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Perturbation and renormalization in thermo field dynamics
ANNALS
OF PHYSICS
Perturbation
152, 348-375
(1984)
and Reno.rmalization H. MATSUMOTO,
Departmen!
of Phwics,
University
I. OJIMA,* of Alberta,
Received
January
in Therm0
Field Dynamics
AND H. UMEZAWA Edmonton,
Alberta.
T6G 2JI.
Canada
31, 1983
A systematic renormalization procedure used in the perturbative calculation of the real-time Green’s functions at finite temperature is presented. The formalism of therm0 field dynamics is employed throughout, permitting the use of Feynman diagram techniques. The renormalizability by means of the temperature-independent counterterms is proved. causal
I. INTRODUCTION The formalism of therm0 field dynamics developed in Refs. [ 1, 21 is a technique whereby the usual field theory, defined in real space-time coordinates, can be generalized to the case with finite temperature. It has been applied to various problems in the physics of condensed matter [3] and recently has been extended to deal with gauge theories containing indefinite metric in their covariant formulation [4]. The advantage of the therm0 field dynamics formalism can be seen not only in its intimate relationship [4] with the algebraic formulation of statistical mechanics (51 but also in the fact that all the operator relations in the conventional field theory are retained. Furthermore, the Feynman diagram method can be formulated very easily by means of the real-time causal Green’s functions [2, 31, which are expressed in terms of “temperature-dependent vacuum” expectation values. The best merit of this formalism lies in the fact that it allows us to discuss both the time and temperature dependence of physical quantities in a very straightforward manner. Thus, it enables us to treat dynamical phenomena at equilibrium as well as inequilibrium (at least, within a certain limited region around equilibria). A particularly significant application of the therm0 field dynamics is in the study of (not only static but also) dynamical effects in phase transitions and critical phenomena. To prepare for this program, we need to reformulate the standard machinery of the renormalization group in the framework of the therm0 field dynamics. With this motivation, we discuss first in the present paper the Feynman diagram method in the therm0 field dynamics and the problem of the renormalization procedure. Therm0 field dynamics provides us with a relatively straightforward method whereby one can calculate any dynamical quantity through the use of perturbation theory, which is sometimes difficult in the Matsubara method 161. * Research
Institute
for Mathematical
Sciences.
Kyoto
348 0003-49
16/84 $7.50
Copyright All rights
$1 1984 by Academic Press. Inc. of reproduction in any form reserved.
University,
Kyoto
606, Japan.
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AND RENORMALIZATION
INTFD
349
After explaining perturbative calculations in thermo‘ field dynamics in Sections 2 and the renormalization procedure in Section 3, we prove in Section 4 that all the ultraviolet divergences of the Feynman diagrams in perturbation expansions of realtime causal Green’s functions at finite temperature can be eliminated by the counterterms prepared at zero temperature provided the theory is renormalizable at T = OK. Though it might be expected intuitively that an increase in temperature does not upset the finiteness of matrix elementsof observables, this is not at all obvious in the usual temperature-dependent Green’s function formalism [6 ] since each term in the Matsubara frequency expansions seems to produce temperature-dependent divergences arising from the higher-order loop corrections. The temperature independence of divergences is shown only after the summation over the Matsubara frequencies is performed. This leads to some difficulty in constructing a systematic proof of the temperature independenceof the divergences in every order of the perturbation expansion. The situation is different in therm0 field dynamics since the Matsubara-frequency expansions are not required. As will be shown in this paper, the finiteness of the temperature-dependent parts can be seen transparently through a systematic formulation of the renormalization procedure. The result is rephased as
renormalizations at different temperatures are related through a finite renormalization. This result may play a crucial role in comparing physical quantities at different temperatures and in discussing the renormalization group equation at finite temperature. The cancellation mechanism of temperature-dependent infinities is illustrated explicitly up to the order of two loops for the simple case of the A#” theory in Section 5. In Section 6, the renormalization procedure at finite temperature which does not require any zero-temperature calculation is presentedexplicitly by meansof the same model as the one in Section 5. Section 7 is devoted to some concluding remarks.
II. FEYNMAN
DIAGRAM
METHOD
In this section, we briefly present the essential points of therm0 field dynamics relevant to our purpose. To avoid unnecessary complications, we restrict ourselves to a real scalar field Q(X) whose dynamics is specified by a Lagrangian density P(W)>. In therm0 field dynamics / 1 - 41, a field theory at the finite temperature is formulated by doubling the freedom of fields. To each field Q(x) is associated another field called a tilde field 6(x). Then form a thermal doublet Q”(x) as
5951152/2-S
@“(x) = G(x)
(a= 1)
= 6+(x)
(a = 2).
(2-l)
350
MATSUMOTO
The dynamics of @“(x)
ET AL.
is governed by the thermal Lagrangian
2(x)
given by
L?(x)=x &L”9m(X),
(2.2)
a
where
K?(x) = W@(x)) = L22)
(a= = P*(cqx))
1) (2.3)
(a = 2)
and Ea= 1 z-
(a= 1
1) (2.4)
(a = 2).
In (2.3), the asterisk indicates the complex conjugate. The representation is requred to satisfy the tilde operation rule, which is defined for arbitrary operators 0, and 0, (or 0) by (2.5a)
0~~==~,8,, c~o-~=cI*o,
+c2*6,,
(2.5b)
where c, and c2 are c-number and o==qo
(2.6)
with q being a phase factor. The temperature-dependent vacuum 10(/3)) is annihilated by the physical annihilation operators a@ k) and a’& k):
4 These physical operators
k) IWO) = a’@,k) INO> = 0.
are related to the quasi-particle
(2.7)
field Y”
satisfying
P-“(x)= 0 [$-cd--iv) I through the Bogoliubov Y”(X)
a”(k)
transformation:
= (2n:3,2 1 Jt:k)
= Ugy(w(k))
[a”(k)
aY(j, k), sinh B(k) cash B(k)
eik’x-iw(k)t
+ a”+(k)
.
emik xtrw(k)f],
(2.9)
(2.10)
PERTURBATION
The Bogoliubov
AND
RENORMALIZATION
351
IN TFD
parameter B(k) is related to the temperature through the relation (2.12)
sinh’ 0(k) = eQWC,!)_ I. A master formula controlling the relation between the tilde operation temperature-dependent vacuum is the tilde substitution rule [ l-41 :
l?fb(H-g)d )O(p))= ot )O@)),
and the
(2.13)
where H is the Hamiltonian of the system, r? is its tilde conjugate and 0 is an arbitrary operator. It has been shown [ 1,3,4] that the substitution rule (2.13) is equivalent to the Kubo-Martin-Schwinger (KMS) condition [ 71. It is straightforward to formulate the perturbation theory at finite temperature. The perturbation theory can be developed by regarding 5?(x) as the given Lagrangian density for the field W(x) and by considering the condition of the tilde operation rules and substitution rules (i.e., KMS condition). In the interaction picture, the Lagrangian is separated into a free part &x) and an interaction part g(x): (2.14)
52(x) = L&(x) + s&x) with
where Y”(x) is the field in the interaction picture and w(k) is the physical energy of quasi-particles (which is usually temperature dependent). The time-ordered product of Q”(x) is given by the usual formula
= (Team
... Y”n(x,) exp(i J d4x @(x)})4 Vexpti S d4x ~GW, ’
(2.16)
where ( )D is the temperature-dependent vacuum expectation value in the interaction picture. The Gibbs statistical average of physical quantities is given by the (ai = 1) components of the above expectation values. The free propagator for -i”“(x) satisfying (2.8) is evaluated by the use of (2.9~(2.11) [2,3]:
(T,‘%“(x) sayy))il =&J day(k) = [U,(w(k))
d4k eik-y) Llyk), r[ki - (w(k) - itk)‘]-’
(2.17) U,(w(k))]“Y,
(2.18)
352
MATSUMOTO
where U,(o(k))
ET AL.
is given by (2.11) and r is given by (2.19)
We can write (2.18) in the following A”b(k)
form: = A;“(k)
(2.20)
+ A;“(k),
where A,,(k) = r[ki - (o(k) 4@)
= -2rci4ki
- i &)‘I-
- W(k)2)
(2.21)
e51k,l/2 1 )’
,,,k,,; _ 1 ( ,,,&2
(2.22)
Note that A,,(k) is diagonal and depends on temperature only through w(k), while A,(k) has nonvanishing off-diagonal components and a factor damping exponentially at high energy. Other expressions for A*“(k) which we wil find useful in the following work are (2.23) A(k) = UAkJ d(k) U,(k,) = WlkI)~W
(2.24)
u,(lkI)
with (2.25) (2.26) U(K,
k) =
E(K)
6(K*
-
co(k)‘).
(2.27)
The Feynman diagrams are constructed by means of the propagator 4”’ and vertices associated with 2, in the usual manner. This rule can be extended to cases of complex fields, fields with spin and gauge fields. It should be noted that, contrary to Matsubara’s method, the Wick theorem holds true in the therm0 field dynamics in a manner precisely analogous to conventional field theory. A close relation does exist, however, between therm0 field dynamics and imaginary-time formalism of Matsubara treated by means of perturbation theory. This relationship was discussed in Ref. [S]. The perturbative calculation method and the renormalization prescription based on the Feynman diagram method in the therm0 field dynamics is analogous to that employed in conventional quantum field theory. ’ However, there appears a new situation in the case of therm0 field dynamics for the following reason. The quasiparticle states constructed at finite-temperature turn out to be unstable due to the ’ At T= OK (p= l/k, T+ co), the therm0 standard quantum field theory.
field
dynamics
reduces
simply
to the duplication
of the
PERTURBATION
ANDRENORMALIZATION
353
INTFD
thermal instability. Since the free propagator for the internal line should be that of a stable particle, the instability shows up only through the full propagator A’“Y(k). Another difference between therm0 field dynamics and the usual quantum field theory appears when one considers chains of propagator A(k) and self-energy C(k) with the same energy momenta k. This creates powers of A(k), causing multiple pole singularities. Particular care must be taken in treating in order to demonstrate that these multiple pole singularities do not result in any catastrophic behavior. This will be proved later.
III.
RENORMALIZATION
In this section, we illustrate, using the A@-model, how the renormalization procedure is performed in the perturbative calculation of the therm0 field dynamics. The Lagrangian for the scalar field Q(x) is
Y(x)=+- [(a,@(X))‘-p’@“(x)] --&A@“(x).
(3.1)
The thermal Lagrangian is given by (2.2) with Pa(x)=* The thermal Lagrangian given by
[(a,@a)2-p”@a*(x)] g(x)
+P”(x).
(3.2)
is separated as (2.14); 5&x) is the free Lagrangian
~o(x)=~&,~[(a,Y(X))2-m:~a2(x)]
c% and g,(x) is the interaction Lagrangian including the renormalization m
(3.3) counter terms:
= - s E, g, Y4(x)/4!
a
+ x - + Pyx) 6p2y’“(x) + + sz((a, 9’“(x))” - m:,P’(x)) a 1
-& SZ,sc”“(x> I. Here mR is the renormalized and Y”(x) the renormalized
(3.4)
mass parameter, g, the renormalized coupling constant field. The renormalization constants are defined as
@“(x) = z”2P(x), A =g,z,z-2,
(3Sa) (3Sb)
354
MATSUMOTO
ET
AL.
6/L*=[p*-m;]z,
(3Sc)
6Z=Z-
(3Sd)
1,
6Z,=Z,-1.
