Haibo Li Institute of High Energy Physics, Beijing . International Workshop on Heavy Quarkonium
Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond...
Transcript of Heavy-quarkonium production: Beyond the static and on ... · Heavy-quarkonium production: Beyond...
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Application to other processes
ë Inelastic photo-production →
J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 11/12
Application to other processes
ë Inelastic photo-production →
ë Inelastic electro-production →
J-Ph. LANSBERG, Ecole Polytechnique QWG4 – 28-06-2006 11/12
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
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