1051

Nonplanar graphs and anomaliesin chiral noncommutative gaugetheories

Marie Gagne-Portelance and D.G.C. McKeon

Abstract: The AV (n) one-loop graphs are examined in a 2n-dimensional massless noncom-mutative gauge model in which both a U(1) axial gauge field A and a U(1) vector gaugefield V have adjoint couplings to a Fermion field. A possible anomaly in the divergence ofthe n + 1 vertices is examined by considering the surface term that can possibly arise whenshifting the loop momentum variable of integration. It is shown that despite the fact that thegraphs are nonplanar, surface terms do arise in individual graphs, but that in 4n dimensions,a cancellation between the surface term contribution coming from pairs of graphs eliminatesall anomalies, while in 4n + 2 dimensions such a cancellation cannot occur and an anomalynecessarily arises.

PACS No.: 11.30.Rd

Résumé : Nous étudions les graphes AV (n) à une boucle dans un modèle de jauge sans massenon commutant à 2n dimensions dans lequel et le champ de jauge U (1) axial A et le champde jauge U (1) vecteur V ont un couplage adjoint à un champ de fermion. Nous examinonsune anomalie possible dans la divergence des n + 1 vertex en considérant les termes desurface qui peuvent survenir en déplaçant la variable d’intégration du moment de boucle.Nous montrons que des termes de surface peuvent survenir, même si les graphes sont nonplans, sauf en 4n dimensions où une annulation entre les contributions venant de paires degraphes annihile toute anomalie, alors qu’en 4n + 2 dimensions une telle annulation ne peutse produire, générant nécessairement une anomalie.

[Traduit par la Rédaction]

1. Introduction

The anomaly in the divergence of the chiral current has played a significant role in conventionalquantum field theories [1–6]. The possibility of having anomalies in models in noncommutative spacehas been extensively discussed [7–17]. In this paper, we examine anomalies in 2n-dimensional non-commutative space using Feynman diagrams, the approach employed originally in refs. 2 and 3.

We consider so-called “adjoint” couplings in which a Fermion couples to gauge fields through aMoyal commutator. Regulating naively divergent diagrams through dimensional regularization [18] is

Received 25 April 2005. Accepted 28 July 2005. Published on the NRC Research Press Web site athttp://cjp.nrc.ca/ on 30 September 2005.

M. Gagne-Portelance and D.G.C. McKeon.1 Department of Applied Mathematics, University of WesternOntario, London, ON N6A 5B7, Canada.

1 Corresponding author (e-mail: dgmckeo2@uwo.ca).

Can. J. Phys. 83: 1051–1061 (2005) doi: 10.1139/P05-050 © 2005 NRC Canada

1052 Can. J. Phys. Vol. 83, 2005

problematic due to difficulties associated with defining the totally antisymmetric tensor outside of aninteger number of dimensions. (Indeed, these diagrams are in fact finite [19].) Anomalies can be bestanalyzed by considering surface terms arising when shifting the variable of integration [2, 3, 20]; in factsurface terms can themselves be used as regulating parameters [21].

We first consider how surface terms can arise when shifting the variable of integration in linearlydivergent integrals, and how these surface terms vanish when these integrals are modified by oscillatoryfunctions that arise in noncommutative models. Next, the nature of surface terms occurring in thedivergence of one-loop AV (n)(n + 1)-point diagrams is considered. We show how in 4n-dimensionalmodels, surface terms can be chosen to ensure that there is no anomaly, while in 4n + 2 dimensions,this is no longer the case. This is consistent with the work in ref. 16 where a functional method has beenused to derive this result.

