Comment on the 2nd Order Seiberg Witten Maps

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1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Comment on the 2nd Order Seiberg Witten Maps Josip Trampetić Michael Wohlgenannt ienna, Preprint ESI October 11, 2007 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via

2 Comment on the 2nd order Seiberg-Witten maps Josip Trampetić 1 and Michael Wohlgenannt 2 1 Theoretical Physics Division, Rudjer Bošković Institute, Zagreb, Croatia 2 Erwin Schrödinger International Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria Dated: October 11, 2007 In this comment, we discuss the Seiberg-Witten maps up to the second order in the noncommutative parameter θ. They add to the recently published solutions in [1]. Expressions for the vector, fermion and Higgs fields are given explicitly. PACS numbers: Nx, q, i The main purpose of this comment is to complete the second order Seiberg-Witten SW maps constructed in [1, 2]. We consider canonically deformed space-time. The commutator of coordinates is given by the constant antisymmetric matrix θ µν, [x µ, x ν ] x µ x ν x ν x µ = iθ µν, 1 where we have used the Weyl-Moyal star product { i f gx = exp } 2 θµν fygx y µ x ν. 2 y x A prescription for constructing arbitrary gauge theories on a NC space-time was presented in [3]. Seiberg- Witten SW maps [3, 4] relate noncommutative NC gauge fields and ordinary fields in commutative theory via a power series expansion in θ. In simplest possible approach to the construction of NC gauge field theories all field products are replaced by -products. This approach, however, fails for general gauge theories. For example for SUN gauge theories, the -commutator of two infinitesimal gauge transformations does not close in the SUN Lie algebra. This is the reason one has to go to the enveloping algebra [5] of the Lie algebra of a given group. Higher-order SW terms are now expressed in terms of the zeroth-order commutative fields, and as a consequence we do have the same number of degrees of freedom as in the commutative case. The SW maps are not unique. The free parameters are chosen such that the non-commutative gauge fields are hermitian and the action is real. Still, there is some remaining freedom including the freedom of classical field redefinition and noncommutative gauge transformation. As already remarked in [1], the second order solution for the gauge field - and therefore also for the field strength - given in [2] are not correct. We will provide the corrected expressions. Most importantly, in this comment we want to add the second order expansion of the hybrid SW map for the Higgs field to the work [1]. This is necessary if one wants to consider the Noncommutative Standard Model NCSM [6 8] including the Yukawa couplings. We have computed a special solution and not the most general one, because of the complexness. The SW map of the gauge parameter is a solution of the relation i α Λ i Λ α + [ Λ α, Λ ] = i Λ α. 3 This relation has to be solved order by order. Therefore, we expand the noncommutative gauge parameter Λ α in θ and in terms of the commutative gauge parameter α, Λ α = α + Λ θ α[ ] + Λ θ2 α [ ] + Oθ 3. Noncommutative fields and gauge parameters are denoted by a hat throughout the paper. To first order in θ, the equivalence condition 3 reads i α Λ θ Λ θ + [α, Λθ ] + [Λθ α, ] iλθ α 4 and to second order = i 2 θµν { µ α, ν }, Λ 2 = 1 8 θµν θ κλ [ µ κ α, ν λ ] [Λ θ α, Λ θ ] i 2 θµν { µ Λ θ α, ν} { ν α, µ Λ θ }, 5 with α = i[α, ]. This is an inhomogeneous equation. The homogeneous part to order k is given by Λ k := i α Λ θk Λ θk α +[α, Λ θk ]+[Λ θk α, ] iλ θk α = 0. 