Mass deformed Lorentzian-Bagger-Lambert -Gustavsson theoryfrom Aharony-Bergman-Jafferis theory

Tanay K. Dey*

Institute of Physics, Bhubaneswar 751 005, India

Kamal Lochan Panigrahi†

Department of Physics, Indian Institute of Technology Ropar, Rupnagar 140 001, Indiaand Department of Physics and Meteorology, Indian Institute of Technology Kharagpur, Kharagpur 721 302, India

(Received 26 October 2012; published 28 December 2012)

We construct mass deformed SUðNÞ Lorentzian-Bagger-Lambert-Gustavsson theory together with

UðM� NÞk Chern-Simons theory. This mass deformed Lorentzian-Bagger-Lambert-Gustavsson theory is

a low energy world volume theory of a stack of N number of M2-branes far away from C4=Zk singularity.

We carry out this by defining a special scaling limit of the fields of this theory and simultaneously sending

the Chern-Simons level to infinity.

DOI: 10.1103/PhysRevD.86.126007 PACS numbers: 11.25.Hf

I. INTRODUCTION

In the last few years there has been a huge interestin constructing three-dimensional low energy worldvolume theories of coincident M2-branes with the aimof finding out the dual boundary gauge theory of a11-dimensional gravity theory with geometric structureAdS4 � S7. With this aim in view Bagger, Lambert[1–3], and Gustavsson [4,5] (BLG) first constructed aChern-Simons matter theory with N ¼ 8 supersymmetryand SOð8Þ global symmetry based on 3-algebra. The gaugegroup of this theory is restricted to SOð4Þ. By complex-ifying the matter fields, the BLG theory can be rewritten asa Chern-Simons gauge theory with the gauge groupSUð2Þ � SUð2Þ generated by ordinary Lie algebra. Oneof the gauge groups is associated with Chern-Simons levelk and the other with�k [6]. Immediately, it was discoveredthat the moduli space of BLG theory describes worldvolume theory of only two coincident M2-branes [6–12].In order to go beyond these two brane theories, it wassuggested in Refs. [10,13–18] to use a 3-algebra with ametric having indefinite Lorentzian signature. The result-ing Lorentzian-BLG (L-BLG) theory still preservesN ¼ 8 supersymmetry at the classical level, though itsquantum version lacks proper interpretation.

Inspired by BLG theory and subsequent developments,Aharony, Bergman, Jafferis and Maldacena (ABJM) con-structed the low energy effective theory for N number ofmultiple M2-branes sitting at the singularity of the C4=Zk

orbifold [19]. Instead of the 3-algebra, this construction isfully based on an ordinary Lie algebra. The gauge group ofthis theory is UðNÞk �UðNÞ�k with N ¼ 6 explicit su-persymmetry and SUð4ÞR �Uð1Þ global symmetries [20].

It has two parameters, N and k which take only integervalues. The field content of the theory is N � N matricestransforming in the bifundamental representation of thegauge group. The dual gravity theory is the M theory livingon AdS4 � S7=Zk. Although it seems to be different fromthat of the original L-BLG theory, this gauge theory alsoadmits a 3-algebra interpretation [21,22].In Ref. [23] (also in Ref. [24]), a suggestion was put

forward to produce the L-BLG theory from ABJM theory.According to the authors’ proposal, L-BLG theory can beobtained from the ABJM theory by sending the Chern-Simons level k to infinity and setting some of the fieldsof the theory to zero at the same time so that they decouple.In Ref. [25], the suggestion was more refined and it waspointed out that the scaling limit alone is not sufficient toproduce the L-BLG theory correctly from ABJM theory.One needs to associate an extra ghost multiplet also withABJM theory which is decoupled from the theory itself buteffectively couples with it on redefinition of the fields.With the same spirit in Ref. [26], the extended L-BLGtheory with two pairs of Lorentzian generators is derivedby taking a scaling limit of several quiver Chern-Simonstheories obtained from different orbifoldings of theABJM action.Shortly after ABJM theory, as a further generalization,

another model was proposed by Aharony, Bergman,Jafferis [27]), named the ABJ model. This model involvesmodification of ABJM gauge group UðNÞk �UðNÞ�k toUðMÞk �UðNÞ�k with M � N in the Chern-Simons mat-ter fields kept unchanged. In the gravity picture, the braneconstruction is equivalent to the low energy (M� N) frac-tional M2-branes sitting at the C4=Zk singularity in onesector andNM2-branes freely moving around on the other.The geometric structure of the gravity theory remains thesame as that of the ABJM theory but with an additionaltorsion flux that takes values in H4ðS7=Zk;ZÞ ¼ Zk. SinceABJ theory is slightly more general than ABJM theory it

*tanay.dey@gmail.com†panigrahi@iitrpr.ac.in,

panigrahi@phy.iitkgp.ernet.in

PHYSICAL REVIEW D 86, 126007 (2012)

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deserves a similar study. More precisely, one should takethe same scaling limit as taken in ABJM theory and checkwhether this theory also reproduces L-BLG theory. Withthis aim in view, a study is initiated in Ref. [28] with aredefined scaling limit of fields of the theory. This studywas semicomplete, as it concentrated only in the masslesssector. However in order to go towards a nonrelativisticlimit of this theory with some of the supersymmetries ofthe relativistic theory still preserved, we do require a massterm. The nonrelativistic version of the three-dimensionL-BLG theory could be useful to study the stronglycoupled condensed matter system. With this motivation,in this paper we extend the study further with the inclusionof a mass term in the theory. Although our aim is to producethe mass deformed L-BLG theory in this work, one can alsoextend this study to construct the full nonrelativistic versionof the theory, which we leave for future work.

In this work, we begin with the Lagrangian of the ABJtheory and introduce a suitable supersymmetric massdeformation as in Refs. [29–31] and then take the appro-priate scaling limit on the fields and at the same time sendthe Chern-Simons level k to infinity. In this process, we geta mass deformed SUðNÞ L-BLG theory together withUðM� NÞk Chern-Simons theory. This L-BLG theory isa low energy world volume theory of a stack of N numberof M2-branes far away from C4=Zk singularity.

The rest of this paper is organized as follows. In Sec. II,we briefly review the main aspects of the ABJ theory. InSec. III, we introduce the maximally supersymmetric massdeformation term in to the theory. Section IV is devoted tothe scaling limits on the fields of the theory and sending thelevel k to infinity to finally arrive at theUðNÞmass deformedL-BLG theory together with UðM� NÞk Chern-Simonstheory. In Sec. V, we present our conclusions.

