Gell-Mann Oakes Renner Relation Research Articles

Nonperturbative determination of the N=1 supersymmetric Yang-Mills gluino condensate at large N

We present the first nonperturbative large N calculation of the N=1 supersymmetric SU(N) Yang-Mills gluino condensate obtained by means of numerical simulations of the lattice-discretized theory, exploiting large-N twisted volume reduction. We present two different determinations based, respectively, on the Banks-Casher formula and on the Gell-Mann–Oakes–Renner relation, both giving perfectly consistent results. By expressing the lattice results in the Novikov-Shifman-Vainshtein-Zakharov (NSVZ) scheme, we are able for the first time to compare numerical and analytic computations. Our most accurate determination of the renormalization group invariant (RGI) gluino condensate gives ΣRGI/ΛNSVZ3=[1.18(08)stat(12)syst]3=1.64(33)stat(50)syst=1.64(60), in agreement with the N dependence and the value predicted by the weak coupling instanton-based approach ΣRGI/ΛNSVZ3=1. Published by the American Physical Society 2024

GMOR relation for a QCD-like theory from S-duality

Following [1] we study a QCD-like gauge theory using a non-supersymmetric setup in type IIB string theory. The setup includes an O3 plane and N D3 anti-branes and it realises a USp(2N) ‘electric’ gauge theory with four “quarks” in the two-index antisymmetric representation and six heavy scalars in the adjoint representation. Using S-duality we obtain a dual ‘magnetic’ theory that includes SO(2N−1) gauge theory with six scalars in the adjoint representation and four heavy “quarks” in the two-index symmetric representation. The dual theory provides a description of confinement and dynamical symmetry breaking of the form SU(4)→SO(4). We extend the results of [1] by adding masses to the quarks in the electric side and deriving parts of the chiral Lagrangian using the dual magnetic theory. In particular, we derive the Gell-Mann-Oakes-Renner (GMOR) relation for Nambu-Goldstone bosons (pions) using the magnetic theory.

Goldstino spectrum in an ultracold Bose-Fermi mixture with explicitly broken supersymmetry

We theoretically investigate a supersymmetric collective mode called Goldstino in a Bose-Fermi mixture. The explicit supersymmetry breaking, which is unavoidable in cold atom experiments, is considered. We derive the Gell-Mann--Oakes-Renner (GOR) relation for the Goldstino, which gives the relation between the energy gap at the zero momentum and the explicit breaking term. We also numerically evaluate the gap of Goldstino above the Bose-Einstein condensation temperature within the random phase approximation (RPA). While the gap obtained from the GOR relation coincides with that in the RPA for the mass-balanced system, there is a deviation from the GOR relation in the mass-imbalanced system. We point out the deviation becomes large when the Goldstino pole is close to the branch point, although it is parametrically a higher order with respect to the mass-imbalanced parameter. To examine the existence of the goldstino pole in realistic cold atomic systems, we show how the mass-imbalance effect appears in $^6$Li-$^7$Li, $^{40}$K-$^{41}$K, and $^{173}$Yb-$^{174}$Yb mixtures. Furthermore, we analyze the Goldstino spectral weight in a $^{173}$Yb-$^{174}$Yb mixture with realistic interactions and show a clear peak due to the Goldstino pole. As a possibility to observe the Goldstino spectrum in cold atom experiments, we discuss the effects of the Goldstino pole on the fermionic single-particle excitation as well as the relationship between the GOR relation and Tan's contact.

Goldstonic pseudoscalar mesons in Bethe–Salpeter-inspired setting

For a two-particle bound-state equation closer to its Bethe–Salpeter origins than Salpeter’s equation, with effective interaction kernel deliberately forged such as to ensure, in the limit of zero mass of the bound-state constituents, the vanishing of the arising bound-state mass, we scrutinize the emerging features of the lightest pseudoscalar mesons for their agreement with the behavior predicted by a generalization of the Gell-Mann–Oakes–Renner relation.

