On the behavior of symmetry and phase transitions in a strong electromagnetic field

ANNALS

OF PHYSICS

208, 470508

(1991)

On the Behavior of Symmetry and Phase Transitions a Strong Electromagnetic Field HIDEO SUGANUMA Depurtmenr

sf‘ Physics,

Received

April

AND TOSHITAKA Kwto

University,

3, 1990: revised

Kyolo

October

in

TATSUMI 606, Jupan 15, 1990

Symmetry behavior and phase transitions in a strong electromagnetic field are investigated by using the one-loop effective potential. The one-loop effective potential in a uniform external field is constructed by virtue of the j-function regularization method and general discussions are given concerning the symmetry behavior in the strong electromagnetic field by using it. It is emphasized that nonperturbative effects about the field strength are essential in the strong electromagnetic field and they qualitatively modify the results obtained by using the perturbative formula. Furthermore, even qualitative differences between fermion and scalar contributions are seen in the strong electromagnetic field, although no qualitative differences are seen at finite temperatures. Subsequently. two typical subjects are discussed within this approach; the chiral phase transition in the linear o-model and the vanishing of the Cabibbo angle in a toy model. It is found that in both cases, the electric field restores the symmetry. while the magnetic field breaks it further; the critical electric field of the chiral phase transition is eE, z (480 MeV)’ when the magnetic field is absent, H = 0. The characteristic field strengths for the Cabibbo-angle problem are H* 2 1.5 x IO” G and E* Y 5 x 10’” G. 1-1 1991 Academic Press. Inc

1. INTRODUCTION Possibility of phase transitions in nuclear mediums or various environments, e.g., high density or high temperature, has been an important and interesting issue. In particular, symmetry restoration in the models endowed with spontaneous symmetry breaking (SSB) is interesting and has been studied by many authors under high temperature or high density [ 11. In nuclear physics, pion condensations [2], which means non-vanishing expectation value (xi) # 0 in the ground state, and the chiral phase transition [3], the change of the expectation value of quark-antiquark pairs (44) in the vacuum, are two typical examples. They have been frequently treated within the mean-field approximations’ analogous to the Ginzburg-Landau theory. An important aspect of such phase transitions is that they lead to some symmetry change of the system; pion condensations break SU( 3 ) &,, whereas the chiral phase transition restores SU(2), x SU(2), (or SU(3), x SU(3),). ’ There

are some exceptions,

where

thermal

fluctuations

470 0003-4916/91

$7.50

Copyright ( 1991 by Academic Press. Inc. All rights of rcproductmn m any form reaervrd.

become

important

141.

SYMMETRY

AND

PHASE

TRANSITIONS

471

We shall here consider another type of phase transition due to a strong electromagnetic (EM) field: the vacuum is assumed to be in the SSB phase without the EM field and an order parameter relevant to this phase transition is neutral and scalar. In this case, the order parameter cannot directly couple with the EM field in the mean-field approximation level, but does so only through quantum fluctuations (loops) of charged fields, in contrast with the Ginzburg-Landau theory. Thus vacuum polarizations are the most important ingredients in our work [S]. In this paper we only consider a constant uniform field for the external EM one. We utilize the effective potential to the one-loop order to analyze the phase transitions in a general situation, where external electric (E) and magnetic (H) fields with any strength exist at the same time. Relevant effective potential has been already given by way of the proper time method [6] or the method of Bogoliubov transformation [7, S]. In Section 2, we shall derive a similar result by another method, the c-function regularization method, which has been developed by Salam and Strathdee [9] to analyze the symmetry behavior in a strong magnetic field. We extend their method to include the electric field as well as the magnetic one. This method gives us an intuitive physical picture in deriving the effective potential, besides giving a well-defined result [lo]. It involves highly nonlinear terms about the field strength, which stem from the nonperturbative effects by the EM field, and can be expressed in a closed form in terms of gauge and Lorentz invariant quantities, J = (N’E’)/2 and ?? = E. H. We shall also see that an imaginary part, which means a pair creation, naturally arises in the effective potential. This type of a pair creation [6], which is called Schwinger’s mechanism, is a genuine nonperturbative process and has been studied in relation to various phenomena, especially the relativistic heavy-ion collisions [ 111. A real part of the effective potential is responsible to the symmetry behavior and also involves full nonperturbative effects about the field strength. We shall study these nonperturbative effects of 9 as well as 9 on the symmetry behavior in the presence of a strong EM field in some detail. For the quantity 9, little attention has been paid so far due to the following reasons: one frequently considered only special situations C!Y= 0, i.e., E = 0 or E I H, so that we can never arrive at the general case 9 # 0 by any Lorentz transformation. Most authors have used the perturbative formula only, where 9 is dominant and the contribution of %’ is the higher order effect. We, however, maintain this effect carefully throughout this paper because of its relative importance in the strong EM field. Indeed we shall see that the contribution by 3 is indispensable and even becomes comparable with the one by F in some situations. On the basis of these considerations, we shall then study some phase transitions in the EM field. One example in this context is the instability of W+ W condensates and inhomogeneous vacuum structures before the symmetry restoration in the Weinberg-Salam model [12, 131; we shall not touch on this subject here. This is also the case for the chiral phase transition, where the neutral and scalar order parameter comes out. Recently it has been demonstrated within the Nambu Jona-Lasinio model that the chiral phase transition can occur at a critical value of

472

SUGANUMA

AND

TATSUMI

the electric field, while the magnetic field, on the contrary, inhibits the phase transition by stabilizing the chirally-broken vacuum state [ 141. In their treatment, it was implicitly assumed that every hadron is made of constituent quarks and they only considered the quark loop in response to the external EM field. However, studies from the hadronic level are also indispensable, because the critical point of the chiral phase transition would be nearly equal to that of the deconfinement transition, as suggested in the lattice gauge calculations [15]. In this connection some authors have calculated hadron loop contributions by way of the o-model [16] or the chiral perturbation theory [ 173 in finite temperature or density. We shall here consider this phase transition in a strong EM field along this line. As another example of such phase transition, which would be more interesting, we bear in mind the problem of the vanishing of the Cabibbo angle in a strong magnetic field as indicated by Salam and Strathdee [lS]. They argued restoration of symmetries in the purely magnetic field by analogy with the superconductivity and indicated a possibility of the vanishing of the Cabibbo angle, which, in turn, means strangeness conservation in the weak processes in some nuclei, e.g., 35Ar, 93Nb where a strong magnetic field would be expected. In their model, the Cabibbo angle 8, is not considered to be a constant but it arises as a consequence of SSB from a scalar (Higgs) field. On the other hand, in the standard electroweak model based on the gauge group SU(2), x U( l)Y, the Cabibbo angle or, generally, the KobayashiiMaskawa matrix is regarded as constant, while it is reasonable to expect that angle 8,. may have a dynamical origin if we regard vacuum to be a physical medium [19]. In the previous paper [20], we have reexamined this problem and concluded that naive analogy with superconductivity cannot hold in this case: the Cabibbo angle grows larger in a sufficiently strong magnetic field of m lOI G, while it decreases in a weak one where we recover the Salam-Strathdee observation. Then it is interesting to see whether this conclusion should be changed in the general situation, where both magnetic and electric fields coexist, since the electric case even may become dominant in certain circumstances, e.g., in nuclei or inside hadrons [21]. In this paper we shall discuss this point. This phenomenon, if observed, not only has many consequences on physics including strangeness but also is important to get deep insight into the origin of the Cabibbo angle. Some experiments already have been done, but any signs have not been observed as yet [22]. This may be clearly tested in future experiments using hypernuclei. In Section 2 we derive the effective potential in a uniform EM field with the use of the i-function regularization method. We shall see general features of the contributions from scalar and fermion loops to the symmetry behavior separately in Section 3. Subsequently, we shall apply this approach to some specific problems: restoration of the chiral symmetry will be discussed in Section 4 and vanishing of the Cabibbo angle within a toy model a la Salam and Strathdee in Section 5. In the course of discussion on the former, we also pay attention to the role of the contributions of 9 besides the interplay of fermion and boson contributions. In relation to the latter problem, we shall reexamine their idea in some detail, including both