In Appendix A, it is shown that (3.3) is the only possible choice for @ in the perturbative calculation. The renormalization conditions are given by
Red’-‘(k)“” Ik;=m;+~.k=k8 = 0,
(3.6)
=& a )
(3.7)
$Red’-‘(k)“” 0
k;=m;+k;,k=k,
and
where F~,‘~2~~~~4is the irreducible four point vertex. Conditions (3.6F(3.8) are straightforward generalizations of the renormalization conditions used in the usual zero-temperature field theory. The perturbative calculations thus become similar to those in zero temperature. The differences occur due to the appearance of the thermal distribution functions in propagators and the fact that the renormalizaton conditions refer only to the real parts of the vertex functions, although, due to the thermal effects, the vertex functions are in general complex. Since +. in (3.3) gives a free propagator with the same high momentum behavior as the corresponding zero-temperature propagator (the thermal distribution functions do not modify the high momentum behavior), the renormalizability can be determined by the usual power counting method used in the estimation of the ultraviolet divergences. Therefore, it is obvious that when the theory is renormalizable at zero temperature, the theory is renormalizable also at any finite temperature. What is not obvious is the statement that the divergences appearing in the theory are inependent of temperature. If this statement is true, it means that the characteristic prameters at certain temperature such as mR and g, can be written in terms of those parameters determined at another temperature and that these two sets
of parameters are related to each other through a finite renormalization. This property may play a crucial role in the renormalization group aproach to the critical phenomena. Also, it means that, in considering a system at temperature T, instead of renormalization conditions (3.6)-(3.8) at T one can use the renormalization conditions calculated at a certain fixed temperature To in order to obtain finite answers for the various observable quantities at T. We will refer to the former renormalization procedure as the T-renormalization and the latter one as the Torenormalization (zero-renormalization when To = 0). The temperature independence of divergences will be discussedin the next section. We end this section by noting that although the energy of the free field associated with the free Lagrangian (3.3) has the covariant form dk* + rni , this energy does not necessarily be equal to the physical energy E(k) determined from Re A’-‘(~c)~~ = 0; the quasi-particle energy is not covariant as will be shown later.
PERTURBATION
IV.
AND
THE TEMPERATURE
Elimination of Ultraviolet Temperature
RENORMALIZATION
INDEPENDENCE
35.5
IN TFD
OF DIVERGENCES
Divergences by the Counterterms
determined at Zero
Due to the loop corrections in the perturbative calculation, the Green’s functions given by (2.16) suffer from ultraviolet divergences. In this section, we will show that divergences at finite temperatures can be eliminated by means of the counter terms which are prepared by the renormalization at zero temperature. In other words, we can renormalize divergences in the finite-temperature theory by means of the temperature-independent renormalization counter terms. We try to express the Green’s functions at finite temperature in terms of the renormalized coupling constant g and the renormalized mass m calculated at zero temperature and to expand them in power seriesof g. This calculation is essentially the perturbation calculation with
rni = m2,
g,
=
g’
(4.1)
in (3.3) and (3.4). The renormalization conditions (3.6)-(3.8) are required at zero temperature and the renormalization factors 62, dZ, and 8~’ are those at zero temperature. The free Lagrangian &$ leads to the propagator (2.20) with w(k)’ = k2 + m2; ~~~~=U,~I~,o~2~m:+i6rUx~/~,o =4(k)
(4.2)
+ d,(k)
(4.3)
with
d,(k)=
5
(4.4)
k2-m’+i&
d,(k) = -2ni 6(k2 - m2) e,,k0;
_ 1
1
eDlkol12
eDlkollZ
1
1 .
(4.5)
In this way, we can formulate a perturbation theory at T # 0 in which the internal lines are given by d(k) in (4.3) and interaction vertices given by @, in (3.4). As will be demonstrated later, the physical quantities at finite temperature can be shown to be finite. This renormalization method is the zero renormalization (i.e., T,renormalization with T, = 0). It is evident from (4.2) - (4.5) and 2, in (3.4) that the usual power counting method [9] for the estimation of the ultraviolet divergences should also work in the present situation. Therefore, we need to examine the ultraviolet divergences only in
356
MATSLJMOTO
ET
AL.
the one-particle irreducible (lP.1.) diagrams. Our proof makes use of mathematical induction with respect to the number L of loops.’ In the case of L = 0, the Feynman diagrams have no loops and, therefore, no infinities. Assume now that all of the n-point lP.1. vertices I-, have been made finite up to (L - l)-loop order by means of the counter terms determined by the primitively divergent graphs at zero temperature. We then consider r,, at L-loop order. Using the skelton expansion and the power counting method [9], we can easily see that the m’s other than primitively divergent ones may contain only divergences arising from rm’s (m < n) embedded in r,, as their (proper) subdiagrams which should have loops less than L and hence which have already been made finite owing to the above assumption. Therefore, the only new divergences at the L-loop order are those created by the final integration of the primitively divergent T,,‘s. According to (4.3), the internal lines in these m’s are d, + A,, all of which should belong to certain loop integrals because our diagrams are one-particle irreducible. Let us make a functional expansion of the T,‘s in terms of A,. Since A, given in (4.5) has an exponential damping factor and it is restricted to the mass shell (i.e., k2 = m’), loops containing A, do not diverge at high momenta. This means that the substitution of A, into a line in a Feynman diagram is equivalent to cutting and opening this line as far as the divergence of the loops is concerned. Owing to this observation as well as the fact that we consider here only the primitively divergent m’s, we need to consider only the L-loop order diagrams which are constructed using the propagator A,. The divergences created by A, can be renormalized by the counter terms set at zero temperature since A, and the coupling constant g are independent of temperature. The renormalization prescription now becomes the same as the one at zero temperature. Thus, we have proved that all the ultraviolet divergences in the Green’s functions are renormalized by the counter terms introduced at zero temperature. There remains only one problem originating from the multiple-pole singularities in the propagator A’(k)a4. In the perturbative expansion, there can appear chains consisting of the propagator A(k) and the propagator self-energy Z(k) with fourmomentum k. This creates multiple poles at k2 = m2. In the zero-temperature case, are circumvented by the presence of the those multiple-pole singularities infinitesimally small imaginary constant ic in [k2 - m2 + i&l-‘. which allows the Wick rotation of the integration contour without introducing any pinching of the contour. However, in the present case, [k2 - m2 + ie] -’ and jk2 - m” - ie] -’ are mixed through the thermal distribution matrix U,(i k, I) and, naively, it would seem that pinching of the integration contour must occur, resulting in new divergences due to the coincidence of singularities. Since these divergences originate from the expansion of A’ in powers of A, these are, of course, false divergences; they should
’ Note that, in (3.4), we do not make the normal ordering prescription loop-wise renormalization. This is because the operator ordering made additional contact contribution at finite temperature.
for L$ in order to assure the at T= OK gives rise to an
PERTURBATION
AND
RENORMALIZATION
IN
351
TFD
not show up in the final expression for A’. The proof of this is presented in Appendix B.
V. THE ZERO RENORMALIZATION In this section, by using the 1Q4-model we illustrate explicitly the cancellation of the temperature-dependent infinities. The interaction Lagrangian 2, is given by (3.4) with g, = g, mR = m. Here g and m are the renormalized coupling constant and mass at zero temperature, respectively. The zeroth-order propagator is given by (4.2F(4.5). The Feynman diagrams are constructed from the propagator A”?(k) and the fourpoint vertex gsa@l’ “u4, where 6al”‘u4 = 1 only for a, = a, = a3 = a4 and otherwise aal”‘a4 = 0. The two point lP.1. vertices r, is the self-energy E. At the one loop order it is given by d4k (A;“(k)
+ A;“(k))
- S,i.& - (p’ - m*) 6Z,,,
1
rn); (5.1)
which can be made finite by the choice
a& = “&j 2 P7c) d4kk2_ A2+ iE, dZ(,, = 0.
(5.2b)
Note that this choice is independent of temperature. We obtain the finite result
=
r"ygm*F(pm)(Er"ym;)
(5.3)
with (5.4)
In (5.3) o(k) is given by w(k) = v&%? and f,(w) is the boson distribution function f,(w) = l/(e”” - 1). Next we calculate four point vertex r,(=r~;.‘.‘.‘~) at the one loop order. By writing r4 as
358
MATSUMOTO
ET AL.
where y is illustrated diagramatically in Fig. 1. Since the interaction (3.4) does not mix 9’ and .Y* and since the off-diagonal mixing terms of the propagator, A,, =A,, =A,mv damps exponentially at high momentum, the possible divergent contribution to y in the one loop order comes only from the diagonal part, the origin of which is the loop constructed by A,,. This divergence is cancelled by the counter term g6ZIca1PI’~‘aa:
-&T&
2m
i
y
. d4q
ca A;“(q)
A;“(q
- kj - ki) lkz=,+
(i,j)(k.O
(5.7)
where the summation C,i,j)(k,l, is over all possible pair that can be constructed from (i;j; k; 1= 1, 2, 3,4). The real part of y is given by .d4q
ca A;“(q)
%
A;“(q
- ki - kj)
(i,]yk./)
2~~ A;“(q)
A;“(q
- ki - k,J + dZ,
I
. (5.8)
We denote the integrals in (5.8) by
Mk)= Rec2Lj4 I d4q 6’
FIG.
1.
Feynman
diagrams
A;“(q)
contributing
A,““(q - k)
to the vertex
correction.
(5.9)
PERTURBATION
AND
RENORMALIZATION
359
IN TFD
and
Z,(k)= Re-c2i)4I dq
Fa d;“(q)
Ly(q
- k).
(5.10)
Note that I,(k) is nothing more than the one loop correction at T = OK. In (5.8) the divergences in I,(k) are cancelled by the counter term 6Z, : (5.11)
-6Z, = $ (W, + k,) + Z,(k, + kJ + I,,@, + k4))I++ We now define &(k*)
(5.12)
= Z,(k) - Z,(O).
This in the region k2 < 0 gives k2 - 4m2
2 dk2(k2
~Fyim2j)j
- 4m*)
1. (5.13)
The function lb(k) is a finite integral: 1 ‘dk) = (2# .* 1 = 16rc* Ik\ I ,,, &J&J
log
(k2 + 2 Ikl dm)*
(k2-2)kJd&%?)2-4q:,k;
For the later convenience, we choose the subtraction -k, = -k, = (m, 0). We can rewrite the real part as Re $I.“‘,“”
I .. 4
= Ea~S”I”’ a4 f
- 4q;k:, ’
(5.14)
points in (5.8) as k, = k, =
(f(k, + k2) + f(k, + k3) + ?(k, + k,) - fo(4m’)), (5.15)
where T(k) = fo(k2) + 21,(k).
The imaginary
part of y is finite.
(5.16)
360
MATSUMOTO
FIG.
2.
Feynman
diagrams
ETAL.
contributing
to the self-energy.
We now proceed to the two loop order. The two loop contribution energy part is shown in Fig. 2. We have d4q d41A”O(q) AaD d4q d‘vdyq)
to the self-
Au4(k - q - I) Ayq)
ea doa
- [d/i;,, + 6Zo,(k2 - m”)] B“ P”
LjdqAaacq)
+ Eniv +Z -;
gc”
p
l(I) (2??)4 -
(2:)"
. d4qAaa(q)Aao(q) J
Ea scl:,,,
(5.17)
where the index a should be summed over. We can rewrite (5.17) using (4.3), (5.2) and (5.7) in the following form: ,Et2,(k)“4 = rmb
[kg2
(pkj-4
j21
- @P:,, + (k* -m”> =d
d4q d41A;*(q) A;“(Z) A;*(k - q - f) ca
+ ~Zu,, k:u
1 d4ZA;“(1 - q - k) A;“(l)
+Gg
(2i)4
I
d41ca A;“(Z) A;“(l) E’ + 6Z(,, EO I
+r”“&?’[&J’j
d4q d41A;“(q) A;“(Z) A;“(k - q - I) ca
PERTURBATION
AND
RENORMALIZATION
d4q d4rA;b(q) d4q (ED d;“(q)d;“(q)
361
IN TFD
Ll;“(l)Ll;b(k
- q - I)
++lj’“(q)A;a(q)
PI.
(5.18)
The divergence in the first term of (5.18) is cancelled by 8,~:~) and 6Z,,, since it is the same as the ones appearing at zero temperature. The second term can be amalgamated into the integration over the vertex function at T = OK:
1 (2i)J
1 Py;Y&-JT= OK). Id4qA;“(q) 1
This is finite. The last integration
d41 e“ A;“(l)A;“(l)
F’ + 6Z,,, E’=
(5.19)
in (5.18) is calculated as (see Appendix C).
