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On the Meissner effect in gauge theories
ANNALS
OF PHYSICS
134, 259-285 (1981)
On the Meissner
Effect in Gauge
Theories*
GRAHAM M. SHORE
Cambridge,
Lyman Laboratory of Physics, Harvard University, Massachusetts 02138, and Newman Laboratory of Nuclear CorneN University, Ithaca, New York 14853
Studies,’
Received December 22, 1980
The phase structure of spontaneously broken scalar electrodynamics in an external electromagnetic field is analyzed. With no external field, the spectrum comprises a scalar boson of mass mH and a vector boson of mass m,. If mH < m,, it is shown that in the tree approximation, as the external field is increased, a first order phase transition to a restored symmetry phase occurs, and the critical field strength is calculated. Below the critical point the external field is completely screened, this being the analogue of the Meissner effect in superconductivity. If mH > m,, a, third phase, characterized by vortex solutions of the field equations, occurs. Quantum effects, such as pair production in an electric field, are considered at the one (and two) loop level in the massless theory (the Coleman-Weinberg model). The leading correction to the critical magnetic field strength is calculated, and it is shown that for an external electric field the phase transition does not exist.
1. INTRODUCTION This paper is an investigation of the phase structure of scalar electrodynamics in the presence of an external electromagnetic field. This theory is known to be the relativistic quantum field theory analogue of the Ginzburg-Landau model of superconductivity [I], and so we expect many of the interesting physical effects associated with superconductivity to occur also in scalar electrodynamics. In particular, if a sample of superconducting material is placed in a sufficiently small, constant, magnetic field, the external field does not penetrate the sample. This screening of the external field is known as the Meissner effect. However, if the field is increased beyond a critical value, a phase transition occurs from the superconducting to the normal state, and the magnetic field takes a constant, non-zero value within the sample. In addition, for certain types of superconductor (type II), a third phase is possible in which the magnetic field forms vortices within the sample. In this paper we demonstrate the analogous results in scalar electrodynamics. We also make some remarks clarifying the distinction between the phase transition in scalar electrodynamics and the symmetry restoring phase transition which occurs in * Research supported in part by the National Science Foundation under Grant PHY 77.22864. ’ Permanent address.
2.59 0003.4916/81/080259-27$05.00/O Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.
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spontaneously broken, non-Abelian gauge theories in a background electromagnetic field. As is well known, for a certain range of parameters scalar electrodynamics exhibits spontaneous symmetry breakdown [2,3] with the vector bosons acquiring mass through the Higgs mechanism. If we now introduce a source which would in the absence of symmetry breaking give rise to a constant electromagnetic field, we find that it is screened, the measured field within the system being zero. This is the gauge theory analogue of the Meissner effect, and occurs because the vector bosons corresponding to the external field are massive. As the source is increased, the tree approximation indicates that a first order phase transition occurs to a restored symmetry phase in which the vector bosons are massless, and the electromagnetic field jumps to a constant value. In the case where the external field is electric however, pair production of charged scalar particles from the vacuum occurs once the critical point is reached. The effect of this phenomenon on the existence of the phase transition is discussed in Section 4. For the range of parameters such that the mass of the scalar boson mH is less than or equal to that of the vector boson m, we conclude that only these two phases are possible. If m, > m w, however, we find an inconsistency in the form of the scalar propagator which indicates that a third phase, characterized by a spatially varying vacuum expectation value of the scalar field, occurs. In this phase the ground state is described by the Nielsen-Olesen vortex solutions of the field equations. We restrict our explicit calculations to the type I case, mH < m,. The question of phase transitions in scalar electrodynamics has been discussed before in the literature. In particular, Harrington and Shepard [4], and Kirzhnits and Linde [5,6], have exploited the analogy with the Ginzburg-Landau model to deduce to lowest order the critical field strengths from known results in superconductivity. In contrast, we present a self-contained discussion of the phase transition in the relativistic field theory, then go on to analyze the one (and two) loop quantum effects. The computation of the rate of pair production in a background electric field was performed long ago by Schwinger [ 71, and indeed our methods of calculation owe much to his work. The paper is organized as follows. Section 2 contains a brief review of the calculation of critical field strengths in the Ginzburg-Landau model. Guided by these results, we present in Section 3 a discussion of the phase transition in scalar electrodynamics in the semi-classical, or tree, approximation, and find the critical field strengths. In Section 4, we extend the analysis to include quantum effects at the one (and for the broken phase, two) loop level. These calculations are performed using the heat kernel method for evaluating functional determinants modified to allow the use of dimensional regularization [8, 91. Finally, Section 5 contains a brief discussion of our results and some remarks on the distinction between the phase transition in a background electromagnetic field for Abelian and non-Abelian gauge theories [6, 10, 111.
261
MEISSNER EFFECT IN GAUGE THEORIES
2. THE GINZBURG-LANDAU
MODEL
OF SUPERCONDUCTIVITY
In order to motivate our discussion of the Meissner effect in relativistic scalar electrodynamics, we begin in this section with a very simple derivation of the critical magnetic field strength in the Ginzburg-Landau model of superconductivity [ 11. This is a non-relativistic phenomenological theory, which serves as a good approximation to the standard BCS theory in the neighborhood of the critical temperature. In this model, the Helmholtz free energy density may be expressed as F = jBZ + (1/2m*)(D#)*
. (D)) + I’(#*$),
(2-l)
where #(x) is a complex order parameter, B = V x A is the (microscopic) field, D = -iV + e*A, and w*$)=
J%+4~12+f~ld14,
magnetic (2.2)
e* and m* are the effective charge and mass, I’,, is the free energy density of the normal state in zero magnetic field, and Q and b are real, temperature dependent, phenomenological constants. For a < 0, V(#*$) may have a nontrivial minimum at Iqq*=-a/b=u*.
(2.3)
Now suppose an external magnetic field H is applied to the system. In this case, the appropriate thermodynamic potential to be considered is not the Helmholtz but the Gibbs free energy density, which may be defined as G=F-B-H.
(2.4)
The requirement that G be stationary with respect to variations parameter and vector potential yields the equations
(1/2m*)D*$+
vl(#*d)#=O,
$*fj$+Vx(B-H)=O,
of the order (2.5) P-6)
while the corresponding surface terms vanish if the following boundary conditions are satisfied n .D#=O, nx (B-H)=O,
(2.7) (2.8)
n being the surface normal. Imposing V x H = 0, these reduce to the standard Ginzburg-Landau equations. There are two simple solutions to these equations,
(4
B = H,
l$l=@
(b)
B = 0,
IdI=u*
(2.9) (2.10)
262
GRAHAM
M. SHORE
Solution (a) clearly corresponds to the normal state, in which the magnetic field permeates the system. Solution (b), which describes a state in which the magnetic field within the system is zero (the Meissner effect), corresponds to the superconducting state. The Gibbs free energies for these states are Ga-- -+H’
t V(O),
(2.11) (2.12)
G b = V(u’).
The system will exist in the state with the lowest Gibbs free energy. To determine the critical external field IHI, at which the phase transition from the superconducting to the normal state occurs, we therefore equate G, and G, evaluated at IHI = IH Ic yielding H, = -2( V(u’) - V(0)) = a2/b.
(2.13)
Clearly, G, = G, + f(H’
- H,2).
(2.14)
For external fields less than JH Jc, G, < G, and so the system is in the superconducting state, while for fields exceeding lHlc the system is in the normal state. Since the order parameter changes discontinuously at the critical point, the phase transition is first order. The phase transition described above is characteristic of a type I superconductor. For certain values of the parameters, however, a further class of solutions may be energetically favorable. These vortex solutions describe a state in which the magnetic field permeates the system in the form of thin flux tubes, and the order parameter ( is spatially varying. Materials which exhibit this mixed state are known as type II superconductors. Whether a given material is of type I or type II is determined by the ratio of the two length scales which characterize the system-the coherence length r, which provides a scale for variations of the order parameter, and the magnetic field penetration depth A. Defining K = A/<, the condition for a type I or type II superconductor is
d-2lc<
1,
type I,
d- 2K> 1,
type II.
(2.15)
In terms of the parameters already introduced, fiu
= 2m*b’/2/e*.
(2.16)
The mixed state is energetically favorable for values of the external magnetic field in the range lHlcl < (H( < [Hlc2, where IHlc2=d5W,
263
MEISSNER EFFECT IN GAUGE THEORIES
FIG. 1. Phase diagram for a superconductor in the Ginzburg-Landau and K with ( fixed.
model as a function of / HI
and ] H ICl is less than ]H(,. However, since a full discussion of the vortex solution and the determination of ]Hlc, involves analyzing the Ginzburg-Landau equations numerically, we shall not present any details here. The phase diagram for the model as a function of ]HI and ic for fixed < is shown in Fig. 1.
3. PHASE TRANSITIONS
IN SCALAR ELECTRODYNAMICS
The Ginzburg-Landau model serves as an analogue of scalar electrodynamics, and all the effects just discussed have their counterparts in this relativistic quantum field theory. In particular, we shall show in this section that as the external electromagnetic field is increased there is a phase transition in scalar electrodynamics from a broken symmetry phase, in which the vector bosons are massive and the external field is screened, to a restored symmetry phase with massless vector bosons and a long range electromagnetic field. Further, depending on a ratio of length scales (in this case the scalar and vector boson masses at zero external field), the model may be of type I or type II, the type II model having an extra phase described by the Nielsen-Olesen vortex [ 121 solution of the field equations. The Lagrangian for scalar electrodynamics is rp = GF,,F~~” where F,,” is the electromagnetic
- $(D,#)*(Dy)
- v(@*#),
field tensor, D, = a, - &I,,
Pq*qq = v, + $n*(P + (A/4!) 4”.
(3.1)
and the potential (3.2)
(Since the discussion in this section is at the semi-classical level, we omit the distinction between bare and renormalized parameters here.) For negative values of m’, this potential has a non-trivial minimum at I# I2 = -6m2/3, = v*.
(3.3)
264
GRAHAM
M. SHORE
To incorporate the effects of an external electromagnetic source term coupled to F,,,, viz.,
field, we introduce
a
The source is chosen so that in the absence of scalar fields it gives rise to a specified field stength F$. (Notice that this reduces to the familiar source term LJpA,, , with J” = a,F$‘ for fields F”$ which vanish at infinity.) This extra term is the analogue of the additional B . H term included in the Gibbs free energy in the Ginzburg-Landau model. The vacuum expectation values of the fields are determined from the requirement that the modified action be stationary with respect to variations of A, and 4. This yields the equations of motion D*# - V’(#*# i?,(F“” -F;,“)
= 0,
(3.5)
+ i#*@‘# = 0.
(3.6)
As before, there are two simple solutions to these equations:
The first solution (a) describes the conventional symmetric phase. The source gives rise to a long range electromagnetic field, the vacuum expectation values of the scalar field is zero, and the vector bosons are massless. Solution (b) describes the phase with broken gauge symmetry. The scalar ‘field develops a vacuum expectation value and the vector bosons acquire a mass by the Higgs mechanism, thus screening the external electromagnetic field. The measured electromagnetic field in this phase is zero. This is the Meissner effect in scalar electrodynamics. Which phase is actually realized for a given external field F”,‘” is determined by minimizing the energy density E of the fields. This is given by the TO, component of the energy-momentum tensor T,, (our metric convention is (-+++)), where
T,, = --+@U)*U’,/) - f(D,O*(D,O
In this paper we are concerned with the effects of a constant external field in scalar electrodynamics. In fact we will restrict ourselves to the special cases of a pure electric or pure magnetic field, since these cases are sufficient to illustrate most of the interesting physical effects, and yet the one loop corrections can still be calculated analytically.
