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1 hep-th/ REF. TUW REF. UWThPh Deformed QED via Seiberg-Witten Map arxiv:hep-th/ v1 16 Feb 2001 A. A. Bichl 1, J. M. Grimstrup 2, L. Popp 3, M. Schweda 4, R. Wulkenhaar 5 1,2,3,4 Institut für Theoretische Physik, Technische Universität Wien Wiedner Hauptstraße 8-10, A-1040 Wien, Austria 5 Institut für Theoretische Physik, Universität Wien Boltzmanngasse 5, A-1090 Wien, Austria Abstract. With the help of the Seiberg-Witten map for photons and fermions we define a θ-deformed QED at the classical level. Two possibilities of gauge-fixing are discussed. A possible non-abelian extension for a pure θ-deformed Yang-Mills theory is also presented. 1 Work supported in part by Fonds zur Förderungder Wissenschaftlichen Forschung FWF under contract P14639-TPH. 2 Work supported by The Danish Research Agency. 3 Work supported in part by Fonds zur Förderungder Wissenschaftlichen Forschung FWF under contract P13125-PHY. 5 Marie-Curie Fellow.

2 1 Introduction The Seiberg-Witten map was originally discussed in the context of string theory, where it emerged from a 2D-σ-model regularized in different ways [1]. It was argued by Seiberg and Witten that the ordinary gauge theory should be gauge-equivalent to a noncommutative counterpart. The Seiberg-Witten map may be also introduced with the help of covariant coordinates. In this way the deformed gauge theory emerges as a gauge theory of a certain noncommutative algebra [2], [4] leading to the same results as in ref.[1]. In this note we apply the idea of the Seiberg-Witten map in order to define a θ-deformed QED at the classical level, including the corresponding Seiberg-Witten map for the infinitesimal local gauge transformation of the fermions. Using the definitions of the star-product [4] and the Seiberg-Witten map for the Abelian gauge field and the fermions we define a θ-deformed QED in terms of the deformation parameter θ of a noncommuting flat space-time. This deformation parameter, which is treated as a constant external field with canonical dimension 2, leads to a field theory with infinitely many θ-dependent interaction vertices. In order to prepare the quantization of this deformed QED and possible non-abelian extensions we discuss the gauge-fixing procedure of the photon sector in the Seiberg-Witten framework. It is interesting to note that there are two possibilities of gauge-fixing. The linear one corresponds to the usual gauge-fixing of the undeformed Abelian model, whereas the second possibility is induced by a nonlinear gauge-fixing emerging from the noncommutative structure of the deformed QED. It is important to observe that the θ-deformed QED is still characterized by an Abelian gauge symmetry. The symmetry of the model is described by a linear BRST-identity. The letter is organized as follows. Section 2 is devoted to a definition of the Seiberg-Witten map for photons and fermions. With these θ-expansions for the basic field variables of the model and the usual definition of the star-product [4] an Abelian gauge invariant deformed QED action is defined. The gauge-fixing procedure à la BRST [5] will be discussed in section 3. Higher derivative gauge invariant monomials which are allowed due to the presence of θ are analyzed in section 4. In fact such terms are required if one studies radiative corrections [6]. In a last step we present also a non-abelian extension of the photon sector, the pure θ- deformed noncommutative Yang-MillsNCYM theory, involving the discussion of the invariant action, the gauge-fixing procedure and the higher derivative gauge invariant extensions of the action. 2 Deformed QED via Seiberg-Witten Map The starting point is the undeformed QED with its local gauge invariance S inv = d 4 x [ ψiγ µ D µ mψ 14 ] F µνf µν, 1 where D µ ψ = µ ia µ ψ, 2 F µν = µ A ν ν A µ. 3 1

