Massive Operator Matrix Elements Research Articles

The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements [formula omitted] and [formula omitted]

The non-first-order-factorizable contributions1 to the unpolarized and polarized massive operator matrix elements to three-loop order, AQg(3) and ΔAQg(3), are calculated in the single-mass case. For the F12-related master integrals of the problem, we use a semi-analytic method based on series expansions and utilize the first-order differential equations for the master integrals which does not need a special basis of the master integrals. Due to the singularity structure of this basis a part of the integrals has to be computed to O(ε5) in the dimensional parameter. The solutions have to be matched at a series of thresholds and pseudo-thresholds in the region of the Bjorken variable x∈]0,∞[ using highly precise series expansions to obtain the imaginary part of the physical amplitude for x∈]0,1] at a high relative accuracy. We compare the present results both with previous analytic results, the results for fixed Mellin moments, and a prediction in the small-x region. We also derive expansions in the region of small and large values of x. With this paper, all three-loop single-mass unpolarized and polarized operator matrix elements are calculated.

The first–order factorizable contributions to the three–loop massive operator matrix elements [formula omitted] and [formula omitted

The unpolarized and polarized massive operator matrix elements AQg(3) and ΔAQg(3) contain first–order factorizable and non–first–order factorizable contributions in the determining difference or differential equations of their master integrals. We compute their first–order factorizable contributions in the single heavy mass case for all contributing Feynman diagrams. Moreover, we present the complete color–ζ factors for the cases in which also non–first–order factorizable contributions emerge in the master integrals, but cancel in the final result as found by using the method of arbitrary high Mellin moments. Individual contributions depend also on generalized harmonic sums and on nested finite binomial and inverse binomial sums in Mellin N–space, and correspondingly, on Kummer–Poincaré and square–root valued alphabets in Bjorken–x space. We present a complete discussion of the possibilities of solving the present problem in N–space analytically and we also discuss the limitations in the present case to analytically continue the given N–space expressions to N∈C by strict methods. The representation through generating functions allows a well synchronized representation of the first–order factorizable results over a 17–letter alphabet. We finally obtain representations in terms of iterated integrals over the corresponding alphabet in x–space, also containing up to weight w = 5 special constants, which can be rationalized to Kummer–Poincaré iterated integrals at special arguments. The analytic x–space representation requires separate analyses for the intervals x∈[0,1/4],[1/4,1/2],[1/2,1] and x>1. We also derive the small and large x limits of the first–order factorizable contributions. Furthermore, we perform comparisons to a number of known Mellin moments, calculated by a different method for the corresponding subset of Feynman diagrams, and an independent high–precision numerical solution of the problems.

The inverse Mellin transform via analytic continuation

We present a method to calculate the x-space expressions of massless or massive operator matrix elements in QCD and QED containing local composite operator insertions, depending on the discrete Mellin index N, directly, without computing the Mellin-space expressions in explicit form analytically. Here N belongs either to the even or odd positive integers. The method is based on the resummation of the operators into effective propagators and relies on an analytic continuation between two continuous variables. We apply it to iterated integrals as well as to the more general case of iterated non-iterative integrals, generalizing the former ones. The x-space expressions are needed to derive the small-x behaviour of the respective quantities, which usually cannot be accessed in N-space. We illustrate the method for different (iterated) alphabets, including non-iterative 2F1 and elliptic structures, as examples. These structures occur in different massless and massive three-loop calculations. Likewise the method applies even to the analytic closed form solutions of more general cases of differential equations which do not factorize into first-order factors.

Formula omitted]) polarized heavy flavor corrections to deep-inelastic scattering at Q2 ≫ m2

We calculate the quarkonic O(αs2) massive operator matrix elements ΔAQg(N),ΔAQqPS(N) and ΔAqq,QNS(N) for the twist–2 operators and the associated heavy flavor Wilson coefficients in polarized deeply inelastic scattering in the region Q2≫m2 to O(ε) in the case of the inclusive heavy flavor contributions. The evaluation is performed in Mellin space, without applying the integration-by-parts method. The result is given in terms of harmonic sums. This leads to a significant compactification of the operator matrix elements and massive Wilson coefficients in the region Q2≫m2 derived previously in [1], which we partly confirm, and also partly correct. The results allow to determine the heavy flavor Wilson coefficients for g1(x,Q2) to O(αs2) for all but the power suppressed terms ∝(m2/Q2)k,k≥1. The results in momentum fraction z-space are also presented. We also discuss the small x effects in the polarized case. Numerical results are presented. We also compute the gluonic matching coefficients in the two–mass variable flavor number scheme to O(ε).

