Domain walls at finite temperature

Nuclear Physics B265 [FS15] (1986) 45-64 © North-Holland Publishing Company

DOMAIN

W A L L S AT F I N I T E T E M P E R A T U R E * C. ARAGAO de CARVALHO

Departamento de Ffsrca, Pontlficta Umverszdade Cat6hca, Cx P 38071, CEP22453, Rto de Janezro, R J, Brasd

G.C. MARQUES, A.J. da SILVA and I. VENTURA Departamento de Ffstca-Matemtltzca, Unwerstdade de S~o Paulo, Cx P. 20516, CEP 01498, S~o Paulo, SP, Brasd

Received 10 January 1984 (Revised 8 January 1985) We suggest that the phase transmon of A~b4 theory as a function of temperature coincides with the spontaneous appearance of domain walls Based on one-loop calculations, we estimate Tc ~ 4M/x/-~ by lmposmg that their surface tension vanishes, to be compared with T~ ~ 4.9M/,/-~ from ettecttve potential calculations (which we perform directly m the broken phase). Domain walls, as well as other types of fluctuations, disorder the system above To, leading to (~b)= 0.

1. Introduction

The description o f the evolution o f the universe in the standard (big bang) cosmological m o d e l requires the study o f field theories at finite temperature [1]. In addition, heavy-ion collisions provide an adequate experimental setting for testing finite-temperature effects, up to temperatures o f the order o f 100 MeV, so far [2]. It is, then, not at all surprising that a lot o f progress has been m a d e in the subject, as evidenced by the investigation o f such interesting p h e n o m e n a as pion condensation [3] a n d d e c o n f i n e m e n t o f quarks [4]. Temperature gives us access to new phases o f matter t h r o u g h phase transitions such as the two just mentioned. The investigation o f one such transition, in the case o f a scalar field theory, will be the main interest o f this paper. The A&4 t h e o r y has been used extensively [1, 5] in studies o f finite-temperature ettects. Being a p r o t o t y p e o f the self-interactions o f the Higgs sector in the electroweak a n d in unified theories, as well as a model for pion condensation, its investigation is not d e v o i d o f p h e n o m e n o l o g i c a l interest. We shall concentrate our attention on the phase transition that takes place when, starting from the lowtemperature ordered phase, we increase T up to a critical To, b e y o n d which the system is in a disordered (high-T) phase. The two phases m a y be distinguished by an order parameter, (d~(x))r, the expectation value o f the scalar field at finite * Work partially supported by FINEP, CNPq and CAPES. 45

46

C. Aragao de Carvalho et a l / Domain walls at finite temperature

temperature. For T ~> Tc this parameter vanishes, whereas it is different from zero in the ordered phase ( T < To). As far as we know, this phase transition has been studied by means of the effective potential [5]. The potential is just the Gibbs free energy per unit volume for systems which, being translationally invariant, have an order parameter independent of position, i.e. q~(T)-= (d~(x))r [6]. As a consequence, the study of its minima yields the picture of the transition - above T~, ~ ( T ) = 0; below ~ ( T ) ~ 0. If one could compute the potential exactly, this would lead to a precise determination of To. The problem of calculating T¢ lies in the fact that no exact computation of the effective potential is available. Previous work has concentrated on approximating the potential via semiclassical expansions around uniform backgrounds. These calculations are performed up to one-loop terms in the expansion, the first to reveal any temperature dependence. Thus, at best, they should provide some sort of bound on T¢. Nevertheless, they are plagued with difficulties which, in our view, raise some questions about the reliability of such estimates. The most serious difficulty is that the one-loop effective potential in the broken (low-T) phase, if calculated via an expansion around a uniform background, ~b¢l= qg, becomes complex for 0 ~< I~1<~~or. Since the transition is second order, its order parameter should be driven continuously down to zero as we approach Tc from the low-temperature phase or, equivalently, the minima of the effective potential should vanish as we go through T¢. In the case of a semiclassical approximation, the minima can be related to the background field so that, to reach qg(T) = (~b(x))T = 0, we have to go into a "forbidden" region (d~- 0), where the potential has an imaginary part. In fact, the work of ref. [7] illustrates that, in the broken phase, the loop expansion around a uniform background will only yield a reasonable approximation for the region outside the minima of the potential. As these minima, which change with temperature, get close to the region 1~1 <~ ~ r , one should not trust it any more. Most likely, other types of background should be included in a semiclassical approach, with the net effect of disordering the system more and more, yielding (~b) = 0 for a lower value of T. Before moving on to discuss our proposal for estimating T¢, a couple of comments are in order: (i) normally, in the literature, the effective potential in the broken phase is obtained by just continuing the result of the symmetric phase to negative values of m 2. This is, in principle, perfectly valid because the potential should be the same no matter which phase is used. However, a one-loop calculation is just the first-order term in a semiclassical expansion. Thus, the background field around which we expand should better be a minimum of the action or else we will have to face negative eigenvalues for the determinant of the fluctuations. Although the calculation of the potential can be done in either phase, the approximation used has it validity closely tied to the particular phase we are considering; (ii) numerical estimates of T¢ via the one-loop result around a uniform background are obtained in the so-called high-temperature limit, i.e. T >>m. This limit is used for the real

