Discrete symmetry breaking and restoration at finite temperature in 3D Gross-Neveu model

Transcription

1 1 Discrete symmetry breaking and restoration at finite temperature in 3D Gross-Neveu model arxiv:hep-th/981199v1 11 Nov 1998 Bang-Rong Zhou Department of Physics, Graduate School at Beijing University of Science and echnology of China, Academia Sinica, Beijing 139, China and he Abdus Salam International Centre for heoretical Physics, rieste 341, Italy Abstract Dynamical spontaneous breaking of some discrete symmetries including special parities and time reversal and their restoration at finite temperature are researched in 3D Gross-Neveu model by means of Schwinger-Dyson equation in the real-time thermal field theory in the fermion bubble diagram approximation. When the momentum cut-off Λ is large enough, the equation of critical chemical potential µ c and critical temperature c will be Λ-independent and identical to the one obtained by auxialiary scalar field approach. he dynamical fermion mass m m(,µ), as the order parameter of symmetry breaking, has the same ( c ) 1/ behavior at < c as one in 4D NJL-model and this shows the second-order phase transition feature of the symmetry restoration at > c. It is also proven that no scalar bound state could exist in this model. PACS numbers: 11.1.Kk, 11.1.Wx, 11.3.Er, 11.3.Qc Key words: 3D Gross-Neveu model, discrete symmetry breaking and restoration, real-time thermal field theory, scalar bound state Regular Associate of he Abdus Salam ICP.

2 he Gross-Neveu (GN) model [1] in +1 space-time dimension is a model with fourfermion interactions. As a well-known and applicable model to codensed matter physics, it has been extensively researched [-6]. he coventional approach is to introduce an auxialiary scalar field, then to replace the four-fermion interactions by the Yukawa couplings between the auxialiary scalar field and the fermion fields. Alternatively, one can also use Schwinger-Dyson equation directly to research the fermion condensates induced by the four-fermion interactions and relevant dynamical behavier. In the latter approach, no effective scalar field is introduced and, as far as some problems, for example, the calculation of the propagators for the bound states composed of fermions at finite temperature, are concerned, this approach could be more straightforward and convenient. On the other hand, almost all the discussions about the finite temperature behavior of the model were conducted in the imaginary-time formalism of thermal field theory. However, we also want to examine the application of the real-time formalism of thermal field theory to the model. herefore, in this letter, instead of using the auxialiary scalar field and the imaginary-time thermal field theory, we will research the 3D GN model at finite temperature by means of Schwinger-Dyson equation in the real-time thermal field theory. All the discussions will be made in the fermion bubble diagram approximation. his approximation amounts to the leading order of 1/N expansion and is sometimes also called the mean field approximation. Since 3D GN model is renormalizable in 1/N expansion [5] and in +1 dimension, there is no the kink effect which could appear in a 1+1 dimension system [7], the mean field approximation can be considered as a reliable one. he results will also be compared with the previous works ones based on the auxialiary field and the imaginary-time thermal field theory. he Lagrangian of the model can be written by L(x) = N k=1 ψ k (x)iγ µ µ ψ k (x) + g N N [ ψ k (x)ψ k (x)]. (1) k=1 In 3 dimensions, the coupling constant g has mass dimension -1 thus the model is perturbatively non-renormalizable. However, this fact does not affect our discussions based on the 1/N expansion. he γ µ (µ =, 1, ) can be taken as matrices and related to the Pauli matries σ i (i = 1,, 3) by γ = σ 3 = which satisfy the Dirac algebra ( ) ( ) 1 i, γ 1 = iσ 1 =, γ = iσ = 1 i ( ) 1, () 1 {γ µ, γ ν } = g µν, µ, ν =, 1,. (3) In this representation ψ k (x) is two-component complex spinor with N color degrees of freedom. Notice that there is no matrix which anti-commutes with all the three γ µ, i.e. no γ 5 matrix exists in 3 dimensions. Consequently L(x) has no chiral symmetry, though ψ k (x) are zero-mass fields. However, it is easy to verify that the action of system is invariant for the discrete transformation (omitting the index k of ψ) ψ 1 (x) Z ψ (x), if ψ(x) = the special P 1 and P parity transformations ( ψ 1 ) (x) ψ, (4) (x) ψ(t, x 1, x ) P 1 γ 1 ψ(t, x 1, x ), (5) ψ(t, x 1, x ) P γ ψ(t, x 1, x ), (6)

