Noncommutative N=1 super Yang-Mills, the Seiberg-Witten map and UV divergences

Abstract

Classically, the dual under the Seiberg-Witten map of noncommutative U(N),
{\cal N}=1 super Yang-Mills theory is a field theory with ordinary gauge
symmetry whose fields carry, however, a \theta-deformed nonlinear realisation
of the {\cal N}=1 supersymmetry algebra in four dimensions. For the latter
theory we work out at one-loop and first order in the noncommutative parameter
matrix \theta^{\mu\nu} the UV divergent part of its effective action in the
background-field gauge, and, for N>=2, we show that for finite values of N the
gauge sector fails to be renormalisable; however, in the large N limit the full
theory is renormalisable, in keeping with the expectations raised by the
quantum behaviour of the theory's noncommutative classical dual. We also obtain
--for N>=3, the case with N=2 being trivial-- the UV divergent part of the
effective action of the SU(N) noncommutative theory in the enveloping-algebra
formalism that is obtained from the previous ordinary U(N) theory by removing
the U(1) degrees of freedom. This noncommutative SU(N) theory is also
renormalisable.

Full text

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arXiv:0907.2437v4 [hep-th] 30 Jul 2009
NSF-KITP-09-112
FTI/UCM 90-2009
Noncommutative N = 1 super Yang-Mills, the
Seiberg-Witten map and UV divergences.
C.P. Mart´ ın†, 1and C. Tamarit††, 2
†Departamento de F´ ısica Te´ orica I, Facultad de Ciencias F´ ısicas
Universidad Complutense de Madrid, 28040 Madrid, Spain
††Kavli Institute for Theoretical Physics, University of California
Santa Barbara, CA, 93106-4030, USA
Classically, the dual under the Seiberg-Witten map of noncommutative
U(N), N = 1 super Yang-Mills theory is a field theory with ordinary
gauge symmetry whose fields carry, however, a θ-deformed nonlinear real-
isation of the N = 1 supersymmetry algebra in four dimensions. For the
latter theory we work out at one-loop and first order in the noncommu-
tative parameter matrix θµνthe UV divergent part of its effective action
in the background-field gauge, and, for N ?= 1, we show that for finite
values of N the gauge sector fails to be renormalisable; however, in the
large N limit the full theory is renormalisable, in keeping with the expec-
tations raised by the quantum behaviour of the theory’s noncommutative
classical dual. We also obtain –for N ≥ 3, the case with N = 2 being
trivial– the UV divergent part of the effective action of the SU(N) non-
commutative theory in the enveloping-algebra formalism that is obtained
from the previous ordinary U(N) theory by removing the U(1) degrees
of freedom. This noncommutative SU(N) theory is also renormalisable.
PACS: 11.10.Gh, 11.10.Nx, 11.15.-q, 11.30.Pb.
Keywords:
Renormalization,
commutative geometry.
Regularization and Renormalons, Supersymmetry, Non-
1E-mail:carmelo@elbereth.fis.ucm.es
2E-mail: tamarit@kitp.ucsb.edu

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1 Introduction
Noncommutative gauge theories are known to arise as low energy limits of (super)string theory
[1, 2], and they are interesting on their own as examples of nonlocal theories. One of their
intriguing features is that noncommutative U(N) gauge theories, considered as effective de-
scriptions of the dynamics of D-branes with Neveu-Schwarz backgrounds, are known to have a
dual description in terms of fields with ordinary gauge invariance [1]. This equivalence, which
can be traced back to the possibility of choosing different yet equivalent regularisations of the
D-Brane effective action, can be formulated by means of a map which relates noncommutative
and ordinary gauge fields in a way consistent with their respective gauge symmetries, so that
orbits of noncommutative gauge transformations are mapped into orbits of ordinary gauge
transformations. These maps are called Seiberg-Witten maps. Their role linking different DBI
actions has also been shown to hold, at least to a certain approximation, in the N = 1 super-
symmetric case [3]. In principle, this equivalence holds for the D-Brane effective actions, but
one may wonder whether it also holds, at the quantum level, for the noncommutative gauge
theories that do not involve the higher order terms present in the DBI actions.
The idea of mapping noncommutative to ordinary gauge symmetries was the starting point
for the formulation of noncommutative gauge theories for arbitrary gauge groups by means
of Seiberg-Witten maps pioneered in refs. [4, 5, 6]. In the “standard” formalism, closure
under gauge transformations restricts the gauge groups to be U(N) and the representations
to be (anti-)fundamental or bi-(anti)-fundamental, while the formalism which makes use of
Seiberg-Witten maps, also referred to as the enveloping algebra formalism, makes it possible
to consider arbitrary gauge groups and representations by mapping the enveloping-algebra
valued noncommutative gauge fields to ordinary Lie-algebra valued gauge fields.
