D.DjukanovicJ.Gegeliaand Ulf-G.Mei?ner

1Helmholtz Institute Mainz,University of Mainz,D-55099 Mainz,Germany

2Institute for Advanced Simulation,Institut für Kernphysik and Jülich Center for Hadron Physics,Forschungszentrum Jülich,D-52425 Jülich,Germany

3Tbilisi State University,0186 Tbilisi,Georgia

4Helmholtz Institut für Strahlen-und Kernphysik and Bethe Center for Theoretical Physics,Universit?t Bonn,D-53115 Bonn,Germany

The concept of triviality in quantum field theories originates from papers by Landau and collaborators studying the asymptotic behaviour of the photon propagator in quantum electrodynamics(QED)[1?2](for a review see e.g.Ref.[3]).Resumming the leading logarithmic contributions they found that the photon propagator has a pole at large momentum transfer.If this pole persists also in non-perturbative calculations then to avoid the apparent inconsistency QED has to be a non-interacting,i.e.trivial,theory.In calculations applying a finite cuto ffthis problem manifests itself as a singularity in the bare coupling for a finite value of the cuto ff.It is therefore impossible to remove the cuto ffunless the renormalized coupling vanishes.The issue of triviality has been investigated in Ref.[4]in a lattice version of QED.It was found in that work that while the Landau pole lies beyond the accessible region of the parameter space due to spontaneous chiral symmetry breaking,spinor QED(with four fl avors)does not exist as an interacting theory.

The standard model,in a modern point of view,is a leading order approximation to an effective field theory(EFT).[5]While the effective Lagrangian contains an in finite number of local interactions consistent with the underlying symmetries,at low energies the contributions in physical quantities of the interactions with coupling constants of negative mass dimensions(i.e.nonrenormalizable interactions in the traditional sense)are suppressed by powers of the energy divided by a large scale characterising those degrees of freedom,which are not explicitly taken into account in the effective theory.In the framework of EFT the solution of the leading order Wilson renormalization group equations might be obstructed at very large cutoffs,however,this should not be asevereproblem because of irrelevant interactions or omitted fields being important at short distances.[5]Therefore inconsistencies in the renormalization group analysis of renormalizable quantum field theories,like QED or?4theory of self-interacting scalars,might be absent in the corresponding EFT framework.

In this letter we address the consequences of treating QED as a leading order approximation of an EFT for the problem oftriviality.To that end we analyse the contributions of the next-to-leading order interaction,i.e.dimension five operator,the well-known Pauli term.We start with the most general U(1)locally gauge invariant effective Lagrangian of the electron(and the positron) fieldψinteracting with the electromagnetic fieldAμ

wheremis the(bare)electron mass,eis the(bare)electromagnetic charge,andLhocontains an in finite number of terms with operators of dimension six and higher.We assume that the contributions of these terms in the photon self-energy are suppressed compared to those of the Pauli term.We remark here that the standard QED Lagrangian given by the first line of Eq.(1)describes the experimental data very well.From the modern point of view this is because the contributions of terms in the second line are beyond the current accuracy of the data.In particular,the anomalous magnetic moment of the electron in standard QED gets contributions only from loop diagrams and its calculated value agrees with the experiment very well suggesting that the Pauli term is suppressed by a scale larger than 4×107GeV.[5]

The scale-dependent renormalized running couplingeR(q2)can be de fined by the following relation??We carried out all calculations in Landau gauge,however,the results are gauge independent.

Here,eris the renormalized coupling atq2=?m2forcR=0,wherecRis a renormalized coupling constant of the higher-order LagrangianLho.It is suppressed by two orders of some large scale.Forκ=cR=0 the running coupling has the well-known pole singularity at(the Landau pole)

While this pole appears at extremely high energies,reducing its practical importance to nil,it is still a problem if present in the full theory.[3]For reasonable values ofthe Landau pole is absent remedying the inconsistency at the level of an EFT.

