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تسجيل مستخدم جديدMultiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
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تأليف
M. Yu. Kalmykov
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We continue the study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results: Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth (see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions. Theorem B: The epsilon expansion of a hypergeometric function with one half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are ratios of polynomials. Some extra materials are available via the www at this http://theor.jinr.ru/~kalmykov/hypergeom/hyper.html
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We prove the following theorems: 1) The Laurent expansions in epsilon of the Gauss hypergeometric functions 2F1(I_1+a*epsilon, I_2+b*epsilon; I_3+p/q + c epsilon; z), 2F1(I_1+p/q+a*epsilon, I_2+p/q+b*epsilon; I_3+ p/q+c*epsilon;z), 2F1(I_1+p/
q+a*epsilon, I_2+b*epsilon; I_3+p/q+c*epsilon;z), where I_1,I_2,I_3,p,q are arbitrary integers, a,b,c are arbitrary numbers and epsilon is an infinitesimal parameter, are expressible in terms of multiple polylogarithms of q-roots of unity with coefficients that are ratios of polynomials; 2) The Laurent expansion of the Gauss hypergeometric function 2F1(I_1+p/q+a*epsilon, I_2+b*epsilon; I_3+c*epsilon;z) is expressible in terms of multiple polylogarithms of q-roots of unity times powers of logarithm with coefficients that are ratios of polynomials; 3) The multiple inverse rational sums (see Eq. (2)) and the multiple rational sums (see Eq. (3)) are expressible in terms of multiple polylogarithms; 4) The generalized hypergeometric functions (see Eq. (4)) are expressible in terms of multiple polylogarithms with coefficients that are ratios of polynomials.
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We discuss properties of the $epsilon$-expansion in $d=4-epsilon$ dimensions. Using Lagrange inversion we write down an exact expression for the value of the Wilson-Fisher fixed point coupling order by order in $epsilon$ in terms of the beta function
coefficients. The $epsilon$-expansion is combinatoric in the sense that the Wilson-Fisher fixed point coupling at each order depends on the beta function coefficients via Bell polynomials. Using certain properties of Lagrange inversion we then argue that the $epsilon$-expansion of the Wilson-Fisher fixed point coupling equally well can be viewed as a geometric expansion which is controlled by the facial structure of associahedra. We then write down an exact expression for the value of anomalous dimensions at the Wilson-Fisher fixed point order by order in $epsilon$ in terms of the coefficients of the beta function and anomalous dimensions. We finally use our general results to compute the values for the Wilson-fisher fixed point coupling and critical exponents for the scalar $O(1)$ symmetric model to $O(epsilon^7)$.
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