UV/IR mixing in noncommutative QED defined by Seiberg-Witten map
aa r X i v : . [ h e p - t h ] J u l Preprint typeset in JHEP style - HYPER VERSION hep-th/yymmnnn
UV/IR mixing in noncommutative QEDdefined by Seiberg-Witten map
Peter Schupp, Jiangyang You
School of Engineering and ScienceJacobs UniversityCampusring 128759 Bremen, Germany
E-mail: [email protected], [email protected]
Abstract:
Noncommutative gauge theories defined via Seiberg-Witten map havedesirable properties that theories defined directly in terms of noncommutative fieldslack, covariance and unrestricted choice of gauge group and charge being amongthem, but nonperturbative results in the deformation parameter θ are hard to obtain.In this article we use a θ -exact approach to study UV/IR mixing in a noncommutativequantum electrodynamics (NCQED) model defined via Seiberg-Witten map. Thefermion contribution of the one loop correction to the photon propagator is computedand it is found that it gives the same UV/IR mixing term as a NCQED model withoutSeiberg-Witten map. ontents
1. Introduction 12. The model 33. One-loop computation 54. Conclusion 7A. Feynman rules for photon-fermion interaction 8
1. Introduction
Noncommutative quantum electrodynamics (NCQED) is usually defined in analogyto Yang-Mills theory with matrix multiplication replaced by star products. Theresulting action is invariant under noncommutative gauge transformations. Suchmodels appear quite naturally in certain limits of string theory in the presence ofa background B -field [1], they can also be used to gain some understanding aboutphenomenological implications of a quantum structure of spacetime. One of theparticularly intriguing effects is UV/IR-mixing, an interrelation between short andlong-distance scales that is absent in ordinary quantum field theory. There are how-ever some problems with this simple definition of NCQED: (1) The possible choicesof charges for particles are restricted to ± U ( N ) in the funda-mental representation. (2) An ordinary gauge field a µ ( x ) transforms like a vectorunder a change of coordinates, a ′ µ ( x ′ ) = ∂x ν /∂x ′ µ a ν ( x ), while for the fields A µ ( x ) ofNCQED this holds only for rigid, affine coordinate changes [2, 3]. The solution toboth problems is an alternative approach to noncommutative gauge theory based onSeiberg-Witten maps. This approach to noncommutative gauge theory (especially,the noncommutative extension of nonabelian gauge theories) has been establishedfor quite some time [4–6]. The idea is to consider noncommutative gauge fields A µ and gauge transformation parameters Λ that are valued in the enveloping algebra ofthe gauge group and can be expressed in terms of the ordinary gauge field a µ , gaugeparameter λ and the noncommutative parameter θ µν in such a way that an ordi-nary gauge transformations of a µ induces a non-commutative gauge transformationof A µ [ a ] with non-commutative gauge parameter Λ[ λ, a ].– 1 –omparing to the simpler formulation in which the gauge group is directly de-formed by replacing the normal product of group elements with the Moyal-Weyl starproduct, the Seiberg-Witten map approach removes the restrictions on the gaugegroup and charge and allows the construction of more realistic models. The noncom-mutative action can be treated as a complicated action written in terms of ordinaryfields, which when expanded in the powers of the noncommutativity parameter θ gives the usual commutative action (both free and interacting parts) at zeroth orderin θ plus higher order