The quantum properties of noncommutative gauge theories, both in the standard and
enveloping algebra approaches, have been analysed in many works. Concerning the stan-
dard approach, nonsupersymmetric noncommutative U(N) Yang-Mills theories are plagued by
pathological IR divergences coming from the UV/IR mixing effect [7], which are suppressed
in the large N limit, in which only planar diagrams contribute and the sole effect of noncom-
mutativity is producing phase factors depending on the external momenta which can be taken
out of the loop integrals. Noncommutative supersymmetric gauge theories [8] exhibit a bet-
ter behaviour in the infrared, as the problematic divergences are milder or altogether absent
[9, 10, 11]. These milder noncommutative IR divergences are logarithmic and can be integrated
leading to a consistent renormalisable supersymmetric noncommutative Wess-Zumino [12] and
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most likely to consistent renormalisable, or even UV finite, supersymmetric noncommutative
U(N) theories [13, 14]. A noncommutative extension of the MSSM has been put forward in
ref.[15], which contains more “particle” states than the ordinary MSSM due to the noncom-
mutative anomaly cancellation conditions [16, 17] and other noncommutative requirements.
On the other hand, concerning the theories defined by means of Seiberg-Witten maps,
they are known to have gauge anomaly cancellation conditions identical to their commutative
counterparts [18], and their renormalisability properties have been studied in a wide number
of papers [19, 20, 21, 22, 23, 24, 25, 26, 27]. The results can be summarised as follows:
pure gauge theories, U(1) or SU(N), are one-loop renormalisable at least to first order in
the noncommutativity parameters. The introduction of matter fields in the form of Dirac
fermions or complex scalars in arbitrary representations (but such that the matter Lagrangian
in terms of noncommutative fields does not involve a covariant derivative with a star-product
commutator), does not spoil the renormalisability of the gauge sector of the theory; however,
the full theory seems to be nonrenormalisable in all cases analysed. These cases for which the
renormalisability of the matter sector has been addressed are: Dirac fermions with gauge groups
U(1) [20, 21] or SU(2) in the fundamental representation [22], and U(1) complex scalars [25].
Renormalisability is spoilt by the appearance of divergences in matter field Green functions
which cannot be removed by multiplicative renormalisations or field redefinitions. There is still
no definitive answer concerning whether other types of matter fields or representations could
overcome this problem, despite promising results concerning chiral fermions [27]. Still, the
renormalisability properties of theories with Majorana fermions or/and covariant derivatives
involving a star-product commutator have not been studied. Moreover, supersymmetry could
be expected to make some divergences go away. However, though generally supersymmetry is
associated with a cancellation of divergences between bosonic and fermionic degrees of freedom,
and noncommutative U(N) theories defined by means of Seiberg-Witten maps have been shown
to be compatible with supersymmetry, it turns out that the latter is realised nonlinearly in
the ordinary fields [3], and thus it is not clear how it will affect divergences.
Comparing the quantum properties of noncommutative theories in both the standard and
enveloping algebra approaches raises interesting questions regarding their equivalence for U(N)
gauge groups, for which the Seiberg-Witten map establishes a classical equivalence.
different gauge anomaly cancellation conditions makes this equivalence doubtful in the presence
of chiral fermions, at least when noncommutativity is treated perturbatively. In the case of
theories without matter, the equivalence has been found to hold for noncommutative Chern-
Simons [28] –a theory which is UV finite–, whereas for other gauge theories with or without
The
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matter there is no concluding evidence, since on the side of the enveloping algebra approach
the theories studied have exclusively U(1) and SU(N) gauge groups, while to make contact
with the standard formalism one should consider U(N) in the large N limit, in which the
theories, at least at the one-loop level, are supposed to be well behaved and renormalisable for
infinitesimal noncommutativity.
We have so far identified several issues that needed further investigation. On one hand, the
renormalisability properties, both for the gauge sector and the full theory, of noncommutative
theories defined by means of Seiberg-Witten maps with Majorana fermions and/or involving
a covariant derivative with star-product commutators and/or supersymmetry. On the other
hand, the equivalence at the quantum level of the standard and enveloping algebra approaches
for supersymmetric noncommutative U(N) gauge theories in the large N limit,i.e., the quantum
duality of supersymmetric noncommutative U(N) formulated in terms of noncommutative fields
and the supersymmetric theory, whose fields are ordinary gauge fields carrying a nonlinear
realisation of supersymmetry, obtained from the former by using the Seiberg-Witten map.