3 S inv is invariant with respect to infinitesimal gauge transformations: δ λ A µ = µ λ, δ λ ψ = iλψ, δ λ ψ = i ψλ. 4 In order to get a θ-deformed QED in the sense of Seiberg and Witten [1] one defines  µ = A µ +A µa ν ;θ ρσ +Oθ 2, ˆψ = ψ +ψ ψ,a ν ;θ ρσ +Oθ 2, with the corresponding Seiberg-Witten maps given by ˆλ = λ+λ λ,a ν ;θ ρσ +Oθ 2, 5  µ A ν + ˆδˆλ µ A ν = µA ν +δ λ A ν, 6 ˆψψ,A ν + ˆδˆλˆψψ,Aν = ˆψψ +δ λ ψ,a ν +δ λ A ν. 7 The deformed infinitesimal gauge transformations are defined as usual [7]: ] ˆδˆλ µ = µˆλ+i [ˆλ,  µ = µˆλ+iˆλ  µ iâµ ˆλ, ˆδˆλˆψ = iˆλ ˆψ, ˆδˆλ ˆ ψ = iˆ ψ ˆλ, 8 where we have used the star-product [4]. Expanding 8 in powers of θ one gets ˆδˆλ µ = µˆλ θ ρσ ρ λ σ A µ +Oθ 2, With 6 8 one gets the following solutions of 5: ˆδˆλˆψ = iˆλˆψ 1 2 θρσ ρ λ σ ψ +Oθ 2. 9 A µ = 1 2 θρσ A ρ σ A µ +F σµ, ψ = 1 2 θρσ A ρ σ ψ, λ = 1 2 θρσ A ρ σ λ. 10 In terms of the commutative fields 3 the deformed field strength becomes ˆF µν = µ  ν ν  µ +θ ρσ ρ A µ σ A ν +Oθ 2 = F µν +θ ρσ F µρ F νσ A ρ σ F µν +Oθ The infinitesimal gauge transformation of ˆF µν is calculated with δ λ A µ = µ λ from now on we omit the symbol Oθ 2 ˆδˆλ ˆFµν = θ ρσ ρˆλ σ ˆFµν = θ ρσ ρ λ σ F µν. 12 2

4 Now we are able to write down the analogue of 1, leading to the following generalization [ ] ˆΣ inv = d 4 x ˆ ψ iγ µ 1 ˆDµ m ˆψ 4 ˆF µν µν ˆF [ ] = d 4 x ˆ ψ iγ µ 1 ˆDµ m ˆψ 4 ˆF ˆFµν µν, 13 with Using the above expansions in θ one gets Σ inv = ˆD µˆψ = µˆψ i  µ ˆψ. [ d 4 x { ψiγ µ D µ ψ 1 2 θρσ µ A ρ σ ψ +A ρ µ σ ψ + i ] 2 θρσ A ρ σ A µ +F σµ ψ +A µ A ρ σ ψ i ρ A µ σ ψ 1 2 θρσ A ρ σ ψiγ µ D µ ψ m ψψ mθρσ ρ A σ ψψ 1 4 F µνf µν 1 2 θρσ F µρ F νσ F µν 1 4 F ρσf µν F µν }. 14 Equation 14 contains non-renormalizable vertices of dimension 6. The quantity θ ρσ will be considered as a classical unquantized external field with dimension 2. The θ-deformed action 14 is invariant with respect to 4. 3 Gauge-Fixing à la BRST In order to quantize the system 14 we need a gauge-fixing term which allows of the calculation of the photon propagator. This can be done in a twofold manner: Corresponding to the usual BRST-quantization procedure one replaces the infinitesimal gauge parameters λ and ˆλ by Faddeev-Popov ghost fields c and ĉ leading to the following BRST-transformations for the Abelian structure Corresponding to the noncommutative structure of 9 one gets Nilpotency of 15 and 16 implies sa µ = µ c, sψ = icψ. 15 ŝâµ = µ ĉ θ ρσ ρ c σ A µ, ŝˆψ = iĉ ˆψ 1 2 θρσ ρ c σ ψ. 16 sc = 0, 17 ŝĉ = 1 2 θρσ ρ c σ c. 18 3