473

SYMMETRY AND PHASE TRANSITIONS

magnetic and electric fields. Section 6 is devoted to summary and concluding remarks. Some analyses are given in Appendix A concerning these models at high temperatures to supplement our argument. 2. EFFECTIVE POTENTIAL 2.1. C-Function Regularization Method It is useful to apply the effective potential approach in analyzing the behavior of the symmetry in external fields; unlike the superconductor, an order parameter 4 cannot directly couple with the EM field, but does so only through the quantum fluctuations given by various loops. The one-loop effective potential (or the effective action) is given by the logarithm of a held-dependent operator H, e.g., the KleinGordon operator for a boson field or the Dirac operator for a fermion field. We, in the following, use the i-function regularization method [lo] to evaluate the effective potential. In the [-function regularization method, the logarithm of an operator H is represented in an integral form, lnH=!ir+n[

-$HPv]=!%[

--${&I:

dssvP1exp(-Hs)}]

(2.1)

and one regularizes the functional determinant by way of lndetH=trlnH=l%[

---$cH(v)],

(2.2)

where the [ function associated with H is defined by ~,(v,=lj’dss”~ r(v) 0

1tr exp( - Hs).

The integral [,, diverges unless v > 2 in most cases(see Eq. (2.29)); however, it can be analytically continued to the region Re v d 2 to be regular at v = 0. We first consider the one-loop contribution of a fermion field to the effective potential in a uniform EM field under the general situation,’ 9 = E. H # 0. It is given by -

s

dpfir$” =

I

&V:q;f;;

=ihTrlnf-iD-‘($

E, H)),

(2.4)

where the matrix elements of the Dirac operator DP1 are (xl D-’

Iy)=

-i[i$--e&M+iE]fi(X-y)

z Conversely, when $4= 0 we can choose a frame where E= 0 (.S < 0) or case has been already discussed in Ref. 191.

(2.5) H = 0 (3

> 0).

The former

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SUGANUMA

AND

TATSUMI

for the fermion field with charge e( > 0) and mass M, which may be a function of an order parameter 6 and coupling constants3 and Tr indicates the complete diagonal sum over the Dirac indices and the continuous space-time variables.4 We can rewrite Eq. (2.4) into the form

by virtue of the property

such that

Trln(&-++s)=~Trln(-fl’+M2-iiE), with pP E i 8’ and LIP = p” - eAp satisfying -ieF@‘. Then Eq. (2.6) leads to

(2.7 1 the commutation

relation

[ZZI’, ZI”] =

(2.8) by way of Eqs. (2.1) and (2.2). In this case, [ function written clss”~’ tr(xl

associated with G ~ ’ can be

exp( -G-Is)

IX),

(2.9 1

where tr only refers to the Dirac indices. It is to be noted that is prescription ensures the convergence of the integral for s -+ co. For $Y#O, we can always choose a frame in which E//H; let us consider a constant EM field, F03

=

-,?j

-F30=

F’* = -F2’

= - H

others = 0

(2.10)

with the special choice of gauge

Thereby trexp($n,,.,FY”) Therefore the integrand

= 4 cos(eHs)

(2.12)

cosh(eEs).

in Eq. (2.9) is now reduced to the form,

tr(.ul exp( -G = 4 cos(eHs)

‘3) 1.x) cosh(eEs)(.xyl

x (tzl exp( -2ieEsKy)

exp( -2ieHsK,) It:) exp(i( -M’+

Ix.v) iE)s),

’ In this paper we only deal with this type of theories. 4 Throughout this paper we use the natural units, h = c = 1, and k, = 1. except want to stress their roles.

(2.13)

in the place where

we

SYMMETRY

where two kinds of Hermite

AND

PHASE

475

TRANSITIONS

operators appear (2.14)

New canonical variables, (X, P.y) and ( Y, Pt.), are defined by JZXd7’,

&?P,=lT’;

a

Y-no,

@P&I3

>

(215) .

subject to the commutation relations, [PA., X] = [P y, Y] = -i. Thus evaluation of the effective potential is reduced to the eigenvalue problem for the operators, K, and K,, since matrix elements appeared in Eq. (2.13) can be written as the sum over eigenvalues of the operators.’ 2.2. Eigenvalue

Problem

for

Kx and K,

The operator Kx is a familiar harmonic oscillator and its eigenvalues are discrete; they consist of the well-known Landau levels. Therefore the matrix element relevant to K,y can be written (-q’

exp(-2ieHsK,)

IV>

=g,zfexp

[ -is (n+i)

(2.16)

(2eH)],

where the pre-exponential factor comes from the degeneracy of each Landau level. On the other hand, operator Ky is an inverted oscillator [23], of which eigenvalues 1, are continuous and can take values from -cc to + co, so that the sum over the eigenvalues is not so trivial. Eigenfunctions are given by linear combination of the parabolic cylinder functions; e.g., two independent real functions @‘(I*, q = t/z Y) and W(A, -4) of which the asymptotic forms are

(2.17) W(A,q+

--co)-

,

with k=(l

+,-~ni)l/Z-,-xi,

cj(A)=argr(i-iA).

These formulae are anticipated from the WKB element relevant to the operator K, reads (tzl

exp( -2ieEsK,)

I tz) = g

approximation.

(2.18) Then, the matrix

s dApzp(%) exp( - i2eEAs),

(2.19)

5 The explicit forms of 17” may be changed by a gauge transformation, which corresponds to a unitary transformation on 17”. Nevertheless eigenvalues of the operators KY and K, are unchanged. Thus the effective potential is gauge invariant.

595,208/2-I5

476

SUGANUMA

AND

TATSUMI

where factor eE/2n is the same as the Landau degeneracy one; the density of state p(j) is given by the formula [23], p(l, L)= jln

(2.20)

L+o(A),

as a result of imposing a fictitious boundary condition The A dependent component a(jb) is given as

at large L, W(A, + L) = 0.

cT(l.)=a(-A)=n-‘(LY(A) z-n

-’ Re tj(i+U).

The digamma function, I/(Z) = din r(z)/&, is meromorphic z = 0, - 1, - 2, ... and its series representation reads II/(=)= ->:‘$

f 1-L ,,=o ( n+l

(2.21 ) with simple poles at

=+n >

with y being Euler’s constant. Using Eqs. (2.21) and (2.22) we finally obtain (2.23)

besides i independent terms, which have no effect on the effective potential Section 2.3). Thus the matrix element (2.19) can be evaluated by virtue of Eq. (2.23):
It;) =gJ

z-

dA A,,gO iz +n(~~‘~,2)z

eE 1 471sinh(seE)’

(see

exp( -i2eEsi,) (2.24)

2.3. Expression for the IZffectitte Potential

By way of Eqs. (2.16) and (2.24) the c function associated with G -’ (2.9) leads to

(2.25)

From the fact that if we perform the s-integral in Eq. (2.25) over a contour C in the complex s plane, as shown in Fig. 1, it gives zero and the one along C, gives

SYMMETRY

AND

PHASE

477

TRANSITIONS

s-plane

FIG. 1. are located

Contour C in the complex s-plane. C, is a quarter at s = nni/et” (12 # 0) on the imaginary axis.

no contribution, obtain

of the circle with

a infinite

radius.

