Phase Transitions in Field Theories

Nuclear Physics BI70 [FSI] (1980) 388-408 North-Holland Publishing Company

FIRST AND SECOND ORDER PHASE TRANSITIONS IN GAUGE THEORIES AT FINITE TEMPERATURE

Paul GINSPARG 1

CEN-Saclay, Bofte Postale No. 2, 91190 Gif-sur-Yvette, France

Received 31 March 1980

We consider in general the nature of the phase transition which occurs in 4D gauge theories coupled to scalar and spinor fields at finite temperature. It is shown that the critical behavior can be isolated in an effective 3D theory of the zero frequency mode whose lagrangian may be calculated explicitly in weak coupling perturbation theory. This lagrangian, in turn, may be investigated by means of standard e-expansion techniques. Theories with an asymptotically free gauge coupling constant possess no stable fixed point in the e-expansion and are inferred to have weakly first-order phase transitions; theories not satisfying this condition may have second-order transitions.

I. Introduction

Arguing in analogy with superconductivity, Kirzhnits and Linde [!] suggested that a spontaneously broken symmetry in a relativistic field theory coupled to a finite temperature heat bath would be restored above some critical temperature Tc __/3-1. Analyses by Weinberg [2] and Dolan and Jackiw [3] then established this effect on a quantitative basis. Considering a general renormalizable field theory of gauge fields gauge-invariantly coupled to scalar and spinor fields, Weinberg [2] has given a formula for the transition temperature in terms of the gauge, scalar quartic, and scalar-spinor Yukawa coupling constants. However, the question of whether the phase transition is first order, proceeding with the scalar field expectation value jumping discontinuously to zero at ~, or second order, the scalar field expectation value vanishing continuously at T c, is left open by the above authors since the validity of their calculations does not extend to the immediate vicinity of To. For temperatures too close to To, Weinberg [2] notes that the coefficients in his expansion grow large, invalidating his procedure, and Dolan and Jackiw [3] find a spurious imaginary part in their effective potential, rendering it inapplicable. The problem, as emphasized in ref. [2], is that standard perturbation theory breaks down due to the infrared divergences associated with vanishing masses (or, equiva- lently, long wavelength fluctuations) as the phase transition is approached.

N.S.F. predoctoral fellow, ADW fellow. Permanent address: Lab. of Nuclear Studies, Cornell Univ. Ithaca, N.Y. 14853, USA.

388

P. Ginsparg / Phase transitions 389

Powerful methods for dealing with such infrared divergences in 3D theories have developed from the application of renorrnalization group methods to statistical mechanics, in particular from the e-expansion of Wilson and Fisher [4]. In this paper, we formulate a procedure whereby these methods can be brought to bear on the critical region associated with the finite temperature phase transition in theories of the general r1~s considered in ref. [2]. The argument here centers upon the fact that in weakly coupled field theories, i.e., those for which a perturbative expansion is justified, the transition temperature is parametrically large. Specifically, one has fl:~gZ/p2, where g2 denotes the largest in order of magnitude of the scalar quartic, squared gauge, and squared Yukawa coupling constants (all taken

390 P. Ginsparg / Phase transitions

then the Meissner effect [Higgs Mechanism] operates to cut off the long wavelength fluctuations of the formerly massless gauge field which had tended to disorder the system. The system is thus rendered unstable with respect to the acquisition of an infinitesimal order parameter and is driven to a first-order transition. This argument, at least when the fluctuations are still dominated by gauge field loops*, applies perhaps even more forcefully to relativistic gauge theories where there are typically many gauge fields which have their long wavelength fluctuations cut off at the phase transition as a result of picking up masses via the Higgs mechanism.

Examining certain special cases, other authors have found examples of first-order phase transitions in the theories we consider here. One can, for example, easily produce a first-order transition when there exists a cubic invariant which can be added to the scalar interaction lagrangian [7] (provided, of course, that the cubic terms are taken large enough to insure that the phase transition occurs outside of the critical region where scalar field fluctuations become important). However, since this is not always possible and since, moreover, most authors prefer to eliminate this type of term by imposing a discrete inversion (~--~ -~p) symmetry, we find it useful and interesting to study the possibility of fluctuation induced first-order transitions in the absence of such terms.

