The structure of chiral symmetry restorations at finite temperature is
thoroughly investigated in the supersymmetric Nambu-Jona-Lasinio model with a
soft supersymmetry breaking term. It is found that the broken chiral symmetry
at vanishing temperature is restored at sufficiently high temperature in two
patterns, i. e., the first order and second order phase transition depending on
the choice of the coupling constant $G$ and the supersymmetry breaking
parameter $\Delta$. The critical curves expressing the phase boundaries in the
$G-\Delta$ plane are completely determined and the dynamically generated
fermion mass is calculated as a function of temperature.Comment: 12 pages, 4 figures, REVTe

Hashida, J.

Muta, T.

Ohkura, K.

Physics Letters B

English

arXiv

The structure of chiral symmetry restorations at finite temperature is thoroughly investigated in the supersymmetric Nambu-Jona-Lasinio model with a soft supersymmetry breaking term. It is found that the broken chiral symmetry at vanishing temperature is restored at sufficiently high temperature in two patterns, i.e., the first order and second order phase transition depending on the choice of the coupling constant G and the supersymmetry breaking parameter Δ. The critical curves expressing the phase boundaries in the G-Δ plane are completely determined and the dynamically generated fermion mass is calculated as a function of temperature

Muta, Taizo

Name not available

arXiv:hep-ph/9808266 v2   23 Feb 1999HUPD-9821Pattern of Chiral Symmetry Restoration at Finite Temperaturein A Supersymmetric Composite ModelJ. Hashida, T. Muta and K. OhkuraDepartment of PhysicsHiroshima UniversityHigashi-Hiroshima, Hiroshima 739-8526(March 15, 2006)AbstractThe structure of chiral symmetry restorations at finite temperature isthoroughly investigated in the supersymmetric Nambu-Jona-Lasinio modelwith a soft supersymmetry breaking term. It is found that the broken chiralsymmetry at vanishing temperature is restored at sufficiently high tempera-ture in two patterns, i. e., the first order and second order phase transitiondepending on the choice of the coupling constant G and the supersymmetrybreaking parameter ∆. The critical curves expressing the phase boundariesin the G−∆ plane are completely determined and the dynamically generatedfermion mass is calculated as a function of temperature.PACS numbers: 11.10.Wx, 11.15.Pg, 11.30.Qc, 11.30.Rd, 12.60.JvTypeset using REVTEX1The supersymmetry is an essential ingredient of unified field theories for elementaryparticles so that it is worth trying to study a supersymmetric version of composite Higgsmodels such as Technicolor model. We shall consider a supersymmetric composite modelin the present communication with an interest in the early universe and take into accountfinite temperature effects. As a prototype of the composite model we pick up the Nambu-Jona-Lasinio (NJL) model which is known to be useful to investigate the mechanism ofthe dynamical symmetry breaking [1]. The supersymmetric version of the NJL model iseasily constructed. Unfortunately, however, in the supersymmetric NJL model the chiralsymmetry is strongly protected to keep the boson-fermion supersymmetry (SUSY) and hencethe dynamical chiral symmetry breaking does not take place [2]. If a soft SUSY breaking termis added to the SUSY NJL Lagrangian, the dynamical breakdown of the chiral symmetryis brought about for sufficiently large SUSY breaking parameter ∆ [3]. The reason for thisis simple: The large ∆ implies the large effective mass of the scalar components of thesuperfields so that quantum effects due to the scalar components get suppressed comparedwith that of the spinor components. Thus the model becomes closer to the original NJLmodel which allows the dynamical fermion mass generation. Stating the same substance ina different way we realize that the boson by acquiring its mass term forces supersymmetryto make balance so that