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Crossover at a bicritical point: Asymptotic behavior of a field theory with quadratic symmetry breaking
ANNALS OF PHYSICS114, 356409 (1978)
Crossover at a Bicritical Point: Asymptotic Field Theory with Quadratic Symmetry DANIEL J. AMIT
AND YADIN
Behavior Breaking
of a
Y. GOLDSCHMIDT
Racah Institute of Physics, Hebrew University, Jerusalem, Israel Received August 1, 1977
We discuss the case of an O(N)-symmetric theory whose symmetry is broken by a quadratic term which produces two different masses (correlation lengths) one associated with M components and one with N - M. This theory describes physical systems having a bicritical point. To this problem we contribute by showing how within renormalized field theory the calculation of the shift in the critical temperature (leading to a crossover exponent) is separated from the calculation of the crossover function. The computation of the latter is then carried out in terms of the natural temperature variable, relative to the anisotropic critical temperature. This feature allows one to bypass the complicated corrections to scaling generated in earlier approaches by the use of a different variable. Within this framework which allows many tests of universality the results are explicit and readily generalizable to the ordered state and to higher orders in E. To field theory we contribute by showing how the mass-dependent renormalization procedure allows the interpolation between asymptotic behaviors with internal O(N)- and O(M)-symmetries. The interpolating parameter is the ratio between the momentum scale and the mass difference breaking the symmetry.
1. INTR~D~JcTI~N The study of the bicritical point-a point at which two critical lines merge- is not new. Three recent papers [I] describe the relevance of precise theoretical studies of the bi-critical behavior to present day experimental technology in solid-state physics. A study of the problem reveals that very interesting theoretical questions abound as well. It is clear, of course, that in any experimental situation anisotropies are present. Exact global symmetries accompany conservation laws only, as they do in particle physics, and in situations such as superfluids and superconductors. In a three-component ferromagnet, or in a four-component antiferromagnet [2], the discrete structure of the lattice will never render and exact continuous global gauge symmetry. If the symmetry is broken in the quartic terms, then asymptotically the anisotropies are suppressed if the number of components is smaller than 4 [3]. In the language of the renormalization group one says that for low numbers of components the isotropic fixed point is the stable one. On the other hand, the symmetry is also generically broken at the quadratic or soft level. Here the asymptotic behavior is eventually dominated by the fixed point with a lower number of components-those corresponding to the lower bare mass (coefficient of the quadratic term). 356 OOO3-4916/78/1142-O356$05OO/O Copyright 0 1978 by AcademicPress,Inc. AU rights of reproduction in any form reserved.
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Nevertheless, there is a range of parameters- scale of momenta, temperature, etc.in which the fixed point of high symmetry dictates the asymptotic behavior. The reason is that because of the presence of an additional length the asymptotic region here is much richer than in the usual situations with a single length. Such an extra length is, for example, the mass-inverse correlationlength-of the N --M components at the temperature at which the M components become critical (massless)-a quantity that is directly related to the anisotropy. If this mass, m, is much smaller than a microscopic momentum K, there is a range of the parameter a-the overall momentum scale or the temperature-such that
but the ratio cx/m is arbitrary. Moving from u/m < 1 to a/m > 1 introduces a structure into the asymptotic region. And, since it all happens for 01< K, much of this structure is universal. The possibility of obtaining this complex asymptotic behavior in a universal way, and hence in a controlled E-expansion, is a source of much theoretical interest. Furthermore, the study of field theories with quadratically broken symmetry has known few detailed applications, though the general framework had all been set by Symanzik [4]. On the other hand, the knowledge of this behavior is of direct importance to the analysis of experiments. As the temperature is lowered, at fixed anisotropy, towards the low-symmetry critical temperature, there may be a wide temperature range in which the behavior of the thermodynamic quantities will change from one power law to a second one---both nonclassical. Only when the temperature difference-corresponding to ,x2-becomes much smaller than m2, will a uniform power behavior set in; the expbonents will be those of the system with low symmetry. To be more specific, a reiteration of the phenomenological expectations concerning the behavior of a system near a bicritical point [5] will be helpful. Consider a bare Lagrangian of the form 27 = P412
+ &P12412 +
Bp22422
+ (V49(92)2,
=
4i2,
+2 = 412 +
(1.1)
where d12 = ; $v, i-l
922
5
422.
(1.2)
i=rvf+1
Both p12 and ,p22are linear functions of the temperature [6]. If l-h2 -=I P22, which implies that the mean-field transition temperature of the M compoents of type 1 is higher, then one expects the first M components to become critical before the rest do. The fluctuations of the remaining (N - M) components grow, but never diverge. One therefore expects a critical behavior of an O(M)symmetric theory. For p22 < p12 one expects an O(N - M)-symmetric critical behavior of the type 2 components. When p12 = p22 the theory is O(N)-symmetric. This is the bicritical point at which the two critical lines, the O(M)- and O(N - M> symmetric lines, join. This situation is depicted qualitatively in Fig. 1.
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FIG. 1. A qualitative phase diagram of a bicritical point g = pCLa2 - p12. The heavy line is a line of first-order transitions.
If the anisotropy is denoted by g, namely,
g x Ps2- A29 the expected behavior of a quantity such as the longitudinal susceptibility along an essy axis-is [5]
(1.3)
susceptibility-the
where t = (T - T&/T,, , with TcN the critical temperature of the O(N)-symmetric system. The exponent +-the cross-over exponent-describes the departure of the two critical lines from the bi-critical point. As is well known, C$is a universal quantity, whose value was calculated by Wilson to second order in E [7]. Its calculation implies an expansion about the O(N)-symmetric fixed point, namely, for g/t < 1. This exponent is completely determined by the O(N)-symmetric fixed point. As will be shown in Section 2, it is independent of M. This is not the case for the functionf: One expects thatf(z) will vanish for some z0 # 0 which defines the critical lines via
g = z,t*,
(1.5)
giving TCM(g), the critical temperature at fixed anisotropy. If t -+ 0 but g/t < 1 then x~l N tv. On the other hand, if g is fixed then as t + gl/*/z, ,
(1.6)
tY in (1.4) is a constant and the singularity comes fromf, which develops a behavior like
f(z) - (z -
z$
cc [T - TcM( g)l; oc 73:
where + is the exponent for the M-component
theory.
(1.7)
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The cross-over between these two behaviors is conventionally described by an effective exponent which depends on the temperature variable T. Namely, one defines
yefr(T) = d ln X?(T) dln7
’
The relationship between x-l, g, and t has a form analogous to that of an equation of state; see Eq. (2.17) below. To obtain the form of the latter to any given order in E resuires simply the calculation of the external field to a given number of loops [8]. But here the situation is complicated by several factors [9]. One reason is that in order to obtain form (1.4) one has to substitute in the result of the perturbation calculation u=u$, where U$ is the fixed point of the isotropic N-component system. But the true, and only,, fixed point for the problem is the O(M)-symmetric one, u$, . Therefore, there are corrections to scaling to the O(M)-symmetric asymptotic behavior which will show as powers of logarithms and will not permit a direct exponentiation of the true asymptotic behavior. Another source of difficulties is the inadequate temperature variable, which is different from the variable 7 = T - T,(g) characterizing the O(M) behavior. In order to proceed, one has to make several assumptions from the beginning, which are not dictated by the perturbation calculation. One has to a,s,sumea priori that the behavior of the theory in the critical region is that of a system with an O(M)symmetry. Then one has to compare the Eexpansion with the corresponding expansion for the O(M)-symmetric system, and try to match the two expansions even though they are expressed with different variables-a procedure which is not devoid of ambiguities. Rloreover, one has to use results of the renormalization group calculations for the O(M)-symmetric theory with corrections to scaling (see section 3 below) in order to make the matching possible and to exponentiate the true behavior. This is just an outline of the strategy adopted by Bruce and Wallace [9]. It is just thi:s difficulty which impelled Nelson and Domany as well as Horner and Kosterlitz [l] to adopt special renormalization group strategies. Nevertheless, the traces of the C)(N)-symmetric critical temperature are still present in all these treatments, and thle special methods used make it hard to envisage how higher orders in E are to be calculated systematically. Here it will be shown that, in the framework of renormalized field theory, it is possible to formulate the whole problem in two equivalent forms. Once as a soft expansion-in temperature and in quadratic anisotropy-about an O(N)-symmetric theory. Then as an expansion in temperature about an asymmetric theory in which M components are massless (critical) and (N - M) have a fixed mass (anisotropy). The first formulation is natural for the calculation of 4, and some studies of universality. It is presented in Section 2. The second formulation, which will constitute the bulk of this article, provides a description which is ab initio in terms of the natural variable T - TCM( g). In other words, we return to the original variable of Riedel and Wegner [5], formalize it, and compute with it. Within this formulation the extensions to higher orders in E, as well as to ordered states, are absolutely standard. Furthermore, many
AhlIT AND GOLDSCHMIDT
360
additional questions of universality can be studied. In this formulation one does not have to assume the asymptotic O(M)-symmetric behavior. It comes out automatically. The plan of the paper is as follows: In section 2 we present the expansion about the symmetric theory and repeat the calculation of the cross-over exponent 4. Then, in Section 3, the difficulties encountered in the derivation of the cross-over function from the expansion about the isotropic theory are discussed. Here the scene is changed, and the Lagrangian (1.1) is rewritten so that the symmetry breaking is fully contained in the free part. In Section 4 we show, following Symanzik [4], that the theory can be renormalized in such a way that m, the mass of the (N - M) non-critical components, can be held fixed while the renormalization momentum scale is varied and the renormalization group equations are derived. Then in Section 5 we argue that in order to be able to describe the infrared asymptotic behavior of the theory, at hxed anisotropy, the renormalization constants must depend on m. Sections 6 and 7 describe two different procedures for the calculation of the renormalization constants and Wilson functions are presented. The first utilizes normalization conditions. The second generalizes the minimal subtraction method of ‘t Hooft and Veltman [lo] to include m-dependence. Next, in Section 8, the renormalization group equations are solved in the presence of two lengths. This requires some extension of existing techniques. In Section 9 the temperature is introduced as a soft insertion in a non-symmetric theory and renormalized. This is the temperature difference relative to TCM( g). The renormalization group equations for functions which depend on T,
(1.5) are derived. They are formally solved in Section 10. Section 11 describes two different procedures for calculating the renormalizatione constant of the temperature. One is minimal subtraction of the 42 insertion; the second is fixing the value of the two-point function at some non-zero value of the temperature. All these functions are computed to O(E). The results are assembled in Section 12 to give an explicit expression for the cross-over function. In section 13 we derive the RGEs for the specific heat and for the non-ordering sysceptibility, and it is shown that they fall into the standard pattern solved in previous sections. This is not the case for the transverse susceptibility which is discussed qualitatively. Finally, in Section 14 we discuss various aspects of universality, and in Section 15 we list extensions of the present calculations which are in progress. 2. SOFT EXPANSION
ABOUT
THE SYMMETRIC
THEORY: THE CROSSOVER EXPONENT
The main results of this section are not new, although some are shown to be true to all orders in renormalized perturbation theory and in E(= 4 - d). Otherwise this section serves to recast the calculation of the shift in the critical temperature due to anisotropy in the language of renormalized field theory.
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This, on the one hand, emphasizes the similarity with the calculation of the equation of state and in this way helps us establish later, in Section 14, some aspects of universality. On the other hand it serves to underline the contrast between the computation of the cross-over exponent-a phenomenon belonging to the realm of the symmetric theory, and that of the cross-over function to which the bulk of the paper is devoted. As we shall see, the latter is grounded in the asymmtric theory, and in the shift of reference system the natural variable becomes the temperature difference from the critical temperature of the system with low symmetry. The critical temperature of the symmetric theory disappears. We thus considered it instructive to present, in a common language, thes’e two complementary calculations for the same system. The Lagrangian (1.1) can be rewritten in the form 9 = HV4)2 + +pc242 + (h/49(+2)2 + ii&242 - QgB, where pC2 is the value of the vare mass which gives an O(N)-symmetric (massless) theory when 6~~ = g = 0. The operator B is given by B = (l/N)[(N
- M) +12 - Mc#~~].
(2.1)
critical
(2.2)
Translated into the old parameters of Eq. (1 .l), the new ones read PC2 + 8p2 = (1/N)[MpL,2 + (N - M) &I
(2.3)
and g =
P22
- P12.
In other words, g is proportional to the anisotropy. It is a proper symmetry breaking parameter. The special choice of the operator B is dictated by the requirement that it be invariant under the subgroups O(M) and O(N - M), whose symmetry was not broken, and that it remain orthogonal to C2 under renormalization transformations, namely, that the counterterms generated by +2 will be proportional to 42, and those generated by tl be proportional to B. Any other combination of +12 and 622 cannot be renormalized separately. That this is the case we show in Appendix A. The renormalized form of the Lagrangian can be written as 9 = $(V+)” + (zM/~!)(+~)~ + -&t+2 - QyB + counter-terms, counter-terms = &(Z, - 1)(V+)2 + &[(pC2 + 8~~) Z, - t] 42
(2.4)
- y)B + (1/4!)@2,” - u/F)(+~)~,
(2.5)
- +( gZ,
where t and y ,are related to 8p2 and to g via a2=z P
d?t ’
(2.6)
P = ZBY.
(2.7)
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Both 2~ and Z, are functions of u and E (= 4 - d). They are the renormalization constants which render finite the insertions of @ and B operators into vertices. In other words, let rLN9”*“‘( pi , qi , ri; 6p2, g, A, A) be the bare vertex with N external legs, L insertions of +2 and M insertions of B, then the renormalized vertex is given by FiNsLeM)(pi , qi , ri ; t, y, 24,fc) = Z~‘2Z,L,ZBMFiN*L*M)~i
, qi , ri ; 6j~‘, g, A, A). (2.8)
The parameter p C2was suppressed, since it is determined in terms of X and A by the requirement that the theory be critical when 6~~ = g = 0. The functions A. Z, , Z,O , and Z, of u and E are calculated in the standard way [3, 131 by demanding that Fk”’ (p = 0) dqf’/dp2 Isp , r$O), and Q~~7” be finite as (1 -+ co (or as E -+ 0 for a theory that was dimensionally regularized [12]). The subscript R indicates that the function is renormalized, and sp indicates that the renormalization was carried out at a symmetric momentum point. Here we will follow the second alternative and calculate the renormalization constants by minimal subtraction of dimensional poles [IO, 121. But first we derive the renormalization group equations. Equation (2.Q or alternatively (2.5), implies that the vertex PQ, at arbitrary values of 6$ and y, is renormalizable via [13] FAN’<& ; t, y, u, K) = zy2riN)(pi
; sp2, g, A, A),
(2.9)
The coefficient functions are the standard ones in the O(N)-symmetric be read off from the literature [14]. The only new function is yB . For completeness we give the list to order two loops:
theory. They can
which leads directly to the differential equation
B(u) = (
a ln(kc) au
y@(u) = -/3(u)q@ =
-/3(u)
ya will be computed below.
--1
= -u (6 - y
N+2 = 72
y&4) = /3(u) y
yB
)
q.
24+ 3N;
l4 .a),
(2.11)
242,
=+4(1-&), (2.14)
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If we restrict ourselves to N = 2, p = 0 and choose both external legs of I’@) to be of type I-belonging to the group with index i ,( M. Then
and Eq. (2.10) reads [ 151 (2.16)
[
The solution of this equation is standard [16]. The equation is completely analogous to that for the equation of state, and so its solution is
The exponents y and v are the conventional exponents of the susceptibility and the correlation length, respectively, in the O(N)-symmetric state. They, as well as
rB* = yEt(u= u”), are calculated at the O(N)-symmetric fixed point (the zero of /3) which dominates the asymptotic behavior of the solutions of Eq. (2.16). The functions X, Y, and 2 are the non-universal parts of the solution-depending on the initial value of U, rather than only on u*. They vary slowly, as functions of t, in the asymptotic region and thus they contribute only to corrections to scaling. The expressions for these functions are given in Appendix B. @ is an arbitrary function which is computed in perturbation theory. The exact result, Eq. (2.17), is an expression of the fact that apart from a normalization of the temperature, the anisotropy and the scale of the susceptibility the form of the latter is universal, i.e., independent of the initial value of u, and hence X. The statement is true to all orders. Furthermore, comparing the solution (2.17) to the phenomenological ansatz for the cross-over behavior, Eq. (1.4), one identifies the cross-over exponent 4 to be + = v(2 - yB*). Consequently, to obtain 4 to any order in E it suffices to compute ye, via Eq. (2.14). Turning to the actual calculation of 4, we show first that to all orders in E 4 on N only-it is a property of the isotropic theory and not of the particular which the symmetry was broken [5, 171. To see this consider a determination of Z, via a normalization condition
595/114/I/=4
(2.18) 2, and depends
form in such as
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For our purposes it is sufficient to prove that the Z, which ensures (2.19) is independent of M because any other renormalization procedure differs only by a finite renormalization. Using Eq. (2.8) we can rewrite (2.19) in the form (2.20)
The insertion of an operator &” (no sums) into a vertex I’::), calculated in the O(N)symmetric theory, depends only on whether i = 1 or i # 1. Denoting such an insertion by I’& , one can rewrite Eq. (2.20) as & Z,Z,[(N
- M) r$
+ (N - M)(M
- 1) r;;!2 - (N - M) Mr~;,] (2.21)
Z, , as well as the brackets, is independent of M, and the explicit factor of N will disappear when 2, is substituted in Eq. (2.14) to obtain yB . The graphs needed for the computation of Z,Z, to order two loops are drawn in Figs. 2 and 3. The expressions corresponding to these graphs are
(2.22)
where I and I4 are the integrals corresponding to graphs (b) and (d) in Fig. 2. They are defined in Appendix C, where their Eexpansion is also reproduced. Inserting (2.22) in (2.21), substituting the E-expansion of the integrals from Appendix C, and for h the expression [13] (2.23)
(a) FIG.
(b)
Cc)
2. Graphs of r$fl.
FIG. 3. Graphs r$.
Cd)
The x indicates an insertion of g18.
The circle indicates an insertion of gz2 .
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subtraction. To obtain ‘ye one writes
yB= -/qu)aln@$‘4) + Y&4
(2.24)
Then using yB(u) from Eq. (2.12), and setting [I 31
u=u*=-LN+8’+
18(3N + 14) E2 (N+8)3 ’
where U* is the zero of /3 (Eq. (2.1 l)), one obtains 2 YB*=N+8E+
-N2
+ 18N + 88 E2 2(N + 8)3 '
(2.26)
Together with the E expansion of v [I 31 for the isotropic theory one finds from (2.18) and (2.26) N3 + 24N2 + 68N $=1-t N E2+ O(E3), (2.27) 2(N + 8) ’ + 4(N + 8)’ in agreement with previous calculations [7].
.3. DIFFICULTIES IN OBTAINING
THE CROSSOVER FUNCTION
The standard way of proceeding would be to expand xc1 to some given number of loops, and to use the general form, Eq. (2.17), to identify CDto a correspondint order in E-just as one does for the equation of state. One knows that Q(z) must vanish for some z0 # 0. This is the phenomenological identification of the critical lines in Fig. 1. Namely, td KY. (3.1) Then the temperature is normalized to fix the coefficient in this equation as unity. If in addition one demands that at J’ = 0, the isotropic behavior
all non-universality is eliminated. The rest can be computed at the fixed point as a power series in E. But here one encounters the difficulties outlined in the Introduction. In terms of the variables
one expects @ = @i(z)to have a singularity of the form (1 - z)i, i.e., Q(z) = X(1 - Z)+(Z),
(3.3)
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where 7 = y(M) [l]. One cannot obtain the correct singularity from the perturbation calculation because of the presence of corrections to scaling. These are generated by the fact that the true fixed point is the O(M)-symmetric one and the choice II = UX made above suppresses the transients with respect to the unstable O(N)-symmetric fixed point, thus giving rise to corrections to scaling about the asymptotic O(M)symmetric behavior. These corrections to scaling appear in the calculation as powers of logarithms and prevent the exponentiation of the dominant singularity. In order to proceed, one has to assume the form (3.3) a priori, i.e., assume that Q(z) vanishes like (1 - z>*, where 3 = y(M). To make this point clearer, let us consider an isotropic M-component system. The full solution of the RGE, including corrections to scaling, is given by Eq. (2.17) withy = 0. The factors Xand Z describing these corrections are defined in Appendix B, and calculated to order E in Appendix D. The result is -(M+2)/(M+S)
-1
=
[ 4’)1
T1+((h4+2)/2(M+8)k
u
XL
where T is the temperature
difference of the M-component
u-y’)=v Substituting
theory, and
+ TF/2 .(u-1 - q&E).
(3.5)
u = u * = 6e/(N + 8) one obtains xi1
=
T1+((M+2)/2(M+8)k
Expanding this expression,
M+8 N-t8
E,2N _ M
~
+T’
N+8
-(M+B)/(M+W (3.6)
one arrives at M+2+ 8) EIn T xp = 7 1 + 2tN ( 1,
(3.7)
and one could think this comes from the expansion of
which of course is wrong. [Notice that it is easy to find the generalization of (3.4) to 0(c2) by calculating X and Z of Appendix B to U(E~) and substituting in (B.l) with y = 0. This leads to the result of Bruce and Wallace [9] for the Gaussian-Heisenberg cross-over function. But problems remain even after one assumes the form (3.3). In matching the perturbation series to an expressions such as (3.4) one has to insert the correspondence between 7 and z. For z N 1 it is simply 7 t+ 1 - z. Otherwise 7 t-t f(z)(l - z) and f(z) has to be determined by the matching procedure itself. All of these are the results of an unfortunate choice of variables, which can be completely avoided, as we show below.
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4. FORMULATION
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The way to avoid the difficulties discussed in the previous section is to formulate the theory against a background which is intrinsically asymmetric rather than the O(N)-symmetric theory used in Section 2. We will now proceed to formulate the problem in a form in which the anisotropy is fully included in the free part. This will enable us to write RGE directly for the anisotropic theory, and then to express xL in terms of the natural parameter T (= T - Tch,( g)). This will be the parameter which will determine the flow of coupling constants, etc., into the proper, O(M)-symmetric, fixed point without any corrections to scaling. This is a project which falls well within the scope of renormalized field theory, although at certain points we will have to extend the lore of results [18] in order to deal with the: presence of a second length. Symanzik [4] proved that a (+2)2-theory with quadratic symmetry breaking is renormalizable. This he did by deriving generalized Ward-Takahashi identities for non-linear symmetry breaking. The statement that was proved by Symanzik contains the assertion that the number of renormalized parameters-masses and coupling constants-equals the number of bare parameters. The original proof is transcribed in Appendix H in the language of functional integrals over c-number fields, which is more familiar to statistical physicists. The same result can be reached by considering the counter-tmerms which are generated interatively in perturbation theory [19]. For clarity, we start with a theory with T = 0. Namely, M components are critical and (N - M) have a mass (inverse correlation length) m jixed. We return to the Lagrangian (1.1) but choose p12 = & and p22 = ,ui, such that the vertices generated by it after renormalization satisfy the conditions r&(p
= 0; 112,u, K) = 0,
(4.la) (4.lb) (4.lc)
(4.1 e) The index i stands for external legs in the critical group, while j stands for the noncritical ones. K is a renormalization momentum scale, and sp stands, as before, for a symmetric momentum point. With these definitions of pIc and pzc the Lagrangian (1.1) can be written in the form 2 = w992 + &4cA2 + &&A2
+ (h/4!)(&)”
+ Bto4”
(4.2),
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AND
GOLDSCHMIDT
form
9 = i(v+)’
+ $m2&2 + (l/4!) z@(@)~ + &T@ + C.T.
