1. New Observables in Quarkonium Production
Quarkonium polarisation in (p-A and) nucleus-nucleus collisions from SPS to the LHC
A long standing puzzle
General remarks on the measurement method
A rotation-invariant formalism to measure vector polarizations and parity asymmetries
Quarkonium polarization
Heavy Ion applications
João Seixas
(LIP & Physics Dep. IST Lisbon) in collaboration with Pietro Faccioli, Carlos Lourenço, Hermine Wöhri, Valentin Knünz, Ilse Kratschmer
New Observables in Quarkonium Production
ECT* Trento, 28 Fev-4 Mar 2016

2. A varied menu for the LHC (and AFTER)
Measure polarization = determine average angular momentum composition of the particle, through its decay angular distribution
It offers a much closer insight into the quality/topology of the contributing production processes wrt to decay-averaged production cross sections
Polarization analyses are particularly important to (for example):
understand still unexplained production mechanisms [J/ψ, χc , ψ’, , χb]
characterize the spin of newly (eventually) discovered resonances [X(3872), Higgs, Z’, graviton, ...]
Understand the properties of dense and hot matter

3. Task list
One assumes that the production of quark-antiquark states can be described using perturbative QCD, as long as we “factor out” long-distance bound-state effects
A seemingly inescapable prediction of NRQCD approach is that “high” pT quarkonia come from fragmenting gluons and are fully tranversely polarized
However, despite good success in describing cross sections...

4. Task list
One assumes that the production of quark-antiquark states can be described using perturbative QCD, as long as we “factor out” long-distance bound-state effects
A seemingly inescapable prediction of NRQCD approach is that “high” pT quarkonia come from fragmenting gluons and are fully tranversely polarized
the first comparisons with polarization data were not promising…
HX frame
TeV
NRQCD factorization Braaten, Kniehl & Lee, PRD62, (2000)
CDF Run II
CDF Coll., PRL 99, (2007)
But:
Until recently the experimental situation was contradictory and incomplete, as it was emphasized in Eur. Phys. J. C69, 657 (2010)  improve drastically the quality of the experimental information
maybe the theory is only valid at asymptotically high pT  extend measurements to pT >> M
contributions of intermediate P-wave states have not been fully calculated yet and are still unknown experimentally (remember Ilse’s talk)  measure polarizations of directly produced states, ψ’ and (3S)  measure polarizations of P-wave states, χc and χb, and their feeddown to S states

5. Task list
There are problems actually on the theory side which are becoming quite evident
Remember that in NRQCD the factorization hypothesis implies that the cross-section for the inclusive production of a meson H in a A+B collision is
𝜎 𝐴+𝐵→𝐻+𝑋 = 𝑆,𝐿,𝐶 𝒮(𝐴+𝐵→𝑄 𝑄 2𝑆+1 𝐿 𝐽 𝐶 +𝑋)×ℒ( 𝑄 𝑄 2𝑆+1 𝐿 𝐽 𝐶 →𝐻)
SDC
LDME

6. Task list
There are problems actually on the theory side which are becoming quite evident
Remember that in NRQCD the factorization hypothesis implies that the cross-section for the inclusive production of a meson H in a A+B collision is
𝜎 𝐴+𝐵→𝐻+𝑋 = 𝑆,𝐿,𝐶 𝒮(𝐴+𝐵→𝑄 𝑄 2𝑆+1 𝐿 𝐽 𝐶 +𝑋)×ℒ( 𝑄 𝑄 2𝑆+1 𝐿 𝐽 𝐶 →𝐻)
SDC
LDME

7. Task list
There are problems actually on the theory side which are becoming quite evident
Remember that in NRQCD the factorization hypothesis implies that the cross-section for the inclusive production of a meson H in a A+B collision is
𝜎 𝐴+𝐵→𝐻+𝑋 = 𝑆,𝐿,𝐶 𝒮(𝐴+𝐵→𝑄 𝑄 2𝑆+1 𝐿 𝐽 𝐶 +𝑋)×ℒ( 𝑄 𝑄 2𝑆+1 𝐿 𝐽 𝐶 →𝐻)
SDC
LDME
𝜆 𝜃 = 𝒮 𝑇 + 𝒮 𝐿 𝒮 𝑇 − 𝒮 𝐿

