403

Operator regularization andnoncommutative Chern–Simonstheory

D.G.C. McKeon

Abstract: We examine noncommutative Chern–Simons theory using operator regularization.Both theζ function and theη function are needed to determine one-loop effects. Thecontributions to these functions coming from the two-point function is evaluated. TheU(N) noncommutative model smoothly reduces to theSU(N) commutative model as thenoncommutative parameterθµν vanishes.

PACS No.: 11.10.–z

Résumé : Nous étudions la théorie non commutative de Chern–Simons en utilisant larégularisation d’opérateurs. Nous avons besoin des fonctionsς et η pour déterminer l’effetdes boucles. Nous évaluons la contribution à ces fonctions qui provient de la fonction à deuxpoints. Le modèle non commutatifU(N) se réduit doucement au modèle commutatif SU(N)

lorsque le paramètre de non commutativité,θµν disparaît.

[Traduit par la Rédaction]

1. Introduction

Normally, Chern–Simons theory is a purely topological theory as it is metric independent [1–3].However, if we consider Chern–Simons theory in a noncommutative space [4, 5] in which

[xµ, xν

] = −iθµν (1)

metric dependence inevitably arise. The consequences of the presence ofθµν in noncommutative Chern–Simons theory are considered in ref. 6. In particular, the one-loop contribution to the two-point functionis computed in ref. 7.

In this paper, we would like to examine this one-loop, two-point function using operator regular-ization [8], a generalization ofζ -function regularization [9]. The modulus and phase of the functionaldeterminant associated with the effective action at one-loop order are associated with theζ function [8,9] andη function [1, 10–12], respectively. This approach has been used in conventional Chern–Simonstheory [11–15].

Received 2 April 2002. Accepted 9 March 2004. Published on the NRC Research Press Web site athttp://cjp.nrc.ca/ on 19 May 2004.

D.G.C. McKeon. Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5B7,Canada (e-mail: DGMCKEO2@UWO.CA).

Can. J. Phys.82: 403–409 (2004) doi: 10.1139/P04-017 © 2004 NRC Canada

404 Can. J. Phys. Vol. 82, 2004

2. The ζ and η functions

The usual Chern–Simons action is given by

S =∫

d3x tr εijk

(Ai∂jAk − 2i

3AiAjAk

)(2)

whereAi = Aai Ta with T a being the generator of a Lie groupG. If G is a unitary group,U(N), then

(2) can be generalized by converting the products occurring in (2) to a Moyal product in which [4, 5]

Ai(x) ∗ Aj(x) = exp

[− i

2∂x × ∂y

]Ai(x)Aj (y)

∣∣∣∣x=y

(3)

wherea × b = θij aibj .To use operator regularization to compute radiative effects, it is necessary to employ background-

field quantization [16–18]. We begin by splittingAi into the sum of background and quantum fields

Ai → Ai +Qi (4)

A gauge-fixing Lagrangian

Lgf = tr (NDi(A)Qi) (5)

is chosen so as to leave the symmetry

Ai → Ai +Di(A)� (6a)

Qi → Qi − i (Qi ∗�−� ∗Qi) (6b)

≡ Qi − i [Qi,�]

present in (2) unbroken. In (5),N is a Nakanishi–Lautraup field andDi(A) is the covariant derivative

Di(A)f = ∂if − i [Ai, f ] (7)

The ghost Lagrangian associated with the gauge fixing of (5) and the gauge transformation

Ai → Ai +Di(A+Q)� (8a)

Qi → Qi (8b)

is

Lghost = CDi(A)Di(A+Q)C (9)

From (2), (5), and (9), it is evident that the terms in the effective LagrangianL+ Lgf + Lghost that arebilinear in the quantum fields are

L(2) = (Qai ,N

a) ( εipjD

abp (A) −Dab

i (A)

Dabj (A) 0

)(Qbj

Nb

)+ C

a(Dabi (A)Dbc

i (A))Cc (10)

so that the one-loop generating functional is given by

W(1)(A) =(ln det−1/2HI

)+ (ln detHII ) (11)

© 2004 NRC Canada

McKeon 405

whereHI andHII are the two operators in (10). AsHI is linear in derivatives, it is necessary to makethe replacement

ln det−1/2HI → ln det−1/4H 2I (12)

(A possible loss of phase in this replacement will be considered below.) Since

εijkDj (A)Dk(A)f = − i

2εijk

[Fjk, f

](13)

where

Fjk = ∂iAj − ∂jAi − i(Ai ∗ Aj − Aj ∗ Ai

)(14)

it is evident that

H 2I =

( −D2δij −εimnFmnεmnjFmn −D2

)(15)

We note that

[M,N ] = MaT a ∗NbT b −NbT b ∗MaT a

= 1

2

([T a, T b

] {Ma,Nb

}+{T a, T b

} [Ma,Nb

])≡(if abc

{Ma,Nb

}+ dabc

[Ma,Nb

])T c (16)

and this becomes, on account of the Moyal product defined in (3),

= 2i

[f abc cos

(p × q

2

)+ dabc sin

(p × q

2

)](MaNbT c) (17)

wherep andq are the momenta ofMa andNb, respectively. This is to be utilized when employing aperturbative expansion ofW(1)(A) in powers ofA. (Conventions forf abc anddabc are those of ref. 19.)

