Unraveling duality violations in hadronic tau decays Page: 4 of 20
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(i) The truncation of the perturbation theory series in as. This is sometimes considered to
be the largest uncertainty, and it is usually estimated by applying different resummation
techniques, such as fixed-order perturbation theory (FOPT), contour-improved fixed-
order perturbation theory (CIPT) or renormalon chain resummation (RCPT). This has
been studied, e.g., in Refs. [3, 6, 8, 9, 10, 11].
(ii) The truncation of the Operator Product Expansion (OPE). The OPE is an expansion
in inverse powers of the momentum, each multiplied by a vacuum condensate of in-
creasingly higher dimension, which in any practical application has to be truncated at
a certain order [12, 13]. Furthermore, the divergence of the perturbative series in as
may be linked to the OPE through renormalons [14]. This suggests a strong correlation
between perturbation theory and the OPE. In particular, the gluon condensate depends
on the renormalization scheme and may also depend on the order at which the series in
as is truncated [15]. However, if the same prescription for the perturbative expansion
is used in both vector and axial-vector channels, the gluon condensate should also be
the same in these two channels. The analysis of Refs. [2, 3] currently finds two different
values for the gluon condensate from the vector and the axial-vector channels, given,
e.g. in CIPT, by
as (GG) = (0.4 0.3) x 10-2 GeV4 , (1.6)
7F Vector
a-(GG) = (-1.3 0.4) x 10-2 GeV4 ,
7f Axial
which are incompatible within about 3 standard deviations. This suggests that the
systematic errors may not be fully understood. The situation concerning condensates
of higher order is also unclear. See for example the recent analyses in Refs. [16, 17, 18,
19, 20, 21, 22, 23], and references therein.
(iii) The OPE is valid in the euclidean region, where there are no physical states. In contrast,
on the Minkowski axis the expansion is not guaranteed to work. This lack of convergence
of the OPE on the Minkwoski axis is usually referred to as due to "Duality Violations"
(DVs). To our knowledge, the best way to understand the problem is at N, = oc,
where the spectrum becomes an infinite set of poles. Even in this limit the OPE has
a cut due to the logarithms from the anomalous dimensions that survive in the large-
N, limit, and does not reproduce the set of infinite poles. Nevertheless, the standard
analysis of tau decays involves an integration over a circle in the complex momentum
plane which includes the Minkowski axis as well (see Eq. (1.8) below). Of course, it is
not known how much things might improve in the real-world case, in which N, = 3,
relative to the limit N, -> oc, because at N, = 3 the infinite set of poles becomes a
cut. However, it is unlikely that the OPE will reproduce this cut systematically. Of
course, this problem has been widely recognized, and strategies have been introduced
to ameliorate the problem. One of those strategies employed in both the ALEPH and
OPAL analyses is the use of so-called "pinched weights," in which zeroes in the weight
function suppress the region near the Minkowski axis [24] (see below). While this is
expected to help, it is unknown to what degree, because there is no systematic theory
of DVs [25]. This makes it very difficult to estimate the associated systematic error.
In the method of analysis that has become standard for tau decays DVs are set to zero by
fiat [1]. We consider this situation to be unsatisfactory; if eventually DVs are to be dismissed,2
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Cata, Oscar; Cata, Oscar; Golterman, Maarten & Peris, Santiago. Unraveling duality violations in hadronic tau decays, article, March 3, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc899116/m1/4/: accessed January 22, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.