Poles

we can then change the path along the real axis into C’. Thus we

where p.v. refers to the Cauchy principal value. The integrand in each integral in Eq. (2.26) is space-time independent, so that four-dimensional volume V is solely factored out after the space-time integration. The second term, which gives the imaginary part of the effective potential, comes out of the one-half of the residues at infinite poles lying on the imaginary axis, s = - nni/eE (n = 1 to cc, ); it can be interpreted as the probability of fermionantifermion pair creation. By virtue of the property such that (2.27) if a function J’(Y) is already finite at v = 0, we find

(2.28)

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SUGANUMA

AND

TATSUMI

for the pair creation rate per unit time-volume, which is reduced to the standard form if H+ 0 [24]. It is well known that this mechanism owing to Schwinger [6] is attributed to a genuine nonperturbative effect by the electric field. The first term in Eq. (2.26) gives the real part of the effective potential, which governs the symmetry behavior in the EM field. The coefficient of ( - i)‘+ ’ V/f(v) of the first term is further decomposed into finite and infinite terms at v = 0; that is,

s =s‘XI J; s

e’EH p.v. ,z dss’~ I coth(eHs) 4rr2 dss+’

$

cot(eEs)e

msMz

coth(eHs)e-I”“*

0

+ p.v.

dss’-‘-$

(eEscot(eEs)-

l)(eHscoth(eHs)-

dss’p3 $

(eEs cot(eEs) - 1 + f (eEs)‘)ephM’

l)e-“‘fZ’Z

0

+ p.v.

-

CL, , (e-V2 -JAf~ I 0 dss‘- 12nZe .

(2.29)

The second and third terms in Eq. (2.29) are already finite at v = 0, so that we can use Eq. (2.27) again to get the effective potential. On the other hand, the first and fourth terms in Eq. (2.29), which represent purely magnetic and electric contributions, respectively, need some analytical continuation to be finite. Indeed, they can be analytically continued in the complex v plane and expressed in terms of the generalized Riemann zeta function [(z, c() and the gamma function Z-(Z): ir, s0

d.wp2 $

coth(eHs)e

s”’

M’+2eH 2eH

>

+eHM-2c’-” \‘- 1

1

(2.30)

and

-

= -gM(e-0’

-=I,

F’(v).

(2.31)

Substituting Eqs. (2.30) and (2.31) into Eq. (2.29), we finally get a finite and welldefined result.

SYMMETRY

AND

PHASE

479

TRANSITIONS

Re V

3(eE.~ cot(eEs) - 1 )(eHs coth(eHs)

- l)e--““”

+ p.v.

+ (W’ zln

M’,

which leads to the perturbative

(2.32)

form, (E2-H2)lnM2

(2.33)

in the weak-field limit,’ eE/M’ or eH/M’ 6 1 [ 181. It is to be noted that Eq. (2.32) is gauge and Lorentz invariant in spite of the appearance; in particular, it is an even function about both H and E, so that odd parity terms never exist there. 2.4. Scalar Case Next we consider mass m, given by

the one-loop contribution

I

of a scalar field with charge e and

d4.xJ’.,oop = -%trln{-ii-‘(J, = -i#z !lo-z

where the Klein-Gordon

operator

(xl A-’

Iv)=

H, E)f [( -i)F’

i_l-l(v)l,

(2.34)

A ’ is -i[Z7-m’+i&]6(.u-y)

(2.35)

with IZP = i P-eAp and tr only means the sum over the space-time variables. A parallel argument with the fermion case can be applied and here we give only the final result for the effective potential,

6 We can see the standard Cell-Mann-Low p-function [25] in the coefficient of the second term in Eq. (2.33).

coefficient

+1/127-r’

for the fermionic

QED

480

SUGANUMA

V scalar I-IOOP

AND

TATSUMI

(ln(2eH)-1)+(2eH)‘[’

-1,x

m2 + eH

)I

(eE)Z In ,n2 + 96~’

+i

z e’EH 1 C --(-I)” ,,=I 8x2 n

em’ mmZ~?6) sinh(nrcH/E)’

(2.36)

The imaginary part (the fifth term in Eq. (2.36)) also represents the pair creation and is a manifestly nonperturbative effect. The real part of the effective potential is reduced to the perturbative form Re VsCa’ar - h n24 In ~12 + $(E’--H’)lnin I-loop - 32n2 in the weak-field

limit [18].’

2.5. Effects of Higher Loops So far we have concentrated our attention on the one-loop contribution in the presence of the uniform EM field. It is worth noting again that the external EM field couples with the order parameter only through the fermion or scalar loop, so that the one-loop contribution is the leading order in our case and indispensable to discuss the symmetry behavior in the EM field; in this sense we may say that discussions within the one-loop level correspond to the classical (mean-held) approximation in cases such as the superconductivity, where external fields directly couple with the order parameters. Further, the Schwinger formula for the pair production is given as the imaginary part of the effective potential in the one-loop level, so that we can discuss the symmetry behavior, which is governed by the real part of the effective potential, and the pair creation in the same level by way of the one-loop expression. In another respect, discussion within the one-loop level becomes useful to compare the phase transitions in the strong EM field with those in high temperature, which usually have been discussed in the one-loop level. Actually we will give some discussion about this point in Appendix A. Higher-loop contributions beyond the one-loop level should be considered as higher-order corrections to the leading order one. Unfortunately, general analyses ‘The factor + 1/48x’ p-function for the scalar

in the coeffkient QED 1251.

of the second

term

in Eq. (2.37)

is also the coefficient

of

SYMMETRY

AND

PHASE

TRANSITIONS

481

could not be done for higher loops and some definite models are needed to estimate them, while the one-loop contribution can be expressed only in terms of the particle masses. Given such models, they would include other coupling constants besides the EM one, e. Therefore the loop expansion becomes a complicated one with respect to these coupling constants: each higher-loop contribution should include full-order effects with respect to them. We have already seen that even the one-loop effective potential includes full-order effects with respect to not only the change e but also to the other coupling constants through particle masses (see Eq. (2.5)). Thus the expansion parameter relevant to the loop expansion is not trivial unlike the usual perturbation scheme; it would be formidable to estimate the order of higher-loop effects systematically and to examine the validity of the loop expansion. For the fermion-loop contributions in the fermion-scalar interacting cases like the a-model (see Sections 4, 5) we can barely estimate the order of such contributions for the specified type of diagrams, the daisy or ring diagrams, which must correspond to the Hartree approximation or the large N approximation [28]. Since the higher-loop contribution is clearly small in the weak-field case, eE/M’ or eH/M’< 1, we consider the strong-field case s.t. eE/M’ or eH/M’$ 1, with M being the fermion mass. We only quote some results here leaving the detailed calculations in Appendix B. The ?I-loop effective potential will have terms of order (2.38) for the pure electric case and (2.39) for the pure magnetic case, where g denotes the fermionscalar coupling constant and m the scalar mass included in a model. In order for the loop expansion to be reliable, the expansion parameter must be smaller than one. It is sufficient for the parameter h g2& 87~m2

(2.40 )

or (2.41) with x = M/Xii2 to be sufficiently small. It then seemsreasonable to conclude that the loop expansion works even in a strong EM field as long as the values of these factors do not exceed O( 1).

482

SUGANUMA AND TATSUMI

We will see in Sections 4, 5 that the critical strengths of the phase transitions satisfy this condition except in the very vicinity of the critical point. In the remaining part of this paper, we will mainly discussthe one-loop contribution. 3. NONPERTURBATIVE

EFFECTS OF F

AND $9 ON THE SYMMETRY BEHAVIOR

3.1. General Features Here we consider general features about nonperturbative effects of the electromagnetic field on symmetry behavior. In the previous section we have derived the effective potentials, which involve essentially full-order effects about the field strength. These formulae (2.32) and (2.36) are already finite and can be renormalized by relevant conditions. We write these one-loop contributions in the form, v:S,,td,

H, E) = v:!;:,,($,

(i = fermion or scalar), (3.1)

+ vk’;($, K El

where Vy!{&,(&) is the usua1 one-loop contribution without any external field [24]. Total effective potential can be obtained by adding the classical one,” V:‘(d): V;;‘,(q%H, E) = V;;’ + V:!;&, + Vl-‘;.

