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Dynamical symmetry breaking in the external gravitational and constant magnetic fields



KOBE-TH-97-02 September 1997

Dynamical symmetry breaking in the external gravitational and constant magnetic ?elds

arXiv:hep-th/9709077v1 10 Sep 1997

T. Inagaki ? , Department of Physics, Kobe University, Rokkoudai, Nada, Kobe 657, Japan S. D. Odintsov ? , Dept. Theor. Phys., Tomsk Pedagogical University, 634041 Tomsk, Russia and Department de Fisica, Universidad del Valle, A.A.25360 Cali, Colombia Yu. I. Shil’nov ? Institut D’Estudis Espacials De Catalunya, Edif. Nexus-104, Gran Capita 2-4, 08034, Barcelona, Spain and Department of Theoretical Physics, Faculty of Physics, Kharkov State University, Svobody Sq. 4, 310077, Kharkov, Ukraine
Abstract We investigate the e?ects of the external gravitational and constant magnetic ?elds to the dynamical symmetry breaking. As simple models of the dynamical symmetry breaking we consider the Nambu-Jona-Lasinio (NJL) model and the supersymmetric Nambu-Jona-Lasinio (SUSY NJL) model non-minimally interacting with the external gravitational ?eld and minimally interacting with constant magnetic ?eld. The explicit expressions for the scalar and spinor Green functions are found up to the linear terms on the spacetime curvature and exactly for a constant magnetic ?eld. We obtain the e?ective potential of the above models from the Green functions in the magnetic ?eld in curved spacetime. Calculating the e?ective potential numerically with the varying curvature and/or magnetic ?elds we show the e?ects of the external gravitational and magnetic ?elds to the phase structure of the theories. In particular, increase of the curvature in the spontaneously broken chiral symmetry phase due to the ?xed magnetic ?eld makes this phase to be less broken. On the same time strong magnetic ?eld quickly induces chiral symmetry breaking even at the presence of ?xed gravitational ?eld within nonbroken phase.
?

?

e-mail : inagaki@hetsun1.phys.kobe-u.ac.jp e-mail : sergei@ecm.ub.es, odintsov@quantum.univalle.edu.co ? e-mail : visit2@ieec.fcr.es

1

1

Introduction

The idea of the dynamical symmetry breaking in quantum ?eld theory has been introduced quite long ago [1]. In the attempts to realize this idea in the early universe one should study the external gravitational ?eld (or curved spacetime). The investigation of the dynamical symmetry breaking in four-fermion models [1, 2] in four and three dimensional curved spacetime has been started in Refs.[3, 4] using the weak curvature approximation (for a review and list of references, see [5]). A validity of the weak curvature approximation is discussed in Ref.[6]. In the case when the external gravitational ?eld is treated exactly (de Sitter or anti-de Sitter or Einstein universe background) the phase structure of the four-fermion models has been studied in Refs.[7]. (for a renormalization group approach, see also [8]). There appeared recently some indications that early universe may contain large primordial magnetic ?elds. The role of these ?elds in cosmology (in particularly, in the in?ationary models of the universe) has been discussed in Refs.[9]. The presence of magnetic ?eld in the early universe may lead to di?erent e?ects. In particular, it is reasonable then to investigate the dynamical symmetry breaking in curved spacetime with magnetic ?elds§ . Such study for 3D and 4D four-fermion models has been undertaken in Refs.[13, 14] in the approximation when the e?ects of the gravitational and magnetic ?eld maybe simply summarized (in other words, the correspondent terms in the e?ective potential are evaluated with the neglecting of the interaction between gravitational and magnetic ?elds). However, the coherent e?ect of combined gravitational and magnetic ?eld may become very relevant. That is the purpose of the present work to investigate the phase structure of fourfermion models in weakly curved spacetime with constant magnetic ?eld. We develop the approximation where the e?ective potential and Green function (GF) of the theory maybe represented as the expansion in the powers of curvature invariants. In each order of this expansion the external magnetic ?eld is treated exactly, i.e. we take into account the coherent e?ect of combined gravitational and magnetic ?elds. The paper is organized as follows. In the next section we present new form of the local momentum representation of propagators [15] with account of the constant magnetic ?eld. In each order on the expansion in curvature invariants for Green functions the magnetic ?eld is included exactly. The detailed calculation is presented for scalar and spinor Green functions. Section 3 is devoted to the study of phase structure in the NJL model in curved spacetime with magnetic ?eld. We calculate the e?ective potential with account of linear curvature terms for 3D and 4D models. The numerical investigation of the e?ective potential for 4D NJL model is given. In section 4 we discuss the SUSY NJL model non-minimally interacting with the external gravitational ?eld and minimally interacting with constant magnetic ?eld. The e?ective potential is evaluated and the correspondent phase structure is given. Some discussion and outlook are presented in
For the works where four-fermion models have been discussed in the presence of constant electromagnetic ?elds, see [10, 11, 12]
§

2

the last section.

2

Scalar and spinor Green functions in the magnetic ?eld in curved spacetime

One of the fundamental objects in the ?eld theory is a Green function. It is a basic quantity for a number of calculations of quantum e?ects. Here we calculate the scalar and spinor Green functions in the magnetic ?eld in curved spacetime to study the phase structure of the NJL and SUSY NJL model in the external gravitational and constant magnetic ?elds.

