Three loop massive operator matrix elements and asymptotic Wilson coefficients with two different masses

Abstract

Starting at 3-loop order, the massive Wilson coefficients for deep-inelastic scattering and the massive operator matrix elements describing the variable flavor number scheme receive contributions of Feynman diagrams carrying quark lines with two different masses. In the case of the charm and bottom quarks, the usual decoupling of one heavy mass at a time no longer holds, since the ratio of the respective masses, η = m 2 c /m 2 b ∼ 1/10, is not small enough. Therefore, the usual variable flavor number scheme (VFNS) has to be generalized. The renormalization procedure in the two-mass case is different from the single mass case derived in [1]. We present the moments N = 2, 4 and 6 for all contributing operator matrix elements, expanding in the ratio η. We calculate the analytic results for general values of the Mellin variable N in the flavor non-singlet case, as well as for transversity and the matrix element A(3) gq . We also calculate the two-mass scalar integrals of all topologies contributing to the gluonic operator matrix element A gg . As it turns out, the expansion in η is usually inapplicable for general values of N . We therefore derive the result for general values of the mass ratio. From the single pole terms we derive, now in a two-mass calculation, the corresponding contributions to the 3-loop anomalous dimensions. We introduce a new general class of iterated integrals and study their relations and present special values. The corresponding functions are implemented in computer-algebraic form.(2),NS qq,Q (N F + 2) δ 2 +Ĉ(2),NS q,(2,L) (N F ) 5 The sum in Eq. (2.19) is over the words w given by the different orderings of the Lorentz indices. For example, for M = 3 one obtains, Sf µ1,µ2,µ3 = 1 6 (f µ1,µ2,µ3 + f µ1,µ3,µ2 + f µ2,µ1,µ3 + f µ2,µ3,µ1 + f µ3,µ1,µ2 + f µ3,µ2,µ1 ).

The three-loop single mass polarized pure singlet operator matrix element

“…Related to it, the massless parton distributions are unfolded, requiring a correct description of the heavy flavor effects. On the other hand, in order to describe the transition of massive partons becoming effectively massless, the variable flavor number scheme can be used [6][7][8]. This transition is described by massive operator matrix elements (OMEs), and after its application, effective calculations for scattering reactions at hadron colliders are possible, based also on heavy quark parton distributions.…”

“…Several of these transition matrix elements have been already computed in the unpolarized and polarized case in the single, cf. [9][10][11][12][13][14][15][16], and two-mass case [7,8,[17][18][19][20].…”

The three-loop polarized pure singlet operator matrix element with two different masses

“…In the unpolarized case the two-mass corrections have been calculated for all OMEs to three-loop order in Refs. [3][4][5][6][7]. For the OME A In the polarized case, the flavor non-singlet three loop OME A (3),NS,tm qq,Q [4] and A (3),tm gq,Q [7] have been calculated.…”

“…[3][4][5][6][7]. For the OME A In the polarized case, the flavor non-singlet three loop OME A (3),NS,tm qq,Q [4] and A (3),tm gq,Q [7] have been calculated. In the present paper we compute the two mass contributions to the pure singlet massive OME A (3),PS,tm Qq .…”

“…Qq (z, Q 2 ) and is one of the contributions of the two-mass variable flavor number scheme [4,10] in the polarized case, describing the respective transitions of the polarized parton densities in the case the heavy quarks become light. They contribute in particular also to the charm-and bottom quark distributions.…”