We derive analytic results for scalar massless bosonic vacuum sum-integrals at two loops. Building upon a recent factorization proof of massive two-loop vacuum integrals, we are able to solve the corresponding Matsubara sums and map the result onto one-loop structures, thereby proving factorization also in the sum-integral setting. Analytic results are provided for generic integer-valued propagator- and numerator-powers of the class of sum-integrals under consideration, allowing to eliminate them from any perturbative expansion, dramatically simplifying the evaluation of some observables encountered e.g. in hot QCD.

We investigate the structure of a particular class of massive vacuum Feynman integrals at two loops. This class enjoys the linear relation m1 + m2 = m3 between its three propagator masses, corresponding to zeros of the associated Källén function. Apart from having applications in thermal field theory, the integrals can be mapped onto one-loop three-point functions with collinear external momenta, suggesting the term “collinear” masses. We present a closed-form solution for these integrals, proving that they can always be factorized into products of one-loop cases, for all integer-valued propagator powers.

It is shown how the geometrical splitting of N-point Feynman diagrams can be used to simplify the parametric integrals and reduce the number of variables in the occurring functions. As an example, a calculation of the dimensionally-regulated one-loop four-point function in general kinematics is presented.

A geometrical approach to the calculation of N-point Feynman diagrams is reviewed. It is shown that the geometrical splitting yields useful connections between Feynman integrals with different momenta and masses. It is demonstrated how these results can be used to reduce the number of variables in the occurring functions.

The L-loop 4-point ladder diagram of massless phi^3 theory is finite when all 4 legs are off-shell and is given in terms of polylogarithms with orders ranging from L to 2L. We obtain the exact solution of the linear Dyson-Schwinger equation that sums these ladder diagrams and show that this sum vanishes exponentially fast at strong coupling.

FIELD: oil and gas production. SUBSTANCE: inductive logging is performed along borehole of well with instrument containing at least one transmitter and multitude of receivers spaced from transmitter along borehole of well. Efficient permeability of formation is determined directly near the borehole of the well by measurements made with the inductive receiver nearest to the inductive transmitter. Efficient specific permeability of formation is used for correction of measurements of apparent specific permeability made at least with one inductive receiver in the instrument. EFFECT: simplification. 21 cl, 7 dwg

A geometrical way to calculate N-point Feynman diagrams is reviewed. As an example, the dimensionally regulated three-point function is considered, including all orders of its ɛ-expansion. Analytical continuation to other regions of the kinematical variables is discussed.

When calculating higher terms of the ɛ-expansion of massive Feynman diagrams, one needs to evaluate particular cases of multiple inverse binomial sums. These sums are related to the derivatives of certain hypergeometric functions with respect to their parameters. Exploring this connection and using it together with an approach based on generating functions, we analytically calculate a number of such infinite sums, for an arbitrary value of the argument which corresponds to an arbitrary value of the off-shell external momentum. In such a way, we find a number of new results for physically important Feynman diagrams. Considered examples include two-loop two- and three-point diagrams, as well as three-loop vacuum diagrams with two different masses. The results are presented in terms of generalized polylogarithmic functions. As a physical example, higher-order terms of the ɛ-expansion of the polarization function of the neutral gauge bosons are constructed.

We derive expressions for three-body phase space that are explicitly symmetrical in the masses of the three particles. We study geometrical properties of the variables involved in elliptic integrals and demonstrate that it is convenient to use the Jacobian zeta function to express the results in four and six dimensions.