2.1

Green function for a scalar ?eld

First we study GF for the scalar ?eld G(x, y ; m) in an external electromagnetic and gravitational ?eld. We assume that the spacetime curves slowly and neglect the terms involving the metric derivatives higher than third (weak curvature approximation). However we treat the external magnetic ?eld exactly. The GF for the scalar ?eld G(x, y ; m) satis?es the Klein-Gordon equation : 1 (iD ? iD? ? m2 ? ξR)G(x, y ; m) = √ δ D (x ? y ), ?g (1)

where ξ represents the non-minimal coupling constant with external gravitational ?eld(see [24]), m is the scalar ?eld mass and D ? = ?? ? ieA? . In the present paper the vector potential of the external electromagnetic ?eld is chosen in the form 1 A? (x) = ? F?ν xν , 2 (2)

where F?ν is constant matrix of electromagnetic ?eld strength tensor. The covariant derivative for a scalar ?eld is only an ordinary derivative ?? = ? ? . For a vector ?eld v ? the covariant derivative is rewritten as ?? v ? = ?? v ? + Γν ν? v ? √ 1 = √ ?? ?gv ? . ?g Thus the ?rst term of the left hand side in Eq.(1) is rewritten as √ √ ?gD ? D? G(x, y ; m) = ?g (?? ? ieA? )(?? ? ieA? )G(x, y ; m) √ = (?? ? ieA? ) ?gg ?ν (?ν ? ieAν )G(x, y ; m).

(3)

(4)

We want to expand Eq.(1) around R = 0. For this purpose we introduce the Riemann normal coordinates [16]. In the Riemann normal coordinates framework the metric tensor 3

is expanded as 1 (0) (y ? x)ρ (y ? x)σ , g?ν (y ) = η?ν + R?ρσν 3 1 (0) g (y ) = ?1 ? R?ν (y ? x)? (y ? x)ν , 3

(5)

(0) (0) where the su?x (0) for R?ρσν and R?ν designates the curvature tensor at the origin x. Substituting Eq.(5) into Eq.(4) we obtain



?gD ? D? G(x, y ; m) =

1 η ?ν + R(0) αβ (y ? x)α (y ? x)β η ?ν 6 1 ? ν ? R(0) α β (y ? x)α (y ? x)β (?? ? ieA? ) 3 2 ν ? R(0) α (y ? x)α (?ν ? ieAν )G(x, y ; m), 3

(6)

where we keep only terms independent of curvature or linear in curvature. We expand the GF, G(x, y ; m), as G(x, y ; m) = G(0) (x, y ; m) + G(1) (x, y ; m) + O(R2 ), (7)

where G(0) and G(1) represent the terms independent of R and the terms linear in R respectively. Substituting Eqs.(6) and (7) into (1) we can perturbatively solve the Klein-Gordon equation (1) about the spacetime curvature R. The piece independent of curvature in Eq.(1) is given by η ?ν (?? ? ieA? )(?ν ? ieAν ) + m2 G(0) (x, y ; m) = ?δ D (x ? y ). (8)

Thus G(0) (x, y ; m) satis?es the Klein-Gordon equation in ?at spacetime. We will give the explicit expression for G(0) (x, y ; m) below. The piece linear in R in Eq.(1) becomes 1 (0) R αβ (y ? x)α (y ? x)β η ?ν 6 1 ? ν ? R(0) α β (y ? x)α (y ? x)β (?? ? ieA? )(?ν ? ieAν ) 3 2 ν ? R(0) α (y ? x)α (?ν ? ieAν ) + ξR G(0) (x, y ; m) 3 + η ?ν (?? ? ieA? )(?ν ? ieAν ) + m2 G(1) (x, y ; m) = 0.

(9)

Hence the piece G(1) involving the terms linear in R is expressed with the help of the GF G(0) (x, y ; m) in ?at spacetime (for more details of local momentum representation, see [5, 15, 16, 17]). 4

It is more convenient to introduce the new variable de?ned by ? (x ? y ; m), G(x, y ; m) ≡ Φ(x, y )G where Φ(x, y ) satis?es Inserting Eq.(10) into Eq.(9) we obtain (?? ? ieA? )Φ(x, y ) = 0. (11) (10)

1 (0) R αβ (y ? x)α (y ? x)β ? ? ?? 6 1 ? ν ? R(0) α β (y ? x)α (y ? x)β ?? ?ν 3 2 ν ? (0) (y ? x; m) ? R(0) α (y ? x)α ?ν + ξR(0) G 3 ? (1) (x ? y ; m) = 0. + ?? ?? + m2 G

(12)

? (0) (y ? x; m) and G ? (1) (x ? y ; m). A? -dependence disappears in the relationship between G (1) ? (x ? y ; m) is given by Therefore the GF G ? (1) (x ? y ; m) = G × dD zG00 (x ? z ; m) 1 (0) R αβ (z ? y )α(z ? y )β ? ? ?? 6

1 ? ν ? R(0) α β (z ? y )α (z ? y )β ?? ?ν 3 2 ν ? (0) (z ? y ; m), ? R(0) α (z ? y )α ?ν + ξR(0) G 3 where the function G00 (x ? y ; m) satis?es (? ? ?? + m2 )G00 (x ? y ; m) = ?δ D (x ? y ).

(13)

(14)

The linear curvature correction terms of the scalar GF are expressed by the GF in ?at spacetime. In the constant curvature spacetime, R?νρσ = the equation (13) simpli?es to R (η?ρ ηνσ ? η?σ ηνρ ), D (D ? 1) (15)

? (1) (x ? y ; m) G R = dD zG00 (x ? z ; m) D (D ? 1) 1 D?3 ηαβ (z ? y )α(z ? y )β ? ? ?? + (z ? y )? (z ? y )ν ?? ?ν × 6 3 2 ? (0) (z ? y ; m). ? (D ? 1)(z ? y )??? + D (D ? 1)ξ G 3 5

(16)

(17)

As is shown in Eq.(8), G(0) (x, y ; m) satis?es the Klein-Gordon equation in ?at spacetime. According to the Schwinger proper-time method the Klein-Gordon equation is exactly solvable in the constant electromagnetic ?eld [18, 19]? . To calculate G(0) (x, y ; m) we introduce the proper-time Hamiltonian which is de?ned by H (x? , i?? )G(0) (x, y ; m) ≡ (i? ? + eA? )(i?? + eA? ) ? m2 G(0) (x, y ; m). (18)

For this proper-time Hamiltonian the time evolution operator U (x, y ; s) is de?ned by i ? U (x, y ; s) = H (x, i?? )U (x, y ; s), ?s (19)

with the boundary conditions lims→0 U (x, y ; s) = δ D (x ? y ), lims→?∞ U (x, y ; s) = 0. Comparing Eq.(19) with Eq.(18) we ?nd G(0) (x, y ; m) = ?i
0 ?∞

(20)

dsU (x, y ; s).