C. Aragao de Carvalho et al. / Domain walls at finite temperature

47

part of the potential and the resulting expression masks the difficulty mentioned in the previous paragraph, as no trace of the imaginary part appears in it. Our proposal [8-10] for computing the critical temperature also relies on a semiclassical approximation. However, instead of using a uniform background, we use an "interface"-type background. This interface is just a domain wall separating regions where the field has opposite sign. Within a given region the field is essentially constant and takes one of the values, + ~bv, of the degenerate minima of the classical potential in the broken phase. The critical temperature is then computed by establishing when the surface tension of such a wall vanishes. Technically, this amounts to taking an infinite two-dimensional surface whose profile is a one-dimensional kink and computing where its Gibbs free energy difference per unit area (with respect to a uniform situation) vanishes. In the Ising model of classical statistical mechanics the scheme just described would be related to Peierls' argument to estimate the critical temperature, There, such a temperature is obtained by computing the free energy difference between: (i) the system with a boundary of half the spins up and the other half down; (ii) the system with a boundary of spins of the same sign. The temperature at which this difference vanishes can be interpreted as that for which an interface separating regions of opposite spins becomes energetically favored. Equivalently, it is the temperature at which the surface tension of such an interface vanishes. It can be rigorously shown [11] that this "surface tension" temperature, Ts, coincides with the critical temperature, To, for Ising-like models. Our suggestion is an attempt to obtain Ts and, thus, Tc by means of a semiclassical calculation. It is true that we do more than fix the boundary of our interface, as we choose one particular interface as background field in our approach. As a consequence, we do not allow for fluctuations of the interface whose cost in energy would tend to yield a value of Ts somewhat higher than the one we estimate (we shall come back to this point in our concluding remarks). However, our estimate is the by-product of a well-defined calculation which can be systematically improved and is not afflicted with the difficulties of the usual approach. It provides an alternative which has the advantage of emphasizing the relevance of certain configurations in signalling the transition and helps make contact with ideas of statistical mechanics. If we are willing to accept the usual estimate of 4.9m/x/-A (in spite of the questions raised) as some kind of bound on To, our value of 4.0m/x/~ seems to indicate that the true value of T~ should lie somewhere in this range. The paper is organized as follows: sect. 2 exhibits the calculation of the free energy per unit area (surface tension) of a domain wall; sect. 3 shows the calculation of the effective potential directly in the broken phase and the evaluation of effective couplings as functions of temperature; sect. 4 discusses the phase transition in the high-temperature ( T ~ m) limit, making use of the very simple graphical expansion introduced in sect. 2 and extended in appendix C; sect. 5 exhibits our concluding remarks. Details of our renormalization procedure were left for appendices A, B

C. Aragao de Carvalho et al. / Domain walls at fimte temperature

48

and C. We stress that the same graphical expansion which simplifies the high T ( T ~- m) limit proves helpful in relating the renormalization in the broken phase to that of the unbroken one. 2. The surface tension

We shall consider A~b4 theory at finite temperature. Its hamiltonian density is given by = ½~r2+½(V4Q2 + ½m2~b2+ ~.T~b4.

(2.1)

We may obtain the partition function, Z(/3), by taking the trace Z(/3) = Tr (e-~U),

H = I d3x Y(.

(2.2)

Z(fl) may be expressed [6] as a functional integral over the fields and their canonical momenta, defined in euclidean four-space:

it

where ~"= is the euclidean time, the integral over momenta is unrestricted whereas the integral over fields only includes those which satisfy periodic (since we are dealing with bosons) boundary conditions in ~': ~b(x, fl) = ~b(x, 0).

(2.4)

The normalizing constant N may be chosen so that Z(oo)= 1. Performing the (quadratic) momentum integral yields

Z(fl)=N-l(fl)~[Dq~]exp(-½I; d~"I dax[(Otzq~)2+m2dp2+~Aq~4]). (2.5) The expression inside the square brackets is just the euclidean lagrangian, LPE, and it leads to the equation of motion [~b - mE~b-~A~b3 = 0,

02 [] -- 0,0~, = ~ 5 + V 2 .

(2.6)

If we take m E "< 0, the preceding equation possesses nontrivial solutions, independent of r, which correspond to one-dimensional solitons (kinks). If we introduce the notation x - (XL, X-r) for longitudinal and transverse components, respectively, one such solution (located at the origin for simplicity) will be

Cs(XI3 --- x/g~AI tanh (½lmlxL) •

(2.7)

The situation m2< O, typical of the broken symmetry phase, leads to an effective

C. Aragao de Carvalho et al. / Domain walls at finite temperature

49

potential, at tree level, which has degenerate minima at ~bv= +x/6[ m I/x~-, if we take the temperature to be zero. If we compute the classical action of ~bs minus that of one of the minima, in just one dimension (XL), we obtain a finite result:

AS¢l-- Sd(soliton) -

S¢l(minimum)

= f~oodXL{~eE(~) - ze~(~v)} =

442 Iml3.

(2.8)

A However, in three spatial dimensions the analogous quantity will diverge like an area. We shall regard such a structure as being a two-dimensional domain wall, immersed in three-space, that separates, along the longitudinal direction, the two distinct vacua. Being independent of T, this solution satisfies the periodicity condition trivially. In fact, a simple rescaling ~-~ ~'/fl shows that the kinectic term in ~E is more and more suppressed as we increase the temperature, so that static solutions are most relevant. We shall compute, in semiclassical approximation, the contribution to the partition function coming from the soliton sector, normalized by that of the vacuum sector ( T = 0). This amounts to taking as background classical field, 4~(x, z), the soliton solution (4~(XL)) and one of the (position-independent) degenerate minima (4~v), respectively. We then write ~b(x, T)

