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Published for SISSA by Springer

Received: December 4, 2017

Revised: February 5, 2018

Accepted: April 9, 2018

Published: April 12, 2018

Quantum noncommutative ABJM theory: first steps

Carmelo P. Martin,a Josip Trampeticb,c and Jiangyang Youd

aDepartamento de F́ısica Teórica I, Facultad de Ciencias F́ısicas,

Universidad Complutense de Madrid,

28040-Madrid, Spain
bRudjer Bošković Institute, Division of Experimental Physics,

HR-10002 Zagreb, Croatia
cMax-Planck-Institut für Physik, (Werner-Heisenberg-Institut),

Föhringer Ring 6, D-80805 München, Germany
dRudjer Bošković Institute, Division of Physical Chemistry,

HR-10002 Zagreb, Croatia

E-mail: carmelop@fis.ucm.es, josip.trampetc@irb.hr,

jiangyang.you@irb.hr

Abstract: We introduce ABJM quantum field theory in the noncommutative spacetime

by using the component formalism and show that it is N = 6 supersymmetric. For the

U(1)κ × U(1)−κ case, we compute all one-loop 1PI two and three point functions in the

Landau gauge and show that they are UV finite and have well-defined commutative limits

θµν → 0, corresponding exactly to the 1PI functions of the ordinary ABJM field theory.

This result also holds for all one-loop functions which are UV finite by power counting. It

seems that the noncommutative quantum ABJM field theory is free from the noncommu-

tative IR instabilities.

Keywords: Chern-Simons Theories, Non-Commutative Geometry

ArXiv ePrint: 1711.09664

Open Access, c© The Authors.

Article funded by SCOAP3.
https://doi.org/10.1007/JHEP04(2018)070

mailto:carmelop@fis.ucm.es
mailto:josip.trampetc@irb.hr
mailto:jiangyang.you@irb.hr
https://arxiv.org/abs/1711.09664
https://doi.org/10.1007/JHEP04(2018)070


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Contents

1 Introduction 1

2 Classical NCABJM field theory 3

2.1 Noncommutative BRST transformations 4

2.2 Noncommutative generalization of the action 5

2.3 Noncommutative supersymmetric transformations 6

3 Feynman rules of the U(1)κ ×U(1)−κ NCABJM quantum field theory 7

4 Power counting and the limit θµν → 0 8

5 Gauge field
〈
AµAν

〉
and hgauge field

〈
µÂν

〉
two-point functions 9

5.1 Gauge field bubble and tadpole diagrams 9

6 Mixed gauge field – hgauge field,
〈
AµÂν

〉
, two-point functions 11

6.1 Gauge field – hgauge field bubble and tadpole: scalar and fermion loops 11

7 Gauge field
〈
Aµ1Aµ2Aµ3

〉
and hgauge field

〈
µ1µ2µ3

〉
, three-point

functions 12

8 Mixed gauge field – hgauge field,
〈
Aµ1Aµ2µ3

〉
,
〈
µ1µ2Aµ3

〉
, three-

point functions 15

8.1 Loop integrals contributing to the
〈
Aµ1Aµ2µ3

〉
3-point function 16

8.2 Computations of the
〈
Aµ1Aµ2µ3

〉
3-point functions 18

9 Scalar | fermion,
〈
XAX

B
〉
|
〈
ΨAΨ̄B

〉
, two-point functions 20

9.1 One-loop scalar
〈
XAX

B
〉

2-point function 20

9.2 One-loop fermion
〈
ΨAΨ̄B

〉
2-point function 21

10 Fermion – gauge field | – hgauge field,
〈
ΨAΨ̄BA

µ
〉
|
〈
ΨAΨ̄BÂ

µ
〉
, three-

point functions 22

11 Scalar – gauge field | – hgauge field,
〈
XAXBA

µ
〉
|
〈
XAXBÂ

µ
〉
, three-

point functions 24

12 Summary and discussion 26

A SU(4)R supersymmetric invariance of U(1)κ ×U(1)−κ 28

A.1 Variations of the action with respect to gauge and scalar fields 28

A.2 Variations of the action with respect to fermion fields 30

A.3 Cancellations between SCS, Skin and S4 variations 31

A.4 Classical SUSY invariance regarding S6 terms 33

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B Two point functions:
〈
AµAν

〉
34

C Integrals from two point functions 35

C.1 Master scalar integral I(p, θ) 36

C.2 An integral with a bounded but ill-defined p̃µ → 0 limit 37

D Feynman rules 38

D.1 Gauge fields and ghosts-gauge field vertices 38

D.2 Scalar-gauge fields vertices 38

D.3 Fermion-gauge field vertices 39

1 Introduction

ABJM field theory at the level κ was introduced in [1] to provide a holographic dual of

the M theory on the AdS4 × S7/Zk, thus furnishing a concrete realization of the famous

gauge/gravity duality conjecture [2]. From the point of view of Quantum Gravity, ABJM

quantum field theory deserves being analyzed thoroughly since it affords possibility of

studying gravity on four dimensional spacetime at the quantum level [3, 4]. Besides, the

ABJM theory may be useful in the effective field theory description of a certain condensed

matter systems where the Chern-Simons action arises naturally [5, 6].

That the ABJM field theory, at the level κ, carries the N = 6 super-conformal symme-

try was explicitly verified in the component formalism formulation of the theory [7]. This

super-conformal symmetry is enhanced to N = 8, when κ = 1 or κ = 2 [8]. In ref. [9]

ABJM theory was formulated in the N = 3 harmonic superspace. Such formulation was

used to show that the super-field perturbation theory, obtained in the background field

formalism for the background field gauge, is UV finite [10].

Modern on-shell techniques have been used to work out some tree-level [11] and one-

loop [12, 13] scattering amplitudes in the ABJM field theory. These computations have

unveiled beautiful algebraic — the Yangian of the corresponding super-conformal alge-

bra [11, 14] — and geometric — the orthogonal Grassmannian [15] — structures that play

an important role to the analysis of the theory: its integrability in particular [16].

Noncommutative field theory — see [17], for a review — is a well-established area of

research in High Energy Theoretical Physics. Surprisingly, to the best of our knowledge,

no formulation of the ABJM field theory on noncommutative spacetime can be found in

the existing literature. This state of affairs should not continue, since quantum ABJM field

theory on the noncommutative spacetime could be helpful — through the gauge/gravity

correspondence — in studying noncommutative gravity in four dimensions and, on the

other hand, noncommutative Chern-Simons theory naturally arises in the study of the

Fractional Quantum Hall effect [18]. Further, noncommutative spacetime as defined by

the Moyal product breaks conformal invariance, so that one may discuss in a well-defined

setting wether or not the beautiful structures and effects that occur in the ABJM field

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theory disappear together with the disappearances of the super-conformal invariance of it

or, perhaps, are replaced by some noncommutative ones.

Main purpose of this paper is to formulate the complete ABJM quantum field theory

on the noncommutative spacetime as defined by the so-called Moyal star-product, via star

commutator of the coordinates [xµ ?, xν ] = iθµν , with θµν being the noncommutativity

matrix. We shall do this in the component formalism and show that both the classical

ABJM action and the N = 6 ordinary transformations [7] can be nicely generalized to

the Moyal noncommutative spacetime to define a noncommutative ABJM quantum field

theory with N = 6 supersymmetry.

As already mentioned, ABJM theories are proposed as the holographic dual of M2

brane in appropriate backgrounds. The noncommutative deformation of the gravity dual

of the ordinary ABJM theory was worked out in [19]. Recently, it has also been shown

in [20] that this B-field charged IIA supergravity background, for generic κ’s, poses the same

amount of supersymmetry as its ordinary N = 6 counterpart does. Therefore, as will be

shown below, by possessing six supersymmetries our noncommutative ABJM (NCABJM)

action does fulfill the necessary condition to become dual to the superstring/supergravity

theory on the deformed background constructed in [19].

Another important aim of this paper is to check on the quantum level, whether the limit

θµν → 0 of the noncommutative ABJM theory restores back the ordinary/commutative

ABJM theory introduced in [1]. We shall do this by computing all one-loop 1PI functions

involving fewer than four fields in the noncommutative variant of the U(1)κ ×U(1)−κ the-

ory. This is a nontrivial issue for the following reasons: in the component formalism the

1PI Green functions are not UV finite by power counting and, therefore, one cannot use

Lebesgue’s dominated convergence theorem to take limit θµν → 0 inside the integral. Ac-

tually,the expected UV finiteness of the theory comes as a result of cancellations that occur

upon summing over all the planar parts of the UV divergent Feynman diagrams contribut-

ing to a given 1PI Green function. Now, due to the UV/IR mixing the nonplanar part

of each UV divergent Feynman diagram contributing to a given 1PI function develops, in

general, a noncommutative IR divergence; only upon adding up all those noncommutative

IR divergent contributions one may expect that the final noncommutative IR divergence

goes away completely. Of course, when cancellation of infinities takes place by summing up

all contributions, local finite parts of the 1PI Green functions may not be uniquely defined.

What is more, Moyal phases act as UV regulators of the nonplanar contributions — trading

an UV divergence for an IR one — but they are regulators which break Lorentz invariance,

so that structures of the finite contributions arising from them are not given by the stan-

dard results in renormalization theory. Actually, values of some integrals contributing to

a certain Feynman diagram — see appendix C.2, for example — remains bounded as one

approaches θµν = 0 point, but the θµν → 0 limit does not exist. Putting it all together,

we conclude that it is far from clear that the limit θµν → 0 of the 1PI Green functions in

the noncommutative formulation of the ABJM quantum field theory are the corresponding

functions in the commutative ABJM quantum field theory.

Layout of this paper is as follows: in section 2, we describe the field content of the

ordinary/classical U(1)κ×U(1)−κ ABJM field theory action to set the notation and conven-

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tions regarding the global SU(4) R-symmetry of the ABJM theory without the notation

complications due to the use of the U(N) groups. Classical action of noncommutative

U(N)κ × U(N)−κ ABJM field theory is introduced next, along with the noncommutative

BRST transformations which leave that action invariant — subsections 2.1 and 2.2. Non-

commutative N = 6 supersymmetry transformations which leave the classical action of the

ABJM theory invariant are introduced in subsection 2.3. In the appendix A we display a

detailed proof that supersymmetric transformations introduced in subsection 2.3 do indeed

leave the classical noncommutative ABJM action invariant. Also in the appendix A we

consider only the U(1)κ × U(1)−κ case since the generalization to the U(N)κ × U(N)−κ is

straightforward and, besides, it is for the U(1)κ ×U(1)−κ case that the difference between

the classical action of the noncommutative ABJM theory and the ordinary ABJM theory

is more conspicuous, due to the fact that the Moyal star-product is not commutative and

generates nonabelian gauge symmetry. Feynman rules the noncommutative U(1)κ×U(1)−κ
ABJM quantum field theory in Landau gauge are given section 3. Power counting rules

and limit θµν → 0 were discussed in section 4, while remaining rules relevant to our com-

putations are given in the appendix D. Let us point out that we quantize the theory in

the Landau gauge for two reasons: i) the Chern-Simons propagator is simpler and ii) it

does not contain contributions with a dangerous IR behaviour — see section III of ref. [21].

In sections 5 to 12 we show and discuss that, at the one-loop level all the 1PI two and

three point functions of the noncommutative U(1)κ×U(1)−κ ABJM quantum field theory

are UV finite and have well-defined limits when θµν → 0, and that those limits are equal

to the corresponding Green functions of the commutative ABJM quantum field theory.

Remaining appendices are needed for properly understanding of the main text.

2 Classical NCABJM field theory

We begin our construction for NCABJM field theory from its field contents, which is

identical to the commutative theory, although the fields are noncommutative. For this

reason and the convenience of comparison we briefly summarize the known results on the

commutative ABJM theory first. Our conventions follow exactly those in [7]. We start

with the U(1)κ×U(1)−κ theory since it has less indices and thus it is simpler with respect

to the general U(N)κ ×U(N)−κ field theory.

The pair of the U(1)κ × U(1)−κ vector gauge fields are denoted as Aµ and µ, i.e.

gauge and hgauge fields, respectively. Scalars XA and fermions ΨA have U(1) charges

(+,−), while their adjoints have charges (−,+), respectively. As in constructing the full

U(N)κ × U(N)−κ theory with above convention we choose to normalize fields so that the

κ-level Lagrangian is κ times the level-1 Lagrangian. Thus the N=1 action is as given below:

S =
κ

2π

∫
d3x

(
−DµXADµXA + iΨ̄A /DΨA +

1

2
εµνλ

(
Aµ∂νAλ − µ∂νÂλ

))
, (2.1)

with four complex scalars XA and their adjoints XA, where a lower index labels the 4

representation and an upper index labels the complex-conjugate 4̄ representation of the

global SU(4) R-symmetry, respectively. Covariant derivative acting on scalar fields XA and

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XA respectively reads:

DµX
(A)
A = ∂µX

(A)
A

(−)
+ i(Aµ − µ)X

(A)
A . (2.2)

The above pair of two-component fermi fields with notation Ψ̄A or Ψ̄A, in (2.1), implies

transposing the spinor index of ΨA and ΨA, respectively and right multiplication by γ0

respectively, though that index is not displayed. In this definition there is no additional

complex conjugation, since the lower index indicates the 4 and an upper index indicates the

4̄ representation, respectively. With these conventions identities that hold for Majorana

spinors shall be used for our spinors, as well, even though they are Dirac-complex fields:

Ψ̄AΨB = Ψ̄BΨA. Considering Pauli-Dirac algebra conventions our 2 × 2 Dirac matrices

satisfy {γµ, γν} = 2ηµν . Here index µ = 0, 1, 2 is 3-dimensional Lorentz index with sig-

nature (−,+,+). Using a Majorana representation implies that γµ is real, while choices

γ0 = iσ2, γ1 = σ1, γ2 = σ3 and γµνλ = εµνλ, gives γ0γ1γ2 = 1.

