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D. Toublan

Publications and source records attributed to D. Toublan.

33 records · Page 2Linked to original sources
arXiv2001

Kaon Condensation and Goldstone's Theorem

We consider QCD at a nonzero chemical potential for strangeness. At a critical value of the chemical potential equal to the kaon mass, kaon condensation occurs through a continuous phase transition. We show that, when the isospin symmetry is exact in the Lagrangian, a Goldstone boson with a dispersion relation $E \sim p^2$ appears in the kaon condensed phase. At the same time, the number of the Goldstone bosons is less than the number of broken generators. Both phenomena are familiar in non-relativistic systems. We interpret our results in terms of a Goldstone boson counting rule found previously by Nielsen and Chadha. We also formulate a criterion sufficient for the equality between the number of Goldstone bosons and the number of broken generators.

hep-ph↗
arXiv2001

Diquark Condensate in QCD with Two Colors at Next-to-Leading Order

We study QCD with two colors and quarks in the fundamental representation at finite baryon density in the limit of light quark masses. In this limit the free energy of this theory reduces to the free energy of a chiral Lagrangian which is based on the symmetries of the microscopic theory. In earlier work this Lagrangian was analyzed at the mean field level and a phase transition to a phase of condensed diquarks was found at a chemical potential of half the diquark mass (which is equal to the pion mass). In this article we analyze this theory at next-to-leading order in chiral perturbation theory. We show that the theory is renormalizable and calculate the next-to-leading order free energy in both phases of the theory. By deriving a Landau-Ginzburg theory for the order parameter we show that the finite one-loop contribution and the next-to-leading order terms in the chiral Lagrangian do not qualitatively change the phase transition. In particular, the critical chemical potential is equal to half the next-to-leading order pion mass, and the phase transition is second order.

hep-ph↗
arXiv2001

QCD at small non-zero quark chemical potentials

We study the effects of small chemical potentials associated with the three light quark flavors in QCD. We use a low-energy effective field theory that solely relies on the symmetries of the QCD partition function. We find three different phases: a normal phase, a pion superfluid phase and a kaon superfluid phase. The two superfluid phases are separated by a first order phase transition, whereas the normal phase and either of the superfluid phases are separated by a second order phase transition. We compute the quark-antiquark condensate, the pion condensate and the kaon condensate in each phase, as well as the isospin density, the strangeness density, and the mass spectrum.

hep-ph↗
arXiv2001

Statistical properties of the spectrum of the QCD Dirac operator at low energy

We analyze the statistical properties of the spectrum of the QCD Dirac operator at low energy in a finite box of volume $L^4$ by means of partially quenched Chiral Perturbation Theory (pqChPT), a low-energy effective field theory based on the symmetries of QCD. We derive the two-point spectral correlation function from the discontinuity of the chiral susceptibility. For eigenvalues much smaller than $E_c=F^2/ΣL^2$, where $F$ is the pion decay constant and $Σ$ is the absolute value of the quark condensate, our result for the two-point correlation function coincides with the result previously obtained from chiral Random Matrix Theory (chRMT). The departure from the chRMT result above that scale is described by the contribution of the nonzero momentum modes. In terms of the variance of the number of eigenvalues in an interval containing $n$ eigenvalues on average, it results in a crossover from a $\log n$-behavior to a $n^2 \log n$-behavior.

hep-th↗
arXiv2000

Dirac Spectrum in Adjoint QCD

In this lecture we discuss some exact results for the low-lying spectrum of the Dirac operator in adjoint QCD. In particular, we find an analytical expression for the slope of the average spectral density. These results are obtained by means of a generating function which is an extension of the QCD partition function with fermionic and bosonic ghost quarks. The low-energy limit of this generating function is completely determined by chiral (super-)symmetries. Our results for the slope of the average spectral density are consistent with the results for the scalar susceptibility which can be obtained from the usual chiral Lagrangian.

hep-th↗
arXiv2000

QCD-like Theories at Finite Baryon Density

We study QCD-like theories with pseudoreal fermions at finite baryon density. Such theories include two-color QCD with quarks in the fundamental representation of the color group as well as any-color QCD with quarks in the adjoint color representation. In all such theories the lightest baryons are diquarks. At zero chemical potential $μ$ they are, together with the pseudoscalar mesons, the Goldstone modes of a spontaneously broken enlarged chiral symmetry group. Using symmetry principles, we derive the low-energy effective Lagrangian for these particles. We find that a second order phase transition occurs at a value of $μ$ equal to half the mass of the Goldstone modes. For values of $μ$ beyond this point the scalar diquarks Bose condense and the diquark condensate is nonzero. We calculate the dependence of the chiral condensate, the diquark condensate, the baryon charge density, and the masses of the diquark and pseudoscalar excitations on $μ$ at finite bare quark mass and scalar diquark source. The relevance of our results to lattice QCD calculations and to real three-color QCD at finite baryon density is discussed.

