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Heavy-quarkonium production: Beyond the static and on-shell approximations 4 th Quarkonium Working Group Workshop Brookhaven National Lab, USA June 27-30, 2006 Jean-Philippe LANSBERG CPhT, ´ Ecole Polytechnique & PTF, Li` ege in collaboration with: J.R. Cudell, Y. Kalinovsky

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Page 1: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Heavy-quarkonium production:

Beyond the static and on-shell approximations

4thQuarkonium Working Group WorkshopBrookhaven National Lab, USA

June 27-30, 2006

Jean-Philippe LANSBERGCPhT, Ecole Polytechnique & PTF, Liege

in collaboration with: J.R. Cudell, Y. Kalinovsky

Page 2: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Naive pQCD approach: Colour Singlet Model (CSM)

One supposes factorisation between the hard part and the soft part

ë The hard part consists in the creation of two quarks Q and Q BUT

ß on-shell (×)ß in a colour singlet state (we want a physical state thereafter)

ß with a vanishing relative momentumß in a 3S1 state (for J/ψ, ψ′ and Υ)

ë For the soft part, the amplitude of probability that the quarks bind is given bya Schrodinger wave function

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 01/12

Page 3: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Naive pQCD approach: Colour Singlet Model (CSM)

One supposes factorisation between the hard part and the soft part

ë The hard part consists in the creation of two quarks Q and Q BUT

ß on-shell (×)ß in a colour singlet state (we want a physical state thereafter)

ß with a vanishing relative momentumß in a 3S1 state (for J/ψ, ψ′ and Υ)

ë For the soft part, the amplitude of probability that the quarks bind is given bya Schrodinger wave function

ë This description seems correct and compatible with all experiments until the CDFmeasurements of the J/ψ and ψ′ direct production at

√s = 1.8TeV,

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 01/12

Page 4: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Naive pQCD approach: Colour Singlet Model (CSM)

One supposes factorisation between the hard part and the soft part

ë The hard part consists in the creation of two quarks Q and Q BUT

ß on-shell (×)ß in a colour singlet state (we want a physical state thereafter)

ß with a vanishing relative momentumß in a 3S1 state (for J/ψ, ψ′ and Υ)

ë For the soft part, the amplitude of probability that the quarks bind is given bya Schrodinger wave function

ë This description seems correct and compatible with all experiments until the CDFmeasurements of the J/ψ and ψ′ direct production at

√s = 1.8TeV,

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 01/12

Page 5: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

COM and the polarisation at the Tevatron . . .

One of the solutions proposed is the Color Octet Mechanism:Physical states can be produced by coloured pairs

Within COM,ë J/ψ, ψ′ and Υ can be produced by a single –coloured– gluon

ë Gluon fragmentation therefore dominates

ë Since Pgluon �, the gluon is nearly on-shell and transversally polarised

ë Due to NRQCD spin symmetry, the Q is to have the same polarisation

ë Experimentally, one can study α such that:

α = +1⇔ Transverse α = 0⇔ Unpolarised α = −1⇔ Longitudinal

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 02/12

Page 6: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

COM and the polarisation at the Tevatron . . .

One of the solutions proposed is the Color Octet Mechanism:Physical states can be produced by coloured pairs

Within COM,ë J/ψ, ψ′ and Υ can be produced by a single –coloured– gluon

ë Gluon fragmentation therefore dominates

ë Since Pgluon �, the gluon is nearly on-shell and transversally polarised

ë Due to NRQCD spin symmetry, the Q is to have the same polarisation

ë Experimentally, one can study α such that:

α = +1⇔ Transverse α = 0⇔ Unpolarised α = −1⇔ Longitudinal

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 02/12

Page 7: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

COM and the polarisation at the Tevatron . . .