2. Surface terms

In noncommutative models, one has a nonvanishing commutation relation between xµ and xν ; thisresults in oscillatory functions being present in the Feynman rules associated with vertices [22, 23]. Weare thus led to consider the possibility of surface terms arising from integrals of the form

Iµ =∫

d2nk

(2π)2n

[ei(k+p)·q̃ (k + p)µ

[(k + p)2 +m2]n − eik·q̃ kµ

[k2 +m2]n]

(1)

where q̃µ = θµνqν with θµν = −θνµ being the antisymmetric tensor arising because of the noncom-mutative nature of space-time. Equation (1) is taken to be in Euclidean space. In the first instance, wewill set q = p so that upon using the identity

a−n − b−n = −n(a − b)

∫ 1

0dx [xa + (1 − x)b]−n−1 (2)

then (1) reduces to

Iµ =∫

d2nk

(2π)2

∫ 1

0dx eik·p̃

[nkµ(−2k · p − p2)[

(k + xp)2 + x(1 − x)p2 +m2]n+1 + pµ[

(k + p)2 +m2]n]

(3)

Each integral in (3) is now at most logarithmicly divergent and hence a shift of integration variable doesnot result in a surface term, so that

Iµ =∫

d2nk

(2π)2n

∫ 1

0dx eik·p̃

[n(k − xp)µ(−2(k − xp) · p − p2)[

k2 + x(1 − x)p2 +m2]n+1 + pµ[

k2 +m2]n]

(4)

The denominators in (4) can now be exponentiated, leading to

Iµ = 1

�(n)

∫d2nk

(2π)2n

∫ 1

0dx∫ ∞

0dt tn eik·p̃−(k2+m2)t

×[(k − xp)µ(−2(k − xp) · p − p2) e−x(1−x)p2t + pµ

t

](5)

Completing the square in the argument of the exponential, shifting the variable of integration k, andperforming symmetric integration reduces Iµ to

Iµ = 1

�(n)

∫d2nk

(2π)2n

∫ 1

0dx∫ ∞

0dt tn e−(k2+m2)t e−p̃2/4t

×[(

−2k2

npµ +

(ip̃

2t− xp

)µ(2x − 1)p2

)e−x(1−x)p2t + pµ

t

](6)

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Gagne-Portelance and McKeon 1053

An integration by parts shows that

∫ 1

0dx x(1 − 2x)e−x(1−x)p2t = 1

p2t

(−it

∫ 1

0e−x(1−x)p2t

)

so that (6) reduces to

Iµ = pµ

�(n)

∫d2nk

(2π)2n

∫ 1

0dx∫ ∞

0dt tn e−(k2+m2)t e−p̃2/4t

[−k

2

n+ 1

t

]e−x(1−x)p2t (7)

If we were to first integrate over k in (7) at this stage, we would formally obtain Iµ = 0 even if p̃µ = 0.This would not be correct though, as the integral over t has served to regulate the integral over k andhence the linear divergence in the k integration that is responsible for surface term contributions [2] hasbeen lost.

Integrating by parts we find that

∫ ∞

0dt tnk2 exp

[−(k2 +�2)t − p̃2

4t

]=∫ ∞

0d(−e−k2t

)tn exp

[−(�2t + p̃2

4t

)]

=∫ ∞

0dt

[ntn−1 + tn

(−�2 + p̃2

4t2

)]exp

[−(k2 +�2)t − p̃2

4t

](8)

so that (7) becomes

Iµ = pµ

�(n+ 1)

∫d2nk

(2π)2n

∫ 1

0dx∫ ∞

0dt tn

(�2 − p̃2

4t2

)exp

[−(k2 +�2)t − p̃2

4t

](9)

provided �2 = x(1 − x)p2 +m2.It is now possible to integrate over kµ, so that

Iµ = pµ

�(n+ 1)

1

(4π)n

∫ 1

0dx∫ ∞

0dt

(− d

dt

)[(x(1 − x)p2 +m2

)t + p2

4t

]

× exp

[−(x(1 − x)p2 +m2)t − p̃2

4t

](10)

= −pµ(4π)n�(n+ 1)

exp

{−[x(1 − x)p2 +m2]t − p̃2

4t

∣∣t=+∞t=0+

}(11)

If p̃2 �= 0, it is apparent from (11) that Iµ = 0; otherwise

Iµ = pµ

(4π)n�(n+ 1)(12)

We have not been able to generalize (2) to accommodate (1) when q̃ �= p̃. However, a more heuristicargument can be used to demonstrate that even in this case, Iµ = 0.

