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Scientific Report 2007-2009
Theoretical physics
T2. B decay spectra in resummed QCD calculations
At present, all experimental data in particle physics
are compatible with the current theory of strong and
electroweak interactions — the so-called standard model
(SM). All the particles predicted by the SM have been
discovered, with the exception of the Higgs boson, which
will probably be discovered in the next decade at the
Large Hadron Collider (LHC) at Cern, Geneve — currently
in the first stages of operation. Apart from the
Higgs search, a good fraction of present-day research
in particle physics involves detailed comparison of experimental
data with theoretical distributions of known
particles with standard interactions. Weak and electromagnetic
interactions of quarks are, in general, largely
affected by the accompanying strong interactions effects,
described by Quantum Chromodynamics (QCD). Even
if someone is not primarily interested in strong interactions,
he has to face them in any case as background
effects, as for example in Higgs physics at LHC.
In order to measure the strenght of the weak coupling
of a beauty quark to an up quark — the CKM matrix element
V ub — one has to compute for example the lepton
energy spectrum in the semileptonic decay B → X u lν l ,
where X u is any hadron state coming from the fragmentation
of the u quark. Let us note that at present there
is no theory for any CKM element, but only some consistency
relations among them coming from unitarity of
the matrix itself, as implied by the SM. Such free parameters
are therefore measured by comparing theoretical
distributions containing them as unknown quantities,
to experimental data. The only analytic tool available
at present to compute QCD effects is perturbation theory:
the corrections are computed with Feynman diagrams
as truncated series in α S , the strong coupling.
The latter decreases logarithmically with the energy of
the process — asymptotic freedom; for beauty decays
α S (m b ) ≃ 0.21. In some regions of phase space of the
above decays, experimentally relevant, many-body effects
related to infrared divergencies become important:
they manifest themselves in an enhancement of the coefficients
of αS n in the perturbative series. This effect
renders the fixed-order expansion completely unreliable
and forces the resummation to all orders in α S of the
enhanced terms.
A general problem of all-order resummation is that an
integration of the QCD coupling constant in the low energy
region is involved, in which α S is large and outside
the perturbative domain — the old problem of the Landau
ghost makes a specific appearence here. As shown in
the eighties, this problem cannot be solved inside perturbation
theory, because of missing dynamical input from
the perturbative phase, and therefore it is necessary to
introduce an arbitrary prescription from outside in order
to have well-definite predictions for resummed spectra.
These conclusions also apply to resummation in heavy
flavor decays, as we explicitly found, but are in disagreement
with those reached in the past few years by other
groups (M. Neubert and coll. for instance). In our modelization
of non-perturbative effects, we also equate the
on-shell mass of the b quark to the mass of the B meson:
m b = m B . That is consistent because the mass of a
heavy quark has a much weaker physical meaning that is
generally believed nowadays: quarks are confined. If one
aims at describing all QCD dynamics at a fundamental
level, he has to abandon perturbative QCD, because the
latter has always to be supplemented with phenomenological
models connecting the fictitious “parton world”
with the real “hadron world”. QCD can be computed
exactly only with numerical Monte-Carlo methods after
regularization on a lattice. In that case, one only deals
with bare, regularization dependent, quark masses and
with observable hadron masses. A renormalized beauty
mass can certainly be defined but it is just conventional
and bears to relation to dynamics. A consequence is that
one has the freedom to fix m b in a rather arbitrary way
and a simple way to implement hadron kinematics in a
parton computation is just to identify the unphysical b
mass with the physical B mass. Also these assumptions
are in disagreement with a large part of the B physics
community (M. Neubert and E. Gardi). Many colleagues
object that the Operator Product Expansion (OPE) involves
m b and not m B and that identifying them would
introduce non-vanishing 1/m b corrections. Our reply is
that implementing the OPE from first principles, i.e. on
a lattice, would give similar problems in the definition of
the quark mass as those discussed above, augmented by
the specific ultraviolet power divergencies brought in by
the small-momentum expansion.
We have used the accurate experimental data from
B fragmentation to fix the prescription for the QCD
coupling constant at low energy [1], from which we
have derived our resummed B decay spectra [2]. By
comparing the latter with experimental data, we have
found some disagreement with electron spectra from
the B-factories in the low-energy region, which we interpret
as originating from an undersubtracted b → clν l
background [2]. We have obtained for V ub the value
(3.76 ± 0.13 ± 0.22)10 −3 [3], smaller than the values
obtained by the other groups and in good agreement
with the determinations coming from lattice QCD
(exclusive channels) or from global fits to the SM of the
UTfit collaboration.
References
1. U. Aglietti et al., Nucl. Phys. B775, 162 (2007).
2. U. Aglietti et al, Nucl. Phys. B768, 85 (2007).
3. U. Aglietti et al., Eur. Phys. J. C59, 831 (2009).
4. U. Aglietti et al., Phys .Lett. B653, 38 (2007).
Authors
U. Aglietti, G. Corcella, G. Ferrera, A. Renzaglia.
Sapienza Università di Roma 25 Dipartimento di Fisica