(4.3)
sYman..Zik’S theorem assures us that function ,ulc , pFlzc , h, Z&l), and Zi2) can be found such that [20] riNN)(pi
; 177, U, K)
=
(z~‘)N1’“(z~‘)N2’”
riN’(&
; &c
, /&c
, h,
A)
(4.4)
are renormalized, and satisfy Eqs. (4.1). P’) has N1 legs of type i and N2 (= N - NJ legs of type j. The field renormalization constants ZL” and Zh2’ are determined by conditions (4. lc) and (4. Id). They differ by a finite factor. Recall that plc , pzc are the bare masses of particles of types i andj, respectively. h is the bare coupling constant and A is a momentum cutoff. A few comments on these normalization conditions are in order. First, from Symanzik’s theorem again, it follows that once (4.le) is satisfied, the other components of P), namely, rj&\ and rj$ , are all finite. (See, e.g., Appendix H.) The values of these other parts of P4) are completely determined. This is equivalent to saying that an O(N)-symmetric counter-term will suffice to make finite a theory with a quadratically broken symmetry (4.19). One should notice that the number of renormalized parameters needed, masses and couplings, equals the number of bare parameters. The normalization conditions (4.1) are so chosen as to ensure that for a fixed bare theory (fixed p1 , p2 , h, fl), m will remain fixed. This is not a trivial matter, and it is here that the first extension of previous applications of field theory to critical phenomena enters. Notice that Eq. (4. la) implies a relation of the form ml(plc , kc , A 4 = 0
(4.5a)
which can be kept intact as K is varied at a fixed bare theory. This depends on the fact that ZL’) cancels out of Eq. (4.la). This is much less obvious with regards to (5.1 b). Can one deduce from it that m
=
m(h,
p2c
(4.5b)
,&4?
This time Zh2’ does not cancel. Usually, in a massless theory Z, depends on K to prevent the infrared divergence in quantities like P2)/+P. Here this can be avoided by noting that in r,‘i’ the internal progagators of typej have a mass, and the i-propagators are too few to cause infrared problems. Thus in Eq. (4.ld) we could legitimately set pz = 0. It can then be solved for Z,J’) without introducing K, so that
zy z zp
3 P2c
, A, 4.
All this implies that (4.ld) leads to an equation like (4.5b), and therefore K can be varied at a fixed bare theory holding m-the anisotropy-constant. This is essential for the derivation of the RGE.
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To summarize:
The normalization
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conditions (4.1) imply (4.6a) (4.6b) (4.6~)
(4.6d) (4.6e) which can be inverted to give prc , p2c , h, and Z$” as a function of m, u, K, and (1. If now in ESq.(4.4) all external legs are chosen to be of type i and ~(a/a~) is applied at fixed bare theory, the resulting RGE is
(4.8)
Y6 = K
a In Z:’ aK
A.rn
(4.9)
This is all very similar to the usual case of a critical O(M)-symmetric critical theory, all except for the dependence of the functions /3 and yb on m/K. At this point m, the inverse of the transverse correlation length at TCM , becomes the new measure of the anisotropy. That this is a very natural measure is corroborated by the rest of this article.
5. MUST fl AND ym DEPEND ON m/K ?
At this point the second extension of the renormalization group treatments enters. We wrote in Eq. (4.7) an explicit dependence of the functions /3 and yd on m/K. This is not dictated by general renormalization requirements. The special renormalization conditions (4.la)-(4.le) do imply such a dependence, as is shown in the next section. However, as we saw in the discussion of the expansion about the symmetric theory, the renormali.zation constantsof the massless symmetric theory can render the asymmetric theory finite as well. These would have been obtained had renormalization been carried through using “naive” minimal subtraction of poles in (4 - d) [lo, 121.
Some additional comments on this subject are reserved for Section 7. If the rermrmalization constants, or the counter-terms, are taken independent of m/K the critical behavior will be that of an O(N)-symmetric theory. One is, however, intuitively convinced that as the critical point is approached (momentum scale much
370
AMIT
AND
GOLDSCHMIDT
smaller than m or 7 < m2) the theory should be dominated asymptotically by the M-component critical behavior. Clearly, were m2 to become very large, all N - M components of type j, which are associated with m2, would be frozen out, and M-component behavior would ensue. This fact is obscured in a “naive” E-expansion. Graphs whose contribution vanishes as m -+ co may have an m-independent pole in E, and will not vanish in the limit. As an example, consider the expression J = (~/E)(K~ + m2)-6/2.
(5.1)
Such a form may be associated with a one-loop integral of rc4). Expanding in E, one has J = t - i ln(K2 + m”) + O(&+).
(5.2)
Minimal subtraction would keep l/c in the renormalization constants [12] and drop the In. This is fine as long as m is finite, but it brings about a situation in which the contribution of J to the function, which is being renormalized, vanishes as m -+ CO, but the subtraction induced by the renormalization constant does not. Hence in this limit the theory will not be finite. See also the explicit calculations in Section 6 and 7. In other words, Symanzik’s theorem breaks down in the limit of injinite quadratic symmetry breaking. But what if m2 is kept fixed ? One runs into the same type of difficulty,
as one approaches the critical region. The way to see this is to note that if, for example, the momenta in FN) are scaled by a factor p then dimensional analysis implies
In the asymptotic region p + 0, which is the region of interest, the theory is equivalent to one with a very large mass. Thus, unless the renormalization has been so carried out as to include the limit m + co, one is sure to obtain nonsensical results for fixed m as the momentum, or temperature, tends to zero. In the following sections we describe two different renormalization procedures which satisfy the requirement that the c-expansion remains meaningful when m + co. The two procedures lead to an identical cross-over function which one would expect on the basis of universality. The fact that the Wilson functions /I, y* , etc., depend on m/K makes the solution of the renormalization group equation more complicated. This is another extension to which we turn in Sections 8 and 10. It is important to realize that the above considerations are not additional assumptions. That all vertices with external legs of type i tend, as m -+ co, to their Mcomponent counterparts follows directly from perturbation theory. This fact is automatically taken into account by the normalization conditions (4.1), which would provide m-dependent renormalization constants.
TREATMENT
6.
CALCULATION
OF BICRITICAL
OF p USING
371
Pt
NORMALIZATION
CONDITIONS
In the first renormalization procedure we use the normalization:conditions, Eqs. (4.1) together with Eqs. (4.6), to obtain /I to the order of one loop. The Lagrangian (4.2) is used to generate the graphs directly in terms of m2. In Fig. 4 the graphs’ contrbution to J’,‘$‘, at this order are drawn. Equation (4.le) reads UK’
FIG. 4. separately.
Graphs
of I’&
=
h -
y
~2&,(o)
to one loop.
Graphs
-
with
v
~2&p(fTZ).
massless
and
massive propagators
are drawn
The integrals are defined in Appendix C, where their +expansion-is given as well. To obtain Isi,(m) one setsP2 = ~~ in Eq. (C.6). Equation (6.1) can bkwritten as x2K-2’
Notice that “n.aive” minimal subtraction would have us discard the second terms in all parentheses in Eq. (6.2). Calculating ,D from definition (4.9, one finds
1 _
(1 +
4m2/K2)-1/z
In the limit m .+ 00 (6.4) as one expects for a theory of Mmassless one has
particles (Mcritical:components).-As
p+-al+-U2 which is the proper /3 for an O(N)-symmetry
N-t8 6 theory.
’
m/K + 0
AMIT AND GOLDSCHMIDT
372
An expression such as Eq. (6.3) is not very convenient for further calculations. We therefore proceed to an alternative procedure which will yield much simpler expressions without foregoing the essential content of the cross-over phenomenon.
7. RENORMALIZATION
BY GENERALIZED
MINIMAL
SUBTRACTION
In describing a second procedure of renormalization we have two objectives in mind. First, the procedure of generalized minimal subtraction (GMS) to be presented enjoys many of the simplifying features of “naive” minimal subtraction. This becomes a weighty consideration as the calculation proceeds. Second, using two methods provides a test of universality. Below the method will be presented on the level of one loop. That it works on the two-loop level will be shown in Appendix F. The requirement we impose is that for finite m the renormalization constants, or counter-terms, cancel the poles in E in all vertices. In addition, we demand that as m + 00 all vertices with external legs of type i be finite order by order in u and E. Starting with I’$& we write U,,
=
hK-f
=
U +
q(m/K)
(7.1)
U2.
On the one-loop level Z, = 1, and al(m/K) is chosen so as to make ri(i4i)iR finite as E --f 0 and us m --f co to order u2. Symanzik’s theorem assures us that for finite m the same u0 will make all of P4) finite [21]. One such choice is m
4 c-1K
Introducing adP1
=- M+&+N--M 6E 6~
[I - :ln (1 + $)I.
it into J’:;l:, , one has ,..., p4
=
= K”
I
; m, 4 K)
U +
UlU2
-
u2y
u
-
Mt8
U2 ___
18
I
(7.2)
TREATMENT
313
OF BJCRITJCAL Pt
where we denoted p1 + pz = P. The expression in the first set of square brackets is finite as in the usual dimensional regularization. The expression in the second set of square brackets can be written, using Eq. (C.6) for I( p, m), as [ ] = f (1 - g I1 dx In [x(1 - x) $ + $-I)
= -.:rdx;n[ In “naive”
x(1 ;2xy2+
dimensional
- f (1 - z In (1 + $))
m2 ] + O(E).
regularization
(7.4)
there would be no denominator
in the In.
As long as m is finite the difference, Eq. (7.4), is finite as E -+ 0. The choice (7.2) has,
in addition, the property that as m + cc the finite term vanishes, and it leads to a I’::\, identical to that of the O(M)-symmetric theory. Furthermore, it gives a meaning to the E-expansion for all values of m. The form (7.2) is not unique in satisfying the above resuirements. It is easy to see that if (1 + m2/tc2)-c/2is replaced by (a + m2/K2)-E/2the whole argument goes through. We will return to discuss this arbitrariness in the renormalization procedure in the context of the discussion of universality in Section 14. Calculating p, using Eq. (4.3, one finds p(u,'-)
= --EU + q+
+ yc2
(7.5)
1 + LeiK2 + O(U3,#%),
and the correction terms are finite for all values of m. Clearly, as m + 0 or as m -+ co this /I tends to the limits (6.4) and (6.5), respectively. The difference between the two procedures amounts to a finite renormalization for all ualuesqf m. Universal quantities-inter alia the universal part of the cross-over behavior-should not be affected. The function ‘ymis, of course, zero at the one-loop level. Still, for completeness, we write down. its form at the lowest non-trivial order. Normalization conditions give Yb =
N+2 --u2~??!!u2u,
72
‘dx
36
Kzo
s
’ I
,,d4
Y yx(1 - x) +
rn2/K”
+
O(u3,
U2E).
(7.6) GMS leads to the simpler form
ye = L u2 W + 2) + (M + 2>@“/K”) 72
1+
rn2iK”
.
(7.7)
Both have proper limits as m -+ 0 and as m + co.
8. SOLUTION OF THE RGE IN THE PRESENCE OF Two LENGTHS
The depend#ence of /3, y+ , etc., on m/K brings about the third new feature of the present application. RGEs are usually solved by the method of characteristics. What has made the implementation of this method very simple to date was the fact that
374
AMIT
AND
GOLDSCHMIDT
all the coefficient functions have depended on a single variable-usually the dimensionless renormalized coupling constant u [22]. Here the situation is more complicated, and hence we describe the process in some detail. To the order of one loop the RGE reads
where all external legs of FN) are of type i. The first characteristic
equation is
which, with (7.5) used, reads -=du dK
(N + 8) + (M + 8)(m2/K2) u2 1 + m2/K2
(8.3)
This is a Bernoulli differential equation whose solution is M+8 u-1
zzz
K x-f-1 dx + qnl) s Kg1 + 1)22/X2
N--MKF
6~
6
_ Mf8 6~
j-f
-
M
1
sgp0
12
Kc
dy
J>-d2
y + m2/x2
+
c(t’?l)
Kc.
By dimensional considerations C(m) = Cm-‘, where C is dimensionless. Let us denote the solution of Eq. (8.2) by @(U,
which can be rewritten
m/K)
=
(8.5)
c,
as an expression for u in the form u = F(@, m/K).
(8.6)
In the special case of Eq. (8.3)
and F is given by the inverse of (8.4) with that 0 satisfies the equation
Denoting
C replaced by @. Notice,
m/p by FFiwe define u(p), the flowing @(U,
WZ/K)
=
@(U@),
coupling m/K).
constant,
incidentally,
by the relation (8.9)
TREATMENT
OF
BICRITICAL
375
Pt
Then, using Elq. (8.8), one can readily show that u(p) satisfies the equation
du(p)= P(U(P), g, ’ ___ 4
(8.10)
with the initial condition u(p = 1) = u.
(8.11)
This is a generalization of the equation describing the flow of the coupling constant as a function of the overall scale of length. Using Eqs. (8.9) and (8.7) one obtains an explicit expression for u(p), namely,
tl-l(,O) = p’u-1 +
(8.12)
For m < K the behavior of u-l(p) can be simplified if the interval divided in two. As is shown in Ap&rdix E for 1 > p 3 m/K, N-t8
14-l(p) = 7
N--M 12
- -----In
+ p’ [24-l - v and for
m/K
2:
p
>
(
l+-
+ v
0 < p < 1 is
m2 Kzpe
1
In (I + $j],
(8.13a)
0
PM+ 8 u-‘(p) = -~ v 6~
In (1 + $)
(8.13b) For very sm.allvalues of E one can write a single expression for 21-l(p)in the whole range of p. It is 14-l(p)
=
M+8
6E
-
(8.14) But, for finite E - E = 1, for example, the difference between (8.14) and (8.13) can become quite fsignificant. The latter expressionsare more reasonable, though they are somewhat more complicated. They join smoothly at p = m/K and, what is more important, if the initial value of u is smaller than 216, then u(p) remains below UC throughout. As a consequencethe effective exponent defined below stays between its O(M) and 0(N) value. Some comments on this question will be added in the discussion
316
AMIT
of universality-section 14. Notice, (8.14), have the property
AND
GOLDSCHMIDT
in particular,
6~ u@) ygg ____ Mf8
that both versions, (8.13) and
= UC.
(8.15)
A plot of u(p) for everal initial values of the coupling constant is given in Fig. 5, for N = 2, M = 1, ~~ = 1, m2 = 0.01, and E = 1, using Eq. (8.13) for u(p). In Fig. 6 we compare the use of Eqs. (8.13) and (8.14) for E = I, 0.5 and 0.1. Then in Fig. 7 we plot u(p) for 6~ zPYgq=u; for several values of m2 with N = 2, M = I, ~~ = 1, E = 1. In order to study the solution for r IN) let us switch back for a moment to the more general case which includes yb in the RGE. The complete equation would be (8.16)
FIG. 5. A plot of u(p) for several initial V&ES of the coupling constant for N = 2, M = 1, K* = 1, In2 = 0.01,
E = I.
0.62 0.60 0.68 0.66 0.64 0.62 0.60
FIG. 6. Comparison of Eqs. (8.13) and (8.14) for < = 1, 0.5, and 0.1 with m2 = 0.005, ~~ = 1, N=2,andM=l.
TREATMENT
OF BICRITICAL
lo-lo 10-8 10-6
FIG. l
=
7.
Plot
of u(p) for
II = u* for
m2 = lo-*,
10-4
10-2
377
Pt
100
102 P2
10-4,
1O-B, lo+.
I? =
1, N = 2, M = 1, and
1.
Here there is a second characteristic equation, namely, (8.17) which is solved by inserting solution is
U(K)
along a characteristic curve from Eq. (8.6). The
Next the momenta are scaled by a factor p. Dimensional (8.18) gives in the standard way
analysis together with
and a change of variable x -+ x/p results in T(N) = P d-N&t-1) R
where we have used Eq. (8.9) and the fact that the canonical dimension d - N(d/2 - 1) [23]. Returning to the one-loop approximation, Eq. (8.19) becomes j’$“‘(,pi
; m,
u,
K)
=
pd-N(d’2-1)r;N)(Pi
; @i,
U(p,,
K).
of PN) is
(8.20)
378
AMIT AND GOLDSCHMIDT
As p + 0, 1%becomes very large. All graphs with internal propagators of type j will vanish. u(p) -+ z& and y6 + qM, the anomalous dimension of the field, as one would expect a theory with A4 critical components to behave. In conclusion, Eq. (8.19) serves as a starting point for a calculation of a cross-over function as the scale of momentum crosses the value of the anisotropy mass m. This cross-over which can be described by an effective 7 will be described elsewhere. Here we proceed to calculate the more popular cross-over in temperature.
9. THE THEORY ABOVE THE CRITICAL
TEMPERATURE
As was indicated in Section 4, TV in Eq. (4.2) is proportional to the temperature difference from TCM( g). One should also recall that m the inverse of the transverse correlation length at the critical temperature TCM has become our measure of the anisotropy. From the fact that 2,~ of the symmetric theory renormalizes 6p2, Eq. (2.Q it follows that for finite m the same renormalization constant will renormalize TV. Namely, the transformation from the Lagrangian (4.2) to (4.3) can be affected by the relation To = z&T. (9.1) 2,~ will renormalize multiplicatively the insertions of @ into the vertices of the critical asymmetric theory described in Section 4. This follows from the fact that the superficial (primitive) divergence of PzJ) is logarithmic and hence the difference between a graph with massless propagators and an identical graph with massive propagators is not superficially divergent. In the light of the discussion in Section 5 we prefer not to use the Z,z of the symmetric theory, since the m2 --f co, or m2/r + co, limits will be problematic. The conclusion that there exists a Z,z , which makes the vertex function with TV finite, persists. For finite m the difference between one choice of Z,z and another consists of a finite renormalization. As m -+ -m we require that the insertion of @ into vertices with external legs of type i remain finite and tend to their O(M)-symmetric form, as we did in Section 7. The above discussion can be summarized by saying that there is a
Zb2 = Zd2(u,m/K)
(9.2)
such that
P-3) is finite as fl + co (or E + 0) and as m + co if 7. and 7 are connected via Eq. (9.1), and the parameters on the right hand side expressed in terms of u, K, and m. All this is true order by order in the double expansion in u and E.
TREATMENT
OF BICRITICAL
379
Pt
The RGE for the longitudinal two-point vertex I’& follows directly from the application of ~(aji3~) at constant m, and A, using Eq. (9.1). It reads
with y* defined by Eq. (4.6) and Y& =
-K------
a In .&,2 i3K
. m.h
Recall that I’::) is proportional to the inverse of the longitudinal (easy-axis) susceptibility, &, and that to the order of one loop y+ = 0. There still remains the question as to how 2,~ is to be determined. We have chosen again two different ways. One is to use a modified minimal subtraction as presented in Section 7 requiring that Z,JIlb (‘J) have no poles in E and be finite for all values of m. The other, which we find particularly convenient, is as follows: The only function in which r@ introduces primitive divergence is r c2).Consider the inverse longitudinal susceptibility XLl(T,
The normalization
m,
condition
24, K)
=
rC2)IIF&’
=
0;
=
0.
7, 111, %
K>.
(9.6)
(4. la) implies X;l(o,
To this we add another normalization XL1(K2,
l?‘l,
U, K)
condition, M,
U, K)
(9.7)
namely =
(9.8)
K2.
This determines Z,Z , since it includes the graphs of I’L211). Both ways of determining Z,z lead to the same cross-over function. The special advantage of Eq. (9.8) is the fact that, once the renormalization group equation is used to roll 7 out of the critical region, to the value K2, all the information will be contained in the prefactor. This we proceed to show in the following sections.
10. SOLUTION
OF THE RGE
ABOVE THE CRITICAL
TEMPERATURE
The presence of the temperature term in the RGE, Eq. (9.4), leads to an additional characteristic equation. It is dln T (10.1) K- dK = y,z which is analogous to Eq. (8.17). The solution proceeds, therefore, along the same lines. Namely,, U(K) along the characteristic is substituted into (10.1) using Eqs. (8.5) and (8.6). Equation (10.1) can then be integrated, which gives (10.2) 595/114/I/2-25
380
AMIT AND GOLDSCHMIDT
This leads us to a definition of a flowing temperature T@) as
which satisfies the differential equation P
d In T(P) dP
= y62(u@)'$)
- 2,
T(1) = 7.
In terms of Fi, u(p>, and T(P) the solution of Eq. (9.4) has the property that XL
-I
m2
-ZK2
’
K2, u = p2exp x
XL1
@-
) $,
(10.5)
u(p)).
But so far p is an arbitrary parameter, which we choose by the requirement that 7 on the r.h.s. be out of the critical region. This we ensure by choosing p to satisfy (10.6)
T(p) = K2,
which together with (10.3) gives the following relation between p and the original T: (10.7)
This equation has to be solved to give p(T, m, K). On the one-loop level yB = 0. Thus, Eq. (10.5), after reads
T(p)
is substituted from (10.6), (10.8)
Here the advantage of using Eq. (9.8) to determine 2,~ manifests itself by the fact that (10.8) reduces to
XL’(5, $, u) = P2. Be that as it may, (10.8) or (10.9) gives the cross-over function after p is substituted everywhere in terms of 7 from Eq. (10.7), and u(p) of section 7 is used. This we proceed to do in the next section. One should keep in mind, of course, that beyond the one-loop Ievel the exponential prefactor in Eq. (10.5) has to be taken into account as well. 11. Two CALCULATONS
OF yds
We now compute yQ2 to order one loop in the two procedures mentioned end of Section 9, namely, GMS applied to PJ) and a normalization r,‘$p
= 0;
7
=
K’,
m,
U, k)
=
K2.
at the
Pt
381
Each of these two approaches has its own advantages, both is one more check of universality. Or, rather, it is calculations to a test. The graphs which enter I’:F.l’, in the first procedure, that I’$*l) is the temperature derivative of I’,, (2). According for I’::,*) to become finite is r(eA = zd*zJ$l).
and the implementation of universality which puts the
TREATMENT
OF
BICRITICAL
are given in Fig. 8. Notice to Eq. (2.8) the combination (11.1)
11R
FIG. 8. One-loop graphs of Fit). The 4” insertion is decomposed into components giving different contributions. 4r is the momentum of the 4” insertion.
For I’fiV1) one has a?(p*,p*,q;To
= 1- y
=o,m,q
AZ(q) - v
with Z(q) and Z(q, m) defined in Appendix C. Using the tr-expansion of the dimensionally
XZ(q, m),
(11.2)
regularized integrals and writing
z,, = 1 + c1 : (
1
u,
(11.3)
we see that GMS gives Cl =-+-M+2,
N-M
6~
[
6~
1 -Gln(l
+-$-)I
(11.4)
and zd.=l+[y-
7
In (1 + $)]
24.