(21)

The time evolution operator U (x, y ; s) is obtained by solving the equation of motion for x? and π ? = i? ? ? eA? . For a constant electromagnetic ?eld, x? (s) = U ? (s)x? U (s) and π ? (s) = U ? (s)π ? U (s) satisfy the following equations : dx? (s) = i[H, x? ] = ?2π ? . ds dπ? (s) = i[H, π? ] = 2eF?ν π ν . ds These di?erential equations can be easily solved, π (s) = e2eF s π (0), (22) (23)

(24)

e2eF s ? 1 π (0), (25) x(s) ? x(0) = ? eF where we use the matrix notation. Substituting Eqs.(24) and (25) into Eq.(19) we obtain the di?erential equation for the time evolution operator U (x, y ; s), ? U (x, y ; s) = (π 2 (s) ? m2 )U (x, y ; s) ?s i = (x ? y )K (x ? y ) ? tr[eF coth(eF s)] ? m2 U (x, y ; s), 2 i
?

(26)

For other types of exact solutions of the Klein-Gordon equation in the external electromagnetic ?eld, see [20].

6

1 (27) K ≡ e2 F 2 [sinh(eF s)]?2 . 4 After the integration over s in Eq.(26) the time evolution operator is found to be : U (x, y ; s) = ? 1 sinh(eF s) i Φ(x, y )s?D/2 exp ? tr ln D/ 2 (4π ) 2 eF s i × exp (x ? y )eF coth(eF s)(x ? y ) + im2 s . 4

where K is de?ned by

(28)

Substituting this time evolution operator (28) to Eq.(21) we ?nd the GF, G(0) (x, y ; m), G(0) (x, y ; m) = ?i =?
0 ?∞ 0

dsU (x, y ; s)

1 sinh(eF s) 1 ?D/2 Φ( x, y ) tr ln dss exp ? (4π )D/2 2 eF s ?∞ i × exp (x ? y )eF coth(eF s)(x ? y ) + im2 s . 4

(29)

Taking the limit F → 0 and changing the variable s → ?s in Eq.(29) we get the GF, G00 (x ? y ; m), G00 (x ? y ; m) = ?
∞ 0

ds

1 (4πs)D/2 (30)

i π × exp ?i D ? (x ? y )? (x ? y )? ? im2 s , 4 4s

where the phase factor is determined to satisfy the boundary condition (20). In the present paper we consider the constant magnetic ?eld along the z -axis. For this ? (0) (x ? y ; m), reduces to constant magnetic ?eld, F12 = ?F21 = B , the GF, G ? (0) (x ? y ; m) = ? G 1 (4πs)D/2 0 eBs π i × exp ?i D ? (x ? y )? C?ν (x ? y )ν ? im2 s , sin(eBs) 4 4s


ds

(31)

where

1 ? eBs cot(eBs) . B2 Inserting the Eqs.(30) and (31) into (17) we obtain the GF linear in R C?ν = η?ν + F? λ Fλν
∞ R dt dD z D (D ? 1) (4πt)D/2 0 π i × exp ?i D ? (x ? z )? (x ? z )? ? im2 t 4 4t

(32)

? (1) (x ? y ; m) = G

∞ 0

ds (4πs)D/2

7

×

D?3 ηαβ (z ? y )α (z ? y )β ? ? ?? 6 2 1 + (z ? y )? (z ? y )ν ?? ?ν ? (D ? 1)(z ? y )? ?? + D (D ? 1)ξ 3 3 π i eBs exp ?i D ? (z ? y )?C?ν (z ? y )ν ? im2 s . × sin(eBs) 4 4s

(33)

Thus, we obtain the explicit expression of the scalar GF in the external gravitational and magnetic ?eld. We need only the coincidence limit x → y of the GF to calculate the e?ective potential of the models considered in the present paper. After the Wick rotation s → ?is, t → ?it, z 0 → ?iz 0 and the integration over z SpG(x, x; m) simpli?es to SpG(0) (x, x; m) = ?i (4π )D/2
∞ 0

dss?D/2

eBs exp ?m2 s , sinh(eBs)

(34)

SpG(1) (x, x; m) =

∞ ∞ i R dt ds exp ?m2 (s + t) D/ 2 (4π ) D (D ? 1) 0 0 (2?D )/2 eB (s + t) × 1 + eBt coth(eBs) sinh(eBs) ?2D + 1 (D ? 2)t D?3 (D ? 2 + 2eBs coth(eBs)) + × ? 6 3 s+t D?3 ?2D + 1 + ? (D ? 2 + 2eBs coth(eBs)) + eB coth(eBs) 6s 3 2t × 1 + eBt coth(eBs)

+ +

1 (D ? 3) (eBs)2 coth2 (eBs) + 1 + 4eBs coth(eBs) 3s (D ? 2)t t × + D (D ? 1)ξ . s + t 1 + eBt coth(eBs)

2(D ? 1) ? D (D ? 2) t 3 4 s+t

?

2

+2

eBt coth(eBs) 1 + eBt coth(eBs)

2

? ?

