# UV/IR mixing in noncommutative SU(N) Yang-Mills theory

C.P. Martin  Universidad Complutense de Madrid (UCM), Departamento de Física
Teórica and IPARCOS, Facultad de Ciencias Físicas, 28040 Madrid, Spain
<EMAIL_ADDRESS>Josip Trampetić  Ruđer Bošković Institute, Division of
Experimental Physics, Bijenička 54, 10000 Zagreb, Croatia
<EMAIL_ADDRESS>Max-Planck-Institut für Physik, (Werner-Heisenberg-
Institut), Föhringer Ring 6, D-80805 München, Germany<EMAIL_ADDRESS>Jiangyang You  Ruđer Bošković Institute, Division of Physical Chemistry,
Bijenička 54, 10000 Zagreb, Croatia<EMAIL_ADDRESS>
(today)

###### Abstract

We show that there are one-loop IR singularities arising from UV/IR mixing in
noncommutative SU(N) Yang-Mills theory defined by means of the $\theta$-exact
Seiberg-Witten map. This is in spite of the fact that there are no ordinary
U(1) gauge fields in the theory and this is at variance with the
noncommutative U(N) case, where the two-point part of the effective action
involving the ordinary SU(N) fields do not suffer from those one-loop IR
singularities.

Space-time noncommutativity and Gauge field theory and UV/IR

##  I Introduction

Underlying mathematical structures Kontsevich:1997vb ; Madore:2000en ;
Jurco:2000fb ; Jurco:2001rq ; Jurco:2001kp ; arXiv0711.2965B ; arXiv0909.4259B
in the noncommutative (NC) quantum field theories (NCQFT) lead to observable
consequences for the low energy physics, realised first by the perturbative
loop computation proposed by Filk Filk:1996dm . Second is famous example of
running of the coupling constant in the U(1) NCGFT in the
star($\star$)-product formalism Martin:1999aq .

Since commutative gauge symmetry and its deformed noncommutative gauge
symmetry suppose describe the same physical system –open strings ending on a
stack of Dp-branes with a constant magnetic field on it–, therefore they are
expected to be equivalent. However due to the NC spacetime there are number of
principal problems: Commutative local gauge transformations for the D-brane
effective action do not commute with $\star$-products. Also it is important to
note that the introduction of $\star$-products induces field operator ordering
ambiguities and breaks ordinary gauge invariance in the naive sense, as well
as there is also a question of the charge quantisation.

The problems above were solved by celebrated Seiberg-Witten (SW) maps
Seiberg:1999vs  between ordinary and the noncommutative quantities/fields. SW
map promote not only the noncommutative fields and composite nonlocal
operators of the commutative fields, but also the NC gauge transformations as
the composite operators of the commutative gauge fields and gauge
transformations Okawa:2001mv ; Brace:2001fj ; Barnich:2002tz ; Cerchiai:2002ss
; Barnich:2002pb ; Banerjee:2004rs ; Martin:2012aw ; Horvat:2011qn ;
Trampetic:2015zma . This method, referred as SW map or enveloping algebra
approach avoids both the gauge group and the charge quantization problem
issues Horvat:2011qn . Thus the NC deformed QFT’s can be defined for arbitrary
gauge group representations Martin:2012aw , and building semi-realistic NC
deformed particle physics models are made easier.

One of the particularly intriguing quantum effect of the spacetime
noncommuttivity is the exhibition of fascinating dynamics due to the
ultraviolet/infrared phenomenon, named as the UV/IR mixing, an interrelation
between short and long distance scales absent in ordinary quantum field
theory, in which the ultraviolet and infrared degrees of freedom are mixed in
a way similar to what is expected to happen in a theory of Quantum Gravity. In
NC theory without SW maps Bigatti:1999iz ; Minwalla:1999px ;
VanRaamsdonk:2000rr ; VanRaamsdonk:2001jd ; Hayakawa:1999yt ; Hayakawa:1999zf
; Hayakawa:2000zi ; Matusis:2000jf ; Ruiz:2000hu ; Khoze:2000sy ;
Armoni:2003va ; Ferrari:2004ex  it was shown for the first time how UV short
distance effects, could alter the IR dynamics, thus becoming known as the
celebrated UV/IR mixing. With the SW map included Zeiner:2007 ; Schupp:2008fs
; Horvat:2011bs ; Horvat:2011qg ; Horvat:2013rga ; Horvat:2015aca  we have
found the same UV/IR property. Note here that UV/IR mixing shows up also in
the NC deformation parameter exact Scalar $\phi^{4}$ theories and on the
$\kappa$-Minkowski and Snyder manifolds, respectively Grosse:2005iz ;
Meljanac:2011cs ; Meljanac:2017grw ; Meljanac:2017jyk .

UV/IR mixing as an important quantum effect, connects the noncommutative field
theories with holography in a model-independent way Cohen:1998zx ;
Horvat:2010km .

Using the weak gravity conjecture (WGC) Li:2006jja ; Huang:2006tz  and scalar
fields results from Palti:2017elp , the very notion of UV/IR mixing was
implemented into the idea of scalar fields WGC in the recent Lüst and Palti
article Lust:2017wrl ; it manifests itself as a form of hierarchical UV/IR
mixing and is tied to the interaction between the weak gravity conjecture and
nonlocal (possibly noncommutative) gauge operators. Namely in the scalar field
theory, the mass of the scalar is far from ultra-violet scale. Thus the form
of hierarchical UV/IR mixing restricts the mass of the scalar field to a IR
scale deep below the UV scale which is associated to quantum gravity.
Additionally one can think about naturalness of the scalar mass in a way that
the UV physics knows nothing about the theory in the deep IR. So instead of
introducing the higher dimensions and new symmetries one might hope that
connecting the deep-low IR and far-high UV could solve the naturalness problem
Craig:2019zbn ; Koren:2020biu .

In the literature, all above notions with UV/IR mixing connections are
considered as possible windows to Quantum Gravity.

A chief characteristic of noncommutative U(N) theory is that only the U(1)
part of the two-point function for the gauge field exhibits UV/IR mixing at
the one-loop level. Indeed, the SU(N) part of that two-point function does not
get any contribution from one-loop nonplanar diagrams. This raises the
question of whether noncommutative SU(N) gauge theories experience this UV/IR
mixing, for there is by construction no fundamental U(1) degrees of freedom in
them. The purpose of this paper is to answer this question, which as we shall
see demands to carry out lengthy computations. Let us recall that to formulate
Yang-Mills theories on noncommutative spacetime for arbitrary gauge groups in
arbitrary representations one has to use the noncommutative framework put
forward in references Madore:2000en ; Jurco:2001rq ; Calmet:2001na ;
Aschieri:2002mc . In this formalism the noncommutative gauge fields and
noncommutative gauge transformations are defined solely in terms of the
ordinary counterparts by using the Seiberg-Witten map, which in this case take
values in the universal enveloping algebra of the Lie algebra of the gauge
group. Thus, any –unitary– representation of the Lie algebra is admissible;
but, then, one needs to address the problem of too many degrees of freedom,
since all coefficient functions of the monomials in the generators could a
priori be physical fields. The solution furnished in [2,4,15] is that those
coefficient fields are not all independent: they are functions of the correct
number of ordinary gauge fields via SW maps. Thus, for semisimple Lie
algebras, no fundamental U(1) degrees of freedom occur in the noncommutative
gauge theory. That the Seiberg-Witten map in question exists for any compact
gauge group –for SU(N), in particular– in any representation has been shown in
reference Barnich:2002pb  by using BRST techniques, thus proving that the
Seiberg-Witten exists not only for the U(N) group in the fundamental,
antifundamental or bifundamental representations. The noncommutative Yang-
Mills action for arbitrary compact gauge groups has been introduced in
references Madore:2000en ; Jurco:2001rq ; Calmet:2001na ; Aschieri:2002mc  and
fully studied in reference Barnich:2002pb . It is by using so-called
enveloping algebra formalism that we have just mentioned that the
noncommutative Yang-Mills theory for SU(N) in the fundamental representation
is formulated –see reference Calmet:2001na – without the need of introducing
the interacting U(1) fundamental degrees of freedom which occur in the
noncommutative U(N) case.

Using the enveloping algebra formalism we have just made discussed, the
background-field and path integral methods, from the classical Yang-Mills (YM)
action we shall first construct the BRST exact noncommutative effective action
$S_{BRSTexact}$, with the NC fields spanned on the Moyal manifold and being
expressed in terms of ordinary YM fields by means of $\theta$-exact SW maps.
Let stress that the quantization of the theory will be carried out by
integrating in the path integral over the ordinary SU(N) gauge fields and
ghosts –this is the SU(N) analogue of quantization method succesfully used in
Horvat:2011qg ; Horvat:2013rga ; Martin:2016zon  for the U(1) case. Then, we
investigate the UV/IR mixing phenomena in the gauge sector and for a Dirac
fermions in the fundamental representation of SU(N). Finally we have to state
clearly that this theory is not really the pure Moyal NCYM, but a class of
Moyal deformed gauge theory which is different, and not yet completely
understood.

This article is on the line of our previous works Martin:2016zon ;
Martin:2016hji ; Martin:2016saw ; Martin:2017nhg , and it is organized as
follows. In the next section we introduce classical and the background-field
effective actions. We compute one-loop two-point functions by using DeWitt
method DeWitt1 ; Kallosh:1974yh ; DeWitt2 ; DeWitt3  and in Section three
prove existence of the gauge independent UV/IR mixing phenomena. Section four
is devoted to the fundamental Dirac fermions in the framework of the
noncommutative QCD. Conclusion is given in fifth Section, while details of
$\theta$-exact SW maps for all relevant fields and details of divergent
integral solutions are given in Appendices A and B, respectively.

##  II The classical action and the background-field effective action

Let $a_{\mu}=a^{a}_{\mu}\,T^{a}$, in terms of component fields $a^{a}_{\mu}$,
be an ordinary gauge field, with $T^{a}$ being the generators of SU(N) in the
fundamental representation, normalized so that Tr$(T^{a}T^{b})=\delta^{ab}$.
The symbol $A_{\mu}$ will denote the noncommutative gauge field defined in
terms of the previous SU(N) ordinary gauge field $a_{\mu}$ by means of the
Seiberg-Witten map Seiberg:1999vs .

The classical action, $S_{cl}[A]$, of our noncommutative SU(N) YM theory reads

$S_{cl}[A(a_{\lambda})]=-\frac{1}{4}\int d^{4}x\,{\rm
Tr}\>F_{\mu\nu}[A(a_{\lambda})]\star F^{\mu\nu}[A(a_{\lambda})],$ (1)

where
$F_{\mu\nu}[A(a_{\lambda})]=\partial_{\mu}A_{\nu}(a_{\lambda})-\partial_{\nu}A_{\mu}(a_{\lambda})-ig[A_{\mu}(a_{\lambda}),A_{\nu}(a_{\lambda})]_{\star}$111
We stress here that $F_{\mu\nu}[A(a_{\lambda})]$ in (1) takes values in the
universal enveloping algebra of the SU(N) Lie algebra in the fundamental
representation, thus demands the use of the formalism put forward in
references Madore:2000en ; Jurco:2001rq ; Calmet:2001na ; Aschieri:2002mc ;
Barnich:2002pb . , with star($\star$)-product being the Moyal-Weyl one, and
self-evident notations for $\star$-commutator. Important to note is that the
dynamical field variable in $S_{cl}[A(a_{\lambda})]$ is $a_{\lambda}$. For
details of SW maps see Appendix A.

To quantize the classical theory whose action is, $S_{cl}[A(a_{\lambda})]$, in
(1), we shall use the background field method Khoze:2000sy  and, thus, we
shall split $a_{\mu}$ as follows

$a_{\mu}=b_{\mu}\,+\,q_{\mu},$ (2)

where $b_{\mu}=b^{a}_{\mu}\,T^{a}$ denotes the background field and
$q_{\mu}=q^{a}_{\mu}\,T^{a}$ stands for the quantum field –the field to be
integrated over in the path integral.

Since the noncommutative gauge field, $A_{\mu}$, is a function of $a_{\mu}$
–$A_{\mu}=A_{\mu}(a_{\lambda})$–, the splitting in (2) gives rise to the
following splitting of $A_{\mu}$

$A_{\mu}(b+q)=B_{\mu}(b)\,+\,Q_{\mu}(b,q).$ (3)

Notice that $A_{\mu}(b+q)$ is obtained from $A_{\mu}(a_{\lambda})$ by
replacing $a_{\mu}$ with $b_{\mu}\,+\,q_{\mu}$ in it. Obviously, $B_{\mu}(b)$
is given by the Seiberg-Witten map applied to the ordinary SU(N) gauge field
$b_{\mu}$. Mark that $Q_{\mu}(b,q)$ is a function of both $b_{\mu}$ and
$q_{\mu}$.

The background field effective action $\Gamma[b]$ is given by the following
equation

$e^{i\Gamma[b]}=\int{\cal D}q^{a}_{\mu}{\cal D}F^{a}{\cal D}c^{a}{\cal
D}\bar{C}^{a}\,e^{i{\cal S}-i\int
d^{4}x\,q^{a}_{\mu}(x)\frac{\delta\Gamma[b]}{\delta b^{a}_{\mu}(x)}},$ (4)

with

${\cal S}=S_{cl}[A(b+q)]+S_{BRSTexact}.$ (5)

Classical action, $S_{cl}[A(b+q)]$, is defined in (1) and $S_{BRSTexact}$
stands for the gauge-fixing terms including the ghost’s contributions. To
render the sum $S_{cl}[A(b+q)]+S_{gf+gh}$ as simple as possible we shall
choose the following Feynman type of gauge fixing ($S_{gf}$) and ghost
($S_{gh}$) actions sum $S_{gf+gh}$

$S_{BRSTexact}=s\int d^{4}x\,{\rm
Tr}\;\Big{(}\bar{C}\star\big{(}\frac{1}{2}F+D_{\mu}[B]Q^{\mu}(b,q)\big{)}\Big{)}$
(6)

where $s$ denotes de ordinary BRST operator, and $\bar{C}=\bar{C}^{a}\,T^{a}$
is the antighost, while $F=F^{a}\,T^{a}$ Lautrup-Nakanishi auxiliary field. As
noted the $S_{BRSTexact}$ is BRST-exact.

The BRST operator $s$ acts on the ordinary and noncommutative fields,
respectively,

$sb_{\mu}=0,\quad sq_{\mu}=\partial_{\mu}c-ig[b_{\mu}+q_{\mu},c],\quad
sc=igcc,\quad s\bar{C}=F,\quad sF=0,$ (7)

with $c=c^{a}\,T^{a}$ denoting ordinary ghost field. The notion of Seiberg-
Witten map implies the action of $s$ on the noncommutative fields
$B_{\mu}(b)$, $Q_{\mu}(b,q)$ and $C(b+q,c)$ is the following

$sB_{\mu}(b)=0,\quad sQ_{\mu}(b,q)=s_{NC}Q_{\mu}(b,q),\quad
sC(b+q,c)=s_{NC}C(b+q,c),$ (8)

where $s_{NC}$ is the noncommutative BRST operator whose action on the
noncommutative field thus runs as follows

$\begin{array}[]{l}{s_{NC}B_{\mu}(b)=0,}\\\\[4.0pt]
{s_{NC}Q_{\mu}(b,q)=D_{\mu}[B+Q]C(b+q,c)=\partial_{\mu}C(b+q,c)-ig[B_{\mu}+Q_{\mu},C]_{\star}\,,}\\\\[4.0pt]
{s_{NC}C(b+q,c)=igC(b+q,c)\star C(b+q,c).}\end{array}$ (9)

By construction, we also have $s_{NC}\bar{C}=s\bar{C}=F$ and $s_{NC}F=sF=0$.

Using the definitions above, one gets

$S_{BRSTexact}=\int d^{4}x\,{\rm Tr}\;\Big{(}\frac{1}{2}F\star F+F\star
D_{\mu}[B]Q^{\mu}(b,q)-\bar{C}\star D_{\mu}[B]D^{\mu}[B+Q]C(b+q,c)\Big{)}.$
(10)

##  III The two-point contribution to the effective action at one-loop

The final purpose of this section is the computation of one-loop contribution
to the effective action $\Gamma[b]$, which has been defined in (4). We shall
begin by expanding in powers of $q_{\mu}$ and $b_{\mu}$ the action, ${\cal
S}$, in (5) and, then, dropping the terms with more than two $b_{\mu}$’s and
more than two $g$’s –$g$ being the coupling constant.