Another pertinent special case occurs when the symmetry breaking at zero temperature is chosen in the Coleman-Weinberg [8] mode, i.e., dominated by one-loop gauge field radiative corrections. With the zero temperature theory taken either in the "massless'" form (O2V/Oep2[~o_o=O) [9] or with a small mass (O2V/Ocp2lr_o~ga(cp~2) [10], examination of the finite temperature effective potential suffices to establish the existence of a first-order phase transition in which the discontinuity in the scalar field expectation value is typically of the same order as its zero temperature value (while refs. [9, 10] consider explicitly only U(I) gauge theories, the results are easily extended to the general case). We remark that this situation (scalar quartic coupling


monopole [14] number which require a detailed understanding of the region close to the phase transition could perhaps be more fully understood through application of the ideas presented here to a realistic theory of specified gauge, scalar, and fermionic content.

In sect. 2, we develop our formalism and method of extracting the effective 3D theory using the simplest available example of 4D q~'* scalar field theory. The general case is treated in sect. 3, where we establish the form of the effective 3D theory associated with an arbitrary 4D gauge theory coupled to scalars and fermions. In sect. 4, we analyze the effective 3D theories by means of e-expansion methods and discuss examples of first and second order phase transitions. Sect. 5 presents some concluding remarks.

2. 4 sca la r f ie ld theory

To fix the notation, we start with the generating functional for a zero temperature scalar field theory in euclidean space-time,

[ ~(0~) ~ + ~(9~ ) + C.T.+ f ddxJ '~], (2.1) Z(J')=fexp - f ddx( ' 2 Itt2~2 /~ 4

where q0, _~2, and 2~ are, respectively, the renormalized field, mass squared, and quartic coupling constant, and C.T. indicates renormalization counterterms. We will render the theory finite using dimensional regularization.

At finite temperature f l - I we exchange the causal boundary conditions at real time t = _+ oo for boundary conditions periodic with period fl in euclidean time*. This implies a finite temperature generating functional of the form

f~ [ I ~ f, l{[2~rn) 2 it2) Z(J') = exp - ~ .=-~ [---fi-- +k 2 - p.(k)q)_.(-k)

l Eo 4! f13 n, , ,,

+ C.T.+ fJ'+], (2.2)

* References to the original work on the subject can be found in the standard references of [15]. A heuristic derivation of the finite temperature formalism is given, for example, in ref. [19]. Note that the functional integral acquires a temperature-dependent normalization factor [15, 19] important in calculations of the partition function but which does not enter in calculations of expectation values and Green functions. It will be safely ignored in the remainder of this paper.

392

where

P. Ginsparg / Phase transitions

= 1 (27r)a-1 f da-'k'

1 e ' . . . . . - -=

~,, = 2~rn /#.

The counterterms in (2.2) are defined to be precisely those of (2.1); they are independent of temperature and continue at arbitrary temperature to remove ultraviolet divergences from the theory in all orders of perturbation theory*.

The objective at this point is to integrate out the n ~ 0 modes to leave an effective 3D theory of the remaining n = 0 mode described by some effective lagrangian t~cff. With this in mind, we rescale the fields opt(k) by a factor l /~ / f l in order that the quadratic term conform to standard convention for d - 1 dimensional theories, and write

- - - a ' k" -k ' -k " ) 4! n,n',n"= -ao ,k',k"

+ C.T.+ fJ ]. (2.3)

Next, to facilitate integration over the n # 0 modes, we introduce the following diagrammatic notation: an n = 0 mode is represented by a single line and the sum over n ~ 0 modes is represented by a double line propagator (fig. 1). The interaction vertices resulting from the interaction term in (2.3) are represented diagram- matically as in fig. 2. We note that ~04 field theory in 3 dimensions has a dimensionful coupling constant; the above formalism conveniently chooses it to be given in units of ft.

