Aug 15, 1993 - Mark Burgess. Institute of Physics, University of Oslo, P.O. Box 1048, Blindern, 0816 Oslo 8, Norway. David J.Toms. Physics Department ...
PHYSICAL REVIEW D
15 AUGUST 1993
VOLUME 48, NUMBER 4
Is there a phase transition in Maxwell-Chem-Simons Institute of Physics,
Mark Burgess of Oslo, P. O. Box 1048, Blindern, 0816 Oslo 8, Norway
University
Physics Department,
University
Institute of Physics, University
theory?
David J. Toms of Newcastle Upon Tyne, NE1 AU, England
Nils Tveten of Oslo, P. O. Box 10/8, Blindern, 0818 Oslo 8, Norway (Received 28 January 1993)
We reexamine recent work on the Abelian Higgs model (scalar +ED) in 2+1 dimensions. We compute the effective potential at finite temperature using the Vilkovisky-DeWitt formalism and show that there is spontaneous symmetry breaking at one loop. We use an efFective action formalism to take into account ring corrections and find that classically broken symmetries can be restored at high temperature. The presence of the Chem-Simons parameter can hinder symmetry breaking for values of the self-couplings very close to the critical value normally required for a phase transition.
PACS number(s):
11.15.Ex, 05.70.Fh
I. INTRODUCTION Interest in mechanisms which might explain hightemperature superconductivity has focused attention on (2+1)-dimensional physics. Laughlin s suggestion [1] that fractional statistics (anyons) [2, 3] might play a role in such a mechanism has made Chem-Simons theories a special source of interest. One of the important points which must be demonstrated in any mechanism for superconductivity is the presence of a phase transition, defined by some critical temperature, below which the system superconducts and above which it has finite resistance. In the language of field theory such a model will exhibit symmetry breaking: As the symmetry breaks, gauge invariance is lost and superconductivity sets in. Recent work [4] has attempted to shed some light on this by modeling such a phase transition by use of the Higgs mechanism in a system of charged bosons miniterms. This is mally coupled to Maxwell-Chem-Simons a kind of simulated Bose-Einstein condensation (BEC) as in the Abelian Higgs model. Although the Higgs mechanism is a somewhat artificial way of representing BEC, the AP model is not completely without merit since the self-coupling provides the Bose field P with a repulsive interaction which makes it "hard core" [5]. If one is trying to make contact with anyons, this detail is of crucial importance since anyons represented as bosonic particles coupled to a Chem-Simons field must not be allowed to pass through one another or else their statistics cannot be anything other than bosonic. (Although there is no good reason to suppose that the theory described here has anything to do with anyons, it may still partially describe phase transitions in systems where anyons exist. We shall mention this briefly in Sec. VII. ) Another reason for looking at scalar electrodynamics is that AP theory does not undergo spontaneous symmetry breakdown in 3+1 dimensions, whereas scalar electrodynamics does. The model has some diKculties however: The introduc-
tion of a mass-squared term which can be less than zero m 0 at the classical level presents infrared difIiculties for the loop expansion. In addition, there is another diKculty with gauge theories, noted in Ref. [6] for finitetemperature theories and since noted more generally in the context of the effective action [7]. The problem is namely that of gauge-fixing dependence in the effective potential. The Faddeev-Popov procedure for gauge fixing in the path integral is somewhat subtle when expanding about a state in which gauge invariance is already broken (so-called off-shell states). These are precisely the states one is interested in when dealing with spontaneous symmetry breaking. The results obtained in Ref. [4] do not take account of either of these difFiculties, and it is easy to show that they are wrong. In this paper we wish to discuss the correct resolution of at least some of these problems. The problem of gauge-fixing dependence is now well established [7] and is readily cured. The solution is to use the VilkoviskyDeWitt off-shell efFective action which is invariant unand field reparametrizations der gauge transformations as well as changes in the choice of gauge-fixing condition. As a check on our computations we calculate the one-loop effective potential V( ) both in the unitary gauge keeping track of geometrical corrections and in the LandauDeWitt (orthogonal) gauge, where they may be avoided for scalar electrodynamics. Corresponding calculations have been carried out in 3+1 dimensions in [8], where the gauge dependence of the standard prescription is shown explicitly. The second point, the issue of the infrared divergences, is somewhat unfamiliar in 2+1 dimensions. We shall attempt to comment on a plausible scenario for While these corrections, pending further investigation. writing our paper we received a paper which discusses some related issues in (2 + 1)-dimensional [(2+1)D] AP theory [9]. We begin by discussing the gauge-fixing independent in 2+1 dieffective action for scalar electrodynamics
0556-2821/93/48(4)/1808(13)/$06. 00
1808
(
1993
The American Physical Society
IS THERE A PHASE TRANSITION IN MAXWELL-CHERN-. . . mensions. That we need to consider subtleties in the gauge-fixing prescription has been discussed in [7] and demonstrated explicitly in Ref. [8]. The problem is that the standard Faddeev-Popov procedure assumes that the integrand in the functional integral is gauge invariant (apart from any gauge-Fixing terms, which we shall treat as being part of the functional measure). This is incorrect in perturbation theory. The problem does not arise unless we go off shell, that is, when the background field is no longer a solution of the classical field equations. This is precisely what we are interested in, spontaneous symmetry breaking: Radiative corrections can lead to a nonzero condensate of the field. Strictly speaking, at the minimum, gauge invariance is lost and so there is formally no need to fix a gauge; however, if the effective action is to be sensible in the limit (P) 0, then gauge fixing is required. To solve the diKculty in the general case a redefinition of the standard effective action is required. In scalar electrodynamics this correction may be avoided provided one chooses a very special gauge which maintains the gauge invariance of the functional integrand by construction. (This is the Landau-DeWitt gauge, a special covariant form of Rg gauge [8,10].) As a further point we emphasize that the FaddeevPopov determinant is background Geld dependent in the correct approach. In Ref. [4] it is claimed that choosing the Lorentz-gauge condition 0~A~ = 0 and taking the limit o. 0 exempts the authors from taking the Faddeev-Popov determinant into account. This is incorrect. This gives us another reason for tackling the problem from two points of view. The only gauge in which it is correct to take the o, — + 0 limit is the Landau-DeWitt gauge. A gauge in which there is no ghost determinant is the unitary gauge, in which one parametrizes the theory in such a way as to eliminate gauge invariance explicitly before integrating over the gauge fi.elds. Gauge fixing always requires the removal of two degrees of freedom by whatever method.