MEISSNER
EFFECT
From Eq. (3.9) the corresponding and (3.6) are &=fB'-B.
B,,+
IN GAUGE
THEORIES
265
energy densities for fields satisfying Eqs. (3.5) V(qd*$),
pure B field,
(3.10)
pure E field.
(3.11)
and E =$I?'--*EC1
+ V(#*#),
(Here we simply use the notations E, B for 1E I, 1B I.) It is clear that for a sufficiently large external field, phase (a) will be energetically favorable. To determine the critical field strength B,, we therefore equate E, and et, evaluated at B,, giving Bf = -2( V(u')-
V(O))= 3m4/A
(3.12)
an identical result holding for E,. Scalar electrodynamics in an external magnetic (electric) field therefore undergoes a first order phase transition at B, (E,) from a broken symmetry phase in which the external field is screened to a restored symmetry phase. In the broken symmetry phase the spectrum comprises a vector boson of mass m, and a scalar boson of mass m,. In terms of these masses, we can reexpress the critical field strengths as Ef=B:=(
l/4$)
rnirn;.
(3.13)
The spectrum in the restored symmetry phase requires a little more discussion. First of course there is the massless photon. The propagator for the scalar particles is the inverse of the operator (-0’ + m’). At first sight it appears that with m* negative this operator may’ have negative eigenvalues and the propagator may be tachyonic, indicating some inconsistency. In fact this is not the case. The eigenvalues of (-D* + m’) for a purely magnetic background field (taken along the z-axis) are (kf + m* + (2n + I)&), n = 0, l,..., and so the smallest eigenvalue is (eB + m*). For arbitrarily small fields B this is indeed negative. However, for fields greater than the critical value (3.13), e*B* >+rn&rn&. Recalling that mf,= -2m*, we see that the eigenvalue spectrum of the scalar wave of operator is positive, provided we are in the type I parameter range mH < m,. For values of the parameters in the potential (3.2) such that m,Q m,, only the phases described by solutions (a) and (b) of the equations of motion are possible. For mH > m,, however, a third phase is energetically favorable for a certain range of external field strengths, this new phase being described by the Nielsen-Olesen vortex solutions of the field equations. Since a detailed analysis of this phase, which is analogous to type II superconductivity in the Ginzburg-Landau model, would involve us in numerical calculations involving the vortex solutions, we shall not consider this parameter range in this paper. However, from the arguments above, we may deduce quite simply the value of the
266
GRAHAM
M.
SHORE
01 El
FIG. 2. Phase diagram in the tree approximation and m,Jm, with mH fixed.
BIE)
for scalar electrodynamics as a function of B(E)
I
Broken, Ordered, PlWS-2 0
I
FIG. 3. Phase diagram in the tree approximation and m&q,, with m, fixed.
hi
‘MW
for scalar electrodynamics as a function of B(E)
upper critical field at which a phase transition from the vortex state to the restored symmetry state occurs. This critical field is given by the condition that the scalar wave operator is positive; i.e., B > (1/2e)mi. We can therefore plot a phase diagram for scalar electrodynamics as a function of B (or B) and m&n,, where we take mH fixed. This is shown in Fig. 2. In Fig. 3 we have made the analogous plot with m, held fixed. The upper critical field is
the lower critical field requiring a numerical analysis for its evaluation.
4. QUANTUM
EFFECTS AND CRITICAL
FIELDS
So far we have discussed the phase transitions occurring in scalar electrodynamics with an external electromagnetic field in the tree approximation.We now show how these results are modified when we go beyond the tree level and include quantum corrections.
MEISSNER
EFFECT
IN GAUGE
267
THEORIES
The first major effect is that with zero external field the gauge symmetry is broken for positive, but sufficiently small, values of the scalar mass parameter mR in the Lagrangian. This was first shown by Coleman and Weinberg [3]. In particular the choice mR = 0 gives a theory in which the symmetry is broken due to one loop radiative corrections, and in which the ratio of the scalar and vector boson masses is calculable. This parameter choice is both algebraically simple and of considerable intrinsic interest, e.g., in connection with the gauge hierarchy problem [ 131, and we shall restrict our detailed calculation to this case. A further purely quantum effect, which does not appear in the semiclassical calculation of Section 3, is that for certain values of the external field the effective action develops an imaginary part [7]. In particular, this occurs in the case of a pure electric field. This is interpreted as indicating the pair production of charged scalars from the vacuum due to the externally applied field. In this section we compute the quantum corrections to the effective action. As before, there are two phases corresponding to broken and restored symmetry, and the critical fields are evaluated by equating their energy densities. In the symmetric phase, the leading quantum corrections come from one loop diagrams, and are evaluated using the heat kernel method. In the broken phase however, since we are considering the massless (Coleman-Weinberg) theory, the lowest order contribution to the energy density only arises at the one loop level, and so to compute the leading corrections we work to two loops. The generating functional W[Ft”, 5,] is given by exp W[F&, J,] =
i
G@A, G9#,A[A] I
x exp i d%(L$ + (lpa,)(a,A,)* 1,
+ $I;,,F:,”
- J,4,>
I
, (4.1)
where the Lagrangian % = -+‘,,J“”
- t~~,~W,~),
- @B/41) #4,
(4.2)
with 0;” = a, gab + eB.PbA,, . Here we have expressed Y0 in terms of the real and imaginary parts of the complex scalar field; i.e., 4 = 9, + &, with 4,) a = 1,2, real, and 4’ = d,#,. The ghost Jacobian A[A] may be expressed as A [A] = j GZwj 9% exp(l/&)i
Id”xcZ a*w.
(4.3)
Dimensional regularization and renormalization (minimal subtraction) are used. The renormalized fields occurring in the effective action are defined as follows: 0, = ~;y3$,, A WR=z-“*A 3 NB’ 595/134/24
(4.4) (4.5)
268
GRAHAM
M. SHORE
with
Z, = 1 - (6 - 2~) z,=1+-y-,
2 ei 3 (4n)
e’ pqn-4’
1
(4.6)
1 n -4
while the coupling constant counterterms are given by
2 eB=p2-“/2eR I--+( 1= 2-*2W2e ABEp4-n jl. (l+~\~$,j -&)R7 -411 ei ‘2(4x)2n-4’
l
(4.8)
(4.9)
I
Also, (4.10)
czB =Z,a,
To one loop order, the effective action I’[Ff,,
@] is
(4.11) Here CS is the matrix of second functional derivatives of the action with respect to the fields A, and $; i.e., (4.12) with
and CPGis the ghost operator, QG = (Ma-8’)
4x9 Y).
(4.17)
MEISSNER
EFFECT
IN
GAUGE
THEORIES
269
(Notice that in the definition and evaluation of these determinants we have performed a Wick rotation to the metric (++++).) Again, we expect to find two phases characterized by (a) a non-vanishing constant electromagnetic field with 4 = 0 and (b) zero electromagnetic field with Q taking a constant non-vanishing vacuum expectation value. Both of these field configurations satisfy the constraints 3, F,,” = 0 and D,@ = 0, which we now assume. For a phase (a) configuration, the matrix @ reduces to
g= -f:b -a2gpu + (1 -0
(4.18)
while for a phase (b) configuration,
To evaluate the determinants of these operators, we use the configuration space heat kernel method together with dimensional regularization. While in the simplest cases, such as the analysis of the Coleman-Weinberg mechanism presented below, this is more cumbersome than alternative momentum space approaches, it is ideally suited for calculations involving specified background fields. In particular, we believe that the formalism developed here will be useful in generalizing to other background fields, such as nonconstant electrpmagnetic fields and even gravitational fields 1141. The main points of the method are as follows. Given a differential operator g, the corresponding (heat) kernel Y(x, y; t) is defined as follows: (4.20)
e, Y;0)= 4x9Y).
(4.21)
The Green’s function G(x, y) of g, defined by gG(x, Y) = 6(x, Y),
(4.22)
G(x, y) = 1” df F(x, y; t).
(4.23)
is given in terms of P by
0
It is also straightforward
to show that logdetg=-
m df t - ’ Tr .Y(x, y; t), I0
(4.24)
270
GRAHAM
M. SHORE
where Tr denotes a functional trace over x, y and a trace over all indices on 5. As usual, the ultraviolet divergences appear as poles in (n - 4)-l, and are removed by the conventional minimal subtraction scheme. 4(i). Broken Symmetry Phase We first evaluate the effective action and energy density in the broken symmetry phase (b). As an illustration of the heat kernel method in a familiar example we use it here to perform the one loop calculation. (For a more conventional derivation using momentum space methods and dimensional regularization, see e.g., Ref. [ 141). The extension to two loops is made by analyzing the results of a paper by Kang [ 151 on the gauge independence of the scalar to vector mass ratio to two loop order. Consider the operator .?S of Eq. (4.19). Log det 9 is evaluated as follows: log det g = log det gab + log det gPV + 1% det (b - G,,%,,
(4.25)
W%J,
where G,, and G,, are the Green’s functions for the operators gac and PZ,“, respectively. Defining the projection operator (4.26)
Pab = (62, - (ewe)) and the kernel ~Jx,
y; t) by (4.27) (4.28)
strictly, in the notation of Eq. (4.16),
5d”z~&, 2)‘q&9 y; It follows immediately
0 = -
@&,
,&
Y; 0 1
(4.29)
that
%b(X, y; t) = p,g
(
x, y; t; *fG
1
+ (1 - P),,F
(
x, y; t; $ fG * 1
(4.30)
Here, g(x, y; t; m’) is the kernel for the operator (-a’ + m*), jF(x, y; t; m’) = [ 1/(47r)“/*] t-** e-(x-y)2/4t eCm2’
(4.3 1)
Now, log det gab = -
(4.32)
MEISSNER
EFFECT
IN GAUGE
where the trace is over x, y and a, b. Substituting logdetg,,=-
00dtt-1 I0
271
THEORIES
Eq. (4.30) we therefore find
Tr
$4:)
1 . (4.33)
X,Y
Turning now to gtiV, we find that the corresponding kernel F@‘,,(x,y; t; ei&) be expressed in the form
= {(-L%,, Here R(x,
+ 8, a,) Zx,
y; t) - a,a,X(x,
y; t/aB)} e-‘~@~‘.
may
(4.34)
y; t) is given by the convolution,
R(x, y; t) = d”xG(x, z; 0) S’(z, y; t; 0),
(4.35)
I
where we use the notation G(x, y; m’) for the Green’s function (-8’ + m*), and satisfies the equation a%qx,
of the operator
y; t) = 5(x, y; t; 0).
(4.36)
t) = [ 1/(4n)‘t’2] tl-“2 j1 du ud2-2 e-“(x-y)y4f.
(4.37)
Explicitly, &?yx,
y;
0
It follows that log det @,,” = - O3dt t-‘Tr i0 =--
O”dt t-l z I‘0
F&“(x, y; t; ei&)
(4.38)
{(n - 1) g(x, y; t; ei&)
+ F(x, y; t; a,ej$ fj$)}.
(4.39)
To evaluate the remaining term in Eq. (4.25), we first use the lemmas wtld43~
= -Kpab
(4.40)
and
(4.41)
272
GRAHAM
M. SHORE
to prove
It follows that log
de@,,
-
5,
=logdet
%,,
G,
v %J
]I -asejj)iG
(x, y;*&) (4.43)
.
Finally, the ghost operator gives log det LZ2c= -
dt t-‘F(x,
(4.44)
y; t; 0).
Collecting these results and performing some rearrangement, we find the following expression for the one loop contributions to the effective action, + log det 9 - log det !2G
+ (n - 1) F(x,
y; t; ei&)
- 5(x,
y; t; 0)
I
d”rG(x,z;O)G(z,
y&b;)
1
(4.45)
These integrals may be evaluated using the formula
5codt t-’
g .E?(x, y; t; m’) = &T(-;)
(m’)“‘(d”x.
