E. Gubankova
Using flow equations, we derive an effective quark-quark interaction and obtain the coupled set of gap equations for the condensates of the CFL phase of massless $N_f=3$ dense QCD. The formalism developed here enables one to consider more general case of nonzero $s$-quark mass.
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A supersymmetric composite model of color superconductivity is proposed. Quarks and diquarks are dynamically generated as composite fields by a newly introduced strong gauge dynamics. It is shown that the condensation of the scalar component of the diquark supermultiplet occurs when the chemical potential becomes larger than some critical value. We believe that the model well captures aspects of the diquark condensate behavior and helps our understanding of the diquark dynamics in real QCD. The results obtained here might be useful when we consider a theory composed of quarks and diquarks.
The influence of a constant uniform external chromomagnetic field $H$ on the formation of color superconductivity has been investigated. The consideration was performed in the framework of a (2+1)-dimensional Nambu--Jona-Lasinio model with two different four-fermionic structures responsible for $<\bar qq>$ and diquark $<qq>$ condensates. In particular, it was shown that there exists a critical value $H_c$ of the external chromomagnetic field such that at $H>H_c$ a nonvanishing diquark condensate is dynamically created (the so-called chromomagnetic catalysis effect of color superconductivity). …
In this talk, a supersymmetric (SUSY) composite model of color superconductivity is discussed. In this model, quark and diquark supermultiplets are dynamically generated as massless composites by a newly introduced confining gauge dynamics. It is analytically shown that the scalar component of diquark supermultiplets develops vacuum expectation value (VEV) at a certain critical chemical potential. We believe that our model well captures aspects of the diquark condensate behavior and helps our understanding of its dynamics in real QCD. The results obtained here might be useful when we consider …
We study the instabilities of quark matter in the framework of a generalized Nambu--Jona-Lasinio model, in order to explore possible competition between three-quark clustering to form nucleons and diquark formation leading to color superconductivity. Nucleon and $Δ$ solutions are obtained for the relativistic Faddeev equation at finite density and their binding energies are compared with those for the scalar and axial-vector diquarks found from the Bethe-Salpeter equation. In a model with interactions in both scalar and axial diquark channels, bound nucleons exist up to nuclear matter density.…
We discuss random matrix models for the spontaneous breaking of both chiral and color symmetries at zero chemical potential and finite temperature. Exploring different Lorentz and gauge symmetric color structures of the random matrix interactions, we find that spontaneous chiral symmetry breaking is always thermodynamically preferred over diquark condensation. Stable diquark condensates appear only as SU(2) rotated chiral condensates, which do not represent an independent thermodynamic phase. Our analysis is based on general symmetry arguments and hence suggests that no stable and independent …
We investigate color superconductivity and chiral symmetry restoration at finite temperature and baryon density in the frame of standard two flavor Nambu--Jona-Lasinio model. We derive the diquark mass in RPA, discuss its constraint on the coupling constant in the diquark channel, and find a strong competition between the two phase transitions when the coupling constant is large enough.
We investigate the fluctuations of the diquark-pair field and their effects on observables above the critical temperature $T_c$ in two-flavor color superconductivity (CSC) at moderate density using a Nambu-Jona-Lasinio-type effective model of QCD. Because of the strong-coupling nature of the dynamics, the fluctuations of the pair field develop a collective mode, which has a prominent strength even well above T_c. We show that the collective mode is actually the soft mode of CSC. We examine the effects of the pair fluctuations on the specific heat and the quark spectrum for T above but close to…
We investigate modification of the dilepton production rate by the diquark fluctuations that form well-developed collective modes near the critical temperature of color superconductivity. Through the analysis of the photon self-energy called the Aslamasov-Larkin, Maki-Thompson and density of states terms in the theory of metalic superconductivity, it is shown that the collective mode in the diquark channel affects the photon self-energy significantly and thereby gives rise to an anomalous enhacement of the dilepton production rate in the low invariant-mass region.
