The QED vacuum polarization function at four loops and the anomalous magnetic moment at five loops

The QED vacuum polarization function at four loops and the anomalous magnetic moment at five loops

P.A. Baikov A. Maier P. Marquard Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, 1(2), Leninskie gory, Moscow 119991, Russian Federation Physik Department T31, James-Franck-Straße, Technische Universität München, D–85748 Garching, Germany Deutsches Elektronen Synchrotron DESY, Platanenallee 6, D15738 Zeuthen, Germany

The anomalous magnetic moment of the muon is one of the most fundamental observables. It has been measured experimentally with a very high precision and on theory side the contributions from perturbative QED have been calculated up to five-loop level by numerical methods. Contributions to the muon anomalous magnetic moment from certain diagram classes are also accessible by alternative methods. In this paper we present the evaluation of contributions to the QED corrections due to insertions of the vacuum polarization function at five-loop level.

Perturbative calculations, QED, lepton anomalous moment

1 Introduction

The anomalous magnetic moments of electron and muon are among the best experimentally measured quantities and are also very well understood from theoretical side. In particular, the perturbative QED corrections have been under consideration for a very long time. The leading-order QED corrections have been considered by Schwinger in Ref. Schwinger:1948iu, the next-to-leading order corrections not much later in Refs. Petermann:1957hs; Sommerfield:1957zz. At three loops, i.e. next-to-next-to-leading order, the QED corrections have been first calculated numerically in Ref. Kinoshita:1995ym and later analytically in Refs. Laporta:1996mq; Melnikov:2000qh; Marquard:2007uj. At four and five loops the complete QED contributions have been calculated numerically in Refs. Aoyama:2012wk; Aoyama:2012wj while only partial analytical results exist Lautrup:1974ic; Kinoshita:1990ur; Kawai:1991wd; Faustov:1990zs; Kataev:1991cp; Broadhurst:1992za; Baikov:1995ui; Laporta:1993ds; Baikov:2012rr; Lee:2013sx. For a more thorough review of the current status see e.g. Ref. Jegerlehner:2009ry.

In the case of the muon anomalous moment, next to the universal contribution, contributions from diagrams with closed electron loops are of particular interest. A subgroup of this class of diagrams, contributions from corrections to the photon propagator, can be easily obtained using already available building blocks Lautrup:1974ic; Barbieri:1974nc. At five loops the leading contributions have been obtained from considering the asymptotic form Baikov:2012rr, which resulted in unexpected discrepancies with the results obtained in Ref.Aoyama:2010zp. The aim of this paper is to improve the predictions for the anomalous magnetic moment of the muon made in Ref. Baikov:2012rr and to resolve the discrepancies. To this extend we approximately reconstruct the photon vacuum polarization at four loops using all available information.

This paper is organized as follows: In Section 2 we collect the necessary information to construct an approximation for the vacuum polarization function at four loops. In Section 3 we present the calculation of the contribution to the anomalous magnetic muon from vacuum polarization insertions.

2 QED vacuum polarization at four loops

We define the vacuum polarization function as usual by


with the current and write it as an expansion in the fine-structure constant


At three and four loops the vacuum polarization can only be calculated in a low-energy, high-energy or threshold expansion at the moment. The available information can then be used to construct an approximate function using Padé approximation. In the following we will present the results for the corresponding expansions and construct an approximating function. The renormalization of lepton mass and fine-structure constant is performed in the on-shell scheme. Note that in this scheme the condition is being imposed.

2.1 Low-energy expansion

In the low-energy limit the polarization function can be expanded in a power series in


For the QED case at hand, the result for the non-singlet contribution can in principle be obtained from the QCD results given in Ref. Maier:2009fz. But since the results are not given expressed through colour factors they cannot easily be translated to the case of QED. The low-energy expansion was therefore recalculated for the case at hand. The calculation follows a well-established path. The Feynman diagrams are generated using qgraf Nogueira:1993 and mapped onto six topologies using q2e and exp Seidensticker:1999bb; Harlander:1997zb. Then a FORM vermaseren-form program is used to apply projectors and take traces. The resulting scalar integrals are then reduced to master integrals using Crusher crusher, which implements Laporta’s algorithm Laporta:2001dd for solving integration-by-parts identities. The needed master integrals have been calculated in Refs. Chetyrkin:2006dh; Schroder:2005va; Schroder:2005hy; Laporta:2002pg; Chetyrkin:2004fq; Kniehl:2005yc; Schroder:2005db; Bejdakic:2006vg; Kniehl:2006bf; Kniehl:2006bg. Combining all steps and performing the renormalization of all quantities in the on-shell scheme, the first three moments are obtained


where denotes contributions from closed lepton loops and where we used the abbreviations and . marks contributions from singlet diagrams. These contributions are numerically tiny, amounting to at most of each moment.