To compute Ecff, we need consider graphs with all external lines corresponding to the n = 0 mode and all internal lines corresponding to sums over the n ~: 0 propagators as in the examples of figs. 3-5. These graphs contribute to the mass

* That the zero temperature renormalization counterterms continue to remove the divergences in all orders of the finite temperature theory is implicidy indicated in ref. [2] and explicitly demonstrated in the case of dimensionally regularized theories in ll6].


(a) (b)

Fig. 1. Scalar propagators: (a) n ffi 0 mode, Co) sum of n ~ 0 modes.

XX XX (o) (b) (c) {d)

Fig. 2. Scalar quartic interaction vertices.

Fig. 3. Leading contribution to the mass squared term in 12el t.

V

O QQQ

Fig. 4. Some higher-order contributions to gaf.

8 Q /x Fig. 5. Two-loop contributions to tree approximation mass term in l~eff.

and wave-function renormal izat ions, and to the N-point functions of the effective

theory of the n = 0 mode. Since all n ~ 0 propagators have the infrared cutoff 2~r/fl, fl sets the scale for the estimation of these graphs. We are thus led to a character izat ion of the graphs of the theory by their order in )~ and ft.

A graph with V vertices, N external legs, and superficial degree of divergence I D = - V - ~ N + 3 (calculated using 3 as the dimension of all momentum integra-

tions) has associated to it a factor

(Xl/~)v/~ -" = xv/~ :/'-~ . (2.4a)

Let V b, V, and V d be the number of vertices of the types depicted, respectively, in figs. 2b -d . Then V = V b + V + V d, N = 2V b + V, and the factor (2.4a) becomes

( ~//3 ) v B - n ~ ~(,/2O/2 N- 2) + V, + 2 Vd). (2.4b)


The leading graphs, with V = V d = 0, result in an N-point vertex in E~rt with coupling proportional to XO/2XO/2XN-2)).

This result indicates that, except for the case N = 2 where the contribution goes as 4 / f l2~4 , the effect of the n ~ 0 modes is perturbatively small compared to the leading order of the n = 0 theory. For N = 4, for example, their contribution of X3/2 is down by a factor of 4 from the 4/]3 appearing in the tree approximation to Een" Their contribution to higher N-point functions is even further suppressed relative to the contribution of the n = 0 theory. It is clear, then, that our classification of graphs in the region fl.~XI/2/tt allows one to construct ~ff for the 3D theory systematically in a power series in 4.

Let us now turn attention to the one contribution of the n =# 0 modes which is not suppressed by a power of 4, the mass renormalization graph of fig. 3. Denoting its value by -M2(~,/2) 1 fpqOo(P)CPo(-P), we see that

1X n~Of k l M2(fl'~2) = 2 fl k2 + (2~rn//fl)2 _ g2 (2.5)

Performing the k integration in d - I dimensions by the usual rules [17],

1 7r'd-1'/2F({-- d) E - t tz M2(fl '"2) = 2fl (2or)d-' n=O

Recalling the condition f1292 4, we next expand M2(fl,//.2) in powers of f12p2:

_ A ~"a- ')/2 / 3 d l [ (_d_~_ fl2tx 2 MZ(fl'/'t2) fla-2 (2vr) a - - ' Fk~-~) , , - I ~ (2rrn) 3-d 1 - ) (2~n) 2

f14 4 ]

- , ~ (2~r)d-3f(3-d) 2 (2~r)5_df(5--d) fld_2 (2tr) a- I" -- d - 3 f1292

- 24fl2 ~ + ~' / - ln2~r I/2 + In

h l "at- 4~2~4~(3) ~"'7. + 0(/~4~ 6) + 0(4 - d). 2%r q

(2.6)


[The last line follows from the relations ~'(- 1) = -~ ,~ ' (5 -d )= l / (4 -d )+y+ I ]d )= -2V~ (1 +(4-d) ( - i~ , - ln2+ 1)+O(4-d)2) ] . It O(4 -d) , and F (~- i

is easily verified that the pole term in l / (4 - d) is precisely cancelled by the usual zero temperature mass counterterm of (2.1).