non-commutative corrections. Therefore, such a theory canbe considered to be a minimal noncommutative extension of the corresponding com-mutative model. Following this line, a minimal noncommutative extension of thestandard model has been established [7–9] and influences to particle physics havebeen studied up to loop level in low orders of θ [10, 11]. Besides being useful for phe-nomenology, the θ -expansion was also shown to be improving the renormalizability ofthe noncommutative gauge theory [12–15]. The photon self energy is renormalizableup to any finite order of θ [12] (with the sacrifice of introducing an infinite numberof coupling constant from the freedom/ambiguity within the Seiberg-Witten map).Although the θ -expansion method works nicely in model building, crucial non-perturbative information is lost due to the cut off at finite order of θ . It is longknown that in the noncommutative field theories [16–19], the Moyal-Weyl star prod-uct results in a nontrivial phase factor for the Fourier modes when two functions aremultiplied together. Such a phase, when it appears in loop calculation, regulates theultraviolet divergence in the one loop two point function of both noncommutative φ and noncommutative quantum electrodynamics (NCQED) but introduces an in-frared divergent term of the form 1 / ( pθθp ). As the nontrivial phase factor appearsonly when all orders of θ in the star product are summed over, this effect does notshow up in the noncommutative gauge theories defined by the Seiberg-Witten mapapproach when it is studied using the θ -expansion method, (thus it is sometimesclaimed that such a theory is free of UV/IR mixing). However, as already suggestedin some very early papers [2, 6, 20], the θ -expansion is not the only possible wayof expressing the Seiberg-Witten map. As the noncommutative gauge field A µ is afunction of both the ordinary field a µ and the noncommutativity parameter θ ij , onecan, instead of expanding A µ in power of θ , expand it in powers of a µ . The first sev-eral orders of the expansion can be written in a simple form by introducing certaingeneralized star products [6, 20]. Such an expansion enables us to treat all orders of θ at once in each interaction vertex, thereby allowing us to compute nonperturbativeresults. In this article we are going to use this expansion to compute the fermionone loop correction to the photon two point function of a NCQED model definedby Seiberg-Witten map. We will see that UV/IR mixing will still arise via the non-trivial phase factors, hence the absence of UV/IR mixing in the Seiberg-Witten mapapproach to noncommutative gauge theory so far has been really a technical artifactof the perturbative θ -expansion method, but not a feature of the theory itself.– 2 – . The model For simplicity we consider a NCQED model with a U (1) gauge field A µ and a fermionfield Ψ which lives in the adjoint representation of the noncommutative gauge group U (1) ⋆ . The action is as following S = Z − F µν F µν + i ¯Ψ / D Ψ (2.1)with D µ Ψ = ∂ µ Ψ − i [ A µ ⋆ , Ψ] and F µν = ∂ µ A ν − ∂ ν A µ − i [ A µ ⋆ , A ν ]The θ -exact Seiberg-Witten map can be obtained in several ways: From theclosed formula derived using deformation quantization based on Kontsevich formalitymap [2], by the relationship between open Wilson lines in the commutative andnoncommutative picture [20], or by a direct recursive computation using consistencyconditions. The computation of the one loop two-point function requires fully θ -exactinteraction vertices up to four external legs, i.e. one needs the θ -exact Seiberg-Wittenmap of A µ up to third order in a µ . This has been computed in its inverse form ( a µ in terms of A µ up to A ) in [20]. Here we simply take the inverse of this result bymatching the trivial identity a µ ( A µ ( a µ )) = a µ order by order, resulting in A µ = a µ − θ ij a i ⋆ ( ∂ j a µ + f jµ ) + 12 θ ij θ kl (cid:26)
