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A COMPREHENSIVE COORDINATE SPACE RENORMALIZATION OF

arXiv:hep-ph/9206252v1 26 Jun 1992A COMPREHENSIVE COORDINATE SPACE RENORMALIZATION OFQUANTUM ELECTRODYNAMICS TO 2-LOOP ORDERPeter E. Haagensen1 and Jos´e I. Latorre2Departament d’Estructura i Constituents de la Mat`eriaFacultat de F´ısica, Universitat de BarcelonaDiagonal 647, 08028 Barcelona, SpainABSTRACTWe develop a coordinate space renormalization of massless Quantum Electrodynamics using thepowerful method of differential renormalization. Bare one-loop amplitudes are finite at non-coincidentexternal points, but do not accept a Fourier transform into momentum space.

The method providesa systematic procedure to obtain one-loop renormalized amplitudes with finite Fourier transforms instrictly four dimensions without the appearance of integrals or the use of a regulator. Higher loopsare solved similarly by renormalizing from the inner singularities outwards to the global one.

Wecompute all 1- and 2-loop 1PI diagrams, run renormalization group equations on them and checkWard identities. The method furthermore allows us to discern a particular pattern of renormalizationunder which certain amplitudes are seen not to contain higher-loop leading logarithms.

We finallypresent the computation of the chiral triangle showing that differential renormalization emerges as anatural scheme to tackle γ5 problems.UB-ECM-PF 92-14June 19921 e-mail: hagensen@ebubecm1.bitnet2 e-mail: latorre@ebubecm1.bitnet

0. IntroductionAll one-loop diagrams in Quantum Electrodynamics (QED) are products of propagators whenwritten in coordinate space and, therefore, finite for separate external points.

Apart from tadpoles,this is in fact true for any quantum field theory. This striking observation has been in the mind ofthe physics community since long ago though little practical use of it has been made.

The need forrenormalization comes through the fact that the product of propagators is not a distribution and,thus, has no Fourier transform. In crude terms, renormalization is a procedure to extend products ofdistributions into distributions.Differential Renormalization (DR) [1] provides a prescription to implement this project whichis in a sense minimal.

The basic idea is to write an amplitude in position space and then expressit as derivatives of a distribution less divergent at coincidence points. The typical computation ofprimitively divergent Feynman integrals is substituted for solving trivial differential equations.

Thisprescription is complemented with the instruction of using the derivative in front of the distributionas acting to its left, which is standard in the theory of distributions.We claim that differential renormalization is minimal because it never changes the value of aprimitively divergent Feynman diagram away from its singularities, and neither does it modify thedimensionality of space-time or introduce a regulator. It is simple and addresses straighforwardly theconcept of renormalization, rather than regularization.

In spirit, it is close to BPHZ renormalization[2], in that both prescriptions deliver renormalized amplitudes graph by graph. In the absence of arenormalized lagrangian, DR needs a supplementary effort to prove unitarity.

In [3], this problem hasbeen analyzed for λφ4-theory and perturbative unitarity proven to 3-loop order.A number of theories have been used as a test for differential renormalization.In [4], thesupersymmetric Wess-Zumino model was studied up to three loops with ease. Conformal invariancein QCD was exploited in [5] thanks to the position space nature of differential renormalization.

Themethod also extends to massive theories [6] and to lower dimensional theories [7].Other coordinate space regularization and renormalization methods have occasionally been used inthe past. From Schwinger [8] to all the recent conformal field theory developments [9], the advantagesof postponing loop integration to higher loops and the possibility of exploiting conformal invariancehave led to numerous applications of coordinate space methods.1

In this paper we present a comprehensive study of the differential renormalization of masslessQED up to two loops. There are obvious reasons to undertake such a computation.

QED is a physicaltheory with gauge symmetry. The consistency of the method passes a stringent test because of theexistence of Ward identities (WIs) and potential problems hidden in overlapping divergences.

It alsofaces the problem of the chiral anomaly and the correct treatment of γ5.Our results are in perfect agreement with other computational schemes (e.g., [8],[10],[11],[12],[13])but are obtained in a simpler form. We devote Sections 1 and 2 to presenting the renormalized am-plitudes up to two loops.

Whereas several standard treatments of QED will perform the subtractionson mass-shell, here renormalized amplitudes are subtracted at an arbitrary and unphysical scale M,due to the fact that we are dealing with massless QED; this massless limit exists as a quantum fieldtheory and provides substantial information about massive QED. We pay special attention to the waydifferential renormalization uses the basic WIs of the theory and how they are repeatedly checked intwo-loop computations.

Deeply rooted in the nature of DR lies the idea that each Feynman diagramhas as many hidden scales as singular regions. Some of these scales are fixed by WIs.

The rest arerenormalization-scheme prescriptions. Keeping all this freedom manifest allows the method to “pre-dict” that the renormalization group equations will display a particular organization.

This “structuredrenormalization group”, discussed in Section 3, provides a deep insight into the absence of promotionof leading logs in some diagrams, which is seen as accidental in standard treatments. Finally, wereview the calculation of the chiral anomaly in our regulator-free way in Section 5.

We also computethe absence of infinite renormalizations (i.e., absence of ln’s) at two loops of the triangle amplitude.As differential renormalization never leaves four dimensions, it stands as a good candidate to workwith γ5-related observables. The anomaly is just one instance.1.

One-loop Renormalization1.1 Conventions and definitionsIn d = 4 Euclidean space, massless QED is described by the Lagrangian:L = 14F 2µν + 12a(∂µAµ)2 + ψ(∂/ + ieA/)ψ,(1.1)2

where Aµ(x) and ψ(x) are the usual U(1) gauge and Fermi fields, Fµν(x) = ∂µAν(x) −∂νAµ(x)is the gauge field strength, a is a gauge fixing parameter, and γ-matrices satisfy the usual Cliffordalgebra, {γµ, γν} = 2δµν.Loop diagrams are calculated with the following Feynman rules:i) the gauge field and massless fermion propagators are, respectively,G(0)µν (x −y; a) =14π21 + a2δµν(x −y)2 + (1 −a)(x −y)µ(x −y)ν(x −y)4S(0)(x −y) = −14π2 γµ∂µ1(x −y)2 ,ii) to each vertex is associated a factor ieγµ,iii) internal coordinates are integrated over,iv) closed fermion loops are multiplied by a factor −1, andv) diagrams have no symmetry factors.We also use the convention that for derivative operators O = ∂µ, ∂/,, etc., we take Of(x −y) tomean O(x)f(x −y), unless explicitly stated otherwise.Renormalization of QED entails the renormalization of only three different vertex functions: the2-point vacuum polarization, Πµν(x −y), the 2-point fermion self-energy, Σ(x −y), and the 3-pointvertex, Vµ(x −z, y −z). These will contribute to the effective action in the following way:Γeff=Zd4xd4y12Aµ(x)(1 −1a)∂µ∂ν −δµνδ(4)(x −y) −Πµν(x −y)Aν(y)+ ψ(x)∂/ δ(4)(x −y) −Σ(x −y)ψ(y)o+Zd4xd4yd4z ψ(x)hieγµδ(4)(x −y)δ(4)(x −z) + Vµ(x −z, y −z)iψ(y)Aµ(z) + · · · .

