Unitary gauge and phase transition at finite temperature

Nuclear Physics B362 (1991) 616--04<} North-Holland

UNITARY GAUGE AND PHASE TRANSITION AT FINITE TEMPERATURE M. C H A I C H I A N t.,, E.J. F E R R E R ~-'~ and V. de la 1NCERA ~

IDepartment of High Energy Physics, Unicersity of Helsinki, Siltacuorenpenger 20 C, SF-O0170 Helsinki, Finland 2Research Institute for Theoretical Physics, Unicersity of Helsinki, Siltacuorenpenger 20 C, SF-O0170 Helsinki, Finland 3Lebedec Physical Institute, Academy of Sciences of USSR 117924 Moscow, USSR Received 10 May 1990 (Revised 28 February 1991 )

The unitary gauge in perturbative calculations of the phase transitions at finite temperature by Landau's method is investigated. The leading high*temperature contribution of the two-loop effective potential of the abelian Higgs model in unitary gauge is explicitly calculated. We find that already in the two-loop approximation appears a non-analyticity in the symmetry breaking parameter which makes a significant contribution in the high-temperature asymptotics of the effective potential near the phase-transition point. This fact justifies the remarks made in the literature, which suggest that the critical temperature should be calculated from the leading high-temperature contribution of the one-loop effective potential only in renormalizable gauges. As an additional inadequacy of the unitary gauge to investigate the phase transition at finite temperature, it is also shown that in this approximation the temperature-dependent divergences are not cancelled, contrary to what occurs for renormalizable gauges.

I. Introduction Since the suggestion done by Kirzhnits and Linde [1] about the possibility to restore the symmetry breaking of a relativistic system by heating it up, the study of the phase transitions at finite temperature in quantum field theory has become the matter of interest for an increasing number of experts. The main motivations for the study of the phase transition at finite temperature in field theory are given by their consequences on cosmology and elementary particle physics. In order to stress this fact on an example, we just point out that all the history of the Universe is plagued by the phase transitions that took place due to the cooling of the Universe by evoluting from a very hot initial stage into its present state [2, 3]. Among the field theoretical models where the system can evolve from a highly symmetric phase to a phase with a spontaneously broken symmetry, the most interesting ones are those exhibiting gauge symmetry, because of their physical content. * Present address: CERN, Theory Division, CH-1211 Geneva 23, Switzerland. 0550-3213/91/$03.50 S3 1991 - Elsevier Science Publishers B.V. (North-Holland)

M. Chaichian et al. / Unitary gauge

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Crucial contributions to the mathematical formulation of phase transitions at finite temperature in gauge theories are presented in the classical works of Dolan and Jackiw [4], Weinberg [5] and Kirzhnits and Linde [6]. The essence of the modern formulation of phase transition at finite temperature [4-6] can be interpreted as a generalization to the gauge theories of the original Landau theory of phase transitions [7]. In the last formulation, a fundamental role is played by the effective potential as a function of a symmetry breaking parameter sc and the temperature T. In gauge theories, however, such formulation has some difficulties. The problem is that the effective potential in gauge theories is gauge independent only on the mass-shell, i.e. only after evaluating it on the solution of the extremum equations, OV/O(ch)= 0, taken with respect to all the non-zero average fields (~b) of the theory [8]. However, as we will see later, in order to find the transition temperature by using the effective potential formulation one must take it off-shell. Hence, in principle the obtained results can be gauge dependent. Fortunately, in perturbative calculations when searching the transition temperature we can restrict ourselves to consider only a few terms in the high-temperature perturbation expansion of the effective potential, because the higher-order terms are smaller and can be neglected. The resulting transition temperature obtained within the above approximation is the same for all renormalizable gauges [4, 5], whilst for non-renormalizable gauges transition temperatures different from those obtained for renormalizable gauges are found. It is natural to ask: which one is the correct transition temperature and why? In refs. [4,5] some preliminary arguments against the use of non-renormalizable gauges to calculate the phase transition critical temperature through Landau's procedure are given. However, a quantitative analysis of the problems which emerge when one takes non-renormalizable gauges has not been done in the literature since then, to our knowledge. Recently, the use of non-renormalizable gauges, specifically the unitary one, has regained interest in the context of the phase transitions in highly dense hot electroweak matter [9]. In this system a high lepton-number density gives rise to the appearance of non-zero average values of the zero components of the vector gauge fields [10]; these average fields are responsible for mixture among different fields in the lagrangian with subsequent difficulties in the calculation of the effective potential. The unitary gauge provides an explicit simplification of the above mixtures [9, 11]. In this context the use of the unitary gauge [9] or a renormalizable one [12] showed a different physical picture for the hot superdense electroweak phase. When the unitary gauge is used, the Bose-Einstein condensation of W-mesons which appears at T = 0 by considering a high density of leptons [13], is realized with increasing temperature at lepton densities lower than the critical one found at zero temperature [9]; while taking a renormalizable gauge, such a behaviour is not found [12]: namely, by increasing the temperature, the common situation where

~1~

M. ('haichian el a/. / Unitao' gauge

the temperature serves as a restorer of symmetry (i.e. at some critical temperature, which depends on the lepton-number density, the W-meson condensate evaporates) is gained [12]. The above considerations give evidence about the importance to increase the understanding of the difference between the renormalizable and non-renormalizable gauges in the formulation of phase transitions at finite temperature in gauge theories. Our main purpose in this article is, precisely, to extend the analysis of this problem given originally in papers [4-6], in order to definitively define what kind of gauges must be used to find the correct transition temperature, by using Landau's procedure. We will concentrate our analysis on the abelian Higgs model, because, as we will show, already in this simple case the inadequacy of the unitary gauge to handle the phase transition at finite temperature can be proven. Our basic results, however, will be model-independent and therefore they can be straightforwardly extrapolated to non-abelian systems with and without non-zero fermion-number density. We must also point out that alternative methods to determine the phase-transition critical temperature [14, 15] have been proposed, with the peculiarity that they are gauge independent. We will not analyse them in detail here. The paper is organized as follows. In sect. 2 we summarize the peculiarities of the one-loop effective potential calculated in a renormalizable as well as in the unitary gauge. We emphasize the off-shell characters of " L a n d a u ' s procedure" to determine the transition temperature for a phase transition of second kind. We explicitly show how the unitary limit in the gauge p a r a m e t e r ~1 of the renormalizable R~ gauge is singular and verify the gauge independence of the on-shell high-temperature limit of the effective potential. In sect. 3 a general analysis of the leading temperature contributions to the perturbative expansion of the effective potential at higher orders in renormalizable and unitary gauges is given. The leading temperature contribution to the two-loop effective potential in the unitary gauge is found in sect. 4. It is shown that already in this approximation non-analyticities in the symmetry-breaking p a r a m e t e r proportional to powers of temperature appear. They are comparable to the lower-order contributions and therefore cannot be neglected when searching the critical temperature of the phase transition. We have also found t e m p e r a t u r e - d e p e n d e n t divergences in the two-loop approximation that are not cancelled as is the case for renormalizable gauges.