In operator regularization, one first uses

ln detH = tr lnH = lims→0

− d

ds

(trH−s) = − d

ds

∣∣∣∣0

1

-(s)

∫ ∞

0dt t s−1tre−Ht ≡ −ζ ′(0) (18)

and then employs an expansion due to Schwinger [20]

tr e−(H0+H1)t = tr

[e−H0t + (−t)

1e−H0tH1 + (−t)2

2

∫ 1

0due−(1−u)H0tH1 e−uH0tH1 + . . .

](19)

to effect an expansion in powers of the background field. (The dependence ofH on the backgroundfield resides entirely inH1.)

It is now possible to apply (18) and (19) to compute the contribution of the two-point function totheζ function associated with ln det−1/4H 2

I and ln detHII . We find that

W(1)(A) = − d

ds

∣∣∣∣0

1

-(s)

∫ ∞

0dt t s−1tr

{−1

4exp−t

( −D2δij −εimnFmnεmnjFmn −D2

)

+ exp−t (−D2)}

(20)

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406 Can. J. Phys. Vol. 82, 2004

Employing the expansion of (19) and computing functional traces in momentum space with [20]

< p|f |q >= 1

(2π)3/2f (p − q) (21)

we find that the contribution to the two-point function coming from (20) is

W(1)2 (A) = − d

ds

∣∣∣∣0

1

-(s)

∫ ∞

0dt t s−1

∫dp dq

(2π)3

{−1

4(−t)2

∫ 1

0due−[(1−u)p2+uq2

]t

×[(−εimn)

(f apbf

pmn(p − q) cos

p × q

2+ dapbf

pmn(p − q) sin

p × q

2

)

× (εrsi)

(f bqaf

qrs(q − p) cos

q × p

2+dbqaf qrs(q − p) sin

q × p

2

)]}(f aij ≡ ∂iA

aj − ∂jA

ai

)(22)

Upon making the usual shift in momentum variables, this becomes

= N

2

d

ds

∣∣∣∣0

1

-(s)

∫ ∞

0dt t s+1

∫dp dq

(2π)3

∫ 1

0due−[q2+u(1−u)p2

]t

× f amn(p)fbmn(−p)

(δab − δa0δb0 cos(p × q)

)(23)

The standard integrals [21]∫dnk

(2π)ne−k2t = 1

(4πt)n/2(24a)∫ ∞

0dt tν−1 e−At = -(ν)A−ν (24b)

∫ ∞

0dt tν−1 e−γ t−β/t = 2

(β

γ

)ν/2K±ν(2

√βγ ) (24c)

can be used to reduce (23) to

W(1)2 (A) = N

2

d

ds

∣∣∣∣0

1

-(s)

∫d3p

(4π)3/2

∫ 1

0du f amn(p)f

amn(−p)

[-

(s + 1

2

)(u(1 − u)p2

)−s−(1/2)δab

−2

(p̃2

4u(1 − u)p2

)(s+1/2)/2

Ks+1/2

2

√u(1 − u)p2p̃2

4

δa0δb0

(25)

wherep × q ≡ p̃ · q. It is now possible to evaluated

ds

∣∣∣∣0

in (25) explicitly, leaving us with

W(1)2 (A) = N

2

∫d3p

(4π)3/2

∫ 1

0du f amn(p)f

bmn(−p)

[√π

u(1 − u)p2 δab

−(

4p̃2

u(1 − u)p2

)1/4

K1/2

(√u(1 − u)p2p̃2

)δa0δb0

](26)

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McKeon 407

SinceK±1/2(z) =√

π2z e−z, it is possible to compute both integrals overu in (26) using [21]

∫ 1

0du ua−1(1 − u)b−1 = -(a)-(b)

-(a + b)(27a)

∫ 1

0du

1√u(1 − u)

e−A√u(1−u) = 2

∫ π/2

0dθ e−(A/2) cosθ = πI0

(A

2

)(27b)

leaving us with

W(1)2 (A) = N

16

∫d3p√p2

[f amn(p)f

bmn(−p)

(δab − I0

(√p2p̃2

2

)δa0δb0

)](28)

In the limit θµν → 0 (so thatp̃2 → 0),I0

(√p2p̃2

2

)→ 1 leaving only theSU(N) contribution to (28).