where dV is the (2+1)-dimensional volume element. We work directly in Euclidean space and follow the conventions used in [11]. The standard Lorentz gauge may be chosen by inserting b[o))'A~]det( — ) into the functional measure. The unitary gauge is effected through
the parametrization 1
II. UNITARY GAUGE In the unitary gauge one reduces a gauge theory to a nongauge theory by integrating out the gauge freedom explicitly. If we begin with a gauge theory, enforcing the Lorentz condition, for instance, then integrating out the gauge freedom cancels the usual Faddeev-Popov determinant in the measure, showing that the gauge freedom is explicitly lost at the level of the Lagrangian. The Faddeev-Popov method is then no longer required since there are no physically indistinct Geld modes left. The most general gauge-invariant renormalizable theory in 2+1 dimensions is described by the action 4
~scalar—
dV
iO
(3)
1 B„=A„+ — 8„0, e"'
(4)
so that the action becomes
F""F— „O„Bp + —(0"p) (c))„p) „„+i e" B—
dV
~
~
1809
1 2 2 ~ 1 2 2 A 4 g — +— epBB„+ mp+ — " — 4l p+ 61 p 2 2
The factor in the functional measure now becomes 1 — 8"B„— Qg e
[d0]b
det( — )
= det(e).
(6)
The Faddeev-Popov determinant is seen to cancel as claixned. The functional integral is now of the form
Z=
cp x
Gp
x 8B~ x c
K)g
so that the one-loop effective action is given by I'1
=
l
2
ln det S, ; [p]
—lndet(ep),
(8)
where p is the background field and S.,~ = S;~ + I'; -S y. The exact expression for the connection may be found in Eqs. (3.69) —(3.73) of [7]. The result for the unitary gauge may be found in [11]. The only relevant term is
I&
(
)~
where p(x,
1
1:
e p( )0x&p(
xx)B
x)
p(x
(9)
x') satisfies (
— +e
The components
p
)p(x, x')
= b(x, x').
(10)
of the one-loop operator are now
S ~ = — +m + —p + —p b(x x'), 4! &P
2
+v
A D"C t D„C + m Ct@+ —, CtC
deep
~m
+ —p +
p
x o)"p(x, y)8„'p(x', y), (2)
where ( —
+e
p
)p(x, x')
=
b(x, x'). The only differ-
1810
MARK BURGESS, DAVID
J. TOMS,
ence between these expressions and those of Ref. [11] is the presence of the Chem-Simons term which does not enter into the computation of the Vilkovisky connection. The parameter v is kept so that, in the limit v 0, the
~
AND NII S TVETEN
connection term and the raising of indices in (8) are removed, giving the result in the standard formalism. We are now in a position to trace over the vector indices. After some work it may be shown that
I
~l
detS. ~" ——[det(ep)] [det( — v
e'p'+
2 ——
P
(
-',
S
[det( —
p )]
, +— lm , + — p + 2 6
2
M2 3 ——ep
M45
(
+e
' + 'I pl(p'+ —.
i=2
g
4b
120 p r
+ M, )],
(13)
v ( A , A g 4) 6 , — g 4 l6 — lm , + — llm + 6 p , + 120 p + p +4ep 120 p r ( 2 6 (
'.
4e'p')
(i4)
r
Hence
i lndet I'! = —
5
2
)
& 1 ~ 4& — — + m + —p + — —lndet( — + M;2 ). lndet( — + e 2 p 2 ) + 2 4! p = 2
1
l
l
The free-Beld limit of this expression may be checked + 0, giving by setting e, A, g —
I
!
1
-+
det( —
+m ) + +-'lndet( — + p ), —ln
(i6)
~
z
ln det( —
+vm
)
(i7)
which, provided one takes the correction term v = 1, gives two massive degrees of freedom from the scalar field and one Chem-Simons massive degree of freedom from the gauge field [12,13].
I'!'„1 =
—Tr ln( — + e'P').
(20)
Using the global invariance of the background field, we are free to choose P = Pr and Pz —0 in order to simplify the calculation. After some purely algebraic manipulations, involving completing the square to diagonalize in the quantum fields, we obtain
m'+-—
(
2
2
r
( Trln — +m'+ —P'+ i+ — e'P +— 6 2 ( n) — 2 2 —Tr ln( — + e Q ) + —Tr ln n 2 l
l
III. LANDAU-DeWITT GAUGE The foregoing results can also be achieved by working in an "orthogonal" class of gauges, known as the Landau-DeWitt gauge [8]. This gauge is chosen in such a way that generators of gauge transformations at the background field are orthogonal to the field interval between the background and quantum fields so that gauge invariance of the exponential in the functional integral is preserved for all field configurations which contribute under gauge fixing. One then represents this condition as a proper h function [14] by taking the n 0 limit of the result. We shall parametrize the field by
——Tr ln (Ah" + BB"0 + ip, e" „8"), 2
—i), and A = — + e P, B = —( —e P 4 = [— + mz + —pz + (1+ —) ezra ]. Tracing over vector
where
Q
indices one obtains z
Trln[A
+ (A B —Ay,
—Bp
)
].
(22)
This is seen to factorize into
~
The generators of gauge transformations may be read off straightforwardly by considering the equation bp' = R[y]* ( (see Ref. [11]). The gauge-fixing condition y(cp, P) is then R[P]* y; = 0, where i runs over all of the integration variables in the action. We set g —+ 0 for the scalar field for simplicity. Noting that 8[y] lim o[P (2vrrr. ) ~ exp( —z y )] (see Ref. [14] for a discussion of this; note that these authors use DeWitt's condensed notation), the gauge-fixing part of the action is therefore
(21)
iTrln(A+ B )(A —p
).
(23)
The first of these terms may be factorized again to give two terms z Trln( — + Mz) and z Tr ln( — + Ms); the latter may be manipulated to give
—Tr ln
t'n( —
+ e'P ) — ( — + m'+
— ', P )
+ e'P )
2
.
