Hypergeometric functions

Hypergeometric functions and Feynman Diagrams: reduction and epsilon-expansion
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In many cases, the results of analytical calculation of Feynman diagrams can be represented as combinations of hypergeometric functions. For practical applications, finding a hypergeometric representation is not enough. It is necessary to construct the so-called epsilon-expansion, which we may understand as the construction of the analytical coefficients of the Laurent expansion of hypergeometric functions around rational values of their parameters. In this direction, very limited results are available.

One of the classical tasks in mathematics is to find the full set of parameters and arguments for which hypergeometric functions are expressible in terms of algebraic functions. Quantum field theory makes a quantum generalisation of this classical task: to find the full set of parameters and arguments so that the all-order epsilon-expansion is expressible in terms of known functions or identify the full set of functions which must be invented in order to construct the all-order epsilon-expansion of generalized hypergeometric functions. The complete solution of this problem is still open.

There are three different ways to describe special functions: by a series whose coeffcients satisfy certain recurrence relations; as a solution of a system of differential and difference equations (holonomic approach); and as integrals of Euler or Mellin-Barnes type. For functions of a single variable all these representations are equivalent, but some properties of the function may be expressed more readily in one representation than another. These three different representations of special functions have led physicists to three different approaches to developing the epsilon expansion of hypergeometric functions.

In the Euler integral representation, the most important results are related to the construction of the all-order epsilon-expansion of Gauss hypergeometric functions with special values of parameters in terms of Nielsen polylogarithms [D00,DK00,DK01] . There are several important master-integrals expressible in terms of Gauss (2F1) hypergeometric functions. This set of integrals includes one-loop propagator type diagram with arbitrary values of mass and momentum; two-loop bubble integral with an arbitrary values of masses, and one-loop massless vertex diagram with three non-zero external momenta [BD91,DT93,DT96] .

The series representation is an intensively studied approach. Particularly impressive results involving series representations were derived in the framework of the nested sum approach for hypergeometric functions with a balanced set of parameters in [MUW02,W04] , and in framework of the generating function approach for hypergeometric functions with one unbalanced set of parameters in [DK04,KWY07b,KK08a] .

An approach using the iterated solution of differential equations has been explored in [KWY07a,KWY07c,KK08a] . One of the advantage of the iterated solution approach over the series approach is that it provides a more efficient way to calculate each order of the epsilon-expansion, since it relates each new term to the previously derived terms, rather than having to work with an increasingly large collection of independent sums at each order. This technique includes two steps: (i) the differential reduction algorithm (to reduce a generalized hypergeometric function to basic functions); (ii) iterative solution of the proper differential equation for the basic functions (equivalent to iterative algorithms for calculating the analytical coefficients of the epsilon-expansion).

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Gauss hypergeometric function 2F1(a,b;c;z)

Gauss hypergeometric function 2F1(a,b;c;z) with integer parameters :
It is a particular case of generalized hypergeometric functions with integer parameters.
Gauss hypergeometric function 2F1(a,b;c;z) with rational parameters :
If I1,I2,I3 are arbitrary integers, the Laurent expansions of the Gauss hypergeometric functions

2F1(I1+a*ep, I2+b*ep; I3+p/q+c*ep;z) ,

2F1(I1+p/q+a*ep, I2+p/q+b*ep; I3+p/q + c*ep;z) ,

2F1(I1+p/q+a*ep, I2+b*ep; I3+c*ep;z) ,

2F1(I1+p/q+a*ep, I2+b*ep; I3+p/q + c*ep;z)

are expressible in terms of multiple polylogarithms of arguments being powers of q-roots of unity and new variable, that are algebraic function of variable z with coefficients that are ratios of polynomials [W04,KK08a,KWY07a] .

For these functions, the explicit Laurent series expansion up to functions of weight 4 are constructed. The result of the expansion is expressible in terms of Nielsen polylogarithms only. The results of expansion are available here.

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Generalized hypergeometric function pFp-1(A;B;z)

Generalized hypergeometric function pFp-1(A;B;z) with integer parameters :
If A,B are lists of integers, the Laurent expansions of the generalized hypergeometric functions
pFp-1(A+a*ep; B+b*ep;z)
are expressible via generalized polylogarithms. of argument z with coefficients that are ratios of polynomials [MUW02,KWY07c] .

The first five coefficients of the epsilon-expansion for basis hypergeometric functions are available here.

Generalized hypergeometric function pFp-1(A;B;z) with rational parameters :
If A,B are lists of integers and I, p, q are integers, the Laurent expansions of the generalized hypergeometric functions
pFp-1(A+a*ep, p/q + I; B+b*ep;z) ,
pFp-1(A+a*ep; B+b*ep, p/q + I;z) ,
are expressible in terms of multiple polylogarithms of arguments being powers of q-roots of unity and new variable, that are algebraic function of variable z with coefficients that are ratios of polynomials [W04,KWY07b,KK08a] .

