# Induced Chern-Simons terms

Abstract: We examine the claim that the effective action of four-dimensional $\mathrm{SU}{(2)}_{L}$ 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 calculations can be performed. These calculations demonstrate that the existence of the Chern-Simons term in four dimensions may be rather subtle.

Topics: Hamiltonian lattice gauge theory (69%), Introduction to gauge theory (65%), Gauge theory (65%), Chern–Simons theory (62%), Thermal quantum field theory (59%)

PUBLISHED VERSION

McCarthy, Jim; Wilkins, Andy

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

© 1998 American Physical Society

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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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)

L

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)

L

gauge theory at ﬁnite temperature T51/

b

, described by

S5

E

0

b

d

t

E

d

3

x

~

2

1

2

tr F

2

1

c

¯

L

D”

c

L

!

. ~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”

a

T

a

1

mg

0

, where

m

is the real chemical

potential for the particle-number charge

B

L

5

E

d

3

x

c

¯

L

g

0

c

L

. ~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

m

:

S

eff

5

m

E

0

b

d

t

E

d

3

x

e

ijk

tr

~

A

i

]

j

A

k

2

2

3

gA

i

A

j

A

k

!

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

L

is only quasi-conserved. Then, when the

gauge conﬁgurations tunnel from one vacuum sector to an-

other, baryons will be created or destroyed. Because

m

is

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

m

is

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

c

¯

R

D”

†

c

R

which yields

a theory of Dirac fermions with an axial quasi-conserved

charge

S5

E

c

¯

~

]

” 1igA”

a

T

a

1

mg

0

g

5

!

c

. ~4!

The coefﬁcient of

m

A

l

a

A

d

a

in the Chern-Simons term is

G

l

d

0

~

p,M,T

!

5

E

k

tr

g

l

D

~

k,M

!

g

0

g

5

D

~

k,M

!

g

d

3D

~

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

5

b

21

(

n

d

3

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

2

)(k

2

1M

2

)

21

yields

G

l

d

0

~

p,M,T

!

5C

e

0l

d

a

p

a

1O

~

p

2

/M

!

. ~6!

Since C is mass independent, Pauli-Villars regularization

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

G

PV

l

d

0

~

p,m,T;0

!

[ lim

M→`

@

G

l

d

0

~

p,m,T;0

!

2G

l

d

0

~

p,M,T;0

!

#

501 O

~

m

21

!

. ~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

0556-2821/98/58~8!/085007~7!/$15.00 © 1998 The American Physical Society58 085007-1

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 (

t

,x) where 0<

t

<

b

. Our gamma ma-

trices are Hermitian and satisfy

@

g

m

,

g

n

#

1

52

d

m

n

and

g

5

52i

g

0

g

1

. ~8!

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

Z

@

A,

m

,

h

¯

,

h

#

5

E

@

d

c

¯

d

c

#

e

2S2

*

h

¯

c

2

c

¯

h

, ~9!

with

S5

E

M

c

¯

D”

c

and D” 5

]

” 1m1

mg

0

g

5

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

m

for the Her-

mitian axial charge Q

5

5

*

c

¯

g

0

g

5

c

is real. One can check

this through a derivation of the path-integral representation

of the partition function.

1

The U~1! gauge transformations are

A

m

→A

m

2ie

21

e

i

u

]

m

e

2i

u

,

c

→e

i

u

c

. ~12!

A gauge transformation is called ‘‘small’’ when

u

is well

deﬁned on M, while if only e

i

u

is well deﬁned ~but not

u

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

large gauge transformation is

u

~

x,

t

!

52

p

N

˜

t

/

b

, for N

˜

P Z. ~13!

This shifts A

0

by a constant

A

0

→A

0

22

p

N

˜

/e

b

. ~14!

The Chern-Simons term in this context is

m

E

M

A

1

. ~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

m

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

gator

D

~

k

!

5

1

ik”1m1

mg

0

g

5

5

1

ik”1m2i

mg

1

. ~16!

The second equality holds in two dimensions because of the

identity

g

l

g

5

52i

e

l

d

g

d

and shows that a constant

m

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

G

l

~

m,T,

m

!

5

E

k

tr e

g

l

m2ik”

˜

k

˜

2

1m

2

, ~17!

where k

˜

1

[k

1

2

m

.

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

larization in which a massive spinor

x

is added into the path

integral

2

Z5 lim

M→`

E

@

d

c

¯

d

c

d

x

¯

d

x

#

e

2S

~

c

¯

,

c

,A,m

!

1S

~

x

¯

,

x

,A,M

!

. ~18!

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

gives

G

PV

l

~

m

!

[ lim

M→`

@

G

l

~

m

!

2G

l

~

M

!

#

. ~19!

