格点格林函数围道积分求贝塞尔矩的椭圆积分
Elliptic integral evaluation of a Bessel moment by contour integration of a lattice Green function
摘要：
证明了由2圈4点质量费曼图的角平均值得到的贝塞尔矩$$M:=\int_0^\infty t I_0^2(t)K_0^2(t)K_0(2t) {\rm d}t ={1/12} {\bf K}(\sin(π/12)){\bf K}(\cos(π/12)) =\frac{Γ^6(\frac13)}{64π^22^{2/3}}$$的椭圆积分计算，其中一个内部质量加倍。该评估遵循六方晶格的格林函数的轮廓积分，从而将$M$与两个更易处理的矩的线性组合相关联，一个矩由金刚石晶格的格林函数给出，并且两者都通过使用W. N. Bailey将Appell二重级数约化为椭圆积分的乘积。从等质量Dalitz图的椭圆积分的三次和倍半模变换被证明和广泛使用。推导给出的总和规则$$\int_0^\infty(I_0(a t)K_0(a t)-\frac{2}π K_0(4a t) K_0(t))K_0(t) {\rm d}t=0$$与$a>0$，证明了解析延续的身份从贝利的工作，和$$\int_0^\infty t I_0(a t)(I_0^3(a t)K_0(8t)- \frac{1}{4π^2} I_0(t)K_0^3(t)) {\rm d}t=0$$与$2\ge a\ge0$，证明了费曼图在两个时空维度产生的枚举的楼梯多边形在四维。
英文摘要：
A proof is found for the elliptic integral evaluation of the Bessel moment $$M:=\int_0^\infty t I_0^2(t)K_0^2(t)K_0(2t) {\rm d}t ={1/12} {\bf K}(\sin(π/12)){\bf K}(\cos(π/12)) =\frac{Γ^6(\frac13)}{64π^22^{2/3}}$$ resulting from an angular average of a 2-loop 4-point massive Feynman diagram, with one internal mass doubled. This evaluation follows from contour integration of the Green function for a hexagonal lattice, thereby relating $M$ to a linear combination of two more tractable moments, one given by the Green function for a diamond lattice and both evaluated by using W.N. Bailey's reduction of an Appell double series to a product of elliptic integrals. Cubic and sesquiplicate modular transformations of an elliptic integral from the equal-mass Dalitz plot are proven and used extensively. Derivations are given of the sum rules $$\int_0^\infty(I_0(a t)K_0(a t)-\frac{2}π K_0(4a t) K_0(t))K_0(t) {\rm d}t=0$$ with $a>0$, proven by analytic continuation of an identity from Bailey's work, and $$\int_0^\infty t I_0(a t)(I_0^3(a t)K_0(8t)- \frac{1}{4π^2} I_0(t)K_0^3(t)) {\rm d}t=0$$ with $2\ge a\ge0$, proven by showing that a Feynman diagram in two spacetime dimensions generates the enumeration of staircase polygons in four dimensions.
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