-~tb~(XL)+n(x, z) = L~v+ n(x, ~'),

(2.9)

where B(x, T) denotes quantum fluctuations around the classical background. We expand the action up to terms quadratic in ,/ which corresponds to keeping only one-loop contributions in our calculation. If m and A are to be taken as renormalized parameters we shall have to introduce counterterms to obtain physical quantities [6]. These will remove the ultraviolet infinities that appear in the calculation. Thus, performing the expansion in ,/ we obtain

Zs(fl) exp {--f~ d" f d2x~f?o~dXLo~E(~b~(XL)))(det[-[~+m2+~A~b2(XL)]~)-l/2 Zv(fl)

exp

-

dr

d2XT

}

1 ]1.4k2q $ - 1 / 2 dXL~E(~bv) (det [-[3 + ~. .2. . 1_2,,~,vJOJ

(2.10) where (det [M]~) -1/2 - ~ [ D n ] exp { - j o~ d~" j d3x hiM]n}. The integral above is over functions ~/ which are periodic in r, of period ft. Therefore we may write 1 ~ 71(x,~)=----~ ~ a,(x) e "°"~, n~--oo

2~rn to,-----j~

(2.11)

C. Aragao de Carvalho et al. / Domain walls at fimte temperature

50

Using this, the logarithm of the ratio Zs/Zv may be written ~ I] n°°__o~det [Ms(n)]~ In (Zs/Zv) = -~SAASc~-½ In ~ [l-I, o~det [ M v ( n ) ] J '

(2.12)

where we have taken S d:XT = A to be the transverse area and Ms(n) = 6 ( x - x')[~o~-V2+ m: + ½~t4~(xL)], M,,(n) = 6 ( x - x ,)[~o,-V 2 2 + m 2+~A~bv]. , 2

(2.13)

The eigenvalues of these operators may be obtained by solving [ - V 2 + m2 +½Aqb~l(X)]Vj(X) = E2(j)vj(x) ,

(2.14)

where ~bcl(x) may be either ~bs(XL) or ~bv.Then In ( Z d Z v ) = -[3aAs~-½1n [I-I~=-~I~J' (w2+E2(j~))]

LIIL-_ Fl,v (o,2. + E ) ( L ) ) J

(2.15) "

If we invert the order of the products and use the identity 1+

- - - ,

n= 1

(2.16)

71"2

we obtain In (ZdZv) = -[3AAS¢,-E In [sh (½/3Es(L))] + E In [sh (½/3E~(jv))]. Js

(2.17)

Jv

The sums extend over discrete as well as continuous eigenvalues. Finally we may re-express (2.17) as In (Z~/Z,¢) = -~AAS¢~- I E [½/3E~(j~)+ In (1 - e-aE~u~))] ~- 3s

- E [½/3E,(jv) + In (1 - e-t3E~°*))]].

(2.18)

9

Jv

Clearly, the first term in each sum refers to the zero-point energy contribution and the second is a Bose-Einstein term. The free energy difference per unit area between soliton and vacuum (T = 0) sectors is just the surface tension Af=-f~-f~ = - ~ A In ( Z J Z v ) .

(2.19)

Using f = ~ - To-Y,where ~ and b~ denote internal energy per unit area and entropy per unit area, respectively, we may write Ae = ASd-

4421 ml--------~, h

(2.20)

C. Aragao de Carvalhoet al. / Domain wallsat fimte temperature 1

TA6e = - 2/3A {[Tr In Ms]~ - [Tr In Mv]a },

51 (2.21)

where we have made use of det M = - e x p T r l n M , while Ms, M~ refer to the arguments of the determinants in (2.10). Thus far, we have associated the classical action of the soliton with the internal energy and the fluctuations around the soliton background with the entropy. It will be very useful, later on, to note that the full free energy is connected to the F-functional of field theory, i.e. the generator of one-particle irreducible graphs, via

Af = + ~ A {F(~b~(XL))-F(c~v)}=+~AAF.

(2.22)

In fact, we shall adopt the convention F(~bv)-= 0 so as to measure deviations from the zero-temperature vacuum. Thus

AF( ~b~(x, ~')) =- F( ~b~(x, ~-)).

(2.23)

At this stage, all we need to arrive at the free energy per unit area is to exhibit the eigenvalues for both soliton and vacuum (T = 0) sectors and perform the sums in (2.18). Since neither classical background depends on xT, we put

vj(x) = e '~r XT/)jL(XL)

(2.24)

.

The resulting eigenvalue equations are well known [12]. The sums over the eigenvalues require some caution and they are performed in detail in appendix A. Here we just quote the result (with M = Iml) TA6e = -{s~l + T~:2(T)}, 1 f d2kr f / - ~ -

(2.25)

~

3x/2

(-~ )2 ~x/ k~ + x/ k~ + ~M" - - -

M .J-o~ z'a" gM(kL

M

k2+k2+2M 2 ,

(2.26)

¢2(T) = - ~ l I ~d2kT - ~ 5 ~f", n (1 - e-~'/U0 +In (1 - e - ~ ~ ) dkL gM(kL) Xln (1 --e -~'/k~÷~÷2M2)} , M1 I~oo(--~)

gM(x)=2x/~M2{

1

1

}

2x2+M2 t X2 +-2M ~ .