###  III.1 Removing $O(g^{3})$ terms from the path integral

Let us first integrate out the field $F^{a}$ in the path integral in (4) by
taking advantage of (10). Thus, one obtains

$e^{i\Gamma[b]}=\int{\cal D}q^{a}_{\mu}{\cal D}c^{a}{\cal
D}\bar{C}^{a}\,e^{iS-i\int d^{4}x\,q^{a}_{\mu}(x)\frac{\delta\Gamma[b]}{\delta
b^{a}_{\mu}(x)}},$ (11)

where $S$ is the sum

$S=S_{cl}[A(b+q)]+S_{gf}+S_{gh};$ (12)

while actions $S_{gf}$ and $S_{gh}$ being given by

$\begin{array}[]{l}{S_{gf}=-\frac{1}{2}\int d^{4}x\,{\rm
Tr}\;(D_{\mu}[B]Q^{\mu}(b,q)\star D_{\nu}[B]Q^{\nu}(b,q))}\\\\[4.0pt]
{\phantom{S_{gf}=}+\frac{1}{2N}\int d^{4}x\,({\rm
Tr}D_{\mu}[B]Q^{\mu}(b,q))\star({\rm Tr}D_{\nu}[B]Q^{\nu}(b,q)),}\\\\[8.0pt]
{S_{gh}=-\int d^{4}x\,{\rm Tr}\;(\bar{C}\star
D_{\mu}[B]D^{\mu}[B+Q]C(b+q,c)).}\end{array}$ (13)

To obtain above $S_{gf}$ we have to use the following SU(N) generators
identity

$\sum_{a}(T^{a})^{i_{1}}_{\bar{j}_{1}}(T^{a})^{i_{2}}_{\bar{j}_{2}}=\delta^{i_{1}}_{\bar{j}_{2}}\delta^{i_{2}}_{\bar{j}_{1}}-\frac{1}{N}\delta^{i_{1}}_{\bar{j}_{1}}\delta^{i_{2}}_{\bar{j}_{2}}.$
(14)

We start with definitions

$\tilde{q}^{a}_{\mu}={\rm Tr}\,(T^{a}Q_{\mu}(b,q)),\;{\rm
and}\;\tilde{c}^{a}={\rm Tr}\,(T^{a}C(b,q)),$ (15)

where the Seiberg-Witten maps $Q_{\mu}(b,q)$ and $C(b,q)$ are fully discussed
in Appendix A –see (66) and (68). Next, in the path integral (11), we shall
make the following variable change

$q^{a}_{\mu}\rightarrow\tilde{q}^{a}_{\mu},\quad
c^{a}\rightarrow\tilde{c}^{a},$ (16)

for thus we shall remove from the path integral the lengthy interaction terms
due to contributions to the classical action coming from the SU(N) part –see
(69)– of the Seiberg-Witten maps $Q_{\mu}(b,q)$ and $C(b,q)$ when expressed as
functions of $q_{\mu}$. This is a much welcome simplification since, as we
shall see below, we will still have to deal with the interaction terms the
U(1) part of the former Seiberg-Witten maps introduce. Notice that the
Jacobian of this transformation is trivial in Dimensional Regularization,
since all the momentum integrals it involves vanish –for details see
Martin:2016saw .

Now, let $\Gamma_{2}[b]$ be the two-point contribution to $\Gamma[b]$ in (11).
The change of variables (16) and the use of (75) and (77) from Appendix A,
after some laborious algebra, leads to

$i\Gamma_{2}[b]=Ln\int{\cal D}\tilde{q}^{a}_{\mu}{\cal D}\tilde{c}^{a}{\cal
D}\bar{C}^{a}\;e^{i(S_{0}+S_{1}+S_{2}+S_{1eom}+S_{2eom}+S_{0gh}+S_{1gh}+S_{2gh})}+O(g^{3}),$
(17)

where

$\begin{array}[]{l}{S_{0}=-\frac{1}{2}\int d^{4}x\,{\rm
Tr}\,\,\partial_{\mu}\tilde{q}_{\nu}\partial^{\mu}\tilde{q}^{\nu},}\\\\[8.0pt]
{S_{1}=+ig\int d^{4}x\,{\rm
Tr}\,\,b_{\mu}[\tilde{q}_{\nu},\partial^{\mu}\tilde{q}^{\nu}]_{\star}+2ig\int
d^{4}x\,{\rm
Tr}\,\,\partial_{\mu}b_{\nu}[\tilde{q}^{\mu},\tilde{q}^{\nu}]_{\star}\,,}\\\\[8.0pt]
{S_{2}=+\frac{1}{2}g^{2}\int d^{4}x\,{\rm
Tr}\,\,[b_{\mu},\tilde{q}_{\nu}]_{\star}[b^{\mu},\tilde{q}^{\nu}]_{\star}-\frac{N}{2}g^{2}\int
d^{4}x\,\partial_{\mu}\hat{A}^{(2)\,0}_{\nu}(b,\tilde{q})\partial^{\mu}\hat{A}^{(2)\,0\,\nu}(b,\tilde{q})}\\\\[4.0pt]
{\phantom{S_{2}=}+ig^{2}\int d^{4}x\,{\rm
Tr}\,\,\hat{A}^{(2)}_{\mu}(b,b)[\tilde{q}_{\nu},\partial^{\mu}\tilde{q}^{\nu}]_{\star}+ig^{2}\int
d^{4}x\,{\rm
Tr}\,\,\hat{A}^{(2)\,0}_{\nu}(b,\tilde{q})[\partial_{\mu}\tilde{q}^{\nu},b^{\mu}]_{\star}}\\\\[4.0pt]
{\phantom{S_{2}=}+ig^{2}\int d^{4}x\,{\rm
Tr}\,\,\partial_{\mu}\hat{A}^{(2)\,0}_{\nu}(b,\tilde{q})[b^{\mu},\tilde{q}^{\nu}]_{\star}+g^{2}\int
d^{4}x\,{\rm
Tr}\,\,\big{\\{}\partial^{2}-\partial_{\nu}\partial^{\rho}\hat{A}^{(2)}_{\rho}(b,b)\big{\\}}\hat{A}^{(2)\,0\,\nu}(\tilde{q},\tilde{q})}\\\\[4.0pt]
{\phantom{S_{2}=}+g^{2}\int d^{4}x\,{\rm
Tr}\,\,[b_{\mu},b_{\nu}]_{\star}[\tilde{q}^{\mu},\tilde{q}^{\nu}]_{\star}+2ig^{2}\int
d^{4}x\,{\rm
Tr}\,\,\partial_{\mu}\hat{A}^{(2)}_{\nu}(b,b)[\tilde{q}^{\mu},\tilde{q}^{\nu}]_{\star}}\\\\[4.0pt]
{\phantom{S_{2}=}+2ig^{2}\int d^{4}x\,{\rm
Tr}\,\partial_{\mu}\hat{A}^{(2)\,0}_{\nu}(b,\hat{q})[b_{\nu},\tilde{q}^{\mu}]_{\star}+\frac{N}{2}g^{2}\int
d^{4}x\,\partial_{\mu}\hat{A}^{(2)\,0\,\mu}(b,\tilde{q})\partial_{\nu}\hat{A}^{(2)\,0\,\nu}(b,\tilde{q})}\\\\[4.0pt]
{\phantom{S_{2}=}-\frac{1}{2N}g^{2}\int d^{4}x\,{\rm
Tr}\,\,[b_{\mu},\tilde{q}^{\mu}]_{\star}[b^{\nu},\tilde{q}^{\nu}]_{\star}-ig^{2}\int
d^{4}x\,{\rm
Tr}\,\,\partial_{\mu}\hat{A}^{(2)\,0\,\mu}(b,\tilde{q})[b_{\nu},\tilde{q}^{\nu}]_{\star}\,,}\end{array}$
(18) $\begin{array}[]{l}{S_{1eom}=g\int d^{4}x\,{\rm
Tr}\,\\{\big{(}\partial^{2}b^{\nu}-\partial^{\nu}\partial_{\rho}b^{\rho}\big{)}\hat{A}^{(2)}_{\nu}(\tilde{q},\tilde{q})\\},}\\\\[8.0pt]
{S_{2eom}=+g^{2}\int d^{4}x\,{\rm
Tr}\,\\{\big{(}\partial^{2}b^{\nu}-\partial^{\nu}\partial_{\rho}b^{\rho}\big{)}\hat{A}^{(3)}_{\nu}(\tilde{q},\tilde{q},b)\\}}\\\\[4.0pt]
{\phantom{S_{2eom}=}-2g^{2}\int d^{4}x\,{\rm
Tr}\,\\{\big{(}\partial^{2}b^{\nu}-\partial^{\nu}\partial_{\rho}b^{\rho}\big{)}\hat{A}^{(2)}_{\nu}(\hat{A}^{(2)\,a}(\tilde{q},b)T^{a},\tilde{q})\\}}\\\\[4.0pt]
{\phantom{S_{2eom}=}-g^{2}\int d^{4}x\,{\rm
Tr}\,\big{(}E_{\nu}(b,b)\hat{A}^{(2)}_{\nu}(\tilde{q},\tilde{q})\big{)}+\frac{g^{2}}{N}\int
d^{4}x\,{\rm Tr}\,\big{(}E_{\nu}(b,b)\big{)}{\rm
Tr}\,\big{(}\hat{A}^{(2)}_{\nu}(\tilde{q},\tilde{q})\big{)}}\\\\[4.0pt]
{\quad\quad
E_{\nu}(b,b)=\partial^{2}\hat{A}^{(2)}_{\nu}(b,b)-\partial_{\nu}\partial^{\rho}\hat{A}^{(2)}_{\rho}(b,b)-i\partial^{\mu}[b_{\mu},b_{\nu}]_{\star}-i[b^{\mu},\partial_{\mu}b_{\nu}-\partial_{\nu}b_{\mu}]_{\star}\,,}\end{array}$
(19)

and

$\begin{array}[]{l}{S_{0gh}=-\int d^{4}x\,{\rm
Tr}\,\,\bar{C}\partial^{2}\tilde{c},}\\\\[8.0pt] {S_{1gh}=ig\int d^{4}x\,{\rm
Tr}\,\,b^{\mu}\\{\tilde{c},\partial_{\mu}\bar{C}\\}-ig\int d^{4}x\,{\rm
Tr}\,\,b^{\mu}\\{\partial_{\mu}\tilde{c},\bar{C}\\},}\\\\[8.0pt]
{S_{2gh}=-g^{2}\int d^{4}x\,{\rm
Tr}\,\,[b^{\mu},\tilde{c}]_{\star}[b_{\mu},\bar{C}]_{\star}+ig^{2}\int
d^{4}x\,{\rm
Tr}\,\,\bar{C}[b^{\mu},\partial_{\mu}\hat{C}^{(2)\,0}(\tilde{c},b)]_{\star}}\\\\[4.0pt]
{\phantom{S_{2gh}=}+ig^{2}\int d^{4}x\,{\rm
Tr}\,\,\hat{A}^{(2)}_{\mu}(b,b)\\{\tilde{c},\partial^{\mu}\bar{C}\\}_{\star}-ig^{2}\int
d^{4}x\,{\rm
Tr}\,\,\hat{A}^{(2)}_{\mu}(b,b)\\{\partial^{\mu}\tilde{c},\bar{C}\\}_{\star}}\\\\[4.0pt]
{\phantom{S_{2gh}=}-ig^{2}\int d^{4}x\,{\rm
Tr}\,\,\hat{C}^{(2)\,0}(\tilde{c},b)[\partial_{\mu}\bar{C},b^{\mu}]_{\star}\,,}\end{array}$
(20)

with self-evident notations for equations of motions ($eom$), and
$\star$-anticommutator. Notice that $S_{2}$ and $S_{2gh}$ only involve the
U(1) part of the Seiberg-Witten maps $Q_{\mu}(b,q)$ and $C(b,c)$ as defined in
(66) and (68): the SU(N) parts of those maps have been disposed of by
introducing the SU(N) fields $\tilde{q}$ and $\tilde{c}$–see (75) and (77) in
Appendix A.

Now, let $p^{\mu}$ and $q^{\mu}$ be two arbitrary vectors. We shall use the
following notation

$\begin{array}[]{l}p\wedge q=\theta^{\mu\nu}p_{\mu}q_{\nu}\equiv
p_{\mu}\theta^{\mu\nu}q_{\nu}\equiv p\theta
q=p\tilde{q},\quad\tilde{q}^{\mu}=\theta^{\mu\nu}q_{\nu},\\\\[8.0pt]
{(f\star_{t}g)(x)=e^{i(p_{1}+p_{2})x}e^{-i\frac{t}{2}p_{1}\wedge
p_{2}}f(p_{1})g(p_{2})}.\end{array}$ (21)

Then, $\hat{A}^{(2)\,0}_{\nu}(b,\tilde{q})$, $\hat{A}^{(2)}_{\nu}(b,b)$,
$\hat{A}^{(2)}_{\mu}(\tilde{q},\tilde{q})$,
$\hat{A}^{(3)}_{\mu}(\tilde{q},\tilde{q},b)$,
$\hat{A}^{(2)}_{\nu}(\hat{A}^{(2)\,a}(\tilde{q},b)T^{a},\tilde{q})$ and
$C^{(2)\,0}(\tilde{c},b)$, are given by the following expressions:

$\begin{array}[]{l}{\hat{A}^{(2)\,0}_{\mu}(b,\tilde{q})=\frac{1}{2N}\,e^{i(p_{1}+p_{2})x}\big{(}\frac{e^{-\frac{i}{2}p_{1}\wedge
p_{2}}-e^{\frac{i}{2}p_{1}\wedge p_{2}}}{p_{1}\wedge p_{2}}\big{)}}\\\\[3.0pt]
{\phantom{\hat{A}^{(2)\,0}_{\mu}(b,\tilde{q})=\frac{1}{2N}XXXXXX,}\times[2\tilde{p}_{2}^{\mu_{1}}\delta^{\mu_{2}}_{\mu}+2\tilde{p}_{1}^{\mu
2}\delta^{\mu_{1}}_{\mu}-(p_{2}-p_{1})_{\mu}\theta^{\mu_{1}\mu_{2}}]b^{a}_{\mu_{1}}(p_{1})\tilde{q}^{a}_{\mu_{2}}(p_{2}),}\\\\[8.0pt]
{\hat{A}^{(2)}_{\nu}(b,b)=\frac{1}{2}\,e^{i(p_{1}+p_{2})x}\big{(}\frac{e^{-\frac{i}{2}p_{1}\wedge
p_{2}}-1}{p_{1}\wedge p_{2}}\big{)}}\\\\[3.0pt]
{\phantom{\hat{A}^{(2)}_{\nu}(b,b)=\frac{1}{2}XXXXXx}\quad\quad\times[2\tilde{p}_{2}^{\mu_{1}}\delta^{\mu_{2}}_{\mu}+2\tilde{p}_{1}^{\mu
2}\delta^{\mu_{1}}_{\mu}-(p_{2}-p_{1})_{\mu}\theta^{\mu_{1}\mu_{2}}]b_{\mu_{1}}(p_{1})b_{\mu_{2}}(p_{2}),}\\\\[8.0pt]
{\hat{A}^{(2)}_{\mu}(\tilde{q},\tilde{q};t)=-\frac{i}{4}\,e^{i(p_{1}+p_{2})x}\,\int_{0}^{t}ds\,e^{-i\frac{s}{2}p_{1}\wedge
p_{2}}}\\\\[3.0pt]
{\phantom{\hat{A}^{(2)}_{\mu}(\tilde{q},\tilde{q};t)=\frac{i}{4}XXXXx}\quad\quad\times[2\tilde{p}_{2}^{\mu_{1}}\delta^{\mu_{2}}_{\mu}+2\tilde{p}_{1}^{\mu
2}\delta^{\mu_{1}}_{\mu}-(p_{2}-p_{1})_{\mu}\theta^{\mu_{1}\mu_{2}}]\tilde{q}_{\mu_{1}}(p_{1})\tilde{q}_{\mu_{2}}(p_{2}),}\\\\[8.0pt]
{\hat{A}^{(2)}_{\mu}(\tilde{q},\tilde{q})=\hat{A}^{(2)}_{\mu}(\tilde{q},\tilde{q};t=1),}\\\\[8.0pt]
{\hat{A}^{(2)}_{\mu}(\tilde{q},b;t)=-\frac{i}{4}\,e^{i(p_{1}+p_{2})x}[2\tilde{p}_{2}^{\mu_{1}}\delta^{\mu_{2}}_{\mu}+2\tilde{p}_{1}^{\mu
2}\delta^{\mu_{1}}_{\mu}-(p_{2}-p_{1})_{\mu}\theta^{\mu_{1}\mu_{2}}]}\\\\[4.0pt]
{\phantom{\hat{A}^{(2)}_{\mu}(\tilde{q},b;t)=\frac{i}{4}}\quad\times\int_{0}^{t}ds\,[e^{-i\frac{s}{2}p_{1}\wedge
p_{2}}b_{\mu_{1}}(p_{1})\tilde{q}_{\mu_{2}}(p_{2})+e^{i\frac{s}{2}p_{1}\wedge
p_{2}}\tilde{q}_{\mu_{2}}(p_{2})b_{\mu_{1}}(p_{1})],}\\\\[8.0pt]
{\hat{A}^{(3)}_{\mu}(\tilde{q},\tilde{q},b)=-\frac{1}{4}\theta^{ij}\int_{0}^{1}dt}\\\\[4.0pt]
\phantom{XXX}\times\Big{[}+\\{\tilde{q}_{i},2\partial_{j}\hat{A}^{(2)}_{\mu}(\tilde{q},b;t)-\partial_{\mu}\hat{A}^{(2)}_{j}(\tilde{q},b;t)\\}_{\star_{t}}{+\\{\hat{A}^{(2)}_{i}(\tilde{q},b;t),2\partial_{j}\tilde{q}_{\mu}-\partial_{\mu}\tilde{q}_{j}\\}_{\star_{t}}}\\\\[4.0pt]
{\phantom{XXXXx,}+\\{b_{i},2\partial_{j}\hat{A}^{(2)}_{\mu}(\tilde{q},\tilde{q};t)-\partial_{\mu}\hat{A}^{(2)}_{j}(\tilde{q},\tilde{q};t)\\}_{\star_{t}}+\\{\hat{A}^{(2)}_{i}(\tilde{q},\tilde{q};t),2\partial_{j}b_{\mu}-\partial_{\mu}b_{j}\\}_{\star_{t}}}\\\\[4.0pt]
{\phantom{XX\hat{A}^{(3)}_{\mu}(\tilde{q},\tilde{q},b)=-\frac{1}{4}\theta^{ij}\int_{0}^{1}dt\Big{[}}-i\\{\tilde{q}_{i},[\tilde{q}_{j},b_{\mu}]_{\star_{t}}\\}_{\star_{t}}-i\\{\tilde{q}_{i},[b_{j},\tilde{q}_{\mu}]_{\star_{t}}\\}_{\star_{t}}-i\\{b_{i},[\tilde{q}_{j},\tilde{q}_{\mu}]_{\star_{t}}\\}_{\star_{t}}\Big{]}},\\\\[8.0pt]
{\hat{A}^{(2)}_{\nu}(\hat{A}^{(2)\,a}(\tilde{q},b)T^{a},\tilde{q})=-\frac{1}{16}\int\frac{d^{4}\\!p_{2}}{(2\pi)^{4}}\frac{d^{4}\\!p_{3}}{(2\pi)^{4}}\frac{d^{4}\\!p_{4}}{(2\pi)^{4}}\quad
e^{i(p_{2}+p_{3}+p_{4})x}}\\\\[3.0pt]
{\times[2\tilde{p}_{2}^{\mu_{1}}\delta^{\mu_{2}}_{\nu}+2(\tilde{p}_{3}+\tilde{p}_{4})^{\mu_{2}}\delta^{\mu_{1}}_{\nu}+(p_{3}+p_{4}-p_{2})_{\nu}\theta^{\mu_{1}\mu_{2}}][2\tilde{p}_{4}^{\mu_{3}}\delta^{\mu_{4}}_{\mu_{1}}+2\tilde{p}_{3}^{\mu_{4}}\delta^{\mu_{3}}_{\mu_{1}}+(p_{3}-p_{4})_{\mu_{1}}\theta^{\mu_{3}\mu_{4}}]}\\\\[4.0pt]
{\qquad\quad\quad\quad\times\Big{\\{}\int_{0}^{1}dt\int_{0}^{1}ds\,e^{-i\frac{t}{2}(p_{3}+p_{4})\wedge
p_{2}}e^{-i\frac{s}{2}p_{3}\wedge p_{4}}[{\rm
Tr}\,(b_{\mu_{3}}(p_{3})\tilde{q}_{\mu_{4}}(p_{4})T^{b})]T^{b}\tilde{q}_{\mu_{2}}(p_{2})}\\\\[3.0pt]
{\qquad\quad\quad\quad\phantom{\Big{\\{}}+\int_{0}^{1}dt\int_{0}^{1}ds\,e^{-i\frac{t}{2}(p_{3}+p_{4})\wedge
p_{2}}e^{+i\frac{s}{2}p_{3}\wedge p_{4}}[{\rm
Tr}\,(\tilde{q}_{\mu_{4}}(p_{4})b_{\mu_{3}}(p_{3})T^{b})]T^{b}\tilde{q}_{\mu_{2}}(p_{2})}\\\\[3.0pt]
{\qquad\quad\quad\quad\phantom{\Big{\\{}}+\int_{0}^{1}dt\int_{0}^{1}ds\,e^{+i\frac{t}{2}(p_{3}+p_{4})\wedge
p_{2}}e^{-i\frac{s}{2}p_{3}\wedge p_{4}}\tilde{q}_{\mu_{2}}(p_{2})[{\rm
Tr}\,(b_{\mu_{3}}(p_{3})\tilde{q}_{\mu_{4}}(p_{4})T^{b})]T^{b}}\\\\[3.0pt]
{\qquad\quad\quad\quad\phantom{\Big{\\{}}+\int_{0}^{1}dt\int_{0}^{1}ds\,e^{+i\frac{t}{2}(p_{3}+p_{4})\wedge
p_{2}}e^{+i\frac{s}{2}p_{3}\wedge p_{4}}\tilde{q}_{\mu_{2}}(p_{2})[{\rm
Tr}\,(\tilde{q}_{\mu_{4}}(p_{4})b_{\mu_{3}}(p_{3})T^{b})]T^{b}\Big{\\}},}\\\\[8.0pt]
{C^{(2)\,0}(\tilde{c},b)=-\frac{1}{2N}\theta^{ij}\,e^{i(p_{1}+p_{2})x}\,p_{1i}\,\big{(}\frac{e^{-\frac{i}{2}p_{1}\wedge
p_{2}}-e^{\frac{i}{2}p_{1}\wedge p_{2}}}{p_{1}\wedge
p_{2}}\big{)}\tilde{c}^{a}(p_{1})b_{j}(p_{2}).}\end{array}$ (22)

###  III.2 The one-loop contribution to the two-point function

By evaluating the order $g^{2}$ contribution to the right hand side of eq.
(17), we shall obtain the one-loop contribution to $\Gamma_{2}[b]$, the two-
point bit of the effective action. The result that we obtained, with help of
integral basis given in Appendix B and Grozin:2000cm ; Martin:2016zon , runs
thus

$\begin{array}[]{l}{\Gamma_{2}[b]=\frac{g^{2}}{N}\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,\big{[}b_{\mu}(p)b_{\nu}(-p)\big{]}}\\\\[3.0pt]
{\phantom{.}\times\Big{\\{}\frac{1}{16\pi^{2}}\frac{1}{(\tilde{p}^{2})^{3}}\Big{[}\frac{16}{3}p^{2}\theta^{\mu
i}\tilde{p}_{i}\theta^{\nu
j}\tilde{p}_{j}-\frac{8}{3}\tilde{p}^{2}(\theta^{\mu
i}\tilde{p}_{i}p^{\nu}+\theta^{\nu
j}\tilde{p}_{j}p^{\mu})-\frac{32}{3}(\tilde{p}^{2})^{2}\eta^{\mu\nu}+\frac{32}{3}\tilde{p}^{2}\tilde{p}^{\mu}\tilde{p}^{\nu}\Big{]}}\\\\[3.0pt]
{\phantom{x}-\frac{1}{\pi^{2}}\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\int_{0}^{1}dx\,\Big{[}x(1-x)p^{2}\tilde{p}^{2}K_{2}(\sqrt{x(1-x)p^{2}\tilde{p}^{2}})\Big{]}\Big{\\}}}\\\\[3.0pt]
\phantom{x}+{\frac{g^{2}}{16\pi^{2}}\,\int\frac{d^{4}p}{(2\pi)^{4}}{\rm
Tr}\,\big{[}b_{\mu}(p)\big{(}p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu}\big{)}b_{\nu}(-p)\big{]}\Big{\\{}-\frac{11}{3}\big{(}N-\frac{2}{N}\big{)}\Big{(}\frac{1}{\epsilon}+Ln\frac{p^{2}}{4\pi\mu^{2}}+\gamma-1\Big{)}}\\\\[3.0pt]
{\phantom{\frac{g^{2}}{16\pi^{2}}(N-\frac{2}{N})\,{\rm
Tr}XXXXXXXXXX}+\frac{4}{N}\int_{0}^{1}dx\,(3+2x)\,K_{0}(\sqrt{x(1-x)p^{2}\tilde{p}^{2}})\Big{\\}}}\\\\[3.0pt]
{+\;\Gamma_{2}[b]^{(eom0)}+\;({\rm 2-loop\;order})},\end{array}$ (23)

where $\Gamma_{2}[b]^{(eom0)}$, which obviously vanishes upon imposing the
equations of motion, contains all the contributions on which integral

$\int d^{4}x\,q^{a}_{\mu}(x)\frac{\delta\Gamma[b]}{\delta b^{a}_{\mu}(x)},$
(24)

in (4), is involved. That is, $\Gamma_{2}[b]^{(eom0)}$ is the sum of all the
contributions which involve either $S_{1eom}$ or $S_{2oem}$ from (17), and it
is given by the following long expression

$\begin{array}[]{l}{\Gamma_{2}[b]^{(eom0)}=}\\\\[3.0pt]
{+\frac{g^{2}}{16\pi^{2}}\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,[b_{\mu}(p)(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})b_{\nu}(-p)]}\\\\[3.0pt]
{\phantom{\Gamma_{2}[b]^{(eom0)}=}\times\Big{\\{}N\,[\frac{15}{\epsilon}+15Ln(\frac{p^{2}}{4\pi\mu^{2}})+\frac{1}{15}Ln(p^{2}\tilde{p}^{2})+60Ln\,2-15\gamma-30]}\\\\[3.0pt]
{\phantom{\Gamma_{2}[b]^{(eom0)}=}\phantom{\Big{\\{}N\,[\frac{15}{\epsilon}+15Ln\frac{p^{2}}{4\pi\mu^{2}}}+\frac{1}{N}\,\big{[}16\big{(}-\frac{1}{\epsilon}-Ln\frac{p^{2}}{4\pi\mu^{2}}+\gamma+2\big{)}-34Ln\,2\big{]}\Big{\\}}}\\\\[3.0pt]
{-\frac{1}{N}\,\frac{g^{2}}{16\pi^{2}}\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,[b_{\mu}(p)(p^{2}\delta^{\mu}_{\phantom{\mu}\rho}-p^{\mu}p_{\rho})b_{\nu}(-p)]}\\\\[4.0pt]
{\phantom{\Gamma_{2}[b]^{(eom0)}}\times\Big{\\{}\big{[}4\big{(}-\frac{1}{\epsilon}-Ln\frac{p^{2}}{4\pi\mu^{2}}-\gamma+2\big{)}+8\int_{0}^{1}dx\,K_{0}(\sqrt{x(1-x)p^{2}\tilde{p}^{2}})\big{]}\frac{\tilde{p}_{\rho}\tilde{p}_{\nu}}{\tilde{p}^{2}}}\\\\[4.0pt]
{\phantom{\Gamma_{2}[b]^{(eom0)}=-\frac{1}{N}\,\frac{g^{2}}{16\pi^{2}}Tr[b_{\mu}(p)(p^{2}\quad\quad}+\frac{1}{(\tilde{p}^{2})^{3}}(\frac{28}{3}\tilde{p}^{2}\theta^{\rho\sigma}\theta_{\sigma}^{\phantom{\sigma}\nu}+\frac{84}{3}\tilde{p}^{\rho}\theta^{\nu
i}\theta_{i}^{\phantom{i}\lambda}\tilde{p}_{\lambda})\Big{\\}}}\\\\[3.0pt]
{+\frac{g^{2}N}{16\pi^{2}}\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,[b_{\mu}(p)(p^{2}\delta^{\mu}_{\phantom{\mu}\rho}-p^{\mu}p_{\rho})(p^{2}\delta_{\sigma}^{\phantom{\sigma}n}-p_{\sigma}p^{\nu})b_{\nu}(-p)]}\\\\[3.0pt]
{\quad\quad\times\Big{\\{}\frac{\eta^{\rho\sigma}}{\tilde{p}^{2}}[\frac{3}{2}(\theta^{ij}\theta_{ji}){\cal
C}_{1}-\frac{\tilde{p}^{2}}{p^{2}}{\cal C}_{2}-\frac{\theta^{\lambda
i}\tilde{p}_{i}\theta_{\lambda}^{\phantom{\lambda}j}\tilde{p}_{j}}{\tilde{p}^{2}}{\cal
C}_{4}-\frac{1}{2}{\cal
B}_{1}(\tilde{p}^{2})^{2}]+\frac{\tilde{p}_{\rho}\tilde{p}_{\sigma}}{\tilde{p}^{2}}[{\cal
B}_{1}\tilde{p}^{2}+\frac{1}{p^{2}}{\cal
C}_{2}+\frac{1}{2}\theta^{ij}\theta_{ji}{\cal C}_{4}]}\\\\[3.0pt]
{\phantom{\\{\eta^{\rho\sigma}[\frac{3}{2}(\theta^{ij}\theta_{ji}){\cal
C}_{1}+\frac{\tilde{p}^{2}}{p^{2}}{\cal C}_{2}-\frac{\theta^{\lambda
i}\tilde{p}_{i}\theta_{\lambda}^{\phantom{\lambda}}\tilde{p}_{j}}{}}-3\frac{\theta^{\rho\,i}\,\theta_{i}^{\phantom{i}\sigma}}{\tilde{p}^{2}}{\cal
C}_{1}+\frac{\theta^{\rho\,i}\tilde{p}_{i}\,\theta^{\sigma\,j}\tilde{p}_{j}}{(\tilde{p}^{2})^{2}}{\cal
C}_{4}-\frac{\theta^{\rho\lambda}\theta_{\lambda}^{\phantom{\lambda}i}\tilde{p}_{i}\;\tilde{p}^{\sigma}+\theta^{\sigma\lambda}\theta_{\lambda}^{\phantom{\lambda}i}\tilde{p}_{i}\;\tilde{p}^{\rho}}{(\tilde{p}^{2})^{2}}{\cal
C}_{4}\Big{\\}}}\\\\[3.0pt]
{-\frac{g^{2}}{N}\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,[b_{\mu}(p)(p^{2}\delta^{\mu}_{\phantom{\mu}\rho}-p^{\mu}p_{\rho})(p^{2}\delta_{\sigma}^{\phantom{\sigma}\nu}-p_{\sigma}p^{\nu})b_{\nu}(-p)]}\\\\[3.0pt]
{\quad\quad\times\Big{\\{}\frac{\eta^{\rho\sigma}}{\tilde{p}^{2}}[3(\theta^{ij}\theta_{ji})({\cal
C}_{1}+\frac{1}{2}\tilde{{\cal C}}_{1})-\frac{\tilde{p}^{2}}{p^{2}}(2{\cal
C}_{2}+\tilde{{\cal C}}_{2})-\frac{\theta^{\lambda
i}\tilde{p}_{i}\theta_{\lambda}^{\phantom{\lambda}j}\tilde{p}_{j}}{\tilde{p}^{2}}(2{\cal
C}_{4}+\tilde{{\cal C}}_{4})}\\\\[3.0pt] {\quad\qquad-({\cal
B}_{1}+\tilde{{\cal B}}_{1}+\frac{1}{2}\tilde{{\cal
A}})(\tilde{p}^{2})^{2}]+\frac{\tilde{p}_{\rho}\tilde{p}_{\sigma}}{\tilde{p}^{2}}[2{\cal
B}_{1}\tilde{p}^{2}+\tilde{{\cal B}}_{1}\tilde{p}^{2}+\frac{1}{p^{2}}({\cal
C}_{2}+\tilde{{\cal C}}_{2})+\frac{1}{2}\theta^{ij}\theta_{ji}({\cal
C}_{4}+\frac{1}{2}\tilde{{\cal C}}_{4}])}\\\\[3.0pt]
{\qquad\qquad-3\frac{\theta^{\rho\,i}\,\theta_{i}^{\phantom{i}\sigma}}{\tilde{p}^{2}}(2{\cal
C}_{1}+\tilde{{\cal
C}}_{1})+\frac{\theta^{\rho\,i}\tilde{p}_{i}\,\theta_{\sigma}^{\phantom{\sigma}j}\tilde{p}_{j}}{(\tilde{p}^{2})^{2}}(2{\cal
C}_{4}+\tilde{{\cal
C}}_{4})-\frac{\theta^{\rho\lambda}\theta_{\lambda}^{\phantom{\lambda}i}\tilde{p}_{i}\;\tilde{p}^{\sigma}+\theta^{\sigma\lambda}\theta_{\lambda}^{\phantom{\lambda}i}\tilde{p}_{i}\;\tilde{p}^{\rho}}{(\tilde{p}^{2})^{2}}({\cal
C}_{4}+\frac{1}{2}\tilde{{\cal C}}_{4})\Big{\\}}}\\\\[3.0pt]
{-\frac{i}{2}\frac{g^{2}N}{16\pi^{2}}\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,[b_{\sigma}(p)(p^{2}\delta^{\sigma}_{\phantom{\sigma}\rho}-p^{\sigma}p_{\rho})f_{\mu\nu}(-p)](4\frac{\theta^{\mu\,i}\tilde{p}_{i}\eta^{\rho\nu}}{\tilde{p}^{2}}+2\theta^{\mu\nu}\frac{\tilde{p}^{\rho}}{\tilde{p}^{2}})}\\\\[3.0pt]
{\qquad\qquad\quad\times\big{[}2\big{(}-\frac{1}{\epsilon}-Ln\frac{p^{2}}{4\pi\mu^{2}}-\gamma+2\big{)}-4\int_{0}^{1}dx\,K_{0}(\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2})\big{]}}\\\\[3.0pt]
{+\frac{i}{N}\frac{g^{2}}{16\pi^{2}}\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,[b_{\sigma}(p)(p^{2}\delta^{\sigma}_{\phantom{\sigma}\rho}-p^{\sigma}p_{\rho})f_{\mu\nu}(-p)](4\frac{\theta^{\mu\,i}\tilde{p}_{i}\eta^{\rho\nu}}{\tilde{p}^{2}}+2\theta^{\mu\nu}\frac{\tilde{p}^{\rho}}{\tilde{p}^{2}})}\\\\[3.0pt]
{\qquad\times\big{[}2\big{(}-\frac{1}{\epsilon}-Ln\frac{p^{2}}{4\pi\mu^{2}}-\gamma+2\big{)}-4\int_{0}^{1}dx\,\big{(}K_{0}(\sqrt{x(1-x)p^{2}\tilde{p}^{2}})-2K_{0}(\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2})\big{)}\big{]}.}\end{array}$
(25)