The unpolarized and polarized single-mass three-loop heavy flavor operator matrix elements Agg,Q and ∆Agg,Q

We calculate the gluonic massive operator matrix elements in the unpolarized and polarized cases, Agg,Q(x, μ2) and ∆Agg,Q(x, μ2), at three-loop order for a single mass. These quantities contribute to the matching of the gluon distribution in the variable flavor number scheme. The polarized operator matrix element is calculated in the Larin scheme. These operator matrix elements contain finite binomial and inverse binomial sums in Mellin N-space and iterated integrals over square root-valued alphabets in momentum fraction x-space. We derive the necessary analytic relations for the analytic continuation of these quantities from the even or odd Mellin moments into the complex plane, present analytic expressions in momentum fraction x-space and derive numerical results. The present results complete the gluon transition matrix elements both of the single- and double-mass variable flavor number scheme to three-loop order.

Logarithmic contributions to the polarized O(αs3) asymptotic massive Wilson coefficients and operator matrix elements in deeply inelastic scattering

We compute the logarithmic contributions to the polarized massive Wilson coefficients for deep-inelastic scattering in the asymptotic region $Q^2 \gg m^2$ to 3-loop order in the fixed-flavor number scheme and present the corresponding expressions for the polarized massive operator matrix elements needed in the variable flavor number scheme. The calculation is performed in the Larin scheme. For the massive operator matrix elements $A_{qq,Q}^{(3),\rm PS}$ and $A_{qg,Q}^{(3),\rm S}$ the complete results are presented. The expressions are given in Mellin-$N$ space and in momentum fraction $z$-space.

The polarized transition matrix element Agq(N) of the variable flavor number scheme at [formula omitted

We calculate the polarized massive operator matrix element Agq(3)(N) to 3-loop order in Quantum Chromodynamics analytically at general values of the Mellin variable N both in the single- and double-mass case in the Larin scheme. It is a transition function required in the variable flavor number scheme at O(αs3). We also present the results in momentum fraction space.

The two-mass contribution to the three-loop polarized gluonic operator matrix element [formula omitted

We compute the two-mass contributions to the polarized massive operator matrix element Agg,Q(3) at third order in the strong coupling constant αs in Quantum Chromodynamics analytically. These corrections are important ingredients for the matching relations in the variable flavor number scheme and for the calculation of Wilson coefficients in deep–inelastic scattering in the asymptotic regime Q2≫mc2,mb2. The analytic result is expressed in terms of nested harmonic, generalized harmonic, cyclotomic and binomial sums in N-space and by iterated integrals involving square-root valued arguments in z space, as functions of the mass ratio. Numerical results are presented. New two–scale iterative integrals are calculated.

The O(α2) initial state QED corrections to [formula omitted

We calculate the complete O(α2) initial state radiation corrections to e+e− annihilation into a neutral vector boson in a direct analytic computation without any approximation. The corrections are represented in terms of iterated incomplete (elliptic) integrals over alphabets of square-root valued letters. Performing the limit s≫me2, we find discrepancies with the earlier results of Ref. [1] and confirm results obtained in Ref. [2] where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in me2/s. In this way, we also confirm the validity of the factorization of massive partons in the Drell-Yan process to O(α2). We add non–logarithmic terms at O(α2) which have not been considered in previous calculations. The final results in the limit s≫me2 can be given in terms of Nielsen integrals.

Subleading logarithmic QED initial state corrections to [formula omitted] to O(α6L5)

Using the method of massive operator matrix elements, we calculate the subleading QED initial state radiative corrections to the process e+e−→γ⁎/Z⁎ for the first three logarithmic contributions from O(α3L3),O(α3L2),O(α3L) to O(α5L5),O(α5L4),O(α5L3) and compare their effects to the leading contribution O(α6L6) and one more subleading term O(α6L5). The calculation is performed in the limit of large center of mass energies squared s≫me2. These terms supplement the known corrections to O(α2), which were completed recently. Given the high precision at future colliders operating at very large luminosity, these corrections are important for concise theoretical predictions. The present calculation needs the calculation of one more two–loop massive operator matrix element in QED. The radiators are obtained as solutions of the associated Callen–Symanzik equations in the massive case. The radiators can be expressed in terms of harmonic polylogarithms to weight w = 6 of argument z and (1−z) and in Mellin N space by generalized harmonic sums. Numerical results are presented on the position of the Z peak and corrections to the Z width, ΓZ. The corrections calculated result into a final theoretical accuracy for δMZ and δΓZ which is estimated to be of O(30keV) at an anticipated systematic accuracy at the FCC_ee of ∼100keV. This precision cannot be reached, however, by including only the corrections up to O(α3).

The polarized three-loop anomalous dimensions from on-shell massive operator matrix elements

We calculate all contributions ∝TF to the polarized three–loop anomalous dimensions in the M–scheme using massive operator matrix elements and compare to results in the literature. This includes the complete anomalous dimensions γqq(2),PS and γqg(2). We also obtain the complete two–loop polarized anomalous dimensions in an independent calculation. While for most of the anomalous dimensions the usual direct computation methods in Mellin N–space can be applied since all recurrences factorize at first order, this is not the case for γqg(2). Due to the necessity of deeper expansions of the master integrals in the dimensional parameter ε=D−4, we had to use the method of arbitrary high moments to eliminate elliptic contributions in intermediate steps. 4000 moments were generated to determine this anomalous dimension and 2640 moments turned out to be sufficient. As an aside, we also recalculate the contributions ∝TF to the three–loop QCD β–function.

