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(Open Access) Induced Chern-Simons terms (1998) | J. McCarthy

PUBLISHED VERSION

McCarthy, Jim; Wilkins, Andy

Induced Chern-Simons terms Physical Review D, 1998; 58(8):085007

http://link.aps.org/doi/10.1103/PhysRevD.58.085007

http://link.aps.org/doi/10.1103/PhysRevD.62.093023

http://hdl.handle.net/2440/12758

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3. The right to use all or part of the Article, including the APS-prepared version without

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make copies of all or part of the Article, including the APS-prepared version without

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research purposes.”

15th April 2013

Induced Chern-Simons terms

Jim McCarthy

and Andy Wilkins

†

Department of Physics and Mathematical Physics, University of Adelaide, Adelaide 5005, Australia

~Received 14 October 1997; published 10 September 1998!

We examine the claim that the effective action of four-dimensional SU(2)

gauge theory at high and low

temperature contains a three-dimensional Chern-Simons term which has the chemical potential for baryon

number as its coefﬁcent. The four-dimensional theory has a two-dimensional analogue in which exact calcu-

lations can be performed. These calculations demonstrate that the existence of the Chern-Simons term in four

dimensions may be rather subtle. @S0556-2821~98!07118-5#

PACS number~s!: 11.10.Wx, 98.80.Cq

I. INTRODUCTION

Consider the the four-dimensional Euclidean SU(2)

gauge theory at ﬁnite temperature T51/

, described by

tr F

D”

. ~1!

There are an even number of massless left-handed fermions

to avoid the global SU~2! anomaly @1#, and the Dirac opera-

tor is D” 5

]

” 1igA”

, where

is the real chemical

potential for the particle-number charge

. ~2!

It has been suggested by Redlich and Wijewardhana @2#,

Tsokos @3#, and Rutherford @4#, that — at both high and low

temperature — the effective action obtained by integrating

out the fermions contains a term reminiscent of the three-

dimensional Chern-Simons term with the coefﬁcient

eff

ijk

]

1•••.

~3!

This model has been used @5,6# to describe baryogenesis

by weak interactions at temperatures around the weak scale

in the early universe. The authors note that because of the

U~1! anomaly, B

is only quasi-conserved. Then, when the

gauge conﬁgurations tunnel from one vacuum sector to an-

other, baryons will be created or destroyed. Because

real, the ‘‘Chern-Simons’’ term in Eq. ~3! is not gauge in-

variant, and so breaks the degeneracy of the topological

vacua. Thus the system would be biased to ‘‘fall’’ in one

particular direction resulting in more baryons being created

than antibaryons.

Let us now present a calculation that produces no Chern-

Simons term at low temperature. We use Pauli-Villars regu-

larization which is manifestly gauge invariant. Since

real we are only interested in the real part of the effective

action, log det D” D”

†

. The standard way @2,4,5# to obtain this

is to ‘‘vectorize’’ the model by adding

D”

†

which yields

a theory of Dirac fermions with an axial quasi-conserved

charge

]

” 1igA”

. ~4!

The coefﬁcient of

in the Chern-Simons term is

p,M,T

k,M

k1 p,M

. ~5!

Here D(k,M) is the propagator of a Dirac fermion with mass

M and the integral over momentum space is

(

k for nonzero temperature. Following Refs. @2,4#

we add a mass m for the fermions at low temperature. Ex-

panding the denominator in powers of (2k• p1 p

)(k

)

yields

p,M,T

. ~6!

Since C is mass independent, Pauli-Villars regularization

will yield, in apparent contradiction to @2–4#,

p,m,T;0

[ lim

M→`

p,m,T;0

p,M,T;0

501 O

. ~7!

It is tempting to invoke gauge invariance in order to rule

out the appearance of the Chern-Simons term. However, this

is too naive, because—although the term is not gauge invari-

ant by itself—it is still possible that the entire effective ac-

tion may be invariant @4,7,8#. In later sections we shall

present simple examples of this phenomena.

In light of the apparent contradiction of Pauli-Villars

regularization with the results of Refs. @2–4#, and the

subtlety of gauge invariance, we feel that the problem needs

more study. Fortunately, there is a related model in two di-

mensions in which further calculations can be made more

simply. We believe there is nothing in the following calcu-

lations that suggests our results are particularly speciﬁc to

Electronic address: jmccarth@physics.adelaide.edu.au

†

Electronic address: awilkins@physics.adelaide.edu.au

PHYSICAL REVIEW D, VOLUME 58, 085007

two dimensions. Indeed, in the conclusion we reproduce the

result of Ref. @2# by performing an exact calculation in the

2D model.

II. THE TOY MODEL

We work in a ﬂat two-dimensional ~2D! Euclidean space

M with coordinates (

,x) where 0<

. Our gamma ma-

trices are Hermitian and satisfy

and

52i

. ~8!

The 2D equivalent of the vectorized theory of Eq. ~4! is

2S2

, ~9!

with

D”

and D” 5

]

” 1m1

1ieA” . ~10!

A mass term has been included for generality at this point.

