19 research outputs found
Analytical evaluation of certain on-shell two-loop three-point diagrams
An analytical approach is applied to the calculation of some
dimensionally-regulated two-loop vertex diagrams with essential on-shell
singularities. Such diagrams are important for the evaluation of QED
corrections to the muon decay, QCD corrections to top quark decays t->W^{+}b,
t->H^{+}b, etc.Comment: 2 pages, LaTeX, contribution to proceedings of ACAT2002 (Moscow, June
2002
Equivalence of Recurrence Relations for Feynman Integrals with the Same Total Number of External and Loop Momenta
We show that the problem of solving recurrence relations for L-loop
(R+1)-point Feynman integrals within the method of integration by parts is
equivalent to the corresponding problem for (L+R)-loop vacuum or (L+R-1)-loop
propagator-type integrals. Using this property we solve recurrence relations
for two-loop massless vertex diagrams, with arbitrary numerators and integer
powers of propagators in the case when two legs are on the light cone, by
reducing the problem to the well-known solution of the corresponding recurrence
relations for massless three-loop propagator diagrams with specific boundary
conditions.Comment: 8 pp., LaTeX with axodraw.st
Two-Loop Gluon-Condensate Contributions To Heavy-Quark Current Correlators: Exact Results And Approximations
The coefficient functions of the gluon condensate , in the correlators
of heavy-quark vector, axial, scalar and pseudoscalar currents, are obtained
analytically, to two loops, for all values of . In the limiting
cases , , and , comparisons are made with previous
partial results. Approximation methods, based on these limiting cases, are
critically assessed, with a view to three-loop work. High accuracy is achieved
using a few moments as input. A {\em single} moment, combined with only the
{\em leading} threshold and asymptotic behaviours, gives the two-loop
corrections to better than 1% in the next 10 moments. A two-loop fit to vector
data yields GeV.Comment: 9 page
Small-threshold behaviour of two-loop self-energy diagrams: some special cases
An algorithm to construct analytic approximations to two-loop diagrams
describing their behaviour at small non-zero thresholds is discussed. For some
special cases (involving two different-scale mass parameters), several terms of
the expansion are obtained.Comment: 7 pages, plain latex; talk given at DESY-Zeuthen Workshop "QCD and
QED in Higher Order", Rheinsberg, April 1996, to appear in Proceeding
Analytical Results for Dimensionally Regularized Massless On-shell Double Boxes with Arbitrary Indices and Numerators
We present an algorithm for the analytical evaluation of dimensionally
regularized massless on-shell double box Feynman diagrams with arbitrary
polynomials in numerators and general integer powers of propagators. Recurrence
relations following from integration by parts are solved explicitly and any
given double box diagram is expressed as a linear combination of two master
double boxes and a family of simpler diagrams. The first master double box
corresponds to all powers of the propagators equal to one and no numerators,
and the second master double box differs from the first one by the second power
of the middle propagator. By use of differential relations, the second master
double box is expressed through the first one up to a similar linear
combination of simpler double boxes so that the analytical evaluation of the
first master double box provides explicit analytical results, in terms of
polylogarithms \Li{a}{-t/s}, up to , and generalized polylogarithms
, with and , dependent on the Mandelstam variables
and , for an arbitrary diagram under consideration.Comment: LaTeX, 16 pages; misprints in ff. (8), (24), (30) corrected; some
explanations adde
Asymptotic expansion of Feynman integrals near threshold
We present general prescriptions for the asymptotic expansion of massive
multi-loop Feynman integrals near threshold. As in the case of previously known
prescriptions for various limits of momenta and masses, the terms of the
threshold expansion are associated with subgraphs of a given graph and are
explicitly written through Taylor expansions of the corresponding integrands in
certain sets of parameters. They are manifestly homogeneous in the threshold
expansion parameter, so that the calculation of the given Feynman integral near
the threshold reduces to the calculation of integrals of a much simpler type.
The general method is illustrated by two-loop two-point and three-point
diagrams. We discuss the use of the threshold expansion for problems of
physical interest, such as the next-to-next-to-leading order heavy quark
production cross sections close to threshold and matching calculations and
power counting in non-relativistic effective theories.Comment: 24 pages, LaTeX, 4 figures included via epsf.st
Threshold expansion of the sunset diagram
By use of the threshold expansion we develop an algorithm for analytical
evaluation, within dimensional regularization, of arbitrary terms in the
expansion of the (two-loop) sunset diagram with general masses m_1, m_2 and m_3
near its threshold, i.e. in any given order in the difference between the
external momentum squared and its threshold value, (m_1+m_2+m_3)^2. In
particular, this algorithm includes an explicit recurrence procedure to
analytically calculate sunset diagrams with arbitrary integer powers of
propagators at the threshold.Comment: 26 pages (23 pages in LaTeX and 3 PS figures), a typo in eq.(23)
corrected; final version to appear in Nucl.Phys.
Analytical Result for Dimensionally Regularized Massive On-Shell Planar Double Box
The dimensionally regularized master planar double box Feynman diagram with
four massive and three massless lines, powers of propagators equal to one, all
four legs on the mass shell, i.e. with p_i^2=m^2, i=1,2,3,4, is analytically
evaluated for general values of m^2 and the Mandelstam variables s and t. An
explicit result is expressed in terms of polylogarithms, up to the third order,
depending on special combinations of m^2,s and t.Comment: 10 pages, LaTeX with axodraw.st
Heavy mass expansion, light-by-light scattering and the anomalous magnetic moment of the muon
Contributions from light-by-light scattering to (g_\mu-2)/2, the anomalous
magnetic moment of the muon, are mediated by the exchange of charged fermions
or scalar bosons. Assuming large masses M for the virtual particles and
employing the technique of large mass expansion, analytical results are
obtained for virtual fermions and scalars in the form of a series in (m_\mu
/M)^2. This series is well convergent even for the case M=m_\mu. For virtual
fermions, the expansion confirms published analytical formulae. For virtual
scalars, the result can be used to evaluate the contribution from charged
pions. In this case our result confirms already available numerical
evaluations, however, it is significantly more precise.Comment: revtex4, eps figure
Finite calculation of divergent selfenergy diagrams
Using dispersive techniques, it is possible to avoid ultraviolet divergences
in the calculation of Feynman diagrams, making subsequent regularization of
divergent diagrams unnecessary. We give a simple introduction to the most
important features of such dispersive techniques in the framework of the
so-called finite causal perturbation theory. The method is also applied to the
'divergent' general massive two-loop sunrise selfenergy diagram, where it leads
directly to an analytic expression for the imaginary part of the diagram in
accordance with the literature, whereas the real part can be obtained by a
single integral dispersion relation. It is pointed out that dispersive methods
have been known for decades and have been applied to several nontrivial Feynman
diagram calculations.Comment: 15 pages, Latex, one figure, added reference