Since the momentum integral is now ﬁnite we can shift away

all dependence on

m

. 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

g

m

50. When l50 symmetric summa-

tion ~or integration! gives G

0

(m,T)50. For l51 the answer

obtained depends on the order of integration. Performing the

k

1

integral ﬁrst gives

1

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

tion function Tr exp2

b

(H1

m

Q

5

), 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

c

¯

5

c

†

g

5

, 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!

c

M

[e

2i

p

g

M

0

g

M

5

/4

c

E

,

and

c

M

†

[

c

E

†

e

2i

p

g

M

0

g

M

5

/4

, ~11!

with

g

M

0

[i

g

E

5

,

g

M

i

5

g

E

i

, and

g

M

5

5

g

E

0

, the SO~4! invariant Euclid-

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

Euclidean space spinors as

c

E

→

h

P

g

E

0

c

E

and

c

¯

E

→

h

P

*

c

¯

E

g

E

0

,so

that the

m

Q

5

term breaks parity invariance as required.

2

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

G

1

~

m,T

!

5e

E

k

0

E

2L2

m

L2

m

dk

˜

1

k

˜

1

k

˜

1

2

1m

2

1k

0

2

→

L→`

E

k

0

050.

~20!

However, performing the k

0

summation ﬁrst yields

G

1

~

m,T

!

5

b

2

e

E

dk

1

k

˜

1

b

A

k

˜

1

2

1m

2

p

tanh

~

p

b

A

k

˜

1

2

1m

2

!

52e

p

m

. ~21!

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

answers are mass independent, so Pauli-Villars regulariza-

tion yields

G

PV

l

~

m,T

!

50 ;m,T. ~22!

An alternative treatment is not to put

m

into the propaga-

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

m

and

A. The correlation function of interest is the logarithmically

divergent two-point function

G

l0

~

m,T

!

5

E

k

tr

m2ik”

k

2

1m

2

g

0

g

5

m2ik”

k

2

1m

2

ie

g

l

. ~23!

This method has the advantage that we can easily make

m

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

G

l

d

~

p,m,T50

!

5Y

~

p

2

,m

2

!

e

l

d

1Z

~

p

2

,m

2

!

p

s

e

s

(l

p

d

)

.

~24!

The parentheses indicate symmetrization. Gauge invariance

implies

p

l

G

l

d

50 ⇒ p

1

G

10

5 p

0

G

00

⇒ Y52

1

2

p

2

Z. ~25!

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

case we ﬁnd Z}m

22

1O(p

2

). Then setting p

2

50 gives

Y50 ⇒ G

l

d

~

mÞ0,T5 0

!

50. ~26!

However, for m50 we obtain

G

10

~

p,m5 0,T50

!

5

2e

p

p

0

2

p

0

2

1 p

1

2

. ~27!

Interestingly, this is ambiguous in the zero-momentum limit

G

10

~

m50,T5 0

!

→

H

0 p

0

→0 then p

1

→0,

2e

p

p

1

→0 then p

0

→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,

G

l

d

~

MÞ0,T50

!

5ieM

2

tr

g

d

g

5

g

l

E

k

~

k

2

1 M

2

!

22

522e

p

e

l

d

, ~29!

while

G

l

d

~

M50,T50

!

50. ~30!

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

ing the one-point function,

G

PV

10

~

m,T5 0

!

5

H

0 mÞ0,

2e

p

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:

G

PV

10

~

m,T5 0

!

50 ;m. ~32!

At nonzero temperature there is no IR problem because k

0

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

l

d

can depend on the normal vector in

the p

0

direction. It turns out @11#, that G

10

has the same form

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

0

is quantized, which means

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

plies that p

0

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

m

by functional methods.

3

To make the eigenvalue

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

<

t

<

b

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

position on the background gauge ﬁeld

A

m

5

1

e

]

m

s

1

1

e

e

m

n

]

n

r

1h

m

. ~33!

The ﬁelds

s

and

r

are well deﬁned on M and h

m

is constant.

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

only by the

m

term. However, using the identity

g

0

g

5

3

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

g

1

we can shift the

m

into h

1

. The form of the generat-

ing functional is well known @13#

Z

@

A,

h

¯

,

h

#

5exp

S

E

h

¯

e

2i

s

2

g

5

r

D

0

e

i

s

2

g

5

r

h

1

1

2

p

E

r

h

r

D

det D”

0

. ~34!

Here D”

0

5

]

” 1ieh”2i

mg

1

and has associated propagator D

0

.

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”

0

5

U

1

h

~

iR/

b

!

Q

F

u

f

G

~

0,iR/

b

!

U

2

5

U

q

1/24

)

m51

`

~

12q

m

!

(

nP Z

q

~

n1

u

!

2

/2

e

2

p

i

~

n1

u

!

f

U

2

.

~35!

In this formula

u

52

b

eh

0

/2

p

and

f

5

1

2

1R(eh

1

2

m

)/2

p

and the parameter q5e

22

p

R/

b

.