The pion quasiparticle in the low-temperature phase of QCD

We extend our previous studies [PhysRevD.90.054509, PhysRevD.92.094510] of the pion quasiparticle in the low-temperature phase of two-flavor QCD with support from chiral effective theory. This includes the analysis performed on a finite temperature ensemble of size 20 × 643 at T ≈ 151MeV and a lighter zero-temperature pion mass mπ ≈ 185 MeV. Furthermore, we investigate the Gell-Mann–Oakes-Renner relation at finite temperature and the Dey-Eletsky-Ioffe mixing theorem at finite quark mass.

Properties of mesons in a strong magnetic field.

By extending the Phi -derivable approach in the Nambu–Jona-Lasinio model to a finite magnetic field we calculate the properties of pion, sigma , and rho mesons in a magnetic field at finite temperature not only in the quark–antiquark bound state scheme but also in the pion–pion scattering resonant state scenario. Our calculation as a result makes manifest that the masses of pi ^{0} and sigma meson can be nearly degenerate at the pseudo-critical temperature which increases with increasing magnetic field strength, and the pi ^{pm } mass ascends suddenly at almost the same critical temperature. Meanwhile the rho mesons’ masses decrease with the temperature but increase with the magnetic field strength. We also check the Gell-Mann–Oakes–Renner relation and find that the relation can be violated clearly with increasing temperature, and the effect of the magnetic field becomes pronounced around the critical temperature. With different criteria, we analyze the effect of the magnetic field on the chiral phase transition and find that the pseudo-critical temperature of the chiral phase cross, T_mathrm{c}^{chi }, is always enhanced by the magnetic field. Moreover, our calculations indicate that the rho mesons will get melted as the chiral symmetry has not yet been restored, but the sigma meson does not disassociate even at very high temperature. Particularly, it is the first to show that there does not exist a vector meson condensate in the QCD vacuum in the pion–pion scattering scheme.

Pion and σ-meson Properties in a Strong Magnetic Field**Supported by the National Natural Science Foundation of China under Grant Nos. 10935001, 11175004, and 11435001, and the National Key Basic Research Program of China under Grant Nos. G2013CB834400 and G2015CB856900

With the Nambu–Jona-Lasinio (NJL) model we calculate the properties of pion and σ-meson at finite temperature and finite magnetic field. The obtained temperature and magnetic field strength dependence of the constituent quark mass M, the pion and σ-meson masses and the neutral pion decay constant indicates that, in the simple four fermion interaction model, there exists the magnetic catalysis effect. It also shows that the Gell-Mann–Oakes–Renner relation is violated obviously with the increasing of the temperature, and the effect of the magnetic field becomes pronounced only around the critical temperature. The deviation of the critical temperatures obtained with different criteria indicates that the chiral phase transition driven by the temperature in the magnetic field strength region we have considered is in fact a crossover.

Corrections to the SU(3) × SU(3) Gell-Mann-Oakes-Renner relation and chiral couplings $ L_8^r $ and $ H_2^r $

Next to leading order corrections to the SU(3) × SU(3) Gell-Mann-Oakes-Renner relation (GMOR) are obtained using weighted QCD Finite Energy Sum Rules (FESR) involving the pseudoscalar current correlator. Two types of integration kernels in the FESR are used to suppress the contribution of the kaon radial excitations to the hadronic spectral function, one with local and the other with global constraints. The result for the pseudoscalar current correlator at zero momentum is ψ 5(0) = (2.8 ± 0.3) ×10-3 GeV4, leading to the chiral corrections to GMOR: δ K = (55 ± 5)%. The resulting uncertainties are mostly due to variations in the upper limit of integration in the FESR, within the stability regions, and to a much lesser extent due to the uncertainties in the strong coupling and the strange quark mass. Higher order quark mass corrections, vacuum condensates, and the hadronic resonance sector play a negligible role in this determination. These results confirm an independent determination from chiral perturbation theory giving also very large corrections, i.e. roughly an order of magnitude larger than the corresponding corrections in chiral SU(2) × SU(2). Combining these results with our previous determination of the corrections to GMOR in chiral SU(2) × SU(2), δ π , we are able to determine two low energy constants of chiral perturbation theory, i.e. $ L_8^r=\left( {1.0\pm 0.3} \right)\times {10^{-3 }} $ , and $ H_2^r=-\left( {4.7\pm 0.6} \right)\times {10^{-3 }} $ , both atthe scaleof the ρ-meson mass.