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For generalized hypergeometric functions with integer and /or half-integer values of parameters, and special values of argument (z=1/4,1/2,3/4,1) the epsilon epxansion up to constant of weight 5 is constructed [FK99,DK01] . For discussion of proper basis we refer to Basis for Single scale massive Feynman diagrams In this case, the result of expansion can be written in terms of generalized log-sine functions and calculated with high accuracy by the help of program LSJK

The application of our resuls for calculation of different master-integrals are given in [JKV03,JK04,K06] .

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References

[BD91] E.E.Boos,A.I.Davydychev, Theor. Math. Phys. 89 (1991) 1052,

[D00] A.I.Davydychev, Phys. Rev. D61 (2000) 087701, [hep-ph/9910224]

[DK00] A.I.Davydychev, M.Yu.Kalmykov, Nucl. Phys. Proc. Suppl. 89 (2000) 283, [hep-th/0005287]

[DK01] A.I.Davydychev, M.Yu.Kalmykov, Nucl. Phys. B605 (2001) 266, [hep-th/0012189]

[DK04] A.I.Davydychev, M.Yu.Kalmykov, Nucl. Phys. B699 (2004) 3, [hep-th/0303162]

[DT93] A.I.Davydychev, J.B.Tausk, Nucl. Phys. B397 (1993) 123,

[DT96] A.I.Davydychev, J.B.Tausk, Phys. Rev. D53 (1996) 7381 [hep-ph/9504431]

[FK99] J.Fleischer, M.Yu.Kalmykov, Phys. Lett. B470 (1999) 168 [hep-ph/9910223]

[JKV03] F.Jegerlehner, M.Yu.Kalmykov, O.Veretin, Nucl. Phys. B658 (2003) 49 [hep-ph/0212319]

[JK04] F.Jegerlehner, M.Yu.Kalmykov, Nucl. Phys. B676 (2004) 365 [hep-ph/0308216]

[K04] M.Yu.Kalmykov, Nucl. Phys. B (Proc. Suppl.) 135 (2004) 280, [hep-th/0406269]

[K06] M.Yu.Kalmykov, JHEP 04 (2006) 056, [hep-th/0602028]

[KK08a] M.Yu.Kalmykov, B.A.Kniehl, Nucl. Phys. B809 (2008) 365-405 [hep-th/0807.0567]

[KK10] M.Yu.Kalmykov, B.A.Kniehl, Phys.Part.Nucl. 41 (2010) 942-945 [math-ph/1003.1965]

[KWY07a] M.Yu.Kalmykov, B.F.L.Ward, S.Yost JHEP 02 (2007) 040, [hep-th/0612240]

[KWY07b] M.Yu.Kalmykov, B.F.L.Ward, S.Yost JHEP 10 (2007) 048, [hep-th/0707.3654]

[KWY07c] M.Yu.Kalmykov, B.F.L.Ward, S.Yost JHEP 11 (2007) 009, [hep-th/0708.0803]

[MUW02] S.Moch, P.Uwer, S. Weinzierl, J. Math. Phys 43 (2002) 3363, [hep-ph/0110083]

[W04] S.Weinzierl, J. Math. Phys. 45 (2004) 2656, [hep-ph/0402131]

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Codes for some expressions

Gauss hypergeometric function :
hep-th/0602028 :
Eqs. (4.1), (4.2) 		FORM
- hypergeometric2F1 . 		Mathematica
hypergeom.m ,
hep-th/0612240 :
Eqs. (3.3), (3.4) 		FORM
- hypergeometric2F1 - angle represenation .
hep-th/0612240 :
Eqs. (2.19)-(2.22) 		FORM
- hypergeometric2F1 - integer values of parameters .
Generalized hypergeometric function:
hep-th/0708.0803
Eqs. (2.15), (3.1)-(3.4), (3.5) 		FORM
- hypergeometric pFp-1 - integer values of parameters .
3F2 hypergeometric function:
FORM - hypergeometric 3F2 - with one half-integer parameter .
FORM - hypergeometric 3F2 - with integer parameters .
4F3 hypergeometric function:
FORM - 4F3 - with one half-integer parameter: one set of basis functions .
FORM - hypergeometric 4F3 - with integer parameters .
5F4 hypergeometric function:
FORM - 5F4 - with one half-integer parameter .
FORM - hypergeometric 5F4 - with integer parameters .

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HYPERDIRE

HYPERDIRE: version 1.0
example manual: Chapter 4 of arXiv:0904.0214v1
HYPERDIRE: version 1.2: (supported by Vladimir Bytev)

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