Reconciling the Light-Cone and NRQCD Approaches
to Calculating e+e− → J/ψ + ηc
Geoffrey Bodwin, Daekyoung Kang, Jungil Lee
(hep-ph/0603185)
Outline
• The Conflict Between Theory and Experiment
• A Proposed Solution (Bondar and Chernyak)
• Possible Reasons for the Apparent Discrepancy Between the Light-Cone and NRQCD Calculations
– BC Used an Ad Hoc Model Quarkonium Light-Cone Distribution.
– The High-Momentum Tail of the Light-Cone Distribution Requires Special Treatment.
– BC Include Some Factors Zi.
• Numerical Results
• Conclusions
The Conflict Between Theory and Experiment
• Experiment
Belle: σ(e+e− → J/ψ + ηc) × B>2 = 25.6 ± 2.8 ± 3.4 fb.
BaBar: σ(e+e− → J/ψ + ηc) × B>2 = 17.6 ± 2.8 ± 2.1 fb.
• NRQCD at LO in αs and v
Braaten, Lee: σ(e+e− → J/ψ + ηc) = 3.78 ± 1.26 fb.
Liu, He, Chao: σ(e+e− → J/ψ + ηc) = 5.5 fb.
The two calculations employ different choices of mc, NRQCD matrix elements, and αs.
Braaten and Lee include QED effects.
• Trends
– The Belle cross section has moved down from 33+7
−6 ± 9 fb.
– The BaBar cross section is even lower.
– Braaten and Lee found a sign error in the QED interference term that raised the prediction
from 2.31 ± 1.09 fb.
– Zhang, Gao, Chao: A calculation of corrections at NLO in αs shows that the K factor is
approximately 1.96.
– The trends are in the right direction, but a discrepancy remains.
A Proposed Solution
• Bondar and Chernyak (BC) and Ma and Si have proposed calculating the cross section in the
light-cone formalism.
– Focus on the BC calculation.
• BC find
σ(e+e− → J/ψ + ηc) = 33 fb.
– They attribute the increased cross section to the finite width of the light-cone distribution.
– Takes into account corrections that arise from the relative momentum of the Q and Q̄ in the
quarkonium.
– If the light-cone distribution is replaced by δ(z − 1/2), the cross section reduces to ≈ 3 fb.
• The NRQCD and light-cone approaches are both believed to be valid approximations to QCD in
the limit E >> mc, ΛQCD.
• Why is there such a large discrepancy between the NRQCD and light-cone results?
Possible Reasons for the Apparent Discrepancy
Between the Light-Cone and NRQCD Calculations
BC Used Ad Hoc Model Quarkonium Light-Cone Distributions.
• The model light-cone distributions may not be the true light-cone distributions of QCD.
• We derived a light-cone distribution from potential-model wave functions (Cornell potential).
• If the model potential is the exact static QQ̄ potential, then the errors in the potential model are
of relative order v 2.
– v is the velocity of the Q or Q̄ in the quarkonium rest frame.
– v 2 ≈ 0.3 for charmonium.
• The Cornell potential provides a good fit to lattice data for the static QQ̄ potential.
We computed the wave function by numerical integration
of the Schrödinger Equation.
Calculate the Light-Cone Distribution φ(z)
• Fourier transform the wave function:
ψ(x) → ψ(p).
• Use on-shell kinematics to relate p to the light-cone momentum fraction:
p+
EQ + p3
z= + =
,
P
2EQ
q
EQ =
p2 + m2c .
Error of relative order v 2.
• Integrate out p⊥ to form the light-cone distribution φ(z).
• The integrals converge slowly, and tricks are needed to achieve adequate numerical accuracy.
Checked against analytic forms in special cases.
Result:
Conclusions:
• The BC light-cone distribution agrees approximately
with the shape of potential-model light-cone distribution.
• The BC and potential-model light-cone distributions
give similar results for the cross section.
The High-Momentum Tail of the Light-cone Distribution Requires Special Treatment
• The regions near z = 0 and z = 1 correspond to large p2.