12 ( a k ⋆ ( ∂ l a i + f li )) ⋆ ( ∂ j a µ + f jµ )+ a i ⋆ ( ∂ j ( a k ⋆ ( ∂ l a µ + f lµ )) − ∂ µ ( a k ⋆ ( ∂ l a j + f lj ))) − a i ⋆ ( ∂ k a j ⋆ ∂ l a µ )+[ a i ∂ k a µ ( ∂ j a l + f jl ) − ∂ k ∂ i a µ a j a l + 2 ∂ k a i ∂ µ a j a l ] ⋆ (cid:27) + O ( A ) (2.2)where ⋆, ⋆ , ⋆ are Moyal-Weyl star product and two generalized star products: f ( x ) ⋆ g ( x ) = e i θ µν ∂∂yµ ∂∂zν f ( y ) g ( z ) (cid:12)(cid:12)(cid:12)(cid:12) x = y = z (2.3) f ( x ) ⋆ g ( x ) = sin ∂ ∧ ∂ ∂ ∧ ∂ f ( x ) g ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x = x = x (2.4)[ f ( x ) g ( x ) h ( x )] ⋆ = (cid:2) sin( ∂ ∧ ∂ ) sin( ∂ ∧ ( ∂ + ∂ )2 ) ( ∂ + ∂ ) ∧ ∂ ∂ ∧ ( ∂ + ∂ )2 + { ↔ } (cid:3) f ( x ) g ( x ) h ( x ) (cid:12)(cid:12)(cid:12)(cid:12) x i = x (2.5) We use a Minkowskian signature here. In the next section we allow a Wick rotation, thus theresult is actually on noncommutative R . This procedure is the same as taken in [17–19], but differsfrom a procedure where the action is directly written down in R . – 3 –here ∂ ∧ ∂ = θ ij ∂∂x i ∂∂x j (2.6)The expansion for a matter particle in the adjoint representation of U (1) ⋆ canbe easily obtained by taking the linear part (linear operator acting on a µ ) in theexpansion of A µ , which leads to following result:Ψ = ψ − θ ij a i ⋆ ∂ j ψ + 12 θ ij θ kl (cid:26) ( a k ⋆ ( ∂ l a i + f li )) ⋆ ∂ j ψ + 2 a i ⋆ ( ∂ j ( a k ⋆ ∂ l ψ )) − a i ⋆ ( ∂ k a j ⋆ ∂ l ψ ) − (cid:2) a i ∂ k ψ ( ∂ j a l + f jl ) − ∂ k ∂ i ψa j a l (cid:3) ⋆ (cid:27) + O ( a ) ψ (2.7)Now the action can be expanded as following: S = Z − f µν f µν + i ¯ ψ /∂ψ + L pp + L pf (2.8) L pf and L pp are photon-fermion and photon self interaction terms, in this article weconcentrate on the photon-fermion part, so we write out L pf explicitly: L pf = ¯ ψγ µ [ a µ ⋆ , ψ ] + i ( θ ij ∂ i ¯ ψ ⋆ a j ) /∂ψ − i ¯ ψ ⋆ /∂ ( θ ij a i ⋆ ∂ j ψ ) + ( θ ij ∂ i ¯ ψ ⋆ a j ) γ µ [ a µ ⋆ , ψ ] − ¯ ψγ µ [ a µ ⋆ , θ ij a i ⋆ ∂ j ψ ] − ¯ ψγ µ [ 12 θ ij a i ⋆ ( ∂ j a µ + f jµ ) ⋆ , ψ ] − i ( θ ij ∂ i ¯ ψ ⋆ a j ) /∂ ( θ kl a k ⋆ ∂ l ψ )+ i θ ij θ kl (cid:0) ( a k ⋆ ( ∂ l a i + f li )) ⋆ ∂ j ¯ ψ + 2 a i ⋆ ( ∂ j ( a k ⋆ ∂ l ¯ ψ )) − a i ⋆ ( ∂ k a j ⋆ ∂ l ¯ ψ )+ (cid:2) a i ∂ k ¯ ψ ( ∂ j a l + f jl ) − ∂ k ∂ i ¯ ψa j a l (cid:3) ⋆ (cid:1) /∂ψ + i θ ij θ kl ¯ ψ/∂ (cid:0) ( a k ⋆ ( ∂ l a i + f li )) ⋆ ∂ j ψ +2 a i ⋆ ( ∂ j ( a k ⋆ ∂ l ψ )) − a i ⋆ ( ∂ k a j ⋆ ∂ l ψ )+ 12 θ ij θ kl (cid:2) a i ∂ k ψ ( ∂ j a l + f jl ) − ∂ k ∂ i ψa j a l (cid:3) ⋆ (cid:1) + ¯ ψ O ( a ) ψ (2.9)One noticeable feature of L pf is that it contains vertices identical to NCQED (withoutSeiberg-Witten map) in leading order instead of ordinary QED as the θ -expandedapproach does. This observation holds also for L pp . One thus knows that the one-looptwo point function will contain UV/IR mixing terms coming from those integrals inthe same way as NCQED. The question is only whether there will be new correctionscoming from terms that arise solely due to Seiberg-Witten map or not. As we willsee in the next section, for the fermion loop, the leading order IR divergent result isfully identical for NCQED with and without Seiberg-Witten map.– 4 – . One-loop computation The free part of the action (2 .