(5.20) with
(5.21)
The above example explicitly shows that, due to the exponential damping of A, and the nonmixing nature of the elementary vertices given by @, in (3.4) the divergences apear only in those loops which consist of A, only. Since A,, is independent of temperature, so are those divergences. Thus the divergences at finite temperature are cancelled by the counter terms prepared at zero temperature. The cancellation of divergences in the vertex function including the two loop correction can be shown in a similar manner.
362
MATSUMOTO
ET AL.
The real part of ,X;;‘f(k) is diagonal and may be summarized
Re X;;lf(k) = ta4
2,&k)
1 + + g2 Yl (2x1
as
d4q 6(q2 - m’)
(5.22)
where f0C2j is given by
2,,,,,(k)
= Z&t’)
- &(m2)
- (k2 - m’) &
E,(k’)
lkz-,,,j
(5.23)
with
and &, given by (5.13), I, by (5.14), F by (5.4) and F, by (5.21). The renormalization scheme (the zero renormalization) in this section leads to a renormalized inverse propagator and vertex of the following form: A’-l(k)ay=
(k2 -m*)?‘-P’(k),
p k,. I
(pl.“Q
““Qzg(pl .k4
(5.25) +
y;,‘,yyy.
(5.26)
Then the renormalization conditions (3.6~(3.8) determine the characteristic parameters mR (which will also be denoted by m(T)) and g, (which will also be denoted by g(T)) at finite temperature. If we choose the renormalization point k, = 0, m(7') is the rest energy of quasi-particle state. In the remaining part of this section, we will discuss the relation between (g, m) and (g(T), m(r)). The renormalized propagator and vertex satisfying (3.6t(3.8) with k, = 0 (the Trenormalization) are denoted by d’(T) and Z(T), respectively. If we denote the unrenormalized propagator and vertex by Ah and Z, respectively, then we have (5.27a) and T,(k)
= Z-*l-(k)
= Z(T)m2 I-(T, k),
(5.27b)
PERTURBATION
AND
where Z(T) is the renormalization have the relations A’-‘(T,
RENORMALIZATION
constant
in the T-renormalization.
Therefore we (5.28a)
k) = (Z(T)/Z)Lr’(k),
I-(T, k) = (Z(T)/Z)2 Since the quasi-particle
363
IN TFD
energy o’(T,
(5.28b)
T(k).
k) is determined by
Red’-‘(T,k)=O,
(5.29)
we have from (5.28) w*(T, k) = k* + m2 + Re E”(w’(T,
k), k).
(5.30)
Note that Re C” = -Re ,Z22. From (3.6) - (3.8), we have m(T)* = m2 + Re E”(m’(T), 0) (Z(T)/Z)-’
= MT)/Z)*
(5.3 la)
0),
g(1 + Re Y&‘.‘~~ Iki=0,k,o=k20= -k.
=&Red’-‘(k)” 0
+nuJ~
lk;=m~cr.),k=~.
(5Jlb) (5.3 lc)
From the results of (5.3) and (5.22), we have Re C”(w2(T,
k), k) = gm2F(pm) + g2m2F7’Jm) F,(pm) + g2W
G(k)=l+
d4q.t&W
2 PI + I,(q
t kf - i &(4m2)
l,+,~ + O(g%
4q2 -4
I
I
(5.32)
f&q f k)')
.
(5.33)
The functions & and I, are given in (5.13) and (5.14), respectively. Since G(k,, k) is not a function of k*(= ki - k2) only, G(Sk2 + m2, k) shows a complicated kdependence and therefore o*(T, k) also shows a complicated k-dependence. A low momentum expansion leads to w2(T, k) = m*(T)
t a(T) kZ + O(k4)
(5.34)
with m*(T) = m*[ 1 t gF(pm)
t g’(FQ3m) F,@m) + G,(Pm)) t 0( g3)]
a(r>= 1+ g*F@m)+ G2CB41 + O(g3),
(5.35) (5.36)
MATSUMOTO
364
ET AL.
where (5.37a)
cl(&) =-&G(k)I&z.k=O,
(5.37b)
0
Gz(Pm)
=
Y$
G(k)
(5.37c)
jk6=mz.k=0.
The functions Go and G, have the following integral forms: Go(z)=&+&
I
e
1 cG 647~~1, dx@
+-
(xlog(X+~=I-fi~}
1
1-1’*+2L/(x*-l)(+l)
(ez” _ l)(eZ” _ 1) b
l-y2-2J(x2-
l)(+
1) ’ (5.39)
The ratio Z/Z(T)
is obtained from (5.31~) and (5.32): -W(T)
= 1 -g2 -$
G(k) Ik;=m>.k-o+ W’) 0
= 1 -glG,(pm)
+ O(g’).
(5.40)
The coupling constant g(T) is obtained from (5.151, (5.3 lb) and (5.40) as
g(T) = g( 1+ g(Z,(2m 0) + 21,(0,0)) + m*N.
(5.41)
From (5.14), we have (5.42)
VI.
T-RENORMALIZATION
In the previous section it demonstrated that all of the divergences occurring in the 14” theory are independent of temperature and the renormalization prescriptions outlined in the previous sections were extremely useful in establishing this fact. Once it is establishedthat there is indeed no divergences which depend on temperature we can formulate the renormalization scheme at any temperature. Since it is the finitetemperature renormalized quantities which are related directly to the observable quantities, this latter schemewill be much more useful in practical calculations. Since
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IN TFD
the change of the renormalization conditions from zero temperature to finite temperature does not introduce any new divergences, renormalization schemes at each temperature are related through a finite renormalization. This renormalization scheme may lead us to a useful tool in the renormalization group approach to the study of the temperature dependence of dynamical quantities. In order to perform the renormalization at finite temperature, it is convenient to write $0 and g in (2.14) as 2, = gc $ :I
Ea +
[(a&y”)’
- m*(T)
Y”*]:
a - $)
*I=:y&a n
.ya4
+ SZ(T)i
[(a,Y)’
- m’(T)
YQ2]
1
g(T) -,az,(r)~.“-r~~‘(T)Y”’
:, I
where gC is a c-number and the normal product refers to the temperature-dependent vacuum. The mass counter term is given by
6/f*(T) = /I2-m*(T)f JpZ*(T) z(T)-’ (Ya2)D)Z(T), (
(6.3)
where the vacuum expectation value ( ). refers to the temperature-dependent vacuum. The renormalization conditions are given by (3.6)-(3.8) with rni = m*(T) and g, = g(T). We will choose k, = 0 in this section. The zeroth-order propagator is given by
The perturbative expansion can be performed by the use of the interaction Lagrangian (6.2), regarding g(T) as a small expansion parameter. We calculate the self-energy correction and vertex corrections. The self-energy part E(T, k) and the vertex correction y(T) are defined respectively by A-‘(l’,
k)ay= (k* -m*(T))
ray-Zay(T,
k)
(6.5)
and W-7 ;;:::;
= g(T)[&“’
6”l” ‘*‘I + y(T);;:::ka4”].
(6.6)
Since there exist no tadpole graphs, the lowest order self-energy diagrams and vertex corrections are the second order in g(T) (see Fig. 3). We obtain (c.f. (5.17) and (5.6)) C(T, k)“Y=$g(T)2 - [&*(T)
eagY [&I2
1 d4q d4ZAay(T, q) Amy(T, Z)Aay(T, k - q - 1)
+ 6Z(T)(k2 -m’(T))]
t”Y,
(6.7)
366
MATSUMOTO
FIG.
ET AL.
3. (a) Self-energy diagrams.
(b) Vertex corrections.
x d”Y(T, q)d”Y(T, q - ki - kj) + P’ d=‘“‘=4 6Z,(T).
(6.8)
The counter terms &‘(T>, 6Z(T) and 6Z,(T) are fixed by the renormalization conditions (3.6)-(3.8) and are obtained as functions of g(T) and m(r). Once the counter terms are fixed as functions of g(T) and m(r), the relations of parameters at different temperature can be obtained as follows. Let us consider two temperatures T and T,, and their parameter sets {g(T), m(T)} and (g(T,), m(T,,)}. From (6.3) and (3Sb), we can see that they are related to bare parameters as ,u* = m’(T) -y
Z,(T) Z(T)-‘(Lsp”2)D - Z(T)-’
=m’(T,)-F
Z,(T,)Z(T,)-‘(~*),~--Z(T,)-‘~P*(W
A= S’-7 Z,(T) Z-‘(T)
= g(TJ Z,(TJ
Z-*V’J.
&*(T)
(6.9a) (6.9b)
Since the relations in (6.9) contain divergent integrals, their definition requires a cutoff in the momentum (say/i). From (6.9) we can obtain the following relationship between the parameter sets at T and T,,:
WYWJ
g(T) = F&T, To ; g(To), mV,))>
(6.10a)
W’l
= FmK T, ; @cd, NT,)),
(6. lob)
= FAT, Toi O-o), W’d)-
(6.10~)
PERTURBATION
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367
IN TFD
Since the renormalizations at T and To are related through the finite renormalization, relations (6.10) are well defined in the limit A + co. In order to demonstrate the consistency of these results with those in Sections 5, we calculate the (g, m)-dependence of g(T) and m(T) (i.e., relations between parameters at T # OK and those at T = OK). In Section 5, the mass counter term was defined as (6.11)
6,u2=[m2-p""]Z since the Lagrangian
is not normal ordered. From (6.9) and (6.1 l), we have g(T) + ,-Zl(T)Z(T)-‘(9”‘),
m’(T) = m2 - &‘Z-’
f 6p2(T) Z-‘(T)
(6.12a)
and g(T) = gZ,Z-2Z,(T)-1
Z(T)‘.
(6.12b)
Note that the lowest-order contributions to 6p2(r), SZ(r) are second order in g(T) and those for dZ,(T) are the first order in g(T), while the lowest-order contributions to S,u2 and 6Z, are first order in g and those for 6Z are second order ing. We will denote the order of g by the subscript (n). From (6.12), we have m2(T),,, = m2,
km,,,
(6.13)
=g-
The next order is easily obtained as
m’(T),,, = = -&fl,
g(%,
= d%t,,
+ “Aj 2 PI
d4kd”“(T,
k),,,
(6.14) (6.15)
- =,(T),,,).
Comparing (6.14) with (5.2) and (6.15) and (6.8) with (5.7), we can see that these resuls are simply (5.35) and (5.41). The second-order term, m2(T)c2j, is obtained from m’(T),,,
= -&f2)
mc2, +2
(.;Y’a2)Dco,+ f
(Ly’a2)Dcl, +
&‘PJ,,,
368
MATSUMOTO
ET AL.
where relation (6.15) was used. We have
J. d4k d”“(T,
k)(,,
(6.17)
+ . ..I”.
(6.18)
with d(T, k) given by (6.4). Note that d(T, kY=
[d(k) +d(k)m;,,
td(k)
Therefore
J- d4k d*“(k),
w%,,, = (& I d4k d”“(k)
(6.19) m;,,(T) f’ d”“(k).
(6.20)
Note also that d4kAa”(k)daa(k)
Pmf,,(T)
I (2)
d4k d41d”“(k) A”“(k) PAua(l) --
1
i. d’k d”*(k) d”‘(k) IZ=Sp;,, , 2 g (27q !
(6.21)
where (6.14) was used. Combining (6.7), (6.16) and (6.21), we can see that mf2, (T) obtained from (6.16) is identical with that obtained from (5.17).
