Renormalization and asymptotic states in Lorentz-violating QFT
aa r X i v : . [ h e p - ph ] O c t Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) RENORMALIZATION AND ASYMPTOTIC STATES INLORENTZ-VIOLATING QFT
M. CAMBIASO
Universidad Andres Bello, Departamento de Ciencias F´ısicas,Facultad de Ciencias Exactas, Av. Rep´ublica 220, Santiago, Chile. ∗ E-mail: [email protected]
R. LEHNERT
Indiana University Center for Spacetime Symmetries,Bloomington, IN 47405, USA.E-mail: [email protected]
R. POTTING
CENTRA, Departamento de F´ısica, Universidade do Algarve,8005-139 Faro, Portugal.E-mail: [email protected]
Radiative corrections in quantum field theories with small departures fromLorentz symmetry alter structural aspects of the theory, in particular the defini-tion of asymptotic single-particle states. Specifically, the mass-shell condition,the standard renormalization procedure as well as the Lehmann–Symanzik–Zimmermann reduction formalism are affected.
In the presence of Lorentz breakdown, previous analyses involving theproperties of freely propagating particles have been performed under thetacit assumption that the physics of free particles is determined by thequadratic pieces of the corresponding Lagrangian. However, by analyzingradiative corrections for a sample Lorentz-violating (LV) Lagrangian con-tained in the SME, we claim that the previous line of reasoning fails.We consider the bare gauge-invariant Lagrangian density for single-flavor QED within the minimal SME in the presence of the c µν and˜ k µν only. Both are symmetric and traceless and the latter is definedas a ˜ k µν = ( k F ) µανα . One-loop multiplicative renormalizability of themodel is assumed. For convenience we write the gauge-fixed bare La-grangian together with appropriate IR regularization terms. To under-stand the external fermion states we need to consider radiative corrections roceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) to the fermion two-point function, wherefrom the wave-function renormal-ization “constant” is read-off as the residue of the one-particle pole. Asa zeroth-order system on which perturbation theory is set up, we choosethe full renormalized quadratic Lagrangian together with the LV part.The remaining non-quadratic contributions to the Lagrangian being takenas perturbations. Thus, the proper two-point function adopts the formΓ (2) ( p ) = Γ µ p µ − m − Σ( p µ ), where Γ µ = γ µ + c µν γ ν andΣ( p µ ) = Σ LI ( /p ) + Σ LV ( p , c pγ , ˜ k pγ ) + δ Σ( p µ , c µν , ˜ k µν ) , (1)with c pγ ≡ c µν p µ γ ν and similarly for ˜ k pγ . The first term on the RHS ofEq. (1) denotes the usual Lorentz-symmetric contributions. The rest comefrom LV terms. The last, however, contains terms that are not present in theoriginal Lagrangian. To linear order in LV and given that electromagneticinteractions preserve C, P, and T, we can write:Σ LV ( p , c pγ , ˜ k pγ ) = f c ( p ) c pγ + f ˜ k ( p ) ˜ k pγ , (2) δ Σ( p µ , c pp , ˜ k pp ) = f c ( p ) c pp m + f c ( p ) /pc pp m + f ˜ k ( p ) ˜ k pp m + f ˜ k ( p ) /p ˜ k pp m . (3)The functions f ci ( p ) and f ˜ ki ( p ) are calculable to any order in the fine-structure constant α . Following Ref. 3 we extract the propagator pole: a Γ (2) ( p ) = Z − R ¯ P ( p ) + ¯ P ( p ) Σ (cid:0) /p, c pγ , ˜ k pγ , ( c, k ) pp (cid:1) ¯ P ( p ) , (4)which yields the desired property: Γ (2) ( p ) − = Z R ¯ P ( p ) − + finite.Doing standard perturbative calculations to first order in LV coefficientsand to first order in α , the one-loop corrected wave-function renormalizationand the fermion propagator pole, respectively read: Z − R = ( Z R ) − − α πm h c pp + ˜ k pp i , (5)¯ P ( p ) = /p + ( c phys ) pγ − m phys + α πm (cid:16) c phys ) pp − (˜ k phys ) pp (cid:17) , (6)where: m phys = m + αmπ (cid:20) − γ E −
34 ln (cid:18) m πµ (cid:19)(cid:21) (7)is the usual loop-corrected mass in the minimal-subtraction scheme, and( c phys ) µν = c µν − α π (cid:20) − γ E − ln (cid:18) m πµ (cid:19)(cid:21) (cid:0) c µν − ˜ k µν (cid:1) . (8) a The dependence of Γ (2) ( p ) on c pγ and ˜ k pγ introduce further subtleties. Details of theone-loop calculation together with the necessary Feynman rules can be found in Ref. 4. roceedings of the Sixth Meeting on CPT and Lorentz Symmetry (CPT’13) Equation (8) expresses the radiatively corrected, physical value of theLorentz-violating parameter c µν in terms of its tree-level value. Note thatboth c µν and the mass scale µ are unphysical, renormalization-scheme-dependent quantities, unlike ( c phys ) µν , which is in principle measurable.This is in fact granted by the cancellation between the terms containing µ in Eq. (8) with those coming from the running of c µν . UV divergentquantities have been regularized with dimensional regularization. Equations(6) and (8) show that LV radiative corrections depend only on (2 c pp − ˜ k pp )and c µν phys is indeed infrared finite. Furthermore, it can be shown that the O ( α ) LV radiative corrections to the dispersion relation are proportional to(2 c pp − ˜ k pp ) too. This property is a requirement coming from considerationsof field redefinitions which imply that, in this context, physically observableradiative effects should depend on 2 c µν − ˜ k µν only. Turning to the descrip-tion of asymptotic states, Eq. (6) implies a modified equation of motion forin- and out-spinors. Correspondingly an adapted LSZ reduction formula forthe fundamental scattering amplitude h f | i i , controlled by Eqs. (5) and (6),is obtained.Our results allow us to conclude that asymptotic single-particle states offermions in Lorentz-violating QFT receive concrete modifications due to ra-diative corrections. Specifically, the corresponding Dirac equation turns outto be modified by Lorentz-violating operators not present in the Lagrangian,a novel feature of LV QFT as it does not occur in the Lorentz-symmetriccase. Also, the fermion wave-function renormalization is no longer a con-stant but rather depends on the LV coefficients under consideration and onthe external momentum as well. As shown in Ref. 4 a non-trivial cancella-tion of IR divergences is also achieved in the fermion’s dispersion relationand wave-function renormalization function. Furthermore, IR divergencescancel in the elastic Coulomb scattering cross section when the contributionof soft-photon emission from the corresponding external legs is taken intoaccount. Acknowledgments
This work has been supported in part by: the Portuguese Funda¸c˜ao paraa Ciˆencia e a Tecnologia, the Mexican RedFAE, UNAB Grant DI-27-11/R,FONDECYT No. 11121633 and by the IUCSS.
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