hep-ph↗
arXiv2000

Effective Low Energy Theories and QCD Dirac Spectra

We analyze the smallest Dirac eigenvalues by formulating an effective theory for the QCD Dirac spectrum. We find that in a domain where the kinetic term of the effective theory can be ignored, the Dirac eigenvalues are distributed according to a Random Matrix Theory with the global symmetries of the QCD partition function. The kinetic term provides information on the slope of the average spectral density of the Dirac operator. In the second half of this lecture we interpret quenched QCD Dirac spectra at nonzero chemical potential (with eigenvalues scattered in the complex plane) in terms of an effective low energy theory.

hep-th↗
arXiv1999

On two-color QCD with baryon chemical potential

We study SU(2) color QCD with even number of quark flavors. First, using QCD inequalities we show that at finite baryon chemical potential mu, condensation must occur in the channel with scalar diquark quantum numbers. This breaks the U(1) symmetry generated by baryon charge (baryon superconductivity). Then we derive the effective Lagrangian describing low lying meson and baryon excitations using extended local chiral symmetry of the theory. This enables us to determine the leading term in the dependence of the masses on mu exactly.

hep-ph↗
arXiv1999

The Spectral Density of the QCD Dirac Operator and Patterns of Chiral Symmetry Breaking

We study the spectrum of the QCD Dirac operator for two colors with fermions in the fundamental representation and for two or more colors with adjoint fermions. For $N_f$ flavors, the chiral flavor symmetry of these theories is spontaneously broken according to $SU(2N_f)\to Sp(2N_f)$ and $SU(N_f)\to O(N_f)$, respectively, rather than the symmetry breaking pattern $SU(N_f) \times SU(N_f) \to SU(N_f)$ for QCD with three or more colors and fundamental fermions. In this paper we study the Dirac spectrum for the first two symmetry breaking patterns. Following previous work for the third case we find the Dirac spectrum in the domain $λ\ll Λ_{\rm QCD}$ by means of partially quenched chiral perturbation theory. In particular, this result allows us to calculate the slope of the Dirac spectrum at $λ= 0$. We also show that for $λ\ll 1/L^2 Λ_{QCD}$ (with $L$ the linear size of the system) the Dirac spectrum is given by a chiral Random Matrix Theory with the symmetries of the Dirac operator.

hep-th↗
arXiv1998

The Microscopic Spectral Density of the QCD Dirac Operator

We derive the microscopic spectral density of the Dirac operator in $SU(N_c\geq 3)$ Yang-Mills theory coupled to $N_f$ fermions in the fundamental representation. An essential technical ingredient is an exact rewriting of this density in terms of integrations over the super Riemannian manifold $Gl(N_f+1|1)$. The result agrees exactly with earlier calculations based on Random Matrix Theory.

hep-th↗
arXiv1998

From chiral Random Matrix Theory to chiral Perturbation Theory

We study the spectrum of the QCD Dirac operator by means of the valence quark mass dependence of the chiral condensate in partially quenched Chiral Perturbation Theory (pqChPT) in the supersymmetric formulation of Bernard and Golterman. We consider valence quark masses both in the ergodic domain ($m_v \ll E_c$) and the diffusive domain ($m_v \gg E_c$). These domains are separated by a mass scale $E_c \sim F^2/Σ_0 L^2$ (with $F$ the pion decay constant, $Σ_0$ the chiral condensate and $L$ the size of the box). In the ergodic domain the effective super-Lagrangian reproduces the microscopic spectral density of chiral Random Matrix Theory (chRMT). We obtain a natural explanation of Damgaard's relation between the spectral density and the finite volume partition function with two additional flavors. We argue that in the ergodic domain the natural measure for the superunitary integration in the pqChPT partition function is noncompact. We find that the tail of the two-point spectral correlation function derived from pqChPT agrees with the chRMT result in the ergodic domain. In the diffusive domain we extend the results for the slope of the Dirac spectrum first obtained by Smilga and Stern. We find that the spectral density diverges logarithmically for nonzero topological susceptibility. We study the transition between the ergodic and the diffusive domain and identify a range where chRMT and pqChPT coincide.

hep-th↗
arXiv1997

Pion Dynamics at Finite Temperature

The pion decay constant and mass are computed at low temperature within Chiral Perturbation Theory to two loops. The effects of the breaking of Lorentz Symmetry by the thermal equilibrium state are discussed. The validity of the Gell-Mann Oakes Renner relation at finite temperature is examined.

hep-ph↗
arXiv1994

Consistency of the Chiral Pion-Pion Scattering Amplitudes with Axiomatic Constraints

The pion-pion scattering amplitudes provided by second-order chiral perturbation theory are confronted with known rigorous constraints derived from the axioms of quantum field theory. We mainly test constraints restricting the $π^0$-$π^0$ $S$- and $D$-wave amplitudes in the unphysical interval $0\leq s\leq 4m_π^2$. These constraints impose significant lower bounds for a linear combination of coupling constants specifying the second order chiral Lagrangian. The accepted value of this combination is consistent with these bounds. The $π^0$-$π^0$ $S$- and $D$-waves are strongly correlated by a set of constraints.

hep-ph↗