II. A BRIEF SUMMARY OFABJ THEORY

In this section we briefly recapitulate a few basic factsabout the ABJ theory. As mentioned in the introduction,the ABJ theory is the generalized version of the ABJMtheory. The gauge group of this theory isUðMÞk �UðNÞ�k

with SUð4Þ global symmetry. The quantum consistencydemands jM� Nj � k [27]. Unlike the ABJM theory,which has two parameters, ABJ theory contains one extraparameter due to the addition of a fractional M2-brane.These parameters areM, N and k, all of which take integervalues. The field content of this theory is

(i) two gauge fields, AðLÞ� and AðRÞ

� , associated with thegroups UðMÞ and UðNÞ, respectively,

(ii) four complex scalar fields, YAðA ¼ 1; 2; 3; 4Þ, whichare M� N matrix valued, together with their

Hermitian conjugates YyA that are N �M matrix

valued,(iii) M� N matrix valued fermions c A and their

Hermitian conjugates c Ay given by the N �Mmatrix.

Fields with upper indices, as specified above, transform inthe 4 of R symmetry SUð4Þ group and those with lowerindices transform in the �4 representations. The Lagrangianfor these fields has the following form [20],

L ¼ �TrðD�YyAD�Y

AÞ � iTrð �c A��D�c AÞ þ Vpot

þLCS � i2�

kTrðYy

AYA �c Bc B � YAYy

Ac B�c BÞ

þ 2i2�

kTrðYy

AYB �c Ac B � YAYy

Bc A�c BÞ

þ i2�

k�ABCDTrðYA �c BYC �c DÞ

� i2�

k�ABCDTrðYy

Ac BYyCc DÞ; (2.1)

where LCS given in (2.1) is a Chern-Simons termand Vpot is a sextic scalar potential and they have the

following forms:

LCS ¼ k

4�����Tr

�AðLÞ� @�A

ðLÞ� þ 2i

3AðLÞ� AðLÞ

� AðLÞ�

�

� k

4�����Tr

�AðRÞ� @�A

ðRÞ� þ 2i

3AðRÞ� AðRÞ

� AðRÞ�

�;

Vpot ¼ 4�2

3k2Tr½YAYy

AYBYy

BYCYy

C þ YyAY

AYyBY

BYyCY

C

þ 4YAYyBY

CYyAY

BYyC � 6YAYy

BYBYy

AYCYy

C�: (2.2)

Further, the covariant derivatives for the scalars aredefined as

D�YA ¼ @�Y

A þ iAðLÞ� YA � iYAAðRÞ

� ;

D�YyA ¼ @�Y

yA � iYy

AAðLÞ� þ iAðRÞ

� YyA;

(2.3)

and those for fermions are defined as

D�c A ¼ @�c A þ iAðLÞ� c A � ic AA�;

D�cyA ¼ @�c

Ay � ic yAAðLÞ� þ iAðRÞ

� c yA:

(2.4)

The three-dimensional gamma matrices are

�0 ¼ i�2; �1 ¼ �1; �2 ¼ �3;

where the �’s are the Pauli matrices.In Ref. [21], the general form for the action for a three-

dimensional scale-invariant theory withN ¼ 6 supersym-metry, SUð4Þ R symmetry and Uð1Þ global symmetry wasfound by starting with a 3-algebra in which the tripleproduct is not antisymmetric. The field content is thesame as described above with the matter fields now takingvalues in the 3-algebra:

YA ¼ T�YA�c A ¼ T�c A�A� ¼ A��T

� � T: (2.5)

In what follows, we use the notation as in Ref. [32]. Thecomplex scalar YA has the explicit form

ðYAÞ�� 2 ðM; �NÞ; ðYyA Þ�� 2 ð �N;MÞ; (2.6)

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where � ¼ 1; . . . ;M, � ¼ 1; . . . ; N. The gauge fields

ðAðLÞ� Þ� and ðAðRÞ

� Þ�

are Hermitian matrices of UðMÞand UðNÞ, respectively. Then clearly

ðD�YAÞ�� ¼ @�ðYAÞ�� þ iðAðLÞ

� Þ�ðYAÞ�

� iðYAÞ�ðAðRÞ

� Þ�;

ðD�YyAÞ�� ¼ @�ðYy

AÞ�� � iðYyA Þ�ðAðLÞ

� Þ�

þ iðAðRÞ� Þ�

ðYy

A�: (2.7)

So far we have not introduced the mass term in the theory.However, as mentioned in the introduction, one of the waysto get to the nonrelativistic limit of ABJ theory is theintroduction of a suitable mass term which preservessome of the supersymmetries of the original relativisticversion. Though we are not going to study the nonrelativ-istic theory in this paper, we will introduce it in the nextsection to produce the mass deformed L-BLG theory fromthe mass deformed ABJ theory. This construction can beuseful to construct the nonrelativistic version of the theorywhich we leave for future work.

III. MASS DEFORMED ABJ THEORY

In order to introduce the mass term, we focus on maxi-mally supersymmetric mass deformation. It turns out thatthe suitable form of this mass term is [29–31,33]

Vmass ¼ TrðMCAM

BCY

yBY

AÞ þ iTrðMBA�c Ac BÞ

� 4�

kTrðMC

BYAYy

AYBYy

CÞ¼ m2TrðYy

AYAÞ þ imTrð �c ac aÞ � imTrð �c a0c a0 Þ

� 4�

kmTr

�YaYy

½aYbYb� � Ya0Yy

½a0Yb0Yy

b0��: (3.1)

Here m is the mass parameter and the diagonal form ofmass matrix MA

B ¼ mdiagð1; 1;�1;�1Þ. The matrix MBA

satisfies the following constraints:

My ¼M; TrM¼ 0 and M2 ¼m21;

MABY

B ¼mYa�mYa0 ; MBAY

yB ¼mYy

a �mYya0 ;

MABc A ¼mc b�mc b0 ; MB

A�c A¼m �c b�m �c b0 :

(3.2)

The introduction of a mass term breaks explicitly theoriginal SUð4Þ �Uð1Þ R symmetry and down to SUð2ÞL �SUð2ÞR �Uð1Þ [29–31]. We set A ¼ ða; a0Þ, where a anda0 are two SUð2Þ indices that take values 1, 2 and 3, 4,respectively. We used the following convention:

X½aXb� ¼ XaXb � XbXa:

In other words we can say that the mass term of the ABJtheory can be introduced in the same way as in the ordinaryABJM theory. On the other hand, our realization is that the

gaugegroupUðMÞ �UðNÞmakes an impact on the structureof the vacuum manifold.