VII. CONCLUDING
REMARKS
Making use of therm0 field dynamics, we have shown that when a quantum field theory at T = OK is renormalizable with certain counter terms, it is renormalizable at T# OK with the same counter terms. We also presented a scheme for the Trenormalization, which helps us to calculate the temperature-dependent energy and coupling constants. The renormalizations at different temperature are related through a finite renormalization. In contrast to the usual imaginary time formulation for the temperature Green’s functions [6], the proof of the renormalizability in the therm0 field dynamics is both transparent and systematic. In the case of gauge theories, therm0 field dynamics helps us to derive, from the BRS symmetry obtained in Ref. [4], the Ward-Takahashi identities at finite temperature. Since these identities have the same form as those at zero temperature,
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INTFD
all the considerations in this paper can be extended to gauge theories at finite temperature. Though we started from a Lorentz invariant Lagrangian, the result of the perturbative calculation are not Lorentz invariant because the energy of quasi-particles at T # OK is not of the form dm (see (5.30)-(5.34)), implying that the wave equation of a quasi-particle field is not Lorentz covariant. We expect that this noncovariant nature originates from the thermal instability of the quasi-particle state. Note that this instability is caused by the fact that in therm0 field dynamics the continuum states extend into the negative energy region [lo]. Let us close this paper by the following note. The temperature-dependent mass m(T) is expected to become soft at a critical temperature, m(T) -+ 0 for T+ T,. This is becausethe massscale should disappear and a long-range force appearsat T,. This mechanismwill be studied in a forthcoming paper in conjunction with a study of the renormalization group in therm0 field dynamics.
APPENDIX
A: CHOICE OF THE UNPERTURBED
PART.?&
Let us first chooce p0 as (1 a 2 [ (f
-,4”‘“(x)
w;(4)
(Al)
?‘(x)],
Then gr(x) is given by 9,(x) = - 2 E, ; a
g, Y”(x)
+ s E, - ; i a
.9”(x)
Sp’(-iv>
++z ((-$yx))’- .!P(x) 0;(-iv)
Z.
(‘43)
One may be tempted to use as the above uR(k) the quasi-particle energy E(k) defined by Re A’-‘(k)aO Ik;~E2ck,= 0.
(A4)
(Here the real part of A’-‘(k) is taken because it generally has a finite imaginary part due to the thermal effect.) We will show that the above choice wR(k) = E(k) is not a convenient choice in
370
MATSUMOTO
ETAL.
perturbation theory. In the course of the calculation of d’(,r~)“~, &‘(k) perturbatively from condition (A4). Then from (A3), E*(k) is given by E*(k) =p* t k* -Z-l
is obtained
l@*(k).
WI
If E2(k) is considered to be the zeroth-orders contribution (i.e., OR(k) = E(k)), then all of higher-order contributions to the counter term Z-’ d,u*(k) must be cancelled by $, which is possible only when 6p*(k) is independent of k. When &*(k) dependson k, a convenient way is to fix the zeroth-order massparameter from the energy at a momentum k,, i.e., k; + m:, = E*(k,),
646)
and use wR(k) = dm. Then we can calculate E*(k) iteratively as a function of g, and mRby use of (A5). Denoting the nth order of g, by the superscript (n), we have E*(k,)‘“’
= 6,,(m: + k:),
647)
E*(k)‘“’ = p2(“) t 6,,k2 - (Z-l d,u*(k))‘“‘.
(A81
p*(") = d,,rni t (Z-’ 6,u2(kR))‘“),
649)
From (A7), we have
which leads to E*(k)‘“’ =
6,,(mf, + k2) - (Z-’ 6p2(k) - Z-’ 6p2(kR))‘“).
(AlO)
E*(k) appearing in the expression for Z-’ &*(k) is always one order lower than the left-hand side of (AlO). Therefore E2(k) can be obtained perturbatively. In this way we can write all physical quantities in terms of g, and mR, which are the physical parameters at finite temperature. The k-dependenceof the zeroth-order approximation of E*(k) is always k2. This is due to the fact that the original Lagrangian is Lorentz invariant and that the mass term does not depend on k. When E*(k) is expanded in terms of g,, the perturbative expansion of amplitudes is given by an expansion in powers of (ki - (k2 t mi))-'. This is the perturbative calculation based on the free Lagrangian (3.3) with the interaction Lagrangian given by (3.4).
APPENDIX
B: CANCELLATION
OF THE DIVERGENCES THE PINCHING OF INTEGRAL PATHS
ARISING
FROM
We will show in the following how the divergences caused by the pinching of the integration contour cancel. The divergences under consideration arise only from the self-energy diagrams
PERTURBATION
AND
RENORMALIZATION
IN TFD
shown in Fig. 4, where Zi is the proper self-energy part. The integral following form: -
(2i)4 i
d’kF”‘(k;
. ..)[A(k)L.(k)A(k)Z,(k)
. . . s?Z,(k)A(k)]“D,
371
has the
(Bl)
where F represents the additional part of the Feynman diagrams which do not contain one particle lines with the momentum variable k. When the part shown in Fig. 4 is of mth order (m > n), the amplitude summed over all of the mth order diagrams can be written as &j
d4kP’(k;
. ..)[A(k)
C(k)A(k)
... C(k)A(k)]“4’“‘.
032)
Here superscript (m) indicates gmth order and C(k) = f
C(k)“’
033)
i=l
with Z(k)“’ being the ith order contribution to the proper self-energy part and is assumed to be free of divergences. We have already shown that there exists no ultraviolet divergences. Therefore we investigate only the singularities casued by the multiple-pole arising from powers of AaD( i.e., multiple-pole singularities around k2 N m2.
Since 2?(k)
is related to Arab(k) A’ -I(k)efi
C(k)
through the relation = (k2
-
m’)
rob
- P’(k),
(B4)
has the following spectral representation
z(k) = r j dK0, k) U,(K) k, _ ;+ i 6r u&) 5. Here we used the fact that A’(k)
has the spectral representation
A’(k) = 1 dK C’(K>k) UB(K> k, _ :+
FIG.
(W
4.
i 67 u*(K).
Self-energy-typediagrams.
(J36)
372
MATSUMOTO
ET AL.
p and u are related through the relation
CJ(K,k) P(K,k) = R *(K, k)’ + ~*u(K, k)*
(B7)
with R(k,, k) = J’ & CJ(IC, k) A. 0
Note that (B6) is the result of the KMS condition. For real k,, we use the following expressions: d(k)“” = I U,(ko) d(k) U,(ko)l”5
Pf3)
C(k)a4 = {r[U,(k,)~(k)
(B9)
and U,(k,)] 5)“‘:
where d(k) = (k, + i 6~)~y (k* + m”) ’ F(k) = ,f d/cPOGk) k _ ;+
@lo)
i 6r .
0
Then (B2) is rewritten as
(2:)” I
-e-)[ U,(k,) F d(k)@(k) ii(k))’ U,(k,) 1(m)n4 /2 (tPllCl5 , (B12) d4kFu5(k; *.*I U,(ko) d(k)- ,I- C(k) U,(k,) I
. d4k F”‘(k;
= j&j
where we have used the relation U,(ko) rU,&o) = 5.
(B13)
The integrand without F is nothing but the full propagator: (B14) Expression (B14) has no singularity at k* = mz (if Im s(k) # 0), indicating that the singularities arising from the pinching of the integration contour are cancelled out. This can be seenalso from the following considerations; the poles of I/[k’ - m2 + i&l
PERTURBATION
AND RENORMALIZATION
373
IN TFD
and I/[k’ - m2 - k] are never mixed in the integration, when the KMS condition is satisfied in each order of the perturbation calculation. In fact, we have G”‘(k) = d(k) C,(k) +.. C,(k) d(k) = U,(k,) d(k) C,(k) . *. C,(k) d(k) U,(k,) = (d+(k))“C,+(k)
... c,+(k)
+ (d_(k))” C,-(k)
(i,(k”)q
.a. C,-(k)
U,(ko)
U,(k,) $
Ulf(ko)
(B15)
with fl (kofj6)*-(kZ+m2)’
-
di(k)=
0316) 0317)
APPENDIX
C: CALCULATION
OF (5.20)
Equation (5.20) can be obtained as follows. We have from (4.4) and (4.5)
(Cl) Here P indicates the principal part. Since P q*-m*
@?* - m2) ealq,: _ 1
P = (\qo/ - o(q))(jq,j + w(q))> s(‘qo’ -0(q))
1
P
=4wo2
lqol-44
4qol -u(q))
[ 1-
pe4ww 1 X @4l) _ 1 - (@J(q) - 1)’ (Is01-e-l)) L
,,,qb - 1
‘qOi;q;(q) + **.
+ ***I
1
(C2)
374
MATSUMOTO
ETAL.
and
440- 47))= 07 Jhoq -pm(q)
(C3)
0
J
470
_pw(q)
(40
-
w(q))
440
-
4q))
=
(C4)
+
q 0
we have
This is (5.20). Another way of deriving result (C5) is to use relation (6.20). We have
Jd4k [Age(k) + dFa(k) E* d;“(k)]
E= d;“(k)
=&
I J
d4k d”“(T, d4k
E= d;“(k)
m:,,(T).
A is the part which contains the thermal distribution (,~=2)B = 12;j4
+ 2&“(k)
(W function. We have
k)
Ea k2 - m(T)’ + i&P
1 1 + (27r)3 d3k (k2 + m(Q2)1’2
J
1 ,oJ~
_ 1’
(C7)
Then A=i 2 &
[(-iy’“‘>o - W”‘>B
Ih=olm:,,=o
1 1 1 4 w(k)-’ ebwck)- 1 -Tw(k)2
peDdd (@w(k) - I)* I .
ACKNOWLEDGMENTS One of the authors (1.0.) would like to thank Professor W. Israel and the Theoretical Physics Institute at The University of Alberta for their hospitality during the summer of 1981. This work was supported by the Natural Science and Engineering Council, Canada and the Dean of the Faculty of Science, The University of ALberta.
PERTURBATION
AND RENORMALIZATION
IN TFD
Note added in proof: The boundary conditions independence of renormalizability K. Symanzik in Nucl. Phys. B190 [FS3] (1981), 1.
375 was discussed by
REFERENCES 1. Y. TAKAHASHI AND H. UMEZAWA, Collective Phenomena 2 (1975), 55; a preliminary form was first presented in L. LEPLAE, F. MANCINI, AND H. UMEZAWA. Phys. Reports 1OC (1974), 151. 2. H. MATSUMOTO, Forts&r. Physik 25 (1977), 1. 3. H. UMEZAWA, H. MATSUMOTO, AND M. TACHIKI, “Therm0 Field Dynamics and Condensed States,” North-Holland, Amsterdam, 1982. 4. I. OJIMA, Ann. Phys. (N.Y.) 137 (1981), 1. 5. R. HAAG, N. M. HUGENBOLTZ, AND M. WINNIHK, Comm. Math. Phys. 5 (1967), 215. 6. See, for example, A.A. ABRIKOSOV, L. P. GORKOV, AND J. E. DZYALOSHINSKI,“Method of Quantum Field Theory in Statistical Physics,” Pergamon, Oxford, 1965. 7. R. KUBO, J. Phys. Sot. Japan 12 (1957), 570; P. MARTIN AND J. SCHWINGER,Phys. Rev. 115 (1959), 1342. 8. H, MATSUMOTO, Y. NAKANO, H. UMEZAWA, F. MAN~XNI, AND M. MARINARO. Prog. Theor. phys. 70 (1983), 599. 9. H. UMEZAWA, “Quantum Field Theory,” North-Holland, Amsterdam Interscience, New York, 1956); N. N. BOGOLIUBOV AND P. V. SHIRKOV, “Introduction to the Theory of Quantized Fields,” 3rd Ed., Interscience, New York, 1980); S. S. SCHWEBER.“An Introduction to Relativistic Field Theory,” Harper and Row, London, 1961; D. LURIE, “Particles and Fields,” Interscience, New York, 1969); K. NISHIJIMA, “Fields and Particles,” Benjamin, New York, 1969); C. G. CALLAN, JR., in “Methods in Field Theory” (R. Balin and Jean Zinn-Justin, Eds.), North-Holland. Amsterdam, 1976. IO. I. OJIMA, unpublished; talk at International Symposium on Gauge Theory and Gravitation (Nara, Japan, August 1982), Proceedings will be published in Lecture Notes in Physics, Springer).