6 The most general solution to first order, is given by Λ θ α = 1 2 θµν { ν, µ α} c, 7 where {A, B} c ca B + 1 cb A. The requirement of hermiticity fixes the free parameter c to c = 1/2. In this case, the general solution 7 becomes Λ θ α[ ] = 1 4 θµν { ν, µ α}. 8 A special solution for the second order equation reads: Λ θ2 α [ ] = 1 32 θµν θ κλ 9 { µ, { ν κ, λ α}} + { µ, { κ, ν λ α}} + {{ µ, ν κ }, λ α}} {{F µκ, ν }, λ α} 2i[ µ κ, ν λ α],

3 with the field strength F α = α α i[ α, ]. The NC gauge field transforms as α µ = µ Λ α i[ µ, Λ α ]. 10 The enveloping algebra valued gauge potential is therefore determined by the following consistency relations in first and second order α θ σ = σλ θ α i[ σ, Λ θ α ] θµν { µ σ, ν α}, 11 α θ2 σ = σ Λ θ2 α i[ σ, Λ θ2 α ] i[ θ σ, Λ θ α] θµν { µ θ σ, να} θµν { µ σ, ν Λ θ α } + i 8 θµν θ κλ [ µ κ σ, ν λ α], 12 where we have again used α σ θk = α σ θk i[α, σ θk ] and the expansion of the noncommutative gauge field µ [ ] = µ + µ θ θ2 [ ] + µ [ ] + Oθ3. The general solution in first order is θ µ = 1 2 θα {, α µ } c θα {, F αµ } c. 13 Choosing a hermitian gauge parameter Λ α, we obtain θ µ [ ] = 1 4 θα { α µ + F αµ, } 14 and µ θ2 [ ] = 1 64 θα θ 4[, [ µ, α ]] + 8{ α, {F µ, F }} + 8{ α, { F µ, }} + 2i{ α, { µ, }} 2{ α, { µ, }} { µ, {F α, F }} + 8{ α µ, {, }} + 2{ µ, { α, }} + 2{ α, {, µ }} 2{ µ α, {F, }} 2{ α, { µ, }} 4{ α, { µ, }} + 8{ α µ, } + 8i[ α µ, ] 2i[ µ F α, F ] 4i[ α µ, ] 4 α µ 4 µ α + 2i α µ 4i α µ 2i α µ 2i α µ + 4i α µ 2i µ α + 2i µ α + 2i µ α 2i µ α + 2i α µ + 4F α µ F. 15 Concerning Eq. 15, the second order of the SW map, we disagree with ref. [2]. The solution given there, does not satisfy the gauge equivalence relation 12. Next, by using the SW map of the gauge parameter, the expansion of the noncommutative gauge transformation for the NC fermion fields reads where α ψ = i Λ α ψ, 16 ψ[ψ, ] = ψ + ψ θ [ψ, ] + ψ θ2 [ψ, ] + Oθ 3. This leads to the first order in θ consistency relation, α ψ θ = iλ θ ψ 1 2 θµν µ α ν ψ, 17 where α ψ θk α ψ θk iαψ θk. To second order, we obtain α ψ θ2 = iλ θ2 α ψ + iλ θ ψ θ 1 2 θµν µ Λ θ α ν ψ θµν µ α ν ψ θ i 8 θµν θ κλ µ κ α ν λ ψ. The general solution to first order is given by ψ θ = 1 2 θµν ν µ ψ + 1 c 2 θµν µ ν ψ + d θ µν F µν. 19 The hermicity requirement c = 1/2, and choice d = 1/8, leaves us with ψ θ [ψ, ] = 1 2 θα α i 4 [ α, ] ψ. 20 A solution to the second order consistency relation is given by [2] ψ θ2 [ψ, ] = 1 32 θµν θ κλ 4i κ µ ν λ + 4 κ µ ν λ 4 κ µ ν λ + 4F κµ ν λ 4 ν κ µ λ + 8 ν F κµ λ 8i µ κ ν λ + 4i µ ν κ λ 2 κ µ λ ν + κ λ µ ν + 2i κ µ λ ν 2i ν λ κ µ 2 κ µ ν λ i[[ κ µ, ν ], λ ] 4i ν F κµ λ ψ. 21 Finally we consider NC noncommutative Higgs field Φ, which is related to the commutative ones by the hybrid SW map expansion Φ Φ[Φ,, ] = Φ + Φ θ [, ] + Φ θ2 [, ] + Oθ 3. This generalizes the Seiberg-Witten maps of both gauge bosons and fermions. Φ is a functional of two gauge fields and, and it transforms covariantly under the following gauge transformations: Φ[Φ,, ] = i Λ Φ i Φ Λ, 22