Scaling relations for the entire spectrum in mass-deformed conformal gauge theories

We consider mass-deformed conformal gauge theories (mCGT) and investigate the scaling behaviour of hadronic observables as a function of the fermion mass. Applying renormalization group arguments directly to matrix elements, we find mH∼m1/(1+γ⁎) and F∼mηF(γ⁎) for given ηF(γ⁎), for the hadronic masses and the decay constants respectively, thereby generalizing our results from a previous paper to the entire spectrum. Applying the Hellmann–Feynman theorem to the trace anomaly we obtain the hadron mass scaling independent of renormalization group arguments. From the trace anomaly we obtain a relation reminiscent of the Gell-Mann–Oakes–Renner relation in QCD. Using the new results we discuss the scaling of the S-parameter inside the conformal window. Finally, we discuss how spectral representations can be used to relate the mass and decay constant trajectories.

Chiral symmetry breaking in monolayer graphene by strong coupling expansion of compact and non-compact U(1) lattice gauge theories

Due to effective enhancement of the Coulomb coupling strength in the vacuum-suspended graphene, the system may turn from a semimetal into an insulator by the formation of a gap in the fermionic spectrum. This phenomenon is analogous to the spontaneous breaking of chiral symmetry in the strong-coupling relativistic field theories. We study this “chiral symmetry breaking” and associated collective excitations on graphene in the strong coupling regime by taking U(1) lattice gauge theory as an effective model for graphene. Both compact and non-compact formulations of the U(1) gauge action show chiral symmetry breaking with equal magnitude of the chiral condensate (exciton condensate) in the strong coupling limit, while they start to deviate from the next-to-leading order in the strong coupling expansion. Phase and amplitude fluctuations of the order parameter are also investigated: in particular, a mass formula for the pseudo-Nambu–Goldstone mode ( π-exciton), which is analogous to Gell–Mann–Oakes–Renner relation for the pion in quantum chromodynamics (QCD), is derived from the axial Ward–Takahashi identity. To check the applicability of the effective field theory description, typical energy scales of fermionic and bosonic excitations are estimated by identifying the lattice spacing of the U(1) gauge theory with that of the original honeycomb lattice of graphene.

The chiral quark condensate and pion decay constant in nuclear matter at next-to-leading order

Making use of the recently developed chiral power counting for the physics of nuclear matter (Oller et al 2010 J. Phys. G: Nucl. Part. Phys. 37 015106, Lacour et al Ann. Phys. at press), we evaluate the in-medium chiral quark condensate up to next-to-leading order for both symmetric nuclear matter and neutron matter. Our calculation includes the full in-medium iteration of the leading order local and one-pion exchange nucleon–nucleon interactions. Interestingly, we find a cancellation between the contributions stemming from the quark mass dependence of the nucleon mass appearing in the in-medium nucleon–nucleon interactions. Only the contributions originating from the explicit quark mass dependence of the pion mass survive. This cancellation is the reason of previous observations concerning the dominant role of the long-range pion contributions and the suppression of short-range nucleon–nucleon interactions. We find that the linear density contribution to the in-medium chiral quark condensate is only slightly modified for pure neutron matter by the nucleon–nucleon interactions. For symmetric nuclear matter, the in-medium corrections are larger, although smaller compared to other approaches due to the full iteration of the lowest order nucleon–nucleon tree-level amplitudes. Our calculation satisfies the Hellmann–Feynman theorem to the order worked out. Also we address the problem of calculating the leading in-medium corrections to the pion decay constant. We find that there are no extra in-medium corrections that violate the Gell-Mann–Oakes–Renner relation up to next-to-leading order.