General U(N)κ × U(N)−κ ABJM theory consists of four N × N matrices of complex

scalars (XA)aȧ and their adjoints (XA)ȧa, as well as the spinor field matrices (ΨA)aȧ and

their adjoints (ΨA)ȧa, respectivly. They both transform as (N̄,N) and (N, N̄) representa-

tions of the gauge group, respectively. Pair of the U(N) gauge fields are hermitian matrices

(Aµ)ab and (µ)ȧḃ, respectively. In matrix notation, the covariant derivatives for scalars are

DµX
(A)
A = ∂µX

(A)
A

(−)
+ i
(
AµX

(A)
A −X(A)

A µ

)
, (2.3)

while for spinor fields we have equivalent expressions. Infinitesimal gauge transformations

are given by

δAµ = DµΛ = ∂µΛ+i[Aµ,Λ], δµ = DµΛ̂ = ∂µΛ̂+i[µ, Λ̂], δXA = −iΛXA+iXAΛ̂, (2.4)

and so forth. For the general action see the subsections below as well as [7, 8].

2.1 Noncommutative BRST transformations

We now move on to the noncommutative theory by specifying its gauge symmetry in the

BRS convention. Let us first introduce space spanned by the Moyal star(?)-product

(f ? g)(x) = f(x) ? g(x) = f(x)e
i
2

←
∂µθµν

→
∂νg(x), (2.5)

and the following multiplication consistency relations,

XA ? X
B −→ (XA)aḃ ? (XB)ḃb , and XB ? XA −→ (XB)ȧa ? (XA)aḃ, (2.6)

hence the Moyal star-product of four X’s reads as

XA ? X
B ? XC ? X

D, and XA ? XB ? X
C ? XD. (2.7)

It is also worth noting that the maximum (nondegenerate) rank of the matrix, θµν , is 2,

since we are in three dimensions. To avoid unitarity problems — see [22, 23], we shall

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assume that θ0i = 0, i.e., the time-space coordinate commutes. This assumption in three

dimensions constrains nontrivial components of θµν to θ12( 6= 0) component only.

Now we define all noncommutative BRST transformations we need in the rest of this ar-

ticle:

(sAµ)ab = (DµΛ)ab = (∂µΛ)ab + i[Aµ ?, Λ]ab,

(sµ)ȧḃ = (DµΛ̂)ȧḃ = (∂µΛ̂)ȧḃ + i[µ ?, Λ̂]ȧḃ,

(sXA)aȧ = −iΛab ? (XA)bȧ + i(XA)aḃ ? Λ̃ḃȧ,

(sXA)ȧa = i(XA)ȧb ? Λba − iΛ̃ȧḃ ? (XA)ḃa,

(sΨA)aȧ = −iΛab ? (ΨA)bȧ + i(ΨA)aḃ ? Λ̃ḃȧ,

(sΨA)ȧa = i(ΨA)ȧb ? Λbb − iΛ̃ȧḃ ? (ΨA)ḃa,

sΛ = −iΛ ? Λ, sΛ̂ = −iΛ̂ ? Λ̂,

(2.8)

with covariant derivatives being as follows

(DµXA)aȧ = ∂µ(XA)aȧ + i(Aµ)ab ? (XA)bȧ − i(XA)aḃ ? (µ)ḃȧ ,

(DµX
A)ȧa = ∂µ(XA)ȧa + i(µ)ȧb ? (XA)ba − i(XA)ȧḃ ? (Aµ)ḃa ,

(DµΨA)aȧ = ∂µ(ΨA)aȧ + i(Aµ)ab ? (ΨA)bȧ − i(ΨA)aḃ ? (µ)ḃȧ ,

(DµΨA)ȧa = ∂µ(ΨA)ȧa + i(µ)ȧb ? (ΨA)ba − i(ΨA)ȧḃ ? (Aµ)ḃa .

(2.9)

2.2 Noncommutative generalization of the action

Our next step is to present the classical action of NCABJM field theory. From now

on we restrict ourselves to U(1)κ×U(1)−κ theory for simplicity, since, generalization to

U(N)κ×U(N)−κ is straightforward because of the multiplication consistency relations (2.6).

This action consists of terms that are generalizations of those of ordinary U(1)κ×U(1)−κ
ABJM field theory, as well as the new interaction terms that are analogous to the commu-

tative U(N)κ×U(N)−κ theory yet vanish for N=1. The noncommutative Chern-Simons,

kinetic and additional terms having four and six fields respectively, are

S = SCS + Skin + S4 + S6, (2.10)

SCS =
κ

2π

∫
d3x εµνλtr

(
1

2
Aµ ? ∂νAλ +

i

3
Aµ ? Aν ? Aλ −

1

2
µ ? ∂νÂλ −

i

3
µ ? Âν ? Âλ

)
,

(2.11)

Skin =
κ

2π

∫
d3x tr

(
−DµXA ? DµXA + iΨ̄A ? /DΨA

)
, (2.12)

S4 = S4a + S4b + S4c, (2.13)

S4a =
iκ

2π

∫
d3x tr

[
εABCD(Ψ̄A ? XB ?ΨC ? XD)− εABCD(Ψ̄A ? XB ?ΨC ? XD)

]
, (2.14)

S4b =
iκ

2π

∫
d3x tr

[
Ψ̄A ?ΨA ? XB ? X

B − Ψ̄A ?ΨA ? XB ? XB

]
, (2.15)

S4c =
iκ

2π

∫
d3x tr

[
2(Ψ̄A ?ΨB ? XA ? XB)− 2(Ψ̄A ?ΨB ? XA ? X

B)
]
, (2.16)

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S6 = −1

6

κ

2π

∫
d3x tr(N IA ? N I

A)

=
1

3

κ

2π

∫
d3x tr

[
XA ? XA ? X

B ? XB ? X
C ? XC +XA ? X

A ? XB ? X
B ? XC ? X

C

+ 4XA ? X
B ? XC ? X

A ? XB ? X
C − 6XA ? XB ? X

B ? XA ? X
C ? XC

]
, (2.17)

where

N IA = Γ̃IAB
(
XC ? X

C ? XB −XB ? X
C ? XC

)
− 2Γ̃IBCXB ? X

A ? XC ,

N I
A = ΓIAB

(
XC ? XC ? X

B −XB ? XC ? X
C
)
− 2ΓIBCXB ? XA ? XC , (2.18)

with ΓIAB being 4× 4 matrices, the generators of the SO(6) group, satisfying:

ΓIAB = −ΓIBA, ∀I = 1, . . . , 6; ΓIΓJ + ΓJΓI = 2δIJ ,

Γ̃I = (ΓI)† ⇐⇒ Γ̃IAB = (ΓIBA)∗ = −(ΓIAB)∗ =
1

2
εABCDΓICD,

N I
A =

(
N IA

)†
. (2.19)

The coefficients in three possible structures for the Ψ2X2 terms are chosen so that they give

correct result required by supersymmetry. Some points are discussed and demonstrated in

details in the main text and the appendix of ref. [7].

Next we give the noncommutative gauge-fixing plus ghost terms explicitly:

Sgf+ghost = − κ

2π

∫
d3x

[
1

2ξ
∂µA

µ ? ∂νA
ν − Λ̄ ? ∂µD

µΛ− 1

2ξ
∂µÂ

µ ? ∂νÂ
ν +

¯̂
Λ ? ∂µD

µΛ̂

]
,

(2.20)

where covariant derivative is defined as in (2.9): DµΛ = ∂Λ + i[Aµ ?, Λ].

Note that the additional interaction terms of the schematic forms X2Ψ2 and X6 are

not required to deduce the equations of motion of the gauge fields, which are

Jµ =
1

2
εµνλFνλ and Ĵµ = −1

2
εµνλF̂νλ, (2.21)

where

Jµ = iXAD
µXA − iDµXAX

A − Ψ̄AγµΨA, (2.22)

and

Ĵµ = iXADµXA − iDµXAXA − Ψ̄Aγ
µΨA. (2.23)

In the special case of the U(1)κ×U(1)−κ theory one has Jµ = −Ĵµ, and hence the equations

of motion imply Fµν = F̂µν .

2.3 Noncommutative supersymmetric transformations

Next, using notations of previous subsection, we give the supersymmetric transformation

for the U(1) fields: Aµ and µ gauge fields, scalar fields XA, complex fermion fields ΨA,

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←− ←−

gauge propagator hgauge propagator

Aµ(p) Aν(p) µ(p) Âν(p)

Λ̄(p) Λ(p)

ghost propagator

¯̂
Λ(p)

hghost propagator

Λ̂(p)

XA(p)

scalar propagator

XB(p) Ψj
B(p)

fermion propagator

Ψ̄iA(p)

Figure 1. Notations and the propagators of the relevant fields.

and their adjoints as well, respectively:

(δAµ)ab =
(

ΓIAB ε̄
IγµΨA ? X

B − Γ̃IABXB ? Ψ̄Aγµε
I
)a

b ,

(δµ)ȧḃ =
(

ΓIABX
B ? ε̄IγµΨA − Γ̃IABΨ̄Aγµε

I ? XB

)ȧ
ḃ ,

(δXA)aȧ =
(
iΓIAB ε̄

IΨB
)a

ȧ ,

(δXA)ȧa =
(
− iΓ̃IABΨ̄Bε

I
)ȧ

a ,

(δΨA)aȧ =
(
− Γ̃IABεI /DXBε

I +N IAεI
)a

ȧ ,

(δΨA)ȧa =
(

ΓIAB /DX
BεI +N I

Aε
I
)ȧ

a ,

(δΨ̄A)ȧa = (δΨT
Aγ

0)ȧa =
(
− ΓIAB ε̄

I /DXB +N I
Aε̄

I
)ȧ

a , (2.24)

with ε̄I = εIγ0 = (εI)Tγ0, and (N I
A)T = N I

A. Detailed verification of the invariance of the

NCABJM action under these transformations is presented in the appendix A.

3 Feynman rules of the U(1)κ×U(1)−κ NCABJM quantum field theory

Our next task is to derive the Feynman rules needed for checking the properties of the

one loop quantum corrections. In this paper we follow the usual BRST quantization,

with relevant presetting given in previous sections. We shall use a Landau gauge which

amounts to the following setting of the gauge parameter: ξ = 0, after having worked out

free gauge propagators.

Diagramatic notations of the relevant fields in our theory in accord with figure 1, like

free gauge field Aµ(ξ = 0), hgauge field µ(ξ = 0), ghost Λ and hghost Λ̂, scalar XA,

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and finally fermion ψAi field, together with their propagators in momentum space are given

next, respectively:

Aµ → Aν : =⇒ 2π

κ

(−εµνρpρ
p2

)
, µ → Âν : =⇒ 2π

κ

(
εµνρp

ρ

p2

)
, (3.1)

Λ→ Λ̄ : =⇒ 2π

κ

(−i
p2

)
, Λ̂→ ¯̂

Λ : =⇒ 2π

κ

(−i
p2

)
, (3.2)

XB → XA : =⇒ 2π

κ

(−i
p2

)
δA

B, Ψ̄Ai → Ψj
B : =⇒ 2π

κ

(−i/pij
p2

)
δA

B. (3.3)

The interaction vertices are derived following the conventional procedure. Results are listed

in the appendix D.

4 Power counting and the limit θµν → 0

With the relevant Feynman rules derived, we are now ready for the consistency tests of

the perturbative NCABJM field theory at loop level. Before starting the computations we

would like to analyze some general properties. Let’s focus on an arbitrary 1PI Feynman

diagram obtained from the action (2.10) in the case of Landau gauge. Assume that the

Feynman diagram in question has EG external gauge fields, EF external fermions, EX
external scalars and no external ghosts. Then, it is not difficult to show that degree of the

UV divergence D for such diagram reads

D = 3− EG − EF −
1

2
EX . (4.1)

Hence, all one-loop diagrams with EG + EF > 3 are UV finite by power counting. Each

of these diagrams is also IR finite by power counting for non-exceptional momenta, so

that one can apply Lebesgue’s dominated convergence theorem and compute the limit

θµν → 0 of each diagram by setting θµν = 0 before the loop momentum integration.

It is thus plain that all one-loop 1PI Green functions of the noncommutative ABJM

quantum field theory in the Landau gauge with EG + EF > 3 transform into the cor-

responding Green functions of the ordinary ABJM quantum field theory in the limit

θµν → 0. The same conclusion is reached for EG = 0 = EF and EX > 6, EG +

EF = 1 and EX = 6, EG + EF = 2 and EX ≥ 4, and finally for EG + EF = 3

and EX ≥ 2, respectively. However for the following combinations of triplet of number

of fields: (EG, EF , EX) = (0, 0, 4), (1, 0, 4), (0, 0, 6), (1, 0, 2), (1, 2, 0), (0, 0, 2), (0, 2, 0), the

power counting formula (4.1) shows that D ≥ 0, i.e. it always shows the presence of UV

divergence, respectively. So, the remaining 1PI Green functions fail to be UV finite by

power counting and thus its limit θµν → 0 cannot be computed as we have just done. In

the sections that follow, we shall work out the limit θµν → 0 of the one-loop 1PI functions

with fewer than four fields.

Let us point out that the number of scalar fields in each interaction term in the ac-

tion (2.10) is even. Hence, straightforward application of Wick’s theorem leads to the

– 8 –



J
H
E
P
0
4
(
2
0
1
8
)
0
7
0

µ ν←
p

←
p

µ1
ν1

µ2 ν2

ℓ

ℓ− p

Figure 2. Gauge field bubble-loop contribution to the gauge field 2-point function Pµνbub.

ℓ

ℓ− p

µ
←
p ν

←
p

Figure 3. Gauge field bubble, ghost-loop contribution to the 2-point function Gµνbub.

conclusion that any correlation function involving an odd number of scalar fields van-

ishes and that, if number of XA and XA fields in the correlation function is not equal it

also vanishes.

5 Gauge field
〈
AµAν

〉
and hgauge field

〈
µÂν

〉
two-point functions

We would like to remind the reader that not all the integrals that we shall deal with in the

sequel are UV finite by power-counting; so to define them and manipulate them properly, we

shall use Dimensional Regularization — this is why they are defined in D dimensions. Only

after we have made sure that the UV divergences cancel out upon adding up contributions,

we shall take the limit D → 3.

Generally speaking the total
〈
AµAν

〉
one-loop 1PI two-point function Πµν

AA(p) is the

sum of the following contributions

Πµν
AA(p) = (Pµνbub + Pµνtad) + (Fµνbub + Fµνtad) + (Sµνbub + Sµνtad) + (Gµνbub +Gµνtad), (5.1)

where Pµν , Fµν , Sµν , Gµν denotes gauge field, fermion, scalar and ghost running in the

bubble and/or tadpole loop, respectively. Number of contributions from (5.1) vanish due

to the absence of relevant terms in the action, i.e.