3 3 and the time reversal ψ(t, x 1, x ) γ ψ( t, x 1, x ). (7) We indicate that the mass term m ψ(x)ψ(x) (if it exists) is not invariant under Z, P 1, P and. Hence, if the four-fermion interactions could lead to generation of a non-zero dynamical mass of the ψ-fields, then the above Z, P 1, P and discrete symmetries will be spontaneously broken. Since the broken symmetries are discrete, no Goldstone bosons are expected to appear. In the following we will research generation of the dynamical mass of ψ(x) fields and its finite temperature behavior in the fermionic bubble graph approximation. his approximation, as the leading order of the 1/N expansion, will be able to give us a clear insight into the breaking and restoration of above descrete symmetries. At zero-temperature =, assume the four-fermion scalar interactions g N k=1 ( ψ k ψ k ) /N can lead to the fermion condensates N k=1 ψ k ψ k, then we will get the dynamical fermion mass m() = (g/n) N k=1 ψ k ψ k which satisfies the gap equation 1 = g id 3 l (π) 3 1 l m () + iε After Wick rotation, angular integration and introduction of 3D Euclidean momentum cut-off Λ, Eq. (8) becomes 1 = gλ π [ 1 m() Λ (8) arctan Λ ]. (9) m() We indicate that since Λ corredponds to the square root of the 3d squared Euclidean momentum, its introduction does not explicitly break the discussed discrete symmetries including P 1, P and. It is seen from Eq.(9) that the necessary condition for formation of the fermion condensates is sufficiently strong four-fermion coupling g that gλ/π > 1, similar to the one in 4D NJL-model case [8]. In both cases, one can consider the fourfermion couplings as an effective interactions below some high momentum scale Λ. For the sake of satisfying the gap equation Eq. (8), the fine-tuning of the coupling constant g is also required. When, we must replace the vacuum expectation value N k=1 ψ k ψ k by the thermal expectation value N k=1 ψ k ψ k. In the real-time thermal field theory, this implies the substitution of the fermion propagator [9,1] with i l m + iε sin θ(l, µ) = i l m + iε πδ(l m )( l + m) sin θ(l, µ) (1) θ(l ) exp[β(l µ)] θ( l ) exp[β( l + µ)] + 1, (11) where m m(, µ) is the dynamical fermion mass at finite temperature and finite chemical potential µ and the denotation β = 1/. As a result, the gap equation at becomes 1 = gi (1) I = d 3 [ ] l i (π) 3 l m + iε πδ(l m ) sin θ(l, µ) = Λ π { 1 m Λ arctan Λ m π Λ [I (y, r) + I (y, r)] } (13) (14)

4 4 where we have used the denotations I (y, r) = dxx 1 x + y exp( x + y r) + 1 = ln [ ] 1 + e (y r) (15) with x = β l, y = βm and r = βµ. (16) Considering the gap equation (9) at =, Eq.(1) may be changed into m() arctan Λ m() = m arctan Λ m + π f() (17) with f() = ln {[ 1 + e (m µ)/ ] [ 1 + e (m+µ)/ ]}. (18) It is easy to see that no matter m µ or m µ < we always have f () = df()/d >, i.e. f() is a monotone increasing function of temperature. o keep the left-handed side of Eq.(17) to be a constant, the first term in the right-handed side must decreases as increases and finally has m in it at a critical temperature c and a critical chemical potential µ c. Hence taking m = in Eq.(17) we will obtain the equation to determine c and µ c m() arctan Λ m() = π [ c ln(1 + e µ c/ c ) + ln(1 + e µc/c ) ]. (19) Here the explicit momentum cut-off Λ is not important. In fact, as long as the ratio Λ/m() 1 4, then the value of arctan[λ/m()] is almost equal to π/. Assume that is the case then the momentum cut-off Λ will disappear from the equation and we obtain m() = c [ ln(1 + e µ c/ c ) + ln(1 + e µc/c ) ]. () We indicate that the µ criticality equation () comes from the two-point function of frmions which corresponds to O(1) order in 1/N expansion, hence no explicit N-dependence is contained in it. It follows from Eq.() that c /m() = 1/ ln, if µ c = µ c /m() = 1, if c = (1) If scaled in m() by defining u c = µ c /m() and t c = c /m() then the equation () of criticality curve of chemical potential and temperature can be rewritten as u c = 1 + t c ln[( e 1/tc )/]. () hese results obtained under the assumption Λ/m() 1 are identical to the ones given by effective scalar field approach in the leading order of 1/N expansion [5] and also cosistent with the calculations of Ref. [4]. By means of the dynamical fermion mass, we may have more clear understanding of the feature of symmetry restoration phase transition. Assuming Λ/m() 1 and in view of m m(), we will have arctan[λ/m()] arctan[λ/m] π/. hus Eq.(17) will become Λ-indepedent, i.e. m() = m + { ln [ 1 + e (m µ)/ ] + ln [ 1 + e (m+µ)/ ]}. (3)