We investigate the effect of the electric-charge neutrality in $β$ equilibrium on the chiral phase transition by solving the chiral and diquark condensates in the two-flavor Nambu--Jona-Lasinio model. We demonstrate that the electric-charge neutrality plays a similar role as the repulsive vector interaction; they both weaken the first-order chiral phase transition in the high-density and low-temperature region. The first-order chiral phase transition is not affected, however, at finite temperatures where the diquark condensate melts. In this way the chiral phase transition could be second-orde…
The effect of electric and color neutrality on 2-flavor color superconductivity has been investigated. It has been found that the effect of the charge neutrality on a 2-flavor quark system is very different from that on a 3-flavor system. The BCS diquark pair in 2-flavor color superconducting phase has been largely reduced by the large Fermi momentum of electron, the diquark condensate first increases with the increasing of quark's chemical potential $μ$,then decreases rapidly and finally disappears at about $μ=535MeV$, at which the thermodynamic potential equals to that in the neutral normal …
Mesons and diquarks in a color superconducting regime
本研究采用PNJL模型，在三味三色、同位旋对称且奇异数为零的条件下探究色超导区域的介子与双夸克，发现2SC相对其质量有显著影响，证实色超导对描述一阶手征相变附近相关粒子的重要性。
This article focuses on the two-flavor color superconducting phase at moderate baryon density. In order to simultaneously investigate the chiral phase transition and the color superconducting phase transition, the Nambu-Gorkov formalism is extended to treat the quark-antiquark and diquark condensates on an equal footing. The competition between the chiral condensate and the diquark condensate is analyzed. The cold dense charge neutral two-flavor quark system is investigated in detail. Under the local charge neutrality condition, the ground state of two-flavor quark matter is sensitive to the c…
We study the competition of quark-antiquark and diquark condensates under the influence of an external chromomagnetic field modelling the gluon condensate and in dependence on the chemical potential and temperature. As our results indicate, an external chromomagnetic field might produce remarkable qualitative changes in the picture of the color superconducting (CSC) phase formation. This concerns, in particular, the possibility of a transition to the CSC phase and diquark condensation at finite temperature.
The interaction of light quarks and instanton liquid is analyzed at finite density of quark/baryon matter and in the phase of nonzero values of diquark (colour) condensate. It is shown that instanton liquid perturbation produced by such an interaction results in an essential increase of the critical value of quark chemical potential $μ_c$ which provokes the perceptible increase of quark matter density around the expected onset of the colour superconductivity phase.
The effect of color breaking on colored quarks' chiral condensates has been investigated at zero temperature and moderate baryon density. It is found that the influence of the diquark condensate on different colored quarks is very small.
We study the phase structure of the two-flavor Nambu--Jona-Lasinio (NJL) model in the chiral limit, extending a previous study of the competition of an inhomogeneous chiral phase and a two-flavor color-superconducting (2SC) phase [1, 2]. There, an analytic expression for the dispersion relations for quasiparticle excitations in the presence of both a particular inhomogeneous chiral condensate, the so-called chiral density wave (CDW), and a homogeneous 2SC condensate was found. In this work we show how to determine the dispersion relations for arbitrary modulations of the chiral condensate in t…
Bose-Einstein condensation (BEC) of composite diquarks in quark matter (the color superconductor phase) is discussed using the quasi-chemical equilibrium theory at a relatively low density region near the deconfinement phase transition, where dynamical quark-pair fluctuations are assumed to be described as bosonic degrees of freedom (diquarks). A general formulation is given for the diquark formation and particle-antiparticle pair-creation processes in the relativistic flamework, and some interesting properties are shown, which are characteristic for the relativistic many-body system. Behavior…
Some consequences of a classical vector field (chromo-electromagnetic field) coupled to quarks, which undergo to superfluid and/or superconductive states with diquark / diantiquark condensation, are investigated. For this, one scalar field exchange is considered in the lines investigated by Pisarski and Rischke \cite{PISARSKI-RISCHKE} in the mean field approach. Some effects and possible consequences are discussed.
Color superconductivity and speed of sound in the two-flavor quark-meson diquark model
本文研究两味夸克-介子二夸克模型在2SC相中的性质，通过平均场近似和重整化群方程改进热力学势，计算相图与声速，发现声速趋近共形极限，与硬密环微扰理论一致。
The equation of state for quark matter is derived for a nonlocal, chiral quark model within the mean field approximation. We investigate the effects of a variation of the form factors of the interaction on the phase diagram of quark matter under the condition of beta-equilibrium and charge neutrality. Special emphasis is on the occurrence of a diquark condensate which signals a phase transition to color superconductivity and its effects on the equation of state. We calculate the quark star configurations by solving the Tolman- Oppenheimer- Volkoff equations and obtain for the transition from a…
Employing an extended three flavor version of the NJL model we discuss in detail the phase diagram of quark matter. The presence of quark as well as of diquark condensates gives raise to a rich structure of the phase diagram. We study in detail the chiral phase transition and the color superconductivity as well as color flavor locking as a function of the temperature and chemical potentials of the system.