(2.27)

(2.28)

As we had already indicated, this result is meaningless as it stands, since it diverges in the ultraviolet. This was to be expected as we have not yet made use of the counterterms. These are the same as appear in the theory at T = 0 because the

52

C. Aragao de Carvalho et al. / Domain walls at finite temperature

temperature just modifies the infrared, not the ultraviolet, structure of the problem. Thus, in practice, only ~ is affected. Once the countcrterms are identified and their contribution subtracted from (2.26) we obtain a finite result: T AS~= -{~3 "l- T~2( T)},

(2.29)

f3 1 3 ~3 = - ~ / ~ 4 - - ~ M •

(2.30)

We have used rcnormalization conditions at zero momentum and the stcps leading to (2.29) are shown in appendix B. The free energy difference per unit area in the one-loop approximation, with respect to that of the vacuum at T = 0, is finally given by I-

Af=A~_ TAbP=4~M3_x/3 A

1 M3+ T~2(T).

(2.31)

"42 48~

As we shall see in sect. 4 this quantity can be interpreted as a surface tension. The temperature where it vanishes corresponds to the critical temperature, To where symmetry is restored [11]. In the next section we shall try to estimate Tc from the effective potential. The difficulties we will encounter arc the main argument for using the surface tension as an alternative. 3. The effective potential

We shall now compute the effective potential for the theory at finite temperature, in the one-loop approximation, directly in the broken phase. This allows for a comparison with the results of the preceding section and will lead to the effective (temperature-dependent) couplings mentioned in sect. 1. Calculating the potential amounts to finding the generating functional F(&c(x, ~')) in the special case where t~c = ~ is independent of the coordinates (x, ¢). Indeed, the effective potential, U, is just an ordinary function of ~ obtained from F by simply dividing it by fl V ( V is the spatial volume). Thus 1 U ~ ( ~ ) = ~-~r(~b).

(3.I)

Starting from a functional Taylor expansion for F:

F(6o(x, ~)) = ~

d71

daxl • • •

d~'.

daxnF(")(xl, ~'1;... ; XnZn)

×[&¢(Xl,rl)- ~ v ] " " ' [~¢(X,, ¢ , ) - 4~v],

(3.2)

where we have F ( ~ v ) = 0 (as we had done in sect. 2) we arrive at U~ (4~) = ~ 1/~(")(0, 0 ; . . . ;0, 0 ) [ ~ - 4~v]n. n=l

f#~

(3.3)

C. Aragao de Carvalho et al. / Domain walls at finite temperature

53

/~(") denotes the Fourier transform of F("):

F(")(xl, rl;...;x,,r~)=;--';

~. N,. N.=-oo

I d3kl . . d3k"./~(~) (Err) 3 (2~r) 3

x(k,,to,;...;k,,to,)exp[-i

~ (k, xj+to, r j ) ] ,

(3.4)

J=l

where % = 2~rNJl3. The F (") are nth-order functional derivatives of F. In perturbation theory they correspond to all the one-particle irreducible graphs with their n external legs amputated. In fact, the effective potential defined in (3.1) relates to the free energy per unit volume, a~2, of the system, for situations where one h~/s translational invariance: A~2 = min [ U s ( g ) - Uo(&v)] •

(3.5)

We may, then, characterize the minima of (3.5) which correspond to the expectation value (cb(x))r at finite temperature. These minima will be denoted by &v(T) so that ~bv(0) = cbv is the zero-temperature vacuum. The strategy for computing the effective potential in the broken phase, up to one-loop order, is rather simple. It amounts to replacing 4~s(XL) with a constant in (2.10), taking a logarithm and performing a Legendre transform. The calculation o f the determinants in (2.10) can be done either by computing eigenvalues (as in sect. 2) or by making use of a graphical expansion. To see how this goes we identify 1 det (-[-1 + m2+ 1/~2) 2fl'-VIn det ( - D + m2+½A~b2) "

Tzlo- =

(3.6)

tr denotes entropy per unit volume. Using the value of ~b2 = 16m2l/A, we may rewrite (3.6) as 1

TAtr = -2flV

In

det ( - 0 + 2M2+½A(~ 2 - ~bvE))t3 det (-[:] + 2 M 2) '

(3.7)

where we have made use of M 2= [m2l. Then

TAtr= -

1

2BV

Tr In (1 + G,[½A (4~2 - ~b~)]).

(3.8)

The operato~_G_g ~ - ( - I - q + 2 M 2 ) ~ 1 is just a free propagator at finite temperature, with mass x/2M 2. This mass is associated with the excitations around the minima, :i:~bv, o f the broken phase. If we use B--½Atb 2, then Tr In (1 + G ~ ( / ~ - By)) = - -<::> * --<:2>- - - ÷pt~ ~ +

(3.9)

The dashed lines correspond to the "background field" (/? - By), whereas the internal

54

C. Aragao de Carvalho et al. / Domain walls at finite temperature

lines denote propagators G o whose Fourier transform is ~=

1

2rrj

(3.10)

The momentum integrals in Feynman graphs involve (1/~8) ~j 5 d3p/(2~r) 3. Thus, the expansion in (3.9) is just

discrete

sums,

Tr In (1 + G~ ( / ~ - B0) (/~_ (-1)"+' n

,1 oo d3p 1 Bv) ~,=~_~ f (2rr)3 [to2+p2+2M2] . ,

(3.11)

since the background carries no momentum. Expression (3.6) yields the one-loop contribution to the effective potential. The zero-loop term is just the classical action for ~ minus that of ~bv, which can be regarded as an internal energy difference per unit volume: 1 s o , ( 6- ) a ~ = ~-~[

s ~ , ( 6 0 ] = ~a. ( 6 _~ - 6 v ~ ) ~ = l ( B _ K ) ~

(3.12)

