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Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 900 (2015) 80–103

www.elsevier.com/locate/nuclphysb

On effective Kähler potential in N = 2, d = 3SQED

I.L. Buchbinder

a,b

, B.S. Merzlikin

a,c,∗

Department of Theoretical Physics, Tomsk State Pedagogical University, 634061, To msk, Russia

National Research Toms k State University, 634050, Tomsk , Russia

Department of Higher Mathematics and Mathematical Physics, Tomsk Polytechnic University, 634050, Tomsk , Russia

Received 4

June 2015; received in revised form 2 September 2015; accepted 6 September 2015

Available

online 10 September 2015

Editor: Herman

Verlinde

Abstract

compute the two-loop effective Kähler potential in three-dimensional N = 2 supersymmetric elec-

trodynamics

with Chern–Simons kinetic term for the gauge superﬁeld. The effective action is constructed

on the base of background ﬁeld method with one parametric family of gauges. In such an approach, the

quadratic part of quantum action mixes the gauge and matter quantum superﬁelds yielding the complica-

tions

in the computations of the loop supergraphs. To avoid this obstacle and preserve dependence on the

gauge parameter we make a non-local change of quantum matter superﬁelds after which the propagator is

diagonalized, however the new vertices have appeared. We ﬁx the suitable background and develop the ef-

ﬁcient

procedure of calculating the two-loop supergraphs with the new vertices. We compute the divergent

and ﬁnite parts of the superﬁeld effective action, ﬁnd the two-loop effective Kähler potential and show that

it does not depend on the gauge parameter.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

The modern interest to study three-dimensional Chern–Simons-matter models is caused by the

construction of N = 8 and N = 6 superconformal models, known as the BLG [1–3] and ABJM

[4] ones, which are closely related to the AdS

/CFT

correspondence. As it was mentioned in the

Corresponding author.

E-mail

addresses: joseph@tspu.edu.ru (I.L. Buchbinder), merzlikin@tspu.edu.ru (B.S. Merzlikin).

http://dx.doi.org/10.1016/j.nuclphysb.2015.09.002

The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

I.L. Buchbinder, B.S. Merzlikin / Nuclear Physics B 900 (2015) 80–103 81

work [5], the description of the low-energy dynamics of probe M2 brane in the AdS

background

can be realized from the quantum ﬁeld theory side on the base of quantum effective action. The

structure of the effective action in the three-dimensional extended supersymmetric models was

studied in the series of the works [6–10] mainly in the sector of gauge ﬁeld. In the present paper

we consider the aspect, which has not been studied before and which concerns the structure of

the ef

fective action in the matter sector.

It is well kno

wn that the leading part of the low-energy effective action in the supersymmetric

ﬁeld models with chiral matter superﬁelds is described by effective Kähler potential (see, e.g.,

[11]). The effective Kähler potential is responsible for the structure of the quantum moduli space

of 4D, N = 1 gauge-matter theories in the Higgs branch and is closely related to supersymmetric

sigma-models. Computations of the ef

fective Kähler potential in the 4D, N = 1 supersymmetric

models have been carried out in many papers (see e.g. [12–20] for one-loop calculations and [21]

for two-loop calculations).

In three-dimensional supersymmetric theories the structure of effective Kähler potential is

much less well understood. The effective superpotential in N = 1 gauge-matter theories was

studied in [23,24], but it does not correspond to Kähler sigma-models for component scalar

ﬁelds due to an insufﬁcient number of supersymmetries. The two-loop effective Kähler potential

was computed for the three-dimensional W

ess–Zumino model in N = 2 superspace [25] but it

has not been studied in gauge-matter models which have much more interesting classical and

quantum properties. Note that there is a broad discussion of the structure of moduli space of

three-dimensional gauge theories with N = 2 supersymmetry including its Higgs branch (see,

e.g., [26–29]), bu

t the corresponding Kähler potential has never been computed explicitly in per-

turbation theory. In our previous work [6] we discussed the structure of the two-loop low-energy

effective action in N = 2, d = 3SQED with the Chern–Simons kinetic term for the gauge su-

perﬁeld. Within the background ﬁeld method in N = 2, d = 3

superspace [6–10] we compute

two-loop low-energy effective action for gauge superﬁeld in this model up to the four-derivative

order.