the fermion mass is generated dynamically.The SUSY NJL model with soft SUSY breaking is useful to study the mechanism ofthe dynamical chiral symmetry breaking within the framework of supersymmetry. If wetake the model seriously as a prototype of the unified field theory, it is natural to extendthe argument to take into account circumstances of the finite temperature and spacetime-curvature as in the early universe. It should, however, be noted that SUSY is not a goodsymmetry for finite temperature [4]. This fact may discourage studies of SUSY modelsat finite temperature. In our argument we consider the supersymmetry of the model inthe strict sense only at vanishing temperature and focus our attention on the dynamicalbreaking of the chiral symmetry in the same model at finite temperature. The breaking ofSUSY at finite temperature is kinematical since it is caused essencially by the difference of2the statistical distribution functions for bosons and fermions.A natural SUSY extension of the NJL model is characterized by the following Lagrangian[2],LSNJL =∫d4θ[Φ¯+Φ+ + Φ¯−Φ− +GNΦ+Φ−Φ¯+Φ¯−−∆2θ2θ¯2(Φ¯+Φ+ + Φ¯−Φ−)], (1)where d4θ = d2θd2θ¯. The chiral superfield Φ± is represented by component scalar (φ±)and spinor (ψ±) fields and auxiliary field F± so that Φ±(y) = φ±(y) +√2θψ±(y) + F±(y)θ2with y = x+ iθσµθ¯. The chiral superfields Φ+ and Φ− belong to the multiplet N and N¯ inSU(N) respectively. Note that a soft SUSY breaking term is introduced in Eq. (1) followingBuchmueller and Ellwanger [3].In the following arguments it is convenient to introduce auxiliary chiral superfields Hand S and use a Lagrangian equivalent to Eq. (1)L =∫d4θ[(Φ¯+Φ+ + Φ¯−Φ−)(1−∆2θ2θ¯2) + NGH¯H ]+ [∫d2θ (NGHS − SΦ+Φ−) + h.c.]. (2)Written in terms of component fields the Lagrangian takes the following form where we havekept only relevant terms for calculating our effective potential in the leading order of the1/N expansion:L = −φ¯+(2 + |φS|2 +∆2)φ+ − φ¯−(2 + |φS|2 +∆2)φ−+Ψ¯(i∂µγµ − σ + iγ5pi)Ψ− NG|φS|2. (3)The field φS is related to the ordinary scalar (σ) and pseudoscalar (pi) field such that φS =σ + ipi, and Ψ = (ψ+, ψ¯−)T .We would like to calculate the effective potential for Lagrangian (3) in the leading order ofthe 1/N expansion at finite temperature. We adopt the Matsubara formalism [5] to introducetemperature effects in our calculation of the effective potential. We simply replace theintegral in energy variable by the summation where we apply the periodic boundary condition3for boson fields and the anti-periodic boundary condition for fermion fields respectively.The calculation of the effective potential is essentially the same as the one described in ourprevious paper [6]. The summation can be easily performed and after some algebra thefollowing result is obtained,V (|φS|2) = V0(|φS|2) + Vβ(|φS|2), (4)where V0(|φS|2) is the effective potential for vanishing temperature,V0(|φS|2) = |φS|2G+ 2∫dD−1k(2pi)D−1(√k2 + |φS|2 +∆2−√k2 + |φS|2 −√k2 +∆2 +√k2 ), (5)which agrees with the one obtained by performing the k0 integration after Wick rotationin the effective potential given at vanishing temperature, and Vβ(|φS|2) is the temperaturedependent part of the effective potential that is given byVβ(|φS|2) = − 4β∫ dD−1k(2pi)D−1ln(1 + e−β√k2+|φS |21 + e−β√k2× 1− e−β√k2+∆21− e−β√k2+|φS |2+∆2), (6)where D is the space-time dimension (Although D = 4, we keep D to be arbitrary in Eqs.