807 research outputs found
Analytic treatment of the two loop equal mass sunrise graph
The two loop equal mass sunrise graph is considered in the continuous
d-dimensional regularisation for arbitrary values of the momentum transfer.
After recalling the equivalence of the expansions at d=2 and d=4, the second
order differential equation for the scalar Master Integral is expanded in (d-2)
and solved by the variation of the constants method of Euler up to first order
in (d-2) included. That requires the knowledge of the two independent solutions
of the associated homogeneous equation, which are found to be related to the
complete elliptic integrals of the first kind of suitable arguments. The
behaviour and expansions of all the solutions at all the singular points of the
equation are exhaustively discussed and written down explicitly.Comment: 33 pages, LaTeX, v2: +1 figure; v3: changes in the conclusions;
simplifications in the recurrences (6.3) and (6.9
Analytic evaluation of Feynman graph integrals
We review the main steps of the differential equation approach to the
analytic evaluation of Feynman graphs, showing at the same time its application
to the 3-loop sunrise graph in a particular kinematical configuration.Comment: 5 pages, 1 figure, uses npb.sty. Presented at RADCOR 2002 and Loops
and Legs in Quantum Field Theory, 8-13 September 2002, Kloster Banz, Germany.
Revised version: minor typos corrected, one reference adde
The analytic value of a 3-loop sunrise graph in a particular kinematical configuration
We consider the scalar integral associated to the 3-loop sunrise graph with a
massless line, two massive lines of equal mass , a fourth line of mass equal
to , and the external invariant timelike and equal to the square of the
fourth mass. We write the differential equation in satisfied by the
integral, expand it in the continuous dimension around and solve the
system of the resulting chained differential equations in closed analytic form,
expressing the solutions in terms of Harmonic Polylogarithms. As a byproduct,
we give the limiting values of the coefficients of the expansion at
and .Comment: 9 pages, 3 figure
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
and the external invariant equal to the square of the third mass in the
usual -continuous dimensional regularization. We write a second order
differential equation for the amplitude in and show as solve it in
close analytic form. As a result, all the coefficients of the Laurent expansion
in of the amplitude are expressed in terms of harmonic polylogarithms
of argument and increasing weight. As a by product, we give the explicit
analytic expressions of the value of the amplitude at , corresponding to
the on-mass-shell sunrise amplitude in the equal mass case, up to the
term included.Comment: 11 pages, 2 figures. Added Eq. (5.20) and reference [4
The analytical value of the electron (g-2) at order alpha^3 in QED
We have evaluated in closed analytical form the contribution of the
three-loop non-planar `triple-cross' diagrams contributing to the electron
(g-2) in QED; its value, omitting the already known infrared divergent part, is
a_e(3-cross) = 1/2 pi^2 Z(3) - 55/12 Z(5) - 16/135 pi^4
+ 32/3 (a4 + 1/24 ln(2)^4) + 14/9 pi^2 ln(2)^2
- 1/3 Z(3) + 23/3 pi^2 ln(2) - 47/9 pi^2 - 113/48.
This completes the analytical evaluation of the (g-2) at order alpha^3,
giving
a_e(3-loop) = (alpha/pi)^3 { 83/72 pi^2 Z(3) - 215/24 Z(5)
+ 100/3 [( a4 + 1/24 ln(2)^4 ) - 1/24 pi^2 ln(2)^2 ]
- 239/2160 pi^4 + 139/18 Z(3) - 298/9 pi^2 ln(2)
+ 17101/810 pi^2 + 28259/5184 } = (alpha/pi)^3 (1.181241456...).Comment: plain TeX, 16 pages, 2 figures. Submitted to Physics Letters B. Two
postscript files (z2.ps, zx1gen.ps) are include
Precise numerical evaluation of the two loop sunrise graph Master Integrals in the equal mass case
We present a double precision routine in Fortran for the precise and fast
numerical evaluation of the two Master Integrals (MIs) of the equal mass
two-loop sunrise graph for arbitrary momentum transfer in d=2 and d=4
dimensions. The routine implements the accelerated power series expansions
obtained by solving the corresponding differential equations for the MIs at
their singular points. With a maximum of 22 terms for the worst case expansion
a relative precision of better than a part in 10^{15} is achieved for arbitrary
real values of the momentum transfer.Comment: 11 pages, LaTeX. The complete paper is also available via the www at
http://www-ttp.physik.uni-karlsruhe.de/Preprints/ and the program can be
downloaded from http://www-ttp.physik.uni-karlsruhe.de/Progdata
Numerical evaluation of the general massive 2-loop sunrise self-mass master integrals from differential equations
The system of 4 differential equations in the external invariant satisfied by
the 4 master integrals of the general massive 2-loop sunrise self-mass diagram
is solved by the Runge-Kutta method in the complex plane. The method, whose
features are discussed in details, offers a reliable and robust approach to the
direct and precise numerical evaluation of Feynman graph integrals.Comment: 1+21 pages, Latex, 5 ps-figure
Two-Loop Heavy-Flavor Contribution to Bhabha Scattering
We evaluate the two-loop QED corrections to the Bhabha scattering cross
section which involve the vacuum polarization by heavy fermions of arbitrary
mass m_f >> m_e. The results are valid for generic values of the Mandelstam
invariants s,t,u >> m_e^2.Comment: 13 pages, 6 figures. Equations in the appendix generalized to the
heavy-quark cas
Electroweak Fermion-loop Contributions to the Muon Anomalous Magnetic Moment
The two-loop electroweak corrections to the anomalous magnetic moment of the
muon, generated by fermionic loops, are calculated. An interesting role of the
top quark in the anomaly cancellation is observed. New corrections, including
terms of order , are computed and a class of diagrams
previously thought to vanish are found to be important. The total fermionic
correction is which decreases the electroweak
effects on , predicted from one-loop calculations, by 12\%. We give an
updated theoretical prediction for of the muon.Comment: Corrected versio
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