Sum Rules for Leading and Subleading Form Factors in Heavy Quark Effective Theory using the Non‐forward Amplitude Available to Purchase

Within the OPE, we formulate new sum rules in Heavy Quark Effective Theory in the heavy quark limit and at order 1/m_Q, using the non‐forward amplitude. In the heavy quark limit, these sum rules imply that the elastic Isgur‐Wise function ξ(w) is an alternate series in powers of (w − 1). Moreover, one gets that the n‐th derivative of ξ(w) at w = 1 can be bounded by the (n − 1)‐th one, and the absolute lower bound for the n‐th derivative $(−1)nξ(n)(1) ⩾ (2n+1)!!22n$⁠. Moreover, for the curvature we find $ξ″(1) ⩾ 15[4ρ2 + 3(ρ2)2]$ where ρ² = −ξ′(1). These results are consistent with the dispersive bounds, and they strongly reduce the allowed region of the latter for ξ(w). The method is extended to the subleading quantities in 1/m_Q. Concerning the perturbations of the Current, we derive new simple relations between the functions ξ₃(w) and Λ̄ξ(w) and the sums $∑ n ΔEj(n)τj(n)(1)τj(n)(w) (j = 12, 32)$⁠, that involve leading quantities, Isgur‐Wise functions $τj(n)(w)$ and level spacings $ΔEj(n)$⁠. Our results follow because the non‐forward amplitude depends on three variables (w_i, w_f, w_if) = (v_i ⋅ v′, v_f ⋅ v′, v_i ⋅ v_f), and we consider the zero recoil frontier (w, 1, w) where only a finite number of j^P states contribute $(12+, 32+)$⁠. We also obtain new sum rules involving the elastic subleading form factors χ_i(w) (i = 1, 2, 3) at order 1/m_Q that originate from the ℒ_kin and ℒ_mag perturbations of the Lagrangian. To the sum rules contribute only the same intermediate states $(jP, JP) = (12−, 1−),(32−, 1−)$ that enter in the $1/mQ2$ corrections of the axial form factor h_A1(w) at zero recoil. This allows to obtain a lower bound on $−δ1/m2(A1)$ in terms of the χ_i(w) and the shape of the elastic IW function ξ(w). An important theoretical implication is that $χ1′(1)$⁠, χ₂(1) and $χ3′(1) (χ1(1) = χ3(1) = 0$ from Luke theorem) must vanish when the slope and the curvature attain their lowest values $ρ2 → 34, σ2 → 1516$⁠. These constraints should be taken into account in the exclusive determination of |V_cb|.