(1.2)As defined above, these renormalization parts will contain only 1PI loop contributions and Πµνand Σ will, furthermore, lead to the following full photon and electron propagators:Gµν = G(0)µν + G(0)µρ · Πρσ · G(0)σν + G(0)µα · Παβ · G(0)βρ · Πρσ · G(0)σν + · · · = (G(0)−1µν−Πµν)−1S = S(0) + S(0) · Σ · S(0) + S(0) · Σ · S(0) · Σ · S(0) + · · · = (S(0)−1 −Σ)−1,(1.3)3

where the dot indicates a convolution. To all loop orders, renormalized vertex parts satisfy renor-malization group (RG) equations [14] and, as a consequence of gauge invariance, the following Wardidentities:∂µΠµν(x −y) = 0,(1.4)∂∂zµ Vµ(x −z, y −z) = −ie[δ(4)(z −x) −δ(4)(z −y)]Σ(x −y).

(1.5)The specific coefficients in the second Ward identity are uniquely fixed by the tree-level identity solvedby V (0)µ= ieγµδ(4)(x −y)δ(4)(x −z) and S(0)−1 = ∂/ δ(4)(x −y).In what follows, we shall find the 1- and 2-loop corrections to Πµν, Σ and Vµ. We will verify theabove Ward identities, and the requirement that renormalization parts satisfy renormalization groupequations will allow us to calculate the renormalization group functions of QED (beta-functions andanomalous dimensions).1.2 RenormalizationThe bare 1-loop vacuum polarization (Fig.

1a) reads:Π(1)bareµν(x −y) = − ie4π22Trγµ∂/1(x −y)2 γν∂/1(y −x)2= −α3π3 (∂µ∂ν −δµν)1(x −y)4 ,(1.6)with α = e2/4π the fine structure constant. Using the basic DR identity (cf.

Appendix A for all theidentities needed in renormalizing amplitudes up to two loops):1x4 = −14ln x2M 2x2,(1.7)we find the renormalized value of the one-loop vacuum polarization:Π(1)µν (x −y) =α12π3 (∂µ∂ν −δµν)ln(x −y)2M 2Π(x −y)2,(1.8)with the proviso that the total derivatives above, as well as in all other renormalized amplitudesthroughout, are to be understood as acting to the left. Here and below we adopt the conventionof appending a subscript (Π, Σ or V ) to the renormalization scales appearing in the different renor-malization parts, since these scales are a priori independent (but eventually related through Wardidentities, as we shall see).

This quantity gives us then the renormalized AA 2-point function:ΓAAµν (x −y) =(1 −1a)∂µ∂ν −δµνδ(4)(x −y) −Π(1)µν (x −y). (1.9)4

We now impose that it satisfy the usual RG equation:MΠ∂∂MΠ+ β(α) ∂∂α + γa(α) ∂∂a −2γA(α)ΓAAµν (x −y) = 0,(1.10)where β(α) is the QED β-function, γA(α) is the anomalous dimension of the gauge field, and γa(α)a “β-function” associated to the running of the gauge parameter a. This is then easily seen to leadto the 1-loop values:γA(α) = α3πandγa(α) = −2aα3π(1.11)(β(α) has dropped out above because it is of higher order in α).It is well-known that the above computation can be used in the framework of the backgroundfield method to find the beta-function of QED [15].

It turns out that the relation β(α) = 2αγ(α)holds to all orders. Therefore, we can anticipate that the beta function at one-loop isβ(α) = 2α23π .

(1.12)This result will later be confirmed by an independent computation of the one-loop vertex.The bare 1-loop fermion self-energy (Fig. 1b) is:Σ(1)bare(x −y) = e4π22γµ ∂/1(x −y)2 γν1 + a2δµν(x −y)2 + (1 −a)(x −y)µ(x −y)ν(x −y)4.

(1.13)The renormalization of this amplitude is straightforward, apart from the following important subtlety:if we use the same mass scale MΣ in renormalizing the two parts coming from the two different piecesin the photon propagator, we will end up with a renormalized amplitude which is directly proportionalto the gauge parameter a; now, in Landau gauge (a = 0), it would then vanish entirely, and it is easyto see this would be inconsistent with the Ward identity. It is in fact a known property of QED that,in Landau gauge, although the bare 1-loop fermion self-energy vanishes, the renormalized self-energydoes not.

In momentum space, it is equal to a finite constant times q/ , and this is a reflection ofthe ambiguity inherent in the linearly divergent integral defining the amplitude. This turns out to betransparent in differential renormalization: in renormalizing the two pieces in question, no one tellsus we should take the same mass scale MΣ, and so we do not.

We renormalize the first piece (the5

Feynman gauge part) with a scale MΣ, and the second piece with a scale M ′Σ related to the first oneby ln M ′Σ = ln MΣ −λ. This then leads to the following renormalized 1-loop fermion self-energy:Σ(1)(x −y) =α16π3 ∂/a ln(x −y)2M 2Σ + λ(1 −a)/2(x −y)2= aα16π3 ∂/ln(x −y)2M 2Σ(x −y)2−λα(1 −a)8π∂/ δ(4)(x −y) .

(1.14)As expected, in Landau gauge, all that survives is precisely the contact term mentioned above. Wewill see shortly that the Ward identity will not only determine a relation between the self-energy andvertex mass scales, but also a unique value for λ.

The renormalized ψψ 2-point function,Γψψ(x −y) = ∂/ δ(4)(x −y) −Σ(1)(x −y),(1.15)then satisfies the RG equation:MΣ∂∂MΣ+ β(α) ∂∂α + γa(α) ∂∂a −2γψ(α)Γψψ(x −y) = 0,(1.16)and this will lead to the following 1-loop fermion anomalous dimension:γψ(α) = aα4π(1.17)(here, both β and γa drop out).Finally, we now renormalize the 1-loop vertex (Fig. 1c).

Its bare expression is:V (1)bareµ(x −z, y −z) == ie4π23(γργaγµγbγσ) ∂a1(x −z)2 ∂b1(z −y)21 + a2δρσ(x −y)2 + (1 −a)(x −y)ρ(x −y)σ(x −y)4=−i32π3απ3/2(γργaγµγbγσ) ∂a1(x −z)2 ∂b1(z −y)2 [(a −1)∂ρ∂σ + δρσ] ln(x −y)2µ2. (1.18)Like for the fermion self-energy, the photon propagator in a generic gauge leads to two pieces tobe renormalized, in principle with mass scales MV and M ′V .

However, it turns out that this need not bedone here, that is, the choice of the same MV throughout will not lead to any inconsistencies (still, onemay choose different scales at will; this would simply correspond to different choices of scheme). Webriefly sketch the procedure involved in renormalizing the expression above.

In Feynman gauge (a = 1)it is most transparent [1]: there are three propagators 1/(x −x′)2 forming a triangle, with derivativesacting on two of them. This has dimension L−8 and therefore it is log divergent in the ultraviolet.One first integrates by parts until the two derivatives are acting on the same leg, say, ∂a∂b1(x−y)2 ,6

and then subtracts and adds a trace piece thus: (∂a∂b −14δab)1(x−y)2 + 14δab1(x−y)2 . The surfaceterms, with total derivatives acting outside, is power counting L−7 and thus finite, and the tracelesscombination of derivatives is also finite (due to tracelessness).

