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On The Expected Photon Spectrum in B -> X_s + gamma and Its Uses

Ikaros Bigi, Nikolai Uraltsev

TL;DR

The paper investigates how the photon energy spectrum in B → X_s γ encodes information about heavy-quark motion and can be used to control theoretical uncertainties in semileptonic B decays, especially for extracting |V_ub| from B → X_u ℓν. It introduces universality relations that relate hadronic-mass distributions in B → X_u ℓν to the photon spectrum in B → X_s γ, and proposes a practical framework to include leading 1/m_b corrections via a scaled μ^2 parameter, thereby improving the reliability of using kinematic cuts to measure |V_ub|. A key result is that the low-E_γ tail (E_γ < 2 GeV) is predicted to contain about 12% of events, making this region experimentally challenging but theoretically informative for testing duality and calibrating heavy-quark parameters. The authors also caution about potential sizable corrections when relating end-point lepton spectra to photon spectra, emphasizing the need for higher-order and nonperturbative considerations in precise |V_ub| determinations.

Abstract

Measuring the photon energy spectrum in radiative B decays provides essential help for gaining theoretical control over semileptonic B transitions. The hadronic recoil mass distribution in B -> X_u \ellνpromises the best environment for determining |V_ub|. The theoretical uncertainties are largest in the domain of low values of the lepton pair mass q^2. Universality relations allow to describe this domain reliably in terms of the photon spectrum in B -> X_s + γ. A method is proposed to incorporate 1/m_b corrections into this relation. The low-E_γtail in radiative decays is important in the context of extracting |V_ub|. We argue that CLEO's recent fit to the spectrum underestimates the fraction of the photon spectrum below 2 GeV. Potentially significant uncertainties enter in the theoretical evaluation of the integrated end-point lepton spectrum or the B -> X_u \ellνwidth with a too high value of the lower cut on q^2 in alternative approaches to |V_ub|.

On The Expected Photon Spectrum in B -> X_s + gamma and Its Uses

TL;DR

The paper investigates how the photon energy spectrum in B → X_s γ encodes information about heavy-quark motion and can be used to control theoretical uncertainties in semileptonic B decays, especially for extracting |V_ub| from B → X_u ℓν. It introduces universality relations that relate hadronic-mass distributions in B → X_u ℓν to the photon spectrum in B → X_s γ, and proposes a practical framework to include leading 1/m_b corrections via a scaled μ^2 parameter, thereby improving the reliability of using kinematic cuts to measure |V_ub|. A key result is that the low-E_γ tail (E_γ < 2 GeV) is predicted to contain about 12% of events, making this region experimentally challenging but theoretically informative for testing duality and calibrating heavy-quark parameters. The authors also caution about potential sizable corrections when relating end-point lepton spectra to photon spectra, emphasizing the need for higher-order and nonperturbative considerations in precise |V_ub| determinations.

Abstract

Measuring the photon energy spectrum in radiative B decays provides essential help for gaining theoretical control over semileptonic B transitions. The hadronic recoil mass distribution in B -> X_u \ellνpromises the best environment for determining |V_ub|. The theoretical uncertainties are largest in the domain of low values of the lepton pair mass q^2. Universality relations allow to describe this domain reliably in terms of the photon spectrum in B -> X_s + γ. A method is proposed to incorporate 1/m_b corrections into this relation. The low-E_γtail in radiative decays is important in the context of extracting |V_ub|. We argue that CLEO's recent fit to the spectrum underestimates the fraction of the photon spectrum below 2 GeV. Potentially significant uncertainties enter in the theoretical evaluation of the integrated end-point lepton spectrum or the B -> X_u \ellνwidth with a too high value of the lower cut on q^2 in alternative approaches to |V_ub|.

Paper Structure

This paper contains 9 sections, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Theoretical expectation for the photon spectrum
  3. The perturbative spectrum
  4. Combined spectrum
  5. Skeptical remarks
  6. Improving the $M_X^2$ distribution in $B \to X_u \,\ell \nu$
  7. Scaling behavior and universality relations
  8. Leading $1/m_b$ corrections
  9. Conclusions and outlook

Figures (6)

  • Figure 1: CLEO's photon spectrum and the model fit to the data from Ref. CLEO1.
  • Figure 2: Perturbative fraction of decay events $\Phi_\gamma^{\rm pert}(E)$ having $E_\gamma\!<\!E$.
  • Figure 3: Fraction of decay decay events $\Phi_\gamma(E)$ including both perturbative bremsstrahlung and primordial Fermi motion. Solid and dashed lines correspond to ansätze Eqs. (\ref{['n32']}) and (\ref{['n36']}) for the light-cone distribution function, respectively.
  • Figure 4: Expected photon spectrum for the two ansätze for $F(k_+)$. The relative normalization of experimental points and theoretical predictions is not fixed.
  • Figure 5: Photon spectrum in the lab frame where $\Upsilon(4S)$ are at rest, including Doppler smearing with $\beta_B\!=\!0.067$. The relative normalization is arbitrary.
  • ...and 1 more figures