(11.5)
Inserting 2~ in y,g - --K
2 In Z,z 2K
(11.6)
3 m.h
we find 1
N-M
M+2 (1) Y& = -u+7u 6
1+
rn2/K2
(11.7)
*
The superscrilpt 1 indicates the use of the first renormalization procedure. Next we co:mpute x;~(T, m, u, K) to order one loop. To this order the Z4’s are equal to unity. As is shown in Appendix G xL1
=
7
zd2
-
UK’
w
J(0,
T) +
I
The integral
J(m12,
mz2)
is defined in Appendix C.
vJ((m2,m2
+
T)]/.
(11.8)
382
AMIT AND GOLDSCHMIDT
First notice that Z$ of Eq. (11.5) would render (11.S) finite as E+ 0 for all values of m. In fact, substituting J from Eq. (C.8) in (11.8) and (11.5) for Z&Z,one finds
-vu[ln(l
+$)-(I
+g)ln(l ++-)]I (11.9)
Equation (11.8) can be used to calculate ydz in the second procedure, Eq. (9.8). Setting T = ~~ and using Eq. (C.8) for the integrals, one easily finds Z@
=
1 +
Kg
u +
EI&E
[ (1
+
zJ1-f’2
-
($)1-r’2]
U,
(11.10)
which leads to (11.11) It is easily verified that both y(l) d2 and yjb22) have the proper limits as m + 0 and as in ---f co. Substitution of Z$’ in x~l, Eq. (11.S), gives x;l(-$,:,u)
= 7 11 +~Uln$--~U X[(l+$)ln(l+-$-)-(l+$)ln(l++)]l. (11.12)
This XL’ is finite for all values of m, and at T =
12. CALCULATION
K2,
xL1
=
K2.
OF THE CROSSOVER FUNCTION
It was mentioned in the Introduction that the cross-over function can be described by an effective exponent
yeff(T) = din xLY4 dln7
*
(12.1)
It is a temperature-dependent exponent which mediates between the approximate O(N)-behavior for T > m2 and the exact O(M)-behavior as T + 0. We will proceed in two ways, using the two renormalization procedures described in the previous section.
TREATMENT
(a)
OF
BICRITICAL
383
Pt
In the first instance, one has from Eq. (10.8)
and when (11.9) is substituted
-1 z_ ( K2
XL
K '
K2
'U
)
on the r.h.s. the result is =p2
1+
1
N--M 12
___
U(f)
-$ln(l
I-$)].
(12.2)
In this equation p is P(T), according to (10.6). Thus to complete the task, P(T) has to be solved from (10.6) and (10.3) and substituted in Eq. (12.2). The equation to be solved is (12.3) The exponent on the right hand side will behave as In p with a coefficient of order E. Therefore, to leading order one can substitute p2 = T/K2 in the lower limit of the integral to obtain (12.4) By differentiating
with respect to log T we immediately
obtain
(12.5) From now on we will denote From (12.2)1 one obtains d In xc1 -dlnP2 dlnTdh
7
u(T”~/K)
by
I N--M ___ 12
‘(‘)
U(T).
[, +
l,,M2
-_ nr * ln (1 + +)]
(12.6)
after again setting p2 = T/K in the square brackets of Eq. (12.2). Note that in differentiating this expression with respect to ln~, U(T) was held constant. This is justified by the fact that
d4d dU(T) rT-=-q.4
dln r
dr
x [u-l - qp
I-12
N--M
+ y
but this is of order e2 because of the factor
u”(T).
&
+ ; (+‘2E
ln(1 + n~-~)]!,
(12.7)
384
AMIT
AND
GOLDSCHMIDT
Finally, using (12.5) and (11.7) we arrive at yeff=l+
Mf2
____ ,2
4’)
7
+
U(T) [ 1 - f
(12.8)
In (1 -t $)I.
(b) In the second method we have simply -&’ = p”. Therefore from (12.5) we obtain
and using Eq. (11.11) we arrive imediately at Eq. (12.8). In Fig. 9 we plot a few graphs of yen(T) for different values of m2 with N = 2, M = 1 and E = 1. The graphs compare well with the results of series expansions [24]. In Fig. 10 we plot Yerrfor N = 3 and M = 2. In Fig. 11 we plot (yen - 1)/e for various values of E. In all these figures we used the split expression for u(r), Eq. (8.13). In Fig. 12 we plot (yen - 1)/e with the unique expression for U(T), Eq. (8.14), for various values of E. For large values of Ethe overshoot of Yen becomes quite marked.
FIG.
FIG.
9.
Graphs
10.
Graphs
of Y&T)
of yeIf(~)
for
for
m2 = 10-e,
m = 1O-8,
10m4, 1O-B, lo-*
with
lo-“,
N = 3, M = 2 and
10e6 with
N = 2, A4 = 1 and
c = 1.
E = 1.
TREATMENT
OF BICRITICAL
385
Pt
T
FIG. 11. Graphs of (yen - 1)/c for n? = 10-a,
0.10
l
= 0.2(a), 0.5(b), l(c) with N = 2, M = 1.
--.A 1o-1o
1o-8
1o-6
1o-4
lo-’
10 I
FIG. 12. Graphs of (ye!! - 1)/c for me = 1O-s, E = 1, 0.5, 0.2 with N = 2, M = 1 and the unique expression for U(T).
13. THE SPECIFIC HEAT
AND THE NONORDERING
SUSCEPTIBILITY
The computation of the cross-over functions of the specific heat and of the nonordering suslceptibility is straight-forward. The first does not even require any new renormalization functions. We proceed to discuss these two functions briefly. As usual, the specific heat is related to the vertex P.2). (13.1)
C a Tts2)(q = 0; 7, m, u, K), with I’(“*2) the vertex associated with the average of 4”(x) +2( -JJ). Apart from a T-independent constant P”,2) is multiplicatively renormalizable 2~ of Eq. (I 1.5). In other words
by
Ffs2)(q; 7, m, u, /c) = z$[rp2)(q;
To 3 PlC
9 P2c
is finite as fl -+ co, or as E + 0 [25].
3
A, 4 - rP)cq;
0, plc , P2c , A, 4
(13.2)
386
AMIT AND GOLDSCHMIDT
For finite m either the 2,~ of the symmetric massless theory of Section 2 or the m-dependent Z,z of Eq. (11.5) will do. But according to the argument of Section 5 the behavior as 7 -+ 0 is equivalent to the behavior as m + co. In this limit graphs with internal propagators of type j will vanish, and the vertex corresponding to +2~2 will tend to the vertex corresponding to +12+12. Therefore the m-dependent Z63 of Eq. (11.5) which tends to the O(M)-symmetric Z, 2 is the proper renormalization constant to keep r c”s2)finite in the infrared limit. Furthermore, in that limit all internal interaction vertices will have four legs of type i, which implies that h of Eq. (7.1) and /3 of Eq. (7.5) remain the same. In order to avoid stray constants we consider P”*3), which is proportional to the derivative of the specific heat with respect to T. It satisfies the RGE
The functions p and ~~2in Eq. (13.3) are given by Eqs. (7.5) and (11.7), respectively. The characteristic equations are of the same type as before except for the obvious change of replacing Eq. (8.17) by (13.4)
K
The rest is a repetition of the steps of sections 8 and 10. Z,z + Z&M) as m -+ co, and /3(u, m/K) -+ PM in this limit. It follows that when T < m2 the specific heat exponent (Y*--+ Q,,, , the specific heat exponent of the M-component theory. Next we turn to the non-ordering susceptibility which is defined by xg =
-
a"F w
=
po.o*2)
(13.5)
3
where g is the anisotropy parameter of Eq. (2.1) and P”*o,2) is the vertex associated with the product of two operators B. In order to avoid once more the constant divergence, which appears in r~“~o~z)just as in rCo92), we consider its T-derivative, nemely, P”Js2), whose general structure, as well as some graphs of low order, is given in Fig. 13.
--&-- _b_---&J-- --A.(al
(61
(cl
IdJ
FIG. 13. Graphs of P’J~Z~. Wavy lines designate +” insertions; dashed lines designate insertions of B. Graph (c) includes a renormalization of B, and graph (d) a renormalization of 4”.
TREATMENT
The renormalization
OF
BICRITICAL
Pt
387
of F”J*2) is affected via I$y2)(q,
Y1 ) r2 ; 7, m,
24, K)
=
z*&J2rp?
(13.6)
As long as m is finite this can be achieved by the renormalization constants of the symmetric theory. But those will not do, and one should ensure that the limit m2 -+ co is properly renormalized. To accomplish this one simply has to choose ZB(u, m/K) so that r$O,l) becomes finite-namely, the vertex with two external legs of type i and one B-insertion--either by normalization conditions imposed on I’$‘,‘) or by generalized minimal subtraction. The reason this is sufficient is that, apart from a constant factor, as nz -+ co (0,l.Z)
-
r(o,3)
pbo’l)
-
J+Z$)
rb
(13.7)
and 11
9
(13.8)
where on the r.h.s. of (13.7) and (13.8) the vertices are precisely those of the O(M)symmetric theory. Thus, h of Eq. (7.1) will do in this limit; owing to (13.8), Z, + Z,Z of the O(M)-symmetric theory; and (13.6) will be finite in this limit because Z$ renormalizes 1’Co,3). The RGE satisfied by f9 = dxg/dT is
(13.9) which falls once again into the category of equations we have been solving all along. The new characteristic is dln f9 K ~ die = -(yg
+ 2YB),
from which the explicit form of the cross-over function follows directly. We will forego the full solution and will comment on the limiting behaviors only. As was pointed out above, when m + co, ye -+ ~$2 of the O(M)-symmetric theory, as does ~~2 . Thus, in tlhe limit r/m2 ---f 0, (13.10) becomes simply (13.4), and xs
-
T-S”.
(13.11)
On the othelr hand, when m + 0, ‘y*z -+ y$ = 2 - v;’ and ye + 7; = 2 - 4~;;’ (see Eq. (2.18))1, provided U(T) + uj$ in this limit. As is discussed in Sections 8 and 14 this is indeed the case, and hence 2--n,N-26 xu
-
T
(13.12)
388
AMIT
AND
GOLDSCHMIDT
Consequently xg crosses over from a behavior very close to that of (13.12) to that of (13.11). In order to solve (13.10) and (13.9) in full, one needs ys(u, WZ/K). This we calculate by minimal subtraction from I’,,(2~o~1).The graphs of this function to order one loop are given in Fig. 14. To make the correspondence of the graphs with Eq. (13.13)
FIG.
14. Graphs of Ti:Po*ll to one loop.
clearer, the insertion of B was separated into three parts-the one with $12,the one with xr $i2, and the one with CL+, &“. The corresponding expression for the bare vertex is r(2.0.1) 11
=yq
(13.13)
-~hl(q)--~hl(q)+~hltg,m)l.
The discussionof Section 7 then leadsdirectly to u-I!J+
z,=1+-
+?!J’:’
(13.14)
and Mf2 YB
=
6
M ’
-
d
1 ’
1 +
m2/K2
.
(13.15)
When m -+ 0, ys tends to the one-loop part of Eq. (2.24). On the other hand, asm + co, yB tends to ~~aof the O(M)-symmetric theory. It is just the kind of function which leads to the asymptotic behavior of xs discussedabove. Inserting ‘ye and ~~2in Eq. (13.10), one can obtain the explicit cross-over function. Finally, we make a few brief comments concerning the transverse susceptibility, namely, $(T,
m, u, K)
=
r$(p
=
0;
7,
m, u, K).
(13.16)
In this casewe find that the full crossover behavior is lesssimple, since, even in the limit m -+ co, more than one coupling constant enters, in contrast to all the previous cases. To seethis consider the graphs in Fig. 15, where graphs with internal propagators which are masslesshave been distinguished from those with massive propagators. Clearly, as m + 00 only graphs (a), (b), and (f) will survive. The graphs (b) and (f) include the coupling constant which connects ii to jj, and this is the problem. This coupling enters only once in these graphs, and there are no divergences associated with it. But since the counter-terms were chosen O(N)-symmetric, i.e., their form is
TREATMENT
OF BICRITICAL
Pt
389
FIG. IS. Graphs of rj,". (hZ,z - UK~)(~)“, there will be an infinite counter-term renormalizing this coupling constant. In the limit m + XI this term will have nothing to subtract, and rjf) will be infinite. This situation can be treated if one allows for more than one coupling constant, a situation that will be treated elsewhere.Nevertheless one can make somequalitative statements concerning the asymptotic behavior of the transverse susceptibility. First, in the deep asymptotic region, 7 < m2,the graphs that dominate the behavior are, apart frolm the free term (a), proportional to r (OJ)of the O(M)-symmetric theory. The constant of proportionality is just that coupling constant, which appears only once and should not be renormalized, multiplied by a combinatorial factor. But P”J) is a familiar function, whose temperature derivative is the specific heat. Thus, as r/m2 + 0, (13.17) XT’ N ryp f A+%M On the other hand when T > m2, but r < as rYN w P’N.
K2,
x$ is the sameas x;l, which behaves
14. THE PROBLEM OF UNIVERSALITY There are .two aspects of universality of the crossover function which have to be investigated. One is the effect of changing the initial value of the coupling constant on the form of the crossover function. The other is the effect of a change in the renormalization procedure-i.e., changing the renormalization constants (Z-functions) by finite renormalizations. To consider the first aspect let us return first to the symmetric formulation Eq. (2.17). If u = u;Cthen u(p) = u;I: and the functions X, Y, and Z defined in Appendix B are all equal to unity. But for u # u$ thesefunctions depend on the temperature and give corrections to scaling around the O(N)-symmetric fixed point. To the order of one loop, Z in Eq. (2.17) is equal to unity, and one has
where
u(p) = [qg
+ p’ (u-1 - +)]-l,
390
AMIT
AND
GOLDSCHMIDT
and to the same order one has to take p = t1J2 inside u(p). Under these conditions one can claim that Eq. (2.17) is universal in the sense that it has the same form but with redefined t and y, namely, t = Xt, 7 = Yy. This is in fact exactly the assertion of Nelson and Domany( When y < K’, and in the whole critical region T < K’, one can neglect the temperature dependence of X, Y, and 2. Universality will then mean that the form of the crossover function remains unchanged if t, g, and xe are scaled by non-universal constants. This will be true to all orders. Nelson and Domany take the scaling factors X and Y to be temperature dependent. This suffices at the level of one loop, at which they calculate. In higher orders 2 enters as well, and it also depends on temperature. It has the form
+A$)
-(lN+2)/2W+8% [ 1 -
2$$3)
MP~- u>+ OG2)]9
which agrees with Bruce and Wallace [9]. But in the asymptotic region this temperature dependence is irrelevant. Returning to the nonsymmetric formulation, we find that universality holds in a rather spectacular fashion. There is no need to redefine either 7 or m, provided only that the crossover takes place for m2 and 7 much smaller than K~. Namely, the expectations raised in the Introduction, concerning the universality of the whole crossover structure, are fully confirmed by Figs. 16 and 17. First we show in Fig. 16 that for m < Kz (= 1). U(T) spends a long while near us * in a way that is totally independent of the initial value of u. and then goes to u,+, (Compare with Fig. 5, where the mass is much larger.) Then, in Fig. 17, the corresponding Yen is drawn, once again for a wide range of u’s. In this form universality will clearly persist to higher orders in c. The other aspect of universality is the dependence of the crossover function on changes in the renormalization procedure. This problem arises also for the isotropic massless theory on switching, for example, from normalization conditions to minimal subtraction. When one calculates to order two loops one finds that, although the
FIG.
16.
Plot
of U(T) for m2 = 10m8, K? = 1, N = 2, A4 = 1 and E = 1 for various
values
of u.
TREATMENT
OF
BICRITICAL
391
Pt
functions /3, y,+ , ybz , and u * change when the renormalization procedure is changed, the critical exponents are invariant. Returning to our problem, one should notice that a possible finite renormalization within the GMS is a change inside the logarithm in Eq. (7.2). One can replace the 1 by an arbitrary a (> 0), as is shown in Appendix F. Including this constant, one obtains for p F’ (u, +)
= -a
+ y
22 + y
22 1 + t;i2iaK2 .
(14.1)
In U(T) this amounts to a change U(T) =
Mf8 7
x [u-l - L!!C$K + C!T!$!Z ln (1 + -!I?)]/-’ at2
’
(14.2)
FIG. 17. Plot of year for m2 = 1O-8, G = 1, N = 2, A4 = 1 and 4 = 1 for various values of u: (a) 0.7, (b) 0.65, (c) 0.6, (d)0.55, (e) O.S,(f)O.45.
On expanding in powers of E one finds that the terms containing a appear in U(T) as terms of order 8. Therefore in order to check universality, i.e., to test the actual dependence of yefr on a, one has to calculate Yefr systematically to order c2. This is the same situation as was mentioned before concerning the isotropic theory. There, too, everything straightens out when one calculates systematically to the order of two loops. Notice that the solution of Eq. (8.10) forced us to include in u(p) terms of O(+, which ;are terms of O(1) in U-‘(T). Otherwise the solution is not correct even to lowest order. In fact these terms are the ones which transfer u(p) between U; and z& . If we want to check universality we have to include all terms of O(E~)in yen . What is reassuring is the fact that if one redefines the connection between p and 7 by the condition T(p) = UK2 (14.3) instead of Eq. (10.6), yerr returns to its original form (for u = u,$). A redefinition of p in terms of T should have no effect on Yerr . Here the effect is also on terms of O(@) and has to disappear when one includes all such terms.
AMIT AND GOLDSCHMIDT
392
Another possibility is to change C, of Eq. (11.4) and to take instead Zb2 = 1 + [v
+ v
In (a + $)]
u.
Such a change has two effects. It will change x;l to be
and therefore Eq. (12.6) will become d In xi1 -=dln+ dln7
dlnp2
$1,
(1 + s)].
It will also change ‘ybzto be Y@
=
N+2 6
-u---#
N-M 6
rn2/K” a +
m2/K2
’
which is an effect of order e But one can easily verify that the form (12.8) for Yen is unchanged.
15. CONCLUSION
We have shown in great detail the advantage of including quadratic symmetry breaking in the renormalization transformation compared with an expansion about the high symmetry (isotropic) behavior. While the latter requires special techniques, or a priori assumptions combined with a matching procedure, the former is a systematic technique which employs the natural variables for the description of the cross-over phenomenon. The present technique falls well within the framework of renormalized field theory, though the advantage of using renormalization constants which depend on the symmetry-breaking mass was not generally appreciated. Once within this framework, including the extensions described above, the theory can be applied to many other problems without much difficulty. Such problems as the ordered state [26] or the extension to higher orders in E [9] can be treated in a straightforward way with as many advantages as were described here. Our approach can deal with the quadricritical problem by simply allowing a greater number of coupling-constant renormalization parameters [26]. All these results will be presented in future publications.
393
TREATMENT OF BICRITICAL Pt APPENDIX A
Consider a Lagrangian of the form
2 ==fr(v+)’ + i$mV + (l/4!) ZM’($~)” $ $rl&” + gt2tp22 + C.T.
(Al)
We claim that. the counterterms can be written as 1)(V$)2 + +(Z4p2 - m2) (P2-I- $(Z+‘X - UK’)(~~)~
C.T. = +(Z, -
+ +.(r1[(Z1 - 1) + (M - 1) z21 + t2w - MI z21 A2 + +Jt,Z,M
+ t2[@1 -
1) + w - M -
1) z211 42
(0
to all orders in U. The vertices t,+,* and t2+22 are treated as insertions of +12 and +22 operators, res,pectively, with couplings t, and t, . To order one loop there are two types of infinities coming from the graphs shown in Fig. Al. These graphs represent the insertion of a single &2 into rt2). Graph (a) is r,(F$ to order one loop. This graph is independent of I, owing to the symmetry of the first part of the Lagrangian (Al) not including t, and t, . Graph (b) is r$$ at order one loop, which is again independent of I and k.
L
&
L (bl
Ial FIG.
Al.
;e
One-loop graphs of rj,Z& ’ m1.
We will denote the infinite parts of these graphs by Z:l’ - 1 and Z$, respectively. These have to be multiplied by the corresponding couplings. For 1 < i < M the counter-terms will be the infinite parts of the graphs appearing in Fig. A.2, which are of the form
+{tJ(zl’) -
i FIG.
1) + (M - 1) Z$‘] + t,(N - M) z?‘; &“.
i i
i
i
i
A2. One-loop graphs contributing to the q&* counterterm.
For M + 1 :< j < N the counter-terms are the infmite parts of the graphs appearing in Fig. A3, which are of the form
+{t,ZpM + t,[Zy - 1 + (N - M - 1) ZF’]} 4;.
394
AMIT
FIG.
AND
GOLDSCHMIDT
One-loop graphs contributing
A3.
to the &2 counterterm.
Hence, we proved the form (A2) to order one loop. Let us supposethat the form (A2) is true to order N loops with ZiN’, ZiN’, etc. Then thesecounter-terms are added to the Lagrangian to give 2 + gz~“‘(v+)>” + $Z;N)p(N)2+2+ (l/4!) h(N)Z$N)2(l+2)2 + &‘)+12 + q(N)+ 2 22 2'
(A3)
where
tiN) =
t,[Z;N) + (A4 - 1) Zi”‘] + t,(N - M) zjN’,
tiN) = t&fZ;N)
+
t2[Z;N)+
(A4) (N - M - 1) z:“‘].
This Lagrangian without the vertices t(N)4l2 and t(N)+22 is again O(N)-symmetric. Therefore, all infinities coming from r jF*:) to order N + 1 loops are equal to each other. They are denoted by ZiN+l) - 1. All infinities coming from r,(F,$) are equal to each other and are denoted by Zk”+‘). The new counter-terms for the Cl2 and C22insertions are ;N+l)
C. T . = ‘{t’“‘[(Z 2 1
- 1) + (M - 1) ZiN+l) ] + tjN’(iv - M) .gN+‘)} gJ12
+ &“)$N+l)M = &{tJ(z;“”
+ t;N)[Z$“+l) _ 1 + (N - M - 1) Z$‘+l)]} 922
- 1) + (M - 1) ZZ(N+l)]+ t&v - M) ZiN+l}) CjJ1
+ ${tlZ;N+lk
+ t2[(ziN+l) - 1) + (N - M - 1) Z;N+l)]} +Z2,
where &N+l)
Z(N+') 2
=
I
-
=
q--N)
ZCN) 1
+ +
z
;N)-;N+i) 2
z(N)Z(N+l) 12
+
+
cN
_
ZpJ+l)p
1) z;Np$v+~),
+
(N
-
2) z$“)ZjN+l).
This proves (A2) by induction. The vertices t, and t, are not separately multiplicatively renormahzed becausethe coefficients oft, and t, are different in the last two lines of (A2). But if we define
B = $ [(N - M) &2 - hf+,y, then +,2
+
;t,+,2
=;
Mb+
c;-
mt2
42
+
f
(t1
-
t&
TREATMENT
OF BICRITICAL
Pt
395
and the counterterms in (A2) are ; “““‘;-
M) I2 [Z, - 1 + (N - 1) Z,] 62 + ; (t1 - t2)(Zr - 1 - Z&B.
Thus the vertices t and y given by
t= J&+W--M)t,
and
N
y = t, - t,
are multiplicatively renormalizable, and the operators @ and B are orthogonal under the renormalization transformation.
APPENDIX
B:
EXPLICIT FORM OF THE SOLUTION OF THE RGE O(N)-SYMMETRIC FIXED POINT
NEAR THE
The solution of Eq. (2.16) can be written in terms of an arbitrary parameter, p, which scales all lengths [13]. It reads x-l(t, y, u, K) = p2exp
- x-V@),
U(P)>
U(P),
K>,
(Bl)
with Py
U(1) = U,
= FWPN,
032)
In t(p) = In t - /U’“’ 2 >t$i(U) u
,
(B3)
In y(p) = In y - Jl”’
.