Following refs. 5 and 24, we expand the first term of (1) in powers of pµ, keeping only the first termas it alone can possibly give a nonvanishing contribution when integrating over kµ. This results in

Iµ =∫

d2nk

(2π)2n∂

∂kλ

[pλ eik·q̃ kµ

(k2 +m2)n

](13)

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1054 Can. J. Phys. Vol. 83, 2005

which, by the divergence theorem, becomes

= limk2→∞

k2n−1

(2π)2n

∫d�2n

k̂ · p kµ eik·q̃

(k2 +m2)n(14)

If q̃ �= 0, the angular integral vanishes due to the oscillatory nature of the exponential, while if q̃ = 0,then we obtain upon using symmetric integration

Iµ = 1

(2π)2n

∫d�2n

(pµ

2n

)(15)

which is consistent with (12).

3. Potentially anomalous diagrams

We consider a model in noncommutative 2n-dimensional Euclidean space in which a U(1) vectorfield Vµ and a U(1) axial vector field Aµ couples to a spinor ψ through an “adjoint” coupling in whichthe covariant derivative is

Dµψ = ∂µψ − ie(Aµ ∗ γ2n+1ψ − γ2n+1ψ ∗ Aµ)− ig

(V µ ∗ ψ − ψ ∗ V µ) (16)

with “*” denoting the usual Moyal product.The potentially anomalous diagrams we analyze are one-loop diagrams in which a Fermion loop

is coupled to one external Aµ and n external V µ. In computing these diagrams, we use the Fermionpropagator

p · γp2 ≡ 1

p

and the vertices

2iγ µ sin(p × q)

2, 2iγ µγn+1 sin

(p×q2

)for V µ −ψ −ψ and Aµ −ψ −ψ , respectively. We have scaled e and g to one, p is the momentum ofthe incoming Fermion, q the momentum of the outoing Boson, and p × q ≡ pµθµνqν .

In two dimensions (n = 1) there is one Feyman diagram to consider; it leads to the Feynman integral

Iλα(p) = −T r∫

d2k

(2π)2(2i)2 sin

[(k′ − p)× p

2

]sin

(k′ − p

2

)(γ λγ3

1

k′ γα 1

k′ − p

)(17)

where k′ = k + s with s being an arbitrary momentum. If we now consider the divergence in the axialvertex, we find that

pλIλα = +2T r∫

d2k

(2π)2[1 − cos(k′ × p)]

(γ3γ

α 1

k′ − γ3γα 1

k′ − p

)(18)

A shift in the variable of integration in the term in (18) that is proportional to cos(k′ × p) does not leadto a surface term and hence it vanishes. Using the formula

T r(γ α1 . . . γ α2nγ2n+1

) = 2nεα1...α2n (19)

with n = 1 then reduces (18) to

pλIλα = 4∫

d2k

(2π)2εακ

(k′

k′2 − (k′ − p)

(k′ − p)2

)κ(20)

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Gagne-Portelance and McKeon 1055

which by (12) becomes

pλIλα = εακpκ

π(21)

which is consistent with the presence of an axial anomaly in two dimensions [25]. It would appear thatan anomaly would also arise in the divergence of the vector vertex; however, a finite counter term canbe used to remove this anomaly [26].

