Inclusive spectra in charmless semileptonic B decays by Dressed Gluon Exponentiation

TL;DR

This work implements Dressed Gluon Exponentiation to predict the fully differential width for inclusive charmless semileptonic B decays, addressing the large Sudakov logarithms in the small hadronic-mass region via a renormalon-aware, principal-value Borel resummation. By matching to NNLO fixed-order results and converting the on-shell b-quark spectrum to hadronic variables, the authors minimize reliance on leading nonperturbative inputs and extract |V_ub| from Belle measurements with a quantified uncertainty budget. Key contributions include a detailed resummation in moment space, a careful treatment of renormalons and pole-mass ambiguities, and a practical evaluation of event fractions under experimentally relevant cuts (M_X and P^+). The results demonstrate that perturbative control with DGE yields reliable partial widths and an extraction of |V_ub| with competitive theoretical uncertainties, highlighting the potential of NNLO+resummed predictions for precision flavor physics.

Abstract

The triple differential spectrum in \bar{B} -> X_u l \barν is computed by Dressed Gluon Exponentiation (DGE). In this framework the on-shell calculation, converted into hadronic variables, can be directly used as an approximation to the meson decay spectrum, without involving a leading-power non-perturbative function. Sudakov resummation for the fully differential \bar{B} -> X_u l \barν width is formulated in moment space, where moments are defined using the ratio between the lightcone momentum components of the partonic jet p^+/p^- and the hard scale is p^-. In these variables the correspondence with the \bar{B} -> X_s γcase is transparent. The Sudakov exponent is known to next-to-next-to-leading logarithmic accuracy. Further constraints are put on its Borel sum using the cancellation of the leading renormalon ambiguity and the absence of the next-to-leading one, which was proven in the large-beta_0 limit and assumed here to be general. Based on the resummed spectrum, matched to the fully differential NLO result, we calculate the event fraction associated with experimental cuts on the hadronic mass (or the small lightcone component) as well as on the lepton energy. Finally, we extract |V_ub| from recent measurements by Belle and analyze the theoretical uncertainty.

Figures (13)

• Figure 1: The differential spectrum based on the fixed--logarithmic--accuracy formula of Eq. (\ref{['NNLL_x_space']}), which is matched to NLO. The LL, NLL and NNLL accuracy results are plotted as dotdashed, dashed and full line, respectively. The three curves end at $r\simeq 0.028$, where the resummed results become complex owing to the Landau singularity at $\omega=1/2$.
• Figure 2: The function $B_{\cal S}(u)$ with vertical lines indicating the positions of possible renormalons owing to the poles of $\Gamma(-2u)$ in Eq. (\ref{['Gamma_n_large_n']}). The thin line shows the large--$\beta_0$ limit of Eq. (\ref{['B_DJ_large_beta0']}) while the thick lines show $B_{\cal S}(u)$ according to Eq. (\ref{['B_DJ']}) with $N_f=4$ and with different assignments of the parameter $C$ that controls the residue of the Borel singularity in the Sudakov exponent at $u=\frac{3}{2}$: $C=0.01$ is shown as a dotdashed line while the default value $C=1$ and $C=2,4,6,8,10,12$ are shown as full lines.
• Figure 3: The fully differential width (in units of $\Gamma_0$) computed based on Eq. (\ref{['Gamma_n_lnR_matched_w_constant']}), plotted as a function of the small lightcone component $P^+$ at $P^-=3.5$ GeV and at three different values of $E_l=1,\,1.5$ and $2$ GeV. The maximal $P^+$ endpoint ($M_B-2E_l$), denoted by a vertical line, should be: $3.3$, $2.3$ and $1.3$ GeV, respectively. The $u=3/2$ renormalon residue parameter in Eq. (\ref{['B_DJ']}) (see Fig. \ref{['fig:C_dependence']}) is $C=1$. Note that the perturbative support properties, namely $\bar{\Lambda} <P^+<M_B-2E_l$ are violated at both ends. At the low $P^+$ end this is of course a desired consequence of the resummation: the physical support properties $P^+>0$ are approximately recovered. At the high $P^+$ end this is an ${\cal O}(\alpha_s^2/n)$ artifact, which is non-negligible only for large $E_l$.
• Figure 4: The full phase space in the ($P^-$, $E_l$) plain (upper left triangle) after the $P^+$ integration has been performed. The dotted horizontal line $E_l=m_b/2$ starting at $P_H^-=\bar{\Lambda}$ is the upper boundary of the partonic phase space. A lower cut on the charged lepton energy (here $E_l>E_0=1 \,{\rm GeV}$) removes the small triangular region below the horizontal dashed line. An upper cut on the hadronic invariant mass (here $P^+P^-<M_X^2=(1.7 \,{\rm GeV})^2$) affects the distribution for $E_l<\frac{1}{2}(M_B-M_X^2/P^-)$. A full line separates between this region and the rest of the phase space where this cut is irrelevant. On this line we denoted three special points by $D$$(P^-=M_X)$, $C$ ($P^-=M_B-2E_0$) and $F$ ($P^-=M_B$), respectively.
• Figure 5: The event fraction $R_{\rm cut}$ within the range $P^+P^-<M_X^2$ and $E_l>1$ GeV, plotted as a function of the cut value $M_X$. Three different matching schemes are used: (a) the partially integrated distribution with respect to $E_l$, namely Eq. (\ref{['int_M_X_cut']}) and Eq. (\ref{['double_diff_matched']}), numerically integrated over $P^-$ (full line); (b) the fully differential distribution computed by Eq. (\ref{['diff_M_X_cut']}) and the matching of Eq. (\ref{['Gamma_n_lnR_matched_w_constant']}), numerically integrated over the $E_l$ and $P^-$ (dashed line); and (c) same as in (b) but with the matching of Eq. (\ref{['Gamma_n_lnR_matched']}) (dotdashed line).