(B4)
2 7($(a)
As t + 0, t(p) -+ 0 and x-l becomes singular. But if t + 0 on the 1.h.s. of Eq. (Bl), p can be chosen to have t(p) outside the critical region. We choose p by the condition t(p)
=
(B5)
K2.
Then p becomes a function oft through Eq. (B3). In particular t = p+ exp
u(o) y&4) - y,*2 _ I -I! u ______ PW - p”-‘x-l*
Clearly, if 24(p)+ u* as t --t 0 the function X will vary in a gentle way, since there is no pole in the integrand of (B6). In terms of this particular p y(p) = J,p-‘2-Yg) exp
,y
..A;(,
is
Y again is a slowly varying non-universal function.
6
1 G yp-‘2-~8y.
396
AMIT AND GOLDSCHMIDT
Finally, the exponential special p. Namely,
APPENDIX
factor in Eq. (Bl) can also be rewritten in terms of the
C: INTEGRALS IN DIMENSIONAL REGULARIZATION
The two integrals entering Eqs. (2.22) are [13]
w = j- 4 q2(q : p)2- ; [1- ; - ; UP)]+ O(4, L(p)
= 1’ dx In[x(l
- x)p2/~2],
(Cl) W)
0
UP1 ,.“, A) = &
[ 1 - ‘2 - GPl
(C3)
+ P,,] + O(l),
with L again given by Eq. (C2). At the symmetry point with P2 =
(PI
+
P2)2
= K2,
L = -2 and
Z*p(O) = 7 (1+ 5).
(C4)
In the massive case z(p,m)
=
.I- dq
(q2
+
m2)[(q1-p)2
+
m2]
'
(C5)
whose c-expansion is (C6)
Finally,
Jh2,=s4(q2 + +. m2”)
m,2;(q2
m22)
cc71
The m-dependent E-expansion of this integral is Jcm
2
m22)
=
1
(m22)1-“‘2
-
(m12)1-E’2 +
13
O(1).
E
APPENDIX
W
D: CORREC~ONS TO SCALING IN AN O(M)-SYMMET RIG THEORY
For an isotropic M-component
system u(p) is a solution of the equation
Py =PW));
U(1) = U,
Pl)
TREATMENT
OF BICRITICAL
397
Pt
where,up to order one loop [3b] /3(u) = -•EU + y
242.
W9
The solution of Eq. (DI) with fl of Eq. (D2) is u(p) = u*[l + p’(u*u-1 - 1)1-l,
(D3)
where, to order E, u* = &/(M + 8). The RGE1 for the isotropic system is just Eq. (2.16) with y = 0. The solution is a special case of (2.17). In lowest order x-1 s (my, (D4) since Z starts at 0(e2). Using definition (B6) for X and ~~2from Eq. (2.13), one finds X = exp
t~'o) ((M
su
+ 2)/6) u' - ((A4 + 2)/(M --a' + ((M + 8)/6)d2
+ 8))~ du, =
u(p)
("+2)l(M+8)
[ u1
(W
To lowest order, one can substitute r1j2 for p, which leads to Eq. (3.4).
APPENDIX
E: THE FLOW OF THE COUPLING CONSTANT
The solution of the equation
032)
is
u-‘(p) = pQ.-l + Kg
(1 - f) - qg!!
PC1% x-y-~K2 .
(E3)
We assume all along that m2 < ICY. For p > m/K we write 2 x-s/2-1 dx
x-~ dx .Y +
rn2jK’
=
PC f 1
1 +
m2/K2X
= p'"--E/2[- ; + f% =-i(l
-~c)+ln(l
($!c)"]lo2 1
+--Y&-~~In(l
+$).
054)
398
AMIT AND GOLDSCHMIDT
For p < m/K we write o* x-J2 dx P' s 1
x
+
rn2/K”
K2X 12 2
m (-)%+I 1 ~
+ p”x-+
11=1 n - 42
__ (
m2
)I
d/i?
(E5)
Hence one obtains for u(p) when 1 > p 3 m/K: NS8 u-‘(p) = -gy
m2
-vln(l+_iz)
KP
+ pE [u-l
- q&J
+ yln(1
+$)I
and when m/K 3 p > 0: M-4-8 u-w = -gy-+
-vln(l+-$$-)
/-p p-!!g!
+
kg!
In (1 + $)]
+ v
($)e’2. W)
APPENDIX
F: GENERALIZED
MINIMAL
SUBTRACTION
AT THE TWO-LOOP LEVEL
In this appendix we renormalize the coupling constant by GMS up to order u3. As an example we consider I’$], in order to show that the requirements placed on GMS in Section 7 are fulfilled at the level of two loops, namely, that the poles in E are eliminated order by order in U, and that as m - co the vertices of the theory have a well-defined expansion in u and E. First we calculate I’&, using the normalization conditions (4.1). The field renormalization, giving a factor of Zb2, is ignored at this stage, since at the level of two loops it takes care of itself anyway. The graphs contributing to r(*) are shown in Fig. Fl.
468
JACK
COHN
of Maxwell’s equations for an arbitrary distribution of charges and currents embedded in a uniform, isotropic, and transparent medium with constant electric and magnetic permeabilities E and p. In addition, the corresponding expression for the stress tensor {which turns out to be Marx’s stress tensor) will be found and this will also he confirmed by the use of Noether’s theorem together with a suitable action function. Finally, we shall develop the covariant expression for the 4-vector potential and field of an arbitrarily moving point charge in the medium.
2. COVARIANT FORMULATION In this section we begin by developing a manifestly covariant description of Maxwell’s equations for a system of charges and currents in a uniform, isotropic, transparent, and boundaryless medium with constant permeabilities E and p. 2.A. MaxweN’s
Equations
We begin with Maxwell’s covariant equations in the form V x H = (477/Q + (l/c@,
(2.1)
V x E = -(l/c)&
(2.2)
V*B=O,
(2.3)
V . D = 47rp,
(2.4)
which presumably hold in any inertial frame, and the constitutive relations H = B - 47rM = (I/p)B
(2.5)
D=E+4rP=cE
(2.6)
and which hold in the medium rest frame. Combining Eqs. (2.5) and (2.6) with Eqs. (2.1)-(2.4), we have in the medium rest frame, the relations V x B = (4r~Llc)T + Wc)fi, (2.7) V x E = -(l/c)& V*B=O, V . E = (47+)p,
(2.8)
(2.9) (2.10)
and these equations form the basis of our inquiry. We shall now make these equations covariant by expressing them (in the medium rest frame) in terms of quantities (originally defined in the medium rest frame) which transform as tensors. We proceed as follows:
400
AMIT
AND
GOLDSCHMIDT
The expressions for the various functions ZqSpare obtained by taking the external momenta at the symmetry point (ZQ = $K3 2, kikj = (K2/4)(&jij - l), p2 = K2), and suppressing the first two variables in the notation. Using Eq. (Fl), one has K-Erfi;lR
=
24 -
-
U2 q
[I
(f
, 0)
-
u2qqz(p)
&,(o)]
-zsp(g]
+u,M2+6M+20 36
[I(+
7 ‘)
-
zdo)]2
+U3(N36M)2[z(~,~)-zsp(~)]2
+,w--M)w+4)
[I (+ 9 C)
36 ~p--M)u4+4) 36
[I,, (?)
- ZSPCO)]”
36
[z(~,o)
-zsP($)]2
[z(p)
-z($,gIP
+*,w--w+2) +wwM)(M+2) 36 +$
- zsP(o)]2
5M+22 9
[z4 (C 92 , 0, (40) - z (+,
0) &P(O)
+ Zs2P@) - Z48PK4 0, o)]
+ 28~[14(e,~,o,~,~)-z($,o)z*,(~) + 4%43ZSP (C) - Z4SP (0,;, +)I
+U~2(~p~~4(~,~,~,~,0)-z(~,~)zsp(0) +
ZSP (F)
ZSP(O)
-
&SP
($,
f
, q.
(F3)
Notice that in order to simplify the expression (F3) we assumed that the external momenta of D4) are also at some symmetric momentum point, so as to avoid sums over permutations. One can convince oneself, not only that there are no poles in E in Eq. (F3), but also that as m + co the expression remains finite order by order in u and E and in fact reduces to the value of P4) in a theory with M massless particles only.
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Using the notation L&y
+ln(+
-X)+$),
L, ($ , :)
= f dx ln2 ($x(1
(F4) - x) + $-),
one can write the expansion of the one-loop integrals I($+-)
=f[l
-~-~L1(~,a)+$L1(~,~)+$L2(~,~)] + O(E2),
(F5)
from which one can derive expansions for I( P/K, O), &&,v/K), &,(O). Next we expand the general I4 in Eq. (F2) I, (f
, $ ; ?,
?,
?)
= ‘4 r2 (2 - ;) I-(E) 1’ dx[x(l
- x)]-‘/~
0 1
X
dy( 1 - JJ)JJ’/~-~ ‘dz
s0
$ (1 - y)(z(l - z) + YZ”)
s0
+ 2y(l - y)z q
1
+ Y(1 - u> $
+ (1 - VI y w5)
In order to obtain the pole terms we write
{u>’ = {Y = o>-+ [{y>-’- (y = 0)-q; i.e., we add an subtract the brackets with y equal to zero, where the singularity lies. {y}-’ - {y = O}-c = --E In &
+ O(E~).
(F7)
This term will contribute only to the finite part of (F6). The pole terms for a general distribution of masses are given by P.T. = &
[ 1 - ; - E& (f
, ?I).
(W
The finite part depends on the specific values of internal masses. We therefore specialize to each case. First, z*(+,~.o,o,o)
=&[l
-+‘($‘o)]
+0(l);
here we da1 not care about the finite terms, since they are independent of the mass.
402
AMIT
AND
GOLDSCHMIDT
In order t0 Caldate &(p/K, &/K, 0, m / K, m/K), we substitute expression (F7) for the curly brackets in (F6). We are only interested in the limit of this finite part for very large values of m/K. We therefore obtain as m + co : finite part of I4 (f-,2,
O,c,$)
Fz - ; r2 (2 - 5) r(E) * E J dx x-E/2 1 dy(1 - y) y+1 x S~41
+g
x(1 -x;z(*
= -$-q)+n2p4r(l
-z))
-:;z(l m2
- In $x(1 - x) z(1 - z) 1
-Z)
+ O(E) = - f In2 $
- i In -$- + O(E).
(F9) Terms which are finite in the limit m ---t co were dropped, and use was made of
Therefore one obtains
where
r($+)
Asfor&(P/K, kdK, --E In
m/K,
- $ln2!$ m-rco’
i In $
+ O(E).
(F10)
m/K, 0), Eq. (F7) gives in the limit m -+ 00
3 z(l - z)+ (I - y + -&)
$ -t-•Eln
$ z(l - z) + 5 and this gives a mass-independent
contribution
(
l-u+-+,
to the corresponding
+ finite terms in m + O(E).
1 14. Therefore
(Fll)
We can now change our point of view. Instead of regarding I&O), I&m/K), I,,,(O, 0, 0), etc., as the corresponding functions at the symmetry point, we will treat
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them as unknown functions in the expansion of u,, in powers of u (Eq. (Fl)) to be determined from Eq. (F3) by GMS. In order to avoid confusion we will denote these unknown functions by J(O), J(~/K), J&O, 0, 0), etc. From the terms proportional to ~2 in (F3) we obtain J(0) = l/e, W)
including the poles in E and the infinite m limit of the finite terms in the E expansion. We therefore take J (c)
== f - i ln (a + $)
+ $ In (a + $)
+ i ln* (n + $1
+ O(2), (F13)
where a is a positive constant. The difference I(P/K, poles in Eand. will vanish as m + co up to order E. From the terms proportional to u3 one obtains
m/K) - J(m/K)
will not have
r4 (f ) 2) 0, 0,o) - I($ ) 0) ; + f - J,(O,0,O) yg 7; 2;‘, [ 1 - ; - EL1(f- , o)] - $ [ 1 - ; - ; L, ($ , o)] + ; - J,(O,070) =---- 1 2G
1
4-z - J&4 0, 0).
Therefore we take J&4 0, 0) = &
(1 + ;).
and J4(m/K, HI/K, 0) are determined Eq. (F3), respectively. Consider first J4(0, m/K, m/K):
J,(O, m/K, m/K)
(F14)
from the last two brackets in
404
AMIT AND GOLDSCHMIDT
and the proper choice is J4(0,~,~)
=&l
+;)
-;ln+++;ln(a+$)
+ O(E). (FL5)
In a similar way one obtains
J4(y))
=&[
1 + ; - E In (~2 + $)I
+ O(E).
w3.l
The values of the functions J we found have to be substituted in (Fl) to give u,, in terms of U. As for the third and fourth lines of Eq. (F3) it is obvious that the poles in E cancel in each square bracket separately. However, as m -+ co it is the interplay between the two terms which leads to a finite result. Namely, the O(1) terms of these two lines become u3
(N
-
M)(M
+
4,
In2
36
!
vJ2
ln2
Et
K2
K2 1
+,(N--M)(M+2) 36
(
ln2 $!. - In2 f)
= 0.
The effect of the field renormalization (2,~ in the case of rf4)), as well as the calculation of the other Z-functions, is handled in a similar way.
APPENDIX
G: CALCULATION
OF xc1 TO ORDER ONE LOOP
We use the Lagrangian (4.3) to calculate I’:;) (p = 0; 7, m, U, K). At the order of one loop one needs counter-terms which renormalize the mass and the temperature only. The first take care of the quadratic divergences, the second of the logarithmic divergences. Without counter-terms, (4.3) gives, to order one loop, p
= T 1 hit + 2 UKF 11
6
1
N--M
1 -
s q2 + 7
+
6
UKE s
q2 + m2 + T ’
(Gl)
The counter-terms which are needed here are of the form CT. = Qc,#~ + &T#~.
(G2)
They add to the r.h.s. of (Gl) a term c, + c27. The function c1 is determined by requiring that at 7 = 0 the frrst M components be critical, namely, that &(T
=
0,
P?Z, U, K)
=
0.
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This gives 1 UK” 2--
M+2
cl = - ~ 6
N--M6
4
UKf
1
(G3)
q2.
Inserting in I’:,“‘, we have r;;)
=
7
(C2
+
1)
-
y
UK’
I
q2(qe1+
~)
I
-
N-M
1
UKF
6
s
(q2 + m2)(q2+ m2 + T)
which in the notation of Appendix C can be written as ~
#K’~(o,
T)
N---M
-
____
6
UKE.@2,
m2
f
T)
(G4)
after 2,~ is dIEfined by
to order one loop. This is Eq. (11.8).
APPENDIX
H
Here we paraphrase Symanzik’s derivation [4] of the Ward-Takahashi identities induced by quadratic symmetry breaking, using the method of functional integrals. We consider a Lagrangian with N Hermitian scalar fields of the general form
where the first term is O(N)-invariant and the quadratic symmetry-braking term is defined in terms of the symmetric matrix M. The generating functional of the complete Green functions Z(J) corresponding to the Lagra.ngian is given by
in terms the term J Wi(4 Making
of a linear source (external field) J. If one expands Z(J) in terms of Mij, linear in Mij will consist of symmetric Green functions with an insertion of hi< x > w h ic h are not finite. New renormalizations are required. am infinitesimal change of variable
406
AMIT
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GOLDSCHMIDT
where ra are the generators of the group O(N) and E, infinitesimal
parameters, one has
Since the 6s are dummy variables Z(J) does not change. This implies that the firstorder change in E must vanish, which can be written in terms of functional derivatives with respect to J as 035) These are the W-T identities for the generating functional. They relate the nonsymmetric part of Green functions to Green functions into which J dx q$(x) ySj(x) is inserted. In order to derive a linear differential equation, let us introduce a source term for the bi4i insertions.
All quantities have to be calculated with Ji = 0 and Kij = Mij . The W-T identities take the form (H7) The same equation is satisfied by F(J, K) = In Z(J, K), the generating functional of the connected Green functions. The generating functional for the vertex functions, I’, is defined by a Legendre transform, namely, T(@, K) + F(J, K) = j- dx tDi(x) Ji(x)
(W
with
WJ, K) j 2
@i(X) = w Therefore
and one obtains
%=JfJo
*
(H9)
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Setting Ki, = Mij , one obtains W-T identities in the form T;$~(x)
-
2&&~&
W@‘,K) 6K&)
r+,wij = OS
(HW
where the full notation was reintroduced. Taking the functional derivative of (H12) with respect to @, , CD, , where the lower indices include the space time points, one has after setting K = M:
If all Q’s are :set to zero the various derivatives of I’ are just the usual vertex function: one-particle irreducible parts of the Green functions. When a Fourier transformation is performed, (H13) reads
where the function I’$‘,l/ij,(p; 0) is P) with an insertion zero momentum. Specializing to 0(2)-symmetry, we have simply TkZ = ~ld,Z - w&k
of the operator
*
+i4j with
W5)
Taking ww and choosing I = 1, m = 2, we arrive at
I-,',"' (p) - r,;' (p) = 2(m,2 - m2y r,::$,,(p,
--p; O),
(H17)
which related the nonsymmetrix part of the two-point function to a two-point function with one insertion of $r&. Note that the right hand side vanishes for
m, = m3. Taking functional derivative of Eq. (H13) with respect to @t , @t and setting all Q’s to zero, one obtains at a symmetric momentum point (Y = s = 1)
The right hand side is a four-point vertex with external legs of types 1, 1, 1, 2 with an insertion of &c$, at zero momentum. One obtains a similar connection between I’&?i, and I’$, .
408
AMIT AND GOLDSCHMIDT
Equations (H17) and (H18) are the W-T identities needed in order to find all the renormalization conditions and to renormalize the theory by the BPHZ method. The proof by induction was outlined by Symanzik [4]. From Eq. (H17) it is obvious that the same Z, (the wavefunction renormalization) can serve for both I’::) and I’$. Taking the derivative of (H17) with respect to p2 at p2 = 0, we obtain
P2=0
d = 2(m,2 - m22>_ ct;2, dp2
. PLO
0319)
The right hand side is superficially convergent because 1’$$12, is only logarithmically divergent, and hence the assertion follows. Similarly, because the right-hand side of (H18) is superficially convergent, it is evident that one counterterm can subtract the infinities from all the different fourpoint functions. Only the mass terms need different counterterms. REFERENCES 1. D. R. NELSON AND E. DOMANY, Phys. Rev. B 13 (1976), 236; H. HORNER, 2. Physik B 23 (1976), 183; J. M. KOSTERLITZ, .I. Phys. C 9 (1976), 497, and many references to previous work therein. 2. D. MUKAMEL, Phys. Rev. Lerr. 34 (1975), 482. 3. See, e.g., (a) A. AHARONY and (b) E. BREZIN, J. C. LE GUILLOU, AND J. ZINN JUSTIN, in “Phase Transitions and Critical Phenomena” (C. Domb and M. S. Green, Eds.), Vol. VI, and references cited therein, Academic Press, New York, 1976. 4. K. SYMANZIK, Renormalization of theories with broken symmetry, in “Cargtse Lectures 1970” (D. Bessis, Ed.), Gordon & Breach, New York, 1972. See also J. ZINN-JUSTIN, in “Trends in Elementary Particle Physics” (H. Rolnik and K. Dietz, Eds.), Springer-Verlag, Berlin, 1975. 5. E. K. RIEDEL AND F. J. WEGNER, Z. Physik 225 (1969), 195. Notice that in this original work it was T - Tc~ which was used, and which we show below to be the natural variable. The present form is used in Ref. [l]. 6. The physical systems described by an 2 of this type have been discussed by M. E. FISHER AND D. R. NELSON, Phys. Rev. Left. 32 (1974) 1350 and by A. AHARONY AND A. D. BRUCE, Phyx Rev. Lerr. 33 (1974), 427. 7. K. G. WILSON, Phys. Rev. Lett. 28 (1972), 548. 8. E. BREZIN, D. J. WALLACE, AND K. G. WILSON, Phys. Rev. B 7 (1973), 232. See also Ref. [3, sect. 7.11. 9. A. D. BRUCE, J. Phys.C’. 8 (1975), 2992, and A. D. BRUCE AND D. J. WALLACE, J. Phys. A 9 (1976), 1117. 10. G. ‘T H~~FT AND M. VELTMAN, Nucl. Phys. B 44 (1972), 189. 11. See, e.g., Ref. [l]. 12. In the context of critical phenomena see, e.g., J. D. LAWRIE, J. Phy.s. A 9 (1976), 961; D. J. AMIT, J. Phys. A 9 (1979, 1441. 13. See, e.g., D. J. AMIT, “Field Theory, the Renormalization Group and Critical Phenomena,” Chap. 7, McGraw-Hill, New York, 1978. 14. See Ref. [3b] or Ref. [13, Chap. 91. 15. We are being slightly cavalier in writing (2.16), since the proportionality factor between rrr) and x-’ depends on u (it is Z,). But as far as the dependence on t and y is concerned, this is irrelevant. 16. See, e.g., Ref. [3b, Sect. 7.11 or Ref. [13, Sect. 10.71. 17. This is true for the result of Wilson-Ref. [7] above. Here the statement is proved to all orders.
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18. See, e.g., Relfs. [3] and [22] below. 19. See, e.g., S. COLEMAN, “Renormalization and Symmetry: A Review for Non-Specialists,” International School Etore Majorana (1971), Editrice Compositori, Bologna, 1973; see also Ref. [13, Chaps. 7, 101. 20. Here we imply that the regularization is affected by a momentum cutoff A, though usually, as in Section 2, w’e will use dimensional regularization. 21. If one insists on an O(N)-symmetric four-point counterterm then r”) with external legs of type i will not be finite as m --L 00. 22. K. SYMANZIK, in “Particles, Quantum Fields and Statistical Mechanics” (M. Alexanian and A. Zepeda, Eds.), Springer-Verlag, Berlin, 1975. See also Ref. (31; F. JEGERLEHNER AND B. SCHROER, Acta Phys. Austr. Suppl. 11 (1973), 389. 23. The fact that the integrals are over dx/x, rather than the conventional du/j3, is just a matter of convenience.. These two measures are identical, owing to the first characteristic equation, Eq. (8.2). 24. S. SINGH AND D. JASNOW, Phys. Rev. B 11(1975), 3445; Phys. Rev. B 12(1975), 493. 25. S. COLEMAN AND R. JACKIW, Ann. Phys. (N.Y.) 67 (1971), 552. See also K. Symanzik, DESY report 73/6 (1973), Ref. [3, Sect. 3.41, Ref. [13, Sect. 7.41. 26. E. DOMANY, D. R. NELSON, AND M. E. FISHER, Phys. Rev. B 15 (1977), 3493; E. DOMANY AND M. E. FISHER, Phys. Rev. B 15 (1977), 3510. 27. See, Ref. [1:3, Appendixes 9.2, 9.31.