The interaction of light quarks and instanton liquid is analyzed at finite density of quark/baryon matter and in the phase of nonzero values of diquark (colour) condensate. It is shown that instanton liquid perturbation produced by such an interaction results in an essential increase of the critical value of quark chemical potential $μ_c$ which provokes the perceptible increase of quark matter density around the expected onset of the colour superconductivity phase.
A quantum field theoretical approach to the thermodynamics of dense Fermi systems is developed for the description of the formation and dissolution of quantum condensates and bound states in dependence of temperature and density. As a model system we study the chiral and superconducting phase transitions in two-flavor quark matter within the NJL model and their interrelation with the formation of quark-antiquark and diquark bound states. The phase diagram of quark matter is evaluated as a function of the diquark coupling strength and a coexistence region of chiral symmetry breaking and color s…
In this article, we propose a new approach to calculate the Nambu-Gorkov propagator intuitively with some linear algebra techniques in presence of the scalar diquark condensates. With the help of energy projective operators, we can obtain relatively simple expressions for the quark propagators, which greatly facilitate the calculations in solving the Schwinger-Dyson equation to obtain the gap parameters.
We compute the modification of the photon self-energy due to dynamical diquark fluctuations developed near the critical temperature of the color superconductivity through the Aslamasov-Larkin, Maki-Thompson and density of states terms, which are responsible for the paraconductivity in metals at vanishing energy and momentum. It is shown that the rate has a significant enhancement at low invariant-mass region over a rather wide range of temperature in the normal phase. This enhancement is worth exploration in the relativistic heavy-ion collisions, which may thereby reveal the significance of th…
Since the birth of the quark model, the diquark which is composed of two quarks has been considered as a substantial structure of color anti-triplet. This is not only a mathematical simplification for dealing with baryons, but also provides a physical picture where the diquark would behave as a whole object. It is natural to ask whether such a structure is sufficiently stable against external disturbance. The mass spectra of the ground states of the scalar and axial-vector diquarks which are composed of two-light (L-L), one-light-one-heavy (H-L) and two-heavy quarks (H-H) respectively have bee…
We review efforts to study, using the methods of lattice field theory, the phenomenon of diquark condensation via BCS pairing at a Fermi surface, which has been proposed as a mechanism for color superconductivity in dense quark matter. The particular models studied are the Gross-Neveu model and SU(2) lattice gauge theory; in both cases evidence for superfluidity at high density is presented. The behaviour expected for quarks in both fundamental and adjoint representations of SU(2) is contrasted.
The possibility of the crossover from the BCS pairing to the Bose-Einstein condensate (BEC) of diquarks with going down in density is discussed in the framework of in the Nambu Jona-Lasinio (NJL) model. We find that the quark matter at moderate density may be close to the intermediate of the crossover, the precursory regime to the BEC phase.
High density YM-theory is addressed by means of an effective theory where a Higgs type field corresponds to the diquark order parameter. The effective Higgs theory mimics the Cooper instability at the Fermi surface. After Landau gauge fixing, the theory possesses a global color degree of freedom. Color superconductivity corresponds to a spontaneous breakdown of the residual global color symmetry. Numerical simulations illustrate this mechanism of spontaneously broken global color for the case of the standard electro-weak phase transition and for the case of the transition between the confineme…
At high nuclear density and small temperature, due to the asymptotic freedom property of Quantum ChromoDynamics and to the existence of an attractive channel in the color interaction, diquark condensates might be formed. Since these condensates break the color gauge symmetry, this phenomenon has the name of color superconductivity. In the last few years this has become a very active field of research. While a direct experimental test is still missing, color superconductivity might have implications in astrophysics because for some compact stars, e.g. pulsars, the baryon densities necessary for…
We investigate the effects of isospin asymmetry on the competition between color-superconductivity and inhomogeneous chiral symmetry breaking in dense two-flavor quark matter using an extended Nambu--Jona-Lasinio model. We confirm the appearance of a coexistence window where chiral symmetry is inhomogeneously broken and a nonzero spatially homogeneous diquark gap is present, consistently with previous works, and show that such a phase survives at nonzero isospin chemical potentials. We also discuss how the model phase structure becomes modified as large isospin asymmetries are considered.
