LIV Dimensional Regularization and Quantum Gravity effects in the Standard Model Jorge Alfaro Facultad de F´ısica, Pontificia Universidad Cat´olica de Chile Casilla 306, Santiago 22, Chile. [email protected] (Dated: February 7, 2008) Recently,wehaveremarked that themain effect of QuantumGravity(QG)will betomodify the measureofintegrationofloopintegralsinarenormalizableQuantumFieldTheory. IntheStandard Model this approach leads to definite predictions, depending on only one arbitrary parameter. In particular,wefoundthatthemaximalattainablevelocity forparticlesisnotthespeedoflight,but depends on the specific couplings of the particles within the Standard Model. Also birrefringence occurs for charged leptons, but not for gauge bosons. Our predictions could be tested in the next generation of neutrinodetectors such as NUBE.In this paper, we elaborate more on this proposal. 5 In particular, we extend the dimensional regularization prescription to include Lorentz invariance 0 violations(LIV) of themeasure, preserving gauge invariance. Then we comment on theconsistency 0 of our proposal. 2 n Recently, there has been a growing consensus that d(d+2) a (3) J smallLIVtermscouldbe the clue to obtainphenomeno- 4 7 logical predictions from Quantum Gravity [1, 2, 3, 4, 5, 1 ddk kµkν (−1)n−1igµν Γ(n−1−d/2) = 1 6,7,8,9]. ThereasonforthisisthataverytinyLIVpro- g Z (2π)d(k2−∆)n (4π)d/2 2 Γ(n) duces a great effect on processes happening at energies 1 v1 much below the Planck scale. ∆n−1−d/2(4) 9 In a previous letter[10] , we have remarked that the 1 ddk k2 (−1)n−1idΓ(n−d/2−1) 2 main effect of QG is to deform the measure of integra- g Z (2π)d(k2−∆)n = (4π)d/2 2 Γ(n) 1 tion of Feynman graphsat largefour momenta by a tiny 1 1 LIV. Within the Standard Model, such LIV implies sev- (5) 0 eralremarkablepredictions,whicharewhollydetermined ∆n−d/2−1 5 up to an arbitrary parameter.Our predictions could be 1 ddk 1 (−1)ni Γ(n−d/2) 0 = / tested in the next generation of neutrino detectors such g Z (2π)d(k2−∆)n (4π)d/2 Γ(n) h as NUBE[12, 13] 1 t (6) - LIVDimensionalRegularizationWegeneralizedi- ∆n−d/2 p mensionalregularizationtoaddimensionalspacewithan e h arbitrary constant metric gµν. We work with a positive We follow the notation of [11]. : definite metric first and then Wick rotate. We will illus- These definitions preserve gauge invariance, because v trate the procedure with an example. Here g =det(g ) the integration measure is invariant under shifts. As a i µν X and ∆>0. check, consider: r a 1 ddk k +l k 1 ddk k k K = [ µ µ − µ ] µ ν = µ g Z (2π)d [(k+l)2−m2+i0]n [k2−m2+i0]n g Z (2π)d(k2+∆)n (7) 1 ∞dttn−1 ddk k k e−t(gαβkαkβ+∆) = to first order in lµ. With the definitions stated above, it gΓ(n)Z0 Z (2π)d µ ν is trivial to check that Kµ vanishes to first order in lµ. g ∞ To get a LIV measure, we assume that µν dttn−2−d/2e−t∆ = 2(4π)d/2Γ(n)Z 0 g =η +aη η ǫ (8) 1 g Γ(n−1−d/2) 1 µν µν µ0 ν0 µν (1) (4π)d/2 2 Γ(n) ∆n−1−d/2 whereǫ=2−d. Aformerlydivergentintegralwillhavea 2 In the same manner, we obtain, after Wick rotation: poleatǫ=0,sowhenwetakethephysicallimit,ǫ−>0, the answer will contain a LIV term, precisely of the sort 1 ddk k k k k (−1)ni Γ(n−d/2−2) µ ν ρ σ = we obtain in [10]. g Z (2π)d(k2−∆)n (4π)d/2 Γ(n) That is, LIV dimensional regularizationconsists in: 1 1 1)Calculating the d-dimensional integrals using a gen- (g g +g g +g g )(2) ∆n−d/2−24 µν ρσ µρ νσ µρ νσ eral metric g . µν 1 ddk (k2)2 (−1)ni Γ(n−d/2−2) 1 2) Gamma matrix algebra is generalized to a general = g Z (2π)d(k2−∆)n (4π)d/2 Γ(n) ∆n−d/2−2 metric gµν. 2 3) At the end of the calculation, replace g = η + in addition there will be also tiny LIV modifications to µν µν aη η ǫ and then take the limit ǫ−>0. finite Standard Model predictions, as in [6]. µ0 ν0 Asaconcreteexample,letus