Multiloop massive single-scale diagrams: basis of epsilon-expansion
Under construction

One of the problem of multiloop calculation of dimensional regulated Feynman diagrams is the extraction of analytical coefficients in epsilon-expansion. The important examples are massless propagator-type diagrams existing in the renormalization group calculations: the occurring transcendental numbers can be expressed in terms of multiple Euler-Zagier sums [1]:

multiple zeta-function

For lower cases, these sums correspond to ordinary zeta-functions. In the 6-, 7-loop calculations of the beta-function in scalar theory [2] the non-zeta, transcendental numbers, arises (Example of the 6-loop graph giving non-zeta counterterm - picture from [3]). Another example is the calculation of anomalous dimensions [3], where another constant, is appeared. A remarkable property of these constants, established by Dirk Kreimer , is their connection with knots [4].

The transcendental structure of massive single-scale diagrams (when all parameters are equal to unity or zero), arising, for example, in multiloop massive calculations in QED or QCD within the on-shell scheme [5], is less investigated. To classify new constants appearing in this case, David Broadhurst has introduced the ``sixth root of unity'' basis [6] connected with

As alternative approach, an ansatz was elaborated in [7] for constructing the ``irrationalities'' occurring in the epsilon-expansion of massive single-scale diagrams. This construction is closely related with Andrei Davydychev's results conserning the one-loop propagator-type diagrams [8]. The procedure of constructing the ansatz is as follows: for each given weight j the set of the basic transcendental numbers contains (i) all products of the lower-weight elements (ii) a set of new (non-factorizable) elements which can be expressed in terms of the Clausen function, log-sine integrals and generalized log-sine integrals: of two possible angles, where
,
After excluding all linearly-dependent terms , the basis contains the following non-factorizable constants:
This set of elements will be called the odd basis .

By analogy with the odd basis introduced in [7], it is possible to consider the even basis [9] connected with the angles . Apart from the well-known elements (see details in [1]), like zeta-function, the Catalan's constant G, this basis also contains (up to the weight 5)

Later, in [10] a connection has been examined between our basis and binomial sums

where , is harmonic sum. In particular, these sums are connected with epsilon-expansion of hypergeometric function of argument 1/4 (see Appendix B of Ref.[9]).

It was demonstrated [9], that one extra term should be added to the odd weight-5 basis. This new constant can be related to a special case of the multiple binomial sums. For the even basis one needs to add two additional constants of weight 5: one is and second is is the generalization of the log-sine integrals.

For high precison numerical calculation of the proper constant the program LSJK has been developed [14].

References

[1] D.J.Broadhurst, hep-th/9604128

[2] D.J.Broadhurst, D.Kreimer, Int. J. Mod. Phys. C6 (1995) 519, hep-ph/9504352

[3] D.J.Broadhurst, J.A.Gracey, D.Kreimer, Z. Phys. C75 (1997) 559 hep-th/9607174
D.J. Broadhurst, A.V. Kotikov, Phys.Lett. B441 (1998) 345 hep-th/9612013

[4] D.Kreimer, Phys. Lett. B354 (1995) 117, hep-th/9503059
J. Knot Th. Ram. 6 (1997) 479, q-alg/9607022

[5] N. Gray, D.J. Broadhurst, W. Grafe, K.Schilcher, Z. Phys. C48 (1990) 673
S. Laporta, E. Remiddi, Phys. Lett. B379 (1996) 283 hep-ph/9602417
K. Melnikov, T. van Ritbergen, Phys. Lett. B482 (2000) 99 hep-ph/9912391
Nucl. Phys. B591 (2000) 515 hep-ph/0005131

[6] D.J.Broadhurst, Eur. Phys. J. C8 (1999) 311, hep-th/9803091

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

[8] A.I.Davydychev, hep-th/9908032
Phys. Rev. D61 (2000) 087701, hep-ph/9910224

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

[10] J.Fleischer, M.Yu.Kalmykov, A.V.Kotikov, Phys. Lett. B462 (1999) 169; 467 (1999) 310(E), hep-ph/9905249

[11] M.Yu.Kalmykov, O.Veretin, Phys. Lett. B483 (2000) 315, hep-th/0004010

[12] A.I.Davydychev, M.Yu.Kalmykov, hep-th/0203212

[13] M.Yu.Kalmykov, Nucl. Phys. B718 (2005) 276, hep-ph/0503070

[14] M.Yu.Kalmykov, A.Sheplyakov Comput.Phys.Commun. 172 (2005) 45, hep-ph/0411100