The finite terms in (2.5) appear as corrections to the mass squared in the tree approximation to the effective 3D theory:

~ff=~ + +. - . %(k)%(-k)+h/f l .v~ +' ' " (2.7) 4!

It is now evident that there occurs a transition between broken and unbroken symmetry states at a temperature which to leading order in ~, corresponds to f12 = ~/24/t2 (in agreement with refs. [2, 3]), self-consistently justifying use of fl~)~/2/# as a means of accessing the critical region of the theory. It is perhaps helpful to point out that our consideration of the n = 0 mode alone is sufficient to understand the symmetry behavior of the original 4D theory since space-time independent source terms, used for example in defining the effective potential for the 4D theory, couple only to the n -- 0 mode anyway. We moreover note that the n = 0 mode also embodies in its entirety the static critical behavior of the original theory (2.2). This behavior, in turn, is none other than the critical behavior of ~4 field theory in 3 dimensions already well studied via the e-expansion, and to which we shall return in sect. 4.

We pause here briefly to illustrate how higher-order corrections to ~en are calculated using as a specific example the next leading corrections to the transition temperature. O(~ 2) corrections to fl~ arise from the non-dominant terms of the graph in fig. 3, i.e., the until now neglected finite ~x/t 2 terms of eq. (2.6), and in addition terms of the form 2x2/fl 2 coming from the two-loop graphs of fig. 5. The crosses denote the one-loop mass and coupling constant renormalization counterterms (C.T.) of (2.1), fixed by some renormalization prescription for the theory. The pole terms from the graphs in fig. 5 are easily calculated and found to leave l/(3.26r2)()~2/f12)l/(4-d), which acts as a counterterm in ~eff, ultimately cancelling against a pole term of the same absolute value from fig. 6. The finite parts of these graphs, dependent upon the chosen renormalization prescription, will imply a transition temperature of the form

24/~2 (1 + O(A) + O(h lnh)) To2= l/tic2--- )~2

(3 Fig. 6. Two-loop mass renormalization calculated within ~eff'


3. Gauge theory coupled to scalars and fermions

We now proceed to redo the analysis of sect. 2 in the general case of gauge fields coupled to scalar and spinor fields. To facilitate comparison, we adhere roughly* to the conventions of ref. [2] and start with a theory whose zero temperature generating functional is defined by

Z( J ) - - f expl- f dax( F2+ ~l (~- igA:Oa)cP] 2+ ~(~'+ m- igt~f)~k ~,A,~ I.

+ ~/Fiq,,ep , + P(~0) } + C.T. + gauge-fixing terms + source terms].

(3.1)

a a a abc b c A~ is a set of gauge fields, F~=O~,A~-O~A~,+g[ A~,A,, is their field strength ' fields. Oq tensor, cp i is a set of hermitian spin-zero fields, and ~k,~ is a set of spin ~ a

and t~,, are the hermitian matrices representing the gauge generators on the scalar and fermion multiplets, respectively, and the mass matrix m and Yukawa coupling matrix F i are gauge invariant. Finally, the scalar field potential P(q~) is a gauge- invariant quartic polynomial, even in q0.

Again, some sort of weak coupling condition is necessary to justify the use of perturbation theory. For definiteness, we assume the lagrangian to be characterized by a small gauge coupling constant g >2~ >>g4, arbitrary F


the reasonable assumption that methods such as those of ref. [9] can be used to demonstrate the gauge independence of observable quantities.

The finite temperature version of the theory is obtained from (3.1)just as (2.2) was obtained from (2.1). In momentum space, we replace all integrals [ l / (2r)al /ddk with

1 1 1 d'-'k,

and write the energy components k as ~, = 2~rn/~ and (2n + l)~r/fl for bosons (including Fadeev-Popov ghosts) and fermions, respectively [15, 19]. Following the reasoning applied prior to (2.3), we next rescale the fields ~o, A, and ~ by a factor l /~fl so as to work with propagators normalized in accordance with d - l dimensional theories. This rescaling leaves a set .of theories, indexed by the frequency mode n, interacting via gauge coupling g~ ~/fl, Yukawa coupling Fi/X/fl, and scalar quartic coupling )~ijkl/fl" Again, the couplings become dimensionful with a scale set by ft.