IV. SCALING LIMIT OF MASS DEFORMEDABJ THEORY

In this section we construct the mass deformed L-BLGtheory from the mass deformed ABJ theory following thesuggestions given in Refs. [23,25]. Basically we take thescaling limit of the fields of the ABJM theory. Therefore as

the first step we presume that AðLÞ� takes the following form:

AðLÞ� ¼

AðLÞ11� AðLÞ

12�

AðLÞ21� AðLÞ

22�

0@

1A: (4.1)

Here AðLÞ11� is the ðM� NÞ � ðM� NÞ matrix. AðLÞ

12� and

AðLÞ21� are ðM� NÞ � N andN � ðM� NÞmatrices, respec-

tively. AðLÞ22� is the N � N square matrix. We further intro-

duce the field B� defined as

AðLÞ22�¼A��1

2B�; AðRÞ

� ¼A�þ1

2B�: (4.2)

The significance of this notation is that theB� is an auxiliary

field. To see this note that the Chern-Simons term of (2.2)implies

k

4�

Zd3x����

hTr�AðLÞ� @�A

ðLÞ�

��Tr

�AðRÞ� @�A

ðRÞ�

�i¼ k

4�

Zd3x����

hTr�AðLÞ11�@�A

ðLÞ11�

�þ2Tr

�AðLÞ12�@�A

ðLÞ21�

��TrðB�ð@�A��@�A�ÞÞ� (4.3)

and

k

4�

2i

3

Zd3x����

hTr�AðLÞ� AðLÞ

� AðLÞ� � AðRÞ

� AðRÞ� AðRÞ

�

�i¼ k

4�

2i

3

Zd3x����Tr

hAðLÞ11�A

ðLÞ11�A

ðLÞ11�

þ 3AðLÞ11�A

ðLÞ12�A

ðLÞ21� þ 3AðLÞ

22�AðLÞ21�A

ðLÞ12�

þ AðLÞ22�A

ðLÞ22�A

ðLÞ22� � AðRÞ

� AðRÞ� AðRÞ

�

i(4.4)

and together we obtain

LCS¼ k

4�����Tr

hAðLÞ11�@�A

ðLÞ11�þ

2i

3AðLÞ11�A

ðLÞ11�A

ðLÞ11�

þ2AðLÞ12�@�A

ðLÞ21�þ2i

�AðLÞ11�A

ðLÞ12�A

ðLÞ21�þAðLÞ

22�AðLÞ21�A

ðLÞ12�

��B�ð@�A��@�A�þi½A�;A��Þ� i

6B�B�B�

i:

(4.5)

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Since in the above form there is no kinetic term for theB� field, it confirms the claim that B� is an auxiliary

field. Note that the N � N matrix gauge field A22� is not

decoupled from the massive N � ðM� NÞ matrix gaugefield A21� which is much needed to construct the mass

deformed L-BLG theory on N number of M2-branes.To decouple this we consider the scaling limit as inundeformed ABJ theory [28]. The form of the scalinglimit is

AðLÞ11� ¼ AðLÞ

11�; AðLÞ12� ¼ �2 ~A12�; AðLÞ

21� ¼ �2 ~A21�;

AðLÞ22� ¼ A� � 1

2�B�; AðRÞ

� ¼ A� þ 1

2�B�; (4.6)

where � is the small parameter which controls thescaling limit and finally we take � ! 0. Note that A�

and B� belong to the algebra of UðNÞ.We then plug in redefined gauge fields of (4.6) into the

Chern-Simons Lagrangian of (4.5) and we get

LCS ¼ k

4�����Tr

AðLÞ11�@�A

ðLÞ11� þ

2i

3AðLÞ11�A

ðLÞ11�A

ðLÞ11�

þ 2�4 ~A12�@� ~A21� þ 2i�4AðLÞ11�

~A12�~A21�

þ 2i�4AðLÞ22�

~A21�~A12�

!þ k

4�����½��B�ð@�A�

� @�A� þ i½A�; A��Þ þOð�3Þ�� Lð1Þ þLð2Þ; (4.7)

where

Lð1Þ ¼ k

4�����Tr

�AðLÞ11�@�A

ðLÞ11 þ

2i

3AðLÞ11�A

ðLÞ11�A

ðLÞ11�

�þ 2�4 ~A12�@� ~A21� þ 2i�4AðLÞ

11�~A12�

~A21�

þ 2i�4AðLÞ22�

~A21�~A12�Þ (4.8)

and

Lð2Þ ¼ � k�

4�����½TrB�ð@�A � @�A� þ i½A�; A��Þ

�Oð�2Þ�: (4.9)

Notice that in order to decouple the massive states weshould keep k unscaled in the first part of the

Lagrangian, Lð1Þ. However, in the second part we should

scale k ¼ 1�~k and keep ~k finite in the limit � ! 0 and send k

to infinity. Finally in the limit � ! 0 we end up with thefollowing form:

Lð1Þ ¼ k

4�����Tr

�AðLÞ11�@�A

ðLÞ11� þ

2i

3AðLÞ11�A

ðLÞ11�A

ðLÞ11�

�;

Lð2Þ ¼ �~k

4�����TrB�F��;

F�� ¼ @�A� � @�A� þ i½A�; A��:

(4.10)

We now try to find out the physical interpretation of theabove results. The gauge fields of the Lagrangian density,

Lð1Þ, are ðM� NÞ � ðM� NÞ matrices. Therefore, thisLagrangian should describe UðM� NÞk Chern-Simonstheory living on the world volume of fractional M2-branesthat are localized at the origin ofC4=Zk. On the other hand

the fields of the Lagrangian density, Lð2Þ, are N � Nmatrices. Thus we have UðNÞ Chern-Simons gauge theoryfor the stack of N number of M2-branes far away from theC4=Zk. For further analysis of this theory we split B� and

A� gauge fields into Uð1Þ and SUðNÞ parts as

A� ¼ A0�IN�N þ ~A�; Tr ~A� ¼ 0;

