675 research outputs found

### 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

### Using differential equations to compute two-loop box integrals

The calculation of exclusive observables beyond the one-loop level requires
elaborate techniques for the computation of multi-leg two-loop integrals. We
discuss how the large number of different integrals appearing in actual
two-loop calculations can be reduced to a small number of master integrals. An
efficient method to compute these master integrals is to derive and solve
differential equations in the external invariants for them. As an application
of the differential equation method, we compute the ${\cal O}(\epsilon)$-term
of a particular combination of on-shell massless planar double box integrals,
which appears in the tensor reduction of $2 \to 2$ scattering amplitudes at two
loops.Comment: 5 pages, LaTeX, uses espcrc2.sty; presented at Loops and Legs in
Quantum Field Theory, April 2000, Bastei, German

### 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

### Progress on two-loop non-propagator integrals

At variance with fully inclusive quantities, which have been computed already
at the two- or three-loop level, most exclusive observables are still known
only at one loop, as further progress was hampered up to very recently by the
greater computational problems encountered in the study of multi-leg amplitudes
beyond one loop. We discuss the progress made lately in the evaluation of
two-loop multi-leg integrals, with particular emphasis on two-loop four-point
functions.Comment: 9 pages, LaTeX, Invited talk at 5th International Symposium on
Radiative Corrections (RADCOR-2000), Carmel CA, USA, 11--15 September, 200

### Two-Loop Form Factors in QED

We evaluate the on shell form factors of the electron for arbitrary momentum
transfer and finite electron mass, at two loops in QED, by integrating the
corresponding dispersion relations, which involve the imaginary parts known
since a long time. The infrared divergences are parameterized in terms of a
fictitious small photon mass. The result is expressed in terms of Harmonic
Polylogarithms of maximum weight 4. The expansions for small and large momentum
transfer are also givenComment: 13 pages, 1 figur

### 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
$M$ and the external invariant equal to the square of the third mass $m$ in the
usual $d$-continuous dimensional regularization. We write a second order
differential equation for the amplitude in $x=m/M$ and show as solve it in
close analytic form. As a result, all the coefficients of the Laurent expansion
in $(d-4)$ of the amplitude are expressed in terms of harmonic polylogarithms
of argument $x$ and increasing weight. As a by product, we give the explicit
analytic expressions of the value of the amplitude at $x=1$, corresponding to
the on-mass-shell sunrise amplitude in the equal mass case, up to the $(d-4)^5$
term included.Comment: 11 pages, 2 figures. Added Eq. (5.20) and reference [4

### Schouten identities for Feynman graph amplitudes; the Master Integrals for the two-loop massive sunrise graph

A new class of identities for Feynman graph amplitudes, dubbed Schouten
identities, valid at fixed integer value of the dimension d is proposed. The
identities are then used in the case of the two loop sunrise graph with
arbitrary masses for recovering the second order differential equation for the
scalar amplitude in d=2 dimensions, as well as a chained sets of equations for
all the coefficients of the expansions in (d-2). The shift from $d\approx2$ to
$d\approx4$ dimensions is then discussed.Comment: 30 pages, 1 figure, minor typos in the text corrected, results
unchanged. Version accepted for publication on Nuclear Physics

### 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 $M$, a fourth line of mass equal
to $Mx$, and the external invariant timelike and equal to the square of the
fourth mass. We write the differential equation in $x$ satisfied by the
integral, expand it in the continuous dimension $d$ around $d=4$ 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 $(d-4)$ expansion at
$x=1$ and $x=0$.Comment: 9 pages, 3 figure

### 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

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