One of the solutions proposed is the Color Octet Mechanism:Physical states can be produced by coloured pairs

Within COM,ë J/ψ, ψ′ and Υ can be produced by a single –coloured– gluon

ë Gluon fragmentation therefore dominates

ë Since Pgluon �, the gluon is nearly on-shell and transversally polarised

ë Due to NRQCD spin symmetry, the Q is to have the same polarisation

ë Experimentally, one can study α such that:

α = +1⇔ Transverse α = 0⇔ Unpolarised α = −1⇔ Longitudinal

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 02/12

Page 8: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Our modelJ.P.L., J.R. Cudell, Yu.L. Kalinovsky [hep-ph/0507060] PLB 633 301

J.P.L, PhD thesis, [hep-ph/0507175]

ë Beyond the static and on-shell approximations of the CSM:∫ψ(prel)A(prel)dprel= A(0)φ(0) + . . .

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 03/12

Page 9: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Our modelJ.P.L., J.R. Cudell, Yu.L. Kalinovsky [hep-ph/0507060] PLB 633 301

J.P.L, PhD thesis, [hep-ph/0507175]

ë Beyond the static and on-shell approximations of the CSM:∫ψ(prel)A(prel)dprel= A(0)φ(0) + . . .

ÞFrom the Landau equations, we have two cuts contributing to Disc A

+diag. croises & +diag. croises

give back the CSMin the static limit

New Contributions withone off-shell quark

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 03/12

Page 10: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Our modelJ.P.L., J.R. Cudell, Yu.L. Kalinovsky [hep-ph/0507060] PLB 633 301

J.P.L, PhD thesis, [hep-ph/0507175]

ë Beyond the static and on-shell approximations of the CSM:∫ψ(prel)A(prel)dprel= A(0)φ(0) + . . .

ÞFrom the Landau equations, we have two cuts contributing to Disc A

+diag. croises & +diag. croises

give back the CSMin the static limit

New Contributions withone off-shell quark

ÞThe soft part (non-perturbative) is given by a phenomenological vertex function:p2rel −

(prel.P )2

M2 = −~p 2rel(in the CM frame)

ψ(p, P ) =N

(1 + ~p 2rel

Λ2 )2or N exp[

−~p 2rel

Λ2] (in the CM frame)

Þ If we choose m > M2, we switch off the CSM-like contribution; this avoids interferences for a first study.

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 03/12

Page 11: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Problem with gauge invariance

ë To change gauge amounts to the shift: εν(k)→ εν(k) + λkν

ë Gauge invariance states that this cannot affect the final result: OK if Aνkν = 0ë Let us consider QQ→ γγ:

Gauge invariance: Aµνkν4 +Bµνkν4 = 0

ë and now QQ→ Qγ:

Gauge invariance: Γ1Aµνkν4 + Γ2Bµνkν4 = (Γ1 − Γ2)Aµνkν4 6= 0

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 04/12

Page 12: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Problem with gauge invariance

ë To change gauge amounts to the shift: εν(k)→ εν(k) + λkν

ë Gauge invariance states that this cannot affect the final result: OK if Aνkν = 0ë Let us consider QQ→ γγ:

Gauge invariance: Aµνkν4 +Bµνkν4 = 0

ë and now QQ→ Qγ:

Gauge invariance: Γ1Aµνkν4 + Γ2Bµνkν4 = (Γ1 − Γ2)Aµνkν4 6= 0

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 04/12

Page 13: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Restoring Gauge Invariance . . .

ë Adding a new term Cµν, which accounts for missing contributions like

such that Cµνkν4 ≡ (Γ2 − Γ1)Aµνk

ν4

Gauge Invariance: Γ1Aµνkν4 + Γ2Bµνkν4 + Cµνkν4 = (Γ1 − Γ2)Aµνkν4 + (Γ2 − Γ1)Aµνkν4 = 0 !

ë Constraints:

ß No unphysical singularity.

ß Should VANISH when Γ1 = Γ2.

ß SYMMETRY: γ0V µν†(−p′,−p, q, P,−m)γ0 = −V µν(p, p′, q, P,m) inferred from the bare contributions: Aµνand Bµν.

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 05/12

Page 14: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Restoring Gauge Invariance . . .