Gell-Mann-Oakes-Renner Relation: Chiral Corrections from Sum Rules

The next to leading order chiral corrections to the S U ( 2 ) × S U ( 2 ) Gell-Mann-Oakes-Renner (GMOR) relation are obtained using the light-quark pseudoscalar correlator to five-loop order in perturbative QCD, together with new finite energy sum rules (FESR) incorporating polynomial, Legendre type, integration kernels. These kernels are designed to suppress hadronic contributions in the region where they are least known. We obtain for the corrections to the GMOR relation, δ π , the value δ π = ( 6.2 ± 1.6 ) % . As a byproduct, the chiral perturbation theory (unphysical) low energy constant H 2 r is determined to be H 2 r ( ν χ = M ρ ) = − ( 5.1 ± 1.8 ) × 10 − 3 .

Chiral gap and collective excitations in monolayer graphene from strong coupling expansion of lattice gauge theory

Using the strong coupling expansion of the compact and non-compact U(1) lattice gauge theory for monolayer graphene, we show analytically that fermion bandgap and pseudo Nambu--Goldstone exciton (pi-exciton) are dynamically generated due to chiral symmetry breaking. The mechanism is similar to the generation of quark mass and pion excitation in quantum chromodynamics (QCD). We derive a formula for the pi-exciton analogous to the Gell-Mann--Oakes--Renner (GOR) relation in QCD. Experimental confirmation of the GOR relation on a suspended monolayer graphene would be a clear evidence of chiral symmetry breaking.

Chiral corrections to the SU(2) × SU(2) Gell-Mann-Oakes-Renner relation

The next to leading order chiral corrections to the $SU(2)\times SU(2)$ Gell-Mann-Oakes-Renner (GMOR) relation are obtained using the pseudoscalar correlator to five-loop order in perturbative QCD, together with new finite energy sum rules (FESR) incorporating polynomial, Legendre type, integration kernels. The purpose of these kernels is to suppress hadronic contributions in the region where they are least known. This reduces considerably the systematic uncertainties arising from the lack of direct experimental information on the hadronic resonance spectral function. Three different methods are used to compute the FESR contour integral in the complex energy (squared) s-plane, i.e. Fixed Order Perturbation Theory, Contour Improved Perturbation Theory, and a fixed renormalization scale scheme. We obtain for the corrections to the GMOR relation, $\delta_\pi$, the value $\delta_\pi = (6.2, \pm 1.6)%$. This result is substantially more accurate than previous determinations based on QCD sum rules; it is also more reliable as it is basically free of systematic uncertainties. It implies a light quark condensate $<0|\bar{u} u|0> \simeq <0|\bar{d} d|0> \equiv <0|\bar{q} q|0>|_{2\,\mathrm{GeV}} = (- 267 \pm 5 MeV)^3$. As a byproduct, the chiral perturbation theory (unphysical) low energy constant $H^r_2$ is predicted to be $H^r_2 (\nu_\chi = M_\rho) = - (5.1 \pm 1.8)\times 10^{-3}$, or $H^r_2 (\nu_\chi = M_\eta) = - (5.7 \pm 2.0)\times 10^{-3}$.

Universal holographic chiral dynamics in an external magnetic field

In this work we further extend the investigation of holographic gauge theories in external magnetic fields, continuing earlier work. We study the phenomenon of magnetic catalysis of mass generation in 1+3 and 1+2 dimensions, using D3/D7- and D3/D5-brane systems, respectively. We obtain the low energy effective actions of the corresponding pseudo Goldstone bosons and study their dispersion relations. The D3/D7 system exhibits the usual Gell-Mann–Oakes–Renner (GMOR) relation and a relativistic dispersion relation, while the D3/D5 system exhibits a quadratic non-relativistic dispersion relation and a modified linear GMOR relation. The low energy effective action of the D3/D5 system is related to that describing magnon excitations in a ferromagnet. We also study properties of general Dp/Dq systems in an external magnetic field and verify the universality of the magnetic catalysis of dynamical symmetry breaking.