~gr
=
1
dv 2''
&p —«ab
a
b
)
(19)
(24) 0, using o.L 2 e 2 P . The remairnng term can, once again, be factorized, leaving the one-loop effective action given by
It is now safe to take the limit
I'!r![&P] = —Tr ln( — + e g() +
o.
)
~ z
Tr ln( —
+ M) ), (25)
where
IS THERE A PHASE TRANSITION IN MAXWELL-CHERN-. . .
48
M~ (g
= 0) = m,
(26)
M2s(g=0) =e P +2@ +2~)a~ 4e2$ +@2 —2 A —2 — 1 2 1 4 = 0) = — m4+ m + e2 + M44, 5(g P + 5
— 12
&j&
2
The result obtained here is seen to be in complete agreement with that obtained using the unitary gauge.
(27) 2 2 4m2e2(t)
2
~
m2) for some m2. The techniques for this are well known. See, for instance, [15—17]. There are very many equivalent expressions for this result, some of which are better suited to numerical work than others. We begin by not-
+m)=
Trln( —
The integral over the continuous mass may be performed immediately results of dimensional regularization,
+m
dV
dm
momenta and. the using the standard resulting in
)
dv)
——1
l(l--) n
=
A
(28)
36
— +m
)
=
f f dm
dV
G(a, a),
d V is the volume element for spacetime and G(x, x') is the Green's function (inverse) for the operator. We shall use dimensional regularization. After transforming to momentum space and using the fact that bosonic fields satisfy periodic boundary conditions
in imaginary
).
d" k 1 —
time, we have
k2
+~
/ 2~E'b ~
+m
where T = P and v = P~m~/2vr. What we choose to do with this expression depends very much on what we wish to achieve. A simple change of variable leads to the frequently used expression
Trln( —
+m
)
f dVI
1
I
dk
kin(l —a 4V 4
p
S=
)
= vr& fa(
b)
(n+b) +a
+ 4sin(TrA) fp(a, b),
a
=
f
daa(n
—a
(32)
)
xRe(exp[2m (u
+ ib)] —1)
(33)
This type of expression, originally derived in four dimensions (4D) by Martin and Schwinger [15], is often seen in the literature, but we have found it unsuitable in our case. In the low-())) region, the logarithm in (35) blows up; this introduces large numerical errors particularly at the origin which runs away to negative infinity. The result is often the appearance of spurious minima which cannot be trusted. The function q (a, b) is completely convergent, and so we would expect finite numerical behavior. Better results can be obtained by evaluating the integral in terms of simple convergent power series. This can be achieved analytically, after which the summations present only minor diKculties .numerically. Expanding the denominator of the fraction in q(a, b) as a binomial series we get
f
f
so that Tnln( — + m 2 ) = — dV
6m
+
4n T s ~
f
f, z(n,
O)), (34)
)). (35)
(31) The summation may now be performed using the trusted result of Ford [17]:
(29)
where
~+
~
2
2
—4
ing the formal result that
Tnln(
It remains to compute an expression for Trln( — +
— 2 4
2 m2$ +— + 3 A—e2$ 3
IV. EVALUATION
Trln( —
1811
—
(n, O)
=
f
V
(n' —n'))
a-*-"'.
(36)
E=l
The integration may be performed by changing variables to x = 2vrE(u —v), whereupon one obtains
MARK BURGESS, DAVID
1812
f
i(v, 0)
=
)
the
Recognizing
exp
~ ~~
v
1
function
polylogarithm
(37)
Trln( —
7n T2 lmlT
+m ) = Lisje
~
)).
Evaluation by this expression promises some numerical accuracy without the awkward domain errors from the logarithm.
V. RENORMALIZATION AND SYMMETRY BREAKING We now turn to the issue of whether or not the model we have outlined exhibits symmetry breaking. This is
closely tied in with ideas about renormalization and the The literature for convergence of the loop expansion. four dimensions is extensive, and we cite only the most relevant references here. Enlightened discussions have been given by Coleman and Weinberg [18], Weinberg [19], and Dolan and Jackiw [16]. Some other references of interest are [20—24]. (2+1)D is qualitatively different from (3+1)D in a number of ways. In particular, onegpss) might compare the expressions for the zero-temperature one-loop correction for a massive scalar Beld in (3 + 1 =)4, 3, and 2 dimensions: M4 1 — Trln( — + M )4 —64m. z ln (39) 2 ( p~z ) 2
(M')
l
2 1 —
2
— + M )s
Trln( —
+M
)z
lMl ——
'
——M
ln
12'
8vr
(4o) l
(Mi l+1 z
(41)
M is some mass term, and pR is the renormalization scale, which vanishes to one-loop order in (2+1)D. (This in itself is not necessarily significant, since it does not vanish at two-loops where there is a divergence in the "theta" graph. ) The power dependence on the mass is significant. If we express the effective potential as a power series V(P)
=
)
48
as in the Coleman-Weinberg scenario, is beside the point. The quadratic term is the only possibility for creating an instability since the AP term is necessarily positive in order to save the potential from being unbounded from 0.] below. [In (2+1)D we could also have A 0 and g In (2+1)D this is not the case however. Indeed, an explicit computation shows that linear and cubic terms can also help to induce symmetry breaking. The outcome of this is that we may break the symme0 without any diKculties and without the try for m quartic coupling becoming negative, which was noted by Coleman and Weinberg in (3+1)D, provided the mass is sufficiently small. This is significant because it means that our one-loop approximation really can be a reasonable estimate of what the loop expansion is doing. This is not generally the case in (3+1)D, as noted by Weinberg, Dolan, and Jackiw and others [19,16,25 —27]. There one generally sets m & 0 by hand in the classical theory; this invalidates the loop expansion by inducing severe infrared divergences which destabilize the origin. This is associated with imaginary parts in the effective potential. (At high T there may also be infrared divergences regardless of the mass, but that is a separate problem which we shall consider in the next section. ) The renormalized m only becomes non-negative again at the minimum. We do not need this scenario in (2+1)D. One-loop corrections already have a negative sign and will thus tend to destabilize the tree-level terms if they become suKciently large. This will be true even if the mass is positive provided it is small enough. Moreover, we must conclude that it is simply incorrect to choose m & 0 in (2+1)D, since it implies an imaginary mass which appears explicitly in the (2+1)D case, as may be seen from the counterterms. (See the next section. ) We are not allowed to add imaginary counterterms to the theory. So we are left with the result that the symmetry can break with a non-negative mass. One must then ask the question, is the loop expansion valid at the located minimum? This depends on the model. The answer is certainly "no" for AP theory in (3+1)D, since the one-loop corrections are not strong enough to destabilize the tree-level terms without resorting to large logarithms for assistance [18]. As pointed out by Coleman and Weinberg, higher loops contain higher powers of these logarithms; thus, the loop expansion must be invalid. This is not the case here. To begin with we do not have any renormalization logarithms to one-loop order, since the theory is completely finite after regularization. Moreover, the corrections can be large enough in themselves without invalidating naive power-counting arguments. This is easily seen by turning to AP4 theory. This case has been analyzed extensively by our colleagues [28], and so we shall not repeat their analysis here. Briefly, we see that the one-loop potential for a real scalar Beld is given (at zero temperature) by
)
)
T3
1 — Trln(
AND NILS TVETEN
(
Liz(z)
i &„, we finally get
Pz
J. TOMS,
a„(P,
(42)
then az +i —0 in (3+1)D. These coefficients are nonvanishing in (2+1)D and have a special role to play in symmetry breaking, namely, the following. In (3+1)D the P = 0 minimum of the even-polynomial effective potential can only be removed by the introduction of a negative mass squared in A/4 theory. Whether this mass-squared term is explicit at the tree level (due to forces unknown), as in the Higgs model, or implicit in radiative corrections,
v(y) =
2
2
I
(
&lml&l 8vr
)
&'
~&' — m + '~ 12'
)
4!
