Master integrals for massless three-loop form factors: One-loop and two-loop insertions

The three-loop form factors in massless QCD can be expressed as a linear combination of master integrals. Besides a number of master integrals which factorise into products of one-loop and two-loop integrals, one finds 16 genuine three-loop integrals. Of these, six have the form of a bubble insertion inside a one-loop or two-loop vertex integral. We compute all master integrals with these insertion topologies.     

Nine-propagator master integrals for massless three-loop form factors

We complete the calculation of master integrals for massless three-loop form factors by computing the previously-unknown three diagrams with nine propagators in dimensional regularisation. Each of the integrals yields a six-fold Mellin–Barnes representation which we use to compute the coefficients of the Laurent expansion in ?. Using Riemann ζ functions of up to weight six, we give fully analytic results for one integral; for a second, analytic results for all but the finite term; for the third, analytic results for all but the last two coefficients in the Laurent expansion. The remaining coefficients are given numerically to sufficiently high accuracy for phenomenological applications.     

Master integrals for four-loop massless propagators up to weight twelve

We evaluate a Laurent expansion in dimensional regularization parameter ?=(4−d)/2 of all the master integrals for four-loop massless propagators up to weight twelve, using a recently developed method of one of the present coauthors (R.L.) and extending thereby results by Baikov and Chetyrkin obtained at weight seven. We observe only multiple zeta values in our results. Therefore, we conclude that all the four-loop massless propagator integrals, with any integer powers of numerators and propagators, have only multiple zeta values in their epsilon expansions up to weight twelve.     

Boundary terms for massless fermionic fields

Local supersymmetry leads to boundary conditions for fermionic fields in one-loop quantum cosmology involving the Euclidean normal _e n _A/A to the boundary and a pair of independent spinor fields ^A and . This paper studies the corresponding classical properties, i.e., the classical boundary-value problem and boundary terms in the variational problem. If is set to zero on a 3-sphere bounding flat Euclidean 4-space, the modes of the massless spin–1/2 field multiplying harmonics having positive eigenvalues for the intrinsic 3-dimensional Dirac operator onS ³ should vanish onS ³. Remarkably, this coincides with the property of the classical boundary-value problem when spectral boundary conditions are imposed onS ³ in the massless case. Moreover, the boundary term in the action functional is proportional to the integral on the boundary of ^A _e n _AA ^A .     

Fermionic contributions to the three-loop static potential

We consider the three-loop corrections to the static potential which are induced by a closed fermion loop. For the reduction of the occurring integrals a combination of the Gröbner and Laporta algorithm has been used and the evaluation of the master integrals has been performed with the help of the Mellin–Barnes technique. The fermionic three-loop corrections amount to 2% of the tree-level result for top quarks, 8% for bottom quarks and 27% for the charm quark system.     

Strangeness contributions to nucleon form factors

We review a recent theoretical determination of the strange quark content of the electromagnetic form factors of the nucleon. These are compared with a global analysis of current experimental measurements in parity-violating electron scattering.     

The power counting theorem for Feynman integrals with massless propagators

Dyson's power counting theorem is extended to the case where some of the mass parameters vanish. Weinberg's ultraviolet convergence conditions are supplemented by infrared convergence conditions which combined are sufficient for the convergence of Feynman integrals.     

A convergence theorem for lattice Feynman integrals with massless propagators

It is shown that for non-vanishing lattice spacing, conventional infrared power counting conditions are sufficient for convergence of lattice Feynman integrals with zero-mass propagators. If these conditions are supplemented by ultraviolet convergence conditions, the continuum limit of such a diagram exists and is universal.     

Electromagnetic7Li form factors including meson-exchange contributions

We have studied the transversal and longitudinal form factors of the⁷Li ground-state doublet within a microscopic triton + model allowing for specific distortion effects in the internal triton cluster. Meson-exchange contributions were taken into account in form of the dominant isovector pion- and rho-meson-exchange currents. For the derivation of the respective two-body operators we have used a consistent microscopic formalism developed by Riska which guarantees that the continuity equation is satisfied. The consideration of meson-exchange contributions brings our calculation in better agreement with the data. We find that only the longitudinal inelastic form factor has a diffraction minimum belowq4fm^–1, in agreement with the data. Although our approach reproduces the low-q form factors qualitatively well, we underestimate the data at the higher momentum transfers noticeably. Finally we find that the isoscalar part of the central force gives the dominant contribution to the exchange currents while the tensor force plays only a minor role due to cancellation effects between the pion- and rho-meson-exchange contributions.     