The diagrammatics from here parallels that of sect. 2. We introduce single and double line propagators to denote, respectively, the n --- 0 and n v ~ 0 modes for the gauge and ghost fields (figs. 7a, b), and double directed lines to denote the sum over the fermion frequency modes (fig. 7c). As pointed out in sect. l, fermions, possessing no zero frequency mode (fermions do not Bose condense), can be integrated away entirely and do not appear explicitly in the effective 3D theory of the critical region. The new interaction vertices, as shown in figs. 8, 9, are the familiar 4D vertices depicted with all fermion propagators drawn double lined and some choice of the boson propagators drawn double lined.

>

(o) (b) (c )

Fig. 7. (a) Gauge boson propagators, (b) ghost propagators, (c) fermion propagator.

n T i XA X Y

Fig. 8. Vertices with maximal number of n m 0 propagators used to construct leading contributions to


XXXXXY Fig. 9. Vertices used to construct non-leading contributions to ~cff.

"xx-\x f f f "

Since, as will be shown shortly, the imposed weak coupling conditions imply fl~g2/lt2 (where # is now a typical zero temperature mass scale relevant to that sector of the theory whose order-disorder transition is under consideration), taking f l " tic once again allows the use of/3 as a small parameter which helps identify the leading contributions to t~er f in the critical region. Characterizing graphs according to their order in/3 and the coupling constants, we find, just as in eq. (2.4), that a graph with N~ external scalar lines and N 2 external gauge boson lines has associated to it a factor /3 (N'+N2)/2--3 times powers of g, I]'s and hijk/s coming from the vertices. For a given N~ + N 2, it is clear that the leading graphs contain only those vertices with the maximal number of external lines (fig. 8). Moreover, as N~ + N 2 is increased, this leading contribution is reduced by factors of/3 and g. The result is that, except for the case N~ + N 2 -- 2, where the contribution goes as g2/ /32~gO, the effect of the n :~ 0 modes is perturbatively small compared to the leading order of the n -- 0 theory; Eeft for the 3D theory can thus be constructed in a power series in g whose lowest non-trivial order is calculable from the graphs of figs. l0 and I I.

Denoting the sum of the graphs in fig. 10 by 2 l -Mij(/3)~ fvepo( p)Cpo(-p), we proceed as in eqs. (2.5) and (2.6). The leading contributions to Mi~(/3) from figs. 10a, b and c are, respectively,

1X,jkk ~0 L l =X'Jkk 2 fl k2 + (2rn/fl)2 _ #2 24/32 + O()~),

(3.2a)

"~ fk (((2n + 1)=//3) 2 + ( p + k)2 + m2)(((2n + 1)~.//3)2 + k 2 + m 2)

i - 24flz Tr[ F,-/0Fj70] + O(r2),

-~2 fk 1 _ 3g 2 (0,~0,,)0." (O"O")e(d- l).~oZ k~ + (2~rn//3)~- 12132

(3.2b)

(3.2c)

P. Ginsparg I Phase transitions

C) 0 (a) (b) (c) (d)

Fig. 10. Leading contributions to scalar boson mass squared matrix in err.

399

_0_

__0

{a) (b) (c)

Fig. I 1. Leading contributions to the gauge boson self-energy in ~.tt.

In the Landau gauge, fig. 10d has no I / f l 2 part; itsp 2 dependent pieces, along with those of fig. 10b, act as wave-function renormalizations in higher orders of Eat.