Crossover at a Bicritical Point: Asymptotic Field Theory with Quadratic Symmetry DANIEL J. AMIT
AND YADIN
Behavior Breaking
of a
Y. GOLDSCHMIDT
Racah Institute of Physics, Hebrew University, Jerusalem, Israel Received August 1, 1977
We discuss the case of an O(N)-symmetric theory whose symmetry is broken by a quadratic term which produces two different masses (correlation lengths) one associated with M components and one with N - M. This theory describes physical systems having a bicritical point. To this problem we contribute by showing how within renormalized field theory the calculation of the shift in the critical temperature (leading to a crossover exponent) is separated from the calculation of the crossover function. The computation of the latter is then carried out in terms of the natural temperature variable, relative to the anisotropic critical temperature. This feature allows one to bypass the complicated corrections to scaling generated in earlier approaches by the use of a different variable. Within this framework which allows many tests of universality the results are explicit and readily generalizable to the ordered state and to higher orders in E. To field theory we contribute by showing how the mass-dependent renormalization procedure allows the interpolation between asymptotic behaviors with internal O(N)- and O(M)-symmetries. The interpolating parameter is the ratio between the momentum scale and the mass difference breaking the symmetry.
1. INTR~D~JcTI~N The study of the bicritical point-a point at which two critical lines merge- is not new. Three recent papers [I] describe the relevance of precise theoretical studies of the bi-critical behavior to present day experimental technology in solid-state physics. A study of the problem reveals that very interesting theoretical questions abound as well. It is clear, of course, that in any experimental situation anisotropies are present. Exact global symmetries accompany conservation laws only, as they do in particle physics, and in situations such as superfluids and superconductors. In a three-component ferromagnet, or in a four-component antiferromagnet [2], the discrete structure of the lattice will never render and exact continuous global gauge symmetry. If the symmetry is broken in the quartic terms, then asymptotically the anisotropies are suppressed if the number of components is smaller than 4 [3]. In the language of the renormalization group one says that for low numbers of components the isotropic fixed point is the stable one. On the other hand, the symmetry is also generically broken at the quadratic or soft level. Here the asymptotic behavior is eventually dominated by the fixed point with a lower number of components-those corresponding to the lower bare mass (coefficient of the quadratic term). 356 OOO3-4916/78/1142-O356$05OO/O Copyright 0 1978 by AcademicPress,Inc. AU rights of reproduction in any form reserved.
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Nevertheless, there is a range of parameters- scale of momenta, temperature, etc.in which the fixed point of high symmetry dictates the asymptotic behavior. The reason is that because of the presence of an additional length the asymptotic region here is much richer than in the usual situations with a single length. Such an extra length is, for example, the mass-inverse correlationlength-of the N --M components at the temperature at which the M components become critical (massless)-a quantity that is directly related to the anisotropy. If this mass, m, is much smaller than a microscopic momentum K, there is a range of the parameter a-the overall momentum scale or the temperature-such that
but the ratio cx/m is arbitrary. Moving from u/m < 1 to a/m > 1 introduces a structure into the asymptotic region. And, since it all happens for 01< K, much of this structure is universal. The possibility of obtaining this complex asymptotic behavior in a universal way, and hence in a controlled E-expansion, is a source of much theoretical interest. Furthermore, the study of field theories with quadratically broken symmetry has known few detailed applications, though the general framework had all been set by Symanzik [4]. On the other hand, the knowledge of this behavior is of direct importance to the analysis of experiments. As the temperature is lowered, at fixed anisotropy, towards the low-symmetry critical temperature, there may be a wide temperature range in which the behavior of the thermodynamic quantities will change from one power law to a second one---both nonclassical. Only when the temperature difference-corresponding to ,x2-becomes much smaller than m2, will a uniform power behavior set in; the expbonents will be those of the system with low symmetry. To be more specific, a reiteration of the phenomenological expectations concerning the behavior of a system near a bicritical point [5] will be helpful. Consider a bare Lagrangian of the form 27 = P412
+ &P12412 +
Bp22422
+ (V49(92)2,
=
4i2,
+2 = 412 +
(1.1)
where d12 = ; $v, i-l
922
5
422.
(1.2)
i=rvf+1
Both p12 and ,p22are linear functions of the temperature [6]. If l-h2 -=I P22, which implies that the mean-field transition temperature of the M compoents of type 1 is higher, then one expects the first M components to become critical before the rest do. The fluctuations of the remaining (N - M) components grow, but never diverge. One therefore expects a critical behavior of an O(M)symmetric theory. For p22 < p12 one expects an O(N - M)-symmetric critical behavior of the type 2 components. When p12 = p22 the theory is O(N)-symmetric. This is the bicritical point at which the two critical lines, the O(M)- and O(N - M> symmetric lines, join. This situation is depicted qualitatively in Fig. 1.
358
AMIT
AND
GOLDSCHMIDT
FIG. 1. A qualitative phase diagram of a bicritical point g = pCLa2 - p12. The heavy line is a line of first-order transitions.
If the anisotropy is denoted by g, namely,
g x Ps2- A29 the expected behavior of a quantity such as the longitudinal susceptibility along an essy axis-is [5]
(1.3)
susceptibility-the
where t = (T - T&/T,, , with TcN the critical temperature of the O(N)-symmetric system. The exponent +-the cross-over exponent-describes the departure of the two critical lines from the bi-critical point. As is well known, C$is a universal quantity, whose value was calculated by Wilson to second order in E [7]. Its calculation implies an expansion about the O(N)-symmetric fixed point, namely, for g/t < 1. This exponent is completely determined by the O(N)-symmetric fixed point. As will be shown in Section 2, it is independent of M. This is not the case for the functionf: One expects thatf(z) will vanish for some z0 # 0 which defines the critical lines via
g = z,t*,
(1.5)
giving TCM(g), the critical temperature at fixed anisotropy. If t -+ 0 but g/t < 1 then x~l N tv. On the other hand, if g is fixed then as t + gl/*/z, ,
(1.6)
tY in (1.4) is a constant and the singularity comes fromf, which develops a behavior like
f(z) - (z -
z$
cc [T - TcM( g)l; oc 73:
where + is the exponent for the M-component
theory.
(1.7)
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The cross-over between these two behaviors is conventionally described by an effective exponent which depends on the temperature variable T. Namely, one defines
yefr(T) = d ln X?(T) dln7
’
The relationship between x-l, g, and t has a form analogous to that of an equation of state; see Eq. (2.17) below. To obtain the form of the latter to any given order in E resuires simply the calculation of the external field to a given number of loops [8]. But here the situation is complicated by several factors [9]. One reason is that in order to obtain form (1.4) one has to substitute in the result of the perturbation calculation u=u$, where U$ is the fixed point of the isotropic N-component system. But the true, and only,, fixed point for the problem is the O(M)-symmetric one, u$, . Therefore, there are corrections to scaling to the O(M)-symmetric asymptotic behavior which will show as powers of logarithms and will not permit a direct exponentiation of the true asymptotic behavior. Another source of difficulties is the inadequate temperature variable, which is different from the variable 7 = T - T,(g) characterizing the O(M) behavior. In order to proceed, one has to make several assumptions from the beginning, which are not dictated by the perturbation calculation. One has to a,s,sumea priori that the behavior of the theory in the critical region is that of a system with an O(M)symmetry. Then one has to compare the Eexpansion with the corresponding expansion for the O(M)-symmetric system, and try to match the two expansions even though they are expressed with different variables-a procedure which is not devoid of ambiguities. Rloreover, one has to use results of the renormalization group calculations for the O(M)-symmetric theory with corrections to scaling (see section 3 below) in order to make the matching possible and to exponentiate the true behavior. This is just an outline of the strategy adopted by Bruce and Wallace [9]. It is just thi:s difficulty which impelled Nelson and Domany as well as Horner and Kosterlitz [l] to adopt special renormalization group strategies. Nevertheless, the traces of the C)(N)-symmetric critical temperature are still present in all these treatments, and thle special methods used make it hard to envisage how higher orders in E are to be calculated systematically. Here it will be shown that, in the framework of renormalized field theory, it is possible to formulate the whole problem in two equivalent forms. Once as a soft expansion-in temperature and in quadratic anisotropy-about an O(N)-symmetric theory. Then as an expansion in temperature about an asymmetric theory in which M components are massless (critical) and (N - M) have a fixed mass (anisotropy). The first formulation is natural for the calculation of 4, and some studies of universality. It is presented in Section 2. The second formulation, which will constitute the bulk of this article, provides a description which is ab initio in terms of the natural variable T - TCM( g). In other words, we return to the original variable of Riedel and Wegner [5], formalize it, and compute with it. Within this formulation the extensions to higher orders in E, as well as to ordered states, are absolutely standard. Furthermore, many
AhlIT AND GOLDSCHMIDT
360
additional questions of universality can be studied. In this formulation one does not have to assume the asymptotic O(M)-symmetric behavior. It comes out automatically. The plan of the paper is as follows: In section 2 we present the expansion about the symmetric theory and repeat the calculation of the cross-over exponent 4. Then, in Section 3, the difficulties encountered in the derivation of the cross-over function from the expansion about the isotropic theory are discussed. Here the scene is changed, and the Lagrangian (1.1) is rewritten so that the symmetry breaking is fully contained in the free part. In Section 4 we show, following Symanzik [4], that the theory can be renormalized in such a way that m, the mass of the (N - M) non-critical components, can be held fixed while the renormalization momentum scale is varied and the renormalization group equations are derived. Then in Section 5 we argue that in order to be able to describe the infrared asymptotic behavior of the theory, at hxed anisotropy, the renormalization constants must depend on m. Sections 6 and 7 describe two different procedures for the calculation of the renormalization constants and Wilson functions are presented. The first utilizes normalization conditions. The second generalizes the minimal subtraction method of ‘t Hooft and Veltman [lo] to include m-dependence. Next, in Section 8, the renormalization group equations are solved in the presence of two lengths. This requires some extension of existing techniques. In Section 9 the temperature is introduced as a soft insertion in a non-symmetric theory and renormalized. This is the temperature difference relative to TCM( g). The renormalization group equations for functions which depend on T,
(1.5) are derived. They are formally solved in Section 10. Section 11 describes two different procedures for calculating the renormalizatione constant of the temperature. One is minimal subtraction of the 42 insertion; the second is fixing the value of the two-point function at some non-zero value of the temperature. All these functions are computed to O(E). The results are assembled in Section 12 to give an explicit expression for the cross-over function. In section 13 we derive the RGEs for the specific heat and for the non-ordering sysceptibility, and it is shown that they fall into the standard pattern solved in previous sections. This is not the case for the transverse susceptibility which is discussed qualitatively. Finally, in Section 14 we discuss various aspects of universality, and in Section 15 we list extensions of the present calculations which are in progress. 2. SOFT EXPANSION
ABOUT
THE SYMMETRIC
THEORY: THE CROSSOVER EXPONENT
The main results of this section are not new, although some are shown to be true to all orders in renormalized perturbation theory and in E(= 4 - d). Otherwise this section serves to recast the calculation of the shift in the critical temperature due to anisotropy in the language of renormalized field theory.
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This, on the one hand, emphasizes the similarity with the calculation of the equation of state and in this way helps us establish later, in Section 14, some aspects of universality. On the other hand it serves to underline the contrast between the computation of the cross-over exponent-a phenomenon belonging to the realm of the symmetric theory, and that of the cross-over function to which the bulk of the paper is devoted. As we shall see, the latter is grounded in the asymmtric theory, and in the shift of reference system the natural variable becomes the temperature difference from the critical temperature of the system with low symmetry. The critical temperature of the symmetric theory disappears. We thus considered it instructive to present, in a common language, thes’e two complementary calculations for the same system. The Lagrangian (1.1) can be rewritten in the form 9 = HV4)2 + +pc242 + (h/49(+2)2 + ii&242 - QgB, where pC2 is the value of the vare mass which gives an O(N)-symmetric (massless) theory when 6~~ = g = 0. The operator B is given by B = (l/N)[(N
- M) +12 - Mc#~~].
(2.1)
critical
(2.2)
Translated into the old parameters of Eq. (1 .l), the new ones read PC2 + 8p2 = (1/N)[MpL,2 + (N - M) &I
(2.3)
and g =
P22
- P12.
In other words, g is proportional to the anisotropy. It is a proper symmetry breaking parameter. The special choice of the operator B is dictated by the requirement that it be invariant under the subgroups O(M) and O(N - M), whose symmetry was not broken, and that it remain orthogonal to C2 under renormalization transformations, namely, that the counterterms generated by +2 will be proportional to 42, and those generated by tl be proportional to B. Any other combination of +12 and 622 cannot be renormalized separately. That this is the case we show in Appendix A. The renormalized form of the Lagrangian can be written as 9 = $(V+)” + (zM/~!)(+~)~ + -&t+2 - QyB + counter-terms, counter-terms = &(Z, - 1)(V+)2 + &[(pC2 + 8~~) Z, - t] 42
(2.4)
- y)B + (1/4!)@2,” - u/F)(+~)~,
(2.5)
- +( gZ,
where t and y ,are related to 8p2 and to g via a2=z P
d?t ’
(2.6)
P = ZBY.
(2.7)
362
AMIT
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GOLDSCHMIDT
Both 2~ and Z, are functions of u and E (= 4 - d). They are the renormalization constants which render finite the insertions of @ and B operators into vertices. In other words, let rLN9”*“‘( pi , qi , ri; 6p2, g, A, A) be the bare vertex with N external legs, L insertions of +2 and M insertions of B, then the renormalized vertex is given by FiNsLeM)(pi , qi , ri ; t, y, 24,fc) = Z~‘2Z,L,ZBMFiN*L*M)~i
, qi , ri ; 6j~‘, g, A, A). (2.8)
The parameter p C2was suppressed, since it is determined in terms of X and A by the requirement that the theory be critical when 6~~ = g = 0. The functions A. Z, , Z,O , and Z, of u and E are calculated in the standard way [3, 131 by demanding that Fk”’ (p = 0) dqf’/dp2 Isp , r$O), and Q~~7” be finite as (1 -+ co (or as E -+ 0 for a theory that was dimensionally regularized [12]). The subscript R indicates that the function is renormalized, and sp indicates that the renormalization was carried out at a symmetric momentum point. Here we will follow the second alternative and calculate the renormalization constants by minimal subtraction of dimensional poles [IO, 121. But first we derive the renormalization group equations. Equation (2.Q or alternatively (2.5), implies that the vertex PQ, at arbitrary values of 6$ and y, is renormalizable via [13] FAN’<& ; t, y, u, K) = zy2riN)(pi
; sp2, g, A, A),
(2.9)
The coefficient functions are the standard ones in the O(N)-symmetric be read off from the literature [14]. The only new function is yB . For completeness we give the list to order two loops:
theory. They can
which leads directly to the differential equation
B(u) = (
a ln(kc) au
y@(u) = -/3(u)q@ =
-/3(u)
ya will be computed below.
--1
= -u (6 - y
N+2 = 72
y&4) = /3(u) y
yB
)
q.
24+ 3N;
l4 .a),
(2.11)
242,
=+4(1-&), (2.14)
TREATMENT
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If we restrict ourselves to N = 2, p = 0 and choose both external legs of I’@) to be of type I-belonging to the group with index i ,( M. Then
and Eq. (2.10) reads [ 151 (2.16)
[
The solution of this equation is standard [16]. The equation is completely analogous to that for the equation of state, and so its solution is
The exponents y and v are the conventional exponents of the susceptibility and the correlation length, respectively, in the O(N)-symmetric state. They, as well as
rB* = yEt(u= u”), are calculated at the O(N)-symmetric fixed point (the zero of /3) which dominates the asymptotic behavior of the solutions of Eq. (2.16). The functions X, Y, and 2 are the non-universal parts of the solution-depending on the initial value of U, rather than only on u*. They vary slowly, as functions of t, in the asymptotic region and thus they contribute only to corrections to scaling. The expressions for these functions are given in Appendix B. @ is an arbitrary function which is computed in perturbation theory. The exact result, Eq. (2.17), is an expression of the fact that apart from a normalization of the temperature, the anisotropy and the scale of the susceptibility the form of the latter is universal, i.e., independent of the initial value of u, and hence X. The statement is true to all orders. Furthermore, comparing the solution (2.17) to the phenomenological ansatz for the cross-over behavior, Eq. (1.4), one identifies the cross-over exponent 4 to be + = v(2 - yB*). Consequently, to obtain 4 to any order in E it suffices to compute ye, via Eq. (2.14). Turning to the actual calculation of 4, we show first that to all orders in E 4 on N only-it is a property of the isotropic theory and not of the particular which the symmetry was broken [5, 171. To see this consider a determination of Z, via a normalization condition
595/114/I/=4
(2.18) 2, and depends
form in such as
364
AMIT
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GOLDSCHMIDT
For our purposes it is sufficient to prove that the Z, which ensures (2.19) is independent of M because any other renormalization procedure differs only by a finite renormalization. Using Eq. (2.8) we can rewrite (2.19) in the form (2.20)
The insertion of an operator &” (no sums) into a vertex I’::), calculated in the O(N)symmetric theory, depends only on whether i = 1 or i # 1. Denoting such an insertion by I’& , one can rewrite Eq. (2.20) as & Z,Z,[(N
- M) r$
+ (N - M)(M
- 1) r;;!2 - (N - M) Mr~;,] (2.21)
Z, , as well as the brackets, is independent of M, and the explicit factor of N will disappear when 2, is substituted in Eq. (2.14) to obtain yB . The graphs needed for the computation of Z,Z, to order two loops are drawn in Figs. 2 and 3. The expressions corresponding to these graphs are
(2.22)
where I and I4 are the integrals corresponding to graphs (b) and (d) in Fig. 2. They are defined in Appendix C, where their Eexpansion is also reproduced. Inserting (2.22) in (2.21), substituting the E-expansion of the integrals from Appendix C, and for h the expression [13] (2.23)
(a) FIG.
(b)
Cc)
2. Graphs of r$fl.
FIG. 3. Graphs r$.
Cd)
The x indicates an insertion of g18.
The circle indicates an insertion of gz2 .
TREATMENT
one can compute Z,Z, by minimal
OF BICIUTICAL
Pt
365
subtraction. To obtain ‘ye one writes
yB= -/qu)aln@$‘4) + Y&4
(2.24)
Then using yB(u) from Eq. (2.12), and setting [I 31
u=u*=-LN+8’+
18(3N + 14) E2 (N+8)3 ’
where U* is the zero of /3 (Eq. (2.1 l)), one obtains 2 YB*=N+8E+
-N2
+ 18N + 88 E2 2(N + 8)3 '
(2.26)
Together with the E expansion of v [I 31 for the isotropic theory one finds from (2.18) and (2.26) N3 + 24N2 + 68N $=1-t N E2+ O(E3), (2.27) 2(N + 8) ’ + 4(N + 8)’ in agreement with previous calculations [7].
.3. DIFFICULTIES IN OBTAINING
THE CROSSOVER FUNCTION
The standard way of proceeding would be to expand xc1 to some given number of loops, and to use the general form, Eq. (2.17), to identify CDto a correspondint order in E-just as one does for the equation of state. One knows that Q(z) must vanish for some z0 # 0. This is the phenomenological identification of the critical lines in Fig. 1. Namely, td KY. (3.1) Then the temperature is normalized to fix the coefficient in this equation as unity. If in addition one demands that at J’ = 0, the isotropic behavior
all non-universality is eliminated. The rest can be computed at the fixed point as a power series in E. But here one encounters the difficulties outlined in the Introduction. In terms of the variables
one expects @ = @i(z)to have a singularity of the form (1 - z)i, i.e., Q(z) = X(1 - Z)+(Z),
(3.3)
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where 7 = y(M) [l]. One cannot obtain the correct singularity from the perturbation calculation because of the presence of corrections to scaling. These are generated by the fact that the true fixed point is the O(M)-symmetric one and the choice II = UX made above suppresses the transients with respect to the unstable O(N)-symmetric fixed point, thus giving rise to corrections to scaling about the asymptotic O(M)symmetric behavior. These corrections to scaling appear in the calculation as powers of logarithms and prevent the exponentiation of the dominant singularity. In order to proceed, one has to assume the form (3.3) a priori, i.e., assume that Q(z) vanishes like (1 - z>*, where 3 = y(M). To make this point clearer, let us consider an isotropic M-component system. The full solution of the RGE, including corrections to scaling, is given by Eq. (2.17) withy = 0. The factors Xand Z describing these corrections are defined in Appendix B, and calculated to order E in Appendix D. The result is -(M+2)/(M+S)
-1
=
[ 4’)1
T1+((h4+2)/2(M+8)k
u
XL
where T is the temperature
difference of the M-component
u-y’)=v Substituting
theory, and
+ TF/2 .(u-1 - q&E).
(3.5)
u = u * = 6e/(N + 8) one obtains xi1
=
T1+((M+2)/2(M+8)k
Expanding this expression,
M+8 N-t8
E,2N _ M
~
+T’
N+8
-(M+B)/(M+W (3.6)
one arrives at M+2+ 8) EIn T xp = 7 1 + 2tN ( 1,
(3.7)
and one could think this comes from the expansion of
which of course is wrong. [Notice that it is easy to find the generalization of (3.4) to 0(c2) by calculating X and Z of Appendix B to U(E~) and substituting in (B.l) with y = 0. This leads to the result of Bruce and Wallace [9] for the Gaussian-Heisenberg cross-over function. But problems remain even after one assumes the form (3.3). In matching the perturbation series to an expressions such as (3.4) one has to insert the correspondence between 7 and z. For z N 1 it is simply 7 t+ 1 - z. Otherwise 7 t-t f(z)(l - z) and f(z) has to be determined by the matching procedure itself. All of these are the results of an unfortunate choice of variables, which can be completely avoided, as we show below.
TREATMENT
4. FORMULATION
OF BICRITICAL
OF THE RENORMALIZATION
367
Pt
GROUP FOR A NONSYMMETRIC
THEORY
The way to avoid the difficulties discussed in the previous section is to formulate the theory against a background which is intrinsically asymmetric rather than the O(N)-symmetric theory used in Section 2. We will now proceed to formulate the problem in a form in which the anisotropy is fully included in the free part. This will enable us to write RGE directly for the anisotropic theory, and then to express xL in terms of the natural parameter T (= T - Tch,( g)). This will be the parameter which will determine the flow of coupling constants, etc., into the proper, O(M)-symmetric, fixed point without any corrections to scaling. This is a project which falls well within the scope of renormalized field theory, although at certain points we will have to extend the lore of results [18] in order to deal with the: presence of a second length. Symanzik [4] proved that a (+2)2-theory with quadratic symmetry breaking is renormalizable. This he did by deriving generalized Ward-Takahashi identities for non-linear symmetry breaking. The statement that was proved by Symanzik contains the assertion that the number of renormalized parameters-masses and coupling constants-equals the number of bare parameters. The original proof is transcribed in Appendix H in the language of functional integrals over c-number fields, which is more familiar to statistical physicists. The same result can be reached by considering the counter-tmerms which are generated interatively in perturbation theory [19]. For clarity, we start with a theory with T = 0. Namely, M components are critical and (N - M) have a mass (inverse correlation length) m jixed. We return to the Lagrangian (1.1) but choose p12 = & and p22 = ,ui, such that the vertices generated by it after renormalization satisfy the conditions r&(p
= 0; 112,u, K) = 0,
(4.la) (4.lb) (4.lc)
(4.1 e) The index i stands for external legs in the critical group, while j stands for the noncritical ones. K is a renormalization momentum scale, and sp stands, as before, for a symmetric momentum point. With these definitions of pIc and pzc the Lagrangian (1.1) can be written in the form 2 = w992 + &4cA2 + &&A2
+ (h/4!)(&)”
+ Bto4”
(4.2),
368
AMIT
or in a renormalized
AND
GOLDSCHMIDT
form
9 = i(v+)’
+ $m2&2 + (l/4!) z@(@)~ + &T@ + C.T.