The divergence has been isolated inthe term 14δab1(x−y)2 = −π2δabδ(4)(x −y), which turns out to be easily renormalizable throughthe DR identity, Eq.(1.7). For a generic gauge, the same principle of separating divergent terms intotrace and traceless pieces applies, only in this case the γ-matrix structure in front complicates thingsa little: we integrate ∂a and ∂b above by parts onto the photon leg, and then look for the coefficientA to make the expression(γργaγµγbγσ) (∂a∂b + Aδab ) [(a −1)∂ρ∂σ + δρσ](1.19)a traceless (and thus finite) combination of derivatives.

The appropriate value is A = −a/(a + 3);adding and subtracting that trace piece, we are able to find the renormalized expression for the 1-loopvertex:V (1)µ(x −z, y −z) ==−i32π3απ3/2 (γργaγµγbγσ)∂∂xa1(x −z)2 ∂b1(z −y)2 [(a −1)∂ρ∂σ + δρσ] ln(x −y)2µ2+ 4 [(a −1)γµγb −2γbγµ] ∂∂yb1(x −z)2(z −y)2 ∂/1(x −y)2−16(x −z)2(z −y)2 γσ∂µ∂σ −14δµσ1(x −y)2 + 4π2aγµln(x −z)2M 2V(x −z)2δ(4)(x −y). (1.20)We remark here on the fact that the renormalization piece above (containing ln M 2V ) is directlyproportional to the gauge parameter a, and this is also true of Σ(1)(x −y) found above.

This is areflection of the well-known fact that, apart from vacuum polarization infinities, QED is 1-loop finitein Landau gauge.The renormalized 3-point vertexΓψAψµ(x −z, y −z) = ieγµδ(4)(x −y)δ(4)(x −z) + V (1)µ(x −z, y −z)(1.21)satisfies the RG equation:MV∂∂MV+ β(α) ∂∂α + γa(α) ∂∂a −2γψ(α) −γA(α)ΓψAψµ(x −z, y −z) = 0(1.22)with values found previously for γψ and γA, andβ(α) = 2α23π . (1.23)7

(and, again, γa is of higher order than we are considering).This confirms the result from thebackground field method shown previously. Naturally, in the above equation the a-dependent pieces(M∂∂M and γψ) and the a-independent pieces (β ∂∂α and γA) cancel separately.We now consider the Ward identity for Vµ and Σ.1.3 Ward IdentityFor separate points x ̸= y ̸= z, Eq.

(1.5) is trivially verified. The subtlety involved in a coordinatespace Ward identity is, however, its validity at contact.

In order to verify Eq. (1.5), then, we must becareful in particular not to lose any contact terms due to formal manipulations with delta functions,and the best way to guarantee a correct procedure is to integrate it over one of the external variables.This was done in [1] for Feynman gauge, and we use the same procedure here for an arbitrary gauge.The integrated form of the Ward identity is:Zd4y ∂∂zµ V (1)µ(x −z, y −z) = ieΣ(x −z).

(1.24)Integrating the expression for the renormalized vertex, Eq. (1.20), over y, one finds:Zd4yV (1)µ(x −z, y −z) = −i4παπ3/2 ∂µ∂/ −1 + 2a4γµ1(x −z)2 + a2γµln(x −z)2M 2V(x −z)2.

(1.25)Acting now with the z derivative:Zd4y ∂∂zµ V (1)µ(x −z, y −z) = ie364π4 ∂/a ln(x −z)2M 2V + (3/2 −a)(x −z)2. (1.26)Now, we compare this with:ieΣ(1)(x −z) =ie364π4 ∂/a ln(x −z)2M 2Σ + λ(1 −a)/2(x −z)2.

(1.27)Setting a = 0 gives λ = 3, and then, for any a, we also find:ln M 2ΣM 2V= 12 . (1.28)We see then that a Ward identity relating two renormalized amplitudes will, in the context ofdifferential renormalization, enforce a relation between the scales that renormalize the amplitudes.It is an important point we will analyze further that the converse is also true, namely, mass scalesthat renormalize amplitudes not related by a symmetry are not related, and this in turn also enforcesconstraints in the renormalization of these amplitudes.

At two loops, in particular for the anomaloustriangle, the use of the above mass relation will be crucial for the consistency of our calculations.8

2. Two-loop RenormalizationAt two loops, the 1PI diagrams renormalizing Πµν, Σ and Vµ are depicted in Fig.

2. Throughoutthis section, we will perform all calculations in Feynman gauge (a = 1); since γa ∂∂a acting on any2-loop diagram will be one order higher in α, this will not affect the verification of RG equations.2.1 Vacuum PolarizationWe begin with the simplest of the two vacuum polarization diagrams (Fig.

2a), and from nowon we will use translation invariance to set one external point to zero in all diagrams, for the sake ofsimplicity. Its bare expression is:Π(2a)bareµν(x) = (ie)2(4π2)3Zd4ud4v Trγµ∂/1(x −u)2 Σ(1)(u −v)∂/ 1v2 γν∂/ 1x2.

(2.1)Standard manipulations lead to:Π(2a)bareµν(x) =e448π614(∂µ∂ν −δµν)ln x2M 2Σ −1/6x4−δµνx6. (2.2)We note that this diagram is not transverse by itself; the non-transverse piece will be cancelledwhen we consider the entire 2-loop vacuum polarization.

The renormalization now proceeds with thestraightforward use of the DR identities listed in Appendix A, and we find:Π(2a)µν (x) = −196π2απ2((∂µ∂ν −δµν) ln2 x2M 2Σ + 53 ln x2M 2x2!−δµνln x2M 2x2),(2.3)where M is a new renormalization mass parameter appearing at two loops. The diagram with thefermion self-energy inserted on the lower leg of the loop will have the same value as this one.

Wenote here that the ln M 2Σ coming from the 1-loop self-energy subdivergence has been promoted to aln2 M 2Σ at two loops: this is the self-consistency of the renormalization group at work. We will chooseeverywhere the same 2-loop mass scale M. In renormalization terms, this simply corresponds to somechoice of renormalization scheme: because these appear asln x2M2x2, different values of M will leadto amplitudes that differ by finite contact terms or, in other words, by finite renormalizations (andthis will be true for the new M’s appearing at each loop order).

Of course, once we set the (2-loop)M’s we like for the renormalization of Σ and Vµ in particular, they will subsequently be related bythe Ward identity (but this will not be of concern to us here, as we will not compute the 2-loop mass9

relation). Throughout this section, then, we will only care to distinguish between MΣ, MV or MΠand other mass scales M when the former appear as promoted ln’s, i.e., as ln2 MΣ, etc..The other vacuum polarization diagram, Fig.

2b, is the most difficult integral we have had toperform. The bare expression is:Π(2b)bareµν(x−y) = (ie)4(4π2)5Zd4ud4v Trγµ∂/1(x −u)2 ∂/1(u −y)2 γν∂/1(v −y)2 ∂/1(v −x)21(u −v)2 .

(2.4)There are potential problems related to overlapping divergences in this amplitude, and so we mustexamine how differential renormalization deals with them. There are two subdiagrams which containlog singularities related to the regions u ∼v ∼x and u ∼v ∼0.