5 Clearly, due to 10 one has also Introducing now two BRST-doublets ˆ c = c and ˆB = B with ĉ = c 1 2 θρσ A ρ σ c. 19 ŝˆ c = s c = ˆB = B, ŝˆb = sb = 0, 20 one has two possibilities to construct Faddeev-Popov actions: Σ i = d 4 xs c µ A µ + α 2 cb = d 4 x B µ A µ c 2 c+ α 2 B2, 21 and ˆΣ ii = d 4 xŝˆ c µ Â µ + α = 2ˆ cˆb d 4 x B µ Â µ c µ ŝâµ + α 2 B2. 22 ii An expansion in θ of ˆΣ leads to { Σ ii = d 4 x B µ A µ c 2 c+ α 2 B2 1 } 2 θρσ [B µ A ρ σ A µ +F σµ + c 2 ρ ca σ 2 c µ ρ c σ A µ ]. 23 The case described by 21 is the usual linear µ A µ = 0 gauge, whereas the Faddeev-Popov action of 23 corresponds to a highly nonlinear gauge. However, one has to stress that both cases lead to the same photon propagator A µ A ν 0 = 1 k 2 +iǫ g µν 1 α k µk ν. 24 k 2 In terms of 15, 17 and 20 the symmetry content of 21 and 23 may be expressed by a linearized Slavnov-Taylor identity [ ] SΣ 0 θ = d 4 x µ c δσ0 θ +B δσ0 θ = 0, 25 δa µ δ c with Σ 0 θ = Σ inv + { Σ i Σ ii. 26 Formula 25 implies the following transversality condition for the 2-point 1PI vertex functional x µ δ 2 Σ 0 θ δa µ xδa ν y = 0, 27 4

6 meaning that the vacuum polarization of the photon is transverse in momentum space p µ Π µν p, p = In a companion paper [6] we have studied perturbatively this transversality condition. However, at this stage one has to state that the total action 26 is not the whole story. Because of the deformation parameter θ µν which is treated as a constant external field one can construct further gauge invariant quantities. Such additional terms in the BRST-invariant total action are in fact needed for the one-loop renormalization [6] procedure. 4 Higher Derivative Gauge Invariant Terms in the Action The presence of a constant external field θ allows in principle to add infinitely many gauge invariant terms to the Lagrangian. Due to this fact one can construct the following bilinear BRST-invariant action in the photon sector 1 Σ Max h.d. = with { d 4 x 1 2 F µν 2 Fµν 1 2 µ F µν ρ ν F ρ 1 4 F µν 2 2 F µν 1 4 θ2 F µν 2 2 F µν + 1 µ } 2 ν 1 F µν F µν 2 2 Fµν +higher derivative terms, 29 Alternatively, 29 may be rewritten as Σ Max h.d. = µ = θ µ ν ν, µ = θ µ ρ θρ σ σ, θ 2 = θ ν µ θ µ ν, F µν = θ µρ F ν ρ, F = F µ µ. 30 { d 4 x 1 2 F µν 2 Fµν 1 2 µ F µν ρ ν F ρ 1 4 F µν 2 2 F µν 1 2 Aµ [ g µν + µ ν + ν µ θ ρ µθ ρν ] A ν 1 4 θ2 F µν 2 2 F µν A 2 2 A+higher derivative terms } Note that d 4 x 1 2 F µν 2 Fµν and d 4 x 1 2 µ F µν ρ F ν ρ are of topological nature. Observe also that A is gauge invariant. 5