(4.3)
sYman..Zik’S theorem assures us that function ,ulc , pFlzc , h, Z&l), and Zi2) can be found such that [20] riNN)(pi
; 177, U, K)
=
(z~‘)N1’“(z~‘)N2’”
riN’(&
; &c
, /&c
, h,
A)
(4.4)
are renormalized, and satisfy Eqs. (4.1). P’) has N1 legs of type i and N2 (= N - NJ legs of type j. The field renormalization constants ZL” and Zh2’ are determined by conditions (4. lc) and (4. Id). They differ by a finite factor. Recall that plc , pzc are the bare masses of particles of types i andj, respectively. h is the bare coupling constant and A is a momentum cutoff. A few comments on these normalization conditions are in order. First, from Symanzik’s theorem again, it follows that once (4.le) is satisfied, the other components of P), namely, rj&\ and rj$ , are all finite. (See, e.g., Appendix H.) The values of these other parts of P4) are completely determined. This is equivalent to saying that an O(N)-symmetric counter-term will suffice to make finite a theory with a quadratically broken symmetry (4.19). One should notice that the number of renormalized parameters needed, masses and couplings, equals the number of bare parameters. The normalization conditions (4.1) are so chosen as to ensure that for a fixed bare theory (fixed p1 , p2 , h, fl), m will remain fixed. This is not a trivial matter, and it is here that the first extension of previous applications of field theory to critical phenomena enters. Notice that Eq. (4. la) implies a relation of the form ml(plc , kc , A 4 = 0
(4.5a)
which can be kept intact as K is varied at a fixed bare theory. This depends on the fact that ZL’) cancels out of Eq. (4.la). This is much less obvious with regards to (5.1 b). Can one deduce from it that m
=
m(h,
p2c
(4.5b)
,&4?
This time Zh2’ does not cancel. Usually, in a massless theory Z, depends on K to prevent the infrared divergence in quantities like P2)/+P. Here this can be avoided by noting that in r,‘i’ the internal progagators of typej have a mass, and the i-propagators are too few to cause infrared problems. Thus in Eq. (4.ld) we could legitimately set pz = 0. It can then be solved for Z,J’) without introducing K, so that
zy z zp
3 P2c
, A, 4.
All this implies that (4.ld) leads to an equation like (4.5b), and therefore K can be varied at a fixed bare theory holding m-the anisotropy-constant. This is essential for the derivation of the RGE.
TREATMENT
To summarize:
The normalization
OF
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369
conditions (4.1) imply (4.6a) (4.6b) (4.6~)
(4.6d) (4.6e) which can be inverted to give prc , p2c , h, and Z$” as a function of m, u, K, and (1. If now in ESq.(4.4) all external legs are chosen to be of type i and ~(a/a~) is applied at fixed bare theory, the resulting RGE is
(4.8)
Y6 = K
a In Z:’ aK
A.rn
(4.9)
This is all very similar to the usual case of a critical O(M)-symmetric critical theory, all except for the dependence of the functions /3 and yb on m/K. At this point m, the inverse of the transverse correlation length at TCM , becomes the new measure of the anisotropy. That this is a very natural measure is corroborated by the rest of this article.
5. MUST fl AND ym DEPEND ON m/K ?
At this point the second extension of the renormalization group treatments enters. We wrote in Eq. (4.7) an explicit dependence of the functions /3 and yd on m/K. This is not dictated by general renormalization requirements. The special renormalization conditions (4.la)-(4.le) do imply such a dependence, as is shown in the next section. However, as we saw in the discussion of the expansion about the symmetric theory, the renormali.zation constantsof the massless symmetric theory can render the asymmetric theory finite as well. These would have been obtained had renormalization been carried through using “naive” minimal subtraction of poles in (4 - d) [lo, 121.
Some additional comments on this subject are reserved for Section 7. If the rermrmalization constants, or the counter-terms, are taken independent of m/K the critical behavior will be that of an O(N)-symmetric theory. One is, however, intuitively convinced that as the critical point is approached (momentum scale much
370
AMIT
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smaller than m or 7 < m2) the theory should be dominated asymptotically by the M-component critical behavior. Clearly, were m2 to become very large, all N - M components of type j, which are associated with m2, would be frozen out, and M-component behavior would ensue. This fact is obscured in a “naive” E-expansion. Graphs whose contribution vanishes as m -+ co may have an m-independent pole in E, and will not vanish in the limit. As an example, consider the expression J = (~/E)(K~ + m2)-6/2.
(5.1)
Such a form may be associated with a one-loop integral of rc4). Expanding in E, one has J = t - i ln(K2 + m”) + O(&+).
(5.2)
Minimal subtraction would keep l/c in the renormalization constants [12] and drop the In. This is fine as long as m is finite, but it brings about a situation in which the contribution of J to the function, which is being renormalized, vanishes as m -+ CO, but the subtraction induced by the renormalization constant does not. Hence in this limit the theory will not be finite. See also the explicit calculations in Section 6 and 7. In other words, Symanzik’s theorem breaks down in the limit of injinite quadratic symmetry breaking. But what if m2 is kept fixed ? One runs into the same type of difficulty,
as one approaches the critical region. The way to see this is to note that if, for example, the momenta in FN) are scaled by a factor p then dimensional analysis implies
In the asymptotic region p + 0, which is the region of interest, the theory is equivalent to one with a very large mass. Thus, unless the renormalization has been so carried out as to include the limit m + co, one is sure to obtain nonsensical results for fixed m as the momentum, or temperature, tends to zero. In the following sections we describe two different renormalization procedures which satisfy the requirement that the c-expansion remains meaningful when m + co. The two procedures lead to an identical cross-over function which one would expect on the basis of universality. The fact that the Wilson functions /I, y* , etc., depend on m/K makes the solution of the renormalization group equation more complicated. This is another extension to which we turn in Sections 8 and 10. It is important to realize that the above considerations are not additional assumptions. That all vertices with external legs of type i tend, as m -+ co, to their Mcomponent counterparts follows directly from perturbation theory. This fact is automatically taken into account by the normalization conditions (4.1), which would provide m-dependent renormalization constants.
TREATMENT
6.
CALCULATION
OF BICRITICAL
OF p USING
371
Pt
NORMALIZATION
CONDITIONS
In the first renormalization procedure we use the normalization:conditions, Eqs. (4.1) together with Eqs. (4.6), to obtain /I to the order of one loop. The Lagrangian (4.2) is used to generate the graphs directly in terms of m2. In Fig. 4 the graphs’ contrbution to J’,‘$‘, at this order are drawn. Equation (4.le) reads UK’
FIG. 4. separately.
Graphs
of I’&
=
h -
y
~2&,(o)
to one loop.
Graphs
-
with
v
~2&p(fTZ).
massless
and
massive propagators
are drawn
The integrals are defined in Appendix C, where their +expansion-is given as well. To obtain Isi,(m) one setsP2 = ~~ in Eq. (C.6). Equation (6.1) can bkwritten as x2K-2’
Notice that “n.aive” minimal subtraction would have us discard the second terms in all parentheses in Eq. (6.2). Calculating ,D from definition (4.9, one finds
1 _
(1 +
4m2/K2)-1/z
In the limit m .+ 00 (6.4) as one expects for a theory of Mmassless one has
particles (Mcritical:components).-As
p+-al+-U2 which is the proper /3 for an O(N)-symmetry
N-t8 6 theory.
’
m/K + 0
AMIT AND GOLDSCHMIDT
372
An expression such as Eq. (6.3) is not very convenient for further calculations. We therefore proceed to an alternative procedure which will yield much simpler expressions without foregoing the essential content of the cross-over phenomenon.
7. RENORMALIZATION
BY GENERALIZED
MINIMAL
SUBTRACTION
In describing a second procedure of renormalization we have two objectives in mind. First, the procedure of generalized minimal subtraction (GMS) to be presented enjoys many of the simplifying features of “naive” minimal subtraction. This becomes a weighty consideration as the calculation proceeds. Second, using two methods provides a test of universality. Below the method will be presented on the level of one loop. That it works on the two-loop level will be shown in Appendix F. The requirement we impose is that for finite m the renormalization constants, or counter-terms, cancel the poles in E in all vertices. In addition, we demand that as m + 00 all vertices with external legs of type i be finite order by order in u and E. Starting with I’$& we write U,,
=
hK-f
=
U +
q(m/K)
(7.1)
U2.
On the one-loop level Z, = 1, and al(m/K) is chosen so as to make ri(i4i)iR finite as E --f 0 and us m --f co to order u2. Symanzik’s theorem assures us that for finite m the same u0 will make all of P4) finite [21]. One such choice is m
4 c-1K
Introducing adP1
=- M+&+N--M 6E 6~
[I - :ln (1 + $)I.
it into J’:;l:, , one has ,..., p4
=
= K”
I
; m, 4 K)
U +
UlU2
-
u2y
u
-
Mt8
U2 ___
18
I
(7.2)
TREATMENT
313
OF BJCRITJCAL Pt
where we denoted p1 + pz = P. The expression in the first set of square brackets is finite as in the usual dimensional regularization. The expression in the second set of square brackets can be written, using Eq. (C.6) for I( p, m), as [ ] = f (1 - g I1 dx In [x(1 - x) $ + $-I)
= -.:rdx;n[ In “naive”
x(1 ;2xy2+
dimensional
- f (1 - z In (1 + $))
m2 ] + O(E).
regularization
(7.4)
there would be no denominator
in the In.
As long as m is finite the difference, Eq. (7.4), is finite as E -+ 0. The choice (7.2) has,
in addition, the property that as m + cc the finite term vanishes, and it leads to a I’::\, identical to that of the O(M)-symmetric theory. Furthermore, it gives a meaning to the E-expansion for all values of m. The form (7.2) is not unique in satisfying the above resuirements. It is easy to see that if (1 + m2/tc2)-c/2is replaced by (a + m2/K2)-E/2the whole argument goes through. We will return to discuss this arbitrariness in the renormalization procedure in the context of the discussion of universality in Section 14. Calculating p, using Eq. (4.3, one finds p(u,'-)
= --EU + q+
+ yc2
(7.5)
1 + LeiK2 + O(U3,#%),
and the correction terms are finite for all values of m. Clearly, as m + 0 or as m -+ co this /I tends to the limits (6.4) and (6.5), respectively. The difference between the two procedures amounts to a finite renormalization for all ualuesqf m. Universal quantities-inter alia the universal part of the cross-over behavior-should not be affected. The function ‘ymis, of course, zero at the one-loop level. Still, for completeness, we write down. its form at the lowest non-trivial order. Normalization conditions give Yb =
N+2 --u2~??!!u2u,
72
‘dx
36
Kzo
s
’ I
,,d4
Y yx(1 - x) +
rn2/K”
+
O(u3,
U2E).
(7.6) GMS leads to the simpler form
ye = L u2 W + 2) + (M + 2>@“/K”) 72
1+
rn2iK”
.
(7.7)
Both have proper limits as m -+ 0 and as m + co.
8. SOLUTION OF THE RGE IN THE PRESENCE OF Two LENGTHS
The depend#ence of /3, y+ , etc., on m/K brings about the third new feature of the present application. RGEs are usually solved by the method of characteristics. What has made the implementation of this method very simple to date was the fact that
374
AMIT
AND
GOLDSCHMIDT
all the coefficient functions have depended on a single variable-usually the dimensionless renormalized coupling constant u [22]. Here the situation is more complicated, and hence we describe the process in some detail. To the order of one loop the RGE reads
where all external legs of FN) are of type i. The first characteristic
equation is
which, with (7.5) used, reads -=du dK
(N + 8) + (M + 8)(m2/K2) u2 1 + m2/K2
(8.3)
This is a Bernoulli differential equation whose solution is M+8 u-1
zzz
K x-f-1 dx + qnl) s Kg1 + 1)22/X2
N--MKF
6~
6
_ Mf8 6~
j-f
-
M
1
sgp0
12
Kc
dy
J>-d2
y + m2/x2
+
c(t’?l)
Kc.
By dimensional considerations C(m) = Cm-‘, where C is dimensionless. Let us denote the solution of Eq. (8.2) by @(U,
which can be rewritten
m/K)
=
(8.5)
c,
as an expression for u in the form u = F(@, m/K).
(8.6)
In the special case of Eq. (8.3)
and F is given by the inverse of (8.4) with that 0 satisfies the equation
Denoting
C replaced by @. Notice,
m/p by FFiwe define u(p), the flowing @(U,
WZ/K)
=
@(U@),
coupling m/K).
constant,
incidentally,
by the relation (8.9)
TREATMENT
OF
BICRITICAL
375
Pt
Then, using Elq. (8.8), one can readily show that u(p) satisfies the equation
du(p)= P(U(P), g, ’ ___ 4
(8.10)
with the initial condition u(p = 1) = u.
(8.11)
This is a generalization of the equation describing the flow of the coupling constant as a function of the overall scale of length. Using Eqs. (8.9) and (8.7) one obtains an explicit expression for u(p), namely,
tl-l(,O) = p’u-1 +
(8.12)
For m < K the behavior of u-l(p) can be simplified if the interval divided in two. As is shown in Ap&rdix E for 1 > p 3 m/K, N-t8
14-l(p) = 7
N--M 12
- -----In
+ p’ [24-l - v and for
m/K
2:
p
>
(
l+-
+ v
0 < p < 1 is
m2 Kzpe
1
In (I + $j],
(8.13a)
0
PM+ 8 u-‘(p) = -~ v 6~
In (1 + $)
(8.13b) For very sm.allvalues of E one can write a single expression for 21-l(p)in the whole range of p. It is 14-l(p)
=
M+8
6E
-
(8.14) But, for finite E - E = 1, for example, the difference between (8.14) and (8.13) can become quite fsignificant. The latter expressionsare more reasonable, though they are somewhat more complicated. They join smoothly at p = m/K and, what is more important, if the initial value of u is smaller than 216, then u(p) remains below UC throughout. As a consequencethe effective exponent defined below stays between its O(M) and 0(N) value. Some comments on this question will be added in the discussion
316
AMIT
of universality-section 14. Notice, (8.14), have the property
AND
GOLDSCHMIDT
in particular,
6~ u@) ygg ____ Mf8
that both versions, (8.13) and
= UC.
(8.15)
A plot of u(p) for everal initial values of the coupling constant is given in Fig. 5, for N = 2, M = 1, ~~ = 1, m2 = 0.01, and E = 1, using Eq. (8.13) for u(p). In Fig. 6 we compare the use of Eqs. (8.13) and (8.14) for E = I, 0.5 and 0.1. Then in Fig. 7 we plot u(p) for 6~ zPYgq=u; for several values of m2 with N = 2, M = I, ~~ = 1, E = 1. In order to study the solution for r IN) let us switch back for a moment to the more general case which includes yb in the RGE. The complete equation would be (8.16)
FIG. 5. A plot of u(p) for several initial V&ES of the coupling constant for N = 2, M = 1, K* = 1, In2 = 0.01,
E = I.
0.62 0.60 0.68 0.66 0.64 0.62 0.60
FIG. 6. Comparison of Eqs. (8.13) and (8.14) for < = 1, 0.5, and 0.1 with m2 = 0.005, ~~ = 1, N=2,andM=l.
TREATMENT
OF BICRITICAL
lo-lo 10-8 10-6
FIG. l
=
7.
Plot
of u(p) for
II = u* for
m2 = lo-*,
10-4
10-2
377
Pt
100
102 P2
10-4,
1O-B, lo+.
I? =
1, N = 2, M = 1, and
1.
Here there is a second characteristic equation, namely, (8.17) which is solved by inserting solution is
U(K)
along a characteristic curve from Eq. (8.6). The
Next the momenta are scaled by a factor p. Dimensional (8.18) gives in the standard way
analysis together with
and a change of variable x -+ x/p results in T(N) = P d-N&t-1) R
where we have used Eq. (8.9) and the fact that the canonical dimension d - N(d/2 - 1) [23]. Returning to the one-loop approximation, Eq. (8.19) becomes j’$“‘(,pi
; m,
u,
K)
=
pd-N(d’2-1)r;N)(Pi
; @i,
U(p,,
K).
of PN) is
(8.20)
378
AMIT AND GOLDSCHMIDT
As p + 0, 1%becomes very large. All graphs with internal propagators of type j will vanish. u(p) -+ z& and y6 + qM, the anomalous dimension of the field, as one would expect a theory with A4 critical components to behave. In conclusion, Eq. (8.19) serves as a starting point for a calculation of a cross-over function as the scale of momentum crosses the value of the anisotropy mass m. This cross-over which can be described by an effective 7 will be described elsewhere. Here we proceed to calculate the more popular cross-over in temperature.
9. THE THEORY ABOVE THE CRITICAL
TEMPERATURE
As was indicated in Section 4, TV in Eq. (4.2) is proportional to the temperature difference from TCM( g). One should also recall that m the inverse of the transverse correlation length at the critical temperature TCM has become our measure of the anisotropy. From the fact that 2,~ of the symmetric theory renormalizes 6p2, Eq. (2.Q it follows that for finite m the same renormalization constant will renormalize TV. Namely, the transformation from the Lagrangian (4.2) to (4.3) can be affected by the relation To = z&T. (9.1) 2,~ will renormalize multiplicatively the insertions of @ into the vertices of the critical asymmetric theory described in Section 4. This follows from the fact that the superficial (primitive) divergence of PzJ) is logarithmic and hence the difference between a graph with massless propagators and an identical graph with massive propagators is not superficially divergent. In the light of the discussion in Section 5 we prefer not to use the Z,z of the symmetric theory, since the m2 --f co, or m2/r + co, limits will be problematic. The conclusion that there exists a Z,z , which makes the vertex function with TV finite, persists. For finite m the difference between one choice of Z,z and another consists of a finite renormalization. As m -+ -m we require that the insertion of @ into vertices with external legs of type i remain finite and tend to their O(M)-symmetric form, as we did in Section 7. The above discussion can be summarized by saying that there is a
Zb2 = Zd2(u,m/K)
(9.2)
such that
P-3) is finite as fl + co (or E + 0) and as m + co if 7. and 7 are connected via Eq. (9.1), and the parameters on the right hand side expressed in terms of u, K, and m. All this is true order by order in the double expansion in u and E.
TREATMENT
OF BICRITICAL
379
Pt
The RGE for the longitudinal two-point vertex I’& follows directly from the application of ~(aji3~) at constant m, and A, using Eq. (9.1). It reads
with y* defined by Eq. (4.6) and Y& =
-K------
a In .&,2 i3K
. m.h
Recall that I’::) is proportional to the inverse of the longitudinal (easy-axis) susceptibility, &, and that to the order of one loop y+ = 0. There still remains the question as to how 2,~ is to be determined. We have chosen again two different ways. One is to use a modified minimal subtraction as presented in Section 7 requiring that Z,JIlb (‘J) have no poles in E and be finite for all values of m. The other, which we find particularly convenient, is as follows: The only function in which r@ introduces primitive divergence is r c2).Consider the inverse longitudinal susceptibility XLl(T,
The normalization
m,
condition
24, K)
=
rC2)IIF&’
=
0;
=
0.
7, 111, %
K>.
(9.6)
(4. la) implies X;l(o,
To this we add another normalization XL1(K2,
l?‘l,
U, K)
condition, M,
U, K)
(9.7)
namely =
(9.8)
K2.
This determines Z,Z , since it includes the graphs of I’L211). Both ways of determining Z,z lead to the same cross-over function. The special advantage of Eq. (9.8) is the fact that, once the renormalization group equation is used to roll 7 out of the critical region, to the value K2, all the information will be contained in the prefactor. This we proceed to show in the following sections.
10. SOLUTION
OF THE RGE
ABOVE THE CRITICAL
TEMPERATURE
The presence of the temperature term in the RGE, Eq. (9.4), leads to an additional characteristic equation. It is dln T (10.1) K- dK = y,z which is analogous to Eq. (8.17). The solution proceeds, therefore, along the same lines. Namely,, U(K) along the characteristic is substituted into (10.1) using Eqs. (8.5) and (8.6). Equation (10.1) can then be integrated, which gives (10.2) 595/114/I/2-25
380
AMIT AND GOLDSCHMIDT
This leads us to a definition of a flowing temperature T@) as
which satisfies the differential equation P
d In T(P) dP
= y62(u@)'$)
- 2,
T(1) = 7.
In terms of Fi, u(p>, and T(P) the solution of Eq. (9.4) has the property that XL
-I
m2
-ZK2
’
K2, u = p2exp x
XL1
@-
) $,
(10.5)
u(p)).
But so far p is an arbitrary parameter, which we choose by the requirement that 7 on the r.h.s. be out of the critical region. This we ensure by choosing p to satisfy (10.6)
T(p) = K2,
which together with (10.3) gives the following relation between p and the original T: (10.7)
This equation has to be solved to give p(T, m, K). On the one-loop level yB = 0. Thus, Eq. (10.5), after reads
T(p)
is substituted from (10.6), (10.8)
Here the advantage of using Eq. (9.8) to determine 2,~ manifests itself by the fact that (10.8) reduces to
XL’(5, $, u) = P2. Be that as it may, (10.8) or (10.9) gives the cross-over function after p is substituted everywhere in terms of 7 from Eq. (10.7), and u(p) of section 7 is used. This we proceed to do in the next section. One should keep in mind, of course, that beyond the one-loop Ievel the exponential prefactor in Eq. (10.5) has to be taken into account as well. 11. Two CALCULATONS
OF yds
We now compute yQ2 to order one loop in the two procedures mentioned end of Section 9, namely, GMS applied to PJ) and a normalization r,‘$p
= 0;
7
=
K’,
m,
U, k)
=
K2.
at the
Pt
381
Each of these two approaches has its own advantages, both is one more check of universality. Or, rather, it is calculations to a test. The graphs which enter I’:F.l’, in the first procedure, that I’$*l) is the temperature derivative of I’,, (2). According for I’::,*) to become finite is r(eA = zd*zJ$l).
and the implementation of universality which puts the
TREATMENT
OF
BICRITICAL
are given in Fig. 8. Notice to Eq. (2.8) the combination (11.1)
11R
FIG. 8. One-loop graphs of Fit). The 4” insertion is decomposed into components giving different contributions. 4r is the momentum of the 4” insertion.
For I’fiV1) one has a?(p*,p*,q;To
= 1- y
=o,m,q
AZ(q) - v
with Z(q) and Z(q, m) defined in Appendix C. Using the tr-expansion of the dimensionally
XZ(q, m),
(11.2)
regularized integrals and writing
z,, = 1 + c1 : (
1
u,
(11.3)
we see that GMS gives Cl =-+-M+2,
N-M
6~
[
6~
1 -Gln(l
+-$-)I
(11.4)
and zd.=l+[y-
7
In (1 + $)]
24.