ë Adding a new term Cµν, which accounts for missing contributions like

such that Cµνkν4 ≡ (Γ2 − Γ1)Aµνk

ν4

Gauge Invariance: Γ1Aµνkν4 + Γ2Bµνkν4 + Cµνkν4 = (Γ1 − Γ2)Aµνkν4 + (Γ2 − Γ1)Aµνkν4 = 0 !

ë Constraints:

ß No unphysical singularity.

ß Should VANISH when Γ1 = Γ2.

ß SYMMETRY: γ0V µν†(−p′,−p, q, P,−m)γ0 = −V µν(p, p′, q, P,m) inferred from the bare contributions: Aµνand Bµν.

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 05/12

Page 15: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Restoring Gauge Invariance . . .

ë Adding a new term Cµν, which accounts for missing contributions like

such that Cµνkν4 ≡ (Γ2 − Γ1)Aµνk

ν4

Gauge Invariance: Γ1Aµνkν4 + Γ2Bµνkν4 + Cµνkν4 = (Γ1 − Γ2)Aµνkν4 + (Γ2 − Γ1)Aµνkν4 = 0 !

ë Constraints:

ß No unphysical singularity.

ß Should VANISH when Γ1 = Γ2.

ß SYMMETRY: γ0V µν†(−p′,−p, q, P,−m)γ0 = −V µν(p, p′, q, P,m) inferred from the bare contributions: Aµνand Bµν.

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 05/12

Page 16: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Restoring Gauge Invariance . . .

ë Adding a new term Cµν, which accounts for missing contributions like

such that Cµνkν4 ≡ (Γ2 − Γ1)Aµνk

ν4

Gauge Invariance: Γ1Aµνkν4 + Γ2Bµνkν4 + Cµνkν4 = (Γ1 − Γ2)Aµνkν4 + (Γ2 − Γ1)Aµνkν4 = 0 !

ë Constraints:

ß No unphysical singularity.

ß Should VANISH when Γ1 = Γ2.

ß SYMMETRY: γ0V µν†(−p′,−p, q, P,−m)γ0 = −V µν(p, p′, q, P,m) inferred from the bare contributions: Aµνand Bµν.

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 05/12

Page 17: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Restoring Gauge Invariance . . .

ë Adding a new term Cµν, which accounts for missing contributions like

such that Cµνkν4 ≡ (Γ2 − Γ1)Aµνk

ν4

Gauge Invariance: Γ1Aµνkν4 + Γ2Bµνkν4 + Cµνkν4 = (Γ1 − Γ2)Aµνkν4 + (Γ2 − Γ1)Aµνkν4 = 0 !

ë Constraints:

ß No unphysical singularity.

ß Should VANISH when Γ1 = Γ2.

ß SYMMETRY: γ0Cµν†(−p′,−p, q, P,−m)γ0 = −Cµν(p, p′, q, P,m) inferred from the bare contributions: Aµνand Bµν.

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 05/12

Page 18: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Some possible choices . . .

By analysing the different possible Dirac structures, one can derive

Cµνeff =

Γ1 − Γ2

(p− p′).qγµ(p− p′)ν ; going further, we have Cµν

eff =Γ1 − Γ2

k.qγµkν

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 06/12

Page 19: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Some possible choices . . .

By analysing the different possible Dirac structures, one can derive

Cµνeff =

Γ1 − Γ2

(p− p′).qγµ(p− p′)ν ; going further, we have Cµν

eff =Γ1 − Γ2

k.qγµkν

ëHowever these vertices introduce new singularities

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 06/12

Page 20: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Some possible choices . . .

By analysing the different possible Dirac structures, one can derive

Cµνeff =

Γ1 − Γ2

(p− p′).qγµ(p− p′)ν ; going further, we have Cµν

eff =Γ1 − Γ2

k.qγµkν

ëHowever these vertices introduce new singularities

ëChoosing k = 12(p(p.q) + p′(p′.q)),

we can recover the propagators already present in the problem (P1 and P2):

Cµνeff = (Γ1 − Γ2)

pν(p′.q) + p′ν(p.q)

2(p′.q)(p.q)γµ = (Γ1 − Γ2)

(pν

p.q+

p′ν

p′.q

)γµ

= (Γ1 − Γ2)