2
• The potential-model wave function is not valid for p2 >
m
∼ c.
• In NRQCD calculations, contributions from the high-momentum tail of the wave function are
contained in the short-distance coefficients.
– Can use asymptotic freedom to compute them in perturbation theory.
– They first appear at one-loop order (relative order αs).
• We subtract these contributions from the high-momentum regions of the light-cone distribution.
– Removes the part of the light-cone distribution that is not described reliably by the potential
model (or the BC model).
– Without the subtraction, the light-cone calculation would contain part of the order-αs NRQCD
calculation and couldn’t be compared directly with the leading-order NRQCD calculation.
• The subtraction ∆ψ can be computed from the NRQCD matching condition:
R
k
p
(dp) ∆ψ(p)H(p) =
H
H
p=0
−k −p
• ∆φ agrees asymptotically (near z = 0, 1) with φ.
The Coulomb part of the potential dominates in this region.
• ∆φ appears to be singular at z = 1/2, but is actually a
+ distribution satisfying
Z 1
dz ∆φ(z) = 0.
0
• Subtraction of ∆φ shifts contributions from z ≈ 0, 1 to z ≈ 1/2.
• Subtraction of ∆φ reduces the cross section by about a factor of 3.
BC Include Some Factors Zi.
• The Zi correspond to renormalization of mc and the point-like tensor and pseudo-scalar vertices.
• They have no counterpart in the NRQCD calculations.
• They are included in an attempt to resum large logarithms of the momentum transfer divided by
m2c .
• At least some of the Zi factors seem to be inappropriate.
– The vertex-renormalization Zi’s account correctly only for the evolution of the first moments
of the light-cone distributions.
– In BC all of the factors of mc are evolved with Zi factors.
But some of the factors of mc correspond to Mmeson ≈ 2mc and have nothing to do with the
running of mc in off-shell propagators.
• When the Zi are set to unity, the cross section is reduced by about a further factor of 2.
Numerical Results
• σ0: unsubtracted light-cone cross section
(neglecting light-cone distributions of higher
order in v ).
Zi
BC
1
αs
BC
BC
0.21 0.21
mc (GeV)
1.2
1.2
1.2
σ0 (fb)
1
1
1.4
22.51 11.59 7.58 4.95
σ (fb)
7.77
4.30 3.29 2.48
σδ (fb)
5.39
3.04 2.48 1.95
• σ : light-cone cross section after ∆φ has
been subtracted.
• σδ : light-cone cross section with a zero-width
light-cone distribution:
φ(z) ∼ δ(z − 1/2).
• BC allow αs to run inside the integrations
over z and take mc = 1.2 GeV.
• αs = 0.21 and mc = 1.4 GeV are the values used in the Braaten-Lee NRQCD calculation.
• The light-cone cross section, after subtraction of ∆φ and with Zi = 1, is comparable to the LO
NRQCD cross sections.
• The difference between σ and σδ is small (≈ 29%):
After subtraction of ∆φ, the finite width of the light-cone distribution does not produce a large
effect.
• In NRQCD, the corrections of higher order in v to the cross section are known to be large:
approximately 99%.
• Why is this not reflected in the difference between σ and σδ , which is about 29%?
• Our light-cone calculation neglects p⊥ in comparison with p3.
2
2
– The corresponding contributions are suppressed as Mmeson
/Ebeam
≈ 0.34, which makes
the effect small.
• We also drop light-cone distributions that are explicitly of higher order in v .
• These two effects account for 54% of the correction.
• The remaining discrepancy of 99% − 54% − 29% = 16% presumably would be resolved by
corrections to δφ of higher order in αs.
Conclusions
• Once the high-momentum contributions are subtracted and the Zi are set to unity, the light-cone
calculation is compatible with the NRQCD calculations.
• The BC light-cone calculation does not resolve the discrepancy between theory and experiment.
• Most promising theoretical avenues to resolve this discrepancy:
– corrections of higher order in αs,
– corrections of higher order in v .