OF PHYSICS
Perturbation
152, 348-375
(1984)
and Reno.rmalization H. MATSUMOTO,
Departmen!
of Phwics,
University
I. OJIMA,* of Alberta,
Received
January
in Therm0
Field Dynamics
AND H. UMEZAWA Edmonton,
Alberta.
T6G 2JI.
Canada
31, 1983
A systematic renormalization procedure used in the perturbative calculation of the real-time Green’s functions at finite temperature is presented. The formalism of therm0 field dynamics is employed throughout, permitting the use of Feynman diagram techniques. The renormalizability by means of the temperature-independent counterterms is proved. causal
I. INTRODUCTION The formalism of therm0 field dynamics developed in Refs. [ 1, 21 is a technique whereby the usual field theory, defined in real space-time coordinates, can be generalized to the case with finite temperature. It has been applied to various problems in the physics of condensed matter [3] and recently has been extended to deal with gauge theories containing indefinite metric in their covariant formulation [4]. The advantage of the therm0 field dynamics formalism can be seen not only in its intimate relationship [4] with the algebraic formulation of statistical mechanics (51 but also in the fact that all the operator relations in the conventional field theory are retained. Furthermore, the Feynman diagram method can be formulated very easily by means of the real-time causal Green’s functions [2, 31, which are expressed in terms of “temperature-dependent vacuum” expectation values. The best merit of this formalism lies in the fact that it allows us to discuss both the time and temperature dependence of physical quantities in a very straightforward manner. Thus, it enables us to treat dynamical phenomena at equilibrium as well as inequilibrium (at least, within a certain limited region around equilibria). A particularly significant application of the therm0 field dynamics is in the study of (not only static but also) dynamical effects in phase transitions and critical phenomena. To prepare for this program, we need to reformulate the standard machinery of the renormalization group in the framework of the therm0 field dynamics. With this motivation, we discuss first in the present paper the Feynman diagram method in the therm0 field dynamics and the problem of the renormalization procedure. Therm0 field dynamics provides us with a relatively straightforward method whereby one can calculate any dynamical quantity through the use of perturbation theory, which is sometimes difficult in the Matsubara method 161. * Research
Institute
for Mathematical
Sciences.
Kyoto
348 0003-49
16/84 $7.50
Copyright All rights
$1 1984 by Academic Press. Inc. of reproduction in any form reserved.
University,
Kyoto
606, Japan.
PERTURBATION
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INTFD
349
After explaining perturbative calculations in thermo‘ field dynamics in Sections 2 and the renormalization procedure in Section 3, we prove in Section 4 that all the ultraviolet divergences of the Feynman diagrams in perturbation expansions of realtime causal Green’s functions at finite temperature can be eliminated by the counterterms prepared at zero temperature provided the theory is renormalizable at T = OK. Though it might be expected intuitively that an increase in temperature does not upset the finiteness of matrix elementsof observables, this is not at all obvious in the usual temperature-dependent Green’s function formalism [6 ] since each term in the Matsubara frequency expansions seems to produce temperature-dependent divergences arising from the higher-order loop corrections. The temperature independence of divergences is shown only after the summation over the Matsubara frequencies is performed. This leads to some difficulty in constructing a systematic proof of the temperature independenceof the divergences in every order of the perturbation expansion. The situation is different in therm0 field dynamics since the Matsubara-frequency expansions are not required. As will be shown in this paper, the finiteness of the temperature-dependent parts can be seen transparently through a systematic formulation of the renormalization procedure. The result is rephased as
renormalizations at different temperatures are related through a finite renormalization. This result may play a crucial role in comparing physical quantities at different temperatures and in discussing the renormalization group equation at finite temperature. The cancellation mechanism of temperature-dependent infinities is illustrated explicitly up to the order of two loops for the simple case of the A#” theory in Section 5. In Section 6, the renormalization procedure at finite temperature which does not require any zero-temperature calculation is presentedexplicitly by meansof the same model as the one in Section 5. Section 7 is devoted to some concluding remarks.
II. FEYNMAN
DIAGRAM
METHOD
In this section, we briefly present the essential points of therm0 field dynamics relevant to our purpose. To avoid unnecessary complications, we restrict ourselves to a real scalar field Q(X) whose dynamics is specified by a Lagrangian density P(W)>. In therm0 field dynamics / 1 - 41, a field theory at the finite temperature is formulated by doubling the freedom of fields. To each field Q(x) is associated another field called a tilde field 6(x). Then form a thermal doublet Q”(x) as
5951152/2-S
@“(x) = G(x)
(a= 1)
= 6+(x)
(a = 2).
(2-l)
350
MATSUMOTO
The dynamics of @“(x)
ET AL.
is governed by the thermal Lagrangian
2(x)
given by
L?(x)=x &L”9m(X),
(2.2)
a
where
K?(x) = W@(x)) = L22)
(a= = P*(cqx))
1) (2.3)
(a = 2)
and Ea= 1 z-
(a= 1
1) (2.4)
(a = 2).
In (2.3), the asterisk indicates the complex conjugate. The representation is requred to satisfy the tilde operation rule, which is defined for arbitrary operators 0, and 0, (or 0) by (2.5a)
0~~==~,8,, c~o-~=cI*o,
+c2*6,,
(2.5b)
where c, and c2 are c-number and o==qo
(2.6)
with q being a phase factor. The temperature-dependent vacuum 10(/3)) is annihilated by the physical annihilation operators a@ k) and a’& k):
4 These physical operators
k) IWO) = a’@,k) INO> = 0.
are related to the quasi-particle
(2.7)
field Y”
satisfying
P-“(x)= 0 [$-cd--iv) I through the Bogoliubov Y”(X)
a”(k)
transformation:
= (2n:3,2 1 Jt:k)
= Ugy(w(k))
[a”(k)
aY(j, k), sinh B(k) cash B(k)
eik’x-iw(k)t
+ a”+(k)
.
emik xtrw(k)f],
(2.9)
(2.10)
PERTURBATION
The Bogoliubov
AND
RENORMALIZATION
351
IN TFD
parameter B(k) is related to the temperature through the relation (2.12)
sinh’ 0(k) = eQWC,!)_ I. A master formula controlling the relation between the tilde operation temperature-dependent vacuum is the tilde substitution rule [ l-41 :
l?fb(H-g)d )O(p))= ot )O@)),
and the
(2.13)
where H is the Hamiltonian of the system, r? is its tilde conjugate and 0 is an arbitrary operator. It has been shown [ 1,3,4] that the substitution rule (2.13) is equivalent to the Kubo-Martin-Schwinger (KMS) condition [ 71. It is straightforward to formulate the perturbation theory at finite temperature. The perturbation theory can be developed by regarding 5?(x) as the given Lagrangian density for the field W(x) and by considering the condition of the tilde operation rules and substitution rules (i.e., KMS condition). In the interaction picture, the Lagrangian is separated into a free part &x) and an interaction part g(x): (2.14)
52(x) = L&(x) + s&x) with
where Y”(x) is the field in the interaction picture and w(k) is the physical energy of quasi-particles (which is usually temperature dependent). The time-ordered product of Q”(x) is given by the usual formula
= (Team
... Y”n(x,) exp(i J d4x @(x)})4 Vexpti S d4x ~GW, ’
(2.16)
where ( )D is the temperature-dependent vacuum expectation value in the interaction picture. The Gibbs statistical average of physical quantities is given by the (ai = 1) components of the above expectation values. The free propagator for -i”“(x) satisfying (2.8) is evaluated by the use of (2.9~(2.11) [2,3]:
(T,‘%“(x) sayy))il =&J day(k) = [U,(w(k))
d4k eik-y) Llyk), r[ki - (w(k) - itk)‘]-’
(2.17) U,(w(k))]“Y,
(2.18)
352
MATSUMOTO
where U,(o(k))
ET AL.
is given by (2.11) and r is given by (2.19)
We can write (2.18) in the following A”b(k)
form: = A;“(k)
(2.20)
+ A;“(k),
where A,,(k) = r[ki - (o(k) 4@)
= -2rci4ki
- i &)‘I-
- W(k)2)
(2.21)
e51k,l/2 1 )’
,,,k,,; _ 1 ( ,,,&2
(2.22)
Note that A,,(k) is diagonal and depends on temperature only through w(k), while A,(k) has nonvanishing off-diagonal components and a factor damping exponentially at high energy. Other expressions for A*“(k) which we wil find useful in the following work are (2.23) A(k) = UAkJ d(k) U,(k,) = WlkI)~W
(2.24)
u,(lkI)
with (2.25) (2.26) U(K,
k) =
E(K)
6(K*
-
co(k)‘).
(2.27)
The Feynman diagrams are constructed by means of the propagator 4”’ and vertices associated with 2, in the usual manner. This rule can be extended to cases of complex fields, fields with spin and gauge fields. It should be noted that, contrary to Matsubara’s method, the Wick theorem holds true in the therm0 field dynamics in a manner precisely analogous to conventional field theory. A close relation does exist, however, between therm0 field dynamics and imaginary-time formalism of Matsubara treated by means of perturbation theory. This relationship was discussed in Ref. [S]. The perturbative calculation method and the renormalization prescription based on the Feynman diagram method in the therm0 field dynamics is analogous to that employed in conventional quantum field theory. ’ However, there appears a new situation in the case of therm0 field dynamics for the following reason. The quasiparticle states constructed at finite-temperature turn out to be unstable due to the ’ At T= OK (p= l/k, T+ co), the therm0 standard quantum field theory.
field
dynamics
reduces
simply
to the duplication
of the
PERTURBATION
ANDRENORMALIZATION
353
INTFD
thermal instability. Since the free propagator for the internal line should be that of a stable particle, the instability shows up only through the full propagator A’“Y(k). Another difference between therm0 field dynamics and the usual quantum field theory appears when one considers chains of propagator A(k) and self-energy C(k) with the same energy momenta k. This creates powers of A(k), causing multiple pole singularities. Particular care must be taken in treating in order to demonstrate that these multiple pole singularities do not result in any catastrophic behavior. This will be proved later.
III.
RENORMALIZATION
In this section, we illustrate, using the A@-model, how the renormalization procedure is performed in the perturbative calculation of the therm0 field dynamics. The Lagrangian for the scalar field Q(x) is
Y(x)=+- [(a,@(X))‘-p’@“(x)] --&A@“(x).
(3.1)
The thermal Lagrangian is given by (2.2) with Pa(x)=* The thermal Lagrangian given by
[(a,@a)2-p”@a*(x)] g(x)
+P”(x).
(3.2)
is separated as (2.14); 5&x) is the free Lagrangian
~o(x)=~&,~[(a,Y(X))2-m:~a2(x)]
c% and g,(x) is the interaction Lagrangian including the renormalization m
(3.3) counter terms:
= - s E, g, Y4(x)/4!
a
+ x - + Pyx) 6p2y’“(x) + + sz((a, 9’“(x))” - m:,P’(x)) a 1
-& SZ,sc”“(x> I. Here mR is the renormalized and Y”(x) the renormalized
(3.4)
mass parameter, g, the renormalized coupling constant field. The renormalization constants are defined as
@“(x) = z”2P(x), A =g,z,z-2,
(3Sa) (3Sb)
354
MATSUMOTO
ET
AL.
6/L*=[p*-m;]z,
(3Sc)
6Z=Z-
(3Sd)
1,
6Z,=Z,-1.