(pν

P1−p′ν

P2

)γµ

ë Conclusions:Same singularities – Good symmetry – Vanishes when Γ1 = Γ2

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 06/12

Page 21: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Polarised and total cross-sections for J/ψ production at 1.8TeV, with a gaussian vertex function, m=1.87 GeV, Λ =1.8GeV and the MRST gluon distribution with

Cµνeff = (Γ1 − Γ2)

(pν

P2−p′ν

P1

)γµ

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σTOTσL σT

σLO CSM

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 07/12

Page 22: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Polarised and total cross-sections for J/ψ production at 1.8TeV, with a gaussian vertex function, m=1.87 GeV, Λ =1.8GeV and the MRST gluon distribution with

Cµνeff = (Γ1 − Γ2)

(pν

P2−p′ν

P1

)γµ

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σTOTσL σT

σLO CSM

ë The description of pion electroproduction with form factors faces the same problem:

Ý suggested form: Cµνeff =

((2p− q)ν

P2(Γ1 − F ) +

(2p′ + q)ν

P1(Γ1 − F )

)γµ

with F = Γ0 − h(Γ0 − Γ1)(Γ0 − Γ2) h being an arbitrary crossing symmetric function

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 07/12

Page 23: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Polarised and total cross-sections for J/ψ production at 1.8TeV, with a gaussian vertex function, m=1.87 GeV, Λ =1.8GeV and the MRST gluon distribution with

Cµνeff = (Γ1 − Γ2)

(pν

P2−p′ν

P1

)γµ

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σTOTσL σT

σLO CSM

ë The description of pion electroproduction with form factors faces the same problem:

Ý suggested form: Cµνeff =

((2p− q)ν

P2(Γ1 − F ) +

(2p′ + q)ν

P1(Γ1 − F )

)γµ

with F = Γ0 − h(Γ0 − Γ1)(Γ0 − Γ2) h being an arbitrary crossing symmetric function

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 07/12

Page 24: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Going further . . .

ë Study of effects from different choices of Cµν, through autonomous structures, which

ß are gauge invariant ALONE;ß have the good symmetry;ß vanishes when Γ1 = Γ2;ß do no bring new singularity in the problem.

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 08/12

Page 25: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Going further . . .

ë Study of effects from different choices of Cµν, through autonomous structures, which

ß are gauge invariant ALONE;ß have the good symmetry;ß vanishes when Γ1 = Γ2;ß do no bring new singularity in the problem.

ë Three of the simplest possible choices:

1. (Γ1 − Γ2)αγµqν the “α-term”2. (Γ1 − Γ2)β (p+ p′)µ(p+ p′)ν the “β-term”3. (Γ1 − Γ2)ξ gµν the “ξ-term”

1e-08

1e-07

1e-06

1e-05

1e-04

0.001

0.01

0.1

1

10

100

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT.. (GeV)

σL α=1 σT β=1 σL β=1 σT ξ=1

The “α-term” isnaturally big and

longitudinal

Non-vanishing contributions for themultiplicative factors set to one

ß The fact that it is autonomous leaves α, β, ξ free.ß We shall fix them to reproduce the data.

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 08/12

Page 26: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Fitting the data . . .

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σDATA σFRAG σLO CSM σL: α=8 σT: ξ=37.5σTOT

1e-05

0.0001

0.001

0.01

0.1

1

10

2 4 6 8 10

1/(2

πpT)

x B

r x d

σ/(d

ydp T

) (nb

/(GeV

)2 )

PT (GeV)

σDATA σLO CSM σL: α=8 σT: ξ=37.5σTOT

J/ψ @ Tevatron J/ψ @ RHIC

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 09/12

Page 27: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Fitting the data . . .