(3.2)

Then it is enough to renormalize Vii, without any external field,’ since V1.2 involves no more divergences besidesirrelevant divergences independent of 4; the finite expressions for VFA are obtained in the Lorentz invariant manner such as V fermion em

_

- -&(;ylZ-&2)lnM2(fj) h fjg7P.v.

x As - TM= so

Te

x 2s coth(X”s) .bs cot(bs) -

1 + f (X2-

II

B2)s2

(3.3)

II

(3.4)

for the fermion case, and

X s In this paper $6 = fj, # 0. 9 We, hereafter,

we assume that consider

only

2 s

.-- 8s

sinh(X’“s) sin(&s) SSB occurs even in the classical level; i.e., V$‘(d) the real part

of the effective

potential.

has its minimum

at

SYMMETRY

AND

PHASE

483

TRANSITIONS

for the scalar case, where 6=rJ~,

x=rJ~,

(3.5 1

satisfying the relations’” ;(&L(p)+y,

~Y”8=er’

Therefore we take the following renormalization ova!;:,,

19’1.

conditions,

= ~2v~!;:,,td,

0’4

I= 6,

The value of d in the EM field, J,,, condition for Vi%,

(3.6)

= 0.

ap

(3.7)

J=i,

should be determined

by the minimum

that we can roughly see whether symmetry tends to be restored or further broken in response to an external field, by examining the derivative, dVr’/d$, which we call V-derivative: If it is positive, the symmetry tends to be restored and conversely if it is negative, the symmetry tends to be further broken, as depicted schematically in Fig. 2. SO

a

b

FIG. 2. Modification of ~V,,,/C?$ and the following change of the potential minimum due to the external EM field. (a) When V-derivative is positive, total derivative, ?V,,,/&$. is raised and the potential minimum is decreased. and (b) vice versa for the negative V-derivative. ” In the Lorentz

frame

s.t. H//E,

they are reduced

to simple

forms,

X = e IHI, 8 = E 1~1,

respectively.

484

SUGANUMA

AND

For the fermion case, V-derivative

TATSUMI

reads (3.8)

with a characteristic

function,

x [(s cos 0) coth(s cos 0). (s sin 0) cot(s sin 6) - 11, where -yf. = M( $)/( %” + &‘)‘I4

(3.9)

and

cos 0 = 2?/(3P

+ 82)‘/‘2,

sin 0 = &/(P2

+ 8”)“2.

(3.10)

A parameter 8 represents a mixing rate of H and E, e.g.: (i) ,Y # 0, R = 0, i.e., P>O, ??=O for 0=0; (ii) P=&, i.e., 4=0, %#Ofor 0=7c/4; (iii) X=0, d#O, i.e., 9 < 0, 9 = 0 for 0 = 7-r/2. On the other hand, for the scalar field, (3.11) and f=‘ar(&

s cos 8 s sin 9 -1 sinh(s cos 0) ’ sin(s sin 8)

Q) = p.v.

1

(3.12)

with X, = m($)/(X + 8’)‘14. Therefore, if M(J) or nz($) is a monotonous function about 4 as in many cases, e.g., d4-theory, o-model (see Sections 4 and 5), the symmetry behavior in response to the EM field is essentially controlled by the characteristic functions given in Eqs. (3.9) and (3.12). In the special cases, purely magnetic (0 = 0) and purely electric (0 = n/2) cases, the above characteristic functions (3.9) (3.12) can be given by analytic expressions (see Appendix C). 3.2. Fernzion Case Global shape and its contour map of Ffermlo” are shown in Fig. 3 as a function of x1. and 8. In the weak-field limit s.t. s1 $ 1, the function behaves like - ( l/3.$) cos(28), so that Rrmian ~Verll ?f

_ -

h

dM’2J --

241r’ &j

M”

(3.13)

Therefore V-derivative only depends on 9 and this result is also given by virtue of the perturbative formula (2.33). Generally the boundary of the same-sign region of prmion ,F fermion= 0 line, deviates from the straight line .F = 0 due to the nonperturbative effects of Y; in the fermion case it curves to the 9 > 0 region.

SYMMETRY

AND

PHASE

485

TRANSITIONS

In the strong-field limit s.t. xf < 1, F’ferm’on(xl, 0) 2: Ffermlo”(O, 8) and the RHS can be integrated analytically: Fferm“‘“(O, 8) = -27~

t ez,,=,,!,[ II = I

, -i]

sin 8.

(3.14)

This is a monotonous function about the ratio B/s’ and takes - x at 8 = 0 and the value of 0 at tan 0 rr 0.28. Substituting Eq. (3.14) into Eq. (3.8), we obtain 1 -1-i

(3.15)

I

a

map. FIG. 3. (a) Global shape of the characteristic function, Fferm”‘” Is,, 0) and (b) its contour Positive-valued region of Fferm’“” (u,. H) increases for small I,. corresponding to the strong EM field. For constant 9, x, varies with Y along the path shown by the dashed line.

486

SUGANUMAAND

TATSUMI

Here we can see that the RHS in Eq. (3.15) is a complicated function of 9 and 99, which comes out of the nonperturbative nature of the effective potential, and effects of % are never negligible there. Moreover, V-derivative goes to - cc at d = 0 and changes its sign at &/X N 0.28. Thus the symmetry behavior depends not only on the field strength but also on the mixing rate of 2 and 6. We can clearly see the role of Y and the interplay between 9 and 9 from a different point of view. For 9 > 0 ( < 0) fixed, -Y.,-varies along the curve, M l/4 ? x= (2 pq)‘,4 lcos w

(3.16)

as shown in Fig. 3, and it takes a maximum at 8 = 0 (6’= n/2), i.e., 9 = 0. Then the value of I;feermlon varies along the path and even changes its sign for 9 > 0 at the point where the path comes across the line s.t. Frermion = 0. Thus the symmetry behavior is qualitatively modified by inclusion of the nonperturbative effects of 9. 3.3. Scalar Case Similarly, for the charged scalar field, global shape and its contour map of Fscalar are shown in Fig. 4 as a function of X, and 8. Several features of these figures are summarized in the following: in the weak-field limit s.t. x,~ti 1, Fscalarhas an approximate form, Fscalar5 - (6x* ) ~ ’ cos 28, so that the V-derivative reads from Eq. (3.11), (3.17) which recovers the perturbative result (cf. Eq. (2.37)). Thus the scalar-field contribution is qualitatively the same as the fermion one (cf. Eq. (3.13)). In the strong-field limit s.t. x,~6 1, the characteristic function Fscalarasymptotically approaches % (-1)” Fsca’ar(O,19)= 71sin 0 C (3.18) ,)= 1smh(nn tan 19)’ Here Fscalar (O,(3) has a unique sign ( 60) and takes the minimum value of -In 2 at 8 = 0 and zero only at 8 = n/2 (F < 0, 9 = 0). Therefore, the symmetry behavior is determined irrespective of the mixing rate of .# and G. In this case the boundary of the same-sign region of the Fscalarcurves to the 9 < 0 region, so that Fscalarchanges its sign, for 9 < 0, at the boundary given by Eq. (3.16) with m in place of M.

4. CHIRAL SYMMETRY IN THE LINEAR O-MODEL We study the behavior of the chiral symmetry in an external strong EM field by the use of the linear a-model in this section. As is well known, spontaneous

SYMMETRY

AND

PHASE

487

TRANSITIONS

breaking of the chiral symmetry is one of the remarkable features in low energy regime of QCD, and the linear o-model is an effective model of hadrons and includes the mechanism of the spontaneous breaking of the chiral symmetry. An advantageous point of the linear a-model is that the behavior of the chiral symmetry in various environments [ 161 can be easily seen because the order parameter of the symmetry is introduced explicitly in this model, unlike, e.g., the NambuJonaaLasinio model with such a composite order parameter as (44). The Lagrangian of the linear a-model is given by o4P=(kineticterms)-g~,~(cr+~,5”71”)~N--~~((~’+712)--~

;-’ + co,

(4. 1)

a

FIG. 4. (a) Global shape of F”“‘“’ (r,, 0) and (b) its contour map. The positive-valued region of Fsca’ar(.yy, 0) decreases for small I,. and all the values of Fsca’ar(y, 8) are non-positive for X, = 0; IF*ca’ar(.~~. Q)( has a maximum value of In 2 at I, =0, O=O. The dashed line has the same meaning as in Fig. 3.