Pµνtad = Fµνtad = Gµνtad = 0. (5.2)

The remaining Pµνbub, Gµνbub, Sµνbub, Fµνbub, and Sµνtad we comput next.

5.1 Gauge field bubble and tadpole diagrams

Using Feynman rules from the appendix D, in the appendix B we have found that contri-

butions from the gauge field and ghost loops in the gauge field bubble diagrams, figures 2

and 3 respectively, are equal up to the sign:

Pµνbub = −Gµνbub =

∫
dD`

(2π)D

(
2 sin

`θp

2

)2 `µ(`− p)ν
`2(`− p)2

, (5.3)

with definition `θp = `µθ
µνpν .

– 9 –



J
H
E
P
0
4
(
2
0
1
8
)
0
7
0

ℓ

ℓ− p

µ
←
p ν

←
p

B A

C D

Figure 4. Gauge field bubble, scalar-loop contribution to the 2-point function Sµνbub.

µ ν←
p

←
p

ℓ

ℓ− p

B, i1 A, i2

C, i3 D, i4

Figure 5. Gauge field bubble, fermion-loop contribution to the 2-point function Fµνbub.

Since the phase factors cancel, contributions from scalar and fermion loops in the gauge

field bubble diagrams of figures 4 and 5, are:

Sµνbub =
∑
A

∫
dD`

(2π)D
4`µ`ν − 2(`µpν + pµ`ν) + pµpν

`2(`− p)2
,

Fµνbub = −
∑
A

∫
dD`

(2π)D
4`µ`ν − 2(`µpν + pµ`ν) + p2ηµν

`2(`− p)2
,

Sµνbub + Fµνbub =
∑
A

(
pµpν − p2ηµν

)∫ dD`

(2π)D
1

`2(`− p)2
. (5.4)

The contribution from tadpole diagram in figure 6 vanishes:

Sµνtad = 2ηµν
∑
A

∫
dD`

(2π)D
1

`2
= 0, (5.5)

so for AµAν terms in the effective action we finally have the following gauge field polariza-

tion tensor:

Πµν
AA(p) = Sµνbub + Fµνbub = i

∑
A

1

8

1√
p2

(
pµpν − p2ηµν

)
. (5.6)

By inspecting again Feynman rules in the appendix D it is plain that the 1PI 2-point

function, Π̂µν

ÂÂ
, for the noncommutative hgauge fields from µÂν terms in the action reads

Π̂µν

ÂÂ
(p) = 2

∑
A

1

16

i√
p2

(
pµpν − p2ηµν

)
≡ Πµν

AA(p), (5.7)

so that the Π̂µν

ÂÂ
(p) polarization tensor in the limit θµν → 0 is trivially given by the cor-

responding Green function — polarization tensor — of the ordinary/commutative ABJM

quantum field theory.

– 10 –



J
H
E
P
0
4
(
2
0
1
8
)
0
7
0

µ ν←
p

←
p

ℓ

A B

Figure 6. Gauge field tadpole, scalar-loop contribution to the 2-point function Sµνtad.

6 Mixed gauge field – hgauge field,
〈
AµÂν

〉
, two-point functions

For mixed AµÂν type of terms we have the one-loop 1PI two-point function Π̂µν

AÂ
(p) as a

sum of contributions from figures 7, 8, 9

Π̂µν

AÂ
(p) =

(
P̂µνbub + P̂µνtad

)
+
(
F̂µνbub + F̂µνtad

)
+
(
Ŝµνbub + Ŝµνtad

)
. (6.1)

Again number of contributions from (6.1) vanish due to the absence of relevant terms in

the action, i.e.

P̂µνbub = P̂µνtad = Fµνtad = 0. (6.2)

Remaining F̂µνbub, Ŝµνbub, and Ŝµνtad we comput next.

6.1 Gauge field – hgauge field bubble and tadpole: scalar and fermion loops

After some lengthy computations we found that one-loop diagrams which mix different

types of gauge fields (we will call them “mixing terms” in discussions below) always stay

non-planar (i.e. with nontrivial noncommutative phase factors). In this and next section

we evaluate two- and three-point functions of this type.

One more property of mixing terms is that they are generated by the scalar and fermion

fields running in the loop only. Therefore mixed two-point function Π̂AÂ contains three

diagrams from figures 7, 8, and 9:

Ŝµνbub = −
∑
A

∫
dD`

(2π)D
e−i`θp

(2`− p)µ(2`− p)ν
`2(`− p)2

= −
∑
A

[
4I1ηµν +

(
4
(
I2 − I5

)
+ I
)
pµpν + 2

(
2I3 − I6

)(
p̃µpν + pµp̃ν

)
+ 4I4p̃µp̃ν

]
= −

∑
A

[
4I1

(
ηµν −

pµpν
p2

)
+
i

π

1√
p̃2

pµpν
p2

+ 4I4p̃µp̃ν

]
, (6.3)

Ŝµνtad = 2
∑
A

∫
dD`

(2π)D
e−i`θp

`2
ηµν =

∑
A

i

π

1√
p̃2

ηµν
p2
, (6.4)

F̂µνbub =
∑
A

∫
dD`

(2π)D
e−i`θp

tr
(
γµ/̀γν(/̀− /p)

)
`2(`− p)2

=
∑
A

[(
4I1 + 2Ip2 − i

2π

1√
p̃2

)(
ηµν −

pµpν
p2

)
+ 4I4p̃µp̃ν

]
, (6.5)

– 11 –



J
H
E
P
0
4
(
2
0
1
8
)
0
7
0

ℓ

ℓ− p

µ
←
p ν

←
p

Figure 7. Gauge field-hgauge field bubble, scalar-loop contribution to the 2-point function Ŝµνbub.

A, ℓ

µ
←
p ν

←
p

Figure 8. Gauge field-hgauge field tadpole, scalar-loop contribution to the 2-point function Ŝµνtad.

where we denote two structures kµθ
µνpν = kθp and p̃µ = θµνpν , respectively. For the

definitions and details of the above integrals I, I1, . . . . . . , I6, see the appendix C.

Once we sum over all contributions and perform a standard tensor reduction, the

integral boils down to a single tensor structure multiplying one scalar master integral

I(p, θ), which in the Minkovski signature is IM (p, θ). So, from mixed AµÂν terms we

finally have the following polarization tensor:

Π̂µν

AÂ
= Ŝµνbub + Ŝµνtad + F̂µνbub =

∑
A

(
p2ηµν − pµpν

) ∫ dD`

(2π)D
e−i`θp

`2(`− p)2

=
∑
A

(
p2ηµν − pµpν

)
IM (p, θ), (6.6)

with IM (p, θ) for Minkowski signature being given in the appendix C by (C.15) via (C.14).

Taking commutative limit θµν → 0 the above polarization tensor iΠ̂µν

AÂ
from (6.6) takes

very simple form:

lim
θ→0

iΠ̂µν

AÂ
=
∑
A

(
p2ηµν − pµpν

)
lim
θ→0

IM (p, θ) = i
∑
A

(
p2ηµν − pµpν

)
8
√
p2 − i0+

, (6.7)

i.e. IM (p, θ) clearly converges to the commutative value smoothly when θµν → 0, which is

precisely the 1-loop contribution to the iΠ̂µν

AÂ
in the ordinary/commutative ABJM theory.

7 Gauge field
〈
Aµ1Aµ2Aµ3

〉
and hgauge field

〈
µ1µ2µ3

〉
, three-point

functions

From Feynmanm rules in appendix D we have one-loop 1PI three-point function Πµ1µ2µ3
AAA

as a sum of contributions from diagrams in figure 10,

Πµ1µ2µ3
AAA = Pµ1µ2µ3tria1 +Gµ1µ2µ3tria1 +Gµ1µ2µ3tria2 + Sµ1µ2µ3tria1 + Sµ1µ2µ3tria2

+ Fµ1µ2µ3tria1 + Fµ1µ2µ3tria2 + Sµ1µ2µ3bub1 + Sµ1µ2µ3bub2 + Sµ1µ2µ3bub3 , (7.1)

– 12 –



J
H
E
P
0
4
(
2
0
1
8
)
0
7
0

ℓ

ℓ− p

µ
←
p ν

←
p

B, i1 A, i2

C, i3 D, i4

Figure 9. Gauge field-hgauge field buble, fermion-loop contribution to the 2-point function F̂µνbub.

p3 p3 p3 p3 p3

+ + + +

p1 p2 p1 p2 p1 p2 p1 p2 p1 p2

p3

+ + + + +

p3 p3 p1 p2

p1 p2 p1 p1p2 p2 p2 p3 p3 p1

Figure 10. One-loop contributions to the gauge field 3-point function
〈
Aµ1Aµ2Aµ3

〉
.

while for µ1µ2µ3 terms in the effective action S (2.10) we have the one-loop 1PI three-

point functions Π̂µ1µ2µ3
ÂÂÂ

as a sum of contributions from the sum of diagrams in figure 10

where all wavy gauge field lines are replaced by the curly hgauge field lines with relevant

Feynman rules given in the appendix D, for every pair (µi, pi), i = 1, 2, 3:

Π̂µ1µ2µ3
ÂÂÂ

= P̂µ1µ2µ3tria1 + Ĝµ1µ2µ3tria2 + Ĝµ1µ2µ3bub1 + Ŝµ1µ2µ3tria1 + Ŝµ1µ2µ3tria2

+ F̂µ1µ2µ3tria1 + F̂µ1µ2µ3tria2 + Ŝµ1µ2µ3bub1 + Ŝµ1µ2µ3bub2 + Ŝµ1µ2µ3bub3 . (7.2)

In eqs. (7.1) and (7.2), P,G, S and F denote gauge field, ghost, scalar and fermion loops,

respectivly. Other contributions vanish due to the absence of relevant terms in the ac-

tion (2.10). Remaining non-vanishing terms in (7.1) and (7.2) are presented next by looking

into the one-loop corrections to the identical three gauge field vertex.

There are three relevant diagrams: the gauge field triangle (1st diagram in figure 10)

and clockwise/counterclockwise running loop-momenta ghost triangles (2nd and 3d dia-

grams in figure 10), contributing to the Pµ1µ2µ3tria1 , Gµ1µ2µ3tria1 , and to the Gµ1µ2µ3tria2 , respectively.

– 13 –



J
H
E
P
0
4
(
2
0
1
8
)
0
7
0

The gauge field triangle is as follows:

Pµ1µ2µ3tria1 =

∫
dD`

(2π)D
(−2i)3 sin

−`θ(`− p2)

2
sin

(−`+ p1)θ(`− p1 − p2))

2

· sin (−`+ p1 + p2)θ`

2
εσ2µ1σ1εσ1ρ1σ3ε

σ3µ2σ4εσ4ρ2σ5ε
σ5µ3σ6εσ6ρ3σ2

· `
ρ1(`− p1 − p2)ρ2(`− p1)ρ3

`2(`− p1)2(`− p1 − p2)2
,

(7.3)

while the ghost triangles read:

Gµ1µ2µ3tria1 =

∫
dD`

(2π)D
(2i)3 sin

p1θ`

2
sin

p2θ(`− p1)

2
sin

(−p1 − p2)θ(`− p1 − p2)

2

· (`− p1)µ1(`− p1 − p2)µ2`µ3

`2(`− p1)2(`− p1 − p2)2
,

(7.4)

Gµ1µ2µ3tria2 =

∫
dD`

(2π)D
(2i)3 sin

p2θ`

2
sin

p1θ(`− p2)

2
sin

(−p1 − p2)θ(`− p1 − p2)

2

· (`− p2)µ2(`− p1 − p2)µ1`µ3

`2(`− p2)2(`− p1 − p2)2
.

(7.5)

Using a simple transformation ` → −` + p1 + p2 one can turn the denominator and the

phase factor of the Gµ1µ2µ3tria2 to be identical to those in Gµ1µ2µ3tria1 ,

Gµ1µ2µ3tria1 =

∫
dD`

(2π)D
(2i)3 sin

p1θ`

2
sin

p2θ(`− p1)

2
sin

(−p1 − p2)θ(`− p1 − p2)

2

· (`− p1)µ2(`− p1 − p2)µ3`µ1

`2(`− p1)2(`− p1 − p2)2
.

(7.6)

Summing over Pµ1µ2µ3tria1 , Gµ1µ2µ3tria1 and Gµ1µ2µ3tria2 , and reducing the Levi-Civita symbols into

metric contractions, we get

Pµ1µ2µ3tria1 +Gµ1µ2µ3tria1 +Gµ1µ2µ3tria2 = −8i

∫
dD`

(2π)D
sin

`θp1

2
sin

(`− p1)θp2

2
sin

`θ(p1 + p2)

2

· `
µ2(pµ31 pµ12 − pµ11 pµ32 ) + `µ3(pµ21 pµ12 − pµ22 pµ11 ) + `µ1(pµ21 pµ32 − pµ31 pµ22 )

`2(`− p1)2(`− p1 − p2)2
.

(7.7)

If one removes the sin functions from the integrand of the previous integral, one ends up

with an integral which is both UV and IR divergent by power counting. Hence, one can

apply Lebesgue’s dominated convergence theorem and commute the limit θ → 0 with the

integral symbol in (7.7) to conclude that

lim
θ→0

[
Pµ1µ2µ3tria1 +Gµ1µ2µ3tria1 +Gµ1µ2µ3tria2

]
= 0. (7.8)

This is in the full agreement with the fact that in the ordinary abelian ABJM field theory

the first three Feynman diagrams from figure 10 do not exist.

Now, by using Feynman rules one can easy show that the last seven diagrams in

figure 10 do not involve nonplanar contributions, i.e., the Moyal phases in them do not

– 14 –



J
H
E
P
0
4
(
2
0
1
8
)
0
7
0

involve the loop momentum but only the external momenta. Hence the limit θµν → 0 exists

trivially at D = 3, and, if sum of those seven diagrams is UV finite for nonzero θµν , it is

given by the ordinary result. One can show that this is the case. Indeed, the sum of the

contributions to the 4th and 5th diagrams which are not UV finite by power counting reads

−
∑
A

(
e
i
2
p1θp2 + e−

i
2
p1θp2

) ∫ d3`

(2π)3

8 `µ1`µ2`µ3

`2(`+ p1)2(`+ p1 + p2)2
. (7.9)

It can be shown that the sum of contributions to the 6th and 7th diagrams which are

not UV finite by power counting is given by∑
A

(
e
i
2
p1θp2+e−

i
2
p1θp2

) ∫ d3`

(2π)3

8 `µ1`µ2`µ3 − 2`2(`µ1ηµ2µ3 + `µ2ηµ1µ3 + `µ3ηµ1µ3)

`2(`+ p1)2(`+ p1 + p2)2
. (7.10)

By adding contributions of the last three diagrams in figure 10, which are not UV finite

by power counting, one obtains∑
A

(
e
i
2
p1θp2 + e−

i
2
p1θp2

) ∫ d3`

(2π)3

2`2(`µ1ηµ2µ3 + `µ2ηµ1µ3 + `µ3ηµ1µ3)

`2(`+ p1)2(`+ p1 + p2)2
. (7.11)

Finally, the sum of equations (7.9), (7.10) and (7.11) is plain zero. Hence, the sum of the

last seven diagram of figure 10 is indeed UV finite by power counting for non-zero θµν , so

that its θµν → 0 limit is given by the corresponding sum of diagrams of the commutative

ABJM theory.