8. Task list
There are problems actually on the theory side which are becoming quite evident
Remember that in NRQCD the factorization hypothesis implies that the cross-section for the inclusive production of a meson H in a A+B collision is
𝜎 𝐴+𝐵→𝐻+𝑋 = 𝑆,𝐿,𝐶 𝒮(𝐴+𝐵→𝑄 𝑄 2𝑆+1 𝐿 𝐽 𝐶 +𝑋)×ℒ( 𝑄 𝑄 2𝑆+1 𝐿 𝐽 𝐶 →𝐻)
SDC
LDME
IT IS possible to define kinematic discriminants to distinguish 3 𝑃 𝐽 [8] on a statistical basis from the other terms. One could then identify regions of phase space where the quarkonia would have negative cross sections and decay distributions violating angular momentum conservation!

9. Definition of observables
In Quantum Mechanics the study of angular momentum requires a quantization axis (aka “z-axis”)
Many possible (known) choices:
Gottfried-Jackson (GJ)
Collins-Soper (CS)
Helicity (HX)
Perpendicular Helicity (PX)

10. CS  HX for mid rap. / high pT
Frames and parameters z
Collins-Soper axis (CS): ≈ dir. of colliding partons
Helicity axis (HX): dir. of quarkonium momentum
quarkonium rest frame θ
ℓ +
production plane x φ y
λθ = +1
λφ = λθφ = 0
Jz = ± 1
λθ = +1 : “transverse” (= photon-like) pol.
λθ = –1
λφ = λθφ = 0
Jz = 0
λθ = 1 : “longitudinal” pol.
In general 8 parameters, for parity conserving and symmetric detectors, 3 z z′
CS  HX for mid rap. / high pT CS HX
90º x x′ y y′

11. The azimuthal anisotropy is not a detail
Case 2: natural longitudinal polarization, observation frame  to the natural one
Case 1: natural transverse polarization z z
Jx = 0 x
Jz = ± 1 x
λθ = +1
λφ = 0
λθ = +1
λφ =  1 y y
Two very different physical cases
Indistinguishable if λφ is not measured (integration over φ)

12. Frame-independent polarization
The shape of the distribution is obviously frame-invariant.
→ it can be characterized by a frame-independent parameter, e.g.
Note: 𝝀 =𝟏 means Lam-Tung relation z
λθ = –1
λφ = 0
λθ = +1
λφ = 0
λθ = –1/3
λφ = –1/3
λθ = +1/5
λφ = +1/5
λθ = +1
λφ = –1
λθ = –1/3
λφ = +1/3
P. Faccioli, C. Lourenço, J.S., PRL 105, ; PRD 82, ; PRD 83,

13. Positivity constraints for dilepton distributions
P. Faccioli, C.Lourenço, J.S., PRL 105, (2010); PRD 83, (2011)
General and frame-independent constraints on the anisotropy parameters of vector particle decays λφ
Jy V = 0
Jx V = ±V ~
λ = +1
Jz V = 0
Jz V = ±V ~
λ = –1
Jy V = ±V
Jx V = 0 λθ
λθφ
λθφ
physical
domain λθ λφ

14. Positivity constraints for dilepton distributions
P. Faccioli, C.Lourenço, J.S., PRL 105, (2010); PRD 83, (2011)
General and frame-independent constraints on the anisotropy parameters of vector particle decays
F = 𝟏+ 𝝀 𝟑+ 𝝀
physical
domain

15. What polarization axis?1)
helicity conservation (at the production vertex)
→ J =1 states produced in fermion-antifermion annihilations (q-q or e+e–) at Born level have transverse polarization along the
relative direction of the colliding fermions (Collins-Soper axis) 1 . 5
Drell-Yan
Drell-Yan is a paradigmatic case
but not the only one 1 .
(2S+3S)
Remember: DY for Y is a background to deal with λθ . 5
z(HX) .
E866 (p-Cu)
Collins-Soper frame
high pT
90° - . 5
z(CS) 1 2
pT [GeV/c] c 2) g
( )
NRQCD → at very large pT , quarkonium produced from the fragmentation of an
on-shell gluon, inheriting
its natural spin alignment c g
J/ψ
NRQCD g g λθ
CDF
→ large, transverse polarization
along the QQ (=gluon) momentum (helicity axis)
pT [GeV/c]