2. Effective potential in gauge theories and phase transitions In this section we summarize and discuss briefly the main results known in the literature [4-6] which show how the high-temperature asymptotic limit of the off-shell one-loop effective potential behaves differently in renorma|izable and unitary gauges. Cursorily we also show how the unitary limit (i.e. the limit which permits to pass from a renormalizable R~-gauge to the unitary one) of the off-shell

M. Chaichian et al. / Unitary gauge

619

one-loop effective potential does not commute with other asymptotic limits such as the high-temperature limit and even with dimensional regularization. We emphasize the off-shell character of Landau's procedure to determine the critical temperature of a second-kind phase transition.

2.1. LANDAU'S SECOND-KIND PHASE TRANSITION

Before getting involved in the specific problem of the finite-temperature phase transition for the abelian Higgs model, let us begin by recalling the main features of Landau's theory of second-kind phase transitions [7]. The main characteristic of a phase transition of second kind is that the state of the system changes continuously when passing from a less symmetrical phase to a more symmetrical one, or vice versa. The symmetry-breaking phase is characterized by a symmetry-breaking parameter ~: whose extremum value can take arbitrarily small values near the transition point. Due to this fact, the effective potential V(~,T), which is a function of the symmetry breaking parameter ~: and of the temperature T, may be expanded in powers of ~: when the temperature lies in the neighbourhood of its transition value Tc*, V ( ~ , T ) = 1711--~m(T)~ 2 -~- O ( T ) ~ z4 --~ . . . .

(1)

We are considering the cases where V(~:,T) depends only o n ~:2, which will correspond to the situation analysed further. To determine the critical temperature, i.e. the value of the temperature where the phase transition occurs, we have two possibilities starting from (1). These are easily understood from the graphical representation given in fig. 1. There, TI is a temperature corresponding to a symmetrical phase. We can see that the minimum solution of the effective potential V(~, T l) is ~:min(T1) = 0 and A(T I) > 0, while for the non-symmetrical phase with temperature T2, we have that ~Cmin(T2) = so0 4:0 and A(T 2) < 0. Then by the continuity of the transition we have

A(Tc) =

0,

lim ~:min(T) = 0, T---,T~

i.e. at the transition temperature Tc the coefficient A(T~) must be zero,

(2a)

i.e. the limit of the minimum solution aV(£, T)/a£I¢ ~ 0 = 0 when T approaches the critical value Tc, is equal to zero.

(2b)

*In gauge theories the analogue of eq. (1) exists only in the breaking phase, due to the infrared divergences present beyond the phase transition towards the symmetrical phase. Nevertheless, one can still approach the temperature to the transition point starting from the breaking phase, where the expansion coefficients A(T), B(T), etc. are well defined.

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v-v°A>~T~/T2 Fig. 1. Graphical representation of the effective potential V vs the symmetry breaking parameter ~ for a symmetric phase with temperature Tj and for a symmetry-breaking phase with temperature T2 (~¢0 is the minimum solution of the effective potential in the symmetry-breaking phase). "A" denotes the curvature of the effective potential near ~: = 0.

Hence, in relation to gauge theories we must underline that when calculating the critical temperature Tc by Landau's prescriptions (2), we meet immediately with the fact that because neither the off-shell effective potential nor its minimum solution are gauge independent [8], the methods that we have at our disposal to calculate the critical temperature (2), are gauge dependent. 2.2. ONE-LOOP EFFECTIVE POTENTIAL IN A GENERALIZED Ro-GAUGE

We start by considering the abelian Higgs model, whose lagrangian density is given by 1

1

.2,P= --~Fu~F""+ ~(Du6)*(DUcb ) -

m 2

A

-~-~b 2 - ~.Tq~4 ,

(3)

with m 2 < 0 and

F . . = 0.A~ - OvA~ .

(4)

G =

+ igA.,

(S)

= (h, + iq~2.

(6)

As known, this theory is invariant with respect to the gauge transformations

A~.(x) ---,A,(x) + O,w(x),

(7a)

6 ( x ) --* e-/g'°¢x) ~h(x),

(7b)

~b*(x) --* e ig°~{x)4~*(x) •

(7c)

The extremum equation for the scalar field ~b has a non-trivial constant solution. This Higgs solution only fixes the modulus of ~b, keeping arbitrary its phase. By convention we will consider the following form for the vacuum expectation value of

M. Chaichian et al. / Unitary gauge

621

the scalar field:

((Di)() = ~,

(~2)0=0,

(8)

For the sake of simplicity, here we restrict our study of phase transitions to those of second kind. This requirement is satisfied if we confine ourselves to the weak-coupling case (A >> g4) [2]. We choose the renormalizable gauge condition,

F[ A u, & ] = aOUA~, - ½igp[ ck* (&)0 - (&*)0&] - f = 0,

(9)

which is a generalized R,Vgauge with two arbitrary parameters a and p while f , as usual, is an arbitrary function of the 4-dimensional space-time (x, r). Similar to what is known for the Rn-gauge, by taking different relations between a and p we can pass from one to another known gauge condition, i.e. for p = 0, a = 1 (Lorentz gauge), a = p = 1 ('t H o o f t - F e y n m a n gauge), p = 1 / a = ~ (Rn-gauge), p = 1 / a = 0 (Landau gauge) and p ~ 1 / a + m (unitary gauge). The gauge condition (9) generates a gauge lagrangian density