(5) and (6) for later convenience), β is the inverse of the temperature T in unit of kB = 1with kB the Boltzmann constant. Note that the effective potential (4) is normalized so thatV (0) = 0.It is easy to see that the effective potential (4) reduces to V0(|φS|2) in the zero tempera-ture limit β →∞. If we let ∆→ 0 in the zero temperature effective potential (5), we are leftonly with the tree term as is expected by the boson-fermion cancellation in the SUSY model.In the SUSY limit ∆→ 0 the finite temperature effective potential (6) still retains the termof quantum corrections which is composed of the bosonic and fermionic statistical distribu-tion functions. This result reflects the well-known fact that supersymmetry is not a goodsymmetry at finite temperature [4]. At sufficiently low temperature both the Fermi-Dirac4and Bose-Einstein distribution function 1/(exp β√k2 + |φS|2 ± 1) are well approximated bythe Maxwell-Boltzmann distribution function exp(−β√k2 + |φS|2) and hence the quantumcorrection term in Eq. (4) disappears. Thus supersymmetry is a good symmetry not onlyat zero temperature but also in the low temperature region.Although the phase structure of our model at finite temperature is studied by directnumerical estimates of the effective potential (4), it is more convenient to deal with the gapequation V ′(|φS|2) = 0 where the prime denotes the differenciation with respect to |φS|2 :V ′(|φS|2) = ∂V/∂|φS |2,1G−∫dD−1k(2pi)D−1(1√k2 + |φS|2− 1√k2 + |φS|2 +∆2)+2∫ dD−1k(2pi)D−1[1√k2 + |φS|21eβ√k2+|φS|2 + 1− 1√k2 + |φS|2 +∆21eβ√k2+|φS |2+∆2 − 1] = 0. (7)In order to see the situation as clearly as possible we start with the discussion in the caseD = 3 where the gap equation (7) after performing the integration is much simpler than theone for D = 4 and there is no divergence in the integration in Eq. (7). The gap equationfor D = 3 is obtained in an analytical form by performing the integration in Eq. (7):2piG−√|φS|2 +∆2 +√|φS|2 + 2βln1 + e−β√|φS |21− e−β√|φS |2+∆2= 0. (8)By numerical inspections we find that Eq. (8) allows at most two nontrivial solutions for|φS|2 depending on the choice of parameters G, ∆ and β. Thus the effective potential (4)with D = 3 may have zero, one and two extremum (extrema) respectively. Hence, when thetemperature 1/β is increased, we will have possibilities of experiencing no phase transition,the 2nd order phase transition and the 1st order phase transition.The effective potential V (|φS|2) for D = 3 is calculated analytically such thatV (|φS|2) = |φS|2G+13pi[|φS|3 +∆3 − (|φS|2 +∆2)3/2]+Vβ(|φS|2)D=3, (9)5where we have suppressed writing explicitly the temperature dependent term (the last termon the right hand side of Eq. (9)) which is rather lengthy and can be expressed by usingthe polylogarithms.At vanishing temperature the chiral symmetry in the model is broken [3] if sufficientlylarge is the parameter ∆ appearing in the soft SUSY breaking term. In fact by observingthe behavior of the gap equation V ′0(|φS|2) = 0 we can confirm the statement. In threedimensions for vanishing temperature β →∞ the last term on the left hand side of Eq. (8)disappears and we see that only one nontrivial solution for |φS|2 exists when 2pi/G < ∆.The curve on the G−∆ plane given by the equationG∆/(2pi) = 1 (10)divides the whole G − ∆ plane into two regions which represent the unbroken and brokenchiral symmetry respectively.If the parameters G and ∆ are kept in the unbroken region, we remain in the unbrokenregion when temperature is increased. If the parametersG and ∆ are in the broken region, weexperience the chiral symmetry restoration as temperature gets high enough. This situationis easily confirmed by observing the gap equation (8) and the effective potential (9). In thepresent situation there are two types of the chiral symmetry restoration : the first-order andsecond-order phase transition. These two cases can be distinguished from each other by theconditions V ′(0) = 0, V ′′(0) < 0 and V ′(0) = 0, V ′′(0) > 0 respectively. Hence the conditionthat |φS|2 = 0 be a point of inflection for the effective potential, V ′(0) = V ′′(0) = 0,divides these two cases and gives a curve on the G − ∆ plane. The curve is the criticalcurve on the G−∆ plane which divides the region of the broken phase into the regions ofthe first and second order phase transition. Written in parameters G and ∆ the conditionV ′(0) = V ′′(0) = 0 reads2piG∆= 1− 2β∆ln21− e−β∆ , (11)β∆/2 = coth(β∆/2). (12)Here β∆/2 is determined by Eq. (12) and