F Giordano Proton transversity distributions

Francesca GiordanoQCD Frontier 2013

JLab, Newport News, VA

ChasingTransversity

1

Francesca Giordano

Proton Structure

2

1970s:Quantum ChromoDynamics

2013: still not able to describe QCD

fundamental state

Francesca Giordano 3

GPDs (x, bT) TMDs (x, kT)x

zy

kT

S

xp

xz

y

xp

bT

PDFs (x)

∫ TMDs(x,kT) ... dkT∫ GPDs(x,bT) ... dbT

Form Factors (t)

Fourier transform (bT) & ∫ GPDs(x, t) ... dx

Proton Structure


G+

2. Spin-orbit correlations in the nucleon

At leading twist (twist-two) three parton distribution functions characterise momentum and spin ofthe quarks within the nucleon [Jaf97]. In addition to the momentum distribution f q1 (x) 1, introducedin the discussion of the parton model (section 2.1.2) , two spin-dependent functions appear: the helic-ity distribution gq1 (x) and the transversity distribution h

q1 (x) [RS79, AM90, JJ91]. Spin-dependent

parton distribution functions can measured in polarised deep-inelastic scattering processes:

Longitudinally polarised nucleons The helicity distribution can be probed in deep-inelasticscattering of spin-polarised leptons off nucleons spin-polarised in directions longitudinal to the in-coming leptons. The virtual photon, inheriting the lepton polarisation to a degree given by the leptonkinematics, can only interact with quarks polarised in opposite direction. This is a consequence ofhelicity conservation in the absorption of a spin-1 virtual photon by a spin-12 quark (g

⇤q! q). Asthe virtual photon selects only quarks of one polarisation, measurements of the cross section foranti-parallel (�!() or parallel polarisations (�!)) of lepton (!) and target nucleons ()) are sensitiveto number densities q of quarks polarised along or against the nucleon polarisation. In the infinitemomentum frame, these number densities are related to the helicity distribution gq1 (x), defined asthe difference of the probability to find a quark polarised along or against the nucleon in a helicityeigenstate:

gq1 (x) = q�!) (x)�q�!( (x) . (2.11)

The momentum distribution measuring the spin average is given by the sum of these probabilities:

f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)

Transversely polarised nucleons In the basis of transverse spin eigenstates ("+ and "*), thetransversity distribution hq1 (x) measures the difference of the number densities of transversely po-larised quarks aligned along or against the polarisation of the nucleon:

hq1 (x) = q"* (x)�q"+ (x) . (2.13)

The probabilistic interpretation of these parton distribution functions is illustrated in table 2.1.Differences between the helicity and transversity distributions are a consequence of the relativistic

motion of the quarks within the nucleon. Euclidean rotations and Lorentz boosts do not commuteand thus longitudinally polarised nucleons cannot be converted in transversely polarised nucleons atinfinite momentum. Only in case of non-relativistic quarks both distributions would be identical.Another difference emerges from an analysis of helicity amplitudes. Forward quark-nucleon

scattering amplitudes AΛlΛ0l 0 , labelled by the helicities of quarks (l (0) = ±12 ⌘ ±) and nucleons

(Λ(0) = ±12 ⌘ ±), represent the absorption of a quark (l ) from a nucleon (Λ) and the subsequent

emission of the quark (l 0) by the nucleon (Λ0). Due to conservation of helicity, Λ+ l = Λ0 + l

0,parity, AΛlΛ0l 0 = A�Λ�l�Λ0�l

0 and time reversal there are exactly three independent amplitudes:

A++,++, A+�,+� A+�,�+. (2.14)

The optical theorem relates the forward quark-nucleon scattering amplitudes to the cross section ofdeep-inelastic scattering. Parton distribution functions can be considered as imaginary part of theseamplitudes [Jaf97]: The momentum and helicity distributions correspond to amplitudes that conservequark helicity:

f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)

1Here and henceforth, the weak scale dependence of the parton distribution functions is omitted.