2 g = - ½( aOuAU - p g ~ & 2 ) 2 - e(ot3 2 + pg2~qb, )c,

(10)

which must be added to (3) in order to cancel the gauge freedom of the theory. The fields c, e are the F a d d e e v - P o p o v ghost fields. We notice that although we are dealing with the abelian theory (3), the gauge condition (9) gives rise to a ghost interaction term, pgZ~cq~lc (see eq. (10)). Then, after rewriting the lagrangian densities (3) and (10) in terms of the average fields (8), we can write its quadratic part as ..~q =

1 • //~ --I I * ~tA J~,v { ~, P} A v + ~t ~ba ~ab--I { ~, P} &b + ~baM.UA~, + i ? G - l{ ~, p} c,

(11)

where

iaL { :, p}

= ( _ p 2 + M 2) & , , + ( 1 - a 2 ) p ~ , p , ,

i Zb'{ p} = P2t~ ab --"~gb M#{¢,R} = i g ( a p

,

- 1)eah(&o)oP u ,

=ap e -pM 2 , .//l'2b = m21rSaj3bl + m228,,2(~b2 , 1 2 , m 2 = m 2 + ~a¢

1 m 2 = m 2 + gas ~2

M 2 = gZs¢2"

+R2M2,

(12) (13) (14)

(15) (16) (17)

(18) (19)

622

~,1. ( ' h a w h i a n et al. / U u i l a o ' ,~,,au~,,e

Following the method of rcf. [4], we straightforwardly obtain thc effective potential approximatcd up to one loop as

V(~,T)

(20)

= Vo(~) + V ' I } ( ( , T ) ,

where

V{,( ~:)

=

m2

-'g'( ¢ )

A

= __~:-~ + ~ 2

(21)

is the tree-approximation term and

w(l)(~,, T ) _

it,2 E f ( 2d4. ~ ) 4 In det(i'~h'{¢'

p})

_TEf(Tj ir,=) d4p

lndet[iA£,,'{~:,p}

+iN.,,{~,p}]

d4p

+ ih E f (T37 ) lndet[iG-'{e, pl],

(22)

with N.,,{ ~, p} = ME{ ~:, P}-~.b{ ~, p}M/'{ ~,

-p}

= -M2(olp- 1)2p.pJ(p 2-m~_),

(23)

is the one-loop contribution. We have used the notation Y2f d4p/(2~-) 4 = (i//3)32p4fd3p/(27r)3, as is usual for Feynman rules in statistics (13 = 1/T). After the summation over P4, the one-loop correction to the effective potential (22) can be written as

V{~}(~5,T) = V&T(¢) {~} +

V¢'}(¢,T),

(24)

(I) where the unrenormalized one-loop quantum field theory part V6~(~) is given by

droP [In(p2 + m~) + 3In(p2 VO{Iv)T(S¢)= L:¢ (2~-) 4

+ M 2)

+ l n ( p 2 + R 2) + ln(p 2 + R 2) - 21n(p 2 + $2)],

(25)

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M. Chaichian et al. / Unitary gauge

and the one-loop statistical contribution is V¢u(~,T)

1

~

[

2~.2/34~ dxx 2 l n ( l - e - ~ )

+ In(l- e - ~ )

+ 31n(1-e-~)

+ ln(1- e ~

)

- 2In(1 - e - ~ ) ]

. (26)

In the expressions (25) and (26) we have introduced the new notations R 21,2 = b + -- b x / ~- c 2

(27)

(28)

S 2 = pM2/a,

b = ½ ( m 2+aa~: , 2 + 2pM2/a)

(29)

2 + g A1~

(30)

c=m2(m

2+pZm2)/o~2

2.3. HIGH-TEMPERATURE LIMIT AND CRITICAL TEMPERATURE IN RENORMALIZABLE AND UNITARY GAUGES

Once the one-loop effective potential at finite temperature (20) is obtained, we can follow the method (2b) in order to obtain the critical temperature. Then starting from the minimum equation O V ( ~ , T ) ~*o as~ _ = 0,

(31)

we obtain the relation which defines ~ as a function of the temperature:

T+35¢

"

2~t ~4 dXX2L,~2,~/x2+~2m2 1 - e ~

] =0,

(32)

where m 2 runs over the set m 2, M 2, R 2 and S 2, and n i is the corresponding multiplicity to each mode, in this case, 1, 3, 1, 1 and - 2 , respectively. In eq. (32) the masses have been conventionally renormalized taking into account the one-loop QFT contribution [4]. We can observe that the effective potential (24) and specifically its one-loop contributions (25) and (26) are gauge-dependent, as well as the stationary condition (32). All of them depend explicitly on the arbitrary gauge parameters a and p. But if we now consider the high-temperature limit (T >> mi), which is the appropri-

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M. (~haichian et al. / Unitary gauge

ate limit to investigate the phase transitions at finite temperature [2], we can cut the high-temperature series at the terms proportional to ATX, g2T 2. Then from eq. (32) we arrive at

e'4T213g2 + ~-a] : 0 .

, m 2 + .~ASCrnin i 2 + .

(33)

Now, taking into account the condition (2b) we obtain lim ~ : 2 i n ( T ) = O, 7"-, 7~. and thus T2 -

- 12m 2 3g 2 + 2A.

(34)

3 R e c a l l t h a t m 2 ,< 0.

As can be corroborated this critical temperature (34) coincides with the one reported for the Higgs model in refs. [2,4,5]. A surprising fact is that in the high-temperature limit the minimum equation (33) and consequently the critical temperature (34) do not depend on the gauge condition [1,4,5] (i.e. the dependence on the parameters a and p disappears). On the other hand, if in eq. (31) first we take the unitary limit (p ~ 1 / a --* oc) and only after that we take the high-temperature limit as before, we obtain a minimum equation given by 1

2

i

2

~Tm + TSaS/:min - ' } - / T 2 1 3 g 2

q- ½A] = 0 ,

(35)

and consequently the critical temperature [4, 5] -12m 2 Tc2

-

3g2+

½A

.

(36)

As the critical temperatures (34) and (36) are different*, we can conclude that the high-temperature and unitary limits do not commute. This remark may be naturally understood observing that the high-temperature limit establishes a hierarchy between the t e m p e r a t u r e and the different masses of the theory (T >> mi), but precisely the unitary gauge is reached from our gauge condition (9) (and it is also the case for the r / ~ o0 limit in the Rn-gauge) making the masses RI, R E and S infinite. Then, it is clear that the high-temperature limit in this case is not well defined. * It might seem surprising that to a physical quantity such as the critical temperature, could correspond a gauge-dependent value. In the next sections we will show that only eq. (34) must be considered as the true value of the critical temperature of the Higgs phase transition.

