$\n & $[-0.09, +0.04]$ & $[-0.09, +0.08 ]$\n & $[-0.16, +0.11]$ & $[-0.16, +0.17]$\n \\\\\n $\\varepsilon_{e\\tau}^d$ & $+0.023$\n & $[-0.13, +0.14]$ & $[-0.13, +0.14 ]$\n & $[-0.38, +0.29]$ & $[-0.38, +0.35]$\n \\\\\n $\\varepsilon_{\\mu\\tau}^d$ & $-0.001$\n & $[-0.01, +0.01]$ & $[-0.01, +0.01 ]$\n & $[-0.03, +0.03]$ & $[-0.03, +0.03]$\n \\\\\n \\hline\n $\\varepsilon_D^d$ & $-0.145$\n & $[-0.25, -0.02]$ & ${}\\oplus [+0.49, +0.57 ]$\n & $[-0.34, +0.05]$ & ${}\\oplus [+0.42, +0.70]$\n \\\\\n $\\varepsilon_N^d$ & $-0.036$\n & $[-0.14, +0.12]$ & $[-0.14, +0.12 ]$\n & $[-0.28, +0.21]$ & $[-0.28, +0.21]$\n \\\\\n \\hline\n \\end{tabular}\n \\caption{90\\% and $3\\sigma$ allowed ranges for the matter potential\n parameters $\\varepsilon_{\\alpha\\beta}^f$ for $f=u,d$ as obtained from the\n global analysis of oscillation data. The results are obtained\n after marginalizing over oscillation and the other matter\n potential parameters either within the LMA only and within either\n LMA or LMA-D subspaces respectively. The numbers quoted are the\n \\textsc{SNO-poly} variant of the solar analysis. See text for\n details.}\n \\label{tab:final}\n\\end{table}\n\n\\section{Summary}\n\\label{sec:summary}\n\nIn this article we have quantified our current knowledge of the size\nand flavor structure of the matter background effects in the evolution\nof solar, atmospheric, reactor and LBL neutrinos based solely on a\nglobal analysis of oscillation data. It complements the study in\nRef.~\\cite{GonzalezGarcia:2011my} where the analysis of the matter\npotential was perform only considering atmospheric and LBL neutrinos.\n\nAfter briefly presenting the most general parametrization of the\nmatter potential and its connection with non-standard neutrino\ninteractions (NSI), we have focused on the analysis of solar and\nKamLAND data. We have found (see Fig.~\\ref{fig:chisq-sun}) that the\nfit always prefers some non-standard value of the matter potential\nparameters, while the SM potential lies at a $\\Delta\\chi^2 \\sim\n5$--$8$ depending on the details of the analysis. This is consequence\nof the fact that none of the experiments sensitive to \\Nuc{8}{B}\nneutrinos has provided so far evidence of the low energy turn-up of\nthe spectrum predicted in the standard LMA MSW solution (see\nFig.~\\ref{fig:energy-sun}). We have also found in that the present\nanalysis still allows for two disconnected regions in the parameter\nspace, the ``standard'' LMA region and the ``dark side'' LMA-D (see\nFig.~\\ref{fig:region-sun}), and that the statistical difference\nbetween both solutions never exceeds $\\Delta\\chi^2 = 1.4$. Although\nthe LMA-D solution requires rather large values of the matter\nparameters, we have shown (and latter quantified in\nSec.~\\ref{sec:global}) that it is still fully compatible with the\nbounds from atmospheric and LBL oscillation data.\n\nWe have then turned to a global analysis in which the data from solar\nand KamLAND have been combined with those from atmospheric, LBL, and\nother reactor experiments. For what concerns the impact of the\nnon-standard matter potential on the determination of the oscillation\nparameters, we found that the determination of $\\Delta m^2_{21}$,\n$|\\Delta m^2_{31}|$, $\\sin^2\\theta_{23}$, and $\\sin^2\\theta_{13}$ is very\nrobust due to strong synergies between solar+KamLAND and\natmospheric+LBL data. In particular, once the results of solar and\nKamLAND experiments are included in the analysis both the magnitude\nand the flavor structure of NSI are strongly constrained, thus\npreventing the weakening of the $|\\Delta m^2_{31}|$ and $\\theta_{23}$ bounds\nwhich was observed in Refs.