QED contributions to electron g-2

In this paper I briefly describe the results of the numerical evaluation of the mass-independent 4-loop contribution to the electron g-2 in QED with 1100 digits of precision. In particular I also show the semi-analytical fit to the numerical value, which contains harmonic polylogarithms of e iπ 3 , e 2iπ 3 and e iπ 2 , onedimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits. I give also some information about the methods and the program used.

Abstract. In this paper I briefly describe the results of the numerical evaluation of the mass-independent 4-loop contribution to the electron g-2 in QED with 1100 digits of precision. In particular I also show the semi-analytical fit to the numerical value, which contains harmonic polylogarithms of e iπ 3 , e 2iπ 3 and e iπ 2 , onedimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits. I give also some information about the methods and the program used.

Electron g-2
An electron of mass m and spin s has a magnetic moment µ µ = g e 2mc s (1) where g is the gyromagnetic ratio. According to the Dirac's theory [1], the electron has g = 2. The interaction with photons shifts slightly g; we define the anomaly a e = (g − 2)/2 .

Experimental measurements
The current measurements of a e are based on the Penning trap method, developed at the University of Washington. The anomaly is expressed as the ratio of two frequencies which can measured to a very high precision. For the development of this technique, the Noble prize in Physics 1989 was awarded to H. Dehmelt. Their final results were [2]: The most recent value obtained with the same technique by the Harvard group is [3,4]: a e-mail: stefano.laporta@pd.infn.it b Talk given at "Flavour changing and conserving processes" 2017 (FCCP2017), 7-9 September 2017, Villa Orlandi, Anacapri, Capri, Italy. The author thanks the organizers for financially supporting his participation in the workshop.
There are 12672 diagrams which contribute at 5-loop level. They have been computed through a gigantic effort [25]; the most recent value is  We also note the MonteCarlo approach of Ref. [36] which might in future provide an independent value. Concerning the mass-dependent part A 2 (r), A (4) 2 (r) is known in analytical form [37], as well as A (6) 2 (r) [38][39][40][41][42]; the first terms of the expansion for small r of the 4-loop coefficient A (8) 2 (r) are known analytically [43,44] . A (10) 2 (m e /m µ ) and A (10) 2 (m e /m τ ) have been calculated numerically [25]; the first terms of the expansion for small mass ratios of A (6) 3 (m e /m µ , m e /m τ ) and A (8) 3 (m e /m µ , m e /m τ ) are known analytically [44]. The values of the hadronic and weak contributions [45][46][47][48] are shown in table 2, where the most precise determination of the fine structure constant [49,50] α where the first error comes from A (10) 1 , the second one from the hadronic and electroweak corrections, the last one from α. Conversely, assuming the validity of the theory and using the experimental measurement (4), one finds where the errors come respectively from A (10) 1 , hadronic and electroweak corrections, and the experimental measurement. This determination of α is more precise than the experimental one and is that used in the periodic leastsquares adjustements of fundamental constants [48].

Gauge-invariant sets
The contributions to g-2 of single Feynman diagrams may be I.R. or U.V. divergent, and depends on the gauge chosen. Therefore, it is convenient to regroup the diagrams in gauge invariant sets, whose contributions are finite. The number of diagrams at 2, 3 and 4 loops are respectively 7, 72 and 891; they can be arranged in, respectively 3, 9 (see Ref. [51]) and 25 gauge invariant sets. Typical elements of the sets at four loops are shown in Fig.1; the values of the contributions of each gauge set are shown in Ref. [

The numerical fit
By using the integration-by-parts identities [9,52,53] the contributions of all 4-loop vertex diagrams can be reduced to a linear combination of 334 master integrals belonging to 220 topologies. Due to the expected analytical complexity of the result, a completely analytical calculation of all the master integrals (for example similar to that at threeloop) has seemed out of reach. So I have used a different approach: compute very high-precision numerical values of a sufficient number of terms of the expansions of the master integrals in Laurent series of = (4 − D)/2 dimensions; make the right analytical ansatz on the analytical form; fit the rational coefficients of the ansatz by using the PSLQ algorithm [54,55].
As a result, one finds that the analytical expressions of the coefficients of the −expansions of the master integrals contain values of harmonic polylogarithms [57] with argument 1, 1 2 , e iπ 3 , e 2iπ 3 and e iπ 2 , a family of one-dimensional integrals of products of elliptic integrals, and the finite terms of the −expansions of six master integrals. Work is still in progress to fit analytically these six unknown elliptical constants. The result of the analytical fit for A (8) 1 can be written as follows: K(x) is the complete elliptic integral of the first kind. The constants B 3 and C 3 , defined in Ref. [14], admit the hypergeometric representations:    More in detail, the -expansions of the master integrals were first calculated with a precision of 200-300 digits, then recalculated with a precision of 4096 bit (1232 digits). A first run of PSLQ on the coefficients with a trial basis identified which coefficients are linearly independent (that is, which contain new analytical terms). For each of these "unknown constants", the simplest master integral which contains it was identified; these special master integral were recalculated with a precision of 8,16 or 32 kbits (2464, 4932 or 9864 digits). The structure the master integrals was analysed to guess the analytical form of the new elements; several runs of PSLQ with basis containing (all or selections of) these new elements were performed until the PSLQ fit was successful and the correct basis was found.
To give an example, the PSLQ basis needed to fit T of Eq.19 contains F 10 − 1 = 54 terms (F i are the Fibonacci numbers, F 1 = F 2 = 1). Due to the presence of harmonic polylogarithms of complex argument, the general basis needed to perform the fit of V a and V b contains F 17 −1 = 1596 terms. The practical maximum size of basis usable in PSLQ fits turned out to be ∼ 500, so some guessing has been necessary to identify which elements are really needed in the fits (for example, only 93 elements for V a and V b ). The precision of the calculation of elements of the basis is up to 9864 digits. We note that the multipair parallel version [55] of the PSLQ algorithm has been essential to work out these difficult analytical fits in reasonable times.