M. Chaichian et al. / Unitary gauge

625

In the same way, if we first take the high-temperature limit for the one-loop statistical effective potential (26) taken to order T 2, we obtain

V (1) =

7r2T 4 m2T -- _ _ -.1- _ _

90

• ¢

2

12

A~2 Z 2 g2~:2 -1- - ntT 2

36

8

(37) '

while if we first take the unitary limit ( p ~ 1 / a ~ oo) in eq. (26) and then take the high-temperature limit maintaining also the terms to order T 2, we arrive at .

m2V 2

7r2T 4

VB(I)

9~- +

g2~:2

1 ~ 2 T2 + -----~-T 2 .

2---~ + 48

(38)

Expressions (37) and (38) coincide with those obtained in refs. [4, 5]. The expressions (37) and (38) are different, but if we evaluate both of them on the solution of the tree-level minimal equation SC('m~in' =: -- 6m2/A,

(39)

it is easy to see that the same expression for the effective potential is obtained in both cases, V/~')(T) -

,rr2T 4 90

m2 [ 9g 2 12 [ 1 + T ]T2" l

(40)

This feature was interpreted in ref. [11] as evidence for the correctness of both gauges in the high-temperature limit and starting from such conclusion it was argued for the advantages of the unitary gauge compared to others to study the high-temperature phase transitions. In sects. 3 and 4 we will provide definitive evidence which proves that this indeed is not the case, corroborating in this sense the remarks suggested in refs. [4-6]. Let us see finally in more details what we mean by on-shell effective potential. To get the on-shell effective action F (or the effective potential) to order n in powers of h, one has to calculate F up to this order, F (n), and then evaluate it at the minimum solution ~b(n) found from the minimum equation a F ( n ) / & b = O, keeping in F (n)(~bmi
V = V (°) + h V m ,

and from the minimum equation O(V (°) + h v ( l ) ) / O ~ )

=

0, we obtain the minimum

62~

M: ('tlaichiapz el a/. / Unitary ,k,attqc

solution = #&

+

+

{42)

....

Thus, the on-shell effective potential is given, to order h, by r(°)(l~mm-I-(°) ~ d ) (I) ] + h V ( 1 ) ( d ) (0) -I- .AAI} - -r mm / " rain ]1 c/) rain )

= Vm'(cb~l.) + h O& 'I'm,,,'"'mi,1+

hV("(d¢~m~],,) + O(h2) •

(43)

But OV(°)/O(ol~ll,,= 0, because "~mi,, '6~°~ is the tree-level minimum solution. Moreover, (0) V (0) (~bm~ .) is gauge independent because it is the on-shell tree-level effective potential. Therefore, we have that V (~)(4~m~,1) (0) is also gauge independent. At finite temperatures the above analysis is also valid [8]. Now, as in the one-loop approximation, since the tree-level minimum solution does not depend on the temperature, it is evident that the on-shell one-loop effective potential will be gauge independent not only as a whole but even in each order of the high-temperature expansion, because the tree solution (39) does not affect the asymptotics in T. This explains why we obtain the same result (40) when we evaluate the solution (39) in the two different high-temperature asymptotic results (38) and (37). However, this feature cannot solve the difficulty inherent in obtaining two different critical temperatures (34) and (36), because, as we have already pointed out, the procedure which pursues to obtain them, (2b), is not gauge invariant. 2.4. DIMENSIONAL REGULARIZATION AND THE UNITARY LIMIT Similarly to the non-commutativity between the unitary limit and the high-temperature one, there exists a non-commutation peculiarity between the unitary limit and the regularization procedure in the QFT part of the one-loop effective potential (25). This fact was already reported in ref. [16] using an ultraviolet cutoff (k = = A=). Now we will show that the reason which underlies this non-commutativity is the same that we have noticed when considering the high-temperature limit. We use in this case the dimensional regularization technique [17] which is more appropriate for systems with gauge symmetries. After the dimensional regularization of the one-loop Q F T effective potential v~l)(~) of eq. (25) is carried out, we obtain V{RI)(~) = -- {2m 4 + [[~Arrt 2 +

[

2g 2 m 2 ( -~ 2pl

+ ~A2 + 3g4 + -'-3"-

m

ol

a,2

ot2

(:4

+ - -2

+ O(e)

(44)

M. Chaichianet aL / Unitarygauge

627

w h e r e e ~ 0 is the regularization parameter, and 3' = 0.577. If we now take the

unitary limit p--->1/a ~ ~, we observe that infinite terms appear and that this result does not coincide with the one obtained when we first take the unitary limit in (25) and after that carry out the dimensional regularization, i.e. "

=-

)

-+--+O(e)

e

2

.

(45)

As in the high-temperature limit case analysed above, eqs. (44) and (45) coincide only on the mass-shell. In other words, evaluating both eqs. (44) and (45) on the minimum solution (39) we obtain the same expression, given by R ~mi,):

l+---~--]te+---~+O(e

) .

(46)

The non-commutativity of the limiting procedures here, can be understood as follows. If we rewrite eq. (44) in terms of the masses of different modes, we have V~I)(~:) .

[m4+BM4+R4+R4 . . .

2S4](1

+

1-3/

+ O(e)

)

(47)

In the unitary limit R~, R 4 and S 4 tend to infinity. Hence, in a similar way to the case of the high-temperature limit, it is not clear what should be the limit of (R 4 + R 4 - 2 S 4 ) O ( e ) . In fact, we cannot decide whether this term must be neglected due to the smallness of the parameter e, or not. We summarize the above results by stating that the one-loop effective potential in the unitary gauge may be obtained from the one-loop effective potential in the renormalizable gauge (9) only if we take the unitary limit before doing any limiting procedure (such as the regularization or high-temperature limit) which obligates to consider that all the masses of the theory are smaller than some parameter approaching infinity ( 1 / e or T). This observation is easy to interpret, if we recall the non-renormalizable character of the off-shell effective potential in the unitary gauge, which must manifest itself when we try to get any ultraviolet limit (wherein the masses are neglected). In spite of the above remarks, we have not yet given a definitive proof for deciding what is the correct transition temperature ((34) or (36)) when investigating the phase transition at finite temperature in the Higgs model. In the next sections we will present definitive evidence in order to resolve this puzzle. 3. Higher orders of perturbation theory in r e n o r m a l i z a b l e and unitary gauges

There is a basic question that must be answered when solving the conditions (2): Will the critical temperature be affected when additional orders in the perturbation expansion of the effective potential are considered?

