arXiv subjects
Publications and source records attributed to A. De Freitas.
We calculate the polarized massive operator matrix element $A_{gq}^{(3)}(N)$ 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 $O(α_s^3)$. We also present the results in momentum fraction space.
We compute the two-mass contributions to the polarized massive operator matrix element $A_{gg,Q}^{(3)}$ at third order in the strong coupling constant $α_s$ 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 $Q^2 \gg m_c^2, m_b^2$. 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.
Using the method of massive operator matrix elements, we calculate the subleading QED initial state radiative corrections to the process $e^+e^- \rightarrow γ^*/Z^*$ for the first three logarithmic contributions from $O(α^3 L^3), O(α^3 L^2), O(α^3 L)$ to $O(α^5 L^5), O(α^5 L^4), O(α^5 L^3)$ and compare their effects to the leading contribution $O(α^6 L^6)$ and one more subleading term $O(α^6 L^5)$. The calculation is performed in the limit of large center of mass energies squared $s \gg m_e^2$. These terms supplement the known corrections to $O(α^2)$, 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 {\sf 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, $Γ_Z$. The corrections calculated result into a final theoretical accuracy for $δM_Z$ and $δΓ_Z$ which is estimated to be of O(30 keV) at an anticipated systematic accuracy at the FCC\_ee of \sim 100 keV. This precision cannot be reached, however, by including only the corrections up to $O(α^3)$.
We calculate the complete $O(α^2)$ 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 $s \gg m_e^2$, we find discrepancies with the earlier results of Ref.~\cite{Berends:1987ab} and confirm results obtained in Ref.~\cite{Blumlein:2011mi} where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in $m^2_e/s$. In this way, we also confirm the validity of the factorization of massive partons in the Drell--Yan process. We add non--logarithmic terms at $O(α^2)$ which have not been considered in previous calculations. The final results in the limit $s \gg m_e^2$ can be given in terms of Nielsen integrals.
We calculate the massive polarized three-loop pure singlet operator matrix element $A_{Qq}^{(3), \rm PS}$ in the single mass case in the Larin scheme. This operator matrix element contributes to the massive polarized three-loop Wilson coefficient $H_{Qq}^{(3),\rm PS}$ in deep-inelastic scattering and constitutes a three-loop transition matrix element in the variable flavor number scheme. We provide analytic results in Mellin $N$ and in $x$ space and study the behaviour of this operator matrix element in the region of small and large values of the Bjorken variable $x$.
We present the two-mass QCD contributions to the polarized pure singlet operator matrix element at three loop order in $x$-space. These terms are relevant for calculating the polarized structure function $g_1(x,Q^2)$ at $O(α_s^3)$ as well as for the matching relations in the variable flavor number scheme and the polarized heavy quark distribution functions at the same order. The result for the operator matrix element is given in terms of generalized iterated integrals. These integrals depend on the mass ratio through the main argument, and the alphabet includes square--root valued letters.
At $e^+ \, e^-$ colliders the QED--initial state radiation forms a large part of the radiative corrections. Their precise and fast evaluation is an essential asset for the experiments at LEP, the ILC and the FCC-ee, operating at high luminosity. A long standing problem in the analytic calculation of the $O(α^2)$ initial state corrections concerns a discrepancy which has been observed between the result of Berends et al. (1988) \cite{Berends:1987ab} in the limit $m_e^2 \ll s$ and the result by Bl{ü}mlein et al. (2011) \cite{Blumlein:2011mi} using massive operator matrix elements deriving this limit directly. In order to resolve this important issue we recalculated this process by integrating directly over the phase space without any approximation. For parts of the corrections we find exact solutions of the cross section in terms of iterated integrals over square root valued letters representing incomplete elliptic integrals and iterations over them. The expansion in the limit $m_e^2 \ll s$ reveals errors in the constant $O(α^2)$ term of the former calculation and yields agreement with the calculation based on massive operator matrix elements, which has impact on the experimental analysis programs. This finding also explicitly proofs the factorization of massive initial state particles in the high energy limit including the terms of $O(α^2)$ for this process.
We present numerical results on the recently completed $O(α^2)$ initial state corrections to the process $e^+e^- \rightarrow γ^*/Z^*$, which is a central process at past and future high energy and high luminsoity colliders for precision measurements of the properties of the $Z$-boson, the Higgs boson, and the top quark. We observe differences to an earlier result \cite{Berends:1987ab} in the non-logarithmic contributions at $O(α^2)$. The new result leads to a 4 MeV shift in the $Z$ width considering the lower end $s_0 = 4 m_τ^2$ of the radiation region, which is larger than the present accuracy. We present predictions on the radiative corrections to the central processes $e^+e^- \rightarrow γ^*/Z^*$, $e^+e^- \rightarrow Z H$ and $e^+e^- \rightarrow t \overline{t}$ planned at future colliders like the ILC, CLIC, Fcc\_ee and CEPC to measure the mass and the width of the $Z$ boson, the Higgs boson and the top quark, for which the present corrections are significant.