B� ¼ �2B0�IN�N þ ~B�; Tr ~B� ¼ 0;

(4.11)

and then rescale the Uð1Þ part of the B field with �2. Lð2Þthen looks like

Lð2Þ ¼ � �2 ~kN

4�����B0

�F0�� �

~k

4�����Tr ~B�

~F��

¼ �~k

4�����Tr ~B�

~F��;

(4.12)

which is precisely the gauge part of L-BLG theory for the

identification of ~k ¼ �.Inspired by the above result, we now consider the scal-

ing limit of matter fields YA, c A and scale them in thefollowing way:

YA ¼ �2ZA

1� Y

AþIN�N þ ~YA

!; Tr ~YA ¼ 0;

c A ¼ �2�A

1� cþAIN�N þ �A

!; Tr�A ¼ 0;

(4.13)

where ZA, �A are M� N � N matrices and ~YA, �A areN � N matrices. For analyzing the kinetic term of thescalar field we first compute D�Y

A using the definition

of the covariant derivative of (2.3):

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D�YA ¼ �2@�Z

A þ i�2AðLÞ11�Z

A � Zi�2AðRÞ� þ i�2 ~A12�

�1� Y

AþIN�N þ ~YA

�1� @�Y

AþIN�N þ @� ~YA þ i½A�; ~YA� � iYAþB� � i

2 �fB�; ~YAg

0B@

1CA

) Oð�ÞðM�NÞ�N

1� @�Y

AþIN�N þ @� ~YA þ i½A�; ~YA� � iYAþð�2B0

�IN�N þ ~B�Þ � i2 �fB�; ~Y

Ag

!

� Oð�ÞðM�NÞ�N

1� @�Y

AþIN�N þ ~D�~YA � i

2 �f ~B�; ~YAg

!: (4.14)

Here we have defined

~D�~YA ¼ @� ~YA þ i½A�; ~Y

A� � iYAþ ~B�: (4.15)

In a similar way we find that

D�YyA ¼

�Oð�ÞN�ðM�NÞ

1

�@�Y

yþAIN�N þ ð ~D�

~YAÞy

þ i�

2f ~B�; ~Y

yAg�; (4.16)

where

ð ~D�YAÞy ¼ @� ~YyA � i½ ~Yy

A; A�� þ iYyAþ ~B�: (4.17)

Using Eqs. (4.14) and (4.16), we rewrite the kinetic termfor YA in the following form:

TrðD�YyAD

�YAÞ¼N

�2@�Y

yþA@

�YAþþTrð ~D�~YyA~D� ~YAÞ

�i@�YyþATrð ~B� ~YAÞþi@�Y

AþTrð ~B� ~YyAÞ:

(4.18)

Note that the first term diverges in the limit � ! 0.Therefore it seems that the scaling limit is not complete.For this, following the suggestion given in Ref. [25], weadd an extra ghost term for the bosonic fields in the ABJLagrangian. This ghost term is a Uð1Þ multiplet containingfour complex bosonic fields UA. The form of this ghostterm is

Trð@�UyA@

�UAÞ: (4.19)

Note that in (4.19) we have considered a ‘‘wrong’’ signcompared to the kinetic term since UA is a ghost field.Following the argument as in Ref. [25] for the ABJM theory,it is natural to expect that the original ABJ action withsimilar ‘‘ghost’’ term reduces to L-BLG action. The additionof the ghost results in an indefinite kinetic-term signaturearising from a manifestly definite ABJ action through aregular scaling limit. Although the extra ghost term isdecoupled at the level of ABJ theory, it is effectively coupledthrough the following redefinition of the field,

UA ¼ � 1

�YAþIN�N þ �

1

NYA�IN�N; (4.20)

in the process when we implement the scaling limit. Notealso that we have introduced a new scalar field, YA�, through

the redefinition of the field. It will turn out that this new fieldplays a pivotal role in L-BLG theory. Finally, using (4.20),the kinetic term (4.18) together with the ghost term (4.19)reduces to a finite value in the limit of � ! 0 and we get

Trð@�UyA@

�UAÞ � N

�2@�Y

yþA@

�YAþ � Trð ~D�~YyA~D� ~YAÞ

þ i@�YyþATrð ~B� ~YAÞ � i@�Y

AþTrð ~B� ~YyAÞ

¼ �@�YAþ@�Y

y�A � @�Y

yþA@

�YA� � Trð ~D�~YyA~D� ~YAÞ

þ i@�YyþATrð ~B� ~YAÞ � i@�Y

AþTrð ~B� ~YyAÞ: (4.21)

In order to derive the kinetic term for the L-BLG theorywhich only contains a real scalar field, we first split thecomplex scalar field of the ABJ theory into two real scalarfields in the following way:

YA� ¼X2A�1� þ iX2A� ; YyA� ¼X2A�1� � iX2A� ;

~YA¼� ~X2Aþ i ~X2A�1; ~YyA¼� ~X2A� i ~X2A�1:

(4.22)

Note that the reality of the scalar fields, XA�, implies XA�� ¼XA�, ~XAy ¼ ~XA. In the ABJ theory there are four complexfields denoted by gauge indices A running from 1 to 4. In ourcase, we have eight real scalar fields and we denote them bygauge index I which runs from 1 to 8. Finally using theserelations we are able to reproduce the exact form of thekinetic term for the scalar filed of the L-BLG theory:

� 2@�XIþ@�XI� � 2@�X

IþTrð ~B� ~XIÞ� Tr½D�

~XI � ~B�XIþ�2; (4.23)

where we have defined

D�~XI ¼ @� ~XI þ i½A�; ~X

I�: (4.24)

We now consider the scaling limit of the kinetic term forfermions. We follow the same procedure as described in thescalar field.We insert (4.6), (4.13), and (4.11) into the kineticterm for fermion and take the limit � ! 0. This finally yields

�iTrð �c A��D�c AÞ ¼ � iN

�2�c Aþ��@�c Aþ

� iTrð ��A�� ~D��AÞ� �c Aþ��Trð ~B��AÞ: (4.25)

As in the case of the scalar kinetic term, the first term againdiverges in the limit � ! 0. We resolve this issue in a similar

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spirit as in the scalar case, by adding an extra ghost contri-bution in the ABJ Lagrangian of the form

iTrð �VA�� ~D�VAÞ: (4.26)

