Endpoint Logarithms in

Hee Sok ChungKorea University

In collaboration with Geoffrey T. Bodwin (Argonne) and Jungil Lee (Korea U.)

NRF Workshop on Particle Physics and Cosmology June 7-8, 2013, Yonsei University

e+e− → J/ψ + ηc

Outline

• in B factories

• Endpoint double logarithms in

• Summary

e+e− → J/ψ + ηc

e+e− → J/ψ + ηc

Exclusive Double Charmonium Production

3NRF Workshop Hee Sok Chung

• Belle (2004) :

BABAR (2005) : Belle, PRL89, 142001 (2002) and PRD70, 071102(R) (2004)

BABAR, PRD72, 031101(R) (2005)

angle in the J= decay, j cos!lj< 0:9, as shown inFig. 1(a) and 1(c).

The largest backgrounds are due to real J= mesonsfrom QED processes such as J= or !2S" mesons pro-duced via initial state radiation (ISR). J= mesons from Bmeson decay have p# < 2 GeV=c and do not constitute abackground for recoil masses below 6:6 GeV=c2. MostQED backgrounds have low multiplicity, and may haveelectrons or photons escaping detection along the beamline. These backgrounds are suppressed by the requirementof at least five charged tracks and the following require-ment: for each event we calculate the energy deposited inthe EMC plus the energy that can be attributed to anundetected electron or photon,

EQED $ EEMC % pmiss; (2)

where EEMC is the total energy deposited in the EMC, andpmiss is the missing momentum in the lab frame in theevent. We require EQED & Ebeams <&1:0 GeV as shown inFig. 1(b) and 1(d), where Ebeams is the sum of the e%e&

beam energies calculated in the lab frame. We reject theJ= background from !2S" events by vetoing events if theinvariant mass of the J= candidates combined with anypair of oppositely charged tracks with pion mass hypothe-sis is within 15 MeV=c2 of the !2S" mass.

The recoil mass distribution for events in the J= masswindow is shown as points with error bars in Fig. 2. TheISR !2S" background is estimated using a Monte Carlosample of ISR !2S" events. The !2S" feeddown back-ground from continuum production is estimated usingcontinuum !2S" events selected in the data.

The spectrum in Fig. 2 is fit to the sum of signalfunctions representing the "c!1S", #c0, and "c!2S" line-shapes, plus a second-order polynomial background func-tion. The signal line shapes are obtained by convoluting theBreit-Wigner line shape of each resonance with a fixed-width Gaussian representing the recoil mass resolution

function. The widths of the Gaussians are determinedfrom a Monte Carlo simulation of the momentum of thereconstructed J= ; the J= momentum resolution is dif-ferent for the J= ! e%e& and J= ! $%$& samples,but independent of the recoiling system. This shape in turnis convolved with a long radiative tail that is calculated toO!%2" [17] for ISR photons that carry off an energy greaterthan 10 MeV. The free parameters in the data fit are thecoefficients for the background parameterization, the eventyields for each resonance, the masses of the resonances,and the "c!2S" total width. The total widths for the "c!1S"and the #c0 are fixed to their world average values [16] of17:3 MeV=c2 and 10:1 MeV=c2, respectively. The fit isperformed simultaneously to the recoil mass spectra in theJ= ! e%e& and J= ! $%$& samples, and the totalevent yield for each resonance is given by the sum of theyields in each mode.

The fit result is given in Table I and is shown as the solidcurve in Fig. 2. Other known charmonium states may alsobe produced in association with the J= via two virtual-

)2 (GeV/crecM2 2.5 3 3.5

2 c/Ve

M 02 / N

0

10

20

30

2 2.5 3 3.50

10

20

30 (2S)ψ+ ISR

(2S) feeddownψ+

sidebandsψJ/

)2 (GeV/crecM2 2.5 3 3.5

2 c/Ve

M 02 / N

0

10

20

30

FIG. 2 (color online). The fit to the recoil mass distribution isrepresented by the solid curve. The dashed curve is a second-order polynomial representing the background. The points witherror bars refer to the events in the J= mass window. Thehistograms represent different sources of backgrounds.