(4.46)
0
(In evaluating the integral of F(x, y; t; 0), note that
dtt-‘-**=O in dimensional
regularization.)
(4.47)
It may now be checked using Eqs. (4.6) and (4.9) that
MEISSNER
EFFECT
IN
GAUGE
213
THEORIES
the poles at n = 4 in the expression (4.45) are removed by the counterterms in the tree level potential (&/4!) 4;. At this point, however, we follow Coleman and Weinberg in assuming initially that ,I, is of O(ei). Under this assumption, which is eventually justified using the renormalization group, we may to a first approximation drop all but the contribution from the term involving F&X, y; I; ei&), which we see from Eq. (4.46) is of O(ei). The remaining terms are of O(I:) and O(e:L,), and are therefore of the same or higher order in eR as two loop electromagnetic contributions to the effective action. We therefore find the effective action in phase (b),
(4.48) =-
(logy-log4n+y-$)
1 )‘d4x+O(e~).
(4.49)
Extracting the overall volume factor ,I’ d4x yields the negative of the effective potential w, >* As is well known, this expression has an absolute minimum at some non-trivial value of QR, say &, and the vector boson consequently acquires a mass “2, = e,, 4: by the Higgs mechanism. Setting V’(&) = 0, we immediately find the following relation for the scalar coupling, ~ROI> = $ Eliminating
ei@>
(
-log 2
+ log 4x - y + f
)
+ O(ei).
(4.50)
AR, the effective potential is now expressed as V(@x)=&e:q$
(
log%--+
1
+ O(e3
This illustrates the phenomenon of dimensional transmutation. The theory is now parametrized in terms of one dimensionless (e,) and one dimensional (m,) parameter, rather than the two dimensionless couplings evident in the original Lagrangian. Notice that the ‘t Hooft unit of mass p remains unspecified in Eq. (4.51). This is of no significance here since any change in P and hence e,(u) is absorbed into the terms of O(eE). The energy density &b in the broken phase is found by evaluating V(#,) at its minimum +R = (6:. Substituting in Eq. (4.5 I), and expressing the result in terms of the vector boson mass m,, we find Eb= -i [ 1/(4~)~] m4,.
(4.52)
So far we have neglected terms of O(ei) in the effective action (4.49). These will
214
GRAHAM
M. SHORE
give O(ei) corrections to the energy density &b, and must be included in order to calculate the leading corrections to the critical field strengths. The evaluation of these terms, which involve two loop diagrams with internal photon lines as well as the one loop terms of O(e$l,), has been performed by Kang [ 151, and we shall use his results here. Since the renormalization conventions used by Kang are slightly different from those employed here, we shall just quote the result we find for the energy density in the text. Some brief details of the derivation are given in Appendix A. We would however like to emphasize that the O(e$ contribution of the effective potential evaluated at its minimum is indeed gauge independent when expressed in terms of the vector boson mass. This is a non-trivial result, since the effective potential itself is gauge dependent at this order [ 16, 17, 181. We thus find, working to O(ei) in the effective potential, that the energy density in the broken symmetry phase is (4.53) 4(ii). Symmetric Phase We now turn to the evaluation of the energy density in the symmetric phase (a), where the background electromagnetic field is non-zero. Consider the matrix of Eq. (4.18). In this phase, the one loop contribution to the effective action is (4.54)
Ft’ [Fz”, OR= 0] = -f log det ~9 + log det gG =j
I
mdtt-lTI-~ab(X,J’;t)
+$I O”dt t-’
t;
Tr gM,,(x, y;
m dt t-’ Tr L%(x, y;
t;0)
t;0),
t;
(4.55)
where Ffl’,,(x, y; 0) and F(x, y; 0) are as defined in Eqs. (4.34) and (4.31). Their contribution to the effective action is therefore zero by the result (4.47). g=‘,,(x, y; is the kernel for the operator -&,(A:), where the potential A: is such that the corresponding field tensor Fzy is constant; i.e., a,FE, = 0. In fact it is slightly more convenient, and of course entirely equivalent, to revert to using the complex scalar field notation at this point, rather than continue to work with the real components. We therefore evaluate Pi”’ instead as pF)[Ff,,
t)
& = 0] = la dt t-’ Tr 9(x, y; t), 0
where here 9:(x, y;
t)
is the kernel for the operator -0’
with D, = a,, - ie,AE.
(4.56)
MEISSNER
$(x, y; t) is given explicitly presented in Appendix B.
EFFECT
IN GAUGE
215
THEORIES
by the following expression, the derivation of which is
.9(x, y; t) = [l/(471)@] t-w2 @(x, y) X exr+b(x
- y),(eRFR cot eRFRt),,(x--
u),
- t tr log(e,F, t)-’ sin e,F,t},
(4.57)
where Fi denotes the matrix FE,, F:, , and the trace tr is over ,u, V. @(x, y) is the phase factor, 0(x, y) = exp i XAf(z) IY
dz,,
(4.58)
the integral being along the straight line path from y to x. We therefore find I’?) = [ 1/(4n)“l’] jm dt t-‘-w2
e -[trbg(eBFBt)-‘Sin
@sty2
(4.59)
0
The ultraviolet divergences in this expression are contained in the first two terms in an expansion of the exponent in powers of Fit’. Splitting the integral accordingly, and temporarily introducing an infrared regularization, we may write 1 +keit2trFi) dt
e-m*t
t-3(e[trlog(eRFRt)-‘sineRFRtY2
_
1
+ (47$ --seit2trFi)
(4.60)
where in the second integral, all quantities are evaluated in four dimensions. It is now convenient [7] to introduce the scalar and pseudoscalar invariants, Xi and Y2, that can be formed from the field tensor Ft,, , viz., sr; = ;F;,F~,” x22,4
= f(p
*FRPUFR”“=E.
where the dual tensor, *Ff, = $wuabFR”B and defining
- ~~1,
(4.6 1)
B9
(4.62)
and s0123= 1. In terms of these invariants,
x2 = 2(‘5 + W2),
(4.63)
276
GRAHAM
M. SHORE
it may be shown (see Appendix C) that ph Substituting
(eRFRt)-‘sineR~RW= e; t2y*(Im
into Eq. (4.60) and integrating
Pia) = lim
I
Lx4 1 --++
mL0
3
1
(4.64)
then gives 2
1
(;;j2
(4792 n -4 ei t’&
+&~~dtepm2’tp3
cosh eRlx)- 1,
Im cash e, tX
(
1
log$-log4n+y
- 1 +-+e:t2*
d4x.
(4.65)
The pole term is cancelled by the field renormalization counterterm in the tree level action iFB (see Eq. (4.7) for Z,). The complete effective action in the symmetric 4 P” FE“” phase to one loop order is therefore p’)[F;v,
hi = O] =9&j
(4.66)
d4x,
with the effective Lagrangian
P%(m2h2) cc X
I0
dte- dt t -3
ei t ‘R2
Im cash eRtX
- 1% 47r + Y) + +y2 - 1 +feit2*
. )I
(4.67)
Unfortunately, it is not possible to evaluate the integral in Eq. (4.67) analytically for general values of the electromagnetic tield. However, an analytic expression may be obtained in the special cases of pure electric and magnetic fields, and since these cases contain most of the interesting physics we shall restrict ourselves to them. Consider first a pure magnetic field. Then X1 = $B2, X2 = 0, and we find ei t2Y2(Im cash e, tX)-’ Substituting
= e, tB(sinh eRtB)-‘.
in Eq. (4.67) and resealing the integration
Ye,(B) = --fB2 + B . B,, + lim
re:B_1
++I 1 6 (4n)
X
00h I0
e-(m2/e,&xX-3
--
(4.68)
variable to x = eRtB, we find
(log(m2/fi2) - log 471+ y) + $$ 1+$x2
)I
.
(4.69)
MEISSNER
EFFECT
IN GAUGE
277
THEORIES
The integral can now be performed using the formula [ 191,
= --4& (-I,?)
+ (+-:;)
log+g,
(4.70)
where &(z, q) is the generalized Riemann zeta-function. We need the property, c‘Q(z,t)=(2’-
(4.7 1)
1)5&).
Substituting these results into Eq. (4.69), and removing the infrared cutoff (the limit m2 + 0 is well behaved) we thus find Q’&(B) = -;B2 + B . B,,
f$
- log 4n + y - 1 + 12&(-l))
A similar analysis applies for a pure electric field. Here, K = tE2, ei t2S7;(Im cash e, LX- ’ = eRtE(sin eRtE)- ‘.
.
(4.72)
.& = 0, and (4.73)
In this case we find Y&(E)
= fE* -E
eE,,
1 eiE* -xpp
1% +log4n+y-
1 + 12cx(-1)
)
+-&%E’. .
(4.74)
Notice the occurrence of the imaginary part in L&“&E). Now we require the energy density E in these two cases. This is the Too component of the energy - momentum tensor, which to one loop order may be expressed as (4.75) For pure electric or magnetic fields this reduces to (4.76)
E(B) = -%r(f%
(4.77) and we find c(B)=+B2-B.B,,--
1 e:B’ 6(4n)Z
Q 1% 7-c
P
(4.78)
278
GRAHAM
M.
SHORE
and e(E)=+EI-E.
E,,-
1 eiE2 a-&y
(4.79)
1%
where we define c=-log4nty-
(4.80)
1 t 126X(-l).
The value of the field B for which the energy density is minimized calculated, giving eRBcl tct; 1% 7 P
1)
.
is now simply
(4.8 1)
Substituting back into Eq. (4.78) gives the energy density in the symmetric phase (a) expressed in terms of the external source B,,. In fact, it is more instructive to express this result in terms of the actual magnetic field B in the system. We find E,(B) = -+B2+-+
1 e2B2 6 (4a)
log -eRB p2
(4.82)
Similarly, s.(E)=-+E2+-+-
1 e2B2 6 (47r)
log
(4.83)
The presence of an imaginary part in Y&(E) or e,(E) indicates that the vacuum is decaying through the pair production of charged scalar particles in the background electric field E. The probability per unit time per unit volume of a pair being produced is [7] (4.84) 4(iii). Phase Transition
and Critical Fields
Finally we are in a position to demonstrate beyond the tree level the existence of a phase transition from the broken to the restored symmetry phase in massless scalar electrodynamics with an externally applied magnetic field, and to calculate the critical field strength at which the transition occurs. The case of an external electric field is also discussed. We have shown the existence of two possible phases: (a) the symmetric phase, characterized by a non-vanishing constant electromagnetic field with the scalar field vacuum expectation value zero, and (b) the broken symmetry phase, characterized by
MEISSNER
EFFECT
IN GAUGE
279
THEORIES
a vanishing electromagnetic field and a constant scalar field vacuum expectation value. The phase which is actually realized for a given external field is that with the lower energy density. The critical fields are therefore obtained by equating the energy densities E, and et, (in the case of the electric field Re E, is used) given in Eqs. (4.53), (4.82) and (4.83). We find the following results for the critical fields B, and E,: (log$+c)
+O(e;))
(4.85)
and log$+c+
2
(4.86)
+ 2y + log 3 - 2 log 2 - 76/3.
(4.87)
where c = 24[;(-1)
Notice that in the form given, both sides of Eqs. (4.85) and (4.86) are renormalization group invariants, i.e., independent of the unit of mass ,u. The picture of the phase transition which emerges is therefore as follows. With a small external field the theory is in a phase of broken gauge symmetry. The spectrum comprises a massive vector and a massive scalar boson, their masses being related by rn; = (3efJ8n2) rn& + O(ei).