The aim of the present paper is to initiate the study of the perturbati

ve quantum corrections

to the effective Kähler potential in three-dimensional N = 2 supersymmetric gauge theories. We

compute two-loop effective Kähler potential in three-dimensional N = 2 supersymmetric quan-

tum electrodynamics (SQED) with Chern–Simons kinetic term for the gauge superﬁeld.

At the

classical level this model is superconformal, but we show that the conformal invariance is broken

by two-loop quantum corrections. We ﬁnd that the two-loop Kähler potential in the N = 2 super-

symmetric electrodynamics is similar in some aspects to the one-loop effective Kähler potential

in four-dimensional N = 1SQED [19].

In the present paper we study the ef

fective Kähler potential for one particular model: N = 2

SQED with the Chern–Simons kinetic term for the gauge superﬁeld and two chiral superﬁelds.

The following are arguments supporting the study the effective Kähler potential in this model:

The detailed analysis of the 4D superﬁeld effective potentials has been given in the thesis [22].

It is known that the classical action in 3d Chern–Simons theory is obtained by means of dimensional reduction from

4d gauge theory where the vector ﬁeld is described by a topological theta-term. Such 4D theory has no vector ﬁeld

propagator and hence there will be no vector ﬁeld loops. Therefore, the loop contributions to effective action in 3d

Chern–Simons

theory with matter cannot be obtained by means of dimensional reduction from an effective action in 4D

gauge

theory with matter, where the vector ﬁeld is described by the topological theta-term. Thus, the 3D Chern–Simons

theory with matter requires completely independent consideration at quantum level.

82 I.L. Buchbinder, B.S. Merzlikin / Nuclear Physics B 900 (2015) 80–103

• Although this model is quite simple, it possesses a non-trivial effective Kähler potential

which represents the leading part of the low-energy effective action in the Higgs branch. Note

that, in contrast to the four-dimensional case, we need to study the two-loop effective action

since, as we will show further, the one-loop quantum corrections to the ef

fective Kähler

potential are trivial in the sense that they repeat the form of classical Kähler potential. In

general, computation of two-loop quantum corrections is a hard routine, but in the present

case we need to consider just a few two-loop Feynman graphs since the model is Abelian

and, in particular

, ghost superﬁelds do not contribute.

• As we will sho

w further, the form of two-loop quantum corrections to the effective Kähler

potential is in fact dictated by logarithmically divergent supergraphs. Hence, the effective

Kähler potential which is proper to N = 2 Chern–Simons-matter theories it seems can not

appear in three-dimensional models such as N > 2 Chern–Simons-matter gauge theories

which ha

ve no UV divergences [30–32] or the N = 2SQED with Maxwell kinetic term for

the gauge superﬁeld which is superrenormalizable.

• The N = 2S

QED with the Chern–Simons kinetic term is classically superconformal, but,

as we will show, the two-loop quantum corrections to the low-energy effective action break

the conformal invariance. This is analogous to the holomorphic low-energy effective action

in four-dimensional N = 2 gauge theories [33] which is known to be responsible for the

superconformal symmetry breaking.

• We consider the N = 2S

QED with two chiral superﬁelds having different charges with

respect to the gauge superﬁeld. This model is advantageous as compared to similar models

with odd number of chiral superﬁelds which may have parity anomaly [34–36]. Moreover, in

the considered model the effective Kähler potential can be unambiguously computed within

the background ﬁeld method since we can ﬁx the background for chiral matter superﬁelds

which solv

es classical equations of motion. As a result, the obtained effective Kähler poten-

tial corresponds to the gauge-independent part of the effective action.