The presence of two sine functions in (17) leads to having a surface-term contribution to (18). Suchsurface-term contributions can occur in 4n+ 2 dimensions but not in 4n dimensions when consideringthe divergence on vertices of one-loop AV n functions. To illustrate this, we turn to four dimensions. Inthis case, there are two Feynman diagrams to consider; these are associated with the Feynman integrals

Iλα1α2 (p1, p2) = −T r∫

d4k

(2π)4(2i)3

{[sin

(k1 × p1

2

)sin

((k1 − p1)× (−p1 − p2)

2

)

× sin

((k1 + p2)× p2

2

)][1

k1γ α1

1

k1 − p1γ λγ5

1

k1 + p2γ α2

]

+[

sin

(k2 × p2

2

)sin

((k2 − p2)× (−p1 − p2)

2

)sin

((k2 + p1)× p1

2

)]

×[

1

k2γ α2

1

k2 − p2γ λγ5

1

k2 + p1γ α1

]}(22)

In (22), k1,2 = k + s1,2 with s1,2 being arbitrary linear combinations of external momenta p1 andp2. Upon making the change of variable k2 → −k2 in the second integral of (22) we obtain

Iλα1α2 = 8T r∫

d4k

(2π)4

{[sin

(k1 × p1

2

)sin

(−k1 × (p1 + p2)+ p1 × p2

2

)

× sin

(k1 × p2

2

)][1

k1γ α1

1

k1 − p1γ λγ5

1

k1 + p2γ α2

]

+[− sin

(k2 × p1

2

)sin

(−k2 × (p1 + p2)+ p1 × p2

2

)sin

(k2 × p2

2

)][− 1

k2γ α2

1

k2 + p2γ λγ5

1

k2 − p1γ α1

]}(23)

The identity

sin a sin b sin c = 1

4[sin(a + b − c)+ sin(a − b + c)+ sin(−a + b + c)− sin(a + b + c)] (24)

allows us to write

�(4)1 = sin

(k1 × p1

2

)sin

(−k1 × (p1 + p2)+ p1 × p2

2

)sin

(k1 × p2

2

)

= 1

4

[sin

(−k1 × p2 + 1

2p1 × p2

)+ sin

(k1 × (p1 + p2)− 1

2(p1 × p2)

)

+ sin

(−k1 × p1 + 1

2(p1 × p2)

)− sin

(1

2p1 × p2

)]

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1056 Can. J. Phys. Vol. 83, 2005

= 1

4

{sin

(1

2p1 × p2

)[−1 + cos(k1 × p1)+ cos(k1 × p2)

− cos (k1 × (p1 + p2))] + cos

(1

2p1 × p2

)[sin (k1 × (p1 + p2))

− sin(k1 × p1)− sin(k1 × p2)]

}(25)

so that (23) becomes, upon taking the inner product with (p1 + p2)λ,

(p1 + p2)λ Iλα1α2 = 8T r

∫d4k

(2π)4

{�(4)1

(1

k1γ α1γ5

)(− 1

k1 + p2+ 1

k1 − p1

)γ α2

+ �(4)2

(1

k2γ α2γ5

)(1

k2 − p1− 1

k2 + p2

)γ α1

}(26)

Equation (19) now can be used to reduce (26) to

(p1 + p2)λIλα1α2 = 32

∫d4k

(2π)4εµα1να2

[�(4)1kµ1

k21

(− k1 + p2

(k1 + p2)2

k1 − p1

(k1 − p1)2

)ν

−�(4)2kµ2

k22

(− k2 + p2

(k2 + p2)2+ k2 − p1

(k2 − p1)2

)ν](27)

It is evident that if we chose s1 = s2 (i.e., k1 = k2), then (p1 +p2)λIλα1α2 = 0; this choice also ensures

that pα11 I

λα1α2 = pα22 I

λα1α2 = 0. Thus, in four dimensions the VAA “triangle" diagrams are anomalyfree. The cancellation between the two diagrams contributing to this process occurs basically becauseof the odd number of sine functions occurring in each Feynman integral. In fact, the two diagramscontributing to Iλα1α2 in (22) cancel against each other completely in this noncommutative theory andIλα1α2 = 0.