8) is completely identical to ordinary commutativeQED, hence the quantization is straightforward. Vertices coming from the fermion-photon interaction lagrangian (2 .
9) are listed in the appendix. The fermion loopcontribution to the one loop photon two point function contains two diagrams: thenormal vacuum polarization graph as shown in as shown in figure 1(a) and a newfermion tadpole graph in figure 1(b).Figure 1: Fermion loop corrections to the photon self energyDiagram (a) leads to following integral: i Π ij − = − i Z d ki (2 π ) k + p ) ( k − p ) sin p ∧ k (cid:26) [ γ i ( /k + /p γ j ( /k − /p p ∧ k [(˜ p i /k − ˜ k i /p )( /k + /p γ j ( /k − /p γ i ( /k + /p p j /k − ˜ k j /p )( /k − /p p ∧ k ) [(˜ p i /k − ˜ k i /p )( /k + /p p j /k − ˜ k j /p )( /k − /p (cid:27) (3.1)where ˜ p i = θ ij p j .For diagram (b) we get a surprising result: Its contribution can be shown to– 5 –anish: i Π ij = i Z d ki (2 π ) k tr (cid:26) p ∧ k p ∧ k (˜ k i /kγ i + ˜ k i /kγ i ) − p ∧ k ( p ∧ k ) ( /k/p + /k/k )˜ k i ˜ k i +2 /k/k (cid:20) ( − p i ˜ k i + p ∧ kθ i i )+ sin p ∧ k ( p ∧ k ) k − ˜ p ) i ˜ k i + sin p ∧ k p ∧ k θ i i + sin p ∧ k ( p ∧ k ) (2˜ k i ˜ p i + θ i i k ∧ p ) − sin p ∧ k ( p ∧ k ) ˜ k i ˜ k i + (2˜ p i ˜ k i − p ∧ kθ i i ) + sin p ∧ k ( p ∧ k ) k + ˜ p ) i ˜ k i − sin p ∧ k p ∧ k θ i i − sin p ∧ k ( p ∧ k ) (2˜ k i ˜ p i + θ i i k ∧ p ) − sin p ∧ k ( p ∧ k ) ˜ k i ˜ k i (cid:21) − p ∧ k p ∧ k (˜ k i /kγ i +˜ k i /kγ i )+4 sin p ∧ k ( p ∧ k ) ( /k/p − /k/k )˜ k i ˜ k i +2 /k/k (cid:20) (2˜ p i ˜ k i − p ∧ kθ i i )+ sin p ∧ k ( p ∧ k ) k + ˜ p ) i ˜ k i − sin p ∧ k p ∧ k θ i i − sin p ∧ k ( p ∧ k ) (2˜ k i ˜ p i + θ i i k ∧ p ) − sin p ∧ k ( p ∧ k ) ˜ k i ˜ k i − (2˜ p i ˜ k i − p ∧ kθ i i )+ sin p ∧ k ( p ∧ k ) k − ˜ p ) i ˜ k i + sin p ∧ k p ∧ k θ i i − sin p ∧ k ( p ∧ k ) (2˜ k i ˜ p i + θ i i k ∧ p )+ sin p ∧ k ( p ∧ k ) ˜ k i ˜ k i (cid:21)(cid:27) = i Z d ki (2 π ) k tr (cid:26) − p ∧ k ( p ∧ k ) /k/k ˜ k i ˜ k i + 8 sin p ∧ k ( p ∧ k ) /k/k ˜ k i ˜ k i (cid:27) = 0 (3.2)Hence we only need to evaluate the integral (3.1). We work out the trace in (3.1),then write the wedge product in its explicit component form, to obtain: i Π ij = − i Z d ki (2 π ) k + p ) ( k − p ) sin p i θ ij k j (cid:26) [2 k i k j − k g ij −