16. Some remarks on methodology
Measurements are challenging
A typical collider experiment imposes pT cuts on the single muons; this creates zero-acceptance domains in decay distributions from “low” masses:
helicity
Collins-Soper
φHX
φCS
Toy MC with
pT(μ) > 3 GeV/c (both muons)
Reconstructed
unpolarized (1S)
pT() > 10 GeV/c, |y()| < 1
cosθHX
cosθCS
This spurious “polarization” must be accurately taken into account.
Large holes strongly reduce the precision in the extracted parameters

17. Quarkonium polarization: a “puzzle”17
Quarkonium polarization: a “puzzle”
J/𝜓: Measurements at Tevatron , LHC (ALICE) _
J/ψ, pp √s = 1.96 TeV
CDF Run I
CDF Run II
Only 𝜆 𝜃 measured
Only one frame used (HX)
|y| < 0.4
CDF II vs CDF I
→ not known what caused the change
|y| < 0.6
Helicity frame
PRL 85, 2886 (2000)
PRL 99, (2007)
J/ψ, pp √s = 7 TeV
𝜆 𝜃 and 𝜆 𝜙 separately measured
Two frames used (HX & CS)
|cos 𝜃| & |ϕ| dist. fit imposing 𝜆 to be invariant in the two frames
ALICE
2.5 < y < 4,
2 < pT < 8 GeV/c
PRL 108, (2012)

18. Quarkonium polarization: a “puzzle”18
Quarkonium polarization: a “puzzle”
J/𝜓: HERA-B
J/ψ, p-Cu and p-W √s = 41.6 GeV
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured by single variable projections
Three frames used (HX, GJ & CS)
−1.5 < y < 0.8
−0.34 < 𝑥 𝐹 < 0.14
0 < pT < 5.4 GeV/c
EPJ C (2009)

19. Quarkonium polarization: a “puzzle”19
Quarkonium polarization: a “puzzle”
J/𝜓: Other fixed target experiments
𝜆 𝜃
J/ψ, 252 GeV 𝝅 on W
Chicago-Iowa-Princeton Coll (E615).
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
Three frame used (GJ, CS, UC)
Violates Lam-Tung relation
𝜆 𝜃
𝜆 𝜙
PRL 58, 2523 (1987)
2𝜆 𝜃𝜙
PRD 39, (1989)

20. Quarkonium polarization: a “puzzle”
𝜰(nS): Measurements at Tevatron ( ) 20
CDF+D (2002)
Only 𝜆 𝜃 measured
Only one frame used (HX) _
(1S), pp √s = 1.96 TeV
CDF (2012)
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
𝜆 checked
Two frames used (HX & CS)

21. Quarkonium polarization: a “puzzle”21
Quarkonium polarization: a “puzzle”
𝜰(nS): Measurements at LHC (CMS)
(nS), pp √s = 7 TeV
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
Three frames used (HX, CS, PX)
𝜆 checked
Fully multidimensional
|y| < 0.6
0.6<|y| < 1.2
10 < 𝑝 𝑇 < 40 GeV/c
Y(1S), Y(2S), and Y(3S)
Phys. Rev. Lett. 110,
Comparison with CDF results

22. Quarkonium polarization: a “puzzle”22
Quarkonium polarization: a “puzzle”
𝜰(nS): Measurements at LHC (CMS)
(nS), pp √s = 7 TeV
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
Three frames used (HX, CS, PX)
𝜆 checked
Fully multidimensional
|y| < 0.6
0.6<|y| < 1.2
10 < 𝑝 𝑇 < 40 GeV/c
Y(1S), Y(2S), and Y(3S)
Phys. Rev. Lett. 110,
Comparison with CDF results

23. Quarkonium polarization: a “puzzle”23
Quarkonium polarization: a “puzzle”
𝜰(nS): Measurements at LHC (CMS)
(nS), pp √s = 7 TeV
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
Three frames used (HX, CS, PX)
𝜆 checked
Fully multidimensional
Phys. Rev. Lett. 110,

24. Quarkonium polarization: a “puzzle”24
Quarkonium polarization: a “puzzle”
𝜰(nS): Mesurements at LHC (CMS)
(nS), pp √s = 7 TeV
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
Three frames used (HX, CS, PX)
𝜆 checked
Fully multidimensional
Phys. Rev. Lett. 110,

25. A lot of measurements to do...
Measurement of c0(1P), c1(1P) and c2(1P) production cross sections
Measurement of b (1P), b(2P) and b(3P) production cross sections;
Measurement of the relative production yields of J = 1 and J = 2 b states
Measurement of the c1 (1P) and c2(1P) polarizations versus pT and rapidity
Measurement of the b (1P) and b (2P) polarizations …