The aim of this paper is to address some of the open issues mentioned earlier by analysing
the renormalisability properties of N = 1 U(N) super Yang-Mills in the enveloping alge-
bra approach, with the ordinary fields taking values in the fundamental representation of the
gauge group. First, the theory has a Majorana fermion with a covariant derivative involving a
star-product commutator; supersymmetry is also present for the noncommutative fields, and
it is inherited by the ordinary fields albeit in a nonlinear fashion. Secondly, since we have a
U(N) gauge group in the fundamental representation, the theory can also be formulated in the
standard approach, in which case, in the large N limit, it is renormalisable and well-behaved
for small noncommutativity. We will analyse whether one-loop renormalisability in the back-
ground field gauge is achieved at least for large N. Further, in order to complement previous
research regarding theories with simple gauge groups, we will study the renormalisability prop-
erties of the SU(N) model that results from eliminating the U(1) degrees of freedom in the
U(N) theory, with the goal of seeing whether the modified field content and interactions yield
a better behaviour at the quantum level. To tackle these problems, we will compute the diver-
gent part of the one-loop effective action at first order in the noncommutative parameters θµν,
using the background field method in the background field gauge and dimensional regularisa-
tion, and we will study whether the divergences can be removed by appropriate multiplicative
renormalisations of the parameters of the theory plus nonmultiplicative field redefinitions.
The paper is organised as follows. The model and the background field method are in-
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troduced in section 2. Section 3 is devoted to the computation of the full divergent part of
the one-loop effective action: first, a method is outlined which allows to obtain the full result
by calculating a minimum number of diagrams, whose divergent parts are then computed in
dimensional regularisation; following this the full gauge invariant expression is finally recon-
structed. The renormalisability of the theory, both for arbitrary finite and large N, is studied in
section 4, and then conclusions are drawn in section 5. Two appendices are included, the first
one with some Lie and Dirac algebra identities, and the second one displaying the Feynman
rules employed in the computation.
2 The model and the background field method
The action of the model, in terms of noncommutative fields, is the following,
S =
?
d4x−1
2g2TrFµν⋆Fµν+i
g2Tr¯ΛD /⋆Λ,Fµν= ∂µAν−∂νAµ−i[Aµ,Aν]⋆, D⋆,µ= ∂µ−i[Aµ, ]⋆,
(2.1)
where the fields take values in the enveloping algebra of U(N), Aµ= AA
Λ is a Majorana spinor (see appendix A for conventions). The U(N) fields will be taken in the
fundamental representation. The noncommutative product ⋆ is the usual Moyal product,
µTA, Λ = ΛATAand
a ⋆ b = aexp
?ih
2θµν← −
∂µ− →
∂ν
?
b,
with h setting the noncommutative scale. The model has N = 1 supersymmetry in terms of
the noncommutative fields; it can be formulated in terms of a noncommutative vector superfield
in the Wess-Zumino gauge.
The noncomutative fields are defined in terms of U(N) Lie algebra valued ordinary fields,
which we denote by aµ,l, by means of the following Seiberg-Witten maps,
Aµ= aµ−h
Λ =l −h
4θαβ{aα,∂βaµ+ fβµ} + hSµ+ O(h2),
4θαβ{aα,2Dβl + i[aβ,l]} + hL + O(h2),(2.2)
where Dµ= ∂µ− i[aµ, ], fµν= ∂µaν− ∂νaµ− i[aµ,aν], and Sµ, L represent the ambiguities
in the map at order h, given by sums of terms which involve a contraction with θµν, have the
appropriate mass dimensions and transform in the adjoint representation of the gauge group;
they can be argued to be equivalent to field redefinitions, as will be seen in section 4.
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We will work with the following decomposition of the U(N) fields in the fundamental
representation into their SU(N) and U(1) parts:
aµ= aa
µTa+ bµ
1 I
√2N,
1 I
√2N.
fµν= fa
µνTa+ gµν
1 I
√2N,
l = λaTa+ u
(2.3)
This will allow us to study the properties of both the U(N) theory and the SU(N) theory that
results from suppressing the U(1) degrees of freedom bµ,u.
We will argue in the next section that, for the purpose of checking renormalisability, it suf-
fices to compute the divergent part of the effective action ignoring at tree-level the ambiguities
Sµ,L of the Seiberg-Witten maps in eq. (2.2); the ambiguities, however, have to be taken into
account when considering the allowed counterterms. The action in terms of ordinary fields,
after expanding (2.1) with eqs. (2.2) with Sµ= L = 0, turns out to be the following
S =S(0)+ hS(1)+ O(h2),
1
2g2
S(1)=1
4g2
−i
2
S(0)= −
?
d4xTrθαβfµνfµνfαβ−1
d4xTrfµνfµν+
i
g2
?
d4xTr¯lD /l,(2.4)
?
g2
?
d4xTrθαβfαµfβνfαβ−i
4
?
d4xTrθαβ¯lγµ{Dµl,fαβ}
?
d4xTrθαβ¯lγµ{Dβl,fµα}.