(4.88)
In this phase the measured electromagnetic field is zero, the external field being screened. If the external field is magnetic, then as it is increased, a first order phase transition occurs to a restored symmetry phase. The electromagnetic field within the system jumps discontinuously to the critical value given in Eq. (4.85). The spectrum is now that of a massless scalar boson in a background magnetic field, together with the massless photon. However, if the external field is electric, then above the critical value E, given in Eq. (4.86), pair production of massless scalar particles occurs, the rate of production being given in Eq. (4.84). These charged pairs will separate and move to the source of the electric field (e.g., parallel capacitor plates). Unless the electric field strength is artificially maintained despite this current, it will be reduced rapidly. Once the electric field has fallen below E, however, the energetically favorable phase is again that of broken symmetry. In this case therefore there is no observable phase transition. The system exists always in the broken symmetry phase.
280
GRAHAM 5.
M. SHORE
DISCUSSION
This completes our discussion of the phase transitions in scalar electrodynamics. Although we have only performed explicit calculations of the quantum effects for the special case of massless scalar electrodynamics (the Coleman-Weinberg model) we expect the qualitative picture described in the last section to be valid for the whole parameter range mH < m,. The phase diagrams shown in Figs. 2 and 3 remain qualitatively valid for an external magnetic field, except that the region mH/mw= 0 is excluded by the Weinberg lower bound [20] on the mass of the scalar boson. For an external electric field the system is always in the broken symmetry phase. We should emphasize that our results depend on our assumption that for mH< m, only the two phases described exist. In particular, the first order nature of the phase transition is not really a result of our calculation but rather a direct consequence of this assumption. However, we believe the assumption to be true. In the type II case, the presence of a third phase, characterized by vortex solutions of the field equations, is signalled by the inconsistency discussed in Section 3 in the scalar propagator when m,>m,. No such inconsistency is apparent in our treatment of the type I case. Finally, we would like to comment on the distinction between the phase transition discussed here and that occurring in spontaneously broken non-Abelian gauge theories, such as, for example, the Weinberg-Salam model, in a background electromagnetic field. In scalar electrodynamics, in the broken phase the vector bosons corresponding to the external field are massive and the field is screened. However, in the non-Abelian models the U( 1) symmetry of electromagnetism is left unbroken by construction and the photon is always massless. The background electromagnetic field may then be increased continuously until again a first order phase transition occurs to a restored symmetry phase. Notice, however, that in the broken phase below the critical point the background field takes a non-zero value and is not screened. The mechanism underlying the phase transition is therefore quite different from the Abelian case, and is in no sense the analogue of the Meissner effect in superconductivity. The contrast between the two cases is further emphasized by the difference in the critical field strengths. In massless scalar electrodynamics to lowest order we find whereas in a non-Abelian model, e.g., the massless Georgi-Glashow CFfhvLritNrn4W~ SU(2), model, the critical field strength is given by e’(~~,),,it - m&. A full discussion of the phase transition in non-Abelian theories may be found in the subsequent papers in this series [21].
APPENDIX
A
We show here how to extract the result (4.53) for the energy density Ed in the broken symmetry phase from the calculation of Kang [ 151. Notice first that in Kang’s work, logarithms of the coupling constants are absorbed into the definitions of the renormalized parameters, which are defined with renor-
MEISSNER
EFFECT
IN GAUGE
281
THEORIES
malization point M. Also notice that he takes the scalar four point coupling to be (n/6)0” rather than (J/4!W4. To O(e”), he quotes the following result for the effective potential (Eq. (2.15) Ref. [ 151) 1 V(4) = yp4”
1 1 + --(u2Aez+a3e4)fb4(log$-+) (4n)2 8
1 1 ----e6g4
(A-1)
+ (47~)~ 8
(We omit the suffixes on renormalized quantities here.) u2, uj , b,, c, and d, are all known, possibly gauge dependent, constants. The scalar coupling is eliminated using the equation (Eq. (2.16) Ref. [ 151) --f--a3e4 L = (47q -
(2a,a,
+&e6(a,a,-b,-fc,)
(l-log-$)
t
t
c,
d4)
4sfl
log -@ +
(u2u3
-
d4)
4: log2 -@’
64.2)
where do is the position of the minimum of V(4). We therefore find the following form for the energy density, 1 1 E= v(4,) = (4r)2 se”Cf, +&
Substituting
I
1 --a3 3
[ -- 1
2 a2a3-Tc4
1
for the constants, given explicitly
a, = -2a,
u3 = 6,
t
; -a2a3-4
) log%]
1
(A.3)
by
c4 = -24813 t 24a,
d, = 20 - 12a
(A.41
gives a superficially gauge dependent result. However, the energy density is to be expressed in terms of the vector boson mass, rather than the unphysical quantity #,. The relation between these is (Eq. (2.18), Ref. [15]) n&E
22 7e4& e lo + (4:)
I
0, +zJ+
(Y2+z2)logz
4: I
T
(A-5)
with y, = -41/18,
Y2
1 = 57
zl=-$
+a,
z2 = 3 - a.
VW
282
GRAHAM
M. SHORE
We may now reexpress .5 in terms of m, and the renormalized form 1 1 &=-8(4n)2p3+
coupling e in the
2 1 + c (qen)’ ’ ( )
1
where c = -2/a,
- $z*u3 - jc, + u,(y, + z,)
+ ($z2u3 -d,
+ u,(y,
+ z2) log(&/M*)
+ O(e’).
G4.8)
Substituting for the constants, we find that the dependence on log(&M*) cancels, and the remaining contribution to C is independent of the gauge fixing parameter a, as indeed it must be if E is to be interpreted as a physical energy density. We thus find + O(e”) .
(A-9)
Notice finally that we have not so far had to specify the renormalization point M used in the definition of e in Eq. (A.9). Such dependence may be absorbed into the O(e”) corrections, and is therefore of no significance here.
B
APPENDIX
We present here the derivation
of the kernel 9:(x, y; t) for the differential operator
D*, where D, = a, - ieA,, and the potential A,, is such that the corresponding field tensor FM” satisfies a,Fp, = 0.
First define the phase factor, @(x, y) = exp ie
(B-1)
xA,(4dq,, I Y
the integral being along the straight line path from y to x. We may then prove the following result [ 2 11, D,W,
Y) = @(x9 YHGP) eF,,,(u)(x - v), + (i/3) @P,,,(v)(x
- Y)&
- ~9, + 0(x - ~4’1.
03.2)
. . Thus for a field satrsfymg a,F,,, = 0, we find
D, @(x9 v) = (i/2) @(x9 Y) eF&9@
- ~1,.
(B.3)
THEORIES
283
-o?qx, y;t)= - @(x9 aty;t),
(B-4)
MEISSNER
EFFECT
IN
GAUGE
The kernel 9(x, y; t) is defined by
9(x,y;0)=qx,Y). We now make the following Ansafz q-T
[22] for g(x, y; t):
y; t) = [1/(47V’] x expki(x
Substituting
(B-5)
@(x9 Y) - YLA,,(W
- YL + Wb
P-6)
into Eq. (B.4), and using Eq. (B.3) gives
f (x -
Y),
(fffg
+A,nAn,
+ e*F,,tF,,
+ 2ieF,,A,,
(x - y),
(B.7) We therefore look for functions A and C satisfying G!A < +A,,A,,
+ e2FplF,,,, = 0, $++A,,=o.
P.8)
(B.9)
Solving these equations subject to the boundary condition (B.5) (or equivalently, requiring consistency in the limit F,,” = 0 with the expression in the text for P(x, y; t; 0)) yields the solutions A,,(t)
= (eF cot eF&,
C(t) = -f tr log(eFf)- ’ sin eFt - (n/2) log t.
(B. 10) (B.11)
Finally, we must check that with this form for A, the remaining term in Eq. (B.7), -fie(x - y),F,,,A,Jx - y),, vanishes. This is simply seen by observing that A,, contains only even powers of F,,, and therefore that the product E;,A,, is antisymmetric on p, V. We therefore find the following expression for the kernel g(x, y; t): Lqx, y; t) = [ 1/(47r)‘4/2] t-??(X, x
595/l
3412.5
exp{-i(x
y)
- y),(eF cot eRt),,(x - y), - f tr log (eFt)-’ sin eFf). (B. 12)
284
GRAHAM M. SHORE APPENDIX
C
In this appendix we derive the formula &”
log(eFt)-‘sineFtl
=
e2t2Sr;(Im
cosh
&x)-I
(C-1)
quoted in the text, where X= &(T + 6) y2. This was first derived by Schwinger [7]. The first step is to find the eigenvalues f of the matrix (F2)rU = FpnFA,. The following results are not difficult to establish: I;,,, *F.,v = -g,,~z
9
(C-2)
*F WA*FAu - F,tnFnv = %,A.
(C.3)
Squaring Eq. (C.2) and using (C.3), we find s,,,r:
-G&AyFr-’
(C.4)
= 2SSCJL~
which yields the following equation for the eigenvaluesf,
f2+2&f4y=0.
(C.5)
It follows that F2 has two distinct eigenvalues, occurring with degeneracy two, given by f *, where
f * = -jT; f (x: +.Fy.
(C-6)
It is now simple to see that ,-[trlOg(eFe-‘sineFlYZ = ~2f2df+)V2(f-)V2/sin
e(ft)42t
sin edf-)‘12t,
(C-7)
where
(f*)42 = (i/fi)[(&
+ W2)u2 f (6 - i9#2]
W)
and the required result (C.l) follows immediately. ACKNOWLEDGMENTS I would like to thank Ian ARleck, Michael Peskin and Harvey Shepard for interesting discussions in the course of this work. I am also grateful to the U.K. Science Research Council for a postdoctoral fellowship. REFERENCES 1. A. L. FETTER AND J. D. WALECKA, “Quantum Theory of Many Particle Systems,” McGraw-Hill, San Francisco, 1971.
MEISSNER
EFFECT
2. P. W. HIGGS, Phys. Rev. 145 (1966), 3. S. COLEMAN AND E. WEINBERG, Phys.
B. J. HARRINGTON 5. D. A. KIRZHNITS 4.
AND
H. K.
IN GAUGE
285
THEORIES
1156.
Rev. D 7 (1973), 1888. Nucl. Phys. B 105 (1976), 527. Ann. Phys. (N.Y.) 101 (1976), 195.
SHEPARD,
AND A. D. LINDE, Rep. Progr. Phys. 42 (1979) 389. 7. J. SCHWINGER, Phys. Rev. 82 (1951), 664. 8. G. ‘T HOOFT AND M. VELTMAN, Nucl. Phys. B 44 (1972), 9. G. ‘T HOOFT, Nucl. Phys. B 61 (1973), 455. 10. A. SALAM AND J. STRATHDEE, Nature 250 (1974), 569; 6. A. D. LINDE,
189. Nucl.
Phys.
B 90 (1975),
203;
and
“Proceedings, Summer Study Meeting on Kaon Physics” (H. Palevsky, Ed.), BNL, Brookhaven, 11.
1976. S. MIDORIKAWA,
12. H. B.
University of Tokyo preprint INS Rep. 380 (1980).
Phys. B 61 (1973), 45. Phys. Rev. D 13 (1976), 3333. 14. G. M. SHORE, Ann. Phys. (N.Y.) 128 (1980), 376. 15. J. S. KANG, Phys. Reo. D 10 (1974), 3455. 16. R. JACKIW, Phys. Rev. D 9 (1974), 1686. 17. L. DOLAN AND R. JACKIW, Phys. Rev. D 9 (1974), 2904. 18. J. ILIOPOULOS AND N. PAPANICOLAOU, Nucl. Phys. B 105 (1976), 77. 19. W. DI~ICH, W. TSAI, AND K. ZIMMERMAN, Phys. Rev. D 19 (1979), 20. S. WEINBERG, Phys. Rev. Left. 36 (1976), 294. NIELSEN 13. E. GILDENER
21. 22.
