f giordano proton transversity distributions
TRANSCRIPT
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
Francesca Giordano 4
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
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
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
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
6
G
�q
q�q
�G
Transversity distribution
Francesca Giordano
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
7
in helicity basis
helicity flip!
G
�q
q�q
�G
Transversity distributionTransversity is chiral-odd!
Francesca Giordano
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
8
in helicity basis
G
�q
q�q
�G
Transversity distribution
helicity flip!
Transversity is chiral-odd!
Francesca Giordano
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
9
in helicity basis
G
�q
q�q
�G
Transversity distribution
helicity flip!
Transversity is chiral-odd!
Francesca Giordano
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
10
in helicity basis
G
�q
q�q
�G
(No gluon transversity in the puzzle?)
Transversity distribution
helicity flip!
Transversity is chiral-odd!
Francesca Giordano 11
G
�q
q�q
�G
1. Can it be accessed? How?
Transversity is chiral-odd!
Yes! But only in conjunction with another chiral-odd object
Transversity distribution
chiral oddchiral odd }
chiral even observable
h1 x XAccessible in different reactions
=> need of complementary reactions!
Francesca Giordano
Yes! But only in conjunction with another chiral-odd object
12
Transversity distribution
G
�q
q�q
�G
2. Is transversity Universal?-> each reaction covers specific kinematic ranges, and access specific features
1. Can it be accessed? How?
Transversity is chiral-odd!
Accessible in different reactions=> need of complementary reactions!
Francesca Giordano 13
G
�q
q�q
�G
3. How does transversity evolve?-> reactions are at different typical energies
Yes! But only in conjunction with another chiral-odd object
1. Can it be accessed? How?
Transversity is chiral-odd!
Transversity distribution
2. Is transversity Universal?-> each reaction covers specific kinematic ranges, and access specific features
Accessible in different reactions=> need of complementary reactions!
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
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
~?No!
Francesca Giordano 14
G
�q
q�q
�G
no gluon transversity (in proton) and quark and gluon transversities
don’t mix!
3. How does transversity evolve?-> reactions are at different typical energies
Yes! But only in conjunction with another chiral-odd object
1. Can it be accessed? How?
Transversity is chiral-odd!
Transversity distribution
2. Is transversity Universal?-> each reaction covers specific kinematic ranges, and access specific features
?
Accessible in different reactions=> need of complementary reactions!
Francesca Giordano 15
G
�q
q�q
�G
no gluon transversity (in proton) and quark and gluon transversities
don’t mix!
3. How does transversity evolve?-> reactions are at different typical energies
Yes! But only in conjunction with another chiral-odd object
1. Can it be accessed? How?
Transversity is chiral-odd!
Transversity distribution
2. Is transversity Universal?-> each reaction covers specific kinematic ranges, and access specific features
?
Accessible in different reactions=> need of complementary reactions!
Francesca Giordano 16
Transversity distribution(some of the) Possible channelsl p↑ ➛ h X,
h1h2 X, Λ↑ X(SIDIS)
Francesca Giordano 17
chiral oddchiral odd}
chiral even observable
h1 x FF
Transversity distribution(some of the) Possible channelsl p↑ ➛ h X,
h1h2 X, Λ↑ X(SIDIS)
Francesca Giordano 18
h1 x Collinsh1 x IFF0h1 x FF
h1 x 𝛬FF
Transversity distribution(some of the) Possible channelsl p↑ ➛ h X,
h1h2 X, Λ↑ X(SIDIS)
Francesca Giordano 19
h1 x Collinsh1 x IFF
TMD0Collinear
h1 x FF
h1 x 𝛬FFCollinear
Transversity distribution(some of the) Possible channelsl p↑ ➛ h X,
h1h2 X, Λ↑ X(SIDIS)
Francesca Giordano
Transversity distribution(some of the) Possible channels
20
f1 x h1 x FF
p p↑ ➛ h X, h1h2 X, jets
h1 x Collinsh1 x IFF
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)
Francesca Giordano 21
f1 x h1 x FF
h1 x Collinsh1 x IFF
TMD
FF
fq1
0Collinear
h1 x FF
f1 x h1 x Collins
f1 x h1 x IFF0
Collinear
TMD
Transversity distribution(some of the) Possible channels
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
Transversity distribution(some of the) Possible channels
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∝
h1 x Collinsh1 x IFF
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
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
∫ 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
Transversity distributionFirst access in SIDIS
Collins FF
Transversity coupled with a Fragmentation Function (FF)
TMD
Complementary reactions
TMD factorization assumed
FF universality assumedM. Anselmino et al., PRD75:054032, 2007
Francesca Giordano
AUT∝ h1 x H1⊥0
26
Transversity distributionFirst access in SIDIS
Collins FF
Transversity coupled with a Fragmentation Function (FF)
TMD
Very different energies between Hermes/Compass and Belle:
Is TMD evolution different from Collinear?