26. …and a series of questions to answer
Is there a simple composition of processes, probably dominated by one single mechanism, that is responsible for the production of all quarkonia?
Solid curve is a fit to the J/𝜓 CMS data (pT/M>3)
Remaining curves are replicas with normalizations adjusted to the individual datasets
P. Faccioli et al, PLB 736(2014) 98
𝑓 𝑝 𝑇 𝑀 = 𝛽− 𝑝 𝑇 𝑀 2 𝛾 −𝛽
𝛽=3.62±0.07 𝛾=1.29±0.32

27. …and a series of questions to answer
Is there a simple composition of processes, probably dominated by one single mechanism, that is responsible for the production of all quarkonia?
P. Faccioli et al, PLB 736(2014) 98

28. …and a series of questions to answer
Is this mechanism perturbed in the presence of matter at high density and high temperature?

29. Pioneering measurements at SPS: NA60
𝜆 𝜃 and 𝜆 𝜑 measured (p-A); HX and CS frames used.

30. A first step in this program at LHC: polarization as a function of multiplicity
CMS p-p

31. A first step in this program: polarization as a function of multiplicity
CMS p-p

32. Summary
The new quarkonium polarization measurements have many improvements with respect to previous analyses and shed, when combined with cross-section data, a new light on quarkonium production Will we (finally) manage to solve an old puzzle?
General advice: do not throw away physical information! (azimuthal-angle distribution, rapidity dependence, ...)
A new method based on rotation-invariant observables gives several advantages in the measurement of decay distributions and in the use of polarization information
Quarkonium polarization could be used to probe hot and dense matter. A complete program is under way.

33. Backup slides

34. 34
A first step in this program: polarization as a function of multiplicity

36. Quarkonium polarization: a “puzzle”36
Quarkonium polarization: a “puzzle”
(nS): Measurements at LHC (CMS)
(nS), pp √s = 7 TeV
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
Three frames used (HX, CS, PX)
𝜆 checked
Fully multidimensional
Y(1S), Y(2S), and Y(3S)
Phys. Lett. B 727 (2013) 381

37. Quarkonium polarization: a “puzzle”37
Quarkonium polarization: a “puzzle”
(nS): Measurements at LHC (CMS)
(nS), pp √s = 7 TeV
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
Three frames used (HX, CS, PX)
𝜆 checked
Fully multidimensional

38. Quarkonium polarization: a “puzzle”38
Quarkonium polarization: a “puzzle”
(nS): Mesurements at LHC (CMS)
(nS), pp √s = 7 TeV
𝜆 𝜃 , 𝜆 𝜙 and 𝜆 𝜃𝜙 measured
Three frames used (HX, CS, PX)
𝜆 checked
Fully multidimensional

39. J/ψ polarization as a signal of colour deconfinement?λθ
NRQCD
pT [GeV/c]
≈ 0.7
≈ 0.7 λθ
HX frame
CS frame
Starting “pp” scenario:
J/ψ significantly polarized (high pT)
feeddown from χc states (≈ 30%) smears the polarizations Si
Sequential suppression
≈ 30% from χc decays
≈ 70% direct J/ψ
+ ψ’ decays
J/ψ cocktail:
Recombination ? e
As the χc (and ψ’) mesons get dissolved by the QGP, λθ should change to its direct value

40. J/ψ polarization as a signal of sequential suppression?
P. Faccioli, JS, PRD 85, (2012)
B.L. Ioffe and D.E. Kharzeev PRC 68 (2003)
CMS data:
up to 80% of J/ψ’s disappear from pp to Pb-Pb
more than 50% ( fraction of J/ψ’s from ψ’ and χc) disappear from peripheral to central collisions ~
>
CMS PAS HIN
sequential suppression gedankenscenario: in central events ψ’ and χc are fully suppressed and all J/ψ’s are direct
It may be impossible to test this directly:
measuring the χc yield (reconstructing χc radiative decays) in PbPb collisions is prohibitively difficult due to the huge number of photons
However, a change of prompt-J/ψ polarization must occur from pp to central Pb-Pb!
prompt J/ψ polarization in pp
χc-to-J/ψ fractions in pp
χc polarizations in pp
prompt J/ψ polarization in PbPb
Reasonable sequence of measurements:
χc suppression
in PbPb!