In summary, we have shown that the sum of all diagrams in figure 10 involves only

integrals which are UV finite by power counting and that the limit θµν → 0 of the sum

is given by the sum of relevant diagrams in the ordinary ABJM field theory. Hence, the

one-loop 1PI contribution to the
〈
Aµ1Aµ2Aµ3

〉
is UV finite and by taking the limit θµν → 0

of it one obtains the corresponding Green function of the ordinary ABJM quantum field

theory. From Feynman rules in the appendix D it is clear that the same holds for the〈
µ1µ2µ3

〉
three-point function.

8 Mixed gauge field – hgauge field,
〈
Aµ1Aµ2µ3

〉
,
〈
µ1µ2Aµ3

〉
, three-

point functions

For mixed Aµ1Aµ2µ3 type of terms we have the one-loop three-point function Π̂µ1µ2µ3
AAÂ

as a sum of seven contributions, two from clockwise and counterclockwise running scalars,

three running scalars in bubbles and two fermion clockwise and counterclockwise triangles

as shown in figures 11, 12, 13, 14, 15, 16 and 17. We denote them as follows, respectively:

Π̂µ1µ2µ3
AAÂ

= Ŝµ1µ2µ3tria1 + Ŝµ1µ2µ3tria2 + Ŝµ1µ2µ3bub1 + Ŝµ1µ2µ3bub2 + Ŝµ1µ2µ3bub3 + F̂µ1µ2µ3tria1 + F̂µ1µ2µ3tria2 . (8.1)

Other contributions vanish due to the absence of relevant terms in the action.

For µ1µ2Aµ3 type of terms we have the one-loop 1PI three-point function Π̃µ1µ2µ3
ÂÂA

as

a sum of contributions from the same figures 11, 12, 13, 14, 15, 16 and 17, where the wavy

gauge field lines are replaced with curly gauge field lines and vice-versa (wavy ↔ curly).

Π̃µ1µ2µ3
ÂÂA

= S̃µ1µ2µ3tria1 + S̃µ1µ2µ3tria2 + S̃µ1µ2µ3bub1 + S̃µ1µ2µ3bub2 + S̃µ1µ2µ3bub3 + F̃µ1µ2µ3tria1 + F̃µ1µ2µ3tria2 . (8.2)

– 15 –



J
H
E
P
0
4
(
2
0
1
8
)
0
7
0

→
p3, µ3A3

B3

ℓB1A1

ℓ− p1

B2

A2

ℓ− p1 − p2

←
p1, µ1

←
p2, µ2

Figure 11. Scalar triangle-loop contribution1 to the 3-point function Ŝµ1µ2µ3

tria1 .

←
p1, µ1

←
p2, µ2

→
p3, µ3

ℓ− p2

ℓ− p1 − p2

ℓ

A3

B3

B1

A1

B2A2

Figure 12. Scalar triangle-loop contribution2 to the 3-point function Ŝµ1µ2µ3

tria2 .

Other contributions vanish due to the absence of relevant terms in the action. Remaining

terms in (8.1) and (8.2) we compute next by using Feynman rules from the appendix D.

8.1 Loop integrals contributing to the
〈
Aµ1Aµ2µ3

〉
3-point function

Computation of Fynman diagrams from figures 11 and 12, gives, respectively:

Ŝµ1µ2µ3tria1 =
∑
A

e
i
2
p1θp2

∫
dD`

(2π)D
e−i`θ(p1+p2)

· (2`− p1)µ1(2`− 2p1 − p2)µ3(2`− p1 − p2)µ2

`2(`− p1)2(`− p1 − p2)2
, (8.3)

Ŝµ1µ2µ3tria2 =
∑
A

e−
i
2
p1θp2

∫
dD`

(2π)D
e−i`θ(p1+p2)

· (2`− p2)µ2(2`− p1 − 2p2)µ3(2`− p1 − p2)µ1

`2(`− p2)2(`− p1 − p2)2
. (8.4)

Inspecting eqs. (8.3) and (8.4) one finds out that diagrams in figures 11 and 12 transfer

one to each other by simple replacement:

Ŝµ1µ2µ3tria1

∣∣∣∣p1↔p2
µ1↔µ2

= Ŝµ1µ2µ3tria2 . (8.5)

From Fynman diagrams in figures 13 and 14 we have

Ŝµ1µ2µ3bub1 = −
∑
A

e
i
2
p1θp2

∫
dD`

(2π)D
e−i`θ(p1+p2) η

µ2µ3(2`− p1)µ1

`2(`− p1)2
, (8.6)

Ŝµ1µ2µ3bub2 = −
∑
A

e−
i
2
p1θp2

∫
dD`

(2π)D
e−i`θ(p1+p2) η

µ1µ3(2`− p2)µ2

`2(`− p2)2
, (8.7)

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ℓ− p2

µ1,
←
p1 ℓ

µ2,
→
p2

µ3,
←
p3

A

B D

C

Figure 13. Scalar bubble-loop contribution1 to the 3-point function Ŝµ1µ2µ3

bub1 .

ℓµ2,
←
p2

ℓ− p1

µ1,
→
p1

µ3,
←
p3

Figure 14. Scalar bubble-loop contribution2 to the 3-point function Ŝµ1µ2µ3

bub2 .

ℓ

µ2,
←
p2

ℓ− p1 − p2

µ3,
→
p3

µ1,
←
p1

B1

A1 B3

A3

Figure 15. Scalar bubble-loop contribution3 to the 3-point function Ŝµ1µ2µ3

bub3 .

←
p1, µ1

←
p2, µ2

→
p3, µ3

ℓ− p1

ℓ

ℓ− p1 − p2

Figure 16. Fermion triangle-loop contribution1 to the 3-point function F̂µ1µ2µ3

tria1 .

←
p1, µ1

←
p2, µ2

→
p3, µ3

ℓ− p2

ℓ− p1 − p2

ℓ

Figure 17. Fermion triangle-loop contribution2 to the 3-point function F̂µ1µ2µ3

tria2 .

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while in diagram from figure 15 two phase terms combine into the cos function of external

momenta:

Ŝµ1µ2µ3bub3 = Ŝµ1µ2µ3bub3+ + Ŝµ1µ2µ3bub3−

= −2 cos
p1θp2

2

∑
A

∫
dD`

(2π)D
e−i`θ(p1+p2) η

µ1µ2(2`− p1 − p2)µ3

`2(`− p1 − p2)2
. (8.8)

Finally computation of Fynman diagrams from figures 16 and 17, for D = 3 gives:

F̂µ1µ2µ3tria1 = −
∑
A

e
i
2
p1θp2

∫
dD`

(2π)D
e−i`θ(p1+p2)

trγµ1 /̀γµ3(/̀− /p1
− /p2

)γµ2(/̀− /p1
)

`2(`− p1)2(`− p1 − p2)2
, (8.9)

F̂µ2µ2µ3tria2 = −
∑
A

e−
i
2
p1θp2

∫
dD`

(2π)D
e−i`θ(p1+p2)

trγµ2 /̀γµ3(/̀− /p1
− /p2

)γµ1(/̀− /p2
)

`2(`− p2)2(`− p1 − p2)2
. (8.10)

8.2 Computations of the
〈
Aµ1Aµ2µ3

〉
3-point functions

Performing some computations of diagrams in figures 11, 12, 16 and 17 we find that the

opposite loop momenta running generates opposite overall phase factors e±
i
2
p1θp2 . We

then use such phases to decompose the rest of the tensor Π̂µ1µ2µ3
AAÂ

into two groups, the

Π̂µ1µ2µ3
AAÂ+

and the Π̂µ1µ2µ3
AAÂ−

tensors, respectively. There we have three terms from scalar

bubble diagrams, two of them carry fixed running phase while the last one contains two

terms with opposite phases, they are marked as well as the two scalar triangles and the

two fermion triangles.

Next step is to sum over contributions to each phase, for the clockwise running part:

Π̂µ1µ2µ3
AAÂ+

= Ŝµ1µ2µ3tria1 + F̂µ1µ2µ3tria1 + Ŝµ1µ2µ3bub1 + Ŝµ1µ2µ3bub3+ , (8.11)

and, for the counterclockwise running part:

Π̂µ1µ2µ3
AAÂ−

= Ŝµ1µ2µ3tria2 + F̂µ1µ2µ3tria2 + Ŝµ1µ2µ3bub2 + Ŝµ1µ2µ3bub3− . (8.12)

After summing over all terms with loop momenta carrying more than one external index,

i.e. `µ1`µ2`µ3 and `µi`µj , i, j = 1, 2, 3 terms, cancel. Now we use the standard relation

2` · p = (`+ p)2 − p2 − `2 to turn the higher power in ` terms in the triangle integral into

the bubble type of integrals. We also observe that∫
dD`

(2π)D
e
i
2
p1θp2ei`θ(p1+p2)

(`− p1)2(`− p1 − p2)2
=

∫
dD`

(2π)D
e−

i
2
p1θp2ei`θ(p1+p2)

`2(`− p2)2
, (8.13)

therefore such terms after the transformation have to be moved from one group to the

other, and than, as indicated above, the tensor Π̂µ1µ2µ3
AAÂ

boils down to

Π̂µ1µ2µ3
AAÂ

= Π̂µ1µ2µ3
AAÂ+

+ Π̂µ1µ2µ3
AAÂ−

, (8.14)

where

Π̂µ1µ2µ3
AAÂ+

=− e i2p1θp2
(

Πµ1µ2µ3
1 · I(p1 + p2) + Πµ1µ2µ3

2 · Î(p1) + Πµ1µ2µ3
3 · I+

+ Πµ1µ2
4 (p1, p2) · Iµ3+ + Πµ2µ3

4 (p2, p3) · Iµ1+ + Πµ1µ3
4 (p1, p3) · Iµ2+

)
,

(8.15)

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while

Π̂µ1µ2µ3
AAÂ−

= Π̂µ1µ2µ3
AAÂ+

(p1 ↔ p2, µ1 ↔ µ2) . (8.16)

The above master integrals Î(p1), I+ and Iµ+ bear the following forms:

Î(p1) =

∫
dD`

(2π)D
e−i`θ(p1+p2)

`2(`− p1)2
, (8.17)

I+ =

∫
dD`

(2π)D
e−i`θ(p1+p2)

`2(`− p1)2(`− p1 − p2)2
, (8.18)

Iµ+ =

∫
dD`

(2π)D
`µe−i`θ(p1+p2)

`2(`− p1)2(`− p1 − p2)2
, (8.19)

while the tensor structures are given below

Πµ1µ2µ3
1 = 2

(
ηµ2µ3(p1 + p2)µ1 − ηµ1µ3(p1 + p2)µ2

)
,

Πµ1µ2µ3
2 = 2ηµ2µ3pµ12 ,

Πµ1µ2µ3
3 = pµ11 (p1 + p2)µ3(2p1 + p2)µ2 + ηµ1µ2

(
pµ31 (2p1 · p2 + p2

2)− pµ32 p2
1

)
− ηµ1µ3

(
pµ22 p2

1 + pµ21 (2p1 · (p1 + p2) + p2
2)
)

+ ηµ2µ3
(
pµ12 p2

1 − pµ11 (2p1 · p2 + p2
2)
)
,

Πµ1µ2
4 = 2(ηµ1µ2p1 · p2 − pµ21 pµ12 ). (8.20)

Now, by setting θ = 0 in the integrands of Î(p1), I+ and Iµ+, one obtains integrals which

are both UV finite and IR finite by power counting. Therefore one can apply Lebesque’s

dominated convergence theorem to conclude that the limit θµν → 0 of the tensor Π̂µ1µ2µ3
AAÂ+

exists and is given by the corresponding Green function of the commutative ABJM field

theory. It is plain that the analysis carried out for the Π̂µ1µ2µ3
AAÂ+

tensor will apply to the

tensor Π̂µ1µ2µ3
AAÂ−

as well, so that the limit θµν → 0 of the latter is given by the corresponding

Green function in the ordinary ABJM theory too. Putting it all together, one concludes

that the limit θµν → 0 of the Π̂µ1µ2µ3
AAÂ

tensor is given by the ordinary ABJM field theory.

We shall end this subsection by showing explicitly that I+ and Iµ+ have well defined

limit when θ → 0. Both integrals I+ and Iµ+ can be evaluated using the standard Schwinger-

Feynman parametrization, [24]. So, as an example, let us work out I+:

I+ = i

1∫
0

dx

1∫
0

dy (1− y)

∞∫
0

dαα2

∫
dD`

(2π)D
e−α`

2 · eix(1−y)p1θp2 · e−αX− 1
4α

(p̃1+p̃2)2 , (8.21)

where

X = (1− y)
(
x(1− x)p2

1 − y
(
x(1− x)p2

1 − (1− x)(p1 + p2)2 − xp2
2

))
. (8.22)

The integration over variables ` and α then yields Bessel K-functions:

I+ = i (4π)−
D
2

1∫
0

dx

1∫
0

dy(1− y)eix(1−y)p1θp2 · 2 ·X D
4
− 3

2

·
(

(p̃1 + p̃2)2

4

) 3
2
−D

4

K3−D
2

[√
X(p̃1 + p̃2)2

]
.

(8.23)

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In order to analyze the commutative limit we rewrite the D-dimensional Bessel K-function

as sum of two Bessel I-functions

K3−D
2

[√
X(p̃1 + p̃2)2

]
=

π

2 sin
(
3− D

2

)
π

·
(
ID

2
−3

[√
X(p̃1 + p̃2)2

]
− I3−D

2

[√
X(p̃1 + p̃2)2

])
.