Complementary reactions
Collinear evolution assumed
TMD factorization assumed
FF universality assumedM. Anselmino et al., PRD75:054032, 2007
Francesca Giordano
AUT∝ h1 x H1⊥0
27
Transversity distributionFirst access in SIDIS
Collins FF
Transversity coupled with a Fragmentation Function (FF)
TMD
Very different energies between Hermes/Compass and Belle:
Is TMD evolution different from Collinear?
Different energies at Hermes/Compass: how does transversity
evolve?
Complementary reactions
Collinear evolution assumed
No evolution assumed
TMD factorization assumed
FF universality assumedM. Anselmino et al., PRD75:054032, 2007
Francesca Giordano
AUT∝ h1 x H1⊥0
28
Transversity distributionFirst access in SIDIS
Collins FF
Transversity coupled with a Fragmentation Function (FF)
TMD
Very different energies between Hermes/Compass and Belle:
Is TMD evolution different from Collinear?
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?
Complementary reactions
Collinear evolution assumed
No evolution assumed
TMD factorization assumed
Gaussian pT dependence assumed
FF universality assumedM. Anselmino et al., PRD75:054032, 2007
Francesca Giordano
AUT∝ h1 x H1⊥0
29
Transversity distributionFirst access in SIDIS
Collins FF
Transversity coupled with a Fragmentation Function (FF)
TMD
Complementary reactions
Collinear evolution assumed
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
x h1uv !x"! x h1dv !x"#4
PRELIMINARY
• from HERMES data• DiFF analysis
• from BELLE data• f1(x) from MSTW
• PRL107 & arXiv:1202.0323
!
!
!
!
1100
110
0
0.2
0.4
x
x h1uv !x"! x h1dv !x"#4
11% theo. err.
Transversity from Proton data
Transversity from pion pair production SIDIS off transversely polarized target
Band:Torino 2009 transversity
5% theo. err.• from COMPASS data• DiFF analysis
• from BELLE data• f1(x) from MSTW
• new analysis
PRELIMINARY
• from HERMES data• DiFF analysis
• from BELLE data• f1(x) from MSTW
• PRL107 & arXiv:1202.0323
11% theo. err.
Transversity from Proton data
Transversity from pion pair production SIDIS off transversely polarized target
Band:Torino 2009 transversity
5% theo. err.• from COMPASS data• DiFF analysis
• from BELLE data• f1(x) from MSTW
• new analysis
PRELIMINARY
• from HERMES data• DiFF analysis
• from BELLE data• f1(x) from MSTW
• 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
TMD factorization assumed
FF universality assumed
Francesca Giordano 30
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⪦
Francesca Giordano 31
Transversity distributionAccess in pp
X𝜎
Dqh
h
Jets
Single hadron
Looking at the hadron distribution within the jet
TMDAN∝ f1 x h1 x H1
⊥00
AN∝ f1 x h1 x H1⪦
Francesca Giordano 32
Transversity distributionAccess in pp
X𝜎
Dqh
h
Jets
Single hadronTMD
Word of caution: TMD factorization broken in pp➛hadrons
AN∝ f1 x h1 x H1⪦
AN∝ f1 x h1 x H1⊥00
Francesca Giordano 33
Transversity distributionAccess in pp
X𝜎
Dqh
h
Jets
Single hadronTMD
Collinear!
AN∝ f1 x h1 x H1⊥00
AN∝ f1 x h1 x H1⪦
Francesca Giordano 34
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)
Francesca Giordano 35
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