5 5 When < c, we have m/ 1 thus Eq.(3) can be transformed into ( m() = [ ln ( 1 + e ) µ/ + ln ( 1 + e )] µ/ + ln 1 + ( e m/ 1 ) e m/ 1 ) + (1 + e µ/ ) (1 + e µ/ m ) { [ln ( ) 1 + e µ/ + ln ( 1 + e )] µ/ m ( )} m 3 + [1 + cosh(µ/)] + O, (4) 3 where we have made the power series expansion of [exp(m/) 1] in m/ and kept only the terms to the order O(m / ). Combining Eq.(4) with Eq.() (in which µ c is replaced by µ) we obtain m = ( 1 + cosh µ ) { [ c ln (1 + cosh µ c )] [ ( ln 1 + cosh µ )]}. (5) By means of the approximate expression at < c [ ( ln 1 + cosh µ )] [ ( { µ = ln 1 + cosh + µ( })] c ) c c [ ln (1 + cosh µ )] + µ( c ) tanh µ (6) c c c Eq.(5) may be changed into m = ( 1 + cosh µ ) { [ ln (1 + cosh µ c )] µ tanh µ } ( c ) when < c c c (7) We see that the dynamical fermion mass m, as the only order parameter of the discrete symmetry breaking, will have ( c ) 1/ behavior at < c. his situation is similar to the one in 4 dimension NJL-model [1]. Hence, based on the same arguments given in Ref. [1], we can conclude that in the 3D Gross-Neveu model the discrete symmetry restoration at > c will also be a secon-order phase transition. In this model, no continuous symmetry breaking means no Goldstone boson appearing. An interesting question is that whether any (massive or massless) scalar bound state could exist. Such scalar bound state can be thought as the equivalent of a Higgs particle appearing in spontaneous breaking of a continuous symmetry. In present case, it could be the configuration N k=1 ( ψ k ψ k ). However, we will prove that the answer to the above question is negative in 3D Gross-Neveu model. Based on the method taken in Ref.[11] in the real-time formalism of thermal field theory, we can obtain from the scalar four-point function of fermions the formal propagator for the scalar bound state expressed by Γ S (p) = i/n(p 4m + iε) where the functions K(p) = 1 1 4π [ K(p) + H(p) is(p) dx [m p x(1 x)] arctan Λ 1/ [m p x(1 x)] 1/ 1 8π p ln 4m + 4m p + p, if arctan 4m p R ] (p), (8) K(p) + H(p) + is(p) Λ [m p x(1 x)] 1/ π and < p < 4m,

6 6 H(p) = π S(p) = π R(p) = π d 3 { l (l + p) m } + (p p) (π) 3 [(l + p) m ] + ε δ(l m ) sin θ(l, µ), d 3 l (π) 3δ(l m )δ[(l + p) m ] [ sin θ(l, µ) cos θ(l + p, µ) + cos θ(l, µ) sin θ(l + p, µ) ], d 3 l (π) 3δ(l m )δ[(l + p) m ] sin θ(l, µ) sinθ(l + p, µ). (9) It is indicated that when deriving Eq. (8) we have used Eq. (1). As a result, the Λ- dependent divergent sectors in Γ S (p) have cancelled each other and this is just a peculiarity in the leading order of 1/N expansion. Formally p = 4m seems to be a mere simple pole of Γ S (p), but this is not true. First of all we notice that when p 4m, and K(p) has a logarithmic singularity. Next, we can write lim K(p) (3) p 4m H(p) = 1 d l p + ω l p l p cosϕ p ω l p + l p cosϕ {( 8π ω l [p + ω l p l + p cosϕ] + ε [p ω l p + l ) p cosϕ] + ε 1 exp[β(ω l µ)] (p p, µ µ)}, ω l = ( l + m 1/ ). (31) he angular integrations in Eq. (31) are of the following general forms and results: π 1 dϕ =, (3) b + c cosϕ ± ε where b and c are constants. Hence the function H(p) =. As for S(p) and R(p), when p = 4m each of them may be generally expressed by A(p) p =4m = d 3 l δ(l m )δ[(l + p) m ]f(l, p, µ) p =4m = d l l 4ω l π dϕ δ(m ω l p l p cosϕ)f( ω l, p, µ), where f(l, p, µ) is a finite function. In Eq.(33), the condition cosϕ 1 limits l to the only possible value l = p /. Let tan(ϕ/) = t, then the integration of the angle ϕ in Eq. (33) can be written by (33) ( π π ) B = + dϕ δ(m ω l p l p cosϕ) π { = dt δ[(m ω l p + l p )t + (m ω l p l p )] +δ[(m ω l p l p )t + (m ω l p + l } p )]. (34)