(8.24)

The Bessel I-functions can then be expand as power series.1 Next we can observe that

the power series with respect to θ converges for small θ and D < 4, with the leading term

matching the commutative scalar triangle in [24]. Therefore the commutative limit exists.

The integral Î(p1) may be estimated using the same method performed for the integral

I in the appendix C yielding the following result

Î(p1)
∣∣∣
D→3

=

√
2

(4π)
3
2

1∫
0

dx eix(p1θp2)

(
x(1− x)p2

1

(p̃1 + p̃2)2

)− 1
4

K 1
2

[√
x(1− x)p2

1(p̃1 + p̃2)2

]

=
1

4π

1∫
0

dx eix(p1θp2) e
−
√
x(1−x)p21(p̃1+p̃2)2√
x(1− x)p2

1

.

(8.25)

For small θ power series expansion is regular and the commutative limit does exist.

In the view of the computations of the three point function
〈
Aµ1Aµ2µ3

〉
carried out

above, it is apparent that the three point function
〈
µ1µ2Aµ3

〉
also goes to the ordinary

result when the noncommutative tensor θµν → 0.

9 Scalar | fermion,
〈
XAX

B
〉
|
〈
ΨAΨ̄B

〉
, two-point functions

From the four-field (2-scalars-2-fermions) action S4 (2.13) in accord with the given Feyn-

man rules (3d diagram in figure 25, generically representing a number of diagrams as

the one diagram), we have a number of contracted combinations of indices A,B,C,D.

Since we have the following vanishing propagators:
〈
XAXB

〉
=
〈
XAXB

〉
=
〈
ΨAΨ̄B

〉
=〈

ΨAΨ̄B
〉

= 0, the one-loop tadpole contributions to the 2-point functions coming

from (2.14) part of the action vanish. Namely as illustrated in figure 18, we obtain van-

ishing contributions to the relevant tadpole diagrams due to the antisymmtric properties

of Levi-Civita tensor εABCD in (2.14), following filed contractions in the tadpole loops

ΨAΨ̄B=ΨAΨ̄B=XAXB=XAXB = 0.

To work out the one-loop contributions to
〈
XAX

A
〉

and
〈
Ψ̄AΨA

〉
we only need the

vertices coming from (2.15) and (2.16) parts of the action (2.13).

9.1 One-loop scalar
〈
XAX

B
〉

2-point function

Using Feynman rules from appendix D, one can show that the integrand corresponding to

the 1st Feynman diagram of figure 19 vanishes since the epsilon tensor of the gauge field

1Unlike integral I, the integration over the Feynman parameters can only be performed in D-dimension

here, therefore the expansion over θ is performed in D-dimension.

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Figure 18. Vanishing 1-loop contributions due to: ΨAΨ̄B = ΨAΨ̄B = XAXB = XAXB = 0.

XA XB

+

+

XA XB

XA XB

XA XB XA XB

++

Figure 19. Loop contributions to the scalar 2-point function
〈
XAX

B
〉
.

propagator is contracted with two equal momenta. The 2nd diagram vanishes for the same

reason. The integrands of the 3d and 4th diagrams are zero due to the contraction ηµνεµνρ
that occur in each of them. The last diagram — a digram absent in the ordinary theory

— also has a vanishing integrand since it carries factor (see the 3d Feynman rule (D.10)

from figure 25)

sin
[1

2

(
pθp+ `θ`

)]
≡ 0, (9.1)

where p is the external momentum and ` is the loop momentum. Let us point out that S4a

in (2.13) does not contribute to the last diagram in figure 19, since the free propagators〈
ΨAΨ̄B

〉
and

〈
ΨAΨ̄B

〉
vanish, respectively.

Putting it all together we conclude that the one-loop contribution to the scalar two-

point function
〈
XAX

B
〉

(in the Landau gauge) vanishes in both, the noncommutative and

the ordinary ABJM quantum field theories, respectively.

9.2 One-loop fermion
〈
ΨAΨ̄B

〉
2-point function

Again using relevant Feynman rules from appendix D, one can show that the one-loop

contribution to the
〈
ΨAΨ̄B

〉
(in the Landau gauge) vanishes in both, the noncommutative

and the ordinary theories, respectively. Indeed, the integrands of the 1st two diagrams of

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ΨA Ψ̄B

+ +

ΨA Ψ̄BΨA Ψ̄B

Figure 20. Loop contributions to the fermion 2-point function
〈
ΨAΨ̄B

〉
.

p3 p3 p3

p1 p2 p1 p2 p1 p2

+ +

p3

p1 p2

+

Figure 21. Loop contributions to the 3-point functions
〈
ΨA(p1)Ψ̄B(p2)Aµ(p3)

〉
, and p3 = p2− p1.

figure 20 differ by a minus sign, so their sum vanishes. The integrand of the last diagram

of figure 20 vanishes because it contains exactly the same vanishing factor as in eq. (9.1).

Let us point out that the action S4a in (2.13) does not contribute to the last diagram in

figure 20, since the free propagators
〈
XAXB

〉
and

〈
XAXB

〉
vanish — see i.e. figure 18.

10 Fermion – gauge field | – hgauge field,
〈
ΨAΨ̄BA

µ
〉
|
〈
ΨAΨ̄BÂ

µ
〉
, three-

point functions

By using Feynman rules given in the appendix D it can be easily shown that the sum of

the first two diagrams in figure 21 reads

i δAB e
i
2
p2θp3

∫
dD`

(2π)D

[
ei`θp3 − 1

] [γν(/̀+ /p2
)γµ(/̀+ /p1

)γρ
]
ερνσ`

σ

`2(`+ p1)2(`+ p2)2
, (10.1)

where the incoming fermion has momentum p1 and carries index A, the outgoing fermion

has momentum p2 and carries index B and the incoming gauge field has momentum p3 =

p2 − p1 and Lorentz index µ. Now, by expanding the integrand, the integral in (10.1) can

be expressed as the following sum

i δAB e
i
2
p2θp3

∫
dD`

(2π)D

[
ei`θp3 − 1

] [γν /̀γµ/̀γρ]ερνσ`σ
`2(`+ p1)2(`+ p2)2

+

i δAB e
i
2
p2θp3

∫
dD`

(2π)D

[
ei`θp3 − 1

] [γν(/̀γµ/p1
+ /p2

γµ/̀+ /p2
γµ/p1

)γρ
]
ερνσ`

σ

`2(`+ p1)2(`+ p2)2
.

(10.2)

Let us analyze the limit θ → 0 of the second intregral in (10.2), at D = 3. If we remove

the factor e
i
2
p2θp3

[
ei`θp3 − 1

]
from the integral in question, we will end up with an integral

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that is UV finite and IR finite by power-counting for non-exceptional momenta. Hence, we

can use Lebesgue’s dominated convergence theorem and conclude that the limit θ → 0 of

the second integral in (10.2) can be computed by taking such limit under the integral sign;

but this limit is zero. We have thus shown that in the limit θ → 0 in the sum of the first

two diagrams in figure 21 only the first integral in (10.2) contributes. After a little algebra

and by using γνγργµ = ενρµI + ηρµγν − ηνµγρ + ηνργµ, one obtains

i δAB e
i
2
p2θp3

∫
dD`

(2π)D

[
ei`θp3 − 1

] [γν /̀γµ/̀γρ]ερνσ`σ
`2(`+ p1)2(`+ p2)2

=

−2i δAB e
i
2
p2θp3

∫
dD`

(2π)D

[
ei`θp3 − 1

] `µ

(`+ p1)2(`+ p2)2
.

(10.3)

Let us now consider the sum of the last two diagrams in figure 21. Proceeding as above

and after some lengthy algebra, one concludes that, in the limit θ → 0, the sum of these

two diagrams is given by

+ 2i δAB

∫
dD`

(2π)D
[
ei`θp3e−

i
2
p2θp3 − e i2p2θp3

] `µ

(`+ p1)2(`+ p2)2
. (10.4)

Now, adding (10.3) and (10.4), for D = 3 and p3 = p2 − p1 we finally obtain

− 2i δAB (e
i
2
p2θp3 − e− i

2
p2θp3)

∫
d3`

(2π)3
ei`θp3

`µ

(`+ p1)2(`+ p2)2
, (10.5)

which after changing of variables, `→ −`− p1, gives

2i δAB

(
e
i
2
p2θp3 − e− i

2
p2θp3

)
e−ip1θp3

∫
d3`

(2π)3
e−i`θp3

`µ + pµ1
`2(`− p3)2

. (10.6)

Taking into account results presented in subsection C.2 of the appendix C, we conclude

that the integral ∫
d3`

(2π)3
e−i`θp3

`µ + pµ1
`2(`− p3)2

, (10.7)

remains bounded — although its limit does not exist — as θµν approaches zero. Hence,

the vanishing θµν limit of the expression in (10.6) is zero due to the vanishing factor(
e
i
2
p2θp3 − e− i

2
p2θp3

) ∣∣∣
θ→0

= 2i sin
p2θp3

2

∣∣∣∣
θ→0

= 0. (10.8)

To summarize, we have shown that the limit θµν → 0 of the sum of all four diagrams

in figure 21 vanishes, being also UV finite for the nonvanishing θµν .

Let us finally point out that in the ordinary ABJM field theory, with the gauge group

being abelian, the last two diagrams in figure 21 are absent, besides the sum of the first

two is zero. Indeed, this sum is obtained by setting θ = 0 in the expontetials in (10.1), i.e.,

by setting θ = 0 in the Feynman rules from the appendix D.

It is plain that the conclusion we have just reached for the one-loop 1PI contribution

to
〈
ΨAΨ̄BA

µ
〉

will also be valid for
〈
ΨAΨ̄BÂ

µ
〉
, as a sum of contributions from the details

of figures 21, where the wavy gauge field lines are replaced with curly hgauge field lines

and viceversa (wavy ↔ curly).

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p3 p3 p3

p1 p2 p1 p2 p1 p2

+ +

p3 p3

p1 p2 p1 p2

+ +

+ + ++

p3 p3 p3 p3

p1 p2 p1 p2 p1 p2 p1 p2

Figure 22. Loop contributions to the 3-point functions
〈
XA(p1)XB(p2)Aµ(p3)

〉
, and p3 = p2−p1.

11 Scalar – gauge field | – hgauge field,
〈
XAXBA

µ
〉
|
〈
XAXBÂ

µ
〉
, three-

point functions

From Feynman rules in the appendix D we have the one-loop 1PI three-point function

Γµ
XAXBAµ

as a sum of contributions from the first and second line in figure 22, respectively

Γµ
XAXBAµ

= Sµtri1 + Sµtri2 + Pµtri3 + Pµbub1P + Fµbub2F+

+ Sµlegp1 + Sµlegp2 + Sµleghp1 + Sµleghp2. (11.1)

Similarly we have the one-loop 1PI three-point function Γ̂µ
XAXBµ

as a sum of contri-

butions from the following detailed figure 22 where the wavy gauge field lines are replaced

with curly hgauge field lines and vice-versa (wavy ↔ curly)

Γ̂µ
XAXBµ

= Ŝµtri1 + Ŝµtri2 + P̂µtri3 + P̂µbub1P + F̂µbub2F

+ Ŝµleghp1 + Ŝµleghp2 + Ŝµlegp1 + Ŝµlegp2. (11.2)

Other contributions vanish due to the absence of relevant terms in the action. Remaining

terms in (11.1) and (11.2) we compute below.

We concentrate next on the Γµ
XAXBAµ

. The first three diagrams listed in figure 22, i.e.

the triangle diagrams, seem to be superficially logarithmic divergent without NC regulation.

Explicit computation shows, however, that their divergence order are universally reduced

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by one because of the Levi-Civita tensor:

Sµtri1 = iδAB

∫
dD`

(2π)D
e−i

p1θp2
2 e−i`θ(p1−p2)

· (2`+ p1 − p2)µενρσ(`+ 2p1 − p2)ν(`− p2)ρ(`+ p2)σ

`2(`− p2)2(`+ p1 − p2)2

= −4iδAB

∫
dD`

(2π)D
e−i

p1θp2
2 e−i`θ(p1−p2) (2`+ p1 − p2)µενρσ`

νpρ1p
σ
2

`2(`− p2)2(`+ p1 − p2)2
,

(11.3)

similarly

Sµtri2 = 4iδAB

∫
dD`

(2π)D
ei
p1θp2

2
(2`+ p1 − p2)µενρσ`

νpρ1p
σ
2

`2(`− p2)2(`+ p1 − p2)2
, (11.4)

and

Pµtri3 = δAB

∫
dD`

(2π)D
2
εµνρεδηνεσγρ(p2 − `− p1)η(p1 + p2 − `)δ(2p2 − `)σ`γ

`2(`− p2)2(`+ p1 − p2)2

· ei
p1θp2

2 e
i
2
`θ(p1−p2) sin

`θ(p1 − p2)

2

= −4δAB

∫
dD`

(2π)D
ei
p1θp2

2

(
1− e−`θ(p1−p2)

)
ενρσ`

νpρ1p
σ
2 (`µ − pµ2 ).

(11.5)

Lebesgue’s dominated convergence theorem then rules these three integrals as continues at

the commutative limit. The remaining six bubble integrals are given below. The first two

of them are symmetric under the exchange p1 → −p2.

Performing simple variable change `→ −`+p1−p2, we found the following expression

for the 4th diagram in figure 22:

Pµbub1P = −δAB
∫

dD`

(2π)D
2 sin

`θ(p1 − p2)

2
e
i
2
p1θp2

(
e
i
2
`θ(p1−p2) + e−

i
2
`θ(p1−p2)

)
·
ερσµε η

ρδ εηγσ`
δ(`− p1 + p2)γ

`2(`− p1 + p2)2

= −δAB
∫

dD`

(2π)D
2 sin `θ(p1 − p2)e

i
2
p1θp2 ε

µ
ρσ (p1 − p2)ρ`σ

`2(`− p1 + p2)2

= −2iδAB

∫
dD`

(2π)D
e
i
2
p1θp2e−i`θ(p1−p2) ε

µ
ρσ (p1 − p2)ρ`σ

`2(`− p1 + p2)2
,

(11.6)

while for the 5th diagram in figure 22 we have:

Fµbub2F = δAB

∫
dD`

(2π)D
2e−

i
2
`θ(p1−p2)(4− 2) sin

p1θp2 + `θ(p1 − p2)

2

tr(/̀− /p1
+ /p2

)γµ/̀

`2(`− p1 + p2)2
.