Note that $f_{\mu\nu}(-p)=i(p_{\mu}b_{\nu}(-p)-p_{\nu}b_{\mu}(-p))$. ${\cal
B}_{1}$, ${\cal C}_{1}$, ${\cal C}_{2}$, ${\cal C}_{4}$, $\tilde{{\cal A}}$,
$\tilde{{\cal B}}_{1}$, $\tilde{{\cal C}}_{1}$, $\tilde{{\cal C}}_{2}$ and
$\tilde{{\cal C}}_{4}$ in (25) read thus:

$\begin{array}[]{l}{{\cal
B}_{1}=-\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}\Big{[}\frac{\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}}{2}K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}-1\Big{]}},\\\\[3.0pt]
{{\cal
C}_{1}=+\frac{1}{\epsilon}+Ln\frac{p^{2}}{4\pi\mu^{2}}+\gamma+2Ln(p^{2}\tilde{p}^{2})-8Ln2+2\gamma+3-\int_{0}^{1}dx\,K_{0}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}}\\\\[3.0pt]
{\phantom{{\cal
C_{1}}\frac{1}{\tilde{p}}}+p^{2}\tilde{p}^{2}\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}\Big{[}\frac{\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}}{2}K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}-1\Big{]}},\\\\[3.0pt]
{{\cal
C}_{2}=-\frac{9}{\epsilon}-9Ln\frac{p^{2}}{4\pi\mu^{2}}-7\gamma+1-Ln(p^{2}\tilde{p}^{2})+8Ln2+2\int_{0}^{1}dx\,K_{0}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{)}}\\\\[3.0pt]
{\phantom{{\cal
C}_{2}=}-\frac{3}{4}p^{2}\tilde{p}^{2}\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}\Big{[}\frac{\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}}{2}K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}-1\Big{]}},\\\\[3.0pt]
{{\cal
C}_{4}=-\frac{1}{\epsilon}-Ln\frac{p^{2}}{4\pi\mu^{2}}-\gamma-7-2Ln(p^{2}\tilde{p}^{2})+8Ln2+6\int_{0}^{1}dx\,K_{0}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{)}}\\\\[3.0pt]
{\phantom{{\cal
C}_{4}=}-\frac{1}{4}p^{2}\tilde{p}^{2}\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}\Big{[}\frac{\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}}{2}K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}-1\Big{]}},\\\\[3.0pt]
{\tilde{{\cal
A}}=2\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}}\\\\[3.0pt]
{\qquad\qquad\qquad\times\Big{\\{}\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}\Big{[}K_{1}\big{(}\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\big{)}-K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{]}+1\Big{\\}}},\\\\[3.0pt]
{\tilde{{\cal
B}_{1}}=-\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}}\\\\[3.0pt]
{\qquad\qquad\qquad\quad\times\Big{\\{}\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}\Big{[}K_{1}\big{(}\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\big{)}-\frac{1}{2}K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{]}\Big{\\}}},\\\\[3.0pt]
{\tilde{{\cal
C}_{1}}=-\frac{1}{\epsilon}-Ln\frac{p^{2}}{4\pi\mu^{2}}-2Ln(p^{2}\tilde{p}^{2})+4Ln2-3\gamma-3+\int_{0}^{1}dx\,K_{0}(\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2})}\\\\[3.0pt]
{\qquad\;+\frac{p^{2}\tilde{p}^{2}}{4}\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}}\\\\[3.0pt]
{\qquad\qquad\qquad\times\Big{\\{}\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}\Big{[}K_{1}\big{(}\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\big{)}-\frac{1}{2}K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{]}\Big{\\}}},\\\\[3.0pt]
{\tilde{{\cal
C}}_{2}=+\frac{9}{\epsilon}+9Ln\frac{p^{2}}{4\pi\mu^{2}}+7\gamma-1+2Ln(p^{2}\tilde{p}^{2})-4Ln2}\\\\[3.0pt]
{\phantom{\tilde{{\cal
C}}_{2}=}+2\int_{0}^{1}dx\,\Big{[}K_{0}\big{(}\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\big{)}-K_{0}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{]}}\\\\[3.0pt]
{\phantom{{\cal
C}_{2}=}-\frac{3}{4}p^{2}\tilde{p}^{2}\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}}\\\\[3.0pt]
{\qquad\qquad\qquad\times\Big{\\{}\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}\Big{[}K_{1}\big{(}\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\big{)}-K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{]}+1\Big{\\}}},\\\\[3.0pt]
{\tilde{{\cal
C}}_{4}=-\frac{1}{\epsilon}-Ln\frac{p^{2}}{4\pi\mu^{2}}+\gamma+7+2Ln(p^{2}\tilde{p}^{2})-12Ln2}\\\\[3.0pt]
{\phantom{\tilde{{\cal
C}}_{4}=}+6\int_{0}^{1}dx\,\Big{[}K_{0}\big{(}\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\big{)}-K_{0}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{]}}\\\\[3.0pt]
{\phantom{{\cal
C}_{4}=}-\frac{1}{4}p^{2}\tilde{p}^{2}\int_{0}^{1}dx\,x^{-1/2}\int_{0}^{1}dy\,\big{[}xy(1-y)p^{2}\tilde{p}^{2}\big{]}^{-1}}\\\\[3.0pt]
{\qquad\qquad\qquad\times\Big{\\{}\sqrt{xy(1-y)p^{2}\tilde{p}^{2}}\Big{[}K_{1}\big{(}\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\big{)}-K_{1}\big{(}\frac{\sqrt{x(1-x)p^{2}\tilde{p}^{2}}}{2}\big{)}\Big{]}+1\Big{\\}}.}\end{array}$
(26)

###  III.3 Gauge independent UV/IR mixing

It is plane that $\Gamma_{2}[b]$ in (23) develops, as a result of UV/IR
mixing, IR divergences in the region where $\tilde{p}^{\mu}=0$ and that these
divergences do not survive the large $N$ limit. And yet, $\Gamma_{2}[b]$ is a
gauge-fixing dependent quantity so one may ask whether all these IR
singularities are gauge-fixing artifacts. We shall answer this question by
putting the background field $b_{\mu}$ on shell so that our $\Gamma_{2}[b]$
will boil down to the 2-point on-shell DeWitt effective action DeWitt1 ;
Kallosh:1974yh ; DeWitt2 ; DeWitt3 , for we are computing radiative
corrections at order $g^{2}$. It is known that on-shell DeWitt effective
action is a gauge-fixing independent object.

Since we are working at order $g^{2}$, we only demand the $b_{\mu}$ be a
solution, $b^{(0)}_{\mu}$, to the free equation of motion to put $b_{\mu}$ on-
shell:

$\partial^{2}b^{(0)}_{\mu}(x)-\partial_{\mu}\partial^{\nu}b^{(0)}_{\nu}(x)=0.$
(27)

Now, any solution to (27) is of the form

$b^{(0)}_{\mu}(x)=b^{\bot}_{\mu}(x)+\partial_{\mu}\alpha(x),$ (28)

where $\alpha(x)$ is an arbitrary function taking values in the Lie algebra of
SU(N) and $b^{\bot}_{\mu}(x)$ is such that

$\partial^{2}b^{\bot}_{\mu}(x)=0,\quad\partial^{\mu}b^{\bot}_{\mu}(x)=0.$ (29)

Let us replace $b_{\mu}$ in (23) with $b^{(0)}_{\mu}(x)$ in (28) to obtain the
2-point on-shell effective action DeWitt1 ; Kallosh:1974yh ; DeWitt2 ; DeWitt3
:

$\displaystyle\Gamma_{2}[b^{(0)}]$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{N}\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,\big{[}b^{\bot}_{\mu}(p)b^{\bot}_{\nu}(-p)\big{]}\Big{\\{}-\frac{2}{3\pi^{2}}\frac{\eta^{\mu\nu}}{\tilde{p}^{2}}+\frac{2}{3\pi^{2}}\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}$
$\displaystyle-$
$\displaystyle\frac{1}{\pi^{2}}\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\int_{0}^{1}dx\,[x(1-x)p^{2}\tilde{p}^{2}K_{2}(\sqrt{x(1-x)p^{2}\tilde{p}^{2}})]\Big{\\}}+O(g^{3}).$

Notice that $b^{\bot}_{\mu}(p)$ above is the Fourier transform of the
$b^{\bot}_{\mu}(x)$ satisfying the conditions in (29) and that is related to
$b^{\bot}_{\mu}(p)$ by a gauge transformation. Notice that the right-hand side
of equation (III.3) is invariant under arbitrary on-shell gauge
transformations. The latter as defined thus

$b^{\bot}_{\mu}(p)\rightarrow b^{\bot}_{\mu}(p)+p_{\mu}\beta(p),\quad
p^{2}\beta(p)=0.$ (31)

Finally, $\Gamma_{2}[b^{(0)}]$ above develops quadratic IR singularity, in the
limit $\tilde{p}^{\mu}\to 0$, that is given by

$\frac{g^{2}}{N}\frac{-2}{3\pi^{2}\tilde{p}^{2}}\Big{\\{}\eta^{\mu\nu}+2\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{\tilde{p}^{2}}\Big{\\}}{\rm
Tr}\,\big{[}b^{\bot}_{\mu}(p)b^{\bot}_{\nu}(-p)\big{]},$ (32)

which goes away in the large $N$ limit.

Note that obtained (32) has completely different Lie algebra structure with
respect to the famous IR singularity in the one-loop two-point function of the
ordinary gauge field of noncommutative U(N) case VanRaamsdonk:2001jd ,

${\frac{2g^{2}}{\pi^{2}}}\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\big{[}{\rm
Tr}\,b^{\bot}_{\mu}(p)\big{]}\big{[}{\rm Tr}\,b^{\bot}_{\nu}(-p)\big{]}.$ (33)

###  III.4 The UV/IR mixing phenomenon

Let us show now that the IR singularity in (32) which occur when
$\tilde{p}^{\mu}=0$ is a consequence of the UV/IR mixing phenomenon and not
the result of the Seiberg-Witten maps in (22) having denominators which vanish
for specific values of the momenta. We shall display below each and everyone
of the contributions to the path integral in (17) which yield (III.3).

Let us begin with the following definition

$\big{\langle}{\cal
O}(\tilde{q}^{a}_{\mu},\tilde{c}^{a},\bar{C}^{a})\big{\rangle}_{0}=\int{\cal
D}\tilde{q}^{a}_{\mu}{\cal D}\tilde{c}^{a}{\cal
D}\bar{C}^{a}\,e^{i(S_{0}+S_{0gh})}\;{\cal
O}(\tilde{q}^{a}_{\mu},\tilde{c}^{a},\bar{C}^{a}),$ (34)

where $S_{0}$ and $S_{0gh}$ are given in (18) and (20), respectively. Then,

$-ig^{2}\frac{N}{2}\Big{\langle}\,\partial^{\mu}\hat{A}^{(2)\,0}_{\mu}(b^{(0)},\tilde{q})(x)\partial^{\nu}\hat{A}^{(2)\,0}_{\nu}(b^{(0)},\tilde{q})(x)\Big{\rangle}_{0},$
(35)

gives rise to the following contribution

$\displaystyle-$ $\displaystyle\frac{g^{2}}{2N}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\tilde{p}^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{\mu}k^{\nu}}{k^{2}(k\tilde{p})^{2}}(e^{ik\tilde{p}}+e^{-ik\tilde{p}}-2)$
$\displaystyle=$ $\displaystyle i\frac{g^{2}}{N}\frac{1}{16\pi^{2}}{\rm
Tr}\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}\frac{4}{3}\Big{(}\frac{\eta^{\mu\nu}}{\tilde{p}^{2}}-4\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\Big{)}\Big{\\}}\,+\,O(\epsilon),$

with $D=4+2\epsilon$. Now the factor

$\frac{(e^{ik\tilde{p}}+e^{-ik\tilde{p}}-2)}{(k\tilde{p})^{2}},$ (37)

tends to a constant as $k\tilde{p}$ goes to zero. Hence, the singularity at
$\tilde{p}^{2}=0$ of the right hand side of (III.4) is a consequence of the
fact that, when $\tilde{p}^{\mu}\neq 0$ in the integral over $k$ exponentials
$e^{\pm ik\tilde{p}}$ kills the UV divergent behaviour of the rest of the
integrand, rendering a finite result. Of course, these UV divergent behaviour
resurfaces in the form of a IR divergence as $\tilde{p}^{2}\rightarrow 0$.
This is precisely celebrated UV/IR mixing of the U(N) noncommutative gauge
field theories, discovered first in Minwalla:1999px ; Hayakawa:1999yt ;
Hayakawa:1999zf ; Hayakawa:2000zi ; VanRaamsdonk:2000rr ; Matusis:2000jf .

The contribution to (III.3) coming from

$-ig^{2}\frac{N}{2}\Big{\langle}\,\partial^{\mu}\hat{A}^{(2)\,0\,\nu}(b^{(0)},\tilde{q})(x)\partial_{\mu}\hat{A}^{(2)\,0}_{\nu}(b^{(0)},\tilde{q})(x)\Big{\rangle}_{0},$
(38)

reads

$\displaystyle\frac{3g^{2}}{2N}{\rm
Tr}\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}(\theta^{\mu
j_{1}}\theta^{\nu j_{2}}+\theta^{\mu j_{2}}\theta^{\nu
j_{1}})p^{\lambda}p_{j_{2}}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k_{\lambda}k_{j_{1}}}{k^{2}(k\tilde{p})^{2}}(e^{ik\tilde{p}}+e^{-ik\tilde{p}}-2)$
$\displaystyle=i\frac{g^{2}}{N}\frac{1}{16\pi^{2}}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}-8\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\Big{\\}}\,+\,O(\epsilon).$
(39)

In the above equation (39) we meet the very same type of loop integral –the
integral over $k$–, so the IR divergent behaviour as $\tilde{p}^{\mu}$ goes to
zero that occurs on the right hand side of (39) has the UV/IR mixing origin
that we discuss in the paragraph below (III.4).