14 (2 p i p j − p g ij )] − p i θ ij k j [2( p · k )(˜ k i k j + k i ˜ k j ) − ( k + p p i k j + k i ˜ p j +˜ k i p j + p i ˜ k j )+ 12 ( p · k )(˜ p i p j + p i ˜ p j )] + 1( p i θ ij k j ) [( k − ( p · k ) p k p i ˜ p j − ( p k − p · k ) + p k i ˜ k j − ( k − p p · k )(˜ p i ˜ k j + ˜ k i ˜ p j )] (cid:27) (3.3)As expected, we have here in the first square bracket terms that are identical to or-dinary NCQED. In the next pair of square brackets are the new contribution comingfrom the Seiberg-Witten map together with the non-trivial IR-divergent coefficients1 / ( p i θ ij k j ) n , where n equals to one for the second and two for the third term. Theintegral in seems to be not very different to its counterpart in normal NCQED.Previous results [18, 19] suggest that one can rewritesin p i θ ij k j − cos( p i θ ij k j )) (3.4)– 6 –o separate terms with and without nontrivial phase shift (planar and non-planar).However, the IR-divergent term 1 / ( p i θ ij k j ) n introduces unexpected difficulties to theusual renormalization procedure. The term 1 / ( p i θ ij k j ) cannot be removed by intro-ducing a Schwinger parameter as it does not have a fixed sign in R . Furthermore,the term 1 / ( p i θ ij k j ) leads to a complicated Gaussian integral over k µ whose con-vergence in R depends on the explicit choice of p µ (instead of p ). Here, we tryto evaluate the leading order non-planar part by the following trick: We introducean additional variable λ in the sine functions in (3 .
3) to make it sin λ ( p i θ ij k j ), thenone can cancel the negative power of ( p i θ ij k j ) by taking an appropriate number ofderivative over λ , resulting integral is: i Π ′′ ij ( λ ) = − i Z d ki (2 π ) cos( λp i θ ij k j )( k + p ) ( k − p ) (cid:26) ( p i θ ij k j ) [2 k i k j − k g ij −
14 (2 p i p j − p g ij )] − ( p i θ ij k j )[2( p · k )(˜ k i k j + k i ˜ k j ) − ( k + p p i k j + k i ˜ p j + ˜ k i p j + p i ˜ k j )+ 12 ( p · k )(˜ p i p j + p i ˜ p j )]+[( k − ( p · k ) p k p i ˜ p j − ( p k − p · k ) + p k i ˜ k j − ( k − p p · k )(˜ p i ˜ k j + ˜ k i ˜ p j )] (cid:27) (3.5)The computation now proceeds along the lines of the standard dimensional regular-ization method. Taking the derivative with respect to λ the integral (3.5) appears tobe more divergent than (3.5), fortunately the effective UV regulator coming from thecosine decays exponentially and therefore is still effective. Now one integrates over λ and evaluates the resulting function at λ = 1. The free integration constant can befixed by matching the result for the first square bracket to the direct computationin NCQED. Finally we obtained the following (surprisingly simple) results for theleading order IR divergent term:Π ijnon − planar = − π ˜ p i ˜ p j ˜ p (3.6)which is identical to the corresponding result in normal NCQED.