The O(α2) initial state QED corrections to e+e− annihilation to a neutral vector boson revisited

We calculate the non-singlet, the pure singlet contribution, and their interference term, at O(α2) due to electron-pair initial state radiation to e+e− annihilation into a neutral vector boson in a direct analytic computation without any approximation. The correction is represented in terms of iterated incomplete elliptic integrals. Performing the limit s≫me2 we find discrepancies with the earlier results of Ref. [1] and confirm results obtained in Ref. [2] where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in m2/s. In this way, we also confirm the validity of the factorization of massive partons in the Drell–Yan process. We also add non-logarithmic terms at O(α2) which have not been considered in [1]. The corrections are of central importance for precision analyses in e+e− annihilation into γ⁎/Z⁎ at high luminosity.

The two-mass contribution to the three-loop gluonic operator matrix element [formula omitted

We calculate the two-mass QCD contributions to the massive operator matrix element Agg,Q at O(αs3) in analytic form in Mellin N- and z-space, maintaining the complete dependence on the heavy quark mass ratio. These terms are important ingredients for the matching relations of the variable flavor number scheme in the presence of two heavy quark flavors, such as charm and bottom. In Mellin N-space the result is given in the form of nested harmonic, generalized harmonic, cyclotomic and binomial sums, with arguments depending on the mass ratio. The Mellin inversion of these quantities to z-space gives rise to generalized iterated integrals with square root valued letters in the alphabet, depending on the mass ratio as well. Numerical results are presented.

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

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=mc2/mb2∼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 Agq(3). We also calculate the two-mass scalar integrals of all topologies contributing to the gluonic operator matrix element Agg. 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.

Calculating three loop ladder and [formula omitted]-topologies for massive operator matrix elements by computer algebra

Three loop ladder and V-topology diagrams contributing to the massive operator matrix element AQg are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable N and the dimensional parameter ε. Given these representations, the desired Laurent series expansions in ε can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin–Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist–Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product–sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of N are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of V-topologies.

3-loop heavy flavor Wilson coefficients in deep-inelastic scattering

We present our most recent results on the calculation of the heavy flavor contributions to deep-inelastic scattering at 3-loop order in the large Q2 limit, where the heavy flavor Wilson coefficients are known to factorize into light flavor Wilson coefficients and massive operator matrix elements. We describe the different techniques employed for the calculation and show the results in the case of the heavy flavor non-singlet and pure singlet contributions to the structure function F2(x,Q2).

3-loop Massive [formula omitted] Contributions to the DIS Operator Matrix Element [formula omitted

Contributions to heavy flavour transition matrix elements in the variable flavour number scheme are considered at 3-loop order. In particular a calculation of the diagrams with two equal masses that contribute to the massive operator matrix element Agg,Q(3) is performed. In the Mellin space result one finds finite nested binomial sums. In x-space these sums correspond to iterated integrals over an alphabet containing also square-root valued letters.

The logarithmic contributions to the [Formula: see text] asymptotic massive Wilson coefficients and operator matrix elements in deeply inelastic scattering.

We calculate the logarithmic contributions to the massive Wilson coefficients for deep-inelastic scattering in the asymptotic region Q^2 gg m^2 to 3-loop order in the fixed flavor number scheme and present the corresponding expressions for the massive operator matrix elements needed in the variable flavor number scheme. Explicit expressions are given in Mellin N-space.

Calculating massive 3-loop graphs for operator matrix elements by the method of hyperlogarithms

We calculate convergent 3-loop Feynman diagrams containing a single massive loop equipped with twist τ=2 local operator insertions corresponding to spin N. They contribute to the massive operator matrix elements in QCD describing the massive Wilson coefficients for deep-inelastic scattering at large virtualities. Diagrams of this kind can be computed using an extended version of the method of hyperlogarithms, originally being designed for massless Feynman diagrams without operators. The method is applied to Benz- and V-type graphs, belonging to the genuine 3-loop topologies. In case of the V-type graphs with five massive propagators, new types of nested sums and iterated integrals emerge. The sums are given in terms of finite binomially and inverse binomially weighted generalized cyclotomic sums, while the 1-dimensionally iterated integrals are based on a set of ∼30 square-root valued letters. We also derive the asymptotic representations of the nested sums and present the solution for N∈C. Integrals with a power-like divergence in N-space ∝aN,a∈R,a>1, for large values of N emerge. They still possess a representation in x-space, which is given in terms of root-valued iterated integrals in the present case. The method of hyperlogarithms is also used to calculate higher moments for crossed box graphs with different operator insertions.

The [formula omitted] heavy quark corrections to charged current deep-inelastic scattering at large virtualities

We calculate the O(αs2) heavy flavor corrections to charged current deep-inelastic scattering at large scales Q2≫m2. The contributing Wilson coefficients are given as convolutions between massive operator matrix elements and massless Wilson coefficients. Foregoing results in the literature are extended and corrected. Numerical results are presented for the kinematic region of the HERA data.