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σDATA σFRAG σLO CSM σL: α=8 σT: ξ=37.5σTOT

1e-05

0.0001

0.001

0.01

0.1

1

10

2 4 6 8 10

1/(2

πpT)

x B

r x d

σ/(d

ydp T

) (nb

/(GeV

)2 )

PT (GeV)

σDATA σLO CSM σL: α=8 σT: ξ=37.5σTOT

J/ψ @ Tevatron J/ψ @ RHIC

1e-05

0.0001

0.001

0.01

0.1

1

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σDATA σLO CSM σL: α=8 σT: ξ=10 σTOT

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σDATA σFRAG σLO CSM σL: α=27.5

Υ @ Tevatron ψ′ @ Tevatron

In each case, the longitudinal component dominates

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 09/12

Page 28: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

What about fragmentation ?

ë Our approach does not include fragmentation-like processesReminder: a single parton emitted at large pT evolves into a quarkonium

ë We choose to consider this class of contributions through the COMLeibovich

ë We have seen that they give transverse contributionsmight give unpolarised result with our contributions

ë We combine the two contributionswith slightly modified values for the COM matrix elements

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 10/12

Page 29: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

What about fragmentation ?

ë Our approach does not include fragmentation-like processesa single parton emitted at large pT evolves into a quarkonium

ë We choose to consider this class of contributions through the COMLeibovich

ë We have seen that they give transverse contributionsmight give unpolarised result with our contributions

ë We combine the two contributionswith slightly modified values for the COM matrix elements

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σDATA σLO CSM σL: α=4.2 σT: ξ=26.5σTCOM

LDME = 4 10-3

σTOT

-1

-0.5

0

0.5

1

4 6 8 10 12 14 16 18 20

(σT-

2σL)

/(σT+

2σL)

PT (GeV)

CDF DATA (PROMPT)

α=4.2 ξ=26.5 + COMLDME=4 10-3

J/ψ

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 10/12

Page 30: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

What about fragmentation ?

ë Our approach does not include fragmentation-like processesa single parton emitted at large pT evolves into a quarkonium

ë We choose to consider this class of contributions through the COMLeibovich

ë We have seen that they give transverse contributionsmight give unpolarised result with our contributions

ë We combine the two contributionswith slightly modified values for the COM matrix elements

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

10

100

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σDATA σLO CSM σL: α=4.2 σT: ξ=26.5σTCOM

LDME = 4 10-3

σTOT

-1

-0.5

0

0.5

1

4 6 8 10 12 14 16 18 20

(σT-

2σL)

/(σT+

2σL)

PT (GeV)

CDF DATA (PROMPT)

α=4.2 ξ=26.5 + COMLDME=4 10-3

J/ψ

1e-05

0.0001

0.001

0.01

0.1

1

4 6 8 10 12 14 16 18 20

dσ /d

PT

x B

r(nb

/GeV

)

PT (GeV)

σDATA σFRAG σLO CSM σL: α=17.5 σTCOM

LDME = 4.5 10-3

σTOT

-1

-0.5

0

0.5

1

4 6 8 10 12 14 16 18 20

(σT-

2σL)

/(σT+

2σL)

PT (GeV)

α=17.5 + COMLDME=4.5 10-3 CDF DATA

ψ′

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 10/12

Page 31: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Application to other processes

ë Inelastic photo-production →

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 11/12

Page 32: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Application to other processes

ë Inelastic photo-production →

ë Inelastic electro-production →

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 11/12

Page 33: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Application to other processes

ë Inelastic photo-production →

ë Inelastic electro-production →

ë Other states than 3S1:

ß change the vertex function

ß consider the adequate diagrams

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 11/12

Page 34: Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond the static and on-shell approximations 4thQuarkonium Working Group Workshop Brookhaven

Conclusion and perspective

ë It is possible to extend the CSM and the COM;

ë A contribution was missing !

ë Gauge invariance is preserved thanks to the introduction of 4-point vertex;

ë Longitudinal cross sections dominate;

ë Interesting link with pion electroproduction;

ë Ambiguities affecting this 4-point vertex can be exploited to bring theoryin agreement with data from Tevatron and RHIC ;

ë Combination with (COM) fragmentation gives a rather good agreementwith both cross-section and polarisation measurements;

ë Our approach is also applicable to other processes and can be tested.ß Photo- and Electro-production @ HERA;ß B-factories (where there is also a serious problem) ;

J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 12/12