In the previous action,
TrTA{TB,TC} =
bra, for arbitrary representations, has dabc= 0, which means that the SU(N) theory obtained
by eliminating the U(1) degrees of freedom is, to order h, equivalent to its commutative limit.
Therefore, when studying the SU(N) theory we will only consider N ≥ 3. As shown in ref.
[3] (see also [29]) the fields in the action in eq.(2.4) carry a nonlinear realisation of N = 1
supersymmetry which define supersymmetry transformations that leave that action invariant.
all the noncommutative terms involve traces of the type
1
2dABC(see appendix A). For N < 3, the SU(N) part of the Lie alge-
In the enveloping algebra approach, quantisation is performed on the ordinary fields. In
order to compute the effective action with the background field method [30], we split the gauge
field aµ in a background part bµ and a quantum part qµ,
aµ= bµ+ qµ. (2.5)
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A gauge transformation of aµ, δaµ= Dµc, can be generated by two types of transformations
of the fields b,q:
Quantum gauge transformations: δqµ= D[q]µc, δbµ= −i[bµ,c], D[q]µ= ∂µ− i[qµ, ],
Background gauge transformations: δqµ= −i[qµ,c], δbµ= D[b]µc, D[b]µ= ∂µ− i[bµ, ]. (2.7)
(2.6)
In order to quantise q with the path integral formalism, a gauge fixing procedure is needed for
the transformations in eq. (2.6). The background field method relies in a clever choice of the
gauge-fixing function which is covariant under the transformations (2.7). With the gauge-fixing
choice G = D[b]
µ qµ= 0, the gauge-fixing and ghost action are the following
Sgf= −1
2α
?
d4x(D[b]
µqµ)2,Sgh=
?
d4x¯ cD[b]
µD[b+q]µc.(2.8)
Quantising the fields qµ,l,¯l, the generating functional of the background Green functions is
given by
˜Z[˜J, ˜ σ,˜ ¯ σ;b] =
?
[dq][dl][d¯l]exp[i(S[b + q,l,¯l] + Sgf[q;b] + Sgh[c,¯ c,q;b] +˜Jµqµ+ ˜ σl +¯l˜ ¯ σ)],
(2.9)
where ˜J, ˜ σ,˜ ¯ σ are sources for the gauge field and Majorana fermions. Note the use of “˜” to
distinguish the background currents and functional generator ˜Z from the ones defining the
true Green functions of the theory, when the splitting of eq. (2.5) is not used and functional
integration is performed over a. The generator of connected background Green functions is
given by
˜ W[˜J, ˜ σ,˜ ¯ σ;b] = −iln˜Z[˜J, ˜ σ,˜ ¯ σ;b].
Defining the background classical fields as
˜ q =δ˜ W
δ˜J
,
˜l =δ˜ W
δ˜ σ,
˜ ¯l = −δ˜ W
δ˜ ¯ σ,
then by performing a Legendre transformation we get the functional˜Γ which generates the
1PI connected background Green functions:
˜Γ[˜ q,˜l,˜ ¯l;b] =˜ W[J, ˜ σ,˜ ¯ σ;b] −
?
d4x˜Jµ˜ qµ−
?
d4x ˜ σ˜l −
?
d4x˜ ¯l˜ ¯ σ.(2.10)
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In a similar fashion, without using the splitting of eq. (2.5), one can define the true Green
function generators Z[J,σ, ¯ σ] and W[J,σ, ¯ σ] as well as the true classical fields ˆ a,ˆl,ˆ ¯l. Stan-
dard formal manipulations show that the effective action of the theory Γ[ˆ a,ˆl,ˆ ¯l] is related to
˜Γ[˜ q,˜l,˜ ¯l;b] of eq. (2.10) by the following identity [30]:
Γ[ˆ a,ˆl,ˆ ¯l] =˜Γ[0,˜l,˜ ¯l;b]|b=ˆ a,˜l=ˆl,˜ ¯l=ˆ ¯l, (2.11)
where Γ is computed with an unusual gauge-fixing. From the r.h.s. of eq. (2.11) it is clear
that the effective action is obtained by calculating the background effective action for the
Majorana fields after integrating out the quantum fields q, with the background fields bµ
taken as external sources. We thus can write
Γ[ˆ a,ˆl,ˆ ¯l] =
?
d4x
?
