photonphoton spectrumspectrum in in thethe softsoft--energyenergy regionregionPedro D. Ruiz-Femenía

Talk at the Talk at the International Workshop on Heavy QuarkoniumInternational Workshop on Heavy Quarkonium17-20 October 2007, DESY, Hamburg

Nucl.Phys.BNucl.Phys.B (2007),(2007),doidoi:10.1016/:10.1016/j.nuclphysbj.nuclphysb.2007.07.013.2007.07.013

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 2

Motivation

The oPs decay spectrum for

The soft-radiation loop momentum region

spectrum for

Summary

OutlineOutline

NRQED calculation

QED spectrum in the soft-energy region

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 3

MotivationMotivation

NRQED, includes binding effects

satisfies Low´s theorem for[ Manohar, RF ’03 ]

valid for the whole region ?

new NR mode? is the NREFT calculation missing something?

orthopositronium (o-Ps) decay spectrum for

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 4

NRQED NRQED computationcomputation ofof thethe oPsoPs spectrumspectrum

there is a long-distance part in the decay when one ofthe photons in the final state is not hard

long-distance part

short-distance parthard photons

Coulomb Hamiltonian

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 5

DipoleDipole approximationapproximation

dipole approx.Atomic transitions in Ps Multipole expansion

valid for

o-Ps → 3γ decay with one small-energy photon

[ Voloshin ’04 ]

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 6

oPs calculation without multipole expanding the e.m. field

dipole approx.

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 7

EFT EFT oPsoPs spectrumspectrum: : resultsresults

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 8

EFT EFT oPsoPs spectrumspectrum: :

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 9

QED QED computationcomputation

from the NRQED computation in the region

exact phase-space distribution at one-loop is known [ Adkins ’05 ]

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 10

Threshold Threshold expansionexpansion ofof QED 1QED 1--looploop diagramsdiagramsBeneke, Smirnov ‘98

asymptotic expansion of loop integrals involving heavy massiveparticles close to threshold

separation of regions achieved by expanding loop momentaaccording to the v-scaling above (each region contributes to a single power of v)

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 11

LadderLadder graphgraph

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 12

11--looploop QED QED resultresult forfor

agrees with NLO term of NRQED calculation

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 13

RelevanceRelevance forfor Heavy Heavy QuarkoniumQuarkonium

photon spectrum can be writtenas a sum of a direct contrib. anda fragmentation contrib.

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 14

11--looploop QCD QCD correctioncorrection forfor

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 15

ResultsResults forfor thethe directdirect UpsilonUpsilon decaydecay spectrumspectrum

Krämer ‘99

Catani, Hautmann ’95Maltoni, Petrelli ‘99

________________________________________________________International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th P. Ruiz-Femenía - 16

SummarySummary

Application to the soft-energy direct photon spectrum inUpsilon decays is straightforward

Threshold expansion of 1-loop diagrams

NRQED calculation of the photon spectrumvalid for all