Giulia Ricciardi

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• Giulia Ricciardi
• Università di Napoli Federico II, Italy

XLIst Rencontres de Moriond- March 19th, 2006
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Introduction

• Semileptonic B decays spectra
• LD effects due to soft interactions between the
  heavy quark and the light degrees of freedom.
• Large perturbative (PT) contributions at
  threshold
• At threshold different scales the hard scale Q
  is determined by the final total energy EX

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Introduction (2)

• LD effects manifest themselves in PT in the form
  of series of large infrared logs
  
• large logs in PT signal LD effects
• Large logs in PT theory need to be resummed
• universality of LD effects studied by comparing
  the logarithmic structure of different spectra in
  PT
• Universality is looked for into a similar
  factorization structure

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Introduction (3)

• Universality is limited by different kinematics
  of different spectra
• comparison with hadron mass spectrum of
• All semileptonic spectra are divided in two
  classes,
• pure SD relation with the radiative decay (Hadron
  energy spectrum)
• No such pure SD relation (Hadron mass spectrum,
  electron spectrum, spectrum )

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LD effects

• Threshold region characterized by different
  scales
• presence of PT logs terms to be resummed at all
  orders
• Q always indicates the hardest scale Q 2 EX
• Also in the radiative decay B -gt Xs g threshold
  logs to be resummed

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Radiative decay

• Resummed (cumulative) differential distribution
• with
• In the two body decay the hadron energy fixes the
  hard scale Q, by kinematical reasons

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• is the (cumulative) resummed QCD
  form factor
• is a short distance
  coefficient function, independent on ts
• is a remainder function,
  vanishing for ts_? 0

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Semileptonic decays

• In
• one can obtain a general resummation formula for
  the triple
• differential distribution
• where w2EX/mb and x2 El/mb and

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Semileptonic decays

• The infrared logs can be organized in a series
  and factorized
• into the universal form factor S
• where Q 2 EX and by definition
• All single distributions can be obtained by the
  most general triple differential distribution

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Semileptonic decays

• is the universal QCD
  form factor, which now is evaluated for a
  coupling with a general argument
• Q w mb 2 EX
• In the tree body decay at tree level, the hadron
  energy is no more fixed at the b quark mass, as
  in the radiative decay

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Classes of semileptonic spectra

• All distributions can be obtained by integrating
  the triple differential distribution
• can be divided into two classes, according to
  the fact that it is necessary or not to integrate
  over the hadron energy,
• 1) no integration over EX-directly related via
  short distance (SD) coefficients to the photon
  spectrum in the radiative decay.
• the same structure of threshold logs
• f.i. single distribution in EX
• 2) obtained by integrating over EX CANNOT be
  directly related via SD coefficients to the
  photon spectrum in the radiative decay.
• not same structure of threshold logs
• f.i. single distributions in the invariant
  hadronic mass mX2, in the charged electron
  energy, in the lightcone variable p

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Class 1

• Example spectrum in hadron energy w2 EX/mb
• Two integrations from the triple differential
  factorized form, but EX
• not integrated over
• Since the coefficient function does not depend on
  u, the second
• integration only involves the QCD form factor and
  the remainder

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Single distribution in EX

• At EX lt mb/2 there are no large logs
• w 2 EX/mb
• At EX gt mb/2 large logs appear, which can be
  resummed (Sudakov shoulder universality)

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Class 2

• Such distributions do not have the same log
  structure than the radiative decaytherefore
  cannot be simply related via short distance
  coefficients
• F.i. if we compute the NLO distribution in the
  invariant hadron mass squared
• differ to first order with radiative decay,
• that is , f.i. and
  so on

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Check 1 of resummation formula

• In the resummation formula of the triple
  differential spectrum we have that the resummed
  QCD form factor depends explicitly only on
• This is therefore the argument of the factorized
  infrared logarithms
• expanding the resummed expression up to
  we find agreement with previous fixed order
  calculation (De Fazio, Neubert)

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Check 2 of resummation formula

• the argument of the running coupling is the hard
  scale Q, that is the hadronic energy
• The correct argument need to be verified at 2nd
  order in the coupling constant, since
• Only available 2nd order calculation corrections
  to
• order to the distribution in the light cone
  momentum (Hoang, Ligeti and Luke)
• with such scale correctly prediction of
• terms

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Conclusions

• Critical analysis of factorization and
  universality for semi-inclusive B decays
• LD effects manifest themselves in PT series of
  large IR lgsuniversality implies identical
  series of large logs-same factorization structure
  
• Resummation formula for the triple differential
  distribution
• Semileptonic spectra into two groups
• 1) Distributions not integrated over EX?LD
  structure comparable directly with radiative
  decay
• 2) opposite behaviour respect to
• distributions integrated over EX?LD structure
  not directly
• comparable