We remove both divergences bypulling out derivatives in x and y in a symmetric way as we did in the case of the one-loop vertex,Eq.(1.20). More explicitly, the important intermediate step is∂d1(x −u)2 ∂a1(y −u)2 ∂b1(y −v)2 ∂c1(x −v)21(u −v)2 =[ ∂∂xd (1(x −u)2 ∂c1(x −v)2 ) −1(x −u)2 (∂d∂c −δcd4)1(x −v)2 + π2δcdδ4(x −v)(x −u)2 ]1(u −v)2[ ∂∂yb (1(y −v)2 ∂a1(y −u)2 ) −1(y −v)2 (∂b∂a −δba4)1(y −u)2 + π2δabδ4(y −u)(y −v)2 ] .

(2.5)This manipulation makes it evident that each subdivergence is cured separately thanks to the fact thatin coordinate space the external points are kept apart (whereas, in momentum space, the momentumintegral would make the two singularities overlap). The rest of the computation, carried out with thehelp of Feynman parameters, although somewhat lengthy, is conceptually simple since only a globaldivergence needs further correction.

The final renormalized value is:Π(2b)µν (x) = −148π2απ2(−(∂µ∂ν −δµν) ln2 x2M 2V + 173 ln x2M 2x2!+ δµνln x2M 2x2),(2.6)Taking into account the mass relation (Ward identity), Eq. (1.28), the entire 2-loop renormalizedvacuum polarization reads:Π(2)µν (x) = 2Π(2a)µν (x) + Π(2b)µν (x) =116π2απ2(∂µ∂ν −δµν)ln x2M 2x2.

(2.7)This is automatically transverse, and furthermore the ln2 contributions have cancelled. This is animportant point which we will further elaborate in Section 3.10

The 2-loop renormalized AA 2-point functionΓAAµν (x) =(1 −1a)∂µ∂ν −δµνδ(4)(x) −Π(1)µν (x) −Π(2)µν (x)(2.8)will satisfy an RG equation, Eq. (1.10), with MΠ∂∂MΠ substituted for MΠ∂∂MΠ +M∂∂M (in general, anRG equation will include the sum of the derivatives w.r.t.

all masses present in an amplitude). Thiswill yield the 2-loop RG functions:γA(α) = α3π + α24π2andγa(α) = −2aα3π −aα22π2 = −2aγA(α).

(2.9)As mentioned before, the same calculation in the background field method would have led us to the2-loop beta function, which we will derive and present independently through the calculation of the2-loop vertex in Sec. 2.3.2.2 Fermion Self-EnergyNext, we consider 2-loop contributions to the fermion self-energy, shown in Figs.

2c-2e. In orderto eventually verify RG equations and Ward identities, we also give the logarithmic mass derivativeM∂∂M acting on each one of the amplitudes.

The first diagram is that of Fig. 2c:Σ(2c)bare(x) = −(ie)2(4π2)3Zd4ud4v 1x2 γµ ∂/1(x −u)2 Σ(1)(u −v) ∂/ 1v2 γµ .

(2.10)Again, integration by parts, properties of γ-matrices and DR identities lead to the renormalizedvalue:Σ(2c)(x) = −1128π2απ2∂/ln2 x2M 2Σ + ln x2M 2x2,(2.11)where we have also used the identity1x2 ∂aln x2M 2Σx2= 12∂aln x2M 2Σ −1/2x4. (2.12)The mass derivative of this amplitude is:M ∂∂M Σ(2c) = −α2π Σ(1) −α216π2 Σ(0).

(2.13)The next contribution is that of Fig. 2d:Σ(2d)bare(x) = −(ie)4(4π2)5Zd4ud4v 1u21(x −v)2 γµ∂/1(x −u)2 γν∂/1(u −v)2 γµ∂/ 1v2 γν .

(2.14)11

This amplitude is solved by using vertex-like manipulations. It turns out that no new integrals areneeded apart from those appearing in the two-loop vacuum polarization diagram 2b.

We also makeuse of the following trick1(v −x)21v2= −4π2δ(4)(v −x)v2+ δ(4)(v)(v −x)2+ 2∂a1(v −x)2 ∂a1v2 ,which simplifies part of the computation. The renormalized value we finally get is:Σ(2d)(x) =164π2απ2∂/ln2 x2M 2Vx2,(2.15)and the mass derivative of this amplitude is:M ∂∂M Σ(2d) = απ Σ(1) −α28π2 Σ(0),(2.16)where we have used the 1-loop mass relation to express Σ(1) with the mass scale MΣ rather thanMV .We finally present the last of the 2-loop fermion self-energy diagrams (Fig.

2e):Σ(2e)bare(x) = −(ie)2(4π2)3 γµ ∂/ 1x2 γνZd4ud4v1(x −v)2 Π(1)µν (v −u) 1u2 . (2.17)The renormalization of this amplitude is straightforward, and we only point out that while there is inprinciple the possibility that the ln MΠ in Π(1)µν would get promoted to a ln2 MΠ, the γ-matrices inthe amplitude, together with the transverse operator in Π(1)µν , conspire to simply cancel all ln2’s.

Thisis again an instance of the feature we have seen previously in Π(2)µν , namely, the apparently unexpectedcancellation of some particular divergences. The final, renormalized result is:Σ(2e)(x) = −132π2απ2∂/ln x2M 2x2.

(2.18)The mass derivative of this amplitude will be:M ∂∂M Σ(2e) = −α24π2 Σ(0). (2.19)Whereas we would generally expect the mass derivative of a 2-loop amplitude to generate 1-loop andtree-level amplitudes, we see this does not happen here.

In Section 3, we will examine this moreclosely.12

We now add all these amplitudes to find the 2-loop renormalized ψψ 2-point function:Γψψ(x) =∂/ δ(4)(x) −Σ(1)(x) −Σ(2c+2d+2e)(x)=∂/ δ(4)(x) −a16π2απ∂/ln x2M 2Σx2−1128π2απ2∂/ln2 x2M 2Σ −7 ln x2M 2x2. (2.20)We have the 1-loop correction in a generic gauge a, and the 2-loop terms in Feynman gauge.

Thisis written thus because in RG equations we will need∂∂a on the 1-loop term but not on the 2-loopterms. The 2-loop fermion anomalous dimension given by the RG equations is (in Feynman gauge):γψ(α) = α4π −3α232π2 .

(2.21)2.3 VertexWe now turn to the computation of 2-loop corrections to the gauge coupling vertex, shown inFigs. 2f-l.

Many of the integrals are extremely difficult to perform analytically, but fortunately theyneed not be done for the purposes of verifying RG equations and Ward identities (although we willnot present the 2-loop relation between MV and MΣ). The parts of these diagrams that do needto be computed fully are the renormalization pieces from both global and internal divergences, andthose are reasonably simple to calculate.

Because the expressions for the renormalized amplitudes aresomewhat lengthy, we present them in Appendix B. Here, instead, we will explain briefly the differenttechniques we have used and subtleties involved as we go along.

Again, for the purpose of laterverifying RG equations and Ward identities, we also present here the result of the logarithmic massderivative M∂∂M acting on each one of the renormalized amplitudes. We set z = 0 in all amplitudes.The first diagram we consider is Fig.