595.208'2-16

488

SUGANUMA

AND

TATSUMI

where $,,,, 0, rra denote nucleon isodoublets, a-meson, pion isovector fields, respectively; g is the nNN coupling constant with g 2: 10 and rr,, represents the spontaneous breaking of the chiral symmetry in the physical vacuum, go-f, 1: 100 MeV. The last term in Eq. (4.1), err, is the explicitly symmetrybreaking part (P&,). Only c-meson has non-vanishing vacuum expectation value, owing to YsB, which in turn generates a small pion mass, rnz 2: c/a,, and hence c rr m~a, ‘v (140 MeV)’ (100 MeV) 10.25 fm m3. The classical effective potential is expressed in terms of 4 = ( CJ), Timode (6) = $ (($2- ($2 - c$,

(4.2 1

and masses of nucleon, o-meson, and pion are functions of 6 such as m,(8) = id

m:(d) = E,(3$” - CT:),

m:(d) = %(cj’ - ~7:)

(4.3 1

at the tree level. Since the contribution of P’s, is small in the SU(2), sector, 4,. which minimizes Vz;mode’(6) is almost equal to CT~,6,. N cO. Thus nucleon mass is mN 21 gcr, Y 1 GeV, and o-meson mass squared rni ‘v 2ioi in the physical vacuum without external fields. Thereby one finds i. N 25 for m, = 700 MeV. When the system is put in a uniform EM field, the effective potential at the oneloop level is

The second term in RHS of Eq. (4.4) is the effective potential in the case withou t external fields with 6,. as a renormalization point (see Eq. (3.7)),

(4.5)

where subscript (i) denotes species of particles in consideration, i.e., nucleon (N), o-meson (r~), and pion (n), S, their spin, and C,;, counts the number of internal degrees of freedom of each particle; C,,, = 8, C,,, = 1, Ctn, = 3, respectively. The third term in RHS of Eq. (4.4) is the contribution of an external EM field from Eqs. (3.3) and (3.4),

SYMMETRY

AND

PHASE

(tX)coth(tX).(rb)cot

tX

’

sinh(tX)

(ta)-

.-- tf;

sin(tg)

489

TRANSITIONS

i

1-i

1 +;t-(.iy’?-s’)

F(X2--(5i2)

II )

II (4.6)

to which only charged fields, protons, and rc”, contribute. Furthermore, V-derivative, in this case, reads

where x,,, = nrN($)/(X2 + c?‘)‘/~, X, = m,(J)/(X’ + b’)“‘. Unfortunately, there are some difficulties in calculating the effective potential: nr,(d) becomes imaginary and the effective potential is not well defined in the region 141< crO, and similar difficulty occurs for m,(J) in the region 141CO,/,/?. Such difficulties always appear in the case that an effective potential has a nonconvex region, e.g., the 4” theory, Higgs sector of the Weinberg-Salam model [26]. Although a few methods have been proposed in order to avoid these difficulties, they have involved another difficulty; e.g., the Gaussian approximation method, which has succeeded in solving the imaginar)) mass problem for scalar theories, faces what is called the trivialit)> in the 1 + 3 dimension [27]. It is quite desirable to find some methods to evaluate the effective potential for the whole region of 6. In the following we consider the phase transition in two ways around the difficulties. Before doing this, we disregard the contribution of V;:z,“z”‘($), and concentrate our attention on Vzzode’ (& X”, (si), because quantum fluctuation effects should be negligibly small compared with the classical part of the effective potential in the case without external fields, as far as the loop-expansion scheme is reliable. (I)

It is to be noticed that the scalar characteristic function has a bound, Q)l d In 2 over all values of X, (see Fig. 4). Thus the contribution of a charged pion is negligible in Eq. (4.7) compared with that of a proton when IF fcrmlo”(.~N, e)] B (&‘2g’) In 2( = 0.087) is satisfied; this condition is translated into sN < U( 1) (see Fig. 3). Since this condition is almost satisfied in the strong field where the chiral phase transition may occur, we can neglect the pion contribution. Then the effective potential Vz,$“ode’(6. N, 8) can be written with only the proton IF”“‘“‘(s,,

490

SUGANUMA

AND

TATSUMI

degree of freedom, and one can evaluate the effective potential for the whole region of $ and examine their minima. The change of the effective potential due to 6 for X = 0 is shown in Fig. 5, from which one finds that the chiral phase transition in the external EM field is of first order. As the value of ,X“ varies, the maximum at $= 0 grows higher, while the depth of the minimum is almost unchanged. The corresponding phase diagram is shown in Fig. 6; the electric field contributes to the chiral restoration and the magnetic field has a tendency to break it further, on the contrary. There appear three critical fields due to the first-order phase transition: the lower critical field (x., , c$, ), where the local maximum at d = 0 turns into a local minimum, the thermodynamical one (x,, J<.), where V,,, at two local minima become equal, and the upper one (q,z, &), where the original local minimum vanishes. Therefore, in the region above the (x2, 4,z)-curve ground state is in the Wigner phase, while in the region below the (,&, , 4,,)-curve it remains in the NambuGoldstone phase. In the case X = 0, the lower critical field is I$, 21 (250 MeV)‘, the thermodynamical critical one 8‘ h (480 MeV)‘, and the upper critical one gc2 N (540 MeV)“. (II) We include the pion contribution and use the strong field approximation c281? mN> m, G (ST*+ ~?‘)l,~, i.e., .yN, Y, 2: 0, because the external field is sufficiently strong and the masses of nucleons and pions become small as 6 decreases in the critical region of the chiral phase transition. Then we obtain the following

0

2.0

4.0

6.0

1CL 8.0

10.0

LV[fm-‘1 FIG. 5. The contour map of the effective potential denotes the minimum of V,,, for each value of G.

as a function

of 6 and 4 for .X = 0. The thick

line

SYMMETRY

AND

PHASE

491

TRANSITIONS

2.5 NG phase

?f[fm

0

5

2.0 FIG. 6. The phase diagram thermodynamical one and (.x2. including pion contribution.

-‘I

4.0

6.0

in (Iv, &)-plane; (&.,, t:,) is the lower critical field, (~<, JC) the 42) the upper one. The dashed line denotes (&,, J<,) for the case

condition to determine the critical strength of (x,, , gCI), in which the local minimum near fj = 0 turns into a local maximum,

which gives the lower critical field if the phase transition is of first order. It is to be noted that Fscalar (0, 0) is always negative, while 8’fermio”(0,0) can take a negative or positive value depending on 6, Frermi””(0, 6) > 0 for 0 > 0.27. Thus fermion and scalar loops play roles opposite each other in the phase transition. The critical strength in (X, 8) plane is expressed by the dashed line in Fig. 6, where one finds that the pion contribution is rather smaller than the proton one, and does not qualitatively change the results of the case without the pion contribution. Recently it has been indicated within the Nambu-JonaaLasinio model that the chiral phase transition may occur at a critical value of the electric field, eE,. = (270 MeV)‘, and the magnetic field has a tendency to break it further on the contrary [14]. They, however, did not take account of a distinction of the U, d-quark with different electric changes. Our results are qualitatively the same as theirs except for the possibility of a first-order phase transition.

492

SUGANUMA 5.

“CABIBBO

AND

ANGLE”

TATSUMI IN A

TOY MODEL

It is a long standing issue whether the Cabibbo angle vanishes and strangeness is conserved in the weak interactions in such a strong EM field as is expected in some nuclei or hadrons, since Salam and Strathdee first indicated its possibility in the strong magnetic field by analogy with superconductivity [18]. In their model. Cabibbo angle 8,. is not considered to be a constant but arisesas a consequenceof SSB due to a scalar (Higgs) field. In this section we reexamined this problem in the general situation, where both magnetic and electric fields coexist, 3 = E. H # 0 (for which situation we can choose a frame s.t. E//H).