(11.5)
Inserting 2~ in y,g - --K
2 In Z,z 2K
(11.6)
3 m.h
we find 1
N-M
M+2 (1) Y& = -u+7u 6
1+
rn2/K2
(11.7)
*
The superscrilpt 1 indicates the use of the first renormalization procedure. Next we co:mpute x;~(T, m, u, K) to order one loop. To this order the Z4’s are equal to unity. As is shown in Appendix G xL1
=
7
zd2
-
UK’
w
J(0,
T) +
I
The integral
J(m12,
mz2)
is defined in Appendix C.
vJ((m2,m2
+
T)]/.
(11.8)
382
AMIT AND GOLDSCHMIDT
First notice that Z$ of Eq. (11.5) would render (11.S) finite as E+ 0 for all values of m. In fact, substituting J from Eq. (C.8) in (11.8) and (11.5) for Z&Z,one finds
-vu[ln(l
+$)-(I
+g)ln(l ++-)]I (11.9)
Equation (11.8) can be used to calculate ydz in the second procedure, Eq. (9.8). Setting T = ~~ and using Eq. (C.8) for the integrals, one easily finds Z@
=
1 +
Kg
u +
EI&E
[ (1
+
zJ1-f’2
-
($)1-r’2]
U,
(11.10)
which leads to (11.11) It is easily verified that both y(l) d2 and yjb22) have the proper limits as m + 0 and as in ---f co. Substitution of Z$’ in x~l, Eq. (11.S), gives x;l(-$,:,u)
= 7 11 +~Uln$--~U X[(l+$)ln(l+-$-)-(l+$)ln(l++)]l. (11.12)
This XL’ is finite for all values of m, and at T =
12. CALCULATION
K2,
xL1
=
K2.
OF THE CROSSOVER FUNCTION
It was mentioned in the Introduction that the cross-over function can be described by an effective exponent
yeff(T) = din xLY4 dln7
*
(12.1)
It is a temperature-dependent exponent which mediates between the approximate O(N)-behavior for T > m2 and the exact O(M)-behavior as T + 0. We will proceed in two ways, using the two renormalization procedures described in the previous section.
TREATMENT
(a)
OF
BICRITICAL
383
Pt
In the first instance, one has from Eq. (10.8)
and when (11.9) is substituted
-1 z_ ( K2
XL
K '
K2
'U
)
on the r.h.s. the result is =p2
1+
1
N--M 12
___
U(f)
-$ln(l
I-$)].
(12.2)
In this equation p is P(T), according to (10.6). Thus to complete the task, P(T) has to be solved from (10.6) and (10.3) and substituted in Eq. (12.2). The equation to be solved is (12.3) The exponent on the right hand side will behave as In p with a coefficient of order E. Therefore, to leading order one can substitute p2 = T/K2 in the lower limit of the integral to obtain (12.4) By differentiating
with respect to log T we immediately
obtain
(12.5) From now on we will denote From (12.2)1 one obtains d In xc1 -dlnP2 dlnTdh
7
u(T”~/K)
by
I N--M ___ 12
‘(‘)
U(T).
[, +
l,,M2
-_ nr * ln (1 + +)]
(12.6)
after again setting p2 = T/K in the square brackets of Eq. (12.2). Note that in differentiating this expression with respect to ln~, U(T) was held constant. This is justified by the fact that
d4d dU(T) rT-=-q.4
dln r
dr
x [u-l - qp
I-12
N--M
+ y
but this is of order e2 because of the factor
u”(T).
&
+ ; (+‘2E
ln(1 + n~-~)]!,
(12.7)
384
AMIT
AND
GOLDSCHMIDT
Finally, using (12.5) and (11.7) we arrive at yeff=l+
Mf2
____ ,2
4’)
7
+
U(T) [ 1 - f
(12.8)
In (1 -t $)I.
(b) In the second method we have simply -&’ = p”. Therefore from (12.5) we obtain
and using Eq. (11.11) we arrive imediately at Eq. (12.8). In Fig. 9 we plot a few graphs of yen(T) for different values of m2 with N = 2, M = 1 and E = 1. The graphs compare well with the results of series expansions [24]. In Fig. 10 we plot Yerrfor N = 3 and M = 2. In Fig. 11 we plot (yen - 1)/e for various values of E. In all these figures we used the split expression for u(r), Eq. (8.13). In Fig. 12 we plot (yen - 1)/e with the unique expression for U(T), Eq. (8.14), for various values of E. For large values of Ethe overshoot of Yen becomes quite marked.
FIG.
FIG.
9.
Graphs
10.
Graphs
of Y&T)
of yeIf(~)
for
for
m2 = 10-e,
m = 1O-8,
10m4, 1O-B, lo-*
with
lo-“,
N = 3, M = 2 and
10e6 with
N = 2, A4 = 1 and
c = 1.
E = 1.
TREATMENT
OF BICRITICAL
385
Pt
T
FIG. 11. Graphs of (yen - 1)/c for n? = 10-a,
0.10
l
= 0.2(a), 0.5(b), l(c) with N = 2, M = 1.
--.A 1o-1o
1o-8
1o-6
1o-4
lo-’
10 I
FIG. 12. Graphs of (ye!! - 1)/c for me = 1O-s, E = 1, 0.5, 0.2 with N = 2, M = 1 and the unique expression for U(T).
13. THE SPECIFIC HEAT
AND THE NONORDERING
SUSCEPTIBILITY
The computation of the cross-over functions of the specific heat and of the nonordering suslceptibility is straight-forward. The first does not even require any new renormalization functions. We proceed to discuss these two functions briefly. As usual, the specific heat is related to the vertex P.2). (13.1)
C a Tts2)(q = 0; 7, m, u, K), with I’(“*2) the vertex associated with the average of 4”(x) +2( -JJ). Apart from a T-independent constant P”,2) is multiplicatively renormalizable 2~ of Eq. (I 1.5). In other words
by
Ffs2)(q; 7, m, u, /c) = z$[rp2)(q;
To 3 PlC
9 P2c
is finite as fl -+ co, or as E + 0 [25].
3
A, 4 - rP)cq;
0, plc , P2c , A, 4
(13.2)
386
AMIT AND GOLDSCHMIDT
For finite m either the 2,~ of the symmetric massless theory of Section 2 or the m-dependent Z,z of Eq. (11.5) will do. But according to the argument of Section 5 the behavior as 7 -+ 0 is equivalent to the behavior as m + co. In this limit graphs with internal propagators of type j will vanish, and the vertex corresponding to +2~2 will tend to the vertex corresponding to +12+12. Therefore the m-dependent Z63 of Eq. (11.5) which tends to the O(M)-symmetric Z, 2 is the proper renormalization constant to keep r c”s2)finite in the infrared limit. Furthermore, in that limit all internal interaction vertices will have four legs of type i, which implies that h of Eq. (7.1) and /3 of Eq. (7.5) remain the same. In order to avoid stray constants we consider P”*3), which is proportional to the derivative of the specific heat with respect to T. It satisfies the RGE
The functions p and ~~2in Eq. (13.3) are given by Eqs. (7.5) and (11.7), respectively. The characteristic equations are of the same type as before except for the obvious change of replacing Eq. (8.17) by (13.4)
K
The rest is a repetition of the steps of sections 8 and 10. Z,z + Z&M) as m -+ co, and /3(u, m/K) -+ PM in this limit. It follows that when T < m2 the specific heat exponent (Y*--+ Q,,, , the specific heat exponent of the M-component theory. Next we turn to the non-ordering susceptibility which is defined by xg =
-
a"F w
=
po.o*2)
(13.5)
3
where g is the anisotropy parameter of Eq. (2.1) and P”*o,2) is the vertex associated with the product of two operators B. In order to avoid once more the constant divergence, which appears in r~“~o~z)just as in rCo92), we consider its T-derivative, nemely, P”Js2), whose general structure, as well as some graphs of low order, is given in Fig. 13.
--&-- _b_---&J-- --A.(al
(61
(cl
IdJ
FIG. 13. Graphs of P’J~Z~. Wavy lines designate +” insertions; dashed lines designate insertions of B. Graph (c) includes a renormalization of B, and graph (d) a renormalization of 4”.
TREATMENT
The renormalization
OF
BICRITICAL
Pt
387
of F”J*2) is affected via I$y2)(q,
Y1 ) r2 ; 7, m,
24, K)
=
z*&J2rp?
(13.6)
As long as m is finite this can be achieved by the renormalization constants of the symmetric theory. But those will not do, and one should ensure that the limit m2 -+ co is properly renormalized. To accomplish this one simply has to choose ZB(u, m/K) so that r$O,l) becomes finite-namely, the vertex with two external legs of type i and one B-insertion--either by normalization conditions imposed on I’$‘,‘) or by generalized minimal subtraction. The reason this is sufficient is that, apart from a constant factor, as nz -+ co (0,l.Z)
-
r(o,3)
pbo’l)
-
J+Z$)
rb
(13.7)
and 11
9
(13.8)
where on the r.h.s. of (13.7) and (13.8) the vertices are precisely those of the O(M)symmetric theory. Thus, h of Eq. (7.1) will do in this limit; owing to (13.8), Z, + Z,Z of the O(M)-symmetric theory; and (13.6) will be finite in this limit because Z$ renormalizes 1’Co,3). The RGE satisfied by f9 = dxg/dT is
(13.9) which falls once again into the category of equations we have been solving all along. The new characteristic is dln f9 K ~ die = -(yg
+ 2YB),
from which the explicit form of the cross-over function follows directly. We will forego the full solution and will comment on the limiting behaviors only. As was pointed out above, when m + co, ye -+ ~$2 of the O(M)-symmetric theory, as does ~~2 . Thus, in tlhe limit r/m2 ---f 0, (13.10) becomes simply (13.4), and xs
-
T-S”.
(13.11)
On the othelr hand, when m + 0, ‘y*z -+ y$ = 2 - v;’ and ye + 7; = 2 - 4~;;’ (see Eq. (2.18))1, provided U(T) + uj$ in this limit. As is discussed in Sections 8 and 14 this is indeed the case, and hence 2--n,N-26 xu
-
T
(13.12)
388
AMIT
AND
GOLDSCHMIDT
Consequently xg crosses over from a behavior very close to that of (13.12) to that of (13.11). In order to solve (13.10) and (13.9) in full, one needs ys(u, WZ/K). This we calculate by minimal subtraction from I’,,(2~o~1).The graphs of this function to order one loop are given in Fig. 14. To make the correspondence of the graphs with Eq. (13.13)
FIG.
14. Graphs of Ti:Po*ll to one loop.
clearer, the insertion of B was separated into three parts-the one with $12,the one with xr $i2, and the one with CL+, &“. The corresponding expression for the bare vertex is r(2.0.1) 11
=yq
(13.13)
-~hl(q)--~hl(q)+~hltg,m)l.
The discussionof Section 7 then leadsdirectly to u-I!J+
z,=1+-
+?!J’:’
(13.14)
and Mf2 YB
=
6
M ’
-
d
1 ’
1 +
m2/K2
.
(13.15)
When m -+ 0, ys tends to the one-loop part of Eq. (2.24). On the other hand, asm + co, yB tends to ~~aof the O(M)-symmetric theory. It is just the kind of function which leads to the asymptotic behavior of xs discussedabove. Inserting ‘ye and ~~2in Eq. (13.10), one can obtain the explicit cross-over function. Finally, we make a few brief comments concerning the transverse susceptibility, namely, $(T,
m, u, K)
=
r$(p
=
0;
7,
m, u, K).
(13.16)
In this casewe find that the full crossover behavior is lesssimple, since, even in the limit m -+ co, more than one coupling constant enters, in contrast to all the previous cases. To seethis consider the graphs in Fig. 15, where graphs with internal propagators which are masslesshave been distinguished from those with massive propagators. Clearly, as m + 00 only graphs (a), (b), and (f) will survive. The graphs (b) and (f) include the coupling constant which connects ii to jj, and this is the problem. This coupling enters only once in these graphs, and there are no divergences associated with it. But since the counter-terms were chosen O(N)-symmetric, i.e., their form is
TREATMENT
OF BICRITICAL
Pt
389
FIG. IS. Graphs of rj,". (hZ,z - UK~)(~)“, there will be an infinite counter-term renormalizing this coupling constant. In the limit m + XI this term will have nothing to subtract, and rjf) will be infinite. This situation can be treated if one allows for more than one coupling constant, a situation that will be treated elsewhere.Nevertheless one can make somequalitative statements concerning the asymptotic behavior of the transverse susceptibility. First, in the deep asymptotic region, 7 < m2,the graphs that dominate the behavior are, apart frolm the free term (a), proportional to r (OJ)of the O(M)-symmetric theory. The constant of proportionality is just that coupling constant, which appears only once and should not be renormalized, multiplied by a combinatorial factor. But P”J) is a familiar function, whose temperature derivative is the specific heat. Thus, as r/m2 + 0, (13.17) XT’ N ryp f A+%M On the other hand when T > m2, but r < as rYN w P’N.
K2,
x$ is the sameas x;l, which behaves
14. THE PROBLEM OF UNIVERSALITY There are .two aspects of universality of the crossover function which have to be investigated. One is the effect of changing the initial value of the coupling constant on the form of the crossover function. The other is the effect of a change in the renormalization procedure-i.e., changing the renormalization constants (Z-functions) by finite renormalizations. To consider the first aspect let us return first to the symmetric formulation Eq. (2.17). If u = u;Cthen u(p) = u;I: and the functions X, Y, and Z defined in Appendix B are all equal to unity. But for u # u$ thesefunctions depend on the temperature and give corrections to scaling around the O(N)-symmetric fixed point. To the order of one loop, Z in Eq. (2.17) is equal to unity, and one has
where
u(p) = [qg
+ p’ (u-1 - +)]-l,
390
AMIT
AND
GOLDSCHMIDT
and to the same order one has to take p = t1J2 inside u(p). Under these conditions one can claim that Eq. (2.17) is universal in the sense that it has the same form but with redefined t and y, namely, t = Xt, 7 = Yy. This is in fact exactly the assertion of Nelson and Domany( When y < K’, and in the whole critical region T < K’, one can neglect the temperature dependence of X, Y, and 2. Universality will then mean that the form of the crossover function remains unchanged if t, g, and xe are scaled by non-universal constants. This will be true to all orders. Nelson and Domany take the scaling factors X and Y to be temperature dependent. This suffices at the level of one loop, at which they calculate. In higher orders 2 enters as well, and it also depends on temperature. It has the form
+A$)
-(lN+2)/2W+8% [ 1 -
2$$3)
MP~- u>+ OG2)]9
which agrees with Bruce and Wallace [9]. But in the asymptotic region this temperature dependence is irrelevant. Returning to the nonsymmetric formulation, we find that universality holds in a rather spectacular fashion. There is no need to redefine either 7 or m, provided only that the crossover takes place for m2 and 7 much smaller than K~. Namely, the expectations raised in the Introduction, concerning the universality of the whole crossover structure, are fully confirmed by Figs. 16 and 17. First we show in Fig. 16 that for m < Kz (= 1). U(T) spends a long while near us * in a way that is totally independent of the initial value of u. and then goes to u,+, (Compare with Fig. 5, where the mass is much larger.) Then, in Fig. 17, the corresponding Yen is drawn, once again for a wide range of u’s. In this form universality will clearly persist to higher orders in c. The other aspect of universality is the dependence of the crossover function on changes in the renormalization procedure. This problem arises also for the isotropic massless theory on switching, for example, from normalization conditions to minimal subtraction. When one calculates to order two loops one finds that, although the
FIG.
16.
Plot
of U(T) for m2 = 10m8, K? = 1, N = 2, A4 = 1 and E = 1 for various
values
of u.
TREATMENT
OF
BICRITICAL
391
Pt
functions /3, y,+ , ybz , and u * change when the renormalization procedure is changed, the critical exponents are invariant. Returning to our problem, one should notice that a possible finite renormalization within the GMS is a change inside the logarithm in Eq. (7.2). One can replace the 1 by an arbitrary a (> 0), as is shown in Appendix F. Including this constant, one obtains for p F’ (u, +)
= -a
+ y
22 + y
22 1 + t;i2iaK2 .
(14.1)
In U(T) this amounts to a change U(T) =
Mf8 7
x [u-l - L!!C$K + C!T!$!Z ln (1 + -!I?)]/-’ at2
’
(14.2)
FIG. 17. Plot of year for m2 = 1O-8, G = 1, N = 2, A4 = 1 and 4 = 1 for various values of u: (a) 0.7, (b) 0.65, (c) 0.6, (d)0.55, (e) O.S,(f)O.45.
On expanding in powers of E one finds that the terms containing a appear in U(T) as terms of order 8. Therefore in order to check universality, i.e., to test the actual dependence of yefr on a, one has to calculate Yefr systematically to order c2. This is the same situation as was mentioned before concerning the isotropic theory. There, too, everything straightens out when one calculates systematically to the order of two loops. Notice that the solution of Eq. (8.10) forced us to include in u(p) terms of O(+, which ;are terms of O(1) in U-‘(T). Otherwise the solution is not correct even to lowest order. In fact these terms are the ones which transfer u(p) between U; and z& . If we want to check universality we have to include all terms of O(E~)in yen . What is reassuring is the fact that if one redefines the connection between p and 7 by the condition T(p) = UK2 (14.3) instead of Eq. (10.6), yerr returns to its original form (for u = u,$). A redefinition of p in terms of T should have no effect on Yerr . Here the effect is also on terms of O(@) and has to disappear when one includes all such terms.
AMIT AND GOLDSCHMIDT
392
Another possibility is to change C, of Eq. (11.4) and to take instead Zb2 = 1 + [v
+ v
In (a + $)]
u.
Such a change has two effects. It will change x;l to be
and therefore Eq. (12.6) will become d In xi1 -=dln+ dln7
dlnp2
$1,
(1 + s)].
It will also change ‘ybzto be Y@
=
N+2 6
-u---#
N-M 6
rn2/K” a +
m2/K2
’
which is an effect of order e But one can easily verify that the form (12.8) for Yen is unchanged.
15. CONCLUSION
We have shown in great detail the advantage of including quadratic symmetry breaking in the renormalization transformation compared with an expansion about the high symmetry (isotropic) behavior. While the latter requires special techniques, or a priori assumptions combined with a matching procedure, the former is a systematic technique which employs the natural variables for the description of the cross-over phenomenon. The present technique falls well within the framework of renormalized field theory, though the advantage of using renormalization constants which depend on the symmetry-breaking mass was not generally appreciated. Once within this framework, including the extensions described above, the theory can be applied to many other problems without much difficulty. Such problems as the ordered state [26] or the extension to higher orders in E [9] can be treated in a straightforward way with as many advantages as were described here. Our approach can deal with the quadricritical problem by simply allowing a greater number of coupling-constant renormalization parameters [26]. All these results will be presented in future publications.
393
TREATMENT OF BICRITICAL Pt APPENDIX A
Consider a Lagrangian of the form
2 ==fr(v+)’ + i$mV + (l/4!) ZM’($~)” $ $rl&” + gt2tp22 + C.T.
(Al)
We claim that. the counterterms can be written as 1)(V$)2 + +(Z4p2 - m2) (P2-I- $(Z+‘X - UK’)(~~)~
C.T. = +(Z, -
+ +.(r1[(Z1 - 1) + (M - 1) z21 + t2w - MI z21 A2 + +Jt,Z,M
+ t2[@1 -
1) + w - M -
1) z211 42
(0
to all orders in U. The vertices t,+,* and t2+22 are treated as insertions of +12 and +22 operators, res,pectively, with couplings t, and t, . To order one loop there are two types of infinities coming from the graphs shown in Fig. Al. These graphs represent the insertion of a single &2 into rt2). Graph (a) is r,(F$ to order one loop. This graph is independent of I, owing to the symmetry of the first part of the Lagrangian (Al) not including t, and t, . Graph (b) is r$$ at order one loop, which is again independent of I and k.
L
&
L (bl
Ial FIG.
Al.
;e
One-loop graphs of rj,Z& ’ m1.
We will denote the infinite parts of these graphs by Z:l’ - 1 and Z$, respectively. These have to be multiplied by the corresponding couplings. For 1 < i < M the counter-terms will be the infinite parts of the graphs appearing in Fig. A.2, which are of the form
+{tJ(zl’) -
i FIG.
1) + (M - 1) Z$‘] + t,(N - M) z?‘; &“.
i i
i
i
i
A2. One-loop graphs contributing to the q&* counterterm.
For M + 1 :< j < N the counter-terms are the infmite parts of the graphs appearing in Fig. A3, which are of the form
+{t,ZpM + t,[Zy - 1 + (N - M - 1) ZF’]} 4;.
394
AMIT
FIG.
AND
GOLDSCHMIDT
One-loop graphs contributing
A3.
to the &2 counterterm.
Hence, we proved the form (A2) to order one loop. Let us supposethat the form (A2) is true to order N loops with ZiN’, ZiN’, etc. Then thesecounter-terms are added to the Lagrangian to give 2 + gz~“‘(v+)>” + $Z;N)p(N)2+2+ (l/4!) h(N)Z$N)2(l+2)2 + &‘)+12 + q(N)+ 2 22 2'
(A3)
where
tiN) =
t,[Z;N) + (A4 - 1) Zi”‘] + t,(N - M) zjN’,
tiN) = t&fZ;N)
+
t2[Z;N)+
(A4) (N - M - 1) z:“‘].
This Lagrangian without the vertices t(N)4l2 and t(N)+22 is again O(N)-symmetric. Therefore, all infinities coming from r jF*:) to order N + 1 loops are equal to each other. They are denoted by ZiN+l) - 1. All infinities coming from r,(F,$) are equal to each other and are denoted by Zk”+‘). The new counter-terms for the Cl2 and C22insertions are ;N+l)
C. T . = ‘{t’“‘[(Z 2 1
- 1) + (M - 1) ZiN+l) ] + tjN’(iv - M) .gN+‘)} gJ12
+ &“)$N+l)M = &{tJ(z;“”
+ t;N)[Z$“+l) _ 1 + (N - M - 1) Z$‘+l)]} 922
- 1) + (M - 1) ZZ(N+l)]+ t&v - M) ZiN+l}) CjJ1
+ ${tlZ;N+lk
+ t2[(ziN+l) - 1) + (N - M - 1) Z;N+l)]} +Z2,
where &N+l)
Z(N+') 2
=
I
-
=
q--N)
ZCN) 1
+ +
z
;N)-;N+i) 2
z(N)Z(N+l) 12
+
+
cN
_
ZpJ+l)p
1) z;Np$v+~),
+
(N
-
2) z$“)ZjN+l).
This proves (A2) by induction. The vertices t, and t, are not separately multiplicatively renormahzed becausethe coefficients oft, and t, are different in the last two lines of (A2). But if we define
B = $ [(N - M) &2 - hf+,y, then +,2
+
;t,+,2
=;
Mb+
c;-
mt2
42
+
f
(t1
-
t&
TREATMENT
OF BICRITICAL
Pt
395
and the counterterms in (A2) are ; “““‘;-
M) I2 [Z, - 1 + (N - 1) Z,] 62 + ; (t1 - t2)(Zr - 1 - Z&B.
Thus the vertices t and y given by
t= J&+W--M)t,
and
N
y = t, - t,
are multiplicatively renormalizable, and the operators @ and B are orthogonal under the renormalization transformation.
APPENDIX
B:
EXPLICIT FORM OF THE SOLUTION OF THE RGE O(N)-SYMMETRIC FIXED POINT
NEAR THE
The solution of Eq. (2.16) can be written in terms of an arbitrary parameter, p, which scales all lengths [13]. It reads x-l(t, y, u, K) = p2exp
- x-V@),
U(P)>
U(P),
K>,
(Bl)
with Py
U(1) = U,
= FWPN,
032)
In t(p) = In t - /U’“’ 2 >t$i(U) u
,
(B3)
In y(p) = In y - Jl”’
.