The partition function is clearly invariant under small

gauge transformations since e

i

s

h

and its conjugate are in-

variant. It is also invariant under large gauge transformations

in the x and

t

directions

x direction:

d

h

1

5

2

p

N

˜

eR

and

h

¯

→

h

¯

e

2

p

iN

˜

x/R

,

t

direction:

d

h

0

5

2

p

N

˜

e

b

and

h

¯

→

h

¯

e

2

p

iN

˜

t

/

b

. ~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”

0

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”

0

in powers of h

l

5(1/R

b

)

*

A

l

is not gauge in-

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

small

b

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

we can expand, for

u

50,

S

eff

58

A

q

R

b

e

m

E

A

1

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-

m

result. Let us perform the same calcula-

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

in the form

G

1

~

m,T,

m

!

5

E

dk

1

R

C

dz

2

p

i

S

k

1

2

m

2z

2

1

~

k

1

2

m

!

2

1m

2

D

tanh

1

2

b

z,

~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

¯

0

passes up the

imaginary axis and C

¯

1

(C

¯

2

) encircles the RHS ~LHS! of the plane.

JIM McCARTHY AND ANDY WILKINS PHYSICAL REVIEW D 58 085007

085007-4

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21 Jul 2018-

Abstract: Faculty of Science School of Physics Doctor of Philosophy Cosmological Implications of Quantum Anomalies by Neil D. Barrie The aim of this thesis is to investigate the possible implications of quantum anomalies in the early universe. We first consider a new class of natural inflation models based on scale invariance, imposed by the dilaton. In the classical limit, the general scalar potential necessarily contains a flat direction; this is lifted by quantum corrections. The effective potential is found to be linear in the inflaton field, yielding inflationary predictions consistent with observation. A new mechanism for cogenesis during inflation is presented, in which a new anomalous U(1)X gauge group in introduced. Anomaly terms source CP andX violating processes during inflation, producing a non-zero CS number density that is distributed into baryonic and dark matter. The two U(1)X extensions considered in this general framework, gauged B and B−L each containing an additional dark matter candidate, successfully reproduce the observed parameters. We propose a reheating Baryogenesis scenario that utilises the Ratchet Mechanism. The model contains two scalars that interact via a derivative coupling; an inflaton consistent with the Starobinsky model, and a complex scalar baryon with a symmetric potential. The inflatonscalar baryon system is found to act analogously to a forced pendulum, with driven motion near the end of reheating generating an ηB consistent with observation. Finally, we argue that a lepton asymmetric CνB develops gravitational instabilities related to the mixed gravity-lepton number anomaly. In the presence of this background, an effective CS term is induced which we investigate through two possible effects. Namely, birefringent propagation of gravitational waves, and the inducement of negative energy graviton modes in the high frequency regime. These lead to constraints on the allowed size of the lepton asymmetry. These models demonstrate that a concerted approach in cosmology and particle physics is the way forward in exploring the mysteries of our universe.

22 citations

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Abstract: An expansion in the number of spatial covariant derivatives is carried out to compute the ζ-function regularized effective action of 2 + 1-dimensional fermions at finite temperature in an arbitrary non-abelian background. The real and imaginary parts of the Euclidean effective action are computed up to terms which are ultraviolet finite. The expansion used preserves gauge and parity symmetries and the correct multivaluation under large gauge transformations as well as the correct parity anomaly are reproduced. The result is shown to correctly reproduce known limiting cases, such as massless fermions, zero temperature, and weak fields as well as exact results for some abelian configurations. Its connection with chiral symmetry is discussed.

10 citations

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01 Jan 2018-

Abstract: The Cosmic Neutrino Background (C\(
u \)B) contains information from very early times which may help illuminate both the properties of the neutrino sector and the evolution of the universe. Unfortunately, the weakly interacting nature of neutrinos combined with the low temperature of the background today, makes the prospect for detection near impossible in the foreseeable future. Despite this, the dynamics of the C\(
u \)B could have had significant effects on the evolution of the early universe. The prospect of gleaning indirect evidence of the C\(
u \)B is to be explored in this chapter, through considering the possible implications for gravitational wave propagation. Given the dawn of the new era of gravitational wave astronomy, this is an exciting possibility.

2 citations

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Abstract: We point out that the claims made in the paper ``Non-thermalizability of a Quantum Field Theory'' (hep-th/9802008) by C. R. Hagen are irrelevant to our recent results concerning large gauge invariance of the effective action in thermal QED.

### Cites background from "Induced Chern-Simons terms"

...Whatever the validity of the calculation in [8], it is totally irrelevant to the subject of [1] [2] [3] [4] [5] [6], namely the properties of the effective gauge field action obtained by integrating out the charged sources: by the very definition of effective actions, the gauge field there is of…...

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...There has been much recent interest in finite temperature Chern-Simons theories [1] [2] [3] [4] [5] [6] [7], focussing on an inherent incompatibility between large gauge invariance and conventional finite temperature perturbation theory....

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