5.1. A Model oj’ “Cabibbo Angle” We take a toy model for the sake of simplicity, Ip = kinetic terms - $ (d’ - $f)Z - (a,,., where n,,., s,,. are gauge eigenstatesand 4 is a neutral and scalar field having nonvanishing vacuum expectation value, 4 = (4 ). This Lagrangian can be regarded as a simplified version, with the essentialfeatures of the one in Ref. [lS] and may be also interpreted as a model Lagrangian for the generalized WeinberggSalam model with two kinds of Higgs doublets (b would be another doublet besidesthe standard one). Nondiagonal elements in the massmatrix of quarks are generated by 6, and this matrix can be diagonalized by orthogonal matrices U,,, to find the masseigenvalues depending on 6, M,,,,, = iJ(m,

+ ~2,)~ + g2- d2 + 4 J(m,

-m,)’

+ gf+ qJ2

(5.2)

with g+ = g, f gl. Then the eigenstates are written as

(5.3) where we have the two mixing angles Q,,,(d), H,,(d)=:

i

tan-‘-

g+d

m,-m,

Ttan

, ~ g-4

m,+m,

I

(5.4)

Since the weak Lagrangian should be constructed of only the left-hand component, eL can be regarded as the “Cabibbo angle” O<.Thus the “Cabibbo angle” dynamically arisesas a consequenceof SSB. In the following we set g, = g, = g for the sake of simplicity, so that (5.5)

SYMMETRY

AND

PHASE

493

TRANSITIONS

The values of the parameters are chosen such that Q,.(d = 4,) = 0.23, pLd= M,($ = $,.) = 10 MeV, p,, = M,(J = 4, ) = 210 MeV; thereby g$, 2: 0.22 fm ’ without any external field. 5.2. Results We apply the effective-potential method developed in Section 2 and 3 to analyze the behavior of symmetry in an external EM field. Total effective potential is expressed as V;‘,,“(&

H, E) = V;?“(d)

+ V,;.,,“(q?) + V,“,-“(4,

where the classical potential reads P’,s1-S (6) = (A/4)($‘tion without any external field, Vq;&,, is given by

+

H, E),

df)?. One-loop

contribu-

~d-C?)“(121n~(,,+7)

+ (d-s),

(5.6)

by way of the renormalization condition Eq. (3.7), where tti = (nrd + ~2,~)/2. Since Vf.$ has little influence on the effective potential, we can neglect its contribution in the following. The EM contribution is also given by

x (t4)coth(t9).(tZ)cot(tZ)-

+(d-s),

- where 6(s) = &(z@) with electric charge d, s quarks Eq. (3.5)). Then the V-derivative in this case reads

1 ++-8’)

11 (5.7)

e’= - 1/3e in place of e (see

494

SUGANUMA

AND

TATSUMI

with .Yd,* = Md,.s(qJ/(2P + z2p4. p((s, 0) is another characteristic function for the fermion loop and -defined by FExF~~~~~““, where Frermio”(x. 0) is given in Eq. (3.9) with 8 = arc tan(S/S). Noticing that dM,/d$> 0, we can see that symmetry behavior is controlled by the function F shown in Fig. 7. We can see that the function F is always positive (negative) for purely electric (magnetic) fields, and hence effects on the symmetry behavior are opposite for H and E. Moreover, the absolute value of P increases in strong fields and then decreases over a turning point. Hence effects on the symmetry behavior are opposite in the strong (x < 1) and weak (.u 9 I ) field limits. The results of 8,. can be obtained by way of Eqs. (5.5) and (5.8), and are shown in Fig. 8 as a function of d and 2’. We can see that 8,. has a tendency to decrease in the weak case and increase in the strong case for 2, and vice versa for 8. The values 2*, I* of the turning point from the former to the latter tendency are subject to the condition F(-u,*, e*) = F(.(sf, e*), where .x~,~= ~Ld,,,/(%?*2 + (f*2)‘,4 and 8* = arc it is realized as a result of the delicate balance the “Cabibbo” angle would vanish as a result line is shown by the thick line in Fig. 8a. ” It

(5.9)

tan(2*/8*) (dotted lines in Fig. 8a); of electric and magnetic fields. Hence, of the strong electric field; its border is also worth seeing the values of 8,

FIG. 7. Global shape of the function, F(.Y, 0). As a typical case (Y + 0). we separately for purely magnetic case (0 = 0) and electric case (0 = 42) by the thick solid lines.

” For another limit, ?3 # 0, 5 = 0, the behavior Hence we recognize the importance of Y again.

of oC is qualitatively

show F(x,

the same with the pure

0)

d case.

SYMMETRY

AND

PHASE

495

TRANSITIONS

in purely magnetic and electric cases (9 = 0, B # 0) in Fig. 8b. We find that H* 2: 1.5 x lOI G for the E=O case and E* ‘v 5 x 10” G for the H=O case, which are parameter (A, g) independent. Our result for the weak magnetic field qualitatively coincides with that of Salam and Strathdee [ 181, who used the perturbative formula Eq. (2.33). However, nonperturbative effects significantly modify it in the strong field [20] and perturbative treatment is no more verified there. The same features also holds for the electric case. E[fm-‘I-““““““““““““““‘Q,(M ~)[radl

1.2 0.0

0.05

1.0 I-0.15-

0

:

0.1

;

0.2

0.4

0.6

a

1.0

0.8

1.2

IFt[fm-“1

c-

. 0.2

--

O.l-

‘\.-’

-i

---I-

x/g4

= 0.3

-

x/g4=

0.5

------

x/g4=

1.0

1.0

IE’I IEI [lO1’G]

‘\

0

IH*I

l

IHI [lO1’G]

b FIG. 8. (a) The contour map of 0, in the (2, 8)-plane for i,‘g’ = 1.0: the thick line denotes the critical one where 0, vanished. The dotted line denotes (H*, E*). which is independent of the parameter set (i. g). (b) The value of 8, in the typical case (27 = 0, .9 #O): purely magnetic case [H* z 1.5 x lOI G] (right half) or purely electric case [Et z 5 x 10” G] (left half). The dash-dotted line. solid line and dashed line correspond to the cases for E./g” = 0.3, 0.5, 1.0, respectively.

496

SUGANUMA AND TATSUMI

6. SUMMARY AND CONCLUDING

REMARKS

We have discussedthe symmetry behavior in the strong EM field with emphasis on the nonperturbative or nonlinear effects of gauge and Lorentz invariants, 9 and 9. To this end, we first have provided the effective potential by way of the i-function regularization method, which gives an intuitive physical picture for the effective potential, besides naturally giving a gauge and Lorentz invariant result: Landau levels (in the harmonic oscillator potential) for the magnetic field and continuum states (in the inoerse harmonic oscillator potential) for the electric field contribute to it. Subsequently we have investigated general features of the effective potential on the symmetry behavior in detail, which are specified by the characteristic functions for fermion and scalar loops. We have seen that nonperturbative effects about the field strength are essential in the strong EM field, where they qualitatively modify the results given by using the perturbative formulae. We have also seen that the effects of 9, which play a minor role in the weak-field case, are indispensable and even comparable with those of 6. On the basis of above results, issueson the chiral symmetry and “Cabibbo” angle in the EM field have been discussed as typical examples. The former has been examined by the use of the o-model and we have found that the electric field restores the chiral symmetry whereas the magnetic field breaks it further: chiral symmetry will be restored at sufficiently strong electric field irrespective of the magnetic field, eE,. z (480 MeV)’ for H= 0. We have also pointed out the possibility of a first-order phase transition. The issue of the Cabibbo angle vanishing has been investigated within a toy model a la Salam and Strathdee. The symmetry behavior in this model is somewhat complicated. We have found that the naive analogy with the superconductivity cannot hold for the purely magnetic caseand the perturbative treatment of the external field are not verified; indeed %( has a tendency to decrease in the weak field, whereas it increases in the strong field where nonperturbative effects are essential. On the other hand, the electric field works to break the symmetry in the weak case and then restore it in the strong case, i.e., the “Cabibbo angle” vanishes in a strong electric field. Characteristic field strengths (defined in Eq. (5.9)) are independent of the parameters included in this model; H* = 1.5 x 10L9G for the E= 0 case and E* = 5 x lOI G for the H = 0 case. However, it is necessary to construct a realistic model to get the critical value where the Cabibbo angle vanishes. In relation with present problem it is also interesting to compare our observation with that at finite temperature. Generally every symmetry with SSB should be restored at a sufficiently high temperature: some analyses within models treated in this paper are given separately in Appendix A. One finds the chiral phase transition may be of first order and the chiral symmetry is to be restored at T z 212 MeV by the use of the a-model with the same parameters as Section 4. One also finds the symmetry behavior at finite temperature is qualitatively analogous to that in the strong electric field for the model a la Salam and Strathdee; the symmetry breaking

497

SYMMETRY AND PHASETRANSITIONS

is enhanced at low temperature, while the broken symmetry tends to be restored at sufficiently high temperature. The characteristic temperature, a turning point from the former to the latter, is independent of the parameters (EL,8) and T* N 49.4 MeV. Finally it is worth mentioning that our approach may also be applicable for problems in the Yang-Mills field [29, 301 instead of the EM field as an external field. This is a future problem.