Let us now deal with

$-g^{2}\Big{\langle}\,{\rm
Tr}\,\hat{A}^{(2)\,0}_{\nu}(b^{(0)},\tilde{q})(x)[\partial_{\mu}\tilde{q}^{\nu},b^{(0)\,\mu}]_{\star}(x)\Big{\rangle}_{0}.$
(40)

It can be shown that the previous expression is equal to

$\displaystyle\frac{g^{2}}{2N}3{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\tilde{p}^{\mu}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{k^{\nu}}{k^{2}(k\tilde{p})}(e^{ik\tilde{p}}+e^{-ik\tilde{p}}-2)$
(41) $\displaystyle=i\frac{g^{2}}{N}\frac{1}{16\pi^{2}}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}12\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\Big{\\}}\,+\,O(\epsilon).$

Notice that

$\frac{(e^{ik\tilde{p}}+e^{-ik\tilde{p}}-2)}{k\tilde{p}},$ (42)

approaches zero as $\tilde{p}^{\mu}$ goes to zero. Hence, the vanishing, when
$\tilde{p}^{\mu}\rightarrow 0$, of the denominator of the integral over $k$ in
(41) has no bearing on the IR divergence at $\tilde{p}^{\mu}=0$ that occurs on
the right hand side of (41). Indeed, again, this IR divergence at
$\tilde{p}^{\mu}=0$ occurs as a consequence of the fact that $e^{\pm
ik\tilde{p}}$ cuts-off the UV divergent behaviour of the integral over $k$ we
have just mentioned: clearly here $\tilde{p}$ acts as a cut-off giving rise to
the UV/IR mixing phenomenon.

The loop integral –the integral over $k$– which occurs in (41) is also the
only responsible for the IR singularity at $\tilde{p}^{\mu}=0$ of the
following contributions to $\Gamma_{2}[b^{(0)}]$ in (III.3):

$\begin{array}[]{l}{-g^{2}\big{\langle}\,{\rm
Tr}\,\partial_{\mu}\hat{A}^{(2)\,0}_{\nu}(b^{(0)},\tilde{q})(x)[b^{(0)\,\mu},\tilde{q}^{\nu}]_{\star}(x)\big{\rangle}_{0}}\\\\[4.0pt]
{\phantom{g\langle\,{\rm
Tr}\,\partial_{\mu}\hat{A}^{(2)\,0}_{\nu}(b^{(0)}}=i\frac{g^{2}}{N}\frac{1}{16\pi^{2}}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}12\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\Big{\\}}\,+\,\text{additional
terms},}\\\\[8.0pt] {-2g^{2}\big{\langle}\,{\rm
Tr}\,\hat{A}^{(2)\,0}_{\nu}(b^{(0)},\tilde{q})(x)[\partial_{\mu}b^{(0)\,\nu},\tilde{q}^{\mu}]_{\star}(x)\big{\rangle}_{0}}\\\\[4.0pt]
{\phantom{g\langle\,{\rm
Tr}\,\partial_{\mu}\hat{A}^{(2)\,0}_{\nu}(b^{(0)}}=i\frac{g^{2}}{N}\frac{1}{16\pi^{2}}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}8\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\Big{\\}}\,+\,\text{additional
terms},}\\\\[8.0pt] {+g^{2}\big{\langle}\,{\rm
Tr}\,\partial_{\mu}\hat{A}^{(2)\,0}_{\mu}(b^{(0)},\tilde{q})(x)[b^{(0)}_{\,\nu},\tilde{q}^{\nu}]_{\star}(x)\big{\rangle}_{0}}\\\\[4.0pt]
{\phantom{g\langle\,{\rm
Tr}\,\partial_{\mu}\hat{A}^{(2)\,0}_{\nu}(b^{(0)}}=i\frac{g^{2}}{N}\frac{1}{16\pi^{2}}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}-8\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\Big{\\}}\,+\,\text{additional
terms}.}\end{array}$ (43)

There remains to discuss the origin of the singularity at $\tilde{p}^{\mu}=0$
of the following contributions to $\Gamma_{2}[b^{(0)}]$ in (III.3):

$\begin{array}[]{l}{-i\frac{g^{2}}{2N}\big{\langle}\,{\rm
Tr}\,[b^{(0)}_{\mu},\tilde{q}_{\nu}]_{\star}(x)[b^{(0)\,\mu},\tilde{q}^{\nu}]_{\star}(x)\big{\rangle}_{0}=\frac{ig^{2}}{16\pi^{2}N}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}-4\frac{\eta^{\mu\nu}}{\tilde{p}^{2}}\Big{\\}},}\\\\[10.0pt]
{-i\frac{g^{2}}{2}\big{\langle}\,{\rm
Tr}\,[b^{(0)}_{\mu},\tilde{q}^{\mu}]_{\star}(x)[b^{(0)}_{\nu},\tilde{q}^{\nu}]_{\star}(x)\big{\rangle}_{0}=\frac{ig^{2}}{16\pi^{2}N}{\rm
Tr}\,\big{(}b^{(0)\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}-16\frac{\eta^{\mu\nu}}{\tilde{p}^{2}}\Big{\\}},}\\\\[10.0pt]
{-i\frac{g^{2}}{2}\big{\langle}\Big{(}\,{\rm
Tr}\,b^{(0)}_{\mu}[\tilde{q}_{\nu},\partial^{\mu}\tilde{q}_{\nu}]_{\star}\Big{)}^{2}\big{\rangle}_{0}-2ig^{2}\big{\langle}\Big{(}\,{\rm
Tr}\,\partial_{\mu}b^{(0)}_{\nu}[\tilde{q}^{\mu},\tilde{q}_{\nu}]_{\star}\Big{)}^{2}\big{\rangle}_{0}}\\\\[4.0pt]
{\qquad-i\frac{g^{2}}{2}\big{\langle}\Big{(}\,{\rm
Tr}\,b^{(0)}_{\mu}\\{\partial^{\mu}\tilde{c},\bar{C}\\}_{\star}--\,{\rm
Tr}\,b^{(0)}_{\mu}\\{\tilde{c},\partial^{\mu}\bar{C}\\}_{\star}\Big{)}^{2}\big{\rangle}_{0}}\\\\[4.0pt]
{=\frac{ig^{2}}{16\pi^{2}N}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}16\big{\\{}\frac{\eta^{\mu\nu}}{\tilde{p}^{2}}-\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\int_{0}^{1}dx\,[x(1-x)p^{2}\tilde{p}^{2}K_{2}(\sqrt{x(1-x)p^{2}\tilde{p}^{2}})]\Big{\\}},}\\\\[10.0pt]
{-ig^{2}\big{\langle}\,{\rm
Tr}\,[b^{(0)}_{\mu},\tilde{c}]_{\star}(x)[b^{(0)\,\mu},\bar{C}]_{\star}\big{\rangle}_{0}=-\frac{ig^{2}}{16\pi^{2}N}{\rm
Tr}\,\big{(}b^{(0)}_{\,\mu}(p)b^{(0)}_{\,\nu}(-p)\big{)}\Big{\\{}8\frac{\eta^{\mu\nu}}{\tilde{p}^{2}}\Big{\\}}.}\\\\[8.0pt]
\end{array}$ (44)

The singular behaviour at $\tilde{p}^{\mu}=0$ in the expression above comes
uniquely from various types of integrals over $k$, all being given in Appendix
B:

$\,\frac{1}{k^{2}}\,e^{ik\tilde{p}},\;\;\,\frac{(2k+p)_{\mu}(2k+p)_{\nu}}{k^{2}(k+p)^{2}}\,e^{ik\tilde{p}},\;\;......$
(45)

The UV/IR mixing phenomenon that the latter integrals bring about has been
amply discussed in the literature already Minwalla:1999px ; Hayakawa:1999yt .
Notice that one obtains (III.3) by adding the right hand sides of (III.4),
(39), (41), (43) and (LABEL:IR5).

Some final comments are in order. The U(1) part of the Seiberg-Witten map
$Q_{\mu}(b,q)$ as defined in (69), contributes to the singularity at
$\tilde{p}_{\mu}=0$ due to UV/IR mixing –see (III.4–43). The UV/IR mixing
effect the SU(N) bit of $Q_{\mu}(b,q)$ which is involved in, is obtained by
using the field $\tilde{q}_{\mu}$, –see (LABEL:IR5).

##  IV Adding Dirac fermions in the fundamental: Noncommutative QCD

We shall show in this section that the inclusion of a Dirac fermion
transforming under the fundamental representation of SU(N) does not change the
one-loop IR singular behaviour of the two point function of $b_{\mu}(x)$,
$\Gamma_{2}[b]$, that we have unveiled in the previous section.

Let $\psi(x)=\psi_{j}(x)$, $j=1,...,N$–the SU(N) index, be an ordinary Dirac
fermion transforming under the fundamental representation of SU(N). Let
$\Psi(x)=\Psi(a,\psi)_{j}(x)$, with $a_{\mu}(x)=b_{\mu}(x)+q_{\mu}(x)$, be the
noncommutative fermion field obtained from ordinary fields $\psi_{j}(x)$ by
using the Seiberg-Witten map. Then the classical action $S_{Dirac}$, of
$\Psi(a,\psi)_{j}(x)$ coupled to the noncommutative gauge field
$A_{\mu}(x)=B_{\mu}(b)(x)+Q_{\mu}(b,q)(x)$, reads

$S_{Dirac}\,=\,\,\bar{\Psi}(a,\psi)\big{(}i\not{D}\,[A]-m\big{)}\Psi(a,\psi),$
(46)

where $\bar{\Psi}(a,\psi)$ is the Dirac conjugate of $\Psi(x)$, and
$\not{D}\,[A]=i(\not{\partial}-g\not{A}\star)$.

The one-loop contribution, $\Gamma_{Dirac}[b]$, to the background field
effective action $\Gamma[b]$ due to the noncommutative Dirac field
$\Psi(a,\psi)$ is given by the path integral with respect to the ordinary
fields $\psi,\;\bar{\psi}$

$e^{i\Gamma_{Dirac}[b]}=\int{\cal D}\psi{\cal
D}{\bar{\psi}}\;e^{i\,\bar{\Psi}(b,\psi)\big{(}i\not{D}\,[B]-m\big{)}\Psi(b,\psi)}.$
(47)

Now, since $\Psi(a,\psi)_{j}$ is a four component spinor, one can make the
following change of variables

$\psi\rightarrow\Psi=\Psi(b,\psi).$ (48)

in the path integral (47) and obtain:

$e^{i\Gamma_{Dirac}[b]}=\int{\cal D}\Psi{\cal
D}{\bar{\Psi}}\;J_{F}(b)\bar{J}_{F}(b)\,e^{i\,\bar{\Psi}(b,\psi)\big{(}i\not{D}\,[B]-m\big{)}\Psi(b,\psi)}$
(49)

where $J_{F}(b)$ and $\bar{J}_{F}(b)$ are the appropriate Jacobians:

$J_{F}[b]=Det\,\frac{\delta\Psi(b,\psi)_{i}(x)}{\delta\psi_{j}(y)}\quad\text{and}\quad\bar{J}_{F}[b]=Det\,\frac{\delta\bar{\Psi}(b,\psi)_{i}(x)}{\delta\bar{\psi}_{j}(y)}.$
(50)

As with the Seiberg-Witten map for the gauge field, a SW map defining the
noncommutative field $\Psi(b,\psi)$ can be obtained by solving the following
“evolution” problem:

$\displaystyle\frac{d\phantom{t}}{dt}\Psi(b,\psi;t)$ $\displaystyle=$
$\displaystyle-\frac{g}{2}\theta^{\mu\nu}B_{\mu}(b;t)\star_{t}\partial_{\nu}\Psi(b,\psi;t)+\frac{i}{4}g^{2}B_{\mu}(b;t)\star_{t}B_{\mu}(b;t)\star_{t}\Psi(b,\psi;t),$
$\displaystyle\Psi_{\mu}(b;t=0)$ $\displaystyle=$ $\displaystyle\psi,$ (51)

where $B_{\mu}(b,t)$ is given by the “evolution” equation discussed in
Appendix A.

The “evolution” problem in (51) can be solved by expanding in powers of $g$
–see Ref.Martin:2012aw  for details – so that one may show that

$\displaystyle\frac{\delta\Psi(b,\psi)(x)}{\delta\psi(y)}$ $\displaystyle=$
$\displaystyle\mathbb{I}\delta(x-y)+\sum_{n=2}^{\infty}\,g^{n-1}\,\int\prod_{i=1}^{n}\,e^{i\sum_{i=1}^{n-1}p_{i}x}e^{ip_{n}(x-y)}$
$\displaystyle\times$ $\displaystyle{\cal
F}^{\mu_{1}\mu_{2}\cdot\mu_{n-1}}(p_{1},p_{2},...,p_{n-1};p_{n};\theta)\,b_{\mu_{1}}(p_{1})b_{\mu_{2}}(p_{2})\cdots
b_{\mu_{n-1}}(p_{n-1}).$

It can be seen that ${\cal
F}^{\mu_{1}\mu_{2}\cdot\mu_{n-1}}(p_{1},p_{2},...,p_{n-1};p_{n};\theta)$ in
(IV) is a linear combination of functions of the type

${\cal P}(p_{1},....,p_{n}){\cal Q}(p_{i}\wedge p_{j}),$ (53)

where ${\cal P}(p_{1},....,p_{n})$ is a polynomial of the momenta $p_{i}$’s
and ${\cal Q}(p_{i}\wedge p_{j})$ only depends on $p_{i}\wedge
p_{j}=p_{\mu\,i}\theta^{\mu\nu}p_{\nu\,j}$, $i,j=1,...,n$. This fact leads to
the conclusion that all the loop integrals involved in the evaluation of

$Det\,\frac{\delta\Psi(b,\psi)_{i}(x)}{\delta\psi_{j}(y)}$ (54)

vanish in dimensional regularization, and, hence,

$J_{F}[b]=Det\,\frac{\delta\Psi(b,\psi)_{i}(x)}{\delta\psi_{j}(y)}=1.$ (55)

The detailed derivation of (55) is analogous to the one carried out in
Appendices B and C of Ref. Martin:2016saw .

Clearly, in dimensional regularization we also have

$\bar{J}_{F}[b]=Det\,\frac{\delta\bar{\Psi}(b,\psi)_{i}(x)}{\delta\bar{\psi}_{j}(y)}=1.$
(56)

Let us introduce the following notation:

$\langle\cdots\rangle_{f0}=\int{\cal D}\Psi{\cal
D}{\bar{\Psi}}\,\cdots\,e^{i\,\bar{\Psi}(i\not{\partial}-m)\Psi.}$ (57)

Substituting (55) and (56) in (49), one readily obtains, in dimensional
regularization, the following one-loop result

$\displaystyle i\Gamma_{Dirac}(b)$ $\displaystyle=$ $\displaystyle
Ln\sum_{n=0}^{\infty}\frac{(-ig)^{n}}{n!}\,\langle\big{(}\;\bar{\Psi}\gamma^{\mu}B_{\mu}(b)\star\Psi\big{)}^{n}\rangle_{f0}$
$\displaystyle=$
$\displaystyle-\sum\limits_{n=2}^{\infty}\,\frac{g^{n}}{n}\,\prod\limits_{i=1}^{n}\;{\rm
tr}\big{(}B_{\mu_{1}}(b)(x_{1})\cdots
B_{\mu_{n}}(b)(x_{n})\big{)}\,\Gamma^{\mu_{1}\cdots\mu_{n}}(x_{1},...x_{n}),$

where

$\displaystyle{\Gamma^{\mu_{1}\cdots\mu_{n}}(x_{1},...x_{n})=\prod\limits_{i=1}^{n}\,(2\pi)^{D}\delta(\sum\limits_{i=1}^{n}p_{i})\,e^{-i\sum\limits_{i=1}^{n}p_{i}x_{i}}\,e^{-\frac{i}{2}\\!\\!\\!\\!\\!\\!\sum\limits_{1\leq
i<j<n}\\!\\!\\!p_{i}\wedge p_{j}}}$ (59) $\displaystyle\times\,\frac{{\rm
Tr}[(/k+\not{p}_{1}-m)\gamma^{\mu_{1}}(\not{k}-m)\gamma^{\mu_{2}}(\not{k}-\not{p}_{2}-m)\gamma^{\mu_{3}}\cdots\gamma^{\mu_{n-1}}(\not{k}-\sum_{i=2}^{n-1}\not{p}_{i}-m)\gamma^{\mu_{n}}]}{((k+p_{1})^{2}+m^{2})k^{2}((k-p_{2})^{2}+m^{2})\cdots((k-\sum_{i=2}^{n-1}p_{i})^{2}+m^{2})}.$

From (59) we draw the conclusion that the full one-loop $\Gamma_{Dirac}[b]$ in
(47) lacks any singular behaviour when any of the momenta
$\tilde{p}^{\mu}_{i}=\theta^{\mu\nu}\tilde{p}_{\nu\,i}$ vanishes $\forall i$.