k
−i
2k(k!)2˜Γ[ˆ a](k)
i1,..,ik,
A1,..,Ak,
j1,..,jk,
B1,..,Bk
k?
l=1
ˆ ¯lAl
il
k?
p=1
ˆlBp
jp,
where the factor (k!2) takes into account the permutations of the l′s and¯l′s, while the factor
2kcomes from the fact that, since the Majorana fermions are self-conjugate, it is always
possible to interchange one l with an¯l. ˜Γ[ˆ a](k)is nothing but the sum of background 1PI
diagrams with k fermionic legs, k anti-fermionic legs and no quantum gauge field legs, and
with the background field b renamed as ˆ a. Expanding˜Γ[ˆ a](k)in the number of background
gauge fields, one gets
Γ[ˆ a,ˆl,ˆ ¯l] =
?
d4x
?
k
?
n
−i
2k(k!)2˜Γ(n,k)
i1,..,ik,
A1,..,Ak,
j1,..,jk,
B1,..,Bk
µ1,..,µn
C1,..,Cn
k?
l=1
ˆ ¯lAl
il
k?
p=1
ˆlBp
jp
n
?
m=1
ˆ aCm
µm. (2.12)
In the previous formula˜Γ(n,k)is equivalent to a background 1PI diagram with n background
gauge field legs, k fermionic legs and k anti-fermionic legs. Note that our definitions do not
involve any symmetrisation over the background gauge fields. Symmetrising over them we can
make contact with the usual expansion of the effective action in terms of 1PI Green functions:
Γ[ˆ a,ˆl,ˆ ¯l] =
?
d4x
?
k
?
n
−i
n!2k(k!)2Γ(n,k)
i1,..,ik,
A1,..,Ak,
j1,..,jk,
B1,..,Bk
µ1,..,µn
C1,..,Cn
k?
l=1
ˆ ¯lAl
il
k?
p=1
ˆlBp
jp
n
?
m=1
ˆ aCm
µm,
where Γ(n,k), which is obtained from˜Γ(n,k)by summing over the permutations of the back-
ground gauge fields, is the 1PI Green function with n gauge fields and k fermion pairs.
The advantage of using background diagrams coming from the functional generator in
eq. (2.9) is that˜Γ[0,˜l,˜ ¯l;b] is gauge invariant, so that the effective action Γ[ˆ a,ˆl,ˆ ¯l] is indeed
gauge invariant. As explained in the next section, this can be used to simplify the computation
of the divergent part of the effective action.
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3 Computation of the divergent part of the effective action
The aim of this section is to compute the divergent part of the effective action at first order in
hθ, by calculating the background 1PI diagrams˜Γ(n,k)with no external quantum gauge fields
of eq. (2.12) using the Feynman rules associated with the functional generator in eq. (2.9).
These rules can be derived from the expressions for the action, gauge fixing and ghost terms
given in eqs. (2.4), (2.8), keeping in mind the splitting (2.5).
Before plunging into the computation, we will justify a number of simplifications that do not
imply a loss of generality on the final result concerning the regularisation and renormalisation
of the theory.
• We shall carry out our computations in dimensional regularisation with D = 4−2ǫ –it is
always advisable to keep an eye on dimensional reduction. That this regularisation does
not preserve supersymmetry will have no bearing on our conclusions since our compu-
tations are one-loop and the inclusion of the ǫ-scalars of dimensional reduction to turn
our dimensionally regularised theory into a theory regularised by dimensional reduction
–and thus supersymmetric– will not modify the value of UV divergences that we will
compute, but will add new ones which would be subtracted by introducing counterterms
made out of “evanescent” operators and couplings –see ref.[31, 32] for further details.
• Choice of gauge α = 1 in the gauge-fixing term in eq.(2.8). This choice of gauge simplifies
the gauge field propagator.This brings up the question of whether, if problematic
divergences appear for α = 1 that make the theory nonrenormalisable, the consideration
of an arbitrary α might help remove these divergences. The answer is negative whenever
any of the problematic divergences appearing at α = 1 do not go away on the mass shell.
This is due to the results in ref.[33] (see also [34]) which establish that the background
field effective action is independent of the gauge-fixing term if the background fields
are on shell. Thus, when the background fields are on shell any divergent contribution
remaining will be independent of any gauge-fixing term that we chose.
• Setting to zero the tree-level ambiguities Sµ,L of the Seiberg-Witten map of eq. (2.2).
