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

After a term by term Borel resummation, one finds (for
the special case n=5)
Z
0
1
2
35
384
25025
c 5 i
c 5 i
1
e
2
98304
ds
3
2
4
(
(2)
(
8
(1)
3
2
(5)
(
)
35
384
(
1,
(0,
s
2
3
2
4,
3
2
)
385
3072
))
3
2
)
3
385
3072
1
2
1
8
25025
98304
(4)
(2)
( s 1/ 4) ( s 3 / 4) ( s)
(1/ 4) (3 / 4)
s)
The tail of the divergent perturbative series has been
rewritten as a partial sum multiplied by an exponential,
supplemented by a remainder integral written as a Mellin-Barnes
representation. We therefore converted an
infinite sum into a finite sum plus a convergent integral.
We see that the coefficients of this finite sum are the
same as the perturbative ones (up to Euler gamma functions
that we wrote explicitly to emphasize the symmetry
of the formula). This is a resurgence phenomenon.
In our paper, we proved from a hyperasymptotic analysis
of the integral defining Z_0 that the above formal
expression (written here for n=5) is exact for any n and
m as long as n ≥ m, and constitutes what is called the
first level of the hyperasymptotic expansion of Z_0.
Better than that, we also proved that the reinterpreted
tail is, at an optimal value of n, exponentially suppressed
with respect to λ, so that it gives in fact, for this
optimal value of n, the expression of a purely nonperturbative
quantity.
Higher order hyperasymptotic improvement
Now that we gave a meaning to the tail of the divergent
perturbative series, we may recursively use the resurgent
inverse factorial expansion to perform more refined
non-perturbative improvement. This iterative procedure
leads to hyperasymptotic expansions of higher order [2].
To have an idea of the numerical improvement that a
hyperasymptotic expression of Z_0 allows to reach, we
choose λ=1/3.In this case, one finds, from a numerical
integration of the first equation of this note
3
e
3
2
4
1
2i
(
(
(5
1,
3,
3
2
3
2
)
)
(
4
s,
3
2
)
optimal truncation schemes of these partial sums which
have not been developed in this short note. A numerical
analysis has been performed, showing the much more
accurate results that one may obtain with the nonperturbative
hyperasymptotic expansions, compared to
the traditional perturbative approach. Our results are
valid for a wide range of the phase of the complex perturbative
parameter λ, in particular on Stokes lines defined
by arg(λ)=±π, where one may find, from hyperasymptotic
expansions, imaginary contributions that are
of course not obtainable from the truncated perturbative
expansions.
One of the important conclusions concerns the two different
optimal truncation schemes that were studied in
this work but not reported here. Indeed, the best one (i.e.
the one that leads to the best analytical expressions)
implies a truncation of the perturbative series that depends
on the hyperasymptotic level at which the improvement
is performed. This leads to the striking result
that the higher the hyperasymptotic level is reached in
the analysis, the more one has to take into account terms
in the perturbative series. In other words, the more one
wants to increase the exponential improvement, the more
one also has to include perturbative contributions far
in the divergent tail of the perturbative series. This then
implies that the corresponding numerical predictions are
of an amazing accuracy although they involve perturbative
contributions that, taken independently of the nonperturbative
corrections, would lead to desastrous results.
Care should however be taken concerning numerical
instability issues at high hyperasymptotic level.
These results clearly show that hyperasymptotic expansions
are tools that could have a lot of relevance in high
energy physics if, for instance, they may be used in the
theoretical context of precision test of the Standard Model
of particle physics.
References
[1] S. Friot and D. Greynat, ``Non-perturbative asymptotic
improvement of perturbation theory and Mellin-
Barnes representation'', arXiv:0907.5593[hep-th]
[2]R. B. Paris and D. Kaminski, ``Asymptotics and Mellin-Barnes
integrals'', Encyclopedia of Mathematics and
its applications (2001), Cambridge University Press.
Z 0
0.965560481
whereas the purely perturbative series leads to
Z 0
0.96555187
In comparison, the hyperasymptotic expansion at third
level gives
Z 0
0.965560480
Conclusion
We obtained hyperasymptotic expansions directly from
the divergent perturbative expansions of Φ^4 theory.
They are composed of interwoven partial sums whose
coefficients, in our cases of study, are linked together by
a simple resurgence phenomenon. The non-perturbative
interpretation of our results relies crucially on so-called
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