7 In a similar way one can construct with D µ = θ ν µ D ν D µ = µ ia µ also higher derivative terms for the Dirac theory Σ Dir h.d. = { d 4 x m ψ D D +D D 2 +D 2 D ψ + ψγ µ id µ D D+D D 2 +D 2 D ψ + ψγ µ i D µ D 2 1+D D +D D 2 +D 2 D } ψ +h.d.. 32 It is interesting to notice that one can construct also four-fermion vertices. For example one has Σ Dir 4,h.d. = d 4 x { ψ Dµ ψ ψ D µ ψ + ψ D µ D D +D D 2 +D 2 D ψ ψ D } µ ψ +h.d.. 33 However, these terms cannot be obtained from a truly noncommutative action by the Seiberg- Witten map because the corresponding expression would be non-local. Therefore, we expect that contributions of this type, which are possible for individual Feynman graphs, cancel on the level of Green s functions. Equations imply that even at the classical level one has θ-dependent terms. This entails the following form of the vacuum polarization of the photon in momentum space representation Π 0 µνp, p;θ p 2 2 θ 2 p µ p ν g µν p 2 + p 2 p 2 p µ p ν g µν p 2 +p 2 2 p µ p ν +p 2 2 g µν p 2 +p µ p ν +p ν p µ +p 2 θ ρ µθ ρν, 34 in observing that no terms linear in θ µν occur. Due to p p = 0, p p = p 2, 35 it is obvious that 34 fulfills the transversality condition 28, where we have used a notation analogous to 30. In order to be complete the total action is given by { } Σ 0 Σ i θ = Σ inv + Σ ii +Σ Max h.d. +Σ Dir h.d. +ΣDir 4,h.d.. 36 In the limit θ 0 one recovers the ordinary QED with a free Faddeev-Popov sector. One has to stress also that equation 29 is required for the renormalization procedure at the one-loop level for the calculation of Π 1 µνp, p;θ [6]. 5 Non-Abelian Extension: Pure Yang-Mills Case NCYM We start with the non-abelian extension of 5 in considering only the gluon sector. The corresponding gluon field is now Lie-algebra valued: A µ x = A a µxt a, 37 6

8 where the T a are the generators of a UN gauge group with [T a,t b ] = if abc T c, {T a,t b } = d abc T c, Tr T a T b = δ ab, T 0 = 1 N ½. 38 The corresponding non-abelian field strength is defined as usual The infinitesimal gauge transformations are F µν = µ A ν ν A µ i[a µ,a ν ]. 39 δ λ A µ = µ λ+i[λ,a µ ] D µ λ, δ λ F µν = i[λ,f µν ]. 40 As shown in [1] the Seiberg-Witten map for the non-abelian extension is  µ = A µ +A µ = A µ 1 4 θρσ {A ρ, σ A µ +F σµ }+Oθ 2, 41 ˆλ = λ+λ = λ+ 1 4 θρσ { ρ λ,a σ }+Oθ From follows with the help of 41 ˆF µν = µ  ν ν  µ i [µ,Âν] 43 ˆF µν = F µν θρσ 2{F µρ,f νσ } {A ρ,d σ F µν + σ F µν }+Oθ Corresponding to 40 one has ] ˆδˆλ µ = µˆλ+i [ˆλ,  µ, ] ˆδˆλ ˆFµν = i [ˆλ, ˆFµν. 45 In the sense of Seiberg and Witten 45 implies ˆδˆλ µ = µˆλ+i [ˆλ,  µ ] 1 2 θρσ ρ λ σ A µ ρ A µ σ λ+oθ Inserting 41 and 42 one arrives finally at ˆδˆλ µ = D µ λ+ 1 4 θρσ { µ ρ λ,a σ }+{ ρ λ, µ A σ } + i 4 θρσ [{ ρ λ,a σ },A µ ] [λ,{a ρ, σ A µ +F σµ }]+Oθ

9 Now we are ready to construct the gauge invariant non-abelian action. Following [3] one has at the classical level ˆΣ inv = 1 d 4 xtr ˆFµν 4g ˆF µν = 1 d 4 xtr ˆFµν ˆFµν g 2 Using 44 one gets for 48 Σ inv = 1 4g 2 = 1 4g 2 d 4 xtr F µν F µν θρσ 2{F µρ,f νσ }F µν {A ρ,d σ F µν + σ F µν }F µν d 4 x Fµν a Faµν +θ ρσ FµρF a νσf b cµν 1 2 Aa ρd σ F µν b F cµν 1 2 Aa ρ σ FµνF b cµν d abc. 6 Gauge-Fixing for Pure Yang-Mills Theory Similar to the QED case there are again two possibilities of gauge-fixing for the non-abelian generalization. One replaces the infinitesimal gauge parameters by the corresponding ghost fields: The first choice leads to 49 λ c, ˆλ ĉ. 50 sa µ = µ c+i[c,a µ ] = D µ c, With the second possibility of 50 one gets from 42 sc = ic 2 = i {c,c} ĉ = c+ 1 4 θρσ { ρ c,a σ }+Oθ Equations 52 and 41 together with 51 allow now to calculate the BRST-variations for µ and ĉ: ŝâµ = sa µ 1 4 θρσ {sa ρ, σ A µ +F σµ }+{A ρ, σ sa µ +sf σµ }, ŝĉ = sc+ 1 4 θρσ { ρ sc,a σ } [ ρ c,sa σ ]. 53 By explicit calculations one verifies that the equations 53 follow by θ-expansion from [ ] ŝâµ = µ ĉ+i ĉ,âµ, ŝĉ = iĉ ĉ. 54 8