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M. ('haichian et aL / Unitary gauge

In this respect, as we shall see below, a crucial role is played by the renormalizability of the theory. Although the answer to our initial question was definitively given for renormalizable gauges in refs. [4-6], we shall begin our analysis by attending to the high-order behaviour of the effective potential of the abelian Higgs model in the renormalizable gauge (9) in order to make more clear our arguments when studying the effective potential in the unitary gauge. 3.1. RENORMALIZABLE GAUGE From ref. [5] we know that the leading contribution of any divergent Feynman diagram at high temperatures is given by T ° , where D is the superficial degree of divergence of the diagram. Now, as the effective potential can be considered as a vacuum graph expansion, we have that the superficial degree of divergence of any one of such vacuum graphs is given by D = 4

+ Erti6i,

(48)

i where ni is the number of vertices with index of divergence gg. The index of divergence 6~ for the lagrangian densities (3) and (9) can be expressed as 6 i = b i + d i-

4,

(49)

with b i denoting the number of boson a n d / o r ghost lines in the vertex "i", and d, the number of derivatives in such a vertex. The index of divergence of every vertex of the lagrangian densities (3) and (9) are represented in fig. 2. There we can observe that •i ~<0 for all i, from which it follows that D ~< 4. This is precisely a peculiarity of the renormalizable character of the theory. Moreover, writing the superficial degree of divergence D in terms of the number of loops L and the number of internal lines I, we have D = 4L - 21~< 4.

(50)

}- Y X 'Y"

×) 6~= o

6 2 =- ~

63 = o

6~ :-I

6s = o

(a)

(b)

(c)

(d)

(e)

~s :-I

(f)

Fig. 2. Vertex diagrams for the abelian Higgs model in the generalized Rn-gauge (eq. (9)). 6 i denotes the corresponding index of divergence for each vertex. Here wavy lines denote gauge fields, solid lines, scalar fields and dashed lines, ghost fields.

M. Chaichian et aL / Unitary gauge

629

Fig. 3. Graphical representation of an L-loop vacuum diagram of the abelian Higgs model. (Conventions are the same as in fig. 2.)

Thus, taking into account that with the increase of the number of internal lines the number of vertices are increased and consequently the power of the coupling constants too, we can conclude that in a renormalizable weak-coupling theory the increase of the number of loops do not yield contributions to the leading high-temperature behaviour of the effective potential. In fig. 3 a particular vacuum graph is shown for which the increase in the number of loops L produces an attenuation by a factor g4(L-I)~2(L-I)//T2tL-I) to the leading high-temperature one-loop contribution of order T 4.

3.2. U N I T A R Y G A U G E

Let us start from the Higgs unitary lagrangian density 1

1

m2

~

4

~ u = --~FuvF~'~+ -~]Oueh + igAgq~[ 2 - -~-t~ 2 - "~,Tt~ ,

(51)

where $ is a real field with vacuum expectation value given by (th) = ~:. If we analyse the vertices corresponding to this lagrangian density (see fig. 4), we see that not all the indices of divergence 6 i satisfy (contrary to the renormalizable case analysed above) the condition 6i ~< 0. We must keep in mind that in order to find the index of divergences 6~ corresponding to the unitary theory (51) (see fig. 4), we must carry out in eq. (49) the change b i = s i + 2vg (with si denoting the number of scalar lines and v i the number of vector lines in the vertex "i"), since we now deal with a theory where the canonical dimension of the vector field (d(A~,) = 2) differs from that of the scalar field (d(~b) = 1), while in the renormal-

X'YYX

2ig2 t~

(a)

2ig2e,t ~

-ix~,

-ix

(b)

(c)

(d)

Fig. 4. Vertex diagrams for the abelian Higgs unitary lagrangian (61). t~i denotes the corresponding index of divergence for each vertex. (Conventions are the same as in fig. 2.)

6311

M. ('haichian et al. i" Unitao, gauge

izable case analysed in subscct. 3.1 they were equivalent ( d ( ( b ) = d ( A ~ ) = I). Hence, the superficial degree of divergence D will not be limited from abovc. Thus, it follows that the higher orders of perturbation theory of the effective potential corresponding to the unitary theory (51) give rise to leading high-temperature behaviour determined by powers of temperature higher than four (as we see from (38), the high-temperature leading term in the one-Mop approximation of the effective potential is proportional to T4). At the same time, the euclidean Green functions corresponding to the scalar field 4' and to the vector field A u in this unitary theory, are given respectively by

-i ~ { ~ , P } = p2 + m~ '

A

- i m26u~, +PUP, {,~, p} - m 2 p2 + m 2

(52)

(53)

The Green function corresponding to the vector field (53) has the peculiarity that when considering the high-temperature leading contribution of any diagram, it enters in the internal lines with an asymptotic behaviour given by -i A~,, ~ ~ - 5 0 ( 1 ) ,

(54)

in contrast to the corresponding high-temperature leading behaviour of the Green functions in a renormalizable gauge, ,_4R ~ - - i / 3 2 0 ( p - 2 ) .

(55)

Hence, we observe that increasing the number of vector-field internal lines in a diagram does not yield a decrease in the power of the temperature for the high-temperature leading contribution of the diagram, as it is accustomed in the renormalizable case, where as we see from eq. (55) each vector-field internal line contributes with T 2. Furthermore, even in the high-temperature asymptotia the mass contribution is not restricted to the vertices, but each internal line of the vector field brings about a factor M -2 (see eq. (54)). This last feature, together with the unlimited character of the superficial degree of divergence of the diagrams in this theory, make possible the appearance of high-temperature leading behaviour for the higher-order vacuum graphs of the kind T 4 + 2 ( L - 1)/~2(/--1) (see fig. 3) where L denotes the number of loops in the diagram.

