High-precision calculation of the 4-loop contribution to the electron g-2 in QED

TL;DR

This paper presents a highly precise numerical evaluation of the 4-loop quantum electrodynamics contribution to the electron g-2, including a semi-analytical expression involving advanced mathematical functions.

Contribution

It provides the most precise calculation to date of the 4-loop contribution to electron g-2, with a detailed semi-analytical expression fitting the numerical data.

Findings

Numerical value of the 4-loop contribution: -1.912245764926445574152647167439830054060873390658725345

Semi-analytical expression involving harmonic polylogarithms and elliptic integrals

Evaluation of master integrals up to 4800 digits

Abstract

I have evaluated up to 1100 digits of precision the contribution of the 891 4-loop Feynman diagrams contributing to the electron $g$-$2$ in QED. The total mass-independent 4-loop contribution is $ a_e = -1.912245764926445574152647167439830054060873390658725345{\ldots} \left(\frac{\alpha}{\pi}\right)^4$. I have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument $e^{\frac{i\pi}{3}}$, $e^{\frac{2i\pi}{3}}$, $e^{\frac{i\pi}{2}}$, one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits.

Tables

Table 1. Table 2: Contribution to a e ( 4 ) superscript subscript 𝑎 𝑒 4 a_{e}^{(4)} of the 25 gauge-invariant sets of Fig.2.

1	- 1.971075616835818943645699655337264406980
2	- 0.142487379799872157235945291684857370994
3	- 0.621921063535072522104091223479317643540
4	+ 1.086698394475818687601961404690600972373
5	- 1.040542410012582012539438620994249955094
6	+ 0.512462047967986870479954030909194465565
7	+ 0.690448347591261501528101600354802517732
8	- 0.056336090170533315910959439910250595939
9	+ 0.409217028479188586590553833614638435425
10	+ 0.374357934811899949081953855414943578759
11	- 0.091305840068696773426479566945788826481
12	+ 0.017853686549808578110691748056565649168
13	- 0.034179376078562729210191880996726218580
14	+ 0.006504148381814640990365761897425802288
15	- 0.572471862194781916152750849945181037311
16	+ 0.151989599685819639625280516106513042070
17	+ 0.000876865858889990697913748939713726165
18	+ 0.015325282902013380844497471345160318673
19	+ 0.011130913987517388830956500920570148123
20	+ 0.049513202559526235110472234651204851710
21	- 1.138822876459974505563154431181111707424
22	+ 0.598842072031421820464649513201747727836
23	+ 0.822284485811034346719894048799598422606
24	- 0.872657392077131517978401982381415610384
25	- 0.117949868787420797062780493486346339829

Table 2. Table 3: Numerical values of the constants of Eq. 7 and Eq. 48 .

$T_0$	9.515906781243876151283558690966098373
$T_2$	1915.310648253997777888130354499120276542
$T_3$	-3485.275086789599708317057778907752410742
$T_4$	3504.090225594272699233395974800847330934
$T_5$	-725.569913602974274507866288615667084989
$T_6$	1381.628304197738147258897402093908402776
$T_7$	1692.786400388934476652564199811210670453
$V_{4a}$	-223.655742930151691157141102901111870825
$V_{6a}$	14.029138087062071859189974573196626739
$V_{6b}$	842.150210099809624937684343426149287354
$V_{7b}$	463.951882993580804359224932846794527895
$W_{6b}$	-1560.934864680405790411777238139658336036
$W_{7b}$	-1024.004093725178841133583200254534168436
$E_{4a}$	-856.605968292200108497784694038000040595
$E_{5a}$	601.136193120690233763409588135510244820
$E_{6a}$	-457.790342894702531083496436277945999328
$E_{6b}$	-89.049936952630079330356943951138211140
$E_{7a}$	548.453177743013238987339022298522918205
$E_{7b}$	-2145.946406417837479874008380333397996999
$U$	- 132.027597619729495491707871522090745221
$C_{81a}$	116.694585791186600526332510987652818034
$C_{81b}$	- 8.748320323814631572671010051472284815
$C_{81c}$	- 0.236085277120339887503638687666535683
$C_{83a}$	2.771191986145520146810618363218497216
$C_{83b}$	- 0.807847353263827557176395243854200179
$C_{83c}$	- 0.434702618543809180642530601495074086