We now turn to the AV 3 Green’s function in six dimensions. There are 3! = 6 diagrams thatcontribute to this process; we examine the contribution of a matched pair of these diagrams, one ofwhich can be derived from the other by reversing the momentum flow. The corresponding Feynmanintegrals are

Iλα1α2α3 (p1, p2, p3) = −T r∫

d6k

(2π)6(2i)4

{[sin

((k1 − p1 − p2 − p3)× (−p1 − p2 − p3)

2

)

× sin

(k1 × p1

2

)sin

((k1 − p1)× p2)

2

)sin

((k1 − p1 − p2)× p3)

2

)]

×[γ α1

1

k1 − p1γ α2

1

k1 − p1 − p2γ α3

1

k1 − p1 − p2 − p3γ λγ7

1

k1

]

×[

sin

(k2 × (−p1 − p2 − p3)

2

)sin

((k2 + p1 + p2 + p3)× p3

2

)

× sin

((k2 + p1 + p2)× p2

2

)sin

((k2 + p1)× p1

2

)]

×[γ α3

1

k2 + p1 + p2γ α2

1

k2 + p1γ α1

1

k2γ λγ5

1

k2 + p1 + p2 + p3

]}(28)

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Gagne-Portelance and McKeon 1057

Again, we have set k1,2 ≡ k + s1,2. In the second part of (28), we let k2 → −k1, as in (22), so that

Iλα1α2 = −16T r∫

d6k

(2π)6

{[sin

(k1 × (−p1 − p2 − p3)

2

)sin

(k1 × p1

2

)

× sin

((k1 − p1)× p2

2

)sin

((k1 − p1 − p2)× p3)

2

)][γ α1

1

k1 − p1γ α2

1

k1 − p1 − p2

× γ α31

k1 − p1 − p2 − p3γ λγ7

1

k1

]+[(+1) sin

(k2 × (−p1 − p2 − p3)

2

)

× sin

((k2 × p1

2

)sin

((k2 − p1)× p2

2

)sin

((k2 − p1 − p2)× p3

2

)]

×[(+1)γ α3

1

k2 − p1 − p2γ α2

1

k2 − p1γ α1

1

k2γ λγ7

1

k2 − p1 − p2 − p3

]}(29)

We now use the identity

sin a sin b sin c sin d = 1

8[cos(a + b + c + d)+ cos(a + b − c − d)

+ cos(a − b + c − d)+ cos(a − b − c + d)

− cos(−a + b + c + d)− cos(a − b + c + d)

− cos(a + b − c + d)− cos(a + b + c − d)] (30)

to write

�(6)1 = sin

(k1 × (−p1 − p2 − p3)

2

)sin

(k1 × p1

2

)

× sin

((k1 − p1)× p2

2

)sin

((k1 − p1 − p2)× p3

2

)

= 1

8

{cos

[1

2(−p1 × p2 + p3 × p1 − p2 × p3)

+ cos

(−k1 × (p2 + p3)+ 1

2(p1 × p2 − p3 × p1 + p2 × p3)

)]

+ cos

(−k1 × (p1 + p3)+ 1

2(−p1 × p2 − p3 × p1 + p2 × p3)

)

+ cos

(−k1 × (p2 + p1)+ 1

2(p1 × p2 + p3 × p1 − p2 × p3)

)

− cos

(k1 × (p1 + p2 + p3)+ 1

2(−p1 × p2 + p3 × p1 − p2 × p3)

)

− cos

(−k1 × p1 + 1

2(−p1 × p2 + p3 × p1 − p2 × p3)

)

− cos

(−k1 × p2 + 1

2(p1 × p2 + p3 × p1 − p2 × p3)

)

− cos

(−k1 × p3 + 1

2(−p1 × p2 − p3 × p1 + p2 × p3)

)}(31)

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1058 Can. J. Phys. Vol. 83, 2005

with this we find that

(p1 + p2 + p3)λIλα1α2α3 = −16T r

∫d6k

(2π)6

{�(6)1

(γ α1

1

k1 − p1γ α2

1

k1 − p1 − p2γ α3

)