Here VA is a fermionic field and we redefine this as

VA ¼ � 1

�cþAIN�N þ �

Nc�AIN�N: (4.27)

Just like in the scalar case, we introduce a new fermion fieldc�A through the redefinition of the fermionic ghost field.Finally, we find that the kinetic term for fermions with (4.26)taken into account is nowwell defined even in the limit �!0and it takes the form

� iTrð ��A�� ~D��AÞ � �c Aþ��Trð ~B��AÞ � i �c Aþ��@�c�A

� i �c A���@�cþA: (4.28)

In order to get to the form of the L-BLG theory we nowwritedown the complex fermion field as a combination of two realfermions:

�c A� ¼ �c�2A�1þ i �c�2A; c�A¼��2A�1þ i��2A;

��A ¼� �c 2A� i �c 2A�1; �A¼��2Aþ i�2A�1: (4.29)

Using the above relations, the expression of Eq. (4.28) cannow be rewritten as

� iTrð �c I��D��IÞ � 2i �cþI�

�Trð ~B��IÞ� 2i �c�I�

�@��þI: (4.30)

This final expression is exactly theSOð8Þ invariant fermionickinetic term of the L-BLG model.Having the analysis on kinetic terms, we now consider

the scaling limit in the potential terms in (2.1) and (2.2). Inorder to take the scaling limit we follow the same approachas in Ref. [23]. The extra job in our case, due to ourconvention, is that we have to explicitly show that themodes ZA decouple in the scaling limit. For an example,consider the following term of the bosonic potential (2.2):

1

k2TrðYBYy

BYCYy

CYAYy

A Þ: (4.31)

By then using the scaling limit of scalar fields of Eq. (4.13),we compute

1

k2TrðYBYy

BYCYy

CYAYy

AÞ ¼�2

~k2Tr

�4ZBZyB �ZBYy

þBIN�N þ �2ZB ~YyB

�YBþZyB þ �2 ~YBZy

B1�2YBþY

yþBIN�N þ 1

� ðYBþ ~YyB þ ~YBYy

þBÞ þ ~YB ~YyB

0@

1A3

¼ �2

~k2Tr

�1

�2YAþY

yþAIN�N þ 1

�ðYAþ ~Yy

A þ ~YAYyþAÞ þ ~YA ~Yy

A

�3 þOð�4Þ: (4.32)

Note that the modes ZA really decouple in the limit � ! 0. Further, the final expression is exactly in the same form as in thepotential ofUðNÞ �UðNÞABJM theory and diverges as usual in the limit � ! 0. Again we can resolve this issue followingthe analysis of Ref. [23]. We write the potential as a sum of VðnÞ

B , VB ¼ P6n¼0 V

ðnÞB , where VðnÞ

B contains n Yþ fields and(6� n) ~Y fields and also VðnÞ

B scales as �2�n in the limit � ! 0. Then it is obvious that potential terms with n < 2 vanish inthe limit � ! 0 and to avoid the divergences the coefficients of the terms, VðnÞ

B , should vanish for n � 3. Therefore, finallythe nonzero contribution comes only from the Vð2Þ

B part of the potential and these contributing terms combine as

4�2

3~k2Tr½YAþY

yþA

~YB ~YyB~YC ~Yy

C þ YBþYyþB

~YC ~YyC~YA ~Yy

A þ YCþYyþC

~YA ~YyA~YB ~Yy

B þ ðYAþ ~YyA þ ~YAYy

þAÞðYBþ ~YyB þ ~YBYy

þBÞ ~YC ~YyC

þ ðYCþ ~YyC þ ~YCYy

þCÞðYAþ ~YyA þ ~YAYy

þAÞ ~YB ~YyB þ ðYBþ ~Yy

B þ ~YBYyþBÞðYCþ ~Yy

C þ ~YCYyþCÞ ~YA ~Yy

A þ YyþAY

Aþ ~YyB~YB ~Yy

C~YC

þ YyþBY

Bþ ~YyC~YC ~Yy

A~YA þ Yy

þCYCþ ~Yy

A~YA ~Yy

B~YB þ ðYy

þA~YA þ ~Yy

AYAþÞðYy

þB~YB þ ~Yy

BYBþÞ ~Yy

C~YC þ ðYy

þC~YC þ ~Yy

CYCþÞ

� ðYyþA

~YA þ ~YyAY

AþÞ ~YyB~YB þ ðYy

þB~YB þ ~Yy

BYBþÞðYy

þC~YC þ ~Yy

CYCþÞ ~Yy

A~YA þ 4ðYAþY

yþB

~YC ~YyA~YB ~Yy

C þ YCþYyþA

~YB ~YyC~YA ~Yy

B

þ YBþYyþC

~YA ~YyB~YC ~Yy

A þ ðYAþ ~YyB þ ~YAYy

þBÞðYCþ ~YyA þ ~YCYy

þAÞ ~YB ~YyC þ ðYBþ ~Yy

C þ ~YBYyþCÞðYAþ ~Yy

B þ ~YAYyþBÞ ~YC ~Yy

A

þ ðYCþ ~YyA þ ~YCYy

þAÞðYBþ ~YyC þ ~YBYy

þCÞ ~YA ~YyB � 6½YAþY

yþB

~YB ~YyA~YC ~Yy

C þ YBþYyþA

~YC ~YyC~YA ~Yy

B þ YCþYyþC

~YA ~YyB~YB ~Yy

A

þ ðYAþ ~YyB þ ~YAYy

þBÞðYBþ ~YyA þ ~YBYy

þAÞ ~YC ~YyC þ ðYCþ ~Yy

C þ ~YCYyþCÞðYAþ ~Yy

B þ ~YAYyþBÞ ~YB ~Yy

A þ ðYBþ ~YyA þ ~YBYy

þAÞ� ðYCþ ~Yy

C þ ~YCYyþCÞ ~YA ~Yy

B�: (4.33)

As earlier, by decomposing the complex scalar fields into two real scalar fields by using Eq. (4.22) we can get to the form ofthe L-BLG theory. This analysis has already been done in Ref. [23] for ABJM. We are not going to repeat this here butwrite down the final form of this sextic potential:

TANAY K. DEY AND KAMAL L. PANIGRAHI PHYSICAL REVIEW D 86, 126007 (2012)