4. Conclusion
By explicit computation we have shown that NCQED defined via Seiberg-Wittenmap still exhibits UV/IR mixing in its photon one-loop two-point function, whenthis theory is treated nonperturbatively in θ . The proof of principle that this non-perturbative computation can be done at all is perhaps the most important resultof this work. To find the full expression for the UV/IR mixing term one needsto compute also the photon self interaction loop corrections, which can be done– 7 –y a procedure practically identical to the computation of the fermion loop. By thearguments given in section 2, we know that there exists in general also UV/IR mixingterms in the photon loop correction. Hence it is quite safe to say that UV/IR mixingstill exists in noncommutative quantum gauge theories constructed using Seiberg-Witten maps and one still needs to worry about unusual large modifications to thevery low energy physics from arbitrarily small θ since the θ → θ -exact approach gives rise to a regularization prob-lem, which requires some improvement in the renormalization procedure. From thisview point the θ -expansion method in [12] seems to be more convenient. Another pos-sible candidate is the Hamiltonian approach to the renormalization, which has suc-cessfully achieved finite results for noncommutative scalar field theory in Minkowskispace-time [21], while a related approach [22] to NCQED based on the Yang-Feldmanequation encountered similar problems for the photon two point function as we en-countered here . Acknowledgments
Helpful discussions with Robert C. Helling are gratefully acknowledged.
A. Feynman rules for photon-fermion interaction i k k V ipff ( k , k ) = 2 γ i sin k ∧ k k i /k − ˜ k i /k ) sin k ∧ k k ∧ k (A.1) It is worth also to mention that in [22] an inexplicit expansion of open-Wilson lines is constructedup to arbitrary formal order of the gauge field, while the author probably did not notice theconnection between the expansion of open-Wilson lines and Seiberg-Witten maps and erroneouslyclaims that the Seiberg-Witten map is only valid in an θ -expanded way. – 8 – k p p i i V i i ppff ( p , p , k , k ) = (cid:26) i sin p ∧ k sin p ∧ k p ∧ k ˜ k i γ i − i sin p ∧ k sin p ∧ k p ∧ k ˜ k i γ i − i sin k ∧ k sin p ∧ p p ∧ p (2 γ i ˜ p i − /p θ i i ) − i sin p ∧ k sin p ∧ k p ∧ k p ∧ k ( /p + /k )˜ k i ˜ k i + 2 i/k (cid:20) sin k ∧ k sin p ∧ p p ∧ p k ∧ k ( p ∧ k θ i i − p i ˜ k i ) − sin p ∧ k sin p ∧ k p ∧ k p ∧ k p − ˜ k ) i ˜ k i + sin p ∧ k sin p ∧ k p ∧ k θ i i + (cid:18) sin p ∧ k sin p ∧ k p ∧ k p ∧ k + sin p ∧ p sin k ∧ k p ∧ k k ∧ k (cid:19) (2˜ k i ˜ p i + θ i i k ∧ p − ˜ k i ˜ k i ) (cid:21) + 2 i/k (cid:20) sin k ∧ k sin p ∧ p p ∧ p k ∧ k (2˜ p i ˜ k i − p ∧ k θ i i )+ sin p ∧ k sin p ∧ k p ∧ k p ∧ k p + ˜ k ) i ˜ k i − sin p ∧ k sin p ∧ k p ∧ k θ i i − (cid:18) sin p ∧ k sin p ∧ k p ∧ k p ∧ k + sin p ∧ p sin k ∧ k p ∧ k k ∧ k (cid:19) (2˜ k i ˜ p i + θ i i k ∧ p + ˜ k i ˜ k i ) (cid:21) + { p ↔ p and i ↔ i } (cid:27) δ ( k − k − p − p ) (A.2) References [1] N. Seiberg and E. Witten,
String theory and noncommutative geometry , JHEP (1999) 032, [ hep-th/9908142 ].[2] B. Jurco, P. Schupp, and J. Wess, Nonabelian noncommutative gauge theory vianoncommutative extra dimensions , Nucl. Phys.