G. M. SHORE, M. R. BROWN
AND P. OLESEN, AND S. WEINBERG,
Nucl.
in preparation. AND
M. J. DUFF,
Phys. Rev.
D 11 (1975),
2124.
2929.
OF PHYSICS
134, 259-285 (1981)
On the Meissner
Effect in Gauge
Theories*
GRAHAM M. SHORE
Cambridge,
Lyman Laboratory of Physics, Harvard University, Massachusetts 02138, and Newman Laboratory of Nuclear CorneN University, Ithaca, New York 14853
Studies,’
Received December 22, 1980
The phase structure of spontaneously broken scalar electrodynamics in an external electromagnetic field is analyzed. With no external field, the spectrum comprises a scalar boson of mass mH and a vector boson of mass m,. If mH < m,, it is shown that in the tree approximation, as the external field is increased, a first order phase transition to a restored symmetry phase occurs, and the critical field strength is calculated. Below the critical point the external field is completely screened, this being the analogue of the Meissner effect in superconductivity. If mH > m,, a, third phase, characterized by vortex solutions of the field equations, occurs. Quantum effects, such as pair production in an electric field, are considered at the one (and two) loop level in the massless theory (the Coleman-Weinberg model). The leading correction to the critical magnetic field strength is calculated, and it is shown that for an external electric field the phase transition does not exist.
1. INTRODUCTION This paper is an investigation of the phase structure of scalar electrodynamics in the presence of an external electromagnetic field. This theory is known to be the relativistic quantum field theory analogue of the Ginzburg-Landau model of superconductivity [I], and so we expect many of the interesting physical effects associated with superconductivity to occur also in scalar electrodynamics. In particular, if a sample of superconducting material is placed in a sufficiently small, constant, magnetic field, the external field does not penetrate the sample. This screening of the external field is known as the Meissner effect. However, if the field is increased beyond a critical value, a phase transition occurs from the superconducting to the normal state, and the magnetic field takes a constant, non-zero value within the sample. In addition, for certain types of superconductor (type II), a third phase is possible in which the magnetic field forms vortices within the sample. In this paper we demonstrate the analogous results in scalar electrodynamics. We also make some remarks clarifying the distinction between the phase transition in scalar electrodynamics and the symmetry restoring phase transition which occurs in * Research supported in part by the National Science Foundation under Grant PHY 77.22864. ’ Permanent address.
2.59 0003.4916/81/080259-27$05.00/O Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in any form reserved.
260
GRAHAM
M. SHORE
spontaneously broken, non-Abelian gauge theories in a background electromagnetic field. As is well known, for a certain range of parameters scalar electrodynamics exhibits spontaneous symmetry breakdown [2,3] with the vector bosons acquiring mass through the Higgs mechanism. If we now introduce a source which would in the absence of symmetry breaking give rise to a constant electromagnetic field, we find that it is screened, the measured field within the system being zero. This is the gauge theory analogue of the Meissner effect, and occurs because the vector bosons corresponding to the external field are massive. As the source is increased, the tree approximation indicates that a first order phase transition occurs to a restored symmetry phase in which the vector bosons are massless, and the electromagnetic field jumps to a constant value. In the case where the external field is electric however, pair production of charged scalar particles from the vacuum occurs once the critical point is reached. The effect of this phenomenon on the existence of the phase transition is discussed in Section 4. For the range of parameters such that the mass of the scalar boson mH is less than or equal to that of the vector boson m, we conclude that only these two phases are possible. If m, > m w, however, we find an inconsistency in the form of the scalar propagator which indicates that a third phase, characterized by a spatially varying vacuum expectation value of the scalar field, occurs. In this phase the ground state is described by the Nielsen-Olesen vortex solutions of the field equations. We restrict our explicit calculations to the type I case, mH < m,. The question of phase transitions in scalar electrodynamics has been discussed before in the literature. In particular, Harrington and Shepard [4], and Kirzhnits and Linde [5,6], have exploited the analogy with the Ginzburg-Landau model to deduce to lowest order the critical field strengths from known results in superconductivity. In contrast, we present a self-contained discussion of the phase transition in the relativistic field theory, then go on to analyze the one (and two) loop quantum effects. The computation of the rate of pair production in a background electric field was performed long ago by Schwinger [ 71, and indeed our methods of calculation owe much to his work. The paper is organized as follows. Section 2 contains a brief review of the calculation of critical field strengths in the Ginzburg-Landau model. Guided by these results, we present in Section 3 a discussion of the phase transition in scalar electrodynamics in the semi-classical, or tree, approximation, and find the critical field strengths. In Section 4, we extend the analysis to include quantum effects at the one (and for the broken phase, two) loop level. These calculations are performed using the heat kernel method for evaluating functional determinants modified to allow the use of dimensional regularization [8, 91. Finally, Section 5 contains a brief discussion of our results and some remarks on the distinction between the phase transition in a background electromagnetic field for Abelian and non-Abelian gauge theories [6, 10, 111.
261
MEISSNER EFFECT IN GAUGE THEORIES
2. THE GINZBURG-LANDAU
MODEL
OF SUPERCONDUCTIVITY
In order to motivate our discussion of the Meissner effect in relativistic scalar electrodynamics, we begin in this section with a very simple derivation of the critical magnetic field strength in the Ginzburg-Landau model of superconductivity [ 11. This is a non-relativistic phenomenological theory, which serves as a good approximation to the standard BCS theory in the neighborhood of the critical temperature. In this model, the Helmholtz free energy density may be expressed as F = jBZ + (1/2m*)(D#)*
. (D)) + I’(#*$),
(2-l)
where #(x) is a complex order parameter, B = V x A is the (microscopic) field, D = -iV + e*A, and w*$)=
J%+4~12+f~ld14,
magnetic (2.2)
e* and m* are the effective charge and mass, I’,, is the free energy density of the normal state in zero magnetic field, and Q and b are real, temperature dependent, phenomenological constants. For a < 0, V(#*$) may have a nontrivial minimum at Iqq*=-a/b=u*.
(2.3)
Now suppose an external magnetic field H is applied to the system. In this case, the appropriate thermodynamic potential to be considered is not the Helmholtz but the Gibbs free energy density, which may be defined as G=F-B-H.
(2.4)
The requirement that G be stationary with respect to variations parameter and vector potential yields the equations
(1/2m*)D*$+
vl(#*d)#=O,
$*fj$+Vx(B-H)=O,
of the order (2.5) P-6)
while the corresponding surface terms vanish if the following boundary conditions are satisfied n .D#=O, nx (B-H)=O,
(2.7) (2.8)
n being the surface normal. Imposing V x H = 0, these reduce to the standard Ginzburg-Landau equations. There are two simple solutions to these equations,
(4
B = H,
l$l=@
(b)
B = 0,
IdI=u*
(2.9) (2.10)
262
GRAHAM
M. SHORE
Solution (a) clearly corresponds to the normal state, in which the magnetic field permeates the system. Solution (b), which describes a state in which the magnetic field within the system is zero (the Meissner effect), corresponds to the superconducting state. The Gibbs free energies for these states are Ga-- -+H’
t V(O),
(2.11) (2.12)
G b = V(u’).
The system will exist in the state with the lowest Gibbs free energy. To determine the critical external field IHI, at which the phase transition from the superconducting to the normal state occurs, we therefore equate G, and G, evaluated at IHI = IH Ic yielding H, = -2( V(u’) - V(0)) = a2/b.
(2.13)
Clearly, G, = G, + f(H’
- H,2).
(2.14)
For external fields less than JH Jc, G, < G, and so the system is in the superconducting state, while for fields exceeding lHlc the system is in the normal state. Since the order parameter changes discontinuously at the critical point, the phase transition is first order. The phase transition described above is characteristic of a type I superconductor. For certain values of the parameters, however, a further class of solutions may be energetically favorable. These vortex solutions describe a state in which the magnetic field permeates the system in the form of thin flux tubes, and the order parameter ( is spatially varying. Materials which exhibit this mixed state are known as type II superconductors. Whether a given material is of type I or type II is determined by the ratio of the two length scales which characterize the system-the coherence length r, which provides a scale for variations of the order parameter, and the magnetic field penetration depth A. Defining K = A/<, the condition for a type I or type II superconductor is
d-2lc<
1,
type I,
d- 2K> 1,
type II.
(2.15)
In terms of the parameters already introduced, fiu
= 2m*b’/2/e*.
(2.16)
The mixed state is energetically favorable for values of the external magnetic field in the range lHlcl < (H( < [Hlc2, where IHlc2=d5W,
263
MEISSNER EFFECT IN GAUGE THEORIES
FIG. 1. Phase diagram for a superconductor in the Ginzburg-Landau and K with ( fixed.
model as a function of / HI
and ] H ICl is less than ]H(,. However, since a full discussion of the vortex solution and the determination of ]Hlc, involves analyzing the Ginzburg-Landau equations numerically, we shall not present any details here. The phase diagram for the model as a function of ]HI and ic for fixed < is shown in Fig. 1.
3. PHASE TRANSITIONS
IN SCALAR ELECTRODYNAMICS
The Ginzburg-Landau model serves as an analogue of scalar electrodynamics, and all the effects just discussed have their counterparts in this relativistic quantum field theory. In particular, we shall show in this section that as the external electromagnetic field is increased there is a phase transition in scalar electrodynamics from a broken symmetry phase, in which the vector bosons are massive and the external field is screened, to a restored symmetry phase with massless vector bosons and a long range electromagnetic field. Further, depending on a ratio of length scales (in this case the scalar and vector boson masses at zero external field), the model may be of type I or type II, the type II model having an extra phase described by the Nielsen-Olesen vortex [ 121 solution of the field equations. The Lagrangian for scalar electrodynamics is rp = GF,,F~~” where F,,” is the electromagnetic
- $(D,#)*(Dy)
- v(@*#),
field tensor, D, = a, - &I,,
Pq*qq = v, + $n*(P + (A/4!) 4”.
(3.1)
and the potential (3.2)
(Since the discussion in this section is at the semi-classical level, we omit the distinction between bare and renormalized parameters here.) For negative values of m’, this potential has a non-trivial minimum at I# I2 = -6m2/3, = v*.
(3.3)
264
GRAHAM
M. SHORE
To incorporate the effects of an external electromagnetic source term coupled to F,,,, viz.,
field, we introduce
a
The source is chosen so that in the absence of scalar fields it gives rise to a specified field stength F$. (Notice that this reduces to the familiar source term LJpA,, , with J” = a,F$‘ for fields F”$ which vanish at infinity.) This extra term is the analogue of the additional B . H term included in the Gibbs free energy in the Ginzburg-Landau model. The vacuum expectation values of the fields are determined from the requirement that the modified action be stationary with respect to variations of A, and 4. This yields the equations of motion D*# - V’(#*# i?,(F“” -F;,“)
= 0,
(3.5)
+ i#*@‘# = 0.
(3.6)
As before, there are two simple solutions to these equations:
The first solution (a) describes the conventional symmetric phase. The source gives rise to a long range electromagnetic field, the vacuum expectation values of the scalar field is zero, and the vector bosons are massless. Solution (b) describes the phase with broken gauge symmetry. The scalar ‘field develops a vacuum expectation value and the vector bosons acquire a mass by the Higgs mechanism, thus screening the external electromagnetic field. The measured electromagnetic field in this phase is zero. This is the Meissner effect in scalar electrodynamics. Which phase is actually realized for a given external field F”,‘” is determined by minimizing the energy density E of the fields. This is given by the TO, component of the energy-momentum tensor T,, (our metric convention is (-+++)), where
T,, = --+@U)*U’,/) - f(D,O*(D,O
In this paper we are concerned with the effects of a constant external field in scalar electrodynamics. In fact we will restrict ourselves to the special cases of a pure electric or pure magnetic field, since these cases are sufficient to illustrate most of the interesting physical effects, and yet the one loop corrections can still be calculated analytically.