4 where Λ and Λ are the corresponding gauge parameters. Hermitian conjugation yields Φ[Φ,, ] = Φ[Φ,, ]. The covariant derivative for the noncommutative Higgs field Φ is given by D µ Φ = µ Φ i µ Φ Φ µ. 23 As explained in [6], the precise representations of the gauge fields and in the Yukawa couplings are inherited from the fermions on the left ψ and on the right side ψ of the Higgs field, respectively. The hybrid SW map for the Higgs boson up to second order is of course only unique up to a solution of the homogeneous equation. The most general solution to first order reads Φ θ [Φ,, ] = 1 2 θα 24 [ α Φ i 2 a αφ Φ α + α Φ i 2 αφ bφ α a α Φ + 1 ] 4 1 bφ α Conventionally, we choose a = b = 1 and obtain to the first order in θ Φ θ [Φ,, ] = 1 2 θα 25 [ α Φ i 2 αφ Φ α + α Φ i2 ] αφ Φ α, while for the second order we have found the following lengthy expression: Φ θ2 Φ θ2 [Φ,, ] = i 32 θα θ { α 4 Φ 3i Φ + 4iΦ + 4 Φ 2iΦ [ + 4i Φ + 4 Φ + 2i Φ 2Φ + 4 Φ + 4iΦ + 3 Φ 4 Φ 4 Φ + Φ i 2 ] + 8i Φ + 5 Φ 8Φ + 4i Φ 3 Φ + Φ 4 i + i i Φ + 4 Φ + 4Φ i + i i } [ + α 4 Φ + 4i Φ + 4Φ Φ + 4Φ + 4i Φ 2i Φ 4i Φ + Φ 4i ] [ + α Φ 4i ] 8i 4i 4 + α Φ 4i 4 [ [ + Φ + 2i α i ] + α 2i 3 ]. 26 The above expression may be written in a more convenient way as Φ θ2 [Φ,, ] = Φ θ2 [ ] + Φ θ2 r [ ] i 8 θµν θ κλ i κ λ µ Φ ν κ λ µ Φ ν i κ µ λ Φ ν +i κ µ ν Φ λ κ µ λ Φ ν κ µ ν Φ λ 2 κ µ Φ λ ν + κ µ Φ ν λ + i κ µ Φ λ ν 2i κ µ Φ ν λ 2 κ λ µ Φ ν 4i κ µ Φ λ ν + i κ µ Φ ν λ + κ µ Φ λ ν + κ µ Φ ν λ 2i κ λ µ Φ ν + κ λ Φ µ ν + κ Φ µ λ ν κ Φ µ ν λ i κ Φ µ λ ν + i κ Φ µ ν λ +2 κφ µ λ ν + i κ Φ λ µ ν + κ µ λ Φ ν κ µ ν Φ λ + i κ µ λ Φ ν i κ µ ν Φ λ + i κ µ Φ ν λ κ µ Φ λ ν + 2 κ µ Φ ν λ 1 32 θµν θ κλ 2 κ λ Φ µ ν, 27 where Φ θ2 [ ] and Φ θ2 r [ ] denote the second order expansion for fermion fields 21, Φ θ2 [ ] = ψ θ2 [ψ, ]ψ Φ, 28 in the latter case the gauge fields are supposed to act from the right, Φ θ2 r [ ] = 1 32 θµν θ κλ Φ 4i κ µ ν λ + 4 κ µ ν λ

5 4 κ µ ν λ + 4F κµ ν λ 4 ν κ µ λ + 8 ν F κµ λ 8i µ κ ν λ + 4i µ ν κ λ 2 κ µ λ ν + κ λ µ ν + 2i κ µ λ ν 2i ν λ κ µ 2 κ µ ν λ i[[ κ µ, ν ], λ ] 4i ν F κµ λ. 29 The solutions representing SW maps up to second order in θ for fermion fields and Higgs fields, i.e., Eqs. 21 and 26 or 27, are identical for = 0. Higher SW expansions of the NC gauge, fermion and Higgs fields up to second order in the noncommutative parameter θ, are important due to the further extension of previously published results. Specifically those are applications of the enveloping algebra based, θ expanded, approach to higher gauge groups [7] and to particular NCSM gauge sector representations [9]. Proof that SW noncommutative gauge theories are anomaly free and the properties of the gauge anomaly for general SW mapping as well as of the U1 A anomaly in noncommutative SUN theories [10] are certainly very important. This comment is very important regarding further investigations of renormalizability properties of the θ expanded NC field theories in general [11]. Certainly, recent results [12, 13], showing