Massive Propagators Research Articles

Unitarity cuts with massive propagators and algebraic expressions for coefficients

In the first part of this paper, we extend the $d$-dimensional unitarity cut method of Anastasiou et al. to cases with massive propagators. We present formulas for integral reduction with which one can obtain coefficients of all pentagon, box, triangle and massive bubble integrals. In the second part of this paper, we present a detailed study of the phase space integration for unitarity cuts. We carry out spinor integration in generality and give algebraic expressions for coefficients, intended for automated evaluation.

Two-loop amplitudes and master integrals for the production of a Higgs boson via a massive quark and a scalar-quark loop

We compute all two-loop master integrals which are required for the evaluation of next-to-leading order QCD corrections in Higgs boson production via gluon fusion. Many two-loop amplitudes for 2 -> 1 processes in the Standard Model and beyond can be expressed in terms of these integrals using automated reduction techniques. These integrals also form a subset of the master integrals for more complicated 2 -> 2 amplitudes with massive propagators in the loops. As a first application, we evaluate the two-loop amplitude for Higgs boson production in gluon fusion via a massive quark. Our result is the first independent check of the calculation of Spira, Djouadi, Graudenz and Zerwas. We also present for the first time the two-loop amplitude for gg -> h via a massive squark.

``Background field integration-by-parts'' and the connection between one-loop and two-loop Heisenberg-Euler effective actions

We develop integration-by-parts rules for Feynman diagrams involving massive scalar propagators in a constant background electromagnetic field, and use these to show that there is a simple diagrammatic interpretation of mass renormalization in the two-loop scalar QED Heisenberg-Euler effective action for a general constant background field. This explains why the square of a one-loop term appears in the renormalized two-loop Heisenberg-Euler effective action. No integrals need be evaluated, and the explicit form of the background field propagators is not needed. This dramatically simplifies the computation of the renormalized two-loop effective action for scalar QED, and generalizes a previous result obtained for self-dual background fields.

Heavy quark vacuum polarization: first two moments of the ${\mathcal{O}}(\alpha_s^3 n_f^2)$ contribution

The vacuum polarization due to a virtual heavy quark pair and specifically the coefficients of its Taylor expansion in the external momentum are closely related to moments of the cross section for quark-antiquark pair production in electron-positron annihilation. Relating measurement and theoretically calculated Taylor coefficients, an accurate value for charm- and bottom-quark mass can be derived, once corrections from perturbative QCD are sufficiently well under control. Up to three-loop order these have been evaluated previously. We now present a subset of four-loop contributions to the lowest two moments, namely those from diagrams which involve two internal loops from massive and massless fermions coupled to virtual gluons, hence of order \(\alpha_s^3 n_f^2\). The calculation demonstrates the applicability of Laporta’s algorithm to four-loop vacuum diagrams with both massive and massless propagators and should be considered a first step towards the full evaluation of the order \(\alpha_s^3\) contribution.

Master integrals with 2 and 3 massive propagators for the 2-loop electroweak form factor—planar case

We compute the master integrals containing 2 and 3 massive propagators entering the planar amplitudes of the 2-loop electroweak form factor. By this me mean the process f f ¯ → X , where f f ¯ is an on-shell massless fermion pair and X is a singlet under the electroweak gauge group SU ( 2 ) L × U ( 1 ) Y . This work is a continuation of our previous evaluation of master integrals containing at most 1 massive propagator. The masses of the W, Z and Higgs bosons are assumed to be degenerate. The 1 / ε poles and the finite parts are computed exactly in terms of a new class of 1-dimensional harmonic polylogarithms of the variable x = − s / m 2 , with ε = 2 − D / 2 , D the space–time dimension and s the center-of-mass energy squared. Since thresholds and pseudothresholds in s = ± 4 m 2 do appear in addition to the old ones in s = 0 , ± m 2 , an extension of the basis function set involving complex constants and radicals is introduced, together with a set of recursion equations to reduce integrals with semi-integer powers. It is shown that the basic properties of the harmonic polylogarithms are maintained by the generalization. We derive small-momentum expansions | s | ≪ m 2 of all the 6-denominator amplitudes. Comparison with previous results in the literature is performed finding complete agreement.

Numerical evaluation of multi-loop integrals by sector decomposition

In a recent paper [Nucl. Phys. B 585 (2000) 741] we have presented an automated subtraction method for divergent multi-loop/leg integrals in dimensional regularisation which allows for their numerical evaluation, and applied it to diagrams with massless internal lines. Here we show how to extend this algorithm to Feynman diagrams with massive propagators and arbitrary propagator powers. As applications, we present numerical results for the master 2-loop 4-point topologies with massive internal lines occurring in Bhabha scattering at two loops, and for the master integrals of planar and non-planar massless double box graphs with two off-shell legs. We also evaluate numerically some two-point functions up to 5 loops relevant for beta-function calculations, and a 3-loop 4-point function, the massless on-shell planar triple box. Whereas the 4-point functions are evaluated in non-physical kinematic regions, the results for the propagator functions are valid for arbitrary kinematics.

Formula omitted] vs. pole masses of gauge bosons II: two-loop electroweak fermion corrections

We have calculated the fermion contributions to the shift of the position of the poles of the massive gauge boson propagators at two-loop order in the Standard Model. Together with the bosonic contributions calculated previously the full two-loop corrections are available. This allows us to investigate the full correction in the relationship between MS and pole masses of the vector bosons Z and W. Two-loop renormalization and the corresponding renormalization group equations are discussed. Analytical results for the master-integrals appearing in the massless fermion contributions are given. A new approach of summing multiple binomial sums has been developed.

Full two-loop electroweak corrections to the pole masses of gauge bosons

We discuss progress in SM two-loop calculations of the pole position of the massive gauge-boson propagators.