In Appendix A, it is shown that (3.3) is the only possible choice for @ in the perturbative calculation. The renormalization conditions are given by
Red’-‘(k)“” Ik;=m;+~.k=k8 = 0,
(3.6)
=& a )
(3.7)
$Red’-‘(k)“” 0
k;=m;+k;,k=k,
and
where F~,‘~2~~~~4is the irreducible four point vertex. Conditions (3.6F(3.8) are straightforward generalizations of the renormalization conditions used in the usual zero-temperature field theory. The perturbative calculations thus become similar to those in zero temperature. The differences occur due to the appearance of the thermal distribution functions in propagators and the fact that the renormalizaton conditions refer only to the real parts of the vertex functions, although, due to the thermal effects, the vertex functions are in general complex. Since +. in (3.3) gives a free propagator with the same high momentum behavior as the corresponding zero-temperature propagator (the thermal distribution functions do not modify the high momentum behavior), the renormalizability can be determined by the usual power counting method used in the estimation of the ultraviolet divergences. Therefore, it is obvious that when the theory is renormalizable at zero temperature, the theory is renormalizable also at any finite temperature. What is not obvious is the statement that the divergences appearing in the theory are inependent of temperature. If this statement is true, it means that the characteristic prameters at certain temperature such as mR and g, can be written in terms of those parameters determined at another temperature and that these two sets
of parameters are related to each other through a finite renormalization. This property may play a crucial role in the renormalization group aproach to the critical phenomena. Also, it means that, in considering a system at temperature T, instead of renormalization conditions (3.6)-(3.8) at T one can use the renormalization conditions calculated at a certain fixed temperature To in order to obtain finite answers for the various observable quantities at T. We will refer to the former renormalization procedure as the T-renormalization and the latter one as the Torenormalization (zero-renormalization when To = 0). The temperature independence of divergences will be discussedin the next section. We end this section by noting that although the energy of the free field associated with the free Lagrangian (3.3) has the covariant form dk* + rni , this energy does not necessarily be equal to the physical energy E(k) determined from Re A’-‘(~c)~~ = 0; the quasi-particle energy is not covariant as will be shown later.
PERTURBATION
IV.
AND
THE TEMPERATURE
Elimination of Ultraviolet Temperature
RENORMALIZATION
INDEPENDENCE
35.5
IN TFD
OF DIVERGENCES
Divergences by the Counterterms
determined at Zero
Due to the loop corrections in the perturbative calculation, the Green’s functions given by (2.16) suffer from ultraviolet divergences. In this section, we will show that divergences at finite temperatures can be eliminated by means of the counter terms which are prepared by the renormalization at zero temperature. In other words, we can renormalize divergences in the finite-temperature theory by means of the temperature-independent renormalization counter terms. We try to express the Green’s functions at finite temperature in terms of the renormalized coupling constant g and the renormalized mass m calculated at zero temperature and to expand them in power seriesof g. This calculation is essentially the perturbation calculation with
rni = m2,
g,
=
g’
(4.1)
in (3.3) and (3.4). The renormalization conditions (3.6)-(3.8) are required at zero temperature and the renormalization factors 62, dZ, and 8~’ are those at zero temperature. The free Lagrangian &$ leads to the propagator (2.20) with w(k)’ = k2 + m2; ~~~~=U,~I~,o~2~m:+i6rUx~/~,o =4(k)
(4.2)
+ d,(k)
(4.3)
with
d,(k)=
5
(4.4)
k2-m’+i&
d,(k) = -2ni 6(k2 - m2) e,,k0;
_ 1
1
eDlkol12
eDlkollZ
1
1 .
(4.5)
In this way, we can formulate a perturbation theory at T # 0 in which the internal lines are given by d(k) in (4.3) and interaction vertices given by @, in (3.4). As will be demonstrated later, the physical quantities at finite temperature can be shown to be finite. This renormalization method is the zero renormalization (i.e., T,renormalization with T, = 0). It is evident from (4.2) - (4.5) and 2, in (3.4) that the usual power counting method [9] for the estimation of the ultraviolet divergences should also work in the present situation. Therefore, we need to examine the ultraviolet divergences only in
356
MATSLJMOTO
ET
AL.
the one-particle irreducible (lP.1.) diagrams. Our proof makes use of mathematical induction with respect to the number L of loops.’ In the case of L = 0, the Feynman diagrams have no loops and, therefore, no infinities. Assume now that all of the n-point lP.1. vertices I-, have been made finite up to (L - l)-loop order by means of the counter terms determined by the primitively divergent graphs at zero temperature. We then consider r,, at L-loop order. Using the skelton expansion and the power counting method [9], we can easily see that the m’s other than primitively divergent ones may contain only divergences arising from rm’s (m < n) embedded in r,, as their (proper) subdiagrams which should have loops less than L and hence which have already been made finite owing to the above assumption. Therefore, the only new divergences at the L-loop order are those created by the final integration of the primitively divergent T,,‘s. According to (4.3), the internal lines in these m’s are d, + A,, all of which should belong to certain loop integrals because our diagrams are one-particle irreducible. Let us make a functional expansion of the T,‘s in terms of A,. Since A, given in (4.5) has an exponential damping factor and it is restricted to the mass shell (i.e., k2 = m’), loops containing A, do not diverge at high momenta. This means that the substitution of A, into a line in a Feynman diagram is equivalent to cutting and opening this line as far as the divergence of the loops is concerned. Owing to this observation as well as the fact that we consider here only the primitively divergent m’s, we need to consider only the L-loop order diagrams which are constructed using the propagator A,. The divergences created by A, can be renormalized by the counter terms set at zero temperature since A, and the coupling constant g are independent of temperature. The renormalization prescription now becomes the same as the one at zero temperature. Thus, we have proved that all the ultraviolet divergences in the Green’s functions are renormalized by the counter terms introduced at zero temperature. There remains only one problem originating from the multiple-pole singularities in the propagator A’(k)a4. In the perturbative expansion, there can appear chains consisting of the propagator A(k) and the propagator self-energy Z(k) with fourmomentum k. This creates multiple poles at k2 = m2. In the zero-temperature case, are circumvented by the presence of the those multiple-pole singularities infinitesimally small imaginary constant ic in [k2 - m2 + i&l-‘. which allows the Wick rotation of the integration contour without introducing any pinching of the contour. However, in the present case, [k2 - m2 + ie] -’ and jk2 - m” - ie] -’ are mixed through the thermal distribution matrix U,(i k, I) and, naively, it would seem that pinching of the integration contour must occur, resulting in new divergences due to the coincidence of singularities. Since these divergences originate from the expansion of A’ in powers of A, these are, of course, false divergences; they should
’ Note that, in (3.4), we do not make the normal ordering prescription loop-wise renormalization. This is because the operator ordering made additional contact contribution at finite temperature.
for L$ in order to assure the at T= OK gives rise to an
PERTURBATION
AND
RENORMALIZATION
IN
351
TFD
not show up in the final expression for A’. The proof of this is presented in Appendix B.
V. THE ZERO RENORMALIZATION In this section, by using the 1Q4-model we illustrate explicitly the cancellation of the temperature-dependent infinities. The interaction Lagrangian 2, is given by (3.4) with g, = g, mR = m. Here g and m are the renormalized coupling constant and mass at zero temperature, respectively. The zeroth-order propagator is given by (4.2F(4.5). The Feynman diagrams are constructed from the propagator A”?(k) and the fourpoint vertex gsa@l’ “u4, where 6al”‘u4 = 1 only for a, = a, = a3 = a4 and otherwise aal”‘a4 = 0. The two point lP.1. vertices r, is the self-energy E. At the one loop order it is given by d4k (A;“(k)
+ A;“(k))
- S,i.& - (p’ - m*) 6Z,,,
1
rn); (5.1)
which can be made finite by the choice
a& = “&j 2 P7c) d4kk2_ A2+ iE, dZ(,, = 0.
(5.2b)
Note that this choice is independent of temperature. We obtain the finite result
=
r"ygm*F(pm)(Er"ym;)
(5.3)
with (5.4)
In (5.3) o(k) is given by w(k) = v&%? and f,(w) is the boson distribution function f,(w) = l/(e”” - 1). Next we calculate four point vertex r,(=r~;.‘.‘.‘~) at the one loop order. By writing r4 as
358
MATSUMOTO
ET AL.
where y is illustrated diagramatically in Fig. 1. Since the interaction (3.4) does not mix 9’ and .Y* and since the off-diagonal mixing terms of the propagator, A,, =A,, =A,mv damps exponentially at high momentum, the possible divergent contribution to y in the one loop order comes only from the diagonal part, the origin of which is the loop constructed by A,,. This divergence is cancelled by the counter term g6ZIca1PI’~‘aa:
-&T&
2m
i
y
. d4q
ca A;“(q)
A;“(q
- kj - ki) lkz=,+
(i,j)(k.O
(5.7)
where the summation C,i,j)(k,l, is over all possible pair that can be constructed from (i;j; k; 1= 1, 2, 3,4). The real part of y is given by .d4q
ca A;“(q)
%
A;“(q
- ki - kj)
(i,]yk./)
2~~ A;“(q)
A;“(q
- ki - k,J + dZ,
I
. (5.8)
We denote the integrals in (5.8) by
Mk)= Rec2Lj4 I d4q 6’
FIG.
1.
Feynman
diagrams
A;“(q)
contributing
A,““(q - k)
to the vertex
correction.
(5.9)
PERTURBATION
AND
RENORMALIZATION
359
IN TFD
and
Z,(k)= Re-c2i)4I dq
Fa d;“(q)
Ly(q
- k).
(5.10)
Note that I,(k) is nothing more than the one loop correction at T = OK. In (5.8) the divergences in I,(k) are cancelled by the counter term 6Z, : (5.11)
-6Z, = $ (W, + k,) + Z,(k, + kJ + I,,@, + k4))I++ We now define &(k*)
(5.12)
= Z,(k) - Z,(O).
This in the region k2 < 0 gives k2 - 4m2
2 dk2(k2
~Fyim2j)j
- 4m*)
1. (5.13)
The function lb(k) is a finite integral: 1 ‘dk) = (2# .* 1 = 16rc* Ik\ I ,,, &J&J
log
(k2 + 2 Ikl dm)*
(k2-2)kJd&%?)2-4q:,k;
For the later convenience, we choose the subtraction -k, = -k, = (m, 0). We can rewrite the real part as Re $I.“‘,“”
I .. 4
= Ea~S”I”’ a4 f
- 4q;k:, ’
(5.14)
points in (5.8) as k, = k, =
(f(k, + k2) + f(k, + k3) + ?(k, + k,) - fo(4m’)), (5.15)
where T(k) = fo(k2) + 21,(k).
The imaginary
part of y is finite.
(5.16)
360
MATSUMOTO
FIG.
2.
Feynman
diagrams
ETAL.
contributing
to the self-energy.
We now proceed to the two loop order. The two loop contribution energy part is shown in Fig. 2. We have d4q d41A”O(q) AaD d4q d‘vdyq)
to the self-
Au4(k - q - I) Ayq)
ea doa
- [d/i;,, + 6Zo,(k2 - m”)] B“ P”
LjdqAaacq)
+ Eniv +Z -;
gc”
p
l(I) (2??)4 -
(2:)"
. d4qAaa(q)Aao(q) J
Ea scl:,,,
(5.17)
where the index a should be summed over. We can rewrite (5.17) using (4.3), (5.2) and (5.7) in the following form: ,Et2,(k)“4 = rmb
[kg2
(pkj-4
j21
- @P:,, + (k* -m”> =d
d4q d41A;*(q) A;“(Z) A;*(k - q - f) ca
+ ~Zu,, k:u
1 d4ZA;“(1 - q - k) A;“(l)
+Gg
(2i)4
I
d41ca A;“(Z) A;“(l) E’ + 6Z(,, EO I
+r”“&?’[&J’j
d4q d41A;“(q) A;“(Z) A;“(k - q - I) ca
PERTURBATION
AND
RENORMALIZATION
d4q d4rA;b(q) d4q (ED d;“(q)d;“(q)
361
IN TFD
Ll;“(l)Ll;b(k
- q - I)
++lj’“(q)A;a(q)
PI.
(5.18)
The divergence in the first term of (5.18) is cancelled by 8,~:~) and 6Z,,, since it is the same as the ones appearing at zero temperature. The second term can be amalgamated into the integration over the vertex function at T = OK:
1 (2i)J
1 Py;Y&-JT= OK). Id4qA;“(q) 1
This is finite. The last integration
d41 e“ A;“(l)A;“(l)
F’ + 6Z,,, E’=
(5.19)
in (5.18) is calculated as (see Appendix C).