is equal to 1.19968. The critical curve distin-guishing the second order phase transition from the first order one is given by the equationobtained by substituting the above value for β∆/2 in Eq. (11), i.e.,G∆/(2pi) = 2.91631. (13)The whole G−∆ plane is divided by two hyperbolas (10) and (13) into three regions whichcorrespond to the unbroken, broken (2nd order) and broken (1st order) region respectivelyas shown in Fig. 1. Note here that as can be seen in Eq. (9) the effective potential for D = 3reduces to that of the ordinary NJL model by letting ∆→∞. The physical reason for thisresult is simple: In the limit of large ∆ the scalar components of the superfield carry a largeeffective mass and decouple from the spinor components leaving only the contribution ofthe NJL fermions. It should, however, be remarked that the gap equation (8) diverges as∆→∞. This result corresponds exactly to the fact that the gap equation for the ordinaryNJL model diverges linearly for D = 3. In fact it is given by2piG= Λ−√|φS|2 − 2βln(1 + e−β√|φS |2) (14)and corresponds to Eq. (8) with large ∆ replaced by Lambda. In the ordinary NJL modelfor three dimensions the bare coupling constant G is to be kept small for large Λ. This factprecisely reflects the property that G gets smaller as ∆ becomes larger along the criticalcurve.We then go over to the case of the four dimensional space-time. For D = 4 the situationis almost the same as in D = 3. The main difference comes from the presence of thedivergence for D = 4 in the integration of Eqs.(4) and (7). We regularize the divergenceby introducing the three dimensional cut-off Λ in the momentum integration. The effectivepotential for D = 4 at vanishing temperature V0(|φS|2) is calculated analytically while thetemperature dependent part of the effective potential Vβ(|φS|2) is estimated numerically. Inthese calculations all the parameters in the model are scaled by the cut-off Λ. By directnumerical estimates we find that the effective potential (4) with D = 4 represents the chiral7symmetry restoration as temperature gets high enough. In Fig. 2 (a) and (b) we see thatthe chiral symmetry restoration is of the 2nd as well as 1st order phase transition dependingon the choice of the parameters G and ∆ (In our previous paper the existance of the 2ndorder phase transition was overlooked [7]).The gap equation ∂V/∂|φS |2 = 0 reads12α−√1 + x+√1 + x+ δ − (x+ δ) ln 1 +√1 + x+ δ√x+ δ+x ln1 +√1 + x√x+4pi2Λ2∂Vβ(|φS|2)D=4∂|φS|2 = 0, (15)where α, δ and x are defined by α = GΛ2/(8pi2), δ = ∆2/Λ2, x = |φS|2/Λ2. For β →∞ thelast term on the left hand side in Eq. (15) drops out leaving only temperature independentterms and we have the gap equation at zero temperature. It is not difficult to show thatEq. (15) without the temperature dependent term allows a nontrivial solution for x whenthe parameters α and δ satisfy the condition,12α< 1−√1 + δ + δ ln1 +√1 + δ√δ. (16)Thus the dynamical fermion mass is generated and the chiral symmetry is broken dynam-ically at vanishing temperature if the above inequality is satisfied. The boundary of theregion given by Eq. (16) in the α− δ plane is the critical curve dividing the symmetric andbroken phase as shown in Fig. 3.Just in parallel with the case ofD = 3 we find three possibilities in which the gap equation(15) allows no solution, one nontrivial solution and two nontrivial solutions respectivelydepending on the choice of parameters G, ∆ and β. It is easily found by observing thebehaviors of the gap equation and the effective potential that there are two types of phasetransitions, the first order and second order, as temperature increases. As in the case ofD = 3 the critical curve distinguishiding the region of the 1st order phase transition fromthe region of the 2nd order phase transition is given by solving simultaneous equations forG and ∆ corresponding to the following equations,V ′(0) = V ′′(0) = 0. (17)8The critical curve derived from