DIFFERENTIAL RENORMALIZATION OF

arXiv:hep-ph/9203207v1 10 Mar 1992UB-ECM-PF 92/5March 1992DIFFERENTIAL RENORMALIZATION OFMASSIVE QUANTUM FIELD THEORIESPeter E. Haagensen∗and Jos´e I. Latorre†Departament d’Estructura i Constituents de la Mat`eriaFacultat de F´ısica, Universitat de BarcelonaDiagonal 647, 08028 Barcelona, SpainAbstractWe extend the method of differential renormalization to massive quantum fieldtheories, treating in particular λϕ4-theory and QED. As in the massless case, themethod proves to be simple and powerful, and we are able to find, in particular, compactexplicit coordinate space expressions for the finite parts of two notably complicateddiagrams, namely, the 2-loop 2-point function in λϕ4 and the 1-loop vertex in QED.∗e-mail: HAGENSEN@EBUBECM1†e-mail: LATORRE@EBUBECM1

Differential renormalization (DR)[1] is a coordinate space renormalization proce-dure which removes the divergences of bare amplitudes by writing these as derivativesof less singular functions and then prescribing the derivatives to be integrated by partswith all surface terms discarded. The singularities of amplitudes that did not allowfor Fourier transformation into momentum space are eliminated and one ends up withfinite, renormalized amplitudes that satisfy renormalization group equations.

The stan-dard and simplest example of the procedure is for the 1-loop 4-point “bubble” diagramin massless λϕ4, where the following identity is used:1x4 = −14✷ln x2M2x2x ̸= 0. (1)M is an integration constant which will become the subtraction scale of the renormal-ized amplitude.

If we prescribe the laplacian to act only after integration by parts, thenthe r.h.s. above has a finite Fourier transform, and in fact defines the renormalizedvalue of the (divergent) Fourier transform of the l.h.s.. Generally, one can say thatdifferential renormalization provides a prescription to continue distributions defined al-most everywhere (i.e., bare amplitudes, undefined at a finite number of singular points)into bona fide distributions defined everywhere (renormalized amplitudes).The procedure exemplified above has been carried out at higher loops and in dif-ferent models [1,2,3].

One can furthermore check explicitly the form of the divergencesbeing subtracted by examining the surface terms which contain them, thus verifyingthat the method indeed corresponds to a counterterm subtraction procedure [4].In this Letter, we extend the method to allow for the inclusion of masses in an exacttreatment. At the outset, it is clear that the presence of masses should not interfere withthe method since the renormalization procedure is related to short-distance singularitieswhereas masses only change the long-distance behavior of correlators.

In fact, one couldsimply take a pragmatic attitude and expand all massive propagators around zero massand then proceed to use standard massless differential renormalization. However, thequestion remains whether massive field theories are amenable to a treatment exact inthe mass parameter at each order in perturbation theory.

We find that our expectationsare entirely fulfilled. In analogy to the massless case, renormalization is accomplishedthrough the use of (massive) differential renormalization identities and an integrationby parts prescription.

We treat QED at one loop and λϕ4-theory at two loops and,most notably, we are able to give in closed form the full renormalized expressions for2

the 2-loop “setting sun” diagram of λϕ4 and the 1-loop vertex of QED, two results ofconsiderable difficulty of calculation in standard momentum space treatments. In theend, we shall discuss the general features of our modus operandi.First of all, let us recall the euclidean propagator for a massive scalar particle infour dimensions,∆(x, m) =14π2mK1(mx)x(2)where K1 is a modified Bessel function.

Up to permutations of external legs the only1-loop diagram contributing to the 4-point function in λϕ4 theory is:Γ(4)bare(x) = λ22 14π2mK1(mx)x2(3)At short distances, this has the same log divergence as the corresponding masslessdiagram; in the spirit of differential renormalization we look for an expression thatcorresponds to the massive generalization of Eq.(1). We find:m2K21(mx)x2= 12(✷−4m2)mK0(mx)K1(mx)x+ π2 ln¯M2m2 δ(4)(x)(4)where ¯M ≡2M/γ and γ = 1.781072... is the Euler constant.

The contact term has beenadded in order to give a well-defined massless limit to the r.h.s. above, coinciding in factwith Eq.(1).

We shall further analyze this later on, and in fact we will see that Eq. (4)can be seen as a prototype in our treatment, since it presents all the basic guidingelements in working out massive DR identities in general.

The renormalized 1-loop4-point function, Γ(4)R (x, M), is gotten by simply substituting the above in Eq. (3).The one-loop beta function can be obtained from the renormalization group equa-tionM ∂∂M + β ∂∂λ + γmm2∂∂m2 −4γ[−λδ(4)(x) + Γ(4)R (x, M)] = 0,(5)where β(λ) is the β-function, γm(λ) is the anomalous mass dimension, and γ(λ) theanomalous dimension of ϕ.