B604 (2001) 148–180,[ hep-th/0102129 ]. – 9 –
3] R. Jackiw and S. Y. Pi,
Covariant coordinate transformations on noncommutativespace , Phys. Rev. Lett. (2002) 111603, [ hep-th/0111122 ].[4] J. Madore, S. Schraml, P. Schupp, and J. Wess, Gauge theory on noncommutativespaces , Eur. Phys. J.
C16 (2000) 161–167, [ hep-th/0001203 ].[5] A. A. Bichl, J. M. Grimstrup, L. Popp, M. Schweda, and R. Wulkenhaar,
DeformedQED via Seiberg-Witten map , hep-th/0102103 .[6] B. Jurco, L. Moller, S. Schraml, P. Schupp, and J. Wess, Construction ofnon-Abelian gauge theories on noncommutative spaces , Eur. Phys. J.
C21 (2001)383–388, [ hep-th/0104153 ].[7] X. Calmet, B. Jurco, P. Schupp, J. Wess, and M. Wohlgenannt,
The standard modelon non-commutative space-time , Eur. Phys. J.
C23 (2002) 363–376,[ hep-ph/0111115 ].[8] B. Melic, K. Passek-Kumericki, J. Trampetic, P. Schupp, and M. Wohlgenannt,
Thestandard model on non-commutative space-time: Electroweak currents and Higgssector , Eur. Phys. J.
C42 (2005) 483–497, [ hep-ph/0502249 ].[9] B. Melic, K. Passek-Kumericki, J. Trampetic, P. Schupp, and M. Wohlgenannt,
Thestandard model on non-commutative space-time: Strong interactions included , Eur.Phys. J.
C42 (2005) 499–504, [ hep-ph/0503064 ].[10] M. Buric, D. Latas, V. Radovanovic, and J. Trampetic,
Nonzero Z → gammagamma decays in the renormalizable gauge sector of the noncommutative standardmodel , Phys. Rev.
D75 (2007) 097701.[11] A. M. Alboteanu,
The noncommutative standard model: Construction beyond leadingorder in Theta and collider phenomenology . PhD thesis, W¨urzburg, 2007.[12] A. Bichl et. al. , Renormalization of the noncommutative photon self-energy to allorders via Seiberg-Witten map , JHEP (2001) 013, [ hep-th/0104097 ].[13] M. Buric and V. Radovanovic, The one-loop effective action for quantumelectrodynamics on noncommutative space , JHEP (2002) 074, [ hep-th/0208204 ].[14] M. Buric, D. Latas, and V. Radovanovic, Renormalizability of noncommutativeSU(N) gauge theory , JHEP (2006) 046, [ hep-th/0510133 ].[15] M. Buric, V. Radovanovic, and J. Trampetic, The one-loop renormalization of thegauge sector in the noncommutative standard model , JHEP (2007) 030,[ hep-th/0609073 ].[16] S. Minwalla, M. Van Raamsdonk, and N. Seiberg, Noncommutative perturbativedynamics , JHEP (2000) 020, [ hep-th/9912072 ].[17] M. Hayakawa, Perturbative analysis on infrared aspects of noncommutative QED onR**4 , Phys. Lett.
B478 (2000) 394–400, [ hep-th/9912094 ]. – 10 –
18] M. Hayakawa,
Perturbative analysis on infrared and ultraviolet aspects ofnoncommutative QED on R**4 , hep-th/9912167 .[19] A. Matusis, L. Susskind, and N. Toumbas, The IR/UV connection in thenon-commutative gauge theories , JHEP (2000) 002, [ hep-th/0002075 ].[20] T. Mehen and M. B. Wise, Generalized *-products, Wilson lines and the solution ofthe Seiberg-Witten equations , JHEP (2000) 008, [ hep-th/0010204 ].[21] D. Bahns, The ultraviolet-finite Hamiltonian approach on the noncommutativeMinkowski space , Fortsch. Phys. (2004) 458–463, [ hep-th/0401219 ].[22] J. W. Zahn, Dispersion relations in quantum electrodynamics on the noncommutativeMinkowski space , ..