2.2 High-energy expansion

In the high-energy region we write the result in the form


The coefficients of Eq. (7) can be expressed through four-loop massless propagator integrals which, in turn, can be reduced to 28 master integrals. This reduction has been done by evaluating sufficiently many terms of the expansion Baikov:2005nv of the corresponding coefficient functions Baikov:1996rk. The master integrals are known analytically from Baikov:2010hf and numerically from Smirnov:2010hd.

As the result, the leading two coefficients of Eq. (7) are


where denotes contributions from closed lepton loops and where we used the abbreviations and . marks contributions from singlet diagrams. Contrary to the low-energy case these are sizeable and comparable to the contributions from other diagram classes. Naturally the same holds true for the contributions to the anomalous magnetic moment, which are dominated by the high-energy region (see Section 3).

2.3 Threshold expansion

The polarization function in the threshold region can be written as


In non-relativistic quantum field theory the NNLO expression for the threshold cross section is known for an arbitrary gauge group Hoang:2000yr. This means that the derivation of the threshold expansion is essentially the same as for the QCD case, which is described in Refs. Hoang:2008qy; Kiyo:2009gb. The only additional steps consist of setting the group invariants to their values and converting the coupling to the on-shell scheme.

The resulting coefficients read


for and


for , where we again used . The non-logarithmic contributions to and , denoted here by “const”, are not available in the literature.

2.4 Padé approximation

Having all building blocks at hand the polarization function can be reconstructed using Padé approximation Baker:1975; Broadhurst:1993mw; Fleischer:1994ef; Broadhurst:1994qj; Baikov:1995ui; Chetyrkin:1995ii; Chetyrkin:1996cf; Hoang:2008qy; Kiyo:2009gb. For the four-loop contribution we closely follow the procedure outlined in Ref. Kiyo:2009gb. In the three-loop case we introduce slight modifications to accommodate the large amount of information in the available low- and high-energy expansions. We first give a brief review of the method as used in the four-loop case and then discuss the changes for the application to the three-loop contribution in Section 2.4.2.

2.4.1 Approximation procedure at four loops

As in Ref. Kiyo:2009gb, we first split into two parts,


using the ansatz


with the known two-loop polarization function as introduced in Eq. (2) and


The coefficients can be fixed so that all known logarithms in the threshold expansion (Eqs. (10), (14)– (2.3)) are absorbed into the first term on the right-hand side of Eq. (18). Expanding Eq. (19) in the threshold region generates the required logarithms and half-integer powers of , as can be seen from the expansions


In a similar way the factor in Eq. (20) generates logarithms in the high-energy region:


This means we can choose the coefficients with in such a way that these logarithms are also absorbed into . The remaining coefficients are fixed by requiring that has no poles for and, more specifically, .

To estimate the error of the approximation we vary the parameters independently with


In a second step we define


with and construct Padé approximants


using the constraints


and requiring the absence of terms proportional to in the high-energy expansion of . In total there are eight constraints, i.e. we obtain Padé approximants with .

If has poles inside the unit circle the corresponding reconstructed polarization function shows unphysical singularities in the complex plane. We therefore discard such approximants. Furthermore, we require for in order to remove approximants with pronounced additional peaks above the physical threshold at . A notable effect of this cut is the elimination of all Taylor approximants .

Since the approximants are only available in numerical form we show the general features in Fig. 1.

Figure 1: Padé approximation for with muon loops. We show in the top half the approximants and in the bottom half the relative error with respect to the local mean of all approximants obtained.

At the top of the figure we show the envelope of all Padé approximants that have been calculated and compare with the low- and high-energy expansions. In the bottom half we show the maximal deviation from the mean in percent. As can be seen there are only significant deviations of the order of a few percent in the range , which correspond to the uncertainty due to a limited number of terms in the high-energy expansion.