(43)
If we regard the mass as being small for simplicity, then there will be a minimum
at approximately
IS THERE A PHASE TRANSITION IN MAXWELL-CHERN-. . .
(44) In order to justify this minimum we need the coupling constant to be small. Since we have not used any choice of renormalization scale to find the minimum, there is nothing in the way of simply choosing any logarithms occurring at higher loops to be small. Moreover, as long as the mass is sufFiciently small we can justify a nonzero minimum at any small value of the coupling constant. Our colleagues have used higher-loop estimates using the Gaussian effective potential to back this up. Of course, our real interest is in Maxwell-Chem-Simons theory or scalar electrodynamics. Coleman and Weinberg have justified a minimum in the (3+1)D case. We would now like to justify the fact that we may have symmetry breaking in
(2+1)D.
VI. RESULTS FOR SCALAR C}ED
tent.
Using the expressions derived in the previous sections we may compute and plot the efFective potential to one loop. For large enough A we may obtain a minimum at nonzero P at zero T. This minimum is possibly fictitious, since we have not defined the physical mass of the theory. At T = 0 radiative corrections give a contribution to the self-energy. In four dimensions this contribution is divergent, and thus the standard attitude is to cancel off the divergent part and some portion of the finite part by demanding that the physical mass be that point at which the propagator has a pole. This ensures that the theory is parametrized in terms of the physical mass. In (2+1)D our one-loop contributions are purely finite after regularization, but we must still arrange for the pole in the propagator to be at the physical mass. We thus add a counterterm at the tree level which cancels off part of the radiative corrections in such a way that the m in the Lagrangian and all references at higher loops represent the physical mass:
g2~(1)
0
= —bm
2
.
(45)
In A/4 theory a similar condition may be applied to the fourth derivative in order to define the effective coupling constant, though this is not strictly necessary here and we shall omit it (for interest we show the counterterm that would have been): Bm2
=
e2/p[
A/m/
8sr
"2)
—,
(46)
+ 2'/m/ + g~m~ 40+
(47)
Ae
f
+
2'
27r
5As
29m[m
/mf(e
After mass renormalization one finds that the possibilities for minima away from the origin are greatly reduced. All is not lost however: For suKciently large A we may obtain a minimum from the scalar sector, despite the fact that m 0. At first this may appear to be somewhat suspect since a large A implies poor convergence of the loop expansion. A closer investigation reveals that we
)
require A )) m; as long as the mass is small, there be no problem. This minimum is in fact the same present at the level of AP theory for a real scalar [28]. What is the role of the Chem-Simons parameter? Our investigations show that it is a redundant parameter, ex2 2 cept in a very narrow window of values where p2 (( e2$ and the system is close to a critcal point for p = 0. In this highly unstable region, increasing p can tip the balance and restore a broken symmetry, but as p becomes large compared to e P the dependence levels off. Thus switching on p has a slight hindering effect. The effect is not proportional to p, but levels off rapidly, so that the 2 efFect of pz )) e P is only to shift the zero-point energy uniformly. The small window arises essential due to the interplay between the background field and p in M2 and M3 when they are of the same order of magnitude. As soon as one dominates over the other, p becomes impoonly will one field
To examine the role of p more carefully we may expand the p-dependent part of the effective potential in powers of 2eg/p. We have
V(p, eP)
=—ps I+& ( r2eg)' 12ir p g(
(48)
)
This means that the contribution to the effective potential from terms including p is effectively independent of np, , P for small values of P. Hence, inside a region 2eg where n is a constant which is "small enough" (at least less than 1), an increase of p, will not change the form of the effective potential. However, if we start from p = 0 at a critical point, then it is clear that even a very small change in p will have a significant effect. (To be completely accurate we should note that increasing p always has an effect in the sense that it leads to an overall shift of the zero-point energy, but since this is an irrelevant constant, we ignore it. ) We thus conclude that p has a rather insigificant effect on the breakdown of the gauge symmetry, at least to one loop. In the next section we shall attempt to confirm this in a nonperturbative regime. In Ref. [4] the authors claim that symmetry breaking occurs because of a cooperative effect between the negative mass squared and the Chem-Simons parameter. It is straightforward to see that this is incorrect. It is unfortunate that the term which dominates the symmetry breaking is in fact one of several terms which were discarded in Ref. [4] on the grounds that they were complex. Had these authors computed the full set of counterterms, they would have seen where the problem lay. The effect they see due to the Chem-Simons parameter appears because they have only taken half of the one-loop eigenvalues.