10.0 / N

20

40

60 (a) Data

VeG1.0 /

N 20

40(b) Data

lθcos-1 -0.5 0 0.5 1

0

20

40

60 (c) Monte Carlo

(GeV)beams - EQEDE-10 -5 0 5

0

50

100(d) Monte Carlo

FIG. 1 (color online). Distributions of (a) cos!l and (b)EQED & Ebeams in the data, (c) cos!l and (d) EQED & Ebeams inthe signal Monte Carlo. The arrows point to where the selectioncriteria are applied.

TABLE I. Result of the fits to the recoil-mass spectrum. Theerrors are statistical only. Where indicated, the value of thecorresponding parameter is fixed to the current world average[16]. The primary fit is obtained including signals of "c!1S", #c0,and "c!2S". The event yield for the other resonances is deter-mined by including each resonance in the primary fit.

Recoilsystem

Numberof Events

Mass!MeV=c2"

Total width!MeV=c2"

"c!1S" 126' 20 2984:8' 4:0 fixed#c0 81' 20 3420:5' 4:8 fixed"c!2S" 121' 27 3645:0' 5:5 22' 14J= &26' 13 fixed fixed#c1 &5' 16 fixed fixed#c2 &12' 16 fixed fixed !2S" 30' 27 fixed fixed

MEASUREMENT OF DOUBLE CHARMONIUM PRODUCTION . . . PHYSICAL REVIEW D 72, 031101 (2005)

RAPID COMMUNICATIONS

031101-5

PRD72, 031101(R) (2005)

• Factorized form of production amplitude

• are non-perturbative meson distribution amplitudes. These contains all long-distance physics that corresponds to the evolution from pairs to mesons.

• depends only contributions from short distances,and can be calculated using perturbation theory.

• Radiative corrections may involve logarithms in ,which can make a fixed-order calculation unreliable.Factorization theorem allows us to resum such logarithms.

Exclusive Double Charmonium Production

4NRF Workshop Hee Sok Chung

Helicity Selection Rule

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• Asymptotic behavior for is given by the helicity selection rule

That is, processes preserving hadron helicity have slowest asymptotic decrease.

• In , is produced with helicity (zero helicity forbidden by parity invariance)

The suppression comes from quark helicity flip.

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Brodsky and Lepage, PRD24, 2848 (1981)

Helicity Selection Rule

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• Factorization theorem exists for helicity-preserving processes.

• NLO correction to helicity-preserving processes involve single logarithm , which can be resummed using evolution equations.

• In contrast, factorization theorem for helicity-suppressed processes is unavailable.

• NLO corrections to helicity-suppressed processes are known to involve double logarithms (In B factories, ), resummation is necessary.

αs log2(m2/s)

αs log2(m2/s) ≈ 3

NRF Workshop Hee Sok Chung

Jia, Wang and Yang, JHEP10, 105 (2011)

Bodwin, Lee, Garcia i Tormo, PRL101, 102002 (2008),PRD81, 114005 (2010), PRD81, 114014 (2010)

Efremov and Radyushkin, PLB94, 245 (1980)Lepage and Brodsky, PRD22, 2157 (1980)

Double Logarithms in

• Factorization theorem allows one to resum large logarithms.

• A first step in proving factorization is the identification of the momentum regions that give rise to large logarithms.

• In this work we investigate the origin of the double logarithms at NLO in αs by explicit evaluation of the NLO diagrams.

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e+e− → J/ψ + ηc

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Finding Double Logarithms in

• We work at leading order in the relative velocity of of the charmonia.

• is collinear to plus, is collinear to minus.

• The logarithms in m can be identified as would-be soft and collinear singularities regulated by the quark mass m.

• The process requires a quark helicity flip, so that we cannot ignore the quark masses in the numerators.

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e+e− → J/ψ + ηc

p p̄

PJ/ψ = 2p, Pηc = 2p̄, p2 = p̄2 = m2, Q = PJ/ψ + Pηc

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1-Loop Diagrams

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• The pair going up forms the , the rest forms .

• There are also charge conjugation diagrams.

• Diagrams that do not allow logarithmic power counting are ignored.

J/ψ ηc

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Origins of Double Logarithms• The double logarithms in each diagram come from the

Sudakov and endpoint logarithms.

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• Sudakov double logs : the momentum of the gluon is simultaneously soft and collinear.