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Vpot ¼ 1

12TrðXIþ½XJ; XK� þ XJþ½XK; XI� þ XKþ½XI; XJ�Þ2:

(4.34)

In the same way, we can also analyze the potential termsin (2.1) that contain both fermions and bosons. Let us start

with the expression

2�

kTrð �c Ac AY

yBY

B � c B�c BYAYy

AÞ: (4.35)

Then using the scaling of (4.13) and k ¼ 1�~k we obtain that

in the limit � ! 0 (4.35) reduces to

2�

kTrð �c Ac AY

yBY

B�c B�c BYAYy

A Þ!2�~k�3

Trð �c AþIN�Nþ� ��AÞ�ðcþAIN�Nþ��AÞ�ðYyþBIN�Nþ� ~Yy

BÞðYBþIN�Nþ� ~YBÞ

� 2�~k�3

TrðcþBIN�Nþ��BÞð �c BþIN�Nþ� ��BÞðYAþIN�Nþ� ~YAÞðYyþAIN�Nþ� ~Yy

AÞ

¼ 2�~k�3

Trð �c AþcþAYBþY

yþBIN�N�cþB

�c BþYAþYyþAIN�NÞþ 2�

~k�2TrðÞ: (4.36)

The term proportional to 1�3vanishes due to the fermionic

nature of c . In the same way we can show that all termscontaining �c Aþc Aþ vanish. Terms proportional to 1

�2vanish

due to the fact that Trð Þ is zero since it contains either �or ~Y. The coefficient of the term proportional to 1

� is alsozero because this term contains two traceless fields.Therefore the only nonzero terms are proportional to �0.In fact, this analysis is the same as in Ref. [23] and againwe will not repeat the same here. We just write down thepotential term that contains both bosons and fermions. Thislooks like

Vscalef ¼ �cþLX

I½XJ;�IJ�L� � �c LXIþ½XJ;�IJ�L�:

(4.37)

The � matrices are the 8� 8 matrices and are constructedfrom the direct product of ��. The form of these gammamatrices is also given in Ref. [23].

After finishing the discussion on the undeformed partof the Lagrangian, we now move to the scaling limit ofthe mass deformed part of the Lagrangian with the aimof getting a contribution for the mass deformed L-BLGtheory. The mass term we consider here is

Vmass ¼m2TrðYyAY

AÞþ imTrð �c ac aÞ� imTrð �c a0c a0 Þ� 4�

kmTrðYaYy

½aYbYy

b� �Ya0Yy½a0Y

b0Yyb0�Þ: (4.38)

By performing the scaling limit in the first term we find

Vsc:mass ¼ m2TrðYyAY

AÞ ¼ m2

�2NYy

þAYAþ þm2Trð ~Yy

A~YAÞ:(4.39)

We see that in the same way as in the case of the kineticterm here we also need to add the ghost contribution tocancel the divergence of the scalar mass term. The ghostcontribution is of the form

Vghostsc:mass¼�m2TrðUy

AUAÞ

¼�N

�2m2Yy

þAYAþm2ðYAþY

y�AþYy

þAYA�Þ: (4.40)

Finally the mass term with the addition of the ghost termlooks like

Vsc:mass ¼ m2Trð ~YyA~YAÞ þm2ðYAþY

y�A þ Yy

þAYA�Þ: (4.41)

Using the redefinition of the fields introduced in Eq. (4.22),we find Vsc:mass can be rewritten in the form of a scalar massterm of L-BLG theory:

m2Trð ~YyA~YAÞ þm2ðYAþY

y�A þ Yy

þAYA�Þ

¼ 2m2ðX2A�1þ X2A�1� þ X2Aþ X2A� Þþm2Trð ~X2A ~X2A þ ~X2A�1 ~X2A�1Þ

¼ 2m2ðXIþXI�Þ þm2Trð ~XI ~XIÞ: (4.42)

In case of the fermion mass term we find

imTrð �c ac aÞ ¼ iN

�2m �c aþcþa þ imTrð ��a�aÞ (4.43)

and the same for c a0 . Clearly again we have to add theghost contribution to the Lagrangian to get a finite contri-bution. The form of this ghost is

Lm:g:f: ¼ �imTr �VaVa þ im �Va0Va0

¼ �imN

�2�c aþcþa þ imð �c aþc�a þ �c a�cþaÞ

þ imN

�2�c a0þcþa0 � imð �c a0þc�a0 þ �c a0�cþa0 Þ:

(4.44)

Clearly, terms proportional to 1�2cancel and we end up with

the finite result

Vf:mass ¼ imTrð ��a�a � ��a0�a0 Þ þ imð �c aþc�a þ �c a�cþaÞ

� imð �c a0þc�a0 þ �c a0�cþa0 Þ: (4.45)

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126007-7

As in the previous cases, we would again redefine the cand � fields as in Eq. (4.29) and finally we are able to writedown Vf:mass as the mass term of L-BLG theory:

Vf:mass ¼ imTrð ��A�A � ��A0�A0 Þþ 2imð ��þA��A � ��þA0��A0 Þ: (4.46)

Here A runs from 1 to 4 and A0 runs from 5 to 8.As the next step we analyze the first term of mass

deformed potential with the following expression:

Vd:pot1 ¼ � 4�

kTr�YaYy

½aYbYy

b��

¼ � 4�~k

"N

�3YaþY

yþ½aY

bþYyþb� þ

1

�TrðYaþY

yþ½a ~Y

b ~Yyb�

þ ~Ya ~Yy½aY

bþYyþb� þ ðYaþ ~Yy

½a þ ~YaYyþ½aÞðYbþ ~Yy

b�þ ~YbYy

þb�ÞÞ þ TrððYaþ ~Yy½a þ ~YaYy

þ½aÞ ~Yb ~Yyb�

þ ~Ya ~Yy½aðYbþ ~Yy

b� þ ~YbYyþb�ÞÞ

#: (4.47)

Now the first term vanishes:

YaþYyþ½aY

bþYyþb� ¼ YaþY

yþaY

bþYyþb � YaþY

yþbY

bþYyþa

¼ YaþYyþaY

bþYyþb � YaþY

yþaY

bþYyþb ¼ 0:

(4.48)

