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