The 4-loop slope of the Dirac form factor

The 4-loop contribution to the slope of the Dirac form factor in QED has been evaluated with 1100 digits of precision. The value is mF (4)′ 1 (0) = 0.886545673946443145836821730610315359390424032660064745. . . ( α π 4 . We have also obtained a semi-analytical fit to the numerical value. The expression contains harmonic polylogarithms of argument e iπ 3 , e 2iπ 3 , e iπ 2 , one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits. We show the correction to the energy levels of the hydrogen atom due to the slope. 1 The slope of the Dirac form factor In QED the vertex can be written ū(p1) ( γμF1(t) + σμν 2m qνF2(t) ) u(p2) , (1) where m is the electron mass, F1(t) and F2(t) are the Dirac and Pauli form factors. At t = 0, the charge conservation implies that F1(0) = 1 , (2) whereas the value of the Pauli form factor is the g-2 F2(0) = g − 2 2 . (3) The quantity d dt F1(t) ∣


The slope of the Dirac form factor
In QED the vertex can be written where m is the electron mass, F 1 (t) and F 2 (t) are the Dirac and Pauli form factors. At t = 0, the charge conservation implies that whereas the value of the Pauli form factor is the g-2 The quantity d dt F 1 (t) t=0 = F ′ 1 (0) is the slope of the Dirac form factor.

Theoretical expression
We expand perturbatively the slope in powers of α π m 2 F ′ 1 (0) = A 1 α π + A 2 α π 2 + A 3 α π 3 + A 4 α π 4 +. . . . (4) The coefficients known in analytical form are [2][3][4] In this paper we present the result of the calculation of A 4 with a precision of 1100 digits. The first digits of the result are   In table 2 we have listed the known values of the slope and g-2; we see that A 2 , A 3 and A 4 are all positive, in contrast with the alternating signs observed in the g-2.

Shift to the hydrogen levels
Let us now consider the shift to the hydrogen energy levels due to A 4 . We express the energy shift in terms of the frequency shift ∆ f = ∆E/h. For the level nS the frequency shift is [ and is comparable with the experimental error of the extremely precise measurement of 1S − 2S transition [7] f (1S − 2S) = 2466 061 413 187 018 ± 11 Hz . (11) Eq.(10) is the first calculated four-loop correction to the energy levels, of the kind α π 4 (Zα) 4 .

Gauge-invariant sets
There are 891 vertex diagrams contributing to A 4 . These vertex diagrams can be arranged in 25 gauge-invariant sets (Fig.1). The sets are classified according to the number of photon corrections on the same side of the main electron line and the insertions of electron loops (see Ref. [8] Table 4. Separate contributions to the slope from diagrams without and with electron loops.

The analytical fit
By building systems of integration-by-parts identities [9,10] and solving them [11], the contributions to A 4 of all the diagrams are expressed as linear combinations of 334 master integrals, the same ones as appeared in the calculation of 4-loop g-2 [1]. In Ref. [1] these master integrals were calculated numerically with precision ranging from 1100 to 9600 digits; analytical expressions were fit to all these master integrals (single or in particular combinations) by using the PSLQ algorithm [12,13]. We use those results here. Therefore, the analytical expression of A 4 contains the same transcendentals appeared in the g-2 result: values of harmonic polylogarithms [14] with argument 1, 1 2 , e iπ 3 , e 2iπ 3 , e iπ 2 [15,16] , a family of one-dimensional integrals of products of elliptic integrals, and the finite terms of the ǫ−expansions of six master integrals belonging to topologies 24 and 25 of Fig.1. The expression of the analytical fit is written as follows:    Table 5. Numerical values of the addends appearing in Eq.12.
In the above expressions K(x) is the complete elliptic integral of the first kind. The constants B 3 and C 3 have the following hypergeometric representations [1,17]: 4F3 a 1 a 2 a 3 a 4 The numerical values of the constants appearing in Eq. (12) are listed in Table 5. Note the strong numerical cancellations in Eq. (12): the largest term is − 2749470791 387072 ζ(2)ζ(5) = −12115.862 .

Method of calculation
We sketch the method used to obtain A 4 . It is the same used in Ref. [1].
1. Generation of 891 vertex diagrams (C program) from 104 self-mass diagrams. These are the same of the 4-loop g-2 calculation.
2. Extraction of the contribution to A 4 from the amplitude of each diagram by using projectors [18,19] with a FORM program [20,21].
3. Algebraic reduction to master integrals, obtained by building and solving large systems of integrationby-parts identities [9,10] by using the program SYS [11].
4. For the sake of checks we generate a different system for each group of vertex diagrams obtained from the same self-mass diagram.
5. The smallest system contains 10 8 identities, with size of 90GB. The system with the largest number of identities contains 5 × 10 8 , with a size of 170GB.
The largest system has 3 × 10 8 identities with a size of 1.2TB.
6. The ratio between number of independent identities and total number of generated identities is in the range 0.2 − 0.3. The dependent identities become trivial zeroes when substituted into the system, and have been used to check the reliability of hardware and software. No hardware errors were detected. Instead, software errors have been detected in this way (frequency: one every 2-3 weeks), caused by a bug in the OpenMPI message passing library used with the highest level of threads support.
7. We algebraically check that the contribution from a diagram is invariant to the changes in the particular internal routing of the momentum of the external photon.
8. The renormalization is carried out by subtracting suitable counterterms, which are generated with C and FORM programs and calculated numerically with SYS.