This choice simplifies greatly the computation of the diagrams, though when studying
renormalisability one can still contemplate infinite renormalisations of Sµ,L, which tan-
tamounts to consider the most general field redefinitions that cannot be reabsorbed by
gauge transformations, as will be explained in section 4. Again, one may still object that
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considering arbitrary Sµ,L at tree level might be of use to cancel possible pathological
divergences (i.e., that cannot be removed by field redefinitions or multiplicative renormal-
isation) appearing for Stree
µ
= Ltree= 0. This possibility is precluded by the arguments
presented in ref. [35], proven there for a specific model but expected to have general
validity. In this reference the authors claim that, given a theory which is multiplicatively
renormalisable, then by quantising the theory after performing a field redefinition, the
divergences in terms of the new fields can be reabsorbed by the same multiplicative renor-
malisations of physical parameters as in the original case, plus infinite field redefinitions.
In our case, we worry about possible divergences at order hθ for Stree
cannot be removed by infinite field redefinitions. The theory at order h0is known to
be multiplicatively renormalisable, and considering arbitrary Stree
performing finite field redefinitions of order h on the ordinary fields aµ,l,¯l. Thus, the
additional divergences dependent on Stree
to infinite field redefinitions and therefore by assumption would not be useful to cancel
the original problematic divergences at Stree
µ
about the renormalisability of the theory obtained for Stree
validity.
µ
= Ltree= 0 which
µ ,Ltreeis equivalent to
µ ,Ltreethat might appear would be equivalent
= Ltree= 0. It follows that the conclusions
= Ltree= 0 have a general
µ
• Computing a minimum number of diagrams. The use of the background field method
guarantees that the result for the effective action will be gauge invariant. Furthermore,
its divergent part computed in dimensional regularisation will be local. Thus, if one
chooses a basis of all possible local gauge invariant terms up to order h, the divergent
part of the effective action will be a linear combination of these terms. The coefficients in
this linear combination can be determined by identifying its contributions with any given
number and types of fields with the poles in the dimensional regularisation parameter ǫ
of the corresponding 1PI Green functions with the same number and types of external
fields. By appropriately choosing the basis, it can be guaranteed that the contributions
to its elements with a minimum number of fields are also independent of each other,
so that the unknown coefficients in the expansion of the divergent part of the effective
action in terms of the basis can be determined from the diagrams with lowest number of
fields.
We have thus argued that we can determine unambiguously the renormalisability of the
theory by computing the effective action for α = 1, Stree
tions, the Feynman rules relevant to our computations are those given in appendix B; they use
µ
= Ltree= 0. Under these assump-
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a compact notation for the Lie algebra indices, following ref. [36, 37], in which the U(N) field
expansion in the Lie algebra generators in the fundamental representation is taken as
aµ= aµATA,
where TA= {T0,Ta}, with T0=
generators; more details are given in appendix A. This allows to compute simultaneously
diagrams involving both SU(N) and U(1) fields, and the results for the SU(N) theory can also
be easily obtained by setting the external “A” indices to SU(N) indices “a”, and by taking
care to drop the contributions of U(1) indices in terms involving contractions of internal U(N)
Lie algebra indices “A”.
1 I
√2N
the U(1) generator and Tadenoting the SU(N)
Let us start by identifying the diagrams that need to be computed by constructing the
appropriate basis of local gauge invariant terms whose integrals are independent. We use
the decomposition in eq. (2.3). Local gauge invariant terms are then constructed from traced
products of the field strengths and fermion fields and their covariant derivatives; we can classify
them in three sectors: SU(N) sector -only including fields in the Lie algebra of SU(N)- U(1)
sector, and mixed sector. A list follows:
SU(N) sector:
t1= θαβTrfαβfµνfµν,
t3= θαβTr¯λγαD2Dβλ,
t5= θαβTr¯λγµ{fµβ,Dαλ},
t7= θαβTr¯λγα{fβµ,Dµλ},
t9= θαβTr¯λγαρσ{Dβfρσ,λ},
t11= θαβTr¯λγα[Dµfβµ,λ],
t13= θαβTr¯λγαρσ[fρσ,Dβλ],
t15= θαβTr¯λi(γα)ij[{¯λk,(γβλ)k},λj],
t2= θαβTrfαµfβνfµν,
t4= θαβTr¯λγαβµD2Dµλ,
t6= θαβTr¯λγµ{fαβ,Dµλ},
t8= θαβTr¯λγαβµ{Dνfµν,λ},
t10= θαβTr¯λγµ[Dµfαβ,λ],
t12= θαβTr¯λγαβµ[fµν,Dνλ],
t14= θαβTr¯λγαρσ[fβσ,Dρλ],
t16= θαβTr¯λi(γµ)ij[[¯λk,(γµαβλ)k],λj].