×(

1

k1− 1

k1 − p1 − p2 − p3

)γ7

+ �(6)2

(γ α3

1

k2 − p1 − p2γ α2

1

k2 − p1γ α1

) (1

k2− 1

k2 − p1 − p2 − p3

)γ7

}(32)

Using (19) to take the trace in (32), we find that

(p1 + p2 + p3)λIλα1α2α3 = (−16)(−8) εα1µα2να3λ

∫d6k

(2π)6

{�(6)1

[(k1 − p1)

µ(k1 − p1 − p2)ν

(k1 − p1)2(k1 − p1 − p2)2

×(kλ1

k2 − (k1 − p1 − p2 − p3)λ

(k1 − p1 − p2 − p3)2

)]

+ �(6)2

[(k2 − p1)

µ(k2 − p1 − p2)ν

(k2 − p1)2(k2 − p1 − p2)2

(kλ2

k22

− (k2 − p1 − p2 − p3)λ

(k2 − p1 − p2 − p3)2

)]}(33)

In contrast to the situation in four dimensions, the contributions coming from these two diagramsdo not cancel if k1 = k2 (viz. see (27) and (33)). This four point diagram is to be treated as its analoguein the usual six-dimensional field theory where all coordinates commute; surface terms proportionalto the first term in (31) survive. When we combine the contribution of (33) with that coming from thefour remaining diagrams that are part of the AV 3 Green’s function at one-loop order, it is apparent thatthe usual anomalous contribution to the divergence of the axial current [5, 27] is modified by factor ofcos( 1

2 (−p1 × p2 + p3 × p1 − p2 × p3))+ cos( 12 (−p2 × p3 + p1 × p2 − p3 × p1))+ cos( 1

2 (−p3 ×p1 + p2 × p3 − p1 × p2)). The presence of an even number (four) of sine functions occurring in eachFeynman integral is basically responsible for this.

A pattern is apparent; when there are an odd number of sine functions in the one-loopAV n diagram(i.e., in 4n dimensions) then the divergence in each vertex can be made to cancel when considering thecontributions of both a particular Feynman integral and the one in which momentum flow is reversed.When an an even number of vertices occur (i.e., in 4n+ 2 dimensions) this is not the case and the usualaxial anomaly is recovered, modified by a suitable cosine function. We note that this cosine functiongoes to one and does not vanish as θµν → 0, so that in this limit, the anomaly persists even though thecouplings occurring in (16) vanish.

As a final illustration of this, let us consider the one-loop contribution to the AV 4 Green’s functionin eight dimensions. There are 4! = 24 diagrams contributing to this; if we look at one of these as wellas the diagram in which the momentum is reversed, we are left with Feynman integrals

Iλα1α2α3α4 (p1, p2, p3, p4) = −T r∫

d8k

(2π)8(2i)5

{[sin

(k1 × p1

2

)sin

((k1 − p1)× p2

2

)

× sin

((k1 − p1 − p2)× p3

2

)sin

((k1 − p1 − p2 − p3)× p4

2

)

× sin

((k1 − p1 − p2 − p3 − p4)× (−p1 − p2 − p3 − p4)

2

)]

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Gagne-Portelance and McKeon 1059

×[γ α1

1

k1 − p1γ α2

1

k1 − p1 − p2γ α3

1

k1 − p1 − p2 − p3γ α4

1

k1 − p1 − p2 − p3 − p4γ λγ9

1

k1

]

+[

sin

((k1 − p1)× p1

2

)sin

((k2 − p1 − p2)× p2

2

)sin

((k2 − p1 − p2 − p3)× p3

2

)

× sin

((k2 − p1 − p2 − p3 − p4)× p4

2

)sin

(k2 × (−p1 − p2 − p3 − p4)

2

)]

×[γ α4

1

k2 + p1 + p2 + p3γ α3

1

k2 + p1 + p2γ α2

1

k2 + p1γ α1

1

k2γ λγ9

1

k2 + p1 + p2 + p3 + p4

]}(34)