8

q

Momentum distribution

Francesca Giordano







gq1 (x) = q�!) (x)�q�!( (x) . (2.11)


f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)


hq1 (x) = q"* (x)�q"+ (x) . (2.13)








A++,++, A+�,+� A+�,�+. (2.14)


f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)


8

5

14

TABLE IV: Truncated first moments, !f1,[0.001!1]i , and full ones, !f1

i , of our polarized PDFs at various Q2.

x-range in Eq. (35) Q2 [GeV2] !u + !u !d + !d !u !d !s !g !"0.001-1.0 1 0.809 -0.417 0.034 -0.089 -0.006 -0.118 0.381

4 0.798 -0.417 0.030 -0.090 -0.006 -0.035 0.36910 0.793 -0.416 0.028 -0.089 -0.006 0.013 0.366100 0.785 -0.412 0.026 -0.088 -0.005 0.117 0.363

0.0-1.0 1 0.817 -0.453 0.037 -0.112 -0.055 -0.118 0.2554 0.814 -0.456 0.036 -0.114 -0.056 -0.096 0.24510 0.813 -0.458 0.036 -0.115 -0.057 -0.084 0.242100 0.812 -0.459 0.036 -0.116 -0.058 -0.058 0.238

0

0.1

0.2

0.3

0.4

-0.15

-0.1

-0.05

0

0.05

-0.04

-0.02

0

0.02

-0.04

-0.02

0

0.02

-0.04

-0.02

0

0.02

10 -2 10 -1

x(Δu + Δu–) x(Δd + Δd

–)

DSSV

xΔu–

xΔd–

xΔs–

x

Δχ2=1 (Lagr. multiplier)

Δχ2=1 (Hessian)

xΔg

x

Q2 = 10 GeV2-0.1

-0.05

0

0.05

0.1

10 -2 10 -1

FIG. 3: Our polarized PDFs of the proton at Q2 = 10 GeV2

in the MS scheme, along with their !!2 = 1 uncertaintybands computed with Lagrange multipliers and the improvedHessian approach, as described in the text.

tendency to turn towards +1 at high x. The latter be-havior would be expected for the pQCD based models.We note that it has recently been argued [73] that theupturn of Rd in such models could set in only at rela-tively high x, due to the presence of valence Fock states ofthe nucleon with nonzero orbital angular momentum thatproduce double-logarithmic contributions ! ln2(1"x) inthe limit of x # 1 on top of the nominal power behav-ior. The corresponding expectation is also shown in thefigure. In contrast to this, relativistic constituent quarkmodels predict Rd to tend to "1/3 as x # 1, perfectly

AπALL0

pT [GeV]

Δχ2=1 (Lagr. multiplier)Δχ2=1 (Hessian)

-0.004

-0.002

0

0.002

0.004

2 4 6 8

FIG. 4: Uncertainties of the calculated A!0

LL at RHIC in ourglobal fit, computed using both the Lagrange multiplier andthe Hessian matrix techniques. We also show the correspond-ing PHENIX data [23].

consistent with the present data.

Light sea quark polarizations: The light sea quark andanti-quark distributions turn out to be better constrainednow than in previous analyses [36], thanks to the adventof more precise SIDIS data [10, 14, 15, 16] and of the newset of fragmentation functions [37] that describes the ob-servables well in the unpolarized case. Figure 6 shows thechanges in !2 of the fit as functions of the truncated firstmoments !u1,[0.001!1], !d1,[0.001!1] defined in Eq. (35),obtained for the Lagrange multiplier method. On theleft-hand-side, Figs. 6 (a), (c), we show the e"ect on thetotal !2, as well as on the !2 values for the individualcontributions from DIS, SIDIS, and RHIC pp data andfrom the F, D values. It is evident that the SIDIS datacompletely dominate the changes in !2. On the r.h.s. ofthe plot, Figs. 6 (b), (d), we further split up !!2 fromSIDIS into contributions associated with the spin asym-metries in charged pion, kaon, and unidentified hadronproduction. One can see that the latter dominate, closelyfollowed by the pions. The kaons have negligible impacthere. For !u1,[0.001!1], charged hadrons and pions arevery consistent, as far as the location of the minimumin !2 is concerned. For !d1,[0.001!1] there is some slighttension between them, although it is within the tolerance