(3.1)
U(1) sector:
u1= θαβgαβgµνgµν,
u4= θαβ¯ uγαβµ∂2∂µu,
u7= θαβ¯ uγα∂µugβµ,
u2= θαβgαµgβνgµν,
u5= θαβ¯ uγµ∂αugµβ,
u8= θαβ¯ uγαβµu∂νgµν,
u3= θαβ¯ uγα∂2∂βu,
u6= θαβ¯ uγµ∂µugαβ,
u9= θαβ¯ uγαρσu∂βgρσ.
(3.2)
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Mixed sector:
v1= θαβTrgαβfµνfµν,
v4= θαβTrgµνfαβfµν,
v7= θαβTr¯ uγµDµfαβλ,
v10= θαβTr¯ uγαβµDνfµνλ,
v13= θαβTr¯ uγαρσfρσDβλ,
v16= θαβTr¯λγµDµλgαβ,
v19= θαβTr¯λγαρσλ∂βgρσ.
v2= θαβTrgµνfαµfβν,
v5= θaβTr¯ uγµfµβDαλ,
v8= θαβTr¯ uγαfβµDµλ,
v11= θαβTr¯ uγαβµfµνDνλ,
v14= θαβTr¯ uγαρσfβσDρλ,
v17= θαβTr¯λγαDµλgβµ,
v3= θαβTrgαµfβνfµν,
v6= θαβTr¯ uγµfαβDµλ,
v9= θαβTr¯ uγαDµfβµλ,
v12= θαβTr¯ uγαρσDβfρσλ,
v15= θαβTr¯λγµDαλgµβ,
v18= θαβTr¯λγαβµλ∂νgµν,
(3.3)
In the formulae above, “Tr” denotes the trace over the SU(N) generators. The list of terms
spans modulo total derivatives all the possible gauge invariant terms of order hθµνwith the
appropriate dimensions with zero or two Majorana fields. Again, the Majorana properties (A.3)
and (A.4) have been used, so that any term with two Majorana fermions not present above
can be expressed as a linear combination of the ti,ui and vi, again modulo total derivatives.
In the case of terms with four Majorana fermions, t15 and t16 do not span all the allowed
contributions, but the missing ones will play no role in our calculations and we will safely
ignore them.
The contributions to the previous list of terms with a minimum number of fields are inde-
pendent of each other, which, as explained before, allows to fix the coefficients of the expansion
of the divergent part of the effective action, Γdiv, in terms of the ti,ui,vi by computing only
the 1PI diagrams with the least possible number of fields. Let us identify the diagrams that
need to be computed, using the notation in eq. (2.12) for the 1PI background Green functions.
At order h0, the possible gauge invariant terms are Trfµνfµνand Tr¯λD /λ. Thus, using the
notation of eq. (2.12) only the diagrams contributing to˜Γ(2,0)-with two external background
gauge field legs- and˜Γ(0,1)-with two external quantum fermionic legs- need to be computed.
At order h, we have, schematically, the following types of terms:
• Terms of the type Trθfff,Trθgff,θggg, which are spanned by t1,t2, u1,u2 and v1−v4
in eqs. (3.1) ,(3.2) and (3.3), whose contributions with three gauge fields are independent.
Thus it suffices to compute diagrams with three external gauge fields, contributing to
˜Γ(3,0).
• Terms of the type Trθ¯λD3λ,θ¯ u∂3u, which are spanned by t3,t4 and u3,u4 in eqs. (3.1)
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and (3.2). They involve at least two fermionic fields, so that their coefficients in the
expansion of Γdivcan be fixed by computing˜Γ(0,1), which arises from diagrams with
two fermionic legs.
• Terms of the type -neglecting ordering- Trθ¯λDfλ,Trθ¯λfDλ,θ¯ u∂gu,θ¯ ug∂u,Tr¯ uDfλ,
Tr¯ ufDλ, Tr¯λDλg,Tr¯λλ∂g, which are spanned by t5−t14,u5−u9,v5−v19 in eqs. (3.1),
(3.2) and (3.3). Their contributions with one gauge field and two Majorana fields are
again independent, so that it suffices to compute the diagrams contributing to˜Γ(1,1),
i.e., with one background gauge field leg and two quantum fermionic legs.
• Terms of the type Trθ¯λλ¯λλ, such as t15,t16 in eq. (3.1). Though t15,t16 do not span
all possibilities, it is clear that the computation of˜Γ(0,2)(diagrams with four external
fermionic legs) will completely determine the corresponding contribution to the effective
action Γ.
Summarising, at order h the only diagrams that have to be computed are those contributing to
the 1PI Green functions˜Γ(3,0),˜Γ(0,1),˜Γ(1,1)and˜Γ(0,2). We proceed in the next sections, using
dimensional regularisation at D = 4 − 2ǫ dimensions, with the Feynman rules displayed in
appendix B. The calculations are quite involved and were done with the symbolic manipulation
software Mathematica.