Let us now analyze the contributions proportional to 1� ,

TrðYaþYyþ½a ~Y

b ~Yyb� þ ~Ya ~Yy

½aYbþY

yþb�

þ ðYaþ ~Yy½a þ ~YaYy

þ½aÞðYbþ ~Yyb� þ ~YbYy

þb�Þ¼ Trð2YaþY

yþ½a ~Y

b ~Yyb� � YaþY

yþa

~Yb ~Yyb þ YaþY

yþb

~Yb ~Yya

� ðYaþYyþa

~Yb ~Yyb � YaþY

yþb

~Yb ~Yya ÞÞ

¼ Trð2YaþYyþ½a ~Y

b ~Yyb� � 2YaþY

yþ½a ~Y

b ~Yyb�Þ ¼ 0; (4.49)

using

Yaþ ~Yy½aY

bþ ~Yyb� ¼ Yaþ ~Yy

aYbþ ~Yyb � Yaþ ~Yy

bYbþ ~Yy

a

¼ YaþYbþ ~Yya ~Y

yb � YaþYbþ ~Yy

a ~Yyb ¼ 0: (4.50)

Then we find the finite contribution to the potential:

Vd:pot1 ¼ � 4�~kmTrðYaþ ~Yy

a ~Yb ~Yyb � Yaþ ~Yy

b~Yb ~Yy

a þ ~YaYyþa

~Yb ~Yyb � ~YaYy

þb~Yb ~Yy

a Þ þ ~Ya ~YyaYbþ ~Yy

b � ~Ya ~YybY

bþ ~Yya

þ ~Ya ~Yya ~YbYy

þb � ~Ya ~Yyb~YbYy

þaÞ¼ � 4�

~kmTrðYaþ ~Yy

b ð ~Yya ~Yb � ~Yb ~Yy

a Þ þ Yyþa

~Yyb ð ~Ya ~Yb � ~Yb ~YaÞ þ Ybþ ~Yað ~Yy

a ~Yyb � ~Yy

b~Yya Þ

þ Yyþa

~Ybð ~Yyb~Ya � ~Ya ~Yy

b ÞÞ: (4.51)

We now again redefine the field Y’s in the previous way as in Eq. (4.22) and with this redefinition we can write the aboveexpression as

� 4iX2a�1þ ~X2b½ ~X2b�1; ~X2a� � 4iX2aþ ~X2b�1½ ~X2b; ~X2a�1�;� 4iX2a�1þ ~X2b�1½ ~X2a; ~X2b� � 4iX2aþ ~X2b½ ~X2a�1; ~X2b�1�: (4.52)

Let us now analyze these expressions. The expressions in the first term of the first line give

� 4iX2a�1þ Trð ~X2b½ ~X2b�1; ~X2a�Þ for a ¼ b: � 4iTrð ~X2b�1 ~X2b ~X2b � ~X2b�1 ~X2b ~X2bÞ ¼ 0;

for a ¼ 1; b ¼ 2: � 4iX1þTrð ~X4½ ~X3; ~X2�Þ; for a ¼ 2; b ¼ 1: � 4iX3þTrð ~X2½ ~X1; ~X4�Þ; (4.53)

and the expression in the last term of the first line gives

� 4iX2aþ Trð ~X2b�1½ ~X2b; ~X2a�1�Þ for a ¼ b: 0;

for a ¼ 1; b ¼ 2: � 4iX2þTrð ~X3½ ~X4; ~X1�Þ;for a ¼ 2; b ¼ 1: � 4iX4þTrð ~X1½ ~X2; ~X3�Þ:

(4.54)

On the other hand the expressions on the second line give

TANAY K. DEY AND KAMAL L. PANIGRAHI PHYSICAL REVIEW D 86, 126007 (2012)

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� 4iX2a�1þ Trð ~X2b�1½ ~X2a; ~X2b�Þ � 4iX2aþ Trð ~X2b½ ~X2a�1; ~X2b�1�Þ for a ¼ b: 0;

for a ¼ 1; b ¼ 2: � 4iX1þTrð ~X3½ ~X2; ~X4�Þ � 4iX2þTrð ~X4½ ~X1; ~X3�Þ;for a ¼ 2; b ¼ 1: � 4iX3þTrð ~X1½ ~X4; ~X2�Þ � 4iX4þTrð ~X2½ ~X3; ~X1�Þ:

(4.55)

All together we have

� 8iX1þTrð ~X3½ ~X2; ~X4�Þ � 8iX2þTrð ~X4½ ~X1; ~X3�Þ;� 8iX3þTrð ~X1½ ~X4; ~X2�Þ � 8iX4þTrð ~X2½ ~X3; ~X1�Þ: (4.56)

We can write

�6iX1þTrð ~X3½ ~X2; ~X4�Þ ¼ i�1324X1þTrð ~X3½ ~X2; ~X4�Þ þ i�1342X

1þTrð ~X3½ ~X4; ~X2�Þ þ i�1243X1þTrð ~X2½ ~X4; ~X3�Þ

þ i�1234X1þTrð ~X2½ ~X3; ~X4�Þ þ i�1423X

1þTrð ~X4½ ~X2; ~X3�Þ þ i�1432X1þTrð ~X4½ ~X3; ~X2�Þ: (4.57)

In the same way we can proceed for expressions with primed quantities and in summary we obtain the scaling limit of thedeformed potential in the form

Vd:pot ¼ �i16�

3~km�ABCDTrðXAþ ~XB½ ~XC; ~XD�Þ þ i

16�

3~km�A0B0C0D0TrðXA0

þ ~XB0 ½ ~XC0; ~XD0 �Þ: (4.58)

Finally we would like to sum up all the terms of the mass deformed ABJ Lagrangian with the scaling limit defined and thesum gives us

LBLG ¼�2@�XIþ@�XI� � 2@�X

IþTrðB�XIÞ �Tr½D�XI �B�X

Iþ�2 � iTrð ��I��D��IÞ � 2i ��þI�

�TrðB��IÞ� 2i ���I�

�@��þI � 1

4����TrðB�F��Þ þ 1

12ðXIþ½XJ;XK� þXJþ½XK;XI� þXKþ½XI;XJ�Þ2 þ ��þLX

I½XJ;�IJ�L�� ��LX

Iþ½XJ;�IJ�L� þ 2m2ðXIþXI�Þþm2TrðXIXIÞ þ imTrð ��A�A � ��A0�A0 Þ þ 2imð ��þA��A � ��þA0��A0 Þ� 16

3im�ABCDTrðXAþXB½XC;XD�Þ þ 16

3im�A0B0C0D0TrðXA0

þXB0 ½XC0;XD0 �Þ; (4.59)

where we have done away with the tilde over the fields fornotational simplicity. So starting from a mass deformed ABJtheory, we finally arrived at the world volume theory of astack of N number of M2-branes by taking the properscaling limit on the fields and simultaneously sending theChern-Simons level, k, to infinity. This N number of branesis sitting far away from the C4=Zk singularity. The finalexpression of the Lagrangian matches exactly with the massdeformed L-BLG theory formulated based on 3-algebra.