The graphs of fig. I 1 contribute to the gauge boson self-energy H~(p) , where p is the 3-momentum of the external legs. The result takes the form

g2 I I I~ =-~ { ~ trO#O b + l trtat b + ~f=


scalar sector. The scalar sector, in turn, has the tree approximation mass term, from eqs. (3.2a- c),

2 +~l (h,~/** + Tr[ Fi'/0F, y0] +6g2(O~oa)o)}fpg~o(p)~o(-,o). (3.4) -- ~tij 24fl2

The implications of this type of term have been discussed by Weinberg [2]; it typically has the effect of restoring a broken symmetry above some parametrically large critical temperature, 1~tic ~to/g , by changing sign from negative to positive. The advantage of the formalism employed here [besides the ease with which we derive (3.4)] is that it allows, as illustrated in sect. 2, a systematic computation of higher-order corrections to the transition temperature and other interesting properties of the critical theory.

4. e -expans ion ana lys i s

We wish to understand the behavior of 3D theories of the form

i | 2 1 ~eff ~-~ F2+~ V ig AaOacp +~mcPiq~Y+~-v--'fl -q~'qjqkq~l?'' (4.1)

as some of the eigenvalues of myy, the temperature-dependent mass matrix of (3.4), vanish. To study the critical behavior it suffices to consider an effective renormalizable subset of the theory, in which the only fields which appear are those whose masses are small or zero. We will therefore examine here theories of the type (4.1) for which the mass eigenvalues are all degenerate, m 2 = m2--.0 for all i. The fl dependence of the coupling constants, no longer essential to the discussion, will henceforth be absorbed into the A's and g2.

We deal with the infrared divergences which appear as the masses vanish by using the renormalization group [4] to relate a given theory to an equivalent theory with larger masses. A scale-invariant theory, characteristic of a second-order phase transition at the critical point, will correspond to a trajectory in coupling constant space leading to a point fixed under the renormalization group. To find non-trivial fixed points within the framework of perturbation theory, we must work in 4-e dimensions* and look for fixed points of order e, assumed perturbatively small. The physics of interest, of course, occurs for e-- l but nonetheless lowest-order results in e generally agree well enough with experiment that they may be regarded as a faithful description of second-order transitions.

* To avoid any possible confusion, we should note that this continuation in the number of dimensions plays a conceptually different role than that of the dimensional regularization in sect. 2. There, working in the neighborhood of 4 dimensions was used in order to define the original 4D theory; in the present contexL an nssume.~lly well-understood 4D theory is used as the basis for an expansion in towards a 3D theory of interest.


It has been pointed out [20] that theories which possess no stable fixed points within the e-expansion can be reliably identified, on an ad hoc basis, with experi- mental systems which undergo first-order phase transitions. In general, however, this identification is perhaps not entirely well-grounded [21], so we shall assume that we need to supplement the mere absence of stable fixed points in the e-expansion with some demonstrat ion that there is a first-order transition. For- tunately, in all cases to be considered here, trajectories are found to lead to the regime of classical instability, where, as we shall confirm, the demonstrat ion is relatively simple [21, 22].

For a theory in 4 -e dimensions, the renormalization group flows in coupling constant space are generated by the differential equations

d---t- = fix(g 2, X, e) = eh - Bx(g2, X ), (4.2a)

dg2 = Bg(g 2, X, e) = eg 2 - fig(g2, X) (4.2b) dt

where Bx(g 2, X) and Bg(g 2, ~) are the B-functions calculated for 4 dimensions* (the parameter t is defined to increase as trajectories tend towards the infrared rather than the ultraviolet, hence the extra minus sign). We are thus able to make use of the B-functions already well tabulated [24] from investigations of asymptotic freedom for this class of theories. A fixed point is a point (g ,2 ,~, ) in coupling constant space such that

0 = flg(g*2,X*,e) = f lx(g*2,X*,e). (4.3)

As a simple example, let us return to the theory considered in sect. 2 in the generalized form

I 2 I 2 2_i_ ~' /p2~2 ' .~- .~ ) (4.4)

with ~ now an N-component vector. The B-function for this model

f l (X,e) = eX 1 N +____88 X2 + O(XB) ' 8~r 2 6

(4.5)

has a stable fixed point at

6 X* -- (8 7r 2 ) ~ e. (4.6a)

. Eq. (4.2) is true to all orders in g2, h, and e if a minimal subtraction type prescription is used to define the renormalization counterterms of the theory [23]. It is true to one-loop order regardless of the renormalization prescription.