M. Chaichian et al. / Unitary gauge

631

In the same way we are lead to the following general expression for the leading temperature behaviour of the higher-order effective potential:

Vhigher-order

= T4 E L =2

Ct.(T/~)Z(L-1) q_ AT 4

E C.L. .t. l. I. ¢ ) 2(L-2) L~3

+g 2T4 y" C,,L(T/,~)2~L 2), L -3

where L denotes with the number of loops and c L, c~,c~ are numerical coefficients. The above formula can be obtained by taking into account that the main increase in the divergence of a vacuum diagram is reached when a new loop is obtained by the insertion of a vertex (a) of fig. 4 (or equivalently by inserting two vertices (b) of fig. 4). Therefore, we have that the main increase in the temperature, when going from a vacuum graph with L loops to the following one with ( L + 1) loops, can be equal t o T 2, while the contribution of g and ~ is increased by g 2 M - 2 (which is obtained from the dependence on g and ~: for each one of these vertices together with the increase in M 2 brought about by each internal vector line). From the above analysis, it is easy to understand that there is no basis to neglect the higher orders of the unitary effective potential when calculating the transition temperature. In this sense, we answer the question that we raised at the begining of this section. In other words, in the unitary gauge, in contrast to the renormalizable gauges, the leading terms in the high-temperature regime of higher perturbatire orders are comparable with the terms proportional to g2~Z2T2 which appear at the one-loop level and therefore we cannot truncate the high-temperature series in the one-loop approximation. Here in principle, we would need to calculate all the perturbative orders to see how the non-analytical series* in ~ is able to restore the missing O ( A ~ 2 T 2) t e r m s necessary to get in this gauge a transition temperature with identical value to that already found in renormalizable gauges (34). In sect. 4 we will corroborate the conclusions reached here on general grounds, through a specific calculation. We underline once more that the difference between the leading high-temperature behaviour of the higher-order terms of the effective potential in renormalizable gauges (subsect. 3.1) and non-renormalizable gauges (subsect. 3.2) is a consequence of the off-shell character of this magnitude. If we evaluate the effective potential in the minimal solution, as we did in eq. (40) at one-loop level, all the pathologies which appear in the unitary gauge must disappear to coincide with the behaviour in the renormalizable gauge. * We would like to mention that in renormalizable gauges there can appear also non-analyticity in the mass parameter due to the higher orders, but firstly, this non-analyticity is related to the infrared behaviour of the theory, and secondly, it can be shown [2,4,5] that there always exists an interval I T - T c] <~g2Tc,ATc, where Tc is the one-loop critical temperature calculated to order g2T2, AT 2, in which the validity of the perturbation theory can be established and therefore the contribution of the non-analytical terms can be discarded.

632

AI. ('haichian ctal. / Unita/y gauge +

Fig. 5. Graphical representation of the two-loop effective potential of the [tiggs model in the unitary gauge. (We follow the same conventions as in fig. 2.)

4. Leading high-temperature contribution of the two-loop effective potential in the unitary gauge In this section we shall calculate the leading high-temperature contribution of the effective potential for the abelian Higgs model in the unitary gauge at the two-loop approximation, in order to corroborate explicitly within this approximation the general remarks of sect. 3. Two-loop calculations of the temperature effective potential in renormalizable theories have been done [18], and the cancellation of all the temperature-dependent divergences which are typical of higher-order approximations [4] has been explicitly shown without introducing counter terms other than those needed at Y = 0 [18]. In the unitary gauge as we will show further, such a cancellation does not take place, which represents another symptom of the inadequacy of the unitary gauge to investigate the phase transitions at finite temperature. The diagrammatic representation of the two-loop effective potential corresponding to the theory (51) is given in fig. 5. In fig. 6 we present the vacuum graphs which enter in the two-loop effective potential giving also the superficial degree of divergence and the leading high-temperature asymptotic behaviour of each of them.

co D=2

D=4

C>

~~2~2T2

~& T~

D=6 ~ t ; "~ T 6

(a)

(b)

(c)

D=6 ~-2 T~ (d)

Fig. 6. Vacuum diagrams which enter the two-loop effective potential of the abelian Higgs model in the unitary gauge. Each diagram is labelled with its superficial degree of divergence D and with its leading behaviour in the high-temperature limit.

M. Chaichianet al. / Unitarygauge

633

Since we are interested in the high-temperature leading contribution of V(2~(sc, T), we can concentrate our attention on the diagrams (c) and (d) of fig. 6. 4.1. VACUUM DIAGRAM 6c To the Feynman diagram shown in fig. 6c corresponds the integral representation

-4ig'~ 2 V6(if)

M4

~f

d4k d4p (2~r)4 (2~.)4

[MZ3~,+G,P,][M23~,, +G,P,] (p2+M2)(k2+M2)[(p_k)2+m2 ] .

(56)

Performing the product in the numerator of eq. (56), we can write this expression as the sum of the following three terms:

v£,:

~ 8 ig 4~ 2 ~

/.

J

d4k

d4p

1

( 2 ~ ) 4 (2"n') 4 dnk

(p2+M2)(k2+M2)[(p-k)2+m2]

dap

1

+8ig2~ f (2rr)4 (2,n.) 4 (p2+M2)[(p-k)2+m~]

4i d4k ~2 E f (2rr)4

d4p (P'k)2 . (57) (2,/7.)4 (p2 + M 2 ) ( k 2 + M2)[(p_k)2 + m 2]

The high-temperature asymptotic behaviour of each term in eq. (57) is, respectively, proportional to ga~2T2, g2T4 and sC-2T 6. So we can concentrate our attention on the third therm, since in the high-temperature limit the first term can be neglected as compared to the one-loop contributions, and the second one is independent of sc, hence it will not affect the minimum equations which determine the transition temperature. After the summations over P4 and k4, which are indicated in eq. (57) by the convention

E with P4 =

f dak

2rrnT and

dap

(2rr)a(2rr)~k4 =

- 1

d3p

d3k

/32 ~ f ( - ~ ) ~ f ( 2 ~ r ) 3 '

2rrmT, with

n and rn being integers, we obtain

~'6(2) = [~'6Sc t "t- b)st-D" 6c -k [~OFT, 6c ,

(58)

where the statistical Vgc, ^.~t divergent statistical 126~t-o, and quantum field theory

+

[pk-F,(F, i-P,)12

II

_ Ff ] + 4P,(F,

+‘_):-

[pk - (F, + Fz)(P, +

2F&

F2 (f, +t$ [

+

+

l

4F?(f,

+t.+

[PK_“,(“,

[+

-4]’

_

[pk-(E, 4F?(P, -t%)?-

[pk+E,(t.,

+2 4&E,

- FJ

[I:+,

- “,

-h)]2

+ F&F,

_F2)(E, [+(E,

2

+t.$](2t,

[pk-

(P, +F$](2F,+l,)

i F,[(C,-F&Ff]

+F7)]?