that gauge theories in the θ expanded, enveloping algebra based, approach are one-loop renormalizable at first order in θ are very encouraging. They give us hope that it would be possible to investigate higher-loop renormalizability up to the second order in the noncommutative parameter θ. Clearly, one may expect that the renormalizability principle should certainly help to minimize, or even cancel most of ambiguities of SW maps disucssed in [1, 2] and in this comment. Finally, it is necessary to comment that, due to the one-loop renormalizability [10 13], the associated high energy particle physics phenomenology [14, 15] becomes more robust [16]. We want to thank Fabian Bachmaier for contributing to this work and to H. Grosse and J. Wess for many fruitful discussions. M.W. also wants to acknowledge the support from Fonds zur Förderung der wissenschaftlichen Forschung Austrian Science Fund, project P N16. The work of J.T. is supported by the project of the Croatian Ministry of Science, Education and Sport and in part by the ESF, received in the framework of the Research Networking Programme on Quantum Geometry and Quantum Gravity in the form of a short visit grant No [2] L. Möller, JHEP [3] N. Seiberg and E. Witten, JHEP [4] J. Madore, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C [5] B. Jurčo, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C B. Jurčo, L. Möller, S. Schraml, P. Schupp and J. Wess, Eur. Phys. J. C [6] X. Calmet, B. Jurčo, P. Schupp, J. Wess and M. Wohlgenannt, Eur. Phys. J. C [7] P. Aschieri, B. Jurčo, P. Schupp and J. Wess, Nucl. Phys. B [8] B. Melic, K. Passek-Kumericki, J. Trampetic, P. Schupp, and M. Wohlgenannt, Eur. Phys. J. C ; Eur. Phys. J. C [9] W. Behr, N.G. Deshpande, G. Duplančić, P. Schupp, J. Trampetić and J. Wess, Eur. Phys. J. C ; G. Duplančić, P. Schupp and J. Trampetić, Eur. Phys. J. C [10] C. P. Martin, Nucl. Phys. B 652, ; F. Brandt, C. P. Martin and F. R. Ruiz, JHEP 0307, ; C. P. Martin and C. Tamarit, Phys. Rev. D 72, [11] R. Wulkenhaar, JHEP ; J. M. Grimstrup and R. Wulkenhaar, Eur. Phys. J. C ; A. Bichl, J. Grimstrup, H. Grosse, L. Popp, M. Schweda and R. Wulkenhaar, JHEP ; H. Grosse and R. Wulkenhaar, Lett. Math. Phys. 71, ; J. Nonlin. Math. Phys. 11S1, ; Commun. Math. Phys. 256, ;. Rivasseau, F. ignes-tourneret and R. Wulkenhaar, Commun. Math. Phys. 262, ; H. Grosse and M. Wohlgenannt, Eur. Phys. J. C [12] M. Buric and. Radovanovic, JHEP ; JHEP ; Class. Quant. Grav ; M. Buric,. Radovanovic and J. Trampetic, JHEP ; D. Latas,. Radovanovic and J. Trampetic, Phys. Rev. D [13] C. P. Martin, D. Sanchez-Ruiz and C. Tamarit, JHEP ; C. P. Martin and C. Tamarit, arxiv: [hep-th]. [14] J. Trampetić, Acta Phys. Polon. B [hepph/ ]; P. Schupp, J. Trampetic, J. Wess and G. Raffelt, Eur. Phys. J. C ; P. Minkowski, P. Schupp and J. Trampetic, Eur. Phys. J. C ; B. Melic, K. Passek-Kumericki and J. Trampetic, Phys. Rev. D ; Phys. Rev. D [15] T. Ohl and J. Reuter, Phys. Rev. D ; A. Alboteanu, T. Ohl and R. Ruckl, PoS HEP [arxiv:hep-ph/ ]; Phys. Rev. D 74, ; arxiv: [hep-ph]; arxiv: [hepph]. [16] M. Buric, D. Latas,. Radovanovic and J. Trampetic, Phys. Rev. D ; J. Trampetić, arxiv: v1 [hep-ph]. [1] A. Alboteanu, Th. Ohl and R. Rückl, [hep-th].