(
VII. RING CORRECTIONS Let us now apply the technique of computing ring corrections to our (2+1)-dimensional theory. This is a standard technique in four dimensions, and it is therefore interesting to apply it to 2+1 dimensions. It is a differ-
MARK BURGESS, DAVID
1814
ent approximation than the one-loop calculation in Secs. which sums a particular set of loops from the whole loop expansion [29], and thus it should give us a results. Tradisecond opinion on our symmetry-breaking tionally [25,29,27, 26], these corrections are used to show the restoration of symmetry at high temperature; the large-T limit is itself a source of difhculty for the loop expansion. As one increases T infrared divergences set The in, invalidating the usual one-loop approximation. idea is to take care of the temperature dependence of the effective action by hand in such a way that it will be as close to exact as reasonable calculational methods permit. This may be effected by using an effective action in two stages. First one constructs an effective theory for the zero mode [25] in which the temperature dependence is taken into account explicitly. We shall call this effective action I'p[pp]. Then one expands the zero mode around a constant background Geld and computes the one-loop corrections there, using I p[pp] as the "classical action" for the effective theory. This oblique approach has the effect of summing a subset of diagrams from all loops which are the dominant order contributions [25, 29]. The strength of the ring method is that it is a nonperturbative approximation: The subset of loops summed comes from all perturbative orders. This is also its weakness, however, since one does not work a priori to any definite order in a smallness parameter. To illustrate the computation of the temperature dependence we shall first consider a single real scalar field component with the action
II and III,
dt
der
1 — — y( 2
I'p[pp]
A 4 rp + m )p+ — 4I
(49)
—V' + m + (2vrnT)
and expand the logarithm
ln(A
AND NILS TVETEN
where do.
is the volume element on the spatial hypersurface. This example is trivial compared to a gauge theory, but serves to explain the method without much of the complexity. Following Ginsparg [25], we consider the Fourier expansion of the "timelike" Geld modes and separate off the zero mode for which we will Gnd an effective Geld theory:
p(t, x)
)
=
)
(50)
S=
—yp( —V' 2 2
do+
+
+ m 2 )(pp+
)
p„*[—7'
AT 2
p po). n=l
AT
4
4I pp
+ m + (2vrnT) ] p„
n=1
+
v'
+'''I
where the ellipsis refers to contributions which are not of order po. Although higher powers of the action will generate terms of order po, these will be of higher order in our yo perturbation parameters, and thus, provided that our effective theory is well behaved perturbatively, they should not be relevant to our effective one-loop computation. (This still has to be shown. ) Dropping these higher-order interactions we write the effective action for
= —ln
(52)
]G
(x, x')
= 8(x, x')
a UV divergence. This is easily seen by writing (53)
1p
to order pz using
+ B) = ln A + A B + AT
P ' e' '"'~~ p„(x)
and p(t, x) = P ~ pp(x) + p gp(t, x). Since the field is a single-component, real field, we have p„*(x) = p „(x). Substituting in and using the orthonormality of the modes to integrate over t we obtain
—
) AT
(54)
2
the relevant correction term is
1p
48
Po:
If we define the massive Green's function by [
J. TOMS,
do
G„(x, x).
(55)
Note that this correction is one-loop order in the effective action, which we call I'p[pp] for the zero mode, and is computed to leading order in po, this will give the dominant contribution to the full effective action for the background Geld. In 2+1 dimensions we are in fact able to compute the trace of the Green's function exactly although it conceals
AT
do.
pp
do
p'p
G„(x, x)
Gp(x,
x).
(56)
The second term diverges like m /4vr x I'[0], whereas the remainder is Gnite. We shall return to this point presently. The Grst term may be computed by a further application of Ford's formula [17]. Rescaling the integral, we have
)
G„(x, x) =
where A performed
=
)
2
(27rT)
(k + m )/(2vrT) immediately, giving
. The
[A
+n
j,
(57)
sum may now be
IS THERE A PHASE TRANSITION IN MAX%ELL-CHERN-. . .
lim %~ 1
4, T
27l
ir'I'!
d k
— !!A!
g2)
+ 4»n(~A)
~
G„(x, x) =
4vr
Ii + I2
d k [27rT
2T2
I2
d2k
——
gk'+
= —2!m! vr,
m'
d2k
—47rT2
gk'+
T l— n(1 —e
87r
(60)
—1
e
m'
).
~
1
1
27l
47r
=
(m'& ( p~ )
16'
(62)
AT
4'
AT
AT
47r
8~
t'M2
ln
l
(63)
pp.
2
( pR)
+. which has a minimum
Finally A! m!
0—
167r
AT 4ir
ln,(T ) ( p'R )
do
—(pp
(58)
AT
+ Oi
16vr
ln!
(m2(T)
l
q
)j !
at AT
(68)
We now have two renormalization scales Tp and pR. In order for the loop expansion to be well defined we require small logarithms, which implies that the latter scale, here The value for Tp is pR, must vary with temperature. not really arbitrary, but in a sense it defines the critical temperature. Indeed, if m = 0, it is the critical temperature. This is slightly peculiar though since it seems that we may have the critical temperature wherever we like. Although we have eliminated the mass explicitly from the logarithm, there is an implicit assumption behind that step which was that Tp m, holding the logarithms small. Absorbing some irrelevant constants into is defined apTp we have that the critical temperature proximately by 2vrm
+ 3AT ln!
—= 0
(Tp) !
(69)
or
(64)
(pp,
provided we note that the logarithm can never be negative. Thus the effective theory for the zero mode is
9g —
—1
6m'(T)
In the large T limit, the first logarithm goes like T ln(m/T) and so we may rewrite this last result as A!m!