41. J/ψ polarization as a signal of sequential suppression?
Example scenario:
CDF prompt J/ψ
Extrapolated* direct J/ψ
CSM direct J/ψ
prompt-J/ψ polarization in pp: λθ  – 0.15
direct-J/ψ polarization: λθ  – 0.6
helicity frame
(assumed to be the same in pp and PbPb)
* R(χc1)+R(χc2) = 42 % R(χc2)/R(χc1) = 38 % h(χc1) = 0
h(χc2) = ±2
Use a model as shown before and use the expression to calculate the prompt lambdas from the direct and feeddown lambdas

42. J/ψ polarization as a signal of sequential suppression?
If we measure a change in prompt polarization like this... λθ
“prompt”
“direct”
... we are observing the disappearance of the χc
relative to the J/ψ
R(χc) in PbPb
R(χc) in pp
the polarization should become significantly
more longitudinal (in the helicity frame) after the disappearance
of the transversely polarized feed-down contribution
due to c decays. We are assuming that the ‘‘base’’
polarizations of the directly produced S and P states remain
essentially unaffected by the nuclear medium and are,
therefore, not distinguishable
Simplifying assumptions:
direct-J/ψ polarization is the same in pp and PbPb
normal nuclear effects affect J/ψ and χc in similar ways
χc1 and χc2 are equally suppressed in PbPb

43. J/ψ polarization as a signal of sequential suppression?
When will we be sensitive to an effect like this?
pT(μ) > 3 GeV/c,
6.5 < pT < 30 GeV/c, 0 < |y| < 2.4
CMS-like toy MC with
prompt-J/ψ polarization
as observed in pp (and peripheral PbPb)
prompt-J/ψ polarization
as observed in central PbPb
efficiency-corrected
|cosθHX| distribution
precise results
in pp very soon
~20k evts
~20k evts
In this scenario, the χc disappearance is measurable at ~5σ level with
~20k J/ψ’s in central Pb-Pb collisions

44. J/ψ polarization as a signal of sequential suppression?
When will we be sensitive to an effect like this?
CMS-like toy MC

45. J=1 states are intrinsically polarized
Single elementary subprocess:
There is no combination of a0, a+1 and a-1 such that λθ = λφ = λθφ = 0
The angular distribution is never intrinsically isotropic
Only a “fortunate” mixture of subprocesses
(or randomization effects)
can lead to a cancellation of all three observed anisotropy parameters
To measure zero polarization would be (in fact, is) an exceptionally interesting result...

46. Basic meaning of the frame-invariant quantities
Let us suppose that, in the collected events, n different elementary subprocesses yield angular momentum states of the kind
(wrt a given quantization axis), each one with probability
The rotational properties of J=1 angular momentum states
imply that
The quantity
is therefore frame-independent. It can be shown to be equal to
In other words, there always exists a calculable frame-invariant relation of the form
the combinations are independent of the choice of the quantization axis
note: F = 𝟏+ 𝝀 𝟑+ 𝝀

47. Simple derivation of the Lam-Tung relation
Another consequence of rotational properties of angular momentum eigenstates:
for each state
there exists a quantization axis wrt which
→ dileptons produced in each single elementary subprocess have a distribution of the type
wrt its specific “ ” axis.
DY: q q * q * * * q* q* q* _ q q _
Due to helicity conservation at the q-q-* (q-q*-*) vertex,
along the q-q (q-q*) scattering direction _
If all a’s are real then z* belongs to the xz plane
→ for each diagram
sum independent of
spin alignment directions! →
Lam-Tung identity

48. Essence of the LT relation
The existence (and frame-independence) of the LT relation is the kinematic consequence of the rotational properties of J = 1 angular momentum eigenstates
Its form derives from the dynamical input that all contributing processes produce a transversely polarized (Jz = ±1) state (wrt whatever axis)
More generally:
Corrections to the Lam-Tung relation (parton-kT, higher-twist effects) should continue to yield invariant relations. In the literature, deviations are often searched in the form
But this is not a frame-independent relation. Rather, corrections should be searched in the invariant form
For any superposition of processes, concerning any J = 1 particle (even in parity-violating cases: W, Z ), we can always calculate a frame-invariant relation analogous to the LT relation.
The Lam-Tung relation corresponds to the special case
when all processes produce transversely polarized dileptons
with respect to quantization axes belonging to the
production plane