3.1Commutative limit
Here we quote the known commutative result for the dimensionally regularised divergent part
of the effective action:
?
For simplicity, we suppressed the “ˆ” symbols with which we denoted the classical fields in
section 2; we will keep doing so in the rest of the paper. Note that the divergent part only
involves the SU(N) fields a,λ, since the U(1) sector is free in the commutative limit. In fact,
since the U(1) sector is free, in the SU(N) case the result is identical,
Γord,div
[U(N)]=dDx −3g2N
16π2ǫTr
?
−
1
2g2fµνfµν] +
?
dDx
N
16π2ǫ[iTr¯λD /λ]. (3.4)
Γord,div
[SU(N)]= Γord,div
[U(N)]. (3.5)
3.2 Noncommutative contributions to˜Γ(3,0)
The diagrams that contribute are shown in Fig. 1. Note that, though we did not provide in
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Figure 1: Diagrams contributing to˜Γ(3,0)at order h.
appendix B the Feynman rule for the vertex appearing in the first diagram, this diagram is
directly zero since it involves an integral of the type
?
dDl
?
ilµi
(l2)k, (3.6)
which vanishes in dimensional regularisation.
The results for the diagrams are too lengthy to be displayed here individually. We will
quote the final expression for the contribution to the divergent part of the effective action in
position space:
i˜Γ(3,0),NC,div
[U(N)]
µ1,µ2,µ3
A1,A2,A3
aA1
µ1aA2
µ2aA3
µ3=3g2Nh
16π2ǫ
?1
4g2t1−1
g2t2
????
aaa
(3.7)
+2g2Nh
16π2ǫ
?
1
4g2√2N(v1+ 2v4) −
1
g2√2N(v2+ 2v3)
????
baa+ O(h2),
where “|aaa” and |baa” denote the contributions with lowest number of fields, i.e., three SU(N)
gauge fields and one U(1) and two SU(N) gauge fields, respectively. Recall that the ti,ui,vi
are the gauge invariant terms defined in eqs. (3.1), (3.2) and (3.3). To get the SU(N) result,
the external Lie algebra indices of the diagrams have to be set to SU(N) indices, and any U(1)
contributions to internal contractions have to be eliminated. It turns out that all diagrams
involve contractions of the type appearing in eq. (A.1) of appendix A, which, when setting the
uncontracted indices to SU(N) indices, do not involve any contributions from internal U(1)
indices. This is equivalent to saying that the U(1) fields do not run in the loops when the
external fields are the aa
µ. From this we conclude that the SU(N) result is obtained from
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eq. (3.7) by simply setting to zero the U(1) fields:
i˜Γ(3,0),NC,div
[SU(N)]
µ1,µ2,µ3
a1,a2,a3
aa1
µ1aa2
µ2aa3
µ3=3g2Nh
16π2ǫ
?1
4g2t1−1
g2t2
????
aaa+ O(h2). (3.8)
3.3Noncommutative contributions to˜Γ(0,1)
The diagrams contributing to the˜Γ(0,1)Green function at order θ are shown in Fig. 2. The
Figure 2: Diagrams contributing to˜Γ(0,1)at order h.
first diagram is zero as it involves again an integral of the type shown in eq. (3.6). For external
colour indices A, B, it is easily seen that the rest of the diagrams are zero since they are
proportional to either fACDdBCD= 0 or fBCDdACD= 0. To get the SU(N) result one has
to set the external indices to a, b and drop any U(1) contributions in the contractions of
the internal indices. However, since fbCDdaCD= fbcddacd, no U(1) contributions must be
eliminated, and the same argument as before applies. Therefore,
˜Γ(0,1),NC,div
[U(N)]
=˜Γ(0,1),NC,div
[SU(N)]
= O(h2).(3.9)
3.4Noncommutative contributions to˜Γ(1,1)
The diagrams contributing to the˜Γ(1,1)Green function at order θ are shown in Fig. 3. Again,
we will write down the final result of the lengthy computation:
i
2
˜Γ(1,1),NC,div
[U(N)]
i,j,µ
A,B,C
¯lA
ilB
jaC
µ= −
iNh
16π2ǫ
?1
?
?
4t6−1
2t7−1
8t8−1
16t9
????
a¯λλ
+
iNh
16π2ǫ
iNh
16π2ǫ
1
√2N
1
√2N
??
??
v5−3
− v15+1
2v6+ 2v8−1
2v16+3
4v10−1
4v18+3
2v12
????
a¯ uλ
(3.10)
+
4v19
????
b¯λλ+ O(h2).
15