2.4.2 Modifications at three loops

At three loops much deeper expansions can be used for the construction of the Padé approximants. The 30 known coefficients in the low-energy expansion Boughezal:2006uu; Maier:2007yn together with 31 coefficients from the high-energy expansion Maier:2011jd and three coefficients in the threshold expansion lead to a significantly larger system of constraints compared to the four-loop case.

Our strategy will be to incorporate as much of the threshold information as possible into the logarithmic function , so that all constraints are imposed at . The resulting system can be solved very efficiently using well-established techniques of one-point Padé approximation111One-point approximation means that the value of the approximated function and its derivatives at a single point are known..

As in the four-loop case we first split the polarization function into two parts, using


to absorb the logarithms and the threshold singularity. Since we expect to obtain sufficiently many different Padé approximants for a reliable error estimate we refrain from introducing additional parameters at this point.

To map all available information from the low- and high-energy expansions onto , we define


The Taylor approximant is fixed by imposing


We can deduce all further approximants with and using Baker’s recursion formula baker:1970222Note that the denominator in the first equation of Eq. (13) in Ref. baker:1970 contains an obvious typo. The proper expression is .


where the numerator of the Padé approximant is normalised as in Eq. (27).

It should be noted that the known threshold constant at order is not included in the approximants. We find however that the information from the threshold region has virtually no effect on the reconstructed polarization functions in the Euclidean regime .333If not all constraints from the low- and high-energy regions are taken into account, the importance of the threshold information increases. Still, not taking into account any threshold coefficients changes the polarization function by less than one per mille in the euclidean region, even if only three terms each from the low- and the high-energy expansion are taken into account. This is contrary to the four-loop case, where the approximants are far less constrained by the low- and high-energy expansions.

3 The anomalous magnetic moment of the muon at five loops

The QED corrections to the anomalous magnetic moment can be calculated in perturbation theory and can thus be written in form of a power series in the fine structure constant


where can be further decomposed – following the conventions in Ref. Aoyama:2012wk – as


contains the universal contributions, which in case of the muon anomalous magnetic moment only contain muon loops. The diagrams contributing to and have at least one electron or tau loop, respectively. In contributions from diagrams with both electron and tau loops are collected. In this paper we are mainly interested in contributions to without any muon loops.

The contributions to the anomalous magnetic moment of the muon due to photon polarization effects can be calculated (cf. Fig. 2) by using Lautrup:1974ic


This formula can be obtained by considering the one-loop result for for the case of a heavy photon in combination with the dispersion relation for . The right-hand side of Eq. (38) should be expanded in , leading to e.g. at two-loop order

Figure 2: Prototype diagram

The classes of diagrams accessible by this method are shown in Fig. 3.

I(a) I(b) I(c) I(d) I(e) I(f) I(g) I(h) I(i) I(j)

Figure 3: Classes of diagrams accessible by the used method.
this work Ref. Baikov:2012rr Refs. Kinoshita:2005sm; Aoyama:2008hz; Aoyama:2010zp; Aoyama:2008gy
I(a) 20.142 813 20.183 2 20.142 93(23) Kinoshita:2005sm
I(b) 27.690 061 27.718 8 27.690 38(30) Kinoshita:2005sm
I(c) 04.742 149 04.817 59 04.742 12(14) Kinoshita:2005sm
I(d)+I(e) 06.241 470 06.117 77 06.243 32(101)(70) Kinoshita:2005sm
I(e) -1.211 249 -1.331 41 -1.208 41(70) Kinoshita:2005sm
I(f)+I(g)+I(h) 04.391 31 04.446 68(9)(23)(59) Kinoshita:2005sm; Aoyama:2008hz
I(i) 00.252 37 00.087 1(59) Aoyama:2010zp
I(j) -1.214 29 -1.247 26(12) Aoyama:2008gy
Table 1: Results for with pure electronic insertions. The errors listed in the second column are estimated from the spread between different Padé approximants, which is negligible for classes I(a)–I(e). Please note that the authors of Ref. Baikov:2012rr only used the asymptotic form of and did not provide any error estimate.
this work Ref. Aoyama:2012wk
I(a) 22.566 976 22.566 973 (3)
I(b) 30.667 093 30.667 091 (3)
I(c) 05.141 395 05.141 395 (1)
I(e) -0.931 839 -0.931 2 (24)
Table 2: Results for including electronic and muonic contributions.
this work Ref. Aoyama:2012wj
I(a) 0.000 471 0.000 470 94 (6)
I(b) 0.007 010 0.007 010 8 (7)
I(c) 0.023 467 0.023 468 (2)
I(d)+I(e) 0.014 094 0.014 098(5)(4)
I(e) 0.010 291 0.010 296 (4)
I(f)+I(g)+I(h) 0.037 833(20)(6)(13)
I(i) 0.017 47 (11)
I(j) 0.000 397 5 (18)
Table 3: Results for the universal contributions .