[pk+F,(F,

(F,

ft-,)

i

fF,)tJ

+F&P:]

I

-2E?)]l -t.$](2p,)

I

t-,[I -@I

635

M. Chaichian et al. / Unitary gauge

[~k q- El(E l q- E2)] 2

462(81 "{-E2)2 --El[ E32 -- (El q- E2)2](aEl-'[-E2) E1E2[E 2

[pk__(El+E2)(Ei4_282)]2

[pk+ (e 1 - e 2 ) ( 2 8 2 - E 1 ) ]

1 -- e m~

(61 q- E2) 2]

[pk--El(E I + 8 2 ) ] 2

EiE2182--(Ei-{-82)2]

+2 4e2(e, +e2) 2_ e2[e{_ (e, +e2)2](e, + 2E2)

[pk--el(e l-e2)] 2

2

E,Ee[E

[pk +E,(E, +E2)] 2

1

-- (E, -- E2)q

[pk--E~(E, +E2)] 2

+2 4E,(E, +E2)2--E,[E2--(EI +E2)2](2E, +E2) +

1

[~k - El(E I q- E2)] 2

[Pk--(el+E2)(gl+gE2)]2"]"

"

-2

]1

, 1 -}-E2) - - - ~ - ---E2[E - 23 -. . . .(E I +E2)2]{,2E2q-EI) 4E~(E

EiE2[E2--(El +62) 2]

1 _ e~¢,, +~~)_

[pk +El(El --g2)] 2 +2 4e2(el_e2)2_e,[e2_(el_e2)2](2El_e2 [pk - (61 - 82)(61 -

2E2)]2

1

4e2(e,-e2)2-eE[e~ -(e,-e2)2](2e2-el)

QFT

V6¢

=

-2i

d3p

f --

[pk--ele3] 2

d3k [

6162162--(61-}-62) 2]

+

q-

_2[pk 2(E, + Ea)(E1 + 2Ea)] 2

4ca(el + 8 2 ) 2 -

(60)

1 - e ~¢< <') ' [pk --E3(E 3 +E2)] 2

3 [ElE3[(E3--Ei)2--E2]

2[pk--El(E 1-{-E2) ] 2

)

E2162- (E l q-E2)2] (61 q- 2E2)

]

]" (61)

T h e notation E l : ~ q- M 2 , e 2 = V / ( p - k)2 + rn2 and e 3 = ~ + M 2 is used. Finally, writing the t e m p e r a t u r e - d e p e n d e n t expressions (59) and (60) in spherical variables and performing the variable changes Ipl =/3-'x, Ikl =/~-Ly, z = 1 ___ cos 0, 0 being the angle b e t w e e n p and k (the + signs in the definition of z are taken according to the 0 - d e p e n d e n c e s : (1 + cos 0) or (1 - cos 0) in each integral; no integrals with mixed terms of (1 + c o s 0 ) and ( 1 - c o s 0 ) appear in eqs. (59)-(61)), w e can do the expansion of eqs. (59) and (60) in powers of /3. In this

M. Chaichianet al. / Unitarygauge

636

way, we obtain the corresponding high-temperature leading term given by -i 96s~r q "- | ~ ' s t - D

-

• 6c

477.4

[ N r - Ni]

T6 + ~....

(62)

with

Nr=

f~[j~fo2[ _dyy 2

dxx 2

dz 2 +

]

8(X + y ) 2 + 2y2z 2

1

2(x+y)2+(y2+2xy) z]l-ex

1 1--eX+y' (63)

Ni=fo~dXx2f}2dz{f}~dyy 2

4(x+y)2+y2z2

[

2(x+y)Z+(y2+2Xy)Z

1 1] 1------~+ l_eX+,,

(,

+

+

,)

(3_ 1}]} l - e x + 1 - e x+y

"

(64)

4.2. VACUUM DIAGRAM 6d The Feynman diagram of fig. 6d is expressed by

--2ig2 V6(2)-

M2

d4k d4p Y'~f (277.)4 (277.)4

M 2 ~ + PuP~ p2 + m 2

6u. k2 + m2,

(65)

which after doing the product in the numerator, can be separated into the following two terms: d4k

d4p

1

,.,2 v.a, = _6ig2z~ ~ . ;j (2=) 4 (2=) 4 [~2 + M2][~'2 +m,~]

2i d4k d4p 1 ~2 E f (2~r) 4 (2,n-) 4 k 2 + m 2 •

(66)

The high-temperature asymptotic behaviour of each term in (66) is proportional to g2T4 and ~-2T6 respectively. Thus, as we have pointed out above, we have for this vacuum diagram the same high-temperature asymptotics as for V~2). However,

637

M. Chaichian et al. / Unitary gauge

from eq. (66) we see that its asymptotics only gives a temperature-dependent divergence contribution whose divergence is proportional to the infinite statistical volume V t°*)= iT4/(27rEX~,nff; dyy2), i.e. the high-temperature leading contribution of V6(2) can be extracted from 2 i V ~)

t" d4k

EJ

1 (67)

k2 +

to give i T2 I~6a ' = ~ Vt'~ ~--5-+ . . . .