• Endpoint double logs : gluons carry almost all of the collinear momentum from a spectator line to an active line. The double log occurs when the momentum of the spectator line is is simultaneously soft and collinear.

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Origins of Double Logarithms• A change of variables makes the endpoint double log look

like a Sudakov double log.

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Endpoint Double Logs

Sudakov Double Logs

S =

�ddk

(2π)d1

(k2 + iε)[(k − p)2 −m2 + iε][(k + p̄)2 −m2 + iε]

=i

4π2Q2

��1

�IR− log(m2/µ2)

�log(m2/Q2) + 1

2 log2(m2/Q2) + · · ·

�.

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Sudakov Double Logarithms• By applying the soft approximation we can prove that the

Sudakov logarithms cancel between soft gluon insertions into quark and antiquark lines.

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cancel in the sum

... and so on.

S

+ = 0

+ = 0

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Sudakov Double Logarithms• Therefore the Sudakov double logarithms cancel in

the sum of all diagrams.

• The double logarithms in originate solely from endpoint double logarithms.

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e+e− → J/ψ + ηc

NRF Workshop Hee Sok Chung

Endpoint Double Logarithms

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E =

�dd�

(2π)d1

(�2 −m2 + iε)[(�+ p)2 + iε][(�− p̄)2 + iε]

• Endpoint double logs appear when is soft-collinear.

• Helicity flip in the spectator quark line eliminates a factor of in the numerator, giving logarithmic power counting for soft-collinear scaling.

�

�

�

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• The following diagrams potentially contain endpoint logs.

Endpoint Double Logarithms

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Soft quark lineCollinear quark line with momentum containing , but not Collinear quark line with momentum containing , but not

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Absence of Power Divergences

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• The additional collinear lines may give power divergences.

• If is collinear to +, then all momenta surrounding the upper gluon is collinear to +. Then, the numerator vanishes because we can anticommute away any of the momenta so that it is adjacent to another one collinear to the same direction. The numerator must have a factor or .

• Therefore the numerator must have a factor or . produces an endpoint double log, and gives a collinear log, but not a double log.

• The same analysis can be repeated for the remaining diagrams.

� · p �2

�− �⊥

�

� · p�2

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Absence of Endpoint Logs without Helicity Flip• Consider a helicity-preserving process, such as

. In the soft approximation,

• The collinear fermion lines give

The or is eliminated by the equations of motion; the collinear fermion lines do not change the power of the loop momentum in the integrand, giving . Therefore endpoint logs are absent without helicity flip.

• This even holds when the relative velocity of is retained.17

e+e− → hc + ηc

∼�� 1/�

3

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Endpoint Logs with Helicity Flip

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• In the helicity-suppressed processes, we have

• The collinear fermion lines give

• Helicity flip is obtained by either picking up the quark masses or using the equations of motion for the collinear fermion lines.This eliminates a factor of in the numerator.Helicity flip gives the correct logarithmic scaling for endpoint double logarithm.

�

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Summary• We investigated the origin of the double logarithms in

.

• Sudakov double logarithms cancel in the sum over all diagrams, while endpoint double logarithms remain.

• The endpoint double logarithms are re-interpreted as a leading region of loop integration in which a spectator fermion line becomes soft-collinear.

• This re-interpretation may simplify the resummation of logarithms in .

• The factorization for the helicity-flip process is currently under development.

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m2/s

e+e− → J/ψ + ηc

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Supplementary

• The following diagrams do not have double logs

1-Loop Diagrams

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+ charge-conjugation diagrams+ diagrams with counterterms

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Cancellation of Sudakov Logs

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Insertion of soft gluons into collinear-to-minus lines (contribution from collinear-to-plus region).The Sudakov double logs cancel in these combinations.

+

+

+

+

= 0

= 0

= 0

= 0

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Cancellation of Sudakov Logs

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+

+

+

+

Insertion of soft gluons into collinear-to-plus lines (contribution from collinear-to-minus region).The Sudakov double logs cancel in these combinations.

= 0

= 0

= 0

= 0

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Cancellation of Sudakov Logs

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Because the lower meson has C = +1, the paired diagrams are same. The Sudakov double logarithms cancel in each diagram.

=

=

=

= 0

= 0

= 0

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