APPENDIX A.I.

A: THE SYMMETRY BEHAVIOR AT HIGH TEMPERATURE

The Linear o-Model

We study the behavior of the chiral symmetry at finite temperature by the use of the linear o-model with the same parameters as those in Section 4. The one-loop effective potential at finite temperature is

where Vz;mode’( $) and V ;:,t,“z”‘( 4) are the same form as those in Section 4; the last term is the contribution due to the temperature, (4, T)=&jx Fmode’

0

dkk’x

(-1)‘“‘(2S,

x(2Z,+l)ln[l-(-l)‘“‘exp(-w,/T)]

+ 1) (A21

with oi = dm, which involves a sum over nucleon, anti-nucleon, 0 meson, and pion; Si and 1; denote their spin and isospin, respectively. One can also compare the contributions of particles to VTmode’(& T) by evaluating its derivative following the argument given in Section 3,

(A3) with si =nz,(&/T. Here the function F~m’o”csca’ar’(.~)in Eq. (A3) is another characteristic function of the fermion (scalar) at finite temperature,

where ffermiontscak)(k,

_Y) corresponds to the statistical distribution factor for the fermion (scalar) case,~rm’o” (R, x) = l/exp( -6) + 1 and f”‘“‘“‘(k, x) = l/exp( -0) - 1

498

SUGANUMA

FIG.

9.

The shapes of the characteristic

AND

function

TATSUMI

at finite

temperature,

F~~mL“n(.~),

Fyldr(.y)

with W = Jm. The shapesof F~“‘““(,~,-) and FFlar(xs) are depicted in Fig. 9. Since both functions FFlar(x) and Fymi”” (x) are always positive, and decreasing functions of X, they contribute similarly to the restoration of the chiral symmetry. Here we face the imaginary mass problem again; massesof pion and o-meson become imaginary for the small values of d. In order to avoid these difficulties, we disregard the contribution of P’f:~$“($), and consider the phase transition in two ways, similarly to Section 4. (I) Since the contribution of the nucleon would be larger than that of a-meson and/or pion, for g ‘v 10, A 2: 25, in the vicinity of the critical temperature, we first neglect the contributions of mesons, as in Section 4. Then the effective potential can be evaluated for the whole values of 4, and its minimum at the finite temperature, Jr, can be obtained. The chiral phase transition at high temperature is of first order; the change of 4, about T is shown in Fig. 10, and one finds the lower critical temperature to be T,., 2 87 MeV, the thermodynamical one T, 2: 182 MeV, and the upper one Tc2 N 212 MeV. (II) Second, we include meson contributions. Near the critical temperature of the chiral restoration, where nucleon, o-meson, and pion become almost massless, i.e., nz,,,, m,, mrr 4 T, thereby FF’ar(,~,) z FFlar(0) = 7?/6, F~mlo”(~~,) E F~mlon(0) = n2/12. Then the critical temperature T,, , in which the local minimum near 4 = 0 turns into a local maximum, is determined by the formula [28],

SYMMETRY

0.2

PHASE

499

TRANSITIONS

-

0

FIG. 10. The thermodynamical

AND

411

change of 4, about one, T, L 182 MeV;

T: r,, is the lower critical temperature. and T,, the upper one, Tz z 212 MeV.

r,, 4 87 MeV;

T, the

by way of Eq. (A3); numerically T,., z 74 MeV. This corresponds to the lower critical temperature if the phase transition is of first order. Thus, the critical temperature is lowered by including the contributions of mesons. This critical temperature corresponds to that given by Minich and Mohan [16]; they obtained T,. z 80 MeV in the framework of the linear a-model with the high temperature approximation. It, however, is to be mentioned that recently Gerber and Leutweyler [ 171 studied the quark condensate at finite temperature by the use of the partition function in the hadron phase, in which the pion contribution dominates. They also found the phase transition being of second order with T, 1 190 MeV. These results seems somewhat different from ours. A.2. “Cabibbo Angle” in a Toy Model We also see the behavior of “Cabibbo” angle 0, at finite temperature by the use of the Salam-Strathdee model, treated in Section 5. The total effective potential at finite temperature is expressed as (A61 where P’,“,-“(d) and P’;.&(d) are the same as in Section 5, and the last term denotes the contribution due to the temperature, V”,-“(6,

T)=

-??

7cz /OY dkk’ln[l

+(s++d).

+exp(-J-/T)] (A7)

500

SUGANUMA

Here the derivative of k’.;-“($, sv;.-

“(q$ T) afj

AND

TATSUMI

T) reads

4h dM,.(qJ) =7T[x,F=‘mlo”(,Y.,)-(S~d)], ndd

(A8)

with x,,,~ = M,$,,/T. Note that s, d-quark make opposite contributions each other, as in Section 5. The function xF~~~“” (x), which can be read from Fig. 9, increases at low temperature and then decreases at high temperature. Since both of .Y,,~ have large values at low temperature, one finds the relation: .x,sF~mion(x,,) < and thereby aV;PS($, T)/i$< 0. Therefore the symmetry breaking -dY Frermio”(xd), T is enhanced, and eC has a tendency to increase at low temperature. On the contrary, .Y,,~ being small at high temperature, one finds the opposite relations to the above )idFk+mmn (x,), aP’FP”(& T)/d$>O. Thus the broken symcase: x,F kpnion(X,y), metry has a tendency to be restored, and 8, tends to decrease at high temperature. The behavior of Q,.(T) about the temperature is shown in Fig. 11. One can see that 0,. tends to increase at low temperature, while it decreases at high temperature; it finally vanishes at some critical temperature, T,., which depends on the parameters (I., g). One also finds T,. e 74.6 MeV, 81.4 MeV, 93.8 MeV for A/g4 = 0.3, 0.5, 1.0, respectively. The values T* of the turning point from the former to the latter are subject to the condition, .Y,*F~~~~“(.x,~) = .~yd*F~~~~“(x~), where .u:,, = pdh/T*. It is worth noting that T* is independent of the parameters (A, g); T* E 49.4 MeV.

0

20

40

60

80

100

T[MeV] FIG. 11. temperature

The value of 0, ( T) about T. Notations are the same as in Fig. 8(b). A characteristic reads T* -c 49.4 MeV irrespective of the values of the parameter set (i, g).