(35)

Eqs.(34) and (35) correspond to the the vacuum self-energy of the free scalar ?eld with mass m at the one loop level.

2.2

Green function for a spinor ?eld

Next we construct the spinor GF in an external electromagnetic and gravitational ?eld. Let us write the GF, which obeys the Dirac equation: 1 (iγ ? D? ? m)S (x, y ; m) = √ δ D (x ? y ), ?g 8 (36)

where the covariant derivative D? includes the electromagnetic potential A? : 1 (37) D? = ?? ? ieA? + ω ab? σab . 2 The local Dirac matrices γ? (x) are expressed through the usual ?at ones γa and tetrads ea ?: γ ? (x) = γ a e? a (x), 1 (38) σab = [γa , γb]. 4 The spin-connection has the form : 1 1 aν bρ b c a ω ab? = eaν (?? eb ν ? ?ν e? ) + e e ec? (?ρ eν ? ?ν eρ ) 2 4 1 bν 1 bν aρ a a c ? e (?? eν ? ?ν e? ) ? e e ec? (?ρ ec (39) ν ? ?ν eρ ). 2 4 Dimensions of the spinor representation are supposed to be four. Greek and Latin indices correspond to the curved and ?at tangent spacetimes. To calculate the linear curvature corrections the local momentum expansion formalism as in the previous subsection is the most convenient one [5, 15, 17]. In the Riemann normal coordinates framework the tetrads e? a (x) and the spin connection ω ab? σab are expanded as 1 e? a (y ) = δ ? a + R(0)?ρσa (y ? x)ρ (y ? x)σ , 6 (40) 1 (0)ab λ ab ω ? σab = R ( y ? x ) σ . ?λ ab 2 Substituting (39) and (40) into the (37), we obtain the following equation for the GF : 1 iγ a δ ? a + R(0)?ρσa (y ? x)ρ (y ? x)σ 6 ?m S (x, y ; m) = δ D (x ? y ). Ful?lling the expansion on the spacetime curvature degrees where S (0) is the GF in the ?at spacetime, S (1) ? O(R) and so on, we receive the iterative sequence of equations: i ? + e A(x) ? m S (0) (x, y ; m) = δ D (x ? y ) (i ? + e A(x) ? m)S (1) (x, y ; m) i + γ a R(0)?ρσa (y ? x)ρ (y ? x)σ (?? ? ieA? (x)) 6 i (0) + γ a Rbcaλ (y ? x)λ σ bc S (0) (x, y ; m) = 0. 4 9 (43) S = S (0) + S (1) + · · · , (42) 1 (0) ?? + Rbc?λ (y ? x)λ σ bc ? ieA? 4 (41)

(44)

Here and below we can forget about the di?erence between the two kinds of indices (Greek and Latin) because it lies beyond of linear curvature approximation. We assume that just as in the ?at spacetime GF has the form [18]: ?(x ? y ; m), S (x, y ; m) = Φ(x, y )S (45)

where the function Φ(x, y ) is introduced in Eq.(11). Then, we can ?nd the equation which ?(1) (x ? y ; m) function, excluding the evident dependence on A? (x): determines the S ?(1) (x ? y ; m) (i ? ? m)S i ?(0) (x ? y ; m) = ? γ a R(0)?ρσa (y ? x)ρ (y ? x)σ ?? S 6 i (0) ?(0) (x ? y ; m). ? γ a σ bc Rbcaλ (y ? x)λ S 4

(46)

The ?at spacetime GF in the external electromagnetic ?eld is supposed to be known [11, 12, 18]. Denoting as S00 (x ? y ; m) the GF, satisfying the equation (i ? ? m)S00 (x ? y ; m) = δ D (x ? y ), we obtain
?1 ?(1) (z ? y ; m) dD zS00 (x ? z ; m)S

(47)

i ?(0) (x ? y ; m) = ? γ a R(0)?ρσa (x ? y )ρ(x ? y )σ ?? S 6 i (0) ?(0) (x ? y ; m), ? γ a σ bc Rbcaλ (x ? y )λ S 4 or, ?nally, ?(0) (x ? y ; m) = S dD zS00 (x ? z ; m)

(48)

i ?(0) (z ? y ; m) × ? γ a R(0)?ρσa (z ? y )ρ (z ? y )σ ?? S 6 i (0) ?0 (z ? y ; m). ? γ a σ bc Rbcaλ (z ? y )λ S 4

(49)

However, we need only the coincidence limit x → y to calculate the e?ective potential. It provides us the opportunity to simplify (49) especially for the constant curvature ?(1) , in the spacetime with an arbitrary spacetimes (15). Thus the expression for the GF, S dimension D is the following: ?(1) (0; m) = ? S iR 12D (D ? 1) ?(0) (z ; m) dD zS00 (?z ; m) 2 zz ? ?? S (50)

?(0) (z ; m) + 3(D ? 1) z S ?(0) (z ; m) . ?2z 2 γ ? ?? S 10

Now we begin the calculation of the Green function for the D = 4 Gross-Neveu model in the external magnetic ?eld. According to the Schwinger proper-time method as in the previous subsection the ?at spacetime GF is found to be [11, 18]: i π ds exp ?i + sm2 exp ? z? C ?ν zν 2 16(πs) 2 4s 0 1 es e × m + γ ? C?ν z ν ? γ ? F?ν z ν (eBs) cot(eBs) ? γ ? γ ν F?ν , 2s 2 2 ?(0) (z ; m) = ?i S


(51)

where C?ν is de?ned in Eq.(32). For S00 (?z ; m) we have directly from (51): S00 (?z ; m) = ?i
∞ 0

z2 π dt 2 exp ? i + m t + 16(πt)2 2 4t

m?

z . 2t

(52)

Substituting (52) in (50) and calculating the trace over the spinor indices, we have : ?(1) (0; m) = SpS ?iRm 36 d4 zdtds (16π 2 ts)2

× exp ?i (t + s)m2 +

t+s 2 2 1 + eBt cot(eBs) z ? z⊥ 4ts 4t 9 7 ? 4eBs cot(eBs) × eBs cot(eBs) z 2 ? + 2t 2s 4 ? 7eBs cot(eBs) 9 2 + +z⊥ 2t 2s i 7 2 2 ? 2 z 2 z⊥ (1 ? eBs cot(eBs))2 ? e2 B 2 s 2z 2 + z⊥ 2s 2 i 2 2 + z z⊥ (eBs cot(eBs) ? 1) , 2
2 2 2 2 2 2 z⊥ = z1 + z2 , z = z0 ? z3 .

