Preliminary numerical result

We conclude this chapter with the presentation of a numerical result for the real part of the NNLO vertex corrections. We stress again that this corresponds to a preliminary result which treats the c-quark and the b-quark in the closed fermion loops as massless quarks. If we reconsider our results for the imaginary part in this approximation, we find deviations of ∼ 5% of the individual NNLO contributions. As the NNLO terms are subleading for the real part of the colour-allowed tree amplitude, we expect that this approximation will have only a minor impact here. Our preliminary result for the real part of the NNLO vertex corrections reads

Reα1(ππ) = 1.01 ¯ ¯ V(0) + 0.03 ¯ ¯ V(1)+ 0.03 ¯ ¯ V(2) = 1.06, (4.15)

where we have used Wilson coefficients in NLL approximation for simplicity (they are indeed known to the required NNLL accuracy and can be found in [107]). As expected, the contribution is of minor importance in absolute terms. However, the NNLO corrections are found to be as important as the NLO terms which are numerically suppressed by the small Wilson coefficient. Interestingly, the vertex corrections add again constructively and, as can be seen in comparison with the results from [25], come again with the opposite of the spectator interactions. The colour-suppressed tree amplitude α2(ππ) is phenomenologically more interest-

ing as the respective QCD Factorization prediction is rather low for a satisfactory description of the experimental data. In order to derive the NNLO result for α2

we still have to solve some conceptual aspects concerning a Fierz-symmetric defin- ition of (2-loop) evanescent operators. We therefore relegate the discussion of the colour-suppressed tree amplitude to [106].

Chapter 5

Heavy-to-light form factors for

non-relativistic bound states

In this chapter we investigate transition form factors between non-relativistic QCD bound states at large recoil energy. Assuming the decaying quark to be much heavier than its decay product, the relativistic dynamics can be treated according to the factorization formula (1.28) for heavy-to-light form factors obtained from the HQE in QCD. In contrast to the B → π transition, the form factors can be calculated entirely in perturbation theory in the non-relativistic approximation which allows for an explicit analysis of the factorization formula. We perform a NLO calculation based on the methods that we developed in Chapter 2 and look for an interpretation of the results from the viewpoint of QCD Factorization.

The basic idea of this work has already been presented in [108]. We emphasize that the formalism which we develop in this chapter can be applied for Bc →(¯cc) tran- sitions as all these particles can be considered approximatively as non-relativistic bound states. Notice that we treat the charm quark as a light quark in this case,

mc ¿ mb. We will adopt this terminology throughout this chapter although we consider this analysis rather as a toy model for the B → π transition. A phenom- enological study of variousBc decays in QCD Factorization will be given in [109].

5.1

Non-relativistic approximation

The wave function for a non-relativistic (NR) bound state of a quark and an anti- quark with respective masses m1 and m2 can be obtained from the resummation

ψC = ∞ X n=0 1 . . . n w

Figure 5.1: Resummation of potential gluons into a non-relativistic Coulomb wave-function (details of the resummation can be found e.g. in [110]).

of NR (potential) gluon exchange as sketched in Figure 5.1. The solution of the corresponding Schr¨odinger equation with Coulomb potential yields

ψC(~p)∝

κ5/2

(κ2_+~p2₎2, (5.1)

where κ =mrαsCF and mr =m1m2/(m1 +m2) is the reduced mass. The normal-

ization of the wave function gives the (non-relativistic) meson decay constant

fNR = 2√Nc π κ3/2 (m1+m2)1/2 . (5.2)

In this approximation, the Bc meson is entirely dominated by the two-particle Fock state built from a bottom quark with mass M ≡ mb and a charm antiquark with mass m ≡ mc. Consequently to first approximation in the NR expansion, the Bc meson consists of a quark with momentumMwµ and an antiquark with momentum

mwµ, where wµ is the four-velocity of theBc meson (w2 = 1). The spinor degrees of freedom for the Bc meson in the initial state are represented by the Dirac projector

PH = 1₂(1 +w/)γ5.

Similarly, a pseudoscalar ηc meson is interpreted as a c¯c bound state where both constituents have approximately equal momentamw0

µ, wherewµ0 is the four-velocity of the ηc meson (w02 = 1). The Dirac projector of the ηc meson in the final state is given by PL= 1₂(1−w/0)γ5.

In the following we consider heavy-to-light transitions at large recoil energy assuming

M Àm and working in leading power of the HQE inm/M. The QCD dynamics is then described by the SCET degrees of freedom from Table 1.1, where we identify the typical hadronic scale with the mass of the ηc meson Mη ' 2m = O(m) in the NR approach. The relevant scales in the process are thus given by

µh ∼M À µhc∼

√

Mm À µs,c∼m À µNR ∼mv, (5.3)

where the NR scale refers to the virtuality of the potential gluons from Figure 5.1 andv ¿1 is a NR velocity. Notice that all relativistic scales are perturbative in our set-up as M, mÀΛQCD. The relativistic dynamics in the heavy-to-light transition

can therefore be analyzed in perturbation theory. The momentum transfer can be approximated as

q2 _{= (M}

Bw−Mηw0)2 'M2 −4Mm w·w0 (5.4) which implies a large relativistic boost

γ ≡w·w0 ₌ M2 −q2

4Mm =O(M/m) (5.5)