Configuration space based recurrence relations for sunset-type diagrams

We derive recurrence relations for the calculation of multiloop sunset-type diagrams with large powers of massive propagators. The technique is formulated in configuration space and exploits the explicit form of the massive propagator raised to a given power. We write down and evaluate a convenient set of basis integrals. The method is well suited for a numerical evaluation of this class of diagrams. We give explicit analytical formulae for the basis integrals in the asymptotic regime.

Ginsparg-Wilson fermion propagators and chiral condensate

By exploiting the chiral symmetry, we derive analytic formulas for the massless and the massive Ginsparg-Wilson fermion propagators. Using these formulas, we derive an expression for the chiral condensate which is the order parameter for spontaneous chiral symmetry breaking in QCD. These formulas provide the proper way to compute the fermion propagators and may save a significant amount of computing time especially for large lattices in four dimensions.

Configuration space based recurrence relations for sunset-type diagrams

We derive recurrence relations for the calculation of multiloop sunset-type diagrams with large powers of massive propagators. The technique is formulated in configuration space and exploits the explicit form of the massive propagator raised to a given power. We write down and evaluate a convenient set of basis integrals. The method is well suited for a numerical evaluation of this class of diagrams. We give explicit analytical formulae for the basis integrals in the asymptotic regime.

Supergravity and matrix theory do not disagree on multi-graviton scattering

We compare the amplitudes for the long-distance scattering of three gravitons in eleven dimensional supergravity and matrix theory at finite N. We show that the leading supergravity term arises from loop contributions to the matrix theory effective action that are not required to vanish by supersymmetry. We evaluate in detail one type of diagram---the setting sun with only massive propagators---reproducing the supergravity behavior.

Feynman rules in N = 2 projective superspace (II).Massive hypermultiplets

Manifest N = 2 supersymmetric hypermultiplet mass terms can be introduced in the projective N = 2 superspace formalism. In the case of complex hypermultiplets, where the gauge covariantized spinor derivatives have an explicit representation in terms of gauge prepotentials, it is possible to interpret such masses as vacuum expectation values of an Abelian vector multiplet. The duality transformation that relates the N = 2 off-shell projective description of the hyper-multiplet to the on-shell description involving two N = 1 chiral superfields allows us to obtain the massive propagators of the N = 1 complex linear fields in the projective hypermultiplet. The N = 1 massive propagators of the component superfields in the projective hypermultiplet suggest a possible ansatz for the N = 2 massive propagator, which agrees with an explicit calculation in N = 2 superspace.

Automatic computation of three-loop two-point functions in large momentum expansion

We discuss the calculation of two-point three-loop functions with an arbitrary number of massive propagators and one large external momentum. The relevant subdiagrams are generated automatically. The resulting massless two-point integrals and massive tadpoles are transformed on-line to FORM-expressions ready to be used by existing FORM packages which calculate them analytically. As an example we compute the quartic mass corrections to the photon polarization function.

THE SUNSET DIAGRAM IN SU(3) CHIRAL PERTURBATION THEORY

A general procedure for the calculation of a class of two-loop Feynman diagrams is described. These are two-point functions containing three massive propagators, raised to integer powers, in the denominator and arbitrary polynomials of the loop momenta in the numerator. The ultraviolet divergent parts are calculated analytically, while the remaining finite parts are obtained by one-dimensional numerical integration, both below and above the threshold. Integrals of this type occur, for example, in chiral perturbation theory at order p6.

Homogeneous propagators for massless half-integer spin fields in 3+2 De Sitter space

Investigation of the massive propagators allows us to exhibit indecomposable representations appearing with masslessness. By direct examination of the propagators, we prove that massless fields in 3+2 De Sitter space propagate on the light cone. Indeed, it will be shown that a choice of gauge, in which all propagation is confined to the light cone, exists and, in this context, explicit expressions of homogeneous propagators for massless half-integer spin fields are obtained.

Bounds on completely convergent Euclidean Feynman graphs

LetG be a Euclidean Feynman graph containingL(G) lines. We prove that ifG has massive propagators and does not contain any divergent subgraphs its value is bounded byKL(G). We also prove the infrared analogue of this bound.

Gluon condensate and the effective gluon mass

Using massive gauge invariant QCD we show explicity how power like corrections to\(\Pi _{\mu v} \left( q \right) = i\int {dx} e^{iq'x} \left\langle {0\left| {j_\mu ^{em} \left( x \right)\bar j_v^{em} \left( 0 \right)} \right|0} \right\rangle \) arise. Using our result for the 1/q4 contribution, a one to one correspondence is made between the gluon condensate and the effective gluon mass. By relating this mass to,\(\langle 0|\frac{{\alpha _s }}{\pi }G_{\mu v}^2 |0\rangle \) a value ofmgluon=750 MeV is found at −q2=10 GeV2. In addition, within the context of dimensional regularization, a new technique for evaluating two loop momentum integrals with massive propagators is introduced. This method is a derivative of the Mellin transform technique that was applied to ladder diagrams in the days of Reggeisation.

Massive propagators in instanton fields

Green's functions for massive spinor and vector particles propagating in a self-dual but otherwise arbitrary non-Abelian gauge field are shown to be completely determined by the corresponding Green's functions of massive scalar particles.

Renormalization group and infrared behaviour in theories with coupled massless fields

The renormalization-group method is applied to investigate the infrared singularities in gauge theories with Abelian or non-Abelian symmetry, involving both massive and massless fermions. In the Abelian gauge model the infrared structures of massive and massless fermion propagators and of a massive fermion form factor are found. In the non-Abelian gauge model (quantum chromodynamics) the infrared behaviour of a massless gluon propagator and a massive quark form factor is considered in the logarithmic approximation.