(11.7)

Using the fact trγµγνγρ = 2εµνρ, we conclude that the above contribution is

Fµbub2F = 8δAB

∫
dD`

(2π)D
e−

i
2
`θ(p1−p2) sin

p1θp2 + `θ(p1 − p2)

2

ε µ
ρσ (p1 − p2)ρ`σ

`2(`− p1 + p2)2

= 4iδAB

∫
dD`

(2π)D

(
e−

i
2
p1θp2e−i`θ(p1−p2) − e i2p1θp2

)ε µ
ρσ (p1 − p2)ρ`σ

`2(`− p1 + p2)2
.

(11.8)

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Next four asymmetric bubble diagrams from figure 22 are as follows:

Sµlegp1 = −δAB
∫

dD`

(2π)D
2ie

i
2
p1θp2

(
1 + e−i`θ(p1−p2)

) ε µ
ρσ pρ1`

σ

`2(`+ p1)2
, (11.9)

Sµlegp2 = −δAB
∫

dD`

(2π)D
2ie

i
2
p1θp2

(
1 + ei`θ(p1−p2)

) ε µ
ρσ pρ2`

σ

`2(`− p2)2
, (11.10)

Sµleghp1 = δAB

∫
dD`

(2π)D
4ie

i
2
p1θp2 ε µ

ρσ pρ1`
σ

`2(`+ p1)2
, (11.11)

Sµleghp2 = δAB

∫
dD`

(2π)D
4ie

i
2
p1θp2 ε µ

ρσ pρ2`
σ

`2(`− p2)2
. (11.12)

Then, it is not hard to see that integrals Sµleghp1 and Sµleghp2 are planar, while the nonplanar

part of the remaining integrals involve the same master integral ε(p, θq) which is evaluated

in the appendix C.2, with a common q = p1− p2 up to the ± sign. Furthermore, the Levi-

Civita symbols suppress all pµ terms in nonplanar integrals as well as all planar integrals.

Finally we are left with the following sum of the leading order terms from (11.1)

Γµ
XAXBAµ

∼ i

8π
ε µ
ρσ

((
− 2(p1 − p2)ρ − 2pρ1 + 2pρ2

)
ei
p1θp2

2

+ 4(p1 − p2)ρe−i
p1θp2

2

) (p̃1 − p̃2)σ√
(p̃1 − p̃2)2

=
ε µ
ρσ (p1 − p2)ρ

π

(p̃1 − p̃2)σ

|p̃1 − p̃2|
sin

p1θp2

2
,

(11.13)

which clearly vanishes when θ → 0. This concludes our discussion on the existence of the

commutative limit result and its equivalence to the corresponding result obtained by work-

ing within ordinary ABJM quantum field theory. The latter is obtained by setting θµν = 0

in the integrands of each integral above, prior to the integration over the loop momentum.

In view of the computations carried out, it is plain that the limit θµν → 0 of the 1PI

contribution to the 3-point function
〈
XAXBÂ

µ
〉

exists and matches the ordinary result.

12 Summary and discussion

In this paper we have formulated a quantum ABJM field theory on the noncommutative

spacetime as defined by the Moyal star-product. By using component formalism we have

shown that the theory has an N = 6 supersymmetry. We have done so by defining the

supersymmetry transformations of the noncommutative fields which generalize the ordinary

ones and leave the noncommutative classical action invariant. Next, we have considered

the noncommutative Uκ(1)×U−κ(1) field theory — this theory is radically different from

its ordinary counterpart since it is nonabelian — and we have analyzed the existence of

the noncommutativity matrix θµν → 0 limit of each one-loop 1PI function with fewer

— barring ghosts — than four fields. We have shown that this limit exists and it is given

by the corresponding Green function of the ordinary ABJM quantum field theory, a result

which only trivially holds for all one-loop UV convergent (by power counting) 1PI Green

functions. Along the way we have found out that the computed Green functions turned

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out to be not UV divergent, although they were not UV finite by power counting either.

Of course, we have also seen that power counting and Lebesgue’s dominated convergence

theorem immediately lead to the conclusion that, if the UV degree of divergence is negative,

the limit θµν → 0 of the one-loop 1PI functions is given by the ordinary ABJM quantum

field theory results — see section 4, for further details.

As far as our computations can tell the noncommutative ABJM field theory does not

contain any noncommutative IR divergence and, therefore, it has no noncommutative IR

instability. The noncommutative ABJM quantum field theory put forward here makes an

excellent candidate for well defined noncommutauve gauge field theory which turns into

the ordinary ABJM quantum field theory as the noncommutativity matrix θµν approaches

to zero.

Putting it all together, we can conclude that we have introduced a consistent non-

commutative deformation of the ordinary ABJM quantum field theory, this being a chief

asset of the paper. Of course, many properties of the noncommutative theory remain to

be studied. One most essential task in the authors’ minds is to carry out checks which

could verify that indeed the noncommutative quantum field theory of the U(N)κ×U(N)−κ
generalization of our construction will be the gauge dual of the deformed noncommutative

gravity theory constructed in [19].

In this article we have shown that our construction does possess the same N = 6

supersymmetry and, by construction shares the same multiplet as the undeformed theory.

Therefore the next check which must be carried out is to match the correlation functions

by using the standard prescription:〈
exp
[ ∫

d3k φ0(k)O(k)
]〉

= e−SSUGRA

(
φ(k,u)

)
, (12.1)

where φ0(k) is the boundary value (in Fourier space) of the bulk field φ(k, u), and O(k) de-

notes generically the nonlocal composite operators in [25]. See [26] and references therein,

for further details. On the left hand side of equation (12.1) occur the correlation functions

of the nonlocal composite operators, O(k), discussed in [25], which are to be computed in

the noncommutative ABJM quantum field theory. The values and properties of the 1PI

functions studied in this paper is definitely one solid step towards elucidating the prop-

erties and computing the values of the correlation functions on the left hand side of the

equation (12.1). One has to, however, bear in mind some unique difficulties in this pro-

gram: the first and foremost one from the authors’ viewpoint comes from one crucial basic

property of the (Moyal type) noncommutative deformation, which could be called planar

equivalence rule [27, 28]: this rule states that because of the multiplication consistency

relations (2.6), the planar diagrams of the noncommutative field theory, in the sense of the

(star product analogy to the) color ordering [29] sense,2 contain no loop momenta depen-

dent NC phase factor and therefore remain the same as in the commutative theory from

the loop integral perspective. One the other hand the most successful developments in the

2A color decomposition is convenient to show properties like for example figures 7, 8 and 9 are nonplanar.

Yet we did not use it in the computation presented in this article as they are simple enough without it.

Color decomposition can be very beneficial for more complicated amplitudes in NCABJM theory for sure.

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undeformed gauge/gravity duality program are inherently in the planar limit. This makes

a direct comparison uneasy as the planar limit on the field theory side misses the unique

NC features in the quantum corrections as we have seen above,3 while of the information to

nonplanar amplitudes could be uneasy to obtain from the dual gravity/string theory side.

Also, the dual gravity backgrounds constructed for both N = 4 NCSYM and NCABJM

shares the same property that the NC directions of the metric become degenerate at the

r → ∞ boundary, which could raise quite subtle questions in the holographic correlation

function computation [31]. While to carry out the checks that validate (12.1) lies outside

the scope of this paper, we would like to stress that the most decisive check should be

on the correspondence between nonplanar amplitudes obtained from both field theory and

dual gravity/string theory sides. We would be absolutely delighted if some new checks of

the gauge/gravity duality with noncommutative deformation at the amplitude level can be

done in near future.

Other issues that should be addressed are whether there is a supersymmetry enhance-

ment at levels κ = 1, 2, and, of course, whether the results presented in this paper regarding

UV finiteness and the limit θµν → 0 hold at any order in perturbation theory and for the

U(N) gauge groups.

Finally, it would be very interesting to apply nonperturbative methods [32, 33] to the

noncommutative quantum ABJM field theory introduced in this paper.

Acknowledgments

The work by C.P. Martin has been financially supported in part by the Spanish MINECO

through grant FPA2014-54154-P. This work is also supported by the Croatian Science

Foundation (HRZZ) under Contract No. IP-2014-09-9582, and we acknowledge the support

of the COST Action MP1405 (QSPACE). J. You acknowledges support by the H2020

Twining project No. 692194, RBI-T-WINNING, and would like to acknowledge the support

of W. Hollik and the Max-Planck-Institute for Physics, Munich, for hospitality. We also

thank Johanna Erdmenger, Karl Landsteiner and Jun-bao Wu for many discussions on

gauge/gravity duality and/or ABJM theory.

A SU(4)R supersymmetric invariance of U(1)κ ×U(1)−κ

In the following analysis all the volume integrals are like in the action being 3-dimensional.

We integrate over d3x, and denoted it as the integral only, i.e. the notation is
∫
d3x ≡

∫
.

A.1 Variations of the action with respect to gauge and scalar fields

For the noncommutative Chern-Simons term SUSY transformation δ reads

δSCS =
κ

2π

∫
εµνρ

1

2

(
δAµ ? Fνρ − δµ ? F̂νρ

)
, (A.1)

3On the other hand, we have also seen that the planar diagrams still carry the NC phase factors depending

on the external momenta. It has been worked out in the dipole deformation that the phase factor structure,

as it is, can be nontrivial for crucial subject(s) in gauge/gravity duality like integrability [30]. We thank

Jun-bao Wu for pointing this out to us.

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while for the first scalar field kinetic term from (2.12) we have found

δSkinS =
−κ
2π

∫ [
DµδXA ? DµXA +DµXA ? DµδXA

+
(
iδµ ? XA − iXA ? δAµ

)
DµXA +DµXA

(
iδAµ ? XA − iXA ? δµ

)]
,

= δ1SkinS + δ2SkinS + δ1SkinS + δ2SkinS, (A.2)

δ1SkinS =
κ

2π

∫
D2XA ? δXA, (A.3)

δ1SkinS =
(
δ1SkinS

)∗
=

κ

2π

∫
δXA ? D2XA, (A.4)

δ2SkinS =
−iκ
2π

∫ (
XA ? D

µXAδAµ − δµ(DµXA) ? XA

)
,

δ2SkinS =
−iκ
2π

∫ (
δµ ? DµXA − δAµ(DµXA) ? XA

)
, (A.5)

and for the second fermionic kinetic term we finally have

δSkinF = − κ

2π

∫ (
iδΨ̄A ? /DΨA + iΨ̄A ? /DδΨ

A − Ψ̄A ? δ /AΨA + Ψ̄A ? γ
µΨAδµ

)
,

= δ1SkinF + δ1SkinS + δ2SkinS, (A.6)

δ1SkinF =
−κ
2π

∫
δΨ̄A ? /DΨA, (A.7)

δ1SkinF =
−κ
2π

∫
iDµΨ̄A ? γµδΨ

A, (A.8)

δ2SkinF =
κ

2π

∫ (
− Ψ̄A ? δ /AΨA + Ψ̄Aγ

µ ?ΨAδµ

)
. (A.9)

Now from (A.1), (A.2), (A.3), (A.4), (A.7), (A.8) and by using εµνργµ = γνρ we have

δSCS =
κ

2π

∫
1

2

(
ΓIABX

B ε̄I ? γµνΨAXB − Γ̃IABXBΨA ? γµνεI
)
? Fµν

− κ

2π

∫
1

2

(
ΓIAB ε̄

I ? γµνΨA − Γ̃IABXBΨA ? γµνεIXB

)
? F̂µν , (A.10)

δ1SkinS + δ1SkinS =
iκ

2π

∫ (
D2XBΓIAB ε̄

I ?ΨB − Γ̃IABΨB ? ε
ID2XA

)
(A.11)

δ1SkinF + δ1SkinF =
κ

2π

∫ [
− iD2XBΓIAB ε̄

I ?ΨB + iΓ̃IABΨB ? ε
ID2XA

− 1

2

(
ΓIAB ε̄

I ? γµνΨAXB − Γ̃IABXBΨ̄A ? γ
µνεI

)
? Fµν

− 1

2

(
ΓIABX

B ε̄I ? γµνΨA − Γ̃IABΨA ? γ
µνεIXB

)
? Fµν

+ i
(
N I
Aε̄

I ? /DΨA −DµΨ̄A ? γ
µN IAεI

)]
, (A.12)

which gives:

δSCS + δ1SkinS + δ1SkinS + δ1SkinF + δ1SkinF =
iκ

2π

∫ (
N I
Aε̄

I ? /DΨA −DµΨ̄A ? γ
µN IAεI

)
.

(A.13)

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Finally we obtain:

δSCS+δSkinS+δSkinF = δSCS+δ1SkinS+δ1SkinS+δ1SkinF+δ1SkinF

+
κ

2π

∫ [
−
(
iδµ?XA−iXA?δAµ

)
DµXA−DµXA

(
iδAµ?XA−iXA?δÂ

µ
)

−Ψ̄A?δ /AΨA+Ψ̄A?γ
µΨAδµ+iN I

Aε̄
I ? /DΨA−iDµΨ̄A?N

IAγµεI
]
. (A.14)

A.2 Variations of the action with respect to fermion fields

Let us first define two variations with respect to fermion fields as a sum

δΨ = δ1Ψ + δ3Ψ, (A.15)

where both variations acting on fermion fields give, respectively

δ1ΨA = ΓIABγ
µεIDµX

B, δ3ΨA = N I
Aε

I ,

δ1Ψ̄A = −ΓIAB ε̄
IγµDµX

B, δ3Ψ̄A = N I
Aε̄

I ,

δ1ΨA = −Γ̃IABγµεIDµXB, δ3ΨA = N IAεI ,

δ1Ψ̄A = Γ̃IAB ε̄IγµDµXB, δ3Ψ̄A = N IAε̄I . (A.16)

Now we find a variation of the action S4 with respect to the variation δ1Ψ:

δδ1ΨS4 =
κ

2π

∫ [
iΓ̃IBC

(
2iΨ̄A ? γ

µεI ? Dµ(XB ? X
A ? XC) (A.17)

+ Ψ̄B ? γ
µεI ?

(
2XC ? DµX

A ? XA − 2XA ? DµX
A ? XC

+DµXC ? X
A ? XA −XA ? X

A ? DµXC

))
− 2iεABCDδ1Ψ̄A ? XB ?ΨC ? XD + iΨ̄A ? δ1ΨA ? XB ? X

B

− iδ1Ψ̄A ?ΨA ? XB ? XB + 2iδ1Ψ̄A ?ΨB ? XA ? XB − 2iΨ̄B ? δ1ΨA ? XB ? X
A

]
.

