Letter of Intent for KEK Super B Factory Part I: Physics

y QCD factorization [49–51]. In this formalism, the decay matrix elements of B → ππ can be
factorized in the form
〈π(p ′ )π(q)|Qi|B(p)〉 = f B→π (q 2 1
) du T
0
I
i (u)Φπ(u)
1
+ dξdudv T
0
II
i (ξ, u, v)ΦB(ξ)Φπ(u)Φπ(v), (2.60)
for a four-quark operator Qi. The first term represents a factorization of the amplitude into
the B → π form factor and an out-going pion wave function Φπ(u) convoluted with the hard
scattering kernel T I
i (u). Here, the term “factorization” is used for two meanings: one is the
factorization of the diagram to 〈π|V |B〉〈π|A|0〉, while the other is the separation of hard, collinear
and soft interactions. The kernel T I
i (u) describes the hard interaction only and thus is calculable
using perturbation theory. The second term in (2.60) describes a factorization of the amplitude
into three pieces: B → 0, 0 → π, and 0 → π convoluted with a hard interaction kernel
(ξ, u, v). To make the factorization of collinear and soft degrees of freedom more explicit,
T II
i
an effective theory has also been developed, which is called the Soft Collinear Effective Theory
(SCET) [52–55].
The form factor f B→π (q2 ) and the light-cone distribution function (or wave function) ΦB(ξ)
and Φπ(u) contain long-distance dynamics, which has to be treated with non-perturbative methods.
Another method of factorization, Perturbative QCD (PQCD) [56–61], has also been proposed
and is being used for the analysis of various B decay modes. It relies on the Sudakov suppression
of the tail of the wave function, in which the first term (2.60) becomes subleading and the input
of the form factor is unnecessary.
2.5.4 Lattice QCD
Lattice QCD provides a method to calculate non-perturbative hadronic matrix elements from
the first principles of QCD [62]. It is a regularization of QCD on a four-dimensional hypercubic
lattice, which enables numerical simulation on the computer. Since the calculation is numerically
so demanding, one has to introduce several approximations in the calculation, which leads to
systematic uncertainties.
For more than a decade, lattice QCD has been applied to the calculation of matrix elements
relevant to B physics. The best-known quantity is the B meson leptonic decay constant fB,
for which the systematic uncertainty is now under control at the level of 10–15% accuracy. The
important tools to achieve this goal are the following.
• Effective theories for heavy quarks. Since the Compton wave-length of the b quark is
shorter than the lattice spacing a, the discretization error is out of control with the usual
lattice fermion action for relativistic particles. Instead, Heavy Quark Effective Theory
(HQET) [29] or Non-relativistic QCD (NRQCD) [63–65] is formulated on the lattice and
used to simulate the b quark. Another related effective theory is the so-called Fermilab
action [66, 67], which covers the entire (light to heavy) mass regime with the same lattice
action. For the B meson the next-to-leading (1/mQ) order calculation provides < ∼ 5%
accuracy [68].
• Effective theories to describe discretization errors. The discretization effect can be expressed
in terms of the Lagrangian language, i.e. Symanzik effective theory [69, 70]. It
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