Equations

$$a_e=-1.912245764926445574152647167439830054060873390658725345\dots {\left(\frac{\alpha }{\pi }\right)}^4.$$

$$a_e(\text{4-loop})=a_e^{(4)}{\left(\frac{\alpha }{\pi }\right)}^4,$$

$$a_e^{(4)}=-1.912245764926445574152647167439830054060873390658725345171329848\dots .$$

$$a_e^{(4)}(\text{Ref.cite[cite][@@bibrefAoyama:2014sxa]})=-1.91298(84),$$

$${\alpha }^{-1}=137.035999040(90),$$

$$a_e^{\text{th}}=1159652181.664(23)(16)(763)\times 10^{-12},$$

$$a_e^{\text{exp}}=1159652180.73(0.28)\times 10^{-12},$$

$${\alpha }^{-1}(a_e)=137.0359991596(27)(18)(331),$$

$$a_e^{(4)}(\text{no closed electron loops})$$

$$a_e^{(4)}(\text{closed electron loops only})$$

$$a_e^{(4)}=$$

$$+\sqrt{3}\left(E_{4a}+E_{5a}+E_{6a}+E_{7a}\right)+E_{6b}+E_{7b}+U.$$

$$T_0+T_2+T_3=\frac{1243127611}{130636800}+\frac{30180451}{25920}\zeta (2)-\frac{255842141}{2721600}\zeta (3)-\frac{8873}{3}\zeta (2)\ln 2,$$

$$T_4=\frac{6768227}{2160}\zeta (4)+\frac{19063}{360}\zeta (2){\ln }^{2}2+\frac{12097}{90}\left(a_4+\frac{1}{24}{\ln }^{4}2\right),$$

$$T_5=$$

$$+\frac{9656}{27}\left(a_5+\frac{1}{12}\zeta (2){\ln }^{3}2-\frac{1}{120}{\ln }^{5}2\right),$$

$$T_6=$$

$$-\frac{700706}{675}{a}_4\zeta (2)-\frac{26404}{27}{a}_5\ln 2+\frac{26404}{27}\zeta (5)\ln 2-\frac{63749}{50}\zeta (3)\zeta (2)\ln 2$$

$$-\frac{40723}{135}\zeta (4){\ln }^{2}2+\frac{13202}{81}\zeta (3){\ln }^{3}2-\frac{253201}{2700}\zeta (2){\ln }^{4}2+\frac{7657}{1620}{\ln }^{6}2,$$

$$T_7=$$

$$-\frac{142793}{18}{a}_5\zeta (2)-\frac{195848}{21}{a}_7+\frac{195848}{63}{b}_7-\frac{116506}{189}{d}_7$$

$$-\frac{4136495}{384}\zeta (6)\ln 2-\frac{1053568}{189}{a}_6\ln 2+\frac{233012}{189}{b}_6\ln 2+\frac{407771}{432}{\zeta }^2(3)\ln 2$$

$$-\frac{8937}{2}{a}_4\zeta (2)\ln 2+\frac{833683}{3024}\zeta (5){\ln }^{2}2-\frac{3995099}{6048}\zeta (3)\zeta (2){\ln }^{2}2-\frac{233012}{189}{a}_5{\ln }^{2}2$$

$$+\frac{1705273}{1512}\zeta (4){\ln }^{3}2+\frac{602303}{4536}\zeta (3){\ln }^{4}2-\frac{1650461}{11340}\zeta (2){\ln }^{5}2+\frac{52177}{15876}{\ln }^{7}2.$$

$$V_{4a}=-\frac{14101}{480}{\mathrm{Cl}}_4\left(\frac{\pi }{3}\right)-\frac{169703}{1440}\zeta (2){\mathrm{Cl}}_2\left(\frac{\pi }{3}\right),$$