Second, performing the variation with respect to the gauge fields in the kinetic terms of

the X’s and N IAεI we have found

δδA,δÂSkinS =
−κ
2π

∫ [(
iδµ?XA−iXA?δAµ

)
DµXA+DµXA

(
iδAµ?XA−iXA?δµ

)]
=

κ

2π

∫ [
iΓ̃IBCΨ̄B?γ

µεI ?
(
XC ?X

A?DµXA−DµXA?X
A?XC

+XA?D
µXA?XC−XC ?DµX

A?XA

)
−iΓIBC ε̄I ?γµΨB?

(
XA?DµXA?X

C−XC ?DµXA?X
A

+XC ?XA?DµX
A−DµXA?XA?X

C
)]
. (A.18)

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Next after summing (A.17) and (A.18) we have total contribution as

δδ1ΨS4 + δδA,δÂSkinS =
κ

2π

∫ [
iΓ̃IBC

(
2Ψ̄A ? γ

µεI ? Dµ(XB ? X
A ? XC) (A.19)

+ Ψ̄B ? γ
µεI ?

(
Dµ(XC ? X

A ? XA)−Dµ(XA ? X
A ? XC

))
− 2iεABCDδ1Ψ̄A ? XB ?ΨC ? XD + iΨ̄A ? δ1ΨA ? XB ? X

B

− iδ1Ψ̄A ?ΨA ? XB ? XB + 2iδ1Ψ̄A ?ΨB ? XA ? XB − 2iΨ̄B ? δ1ΨA ? XB ? X
A

− iΓIBC ε̄
I ? γµΨB ?

(
XA ? DµXA ? X

C −XC ? DµXA ? X
A

+XC ? XA ? DµX
A −DµXA ? XA ? X

C
)]
,

which should cancel against the variation of SkinF induced by δ3Ψ.

To prove the above statement lets first perform δ3 variation

δδ3ΨSkinF =
κ

2π

∫ (
iδ3Ψ̄A ? /DΨA + iΨ̄A ? /Dδ3ΨA

)
= C.C.+

κ

2π

∫
iΨ̄A ? /Dδ3ΨA

=
κ

2π

∫
iΨ̄A ? /DN

IAεI + C.C. =
κ

2π

∫
iΨ̄A ? γ

µεIDµN
IA + C.C.,

N IA = Γ̃IAB
(
XCX

CXB −XBX
CXC

)
− 2Γ̃IBCXBX

AXC , (A.20)

and then after we sum (A.19) and (A.20) we have our prof verified, i.e.

δδ1ΨS4 + δδ3ΨSkinF + δδA,δÂSkinS = 0, Q.E.D. (A.21)

A.3 Cancellations between SCS, Skin and S4 variations

Let Ψ1,Ψ2, χ3 be spinors, then the integral∫ (
Ψ1i ? Ψ̄2 ? χ3 + Ψ2i ? χ̄3 ?Ψ1 + χ3i ? Ψ̄1 ?Ψ2

)
= 0, ∀i = 1, 2. (A.22)

In our actual computations either of the spinor above may be a ?-product of one of our

Ψ-spinor and one of our scalars X, i.e. χ3 = Ψ3 ? X.

Next we present the simplification to the following contribution from (A.14):

iκ

2π

∫ (
− Ψ̄A ? δ /AΨA + Ψ̄A ? γ

µΨAδµ

)
=
iκ

2π

∫ [
iΨ̄A ?ΨAδXCX

C − iΨ̄A ?ΨA ? δXC ? XC (A.23)

+ 2ΓIBC ε̄
IΨA ?

(
Ψ̄A ?ΨB ? XC −XC ? Ψ̄B ?ΨA

)
+ C.C.

]
.

Now we compute the variation of the 2nd term of S4 induced by δXA:

δXB

[−iκ
2π

εABCD

∫
Ψ̄A ? XB ?ΨC ? XD

]
=

κ

2π

∫ [
− 2iΨ̄A ?ΨAδXBX

B + 2iΨ̄A ?ΨAXBδXB (A.24)

+ 2iΨ̄A ?ΨB ? δXA ? X
B − 2iΨ̄A ?ΨB ? XA ? δXB

− 2ΓIBC ε̄
IΨA ?

(
Ψ̄A ?ΨB ? XC −XC ? Ψ̄B ?ΨA

)]
.

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To work it out we have to use the cyclicality of the ?-product, i.e. employ∫ (
ε̄I ?ΨE ?ΨC

κ + ΨEκ ? ε̄
I ?ΨC + εIκ ? Ψ̄E ?ΨC

)
= 0, ∀κ = 1, 2. (A.25)

Adding up (A.23) and (A.24) with negative C.C. part we have found

δXB

[−iκ
2π

εABCD

∫
Ψ̄A ? XB ?ΨC ? XD

]
+
iκ

2π

∫ [
iΨ̄A ?ΨA ? δXC ? X

C − iΨ̄A ?ΨA ? δXC ? XC

+ 2ΓIBC ε̄
IΨA ?

(
Ψ̄A ?ΨB ? XC −XC ? Ψ̄B ?ΨA

)]
=
iκ

2π

∫ [
− Ψ̄A ?ΨA ? δXB ? X

B + Ψ̄A ?ΨA ? XB ? δXB

+ 2Ψ̄A ?ΨB ? δXA ? X
B − 2Ψ̄A ?ΨB ? XA ? δXB

]
. (A.26)

Now we show that (A.26) cancels against the variations of sum of the 3rd, 4th, 5th

and 6th terms of S4 induced by δXB. After some computations those variations give:

iκ

2π

∫ [
Ψ̄A?ΨA?δXB?X

B−Ψ̄A?ΨA?XB?δXB+2Ψ̄A?ΨB?XA?δXB−2Ψ̄B?ΨA?δXB?X
A
]
,

(A.27)

and it does cancel exactly above expression (A.26), Q.E.D.

Next we show that variations of C.C. terms in (A.23) cancels against the variation δS4

which is a sum of the δXA variation of the first term in S4 and the δXA variation of the

3rd, 4th, 5th and 6th terms of S4

δXA ≡ δXA
[ κ

2π

∫
iεABCDΨ̄A ? XB ?ΨC ? XD

]
=

κ

2π

∫ [
2iΨ̄A ?ΨA ? δXB ? XB − 2iΨ̄A ?ΨA ? XB ? δX

B − 2iΨ̄A ?ΨB ? δXA ? XB

+ 2iΨ̄A ?ΨB ? XA ? δX
B − 2Γ̃IBC ε̄I ?ΨA ?

(
Ψ̄A ?ΨB ? XC −XC ? Ψ̄B ?ΨA

)]
,

(A.28)

δXA ≡ δXA

[
κ

2π

∫
iΨ̄A ?ΨA ? XB ? X

B − iΨ̄A ?ΨA ? XB ? XB

+ 2iΨ̄A ?ΨB ? XA ? XB − 2iΨ̄B ?ΨA ? XB ? X
A

]
=

κ

2π

∫ [
iΨ̄A ?ΨA ? XB ? δX

B − iΨ̄A ?ΨA ? δXB ? XB

+ 2iΨ̄A ?ΨB ? δXA ? XB − 2iΨ̄B ?ΨA ? XB ? δX
A
]
, (A.29)

δS4 = δXA + δXA =
κ

2π

∫ [
iΨ̄A ?ΨA ? δXB ? XB − iΨ̄A ?ΨA ? XB ? δX

B

− 2Γ̃IBC ε̄I ?ΨA ?
(
Ψ̄A ?ΨB ? XC −XC ? Ψ̄B ?ΨA

)]
. (A.30)

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Finally we denote the C.C. terms from eq. (A.23) as δC.C.(A.23) and obtain:

δC.C.(A.23) =
κ

2π

∫ [
iΨ̄A ?ΨA ? XB ? δX

B − iΨ̄A ?ΨA ? δXB ? XB

+ 2Γ̃IBC ε̄I ?ΨA ?
(
Ψ̄A ?ΨB ? XC −XC ? Ψ̄B ?ΨA

)]
, (A.31)

which shows perfect match, i.e. the full cancelation as expected:

δC.C.(A.23) + δS4 = 0, Q.E.D. (A.32)

A.4 Classical SUSY invariance regarding S6 terms

The last step to show the full SUSY invariance of the noncommutative ABJM action is to

confirm that the δ3 transformation of the fermions in the Ψ2X2 terms of the action S4 is

cancelled by the δ = δXA + δXA transformation of X3 ?X3 ∼ X6 order terms in the action

S6. As already given before, the X3 order transformation needed bears the form

δ3ΨA = N IAεI , δ3ΨA = N I
Aε

I , δ3Ψ̄A = N IAε̄I , δ3Ψ̄A = N I
Aε̄

I , (A.33)

where

N I
A = ΓIAB

(
XC ? XC ? X

B −XB ? XC ? X
C
)
− 2ΓIBCX

B ? XA ? X
C , (A.34)

N IA = Γ̃IAB
(
XC ? X

C ? XB −XB ? X
C ? XC

)
− 2Γ̃IBCXB ? X

A ? XC . (A.35)

Notice that Ψ’s are Majorana fermions, i.e. Ψ̄ = ΨTγ0, therefore the variation of Ψ and Ψ̄

can be identified as the same if they carry the identical index.

Taking into account the Majorana nature of the fermions, the overall cyclicality under

the star/matrix product, matrix trace and integration, as well as the definition of fermion

contraction, we have found the following δ3 variation of the action S4

δ3S4 = ∆1+∆2, (A.36)

∆1 = i

∫
−2εABCDtr

(
δ3Ψ̄A?XB?ΨC ?XD

)
(A.37)

+tr
(
δ3Ψ̄A?

(
XB?X

B?ΨA−ΨA?XB?XB+2ΨB?XA?XB−2XB?X
A?ΨB

))
,

∆2 = i

∫
2εABCDtr

(
δ3Ψ̄A?XB?XC ?XD

)
(A.38)

+tr
(
δ3Ψ̄A?

(
ΨA?XB?X

B−XB?XB?ΨA+2XB?XA?ΨB−ΨB?XA?X
B
))
.

Since ∆1 and ∆2 can be handled in practically identical way, we concentrate on the first

one only. Substituting definitions of δ3Ψ̄A and δ3Ψ̄A we have

∆1 = i

∫
−εABCDεAEFGΓIFGε̄

Itr
(
XH ? XH ? XE ? X

B ?ΨC ? XD

−XE ? X
H ? XH ? XB ?ΨC ? XD

)
+ 2εABCDε

EFGHΓIGH ε̄
ItrXE ? X

A ? XF ? X
B ?ΨC ? XD

+ tr
(

ΓIAD
(
XC ? XC ? X

D −XD ? XC ? X
C
)
− 2ΓICDX

C ? XA ? X
D
)

· ε̄I
(
XB ? X

B ?ΨA −ΨA ? XB ? XB + 2ΨB ? XA ? XB − 2XB ? X
A ?ΨB

)
,

(A.39)

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where we used the identity 2Γ̃IAB = εABCDΓICD. Next we recall two fundamental identities

of the Levi-Civita symbols

εABCDε
AEFG = δEFGBCD, εABCDε

EFGH = δEFGHABCD , (A.40)

where the generalized Kronecker δ-symbol is defined as follows

δj1...jni1...in
=
∑
σ∈S4

sign(σ)δ
jσ(1)
i1

. . . . . . δ
jσ(n)
in

. (A.41)

After employment of the generalized Kronecker δ-symbols and some lengthy yet straight-

forward algebra, the transformation (A.37) boils down to

∆1 = iΓIAB ε̄
Itr

∫
−ΨB ?

(
XA ? XC ? X

C ? XD ? X
D

+XC ? XC ? X
D ? XD ? X

A + 4XC ? XD ? X
A ? XC ? X

D
)

+ 2ΨB ?
(
XA ? XC ? X

D ? XD ? X
C

+XC ? XD ? X
D ? XC ? X

A +XC ? XC ? X
A ? XD ? X

D
)
.

(A.42)

We can then easily recognize that all ΨA ’s in the formula above are contracted with ΓIAB
from the SUSY transformation of X: δXA = iΓIAB ε̄

IΨB. All other terms cancel each other,

thus we can rewrite it as

∆1 = −1

3
δXAtr

∫
XA ? X

A ? XB ? X
B ? XC ? X

C +XA ? XA ? X
B ? XB ? X

C ? XC

+ 4XA ? X
B ? XC ? X

A ? XB ? X
C − 6XA ? X

B ? XB ? X
A ? XC ? X

C , (A.43)

and this is exactly ∆1 = −δXAS6. Similarly one can show that ∆2 = −δXAS6, therefore

δ3S4 = −δS6, Q.E.D. (A.44)

B Two point functions:
〈
AµAν

〉
We notice that due to the bi-fundamental nature of the fermions and scalar bosons, they

do not form non-planar contribution to the one-loop identical gauge field amplitudes. This

fact reduces the relevant diagrams for identical gauge fields to pure gauge field theory

(gauge field and ghost loops) only. We use the following convention for the one-loop purely

gauge field diagrams:

• For each vertex, assignment of momenta is in such sequence: outgoing loop momenta,

external momenta, incoming loop momenta.

• Each propagator has the 1st index as outgoing, and the 2nd index as incoming (the

3d index on of the ε-tensor is contracted with the momentum flowing through the

propagator).

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As an example let’s write down the gauge field bubble Pµνbub, figure 2, in this convention

Pµνbub =
1

2

∫
dD`

(2π)D
(−2i)2 sin

`θ(p− `)
2

εµ2µµ1
εµ1ν1ρ1`

ρ1

`2
sin

(p− `)θ`
2

εν1νν2
εν2µ2ρ2(`− p)ρ2

(`− p)2

=

∫
dD`

(2π)D
2 sin2 `θp

2

εµ2µµ1εµ1ρ1ν1`
ρ1εν1νν2εν2ρ2µ2(`− p)ρ2
`2(`− p)2

. (B.1)

We then evaluate the contraction of Levi-Civita symbols in three dimensions as guided by

the dimensional reduction convention, which yields

Pµνbub =

∫
dD`

(2π)D
2 sin2 `θp

2

`µ(`− p)ν + `ν(`− p)µ
`2(`− p)2

. (B.2)

We can then load one more transformation ` → −` + p to turn the first half of the result

above to be identical to the second half and obtain

Pµνbub =

∫
dD`

(2π)D
4 sin2 `θp

2

(`− p)µ`ν
`2(`− p)2

. (B.3)

Next let us turn to the ghost bubble Gµνbub, figure 3,

Gµνbub =

∫
dD`

(2π)D
(−)3(−2i)2 (`− p)µ`ν

`2(`− p)2
sin

pθ`

2
sin
−pθ(`− p)

2

= −
∫

dD`

(2π)D
4 sin2 `θp

2

(`− p)µ`ν
`2(`− p)2

.