Adding (3.11) and (3.12) gives U~(~). Although it is expressed in terms of (/~- Bv) it is a trivial matter to rewrite it as a series expansion in (~-~bv), as in (3.3). We can thus identify the F <") up to one-loop order. The sum in (3.11) is still divergent. The problem comes from the two first graphs in (3.9). Formally, we may sum (3.11) to obtain A -2 U0(q~) = ~..(th. - ~b~)2+2-~,=~" -~

~-~-~3 d3p l n ( l q toZ+p2+2M2 . ½ A ( qq52) ~ z -~]

(3.13)

If we now subtract the contribution of the graphs just mentioned, calculated at zero temperature, the result is finite and given by )t -2 rh2~2+ 1~_ ~ I d3P31n(~)t(q~2-~b2v) ~ U~(q~)=~.(q5 --~v, 2flj=~oo ~ - ~ 14 w2j+p2+2M2] 1

--2

2

f d4p

1

-~A(4~ - ~bv) (27r)4 p2+2M2 +~A2(~2- ~b2)2 I (2~r) d'p 4 P 2 + 12 M 2 '

p_p2+p2.

(3.14)

The two terms that render (3.13) finite are the same one would use at zero temperature since the temperature does not affect the ultraviolet structure of the theory. In appendix C it is shown that they correspond to the contribution of all the counterterms that are required, in the brokenphase, to extract divergences up to one-loop order. Subtractions were performed at zero external momenta, in view of the

55

C. Aragao de Carvalho et al. / Domain walls at fimte temperature

(zero-temperature) renormalization conditions /~(2). r=~( to, = 0 ; P , = 0) = 2 M 2 , ~(4)

(3.15)

+"

(3.16)

r=o~to, = 0 ; p , = 0 ) = A , I

01~(2) YLo { 2

"

cgp l==o.=o - - 1

(3.17)

'

/~(2m/ T=0kwi. . . . 0;p, = 0 ) = 1

(3.18)

Just as we had done before, we may isolate the temperature-independent term in the effective potential. We obtain

A-2

d3 22 1 f ~l~/p2+2M2+½h(d~2 I

-½a(&2-~)

d4p

1

(2,~), e 2 + 2 M 2

4a{)_,/p2+2M 2

1 "~-1)[2(&2 -- ~2v)2 ( d2_~)4 (p2+2M2)2

1 ]f (-g~), d3p In ( 1 e -flx/p2+2M2+(A/2)(4~2d)2) ) + -fl -

-! f ~ l n (1-e-~'/77~).

(3.19)

It is easy to check that the second, third and fifth terms add up to a finite result (they are T-independent). The T-independent part has no divergences. From eq. (3.19) one sees that if ~ 2 ~ ~b2_ 4M2/A the effective potential acquires an imaginary part. This determines the value of ~cr mentioned in sect. 1. Knowledge of the effective potential leads to a natural identification of the effect of temperature on the couplings of the theory. We may define the effective (temperature-dependent) mass and the coupling constant via M2(T) = d ~ 2 ]&=,~v(r}' • _ d 4 U~

(3.20) (3.21)

At zero temperature (~bv= ~b~(0)), we recover M 2 = M=(0) and a = a (0). Eqs. (3.20) and (3.21) may be inverted to yield M 2 and A as functions of M 2 ( T ) and a(T). This may then be used to re-express the results of sects. 2 and 3 in terms of temperature-dependent couplings. One last comment: it turns out that replacing ( m 2 ) with ( - - m 2) in the effective potential o f the symmetric phase, calculated to this approximation, does lead to the correct result we have just obtained. Also, if one is careful to take derivatives at the

56

C. Aragao de Carvalho et al / Domain walls at fimte temperature

appropriate minima (4~ = 0 or ~ = +&v, respectively), the effective couplings are also identical. 4. The phase transition

The results of this section will always refer to the high-temperature limit, T ~, M, in which the thermal wavelength is much smaller than the Compton wavelength of the theory. The leading terms in such a limit are rather easy to obtain if we look at the graphical expansion described previously. Let us begin by examining the effective potential and the usual description of the transition. The first graph of (3.9) involves 1 oo _( d3p

1

C ' ) = 2 ~ = ~,,=_~ _ (2,~) 3 (2,~j/~)2+p2+2M

2"

(4.1)

Performing the sum over j we obtain ( 2 ~ ) 3 2 , / p ~ + 2 M 2 ~-

(2~)' p=+g77T-~(e~'2+2M2-1) .

(4.2)

The first term is the zero-temperature contribution which is cancelled by counterterms. The temperature dependence comes from the second term, R ( T ) , which we rewrite as R ( T) =

1 Io ~ dx x 2 27r2/32 ~/x= + 2 / 3 2 M 2 ( e ~ _

1) .

(4.3)

We may safely take the limit/3M ,~ 1 in (4.3) to arrive at R(T)

T~M

1

2

, r~T •

(4.4)

One can show [13] that the leading behavior of this graph dominates the expansion for T~-M. The other graphs contribute to higher powers of M / T . Therefore, to leading order A - 2 - ~bv)22+~[~X(~bl Ua(4S) =~.(~b 1 -2_&v2)]lT2= 4~" (q~2_~b~)[(q~2 &~)+½T2 ].

