Remarks on Form Factor Bounds

Improved model independent upper bounds on the weak transition form factors are derived using inclusive sum rules. Comparison of the new bounds with the old ones is made for the form factors hA1 and hV in B → D decays. [email protected] 1 A set of model independent bounds has been derived to provide a restriction on the shape of weak transition form factors [1–3]. They have been extensively used to bound weak decay form factors and the decay spectrum of heavy hadrons [2–5]. Here We provide a more stringent upper bound without any further assumptions. This upper bound differs from the one derived previously at order 1/m2Q or αs/mQ. Though this is only a small improvement, it is worth doing because it can give a tighter bound from above if one includes higher order corrections. The bounds are derived from sum rules that relate the inclusive decay rate, calculated using the operator product expansion (OPE) [8,9] and perturbative QCD, to the sum of exclusive decay rates. To be complete, we will derive both the upper and lower bounds, though the lower bound is the same as the previous one. Without loss of generality, we take for example the decay of a B meson into an H meson, with the underlying quark process b → f , where f could be either a heavy or light quark. First, consider the time ordered product of two weak transition currents taken between two B mesons in momentum space, T μν = − i 2MB ∫ dx e 〈B(v)|T (J(x)J(0)) |B(v)〉 = −gT1 + vvT2 + iǫqαvβT3 + qqT4 + (qv + vq)T5, (1) where J is a b → f weak transition current. The time ordered product can be expressed as a sum over hadronic or partonic intermediate states. The sum over hadronic states includes the matrix element 〈H|J |B〉. After inserting a complete set of states and contracting with a four-vector pair aμaν , we obtain: T (ǫ) = 1 2MB ∑ X (2π)δ(~ pX + ~q) |〈X| a · J |B〉| EX − EH − ǫ + 1 2MB ∑ X (2π)δ(~ pX − ~q) |〈B| a · J |X〉| ǫ+ EX + EH − 2MB , (2) See, however, [6,7]for model independent parametrizations of the form factors 2 where T (ǫ) ≡ aμT aν , ǫ = MB − EH − v · q, and the sum over X includes the usual ∫ dp/2EX for each particle in the state X. We choose to work in the rest frame of the B meson, p = MBv, with the z axis pointing in the direction of ~q. We hold q3 fixed while analytically continuing v · q to the complex plane. EH = √ M H + q 2 3 is the H meson energy. There are two cuts in the complex ǫ plane, 0 < ǫ < ∞, corresponding to the decay process b → f , and −∞ < ǫ < −2EH , corresponding to two b quarks and a f̄ quark in the final state. The second cut will not be important for our discussion. The integral over ǫ of the time ordered product, T (ǫ), times a weight function, ǫW∆(ǫ), can be computed perturbatively in QCD [2,3]. For simplicity, we pick the weight function W∆(ǫ) = θ(∆ − ǫ), which corresponds to summing over all hadronic resonances up to the excitation energy ∆ with equal weight. Relating the integral with the hard cutoff to the exclusive states requires local duality at the scale ∆. Therefore, ∆ must be chosen large enough so that the structure functions can be calculated perturbatively. Taking the zeroth moment of T (ǫ), we get M0 ≡ 1 2πi ∫ C dǫ θ(∆− ǫ)T (ǫ) = |〈X|a · J |B〉| 4MBEH + ∑ X 6=H ′ θ(EX − EH −∆)(2π)δ(~q + ~pX) |〈X|a · J |B〉| 2MB , where the primed summation means a sum over all the kinematically allowed states except the H meson. So, |〈X|a · J |B〉| 4MBEHǫ = M0 − ∑ X 6=H ′ θ(EX − EH −∆)(2π)δ(~q + ~pX) |〈X|a · J |B〉| 2MB . (3) On the other hand, the first moment of T (ǫ) gives M1 ≡ 1 2πi ∫ C dǫ ǫ θ(∆− ǫ)T (ǫ) = ∑ X 6=H ′ θ(∆−EX + EH) (EX −EH) (2π)δ(~q + ~pX) |〈X| a · J |B〉| 4MBEX