On the iterative solution of the gap equation in the Nambu-Jona-Lasinio model

Martínez, A.; Raya, Alfredo

doi:10.1088/1742-6596/912/1/012010

Cited by 2 publications

(5 citation statements)

References 21 publications

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“…These findings were generalized by some of us in Ref. [36] for different regularization schemes. In that work, we observed, not surprisingly, that for the three-dimensional (3D) and fourdimensional (4D) hard momentum cut-off regularization schemes, the procedure converges to the dynamical mass in a finite number of iterations and this solution continues to be an attractor even in the super-strong coupling regime.…”

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confidence: 58%

Superstrong coupling NJL model in arbitrary spacetime dimensions

2018

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We study chiral symmetry breaking in the Nambu-Jona-Lasinio model regularized in propertime in arbitrary space-time dimensions through an iterative procedure by writing the gap equation in the form of a discrete dynamical system with the coupling constant as the control parameter. Expectedly, we obtain the critical coupling for chiral symmetry breaking when a nontrivial solution bifurcates away from the trivial one and becomes an attractor. By increasing further the value coupling constant, we observe a second bifurcation where the dynamical solution is no longer an attractor, and observation that holds true in all space-time dimensions. In the super-strong coupling regime, the system becomes chaotic.Quantum chromodynamics (QCD) [1,2] is nowadays accepted to be the theory that describes strong interactions among quarks and gluons. It possesses two important and opposite features, asymptotic freedom [1] in the high energy domain, where quarks move freely at short distances. Perturbative QCD is the appropriate framework to approach this regime. On the other hand, there exists quark confinement [3] in the low-energy and largedistances regime. In this case, the fundamental degrees of freedom bind in hadronic bound-states (mesons, baryons and exotic), the coupling becomes strong and the theory highly non-linear. A number of non-perturbative frameworks have been developed to study low-energy QCD, among which we find Lattice QCD [4], Schwinger-Dyson equations (SDEs) [5,6] (see for reviews of SDEs in hadron physics) and other field-theoretical approaches, like the functional renormalization group [14-16], as well as effective models. Nambu-Jona-Lasinio (NJL) model [17,18] is a favorite one to study the properties of hadrons mostly in connection with chiral symmetry breaking (see, for instance, the reviews in Refs. [19,20]), and it is the general topic of this article. The model is non-renormalizable and thus, there are several schemes in which the regulator and coupling are selected to match static properties of pions (see, for example, Ref. [21]). Here, we select to regulate the gap equation in the proper-time (PT) scheme. Momentum integrals become Gaussian and thus are readily performed; the regulator comes in the PT integration, as opposed to the more frequently employed hard momentum cutoff regularization scheme [22] (the UV regulator in momentum space in the latter case is identified with the IR cut-off in proper-time integrals). Furthermore, by introducing an IR cut-off in momentum (UV in PT), we mimic confinement in the model [23][24][25]; the second cut-off removes quark-antiquark production thresholds, in the deep infra-red region of the quark propagator. This procedure has an added advantage that the resulting quark propagator naturally fulfills the Axial-Vector Ward identities for bound states. Furthermore, the PT regularization scheme allows an easy incorpora-tion of plasma effects, including finite temperature, density and external magnetic fields. Among other things, these extreme conditions make the effec...

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confidence: 58%

Superstrong coupling NJL model in arbitrary spacetime dimensions

2018

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“…This means the non-trivial and non-negative solution of the gap equation can be found through this procedure and is unique. The first example of the iterative procedure failing to converge is shown in Figure 5, corresponding to Equation (12). The first feature that pops to the eye is the non-converging region of the Figure 5, similar to the one found in the logistic map [13] .…”

Section: Results: Gap Equation In Vacuummentioning

confidence: 80%

“…Figure 5. Plot of the iterative procedure to solve the gap equation using Equation (12). As we observed before, if the coupling strength exceeds G c = 1, chiral symmetry is broken.…”

Section: Results: Gap Equation In Vacuummentioning

confidence: 86%

“…This seemingly harmless procedure has in fact proven to be very robust for the hard cut-off regulators even for large values of the coupling constant. However, for other regulation schemes such as the Pauli-Villars and proper time, surprises appear from the behavior of this procedure, as was noticed by [10][11][12] and we revise in deeper detail in this work. To this end, we have organized the remaining of the article as follows: In Section 2 we review the basics of the model.…”

Section: Introductionmentioning

confidence: 70%

“…Cobweb plot of Equation(12) for three values of the coupling: for G = 2.5 (a), the iterative procedure converges as expected and the solution is achieved; for G = 2.8 (b), the iterative procedure fails to converge and orbit around the solution of the gap equation; for G = 3.3 (c), any pattern associated with the trajectories of the iterations has disappeared and becomes chaotic.…”

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confidence: 66%

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Solving the Gap Equation of the NJL Model through Iterations: Unexpected Chaos

Martínez

Raya

2019

Symmetry

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We explore the behavior of the iterative procedure to obtain the solution to the gap equation of the Nambu-Jona-Lasinio (NLJ) model for arbitrarily large values of the coupling constant and in the presence of a magnetic field and a thermal bath. We find that the iterative procedure shows a different behavior depending on the regularization scheme used. It is stable and very accurate when a hard cut-off is employed. Nevertheless, for the Paul-Villars and proper time regularization schemes, there exists a value of the coupling constant (different in each case) from where the procedure becomes chaotic and does not converge any longer.

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On the iterative solution of the gap equation in the Nambu-Jona-Lasinio model

Cited by 2 publications

References 21 publications

Superstrong coupling NJL model in arbitrary spacetime dimensions

Superstrong coupling NJL model in arbitrary spacetime dimensions

Solving the Gap Equation of the NJL Model through Iterations: Unexpected Chaos

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