Method of calculation
All the 104 4-loop self-mass diagrams are generated with a C program; for each self-mass diagram, the corresponding vertex diagrams are generated by inserting a photon in all possible ways. For the sake of subsequent checks, one keeps also the diagrams which give no contribution because of Furry's theorem. The contribution to g-2 of the diagram is extracted from the unrenormalized amplitude of each vertex diagram using suitable projectors, with a FORM [56] program. The results are expressions which typically contain 100-30000 different Feynman integrals. The total number of topologically distinct master integrals is 334. There are 82 topologies which have more than one master integral. For these topologies, products of scalar products are chosen as numerators, the choice depending on the momentum flow of the diagram. For the sake of subsequent checks, one processes separately the contributions from each one of the 104 self-mass diagrams. The master integrals are calculated separately for each selfmass diagram; therefore the total number of the master integrals actually calculated is 4607. Because of differences in momentum routing and therefore in the numerators, topologies with more than one master integral will have a slightly different set of master integrals in different self-mass diagrams. This provides a lot of useful checks. The reduction is exact in D, and is performed by generating large systems of integration-by-parts identities and solving them with the algorithm [9] implemented in SYS. These large systems contain 4·10 6 −50·10 6 identities, with physical disk space ranging from 4GB to 100GB. The reduction to master integrals has been repeated with 32 and 64 bit versions of SYS. In addition, the principal self-mass diagrams were reprocessed using a different momentum flow, checking that reduction to master integrals remained the same (after converting different sets of master integrals).
For the numerical calculation of master integrals I have used combinations of the the difference equation method [9,10] and the differential equation method [59][60][61]. An approach consists in calculating with difference equations (inserting the exponent n of the first electron propagator) the integral obtained putting the photon mass λ equal to the electron mass m, and integrating a differential equation in λ from λ = m to λ = 0. A second approach consists in calculating with difference equations the diagram obtained by putting the the external momentum p = 0, and integrating a differential equation in p 2 from p 2 = 0 to p 2 = −m 2 . The systems of difference and differential equations for master integrals are obtained by building systems of integration-by-parts identities and solving them by using the algorithm [9,10] using rational arithmetic in D. The sizes of the systems of difference or differential equations to be numerically solved are in the range 1MB-3GB. Difference equations are solved using the Laplace transformation method (integral representation of solutions and differential equation of the integrand). Differential equations are solved numerically, by using series expansion with truncated expansions in = (4 − D)/2 as coefficients. The minimum number of terms of the expansion in is 9 ( −4 . . . 4 ). There are cancellations of terms in intermediate steps; no care is used to avoid cancellations, as the corresponding numerical zeroes provide extremely useful checks. In the worst case 37 terms of expansions are needed. The standard precision of calculations is 4 kbit (1232 digits). About 130 digits are lost due to cancellations, so that the result has 1100 digits exact. Note that the analytical fit of some selected master integrals required a precision much higher, up to 16 kbit (9864 digits).
The renormalization counterterms are generated with two procedures written in C and FORM. Their expressions are reduced to master integrals already known in analytical form. I have chosen the Feynman gauge. I checked explicitly the (internal) gauge invariance of the contributions for arbitrary gauge of the photon line going into 1-,2-or 3-loop vacuum polarization diagrams.
In order to perform this calculation, in 1995 I begun writing from scratch a comprehensive C program, SYS [9], now 24000 lines long, containing all the parts necessary to the calculations, like a simplified fast algebraic symbolic manipulator, a numerical solver of systems of difference and differential equations, and a library of arbitrary precision mathematical routines, for integer and floating point numbers, independent of any other existing in the literature.. Another important feature is the thorough protection of large buffers and I/O with checksums. Throughout the calculation, I have discovered several subtle corruptions, due to different causes, like marginal coupling of non-ECC RAM modules (error-rate: 1 bit per week), failing RAID systems (error-rate: one corrupted 64KB block every 100GB of data ), etc. . . ). The prompt identification and elimination of the source of errors has resulted in an highly reliable result.