Finally, it is plain that (IV) and (59) hold whatever the gauge group and
representation provided the noncommutative Dirac field transforms as follows

$\psi(b,\psi)\rightarrow i\Omega(b,\omega)\star\psi(b,\psi)$ (60)

under infinitesimal noncommutative gauge transformations defined by
$\Omega(b,\omega)$, while $\omega(x)$ defines the infinitesimal ordinary gauge
transformations.

##  V Conclusions and outlook

The main conclusion of this paper is that noncommutative SU(N) defined by
means of the $\theta$-exact Seiberg-Witten map has a two-point function for
the gauge field that exhibits UV/IR mixing at the one-loop, in spite of the
fact that there are no fundamental U(1) degrees of freedom. This is at odds
with the noncommutative U(N) case where it is only the U(1) part of the one-
loop two-point function for the ordinary gauge field the one which is affected
by the famous UV/IR mixing. Indeed, in the noncommutative U(N) case the famous
noncommutative IR singularity in the one-loop two-point function of the
ordinary gauge field reads VanRaamsdonk:2001jd

${\frac{2g^{2}}{\pi^{2}}}\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}\big{[}{\rm
Tr}\,b^{\bot}_{\mu}(p)\big{]}\big{[}{\rm Tr}\,b^{\bot}_{\nu}(-p)\big{]}.$ (61)

In Appendix C, we re-derive the above formula using the enveloping algebra
approach together with the background field and the path integral methods
employed in our recent works Martin:2016zon ; Martin:2016hji ; Martin:2016saw
; Martin:2017nhg , as well as in the previous sections of this paper,
confirming eq. (61) and showing this way implicitly the correctness of our
computations.

The noncommutative IR singularity of the noncommutative SU(N) theory reads

$\frac{g^{2}}{N}\frac{-2}{3\pi^{2}\tilde{p}^{2}}\Big{\\{}\eta^{\mu\nu}+2\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{\tilde{p}^{2}}\Big{\\}}{\rm
Tr}\,\big{[}b^{\bot}_{\mu}(p)b^{\bot}_{\nu}(-p)\big{]},$ (62)

–see (32)– which has a novel, as yet unknown, UV/IR mixing tensor and Lie
algebra structure with respect to that in (61). Indeed, in (61), we have the
product of two traces over the generators of the Lie algebra so that only the
U(1) part of the contribution gets affected by the noncommutative IR
divergence. However, in (62), we have only one trace over the product of two
SU(N) generator and, hence, the full SU(N) part of the contribution receives
the noncommutative IR divergence. No $\eta_{\mu\nu}$ occurs in (61).

In subsection 3.4 -see (III.4), (39), (41), etc– we have shown that the
noncommutative IR divergence in (62) occurs because the noncommutative phase
$e^{i\theta^{\mu\nu}k_{\mu}p_{\nu}}$ –$k$ being the loop momenta and $p$ the
external momenta– regularizes, provided $\theta^{\mu\nu}p_{\nu}\neq 0$,
otherwise UV divergent integrals. Thus, the UV divergence is turned into an IR
divergence at $\theta^{\mu\nu}p_{\nu}=0$. This the UV/IR mixing phenomenon and
its origin has nothing to do with the use of the Seiberg-Witten map and its
vanishing denominators at $\theta^{\mu\nu}k_{\mu}p_{\nu}=0$.

We have also shown that this conclusion also holds in noncommutative QCD,
since the addition of Dirac fermions in the fundamental representation to the
noncommutative SU(N) theory does not modify the UV/IR mixing behaviour of the
two-point function at hand.

The phenomenological implications of the UV/IR mixing unveiled in this paper
are worth studying. Notice that this UV/IR mixing affects the noncommutative
Standard Model of Calmet:2001na  and the noncommuative GUTs models of
Aschieri:2002mc ; Martin:2013lba  when defined by means of the $\theta$-exact
Seiberg-Witten map.

In the spirit of the Scattering amplitudes approach Raju:2009yx ; Huang:2010fc
; Arkani-Hamed:2017mur ; Mizera:2019blq ; Latas:2020nji  such a model has been
recently successfully applied to estimate a lower bound on the scale of
noncommutativity Latas:2020nji ; Horvat:2020ycy  by comparison with ATLAS-LHC
collaboration measurements of total cross sections for $\rm
PbPb(\gamma\gamma)\to Pb^{*}Pb^{*}\gamma\gamma$ and $\rm PbPb(\gamma\gamma)\to
Pb^{*}Pb^{*}\ell^{+}\ell^{-}$ reactions in the lead 208Pb ion-ion collision
experiments Aaboud:2017bwk ; Aad:2019ock .

##  Appendix A Seiberg-Witten maps

Let $a_{\mu}$ and $c$ be, respectively, an ordinary gauge field and the
corresponding ghost field taking values in a Lie algebra in a given
irreducible representation. A Seiberg-Witten map which defines a
noncommutative gauge field $A_{\mu}(a)$ and the corresponding noncommutative
ghost field $C(a,c)$ in terms of ordinary $a_{\mu}$ and $c$ is obtained
Barnich:2002tz ; Cerchiai:2002ss ; Barnich:2002pb  by solving, between $t=0$
and $t=1$, the following “evolution” problem

$\begin{array}[]{l}{\frac{d\phantom{t}}{dt}A_{\mu}(a;t)=-\frac{g}{4}\theta^{ij}\\{A_{i}(a,t),\partial_{j}A_{\mu}(a,t)+F_{j\mu}(a,t)\\}_{\star_{t}},\quad
A_{\mu}(a;t=0)=a_{\mu},}\\\\[8.0pt]
{\frac{d\phantom{t}}{dt}C(a,c;t)=\frac{g}{4}\theta^{ij}\\{\partial_{i}C(a,c;t),A_{j}(a,t)\\}_{\star_{t}},\quad
C(a,c;t=0)=c.}\end{array}$ (63)

The product $\star_{t}$ has been defined in (21) and
$F_{\mu\nu}(a;t)=\partial_{\mu}A_{\nu}(a;t)-\partial_{\nu}A_{\mu}(a;t)-ig[A_{\mu}(a;t),A_{\nu}(a;t)]_{\star_{t}}$.

The “evolution” problem in (63) can be solved Martin:2012aw  by expanding in
power of the coupling constant $g$ as befits the definition a field theory in
perturbation theory:

$\begin{array}[]{l}{A_{\mu}(a)=a_{\mu}\,+\,g\,A_{\mu}^{(2)}(a,a;t=1)\,+\,g^{2}\,A_{\mu}^{(3)}(a,a,a;t=1)\,+\,O(g^{3}),}\\\\[8.0pt]
{C(a,c)=c\,+\,g\,C^{(2)}(a,c;t=1)+g^{2}\,C^{(3)}(a,c,a;t=1)\,+\,O(g^{3}),}\end{array}$
(64)

with

$\begin{array}[]{l}{A_{\mu}^{(2)}(a,a;t)=-\frac{1}{4}\theta^{ij}\int_{0}^{t}ds\,\\{a_{i},2\partial_{j}a_{\mu}-\partial_{\mu}a_{j}\\}_{\star_{s}}},\\\\[10.0pt]
{A_{\mu}^{(3)}(a,a,a;t)=-\frac{1}{4}\theta^{ij}\int_{0}^{t}ds\Big{[}\\{a_{i},2\partial_{j}A^{(2)}_{\mu}(a,a;s)-2\partial_{\mu}A^{(2)}_{j}(a,a;s)\\}_{\star_{s}}}\\\\[4.0pt]
{\phantom{A_{\mu}^{(3)}(a,a,a;t)=-\frac{1}{4}\theta^{ij}\int_{0}^{t}ds\Big{[}\\{a_{i},2}+\\{A^{(2)}_{i}(a,a;s),2\partial_{j}a_{\mu}-\partial_{\mu}a_{j}\\}_{\star_{s}}-i\\{a_{i},[a_{j},a_{\mu}]_{\star_{s}}\\}_{\star_{s}}\Big{]}},\\\\[10.0pt]
{C^{(2)}(a,c;t)=\frac{1}{4}\theta^{ij}\int_{0}^{t}ds\,\\{\partial_{i}c,\partial_{i}a_{j}\\}_{\star_{s}}},\\\\[10.0pt]
{C^{(3)}(a,c,a;t)=\frac{1}{4}\theta^{ij}\int_{0}^{t}ds\,\Big{[}\\{\partial_{i}C^{(2)}(a,c;s),a_{j}\\}_{\star_{s}}+\\{\partial_{i}c,A^{(2)}_{j}(a,a;s)\\}_{\star_{s}}\Big{]}.}\end{array}$
(65)

To obtain $B_{\mu}(b)$ and $Q_{\mu}(b,q)$ in (3), we replace $a_{\mu}$ with
$b_{\mu}+q_{\mu}$ in (64) and (65), and then, expand in powers of $a_{\mu}$
and $b_{\mu}$ the resulting expressions:

$\begin{array}[]{l}{A_{\mu}(b+q)=B_{\mu}(b)+Q_{\mu}(b,q),}\\\\[10.0pt]
{B_{\mu}(b)=b_{\mu}+g\,\hat{A}^{(2)}_{\mu}(b,b)+g^{2}\,\hat{A}^{(3)}_{\mu}(b,b,b)+O(g^{3}),}\\\\[8.0pt]
{\hat{A}^{(2)}_{\mu}(b,b)=A_{\mu}^{(2)}(a=b,a=b;t=1),\quad\hat{A}^{(3)}_{\mu}(b,b,b)=A_{\mu}^{(3)}(a=b,a=b,a=b;t=1),}\\\\[12.0pt]
{Q_{\mu}(b,q)=q_{\mu}+g\,\big{(}\hat{A}^{(2)}_{\mu}(b,q)+\hat{A}^{(2)}_{\mu}(q,q)\big{)}}\\\\[8.0pt]
{\qquad\qquad\qquad+g^{2}\,\big{(}\hat{A}^{(3)}_{\mu}(b,b,q)+\hat{A}^{(3)}_{\mu}(q,q,b)+\hat{A}^{(3)}_{\mu}(q,q,q)\big{)}+O(g^{3}),}\\\\[4.0pt]
{\qquad\hat{A}^{(2)}_{\mu}(b,q)=-\frac{1}{4}\theta^{ij}\int_{0}^{1}dt\,\Big{[}\\{b_{i},2\partial_{j}q_{\mu}-\partial_{\mu}q_{j}\\}_{\star_{t}}+\\{q_{i},2\partial_{j}b_{\mu}-\partial_{\mu}b_{j}\\}_{\star_{t}}\Big{]},}\\\\[4.0pt]
{\qquad\hat{A}^{(2)}_{\mu}(q,q)=A_{\mu}^{(2)}(a=q,a=q;t=1),}\\\\[4.0pt]
{\qquad\hat{A}_{\mu}^{(3)}(q,q,b)=-\frac{1}{4}\theta^{ij}\int_{0}^{1}dt}\\\\[8.0pt]
{\qquad\quad\phantom{Xx.}\times\Big{[}\\{q_{i},2\partial_{j}\hat{A}^{(2)}_{\mu}(b,q;t)-2\partial_{\mu}\hat{A}^{(2)}_{j}(b,q;t)\\}_{\star_{t}}+\\{\hat{A}^{(2)}_{i}(b,q;t),2\partial_{j}q_{\mu}-\partial_{\mu}q_{j}\\}_{\star_{t}}}\\\\[4.0pt]
{\phantom{\qquad\hat{A}_{\mu}^{(3)}xx.}+\\{b_{i},2\partial_{j}\hat{A}^{(2)}_{\mu}(q,q;t)-2\partial_{\mu}\hat{A}^{(2)}_{j}(b,q;t)\\}_{\star_{t}}+\\{\hat{A}^{(2)}_{i}(q,q;t),2\partial_{j}b_{\mu}-\partial_{\mu}q_{j}\\}_{\star_{t}}}\\\\[4.0pt]
{\phantom{\qquad\hat{A}_{\mu}^{(3)}xx}-i\\{q_{i},[q_{j},b_{\mu}]_{\star_{t}}\\}_{\star_{t}}-i\\{q_{i},[b_{j},q_{\mu}]_{\star_{t}}\\}_{\star_{t}}-i\\{b_{i},[q_{j},q_{\mu}]_{\star_{t}}\\}_{\star_{t}}\Big{]},}\\\\[4.0pt]
{\qquad\hat{A}_{\mu}^{(3)}(b,b,q)=\hat{A}_{\mu}^{(3)}(q\rightarrow
b,q\rightarrow b,b\rightarrow q),}\\\\[4.0pt]
{\qquad\hat{A}_{\mu}^{(3)}(a,a,a)=A_{\mu}^{(3)}(a=q,a=q,a=q;t=1).}\end{array}$
(66)

Analogously, $C(b+q,c)$ is obtained by setting $a_{\mu}=b_{\mu}+q_{\mu}$ in
$C(a,c)$ in (64):

$\begin{array}[]{l}C(b+q,c)=c+g\,\big{(}\hat{C}^{(2)}(c,b)+\hat{C}^{(2)}(q,c)\big{)}\\\\[10.0pt]
{\phantom{XXXXXXX}+g^{2}\,\big{(}\hat{C}^{(2)}(c,q,q)+\hat{C}^{(2)}(c,b,q)+\hat{C}^{(3)}(c,b,b)\big{)}+O(g^{3})},\\\\[10.0pt]
{\hat{C}^{(2)}(b,c;t)=\frac{1}{4}\theta^{ij}\int_{0}^{t}ds\,\\{\partial_{i}c,b_{j}\\}_{\star_{s}}}\,,\\\\[10.0pt]
{\hat{C}^{(3)}(c,b,b;t)=\frac{1}{4}\theta^{ij}\int_{0}^{t}ds\,\Big{[}\\{\partial_{i}\hat{C}^{(2)}(b,c;s),b_{j}\\}_{\star_{s}}+\\{\partial_{i}c,\hat{A}^{(2)}_{j}(b,b;s)\\}_{\star_{s}}\Big{]}}.\end{array}$
(67)

Since our purpose is to compute $\Gamma_{2}[b]$ at one-loop, we shall not need
the $q$-dependent bits of $C(b+q,c)$:

$C(b,c)=c+g\,\hat{C}^{(2)}(c,b)+g^{2}\,\hat{C}^{(3)}(c,b,b)+O(g^{3}),$ (68)

Assume that $q_{\mu}$ and $b_{\mu}$ take values in the SU(N) Lie algebra in
the fundamental representation. Then $Q_{\mu}(b,q)$ in (66) can be expressed
as the following linear combination:

$Q_{\mu}(b,q)=Q^{(0)}_{\mu}(b,q)\mathbb{I}_{N}\,+\,Q^{a}_{\mu}(b,q)T^{a},$
(69)

where $\mathbb{I}_{N}$ is the $N\times N$ identity matrix and $T^{a}$ are the
SU(N) generators. We shall call $Q^{(0)}_{\mu}(b,q)$ and $Q^{a}_{\mu}(b,q)$
the U(1) component and SU(N) components of the noncommutative field
$Q_{\mu}(b,q)$.