MEISSNER
EFFECT
From Eq. (3.9) the corresponding and (3.6) are &=fB'-B.
B,,+
IN GAUGE
THEORIES
265
energy densities for fields satisfying Eqs. (3.5) V(qd*$),
pure B field,
(3.10)
pure E field.
(3.11)
and E =$I?'--*EC1
+ V(#*#),
(Here we simply use the notations E, B for 1E I, 1B I.) It is clear that for a sufficiently large external field, phase (a) will be energetically favorable. To determine the critical field strength B,, we therefore equate E, and et, evaluated at B,, giving Bf = -2( V(u')-
V(O))= 3m4/A
(3.12)
an identical result holding for E,. Scalar electrodynamics in an external magnetic (electric) field therefore undergoes a first order phase transition at B, (E,) from a broken symmetry phase in which the external field is screened to a restored symmetry phase. In the broken symmetry phase the spectrum comprises a vector boson of mass m, and a scalar boson of mass m,. In terms of these masses, we can reexpress the critical field strengths as Ef=B:=(
l/4$)
rnirn;.
(3.13)
The spectrum in the restored symmetry phase requires a little more discussion. First of course there is the massless photon. The propagator for the scalar particles is the inverse of the operator (-0’ + m’). At first sight it appears that with m* negative this operator may’ have negative eigenvalues and the propagator may be tachyonic, indicating some inconsistency. In fact this is not the case. The eigenvalues of (-D* + m’) for a purely magnetic background field (taken along the z-axis) are (kf + m* + (2n + I)&), n = 0, l,..., and so the smallest eigenvalue is (eB + m*). For arbitrarily small fields B this is indeed negative. However, for fields greater than the critical value (3.13), e*B* >+rn&rn&. Recalling that mf,= -2m*, we see that the eigenvalue spectrum of the scalar wave of operator is positive, provided we are in the type I parameter range mH < m,. For values of the parameters in the potential (3.2) such that m,Q m,, only the phases described by solutions (a) and (b) of the equations of motion are possible. For mH > m,, however, a third phase is energetically favorable for a certain range of external field strengths, this new phase being described by the Nielsen-Olesen vortex solutions of the field equations. Since a detailed analysis of this phase, which is analogous to type II superconductivity in the Ginzburg-Landau model, would involve us in numerical calculations involving the vortex solutions, we shall not consider this parameter range in this paper. However, from the arguments above, we may deduce quite simply the value of the
266
GRAHAM
M.
SHORE
01 El
FIG. 2. Phase diagram in the tree approximation and m,Jm, with mH fixed.
BIE)
for scalar electrodynamics as a function of B(E)
I
Broken, Ordered, PlWS-2 0
I
FIG. 3. Phase diagram in the tree approximation and m&q,, with m, fixed.
hi
‘MW
for scalar electrodynamics as a function of B(E)
upper critical field at which a phase transition from the vortex state to the restored symmetry state occurs. This critical field is given by the condition that the scalar wave operator is positive; i.e., B > (1/2e)mi. We can therefore plot a phase diagram for scalar electrodynamics as a function of B (or B) and m&n,, where we take mH fixed. This is shown in Fig. 2. In Fig. 3 we have made the analogous plot with m, held fixed. The upper critical field is
the lower critical field requiring a numerical analysis for its evaluation.
4. QUANTUM
EFFECTS AND CRITICAL
FIELDS
So far we have discussed the phase transitions occurring in scalar electrodynamics with an external electromagnetic field in the tree approximation.We now show how these results are modified when we go beyond the tree level and include quantum corrections.
MEISSNER
EFFECT
IN GAUGE
267
THEORIES
The first major effect is that with zero external field the gauge symmetry is broken for positive, but sufficiently small, values of the scalar mass parameter mR in the Lagrangian. This was first shown by Coleman and Weinberg [3]. In particular the choice mR = 0 gives a theory in which the symmetry is broken due to one loop radiative corrections, and in which the ratio of the scalar and vector boson masses is calculable. This parameter choice is both algebraically simple and of considerable intrinsic interest, e.g., in connection with the gauge hierarchy problem [ 131, and we shall restrict our detailed calculation to this case. A further purely quantum effect, which does not appear in the semiclassical calculation of Section 3, is that for certain values of the external field the effective action develops an imaginary part [7]. In particular, this occurs in the case of a pure electric field. This is interpreted as indicating the pair production of charged scalars from the vacuum due to the externally applied field. In this section we compute the quantum corrections to the effective action. As before, there are two phases corresponding to broken and restored symmetry, and the critical fields are evaluated by equating their energy densities. In the symmetric phase, the leading quantum corrections come from one loop diagrams, and are evaluated using the heat kernel method. In the broken phase however, since we are considering the massless (Coleman-Weinberg) theory, the lowest order contribution to the energy density only arises at the one loop level, and so to compute the leading corrections we work to two loops. The generating functional W[Ft”, 5,] is given by exp W[F&, J,] =
i
G@A, G9#,A[A] I
x exp i d%(L$ + (lpa,)(a,A,)* 1,
+ $I;,,F:,”
- J,4,>
I
, (4.1)
where the Lagrangian % = -+‘,,J“”
- t~~,~W,~),
- @B/41) #4,
(4.2)
with 0;” = a, gab + eB.PbA,, . Here we have expressed Y0 in terms of the real and imaginary parts of the complex scalar field; i.e., 4 = 9, + &, with 4,) a = 1,2, real, and 4’ = d,#,. The ghost Jacobian A[A] may be expressed as A [A] = j GZwj 9% exp(l/&)i
Id”xcZ a*w.
(4.3)
Dimensional regularization and renormalization (minimal subtraction) are used. The renormalized fields occurring in the effective action are defined as follows: 0, = ~;y3$,, A WR=z-“*A 3 NB’ 595/134/24
(4.4) (4.5)
268
GRAHAM
M. SHORE
with
Z, = 1 - (6 - 2~) z,=1+-y-,
2 ei 3 (4n)
e’ pqn-4’
1
(4.6)
1 n -4
while the coupling constant counterterms are given by
2 eB=p2-“/2eR I--+( 1= 2-*2W2e ABEp4-n jl. (l+~\~$,j -&)R7 -411 ei ‘2(4x)2n-4’
l
(4.8)
(4.9)
I
Also, (4.10)
czB =Z,a,
To one loop order, the effective action I’[Ff,,
@] is
(4.11) Here CS is the matrix of second functional derivatives of the action with respect to the fields A, and $; i.e., (4.12) with
and CPGis the ghost operator, QG = (Ma-8’)
4x9 Y).
(4.17)
MEISSNER
EFFECT
IN
GAUGE
THEORIES
269
(Notice that in the definition and evaluation of these determinants we have performed a Wick rotation to the metric (++++).) Again, we expect to find two phases characterized by (a) a non-vanishing constant electromagnetic field with 4 = 0 and (b) zero electromagnetic field with Q taking a constant non-vanishing vacuum expectation value. Both of these field configurations satisfy the constraints 3, F,,” = 0 and D,@ = 0, which we now assume. For a phase (a) configuration, the matrix @ reduces to
g= -f:b -a2gpu + (1 -0
(4.18)
while for a phase (b) configuration,
To evaluate the determinants of these operators, we use the configuration space heat kernel method together with dimensional regularization. While in the simplest cases, such as the analysis of the Coleman-Weinberg mechanism presented below, this is more cumbersome than alternative momentum space approaches, it is ideally suited for calculations involving specified background fields. In particular, we believe that the formalism developed here will be useful in generalizing to other background fields, such as nonconstant electrpmagnetic fields and even gravitational fields 1141. The main points of the method are as follows. Given a differential operator g, the corresponding (heat) kernel Y(x, y; t) is defined as follows: (4.20)
e, Y;0)= 4x9Y).
(4.21)
The Green’s function G(x, y) of g, defined by gG(x, Y) = 6(x, Y),
(4.22)
G(x, y) = 1” df F(x, y; t).
(4.23)
is given in terms of P by
0
It is also straightforward
to show that logdetg=-
m df t - ’ Tr .Y(x, y; t), I0
(4.24)
270
GRAHAM
M. SHORE
where Tr denotes a functional trace over x, y and a trace over all indices on 5. As usual, the ultraviolet divergences appear as poles in (n - 4)-l, and are removed by the conventional minimal subtraction scheme. 4(i). Broken Symmetry Phase We first evaluate the effective action and energy density in the broken symmetry phase (b). As an illustration of the heat kernel method in a familiar example we use it here to perform the one loop calculation. (For a more conventional derivation using momentum space methods and dimensional regularization, see e.g., Ref. [ 141). The extension to two loops is made by analyzing the results of a paper by Kang [ 151 on the gauge independence of the scalar to vector mass ratio to two loop order. Consider the operator .?S of Eq. (4.19). Log det 9 is evaluated as follows: log det g = log det gab + log det gPV + 1% det (b - G,,%,,
(4.25)
W%J,
where G,, and G,, are the Green’s functions for the operators gac and PZ,“, respectively. Defining the projection operator (4.26)
Pab = (62, - (ewe)) and the kernel ~Jx,
y; t) by (4.27) (4.28)
strictly, in the notation of Eq. (4.16),
5d”z~&, 2)‘q&9 y; It follows immediately
0 = -
@&,
,&
Y; 0 1
(4.29)
that
%b(X, y; t) = p,g
(
x, y; t; *fG
1
+ (1 - P),,F
(
x, y; t; $ fG * 1
(4.30)
Here, g(x, y; t; m’) is the kernel for the operator (-a’ + m*), jF(x, y; t; m’) = [ 1/(47r)“/*] t-** e-(x-y)2/4t eCm2’
(4.3 1)
Now, log det gab = -
(4.32)
MEISSNER
EFFECT
IN GAUGE
where the trace is over x, y and a, b. Substituting logdetg,,=-
00dtt-1 I0
271
THEORIES
Eq. (4.30) we therefore find
Tr
$4:)
1 . (4.33)
X,Y
Turning now to gtiV, we find that the corresponding kernel F@‘,,(x,y; t; ei&) be expressed in the form
= {(-L%,, Here R(x,
+ 8, a,) Zx,
y; t) - a,a,X(x,
y; t/aB)} e-‘~@~‘.
may
(4.34)
y; t) is given by the convolution,
R(x, y; t) = d”xG(x, z; 0) S’(z, y; t; 0),
(4.35)
I
where we use the notation G(x, y; m’) for the Green’s function (-8’ + m*), and satisfies the equation a%qx,
of the operator
y; t) = 5(x, y; t; 0).
(4.36)
t) = [ 1/(4n)‘t’2] tl-“2 j1 du ud2-2 e-“(x-y)y4f.
(4.37)
Explicitly, &?yx,
y;
0
It follows that log det @,,” = - O3dt t-‘Tr i0 =--
O”dt t-l z I‘0
F&“(x, y; t; ei&)
(4.38)
{(n - 1) g(x, y; t; ei&)
+ F(x, y; t; a,ej$ fj$)}.
(4.39)
To evaluate the remaining term in Eq. (4.25), we first use the lemmas wtld43~
= -Kpab
(4.40)
and
(4.41)
272
GRAHAM
M. SHORE
to prove
It follows that log
de@,,
-
5,
=logdet
%,,
G,
v %J
]I -asejj)iG
(x, y;*&) (4.43)
.