$$V_{6a}=$$

$$+19\mathrm{Im}{H}_{0,0,1,0,1,1}\left(e^{i\frac{2\pi }{3}}\right)+\frac{437}{12}\mathrm{Im}{H}_{0,0,0,1,1,1}\left(e^{i\frac{2\pi }{3}}\right)+\frac{29812}{297}{\mathrm{Cl}}_6\left(\frac{\pi }{3}\right)$$

$$+\frac{4940}{81}{a}_4{\mathrm{Cl}}_2\left(\frac{\pi }{3}\right)-\frac{520847}{69984}\zeta (5)\pi -\frac{129251}{81}\zeta (4){\mathrm{Cl}}_2\left(\frac{\pi }{3}\right)$$

$$-\frac{892}{15}\mathrm{Im}{H}_{0,1,1,-1}\left(e^{i\frac{2\pi }{3}}\right)\zeta (2)-\frac{1784}{45}\mathrm{Im}{H}_{0,1,1,-1}\left(e^{i\frac{\pi }{3}}\right)\zeta (2)+\frac{1729}{54}\zeta (3)\mathrm{Im}{H}_{0,1,-1}\left(e^{i\frac{\pi }{3}}\right)$$

$$+\frac{1729}{36}\zeta (3)\mathrm{Im}{H}_{0,1,1}\left(e^{i\frac{2\pi }{3}}\right)+\frac{837190}{729}{\mathrm{Cl}}_4\left(\frac{\pi }{3}\right)\zeta (2)+\frac{25937}{4860}\zeta (3)\zeta (2)\pi$$

$$-\frac{223}{243}\zeta (4)\pi \ln 2+\frac{892}{9}\mathrm{Im}{H}_{0,1,-1}\left(e^{i\frac{\pi }{3}}\right)\zeta (2)\ln 2+\frac{446}{3}\mathrm{Im}{H}_{0,1,1}\left(e^{i\frac{2\pi }{3}}\right)\zeta (2)\ln 2$$

$$-\frac{7925}{81}{\mathrm{Cl}}_2\left(\frac{\pi }{3}\right)\zeta (2){\ln }^{2}2+\frac{1235}{486}{\mathrm{Cl}}_2\left(\frac{\pi }{3}\right){\ln }^{4}2,$$

$$V_{6b}=\frac{13487}{60}\mathrm{Re}{H}_{0,0,0,1,0,1}\left(e^{i\frac{\pi }{3}}\right)+\frac{13487}{60}{\mathrm{Cl}}_4\left(\frac{\pi }{3}\right){\mathrm{Cl}}_2\left(\frac{\pi }{3}\right)+\frac{136781}{360}\mathrm{C}{\mathrm{l}}_2^2\left(\frac{\pi }{3}\right)\zeta (2),$$

$$V_{7b}$$

$$-\frac{87885}{64}\mathrm{Re}{H}_{0,0,0,1,0,1,1}\left(e^{i\frac{2\pi }{3}}\right)-\frac{17577}{8}\mathrm{Re}{H}_{0,0,0,0,1,1,1}\left(e^{i\frac{2\pi }{3}}\right)+\frac{651}{4}{\mathrm{Cl}}_4\left(\frac{\pi }{3}\right)\mathrm{Im}{H}_{0,1,-1}\left(e^{i\frac{\pi }{3}}\right)$$

$$+\frac{1953}{8}{\mathrm{Cl}}_4\left(\frac{\pi }{3}\right)\mathrm{Im}{H}_{0,1,1}\left(e^{i\frac{2\pi }{3}}\right)+\frac{31465}{176}{\mathrm{Cl}}_6\left(\frac{\pi }{3}\right)\pi +\frac{211}{4}\mathrm{Re}{H}_{0,1,0,1,-1}\left(e^{i\frac{\pi }{3}}\right)\zeta (2)$$

$$+\frac{211}{2}\mathrm{Re}{H}_{0,0,1,1,-1}\left(e^{i\frac{\pi }{3}}\right)\zeta (2)+\frac{1899}{16}\mathrm{Re}{H}_{0,1,0,1,1}\left(e^{i\frac{2\pi }{3}}\right)\zeta (2)+\frac{1899}{8}\mathrm{Re}{H}_{0,0,1,1,1}\left(e^{i\frac{2\pi }{3}}\right)\zeta (2)$$

