高阶QED对电子磁矩贡献的数值计算：方法、实现与比较

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本文主要探讨了高阶量子电动力学(QED)对电子异常磁矩的数值计算方法，这是物理学中一个关键且复杂的课题，尤其是在精确测量和理论预测之间的微扰量子场论研究中。作者Sergey Volkov及其团队针对高阶QED中的电子磁矩问题，提出了一个创新的数值评估策略，其核心是利用积分前的费恩曼参数空间中的红外(IR)和紫外(UV)发散进行减法处理，这一过程依赖于Hepp扇区划分的非自适应蒙特卡洛积分技术。他们选择使用图形加速器NVidia Tesla K80来实现这一计算方法，这表明他们充分利用了GPU的并行计算能力，显著提高了计算效率。为了克服数值减法过程中可能出现的舍入误差，他们提出了一种创新的方法，能够在保持计算速度的同时减少这种误差的影响。这种方法的应用范围广泛，包括了2环、3环和4环的QED费曼图，其中不含轻子环，这些都是量子电动力学中的基本图结构。文章详尽地展示了这些高阶图的计算结果，与已知的分析结果进行了深入对比，验证了新方法的精度。特别是，他们展示了六个规范不变的四环图类的贡献，以及78套4环图的具体数值，这些都是对理论预测的重要补充。此外，文章还涉及了5环和6环梯形图的计算，进一步增强了对高阶效应的理解。作者不仅关注计算结果本身，还深入剖析了生成的蒙特卡洛样本的行为，这涉及到统计学的稳健性和误差估计。他们提供了关于图形处理器在这些复杂计算中的性能数据，以及蒙特卡洛积分的收敛性分析，这对于理解计算的稳定性和效率至关重要。这篇文章不仅介绍了高阶QED计算方法的实现细节，还展示了其在实际应用中的性能，以及如何处理相关技术挑战。这对于物理学家们理解和改进高精度的电子磁矩理论预测有着重要的意义，同时对于数值计算方法的发展也具有推动作用。

order to make this subtraction equivalent to the on-shell

renormalization,

it cannot be replaced by L

III. PROBABILITY DENSITY FUNCTIONS

FOR MONTE CARLO INTEGRATION

We use Feynman parameters for calculations. Thus, to

obtain the contribution of a graph G, we need to calculate

the integral

;…;z

Iðz

; …;z

Þδðz

þþz

− 1Þdz

…dz

;

where the function I is constructed by using the known

rules [30].

We use the Monte Carlo approach based on importance

sampling: we generate randomly N samples

; …; z

where

¼ðz

j;1

; …;z

j;n

Þ, using some probability density

function gð

zÞ and approximating the integral value by

j¼1

IðzÞ

gð

zÞ

: ð12Þ

The density g is fixed for a fixed graph G. The speed of

Monte Carlo convergence depends on the selection of g.A

construction of G that gives a good convergence is

described below.

We will use Hepp sectors [37] and functions of the form

that was first used by E. Speer [43] with some modifica-

tions. All the space R

is split

into sectors. Each sector

corresponds to a permutation ðj

; …;j

Þ of f1; 2; …;ng

and is defined by

;…;j

¼fðz

; …;z

Þ ∈ R∶z

≥ z

≥ … ≥ z

We define the function g

ðz

; …;z

Þ on S

;…;j

by the

following relation:

ðz

; …;z

Þ¼

l¼2

ðz

l−1

Degðfj

lþ1

;…;j

gÞ

…z

; ð13Þ

where DegðsÞ > 0 is defined for each set s of internal

lines

of G except the empty set and the set of all internal

lines of G. The probability density function is defined by

gðz

;…;z

ðz

;…;z

ðz

;…;z

Þδðz

þþz

− 1Þdz

…dz

ð14Þ

A fast random samples generation algorithm for a given

DegðsÞ is described in Ref. [29].

Let us describe the procedure of obtaining DegðsÞ. The

following auxiliary definitions repeat the ones from

Ref. [29]. By definition, we set

ωðsÞ¼2N

ðsÞþjeðsÞj=2 − jsj;

where jxj is the cardinality of a set x, eðsÞ is the set of all

electron lines in s, and N

ðsÞ is the number of independent

loops in s.Ifs is the set of all internal lines of a subgraph of

G, then ωðsÞ coincides with the ultraviolet degree of

divergence of this subgraph that is defined above.

The problem of constructing a good gð

zÞ is very close to

the problem of obtaining a simple and close enough upper

bound for jIð

zÞj and proving the integral finiteness; see

Ref. [29]. Feynman parametric expressions for the inte-

grands (without subtraction terms) can be represented as

fractions with denominators that vanish on the boundary of

the integration area, if we are on the mass shell [30].If

we consider the numerators only, we can use the ultraviolet

degrees of divergence themselves; see Ref. [43].Ifwe

take into account the denominators too, the degrees

must be increased, which is performed by I-closures that

are defined below. In addition to vanishing denominators,

the divergence subtraction complicates the problem. The

construction described below is based on both theoretical

considerations

and numerical experiments.

By IClosðsÞ we denote the set s ∪ s

, where s

is the set

of all internal photon lines l in G such that s contains the

electron path in G connecting the ends of l. The set IClosðsÞ

is called the I-closure of the set s.

By definition, we set

ðsÞ¼ωðIClosðsÞÞ:

A graph G

belonging to a forest F ∈ F½G is called a

child of a graph G

∈ F in F if G

⊊ G

and there is no

000

∈ F such that G

000

⊊ G

, G

⊊ G

000

If F ∈ F½G and G

∈ F, then by G

=F we denote the

graph that is obtained from G

by shrinking all children of

in F to points.

We will also use the symbols ω, ω

for graphs G

that are

constructed from G by some operations like those described

above,

and for sets s that are subsets of the set of internal

lines of the whole graph G. We will denote these by ω

ðsÞ

and ω

ðsÞ, respectively. This means that we apply the

operations ω and ω

in the graph G

to the set s

that is the

intersection of s and the set of all internal lines of G

Electron self-energy subgraphs and lines joining them

form chains l

…l

rþ1

, where l

’s are electron

See Sec. IV H and Ref. [30], Appendix B.

Let us remark that the components have intersections on their

boundaries. However, this is inessential for integration.

Note that the sets can be not connected.

Some of the ideas underlying the concept of I-closure and

this procedure of obtaining DegðsÞ will be described in future

papers. (These ideas are quite complicated and are not completely

substantiated mathematically at this moment.)

See the corresponding examples in Ref. [29] .

NUMERICAL CALCULATION OF HIGH-ORDER QED … PHYS. REV. D 98, 076018 (2018)

076018-5

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