1
= —m'(T)P'+
(61)
The divergence in the second term of (56) may be sidestepped by using (-function regularization. This has the effect of absorbing the divergence into a logarithm involving the arbitrary renormalization scale. We get
Gp(x, x)
27ra
derivative of the potential with respect to the field) is less than zero. This is the only possibility for an unstable state. The first-order ring corrections are computed by shifting pp -+ yp + P, integrating out the pp field to one loop. This gives, neglecting P independent terms,
V(P)
Ii
—A2
(
Because of the limit A 1, the sine function selects only the bottommost contribution &om the integral, where there is simple pole. The limit is completely finite and leaves us with
)
X2
1815
—V' +m T pp+
yp,
65
T~
= Tpe
2
2
(70)
This shows that as the mass increases without bound, the critical temperature goes to zero; i.e. , symmetry breaking is lost. Conversely, as A increases it increases up to some upper bound. Setting Tp m enables the critical temperature to be computed; thus, the essential features of the one-loop approximation are regained in this new regime.
where
m'(T) = m' —
8' +
(T& 2vr
( Tp p
(66)
The first term is recognized to be the part of the effective action which would be canceled by one-loop counterterms and may therefore be dropped. We have relabeled the renormalization scale Tp to avoid confusion in the next stage. In this two-dimensional effective theory we have arrived back at a situation where symmetry breaking implies that the effective mass squared (the second
Now we want to find the ring corrections for scalar electrodynamics with a Chem-Simons term. This is considerably more complex and requires an expansion of both P and A„about the zero mode. In the literature this expansion has been performed as an ordinary background field calculation P —+ P + P, A~ + A~ in which P = T2 Pp and A~ = T~ A„and one integrates over time, using the orthonormality of the modes. The propagators are then replaced by the sum over the remaining n states in the Fourier expansion, of a two-dimensional propagator as in (51) above. There are some problems
A„~
J. TOMS,
MARK BURGESS, DAVID with this approach: First, the "background Geld" is not a constant. In [26] the "background" field is implicitly treated as being constant, meaning that the answer is
scalar fields. There are two quartic interactions and the term which can give rise to higher-ring corrections, namely,
only correct in the long-wavelength limit, where Po is effectively x independent. Another problem is that it does = P*, which not necessarily take care of the fact that P means that (P(x)P(x')) g 0 and (P(x)*P(x')*) g 0. We need to know such propagators to find ((S,„t ) ) in order to compute the gauge loops to lowest consistent order e . Let us first do the calculation by the method of [26] to show where things go wrong. The most important contributions still come from the
(
— ~e P
AP,
gQ,
and
e P
A"A„.
(71)
We shall work only to order A, g, e . Since the gauge field does not have any self-interactions, only the treelevel terin will be important there. In Ref. [25] it stated that the p = 0 component of A„decouples from the remainder. In our case this will not happen because of the Chem-Simons term which explicitly couples them. The propagators are defined by
)b„— c)„0 + i—ps„„pcs" (
AND NILS TVETEN
D
" = b„"8(x,x'),
(72)
— + m2) A(x, x') = 8(x, x'),
(»)
(A„(x)A (x')) = D„(x,x'), (&-(x)&b(x')) = ~-b&(x x')
(74) (75)
The interaction terms which give corrections to order P are
(76)
+ —e
l
1+
lz
ebs'
4'gd
—2eF.
4
r4 A"(D„p )
—
ee qp
jb
(— 8"A„)+ —e A„A„Q p
We shall set y, = 0 up to Eq. (84) in order to compare more closely with the existing literature. is then given by
The efFective action
rg] = sg]+r(', &, + ~r, ~r = (s,„',') —'((s,.'„", )') +
(78)
(79)
—,
The only relevant contributions 2
3
are
I& 2 I ( g — + 40 P + — I+ — 2q c) I
~
dCT~dO~I
4|
+4
e2$
B~B A X, X D
~
15 (2 + — e ~) ~
A(xx) X, X
+
2
+—
~)
this reduces to do.
a contribution
3 2) — 2 ——— e ~P + 3
2
C
A X) X
8 8
Dp~ X, X
P cl„A (x, x') O' D~ (x, x')
It may be checked that the —dependence cancels as it must, leaving the limit
which after including
(77)
)
(80) o.
~ 0 regular.
After some manipulations
1 —— g — — P + —e A A~ A(xx), 40 2 4
from the gauge term in I'p
2
p
P,
and the Faddeev-Popov
(81) determinant
leaves the
Our result here difFers from that found in [26] for the gauge terms. Arnold's result was computed in the Landau gauge with 0. In n dimensions this gives a correction —(n —1)e P A(z, z) 2 o. In the gauge-invariant Vilkovisky formalism there are extra contributions both from gauge-fixing terms and from the ghost terms. In (2+1)D the contribution computed in [26] and the ghost contribution cancel exactly, leaving only the residual terms due to covariant gauge fixing. The correction using this approach, for scalar electrodynamics, would be —(n —3)e gPA(z, z) i 0 ——e P A 2 (z, z). o.
~
IS THERE A PHASE TRANSITION IN MAXWELL-CHERN-. . .
1817
correction 2 2 m 2 &pp~ m 2 (T)(pp: m 2 pp2
m,
ApA„pm m (T)ApA&p A(T)= AT+ gT —.
—e
+
~
—3e 2
~
2
)
~
(x, x),
T&ppA
P ApA„p+ e TApA~pA
(82)
(x, x),
(83)
,
(84)
G (x, x) is evaluated at the appropriate mass [see Eq. (61)]. In evaluating (79) one uses the fact (x, x) = T that one of the spacetime integrations results in a b function, which simplifies the structure of the integrals considerably. This is only correct if the fields P are constant. Without this simplification the integrals yield complicated Bessel
g
functions which are hopeless for going any further with. We now reinstate y, and proceed with what we belive to be the correct approach. explicitly for all fields we have that
do-).
Ape:C'+
8
n+0
V
pV»4:4.*+
+ 1+ — e To~is, gypppg„g
*
~)
~
~
* 2 2ee bT— Ap(0, $ )P
—
20& p0:0:*+ 30
+ 4e
vriee ~T2 — nypP
A
*
+
2 ~
in (50)
p4:4.*
V pV'pV
A"* + 4ee
TypA~&g
Writing the expansion
bT2
+ — 2ee
~)
~
pp(BQ )A'„* ~T2 gpss (O, A'*)
(85)
The expansions in P and A„do not decouple, and the result is more complex than the naive application background field method gives. We now have
D„(x,x', n) =
d2k
b„
+ (2n7rT) + p,
k
(2vr)
+
of the
+ (2n7rT)2 —n02 k„k (n —1)[k + (2nvrT) + p, ][k + (2nvrT)2] k2
PE'pv
fk~
+ (2nzT) ~] fk + (2nvrT) + p~]
)
(86)
'
where ko = 2n~T. The work required to evaluate this quantity even to order A, g, e is considerable. own computer algebra tools and eliminating all odd integrals and summations we fi.nd that, to order
((Sl„~i) )
=
do do
)
4e Tpp(x)yp(x')D'~(x, x', n)0, 0'A(x,
Employing
e,
x', n)
n+0
+ —mie T nyp(x)pp(x')[D' (x, x', + 2~ 2+ —~e
~)
n)O, A(x, x',
vr
+ —e
e
n—
)
)].
