Sum Rules for Leading and Subleading Form Factors in Heavy Quark Effective Theory using the Non-forward Amplitude

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 ${\text{(−1)}}^n{\xi }^{\mathrm{(n)}}\text{(1)\hspace{0.17em}⩾\hspace{0.17em}}\frac{\text{(2n+1)!!}}{2^{\text{2n}}}$ . Moreover, for the curvature we find $\text{ξ″(1)\hspace{0.17em}⩾\hspace{0.17em}}\frac{1}{5}{\text{[4ρ}}^2{\text{\hspace{0.17em}+\hspace{0.17em}3(ρ}}^2{)}^2]$ 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 ${\displaystyle \sum _n}{\mathrm{\hspace{0.17em}\Delta E}}_j^{\mathrm{(n)}}{\tau }_j^{\mathrm{(n)}}{\text{(1)τ}}_j^{\mathrm{(n)}}\mathrm{(w)\hspace{0.17em}(j\hspace{0.17em}=\hspace{0.17em}}\frac{1}{2}\mathrm{,\hspace{0.17em}}\frac{3}{2})$ , that involve leading quantities, Isgur‐Wise functions ${\tau }_j^{\mathrm{(n)}}\mathrm{(w)}$ and level spacings ${\mathrm{\Delta E}}_j^{\mathrm{(n)}}$ . 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 $({\frac{1}{2}}^{+}\mathrm{,\hspace{0.17em}}{\frac{3}{2}}^{+})$ . 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 ${\mathrm{(j}}^P{\mathrm{,\hspace{0.17em}J}}^P\mathrm{)\hspace{0.17em}=\hspace{0.17em}(}{\frac{1}{2}}^{-}{\text{,\hspace{0.17em}1}}^{-}\mathrm{),(}{\frac{3}{2}}^{-}{\text{,\hspace{0.17em}1}}^{-})$ that enter in the ${\text{1/m}}_Q^2$ corrections of the axial form factor hA₁(w) at zero recoil. This allows to obtain a lower bound on ${\mathrm{-\delta }}_{{\text{1/m}}^2}^{{\mathrm{(A}}_1)}$ in terms of the χ_i(w) and the shape of the elastic IW function ξ(w). An important theoretical implication is that ${\chi }_1^{\prime }\text{(1)}$ , χ₂(1) and ${\chi }_3^{\prime }{\text{(1)\hspace{0.17em}(χ}}_1{\text{(1)\hspace{0.17em}=\hspace{0.17em}χ}}_3\text{(1)\hspace{0.17em}=\hspace{0.17em}0}$ from Luke theorem) must vanish when the slope and the curvature attain their lowest values ${\rho }^2\mathrm{\hspace{0.17em}\to \hspace{0.17em}}\frac{3}{4}{\mathrm{,\hspace{0.17em}\sigma }}^2\mathrm{\hspace{0.17em}\to \hspace{0.17em}}\frac{15}{16}$ . These constraints should be taken into account in the exclusive determination of...