exotic nuclei structure and reaction noyaux exotiques ... - IPN - IN2P3

Non-perturbative asymptotic improvement of perturbation theory
and Mellin-Barnes representation
IPNO Participation: S. Friot
Collaboration : D. Greynat (IFAE)
Dans ce travail on s’intéresse à un procédé formel permettant l’obtention d’information non-perturbative à
partir de séries perturbatives divergentes. On considère le cas simple de la théorie scalaire Φ^4 en
dimension zéro qui permet un contrôle totalement rigoureux des résultats. Le procédé formel utilisé, qui
repose sur des développements factoriels inverses et sur la représentation de Mellin-Barnes, mène à une
amélioration non-perturbative des séries perturbatives des fonctions à N points de la théorie sous forme de
séries hyperasymptotiques présentant un phénomène de résurgence. Dans ce cas simple on montre que
ces résultats formels peuvent en fait être obtenus rigoureusement à l’aide de la théorie hyperasymptotique
de Mellin-Barnes, ce qui prouve la validité du procédé formel. Ce cadre permet en outre de démontrer le
caractère non-perturbatif de nos résultats. Une étude numérique montre les gains de précision des séries
hyperasymptotiques par rapport aux séries perturbatives.
Introduction
The divergent behavior of a (divergent) asymptotic expansion
does not at all detract from its computational
utility. This statement is corroborated by the fact that, in
what concerns its first few partial sums, a divergent
asymptotic expansion of a given quantity ''converges'' in
general much faster to the exact result than what a
convergent series representation of the same quantity
does. In the case of the Standard Model quantum field
theories, we may therefore say that, regarding the extreme
difficulty to go beyond the first few perturbative orders
when computing observables in QCD or in the electroweak
theory, it is an advantage, for phenomenology,
to deal with a formalism that leads to presumably
asymptotic power series, diverging for all values of the
coupling constants, rather than convergent ones.
However, one has of course to keep in mind that when
dealing with divergent asymptotic perturbative power
expansions, there always remains a finite limit of precision
beyond which the usual asymptotic theory cannot
go, even when the objects that one wants to compute
are well-defined (in the case of the 4D Standard Model
of particle physics, due to the absence of a clear definition
of the theory, already the correct evaluation of the
size of this precision limit is lacking, i.e. theoretical errors
implied by truncations of perturbative expansions, OPE,
etc. are not under control, although it is in principle of
crucial importance in precision test of the Standard Model
if one aims to find new physics effects). All ways to
break open this precision limit are welcome. In the beginning
of the 1990's, new asymptotic objects, which
have in general a larger region of validity (a larger domain
of definition in the complex expansion parameter)
and a greater accuracy than conventional asymptotic
expansions, appeared in the mathematical litterature.
With them, a new asymptotic theory emerged: exponential
asymptotics (or hyperasymptotics). These asymptotic
objects (hyperasymptotic expansions) are very interesting
since they correspond to what we could call in physics
a non-perturbative asymptotic improvement of a
perturbative (asymptotic) power series. Our aim in this
work is to show how hyperasymptotic expansions appear
naturally in the simplest example one may have in
mind, namely zero-dimensional Φ^4 field theory.
sults of our paper [1], focusing only on the case of the 0-
dimensional version of the generating functional of vacuum
to vacuum transitions
Z
0
1
2
e
although in the original paper the general N-point function
case has also been considered.
Leading non-perturbative asymptotic improvement
of perturbative results
The perturbative expansion of Z_0 is given by
Z
0
1
k
This is a divergent series for any value of λ.
For a given choice of λ the first few partial sums begin to
converge to the exact value, but after a certain rank, the
divergence of the perturbative series is of course unavoidable.
This is the usual behaviour of an asymptotic expansion.
Dividing the series into 2 parts, one keeps explicitly n
perturbative terms and the aim now is to give a meaning
to the infinite remaining divergent tail.
This is done formally by applying the following inverse
factorial expansion [2], to (a slightly modified expression
of) the perturbative tail:
( k 1/ 4) ( k
( k 1)
3/ 4)
4!
where c belongs to ]0,1[ and c+m