Finally, the ghost operator gives log det LZ2c= -
dt t-‘F(x,
(4.44)
y; t; 0).
Collecting these results and performing some rearrangement, we find the following expression for the one loop contributions to the effective action, + log det 9 - log det !2G
+ (n - 1) F(x,
y; t; ei&)
- 5(x,
y; t; 0)
I
d”rG(x,z;O)G(z,
y&b;)
1
(4.45)
These integrals may be evaluated using the formula
5codt t-’
g .E?(x, y; t; m’) = &T(-;)
(m’)“‘(d”x.
(4.46)
0
(In evaluating the integral of F(x, y; t; 0), note that
dtt-‘-**=O in dimensional
regularization.)
(4.47)
It may now be checked using Eqs. (4.6) and (4.9) that
MEISSNER
EFFECT
IN
GAUGE
213
THEORIES
the poles at n = 4 in the expression (4.45) are removed by the counterterms in the tree level potential (&/4!) 4;. At this point, however, we follow Coleman and Weinberg in assuming initially that ,I, is of O(ei). Under this assumption, which is eventually justified using the renormalization group, we may to a first approximation drop all but the contribution from the term involving F&X, y; I; ei&), which we see from Eq. (4.46) is of O(ei). The remaining terms are of O(I:) and O(e:L,), and are therefore of the same or higher order in eR as two loop electromagnetic contributions to the effective action. We therefore find the effective action in phase (b),
(4.48) =-
(logy-log4n+y-$)
1 )‘d4x+O(e~).
(4.49)
Extracting the overall volume factor ,I’ d4x yields the negative of the effective potential w, >* As is well known, this expression has an absolute minimum at some non-trivial value of QR, say &, and the vector boson consequently acquires a mass “2, = e,, 4: by the Higgs mechanism. Setting V’(&) = 0, we immediately find the following relation for the scalar coupling, ~ROI> = $ Eliminating
ei@>
(
-log 2
+ log 4x - y + f
)
+ O(ei).
(4.50)
AR, the effective potential is now expressed as V(@x)=&e:q$
(
log%--+
1
+ O(e3
This illustrates the phenomenon of dimensional transmutation. The theory is now parametrized in terms of one dimensionless (e,) and one dimensional (m,) parameter, rather than the two dimensionless couplings evident in the original Lagrangian. Notice that the ‘t Hooft unit of mass p remains unspecified in Eq. (4.51). This is of no significance here since any change in P and hence e,(u) is absorbed into the terms of O(eE). The energy density &b in the broken phase is found by evaluating V(#,) at its minimum +R = (6:. Substituting in Eq. (4.5 I), and expressing the result in terms of the vector boson mass m,, we find Eb= -i [ 1/(4~)~] m4,.
(4.52)
So far we have neglected terms of O(ei) in the effective action (4.49). These will
214
GRAHAM
M. SHORE
give O(ei) corrections to the energy density &b, and must be included in order to calculate the leading corrections to the critical field strengths. The evaluation of these terms, which involve two loop diagrams with internal photon lines as well as the one loop terms of O(e$l,), has been performed by Kang [ 151, and we shall use his results here. Since the renormalization conventions used by Kang are slightly different from those employed here, we shall just quote the result we find for the energy density in the text. Some brief details of the derivation are given in Appendix A. We would however like to emphasize that the O(e$ contribution of the effective potential evaluated at its minimum is indeed gauge independent when expressed in terms of the vector boson mass. This is a non-trivial result, since the effective potential itself is gauge dependent at this order [ 16, 17, 181. We thus find, working to O(ei) in the effective potential, that the energy density in the broken symmetry phase is (4.53) 4(ii). Symmetric Phase We now turn to the evaluation of the energy density in the symmetric phase (a), where the background electromagnetic field is non-zero. Consider the matrix of Eq. (4.18). In this phase, the one loop contribution to the effective action is (4.54)
Ft’ [Fz”, OR= 0] = -f log det ~9 + log det gG =j
I
mdtt-lTI-~ab(X,J’;t)
+$I O”dt t-’
t;
Tr gM,,(x, y;
m dt t-’ Tr L%(x, y;
t;0)
t;0),
t;
(4.55)
where Ffl’,,(x, y; 0) and F(x, y; 0) are as defined in Eqs. (4.34) and (4.31). Their contribution to the effective action is therefore zero by the result (4.47). g=‘,,(x, y; is the kernel for the operator -&,(A:), where the potential A: is such that the corresponding field tensor Fzy is constant; i.e., a,FE, = 0. In fact it is slightly more convenient, and of course entirely equivalent, to revert to using the complex scalar field notation at this point, rather than continue to work with the real components. We therefore evaluate Pi”’ instead as pF)[Ff,,
t)
& = 0] = la dt t-’ Tr 9(x, y; t), 0
where here 9:(x, y;
t)
is the kernel for the operator -0’
with D, = a,, - ie,AE.
(4.56)
MEISSNER
$(x, y; t) is given explicitly presented in Appendix B.
EFFECT
IN GAUGE
215
THEORIES
by the following expression, the derivation of which is
.9(x, y; t) = [l/(471)@] t-w2 @(x, y) X exr+b(x
- y),(eRFR cot eRFRt),,(x--
u),
- t tr log(e,F, t)-’ sin e,F,t},
(4.57)
where Fi denotes the matrix FE,, F:, , and the trace tr is over ,u, V. @(x, y) is the phase factor, 0(x, y) = exp i XAf(z) IY
dz,,
(4.58)
the integral being along the straight line path from y to x. We therefore find I’?) = [ 1/(4n)“l’] jm dt t-‘-w2
e -[trbg(eBFBt)-‘Sin
@sty2
(4.59)
0
The ultraviolet divergences in this expression are contained in the first two terms in an expansion of the exponent in powers of Fit’. Splitting the integral accordingly, and temporarily introducing an infrared regularization, we may write 1 +keit2trFi) dt
e-m*t
t-3(e[trlog(eRFRt)-‘sineRFRtY2
_
1
+ (47$ --seit2trFi)
(4.60)
where in the second integral, all quantities are evaluated in four dimensions. It is now convenient [7] to introduce the scalar and pseudoscalar invariants, Xi and Y2, that can be formed from the field tensor Ft,, , viz., sr; = ;F;,F~,” x22,4
= f(p
*FRPUFR”“=E.
where the dual tensor, *Ff, = $wuabFR”B and defining
- ~~1,
(4.6 1)
B9
(4.62)
and s0123= 1. In terms of these invariants,
x2 = 2(‘5 + W2),
(4.63)
276
GRAHAM
M. SHORE
it may be shown (see Appendix C) that ph Substituting
(eRFRt)-‘sineR~RW= e; t2y*(Im
into Eq. (4.60) and integrating
Pia) = lim
I
Lx4 1 --++
mL0
3
1
(4.64)
then gives 2
1
(;;j2
(4792 n -4 ei t’&
+&~~dtepm2’tp3
cosh eRlx)- 1,
Im cash e, tX
(
1
log$-log4n+y
- 1 +-+e:t2*
d4x.
(4.65)
The pole term is cancelled by the field renormalization counterterm in the tree level action iFB (see Eq. (4.7) for Z,). The complete effective action in the symmetric 4 P” FE“” phase to one loop order is therefore p’)[F;v,
hi = O] =9&j
(4.66)
d4x,
with the effective Lagrangian
P%(m2h2) cc X
I0
dte- dt t -3
ei t ‘R2
Im cash eRtX
- 1% 47r + Y) + +y2 - 1 +feit2*
. )I
(4.67)
Unfortunately, it is not possible to evaluate the integral in Eq. (4.67) analytically for general values of the electromagnetic tield. However, an analytic expression may be obtained in the special cases of pure electric and magnetic fields, and since these cases contain most of the interesting physics we shall restrict ourselves to them. Consider first a pure magnetic field. Then X1 = $B2, X2 = 0, and we find ei t2Y2(Im cash e, tX)-’ Substituting
= e, tB(sinh eRtB)-‘.
in Eq. (4.67) and resealing the integration
Ye,(B) = --fB2 + B . B,, + lim
re:B_1
++I 1 6 (4n)
X
00h I0
e-(m2/e,&xX-3
--
(4.68)
variable to x = eRtB, we find
(log(m2/fi2) - log 471+ y) + $$ 1+$x2
)I
.
(4.69)
MEISSNER
EFFECT
IN GAUGE
277
THEORIES
The integral can now be performed using the formula [ 191,
= --4& (-I,?)
+ (+-:;)
log+g,
(4.70)
where &(z, q) is the generalized Riemann zeta-function. We need the property, c‘Q(z,t)=(2’-
(4.7 1)
1)5&).
Substituting these results into Eq. (4.69), and removing the infrared cutoff (the limit m2 + 0 is well behaved) we thus find Q’&(B) = -;B2 + B . B,,
f$
- log 4n + y - 1 + 12&(-l))
A similar analysis applies for a pure electric field. Here, K = tE2, ei t2S7;(Im cash e, LX- ’ = eRtE(sin eRtE)- ‘.
.
(4.72)
.& = 0, and (4.73)
In this case we find Y&(E)
= fE* -E
eE,,
1 eiE* -xpp
1% +log4n+y-
1 + 12cx(-1)
)
+-&%E’. .
(4.74)
Notice the occurrence of the imaginary part in L&“&E). Now we require the energy density E in these two cases. This is the Too component of the energy - momentum tensor, which to one loop order may be expressed as (4.75) For pure electric or magnetic fields this reduces to (4.76)
E(B) = -%r(f%
(4.77) and we find c(B)=+B2-B.B,,--
1 e:B’ 6(4n)Z
Q 1% 7-c
P
(4.78)
278
GRAHAM
M.
SHORE
and e(E)=+EI-E.
E,,-
1 eiE2 a-&y
(4.79)
1%
where we define c=-log4nty-
(4.80)
1 t 126X(-l).
The value of the field B for which the energy density is minimized calculated, giving eRBcl tct; 1% 7 P
1)
.
is now simply
(4.8 1)
Substituting back into Eq. (4.78) gives the energy density in the symmetric phase (a) expressed in terms of the external source B,,. In fact, it is more instructive to express this result in terms of the actual magnetic field B in the system. We find E,(B) = -+B2+-+
1 e2B2 6 (4a)
log -eRB p2
(4.82)
Similarly, s.(E)=-+E2+-+-
1 e2B2 6 (47r)
log
(4.83)
The presence of an imaginary part in Y&(E) or e,(E) indicates that the vacuum is decaying through the pair production of charged scalar particles in the background electric field E. The probability per unit time per unit volume of a pair being produced is [7] (4.84) 4(iii). Phase Transition
and Critical Fields
Finally we are in a position to demonstrate beyond the tree level the existence of a phase transition from the broken to the restored symmetry phase in massless scalar electrodynamics with an externally applied magnetic field, and to calculate the critical field strength at which the transition occurs. The case of an external electric field is also discussed. We have shown the existence of two possible phases: (a) the symmetric phase, characterized by a non-vanishing constant electromagnetic field with the scalar field vacuum expectation value zero, and (b) the broken symmetry phase, characterized by
MEISSNER
EFFECT
IN GAUGE
279
THEORIES
a vanishing electromagnetic field and a constant scalar field vacuum expectation value. The phase which is actually realized for a given external field is that with the lower energy density. The critical fields are therefore obtained by equating the energy densities E, and et, (in the case of the electric field Re E, is used) given in Eqs. (4.53), (4.82) and (4.83). We find the following results for the critical fields B, and E,: (log$+c)
+O(e;))
(4.85)
and log$+c+
2
(4.86)
+ 2y + log 3 - 2 log 2 - 76/3.
