The supersymmetric version of the Nambu-Jona-Lasinio model is investigated in
connection with the chiral symmetry breaking induced by a soft SUSY breaking
term. It is found that the broken chiral symmetry due to the soft breaking term
is restored at suitably high temperature and the symmetry restoration occurs as
first-order phase transitions. The critical temperature at which the chiral
symmetry is restored is determined as a function of the strength of the soft
breaking term and the field coupling constant. The dynamical fermion mass is
calculated at finite temperature. Some possible applications to the breaking
scenario of unified field theories are discussed.Comment: 9 pages, 2 figure

Hashida, J.

Muta, T.

Ohkura, K.

English

arXiv

The supersymmetric version of the Nambu-Jona-Lasinio model is investigated in connection with the chiral symmetry breaking induced by a soft SUSY breaking term. It is found that the broken chiral symmetry due to the soft breaking term is restored at suitably high temperature and the symmetry restoration occurs as first-order phase transitions. The critical temperature at which the chiral symmetry is restored is determined as a function of the strength of the soft breaking term and the field coupling constant. The dynamical fermion mass is calculated at finite temperature. Some possible applications to the breaking scenario of unified field theories are discussed

Muta, Taizo

Name not available

arXiv:hep-ph/9805483 v1   27 May 1998Thermal Restoration of Chiral Symmetry inSupersymmetric Nambu-Jona-Lasinio Model withSoft SUSY BreakingJ. Hashida, T. Muta and K. OhkuraDeaprtment of PhysicsHiroshima UniversityHigashi-HiroshimaHiroshima 739-8526May 11, 1998HUPD-9814AbstractThe supersymmetric version of the Nambu-Jona-Lasinio model isinvestigated in connection with the chiral symmetry breaking inducedby a soft SUSY breaking term. It is found that the broken chiralsymmetry due to the soft breaking term is restored at suitably hightemperature and the symmetry restoration occurs as first-order phasetransitions. The critical temperature at which the chiral symmetry isrestored is determined as a function of the strength of the soft breakingterm and the field coupling constant. The dynamical fermion mass iscalculated at finite temperature. Some possible applications to thebreaking scenario of unified field theories are discussed.12In various theoretical circumstances the Nambu-Jona-Lasinio model 1)plays a role of the low energy effective theory of the more fundamental uni-fied field theories. Its supersymmetric version is of special interest becausethe model is useful to study the low energy consequences of the supersym-metric grand unified theories. In particular it is important to see whetherthe chiral symmetry posessed by the model is broken dynamically to gen-erate fermion masses. Unfortunately, however, it is well-known 2) that thesupersymmetry of the model protects the chiral symmetry from its breaking.If one introduces a soft SUSY breaking term in the model, one finds thatthe chiral symmetry in the model is broken for a sufficiently large soft SUSYbreaking term in comparison with the coupling constant in the interactionterm .3) This situation gives rise to a possibility of studying the phenomeno-logical consequences of the broken chiral symmetry in the supersymmetrictheory.In connection with the early history of the universe it is especially in-teresting to investigate the finite temperature behavior of the broken chiralsymmetry in the supersymmetric model like the one mentioned above. In thepresent communication we shall examine whether the thermal restoration ofthe chiral symmetry takes place in the supersymmetric Nambu-Jona-Lasiniomodel with the soft SUSY breaking term.The Nambu-JonaLasinio model is characterized by the LagrangianLNJL = iψ¯+σ¯µ∂µψ+ + iψ¯−σ¯µ∂µψ− + GNψ+ψ−ψ¯+ψ¯−, (1)where Weyl spinors ψ± are assumed to consist of N components and weshall work in the leading order of the 1/N expansion throughout the paper.A natural SUSY extension of the model is given by the Lagrangian 2)LSNJL =∫d4θ (Φ¯+Φ+ + Φ¯−Φ− +GNΦ+Φ−Φ¯+Φ¯−), (2)where d4θ = d2θd2θ¯ and chiral superfield Φ± is represented by componentscalar (φ±) and spinor (ψ±) fields and auxiliary field F±,Φ±(y) = φ±(y) +√2θψ±(y) + F±(y)θ2, (3)with y = x+iθσµθ¯. The chiral superfields Φ+ and Φ− belong to the multipletN and N¯ in SU(N) respectively.In this SUSY version of the Nambu-Jona-Lasinio model we introduce thefollowing soft SUSY breaking term 3)3LSB = −∫d4θ ∆2θ2θ¯2(Φ¯+Φ+ + Φ¯−Φ−). (4)It is convenient to introduce auxiliary chiral superfield H and S and use theLagrangian equivalent to Eq.