10 The above results may be summarized by the Seiberg-Witten map for the BRST-transformation  µ A ν +ŝâµa ν = µA ν +sa ν. 55 For practical reasons we choose now the simpler possibility for the gauge-fixing procedure: Σ = d 4 xtr s c µ A µ + α 2 cb = d 4 xtr B µ A µ c µ µ c+i[c,a µ ]+ α 2 B2, where c and B are the corresponding non-abelian antighost and multiplier field. To be complete one also adds further gauge invariant pieces to the total gauge-fixed action. They are obtained from 29 by replacing the derivatives by the covariant derivatives and the Abelian by the non-abelian field strength. The total BRST-invariant action is thus Σ 0 θ = Σ inv +Σ +Σ h.d. = 1 d 4 xtr F 4g 2 µν F µν θρσ 2{F µρ,f νσ }F µν {A ρ,d σ F µν + σ F µν }F µν + d 4 xtr B µ A µ c µ µ c+i[c,a µ ]+ α 2 B2 1 2 F µν D 2 Fµν Dµ F µν D ρ F ν ρ 1 4 F µν D DF µν 1 4 F µν D 2 D2 F µν 1 4 θ2 F µν D 2 2 F µν Dµ D ν F µν F µν D 2 2 Fµν +higher derivative terms. 57 In order to characterize the BRST-symmetry of this model one has to introduce BRST-invariant external sources ρ µ,σ allowing to describe the non-linear BRST-transformations consistently. This leads to Σ ext = d 4 xtrρ µ sa µ +σsc 58 and to the action Σ 0 θ,tot = Σ0 θ +Σ ext. 59 The BRST-symmetry is therefore encoded at the classical level by the non-linear Slavnov-Taylor identity [ ] 0 δσ SΣ 0 θ,tot = d 4 θ,tot δσ 0 θ,tot xtr + δσ0 θ,tot δσ 0 θ,tot δρ µ δa µ δσ δc +B δσ0 θ,tot = δ c 7 Conclusion and Outlook We have defined a θ-deformed QED with the help of Seiberg-Witten maps for photon and fermion fields at the classical level. In the same manner we also discussed a θ-deformed Yang- Mills model. In a next step we focus on the quantization procedure. Some preliminary work has been done in [6], where the quantization of a θ-deformed Maxwell theory is presented. There, oneloop corrections to the 1PI two-point function for the photon are investigated in the presence of non-renormalizable interaction vertices. 9

11 8 Acknowledgement We would like to thank Harald Grosse, Karl Landsteiner, Stefan Schraml, Raymond Stora, and Julius Wess for enlightening discussions. References [1] N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP [hep-th/ ]. [2] J. Madore, S. Schraml, P. Schupp and J. Wess, Gauge theory on noncommutative spaces, Eur. Phys. J. C [hep-th/ ]. [3] B. Jurco, S. Schraml, P. Schupp and J. Wess, Enveloping algebra valued gauge transformations for non-abelian gauge groups on non-commutative spaces, Eur. Phys. J. C [hepth/ ]. [4] T. Filk, Divergencies in a field theory on quantum space, Phys. Lett. B [5] C. Becchi, A. Rouet and R. Stora, Renormalization Of Gauge Theories, Annals Phys [6] A. A. Bichl, J. M. Grimstrup, L. Popp, M. Schweda and R. Wulkenhaar, Perturbative Analysis of the Seiberg-Witten Map, hep-th/ [7] M. Hayakawa, Perturbative analysis on infrared and ultraviolet aspects of noncommutative QED on R 4, hep-th/