—D ~(x, x', n)B' A(x, x', n—
T&pp(x)yp(x')[0;A(x, x', n)0'D*~(x, x',
TAp(x)A~p(x')(A(x, x', n)0, 0'A(x, x', n)
+ 2e
n)
+ O'— A(x, x', n)O, D ~ (x, x', —n)]
—[0;A(x, x', n)][0'A(x, x', n)])
T n pp(x)pp(x') D (x, x', —n)A(x, x, n)
Tvrinpp(x)pp(x')
*-
0'D ~(x, x', —n)] [4(x, x', n)0;D' (x, x', n) —A(x, x', n)—
2
+ 2+ — e'Tp, (x)p, (x')6(x, x', n)083D '(2:, z', n)). ~) ~
our
~
This expression is not generally calculable in this gauge, unless one assumes that pp(x) is slowly varying so that po is approximately constant over the system and that the Chem-Simons parameter p is zero. [This was the answer in (82) and (83).] It is possible that, in a difI'erent gauge, it might be apparent that once spacetime integration falls out, leaving a local result, but in
that case we have to worry about the tions in the particular choice of gauge, new problems in the evaluation. The rameter is another matter however.
(87)
Vilkovisky correcand these present Chem-Silnons paThis introduces a length scale or mass into the gauge Green's functions which eradicates any semblance of calculational simplicity. In the very best case one has integrals of the form
MARK BURGESS, DAVID
1818
I
dk[k + (2n7rT) ]/[k + (2n7rT) + p, ]. The dependence on p might at best be expressed in terms of exponentials or Bessel functions. We choose not to carry the evaluation any further at this time; we shall return to this in a follow-up paper. We write instead the unknown temperature-dependent masses symbolically as m2(T) and m, (T) and extract what information we can from this. It should at least be possible to determine the structure of the efFective potential. The temperaturedependent coupling constant is unaffected by these diKculties. The form of (63) makes it clear that symmetry restora-
+ m2) ( ( ——1PB2 V'2
0„
p,
(
a,
~
™. '+
2
P' +
V'4m! 2 lP I
+ P',
(9o)
with replaced by V'2. It is reassuring to see that the same form for the eigenvalues comes out of this different approach, in spite of the violence we have done to the Chem-Simons term by dimensional reduction. The ringcorrected 2D effective potential is then
daV(g) =
f
do 5
+
)
1 2 — m (T)d 2
—Tr ln( —V'
+, A
T
d
)
+ M„).
n=l
M,
The one-loop terms are proportional to and so we see now directly that the Chem-Simons parameter p yields only a constant contribution which shifts the zero point of the potential (the + parts of M2 3 ~2Dl cancel). If P is the (constant) background field, the explicit expression is
~(y)
1
&(T) ~4
2(T)y2
+
1
+ 2m, 2 (T) + 4m, (T,j P)
16'
p
4f
+-, ~(T)d*)
AND NILS TVETEN
e2
(92)
48
tion will take place. We are therefore interested in finding the main polynomial dependence of the effective action on the two parameters of interest: P and p, . The tree-level action is
S=
1 — V' 2 do. —P[ — 2
—
+ m(T) 2 ]P+ A(T) 4I
—4
1—
+ —A„O""A„ P
(88)
where
—1p, c)i 'LPB2 1 —(v' + m.') —— 'c),2 1 — —— 2+ Bi B2 m.') —-'a,2 ( v
The ring-corrected effective potential is thus found by expanding around a constant background po po + P. The background vector field may be gauged to zero. Note —2 that, to one-loop order, the e 2 (t) terms in O„are just replaced by m, (T) evaluated at the background field, as shown in (89). The expansion is exactly analogous to the one performed in Secs. II and III, with the exception of the vector Geld, whose determinant may be computed directly from (84). The result is M2, 3 (2D)
J. TOMS,
(89)
)
The Grst two terms in the parentheses may be dropped since they are P independent. Here we see that when A or g are large compared to m we may have symmetry breaking, as we found earlier. Thus our nonperturbative calculation backs up the one-loop findings in this respect. The dependence on the Chem-Simons parameter is through the remaining T-dependent masses and coupling constant. Although it is dificult to predict the precise behavior as a function of p, the effect of a mass is typically to reduce the potency of the term concerned [see (61), for instance] and one would therefore expect a large p to have no particular efFect on the symmetry breaking, which is otherwise dominated by the A/4 and gqP terms.
VIII. STATISTICAL GAUGE FIELD The current literature on anyons and their connection with models for the fractional quantum Hall effect and is vast. Some relehigh-temperature superconductivity vant references may be found in [30,31]. It is not our intention here to give any proper discussion of anyons and what phase transitions they may or may not undergo. We only wish to point out that the inclusion of a second gauge field of the usual anyon type does not affect the conclusions we have reached. In anyon theory one normally couples a hard-core matter field to a ChernSimons term which has no associated Maxwell term. The resulting quasiparticles are anyons, at least in the nonrelativistic limit. See [32] for a recent discussion of this. In Ref. [4] the authors reject the statistical gauge field on the grounds that it has no experimental basis. The effect of such a statistical gauge field on the phase transition is either minimal or zero, depending on the particular nature of the Lagrangian. Symmetry breaking in the absence of the Maxwell term has been considered in Refs. [33,34]. We shall not repeat their analysis here. As long as the Maxwell term is present, we may define the most general Lagrangian for a Maxwell-Chem-Simons theory in
IS THERE A PHASE TRANSITION IN MAXWELL-CHERN-. . .