(4.87)
where c = 24[;(-1)
Notice that in the form given, both sides of Eqs. (4.85) and (4.86) are renormalization group invariants, i.e., independent of the unit of mass ,u. The picture of the phase transition which emerges is therefore as follows. With a small external field the theory is in a phase of broken gauge symmetry. The spectrum comprises a massive vector and a massive scalar boson, their masses being related by rn; = (3efJ8n2) rn& + O(ei).
(4.88)
In this phase the measured electromagnetic field is zero, the external field being screened. If the external field is magnetic, then as it is increased, a first order phase transition occurs to a restored symmetry phase. The electromagnetic field within the system jumps discontinuously to the critical value given in Eq. (4.85). The spectrum is now that of a massless scalar boson in a background magnetic field, together with the massless photon. However, if the external field is electric, then above the critical value E, given in Eq. (4.86), pair production of massless scalar particles occurs, the rate of production being given in Eq. (4.84). These charged pairs will separate and move to the source of the electric field (e.g., parallel capacitor plates). Unless the electric field strength is artificially maintained despite this current, it will be reduced rapidly. Once the electric field has fallen below E, however, the energetically favorable phase is again that of broken symmetry. In this case therefore there is no observable phase transition. The system exists always in the broken symmetry phase.
280
GRAHAM 5.
M. SHORE
DISCUSSION
This completes our discussion of the phase transitions in scalar electrodynamics. Although we have only performed explicit calculations of the quantum effects for the special case of massless scalar electrodynamics (the Coleman-Weinberg model) we expect the qualitative picture described in the last section to be valid for the whole parameter range mH < m,. The phase diagrams shown in Figs. 2 and 3 remain qualitatively valid for an external magnetic field, except that the region mH/mw= 0 is excluded by the Weinberg lower bound [20] on the mass of the scalar boson. For an external electric field the system is always in the broken symmetry phase. We should emphasize that our results depend on our assumption that for mH< m, only the two phases described exist. In particular, the first order nature of the phase transition is not really a result of our calculation but rather a direct consequence of this assumption. However, we believe the assumption to be true. In the type II case, the presence of a third phase, characterized by vortex solutions of the field equations, is signalled by the inconsistency discussed in Section 3 in the scalar propagator when m,>m,. No such inconsistency is apparent in our treatment of the type I case. Finally, we would like to comment on the distinction between the phase transition discussed here and that occurring in spontaneously broken non-Abelian gauge theories, such as, for example, the Weinberg-Salam model, in a background electromagnetic field. In scalar electrodynamics, in the broken phase the vector bosons corresponding to the external field are massive and the field is screened. However, in the non-Abelian models the U( 1) symmetry of electromagnetism is left unbroken by construction and the photon is always massless. The background electromagnetic field may then be increased continuously until again a first order phase transition occurs to a restored symmetry phase. Notice, however, that in the broken phase below the critical point the background field takes a non-zero value and is not screened. The mechanism underlying the phase transition is therefore quite different from the Abelian case, and is in no sense the analogue of the Meissner effect in superconductivity. The contrast between the two cases is further emphasized by the difference in the critical field strengths. In massless scalar electrodynamics to lowest order we find whereas in a non-Abelian model, e.g., the massless Georgi-Glashow CFfhvLritNrn4W~ SU(2), model, the critical field strength is given by e’(~~,),,it - m&. A full discussion of the phase transition in non-Abelian theories may be found in the subsequent papers in this series [21].
APPENDIX
A
We show here how to extract the result (4.53) for the energy density Ed in the broken symmetry phase from the calculation of Kang [ 151. Notice first that in Kang’s work, logarithms of the coupling constants are absorbed into the definitions of the renormalized parameters, which are defined with renor-
MEISSNER
EFFECT
IN GAUGE
281
THEORIES
malization point M. Also notice that he takes the scalar four point coupling to be (n/6)0” rather than (J/4!W4. To O(e”), he quotes the following result for the effective potential (Eq. (2.15) Ref. [ 151) 1 V(4) = yp4”
1 1 + --(u2Aez+a3e4)fb4(log$-+) (4n)2 8
1 1 ----e6g4
(A-1)
+ (47~)~ 8
(We omit the suffixes on renormalized quantities here.) u2, uj , b,, c, and d, are all known, possibly gauge dependent, constants. The scalar coupling is eliminated using the equation (Eq. (2.16) Ref. [ 151) --f--a3e4 L = (47q -
(2a,a,
+&e6(a,a,-b,-fc,)
(l-log-$)
t
t
c,
d4)
4sfl
log -@ +
(u2u3
-
d4)
4: log2 -@’
64.2)
where do is the position of the minimum of V(4). We therefore find the following form for the energy density, 1 1 E= v(4,) = (4r)2 se”Cf, +&
Substituting
I
1 --a3 3
[ -- 1
2 a2a3-Tc4
1
for the constants, given explicitly
a, = -2a,
u3 = 6,
t
; -a2a3-4
) log%]
1
(A.3)
by
c4 = -24813 t 24a,
d, = 20 - 12a
(A.41
gives a superficially gauge dependent result. However, the energy density is to be expressed in terms of the vector boson mass, rather than the unphysical quantity #,. The relation between these is (Eq. (2.18), Ref. [15]) n&E
22 7e4& e lo + (4:)
I
0, +zJ+
(Y2+z2)logz
4: I
T
(A-5)
with y, = -41/18,
Y2
1 = 57
zl=-$
+a,
z2 = 3 - a.
VW
282
GRAHAM
M. SHORE
We may now reexpress .5 in terms of m, and the renormalized form 1 1 &=-8(4n)2p3+
coupling e in the
2 1 + c (qen)’ ’ ( )
1
where c = -2/a,
- $z*u3 - jc, + u,(y, + z,)
+ ($z2u3 -d,
+ u,(y,
+ z2) log(&/M*)
+ O(e’).
G4.8)
Substituting for the constants, we find that the dependence on log(&M*) cancels, and the remaining contribution to C is independent of the gauge fixing parameter a, as indeed it must be if E is to be interpreted as a physical energy density. We thus find + O(e”) .
(A-9)
Notice finally that we have not so far had to specify the renormalization point M used in the definition of e in Eq. (A.9). Such dependence may be absorbed into the O(e”) corrections, and is therefore of no significance here.
B
APPENDIX
We present here the derivation
of the kernel 9:(x, y; t) for the differential operator
D*, where D, = a, - ieA,, and the potential A,, is such that the corresponding field tensor FM” satisfies a,Fp, = 0.
First define the phase factor, @(x, y) = exp ie
(B-1)
xA,(4dq,, I Y
the integral being along the straight line path from y to x. We may then prove the following result [ 2 11, D,W,
Y) = @(x9 YHGP) eF,,,(u)(x - v), + (i/3) @P,,,(v)(x
- Y)&
- ~9, + 0(x - ~4’1.
03.2)
. . Thus for a field satrsfymg a,F,,, = 0, we find
D, @(x9 v) = (i/2) @(x9 Y) eF&9@
- ~1,.
(B.3)
THEORIES
283
-o?qx, y;t)= - @(x9 aty;t),
(B-4)
MEISSNER
EFFECT
IN
GAUGE
The kernel 9(x, y; t) is defined by
9(x,y;0)=qx,Y). We now make the following Ansafz q-T
[22] for g(x, y; t):
y; t) = [1/(47V’] x expki(x
Substituting
(B-5)
@(x9 Y) - YLA,,(W
- YL + Wb
P-6)
into Eq. (B.4), and using Eq. (B.3) gives
f (x -
Y),
(fffg
+A,nAn,
+ e*F,,tF,,
+ 2ieF,,A,,
(x - y),
(B.7) We therefore look for functions A and C satisfying G!A < +A,,A,,
+ e2FplF,,,, = 0, $++A,,=o.
P.8)
(B.9)
Solving these equations subject to the boundary condition (B.5) (or equivalently, requiring consistency in the limit F,,” = 0 with the expression in the text for P(x, y; t; 0)) yields the solutions A,,(t)
= (eF cot eF&,
C(t) = -f tr log(eFf)- ’ sin eFt - (n/2) log t.
(B. 10) (B.11)
Finally, we must check that with this form for A, the remaining term in Eq. (B.7), -fie(x - y),F,,,A,Jx - y),, vanishes. This is simply seen by observing that A,, contains only even powers of F,,, and therefore that the product E;,A,, is antisymmetric on p, V. We therefore find the following expression for the kernel g(x, y; t): Lqx, y; t) = [ 1/(47r)‘4/2] t-??(X, x
595/l
3412.5
exp{-i(x
y)
- y),(eF cot eRt),,(x - y), - f tr log (eFt)-’ sin eFf). (B. 12)
284
GRAHAM M. SHORE APPENDIX
C
In this appendix we derive the formula &”
log(eFt)-‘sineFtl
=
e2t2Sr;(Im
cosh
&x)-I
(C-1)
quoted in the text, where X= &(T + 6) y2. This was first derived by Schwinger [7]. The first step is to find the eigenvalues f of the matrix (F2)rU = FpnFA,. The following results are not difficult to establish: I;,,, *F.,v = -g,,~z
9
(C-2)
*F WA*FAu - F,tnFnv = %,A.
(C.3)
Squaring Eq. (C.2) and using (C.3), we find s,,,r:
-G&AyFr-’
(C.4)
= 2SSCJL~
which yields the following equation for the eigenvaluesf,
f2+2&f4y=0.
(C.5)
It follows that F2 has two distinct eigenvalues, occurring with degeneracy two, given by f *, where
f * = -jT; f (x: +.Fy.
(C-6)
It is now simple to see that ,-[trlOg(eFe-‘sineFlYZ = ~2f2df+)V2(f-)V2/sin
e(ft)42t
sin edf-)‘12t,
(C-7)
where
(f*)42 = (i/fi)[(&
+ W2)u2 f (6 - i9#2]
W)
and the required result (C.l) follows immediately. ACKNOWLEDGMENTS I would like to thank Ian ARleck, Michael Peskin and Harvey Shepard for interesting discussions in the course of this work. I am also grateful to the U.K. Science Research Council for a postdoctoral fellowship. REFERENCES 1. A. L. FETTER AND J. D. WALECKA, “Quantum Theory of Many Particle Systems,” McGraw-Hill, San Francisco, 1971.
MEISSNER
EFFECT
2. P. W. HIGGS, Phys. Rev. 145 (1966), 3. S. COLEMAN AND E. WEINBERG, Phys.
B. J. HARRINGTON 5. D. A. KIRZHNITS 4.
AND
H. K.
IN GAUGE
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189. Nucl.
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B 90 (1975),
203;
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“Proceedings, Summer Study Meeting on Kaon Physics” (H. Palevsky, Ed.), BNL, Brookhaven, 11.
1976. S. MIDORIKAWA,
12. H. B.
University of Tokyo preprint INS Rep. 380 (1980).
Phys. B 61 (1973), 45. Phys. Rev. D 13 (1976), 3333. 14. G. M. SHORE, Ann. Phys. (N.Y.) 128 (1980), 376. 15. J. S. KANG, Phys. Reo. D 10 (1974), 3455. 16. R. JACKIW, Phys. Rev. D 9 (1974), 1686. 17. L. DOLAN AND R. JACKIW, Phys. Rev. D 9 (1974), 2904. 18. J. ILIOPOULOS AND N. PAPANICOLAOU, Nucl. Phys. B 105 (1976), 77. 19. W. DI~ICH, W. TSAI, AND K. ZIMMERMAN, Phys. Rev. D 19 (1979), 20. S. WEINBERG, Phys. Rev. Left. 36 (1976), 294. NIELSEN 13. E. GILDENER
21. 22.
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AND P. OLESEN, AND S. WEINBERG,
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