All 4D field theories for which (4.4) serves to model the behavior near a phase transition are thus predicted to have second-order transitions. The critical exponent v, for example, is given by [25]

_1 2 N+2 - - 2 = ----7-~o e + t)te ). (4.6b) v N -1 -o

This result applies to a large class of theories, with arbitrary fermion sector, whose global O(N) symmetry is spontaneously broken at zero temperature. Similar results can be easily obtained for scalars in higher tensor representations and with different global symmetry groups.

Proceeding now to the case of gauge theories, analysis of eq. (4.2) enables an immediate important conclusion. To one-loop order,

f lg(g2,X,e)=eg 2+ b~g 4, 8"n'"

bo= [ ~C2(G ) -~C2(S)], (4.7)

where the constants C2(G ) and C2(S ) are defined in terms of the group structure constants and scalar representation matrices by facafbca_ C2(G)Sab, TrOa0b= C2(S)8 ab [24]. If the gauge coupling constant is asymptotically free, i.e., b o > 0, then the only possible solutions to (4.3) have g.2 = 0. Furthermore, fig(g2, A, e) > 0 for 0


and b 0 < 0 is obtained for m > 22(N- 2). To provide an easily analysable scalar quartic potential, we impose an additional O(m) symmetry among the m N-vectors and write

I 2 1 " a a 2 E~f,--~F +i[(a~,-tgA~,O )~.] +m2(~..cp.)

I + ~x,(%. %)(w~,-,~a) + "x 2(%. ~o~,)(%.,~) (4.9)

(a and fl are summed freely from 1 to m). The equations

dt - e)~l - - - ! ( (Nm + 8)~k2 + 12~.2

16~r 2

+ 4(N+ m + l)X,)k 2 - 3 (N- l)g2h, +]g4}.

d)~2 1 dt = cA2- - (2 (N 16~r 2

+ m + 4)2~22 + 12)~1)~ 2

- 3 (N- 1)g2)~2 +3g4(N - 2)} (4.10)

are found to have a stable fixed point* in the positive definite region of the scalar quartic potential for m>~40N (fig. 13). In the large m limit, we find explicitly

60 X? = 8~'2e(-

7~ _- 8~r2 e + 32- 10N m 2 -t-

967r2e g,2 = (4.1 la)

m-22(N-2) '

, 2 ( 4248N _ o ( ) ) , e I + + (4 . l ib) ~, m m-~ "

We thus predict second-order transitions for all 4D theories whose critical behavior at a transition point is modeled by (4.9). (A similar situation ensues in the case of an SU(N) gauge group coupled to sufficiently many vector representations.) For completeness, we note that in the special case O(N = 2) (scalar QED), there exists another fixed point, stable only in the subspace 2, 2 = 0, for n -~ 2m > 365.9 [5]. This

* I wish to thank S. Hikami for allowing comparison with his unpublished rsults for this model. Eq. (4.10) differs from the corresponding results quoted by Cheng, Eichten, and Li [24].


~2

2-'~-" =

x~-xl/2

Fig. 13. Flow diagram in the g2=g*2 plane in coupling constant space for the model (4.9) with m ~> 4ON. In the shaded region, the scalar quartic form of (4.9) is unbounded below. The fully attractive

fixed point is the one at upper left.

fixed point is given by

g*2 ~_____ 192,r2e

8~r2e [14 36+A] x =.--g-gL ~ -g '

A = (n 2 - 360n - 2160) '/2 , (4.12a)

and has

l 2 F ~'(n + 2) 216 + n + 2 A/. (4.12b) ,, 2 (n+ 8) t n n J

The remarks fol lowing eqs. (4.6b) and (4.1 lb) apply as well, of course, to this fixed

point.

We should now like to say more about the cases which admit no stable fixed

points*. It will be conven ient in the fol lowing to shift to a descr ipt ion of the

coupl ing constants as funct ions of a d imens ionfu l scale parameter ~ [i.e.,

~(3/OK))~(K), etc., replaces 3Mt) /Ot in (4.2)]. We suppose that the ~,,jkt(~)'S have been def ined at some scale M to be generical ly of order g2. Let

X(K) = min ~i),l(~)ninJn*n t (4.13) Inl2-1

* The discussion which follows is similar to that which has appeared in a purely 4-dimensional context [26, 271 .