Let us discuss se

veral technical points concerning two-loop computations in the considered

model. The effective action in quantum ﬁeld theory of gauge ﬁelds is known to be a gauge-

dependent quantity. However, the effective action calculated for background ﬁeld satisfying the

effective equations of motions is gauge independent (see e.g. [37]). When we study the per

turbative quantum corrections to effective action in the frame of loop expansion, the gauge

independent one-loop corrections should be considered on the classical equations of motion

while for the gauge independent two-loop quantum corrections we have to take into account the

effective equations of motion up to one-loop order

. In the N = 2SQED studied in the present

paper it is sufﬁcient to consider constant background chiral superﬁelds to compute the effec-

tive Kähler potential. As we will demonstrate, this background obeys not only classical but also

quantum effective equations of motion up to one-loop order. This guarantees that the tw

o-loop

effective Kähler potential computed in this model is gauge independent. Moreover, in the func-

tional integral we ﬁx the gauge freedom, but we keep the gauge-ﬁxing parameter α arbitrary

throughout all quantum computations. We directly demonstrate that the obtained one- and two-

loop quantum corrections to the effective Kähler potential are independent of α, conﬁrming its

auge independence.

Another technical comment concerns the details of applications of the background ﬁeld

method at the tw

o-loop order. When we perform the background quantum splitting the classi-

cal action acquires a number of terms which mix gauge and matter superﬁelds at the quadratic

order and make the propagator non-diagonal. In quantum computations it is desirable to deal

I.L. Buchbinder, B.S. Merzlikin / Nuclear Physics B 900 (2015) 80–103 83

with the diagonal propagator for quantum superﬁelds. Otherwise the computations become ex-

tremely complicated. There are, in general, two ways to achieve this: (i) to make a non-local

change of ﬁelds to diagonalize the propagator or (ii) to apply a generalized gauge-ﬁxing condi-

tion (R

-gauge) which eliminates the mixed terms at the quadratic order. The latter approach is

usually simpler, but it does not allow one to keep the gauge-ﬁxing parameter arbitrary. Therefore,

in the present work we make a non-local change of quantum superﬁelds to bring the propagator to

the diagonal form. The cost for this is that we get ne

w interaction vertices having non-local form

and playing important role in two-loop quantum computations. This means we should develop a

speciﬁc technique to compute the supergraphs with the new vertices.

The rest of this paper is or

ganized as follows. Section 2 is devoted to some preliminary dis-

cussion concerning the structure of loop quantum corrections to the effective Kähler potential

and specify the background which is suitable for its evaluation. In Section 3 we perform the

background-quantum splitting and derive the form of propagators and interaction v

ertices which

will be employed in loop quantum computations. In the next two sections we calculate one- and

two-loop quantum effective actions and derive the form of effective Kähler potential at the two-

loop order. In the last section we discuss the possible extensions of the results of the present

paper

. In the appendices we collect some technical details of two-loop quantum computations.

Throughout this paper we use the N = 2, d = 3 superspace notations and conventions introduced

in earlier works [6–10].

2. Classical action and speciﬁcation of background

We consider the three-dimensional N = 2 supersymmetric electrodynamics which is de-

scribed by two chiral matter superﬁelds Q

and Q

−

and a gauge superﬁeld V with superﬁeld

strength G =

V . In general at a classical level such a model can have several parameters:

the complex and real mass parameters of the chiral superﬁelds, the topological mass of the gauge

superﬁeld and the Fayet–Iliopoulos term. In the present paper we consider a particular case where

the masses of chiral superﬁelds are v

anishing and the gauge superﬁeld has inﬁnite topological

mass. Moreover, when the Fayet–Iliopoulos term vanishes, the model is superconformal at the

classical level. The only parameter in the classical action is the Chern–Simons level k

S =

2π



zVG−



−

−2V

−

). (2.1)

The corresponding classical equations of motion are

G =

2π

(

−

−2V

−

), (2.2a)

(

±2V

) = 0 ,D

±2V

) = 0 . (2.2b)

The most natural approach to study the quantum effective action is the background ﬁeld

method. For gauge theories in the N = 1 d = 4 superspace this method is discussed in [38].

Basic features of this method for N = 2 d = 3 superspace were formulated in [6–8]. For recent

applications of this method for computing the ef

fective actions in three-dimensional gauge the-

ories in the sector of gauge superﬁeld see [9,10]. Following this method, we split the original

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