Transcript photon spectrum in the soft-energy region Pedro D. Ruiz-Femenía Nucl.Phys.B (2007), doi:10.1016/j.nuclphysb.2007.07.013 Talk at the International Workshop on Heavy Quarkonium 17-20 October 2007, DESY, Hamburg.

photon spectrum in the soft-energy region
Pedro D. Ruiz-Femenía
Nucl.Phys.B (2007),
doi:10.1016/j.nuclphysb.2007.07.013
Talk at the International Workshop on Heavy Quarkonium
17-20 October 2007, DESY, Hamburg
Outline
 Motivation

The oPs decay spectrum for
 NRQED calculation
 QED spectrum in the soft-energy region

The soft-radiation loop momentum region


spectrum for
Summary
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 2
Motivation
 orthopositronium (o-Ps) decay spectrum for
 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?
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 3
NRQED computation of the oPs spectrum
 there is a long-distance part in the decay when one of
the photons in the final state is not hard
short-distance part
hard photons
long-distance part
Coulomb Hamiltonian
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 4
Dipole approximation
dipole approx.
Atomic transitions in Ps
Multipole expansion
valid for
o-Ps  3 decay with one small-energy photon
[ Voloshin ’04 ]
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 5
 oPs calculation without multipole expanding the e.m. field
dipole approx.
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 6
EFT oPs spectrum: results
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 7
EFT oPs spectrum:
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 8
QED computation
 from the NRQED computation in the
region
 exact phase-space distribution at one-loop is known
[ Adkins ’05 ]
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 9
Threshold expansion of QED 1-loop diagrams
Beneke, Smirnov ‘98
 asymptotic expansion of loop integrals involving heavy massive
particles close to threshold
 separation of regions achieved by expanding loop momenta
according to the v-scaling above (each region contributes to a single
power of v)
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 10
Ladder graph
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 11
1-loop QED result for
 agrees with NLO term
of NRQED calculation
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 12
Relevance for Heavy Quarkonium
 photon spectrum can be written
as a sum of a direct contrib. and
a fragmentation contrib.
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 13
1-loop QCD correction for
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 14
Results for the direct Upsilon decay spectrum
Krämer ‘99
Catani, Hautmann ’95
Maltoni, Petrelli ‘99
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 15
Summary

NRQED calculation of the
valid for all
photon spectrum

Threshold expansion of 1-loop diagrams
 Application
to the soft-energy direct photon spectrum in
Upsilon decays is straightforward
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International Workshop on Heavy Quarkonium, DESY, Hamburg, Oct 17th
P. Ruiz-Femenía - 16