In renormalizable theories only logarithmic divergences contribute to the renormalization of coupling constants and therefore there is a direct correspondence between the Gell-Mann-Low[6]and the Wilsonian renormalization group approaches.[7]As a result,the presence of the Landau pole in the expression of the running renormalized coupling automatically leads to the pole in the bare coupling as a function of the cuto ffparameter.However,in an EFT with non-renormalizable interactions the direct link between the two renormalization group equations is lost and therefore the Wilsonian renormalization group approach requires additional study of the cuto ffdependence.We investigate the cuto ffdependence of the bare electromagnetic coupling in our model by applying the higher derivative regularization,[8?9]which preserves the local U(1)gauge invariance.Notice that dimensional regularization is not suitable here as it discards the powerlaw divergences.In addition to the fields in conventional QED,we introduce scalar(i.e.commuting)ghost fieldsˉξandξ,which regulate the one-loop counter term diagrams contributing to the photon self-energy at two-loop order.The effective Lagrangian generating one and two-loop diagrams,contributing to our calculation of the photon selfenergy up to two-loope2κ2order,which are all finite for finite Λ,is given by:

The bare electromagnetic coupling as a function of the cuto ffsatis fies a renormalization group equation,which up to the level of accuracy of our calculation has the form:

whereα(Λ)=e2/(4π),the coefficientA1is given by oneloop diagrams andA2andA3are extracted from twoloop calculations.Notice that there are no power-law divergences at one-loop order and all terms suppressed by powers ofm/Λ have been dropped in our calculations as they are negligible for large values of Λ.

This pole,if remaining in the non-perturbative full expressions of the bare coupling,prevents the Λ→∞limit unlessα0≡0,thus leaving us with a non-interacting theory.In renormalizable theories where only logarithmic divergences contribute in the renormalization of coupling constants,there is a close correspondence between the Gell-Mann-Low and the Wilsonian renormalization group approaches manifesting itself in a direct relation of Eqs.(3)and(8)valid for standard QED.

Using FeynCalc[10?11]and Form[12]and applying the method of dimensional counting of Ref.[13]we have calculated the logarithmically divergent contributions to the photon self-energy generated by one-loop diagrams,and the quadratic divergences generated by the two-loop diagrams,shown in Fig.1,??Notice here that due to the higher derivative regularization there are interaction vertices not present in standard QED.§ We have used HypExp2[14]to expand the hypergeometric functions in a Laurent series in ?.We thank T.Huber and Y.Schr?der for their help in reducing A2to the simple form.and by the corresponding counter term diagrams,shown in Fig.2.Our results read:§?Notice here that due to the higher derivative regularization there are interaction vertices not present in standard QED.§ We have used HypExp2[14]to expand the hypergeometric functions in a Laurent series in ?.We thank T.Huber and Y.Schr?der for their help in reducing A2to the simple form.

whereψ(1)is the trigamma function.For the above values ofA1,A2andA3,and fornaturalvalues ofκ?1/ΛP,in the denominator of Eq.(8)theA3term is larger than theA1andA2terms and the negative sign ofA3guarantees thatα(Λ)has no pole.It would be interesting to check if the exact RG treatment of the Pauli term analogous to that of Ref.[15]leads to the same result.However such an analysis is beyond the scope of this work.

Fig.1 Two-loop diagrams contributing in the photon self-energy.The solid and wiggly lines correspond to the electron and photon propagators,respectively.

Fig.2 One loop counter term diagrams contributing in the photon self-energy at two-loop order.The solid and wiggly lines correspond to the electron and photon propagators,respectively.The crosses denote the counter terms of the one-loop order.Diagrams with ghost propagators are not shown.

To conclude,we have shown that the problem oftrivialityin QED can be attributed to QED being a leading order approximation of an effective field theory.We have shown that already at next-to-leading order,i.e.adding the Pauli term to QED,the Landau pole as well as the pole in the bare coupling as a function of the cuto ffparameter,disappear thus obviating the need for QED to be atrivialtheory.Furthermore,while the triviality of QED is not a settled issue,[3]from the modern point of view,which considers the Standard Model as an EFT,the issue of triviality can be of academic interest only as the higher order operators qualitatively change the UV behaviour of QED.

We are grateful to Holger Gies,Tobias Huber and York Schr?der for constructive remarks.

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