Dropping terms of higher order in λ (viz., γm and γ), theresult for the β-function is:β(λ) = 3 λ16π2(6)with the three permutations of the s, t and u channels having been added. Our resultfor the bubble diagram can be easily Fourier transformed.

The result in momentumspace is:˜Γ(4)R (p) =λ232π2ln¯M2m2 −s1 + 4m2p2 lnq1 + 4m2p2 + 1q1 + 4m2p2 −1(7)3

which agrees with the standard result in textbooks [5]. Note the appearance of the2-particle threshold 4m2 in the operator which was extracted in Eq.(4).

This will bea recurring feature in what follows. We have also worked out this amplitude for twodifferent masses running along the two lines in the loop; we do not present it here,but rather just mention that the result generalizes the one above and, as expected, thethreshold (m1 + m2)2 appears instead of 4m2.A far less trivial example showing the power of differential renormalization is givenby the computation of the 2-loop correction to the 2-point function (often called the“setting sun”).

The complexity of the finite parts of this diagram is such that they arenot presented in standard reviews of λϕ4 renormalization. The bare expression for thediagram is:Γ(2)bare(x) = λ26 14π2mK1(mx)x3.

(8)This bare amplitude is quadratically divergent and thus requires the extraction of twolaplacians in order to have a good Fourier transform. Little effort is needed to verifythe identity that leads to the following renormalized value of the above diagram:Γ(2)R (x, M)=λ296(4π2)3h(✷−9m2)(✷−m2)m2K0(mx)K21(mx) + m2K30(mx)+2π2 ln¯M2m2 (✷+ am2)δ(4)(x)#(9)This result is by itself a remarkable application of differential renormalization.

It corre-sponds to a closed and compact expression for the renormalized 2-loop 2-point function.Its Fourier transform is complicated and we will not work it out here. Again, we notethe appearance of the 3-particle production threshold as a healthy sign of the proce-dure.

Further, the second operator takes the form (✷−m2), which vanishes on massshell. The coefficient of the ✷δ4(x) term is fixed as in the previous example by the re-quirement of a smooth massless limit and this, in turn, fixes γ(λ) in the renormalizationgroup equation to its standard valueγ(λ) = 112 λ16π2.

(10)On the other hand, the coefficient a of m2δ(4)(x) is not fixed by a strict massless limit,which is to be expected, since it corresponds to a mass subtraction, and fixes the(scheme-dependent) γm function in the renormalization group equation. In particular,a can be chosen to be −1 so that the whole 2-loop contribution vanishes on mass shell.4

Though we do not intend to present an exhaustive computation of the perturbativeexpansion of massive λϕ4, let us comment on a few more diagrams. There are onlytwo more diagrams contributing at two loops to the 4-point function.

The first one,usually called the “ice cream cone”, though extremely difficult in momentum space, iseasily calculated along the lines we have sketched above. The second one, consisting oftwo 4-point 1-loop bubbles attached together (and thus, the “double bubble”) is not asstraightforward due to the presence of a convolution.We now turn to the computation of 1-loop diagrams in QED ‡.

The (massive)fermion and photon propagators are, respectively:S(x, m) =14π2 (∂/ −m)mK1(mx)x,(11)and∆µν(x) =14π2δµνx2 ,(12)this latter one being given in Feynman gauge. The ¯ψA/ψ vertex has the value ieγµ.

The1-loop fermion self-energy then reads:Σbare(x) = ie4π2 1x2 γµ(∂/ −m)mK1(mx)xγµ. (13)The mass piece above has a logarithmic short-distance divergence, while the derivativeterm is linearly divergent at short distances.

These divergences are eliminated by thefollowing massive DR identity:mK1(mx)x3= 12(✷−m2)K0(mx)x2+ π2 ln¯M2m2 δ(4)(x),(14)where again the contact term is determined such as to give a well-defined masslesslimit to the r.h.s. (and again equal to the corresponding massless identity).

The final,renormalized expression for the fermion self-energy will then be:ΣR(x, M)=e232π2"(✷−m2) (∂/ + 4m)K0(mx)x2+ m22x/K0(mx)x2!+2π2 ln¯M2m2 (∂/ + bm)δ(4)(x)#. (15)The non-local piece of this amplitude vanishes identically on mass-shell due to theoperator (✷−m2), as expected.

In analogy to the previous case, the ∂/δ4(x) contact‡For a complete set of momentum space techniques to compute one-loop diagrams in gauge theories,see [6].5

term is fixed by the massless limit (and leads to the standard value for the anomalousdimension of the fermionic field), whereas the mδ4(x) term is not. We leave the coeffi-cient b undetermined, and again it is clear that a particular choice, viz.

b = −1, makesthe whole amplitude vanish on mass-shell. The Fourier-transformed, momentum spaceamplitude can be obtained fairly straightforwardly, and we do not present it here.The bare 1-loop vacuum polarization is:Πbareµν (x) = − ie4π22trγµ(∂/ −m)mK1(mx)xγν(−∂/ −m)mK1(mx)x.