$$+\frac{211}{4}\mathrm{Im}{H}_{0,1,-1}\left(e^{i\frac{\pi }{3}}\right){\mathrm{Cl}}_2\left(\frac{\pi }{3}\right)\zeta (2)+\frac{633}{8}\mathrm{Im}{H}_{0,1,1}\left(e^{i\frac{2\pi }{3}}\right){\mathrm{Cl}}_2\left(\frac{\pi }{3}\right)\zeta (2).$$

$$W_{6b}=$$

$$W_{7b}=$$

Full text

[TABLE]

I have fit a semi-analytical expression to the numerical value. The expression contains harmonic polylogarithms of argument $e^{\frac{i\pi}{3}}$, $e^{\frac{2i\pi}{3}}$, $e^{\frac{i\pi}{2}}$, one-dimensional integrals of products of complete elliptic integrals and six finite parts of master integrals, evaluated up to 4800 digits.

17 March 2024

PACS: 12.20Ds; 13.40Em; 06.20Jr; 12.20Fv;

Keywords: Quantum electrodynamics; Anomalous magnetic moment; Feynman diagram; Master integral; High-precision calculation; Analytical fit;

I have evaluated up to 1100 digits of precision the mass-independent contribution to the electron $g$-$2$ anomaly of all the 891 diagrams in 4-loop QED, thus finalizing a twenty-year effort [1, 2, 3, 4, 5, 6, 7] begun after the completion of the calculation of 3-loop QED contribution [8].

Having extracted the power of the fine structure constant $\alpha$

[TABLE]

the first digits of the result are

[TABLE]

The full-precision result is shown in table 1. The result (2) is in excellent agreement ($0.9\sigma$) with the numerical value

[TABLE]

latest result of a really impressive pluridecennial effort[9, 10, 11, 12, 13, 14, 15, 16, 17, 18].

By using the best numerical value of $a_{e}(\textrm{5-loop})=7.795(336){\left(\frac{\alpha}{\pi}\right)}^{5}$(Ref.[18]), the measurement of the fine structure constant[19]

[TABLE]

and the values of mass-dependent QED, hadronic and electroweak contributions (see Ref.[18] and references therein), one finds

[TABLE]

where the first error comes from $a_{e}^{(5)}$, the second one from the hadronic and electroweak corrections, the last one from $\alpha$. Conversely, using the experimental measurement of $a_{e}$[20]

[TABLE]

one finds

[TABLE]

where the errors come respectively from $a_{e}(\textrm{5-loop})$, hadronic and electroweak corrections, and $a_{e}$.

The 891 vertex diagrams contributing to $a_{e}^{(4)}$ are not shown for reasons of space. They can be obtained by inserting an external photon in each possible electron line of the 104 4-loop self-mass diagrams shown in Fig.1, excluding the vertex diagrams with closed electron loops with an odd number of vertices which give null contribution because of the Furry’s theorem. The vertex diagrams can be arranged in 25 gauge-invariant sets (Fig.2), classifying them according to the number of photon corrections on the same side of the main electron line and the insertions of electron loops (see Ref.[21] for more details on the 3-loop classification). The numerical contributions of each set, truncated to 40 digits, are listed in the table 2. Adding respectively the contributions of diagrams with and without closed electron loops one finds

[TABLE]

The contributions of the sets 17 and 18, the sum of contributions of the sets 11 and 12, and the sum of the contributions of the sets 15 and 16 are in perfect agreement with the analytical results of Ref.[22].