~\\cite{Friedland:2004ah, Friedland:2005vy,\n GonzalezGarcia:2011my} from the analysis of atmospheric and LBL data\nalone. In turn, the inclusion of atmospheric+LBL data in the solar\nanalysis severely constrain the allowed range of non-diagonal NSI\ndescribed by the effective parameter $\\varepsilon_N^f$, resulting in the\nstabilization of the $\\Delta m^2_{21}$ and $\\sin^2(2\\theta_{12})$\nbounds. However, unlike for the case of oscillations with the usual SM\npotential, in the presence of non-standard interactions a new solution\nwith $\\sin^2\\theta_{12} > 0.5$ (LMA-D region) becomes allowed. With\nall this, the $3\\sigma$ ranges of the oscillation parameters read:\n\\begin{equation}\n \\label{eq:oscrang}\n \\begin{aligned}\n & \\hspace{-10mm}\\text{Standard Matter Potential}\n & \\hspace{18mm}\n & \\hspace{-10mm}\\text{Generalized Matter Potential}\n \\\\[1mm]\n \\sin^2\\theta_{12} &\\in [0.27, 0.35] \\,,\n & \\sin^2\\theta_{12} &\\in [0.26, 0.35] \\oplus [0.65, 0.75] \\,,\n \\\\\n \\sin^2\\theta_{23} &\\in [0.36, 0.67] \\,,\n & \\sin^2\\theta_{23} &\\in [0.34, 0.67] \\,,\n \\\\\n \\sin^2\\theta_{13} &\\in [0.016, 0.030] \\,,\n & \\sin^2\\theta_{13} &\\in [0.016,0.030] \\,,\n \\\\\n \\Delta m^2_{21} &\\in [6.87, 8.03] \\times 10^{-5}~\\ensuremath{\\text{eV}^2},\n & \\Delta m^2_{21} &\\in [6.86, 8.10] \\times 10^{-5}~\\ensuremath{\\text{eV}^2},\n \\\\\n |\\Delta m^2_{31}| &\\in [2.20, 2.58] \\times 10^{-3}~\\ensuremath{\\text{eV}^2},\n & |\\Delta m^2_{31}| &\\in [2.20, 2.65] \\times 10^{-3}~\\ensuremath{\\text{eV}^2}.\n \\end{aligned}\n\\end{equation}\nThe corresponding bounds on the individual NSI parameters from the\nglobal analysis \\emph{after marginalization from all other oscillation\n and matter parameters} are given in Fig.~\\ref{fig:chisq-eps} and\nTable~\\ref{tab:final}. Comparing the results in the Table with the\nbounds derived in Refs.~\\cite{Davidson:2003ha, Biggio:2009nt} from\nnon-oscillation data we find that, with the possible exception of\n$\\varepsilon_{e\\mu}^{u,d}$, the global oscillation analysis presented here\nyields the most restrictive bounds on the \\emph{vector} NSI\nparameters. This is even more impressive if one considers that the\none-dimensional bounds in Table~\\ref{tab:final} arise as projections\nof a global scan of the entire parameter space, and therefore\ncorrelations among different parameters are properly take into\naccount. Conversely, the bounds from neutrino scattering experiments\nare usually obtained on a one-by-one basis, \\textit{i.e.}\\ varying a\nsingle parameter at a time while keeping all the others set to\nzero. In spite of this, neutrino scattering experiments still provide\ncomplementary information to oscillation experiments, for example for\n$f=d$ they can substantially lifts the degeneracy between the LMA and\nLMA-D solutions. Therefore, although a rigorous combined analysis of\nneutrino oscillations and neutrino scattering experiments is\ntechnically challenging and well beyond the scope of the present work,\nit is certainly worth considering for the future.\n\n\\section*{Acknowledgments}\n\nThis work is supported by Spanish MINECO (grants FPA-2010-20807,\nFPA-2012-31880, FPA-2012-34694, consolider-ingenio 2010 grant\nCSD-2008-0037 and ``Centro de Excelencia Severo Ochoa'' program\nSEV-2012-0249), by CUR Generalitat de Catalunya (grant 2009SGR502), by\nComunidad Autonoma de Madrid (HEPHACOS project S2009\/ESP-1473), by\nUSA-NSF (grant PHY-09-6739) and by the European Union (EURONU project\nFP7-212372 and FP7 Marie Curie-ITN actions PITN-GA-2009-237920\n``UNILHC'' and PITN-GA-2011-289442 ``INVISIBLES'').\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}