(16)By standard manipulations with Bessel functions and the use of a massive DR identity,Eq. (4), we find the following renormalized vacuum polarization at one loop:ΠRµν(x, M)=−e224π4 (∂µ∂ν −δµν✷)(✷−4m2)mK0(mx)K1(mx)x+m22 (K21(mx) −K20(mx))!+ 2π2 ln¯M2m2 δ(4)(x)#.

(17)As expected, the result is automatically transverse and shows the presence of the 2-particle production threshold in the operator coming in front of the non-local piece ofthe amplitude.We finally turn to the 1-loop vertex. Its bare value is:V bareµ(x, y) = ie4π23γρ (∂/ −m)mK1(mx)xγµ (−∂/ −m)mK1(my)yγρ1(x −y)2 .

(18)In the above, the only divergent piece is the one containing the two derivatives:V divµ(x, y) = − ie4π23γρ ∂/mK1(mx)xγµ ∂/mK1(my)yγρ1(x −y)2 . (19)We renormalize this by an identical procedure used in the massless case [1], i.e., byintegrating the two derivatives by parts onto 1/(x −y)2, and separating that into a(finite) traceless piece and a (divergent) trace piece.

The divergence, thus isolated inthe trace, is then renormalized with massive DR identity Eq.(4). The final result reads:V Rµ (x, y, M) = ie4π23 2γbγµγa ∂∂xamK1(mx)x∂∂ybmK1(my)y1(x −y)2−∂∂yb"m2K1(mx)K1(my)xy∂∂xa1(x −y)2#−m2K1(mx)K1(my)xy(∂a∂b −14δab✷)1(x −y)2!−2π2γµ(✷−4m2)mK0(mx)K1(mx)xδ(4)(x −y) −4π4γµ ln¯M2m2 δ(4)(x)δ(4)(x −y)+4m1(x −y)2 ∂∂yµ −∂∂xµ −m2 γµ m2K1(mx)K1(my)xy!#.

(20)6

This concludes our results. We have extended the method of differential renormal-ization to massive λϕ4 and QED, and the results we present here, though not intendedto be exhaustive, should indicate both the feasibility of, and the guidelines for, thetreatment of massive fields at higher loops and in different models.

These guidelinesbasically are: a) the presence of an equal number of K-functions on both sides ofmassive DR identities; b) a direct connection to massless differential renormalization,through the substitution of ln x2M2 in this latter case for K0(mx); c) the appearanceof the appropriate particle production thresholds in the differential operators present inmassive DR identities, and d) the appearance of contact terms, due to the requirementof existence of the massless limit for renormalized distributions. These features havebeen of central importance in helping us find all the renormalized amplitudes presentedhere.Two final comments are in order regarding contact terms in massive DR identi-ties.Firstly, the presence of these terms may seem puzzling, since they vanish atpoints where massive DR identities are valid mathematical equations, i.e., for x ̸= 0.However, insofar as these identities are prescriptions for extending (i.e., renormalizing)distributions, they should be understood as identities everywhere, including at contact,and specific contact terms will then determine precisely and uniquely the value of theextended (i.e., renormalized) distributions at those points.Finally, it is also worthnoting that contact terms encode the entire renormalization group freedom present inthe renormalization procedure.

Our choice has been to fix contact terms in order toagree with the massless case, which corresponds to a wave function renormalizationprescription, and to leave undetermined the mass subtraction, though an on-mass-shellscheme has been indicated.Acknowledgments - We would like to thank D.Z. Freedman for ongoing discussions.This work was supported in part by CAICYT grant AEN90-0033 and by the EEC Sci-ence Twinning Grant SCI-000337.

P.H. also acknowledges a grant from the Ministeriode Educaci´on y Ciencia, Spain.References[1] Freedman, D.Z., K. Johnson and J.I.

Latorre, Nucl.Phys. B, in press.

[2] Haagensen, P.E., Mod.Phys.Lett. A7(1992)893.7

[3] Freedman, D.Z., G. Grignani, K. Johnson and N. Rius, “Conformal Symmetry andDifferential Regularization of the 3-Gluon Vertex”, MIT preprint CTP#1991. [4] Freedman, D.Z., R. Mu˜noz-Tapia and X. Vilas´ıs-Cardona, manuscript inpreparation.

[5] Ramond, P., Field Theory, A Modern Primer, 2nd ed., Addison-Wesley(1990). [6] ’t Hooft, G. and M. Veltman, Nucl.Phys.

B153(1979)365.8

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