(4.5)

Clearly the extrema of this expression occur at 0~2 = {0 4~2v(T) = Wv ,~2 !-rE 4~

(local maximum) (minima at +4~(T))

(4.6)

Since ~b~= ~b~(0) = 6M2/A, the three extrema coincide at a temperature T~ given by ~2 = 24M2. A

(4.7)

Fig. 1 shows the picture of how the effective potential changes with temperature up

C. Aragao de Carvalho et al. / Domain walls at finite temperature

57

r,r,:

I

v13

[ IT=O i [ ,I

\'\~"I

l

./

'

\?L/ -~Ov(Ol'~ "~ *5°v(Ol

l m

Fig. 1. Effectwe potential m one-loop for different values of T<~ To. to To. We note that, by construction, U~(~bv(0)) -= 0, since we chose ~bv to be the zero of the free energy. This amounts to choosing an origin for our free energies, since all we are concerned with are energy differences. It is a trivial subtraction that could be done at any temperature without affecting our results, as long as we are interested in differences. We found it natural to define our vacuum at zero temperature. We now go on to examine our alternative method by looking at the soliton sector. One could just take the high-T limit of (2.31), i.e. examine the limit of ~:2(T). However, our graphical expansion in (3.9) can also be extended to a situation where (~b(x))r is position dependent. In fact this is used in appendix B to take into account the counterterms that render (2.25) finite. All we have to do is to replace t~ with ~bs(XL) in the graphs of (3.9). In the high-T limit the leading behavior is given by the first graph and can be easily calculated:

= -~x/2flAMr

(4.8)

2.

Therefore the contribution to the free energy per unit area for T ~, M is M3 Af=A~-TA~=[4~2---~--+O(I)]-[~/2MT2{I+O(M/T)}].

(4.9)

This quantity will vanish at a temperature Tc given by T 2 = 16

M2 A

(4.10)

I f we go back to the effective potential in the one-loop approximation we will see that it becomes complex for a ~bcr= 16M2/A. This means that even within a --2

58

C. Aragao de Carvalho et al. / Domain walls at fimte temperature

one-loop calculation results obtained from the effective potential cannot be trusted for I~l < 14Sorl.The critical temperature estimated from the effective potential violates this condition since 2~ = 4.9M/x/A'> ~cr ~---4M/x/-A.It is interesting to note that the estimate for T¢ obtained through the surface tension coincides, in the high-T limit, with the value of 4~r. It seems rather suggestive [14] that the temperature at which the semiclassical approximation to the effective potential around a uniform background breaks down should coincide with the estimate for Tc obtained by using a nonuniform background. This is indicative of the relevance of other types of fluctuation in disordering the system- if included in a semiclassical approach, such fluctuations tend, as we have seen, to lower the estimate for Tc. 5. Conclusions We have presented an alternative way of computing the transition temperature on A~b4 theory. Our method is based on a one-loop calculation around an "interface" background (kink). Recent results draw attention to the fact that a one-loop effective potential, computed around a uniform background, besides being complex for I~1 < ~or, is also nonconvex in the region between minima. Although also restricted to one-loop order, our calculation incorporates a novel aspect, typically nonperturbative, that might improve upon the previous result (some of whose problems we have already mentioned). The /~t~ 4 theory in three spatial dimensions at finite temperature can be viewed, in the language of path integrals, as a classical system defined in 4-dimensional euclidean space with one compact direction (the "temperature" axis). However, near the transition temperature, long-range fluctuations are the ones responsible for critical behavior so that the system is effectively 3-dimensional. It is, thus, in the same universality class as the 3D Ising model. That being the case, one should be careful to allow the background field used in a semiclassical approximation to fluctuate. In our proposal this would amount to letting the interface move around as well as vary in thickness, both in a manner compatible with the spin-up-spin-down boundary condition mentioned in sect. 1. The reason for doing so has to do with the so-called roughening temperature, Tr, which, according to ref. [16], signals the restoration of translational invariance in the system. Above Tr, interfaces tend to fluctuate wildly and neglecting such fluctuations in our background would probably lead us to a lower bound for the surface tension temperature, Ts. This is because the cost in energy of a fluctuating surface, being higher, would require a higher value of Ts than the one we have estimated. (For the Ising model T r = 0 in D=2;Tr=O.57Ts in D = 3 ; Tr = Ts in D = 4 and Ts = Tc in all cases [11, 16].) The caveat of the preceding paragraph can easily be taken care of. Furthermore, this type of objection can also be raised with respect to the uniform backgrounds used in approximating the effective potential which, besides being flat and still,

c Aragao de Carvalho et al./Domain walls atfimte temperature

59

have no information about the existence of degenerate minima in the broken phase. The way to circumvent the difficulty is to repeat the calculation of sect. 2 for a bubble of very large radius having different vacua inside and outside. This would be a finite temperature version of the work of Langer [17] and would certainly be a better approximation to Ts, as the bubble surface is allowed to move around (translational modes of its center of mass) as well as get deformed. The result of such a computation yields, for the determinant of the quantum and thermal fluctuations, an expression similar to the one we have obtained in the high-T limit. The translational modes contribute to a prefactor in front of the exponential but the leading behavior in T I M is still given by the graph depicted in (4.8) (now Bs should be replaced with the expression for the bubble background). Details of the calculation will appear elsewhere [18]. The renormalization procedure in the kink sector, up to one loop, has been carried out recently using dimensional regularization [19]. This enables one to have very compact and elegant expressions, valid for arbitrary dimension. A final check on the existing estimates of Tc can be provided by the calculation of the effective potential at finite temperature by a Monte Carlo simulation. This would be an extension of the results of ref. [15]. Work is already in progress in this direction. It is a pleasure to thank J.F. Perez for several interesting discussions and for bringing ref. [16] to our attention. We would also like to thank J.J. Giambiagi and C.G. Bollini for comments and suggestions during the course of this work. Thanks are also due to D. Bazeia and S. Salinas for several discussions that were very helpful. We are indebted to FAPESP (Research Agency of the State of S~o Paulo), Departamento de Ffsica-Matem~itica da Universidade de S~o Paulo, Departamento de Ffsica da Pontificia Universidade Cat61ica do Rio de Janeiro and Centro Brasileiro de Pesquisas Ffsicas for their financial support and kind hospitality during different stages of this project.