The phase structure of hadronic matter at high density relevant to the physics of compact stars and relativistic heavy-ion collisions is studied in a low-energy effective quark theory. The relevant phases that figure are (1) chiral condensation, (2) diquark color condensation (color superconductivity) and (3) induced Lorentz-symmetry breaking (ISB). For reasonable strengths for four-Fermi interactions implied by one-gluon exchange quark interactions as well as instanton-mediated interactions, we find that the ISB phase sets in at a moderate density while the quark condensate vanishes, followed…
The instanton approach to spontaneous chiral symmetry breaking is reviewed, with emphasis on the connection to chiral random matrix theory. We extend the approach to discuss the finite density, zero-temperature behaviour of quark matter. Since the instanton-induced interactions are attractive in both $\bar{q}q$ and $qq$ channels, a competition ensues between phases of matter with condensation in either or both. It results in chiral symmetry restoration due to the onset of diquark condensation, a `colour superconductor', at finite density.
The spectrum of meson and diquark excitations in cold color-superconducting (2SC) quark matter is investigated under local color and electric neutrality constraints with $β$-equilibrium. A 2-flavored Nambu--Jona-Lasinio type model including a baryon $μ_B$, color $μ_8$, and electric $μ_Q$ chemical potentials is used. Two relations between coupling constants $H$ and $G$ in the diquark- and quark-antiquark channels, correspondingly, are treated, $H=3G/4$ and $H=G$. At $H=3G/4$ the gapless- and at $H=G$ the gapped neutral color superconductivity is realized. It is shown that color and electrical n…
We consider a random matrix model which describes the competition between chiral symmetry breaking and the formation of quark Cooper pairs in QCD at finite density. We study the evolution of the phase structure in temperature and chemical potential with variations of the strength of the interaction in the quark-quark channel and demonstrate that the phase diagram can realize a total of six different topologies. A vector interaction representing single-gluon exchange reproduces a topology commonly encountered in previous QCD models, in which a low-density chiral broken phase is separated from a…
Quark-meson diquark model and color superconductivity in dense quark matter
本文通过QMD模型研究稠密夸克物质中的颜色超导性，分析了不同相变下的Goldstone玻色子特性及热力学势计算，揭示了BCS间隙和声速的行为规律。
Since the early days of QCD, it has been argued that inside hadrons quarks organise into substructures. Among those, popular pictures of various phenomena (including exotica and colour superconductivity) give a prominent role to diquarks, i.e. to colour antitriplet combinations of two quarks. Using a gauge-invariant setup, which combines the diquark with a static quark, we study spatial correlations of the two light quarks inside the diquark. After illustrating the setup, we discuss our first results for both the scalar (``good'') and the spin one (``bad'') diquark channels. In a regime in whi…
We investigate the effect of six fermion determinant interaction on color superconductivity as well as on chiral symmetry breaking. Coupled mass gap equations and the superconducting gap equation are derived through the minimisation of the thermodynamic potential. The effect of nonzero quark -- antiquark condensates on the superconducting gap is derived. This becomes particularly relevant for the case of 2-flavor superconducting matter with unpaired strange quarks in the diquark channel. While the effect of six fermion interaction leads to an enhancement of u-d superconductivity, due to nonvan…
The equation of state for quark matter is derived for a nonlocal, chiral quark model within the mean field approximation. Special emphasis is on the occurrence of a diquark condensate which signals a phase transition to color superconductivity and its effects on the equation of state. We present a fit formula for the Bag pressure, which is density dependent in case when the quark matter is color superconducting. We calculate the quark star configurations by solving the Tolman- Oppenheimer- Volkoff equations and demonstrate the effects of diquark condensation on the stability of hybrid stars fo…
We find a (nonlocal) gauge where the wavefunction renormalization constant does not get any corrections for all momenta in the hard-dense loop approximation. In this gauge, we solve the Schwinger-Dyson equations for the diquark condensate in dense QCD to calculate the Cooper pair gap. We determine not only the exponent but also the prefactor of the gap in a gauge independent way. We find that the higher order corrections increase the gap only by about 1.6 times to the leading order gap at Coulomb gauge.