(B4)
2 7($(a)
As t + 0, t(p) -+ 0 and x-l becomes singular. But if t + 0 on the 1.h.s. of Eq. (Bl), p can be chosen to have t(p) outside the critical region. We choose p by the condition t(p)
=
(B5)
K2.
Then p becomes a function oft through Eq. (B3). In particular t = p+ exp
u(o) y&4) - y,*2 _ I -I! u ______ PW - p”-‘x-l*
Clearly, if 24(p)+ u* as t --t 0 the function X will vary in a gentle way, since there is no pole in the integrand of (B6). In terms of this particular p y(p) = J,p-‘2-Yg) exp
,y
..A;(,
is
Y again is a slowly varying non-universal function.
6
1 G yp-‘2-~8y.
396
AMIT AND GOLDSCHMIDT
Finally, the exponential special p. Namely,
APPENDIX
factor in Eq. (Bl) can also be rewritten in terms of the
C: INTEGRALS IN DIMENSIONAL REGULARIZATION
The two integrals entering Eqs. (2.22) are [13]
w = j- 4 q2(q : p)2- ; [1- ; - ; UP)]+ O(4, L(p)
= 1’ dx In[x(l
- x)p2/~2],
(Cl) W)
0
UP1 ,.“, A) = &
[ 1 - ‘2 - GPl
(C3)
+ P,,] + O(l),
with L again given by Eq. (C2). At the symmetry point with P2 =
(PI
+
P2)2
= K2,
L = -2 and
Z*p(O) = 7 (1+ 5).
(C4)
In the massive case z(p,m)
=
.I- dq
(q2
+
m2)[(q1-p)2
+
m2]
'
(C5)
whose c-expansion is (C6)
Finally,
Jh2,=s4(q2 + +. m2”)
m,2;(q2
m22)
cc71
The m-dependent E-expansion of this integral is Jcm
2
m22)
=
1
(m22)1-“‘2
-
(m12)1-E’2 +
13
O(1).
E
APPENDIX
W
D: CORREC~ONS TO SCALING IN AN O(M)-SYMMET RIG THEORY
For an isotropic M-component
system u(p) is a solution of the equation
Py =PW));
U(1) = U,
Pl)
TREATMENT
OF BICRITICAL
397
Pt
where,up to order one loop [3b] /3(u) = -•EU + y
242.
W9
The solution of Eq. (DI) with fl of Eq. (D2) is u(p) = u*[l + p’(u*u-1 - 1)1-l,
(D3)
where, to order E, u* = &/(M + 8). The RGE1 for the isotropic system is just Eq. (2.16) with y = 0. The solution is a special case of (2.17). In lowest order x-1 s (my, (D4) since Z starts at 0(e2). Using definition (B6) for X and ~~2from Eq. (2.13), one finds X = exp
t~'o) ((M
su
+ 2)/6) u' - ((A4 + 2)/(M --a' + ((M + 8)/6)d2
+ 8))~ du, =
u(p)
("+2)l(M+8)
[ u1
(W
To lowest order, one can substitute r1j2 for p, which leads to Eq. (3.4).
APPENDIX
E: THE FLOW OF THE COUPLING CONSTANT
The solution of the equation
032)
is
u-‘(p) = pQ.-l + Kg
(1 - f) - qg!!
PC1% x-y-~K2 .
(E3)
We assume all along that m2 < ICY. For p > m/K we write 2 x-s/2-1 dx
x-~ dx .Y +
rn2jK’
=
PC f 1
1 +
m2/K2X
= p'"--E/2[- ; + f% =-i(l
-~c)+ln(l
($!c)"]lo2 1
+--Y&-~~In(l
+$).
054)
398
AMIT AND GOLDSCHMIDT
For p < m/K we write o* x-J2 dx P' s 1
x
+
rn2/K”
K2X 12 2
m (-)%+I 1 ~
+ p”x-+
11=1 n - 42
__ (
m2
)I
d/i?
(E5)
Hence one obtains for u(p) when 1 > p 3 m/K: NS8 u-‘(p) = -gy
m2
-vln(l+_iz)
KP
+ pE [u-l
- q&J
+ yln(1
+$)I
and when m/K 3 p > 0: M-4-8 u-w = -gy-+
-vln(l+-$$-)
/-p p-!!g!
+
kg!
In (1 + $)]
+ v
($)e’2. W)
APPENDIX
F: GENERALIZED
MINIMAL
SUBTRACTION
AT THE TWO-LOOP LEVEL
In this appendix we renormalize the coupling constant by GMS up to order u3. As an example we consider I’$], in order to show that the requirements placed on GMS in Section 7 are fulfilled at the level of two loops, namely, that the poles in E are eliminated order by order in U, and that as m - co the vertices of the theory have a well-defined expansion in u and E. First we calculate I’&, using the normalization conditions (4.1). The field renormalization, giving a factor of Zb2, is ignored at this stage, since at the level of two loops it takes care of itself anyway. The graphs contributing to r(*) are shown in Fig. Fl.
468
JACK
COHN
of Maxwell’s equations for an arbitrary distribution of charges and currents embedded in a uniform, isotropic, and transparent medium with constant electric and magnetic permeabilities E and p. In addition, the corresponding expression for the stress tensor {which turns out to be Marx’s stress tensor) will be found and this will also he confirmed by the use of Noether’s theorem together with a suitable action function. Finally, we shall develop the covariant expression for the 4-vector potential and field of an arbitrarily moving point charge in the medium.
2. COVARIANT FORMULATION In this section we begin by developing a manifestly covariant description of Maxwell’s equations for a system of charges and currents in a uniform, isotropic, transparent, and boundaryless medium with constant permeabilities E and p. 2.A. MaxweN’s
Equations
We begin with Maxwell’s covariant equations in the form V x H = (477/Q + (l/c@,
(2.1)
V x E = -(l/c)&
(2.2)
V*B=O,
(2.3)
V . D = 47rp,
(2.4)
which presumably hold in any inertial frame, and the constitutive relations H = B - 47rM = (I/p)B
(2.5)
D=E+4rP=cE
(2.6)
and which hold in the medium rest frame. Combining Eqs. (2.5) and (2.6) with Eqs. (2.1)-(2.4), we have in the medium rest frame, the relations V x B = (4r~Llc)T + Wc)fi, (2.7) V x E = -(l/c)& V*B=O, V . E = (47+)p,
(2.8)
(2.9) (2.10)
and these equations form the basis of our inquiry. We shall now make these equations covariant by expressing them (in the medium rest frame) in terms of quantities (originally defined in the medium rest frame) which transform as tensors. We proceed as follows:
400
AMIT
AND
GOLDSCHMIDT
The expressions for the various functions ZqSpare obtained by taking the external momenta at the symmetry point (ZQ = $K3 2, kikj = (K2/4)(&jij - l), p2 = K2), and suppressing the first two variables in the notation. Using Eq. (Fl), one has K-Erfi;lR
=
24 -
-
U2 q
[I
(f
, 0)
-
u2qqz(p)
&,(o)]
-zsp(g]
+u,M2+6M+20 36
[I(+
7 ‘)
-
zdo)]2
+U3(N36M)2[z(~,~)-zsp(~)]2
+,w--M)w+4)
[I (+ 9 C)
36 ~p--M)u4+4) 36
[I,, (?)
- ZSPCO)]”
36
[z(~,o)
-zsP($)]2
[z(p)
-z($,gIP
+*,w--w+2) +wwM)(M+2) 36 +$
- zsP(o)]2
5M+22 9
[z4 (C 92 , 0, (40) - z (+,
0) &P(O)
+ Zs2P@) - Z48PK4 0, o)]
+ 28~[14(e,~,o,~,~)-z($,o)z*,(~) + 4%43ZSP (C) - Z4SP (0,;, +)I
+U~2(~p~~4(~,~,~,~,0)-z(~,~)zsp(0) +
ZSP (F)
ZSP(O)
-
&SP
($,
f
, q.
(F3)
Notice that in order to simplify the expression (F3) we assumed that the external momenta of D4) are also at some symmetric momentum point, so as to avoid sums over permutations. One can convince oneself, not only that there are no poles in E in Eq. (F3), but also that as m + co the expression remains finite order by order in u and E and in fact reduces to the value of P4) in a theory with M massless particles only.
TREATMENT
OF
BICRITICAL
401
Pt
Using the notation L&y
+ln(+
-X)+$),
L, ($ , :)
= f dx ln2 ($x(1
(F4) - x) + $-),
one can write the expansion of the one-loop integrals I($+-)
=f[l
-~-~L1(~,a)+$L1(~,~)+$L2(~,~)] + O(E2),
(F5)
from which one can derive expansions for I( P/K, O), &&,v/K), &,(O). Next we expand the general I4 in Eq. (F2) I, (f
, $ ; ?,
?,
?)
= ‘4 r2 (2 - ;) I-(E) 1’ dx[x(l
- x)]-‘/~
0 1
X
dy( 1 - JJ)JJ’/~-~ ‘dz
s0
$ (1 - y)(z(l - z) + YZ”)
s0
+ 2y(l - y)z q
1
+ Y(1 - u> $
+ (1 - VI y w5)
In order to obtain the pole terms we write
{u>’ = {Y = o>-+ [{y>-’- (y = 0)-q; i.e., we add an subtract the brackets with y equal to zero, where the singularity lies. {y}-’ - {y = O}-c = --E In &
+ O(E~).
(F7)
This term will contribute only to the finite part of (F6). The pole terms for a general distribution of masses are given by P.T. = &
[ 1 - ; - E& (f
, ?I).
(W
The finite part depends on the specific values of internal masses. We therefore specialize to each case. First, z*(+,~.o,o,o)
=&[l
-+‘($‘o)]
+0(l);
here we da1 not care about the finite terms, since they are independent of the mass.
402
AMIT
AND
GOLDSCHMIDT
In order t0 Caldate &(p/K, &/K, 0, m / K, m/K), we substitute expression (F7) for the curly brackets in (F6). We are only interested in the limit of this finite part for very large values of m/K. We therefore obtain as m + co : finite part of I4 (f-,2,
O,c,$)
Fz - ; r2 (2 - 5) r(E) * E J dx x-E/2 1 dy(1 - y) y+1 x S~41
+g
x(1 -x;z(*
= -$-q)+n2p4r(l
-z))
-:;z(l m2
- In $x(1 - x) z(1 - z) 1
-Z)
+ O(E) = - f In2 $
- i In -$- + O(E).
(F9) Terms which are finite in the limit m ---t co were dropped, and use was made of
Therefore one obtains
where
r($+)
Asfor&(P/K, kdK, --E In
m/K,
- $ln2!$ m-rco’
i In $
+ O(E).
(F10)
m/K, 0), Eq. (F7) gives in the limit m -+ 00
3 z(l - z)+ (I - y + -&)
$ -t-•Eln
$ z(l - z) + 5 and this gives a mass-independent
contribution
(
l-u+-+,
to the corresponding
+ finite terms in m + O(E).
1 14. Therefore
(Fll)
We can now change our point of view. Instead of regarding I&O), I&m/K), I,,,(O, 0, 0), etc., as the corresponding functions at the symmetry point, we will treat
TREATMENT
OF
BICRITICAL
403
Pt
them as unknown functions in the expansion of u,, in powers of u (Eq. (Fl)) to be determined from Eq. (F3) by GMS. In order to avoid confusion we will denote these unknown functions by J(O), J(~/K), J&O, 0, 0), etc. From the terms proportional to ~2 in (F3) we obtain J(0) = l/e, W)
including the poles in E and the infinite m limit of the finite terms in the E expansion. We therefore take J (c)
== f - i ln (a + $)
+ $ In (a + $)
+ i ln* (n + $1
+ O(2), (F13)
where a is a positive constant. The difference I(P/K, poles in Eand. will vanish as m + co up to order E. From the terms proportional to u3 one obtains
m/K) - J(m/K)
will not have
r4 (f ) 2) 0, 0,o) - I($ ) 0) ; + f - J,(O,0,O) yg 7; 2;‘, [ 1 - ; - EL1(f- , o)] - $ [ 1 - ; - ; L, ($ , o)] + ; - J,(O,070) =---- 1 2G
1
4-z - J&4 0, 0).
Therefore we take J&4 0, 0) = &
(1 + ;).
and J4(m/K, HI/K, 0) are determined Eq. (F3), respectively. Consider first J4(0, m/K, m/K):
J,(O, m/K, m/K)
(F14)
from the last two brackets in
404
AMIT AND GOLDSCHMIDT
and the proper choice is J4(0,~,~)
=&l
+;)
-;ln+++;ln(a+$)
+ O(E). (FL5)
In a similar way one obtains
J4(y))
=&[
1 + ; - E In (~2 + $)I
+ O(E).
w3.l
The values of the functions J we found have to be substituted in (Fl) to give u,, in terms of U. As for the third and fourth lines of Eq. (F3) it is obvious that the poles in E cancel in each square bracket separately. However, as m -+ co it is the interplay between the two terms which leads to a finite result. Namely, the O(1) terms of these two lines become u3
(N
-
M)(M
+
4,
In2
36
!
vJ2
ln2
Et
K2
K2 1
+,(N--M)(M+2) 36
(
ln2 $!. - In2 f)
= 0.
The effect of the field renormalization (2,~ in the case of rf4)), as well as the calculation of the other Z-functions, is handled in a similar way.
APPENDIX
G: CALCULATION
OF xc1 TO ORDER ONE LOOP
We use the Lagrangian (4.3) to calculate I’:;) (p = 0; 7, m, U, K). At the order of one loop one needs counter-terms which renormalize the mass and the temperature only. The first take care of the quadratic divergences, the second of the logarithmic divergences. Without counter-terms, (4.3) gives, to order one loop, p
= T 1 hit + 2 UKF 11
6
1
N--M
1 -
s q2 + 7
+
6
UKE s
q2 + m2 + T ’
(Gl)
The counter-terms which are needed here are of the form CT. = Qc,#~ + &T#~.
(G2)
They add to the r.h.s. of (Gl) a term c, + c27. The function c1 is determined by requiring that at 7 = 0 the frrst M components be critical, namely, that &(T
=
0,
P?Z, U, K)
=
0.
TREATMENT
OF BICRITICAL
405
Pt
This gives 1 UK” 2--
M+2
cl = - ~ 6
N--M6
4
UKf
1
(G3)
q2.
Inserting in I’:,“‘, we have r;;)
=
7
(C2
+
1)
-
y
UK’
I
q2(qe1+
~)
I
-
N-M
1
UKF
6
s
(q2 + m2)(q2+ m2 + T)
which in the notation of Appendix C can be written as ~
#K’~(o,
T)
N---M
-
____
6
UKE.@2,
m2
f
T)
(G4)
after 2,~ is dIEfined by
to order one loop. This is Eq. (11.8).
APPENDIX
H
Here we paraphrase Symanzik’s derivation [4] of the Ward-Takahashi identities induced by quadratic symmetry breaking, using the method of functional integrals. We consider a Lagrangian with N Hermitian scalar fields of the general form
where the first term is O(N)-invariant and the quadratic symmetry-braking term is defined in terms of the symmetric matrix M. The generating functional of the complete Green functions Z(J) corresponding to the Lagra.ngian is given by
in terms the term J Wi(4 Making
of a linear source (external field) J. If one expands Z(J) in terms of Mij, linear in Mij will consist of symmetric Green functions with an insertion of hi< x > w h ic h are not finite. New renormalizations are required. am infinitesimal change of variable
406
AMIT
AND
GOLDSCHMIDT
where ra are the generators of the group O(N) and E, infinitesimal
parameters, one has
Since the 6s are dummy variables Z(J) does not change. This implies that the firstorder change in E must vanish, which can be written in terms of functional derivatives with respect to J as 035) These are the W-T identities for the generating functional. They relate the nonsymmetric part of Green functions to Green functions into which J dx q$(x) ySj(x) is inserted. In order to derive a linear differential equation, let us introduce a source term for the bi4i insertions.
All quantities have to be calculated with Ji = 0 and Kij = Mij . The W-T identities take the form (H7) The same equation is satisfied by F(J, K) = In Z(J, K), the generating functional of the connected Green functions. The generating functional for the vertex functions, I’, is defined by a Legendre transform, namely, T(@, K) + F(J, K) = j- dx tDi(x) Ji(x)
(W
with
WJ, K) j 2
@i(X) = w Therefore
and one obtains
%=JfJo
*
(H9)
TREATMENT
OF
BICRITICAL
407
Pt
Setting Ki, = Mij , one obtains W-T identities in the form T;$~(x)
-
2&&~&
W@‘,K) 6K&)
r+,wij = OS
(HW
where the full notation was reintroduced. Taking the functional derivative of (H12) with respect to @, , CD, , where the lower indices include the space time points, one has after setting K = M:
If all Q’s are :set to zero the various derivatives of I’ are just the usual vertex function: one-particle irreducible parts of the Green functions. When a Fourier transformation is performed, (H13) reads
where the function I’$‘,l/ij,(p; 0) is P) with an insertion zero momentum. Specializing to 0(2)-symmetry, we have simply TkZ = ~ld,Z - w&k
of the operator
*
+i4j with
W5)
Taking ww and choosing I = 1, m = 2, we arrive at
I-,',"' (p) - r,;' (p) = 2(m,2 - m2y r,::$,,(p,
--p; O),
(H17)
which related the nonsymmetrix part of the two-point function to a two-point function with one insertion of $r&. Note that the right hand side vanishes for
m, = m3. Taking functional derivative of Eq. (H13) with respect to @t , @t and setting all Q’s to zero, one obtains at a symmetric momentum point (Y = s = 1)
The right hand side is a four-point vertex with external legs of types 1, 1, 1, 2 with an insertion of &c$, at zero momentum. One obtains a similar connection between I’&?i, and I’$, .
408
AMIT AND GOLDSCHMIDT
Equations (H17) and (H18) are the W-T identities needed in order to find all the renormalization conditions and to renormalize the theory by the BPHZ method. The proof by induction was outlined by Symanzik [4]. From Eq. (H17) it is obvious that the same Z, (the wavefunction renormalization) can serve for both I’::) and I’$. Taking the derivative of (H17) with respect to p2 at p2 = 0, we obtain
P2=0
d = 2(m,2 - m22>_ ct;2, dp2
. PLO
0319)
The right hand side is superficially convergent because 1’$$12, is only logarithmically divergent, and hence the assertion follows. Similarly, because the right-hand side of (H18) is superficially convergent, it is evident that one counterterm can subtract the infinities from all the different fourpoint functions. Only the mass terms need different counterterms. REFERENCES 1. D. R. NELSON AND E. DOMANY, Phys. Rev. B 13 (1976), 236; H. HORNER, 2. Physik B 23 (1976), 183; J. M. KOSTERLITZ, .I. Phys. C 9 (1976), 497, and many references to previous work therein. 2. D. MUKAMEL, Phys. Rev. Lerr. 34 (1975), 482. 3. See, e.g., (a) A. AHARONY and (b) E. BREZIN, J. C. LE GUILLOU, AND J. ZINN JUSTIN, in “Phase Transitions and Critical Phenomena” (C. Domb and M. S. Green, Eds.), Vol. VI, and references cited therein, Academic Press, New York, 1976. 4. K. SYMANZIK, Renormalization of theories with broken symmetry, in “Cargtse Lectures 1970” (D. Bessis, Ed.), Gordon & Breach, New York, 1972. See also J. ZINN-JUSTIN, in “Trends in Elementary Particle Physics” (H. Rolnik and K. Dietz, Eds.), Springer-Verlag, Berlin, 1975. 5. E. K. RIEDEL AND F. J. WEGNER, Z. Physik 225 (1969), 195. Notice that in this original work it was T - Tc~ which was used, and which we show below to be the natural variable. The present form is used in Ref. [l]. 6. The physical systems described by an 2 of this type have been discussed by M. E. FISHER AND D. R. NELSON, Phys. Rev. Left. 32 (1974) 1350 and by A. AHARONY AND A. D. BRUCE, Phyx Rev. Lerr. 33 (1974), 427. 7. K. G. WILSON, Phys. Rev. Lett. 28 (1972), 548. 8. E. BREZIN, D. J. WALLACE, AND K. G. WILSON, Phys. Rev. B 7 (1973), 232. See also Ref. [3, sect. 7.11. 9. A. D. BRUCE, J. Phys.C’. 8 (1975), 2992, and A. D. BRUCE AND D. J. WALLACE, J. Phys. A 9 (1976), 1117. 10. G. ‘T H~~FT AND M. VELTMAN, Nucl. Phys. B 44 (1972), 189. 11. See, e.g., Ref. [l]. 12. In the context of critical phenomena see, e.g., J. D. LAWRIE, J. Phy.s. A 9 (1976), 961; D. J. AMIT, J. Phys. A 9 (1979, 1441. 13. See, e.g., D. J. AMIT, “Field Theory, the Renormalization Group and Critical Phenomena,” Chap. 7, McGraw-Hill, New York, 1978. 14. See Ref. [3b] or Ref. [13, Chap. 91. 15. We are being slightly cavalier in writing (2.16), since the proportionality factor between rrr) and x-’ depends on u (it is Z,). But as far as the dependence on t and y is concerned, this is irrelevant. 16. See, e.g., Ref. [3b, Sect. 7.11 or Ref. [13, Sect. 10.71. 17. This is true for the result of Wilson-Ref. [7] above. Here the statement is proved to all orders.
TREATMENT
OF
BICRITICAL
Pt
409
18. See, e.g., Relfs. [3] and [22] below. 19. See, e.g., S. COLEMAN, “Renormalization and Symmetry: A Review for Non-Specialists,” International School Etore Majorana (1971), Editrice Compositori, Bologna, 1973; see also Ref. [13, Chaps. 7, 101. 20. Here we imply that the regularization is affected by a momentum cutoff A, though usually, as in Section 2, w’e will use dimensional regularization. 21. If one insists on an O(N)-symmetric four-point counterterm then r”) with external legs of type i will not be finite as m --L 00. 22. K. SYMANZIK, in “Particles, Quantum Fields and Statistical Mechanics” (M. Alexanian and A. Zepeda, Eds.), Springer-Verlag, Berlin, 1975. See also Ref. (31; F. JEGERLEHNER AND B. SCHROER, Acta Phys. Austr. Suppl. 11 (1973), 389. 23. The fact that the integrals are over dx/x, rather than the conventional du/j3, is just a matter of convenience.. These two measures are identical, owing to the first characteristic equation, Eq. (8.2). 24. S. SINGH AND D. JASNOW, Phys. Rev. B 11(1975), 3445; Phys. Rev. B 12(1975), 493. 25. S. COLEMAN AND R. JACKIW, Ann. Phys. (N.Y.) 67 (1971), 552. See also K. Symanzik, DESY report 73/6 (1973), Ref. [3, Sect. 3.41, Ref. [13, Sect. 7.41. 26. E. DOMANY, D. R. NELSON, AND M. E. FISHER, Phys. Rev. B 15 (1977), 3493; E. DOMANY AND M. E. FISHER, Phys. Rev. B 15 (1977), 3510. 27. See, Ref. [1:3, Appendixes 9.2, 9.31.