As in the case of four and six dimensions considered above, we let k1 → −k2 in the second integralof (34); we can then define

�(8)1 = sin

(k1 × p1

2

)sin

((k1 − p1)× p2

2

)sin

((k1 − p1 − p2)× p3

2

)

sin

((k1 − p1 − p2 − p3)× p4

2

)sin

(k1 × (−p1 − p2 − p3 − p4)

2

)(35)

so that

Iλα1α2α3α4 = −32iT r∫

d8k

(2π)8

{�(8)1

(γ α1

1

k1 − p1γ α2

1

k1 − p1 − p2γ α3

1

k1 − p1 − p2 − p3

× γ α41

k1 − p1 − p2 − p3 − p4γ λγ9

1

k1

)+(−�(8)2

)( 1

k2 − p1 − p2 − p3

× γ α31

k2 − p1 − p2γ α2

1

k2 − p1γ α1

1

k2γ λγ9

1

k2 − p1 − p2 − p3 − p4

)}(36)

From (19) and (36), it is apparent that

(p1 + p2 + p3 + p4)λ Iλα1α2α3α4 = −32 · 16i εα1µα2να3ρα4λ

∫d8k

(2π)8

{�(8)1(k1 − p1)

µ

(k1 − p1)2

× (k1 − p1 − p2)ν

(k1 − p1 − p2)2

(k1 − p1 − p2 − p3)ρ

(k1 − p1 − p2 − p3)2

×(kλ1

k21

− (k1 − p1 − p2 − p3 − p4)λ

(k1 − p1 − p2 − p3 − p4)2

)

− �(8)2(k2 − p1)

µ

(k2 − p1)2

(k2 − p1 − p2)ν

(k2 − p1 − p2)2

(k2 − p1 − p2 − p3)ρ

(k2 − p1 − p2 − p3)2

×(kλ2

k22

− (k2 − p1 − p2 − p3 − p4)λ

(k2 − p1 − p2 − p3 − p4)2

)}(37)

so that if k1 = k2, then the right side of (37) vanishes. This also ensures that the divergence of anyvertex of Iλα1α2α3α4 defined by (34) vanishes and thus the eight-dimensional theory is anomaly free.

4. Discussion

We have demonstrated that for models in noncommutative space in which a U(1) gauge fieldcouples to a spinor field through a commutator, (as defined in (16)) there is no axial anomaly in d = 4n

© 2005 NRC Canada

1060 Can. J. Phys. Vol. 83, 2005

dimensions, while in d = 4n+ 2 dimensions, the anomaly persists. This has been illustrated in 2, 4, 6,and 8 dimensions, but the result clearly holds for all n.

This has been done by considering one-loop AV d/2 Feynman diagrams. Due to the nature of thecoupling, the integrals involved are “non-planar", that is, the loop momentum integrals involve trigono-metric functions. (If the coupling were such that only planar diagrams contributed to AV d/2 Green’sfunctions, then the anomaly occurs for all d.)

The explicit functional form of the anomaly in the model we have considered has been determinedusing the nongraphical means described in ref. 16. To determine all the contributions to the anomalythrough graphical considerations would involve considering diagrams with more external vertices, asnow the field strength is given by ∂µAν − ∂νAµ +Aµ ∗Aν −Aν ∗Aµ and the anomaly is proportionalto the product of the field strengths.

It would also be interesting to examine if theAdler–Bardeen theorem holds [28], that is, to examine ifany diagrams beyond one loop can contribute to the anomaly. A surface-term argument in conventionalfour-dimensional space that shows that there is no two-loop contribution to the anomaly [29] wouldindicate that the same is likely true in noncommutative space as well.

Acknowledgements

We would like to thank the Natural Sciences and Engineering Research Council for financial support.R. and D. MacKenzie had a useful suggestion.

References

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© 2005 NRC Canada