501

SYMMETRY AND PHASE TRANSITIONS

APPENDIX

B: HIGHER-LOOP

CONTRIBUTIONS

In what follows, we consider higher-loop contributions for the strong EM-field case by using a fermion-scalar interacting mode1 like the o-model, where the fermion loops play an important role (see Sections 4, 5). It would be formidable to evaluate all multiloop contributions systematically, so we shall limit ourselves to some specific diagrams, “Hartree diagrams,” and only estimate the order of higher-loop contributions; they are composed of fermion loops and scalar internal lines linking them. For practical purposes we only consider the “ring diagrams” (Fig. 12a) and “daisy diagrams” (Fig. l2b). I2 Each p-vertex loop included in the diagrams gives

where A4 denotes the fermion massand the no-vertex loop integral L’“’ corresponds to the one-loop effective potential V,.,OOpgiven in Section 3,

%ds 7 -7 e ‘&‘s 1 +f(%‘-C’).?

x Z~cothZ’s.dscot&s-

1

-f(~‘-~~)lnM’-~M41nM’+a~4

= -$ where x= IV/(%’

(,Z” + 8’) ZL,(.u, 8) + irrelevant terms,

+ cf”‘)““, 8 = tan~‘(d/x)

(B2)

and

x (s cos 19)coth(s cos 8). (s sin 0) cot(s sin 0) -

II

1 + i (~0s’ 8 - sin’ (3)s’ i

1 3 cos2H- sin’ 0) In x’ - - .Y’ In s’ + - .v4. 2 4

(B3)

I2 Similar analysis has been done for the phase transition in high temperature within the 4” model

WI.

502

SUGANUMA

AND

TATSUMI

oo--0-o a

0

b FIG. broken

12. An example of (a) the ring diagram and (b) the daisy diagram. one denote the propagators of fermions and scalars, respectively.

The ring diagram with n loops has two one-vertex

= -$ and n - 2 two-vertex

The

thick

line and the

loops giving L”‘,

(X2 + cf2)3i4XZL2(X,e)

U34)

loops giving L”‘,

ii a (x2+&2)‘!4i?y 1 -$ ?I (X2 + 8’)3’4xl 2(.Y,8)) with I mz(.Y,e) = p.v.

d~s-~e-““~[(~

cos 0) coth(s cos 0)

(s sin 0) cot(s sin 8) - l] + .u’(ln 2 - 1) = -Ffermion(.x, 0) + .u2(ln .I? - l), (v> -2).

(B6)

SYMMETRY

Thus the n-loop contribution

AND

PHASE

503

TRANSITIONS

P’t”” is

v,,ring _- -- 1 g2 “-’ 2m’ (

. (L”‘)‘.

(py,

!

Similarly, the daisy diagram with N loops consists of II - 1 one-vertex loops and an additional (n - 1)-vertex loop giving L’” ~ ” which should be expressed generally in terms of Z,,(.u, 0). Thus the n-loop contribution Vfalsy reads

In the following we examine these contributions case (0 = 7c/2) and the pure magnetic case (0 = 0).

separately

for the pure electric

B.I. Pure Electric Case In this case we find analytic expressions IL,(.U, n/2)=

for I,. by way of Eqs. (B6) and (C4), -;+o(.Y’), (B9)

I- ,(.Y, 7c/2) = 7 -In

I,,(& 7r/2) =

2 + 0(x4)

( - j”, (2m)! 2 ~ ?“i( 2m + 1) + 0(x”) ( - jnz (2m + 2)! 2 ~ 2’n ‘[(2m + 3).x7 + O(P)

for r 3 0. Then we can see that the p-vertex small .Y ( N strong-field case)

for v = 2nz - 1 for v = 2m

loop integral L””

takes the form for

L13) = - &‘1!23(fij i - In 2)~ + O(.u5), 4s

L(4~+ilN-

/j

(-)‘+I-’

471~ 2i+ 1 x[(2i+

1)(&l/2)

(Bll)

2p2’(4i+ 3)! (4i+jI)! (4-j)! 4J-i+4X4-,

(B10)

(i> 1, j=

1, 2, 3, 4).

504

SUGANUMA

Thus the n-loop contribution

AND

TATSUMI

to the effective potential

is estimated as

for the ring diagram and vdaisy 41+,+1-

(Bl3)

except some numerical factor for the daisy diagram. In this case the ring diagram gives a larger contribution than the daisy diagram. B.2. Pure Magnetic

Case

By a way similar to the electric case, we can also find analytic formulae for I,,, Z-,(x, 0) = -In .? + 0(x0),

and T(v + 2) + 0(.x0) x

Z”(X, 0) = r+Z

(v>0)

(B14)

by way of Eqs. (B4), (B5), (B6), and (Cl). Then the p-vertex loop integral takes the following form for small X: L” J= s

h

~33/zx In .x2 + 0(x’),

L(‘) = - -$ 2 In X’ + 0(x0),

W5)

L(“) = -2~2(n-l~(n-2)(.)Y.‘/2)4.~“(~)~~2+0(*”) Thus the n-loop contribution

to the effective potential

rmg VII - -+px’(

(nb3).

reads

--Lln.q($yp

u316)

for the ring diagram and vdw II

-

_

for the daisy diagram (n ~4). It is to be noted that these two diagrams contributions of the same order, opposite to the pure electric case.

(Bl7) give

SYMMETRY AND PHASE TRANSITIONS

APPENDIX

C: CHARACTERISTIC

505

FUNCTIONS

In the limit, purely magnetic case (0 = 0) or purely electric case (0 = z/2), the characteristic functions Ffermion,Fscaiarcan be expressed analytically. C. 1. Fermion Cuse In the purely magnetic case (fI = 0), one can obtain an analytic expression of the characteristic function Fferml“”(x, 0) by way of Binet’s formula [31]: Ffermion(x,0) = - .$: $ Ed “‘(s coth s - 1)

(Cl) Hence one easily finds

FfC~~lOll( x, 0) - In -” - 1/(3x’)

for for

.u+O .Y-+ r(;.

(C-2)

Similarly, the following analytic expression is obtained in the purely electric case (e=71/2), Frerm’o”(x, n/2) = -p.v. jO> f e-“‘(s cot s - 1) x’ dt -““(tcotht-l)+iln(l-e-““‘), =is 0 -se t-

(C3)

by deforming the integration path from the real axis to the imaginary one. We then use Eq. (Cl ) to obtain Ffermion

(.~,,,2)=2i[ln~(~)-(~-~)ln(~)+~-~ln(2~)]

+iln(l

-e-‘-“)

=Re{2i[lnT(~)-(~-~)ln(~)+~-fln(Z.)]} =t+my’(y-ln2)-2

=5+\-‘iy-ln2)+2 + .u’(ln x’ - 1),

+ x’(ln .I? - 1) % (-)’ C -[(2/$ ,=, a+ 1

~)2-‘.?‘+“~\-““+1’

(C4)

506

SUGANUMA

where we have used the formula argT(iy)=

AND

TATSUMI

[31]

-+,v+

k=, f {i-arctan($}.

(C5)

Then we have the following property: Ffermion(?c,

n/2)

_

for

.Y-rO

for

.Y+m.

(C6)

C.2. Scalar Case

One also obtains an analytic formula for the characteristic function Facalar in the purely magnetic case (0 = 0),

=

,z ds

-(scoths-I)+Z[;coth;-1)}

I 0 Y-2 s

i

= Ffermio”(,x,0) - Ffermlon(& x, 0).

It is interesting that the characteristic function terms of the one for fermions. Then it leads to FSCdaI(?c, 0) = 2 In

(C7)

Fsca’ar(,x,0) can be expressed in

x2 In .x2- (x’ - 1 ) In 2 + 9,

(C8)

by way of Eq. (Cl), and one finds Fsca’art~~> 0) -

-In 2 - .Y’ In x1 _ 1,(6MK2)

for for

.u-+O x--t co.

(C9)

A similar formula is obtained in the purely electric case (0 = 7c/2), F s=a’yX., 42) = p.v. = Ffermion(x,7~12)- Ffermlon( $

so that Facalar(x,n/2) = -.u’{ln(2x’) -2~,&(21+

+ y- 1

x, n/2),

(ClO)

SYMMETRY AND PHASE TRANSITIONS

507

by way of Eq. (C4). Also one finds the property for for

x-+0 .Y+ ,;rj.

(Cl21

ACKNOWLEDGMENTS We are grateful to Professor R. Tamagaki for his interest and encouragements. We also thank Dr. T. Muto, Mr. S.-I. Nawa and other members of nuclear theory group in Kyoto for useful discussions. Discussions with Dr. T. Kishimoto on the problem of Cabibbo angle have been helpful and stimulating. This work is supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (No. 01740155).

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AND

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