(53)

where (54) After the Wick rotation and integration over z , one gets: ?(1) (0; m) = SpS iRm 96π 2 (t + s)2 (1

dtds exp[?(t + s)m2 ] + eBt coth(eBs))2

× eBt(eBt + eBs) + 2(eBt + 3eBs) coth(eBs) +2eBt(eBs ? eBt) coth2 (eBs) . (55)

The same program can be done for the 3D case. The GF in the ?at space-time with the external constant magnetic ?eld has the following form [11, 12]: ?(0) (z ; m) = ?i S
∞ 0

i ds ?i(π/4+sm2 ) e exp ? z? C ?ν zν 8(πs)3/2 4s 11

× m+

1 ? e γ C?ν z nu ? γ ? F?ν z ν 2s 2 es ? ν × eBs cot(eBs) ? γ γ F?ν . 2

(56)

For S00 (?z ; m) we have directly from (56): S00 (?z ; m) = ?i
∞ 0

z2 π dt 2 exp ? i + m t + 8(πt)3/2 4 4t

m?

z . 2t

(57)

Substituting (56),(57) in (50), we have: ?(1) (0; m) = SpS Rm t+s 2 d3 zdtds exp ?i (t + s)m2 + z 3 3 / 2 1152π (ts) 4ts 0 2 1 + eBt cot(eBs) ?z⊥ 4t 3 3 ? 2eBs cot(eBs) 2 × eBs cot(eBs) z0 (? + ) t s 1 ? 2eBs cot(eBs) 2 3 +z⊥ ( + ) t s i 2 2 z⊥ (1 ? eBs cot(eBs))2 ? 2 z0 2s
2 2 + z⊥ ) +e2 B 2 ?2s(z0

i 2 2 + z0 z (1 ? eBs cot(eBs)) 2 ⊥ After the Wick rotation and integration over z , one gets: ?(1) (0; m) = SpS iRm 72π 3/2 (t + s)3/2 (1

.

(58)

dtds exp[?(t + s)m2 ] + eBt coth(eBs))2

× 2eBt(eBt + eBs) + (9eBs + 5eBt) coth(eBs) +eBt(eBs ? 3eBt) coth2 (eBs) . (59)

The proper-time integrations remain in our ?nal expressions of the scalar and spinor Green functions in an external magnetic and gravitational ?eld. All remained integrands are exponentially suppressed at the limit s → ∞ and/or t → ∞. There are divergences at s → 0 and/or t → 0 for D ≥ 2.

3

NJL model

Let us discuss now the NJL model [1] in an external magnetic and gravitational ?eld (for a review, see [5]). The NJL model is one of the simplest models where dynamical 12

symmetry breaking is possible. It is well-know that the chiral symmetry of the model is dynamically broken down for a su?ciently large coupling constant. In the present paper we want to know the e?ect of an external electromagnetic and gravitational ?eld to the dynamical symmetry breaking. The NJL model is de?ned by the action which is given by S= √ λ (ψψ )2 + (ψiγ5 ψ )2 dD x ?g iψγ ? (x)D? ψ + 2N , (60)

where N is the number of the fermions. This action has the chiral U (1) symmetry. Introducing the auxiliary ?elds σ=? we can rewrite the action (60) as: S= √ N dD x ?g iψγ ? D? ψ ? (σ 2 + π 2 ) ? ψ (σ + iπγ5 )ψ . 2λ (62) λ λ (ψψ ), π = ? ψiγ5 ψ, N N (61)

In the leading order of the 1/N expansion the e?ective action is given by √ σ2 + π2 1 Γef f (σ, π ) = ? dD x ?g ? i ln det[iγ ? (x)D? ? (σ + iγ5 π )]. (63) N 2λ We can put now π = 0 because the ?nal expression would depend on the combination σ 2 + π 2 only. In this case the e?ective action of the NJL model has the same form as the one of the Gross-Neveu model [2]. Note that Eq.(63) represents the particular example of the e?ective action for composite ?elds [21]. √ De?ning the e?ective potential as Vef f = ?Γef f /N dD x ?g for constant con?gurations of σ and π , we obtain σ2 Vef f (σ ) = + iSp ln x|[iγ ? (x)D? ? σ ]|x . 2λ The second term of the right hand side in Eq.(64) is rewritten as iSp ln x|[iγ ? (x)D? ? σ ]|x = ?iSp
σ

(64)

dm S (x, x; m).

(65)

Thus we can express the e?ective potential of the NJL model by the spinor Green function S (x, x, σ ). Substituting the Green function evaluated in the previous section to Eq.(65) we obtain the e?ective potential in an external magnetic and gravitational ?eld In four-dimensional spacetime the linear curvature correction for the e?ective potential is given by Vef f (σ ) = ?
∞ ∞ dtds R 2 3 2 2 192π 1/Λ 1/Λ (t + s) (1 + eBt coth(eBs))2 × exp[?(t + s)σ 2 ][eBt(eBt + eBs) + 2(eBt + 3eBs) coth(eBs) +2eBt(eBs ? eBt) coth2 (eBs)], (1)

(66)

13

Vef f (σ )/?4 0 -0.005 -0.01 -0.015
eB = 2?2 eB = 0

(0)

Vef f (σ )/?4 0.005 0.004 0.003 0.002 0.001
eB = 0 eB = 2?2

(1)

0 0.8 0 0.2 (b) Vef f
(1) (1)

0

0.2 (a) Vef f
(0)

0.4 0.6 σ/? for R = 0.
(0)

0.4 0.6 σ/? for R = 0.