V. CONCLUSION

In this paper we have started with the undeformed ABJtheory and then added suitable mass terms which preservethe supersymmetry of the theory. Our motivation to addthose mass terms was to obtain the mass deformed L-BLGtheory. In our construction we have introduced two new

parameters, � and ~k. These two parameters are related to the

Chern-Simons term when ~k ¼ �k, where we kept ~k at fixedvalue� by taking � ! 0 at the end of the computation and at

the same time sending k to infinity. We have further dropped

some of the fields of the ABJ theory due the fact that they

are of the order � ! 0. On top of these two new parameters,

we also have added an extra ghost term of Uð1Þ multiplet in

the ABJ theory. Even though the ghost term decoupled at the

level of ABJ theory, it is effectively coupled at the level of

L-BLG theory by the redefinitions of the fields. Finally we

have successfully constructed the mass deformed L-BLG

theory. One can extend the present analysis to write down

the full nonrelativistic ABJ theory and also the L-BLG

theory. Further one can take the same scaling limit, as we

have taken to produce the mass deformed L-BLG theory, on

nonrelativisticABJ theory andcheckwhether this theory also

reproduces the nonrelativistic L-BLG theory. We wish to

return to this problem in the future.

MASS DEFORMED LORENTZIAN-BAGGER-LAMBERT-. . . PHYSICAL REVIEW D 86, 126007 (2012)

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ACKNOWLEDGMENTS

We would like to thank Josef Kluson for participationat an early stage of this work. It is a pleasure to thankSouvik Benerjee, Shankhadeep Chakrabortty, Sudipta Paul

Choudhury, Sumit Das and Sudipta Mukherji for the fruit-

ful discussions. T. K.D. would like to thank the Institute of

Physics, Bhubaneswar, where part of this work was done,

for its hospitality.

[1] J. Bagger and N. Lambert, Phys. Rev. D 75, 045020(2007).

[2] J. Bagger and N. Lambert, Phys. Rev. D 77, 065008(2008).

[3] J. Bagger and N. Lambert, J. High Energy Phys. 02 (2008)105.

[4] A. Gustavsson, Nucl. Phys. B811, 66 (2009).[5] A. Gustavsson, J. High Energy Phys. 04 (2008) 083.[6] M. Van Raamsdonk, J. High Energy Phys. 05 (2008) 105.[7] M.A. Bandres, A. E. Lipstein, and J. H. Schwarz, J. High

Energy Phys. 05 (2008) 025.[8] G. Papadopoulos, J. High Energy Phys. 05 (2008) 054.[9] J. P. Gauntlett and J. B. Gutowski, J. High Energy Phys. 06

(2008) 053.[10] J. Gomis, G. Milanesi, and J. G. Russo, J. High Energy

Phys. 06 (2008) 075.[11] P. De Medeiros, J.M. Figueroa-O’Farrill, and E. Mendez-

Escobar, J. High Energy Phys. 07 (2008) 111.[12] P. de Medeiros, J.M. Figueroa-O’Farrill, and E. Mendez-

Escobar, J. High Energy Phys. 08 (2008) 045.[13] S. Benvenuti, D. Rodriguez-Gomez, E. Tonni, and H.

Verlinde, J. High Energy Phys. 01 (2009) 078.[14] P.-M. Ho, Y. Imamura, and Y. Matsuo, J. High Energy

Phys. 07 (2008) 003.[15] M.A. Bandres, A. E. Lipstein, and J. H. Schwarz, J. High

Energy Phys. 07 (2008) 117.[16] J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk, and

H. Verlinde, J. High Energy Phys. 08 (2008) 094.[17] B. Ezhuthachan, S. Mukhi, and C. Papageorgakis, J. High

Energy Phys. 07 (2008) 041.

[18] H. Verlinde, arXiv:0807.2121.[19] O. Aharony, O. Bergman, D. L. Jafferis, and J. Maldacena,

J. High Energy Phys. 10 (2008) 091.[20] M. Benna, I. Klebanov, T. Klose, and M. Smedback, J.

High Energy Phys. 09 (2008) 072.[21] J. Bagger and N. Lambert, Phys. Rev. D 79, 025002

(2009).[22] M. Schnabl and Y. Tachikawa, J. High Energy Phys. 09

(2010) 103.[23] Y. Honma, S. Iso, Y. Sumitomo, and S. Zhang, Phys. Rev.

D 78, 105011 (2008).[24] Y. Honma, S. Iso, Y. Sumitomo, H. Umetsu, and S. Zhang,

Nucl. Phys. B816, 256 (2009).[25] E. Antonyan and A.A. Tseytlin, Phys. Rev. D 79, 046002

(2009).[26] Y. Honma and S. Zhang, Prog. Theor. Phys. 123, 449

(2010).[27] O. Aharony, O. Bergman, and D. L. Jafferis, J. High

Energy Phys. 11 (2008) 043.[28] J. Kluson, J. High Energy Phys. 04 (2009) 112.[29] K. Hosomichi, K.-M. Lee, S. Lee, S. Lee, and J. Park, J.

High Energy Phys. 07 (2008) 091.[30] K. Hosomichi, K.-M. Lee, S. Lee, S. Lee, and J. Park, J.

High Energy Phys. 09 (2008) 002.[31] J. Gomis, D. Rodriguez-Gomez, M. Van Raamsdonk, and

H. Verlinde, J. High Energy Phys. 09 (2008) 113.[32] M.A. Bandres, A. E. Lipstein, and J. H. Schwarz, J. High

Energy Phys. 09 (2008) 027.[33] O-K. Kwon, P. Oh, C. Sochichiu, and J. Sohn, J. High

Energy Phys. 03 (2010) 092.

TANAY K. DEY AND KAMAL L. PANIGRAHI PHYSICAL REVIEW D 86, 126007 (2012)

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