(5.20) with
(5.21)
The above example explicitly shows that, due to the exponential damping of A, and the nonmixing nature of the elementary vertices given by @, in (3.4) the divergences apear only in those loops which consist of A, only. Since A,, is independent of temperature, so are those divergences. Thus the divergences at finite temperature are cancelled by the counter terms prepared at zero temperature. The cancellation of divergences in the vertex function including the two loop correction can be shown in a similar manner.
362
MATSUMOTO
ET AL.
The real part of ,X;;‘f(k) is diagonal and may be summarized
Re X;;lf(k) = ta4
2,&k)
1 + + g2 Yl (2x1
as
d4q 6(q2 - m’)
(5.22)
where f0C2j is given by
2,,,,,(k)
= Z&t’)
- &(m2)
- (k2 - m’) &
E,(k’)
lkz-,,,j
(5.23)
with
and &, given by (5.13), I, by (5.14), F by (5.4) and F, by (5.21). The renormalization scheme (the zero renormalization) in this section leads to a renormalized inverse propagator and vertex of the following form: A’-l(k)ay=
(k2 -m*)?‘-P’(k),
p k,. I
(pl.“Q
““Qzg(pl .k4
(5.25) +
y;,‘,yyy.
(5.26)
Then the renormalization conditions (3.6~(3.8) determine the characteristic parameters mR (which will also be denoted by m(T)) and g, (which will also be denoted by g(T)) at finite temperature. If we choose the renormalization point k, = 0, m(7') is the rest energy of quasi-particle state. In the remaining part of this section, we will discuss the relation between (g, m) and (g(T), m(r)). The renormalized propagator and vertex satisfying (3.6t(3.8) with k, = 0 (the Trenormalization) are denoted by d’(T) and Z(T), respectively. If we denote the unrenormalized propagator and vertex by Ah and Z, respectively, then we have (5.27a) and T,(k)
= Z-*l-(k)
= Z(T)m2 I-(T, k),
(5.27b)
PERTURBATION
AND
where Z(T) is the renormalization have the relations A’-‘(T,
RENORMALIZATION
constant
in the T-renormalization.
Therefore we (5.28a)
k) = (Z(T)/Z)Lr’(k),
I-(T, k) = (Z(T)/Z)2 Since the quasi-particle
363
IN TFD
energy o’(T,
(5.28b)
T(k).
k) is determined by
Red’-‘(T,k)=O,
(5.29)
we have from (5.28) w*(T, k) = k* + m2 + Re E”(w’(T,
k), k).
(5.30)
Note that Re C” = -Re ,Z22. From (3.6) - (3.8), we have m(T)* = m2 + Re E”(m’(T), 0) (Z(T)/Z)-’
= MT)/Z)*
(5.3 la)
0),
g(1 + Re Y&‘.‘~~ Iki=0,k,o=k20= -k.
=&Red’-‘(k)” 0
+nuJ~
lk;=m~cr.),k=~.
(5Jlb) (5.3 lc)
From the results of (5.3) and (5.22), we have Re C”(w2(T,
k), k) = gm2F(pm) + g2m2F7’Jm) F,(pm) + g2W
G(k)=l+
d4q.t&W
2 PI + I,(q
t kf - i &(4m2)
l,+,~ + O(g%
4q2 -4
I
I
(5.32)
f&q f k)')
.
(5.33)
The functions & and I, are given in (5.13) and (5.14), respectively. Since G(k,, k) is not a function of k*(= ki - k2) only, G(Sk2 + m2, k) shows a complicated kdependence and therefore o*(T, k) also shows a complicated k-dependence. A low momentum expansion leads to w2(T, k) = m*(T)
t a(T) kZ + O(k4)
(5.34)
with m*(T) = m*[ 1 t gF(pm)
t g’(FQ3m) F,@m) + G,(Pm)) t 0( g3)]
a(r>= 1+ g*F@m)+ G2CB41 + O(g3),
(5.35) (5.36)
MATSUMOTO
364
ET AL.
where (5.37a)
cl(&) =-&G(k)I&z.k=O,
(5.37b)
0
Gz(Pm)
=
Y$
G(k)
(5.37c)
jk6=mz.k=0.
The functions Go and G, have the following integral forms: Go(z)=&+&
I
e
1 cG 647~~1, dx@
+-
(xlog(X+~=I-fi~}
1
1-1’*+2L/(x*-l)(+l)
(ez” _ l)(eZ” _ 1) b
l-y2-2J(x2-
l)(+
1) ’ (5.39)
The ratio Z/Z(T)
is obtained from (5.31~) and (5.32): -W(T)
= 1 -g2 -$
G(k) Ik;=m>.k-o+ W’) 0
= 1 -glG,(pm)
+ O(g’).
(5.40)
The coupling constant g(T) is obtained from (5.151, (5.3 lb) and (5.40) as
g(T) = g( 1+ g(Z,(2m 0) + 21,(0,0)) + m*N.
(5.41)
From (5.14), we have (5.42)
VI.
T-RENORMALIZATION
In the previous section it demonstrated that all of the divergences occurring in the 14” theory are independent of temperature and the renormalization prescriptions outlined in the previous sections were extremely useful in establishing this fact. Once it is establishedthat there is indeed no divergences which depend on temperature we can formulate the renormalization scheme at any temperature. Since it is the finitetemperature renormalized quantities which are related directly to the observable quantities, this latter schemewill be much more useful in practical calculations. Since
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the change of the renormalization conditions from zero temperature to finite temperature does not introduce any new divergences, renormalization schemes at each temperature are related through a finite renormalization. This renormalization scheme may lead us to a useful tool in the renormalization group approach to the study of the temperature dependence of dynamical quantities. In order to perform the renormalization at finite temperature, it is convenient to write $0 and g in (2.14) as 2, = gc $ :I
Ea +
[(a&y”)’
- m*(T)
Y”*]:
a - $)
*I=:y&a n
.ya4
+ SZ(T)i
[(a,Y)’
- m’(T)
YQ2]
1
g(T) -,az,(r)~.“-r~~‘(T)Y”’
:, I
where gC is a c-number and the normal product refers to the temperature-dependent vacuum. The mass counter term is given by
6/f*(T) = /I2-m*(T)f JpZ*(T) z(T)-’ (Ya2)D)Z(T), (
(6.3)
where the vacuum expectation value ( ). refers to the temperature-dependent vacuum. The renormalization conditions are given by (3.6)-(3.8) with rni = m*(T) and g, = g(T). We will choose k, = 0 in this section. The zeroth-order propagator is given by
The perturbative expansion can be performed by the use of the interaction Lagrangian (6.2), regarding g(T) as a small expansion parameter. We calculate the self-energy correction and vertex corrections. The self-energy part E(T, k) and the vertex correction y(T) are defined respectively by A-‘(l’,
k)ay= (k* -m*(T))
ray-Zay(T,
k)
(6.5)
and W-7 ;;:::;
= g(T)[&“’
6”l” ‘*‘I + y(T);;:::ka4”].
(6.6)
Since there exist no tadpole graphs, the lowest order self-energy diagrams and vertex corrections are the second order in g(T) (see Fig. 3). We obtain (c.f. (5.17) and (5.6)) C(T, k)“Y=$g(T)2 - [&*(T)
eagY [&I2
1 d4q d4ZAay(T, q) Amy(T, Z)Aay(T, k - q - 1)
+ 6Z(T)(k2 -m’(T))]
t”Y,
(6.7)
366
MATSUMOTO
FIG.
ET AL.
3. (a) Self-energy diagrams.
(b) Vertex corrections.
x d”Y(T, q)d”Y(T, q - ki - kj) + P’ d=‘“‘=4 6Z,(T).
(6.8)
The counter terms &‘(T>, 6Z(T) and 6Z,(T) are fixed by the renormalization conditions (3.6)-(3.8) and are obtained as functions of g(T) and m(r). Once the counter terms are fixed as functions of g(T) and m(r), the relations of parameters at different temperature can be obtained as follows. Let us consider two temperatures T and T,, and their parameter sets {g(T), m(T)} and (g(T,), m(T,,)}. From (6.3) and (3Sb), we can see that they are related to bare parameters as ,u* = m’(T) -y
Z,(T) Z(T)-‘(Lsp”2)D - Z(T)-’
=m’(T,)-F
Z,(T,)Z(T,)-‘(~*),~--Z(T,)-‘~P*(W
A= S’-7 Z,(T) Z-‘(T)
= g(TJ Z,(TJ
Z-*V’J.
&*(T)
(6.9a) (6.9b)
Since the relations in (6.9) contain divergent integrals, their definition requires a cutoff in the momentum (say/i). From (6.9) we can obtain the following relationship between the parameter sets at T and T,,:
WYWJ
g(T) = F&T, To ; g(To), mV,))>
(6.10a)
W’l
= FmK T, ; @cd, NT,)),
(6. lob)
= FAT, Toi O-o), W’d)-
(6.10~)
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Since the renormalizations at T and To are related through the finite renormalization, relations (6.10) are well defined in the limit A + co. In order to demonstrate the consistency of these results with those in Sections 5, we calculate the (g, m)-dependence of g(T) and m(T) (i.e., relations between parameters at T # OK and those at T = OK). In Section 5, the mass counter term was defined as (6.11)
6,u2=[m2-p""]Z since the Lagrangian
is not normal ordered. From (6.9) and (6.1 l), we have g(T) + ,-Zl(T)Z(T)-‘(9”‘),
m’(T) = m2 - &‘Z-’
f 6p2(T) Z-‘(T)
(6.12a)
and g(T) = gZ,Z-2Z,(T)-1
Z(T)‘.
(6.12b)
Note that the lowest-order contributions to 6p2(r), SZ(r) are second order in g(T) and those for dZ,(T) are the first order in g(T), while the lowest-order contributions to S,u2 and 6Z, are first order in g and those for 6Z are second order ing. We will denote the order of g by the subscript (n). From (6.12), we have m2(T),,, = m2,
km,,,
(6.13)
=g-
The next order is easily obtained as
m’(T),,, = = -&fl,
g(%,
= d%t,,
+ “Aj 2 PI
d4kd”“(T,
k),,,
(6.14) (6.15)
- =,(T),,,).
Comparing (6.14) with (5.2) and (6.15) and (6.8) with (5.7), we can see that these resuls are simply (5.35) and (5.41). The second-order term, m2(T)c2j, is obtained from m’(T),,,
= -&f2)
mc2, +2
(.;Y’a2)Dco,+ f
(Ly’a2)Dcl, +
&‘PJ,,,
368
MATSUMOTO
ET AL.
where relation (6.15) was used. We have
J. d4k d”“(T,
k)(,,
(6.17)
+ . ..I”.
(6.18)
with d(T, k) given by (6.4). Note that d(T, kY=
[d(k) +d(k)m;,,
td(k)
Therefore
J- d4k d*“(k),
w%,,, = (& I d4k d”“(k)
(6.19) m;,,(T) f’ d”“(k).
(6.20)
Note also that d4kAa”(k)daa(k)
Pmf,,(T)
I (2)
d4k d41d”“(k) A”“(k) PAua(l) --
1
i. d’k d”*(k) d”‘(k) IZ=Sp;,, , 2 g (27q !
(6.21)
where (6.14) was used. Combining (6.7), (6.16) and (6.21), we can see that mf2, (T) obtained from (6.16) is identical with that obtained from (5.17).