2f:V (2f)bareµ(x, y) = −(ie)3(4π2)41(x −y)2Zd4ud4v γρ ∂/ 1x2 γµ ∂/ 1u2 Σ(1)(u −v)∂/1(v −y)2 γρ . (2.22)The standard integrations by parts and separation into trace and traceless pieces which were usedto renormalize the 1-loop vertex are used here as well, without any added complication.

We also needto consider the diagram identical to the one above, but with the self-energy insertion on the otherfermion leg, Fig. 2g.

The procedure is identical to the one used above, and its contribution to theRG equations at two loops is also the same. From the renormalized expression given in Appendix B,we can calculate the mass derivative of this amplitude:M ∂∂M [V (2f)µ+ V (2g)µ] = −απ V (1)µ−3α28π2 V (0)µ.

(2.23)13

Here and below, we are always taking V (1)µin Feynman gauge, unless otherwise explicitly stated; wemust also be careful to express the result in terms of MV and not MΣ (in the above, for instance,neglecting this would lead to a different coefficient for the contact piece V (0)µ).The second vertex diagram we compute, Fig. 2h, contains a vacuum polarization insertion:V (2h)bareµ(x, y) = (ie)3(4π2)4 (γργaγµγbγσ) ∂a1x2 ∂b1y2Zd4ud4v1(x −u)2 Π(1)ρσ (u −v)1(v −y)2 .

(2.24)To renormalize this, we integrate the derivatives ∂a and ∂b by parts onto the photon leg. The transver-sality of Π(1)µν and the gamma structure in front will arrange things so that the resulting combinationof derivatives will automatically be traceless, thus obviating the need for further renormalization and(again) avoiding the promotion of ln MΠ.

This in turn leads to a peculiar form for the mass derivative:M ∂∂M V (2h)µ= −2α3π V (1)µ(a = 0). (2.25)Now, we are left with the more difficult diagrams, which cannot be computed in closed form.Starting with the diagram of Fig.

2i, its bare value is:V (2i)bareµ(x, y) =−(ie)5(4π2)6 (γργaγσγbγµγcγσγdγρ) ×1(x −y)2Zd4ud4v1(u −v)2 ∂a1(x −u)2 ∂b1u2 ∂c1v2 ∂d1(v −y)2 . (2.26)We first of all renormalize the upper vertex (connecting points z = 0, u, and v), by integrating byparts in z and separating the term integrated by parts into a trace and traceless piece (in the indicesb and c).

The surface term is finite by power counting, and the trace piece easily renormalizes to astructure of the form(γdγµγa)1(x −y)2 ∂a1x2 ∂dln y2M 2Vy2,(2.27)which needs to be renormalized one more time (integration by parts and separation into trace andtraceless pieces), giving a ln2 promotion through the use of a DR identity. This is easily done, andnow we are finally left with the global divergence (x ∼y ∼0) present in the traceless piece (in bc)we got after the first integration by parts.

That term is power counting log divergent, and one mightthink the traceless combination of derivatives then makes it finite, but this is not so: the point isthat there are other free indices in that expression (a and d). The integration is indeed made finiteby tracelessness in the bc indices, because the a and d derivatives can be brought out of the integral,but there remains a global divergence as x ∼y ∼0, because the expression is not traceless in all14

indices. Attempting to separate the trace pieces from four free indices is hopeless because we cannoteven do the integral in u and v, and so we resort to a technique used to renormalize the nonplanar3-loop 4-point diagram in λφ4, valid for primitively divergent expressions in general.

Details can befound in [1], and we do not give them here. The idea consists in writing the factor1(x−y)2 in front as1(x −y)2 = (x −y)21(x −y)4 = −(x −y)24ln(x −y)2M 2(x −y)2.

(2.28)One can verify that the simple substitution above, as is, suffices to renormalize the global di-vergence we had (that is, that term will have a well-defined Fourier transform). The only subtletyin applying the mass derivative to the resulting renormalized amplitude lies in fact in this globallydivergent piece.

M∂∂M on the term above is proportional to δ(4)(x−y) and, like in [1], one must verifythat what multiplies this is a representation of δ(4)(y −z) when x →y. Adding up all contributions,the mass derivative then reads:M ∂∂M V (2i)µ= α2π V (1)µ+ 5α216π2 V (0)µ.

(2.29)The next diagram is that of Fig. 2j, with bare value:V (2j)bareµ(x, y) =−(ie)5(4π2)6 (γργaγµγbγνγcγργdγν) ×∂a1x2Zd4ud4v ∂b1u21(x −v)2 ∂c1(u −v)2 ∂d1(v −y)21(u −y)2 .

(2.30)The first step in renormalizing this diagram is integration by parts of the ∂d and ∂c derivatives,and separation into trace and traceless parts. There will be two of each; the first trace piece is easilyrenormalizable with standard DR identities, and the second one involves the structure(γdγaγµ) ∂a1x2 ∂d1y2 K(x, y) ,(2.31)whereK(x, y) =Zd4u1(x −u)2(y −u)2u2 .

(2.32)This is renormalized one more time by integrating the derivatives by parts onto the K-function andseparating trace and traceless pieces. For the first time, we encounter renormalization structures thatdo not correspond to any lower loop diagrams, and because of consistency with RG equations, thesemust cancel in the only other diagram left to compute, Fig.

2l (the nonplanar diagram). We find thatindeed they do.

The only other divergent pieces remaining are the ones traceless in cb and cd, which15

can have a global divergence for the same reason as the equivalent term in the previous diagram:the presence of extra indices. These are primitively divergent, like in the previous diagram, and aresolved in exactly the same way.

Naturally, the diagram identical to this one, but with the vertexsubdivergence on the opposite fermion leg, Fig. 2k, is solved in the same fashion, and leads to thesame contribution to the 2-loop RG equations.

We present the result of the mass derivative on thesediagrams after the inclusion of the nonplanar diagram, Fig. 2l, precisely because there are structuresin these three diagrams which do not correspond to any lower loop diagram, and which cancel whenthey are added.

So, finally, the nonplanar diagram is:V (2l)bareµ(x, y) = −(ie)5(4π2)6 (γνγaγργbγµγcγνγdγρ) ×Zd4ud4v ∂a1(x −u)2 ∂b1u2 ∂c1v2 ∂d1(v −y)21(x −v)2(u −y)2 . (2.33)To renormalize this, we integrate ∂c by parts around z = 0, and again separate trace and tracelesspieces.

Within the trace pieces, we will find the structureγµ ∂∂xa1x2y2∂∂ya K(x, y)+1x2y2∂∂xa∂∂ya K(x, y),(2.34)and these will precisely cancel with structures found in the two previous diagrams. We can now presentthe mass derivative acting on the sum of these three diagrams:M ∂∂M [V (2j)µ+ V (2k)µ+ V (2l)µ] = απ V (1)µ−α28π2 V (0)µ.

(2.35)We are now ready to consider RG equations. These are easy to verify once we use the massderivatives given above for the 2-loop vertices.

The 2-loop ΓψAψµvertex function,ΓψAψµ(x, y) = ieγµδ(4)(x −y)δ(4)(x) + V (1)µ(x, y) + V (2f+2g+2h+2i+2j+2k+2l)µ(x, y),(2.36)satisfies an RG equation, Eq. (1.22), confirming all the results given previously for different RG func-tions, and yielding also the 2-loop β-function:β(α) = 2α23π + α32π2 .