(2) with the soft breaking term:L =∫d4θ[(Φ¯+Φ+ + Φ¯−Φ−)(1−∆2θ2θ¯2) + NGH¯H]+∫d2θ (NGHS − SΦ+Φ− + h.c.). (5)We write the relevant Lagrangian in terms of component fields to calculatethe effective potential in the leading order of 1/N expansion:L = −φ¯+(2+ |φS |2 +∆2)φ+ − φ¯−(2 + |φS |2 +∆2)φ−+iψ¯+σ¯µ∂µψ+ + iψ¯−σ¯µ∂µψ− + ψ+ψ−φS + ψ¯+ψ¯−φ¯S − NG|φS |2. (6)where we have used equations of motion for the auxiliary fields F± and FH .The field φS is related to the ordinary scalar (σ) and pseudoscalar (pi) fieldsuch thatφS = σ + ipi.The effective potential for this Lagrangian is easily written down in theleading order of the 1/N expansion and readsV0(|φS |2) = |φS |2G+2i∫d4k(2pi)4(ln−k2 + |φS |2−k2 − ln−k2 + |φS |2 +∆2−k2 +∆2 ). (7)After Wick rotation we perform the k0 integration and obtainV0(|φS |2) = |φS |2G+2∫d3k(2pi)3[√k2 + |φS |2 +∆2 −√k2 + |φS |2−(√k2 +∆2 −√k2) ]. (8)Note here that the effective potential is normalized so that V0(0) = 0. Byanalyzing the above effective potential we recognize that the chiral symmetryis broken for sufficiently large ∆. This situation is understood more clearlyby observing the gap equation which is derived from the effective potential(8). The gap equation ∂V0/∂|φS |2 = 0 is found to be41G=∫d3k(2pi)3(1√k2 + |φS |2− 1√k2 + |φS |2 +∆2). (9)Introducing the three dimensional cut-off Λ in the momentum integrationwe perform the integration in Eq. (9) and obtain12α=√1 + x−√1 + x+ δ + (x+ δ) ln1 +√1 + x+ δ√x+ δ−x ln 1 +√1 + x√x, (10)where α, δ and x are defined byα =GΛ28pi2, δ =∆2Λ2, x =|φS |2Λ2. (11)It is not difficult to show that Eq. (10) allows a nontrivial solution for xwith the suitable choice of parameters α and δ. Thus the dynamical fermionmass is generated and the chiral symmetry is broken dynamically. Notingthat the chiral symmetry breaking in the present case is of the second orderphase transition we let the gap parameter x vanish so that we obtain thecritical δ at which the chiral symmetry is broken. The critical δ is given bysolving the following relation as a function of the coupling constant α:12α= 1−√1 + δ + δ ln1 +√1 + δ√δ(12)Now we introduce the temperature. We employ the Matsubara formal-ism 4) to introduce temperature effects in our calculation of the effectivepotential. We simply replace the integral in energy variable by the summa-tion where we apply the periodic boundary condition for boson fields andthe anti-periodic boundary condition for fermion fields respectively. Theenergy variable k0 in the summation is replaced in the following way:∫dk02pif(k0)→ 1β∞∑n=−∞f(ωn), (13)where β is the inverse of the temperature and ωn is given byωn =ωBn ≡ 2nβ pi (boson),ωFn ≡ 2n+1β pi (fermion).5The calculation is essentially the same as the one described in the previouspaper.5) We first calculate the effective potential for the supersymmetricNambu-Jona-Lasinio model without the soft SUSY breaking term and obtainV (|φS |2) = |φS |2G− 2β∑n∫d3k(2pi)3(lnωFn2+ k2 + |φS |2ωFn2 + k2−ln ωBn2+ k2 + |φS |2ωBn2 + k2).(14)The summation on n in Eq. (14) is performed easily to giveV (|φS |2) = |φS |2G− 4β∫d3k(2pi)3ln [cothβ√k2 + |φS |22tanhβ√k22]. (15)The result reflects the well-known fact that the supersymmetry is spoiled atfinite temperature according to the different distribution functions for bosonsand fermions.6) It is, however, interesting to note that the supersymmetry ispreserved in the low temperature region where the Boltzmann distribution isa good approximation. In fact, if we keep only the dominant terms for smallT in Eq.(15), we find that the second term in Eq.(15) disappears leaving thetree expression of the effective potential. We realize that the supersymmetryis a good symmetry only in the low temperature region.We then take into account the soft SUSY breaking term and calculatethe effective potential to obtainV (|φS |2) = |φS |2G− 2β∑n∫d3k(2pi)3(lnωFn2+ k2 + |φS |2ωFn2+ k2− ln ωBn2+ k2 + |φS |2 +∆2ωBn2 + k2 +∆2). (16)The summation on n in the above equation can be easily performed as beforeand results inV (|φS |2) = V0(|φS |2)− 4β∫d3k(2pi)3ln(1 + e−β√k2+|φS |21 + e−β√k21− e−β√k2+∆21− e−β√k2+|φS |2+∆2),(17)where V0 is the effective potential for vanishing temperature already givenin Eq.