In the following, we compare the results obtained in our analysis with results from previous calculations. The numbers shown are obtained by numerically integrating over the best available approximation. In case there are several equivalent approximations the result is obtained by taking the mean of all values obtained. The errors are then calculated by taking the difference between the mean and the smallest and largest values obtained, respectively.

For classes I(a)–I(j) we present our results for the case of purely electronic vacuum polarization insertions in Tab. 1. We compare our results with the values obtained in Ref. Baikov:2012rr, which relies only on the leading asymptotics of , and results obtained by purely numerical calculations.

Since the results for classes I(a)-I(c) are obtained by numerical integrating the exact analytical expression for the one- and two-loop vacuum polarization they are exact.

In the case of classes I(d) and I(e) the used three-loop Padés are highly constrained by a large number of terms in the low- and high-energy expansion. Thus the error from the spread between different approximants is negligible. As can be seen the results are in good agreement with the results obtained in Ref. Kinoshita:2005sm and the analysis in Ref. Baikov:2012rr can clearly be improved by including sub-leading contributions.

In case of classes I(f)-I(j) the used four-loop Padés are less precise but also here we find good agreement within the quoted errors with the results obtained in Ref. Aoyama:2012wk. In all cases one finds a significant improvement when comparing to the leading logarithmic approximation used in Ref. Baikov:2012rr.

For classes I(a)-I(c) and I(e) we can obtain the full result for including muonic contributions. These results are presented in Tab. 2. In Tab. 3 we present our results for the universal corrections and compare with the results given in Ref. Aoyama:2012wj. In both cases the discussion as for the purely electronic contributions can essentially be repeated and also here overall good agreement with results available in the literature is observed. Nevertheless it should be noted that for single diagram classes a certain tension remains.

As a check of our setup we repeated the analysis of Ref. Baikov:1995ui and find good agreement even though in that reference a term in the threshold region proportional to has been omitted (cf. Eq. (12)). Our findings at four-loop order are summarized in Tab. 4.

this work Ref. Baikov:1995ui Ref. Aoyama:2012wk; Kinoshita:2004wi
I(c) 1.440741 1.440744(16) Kinoshita:2004wi
I(d) -0.230337 -0.230362(5) -0.22982(37) Aoyama:2012wk
Table 4: Comparison of four-loop results for obtained in Refs. Baikov:1995ui; Aoyama:2012wk and in this work. The classification of the diagrams corresponds to the notation used in Ref. Aoyama:2012wk.

4 Conclusions

We presented results for a certain set of five-loop contributions to the anomalous magnetic moment of the muon that stem from corrections to the vacuum polarization of the photon. We have shown that an improved treatment of the vacuum polarization, including more than its asymptotic form, leads to a significantly better agreement with results obtained by purely numerical methods. It can be clearly seen that for certain classes of diagrams the asymptotic form of the vacuum polarization function is not sufficient and power suppressed terms play an important role and have to be included in the analysis.


We like to thank K.G. Chetyrkin and J.H. Kühn for initiating the project, fruitful discussions, and reading of the manuscript. In addition, we thank K.G. Chetyrkin for valuable advice on computing the vacuum polarization function in high-energy limit. The work of P.B. was supported by RFBR grant 11-02-01196. P.M. has been supported in part by DFG Sonderforschungsbereich Transregio 9, Computergestützte Theoretische Teilchenphysik, and by the EU Network LHCPHENOnet PITN-GA-2010-264564.


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