7 7 Since the δ-functions have non-zero values only if l = p /, we can set l = ( 1 ± ε) p, ε = + (35) and obtain m ω l p + l p 4 ε p, p + 4m m ω l p l p p (36) which further lead to ε B = lim δ ε l, p dt δ p 4 / t p ε + dt δ p 4 p t p + 4m p + 4m = ( p + 4m ) 1/ p δ lim l, p / ε ε p = ( p + 4m ) 1/ p lim l p / δ l, p / l p / = ( p + 4m ) 1/ p δ( l p /). (37) Substituting Eq. (37) into Eq.(33) we will have A(p) p =4m = 1 p f( l + m, p, µ). (38) l = p / his means that both S(p) p =4m and R(p) p =4m with the form of Eq. (38) are finite. o summarize the above results, we can obtain lim 4m )Γ p 4m (p S (p) =, (39) hence p = 4m is not a simple pole of Γ S (p) and Γ S (p) does not represent the propagator for any scalar bound state. his proves that no scalar bound state could exist in this model. his result, seen from Eqs. (8) and (3), is true at =, and also maintained at a finite temperature. We point out that the above result at = is different from the one obtained in Reference [5]. In conclusion, in 3 dimension Gross-Neveu model, the fermion codensates induced by the scalar four-fermion interactions will spontaneously break the discrete Z, the special parity P 1, P and the time-revesal symmetries and lead to generation of dynamical fermion mass. By means of the real-time thermal field theory in the fermionic bubble diagram approximation, it is clearly shown that the dynamical fermion mass m, as the order parameter of symmetry breakings, will decrease as temperature increases and finally becomes zero at some critical temperature c. his implies restorations of these discrete symmetries at > c. he equation of the criticality curve of chemical potential and temperature will be indepedent of the momemtum cut-off and identical to one obtained by the auxialiary scalar field approach in the 1/N expansion if the ratio between the momentum cut-off and the dynamical fermion mass at = is sufficiently large. Similar to the case in 4D NJL-model, the dynamical mass m has the ( c ) 1/ behavior at < c and this fact indicates the second-order phase transition feature of the symmetry restorations. We have also proven by explicit analysis of the fermionic scalar four-point

8 8 functions that there is no scalar bound state in this model. he author is grateful to Prof. Zhao Bao-Heng for helpful discussions. his work was done (in part) in the frame of Associate Membership Programme of the Abdus Salam International Centre for heoretical Physics, rieste, Italy and partially supported by National Natural Science Foundation of China and by Grant No. LWZ-198 of Chinese Academy of Sciences. References [1] D. J. Gross and A. Neveu, Phys. Rev. D1 (1974) 335. [] J. Zinn-Justin, Quantum Field heory and Critical Phenomena, (Clarendon Press, Oxford, 1993). [3] K. G. Klimenko, Z. Phys. C37 (1988) 457. [4] A. Okopińska, Phys. Rev. D38 (1988) 57. [5] B. Rosenstein, B. J. Warr and S. H. Park, Phys. Rep. 5 (1991) 59. [6] V. E. Rochev and P. A. Saponov, hep-th/9716. [7] R. F. Dashen, S-K, Ma and R. Rajaraman, Phys. Rev. D 11 (1975) [8] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 1 (1961) 345; 14 (1961) 46. [9] N. P. Landsman and Ch. G. van Weert, Phys. Rep. 145 (1987) 141, and the refereces therein. [1] B. R. Zhou, Phys. Rev. D57 (1998) 3171; Commun. heor. Phys., to be published. [11] B. R. Zhou, Institution Report, AS-GS-P-4 (1998).