(2.37)This matches the value gotten by a background field calculation on the 2-loop vacuum polarization.2.4 Ward identity16

At two loops, the method employed previously to verify Ward identities, viz., integrating over anexternal variable, becomes computationally difficult and not very illuminating. For our purposes here,we shall consider instead the following simpler procedure: we apply the logarithmic mass derivativeto both sides of the (2-loop) Ward identity; as we have seen above, this yields 1-loop and tree-levelvertices and self-energies, whose Ward identities have already been verified, so that our problem isreduced to that of matching the coefficients of 1-loop and tree-level quantities on both sides of theidentity.

We have, on the one hand, gained much in simplicity but, of course, on the other hand,some information contained in the Ward identity will thus be lost, namely, the contact terms comingfrom the finite parts of vertices, since these latter have no mass scales and will vanish when the massderivative is applied. Although in this way we cannot derive mass relations like we did at one loop,the match we find represents a highly nontrivial consistency check of our 2-loop computations.Adding up all the contributions from Eqs.

(2.13), (2.16) and (2.19), we find:M ∂∂M Σ(2c+2d+2e) = α2π Σ(1) −7α216π2 Σ(0)(2.38)and from Eqs. (2.23), (2.25), (2.29), (2.35),M ∂∂M V (2f+2g+2h+2i+2j+2k+2l)µ= α2π V (1)µ−3α216π2 V (0)µ−2α3π V (1)µ(a = 0).

(2.39)Once we use the fact that, from Eq. (1.25)Zd4y ∂∂zµ V (1)µ(x, y; a = 0) = 3α8π Σ(0)(x) ,(2.40)we immediately find the amplitudes we have calculated above do indeed satisfy the 2-loop Wardidentity.It is also worthwhile noting, furthermore, that in fact we can divide the above 2-loopamplitudes into three sets that separately verify the Ward identity.

They are:∂∂zµ V (2f+2g+2i)µ(x, y) = −ie[δ(4)(x) −δ(4)(y)]Σ(2c)(x −y) ,(2.41)∂∂zµ V (2j+2k+2l)µ(x, y) = −ie[δ(4)(x) −δ(4)(y)]Σ(2d)(x −y) ,(2.42)∂∂zµ V (2h)µ(x, y) = −ie[δ(4)(x) −δ(4)(y)]Σ(2e)(x −y) . (2.43)The vertex diagrams in each one of these sets are generated by attaching an external photon linein every possible way to an internal fermion line of the corresponding self-energy diagram.

This is avestige of the fact that each bare vertex – individually – formally satisfies a Ward identity with theself-energy gotten by eliminating the external photon line from that vertex diagram.17

3. Structured Renormalization GroupIn this section we point out a certain structure exhibited by the renormalization of the differentrelevant vertex functions of QED.

For that purpose, we gather here the results of the mass derivativeof all 2-loop amplitudes.Vacuum polarization:M ∂∂M [2Π(2a)µν+ Π(2b)µν ] = −α22π2 (∂µ∂ν −δµν) δ(x)(3.1)Self-energy:M ∂∂M Σ(2c) = −α2π Σ(1) −α216π2 Σ(0)(3.2)M ∂∂M Σ(2d) = απ Σ(1) −α28π2 Σ(0)(3.3)M ∂∂M Σ(2e) = −α24π2 Σ(0)(3.4)Vertex:M ∂∂M [V (2f)µ+ V (2g)µ] = −απ V (1)µ−3α28π2 V (0)µ(3.5)M ∂∂M V (2h)µ= −2α3π V (1)µ(a = 0)(3.6)M ∂∂M V (2i)µ=α2π V (1)µ+ 5α216π2 V (0)µ(3.7)M ∂∂M [V (2j)µ+ V (2k)µ+ V (2l)µ] = απ V (1)µ−α28π2 V (0)µ(3.8)The feature the above equations clearly display, which has already been alluded to in Section 2,is the absence of promotion of 1-loop logarithms at two loops for i) the 2-loop vacuum polarization,Eq. (3.1), ii) the fermion self-energy with a vacuum polarization subdiagram, Eq.

(3.4), and iii) thevertex with a vacuum polarization subdiagram, Eq.(3.6). All the rest follow the expected patternof log promotions.Given that the Ward identities provide a relation between MΣ and MV , butnot between MΠ and anything else, we might then understand the above as a manifest, conversestatement to the Ward identity, namely, that the renormalization of the vacuum polarization runsentirely independently of the renormalization of the self-energy and vertex.Thus, for the 2-loopvacuum polarization, although both Π(2a)µνand Π(2b)µνcontain promoted logs of MΣ and MV (cf.Eqs.

(2.3) and (2.6)) coming from their respective subdivergences, in the entire 2-loop amplitudethese promotions cancel. This is as it should be, since otherwise RG equations would imply a relationbetween the mass scales MΣ (or MV ) and MΠ, which cannot happen if the renormalization of18

these amplitudes is to be independent. For the self-energy and vertex with vacuuum polarizationsubdivergences, the same happens: no ln2 M 2Π occur.

The mass derivative of the vertex does givea 1-loop vertex, but it is in Landau gauge, and thus finite. For these amplitudes then, we can statethat there are no genuine 2-loop divergences, and that is a direct consequence of the lack of a Wardidentity relating the vacuum polarization to any other vertex functions.We see then that the renormalization of these amplitudes has, so to speak, split into differentsectors.

A careful study of this structured renormalization pattern should furthermore allow us to makepredictions about the renormalization at higher loops. In [11](cf.

p.423), for instance, the statementis made that all higher-loop vacuum polarization amplitudes with a single fermion loop (i.e., withno lower-loop vacuum polarization subdivergences) do not contain genuine higher loop divergences(ln2’s, or 1/ǫ2’s in dimensional regularization, etc.). This feature, which may seem fortuitous in otherrenormalization methods, appears naturally in differential renormalization.4.

Chiral AnomalyIn this section, we review the computation of the anomalous triangle amplitude ⟨jµ(x)jν(y)j5λ(z)⟩at one loop [1], and carry it on partially at two loops.Some of the main attractive features ofdifferential renormalization are made manifest.We also compare it to another coordinate space,regulator-free computation of the anomaly due to K. Johnson [20] for the sake of completeness.Since differential renormalization is strictly 4-dimensional and does not introduce any unphysicalregulator fields, one is able to avoid the complications introduced by standard methods such asdimensional regularization (in necessitating ad hoc γ5-prescriptions away from d = 4), and Pauli-Villars regularization.Furthermore, differential renormalization does not present the anomaly ascoming from a symmetry that is broken by regularization artifacts, as is usually the case. Instead, weshall see that the bare triangle amplitude, without the γ-matrix trace factor in front, has singularitieswhich bring in two renormalization scales and these, when the trace is included, lead to a finiteamplitude with a continuous one-parameter shift freedom given by the quotient of these scales.

Itturns out that this freedom cannot accomodate simultaneously vector an axial conservation. Therefore,the triangle amplitude is overconstrained by the two Ward identities.

In our scheme, both vector andaxial symmetries appear manifestly on the same footing, and the shift in the anomaly from one to theother is transparent.19

We begin with the 1-loop computation. The basic elements have already been spelled out in [1].Here we simply sketch the key points.