0.8

Figure 1: The behavior of Vef f and Vef f are shown with the varying magnetic ?eld eB (= 0, ?2/2, ?2 , 3?2/2, 2?2 ) for ?xed λ(= 1/2.5) and ?xed Λ(= 10?) in four dimensional (0) (1) ?at spacetime. ? is an arbitrary mass scale. We normalize that Vef f (0) = Vef f (0) = 0. where we introduce the proper-time cut-o? Λ. In four dimensions four-fermion models are not renormalizable. Here we de?ne the ?nite theory by introducing the proper-time cut-o? Λ. Taking into account the proper-time representation for Vef f in four-dimensional ?at spacetime [18], we obtain the e?ective potential with the linear curvature accuracy:
∞ ds 1 σ2 + 2 exp(?sσ 2 )eBs coth(eBs) Vef f (σ ) = 3 2 2λ 8π 1/Λ s ∞ ∞ R dsdt ? 2 3 192π 1/Λ2 1/Λ2 (t + s) (1 + eBt coth(eBs))2 × exp[?(t + s)σ 2 ][eBt(eBt + eBs) + 2(eBt + 3eBs) coth(eBs) +2eBt(eBs ? eBt) coth2 (eBs)].

(67)

It should be emphasised here that for B = 0 we obtain just the same expression for 4D NJL model Vef f in the proper-time representation as it has been found out in [14] (See also [5]). For D = 3 the linear curvature correction for the e?ective potential is given by
∞ ∞ dtds R 3 / 2 5 / 2 144π (1 + eBt coth(eBs))2 1/Λ2 1/Λ2 (t + s) × exp[?(t + s)σ 2 ][2eBt(eBt + eBs) + (9eBs + 5eBt) coth(eBs) +eBt(eBs ? 3eBt) coth2 (eBs)].

Vef f (σ ) = ?

(1)

(68)

14

Vef f (σ )/?4 0.002 0 -0.002 -0.004 -0.006 -0.008 0
eB = 2?2 B=0 eB = ?2 /2 eB = ?2

Vef f (σ )/?4 0.002 0 -0.002 -0.004 -0.006 -0.008 0.8 0 0.2
R = 3?2 R = 2?2 R = ?2 R=0 R = ? ?2 R = ?2?2

eB = 3?2 /2

0.2

(a) Vef f

0.4 0.6 σ/? for R = 3?2 .

(b) Vef f

0.4 0.6 0.8 σ/? for eB = ?2 /2.

Figure 2: The behavior of the e?ective potential Vef f is shown with the varying B or R for ?xed λ(= 1/2.5) and ?xed Λ(= 10?) in four dimensions. For D = 3, four-fermion models are known to be renormalizable in the sense of the 1/N expansion. Thus the cut-o? dependence in the e?ective potential disappears after the usual renormalization procedure. Taking into account the proper-time representation for Vef f in ?at spacetime [11, 12], we can write the e?ective potential with the linear curvature accuracy: Vef f (σ ) = ? R 144π 3/2 σ2 1 + 3/2 2λ 4π
∞ 1/Λ2 ∞ 1/Λ2

ds exp(?sσ 2 )eBs coth(eBs) s5/2 dsdt 5 / 2 (t + s) (1 + eBt coth(eBs))2
∞ 1/Λ2

× exp[?(t + s)σ 2 ] 2eBt(eBt + eBs) + (9eBs + 5eBt) coth(eBs) +eBt(eBs ? 3eBt) coth2 (eBs) . (69)

The e?ective potential (69) gives again the correct expression for Vef f of 3D NJL model in the proper-time representation with B = 0 [13]. Using Eq. (67) we numerically calculate the e?ective potential for D = 4 and show it in Figs.1 and 2. In drawing Fig.1 and 2 the coupling constant λ is kept in the region where the chiral symmetry is broken down for R = B = 0. To see the curvature e?ect in the external constant magnetic ?eld we divide the e?ective potential into the terms (0) (1) independent of the curvature and the terms linear in the curvature, Vef f = Vef f + Vef f . (0) (1) In Fig.1 typical behaviors of the e?ective potential Vef f and Vef f are given for ?xed fourfermion coupling constant λ and proper-time cut-o? Λ. It is clearly seen in Figs.1 and 15

2.(a) that a constant magnetic ?eld B decreases the potential energy in the true vacuum even in the curved spacetime. Curvature e?ects depend on the sign of the spacetime curvature R as is shown in Fig. 2.(b). A negative curvature decreases the potential energy. On the contrary, positive curvature increases the potential energy. The broken chiral symmetry is restored for a su?ciently large and positive curvature even in an external large magnetic ?eld.