We shall see later on that it infrared ~IR! regulates the theory

at zero temperature. The chemical potential

for the Her-

mitian axial charge Q

is real. One can check

this through a derivation of the path-integral representation

of the partition function.

The U~1! gauge transformations are

→A

2ie

]

→e

. ~12!

A gauge transformation is called ‘‘small’’ when

is well

deﬁned on M, while if only e

is well deﬁned ~but not

itself! the transformation is called ‘‘large.’’ An example of a

large gauge transformation is

, for N

P Z. ~13!

This shifts A

by a constant

→A

. ~14!

The Chern-Simons term in this context is

. ~15!

Let us ﬁrst present some perturbative calculations that sug-

gest that this term does not appear in the effective action.

Then we will study the effective action nonperturbatively.

III. PERTURBATIVE RESULTS

Since

is constant, it is efﬁcient to put it into the propa-

gator

ik”1m1

ik”1m2i

. ~16!

The second equality holds in two dimensions because of the

identity

52i

and shows that a constant

simply

shifts the momentum in the loop. Expanding the path integral

in powers of A we ﬁnd the coefﬁcient of the linear term is

the superﬁcially linearly divergent one-point function

m,T,

tr e

m2ik”

, ~17!

where k

To regulate this expression we will use Pauli-Villars regu-

larization in which a massive spinor

is added into the path

integral

Z5 lim

M→`

,A,m

,A,M

. ~18!

This is manifestly gauge invariant and, in the usual fashion,

gives

[ lim

M→`

. ~19!

Since the momentum integral is now ﬁnite we can shift away

all dependence on

. It is possible to go further and explic-

itly calculate each separate term on the right-hand side

~RHS! of Eq. ~19!. The mass term in the numerator of Eq.

~17! gets killed by tr

50. When l50 symmetric summa-

tion ~or integration! gives G

(m,T)50. For l51 the answer

obtained depends on the order of integration. Performing the

integral ﬁrst gives

In the derivation of the path-integral representation of the parti-

tion function Tr exp2

(H1

), we insert a complete basis at

each time slice and then express the action thus derived in terms of

relativistic ﬁelds in Euclidean space. This last part is relatively non-

trivial, but it is found that with the choice

†

, the path inte-

gral of Eq. ~9! correctly calculates the partition function. This care-

ful calculation thereby conﬁrms the recent work of Waldron et al.

@9# who studied the continuous rotation of spinors from Minkowsky

to Euclidean space. It was found that with the deﬁnitions ~subscripts

M and E refer to Minkowsky and Euclidean, respectively!

and

†

[

†

, ~11!

with

, and

, the SO~4! invariant Euclid-

ean action was given by Eq. ~10!. Parity, for example, acts on the

Euclidean space spinors as

→

and

→

,so

that the

term breaks parity invariance as required.

In principle two spinors are needed, however, this is an unneces-

sary notational complication.

JIM McCARTHY AND ANDY WILKINS PHYSICAL REVIEW D 58 085007

085007-2

m,T

2L2

→

L→`

050.

~20!

However, performing the k

summation ﬁrst yields

m,T

tanh

52e

. ~21!

The same result is obtained at zero temperature. However, all

answers are mass independent, so Pauli-Villars regulariza-

tion yields

m,T

50 ;m,T. ~22!

An alternative treatment is not to put

into the propaga-

tor, but to expand the path integral in powers of both

and

A. The correlation function of interest is the logarithmically

divergent two-point function

m,T

m2ik”

. ~23!

This method has the advantage that we can easily make

nonconstant. The momentum p, ﬂowing into the associated

Feynman diagram will then be nonzero, and only after cal-

culating will we set p50. With nonzero p, Adler’s

regularization-independent method @10# can be applied. At

zero temperature, the most general expression with the cor-

rect Lorentz structure and parity is

p,m,T50

)

~24!

The parentheses indicate symmetrization. Gauge invariance

implies

50 ⇒ p

5 p

⇒ Y52

Z. ~25!

However, Z is ﬁnite so we can calculate it. For the massive

case we ﬁnd Z}m

1O(p

). Then setting p

50 gives

Y50 ⇒ G

mÞ0,T5 0

50. ~26!

However, for m50 we obtain

p,m5 0,T50

1 p

. ~27!

Interestingly, this is ambiguous in the zero-momentum limit

m50,T5 0

→

0 p

→0 then p

→0,

→0 then p

→0.

~28!

We attribute this to the IR divergence contained in the

two-point function of Eq. ~23! for M50 and T50. We ﬁnd

a similar problem when naively applying Pauli-Villars regu-

larization at zero temperature. Namely, after taking the trace

over gamma matrices,

MÞ0,T50

5ieM

1 M

522e

, ~29!

while

M50,T50

50. ~30!

This implies, in contradiction to the null result obtained us-

ing the one-point function,

m,T5 0

0 mÞ0,

m50.

~31!

However, this occurs only because the IR divergence has

made the result somewhat arbitrary. In this situation a natural

prescription is to deﬁne the massless theory as the limit of

the massive one:

m,T5 0

50 ;m. ~32!