The contributions of all diagrams can be expressed by means of 334 master integrals belonging to 220 topologies. I have fit analytical expressions to the high-precision numerical values of all master integrals and diagram contributions by using the PSLQ algorithm[23, 24]. The analytical expression of $a_{e}^{(4)}$ contains values of harmonic polylogarithms[25] with argument $1$, $\frac{1}{2}$, $e^{\frac{i\pi}{3}}$ , $e^{\frac{2i\pi}{3}}$, $e^{\frac{i\pi}{2}}$, a family of one-dimensional integrals of products of elliptic integrals, and the finite terms of the $\epsilon-$expansions of six master integrals belonging to the topologies 81 and 83 of Fig.1. Work is still in progress to fit analytically these six unknown elliptical constants. The result of the analytical fit can be written as follows:

[TABLE]

The terms have been arranged in blocks with equal transcendental weight. The index number is the weight. The terms containing the “usual” transcendental constants are:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The terms containing harmonic polylogarithms of $e^{\frac{i\pi}{3}}$, $e^{\frac{2i\pi}{3}}$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The terms containing harmonic polylogarithms of $e^{\frac{i\pi}{2}}$:

[TABLE]

The terms containing elliptic constants:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The term containing the $\epsilon^{0}$ coefficients of the $\epsilon-$expansion of six master integrals (see $f,f^{\prime},f^{\prime\prime},g,g^{\prime},g^{\prime\prime}$ of Fig.3):

[TABLE]

The numerical values of Eqs.(9)-(48) are listed in Table 3. In the above expressions $\zeta(n)=\sum_{i=1}^{\infty}i^{-n}$, $a_{n}=\sum_{i=1}^{\infty}2^{-i}\;i^{-n}$, $b_{6}=H_{0,0,0,0,1,1}\left(\frac{1}{2}\right)$, $b_{7}=H_{0,0,0,0,0,1,1}\left(\frac{1}{2}\right)$, $d_{7}=H_{0,0,0,0,1,-1,-1}(1)$, ${\mathrm{Cl}_{n}}\left({\theta}\right)=\mathrm{Im}{\mathrm{Li}}_{n}(e^{i\theta})$. $H_{i_{1},i_{2},{\ldots}}(x)$ are the harmonic polylogarithms. The integrals $f_{j}$ are defined as follows:

[TABLE]

$K(x)$ is the complete elliptic integral of the first kind. Note that $D_{1}(s)=2J_{2}^{(1,9)}(s)$, with $J_{2}^{(1,9)}$ defined in Eq.(A.12) of Ref.[26]. The integrals $f_{1}(0,0,0)$ and $f_{2}(0,0,0)$ were studied in Ref.[6]. The constants $A_{3}$, $B_{3}$ and $C_{3}$, defined in Ref.[6], admit the hypergeometric representations:

[TABLE]

$A_{3}$ appears only in the coefficients of the $\epsilon$-expansion of master integrals, and cancels out in the diagram contributions. Fig.3 shows the fundamental elliptic master integrals which contains irreducible combinations of $B_{3}$, $C_{3}$ and $f_{m}(i,j,k)$.

The analytical fits of $V_{6b}$, $V_{6a}$, $V_{7b}$, $V_{7i}$ and the master integrals involved needed PSLQ runs with basis of $\sim 500$ elements calculated with $9600$ digits of precision. The multi-pair parallel version[24] of the PSLQ algorithm has been essential to work out these difficult analytical fits in reasonable times.

The method used for the computation of the master integrals with precisions up to 9600 digits is essentially based on the difference equation method[1, 2] and the differential equation method[27, 28, 29]. This method and the procedures used for the extraction of $g$-$2$ contribution, renormalization, reduction to master integrals, generation and numerical solution of systems of difference and differential equations, (all based on upgrades of the program SYS of Ref.[1]) will be thoroughly described elsewhere.

Acknowledgments

The author wants to thank Antonino Zichichi and Luca Trentadue for having provided support to this work, Ettore Remiddi for continuous support and encouragement.

The main part of the calculations was performed on the cluster ZBOX2 of the Institute for Theoretical Physics of Zurich and on the Schrödinger supercomputer of the University of Zurich. The author is deeply indebted to Thomas Gehrmann for having allowed him to use these facilities.

Some parts of the calculations were done on computers of the Department of Physics and INFN in Bologna. The author thanks Michele Caffo, Franco Martelli, Sandro Turrini and Vincenzo Vagnoni for providing suitable desktop computers in Bologna.