Let us introduce the following definitions:

$\begin{array}[]{l}{\hat{A}^{(2)\,0}_{\mu}(b,q)=\frac{1}{N}\,{\rm
Tr}\,\hat{A}^{(2)}_{\mu}(b,q),\;\hat{A}^{(2)\,0}_{\mu}(q,q)=\frac{1}{N}\,{\rm
Tr}\,\hat{A}^{(2)}_{\mu}(q,q),}\\\\[4.0pt]
{\hat{A}^{(3)\,0}_{\mu}(q,q,b)=\frac{1}{N}\,{\rm
Tr}\,\hat{A}^{(3)}_{\mu}(q,q,b),\;\hat{A}^{(3)\,0}_{\mu}(b,q,q)=\frac{1}{N}\,{\rm
Tr}\,\hat{A}^{(2)}_{\mu}(b,q,q),}\\\\[4.0pt]
{\hat{A}^{(3)\,0}_{\mu}(q,q,q)=\frac{1}{N}\,{\rm
Tr}\,\hat{A}^{(2)}_{\mu}(q,q,q),}\\\\[4.0pt]
{\hat{A}^{(2)\,a}_{\mu}(b,q)=\,{\rm
Tr}\,(T^{a}\hat{A}^{(2)}_{\mu}(b,q)),\;\hat{A}^{(2)\,a}_{\mu}(q,q)=\,{\rm
Tr}\,(T^{a}\hat{A}^{(2)}_{\mu}(q,q)),}\\\\[4.0pt]
{\hat{A}^{(3)\,a}_{\mu}(q,q,b)=\,{\rm
Tr}\,(T^{a}\hat{A}^{(3)}_{\mu}(q,q,b)),\;\hat{A}^{(3)\,a}_{\mu}(b,q,q)=\,{\rm
Tr}\,(T^{a}\hat{A}^{(2)}_{\mu}(b,q,q)),}\\\\[4.0pt]
{\hat{A}^{(3)\,a}_{\mu}(q,q,q)=\,{\rm
Tr}\,(T^{a}\hat{A}^{(2)}_{\mu}(q,q,q)).}\end{array}$ (70)

The reader should bear in mind the results in (66). Then, taking into account
(69), one conclude that

$\begin{array}[]{l}{Q^{(0)}_{\mu}(b,q)=g\,\big{(}\hat{A}^{(2)\,0}_{\mu}(b,q)+\hat{A}^{(2)\,0}_{\mu}(q,q)\big{)}}\\\\[4.0pt]
{\phantom{XXXXx}+g^{2}\,\big{(}\hat{A}^{(3)\,0}_{\mu}(b,b,q)+\hat{A}^{(3)}_{\mu}(q,q,b)+\hat{A}^{(3)\,0}_{\mu}(q,q,q)\big{)}+O(g^{3}),}\\\\[4.0pt]
{Q^{a}_{\mu}(b,q)=q^{a}_{\mu}+g\,\big{(}\hat{A}^{(2)\,a}_{\mu}(b,q)+\hat{A}^{(2)\,a}_{\mu}(q,q)\big{)}}\\\\[4.0pt]
{\phantom{XXXXXX,}+g^{2}\,\big{(}\hat{A}^{(3)\,a}_{\mu}(b,b,q)+\hat{A}^{(3)\,a}_{\mu}(q,q,b)+\hat{A}^{(3)\,a}_{\mu}(q,q,q)\big{)}+O(g^{3}).}\end{array}$
(71)

Let us introduce the field $\tilde{q}_{\mu}=\tilde{q}^{a}_{\mu}T^{a}$, where

$\begin{array}[]{l}{\tilde{q}^{a}_{\mu}=Tr(T^{a}Q_{\mu}(b,q))=Q^{a}_{\mu}(b,q)=q^{a}_{\mu}+g\,\big{(}\hat{A}^{(2)\,a}_{\mu}(b,q)+\hat{A}^{(2)\,a}_{\mu}(q,q)\big{)}+}\\\\[4.0pt]
{\phantom{\tilde{q}^{a}_{\mu}=}g^{2}\,\big{(}\hat{A}^{(3)\,a}_{\mu}(b,b,q)+\hat{A}^{(3)\,a}_{\mu}(q,q,b)+\hat{A}^{(3)\,a}_{\mu}(q,q,q)\big{)}+O(g^{3}).}\end{array}$
(72)

The previous expression can be inverted by expanding in powers of $g$:

$q^{a}_{\mu}=\tilde{q}^{a}_{\mu}-g\,\big{(}\hat{A}^{(2)\,a}_{\mu}(b,\tilde{q})+\hat{A}^{(2)\,a}_{\mu}(\tilde{q},\tilde{q})\big{)}+O(g^{2}),$
(73)

where

$\hat{A}^{(2)\,a}_{\mu}(b,\tilde{q})=\hat{A}^{(2)\,a}_{\mu}(b,q=\tilde{q}),\;\hat{A}^{(2)\,a}_{\mu}(q=\tilde{q},q=\tilde{q}).$
(74)

Substituting (73) in (69) and then expanding in powers of $g$, one gets

$\begin{array}[]{l}{Q_{\mu}(b,q)=\tilde{q}_{\mu}+\mathbb{I}_{N}\Big{\\{}g\big{[}\hat{A}^{(2)\,0}_{\mu}(b,\tilde{q})+\hat{A}^{(2)\,0}_{\mu}(\tilde{q},\tilde{q})\big{]}}\\\\[4.0pt]
{\phantom{XXXXXXXX}+g^{2}\,\big{[}\hat{A}^{(3)\,0}_{\mu}(b,b,\tilde{q})+\hat{A}^{(3)\,a}_{\mu}(\tilde{q},\tilde{q},b)+\hat{A}^{(3)\,a}_{\mu}(\tilde{q},\tilde{q},\tilde{q})}\\\\[4.0pt]
{\phantom{XXXXXXXXXx}-\hat{A}^{(2)\,0}_{\mu}(b,\hat{A}^{(2)\,a}_{\sigma}(b,\tilde{q})T^{a})-\hat{A}^{(2)\,0}_{\mu}(b,\hat{A}^{(2)\,a}_{\sigma}(\tilde{q},\tilde{q})T^{a})}\\\\[4.0pt]
{\phantom{XXXXXXXXXx}-\hat{A}^{(2)\,0}_{\mu}(\tilde{q},\hat{A}^{(2)\,a}_{\sigma}(b,\tilde{q})T^{a})-\hat{A}^{(2)\,0}_{\mu}(\hat{A}^{(2)\,a}_{\sigma}(b,\tilde{q})T^{a},\tilde{q})\big{]}\Big{\\}},}\end{array}$
(75)

where

$\begin{array}[]{l}{\hat{A}^{(2)\,0}_{\mu}(b,\tilde{q})=\hat{A}^{(2)\,0}_{\mu}(b,q\rightarrow\tilde{q}),\;\hat{A}^{(2)\,0}_{\mu}(\tilde{q},\tilde{q})=\hat{A}^{(2)\,0}_{\mu}(q=\tilde{q},q\rightarrow\tilde{q})},\\\\[4.0pt]
{\hat{A}^{(3)\,0}_{\mu}(b,b,\tilde{q})=\hat{A}^{(3)\,0}_{\mu}(b,b,q\rightarrow\tilde{q}),\;\hat{A}^{(3)\,a}_{\mu}(\tilde{q},\tilde{q},b)=\hat{A}^{(3)\,a}_{\mu}(q\rightarrow\tilde{q},q\rightarrow\tilde{q},b),}\\\\[4.0pt]
{\hat{A}^{(3)\,0}_{\mu}(\tilde{q},\tilde{q},\tilde{q})=\hat{A}^{(3)\,0}_{\mu}(q\rightarrow\tilde{q},q\rightarrow\tilde{q},q\rightarrow\tilde{q}),}\\\\[4.0pt]
{\hat{A}^{(2)\,0}_{\mu}(b,\hat{A}^{(2)\,a}_{\sigma}(b,\tilde{q})T^{a})=\hat{A}^{(2)\,0}_{\mu}(b,q_{\sigma}\rightarrow\hat{A}^{(2)\,a}_{\sigma}(b,\tilde{q})T^{a}),}\\\\[4.0pt]
{\hat{A}^{(2)\,0}_{\mu}(b,\hat{A}^{(2)\,a}_{\sigma}(\tilde{q},\tilde{q})T^{a})=\hat{A}^{(2)\,0}_{\mu}(b,q_{\sigma}\rightarrow\hat{A}^{(2)\,a}_{\sigma}(\tilde{q},\tilde{q})T^{a}),}\\\\[4.0pt]
{\hat{A}^{(2)\,0}_{\mu}(\hat{q},\hat{A}^{(2)\,a}_{\sigma}(b,\tilde{q})T^{a})=-\frac{1}{4N}\theta^{ij}\int_{0}^{1}dt\,{\rm
Tr}\,\big{(}\\{\tilde{q}_{i},2\partial_{j}\hat{A}^{(2)\,a}_{\mu}(b,\tilde{q})T^{a}-\partial_{\mu}\hat{A}^{(2)\,a}_{j}(b,\tilde{q})T^{a}\\}_{\star_{t}}\big{)},}\\\\[4.0pt]
{\hat{A}^{(2)\,0}_{\mu}(\hat{q},\hat{A}^{(2)\,a}_{\sigma}(b,\tilde{q})T^{a})=-\frac{1}{4N}\theta^{ij}\int_{0}^{1}dt\,{\rm
Tr}\,\big{(}\\{\hat{A}^{(2)\,a}_{i}(b,\tilde{q})T^{a},2\partial_{j}\tilde{q}_{\mu}-\partial_{\mu}\tilde{q}_{j}\\}_{\star_{t}}\big{)}.}\end{array}$
(76)

The notation $q_{\mu}\rightarrow object$ points out that $q_{\mu}$ is to be
replaced with $object$ in the corresponding expression in (70) and (66).

Proceeding in an analogous way, one shows that

$C(b,c)=\tilde{c}+\mathbb{I}_{N}\big{[}g\,\hat{C}^{(2)\,0}(\tilde{c},b)+g^{2}\,\big{(}\hat{C}^{(3)\,0}(\tilde{c},b,b)+\hat{C}^{(2)\,0}(\hat{C}^{(2)\,a}(\tilde{c},b)T^{a},b)\big{)}\big{]}+O(g^{3},b^{3}),$
(77)

where

$\begin{array}[]{l}{\tilde{c}=\tilde{c}^{a}T^{a},\;\tilde{c}^{a}=\,{\rm
Tr}\,\big{(}T^{a}C(b,c)\big{)},}\\\\[4.0pt]
{\hat{C}^{(2)\,0}(\tilde{c},b)=\frac{1}{4N}\theta^{ij}\int_{0}^{1}dt\,{\rm
Tr}\,\big{(}\\{\partial_{i}\tilde{c},b_{j}\\}_{\star_{t}}\big{)},}\\\\[4.0pt]
{C^{(3)\,0}(\tilde{c},b,b)=\frac{1}{4N}\theta^{ij}\int_{0}^{1}dt\,{\rm
Tr}\,\Big{[}\\{\partial_{i}\hat{C}^{(2)}(\tilde{c},b;s),b_{j}\\}_{\star_{s}}+\\{\partial_{i}\tilde{c},\hat{A}^{(2)}_{j}(b,b;s)\\}_{\star_{s}}\Big{]},}\\\\[4.0pt]
{\hat{C}^{(2)\,0}(\hat{C}^{(2)\,a}(\tilde{c},b)T^{a},b)=\frac{1}{4N}\theta^{ij}\int_{0}^{1}dt\,{\rm
Tr}\,\big{(}\\{\partial_{i}\hat{C}^{(2)\,a}(\tilde{c},b)T^{a},b_{j}\\}_{\star_{t}}\big{)},}\\\\[4.0pt]
{\hat{C}^{(2)\,a}(\tilde{c},b)=\frac{1}{4}\theta^{ij}\int_{0}^{1}dt\,{\rm
Tr}\,\big{(}T^{a}\\{\partial_{i}\tilde{c},b_{j}\\}_{\star_{t}}\big{)}.}\end{array}$
(78)

Let us finally recall that (75) and (77) are needed to go from (11) to (17).

##  Appendix B Computing relevant integrals in $D$ dimensions for Euclidean
signature

Here for normalised $D$ dimensional integral we shall use the following
shorthand notations $\int\frac{d^{D}k}{(2\pi)^{D}}\equiv\int$. Now we
categorise integrals by the power of $(k\tilde{p})$ factor in the denominator.
First we have four integrals with power zero:

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle\int\,\frac{e^{\pm
ik\tilde{p}}}{k^{2}},\;\;I_{2}=\int\,\frac{e^{ik\tilde{p}}}{k^{2}(k+p)^{2}},$
$\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\int\,\frac{(2k+p)^{\mu}(2k+p)^{\nu}}{k^{2}(k+p)^{2}}e^{ik\tilde{p}},\;\;I_{4}=\int\,\frac{(2k+p)^{\mu}(2k+p)^{\nu}}{k^{2}(k+p)^{2}}.$
(79)

Then we have six with power one:

$I_{5}=\int\,\frac{1}{k^{2}(k\tilde{p})},\;\;I_{6}=\int\,\frac{k^{\mu}}{k^{2}(k\tilde{p})},\;\;I_{7}=\int\,\frac{k^{\mu}k^{\nu}}{k^{2}(k\tilde{p})},$
(80)
$I_{8}=\int\,\frac{e^{ik\tilde{p}}+e^{-ik\tilde{p}}}{k^{2}(k\tilde{p})},\;\;I_{9}=\int\,\frac{k^{\mu}\big{(}e^{ik\tilde{p}}+e^{-ik\tilde{p}}\big{)}}{k^{2}(k\tilde{p})},\;\;I_{10}=\int\,\frac{k^{\mu}k^{\nu}\big{(}e^{ik\tilde{p}}+e^{-ik\tilde{p}}\big{)}}{k^{2}(k\tilde{p})},$
(81)

and finally comes seven with power two:

$I_{11}=\int\,\frac{1}{k^{2}(k\tilde{p})^{2}},\;\;I_{12}=\int\,\frac{k^{\mu}}{k^{2}(k\tilde{p})^{2}},\;\,I_{13}=\int\,\frac{k^{\mu}k^{\nu}}{k^{2}(k\tilde{p})^{2}},$
(82)
$I_{14}=\int\,\frac{1}{k^{2}(k\tilde{p})^{2}}\big{(}e^{ik\tilde{p}}+e^{-ik\tilde{p}}\big{)},\;\;I_{15}=\int\,\frac{k^{\mu}}{k^{2}(k\tilde{p})^{2}}\big{(}e^{ik\tilde{p}}+e^{-ik\tilde{p}}\big{)},$
(83)
$I_{16}=\int\,\frac{k^{\mu}k^{\nu}}{k^{2}(k\tilde{p})^{2}}\big{(}e^{ik\tilde{p}}+e^{-ik\tilde{p}}\big{)},\;\;I_{17}=\int\frac{k^{\mu}k^{\nu}k^{\rho}}{k^{2}(k\tilde{p})^{2}}\big{(}e^{ik\tilde{p}}+e^{-ik\tilde{p}}\big{)}.$
(84)

Using $k\to-k$ trick we can find immediately that:
$I_{5}=I_{7}=I_{10}=I_{12}=I_{15}=I_{17}=0$. We can also transform $k\to-k-p$
and show that

$I_{3}=\int\,\frac{2k^{\mu}k^{\nu}+k^{\mu}p^{\nu}}{k^{2}(k+p)^{2}}(e^{ik\tilde{p}}+e^{-ik\tilde{p}}),\;\;I_{4}=2\int\,\frac{2k^{\mu}k^{\nu}+k^{\mu}p^{\nu}}{k^{2}(k+p)^{2}}.$
(85)

From history we know the $I_{1,2}$ integrals:

$I_{1}=\frac{1}{(4\pi)^{\frac{D}{2}}}\left(\frac{\tilde{p}^{2}}{4}\right)^{1-\frac{D}{2}}\Gamma\left(\frac{D}{2}-1\right),$
(86)
$I_{2}=\frac{2}{(4\pi)^{\frac{D}{2}}}\int\limits_{0}^{1}dx\big{(}x(1-x)p^{2}\big{)}^{\frac{D}{4}-1}\left(\frac{\tilde{p}^{2}}{4}\right)^{1-\frac{D}{4}}K_{\frac{D}{2}-2}\left[\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\right].$
(87)

Integral $I_{3}$ is a standard nonplanar integral, on which we apply the
decomposition

$I_{3}=\eta^{\mu\nu}p^{2}\cdot
I_{3_{1}}+(p^{\mu}p^{\nu}-\eta^{\mu\nu}p^{2})\cdot
I_{3_{2}}+\tilde{p}^{\mu}\tilde{p}^{\nu}\cdot I_{3_{3}},$ (88)

and after integration over parameter $\alpha$ obtain

$\displaystyle I_{3_{1}}=\frac{2}{\tilde{p}^{2}}I_{1},$ $\displaystyle
I_{3_{2}}=\frac{2}{(4\pi)^{\frac{D}{2}}}\int\limits_{0}^{1}dx(1-2x)^{2}\big{(}x(1-x)p^{2}\big{)}^{\frac{D}{4}-1}\left(\frac{\tilde{p}^{2}}{4}\right)^{1-\frac{D}{4}}K_{\frac{D}{2}-2}\left[\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\right],$
$\displaystyle
I_{3_{3}}=-\frac{2}{(4\pi)^{\frac{D}{2}}}\int\limits_{0}^{1}dx\big{(}x(1-x)p^{2}\big{)}^{\frac{D}{4}}\left(\frac{\tilde{p}^{2}}{4}\right)^{-\frac{D}{4}}K_{\frac{D}{2}}\left[\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\right].$
(89)