At nonzero temperature there is no IR problem because k

is never zero. Pauli-Villars regularization gives zero in

agreement with the one-point function. The Adler argument

is more complicated because the heat bath breaks Lorentz

invariance and so G

can depend on the normal vector in

the p

direction. It turns out @11#, that G

has the same form

as Eq. ~27!. However, this time p

is quantized, which means

it cannot be taken to zero smoothly. We argue that this im-

plies that p

must be set to zero from the very start, and so

the top limit in Eq. ~28! is the correct one.

IV. NONPERTURBATIVE RESULTS

The partition function can also be calculated directly to all

orders in

by functional methods.

To make the eigenvalue

problem well deﬁned, M is chosen to be the torus with 0

and 0<x<R. Here we can make the Hodge decom-

position on the background gauge ﬁeld

]

. ~33!

The ﬁelds

and

are well deﬁned on M and h

is constant.

Our case differs from the Schwinger model @12# on the torus

only by the

term. However, using the identity

We are interested in the trivial sector of the model. The effective

action when the gauge ﬁeld is in a nontrivial winding sector is also

well known @15,16#. Nontrivial sectors may be of interest when

studying baryogenesis in the early universe. A nonzero chemical

potential for the conserved electric charge has also been considered

@17#. In this case the Dirac operator is no longer Hermitian and the

phase in the partition function leads to interesting results.

INDUCED CHERN-SIMONS TERMS PHYSICAL REVIEW D 58 085007

085007-3

52i

we can shift the

into h

. The form of the generat-

ing functional is well known @13#

5exp

det D”

. ~34!

Here D”

]

” 1ieh”2i

and has associated propagator D

The determinant of this operator can be calculated using

zeta-function regularization. The result can be written in

terms of a theta function and Dedekind’s eta function @14,16#

det D”

iR/

0,iR/

1/24

)

m51

12q

(

nP Z

~35!

In this formula

and

1R(eh

)/2

and the parameter q5e

The partition function is clearly invariant under small

gauge transformations since e

and its conjugate are in-

variant. It is also invariant under large gauge transformations

in the x and

directions

x direction:

and

→

x/R

direction:

and

→

. ~36!

The ﬁrst transformation changes the summand in Eq. ~35! by

a phase which is then canceled by the mod squared. The

second transformation can be soaked up by relabeling the

index of summation.

Let us study the partition function as we take the cylin-

drical limit. The determinant ~35! of D”

obtained by zeta-

function regularization is nonlocal in the gauge ﬁeld. Also,

each term in the expansion of the effective action S

eff

5log det D”

in powers of h

5(1/R

)

is not gauge in-

variant. For example, at large R ~the limit to the cylinder! or

small

~high temperature!, the parameter q is small. Then

we can expand, for

50,

eff

1•••, ~37!

where, in the last equality, the Chern-Simons term has been

extracted. The term by itself is not gauge invariant. In the

Appendix we study the one dimensional analogue, det D” on

the circle. Once again zeta-function regularization results in

a nonlocal but gauge-invariant result. Each term in the ex-

pansion in powers of the gauge ﬁeld is not gauge invariant.

We also study the limit to the line. One would not expect the

limit to depend upon whether the boundary conditions on the

circle were initially periodic or antiperiodic. The only

subtlety is that one has to be careful with IR divergences

~zeromodes!. In the 2D model there are no IR problems be-

cause the fermions are antiperiodic along the time direction.

Thus, by setting q5 0 in Eq. ~37!, we see that there is no

induced Chern-Simons term on the cylinder according to

zeta-function regularization.

V. CONCLUSIONS

The effective action of the 2D toy model of baryogenesis

has been calculated in various ways. Because the chemical

potential is real, the Chern-Simons-type term that has been

proposed to appear in the effective action is not gauge in-

variant. As we have seen in one and two dimensions, this

does not rule out its appearance in the effective action. How-

ever, all our gauge-invariant calculations at nonzero tempera-

ture gave no Chern-Simons term. It was only for the mass-

less theory at zero temperature that there was any chance of

getting a term. This was attributed to an ambiguity brought

about through an IR divergence.

How then, did other authors @2# obtain a nonzero result?

The regularization scheme was to subtract off the zero-

temperature, zero-

result. Let us perform the same calcula-

tion in 2D. The one-point function of Eq. ~17! can be written

in the form

m,T,

tanh

~38!

where the contour of integration is shown in Fig. 1~a!. Using

partial fractions and expressing tanh in terms of exponentials

leads to

FIG. 1. Contours of integration in the z-plane. ~a! The contour C

encircles the imaginary axis, and ~b! contour C

passes up the

imaginary axis and C

) encircles the RHS ~LHS! of the plane.

JIM McCARTHY AND ANDY WILKINS PHYSICAL REVIEW D 58 085007

085007-4

Induced Chern-Simons terms

Citations

Cosmological Implications of Quantum Anomalies

Parity breaking in 2 + 1 dimensions and finite temperature

Gravitational Waves and the Cosmic Neutrino Background

Comment on ``Non-thermalizability of a Quantum Field Theory''

References

Table of integrals, series and products

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Q1. What contributions have the authors mentioned in the paper "Induced chern-simons terms" ?