The analytic value of the sunrise self-mass with two equal masses and the external invariant equal to the third squared mass

We consider the two-loop self-mass sunrise amplitude with two equal masses $M$ and the external invariant equal to the square of the third mass $m$ in the usual $d$-continuous dimensional regularization. We write a second order differential equation for the amplitude in $x=m/M$ and show as solve it in close analytic form. As a result, all the coefficients of the Laurent expansion in $(d-4)$ of the amplitude are expressed in terms of harmonic polylogarithms of argument $x$ and increasing weight. As a by product, we give the explicit analytic expressions of the value of the amplitude at $x=1$, corresponding to the on-mass-shell sunrise amplitude in the equal mass case, up to the $(d-4)^5$ term included.Comment: 11 pages, 2 figures. Added Eq. (5.20) and reference [4

Master integrals for massive two-loop Bhabha scattering in QED

We present a set of scalar master integrals (MIs) needed for a complete treatment of massive two-loop corrections to Bhabha scattering in QED, including integrals with arbitrary fermionic loops. The status of analytical solutions for the MIs is reviewed and examples of some methods to solve MIs analytically are worked out in more detail. Analytical results for the pole terms in epsilon of so far unknown box MIs with five internal lines are given.Comment: 23 pages, 5 tables, 12 figures, references added, appendix B enlarge

Differential Geometry applied to Acoustics : Non Linear Propagation in Reissner Beams

Although acoustics is one of the disciplines of mechanics, its "geometrization" is still limited to a few areas. As shown in the work on nonlinear propagation in Reissner beams, it seems that an interpretation of the theories of acoustics through the concepts of differential geometry can help to address the non-linear phenomena in their intrinsic qualities. This results in a field of research aimed at establishing and solving dynamic models purged of any artificial nonlinearity by taking advantage of symmetry properties underlying the use of Lie groups. The geometric constructions needed for reduction are presented in the context of the "covariant" approach.Comment: Submitted to GSI2013 - Geometric Science of Informatio

Light quark mass effects in the on-shell renormalization constants

We compute the three-loop relation between the pole and the minimally subtracted quark mass allowing for virtual effects from a second massive quark. We also consider the analogue effects for the on-shell wave function renormalization constant.Comment: 24 page

On the two-loop contributions to the pion mass

We derive a simplified representation for the pion mass to two loops in three-flavour chiral perturbation theory. For this purpose, we first determine the reduced expressions for the tensorial two-loop 2-point sunset integrals arising in chiral perturbation theory calculations. Making use of those relations, we obtain the expression for the pion mass in terms of the minimal set of master integrals. On the basis of known results for these, we arrive at an explicit analytic representation, up to the contribution from K-K-eta intermediate states where a closed-form expression for the corresponding sunset integral is missing. However, the expansion of this function for a small pion mass leads to a simple representation which yields a very accurate approximation of this contribution. Finally, we also give a discussion of the numerical implications of our results.Comment: Typos corrected and minor changes in Table 2. Published version. 19 pages, 1 figure, 2 table

Special case of sunset: reduction and epsilon-expansion

We consider two loop sunset diagrams with two mass scales m and M at the threshold and pseudotreshold that cannot be treated by earlier published formula. The complete reduction to master integrals is given. The master integrals are evaluated as series in ratio m/M and in epsilon with the help of differential equation method. The rules of asymptotic expansion in the case when q^2 is at the (pseudo)threshold are given.Comment: LaTeX, 13 pages, 1 figur

Two-Loop Planar Corrections to Heavy-Quark Pair Production in the Quark-Antiquark Channel

We evaluate the planar two-loop QCD diagrams contributing to the leading color coefficient of the heavy-quark pair production cross section, in the quark-antiquark annihilation channel. We obtain the leading color coefficient in an analytic form, in terms of one- and two-dimensional harmonic polylogarithms of maximal weight 4. The result is valid for arbitrary values of the Mandelstam invariants s and t, and of the heavy-quark mass m. Our findings agree with previous analytic results in the small-mass limit and numerical results for the exact amplitude.Comment: 30 pages, 5 figures. Version accepted by JHE

Feynman Diagrams and Differential Equations

We review in a pedagogical way the method of differential equations for the evaluation of D-dimensionally regulated Feynman integrals. After dealing with the general features of the technique, we discuss its application in the context of one- and two-loop corrections to the photon propagator in QED, by computing the Vacuum Polarization tensor exactly in D. Finally, we treat two cases of less trivial differential equations, respectively associated to a two-loop three-point, and a four-loop two-point integral. These two examples are the playgrounds for showing more technical aspects about: Laurent expansion of the differential equations in D (around D=4); the choice of the boundary conditions; and the link among differential and difference equations for Feynman integrals.Comment: invited review article from Int. J. Mod. Phys.

Lectures on multiloop calculations

I discuss methods of calculation of propagator diagrams (massless, those of Heavy Quark Effective Theory, and massive on-shell diagrams) up to 3 loops. Integration-by-parts recurrence relations are used to reduce them to linear combinations of basis integrals. Non-trivial basis integrals have to be calculated by some other method, e.g., using Gegenbauer polynomial technique. Many of them are expressed via hypergeometric functions; in the massless and HQET cases, their indices tend to integers at $\epsilon\to0$. I discuss the algorithm of their expansion in $\epsilon$, in terms of multiple $\zeta$ values. These lectures were given at Calc-03 school, Dubna, 14--20 June 2003.Comment: 52 pages, 49 figures. Lectures at Calc-03 school, Dubna, 14--20 June 2003. v2: 2 references added, minor typos corrected. v3: methodical improvements, typo in eq. (3.19) corrected, 2 references adde

Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter

We continue the study of the construction of analytical coefficients of the epsilon-expansion of hypergeometric functions and their connection with Feynman diagrams. In this paper, we show the following results: Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth (see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions. Theorem B: The epsilon expansion of a hypergeometric function with one half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are ratios of polynomials. Some extra materials are available via the www at this http://theor.jinr.ru/~kalmykov/hypergeom/hyper.htmlComment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected and a few references added; v3: few references added