G

�G

�q

�G

q

Helicity distribution

Francesca Giordano







gq1 (x) = q�!) (x)�q�!( (x) . (2.11)


f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)


hq1 (x) = q"* (x)�q"+ (x) . (2.13)








A++,++, A+�,+� A+�,�+. (2.14)


f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)


8

6

G

�q

q�q

�G

Transversity distribution

Francesca Giordano







gq1 (x) = q�!) (x)�q�!( (x) . (2.11)


f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)


hq1 (x) = q"* (x)�q"+ (x) . (2.13)








A++,++, A+�,+� A+�,�+. (2.14)


f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)


8

7

in helicity basis

helicity flip!

G

�q

q�q

�G

Transversity distributionTransversity is chiral-odd!

Francesca Giordano







gq1 (x) = q�!) (x)�q�!( (x) . (2.11)


f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)


hq1 (x) = q"* (x)�q"+ (x) . (2.13)








A++,++, A+�,+� A+�,�+. (2.14)


f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)


8

8

in helicity basis

G

�q

q�q

�G


helicity flip!

Transversity is chiral-odd!

Francesca Giordano







gq1 (x) = q�!) (x)�q�!( (x) . (2.11)


f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)


hq1 (x) = q"* (x)�q"+ (x) . (2.13)








A++,++, A+�,+� A+�,�+. (2.14)


f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)


8

9

in helicity basis

G

�q

q�q

�G


helicity flip!


Francesca Giordano







gq1 (x) = q�!) (x)�q�!( (x) . (2.11)


f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)


hq1 (x) = q"* (x)�q"+ (x) . (2.13)








A++,++, A+�,+� A+�,�+. (2.14)


f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)


8

10

in helicity basis

G

�q

q�q

�G

(No gluon transversity in the puzzle?)


helicity flip!



G

�q

q�q

�G

1. Can it be accessed? How?


Yes! But only in conjunction with another chiral-odd object


chiral oddchiral odd }

chiral even observable

h1 x XAccessible in different reactions

=> need of complementary reactions!

Francesca Giordano


12


G

�q

q�q

�G

2. Is transversity Universal?-> each reaction covers specific kinematic ranges, and access specific features



Accessible in different reactions=> need of complementary reactions!


G

�q

q�q

�G

3. How does transversity evolve?-> reactions are at different typical energies













gq1 (x) = q�!) (x)�q�!( (x) . (2.11)


f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)


hq1 (x) = q"* (x)�q"+ (x) . (2.13)








A++,++, A+�,+� A+�,�+. (2.14)


f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)


8







gq1 (x) = q�!) (x)�q�!( (x) . (2.11)


f q1 (x) = q�!) (x)+q

�!( (x) . (2.12)


hq1 (x) = q"* (x)�q"+ (x) . (2.13)








A++,++, A+�,+� A+�,�+. (2.14)


f q1 (x)⇠ℑ[A++,++ +A+�,+�], (2.15)gq1 (x)⇠ℑ[A++,++�A+�,+�], (2.16)


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~?No!


G

�q

q�q

�G

no gluon transversity (in proton) and quark and gluon transversities

don’t mix!







?