At one loop, the anomalous triangle is (Fig. 3):Tµνλ(x, y) = ⟨jµ(x)jν(y)j5λ(0)⟩= 2 Tr [γ5γλγaγνγbγµγc] ∂a1y2 ∂b1(x −y)2 ∂c1x2 ,(4.1)where the factor of 2 reflects the inclusion of the Bose symmetrized diagram.

The termtabc(x, y) = ∂a1y2 ∂b1(x −y)2 ∂c1x2is power counting L−9 and thus linearly divergent. This is renormalized in the same way we havetreated triangles previously, with the difference that now two derivatives need to be taken out.

Thishas already been done in [1], and we present the final result:tabc(x, y) = Fabc(x, y) + Sabc(x, y) .Fabc is the finite partFabc(x, y) =∂2∂xa∂yb1x2y2 ∂c1(x −y)2+∂∂xa1x2y2∂b∂c −δbc41(x −y)2−∂∂yb1x2y2∂a∂c −δac41(x −y)2−1x2y2∂a∂b∂c −16δ(ab∂c)1(x −y)2 ,(4.2)where () means unnormalized symmetrization in all indices, and Sabc is the renormalization piece:Sabc(x, y) = 14π2δbc∂∂xa−δac∂∂ybδ(x −y)ln M 21 x2x2−13δbc ∂∂xa−∂∂ya+ δac ∂∂xb−∂∂yb+ δab ∂∂xc−∂∂ycδ(x −y)ln M 22 x2x2. (4.3)We note that, like for the 1-loop fermion self-energy, two different mass scales have been used torenormalize the different divergent trace pieces.

Not only are we entitled to do that, but in fact onemore time it will prove crucial that we do so.At this point, we apply the traces outside to get Tµνλ(x, y):Tµνλ(x, y) = Rµνλ(x, y) + aµνλ(x, y) ,(4.4)where Rµνλ comes from the finite part Fabc, andaµνλ(x, y) = −16π4 ln M1M2ǫµνλρ ∂∂xρ−∂∂yρδ(x)δ(y) . (4.5)20

It is fairly simple to see that for x ̸= y ̸= 0, Tµνλ is conserved on all channels. However, justas in the vertex WI studied previously, the subtleties are of course in the contact terms.

It is alsoworthwhile noting that while Sabc (and thus tabc) indeed contains divergences, the precise γ-structurein front arranges things such as to give the final combination ln M1M2 as the only renormalizationmass dependence, implying that aµνλ (and thus Tµνλ) is actually finite (because M∂∂M on thatvanishes). This leads us then to a very clear physical picture: we have a finite anomalous triangle,which however contains an ambiguity – in the choice of Ms – coming from a power counting linearlydivergent Feynman diagram.By Fourier transforming into momentum space, one can verify the conservation laws satisfied byRµνλ on all three channels.

Given the form of aµνλ found above, the vector and axial WIs on Tµνλthen read:∂∂xµ Tµνλ(x, y) = 8π4(1 + 2 ln M1/M2)ǫνλµρ∂∂xµ∂∂yρδ(x)δ(y)∂∂yν Tµνλ(x, y) = 8π4(1 + 2 ln M1/M2)ǫµλνρ∂∂yν∂∂xρδ(x)δ(y)− ∂∂xλ+∂∂yλTµνλ(x, y) = 16π4(1 −2 ln M1/M2)ǫµνλρ∂∂xλ∂∂yρδ(x)δ(y) . (4.6)This is the final and, as it were, a most “democratic” expression of the anomaly: we can tune M1/M2so that Tµνλ is conserved either on the vector channels or on the axial one, but never on both; wehave been able to complete the calculation in a “scheme-free” fashion all the way to the end, withouthaving to commit at any point to conservation on a particular channel.We now proceed to the 2-loop calculations.

The relevant diagrams are indicated in Fig. 4.Writing the amplitude asT (2)µνλ(x, y) = Aµνλ(x, y) + Bµνλ(x, y) ,(4.7)where Aµνλ contains the contributions of the diagrams with fermion self-energy insertions, and Bµνλcontains the contributions of the diagrams with the vertex insertions, we can immediately write, asthe result of a trivial computation:Aµνλ(x, y) =e232π4 Tr [γ5γλγaγνγbγµγc]∂a1y2 ∂b1(y −x)2 ∂cln x2M 2Σx2+∂aln y2M 2Σy2∂b1(y −x)2 ∂c1x2 + ∂a1y2 ∂bln(y −x)2M 2Σ(y −x)2∂c1x2.

(4.8)To compute the diagrams related to vertex corrections we operate as follows. Let us take, for instance,the correction to the vertex at y, which leads to the following integral:Zd4ud4v1(u −v)2 ∂a1u2 ∂b1(u −y)2 ∂c1(y −v)2 ∂d1(v −x)2 ∂e1x2(4.9)21

Treating the vertex in the standard way, we integrate ∂b by parts over y, and separate a trace and atraceless piece, and a surface term. The last two terms do not produce logarithms.

It is only the tracepiece, which is very easy to compute, that brings in a log of the same kind as in Eq.(4.8). Puttingtogether the log pieces of the three vertex correction diagrams we getBµνλ(x, y) = −e232π4 Tr [γ5γλγaγνγbγµγc]∂a1y2 ∂b1(y −x)2 ∂cln x2M 2Vx2+∂aln y2M 2Vy2∂b1(y −x)2 ∂c1x2 + ∂a1y2 ∂bln(y −x)2M 2V(y −x)2∂c1x2+ ... ,(4.10)where the dots indicate the finite pieces not written explicitly here.

Clearly, Aµνλ and Bµνλ cancelexactly except for the fact that the mass scale involved in Aµνλ is MΣ whereas the one in Bµνλ isMV . The one-loop Ward identity shows that the mistmatch is precisely proportional to the 1-loopamplitude, thus showing that the entire 2-loop amplitude is finite.The finite parts of Bµνλ areextremely lengthy to calculate and we have not found a compact way to perform the computation.In fact, the total amplitude is expected to vanish identically since, otherwise, as we now explain, theanomaly would get a finite renormalization.QED is conformally invariant up to the photon propagator and the appearance of renormalizationscales.

Thus, the anomalous triangle is conformally covariant at one loop because it is finite andcontains no photon lines. In any coordinate space treatment, this can be verified explicitly to oneloop, whereas in momentum space calculations, this is obscured because plane waves transform easilyunder translations but not under conformal tranformations.

This conformal property, when conjecturedto all loops, is powerful enough to show the vanishing of all higher-loop triangle amplitudes, in anargument due to Baker and Johnson [16]. From the study of conformal transformations on functionsof three variables, it turns out there is a unique nonlocal, conformal covariant, parity-odd, dimension3, VVA tensor [19], and the 1-loop triangle has precisely this form.

Now, from uniqueness, if thetriangle amplitude is covariant to all orders, it means higher-loop contributions also have to have theform of the 1-loop triangle. But that would mean a renormalization of the very structure which givesrise to the chiral anomaly, and thus a renormalization of the anomaly itself, which is forbidden bythe Adler-Bardeen theorem.

