We calculate the leading Coulomb correction to the energy loss in the electron-nucleus Bremsstrahlung for arbitrary energy of the incoming particle. This correction determines the charge asymmetry, i.e., the difference of electron and positron energy
loss. The result is presented in terms of the classical polylogarithms $mathrm{Li}_2$ and $mathrm{Li}_3$. We use modern multiloop methods based on the IBP reduction and on the differential equations for master integrals. We provide both the threshold and the high-energy asymptotics of the found asymmetry and compare them with the available results.
An analytic formula is given for the total scattering cross section of an electron and a photon at order $alpha^3$. This includes both the double-Compton scattering real-emission contribution as well as the virtual Compton scattering part. When combi
ned with the recent analytic result for the pair-production cross section, the complete $alpha^3$ cross section is now known. Both the next-to-leading order calculation as well as the pair-production cross section are computed using modern multiloop calculation techniques, where cut diagrams are decomposed into a set of master integrals that are then computed using differential equations.
We present a new package for Mathematica system, called Libra. Its purpose is to provide convenient tools for the transformation of the first-order differential systems $partial_i boldsymbol j = M_i boldsymbol j$ for one or several variables. In part
icular, Libra is designed for the reduction to $epsilon$-form of the differential systems which appear in multiloop calculations. The package also contains some tools for the construction of general solution: both via perturbative expansion of path-ordered exponent and via generalized power series expansion near regular singular points.Libra also has tools to determine the minimal list of coefficients in the asymptotics of the original master integrals, sufficient for fixing the boundary conditions.
Within the differential equation method for multiloop calculations, we examine the systems irreducible to $epsilon$-form. We argue that for many cases of such systems it is possible to obtain nontrivial quadratic constraints on the coefficients of $e
psilon$-expansion of their homogeneous solutions. These constraints are the direct consequence of the existence of symmetric $(epsilon+1/2)$-form of the homogeneous differential system, i.e., the form where the matrix in the right-hand side is symmetric and its $epsilon$-dependence is localized in the overall factor $(epsilon+1/2)$. The existence of such a form can be constructively checked by available methods and seems to be common to many irreducible systems, which we demonstrate on several examples. The obtained constraints provide a nontrivial insight on the structure of general solution in the case of the systems irreducible to $epsilon$-form. For the systems reducible to $epsilon$-form we also observe the existence of symmetric form and derive the corresponding quadratic constraints.
This is a sequel of our previous paper where we described an algorithm to find a solution of differential equations for master integrals in the form of an $epsilon$-expansion series with numerical coefficients. The algorithm is based on using general
ized power series expansions near singular points of the differential system, solving difference equations for the corresponding coefficients in these expansions and using matching to connect series expansions at two neighboring points. Here we use our algorithm and the corresponding code for our example of four-loop generalized sunset diagrams with three massive and two massless propagators, in order to obtain new analytical results. We analytically evaluate the master integrals at threshold, $p^2=9 m^2$, in an expansion in $epsilon$ up to $epsilon^1$. With the help of our code, we obtain numerical results for the threshold master integrals in an $epsilon$-expansion with the accuracy of 6000 digits and then use the PSLQ algorithm to arrive at analytical values. Our basis of constants is build from bases of multiple polylogarithm values at sixth roots of unity.
We compute the three-loop QCD corrections to the massive quark form factors with external vector, axial-vector, scalar and pseudo-scalar currents. All corrections with closed loops of massless fermions are included. The non-fermionic part is computed
in the large-$N_c$ limit, where only planar Feynman diagrams contribute.
We compute the three-loop QCD corrections to the massive quark-anti-quark-photon form factors $F_1$ and $F_2$ involving a closed loop of massless fermions. This subset is gauge invariant and contains both planar and non-planar contributions. We perfo
rm the reduction using FIRE and compute the master integrals with the help of differential equations. Our analytic results can be expressed in terms of Goncharov polylogarithms. We provide analytic results for all master integrals which are not present in the large-$N_c$ calculation considered in Refs. [1,2].
We present the Mathematica package DREAM for arbitrarily high precision computation of multiloop integrals within the DRA (Dimensional Recurrence & Analyticity) method as solutions of dimensional recurrence relations. Starting from these relations, t
he package automatically constructs the inhomogeneous solutions and reduces the manual efforts to setting proper homogeneous solutions. DREAM also provides means to define the homogeneous solutions of the higher-order recurrence relations (and can construct those of the first-order recurrence relations automatically). Therefore, this package can be used to apply the DRA method to the topologies with sectors having more than one master integral. Two nontrivial examples are presented: four-loop fully massive tadpole diagrams of cat-eye topology and three-loop cut diagrams which are necessary for computation of the width of the para-positronium decay into four photons. The analytical form of this width is obtained here for the first time to the best of our knowledge.
We formulate a method to find the meromorphic solutions of higher-order recurrence relations in the form of the sum over poles with coefficients defined recursively. Several explicit examples of the application of this technique are given. The main a
dvantage of the described approach is that the analytical properties of the solutions are very clear (the position of poles is explicit, the behavior at infinity can be easily determined). These are exactly the properties that are required for the application of the multiloop calculation method based on dimensional recurrence relations and analyticity (the DRA method).
We describe a strategy to solve differential equations for Feynman integrals by powers series expansions near singular points and to obtain high precision results for the corresponding master integrals. We consider Feynman integrals with two scales,
i.e. nontrivially depending on one variable. The corresponding algorithm is oriented at situations where canonical form of the differential equations is impossible. We provide a computer implementation of our algorithm in a simple example of four-loop generalized sun-set integrals with three equal non-zero masses. Our code provides values of the master integrals at any given point on the real axis with a required accuracy and a given order of expansion in the regularization parameter $epsilon$.