Transversity distribution(some of the) Possible channelsl p↑ ➛ h X,

h1h2 X, Λ↑ X(SIDIS)


chiral oddchiral odd}

chiral even observable

h1 x FF




h1 x Collinsh1 x IFF0h1 x FF

h1 x 𝛬FF




h1 x Collinsh1 x IFF

TMD0Collinear

h1 x FF

h1 x 𝛬FFCollinear



Francesca Giordano

Transversity distribution(some of the) Possible channels

20

f1 x h1 x FF

p p↑ ➛ h X, h1h2 X, jets


TMD

FF

fq1

0Collinear

h1 x FF

f1 x h1 x Collins

f1 x h1 x IFF0

Collinear

TMD

h1 x 𝛬FFCollinear

l p↑ ➛ h X, h1h2 X, Λ↑ X

(SIDIS)


f1 x h1 x FF


TMD

FF

fq1

0Collinear

h1 x FF

f1 x h1 x Collins

f1 x h1 x IFF0

Collinear

TMD


h1 x 𝛬FF

p↑ h↑ ➛ (q↑ q↑➛)l+l-

(Fully polarized Drell-Yan)

h1 x h1 Collinear

Collinearp p↑ ➛ h X, h1h2 X, jets

l p↑ ➛ h X, h1h2 X, Λ↑ X

(SIDIS)

Francesca Giordano


22

They all access transversity viasingle or double spin

asymmetries, f.i.:

AUT∝

AUT∝

ATT∝

How well do we know the unpolarized cross-sections? In particular the TMD

unpolarized ones?

PT∝


TMD

Collinear

Collinear

0

h1 x 𝛬FFp p↑ ➛ h X, h1h2 X, jets f1 x h1 x Collins

f1 x h1 x IFF0

Collinear

TMDAN∝

AN∝

h1 x h1 Collinear

p↑ h↑ ➛ (q↑ q↑➛)l+l-

(Fully polarized Drell-Yan)

l p↑ ➛ h X, h1h2 X, Λ↑ X

(SIDIS)

AUT =�" � �#

�" + �#

Francesca Giordano

H1⊥q

Ph⊥

Ph⊥

Collins FF

23

Transversity distributionFirst access in SIDIS

Interference FF

Collinear

AUT∝ h1 x H1⊥0

AUT∝ h1 x H1⪦

H1⪦q Ph⊥

Ph⊥

Transversity coupled with a Fragmentation Function (FF)

TMD

Francesca Giordano

H1⊥q

Ph⊥

Ph⊥

Collins FF

24


Interference FF

Collinear

AUT∝ h1 x H1⊥0

AUT∝ h1 x H1⪦

H1⪦q Ph⊥

Ph⊥


TMD

∫ h1 (x,pT) H1 (x,kT) dpT dkT⊥

(and same in the denominator for the unpolarized pdfs and ffs)

direct product!(same in the denominator)

Francesca Giordano

AUT∝ h1 x H1⊥0

25


Collins FF


TMD

Complementary reactions

TMD factorization assumed

FF universality assumedM. Anselmino et al., PRD75:054032, 2007

Francesca Giordano

AUT∝ h1 x H1⊥0

26


Collins FF


TMD

Very different energies between Hermes/Compass and Belle:

Is TMD evolution different from Collinear?


Collinear evolution assumed



Francesca Giordano

AUT∝ h1 x H1⊥0

27


Collins FF


TMD



Different energies at Hermes/Compass: how does transversity

evolve?



No evolution assumed



Francesca Giordano

AUT∝ h1 x H1⊥0

28


Collins FF


TMD



Different energies at Hermes/Compass: how does transversity

evolve?Which is the TMD transversity and the TMD Collins pT dependence? And the unpolarized

TMD pT dependence?



No evolution assumed


Gaussian pT dependence assumed


Francesca Giordano

AUT∝ h1 x H1⊥0

29


Collins FF


TMD



No transversity evolution assumedTransversity from Proton data

Transversity from pion pair production SIDIS off transversely polarized target

Band:Torino 2009 transversity

1100

110

0

0.2

0.4

x

x h1uv !x"! x h1dv !x"#4

5% theo. err.• from COMPASS data• DiFF analysis

• from BELLE data• f1(x) from MSTW

• new analysis

!