4

SUSY NJL model

In the present section we consider the supersymmetric NJL model in the external constant magnetic ?eld and external gravity. In ?at space such a model has been introduced in Ref.[22]. Generalization of the model for the case of non-minimal coupling with the external gravitational ?eld (via scalar-gravitational coupling constants) has been presented in Ref.[23]. We consider the model of Ref.[23] in curved spacetime with the external magnetic ?eld. The action in components is given by √ S = d4 x ?g ?φ? (D ? D? + σ 2 + ξ1 R)φ ? φc ? (D ? D? + σ 2 + ξ2 R)φc 1 2 (70) σ , 2λ where σ is an auxiliary scalar as in the original NJL model, ψ is N component Dirac spinor. Note that actually the action (70) represents SUSY NJL model non-minimally interacting with the external gravity and minimally interacting with the external magnetic ?eld. It is evident that supersymmetry of SUSY NJL model is always broken in the external ?elds. So it maybe natural to call the theory with action (70) the extended NJL model. At the leading order of the 1/N expansion the e?ective potential of the SUSY NJL model (70) is given by ?(iγ ? D? ? σ )ψ ? +ψ Vef f (σ ) = 1 2 σ + iSp ln x|[iγ ? (x)D? ? σ ]|x 2λ ?iSp ln x|[D ? D? + σ 2 + ξ1 R]|x ?iSp ln x|[D ? D? + σ 2 + ξ2 R]|x .

(71)

The second term in the right hand side of Eq.(71) corresponds to the radiative correction by spinor ?elds and is rewritten by the spinor GF as is evaluated in the previous section (See Eq.(65)). The third and the fourth terms in the right hand side of Eq.(71) correspond to the radiative correction by scalar ?elds and is represented by the scalar GF ? iSp ln x|[D ? D? + σ 2 + ξR]|x = 2i
σ

m dm G(x, x, m). 16

(72)

Because of the supersymmetry both the radiative correction by spinor ?elds and the one by scalar ?elds are cancelled out in ?at spacetime with vanishing electromagnetic ?elds. Thus the vacuum expectation value of σ disappears for R = B = 0. Substituting Eqs.(34) and (35) into Eq.(72) and integrating over m one gets 2i =?
σ

m dm G(x, x, m)

∞ eB 1 exp ?σ 2 s dss?D/2 D/ 2 (4π ) sinh(eBs) 0 ∞ ∞ 1 R + dt ds exp ?(s + t)σ 2 D/ 2 (4π ) D (D ? 1) 0 0 eB (s + t)?D/2 × 1 + eBt coth(eBs) sinh(eBs) D?3 ?2D + 1 (D ? 2)t × ? (D ? 2 + 2eBs coth(eBs)) + 6 3 s+t ?2D + 1 D?3 (D ? 2 + 2eBs coth(eBs)) + eB coth(eBs) + ? 6s 3 2t × 1 + eBt coth(eBs)

1 (D ? 3) (eBs)2 coth2 (eBs) + 1 + 4eBs coth(eBs) 3s (D ? 2)t t × + D (D ? 1)ξ . s + t 1 + eBt coth(eBs) +

2(D ? 1) ? D (D ? 2) t + 3 4 s+t

?

2

eBt coth(eBs) +2 1 + eBt coth(eBs)

2

? ?

(73)

In some special dimensions Eq.(73) simpli?es. Inserting the Eq.(73) into Eq.(71) we obtain the ?nal expression of the e?ective potential of the SUSY NJL model. Therefore the e?ective potential for D = 4 reads Vef f (σ ) = ds exp(?sσ 2 )eBs coth(eBs) 1/Λ2 s3 ∞ ∞ dsdt R ? 2 3 192π 1/Λ2 1/Λ2 (s + t) (1 + eBt coth(eBs))2 × exp[?(s + t)σ 2 ][eBt(eBt + eBs) +2(eBt + 3eBs) coth(eBs) + 2eBt(eBs ? eBt) coth2 (eBs)] ∞ ds 1 eBs ? 2 exp ?sσ 2 3 2 8π 1/Λ s sinh(eBs) ∞ ∞ R dsdt + 48π 2 1/Λ2 1/Λ2 (s + t)2 (1 + eBt coth(eBs))


1 2 1 ρ + 2 2λ 8π

17

×

eB 4 1 2t exp[?(s + t)σ 2 ] ? ? eBs coth(eBs) sinh(eBs) 3 6 s+t 4 2t 1 + ? ? eB coth(eBs) 6s 3 1 + eBt coth(eBs)
?
2

t +2 ? s+t +

eBt coth(eBs) + 1 + eBt coth(eBs)

2

? ?

1 (eBs)2 coth2 (eBs) + 1 + 4eBs coth(eBs) 3s t t × + 3(ξ1 + ξ2 ) , s + t 1 + eBt coth(eBs)

(74)

where we regularize the e?ective potential by introducing the proper-time cut-o? Λ. The e?ective potential (74) reproduces the expression for Vef f (σ ) in Ref.[23] at the limit B → 0. The proper-time integrations in Eq.(74) are numerically performed below. For D = 3 the e?ective potential reads Vef f (σ ) = ? R 144π 3/2 σ2 1 + 3/2 2λ 4π
∞ 1/Λ2 ∞ 1/Λ2

ds exp(?sσ 2 )eBs coth(eBs) 1/Λ2 s5/2 dsdt 5 / 2 (t + s) (1 + eBt coth(eBs))2


× exp[?(t + s)σ 2 ] 2eBt(eBt + eBs) +(9eBs + 5eBt) coth(eBs) + eBt(eBs ? 3eBt) coth2 (eBs) ?
∞ 1 ds eBs exp(?sσ 2 ) 3 / 2 5 / 2 2 4π sinh(eBs) 1/Λ s ∞ ∞ dsdt R + 24π 3/2 1/Λ2 1/Λ2 (t + s)3/2 (1 + eBt coth(eBs)) eB 5 t 5 2eBt coth(eBs) × exp[?(s + t)σ 2 ] ? ? sinh(eBs) 3 s + t 3 1 + eBt coth(eBs) 2

2 8 eBt coth(eBs) t + + s+t 3 1 + eBt coth(eBs) 4 t eBt coth(eBs) + + 3(ξ1 + ξ2 ) . 3 s + t 1 + eBt coth(eBs)