Eq. (17) is shown by a dotted line in Fig. 3.The full and dotted line divide the whole G−∆ region into three regions of the unbrokenchiral symmetry, the broken chiral symmetry with the 2nd order phase transition and thebroken chiral symmetry with the 1st order phase transition respectively. The dynamicalfermion mass generated by the above chiral symmetry breaking is obtained by numericallysolving the gap equation (15) as is shown in Fig. 4.One of the direct physical consequences of our approach may be found in the field of elec-troweak baryogenesis in early Universe [8]. In this connection it is interesting to see whethercomposite Higgs masses can be taken large enough to account for the present experimentalobservation and to construct realistic composite Higgs models for describing the baryogen-esis [9]. As a physicsl application of our approach we may consider the supersymmetric E6GUT model [10] in which the symmetry breaking patterns can be precisely investigated byusing our results. It is also interesting to note whether our results presented in the paperis subject to any change if we introduce the contribution of order 1/N or higher. Suchcalculation has been done in Ref. [11] in the case of the nonsupersymmetric NJL model inthree dimensions and an extension of their result to the case of the SUSY NJL model shouldbe straightforward. The investigation in this direction will be left for future works.After the completion of our work we became aware of a report discussing the similaranalysis as in the present paper [12]. The conclusion of Ref. [12], however, is quite differentfrom ours.The authors would like to thank K. Kikkawa and C. S. Lim for useful comments onsupersymmetry at finite temperature and T. Inagaki, S. Mukaigawa and M. Tanabashi forenlightening discussions.9REFERENCES[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 and 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] D. Buchholtz and I. Ojima, Nucl. Phys. B498(1997) 228 and references cited therein.[5] T. Matsubara, Prog. Theor. Phys. 14(1955) 351.[6] T. Inagaki, T. Kouno and T. Muta, Intern. J. Mod. Phys. A10(1995) 2241.[7] J. Hashida, T. Muta and K. Ohkura, Mod. Phys. Lett. A13(1998) 1235.[8] J. M. Cline and G. D. Moore, Phys. Rev. Lett. 81(1998) 3315 and references citedtherein.[9] A. I. Bochkarev, S. V. Kuzmin and M. E. Shaposhnikov, Phys. Lett. B244 (1990) 275,K. Funakubo, Prog. Theor. Phys. 96 (1996) 475; hep-ph/9809517, F. Csikor, Z. Fodorand J. Heitger, Phys.Lett. B441(1998) 354.[10] T. Kugo and J. Sato, Prog. Theor. Phys.91(1994)1217.[11] F. P. Esposito, I. A. Shovkovy and L. C. R. Wijewardhana, Phys. Rev. D58(1998)065003.[12] Z. Lalak, J. Pawelczyk and S. Pokorski, Preprint MPI-Ph/93-42(1993).10FIGURES2 4 6 8 10∆0246810GΙΙSΙFIG. 1. Critical curves on G−∆ plane in D = 3: ’S’ represents the region of the parameters Gand ∆ in which the chiral symmetry is not broken at any temperature, ’II’ represents the region ofthe parameters in which the broken chiral symmetry at vanishing temperature is restored at hightemperature as the second order phase transition and ’I’ represents the region of the parametersin which the broken chiral symmetry at vanishing temperature is restored at high temperature asthe first order phase transition.0 0.1 0.2|φS|/Λ−50.0510V/Λ4(a)0 0.2 0.4 0.6 0.8 1.0|φS|/Λ−4−2024(b)X10 X10−7−5FIG. 2. The β dependence of the effective potential (a) for α = 25 and δ = 0.1 and (b) forα = 90 and δ = 0.1.110 0.1 0.2 0.3∆/Λ0102030GΛ2/(8pi2 )sIIIFIG. 3. Critical curves on G −∆ plane in D = 4: The explanation of symbols ’S’, ’II’ and ’I’is the same as the one in Fig. 1.0 10 20 30βΛ00.10.20.30.40.5m/Λ(a)10 15 20 25 30βΛ0.620.6250.63(b)FIG. 4. The dynamical fermion mass as a function of β (a) for α = 25 and δ = 0.1 and (b) forα = 90 and δ = 0.1.12

Pattern of Chiral Symmetry Restoration at Finite Temperature in A Supersymmetric Composite Model

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