VII. CONCLUDING
REMARKS
Making use of therm0 field dynamics, we have shown that when a quantum field theory at T = OK is renormalizable with certain counter terms, it is renormalizable at T# OK with the same counter terms. We also presented a scheme for the Trenormalization, which helps us to calculate the temperature-dependent energy and coupling constants. The renormalizations at different temperature are related through a finite renormalization. In contrast to the usual imaginary time formulation for the temperature Green’s functions [6], the proof of the renormalizability in the therm0 field dynamics is both transparent and systematic. In the case of gauge theories, therm0 field dynamics helps us to derive, from the BRS symmetry obtained in Ref. [4], the Ward-Takahashi identities at finite temperature. Since these identities have the same form as those at zero temperature,
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INTFD
all the considerations in this paper can be extended to gauge theories at finite temperature. Though we started from a Lorentz invariant Lagrangian, the result of the perturbative calculation are not Lorentz invariant because the energy of quasi-particles at T # OK is not of the form dm (see (5.30)-(5.34)), implying that the wave equation of a quasi-particle field is not Lorentz covariant. We expect that this noncovariant nature originates from the thermal instability of the quasi-particle state. Note that this instability is caused by the fact that in therm0 field dynamics the continuum states extend into the negative energy region [lo]. Let us close this paper by the following note. The temperature-dependent mass m(T) is expected to become soft at a critical temperature, m(T) -+ 0 for T+ T,. This is becausethe massscale should disappear and a long-range force appearsat T,. This mechanismwill be studied in a forthcoming paper in conjunction with a study of the renormalization group in therm0 field dynamics.
APPENDIX
A: CHOICE OF THE UNPERTURBED
PART.?&
Let us first chooce p0 as (1 a 2 [ (f
-,4”‘“(x)
w;(4)
(Al)
?‘(x)],
Then gr(x) is given by 9,(x) = - 2 E, ; a
g, Y”(x)
+ s E, - ; i a
.9”(x)
Sp’(-iv>
++z ((-$yx))’- .!P(x) 0;(-iv)
Z.
(‘43)
One may be tempted to use as the above uR(k) the quasi-particle energy E(k) defined by Re A’-‘(k)aO Ik;~E2ck,= 0.
(A4)
(Here the real part of A’-‘(k) is taken because it generally has a finite imaginary part due to the thermal effect.) We will show that the above choice wR(k) = E(k) is not a convenient choice in
370
MATSUMOTO
ETAL.
perturbation theory. In the course of the calculation of d’(,r~)“~, &‘(k) perturbatively from condition (A4). Then from (A3), E*(k) is given by E*(k) =p* t k* -Z-l
is obtained
l@*(k).
WI
If E2(k) is considered to be the zeroth-orders contribution (i.e., OR(k) = E(k)), then all of higher-order contributions to the counter term Z-’ d,u*(k) must be cancelled by $, which is possible only when 6p*(k) is independent of k. When &*(k) dependson k, a convenient way is to fix the zeroth-order massparameter from the energy at a momentum k,, i.e., k; + m:, = E*(k,),
646)
and use wR(k) = dm. Then we can calculate E*(k) iteratively as a function of g, and mRby use of (A5). Denoting the nth order of g, by the superscript (n), we have E*(k,)‘“’
= 6,,(m: + k:),
647)
E*(k)‘“’ = p2(“) t 6,,k2 - (Z-l d,u*(k))‘“‘.
(A81
p*(") = d,,rni t (Z-’ 6,u2(kR))‘“),
649)
From (A7), we have
which leads to E*(k)‘“’ =
6,,(mf, + k2) - (Z-’ 6p2(k) - Z-’ 6p2(kR))‘“).
(AlO)
E*(k) appearing in the expression for Z-’ &*(k) is always one order lower than the left-hand side of (AlO). Therefore E2(k) can be obtained perturbatively. In this way we can write all physical quantities in terms of g, and mR, which are the physical parameters at finite temperature. The k-dependenceof the zeroth-order approximation of E*(k) is always k2. This is due to the fact that the original Lagrangian is Lorentz invariant and that the mass term does not depend on k. When E*(k) is expanded in terms of g,, the perturbative expansion of amplitudes is given by an expansion in powers of (ki - (k2 t mi))-'. This is the perturbative calculation based on the free Lagrangian (3.3) with the interaction Lagrangian given by (3.4).
APPENDIX
B: CANCELLATION
OF THE DIVERGENCES THE PINCHING OF INTEGRAL PATHS
ARISING
FROM
We will show in the following how the divergences caused by the pinching of the integration contour cancel. The divergences under consideration arise only from the self-energy diagrams
PERTURBATION
AND
RENORMALIZATION
IN TFD
shown in Fig. 4, where Zi is the proper self-energy part. The integral following form: -
(2i)4 i
d’kF”‘(k;
. ..)[A(k)L.(k)A(k)Z,(k)
. . . s?Z,(k)A(k)]“D,
371
has the
(Bl)
where F represents the additional part of the Feynman diagrams which do not contain one particle lines with the momentum variable k. When the part shown in Fig. 4 is of mth order (m > n), the amplitude summed over all of the mth order diagrams can be written as &j
d4kP’(k;
. ..)[A(k)
C(k)A(k)
... C(k)A(k)]“4’“‘.
032)
Here superscript (m) indicates gmth order and C(k) = f
C(k)“’
033)
i=l
with Z(k)“’ being the ith order contribution to the proper self-energy part and is assumed to be free of divergences. We have already shown that there exists no ultraviolet divergences. Therefore we investigate only the singularities casued by the multiple-pole arising from powers of AaD( i.e., multiple-pole singularities around k2 N m2.
Since 2?(k)
is related to Arab(k) A’ -I(k)efi
C(k)
through the relation = (k2
-
m’)
rob
- P’(k),
(B4)
has the following spectral representation
z(k) = r j dK0, k) U,(K) k, _ ;+ i 6r u&) 5. Here we used the fact that A’(k)
has the spectral representation
A’(k) = 1 dK C’(K>k) UB(K> k, _ :+
FIG.
(W
4.
i 67 u*(K).
Self-energy-typediagrams.
(J36)
372
MATSUMOTO
ET AL.
p and u are related through the relation
CJ(K,k) P(K,k) = R *(K, k)’ + ~*u(K, k)*
(B7)
with R(k,, k) = J’ & CJ(IC, k) A. 0
Note that (B6) is the result of the KMS condition. For real k,, we use the following expressions: d(k)“” = I U,(ko) d(k) U,(ko)l”5
Pf3)
C(k)a4 = {r[U,(k,)~(k)
(B9)
and U,(k,)] 5)“‘:
where d(k) = (k, + i 6~)~y (k* + m”) ’ F(k) = ,f d/cPOGk) k _ ;+
@lo)
i 6r .
0
Then (B2) is rewritten as
(2:)” I
-e-)[ U,(k,) F d(k)@(k) ii(k))’ U,(k,) 1(m)n4 /2 (tPllCl5 , (B12) d4kFu5(k; *.*I U,(ko) d(k)- ,I- C(k) U,(k,) I
. d4k F”‘(k;
= j&j
where we have used the relation U,(ko) rU,&o) = 5.
(B13)
The integrand without F is nothing but the full propagator: (B14) Expression (B14) has no singularity at k* = mz (if Im s(k) # 0), indicating that the singularities arising from the pinching of the integration contour are cancelled out. This can be seenalso from the following considerations; the poles of I/[k’ - m2 + i&l
PERTURBATION
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IN TFD
and I/[k’ - m2 - k] are never mixed in the integration, when the KMS condition is satisfied in each order of the perturbation calculation. In fact, we have G”‘(k) = d(k) C,(k) +.. C,(k) d(k) = U,(k,) d(k) C,(k) . *. C,(k) d(k) U,(k,) = (d+(k))“C,+(k)
... c,+(k)
+ (d_(k))” C,-(k)
(i,(k”)q
.a. C,-(k)
U,(ko)
U,(k,) $
Ulf(ko)
(B15)
with fl (kofj6)*-(kZ+m2)’
-
di(k)=
0316) 0317)
APPENDIX
C: CALCULATION
OF (5.20)
Equation (5.20) can be obtained as follows. We have from (4.4) and (4.5)
(Cl) Here P indicates the principal part. Since P q*-m*
@?* - m2) ealq,: _ 1
P = (\qo/ - o(q))(jq,j + w(q))> s(‘qo’ -0(q))
1
P
=4wo2
lqol-44
4qol -u(q))
[ 1-
pe4ww 1 X @4l) _ 1 - (@J(q) - 1)’ (Is01-e-l)) L
,,,qb - 1
‘qOi;q;(q) + **.
+ ***I
1
(C2)
374
MATSUMOTO
ETAL.
and
440- 47))= 07 Jhoq -pm(q)
(C3)
0
J
470
_pw(q)
(40
-
w(q))
440
-
4q))
=
(C4)
+
q 0
we have
This is (5.20). Another way of deriving result (C5) is to use relation (6.20). We have
Jd4k [Age(k) + dFa(k) E* d;“(k)]
E= d;“(k)
=&
I J
d4k d”“(T, d4k
E= d;“(k)
m:,,(T).
A is the part which contains the thermal distribution (,~=2)B = 12;j4
+ 2&“(k)
(W function. We have
k)
Ea k2 - m(T)’ + i&P
1 1 + (27r)3 d3k (k2 + m(Q2)1’2
J
1 ,oJ~
_ 1’
(C7)
Then A=i 2 &
[(-iy’“‘>o - W”‘>B
Ih=olm:,,=o
1 1 1 4 w(k)-’ ebwck)- 1 -Tw(k)2
peDdd (@w(k) - I)* I .
ACKNOWLEDGMENTS One of the authors (1.0.) would like to thank Professor W. Israel and the Theoretical Physics Institute at The University of Alberta for their hospitality during the summer of 1981. This work was supported by the Natural Science and Engineering Council, Canada and the Dean of the Faculty of Science, The University of ALberta.
PERTURBATION
AND RENORMALIZATION
IN TFD
Note added in proof: The boundary conditions independence of renormalizability K. Symanzik in Nucl. Phys. B190 [FS3] (1981), 1.
375 was discussed by
REFERENCES 1. Y. TAKAHASHI AND H. UMEZAWA, Collective Phenomena 2 (1975), 55; a preliminary form was first presented in L. LEPLAE, F. MANCINI, AND H. UMEZAWA. Phys. Reports 1OC (1974), 151. 2. H. MATSUMOTO, Forts&r. Physik 25 (1977), 1. 3. H. UMEZAWA, H. MATSUMOTO, AND M. TACHIKI, “Therm0 Field Dynamics and Condensed States,” North-Holland, Amsterdam, 1982. 4. I. OJIMA, Ann. Phys. (N.Y.) 137 (1981), 1. 5. R. HAAG, N. M. HUGENBOLTZ, AND M. WINNIHK, Comm. Math. Phys. 5 (1967), 215. 6. See, for example, A.A. ABRIKOSOV, L. P. GORKOV, AND J. E. DZYALOSHINSKI,“Method of Quantum Field Theory in Statistical Physics,” Pergamon, Oxford, 1965. 7. R. KUBO, J. Phys. Sot. Japan 12 (1957), 570; P. MARTIN AND J. SCHWINGER,Phys. Rev. 115 (1959), 1342. 8. H, MATSUMOTO, Y. NAKANO, H. UMEZAWA, F. MAN~XNI, AND M. MARINARO. Prog. Theor. phys. 70 (1983), 599. 9. H. UMEZAWA, “Quantum Field Theory,” North-Holland, Amsterdam Interscience, New York, 1956); N. N. BOGOLIUBOV AND P. V. SHIRKOV, “Introduction to the Theory of Quantized Fields,” 3rd Ed., Interscience, New York, 1980); S. S. SCHWEBER.“An Introduction to Relativistic Field Theory,” Harper and Row, London, 1961; D. LURIE, “Particles and Fields,” Interscience, New York, 1969); K. NISHIJIMA, “Fields and Particles,” Benjamin, New York, 1969); C. G. CALLAN, JR., in “Methods in Field Theory” (R. Balin and Jean Zinn-Justin, Eds.), North-Holland. Amsterdam, 1976. IO. I. OJIMA, unpublished; talk at International Symposium on Gauge Theory and Gravitation (Nara, Japan, August 1982), Proceedings will be published in Lecture Notes in Physics, Springer).