Appendix A We shall outline the steps leading to eqs. (2.25)-(2.28) of sect. 2. The eigenvalue equation (2.14), for both soliton and vacuum sectors, is given by

--~z2+ V(z) ViE(Z)= e(j)vjL(Z),

(A.1)

where we have used the dimensionless variable

Z=---4~MXL,

F"q/

e(j) =- E ( j ) - L k ~ J /

-2

60

C Aragao de Carvalho et al. / Domain walls at finite temperature

(Pr for soliton and kT for vacuum) and V(z) defined as

V(z)

/3(tanh2 z - 1) = - 3 / c o s h 2 z tzero

(soliton) (vacuum).

(A.2)

The soliton equation is the Schr6dinger equation for the Posch-TeUer potential [10]. It has two bound-states: 1

eo=-2,

V~L(Z)_ cosh 2 z __>E2(0) = p 2

E1

-- sinh z 2 2 3 2 V~L(Z) cosh2 z'-> E~(1)=pT+~M ;

=

1

(A.3)

(A.4)

and a continuum: e ( ~ L ) = ' iPL, -2

s VpL(Z ) = e'PLZ(3tanh 2 z - 1 - / ~ 2 - 3i/~Ltanh z)

-. E2(pL) = p2 +p2L+ 2M 2 ,

(A.5)

x/'2pL/M,

with /~L = PL the longitudinal momentum. The vacuum equation is just a free Schr6dinger equation: E(/~L) __1 "2 --~kL,

v VkL(Z) =

et,~LZ ..> E2(kL)=k2+k2L+2M2

(A.6)

The asymptotic behavior of the continuum eigenvectors in the soliton sector is given by

vpL(z)

> e '(pLz:~a/2) ,

~(PL) = --2

arctan ~ _ - _ - ~ } .

(A.7) (A.8)

If we take a box of side L in the longitudinal direction, periodic boundary conditions yield /~L,.(x/~ML) + ~5(iffL,.) = 2¢rn,

(A.9)

/~L,.(~/~ML) = 2Trn,

(A.10)

for soliton and vacuum sectors, respectively. From that we derive kL=PL+--

t~(pL)

L

(A.11)

Clearly, kT = PT. We can now relate the sums over the continuum eigenstates for soliton and vacuum sectors. Taking the box to infinity, we may replace ( 1 / )L Y~o~ .... with S ~ dkL/27r and use (A.11) to express the sums of (2.18) in terms of k L and

61

C. Aragao de Carvalhoet al./ Domain wallsat finite temperature

kT. Leaving out the classical action and the contribution of bound states, we will have ( TA6e)~o,t = ~1 f.l

d2kT

[~/(kL+_~)2+k2T+2M2_~/k2L_k~Sr2M 2}

Id2kTI~LdkL{ln(l_e-~./(kL+8/L)2+~'+2M 2) +

2--;-

- In (1 - e-aVk~+~+2M2)}. If we expand (kL+

8/L) 2 for

( TA~)cont= f

(A. 12)

L-~ oo this becomes

_d2kT ~ [~

0 f l-" 2 2 2 "~--~8(kL)--~LII~/ kL + kT+ 2M

+--ln (1-e - ~ ~ )

.

(A.13)

Integrating by parts, using (A.8) and adding the contribution of the two bound states will lead to the desired expressions.

Appendix B Let us go through the procedure used in eliminating the divergences of (2.25)(2.28) to arrive at (2.29)-(2.31). We shall concentrate on the T = 0 terms contributing to TA~, which we have named ~ in the text. If we use D=-~/k2L+k2+2M 2, the integrand of the last term in (2.26) may be rewritten as

gD g 3_~At21,.2 2 32M4 2. . . . v I_~D+~M'4 2.,/2M2=2~/2M20 ,D3(2k2 + M 2) D 3 D3(2k2 + M 2) -

3

(B.1)

o

This is the result of a series of manipulations aimed at splitting it into several pieces, each with its particular ultraviolet behavior in kx and kL. All one has to do is divide and multiply by D and rearrange the expression to finally obtain (B.1). The divergences of the third and fourth terms of (B.1) are extracted by graphs coming from the counterterms. These graphs are

{f0° I "-"(~--

Jt/~,~-~ J (2~y

~,~+t~+2M

}

~ '

(B.2)

f~ drx I d3x f ; d'ty f day[B~(XL) - Bv]KO'x, XL, XT; rr, YL, y-r)[Bs(yL) -- B,,],

(B.3)

with K denoting the Fourier transform o f / ( : 1 ~ f d3p / ( ( % ' q) =~j__~_~ J (2~') 3

1 (to,+%)2+(p+q)2+2M2"

(B.4)

62

C Aragao de Carvalho et al. / Domain walls at fimte temperature

In fact, we have taken aj = 0, q = 0 (subtraction at zero momentum). Thus

q=O (27r)3

(to2+p2+2MZ)2}.