48
the Euclidean formulation 4
ther bearing on the dynamical symmetry breaking which
by
we have considered in this paper provided one uses the efFective value of the Chem-Simons mass. We further
e" "A„cl Ag + pe" "A„B ap F— ""F„„+2 —
"a„O ay + J"(A„+a„), + 2 e""— where A& is the electromagnetic field and a~ is the statistical gauge field. The quantum theory is defined by
the partition functional
Z
=
dpAp a~ c
(94)
where S is the action for the above Lagrangian. The coupling term proportional to P is not usually included in the literature, although it is naturally generated by radiative corrections [35,36]. If this term is not present, then the problem breaks up into two distinct parts: a part in A„, which we have considered thus far, and a part in a„, which was considered in [33,34]. If the fields become coupled, then they may be rapidly uncoupled by a simple transformation of the fields P/pA„. This results in two separate phases once again a pure "statistical" gauge sector, with Chem-Simons coefIicient p, and a Maxwell-Chem-Simons sector with effective mass —P/p): Lt = n —Pz/p and efFective charge e(1
a„~ a„+—
4
P'5 I f F" F„„+ "" —— o. ——e"„.„"A„ci Ag
2E ~
r
1
[1] [2]
Leinaas and
i
J. Myrheim,
and therefore has no fur-
Nuovo Cimento
B 37,
1
(1977).
F. Wilczek,
Phys. Rev. Lett. 48, 1144 (1982). J.K. Kim, Phys. Rev. D 46, 1810 (1992). J. Bernstein, K. Benson, and S. Dodelson, Phys. Rev. D 44, 2480 (1991). [6] C.W. Bernard, Phys. Rev. D 9, 3312 (1974). [7] G.A. Vilkovisky, in The Quantum Theory of Gravity, edited by S.M. Christensen (Hilger, London, 1984). [8] E.S. Fradkin and A. A. Tseytlin, Nucl. Phys. B234, 509 [3] [4] [5]
S. Hong and
(1984).
[9] M. B. Einhorn and D.R.T. Jones, "P in three dimensions at finite temperature, " University of Liverpool (DAMTP) report, 1992 (unpublished). [10] A. Rebhan, Nucl. Phys. H288, 832 (1987). [11] I.H. Russel and D.J. Toms, Phys. Rev. D 39, 1735 (1989). [12] J. Schonfeld, Nucl. Phys. B185, 157 (1981). [13] S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. (N. Y.) 140, 372 (1982). [14] H. P. Leivo, S. Huggins, G. Kunstatter, and D.J. Toms, Nucl. Phys. H301, 627 (1987). [15] P. C. Martin and J. Schwinger, Phys. Rev. 115, 1342
(1959).
We have calculated the effective potential for quantized scalar electrodynamics in 2+1 dimensions with a ChernSimons term. We find symmetry breaking due to the scalar sector for A/m or g sufficiently large We .find a very narrow range of values for which the Chem-Simons term can restore a broken symmetry, namely, when the self-couplings are on the brink of the critical values and —2 —2 e 2 p . For Lt 2 )) e 2 p the effect of the Chem-Simons p term is to shift the zero-point energy uniformly for all P in the vicinity of the miriimum.
(
ACKNOWLEDGMENTS
R.B. Laughlin, Science 242, 525 (1988).
J.M.
IX. SUMMARY
[
J"A„, &) a„B„ap + J"a„.
Equation (96) is nondynamical
point out that if one considers models in which the statistical gauge field is generated by some third-party interaction with a parity-breaking field [35], then a kinetic Fz term can be generated by radiative corrections, as well as corrections to the Chem-Simons field. Such a change may lead to a redefinition of some of the coefIicients in our model and could therefore in principle lead to the restoration or breaking of the total gauge symmetry, but again the efFective theory will have the form of the one considered here.
&)
——
2
1819
We would like to thank Roar Aspesaeter Olsen and Finn Ravndal for helpful discussions. We are also grateful to Tor Haugset for confirming our expression for the effective potential in the Landau-DeWitt gauge and for active discussion on AP theory.
[16] L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974). [17] L. Ford, Phys. Rev. D 21, 933 (1980). [18] S. Coleman and E. Weinberg, Phys. Rev. D 7, 1888
(1973). [19] S. Weinberg, Phys. Rev. D 9, 3357 (1974). [20] D. A. Kirzhnits and A. D. Linde, Phys. Lett. 42H, 471
(1972). [21] D. A. Kirzhnits and A. D. Linde, Ann. Phys. (N. Y.) [22]
[23] [24] [25] [26] [27] [28]
[29] [30]
[31]
101,
195 (1976). M. E. Pol, M. G. Amaral, C. Aragao de Carvalho, and R. C. Shellard, Z. Phys. C 32, 609 (1986). A. Wipf, Y'. Fujimota, and H. Y'oneyama, Z. Phys. C 35, 351 (1987). K. Takahashi, Z. Phys. C 26, 601 (1985). P. Ginsparg, Nucl. Phys. H170, 388 (1980). P. Arnold, Phys. Rev. D 46, 2628 (1992). M. E. Carrington, Phys. Rev. D 45, 2933 (1992). R.A. Olsen and F. Ravndal, "Effective potentials for P theory in 2+1 dimensions, " report, 1992 (unpublished). P. Fendley, Phys. Lett. B 196, 175 (1987). F. Wilczek, Fractional Statistics and Anyon Supercon ductivity (World Scientific, Singapore, 1990). Theories of Condensed Matter Systems, Frontiers in Physics (Addisan-Wesley, New York, 1991).
E. Fradkin, Field
1820
MARK BURGESS, DAVID
C.R. Hagen, Phys. Rev. Lett. 68, 3821 (1992). J. Hong, Y. Kim, and P.Y. Pac, Phys. Rev. Lett. 64, 2230 (1990). ]34] R. Jackiw and E. Weinberg, Phys. Rev. Lett. 64, 2234 ]32]
]33]
J. TOMS,
AND NILS TVETEN
(1990). ]35] A. Redlich, Phys. Rev. D 29, 2366 (1984). ]36] S. Randjbar, A. Salam, and J. Strathdee, Nucl. Phys.
B340, 403 (1990).