Sea quark contributions to nucleon electromagnetic form factors with the nonlocal chiral effective Lagrangian

The sea quark contributions to the nucleon electromagnetic form factors of the up,down and strange quarks are studied with the nonlocal chiral effective Lagrangian.Both octet and decuplet intermediate states are included in the one loop calculations.Compared with the strange quark form factors,although their signs are the same,the absolute value of the light quark form factors are much larger.For both the electric and magnetic form factors,the contribution of the d quark is larger than of the u quark.The current lattice simulations of the light sea quark form factors are in between our results for the u and d quarks.     

A superanalog of the Selberg trace formula and multiloop contributions for fermionic strings

An analog of the classical Selberg trace formula is given for discrete groups, acting on the upper complex half-superplane. Applications to the fermionic string measure on the moduli superspace are discussed.     

Master integrals for double real radiation emission in heavy-to-light quark decay

We evaluate analytically the master integrals for double real radiation emission in the \(b \rightarrow u W^*\) decay, where b and u are a massive and massless quark, respectively, while \(W^{*}\) is an off-shell charged weak boson. Since the W boson can subsequently decay in a lepton anti-neutrino pair, the results of the present paper constitute a further step toward a fully analytic computation of differential distributions for the semileptonic decay of a b quark at NNLO in QCD. The latter partonic process plays a crucial role in the study of inclusive semileptonic charmless decays of B mesons. Our results are expressed in terms of multiple polylogarithms of maximum weight four.     

The inelastic contributions to transport collision integrals in the infinite-order sudden approximation

The transport collision integrals of the Wang Chang and Uhlenbeck kinetic theory of polyatomic gases have been evaluated for an anisotropic intermolecular pair potential model for the He-CO₂ interaction. The calculations have made use of the infinite-order sudden approximation to evaluate the state-to-state differential cross-sections, but otherwise involve no approximations.

The results indicate that some earlier approximate methods for the evaluation of the collision integrals are quantiatively correct, although they are given a new interpretation. In particular, the simplifications introduced by Mason and Monchick are shown to yield essentially the same collision integral for viscosity as the more exact treatment. On the other hand, several long established procedures for treating the transport properties of polyatomic gases and gas mixtures are shown to be significantly in error. Specifically the diffusion coefficient for internal energy is some 10 per cent higher than the diffusion coefficient for mass, and the latter itself quite different from that obtained on the basis of the treatment of Monchick and Mason.     

Evidence for strange-quark contributions to the nucleon's form factors at Q2=0.108 (GeV/c)2

We report on a measurement of the parity violating asymmetry in the elastic scattering of polarized electrons off unpolarized protons with the A4 apparatus at MAMI in Mainz at a four momentum transfer value of Q(2)=0.108 (GeV/c)(2) and at a forward electron scattering angle of 30 degrees p)=[-1.36+/-0.29(stat)+/-0.13(syst)]x10(-6). The expectation from the standard model assuming no strangeness contribution to the vector current is A(0)=(-2.06+/-0.14)x10(-6). We have improved the statistical accuracy by a factor of 3 as compared to our previous measurements at a higher Q2. We have extracted the strangeness contribution to the electromagnetic form factors from our data to be G(s)(E)+0.106G(s)(M)=0.071+/-0.036 at Q(2)=0.108 (GeV/c)(2). We again find the value for G(s)(E)+0.106G(s)(M) to be positive, this time at an improved significance level of two sigma.     

An exact treatment of some integrals appearing in the three-body contributions to transport coefficients

The convergence properties of the integral corresponding to the three-body cyclic collisions in the two-dimensional density expansion of transport coefficients are investigated. It is shown exactly that the integral considered by Oppenheim et al. and by Eu diverges logarithmically in the switching parameter ε.