We calculate all contributions $\propto T_F$ 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 $γ_{qq}^{(2),\rm PS}$ and $γ_{qg}^{(2)}$. 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 $γ_{qg}^{(2)}$. Due to the necessity of deeper expansions of the master integrals in the dimensional parameter $\varepsilon = D-4$, 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 $\propto T_F$ to the three-loop QCD $β$-function.
We calculate the massive two--loop pure singlet Wilson coefficients for heavy quark production in the unpolarized case analytically in the whole kinematic region and derive the threshold and asymptotic expansions. We also recalculate the corresponding massless two--loop Wilson coefficients. The complete expressions contain iterated integrals with elliptic letters. The contributing alphabets enlarge the Kummer-Poincaré letters by a series of square-root valued letters. A new class of iterated integrals, the Kummer-elliptic integrals, are introduced. For the structure functions $F_2$ and $F_L$ we also derive improved asymptotic representations adding power corrections. Numerical results are presented.
We calculate the non-singlet, the pure singlet contribution, and their interference term, at $O(α^2)$ 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 $s \gg m_e^2$ we find discrepancies with the earlier results of Ref.~\cite{Berends:1987ab} and confirm results obtained in Ref.~\cite{Blumlein:2011mi} where the effective method of massive operator matrix elements has been used, which works for all but the power corrections in $m^2/s$. 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 $O(α^2)$ which have not been considered in \cite{Berends:1987ab}. The corrections are of central importance for precision analyzes in $e^+e^-$ annihilation into $γ^*/Z^*$ at high luminosity.
We report on recent results on the two-mass corrections for massive operator matrix elements at 2- and 3-loop orders in QCD. These corrections form the building blocks of the variable flavor number scheme. Due to the similar values of the charm and bottom quark masses the two-mass corrections form an important contribution.
We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the $ρ$ parameter. In particular, we present the calculation of the master integrals for which the corresponding differential equations do not factorize to first order. The homogeneous solutions to these differential equations are obtained in terms of hypergeometric functions at rational argument. These hypergeometric functions can further be mapped to complete elliptic integrals, and the inhomogeneous solutions are expressed in terms of a new class of integrals of combined iterative non-iterative nature.
We present the matching relations of the variable flavor number scheme at next-to-leading order, which are of importance to define heavy quark partonic distributions for the use at high energy colliders such as Tevatron and the LHC. The consideration of the two-mass effects due to both charm and bottom quarks, having rather similar masses, are important. These effects have not been considered in previous investigations. Numerical results are presented for a wide range of scales. We also present the corresponding contributions to the structure function $F_2(x,Q^2)$.
We calculate the two-mass QCD contributions to the massive operator matrix element $A_{gg,Q}$ at $\mathcal{O} (α_s^3)$ 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.
We compute the two-loop QCD corrections to the heavy quark form factors in case of the vector, axial-vector, scalar and pseudo-scalar currents up to second order in the dimensional parameter $ε= (4-D)/2$. These terms are required in the renormalization of the higher order corrections to these form factors.
We report on our latest results in the calculation of the two--mass contributions to 3--loop operator matrix elements (OMEs). These OMEs are needed to compute the corresponding contributions to the deep-inealstic scattering structure functions and to generalize the variable flavor number scheme by including both charm and bottom quarks. We present the results for the non-singlet and $A_{gq,Q}$ OMEs, and compare the size of their contribution relative to the single mass case. Results for the gluonic OME $A_{gg,Q}$ are given in the physical case, going beyond those presented in a previous publication where scalar diagrams were computed. We also discuss our recently published two--mass contribution to the pure singlet OME, and present an alternative method of calculating the corresponding diagrams.
Various of the single scale quantities in massless and massive QCD up to 3-loop order can be expressed by iterative integrals over certain classes of alphabets, from the harmonic polylogarithms to root-valued alphabets. Examples are the anomalous dimensions to 3-loop order, the massless Wilson coefficients and also different massive operator matrix elements. Starting at 3-loop order, however, also other letters appear in the case of massive operator matrix elements, the so called iterative non-iterative integrals, which are related to solutions based on complete elliptic integrals or any other special function with an integral representation that is definite but not a Volterra-type integral. After outlining the formalism leading to iterative non-iterative integrals,we present examples for both of these cases with the 3-loop anomalous dimension $γ_{qg}^{(2)}$ and the structure of the principle solution in the iterative non-interative case of the 3-loop QCD corrections to the $ρ$-parameter.