!

!

!"

" ""

" "

"

"

"

1100

110

0

0.2

0.4

x


PRELIMINARY

• from HERMES data• DiFF analysis


• PRL107 & arXiv:1202.0323

!

!

!

!

1100

110

0

0.2

0.4

x


11% theo. err.

Transversity from Proton data





• new analysis

PRELIMINARY



• PRL107 & arXiv:1202.0323

11% theo. err.

Transversity from Proton data





• new analysis

PRELIMINARY



• PRL107 & arXiv:1202.0323

11% theo. err.

Bacchetta, Radici, Courtoy Phys.Rev.Lett.107:012001,2011

Interference FF

AUT∝ h1 x H1⪦

Collinear

Transversity and Collins Gaussian pT dependence assumed


FF universality assumed


Transversity distributionAccess in pp

X𝜎

Dqh

h

AN∝ f1 x h1 x H1⊥00

Single hadronBut! the asymmetry for single

hadron comes mixed with other effects (Sivers, higher twist)TMD

AN∝ f1 x h1 x H1⪦



X𝜎

Dqh

h

Jets

Single hadron

Looking at the hadron distribution within the jet

TMDAN∝ f1 x h1 x H1

⊥00




X𝜎

Dqh

h

Jets

Single hadronTMD

Word of caution: TMD factorization broken in pp➛hadrons


AN∝ f1 x h1 x H1⊥00



X𝜎

Dqh

h

Jets

Single hadronTMD

Collinear!

AN∝ f1 x h1 x H1⊥00



Transversity distributionAccess in fully polarized Drell-Yan

Double Spin asymmetry:ATT∝ h1 x h1 Cleanest theoretical access:

➛ no input needed for fragmentation functions➛ Collinear case

To enhance the signal both quark and anti-quark should come from the valence region- medium-high x region- preferable beam/target combination, f.i.: proton/anti-proton (PAX, GSI) pion/proton (COMPASS, CERN)


Double Spin asymmetry:ATT∝ h1 x h1 Cleanest theoretical access:

➛ no input needed for fragmentation functions➛ Collinear case

But experimentally challenging: ➛ low cross-section➛ background cleaning➛ anti-proton polarization still not proven to work

To enhance the signal both quark and anti-quark should come from the valence region- medium-high x region- preferable beam/target combination, f.i.: proton/anti-proton (PAX, GSI) pion/proton (COMPASS, CERN)

Transversity distributionAccess in fully polarized Drell-Yan

Francesca Giordano

pp IFF medium x high-x, precise Inclusion in measurements global analysis

pp->jets Collins medium x high-x, precise TMD Evolution, measurements TMD factorization breaking

SIDIS Collins medium x high x TMD Evolution measurement of (un)polarized

pdfs and FF pT dependence

SIDIS IFF medium x high x

Present Status

36

experimental input needed

Drell-Yan no data! precise measurements!

Jlab: Hall A&B

STAR

STAR

theoretical input needed

Francesca Giordano

pp IFF medium x high-x, precise Inclusion in measurements global analysis

pp->jets Collins medium x high-x, precise TMD Evolution, measurements TMD factorization breaking

SIDIS Collins medium x high x TMD Evolution measurement of (un)polarized

pdfs and FF pT dependence

SIDIS IFF medium x high x

Present Status

37

experimental input needed

Drell-Yan no data! precise measurements!

Jlab: Hall A&B

STAR

STAR

theoretical input needed

SuperStarfsPHENIX

EIC

Jlab12

Jlab12

BabarBelle/BelleII

SuperStarfsPHENIX

COMPASSII

FERMILABPAX?

Star

Star

... not there yet ...

38

... but we took a few feathers off ...

39

... and planning new experiments ...

40

Stay tuned!

41

Thank you!

F Giordano Proton transversity distributions

Science

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