P. Gimparg / Phase transitions 405

be the minimum value of the quartic form X~jkt(x)q~:pjePkept on the unit sphere, and let n~o(X) be the unit vector along which this minimum is attained. Then the tree approximation to the effective potential of the theory (4.1),

, 2 X,j~t('~) u(cp,) = u(o) +~m ~,~, 4! ~gi~j~k~l' (4.14) has, for m 2 large and negative, a minimum at %= (~0)n~(K), where (~52= -6mZ/~,(x). In order to avoid large log((~o)/x) factors in higher-order radiative corrections which could invalidate the tree approximation (4.14), we may self- consistently choose a renormalization point ~ = (~05.

Let us now assume that the original choice of coupling constants at the scale M defines a second smaller scale K o for which

~(K0) =0. (4.15)

This occurs if and only if the infrared renormalization group trajectories lead to the edge of the classical stability region beyond which the scalar quartic potential no longer has a lower bound. Then when I m21


g2

-x

g2

(o) (b}

Fig. 14. Possible behaviors of infrared trajectories in the absence of fixed points

We will shortly re-express this estimate in terms of more physical parameters. How often is it that renormalization group trajectories with the assumed proper-

ties will exist? Since the condition (4.15) specifies a p - 1 dimensional submanifold of a p-dimensional parameter space, and since scalar quartic couplings intrinsically tend towards negative values in the infrared, there will always be at least some non-trivial set of trajectories for which (4.15) is fulfilled. There might exist as well, however, regions in coupling constant space corresponding to " runaway" trajectories. By this we refer to trajectories which lead outside the perturbative regime without approaching either a fixed point or the edge of the classical stability region, and thus about which we could make no definitive statement. The general form of the fl-functions, flx( )~,g2,e) = e)k - (1/8~r2)(A)k 2 - B~g 2 + Cg 4) (in matrix notation), allows this possibility to be investigated in terms of the variables Rijkt ~ijkl/g 2, for which we have

1 O__RR= _~(AR 2_ (B_bo)g+C) (4.18) g2 dt 8~r 2

With the gauge coupling asymptotical ly free, an attractive fixed point in the space of R's corresponds to runaway trajectories tending asymptotically to infinity at a fixed angle in coupling constant space (fig. 14a).

The only obvious means of assessing this possibility is to analyze the explicit fl-functions for any given model. Following Cheng, Eichten, and Li [24]*, the author has considered SU(N) and O(N) gauge theories (all with gauge coupling maintained asymptotically free) coupled to one vector, two vector, m vector, adjoint, second rank tensor, and adjoint plus one vector representation and found

* The search performed is similar in principle to that performed with respect to ultraviolet trajectories in [24]. A crucial difference, however, is that fermions can not be added here in order to adjust b 0 to arbitrarily small values, b o is fixed uniquely, given the choice of gauge group and scalar representations, by eq. (4.7).


them to behave qualitatively as in fig. 14b, with all trajectories leading to the edge of the stability region, rather than as in fig. 14a. These theories can thus all be predicted to have first-order phase transitions.

It remains to estimate the size of the scale K 0 relative to the scale M. For generic values of order g2 for the )~ijkt(M)'s, we expect f~(M)/g2(M)~ I. The relation (I/g2)x(O/OK)~(K)/gE(x)~l following from (4.18) thus implies x0~e-C/82M, with c some constant. Since the scale M in this problem is naturally given by the zero temperature expectation value (~) r -0 , we predict, from (4.17), that the discontinuity in the expectation value of the order parameter at the first-order transition is exponentially small compared to its zero temperature value,

- c/s2/ - (~0~r- r~e \9~;r-o. (4.19)

If, on the other hand, there were some special relations among the )~uk~(M)'s so that X((cp ~r-0)


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Phase Transitions in Field Theories