Two-flavor color superconductivity in a general Nambu-Jona-Lasinio model with color and charge neutrality
本研究通过一般NJL模型分析色与电荷中性约束下的双味色超导物质，识别关键相互作用通道，发现无能隙相，并计算夸克比例及相图影响。
We discuss the emergence of a low-energy effective theory with quarks, mesons, diquarks and baryons at vanishing and finite baryon density from first principle QCD. The present work also includes an overview on diquarks at vanishing and finite density, and elucidates the physics of transitional changes from baryonic matter to quark matter including diquarks. This set-up is discussed within the functional renormalisation group approach with dynamical hadronisation. In this framework it is detailed how mesons, diquarks, and baryons emerge dynamically from the renormalisation flow of the QCD effe…
We construct the nuclear and quark matter equations of state at zero temperature in an effective quark theory (the Nambu-Jona-Lasinio model), and discuss the phase transition between them. The nuclear matter equation of state is based on the quark-diquark description of the single nucleon, while the quark matter equation of state includes the effects of scalar diquark condensation (color superconductivity). The effect of diquark condensation on the phase transition is discussed in detail.
The chiral phase transition and color superconductivity in an extended NJL model with eight-quark interactions are studied. The scalar-type nonlinear term hastens the chiral phase transition,the scalar-vector mixing term suppresses effects of the vector-type linear term and the scalar-diquark mixing term makes the coexisting phase wider.
We propose to study the superconducting pairing of quarks with the formation of the diquarks as well as the quark-antiquark pairing in QCD by means of the functional integral technique. The dynamical equations for the superconducting order parameters are the nonlinear integral equations for the composite quantum fields describing the quark-quark or quark-antiquark systems. These composite fields are the bi-local fields if the pairing is generated by the gluon exchange while for the instanton induced pairing interactions they are the local ones. The expressions of the free energy densities are …
We investigate the vacuum structure of dense quark matter in strong magnetic fields at finite temperature and densities in a 3 flavor Nambu Jona Lasinio (NJL) model including the Kobayashi-Maskawa-t'Hooft (KMT) determinant term using a variational method. The method uses an explicit structure for the `ground' state in terms of quark-antiquark condensates as well as diquark condensates. The mass gap equations and the superconducting gap equations are solved self consistently and are used to compute the thermodynamic potential along with the charge neutrality conditions. We also derive the equat…
We analyze the fate of the unbroken SU(2) color gauge interactions for 2 light flavors color superconductivity at non zero temperature. Using a simple model we compute the deconfining/confining critical temperature and show that is smaller than the critical temperature for the onset of the superconductive state itself. The breaking of Lorentz invariance, induced already at zero temperature by the quark chemical potential, is shown to heavily affect the value of the critical temperature and all of the relevant features related to the deconfining transition. Modifying the Polyakov loop model to …
In this series of lectures we present the functional integral method for studying the superconducting pairing of quarks with the formation of the diquarks as well as the quark-antiquark pairing in dense QCD. The dynamical equations for the superconducting order parameters are the nonlinear integral equations for the composite quantum fields describing the quark-quark or quark-antiquark systems. These composite fields are the bi-local fields if the pairing is generated by the gluon exchange while for the instanton induced pairing interactions they are the local ones. The expressions of the free…
We study the competence between chiral symmetry restoration and two flavor color superconductivity (2SC) using a relativistic quark model with covariant nonlocal interactions. We consider two different nonlocal regulators: a Gaussian regulator and a Lorentzian regulator. We find that although the phase diagrams are qualitative similar to those obtained using models with local interactions, in our case the superconducting gaps at medium values of the chemical potential are larger. Consequently, we obtain that in that region the critical temperatures for the disappearance of the 2SC phase might …
We investigate the combined effect of the repulsive vector interaction and the positive electric chemical potential on the chiral phase transition by considering neutral color superconductivity (CSC). The chiral condensate, diquark condensate and quark number densities are solved in both two-flavor and two-plus-one-flavor Nambu-Jona-Lasinio(NJL) models with the so called Kobayashi-Maskawa-'t Hooft term under the charge neutrality constraint. We demonstrate that multiple chiral critical-point structures always exist in the NJL model within the self-consistent mean-field approximation and the nu…