Therefore, all higher-loop contributions to the basic triangle must vanishidentically.Let us finally mention a very nice coordinate space computation of the anomaly which makes nouse of an ultraviolet regulator, due to Johnson [20]. His starting point is the conformally covariant22

and manifestly finite form of the triangle gotten by acting out the γ-matrix trace in Eq. (4.1) fromthe beginning.Because there is, apart from this nonlocal structure, a unique contact term withprecisely the same dimension and conformal properties [19], the triangle amplitude can have this localambiguity, and its coefficient can be chosen to give conservation on either channel but not both.

Ifwe want conservation on the x channel, for instance, we have:∂∂xµTµνλ(x, y) + aǫµνλσ ∂∂xσ −∂∂yσδ(x)δ(y)= 0 ,(4.11)where we have shown explicitly the contact term in question, with coefficient a to be determined.To find a, one then integrates ∂µTµνλ in x and y against the “test” function xαyβ. This integral,remarkably, is entirely determined from the long-distance behavior of Tµνλ, and the only cutoffneededto perform it is an infrared one.

Johnson also points out that this situation is reminiscent of thePoisson equation in classical electrostatics, where the coefficient of a contact term in a differentialequation (the charge of a point particle) is also determined by the long-distance behavior of a field(Gauss’ theorem).5. ConclusionIn this paper, we have presented a detailed analysis of the differential renormalization of QED upto two loops.

All computations are reasonably simple when compared to other approaches.Our results can be summarized as follows. We have found explicit expressions for the 1- and 2-loop renormalized amplitudes of the vacuum polarization, self-energy of the fermion and vertex.

Onlysome finite parts of the latter are left in the form of integrals. Ward Identities are verified and provideindependent checks of the computations.

Furthermore, the amplitudes obey renormalization groupequations which yield the beta function and the various anomalous dimensions of all the basic fields inthe theory. This renormalization group equations display a natural organization due to the freedom touse different renormalization subtractions for quantities not related by WIs. We have also presentedthe treatment of the anomalous triangle which, in the very spirit of differential renormalization, isregulator-free.

It sides with other techniques which are based on the interplay between potentialcounterterms and symmetries rather than on explicit computations on a regulated theory.One is rapidly enticed by the ease of computation of differential renormalization. It is our opinionthat the method also deserves further consideration because of its natural treatment of chiral problems.23

AcknowledgmentsWe thank D.Z. Freedman warmly for his active participation in different stages of this paperand K. Johnson for his insight on anomalies.

We would also like to acknowledge discussions with X.Vilas´ıs-Cardona, and a critical reading of the manuscript by R. Tarrach. This work was supported inpart by CAICYT grant # AEN90-0033, and by EEC Science Twinning grant # SCI-000337.

One ofus (PH) also acknowledges a grant from MEC, Spain.Appendix A. Basic Differential Renormalization IdentitiesThe following are the DR identities used to renormalize all of the amplitudes presented in thetext:1x4 = −14ln x2M 2x2(A.1)1x6 = −132ln x2M 2x2(A.2)ln x2M 2x4= −18ln2(x2M 2) + 2 ln x2M ′2x2(A.3)Appendix B. Two-loop renormalized verticesWe present here the final expressions for the 2-loop renormalized vertices.

In the expressionsbelow, a derivative w.r.t. zµ means∂∂zµ = −∂∂xµ −∂∂yµ .V (2f)µ(x, y) =−i16π3απ5/2(γbγµγa) ∂∂zb ln y2M 2y2(x −y)2 ∂a1x2+ln y2M 2y2(x −y)2∂a∂b −δab4 1x2 + π28 δabδ(4)(x)ln2 y2M 2Σ + 2 ln y2M 2y2.

(B.1)V (2h)µ(x, y) =i96π3απ5/2 (γργaγµγbγσ)−∂∂xa 1x2 ∂b1y2 (∂ρ∂σ −δρσ)L(x −y)+∂∂yb1x2y2 ∂a(∂ρ∂σ −δρσ)L(x −y)−16x2y2 γa∂a∂µ −δaµ4 ln(x −y)2M 2Π(x −y)2,(B.2)24

where L(x) = ln x2µ2 −12 ln2 x2M 2Π.V (2i)µ(x, y) =i32π7απ5/2 Zd4ud4v(γdγbγµγcγa) ∂∂zc1(x −y)2v2 ∂b1u21(u −v)2 ∂a1(x −u)2 ∂d1(v −y)2−(γdγcγa) (x −y)22ln(x −y)2M 2(x −y)21v2(u −v)2 (∂c∂µ −δcµ4) 1u2 ∂a1(x −u)2 ∂d1(v −y)2+i16π3απ5/2(γdγµγa) ∂∂zd1(x −y)2 ∂aln y2M 2y2x2+1(x −y)2 (∂a∂d −δad4) 1x2ln y2M 2y2−i64παπ5/2γµδ(4)(x)ln2 y2M 2V + 2 ln y2M 2y2. (B.3)V (2j)µ(x, y) =−i64π7απ5/2(γργaγµγbγdγργc)Zd4ud4v ∂∂zd∂a1x2 ∂b1u21(x −v)2 ∂c1(u −v)21(v −y)21(u −y)2−∂a1x2∂b1u2 (∂c∂d −δcd4)1(u −y)2 + ∂d1(u −y)2 (∂c∂b −δcb4) 1u21(x −v)2(u −v)2(v −y)2+i16π3απ5/2(γbγµγa) ∂∂za 1x2 ∂b1y2ln(x −y)2M 2(x −y)2+ 1x2 (∂a∂b −δab4) 1y2ln(x −y)2M 2(x −y)2−i64παπ5/2γµδ(4)(y)ln2(x −y)2M 2V + 2 ln(x −y)2M 2(x −y)2+i16π5απ5/2(γbγaγµ) ∂∂xa 1x2 ∂b1y2 K(x, y)+∂∂yb1x2y2∂∂xa K(x, y)−1x2y2 ∂∂xa∂∂yb −δab4K(x, y)−i32π3απ5/2γµ14ln x2M 2x2ln y2M 2y2+∂∂xa1(x −y)2y2↼⇀∂∂xaln x2M 2x2+ π2δ(4)(x −y)ln2 x2M 2V + 2 ln x2M 2x2+∂∂ya1(x −y)2x2↼⇀∂∂yaln y2M 2y2.

(B.4)V (2l)µ(x, y) =i128π7απ5/2(γνγaγργbγµγcγνγdγρ)Zd4ud4v∂∂zc∂a1(x −u)2 ∂b1u21v2 ∂d1(v −y)21(x −v)2(u −y)2−i8π7απ5/2γbZd4ud4v ∂a1(x −u)2 (∂b∂µ −14δbµ) 1u2 ∂a1(v −y)2+i8π5απ5/2γµ∂∂xa1x2y2∂∂ya K(x, y)+i16π3απ5/2γµ14ln x2M 2x2ln y2M 2y2+−∂∂za1(x −y)2y2↼⇀∂∂xaln x2M 2x2+ π2δ(4)(x −y)ln2 x2M 2V + 2 ln x2M 2x2. (B.5)25

Figure captions:Figure 1: 1-loop 1PI diagrams of QED.Figure 2: 2-loop diagrams of QED.Figure 3: 1-loop anomalous triangle diagram.Figure 4: 2-loop contributions to anomalous ABJ amplitude.26

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