(8).For almost whole range of the set of parameters α and δ we estimatedthe effective potential (17) by numerical integration. We show a typical ex-ample of its behavior in Fig. 1 for several values of β.6- Fig.1 -Apparently we observe that the broken chiral symmetry at T = 0 is re-stored at critical temperature βΛ = 13 and the chiral symmetry restorationoccurs as a first-order phase transition for which the mass gap is expectedat the critical temperature. In order to see the situation more clearly wewrite down the gap equation1G=∫d3k(2pi)3(1√k2 + |φS |2− 1√k2 + |φS |2 +∆2)−2∫d3k(2pi)3[1√|φS |2 + k21eβ√|φS |2+k2 + 1− 1√|φS |2 + k2 +∆21eβ√|φS |2+k2+∆2 − 1]. (18)By the numerical anlysis of Eq.(18) we recognize that two nontrivial so-lutions for x develop as temperature goes through the critical value signalingthe possibility of the first-order phase transition. The critical temperatureTC and the critical value of x are determined by solving the gap equation(18) and V = 0 simultaneously. The critical temperature TC is obtained atthe same time as in the above analysis and is a function of the soft breakingparameter δ = ∆2/Λ2 for a fixed α = GΛ2/8pi2.In the broken phase x is non-vanishing and hence there develops the non-vanishing vacuum expectation value of the scalar field σ which is responsiblefor generating fermion masses. In Fig. 2 we present the dynamical fermionmass m/Λ =√x as a function of βΛ. We clearly observe the mass gapcharacteristic of the first-order phase transition.- Fig.2-It should be pointed out here that the integration in Eq. (18) is performedrigorously in the case of the three dimensional space-time. We leave theargument in the three dimensional case in the forthcoming publication.The supersymmetric four-fermion theory is considered to be a low-energymanifestation of the more fundamental supersymmetric unified theories.The typical example is the supersymmetric GUT which breaks down to theelectroweak theory. In the early universe scenario this stage of the breakingof the supersymmetric GUT corresponds to the beginning of the inflation7era where the first-order phase transition is required. In this connection ourfinding of the chiral symmetry restoration as the first-order phase transitionis of great interest.The authors would like to thank Dr. T. Inagaki and Dr. S. Mukaigawafor valuable comments and discussions. One of the authors (K. O) would liketo express his sincere gratitude to Dr. T. Onogi for the guidance in the studyof supersymmetric theories. The other of the authors (T. M.) is supportedby the Monbusho Grant-in-Aid for Scientific Research (C) No.08640377 andfor Scientific Research on Priority Areas No.09246105.References[1] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122(1961) 345.[2] W. Buchmueller and S. T. Love, Nucl. Phys. B204(1982) 213;V. Elias, D. G. C. Mckeon, V. A. Miransky, I. A. Shovkovy, Phys. Rev.D54(1996)7884;L. L. Buchbinder, T. Inagaki and S. D. Odintsov, Mod. Phys. Lett. A12(1997)2271.[3] W. Buchmueller and U. Ellwanger, Nucl. Phys. B245(1984) 237.[4] T. Matsubara, Prog. Theor. Phys. 14(1955) 351.[5] T. Inagaki, T. Kouno and T. Muta, Intern. J. Mod. Phys. A10(1995)2241.[6] D. Buchholtz and I. Ojima, Nucl. Phys. B498(1997) 228 and referencestherein.Figure CaptionsFig 1: Effective potential at finite temperature for GΛ2 = 7× 103 and∆/Λ = 0.1Fig 2: Dynamical mass of fermion as a function of βΛ(βCΛ = Λ/TC = 13)80 0.2 0.4 0.6 0.8 1.0σ/Λ−42024V/Λ4  [ ∗ 105  ] βΛ=102 βΛ=16  βΛ=15  βΛ=14 βΛ=13 βΛ=12 βΛ=11 Figure 1:910 15 20 25 30βΛ0.40.450.5m2 /Λ2GΛ2=7∗103, ∆/Λ=0.1Figure 2:

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Thermal restoration of chiral symmetryin supersymmetric Nambu-Jona-Lasinio model with soft SUSY breaking

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THERMAL RESTORATION OF CHIRAL SYMMETRY IN SUPERSYMMETRIC NAMBU–JONA-LASINIO MODEL WITH SOFT SUSY BREAKING

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Thermal Restoration of Chiral Symmetry in Supersymmetric Nambu-Jona-Lasinio Model with Soft SUSY Breaking

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