Chemical Potential Dependence of Vertices

Based on the rainbow-ladder approximation of the Dyson–Schwinger equations and the assumption of the analyticity of the fermion-boson vertex in the neighborhood of zero chemical potential (μ = 0) and neglecting the μ-dependence of the dressed gluon propagator, we apply the method in [Phys. Rev. C 71 (2005) 015205] of studying the dressed quark propagator at finite chemical potential to prove that the general fermion-boson vertex at finite μ can also be obtained from the one at μ = 0 by a simple shift of variables. Using this result we extend the results of [Phys. Lett. B 420 (1998) 267] to the situation of finite chemical potential and show that under the approximations we have taken, the Gell–Mann–Oakes–Renner relation also holds at finite chemical potential.

Hadrons in strong electric and magnetic fields

We use chiral perturbation theory to investigate hadronic properties in strong electric and magnetic fields. A strong-field power counting is employed, and results for pions and nucleons are obtained using Schwinger's proper-time method. In the limit of weak fields, we accordingly recover the well-known one-loop chiral perturbation theory results for the electric and magnetic polarizabilities of pions and nucleons. In strong constant fields, we extend the Gell-Mann–Oakes–Renner relation. For the case of electric fields, we find that non-perturbative effects result in hadron decay. For sufficiently strong magnetic fields, the chiral analysis confirms that the nucleon hierarchy becomes inverted giving rise to proton beta-decay. Properties of asymptotic expansions are explored by considering weak field limits. In the regime where the perturbative expansion breaks down, the first-order term gives the best agreement with the non-perturbative result.

Chiral symmetry breaking as open string tachyon condensation

We consider a general framework to study holographically the dynamics of fundamental quarks in a confining gauge theory. Flavors are introduced by placing a set of (coincident) branes and antibranes on a background dual to a confining color theory. The spectrum contains an open string tachyon and its condensation describes the U ( N f ) L × U ( N f ) R → U ( N f ) V symmetry breaking. By studying worldvolume gauge transformations of the flavor brane action, we obtain the QCD global anomalies and an IR condition that allows to fix the quark condensate in terms of the quark mass. We find the expected N f 2 Goldstone bosons (for m q = 0 ), the Gell-Mann–Oakes–Renner relation (for m q small) and the η ′ mass. Remarkably, the linear confinement behavior for the masses of highly excited spin-1 mesons, m n 2 ∼ n is naturally reproduced.

HEAVY AND LIGHT QUARKS WITH LATTICE CHIRAL FERMIONS

The feasibility of using lattice chiral fermions which are free of O(a) errors for both the heavy and light quarks is examined. The fact that the effective quark propagators in these fermions have the same form as that in the continuum with the quark mass being only an additive parameter to a chirally symmetric anti-Hermitian Dirac operator is highlighted. This implies that there is no distinction between the heavy and light quarks and no mass dependent tuning of the action or operators as long as the discretization error O(m2a2) is negligible. Using the overlap fermion, we find that the O(m2a2) (and O(ma2)) errors in the dispersion relations of the pseudoscalar and vector mesons and the renormalization of the axial-vector current and scalar density are small. This suggests that the applicable range of ma may be extended to ~0.56 with only 5% error, which is a factor of ~2.4 larger than the corresponding range of the improved Wilson action. We show that the generalized Gell–Mann–Oakes–Renner relation with unequal masses can be utilized to determine the finite ma corrections in the renormalization of the matrix elements for the heavy-light decay constants and semileptonic decay constants of the B/D meson.

QCD quark condensate from SUSY and the orientifold large-N expansion

We estimate the quark condensate in one-flavor massless QCD from the known value of the gluino condensate in SUSY Yang–Mills theory using our newly proposed “orientifold” large-N expansion. The numerical result for the quark condensate renormalized at the scale 2 GeV is then given as a function of αs(2 GeV) and of possible corrections from sub-leading terms. Our value can be compared with the quark condensate in (quenched) lattice QCD or with the one extracted from the Gell-Mann–Oakes–Renner relation by virtue of non-lattice determinations of the quark masses. In both cases we find quite a remarkable agreement.