evaluateanintegralthat appears in the calculation of the one loop results of [10]: Acknowledgements ddk kµkν Aµν = = (9) Z (2π)d[k2−m2+i0]3 The work of JA is partially supported by Fondecyt i gµν Γ(2− d2) 1 (10) 1010967 and Ecos-Conicyt C01E05. He wants to thank (4π)d/2 2 2 (m2)2−d2 A.A. Andrianov for several useful remarks. He wants to i ηµν −aδµδνǫΓ(2− d) 1 thank the hospitalityofEcoleNormaleSuperieure,Paris. = 0 0 2 (11) In particular he wants to thank C. Kounnas, C. Bachas, (4π)d/2 2 2 (m2)2−d2 V. Kazakovand A. Bilal for interesting discussions. i ηµν = ( −aδµδν)+a finite LI term (12) 4(4π)2 ǫ 0 0 From now on we take a =α. (4π)2 Using LIV Dimensional Regularization we obtain the [1] Amelino-Camelia, G. et al., Tests of quantum gravity LIV photon self-energy in QED: from observationsofgamma-ray bursts,Nature393, 763 (1998). 4 [2] Gambini, R. and Pullin, J. , Nonstandard optics from LΠµν(q)= e2αq q α β quantum space time, Phys. Rev.D 59, 124021 (1999). 3 [3] Alfaro,J.Morales-T´ecotl,H.A.andUrrutia,L.F.,Quan- (ηαβδµδν +ηµνδαδβ −ηνβδµδα−ηµαδνδβ) (13) 0 0 0 0 0 0 0 0 tum gravity corrections to neutrino propagation, Phys. Rev. Lett.84, 2318 (2000). LIV Dimensional Regularization reinforces our claim [4] Alfaro, J. Morales-T´ecotl, H.A. and Urrutia, L.F., Loop that these tiny LIV’s originates in Quantum Gravity. In quantumgravityandlightpropagation,Phys.Rev.D65, fact the sole change of the metric of space time is a cor- 103509(2002). rection of order ǫ and this is the source of the effects [5] Colladay, D. and Kostelecky, V.A., Lorentz-violating studied in [10]. Quantum Gravity is the strongest can- extension of the standard model, Phys. Rev. D58, 116002(1998). didate to produce such effects because the gravitational [6] Coleman, S.and Glashow, S.L., High-energy tests of field is precisely the metric of space-time and tiny LIV Lorentz invariance, Phys. Rev.D 59, 116008 (1999). modifications to the flat Minkowsky metric may be pro- [7] Collins,J.etal,Lorentzinvarianceandquantumgravity duced by quantum fluctuations. :an additional fine-tuning problem?, Phys.Rev.Lett.93, Consistency of the approach In [6] a Modified 191301(2004). Standard Model with the LIV structure of [10] has al- [8] T.Kifune, Astroph. J. Lett. 518, 21 (1999); G.Amelino- readybeen considered,albeit with arbitraryparameters. Camelia and T.Piran, Phys. Lett. B 497, 265 (2001); J.Ellis, N.E.Mavromatos and D.V.Nanopoulos, Phys. It was shown there that the theory is anomaly free and Rev. D 63, 124025 (2001); G.Amelino-Camelia and renormalizable: By rescaling coordinates and fields, the T.Piran, Phys. Rev. D 64, 036005 (2001); G.Amelino- anomaly can be shown to be the same as in the Lorentz Camelia, Phys. Lett. B 528, 181 (2002). invariant situation, which is anomaly free. These results [9] J.Alfaro and G.Palma, Phys. Rev. D 65, 103516 (2002); (anomaly free and renormalizability) also apply to our J.Alfaro and G.Palma, Phys. Rev.D 67, 083003 (2003). proposal, if we preserve gauge invariance, which is made [10] J. Alfaro, hep-th0412295. easybyLIVDimensionalRegularization. Inthecase[10], [11] Peskin, M. and Schroeder D., An introduction to Quan- tumFieldTheory,Addison-WesleyPublishingCompany, atonelooptheStandardModelpredictionsarenotmodi- New York 1997. fiedexceptfortheLIVintroducedinprimitivelylogarith- [12] Waxman,E.and Bahcall, J.,High energy neutrinosfrom mically divergentintegrals(whichis determineduniquely cosmological gamma ray burst fireballs, Phys. Rev. up to a single constant, that measures the deformation Letts.78 , 2292(1997). of the space-time metric when we approach the physi- [13] Roy,M.,Crawford,H.J.andTrattner,A.,Theprediction caldimension). Starting fromthe two loopcontribution, and detection of uheneutrino bursts,astro-ph/9903231.