We can now consider the loss of phase associated with the replacement of (12). This phase has beendiscussed extensively in refs. 1 and 10–12; it is associated with the so-calledη function

η(s) = 1

-(s+1

2

) ∫ ∞

0dt t (s−1)/2 tr

(HI e−H2

I t)

(29)

If a parameterλ is inserted intoHI so thatHI(λ = 1) = HI , then from (29), it is easily seen that

dηλ(s)

dλ= −s-(s+1

2

) ∫ ∞

0dt t (s−1)/2 tr

(dHI(λ)

dλe−H2

I (λ)t

)(30)

As onlyη(0) is required to determine the phase we are interested in, it is sufficient to compute thepoles arising in the integral overt in (30) on account of the explicit factor ofs arising in front of theintegral.

If now

HI(λ) ≡(εipj

(∂p − iλ[Ap

) − (∂i − iλ[Ai)(∂j − iλ[Aj

)0

)(31)

then it is possible to expand the right side of (30) in powers of the background field using a secondexpansion due to Schwinger [20]

e−(H0+H1)t =[e−H0t + (−t)

∫ 1

0due−(1−u)H0tH1 e−uH0t

+ (−t)2∫ 1

0du u

∫ 1

0dv e−(1−u)H0tH1 e−u(1−v)H0t H1 e−uvH0t + . . .

](32)

From (15), (31), and (32), it is evident that the contribution to (30) that is bilinear in the backgroundfieldAµ is (upon following the approach used above with theζ function)

dη(2)λ (s)

dλ= Nλs

-(s+1

2

) ∫ ∞

0dt t (s+1)/2

∫ 1

0du∫

dp dq

(2π)3e−(q2+u(1−u)p2

)t

×(εijkA

ai (p)f

bjk(−p)

) (δab − δa0δb0 cos(p × q)

)(33)

© 2004 NRC Canada

408 Can. J. Phys. Vol. 82, 2004

The integrals appearing in (33) are identical in form to those in (23); just as we obtain (28) we findthat

dη(2)λ (s)

dλ= Nλs

-(s+1

2

) ∫ 1

0du∫

dp

(4π)3/2εijkA

ai (p)f

bjk(−p)

[δab-

( s2

) (u(1 − u)p2

)−s/2

−2δa0δb0(

p̃2

4u(1 − u)p2

)s/4Ks/2

(√u(1 − u)p2p̃2

)](34)

In the limit x → 0,Kν(x) → 2ν−1-(ν)x−ν and thus whenθµν → 0(p̃2 → 0), (34) reduces to

≈ Nλs

-(s+1

2

) ∫ 1

0du∫

dp

(4π)3/2εijkA

ai (p)f

bjk(−p)[-( s

2

) (u(1 − u)p2

)−s/2] [δab − δa0δb0

](35)

As in the case of theζ function, only theSU(N) contribution to ddληλ(0) survives in the commutative

limit.

3. Discussion

Chern–Simons theory is difficult to regulate on account of the presence of the tensorεijk. Operatorregularization appears, however, to be a suitable way of dealing with this problem as it does not involvealtering the original Lagrangian. This permits one to deal with both the usual commutativeSU(N)

Chern–Simons model and as well noncommutativeU(N) Chern–Simons theory. An analysis of theone-loop two-point function done here has shown that in the commutative limit, the noncommutativeU(N) model smoothly reduces to the commutativeSU(N) model.

It is interesting to compare the results of (28) and (34) with those obtained for the one-loop gluonself energy obtained in ref. 7. In ref. 7, background-field quantization is not employed, and hencethe background-field gauge invariance of (6) is not present; this background-field gauge invariance(respected by (28) and (34)) is equivalent to the Ward identities of conventional quantization procedures.Furthermore, in ref. 7, a complete cancellation of the one-loop contribution to the gluon self energycoming from the gluon and ghost loops occurs. By way of contrast, the two functional determinants of(11) do not result in a complete cancellation; the finite, nonzero results (28) and (34) arise. The absencein ref. 7 of a term in the two-point function proportional to the tensorεijk seems to be particularlysignificant, as this term is related to theη function that in the limitθ → 0 has an importance that isdiscussed in refs. 1 and 2.

Acknowledgements

Most of this work was done while the author was a guest at the University of Sao Paulo; he isparticularly grateful to F. Brandt and J. Frenkel for their hospitality there. R. and D. MacKenzie haduseful suggestions.

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© 2004 NRC Canada