Here $I_{3_{3}}$ can be re-expressed via $I_{3_{1}}$ and
$K_{\frac{D}{2}-2}\left[\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\right]$ as follows

$\begin{split}&I_{3_{3}}=(2-D)I_{3_{1}}\\\
&+\frac{2}{(4\pi)^{\frac{D}{2}}}\frac{p^{2}}{\tilde{p}^{2}}\int\limits_{0}^{1}dx\big{(}4(D-1)x^{2}-D\big{)}\big{(}x(1-x)p^{2}\big{)}^{\frac{D}{4}-1}\left(\frac{\tilde{p}^{2}}{4}\right)^{1-\frac{D}{4}}K_{\frac{D}{2}-2}\left[\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\right].\end{split}$
(90)

Integral $I_{4}$ is a standard planar integral and we have the standard tensor
reduction result:

$\begin{split}I_{4}=\frac{1}{(4\pi)^{\frac{D}{2}}}\big{(}\eta^{\mu\nu}p^{2}-p^{\mu}p^{\nu}\big{)}(p^{2})^{\frac{D}{2}-2}B\left(\frac{D}{2}-1,\frac{D}{2}-1\right)\frac{\Gamma\left(2-\frac{D}{2}\right)}{1-D}.\end{split}$
(91)

Trivially one can see that
$I_{6}=I_{13}\;\tilde{p}_{\nu},\;\,I_{9}=I_{16}\;\tilde{p}_{\nu}$, and
integrals $I_{13}$ and $I_{16}$ can be solved by an NC tensor reduction. Since
the only relevant momentum in these two integrals is $\tilde{p}^{\mu}$, we can
prescribe the following simple tensor structures:

$\displaystyle
I_{13}=\eta^{\mu\nu}\tilde{p}^{2}A_{13}+\tilde{p}^{\mu}\tilde{p}^{\nu}B_{13},$
$\displaystyle
I_{16}=\eta^{\mu\nu}\tilde{p}^{2}A_{16}+\tilde{p}^{\mu}\tilde{p}^{\nu}B_{16},$
(92)

which after contraction with $\eta_{\mu\nu}$ and
$\tilde{p}_{\mu}\tilde{p}_{\nu}$ gives:

$I_{13}=0=I_{6},\;\,I_{16}=\frac{2}{\tilde{p}^{4}}\frac{\eta^{\mu\nu}\tilde{p}^{2}-\tilde{p}^{\mu}\tilde{p}^{\nu}\cdot
D}{1-D}I_{1}.$ (93)

We are now left with $I_{11}$ and $I_{14}$, and they are not as easy as they
seem to be. Priorly we evaluated them in Martin:2016zon  as the following sum
denoted as $T_{-2}$:

$T_{-2}=2I_{11}-I_{14}.$ (94)

Two methods were used to calculate $T_{-2}$ explicitly, one is based on Grozin
parametrization Grozin:2000cm  and the other contour integral parametrization.
We encounter immediately problem with the second method because the poles at
$x=0$ are second order in $I_{11}$ and $I_{14}$ while first order in $T_{-2}$,
which means that we can not define the principle value around this pole for
$I_{11}$ or $I_{14}$. This leaves only the Grozin parametrization producing:

$\begin{split}I_{11}=&-\int\limits_{0}^{\infty}dy\,y\int\limits_{0}^{\infty}d\alpha\,\alpha^{2}e^{-\alpha(k^{2}+iyk\tilde{p})}=\frac{-1}{(4\pi)^{\frac{D}{2}}}\left(\frac{\tilde{p}^{2}}{4}\right)^{\frac{D}{2}-3}\Gamma\left(3-\frac{D}{2}\right)B\left(D-4,4-D\right)=0,\end{split}$
(95)
$\begin{split}I_{14}=-2\int\limits_{0}^{\infty}dy\,y\int\limits_{0}^{\infty}d\alpha\,\alpha^{2}\int
e^{-\alpha(k^{2}+iyk\tilde{p})+ik\tilde{p}}=\frac{1}{(4\pi)^{\frac{D}{2}}}\left(\frac{\tilde{p}^{2}}{4}\right)^{1-\frac{D}{2}}\frac{\Gamma\left(\frac{D}{2}-2\right)}{3-D}.\end{split}$
(96)

###  B.1 Discussing the UV divergent tadpole integral $I_{14}$

In our recent computation a new type of tadpole integral, which is UV
divergent at the $D\to 4-\epsilon$ limit occurred repeatedly. Here we provide
an account of its evaluation. This new tadpole $I_{14}$, in Martin:2016zon
denoted as $T_{-2}$, bears a very simple form

$I_{14}\equiv-
T_{-2}=\int\frac{d^{D}\ell}{(2\pi)^{D}}\,\frac{f_{\star_{2}}(\ell,p)^{2}}{\ell^{2}}.$
(97)

On the other hand, it turns out that $T_{-2}$ is not quite easy to evaluate.
Two usual regularization methods used before, turning tadpole to bubble or
using the $n$-nested zero regulator respectively, did not function here. The
first one produces divergent special function integrals while the second
contains unfavourable powers of the regulator. The parametrization discussed
in the first section of this note offers us an alternative way to handle this
problem, using this parametrization we can express $T_{-2}$ as

$\begin{split}T_{-2}&=\int\frac{d^{D-1}\ell}{(2\pi)^{D-1}}\,\int\limits_{-\infty}^{+\infty}\,\frac{dx}{2\pi}\frac{1}{\ell^{2}+x^{2}}\frac{4\sin^{2}\frac{|\tilde{p}|}{2}x}{x^{2}\tilde{p}^{2}}=\frac{1}{\tilde{p}^{2}}\int\frac{d^{D-1}\ell}{(2\pi)^{D-1}}\,\left(-\frac{1}{|\ell|^{3}}+\frac{2|\tilde{p}|}{|\ell|^{2}}+\frac{e^{-|\ell||\tilde{p}|}}{|\ell|^{3}}\right).\end{split}$
(98)

Here we can only neglect the second term in the last parenthesis because the
first and last exceed the minimal power of $|\ell|$ for massless tadpole to
vanish in the dimensional regularization prescription. One can introduce one
more integrand $y$ to make he first plus the last terms into one:

$\begin{split}T_{-2}&=\frac{1}{\tilde{p}^{2}}\int\frac{d^{D-1}\ell}{(2\pi)^{D-1}}\,\left(-\frac{1}{|\ell|^{3}}+\frac{e^{-|\ell||\tilde{p}|}}{|\ell|^{3}}\right)=\frac{1}{(4\pi)^{\frac{D}{2}}}\left(\frac{\tilde{p}^{2}}{4}\right)^{1-\frac{D}{2}}\frac{\Gamma\left(\frac{D}{2}-2\right)}{D-3}.\end{split}$
(99)

A familiar pattern emerges once we compute the $D\to 4+2\epsilon$ limit

$T_{-2}=\frac{-4}{(4\pi)^{2}\tilde{p}^{2}}\left(-\frac{1}{\epsilon}+Ln\,\tilde{p}^{2}+Ln(\pi\mu^{2})+\Gamma_{E}+2\right)+\mathcal{O}(\epsilon).$
(100)

Here we see the logarithmic UV/IR mixing taking place via a single integral.

###  B.2 The $D\to 4+2\epsilon$ limit of all evaluated integrals

$\displaystyle I_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi^{2}}\frac{1}{\tilde{p}^{2}}+\mathcal{O}(\epsilon),$
(101) $\displaystyle I_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\int\limits_{0}^{1}dx\,K_{0}\left[\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\right]+\mathcal{O}(\epsilon),$
(102) $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\Big{(}\eta^{\mu\nu}\frac{4}{\tilde{p}^{2}}+\big{(}p^{\mu}p^{\nu}-\eta^{\mu\nu}p^{2}\big{)}\int\limits_{0}^{1}dx(1-2x)^{2}K_{0}\left[\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\right]\Big{)}$
(103) $\displaystyle-$
$\displaystyle\frac{1}{\pi^{2}}\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{\tilde{p}^{4}}+\frac{1}{2\pi^{2}}\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{\tilde{p}^{2}}\int\limits_{0}^{1}dx(3x^{2}-1)p^{2}K_{0}\left[\sqrt{x(1-x)p^{2}\tilde{p}^{2}}\right]+\mathcal{O}(\epsilon),$
$\displaystyle I_{4}$ $\displaystyle=$
$\displaystyle\frac{1}{48\pi^{2}}\big{(}\eta^{\mu\nu}p^{2}-p^{\mu}p^{\nu}\big{)}\left(\frac{1}{\epsilon}+\Gamma_{E}+Ln\frac{p^{2}}{4\pi\mu^{2}}-\frac{8}{3}\right)+\mathcal{O}(\epsilon),$
(104) $\displaystyle I_{5}$ $\displaystyle=$ $\displaystyle
I_{6}=I_{7}=I_{8}=I_{10}=I_{11}=I_{12}=I_{13}=I_{15}=I_{17}=0,$ (105)
$\displaystyle I_{9}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi^{2}}\frac{\tilde{p}^{\mu}}{\tilde{p}^{4}}+\mathcal{O}(\epsilon),$
(106) $\displaystyle I_{14}$ $\displaystyle=$ $\displaystyle-
T_{-2}=\frac{1}{(2\pi)^{2}}\frac{1}{\tilde{p}^{2}}\left(-\frac{1}{\epsilon}+Ln\,\tilde{p}^{2}+Ln(\pi\mu^{2})+\Gamma_{E}+2\right)+\mathcal{O}(\epsilon),$
(107) $\displaystyle I_{16}$ $\displaystyle=$
$\displaystyle-\frac{1}{6\pi^{2}}\frac{1}{\tilde{p}^{6}}\big{(}\eta^{\mu\nu}\tilde{p}^{2}-4\tilde{p}^{\mu}\tilde{p}^{\nu}\big{)}+\mathcal{O}(\epsilon).$
(108)

##  Appendix C Rederiving the noncommutative IR divergence for the two-point
function of U(N) in the fundamental representation.

In this Appendix we shall apply the techniques displayed in Sections 2 and 3
to the U(N) case in the fundamental representation and thus rederive the
result in eq. (61). Actually, the techniques in Sections 2 and 3 have been
previously applied in ref. Martin:2016saw  to show perturbatively the duality
of noncommutative U(N) gauge theory under the Seiberg-Witten map.

Let us assume that $a_{\mu}$, $b_{\mu}$ and $q_{\mu}$ in Section 2 –see eq.
(2), in particular– take values in the Lie algebra of U(N) in the fundamental
representation. Then, $A_{\mu}(b+q)$, $B_{\mu}(b)$ and $Q_{\mu}(b,q)$ –see eq.
(3)– as defined by the Seiberg-Witten map in Appendix A also take values in
the Lie algebra of U(N) in the fundamental representation –notice that this
not so for SU(N). Analogously, let us assume that $c$, $\bar{C}$, $F$ in
section 2 –see eq. (8), in particular– are elements of the Lie algebra of U(N)
in the fundamental representation. Then, $C(b+q,c)$ as defined by the Seiberg-
Witten map in Appendix A is an element of U(N) in the fundamental
representation. It is not difficult to convince oneself one can proceed as in
Sections 2 and 3 –by adapting the formulae in those Sections to the U(N) case
at hand and with the help of the results in ref. Martin:2016saw – and compute
the off-shell two-point contribution to effective action –let us call it
$\Gamma^{\rm U(N)}_{2}[b]$– of the DeWitt effective action. The $\Gamma^{\rm
U(N)}_{2}[b]$ reads

$\begin{array}[]{l}{\Gamma^{\rm
U(N)}_{2}[b]=g^{2}\int\frac{d^{4}p}{(2\pi)^{4}}\,[\rm Tr\,b_{\mu}(p)][{\rm
Tr}\,b_{\nu}(-p)]\frac{2}{\pi^{2}}\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{(\tilde{p}^{2})^{2}}}\\\\[3.0pt]
{\phantom{\Gamma^{(U(N))}_{2}[b]}+\frac{g^{2}}{16\pi^{2}}\int\frac{d^{4}p}{(2\pi)^{4}}\,[{\rm
Tr}\,b_{\mu}(p)][{\rm
Tr}\,b_{\nu}(-p)]\big{[}(p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu})\,\frac{11}{3}\,Ln(p^{2}\tilde{p}^{2})\,+\,\Sigma_{\mu\nu}(p,\tilde{p})\big{]}}\\\\[3.0pt]
{\phantom{\Gamma^{(U(N))}_{2}[b]}+\frac{g^{2}}{16\pi^{2}}N\int\frac{d^{4}p}{(2\pi)^{4}}\,{\rm
Tr}\,\big{[}b_{\mu}(p)\big{(}p^{2}\eta^{\mu\nu}-p^{\mu}p^{\nu}\big{)}b_{\nu}(-p)\big{]}}\\\\[3.0pt]
{\phantom{\Gamma^{(U(N))}_{2}[b]+\frac{g^{2}}{16\pi^{2}}N\int\frac{d^{4}p}{(2\pi)^{4}}}\times\Big{\\{}-\frac{11}{3}\Big{(}\frac{1}{\epsilon}+Ln\frac{p^{2}}{4\pi\mu^{2}}\Big{)}+\frac{67}{144\pi^{2}}+{\cal
O}(D-4)\Big{\\}}}\\\\[3.0pt]
{\phantom{\Gamma^{\rm(U(N))}_{2}[b]}+\;\Gamma^{\rm
U(N)}_{2}[b]^{\rm(eom0)}+\;({\rm 2-loop\;order}),}\end{array}$ (109)

where $\Gamma^{\rm U(N)}_{2}[b]^{\rm(eom0)}$, which obviously vanishes upon
imposing the equation of motion, contains all the contributions on which
integral

$\int d^{4}x\,q^{a}_{\mu}(x)\frac{\delta\Gamma^{\rm U(N)}[b]}{\delta
b^{a}_{\mu}(x)},$

in (4), is involved. The $\Gamma^{\rm U(N)}_{2}[b]^{\rm(eom0)}$ is not
physically relevant for it vanishes on-shell, while
$\Sigma_{\mu\nu}(p,\tilde{p})$ is a function such that it remains finite as
$\tilde{p}$ goes to zero.

It is apparent that if, in $\Gamma^{\rm U(N)}_{2}[b]$ above, we replace
$b_{\mu}$ with $b^{\bot}_{\mu}$ taking values in the Lie algebra of U(N) in
the fundamental representation and satisfying the eqs. in (29), we will obtain
eq. (61) as expected. Hence, in the U(N) in the fundamental representation the
color structure of the noncommutative IR divergence of the one-loop two-point
function is $[{\rm Tr}\,b^{\bot}_{\mu}(p)][{\rm Tr}\,b^{\bot}_{\mu}(-p)]$,
i.e, it is purely U(1), whereas in the SU(N) case is quite different: ${\rm
Tr}\,[b^{\bot}_{\mu}(p)b^{\bot}_{\mu}(-p)]$, i.e. there is only one ${\rm
Tr}$.

The $\Gamma^{\rm U(N)}_{2}[b]$ in eq. (109) is the counterpart of
$\Gamma_{2}[b]$ in eq. (23). The fact that the result in eq. (23) has a
notably different tensor structure from that in eq. (109) has to do with the
fact in the U(N) case the change of integration variables in eq. (15) removes
from the classical action and the gauge-fixing terms all the
$\theta$-dependence coming from the Seiberg-Witten map for $Q_{\mu}(b,q)$ and
$C(b,q)$. This is not so in the SU(N) case, for in this case neither
$Q_{\mu}(b,q)$ nor $C(b,q)$ belong to the Lie algebra of SU(N). On the other
hand, both $Q_{\mu}(b,q)$ and $C(b,q)$ belong to the Lie algebra of U(N) in
the fundamental representation if $b_{\mu}$, $q_{\mu}$ and $c$ take values in
the Lie algebra of U(N) in the fundamental representation. The gauge dependent
contributions in $\Gamma^{\rm U(N)}_{2}[b]$ and $\Gamma_{2}[b]$ are removed by
going on-shell as required by DeWitt’s effective action formalism. This
procedure yields eqs. (61) and (62) for U(N) and SU(N), respectively, with
tensor structures and –color structures– quite different.

###### Acknowledgements.

The work by C.P. Martin has been financially supported in part by the Spanish
MINECO through grant PGC2018-095382-B-I00. J.Trampetic would like thank Dieter
Lüst for many discussions and to acknowledge support of Max-Planck-Institute
for Physics, München. The work of J.You has been supported by Croatian Science
Foundation.

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