(B.4)

Thus

Pµνbub +Gµνbub = 0, (B.5)

i.e. all potentially non-planar contributions cancel out.

C Integrals from two point functions

During this work we studied new integrals and found some new relations among them. Here

we present a set of seven integrals I, I1,. . . ,I6 appearing in (6.3) relevant to this work. They

are used to present all loop integral results in the main text. We start with D-dimensions

and for the Euclidian signature:

I =
i

(4π)D/2

∫ 1

0
dx

∫ ∞
0

dλ λ1−D/2 e−λp
2x(1−x)− p̃

2

4λ , (C.1)

I1 =
i

2(4π)D/2

∫ 1

0
dx

∫ ∞
0

dλ λ−D/2 e−λp
2x(1−x)− p̃

2

4λ = iI6, (C.2)

I2 =
i

(4π)D/2

∫ 1

0
dx x2

∫ ∞
0

dλ λ1−D/2 e−λp
2x(1−x)− p̃

2

4λ , (C.3)

I3 =
1

2(4π)D/2

∫ 1

0
dx x

∫ ∞
0

dλ λ−D/2 e−λp
2x(1−x)− p̃

2

4λ , (C.4)

I4 =
−i

4(4π)D/2

∫ 1

0
dx

∫ ∞
0

dλ λ−1−D/2 e−λp
2x(1−x)− p̃

2

4λ , (C.5)

I5 =
i

2(4π)D/2

∫ 1

0
dx x

∫ ∞
0

dλ λ1−D/2 e−λp
2x(1−x)− p̃

2

4λ , (C.6)

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where we recall that p̃µ = θµνpν and p̃0 = θ0νpν = 0. After using∫ ∞
0

dλ λ−ν e−p
2x(1−x)λ− p̃

2

4λ = 2ν
(p2x(1− x)

p̃2

) ν−1
2
Kν−1

(√
p̃2p2x(1− x)

)
, (C.7)

and specifying D = 3, integrals reduce to

I =
i

2 (2π)3/2

∫ 1

0
dx

(
p2x (1− x)

p̃2

)−1/4

K−1/2

(√
p̃2p2x (1− x)

)
. (C.8)

I1 =
i

2 (2π)3/2

∫ 1

0
dx

(
p2x (1− x)

p̃2

)1/4

K1/2

(√
p̃2p2x (1− x)

)
, (C.9)

I2 =
i

2 (2π)3/2

∫ 1

0
dx x2

(
p2x (1− x)

p̃2

)−1/4

K−1/2

(√
p̃2p2x (1− x)

)
, (C.10)

I3 =
1

2 (2π)3/2

∫ 1

0
dx x

(
p2x (1− x)

p̃2

)1/4

K1/2

(√
p̃2p2x (1− x)

)
, (C.11)

I4 =
−i
√

2

(2π)3/2

∫ 1

0
dx

(
p2x (1− x)

p̃2

)3/4

K3/2

(√
p̃2p2x (1− x)

)
, (C.12)

I5 =
i

2 (2π)3/2

∫ 1

0
dx x

(
p2x (1− x)

p̃2

)−1/4

K−1/2

(√
p̃2p2x (1− x)

)
, (C.13)

C.1 Master scalar integral I(p, θ)

Integral I(p, θ) is UV finite when D < 4, therefore polarization tensor Π̂µν

AÂ
has a

smooth commutative limit. To verify this we employ the standard Schwinger-Feynman

parametrization, which yields:

I(p, θ)
∣∣∣
D→3

=

√
2

(4π)
3
2

1∫
0

dx

(
x(1− x)p2

p̃2

)− 1
4

K 1
2

[√
x(1− x)p2p̃2

]

=
1

4π

1∫
0

dx
e−
√
x(1−x)p2p̃2√

x(1− x)p2
.

(C.14)

To get back to the Minkowski signature of our integral expressions (6.3), (6.4), and (6.5)

we apply simple transformations of say (k, p) pair of momenta: k0 → −ik0, and p0 →
−ip0 =⇒ p2 → p2 − i0+, and then under the Wick rotations, performed by making a

change on the righthand side of our integrals (C.1)–(C.6), we obtain:

(IM , IMi ) = (I, Ii)

∣∣∣∣∣ k0→−ik0
p0→−ip0
p2→p2−i0+

, ∀i = 1, . . . ., 6. (C.15)

There are two relations among above integrals which makes results (6.3)–(6.5) simpler:

2IM3 − IM6 = 0, (C.16)

4
(
IM2 − IM5

)
+ IM =

(
− 4IM1 +

i

2π
√
p̃2

)
1

(p2 − i0+)
. (C.17)

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C.2 An integral with a bounded but ill-defined p̃µ → 0 limit

Let us analyze the limit p̃µ → 0 of the following integral:

Iµ(p, p̃) =

∫
d3`

(2π)3
e−i`θp

`µ

`2(`− p)2
, (C.18)

which is, for large loop momenta, the dominant contribution to the diagram in figure 15

— see (8.8). By introducing Schwinger parameters we decompose integral (C.18) into

Iµ(p, p̃) = I5 p
µ + I6 p̃

µ, (C.19)

where integrals I5 and I6 have been defined in (C.2), (C.6) and (C.13), respectively. Taking

into account that

K± 1
2
(z) =

√
π

2

e−z√
z
, (C.20)

one can show that integral (C.18) further boils down to

Iµ(p, p̃) =
1

8π

p̃µ√
p̃2

+
i

16

pµ√
p2

+ fµ(p, p̃), (C.21)

where fµ(p, p̃) vanishes as p̃µ → 0.

Notice that the first summand on the right hand side of (C.21) is bounded as p̃µ → 0,

but this limit depends on the way one approaches p̃µ = 0 point. To conclude, the limit

p̃µ → 0 of integral Iµ(p, p̃) (C.18) is ill-defined, though not divergent.

The discussion above can be generalized to the following D-dimensional integral

ε(p, q̃) =

∫
d3`

(2π)3
e−i`θq

`µ

`2(`− p)2
= Î5 p

µ + Î6 q̃
µ, (C.22)

with

Î5 =
i

2(4π)D/2

∫ 1

0
dx xe−ixpθq

∫ ∞
0

dλ λ1−D/2 e−λp
2x(1−x)− q̃

2

4λ , (C.23)

Î6 =
1

2(4π)D/2

∫ 1

0
dx e−ixpθq

∫ ∞
0

dλ λ−D/2 e−λp
2x(1−x)− q̃

2

4λ . (C.24)

When setting D = 3, integrals Î5 and Î6 boils down to the following forms

Î5 =
i

2(2π)3/2

∫ 1

0
dx xe−ixpθq

(p2x(1− x)

q̃2

)−1/4
K−1/2

(√
q̃2p2x(1− x)

)
, (C.25)

Î6 =
1

2(2π)3/2

∫ 1

0
dxe−ixpθq

(p2x(1− x)

q̃2

)1/4
K1/2

(√
q̃2p2x(1− x)

)
. (C.26)

Expanding ε(p, q̃) over the small q̃’s we have found

ε(p, q̃) =
1

8π

q̃µ√
q̃2

+
i

16

pµ√
p2

+ f̂µ(p, q̃), (C.27)

where f̂µ(p, q̃) vanishes as q̃µ → 0, the expression exactly equivalent to the one in (C.21).

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Aµ1(p1)

Aµ2(p2)

Aµ3(p3)
−→

←−

←−

V µ1µ2µ3

←−

←−

−→

−→ −→

V̂ µ1µ2µ3

µ2(p2)

µ1(p1)

µ3(p3)

Λ̄(q)

Aµ(p)

Λ(k)

¯̂
Λ(q)

Λ̂(k)

µ(p)

V µ V̂ µ

Figure 23. Triple gauge field, -hgauge field, ghost-gauge field, and hghost-hgauge field vertices.

D Feynman rules

D.1 Gauge fields and ghosts-gauge field vertices

Starting with Chern-Simons action (2.11), for triple-gauge field interaction, in accord with

the first two diagrams in figure 23, we extract the following Feynman rules:

V µ1µ2µ3 = −V̂ µ1µ2µ3 = i
κ

2π
2 sin

p1θp2

2
εµ1µ2µ3 , (D.1)

where we recall that pθk = pµθ
µνkν , and pθp = pµθ

µνpν = 0.

From ghost and gauge-fixing field action (2.20), in accord with the second two diagrams

in figure 23, we extract the following Feynman rules:

V µ = −V̂ µ =
κ

2π
qµ 2 sin

pθk

2
. (D.2)

D.2 Scalar-gauge fields vertices

From the kinetic part of the action Skin (2.12), in accord with figure 24, we obtain the

following Feynman rules:

(V µ)BA = i
−κ
2π

e
i
2
kθq(k + q)µδBA, (D.3)

(V̂ µ)BA = i
κ

2π
e−

i
2
kθq(k + q)µδBA, (D.4)

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−→ −→

−→ −→ −→

−→ −→ −→

XA(q) XA(q)

XA(q) XA(q) XA(q)

XB(k) XB(k)

XB(k) XB(k) XB(k)

Aµ(p) µ(p)

Aµ1(p1)

Aµ2(p2)

Aµ1(p1)

µ2(p2)

µ1(p1)

µ2(p2)

(V µ) A
B (V̂ µ) A

B

(V µ1µ2) A
B (V̂ µ1µ2) A

B (Ṽ µ1µ2) A
B

Figure 24. Scalar-gauge field, -hgauge field vertices.

and

(V µ1µ2)BA = i
−κ
2π

ηµ1µ2e
i
2
kθq
[
e−

i
2
p1θ(k−q) + e−

i
2
p2θ(k−q)

]
δBA, (D.5)

(V̂ µ1µ2)BA = 2i
κ

2π
ηµ1µ2

[
e
i
2
qθ(k+p1)e−

i
2
p1θk

]
δBA, (D.6)

(Ṽ µ1µ2)BA = i
−κ
2π

ηµ1µ2e−
i
2
kθq
[
e−

i
2
p1θ(k−q) + e−

i
2
p2θ(k−q)

]
δBA. (D.7)

D.3 Fermion-gauge field vertices

From the kinetic part of the action Skin (2.12), in accord with the first two terms in

figure 25, we obtain relevant Feynman rules,

(V µ
ij )

A
B = i

−κ
2π

γµije
− i

2
qθkδAB, (D.8)

(V̂ µ
ij )

A
B = i

κ

2π
γµije

i
2
qθkδAB, (D.9)

while from the action S4 (2.13), in accord with third diagram in figure 25, we have(
Ṽij

)AC
BD

= i
κ

2π
δij

(
δABδ

C
D − 2δADδ

C
B

)
2 sin

qθk + p1θp2

2
. (D.10)

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−→ −→

Ψ̄Ai(q)

Ψ B
j (k)

Ψ̄Ai(q)

Ψ B
j (k)

µ(p)

(V µ
ij )

A
B (V̂ µ

ij )
A
B (Ṽij)

AC

BD

XD(p1)

Ψ B
j (k)

Ψ̄Ai(q)

XC(p2)

Aµ(p)

Figure 25. Fermion-gauge field, -hgauge field and 2fermions-2scalars vertices.

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

References

[1] O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal

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	Introduction
	Classical NCABJM field theory
	Noncommutative BRST transformations
	Noncommutative generalization of the action
	Noncommutative supersymmetric transformations

	Feynman rules of the U(1)(kappa x U(1)(- kappa) NCABJM quantum field theory
	Power counting and the limit theta**(mu nu) –> 0
	Gauge field < A**(mu) A**(nu) > and hgauge field < hat(A)**(mu) hat(A)**(nu) > two-point functions
	Gauge field bubble and tadpole diagrams

	Mixed gauge field – hgauge field, < A**(mu) hat(A)**(nu) >, two-point functions
	Gauge field – hgauge field bubble and tadpole: scalar and fermion loops

	Gauge field < A**(mu(1)) A**(mu(2)) A**(mu(3)) > and hgauge field < hat(A)**(mu(1)) hat(A)**(mu(2)) hat(A)**(mu(3)) >, three-point functions
	Mixed gauge field – hgauge field, < A**(mu(1)) A**(mu(2)) hat(A)**(mu(3)) >, < hat(A)**(mu(1)) hat(A)**(mu(2)) A**(mu(3)) >, three-point functions
	Loop integrals contributing to the < A**(mu(1)) A**(mu(2)) hat(A)**(mu(3)) > 3-point function
	Computations of the < A**(mu(1)) A**(mu(2)) hat(A)**(mu(3)) > 3-point functions

	Scalar | fermion, < X(A) X**(B) > | < Psi**(A) bar(Psi)(B) >, two-point functions
	One-loop scalar < X(A)X**(B) > 2-point function
	One-loop fermion < Psi**(A) bar(Psi)(B) > 2-point function

	Fermion – gauge field | – hgauge field, < Psi**(A) bar(Psi)(B) A**(mu) > | < Psi**(A) bar Psi(B) hat(A)**(mu) >, three-point functions
	Scalar – gauge field | – hgauge field, < X**(A) X(B) A**(mu) > | < X**(A) X(B) hat A**(mu) >, three-point functions
	Summary and discussion
	SU(4)(R) supersymmetric invariance of U(1)(kappa) x U(1)( - kappa)
	Variations of the action with respect to gauge and scalar fields
	Variations of the action with respect to fermion fields
	Cancellations between S(CS), S(kin) and S(4) variations
	Classical SUSY invariance regarding S(6) terms

	Two point functions: < A**(mu) A**(nu) >
	Integrals from two point functions
	Master scalar integral I(p,theta)
	An integral with a bounded but ill-defined tilde(p)**(mu) –> 0 limit

	Feynman rules
	Gauge fields and ghosts-gauge field vertices
	Scalar-gauge fields vertices
	Fermion-gauge field vertices