We present a novel relativistic density-functional approach to modeling quark matter with a mechanism to mimic confinement. The quasiparticle treatment of quarks provides their suppression due to a large quark selfenergy already at the mean-field level. We demonstrate that our approach is equivalent to a chiral quark model with medium-dependent couplings. The dynamical restoration of the chiral symmetry is ensured by construction of the density functional. Supplemented with the vector repulsion and diquark pairing the model is applied to construct a hybrid quark-hadron EoS of cold compact-star…
Renormalization of the three-flavor quark-meson diquark model
本文研究二味和三味夸克-介子二夸克模型的可重整化性，通过壳上重整化确定参数，并计算CFL相中的能隙和声速，发现能隙趋于常数、声速从上方趋于共形极限。
The equation of state for quark matter is derived for a nonlocal, chiral quark model within the mean field approximation.We investigate the effects of a variation of the formfactors of the interaction on the phase diagram of quark matter. Special emphasis is on the occurrence of a diquark condensate which signals a phase transition to color superconductivity and its effects on the equation of state under the condition of beta- equilibrium and charge neutrality. We calculate the quark star configurations by solving the Tolman- Oppenheimer- Volkoff equations and obtain for the transition from a …
A more general expression for the quark propagator including both chiral and diquark condensates has been derived by using energy projectors. This makes it possible to study the phase transition from hadron phase to color superconductivity phase in the moderate baryon density region by using Feynman diagrammatic method or Green-function method.
We study the emergence of color superconductivity in the theory of the strong interaction at supranuclear densities. To this end, we follow the renormalization group (RG) flow of dense strong-interaction matter with two massless quark flavors from the fundamental quark and gluon degrees of freedom at high energies down to the non-perturbative low-energy regime which is found to be governed by the dynamical formation of diquark states. With the strong coupling at the initial RG scale as the only input parameter, we compute the (chirally symmetric) scalar diquark condensate and analyze its scali…
Utilizing an extended two-flavor Nambu-Jona Lasinio (NJL) model, we review some of the effects of external magnetic fields on two-flavor color superconducting phase (2SC) at moderate baryon densities in the QCD phase diagram. The effective action of the extended NJL model consists of two mass gaps as functions of three intensive quantities, the temperature, the quark chemical potential and the external magnetic field. The nonzero values of the mass gaps, chiral and diquark condensates, induce spontaneous chiral and color symmetry breaking, respectively, and as a result two different phases of …
We study the effects of thermal fluctuations of gluons and the diquark pairing field on the superconducting-to-normal state phase transition in a three-flavor color superconductor, using the Ginzburg-Landau free energy. At high baryon densities, where the system is a type I superconductor, gluonic fluctuations, which dominate over diquark fluctuations, induce a cubic term in the Ginzburg-Landau free energy, as well as large corrections to quadratic and quartic terms of the order parameter. The cubic term leads to a relatively strong first order transition, in contrast with the very weak first …
We investigate the effect of the alignment of the magnetic moments of Cooper pairs of charged quarks that form at high density in three-flavor quark matter. The high density phase of this matter in the presence of a magnetic field is known to be the Magnetic Color-Flavor-Locked (MCFL) phase of color superconductivity. We derive the Fierz identities of the theory and show how the explicit breaking of the rotational symmetry by the uniform magnetic field field opens new channels of interactions and allows the formation of a new diquark condensate. The new order parameter is a spin-1 diquark cond…
We investigate possible precursory phenomena of color superconductivity at finite temperature $T$ with an effective theory of QCD. It is found that the fluctuation of the diquark pair field exists with a prominent strength even well above the critical temperature $T_c$. We show that such a fluctuaiton forms a collective mode, the corresponding pole of which approaches the origin as $T$ is lowered to $T_c$ in the complex energy plane. We discuss the possible relevance of the precursor to the observables to be detected in heavy-ion collisions.
Because of a logarithmic enhancement from soft, collinear magnetic gluons, in dense quark matter the gap for a color superconducting condensate with spin zero depends upon the QCD coupling constant g not as exp(-1/g^2), like in BCS theory, but as exp(-1/g). In weak coupling, the ratio of the transition temperature to the spin-zero gap at zero temperature is the same as in BCS theory. We classify the gaps with spin one, and find that they are of the same order in g as the spin-zero gap.
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