(75)

Since the contribution of scalar ?elds and fermion ?elds are canceled in a supersymmetric theory only a symmetric phase is realized for R = B = 0. The external gravitational and magnetic ?elds considered here break the supersymmetry of the theory. Thus we expect that the chiral symmetry may be broken down. Using Eq.(74) we numerically calculate the e?ective potential of the SUSY NJL model for D = 4 to study the phase structure of the theory. 18

Vef f (σ )/?4 0.004 0.003 0.002 0.001 0 -0.001 -0.002 -0.003 -0.004 0 0.05 (a) Vef f Vef f (σ )/?4 0.004
eB = 15?2
(1) (0)

(0)

Vef f (σ )/?4 0.012

(1)

eB =0

0.01 0.008 0.006 0.004 0.002
eB = 15?2 eB = 0 eB = 15?2

0 0.2 0 (b) Vef f
(1)

0.1 0.15 σ/? for R = 0.

0.4 0.6 0.8 σ/? for R = 0 and ξ1 + ξ2 = 0.

0.2

0.002 0 -0.002 -0.004
eB = 0

-0.006 -0.008 0 0.4 0.6 0.8 σ/? for R = 0 and ξ1 + ξ2 = 1.
(0) (1)

0.2

(c) Vef f

(1)

Figure 3: The behavior of the e?ective potential Vef f and Vef f are shown with the varying magnetic ?eld eB (= 0, 5?2, 10?2 , 15?2) for ?xed λ(= 1/2.5), and ?xed Λ(= 10?) in four dimensions. ? is an arbitrary mass scale. We normalize that Vef f (0) = 0.

19

Vef f (σ )/?4 0.025 0.02 0.015 0.01 0.005 0 0 (a) Vef f 0.05
eB = 15?2 eB = 0

Vef f (σ )/?4 0.025 0.02 0.015 0.01 0.005 0 0 (b) Vef f 0.05
eB = 15?2 eB = 0

0.1 0.15 0.2 σ/? for R = 5?2 and ξ1 + ξ2 = 0.

0.1 0.15 0.2 σ/? for R = 5?2 and ξ1 + ξ2 = 1.

Vef f (σ )/?4 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 0 (c) Vef f
R=0 R = ?5?2 R = 5?2 R = 10?2

Vef f (σ )/?4 0.006 0.004 0.002 0 -0.002 -0.004 -0.006 0 (d) Vef f
R = 5?2 R=0 R = ?5?2 R = 10?2

0.1 0.15 0.2 σ/? for eB = 15?2 and ξ1 + ξ2 = 0.

0.05

0.1 0.15 0.2 σ/? for eB = 15?2 and ξ1 + ξ2 = 1.

0.05

Figure 4: The behavior of the e?ective potential Vef f are shown with the varying B or R for ?xed λ(= 1/2.5), and ?xed Λ(= 10?) in four dimensions.

20

In Figs.3 and 4 we illustrate the typical behaviors of the e?ective potential for D = 4 in the SUSY NJL model. In Fig.3 we draw the terms independent of R, V (0) , and the linear curvature correction to the e?ective potential, V (1) . The linear curvature correction strongly depends on the coupling constant ξ1 + ξ2 . There are large cancellations between the corrections of fermion and scalar ?elds. Thus the terms of the linear curvature correction in the SUSY NJL model are signi?cally smaller than that in the NJL model. As is clearly seen in Fig.4 the chiral symmetry is broken down for a su?ciently large magnetic ?eld and/or a negative curvature for R = 5?2 or B = 15?2 . The external curvature has the opposite e?ects in the case of large ξ1 + ξ2 (> 1/2) for a small B and/or large σ (See Fig.3.(c)).

5

Conclusion

In the present paper we discussed the combined e?ect of the gravitational and magnetic ?elds to the chiral symmetry breaking in NJL and SUSY NJL models. Chiral symmetry is broken at non- zero and non- positive curvature. On the same time, positive curvature acts against chiral symmetry breaking. Nevertheless, the magnetic ?eld e?ects may be signi?cally stronger in realistic situations corresponding to early Universe with primordial magnetic ?elds. It should be noted that the e?ective potential calculated in the present paper depends on how to introduce cut-o? parameter in the proper-time integral, though we do not develop it any further here. Using the results presented in Ref. [5] it is not di?cult to calculate the e?ective potential in above two models exactly on the constant curvature spacetimes (DeSitter or anti- DeSitter, for example [5, 7]). Such calculation shows that curvature e?ects estimated in this work are taken into account qualitatively correctly (one can consider then very strong curvature as well). However it is not clear how to make such calculation exactly both on curvature and on magnetic ?eld. The only possibility to do so is to work on some gravitational- magnetic background where exact solutions of ?eld equations are known (for example, on conformal spacetime with magnetic ?eld). Another interesting proposal could be to study the dynamical symmetry breaking on electromagnetic- gravitational background representing a combination of constant electromagnetic ?eld with gravitational wave (it is known the e?ective Lagrangian in pure electromagnetic or gravitational wave is trivial and no e?ect is expected). Such background may be roughly considered as signal which comes from strongly- graviting object with strong magnetic ?eld. Then one may speculate on the possible use of the dynamical symmetry breaking in gravitational waves detectors. We thank I. L. Buchbinder, S. J. Gates and T. Muta for helpful discussions. T. I. was supported in part by Monbusho Grant-in-Aid for Scienti?c Research Fellowship, No.2616. S. D. O. thanks COLCIENCIES (Colombis) and JSPS (Japan) for partial support of this 21

work. This work (Yu. I. Sh. and S. D. O.) was supported in part by Ministerio de Educacion y Cultura (Spain). Yu. I. Sh. also expresses his deep gratitude to A. Letwin and R. Patov for their kind support.

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