(B.5)

The results (B.2) and (B.5) will cancel the third and fourth term exactly if we just take the T = 0 contributions to the second curly brackets in both expressions (we could have written directly ~ dap/(21r)4). The fifth term is convergent, whereas the two remaining terms, when added to the contribution of the b o u n d states, give a finite result which is quoted in (2.31). The subtraction procedure that we have adopted corresponds to the renormalization conditions shown in sect. 3, eqs. (3.15)-(3.18).

Appendix C The F-functional may be written as a functional Taylor expansion. At T = 0, in D euclidean dimensions, we may write F(d~(x)) =

.=~ In f d°x~" "d°x"F(")(x~" " x") X[~)el(Xl)

Alternatively we may use

B(x)

- -

~)v]" " " [~)cl(Xn)

- -

(~v]"

(C.1)

=~,~d~c~(x) ~ 2 to write

r(B(x))= E ~

d°x, . . . d°x.r~")(x, . . . x.)

n=l

x [ n ( x , ) - Bv]" "" [ B ( x , ) - By].

(C.2)

This last form appears naturally in a graphical expansion analogous to (3.9) for the case of a coordinate-dependent background field tb¢l(x). The F~")(x~ • • • x,) can be explicitly evaluated in one-loop order and we may use them to obtain the F(")(x, ... x,), by using the chain rule of functional differentiation. As an example we may show that

6F

6F

6~b¢,(x) - A~b¢,(x) 6n(x)'

62F 6~/)cl(Xl) ¢~¢~c1(X2)

- A6(xl-

61" 62F x2) 6--B~xl)+A2~b¢l(xl)qbcl(X2)6B(x,)6B(xz)"

(C.3)

(C.4)

Taking the derivatives at 4~ = ~bv(B = By), we obtain

F~l)(x)

= a4,vF~)(x)

=

6Mx/-~F~)(x),

(C.5)

C. Aragao de Carvalho et al. / Domain walls at finite temperature

63

r(2)(Xl, x2) = aS(x, - xgr~')(x,) + X % ~vr~)(x,, x 9 = Z[8(x, - x2)F~l)(x,) + 6M2F~2)(x,, x:)].

(C.6)

The two preceding equations, as well as the analogous ones for F (3) and F (4), can be viewed graphically as F ('> = ~ r(~)= ~ • /-(3)=

taC~

/'(4)----" O

~ O(A'/2),

(C.7)

~-o(a),

(c.8)

,

c~

~ O(A3/2),

*"

~

"* ~

(C.9) ~O(A2).

(C.10)

Starting with the euclidean lagrangian A (6 2 -- 6v~)2 , ~eE = ~(a~6)~+ ~.,

6~ = 6~M - -2,

(C.11)

and using qo = (~b- ~bv), we obtain ~E = ½(0/~)2 + ½(2M2) ~2 + ~.W( 6 x / - ~ ) @3 + I ( A ) @4.

(C.12)

The contributions to F ("), n = 1, 2, 3, 4, coming from this lagrangian, are identical to the ones depicted in (C.7)-(C.10). Therefore, the use of the graphical expansion corresponds to including counterterms like ~3, typical of the broken phase, with no more effort than the one involved in calculating in the symmetric phase! References [1] [2] [3] [4] [5]

[6] [7] [8] [9] [10] [11] [12] [13]

A.D. Linde, Rep. Prog. Phys. 42 (1979) 389 H. Satz, University of Blelefeld preprint BI-TP 82/29 A.B Migdal, Rev. Mod. Phys. 50 (1978) 107 L.D. McLerran and B. Svetitsky, Phys Lett 98B 195 (1981), Phys Rev. D24 (1981) 450; J Kuti, J Polonyl and K Szlachanyi, Phys. Lett. 98B (1981) 199 D.A. Kirzhnlts and A D Lmde, Phys. Lett. 42B (1972) 471; L. Dolan and R. Jacklw, Phys. Rev D9 (1974) 3320; C Bernard, Phys. Rev D9 (1974) 3312; S Weinberg, Phys. Rev D9 (1974) 3357 D Amlt, Field theory, the renormahzatlon group, and critical phenomena (McGraw-Hill, New York, 1978) C M Bender and F. Cooper, Nucl Phys B224 (1983) 403 I. Ventura, Phys Rev B24 (1981) 2812 G.C. Marques and I. Ventura, Univ of Silo Paulo preprint IFUSP/P-342 (1982) D. Bazeia, G C. Marques and I. Ventura, Rev. Bras. Fis 13 (1983) 253 J L Lebowltz and C.E Pfister, Phys Rev Lett 46 (1981) 1031 R. Rajaraman, Phys. Reports 21C (1975) 227 H.E. Haber and H A Weldon, Phys Rev D25 (1982) 502

64

(7.. Aragao de Carvalho et al. / Domain walls at finite temperature

[14] C. Aragfio de Carvalho, D. Bazeia, O.J.P. l~boli, G.C. Marques, A J. da Silva and I. Ventura, Phys Rev. D31 (1985) 1411 [15] D.J. Callaway and D.J. Maloof, Phys. Rev. D27 (1983) 406 [16] J. Bncmont, J.R. Fontaine and J.L. Lebowitz, J. Stat Phys. 29 (1982) 193 [17] J.S. Langer, Ann. Phys. 41 (1967) 108 [18] C. Aragfio de Carvalho and G.C. Marques, in preparation [19] C.G. Bollini and J.J. Giambiagl, Rev. Bras. Fis., volume in honor of the 70th birthday of Mano Sch/Snberg (July 1984)