(68)

From eqs. (62) and (68) we have that the high-temperature asymptotic behaviour of the two-loop effective potential Vt2~(~:, T) is given by

V~~o(~ ,(2)

h2 T6 T ) = -7---7I Nf - Ni] 47r . . ~:2

h2

T2 V( ~ ) 12 ~2"

(69)

Thus, we have corroborated the fact that already in the two-loop approximation of the effective potential for the Higgs model in the unitary gauge, the non-analyticity in the dependence on the symmetry parameter ~:, which we have argued for in the general analysis of sect. 3 has shown up. We have found, moreover, the following important fact. Together with the pure statistical contribution h2NfT6/(47r4~2), we obtain a temperature-dependent divergence, - h 2 / ( 4 7 r 4 ~ 2 ) ( N i T 6 + l'n'4V(°*)T 2) (notice that the coefficient N i is not finite). However, contrary to the renormalizable cases [4, 18], where, as we have pointed out above, these temperature-dependent divergences can be cancelled taking into account the renormalization counter-terms (i.e. in a renormalizable theory the temperature-dependent divergences which appear in the two-loop approximation of the effective potential are cancelled considering the one-loop approximation with the renormalization counter-terms to the mass h 6 m and coupling constant h6A included in the masses of the modes; see ref. [4] for details), we have now the situation that such temperature-dependent divergences cannot be cancelled, because the temperature part of the one-loop effective potential in the unitary gauge,

f(

xx2[,n/,

o

e

has a high-temperature leading contribution of order T 4, which is not enough to cancel the ~-2T6 leading contribution obtained here. This result should not be surprising, because we have to bear in mind that we are working with an unrenormalizable theory.

638

M. ('haichian et al. / Unitary gattgc

5. Concluding remarks In this p a p e r we have investigated the high-temperature behaviour of the Higgs model in the unitary gauge. We found that the higher orders in the perturbation expansion cannot be neglected when considering the high-temperature leading contribution to the effective potential. That is, the high-temperature perturbative expansion of the effective potential in the unitary gauge is broken. As known [4-6], in renormalizablc gauges the high-temperature expansion of the effective potential may be truncated at the one-loop level with terms of O(g2f2T2), because the remaining terms that come from this loop as well as from higher orders, are smaller and can be neglected. On the contrary, in the unitary gauge, as we have shown here, the higher orders contribute with terms non-analytical in the symmetry breaking p a r a m e t e r ~. These terms near to the phase transition point are comparable in order with those proportional to g2~XTewhich appear in the one-loop approximation. For this reason we have no right to truncate the high-temperature expansion at the one-loop level as before. This explains why one obtains a different critical temperature in the unitary gauge when compared with that found in renormalizable gauges and makes it definitely clear that the correct self-consistent transition temperature is only the one calculated by using renormalizable gauges. In addition to the just-mentioned problem in the perturbative calculation of transition temperatures in the unitary gauge, we also have shown that the counter terms which provide the renormalizability of the zero-temperature theory [4] are unable to cancel the temperature-dependent divergences that appear for the unitary gauge in higher-order approximations of the effective potential contrarily to what occurs for renormalizable gauges [4]. Such a non-cancellation is sufficient to invalidate the use of the unitary gauge in the calculation of the critical temperature through the effective potential. Note that this observation does not mean in any way that the unitary gauge is in general a bad gauge condition. We know that if we pass from the effective potential, which is a typical off-shell quantity, to the corresponding on-shell one, i.e. the thermodynamical potential, all this pathological behaviour disappears. The thermodynamical potential, as a welldefined physical quantity, will be the same, notwithstanding the kind of gauge condition that has been employed (renormalizable or unitary). We wish to point out that although our calculations here were carried out for an abelian model, all our statements and outcomes can be straightforwardly extrapolated to non-abelian models, as well as to abelian and non-abelian models at finite densities. It is at this time proper to stress that recently still another different approach, namely the Vilkovisky-De Witt (VD) effective action formalism [19], has been used in order to obtain a gauge-independent critical temperature in the Higgs model [15]. Thus, replacing the effective potential by the VD effective action in

M. Chaichian et al. / Unitary gauge

639

L a n d a u ' s m e t h o d it is possible to work in a g a u g e - i n d e p e n d e n t way. H o w e v e r , we w o u l d like to p o i n t o u t that t h e V D effective action, a l t h o u g h gauge- a n d p a r a m e t r i z a t i o n - i n d e p e n d e n t , d e p e n d s , in principle, on t h e specific c h o s e n m e t r i c [19, 20]. T h e r e f o r e , t h e p r o b l e m of d e p e n d e n c e of the critical t e m p e r a t u r e on the g a u g e in the effective p o t e n t i a l a p p r o a c h could be t r a n s f e r e d into the p r o b l e m of the d e p e n d e n c e on the a r b i t r a r y m e t r i c in this new formalism. In p r a c t i c e the choice of m e t r i c in g a u g e t h e o r i e s must be p e r f o r m e d u n d e r the c o n d i t i o n s of ultralocality, d i f f e o m o r p h i s m invariance a n d the r e q u i r e m e n t that the g a u g e g e n e r ators m u s t be the Killing vectors of the metric. Certainly, a l t h o u g h in g a u g e t h e o r i e s only one m e t r i c s e e m s to fulfill these r e q u i r e m e n t s , n e v e r t h e l e s s this result c a n n o t be c o n s i d e r e d as a g e n e r a l p r o o f of the uniqueness.

T h e a u t h o r s a r e grateful to I.A. Batalin, E.S. F r a d k i n , I.V. Tyutin and G . A . Vilkovisky for useful discussions. Two of us (E.J.F. a n d V. de la I.) w o u l d like to t h a n k the R e s e a r c h I n s t i t u t e for T h e o r e t i c a l Physics, University of Helsinki, w h e r e p a r t o f this w o r k was a c c o m p l i s h e d , for hospitality.

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[17] G. "t Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189 [18] M.B. Kislinger and P.D. Morley, Phys. Rev. DI3 (1976) 2771: A.J. Niemi and G.W. Semenoff. Nucl. Phys. B23(I (1984) 181: Y. Fujimoto, R. Grigjanis and R. Kobes, Prog. Theor. Phys. 73 (1985) 434 [19] G.A. Vilkovisky, Nucl. Phys. B234 (1984) 125: m Quantum theory of gravity, ed. S.M. Christensen (Adam Hilger, Bristol, 1984) p. 169; B.S. DeWitt. The effective action, in Quantum field theory and quantum statistics: Essays in Honour of the sixtieth birthday of E.S. Fradkin, ed. I.A. Batalin, C.J. lsham and G.A. Vilkovisky (Adam Hilger, Bristol, 1987) [20] B.S. DeWitt, in Proc. 12th Johns Hopkins Workshop, Baltimore, ed. G. Domokos and S. Kovesi-Domokos (1988)