First simultaneous extraction of spin PDFs and FFs from a ...
Transcript of First simultaneous extraction of spin PDFs and FFs from a ...
First simultaneous extraction of spin PDFs and FFs from a global
QCD analysis
Jacob Ethier
w/ Jefferson Lab Angular Momentum (JAM) members: Wally Melnitchouk, Nobuo Sato
Hadron Physics with Lepton and Hadron Beams Workshop September 6th, 2017
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arXiv:1705.05889 accepted to PRL
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QCDObservables
• Collinearfactoriza0on!QCDobservablessplitintohardsca9eringcrosssec0onanduniversal,non-perturba0vefunc0ons
• Deep-inelas0csca9ering(DIS)
• Semi-inclusiveDIS(SIDIS)
• Single-inclusiveannihila0on(SIA)
d� =X
f
Zd⇠f(⇠)d�
d� =X
f
Zd⇠d⇣f(⇠)D(⇣)d�
d� =X
f
Zd⇣D(⇣)d�
e+e� ! hX
e�N ! e�X
e�N ! e�hX
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• Collinearfactoriza0on!QCDobservablessplitintohardsca9eringcrosssec0onanduniversal,non-perturba0vefunc0ons
• Deep-inelas0csca9ering(DIS)
• Semi-inclusiveDIS(SIDIS)
• Single-inclusiveannihila0on(SIA)
d� =X
f
Zd⇠f(⇠)d�
d� =X
f
Zd⇠d⇣f(⇠)D(⇣)d�
d� =X
f
Zd⇣D(⇣)d�
Partondistribu0onfunc0ons(PDFs)
QCDObservables
e+e� ! hX
e�N ! e�X
e�N ! e�hX
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• Collinearfactoriza0on!QCDobservablessplitintohardsca9eringcrosssec0onanduniversal,non-perturba0vefunc0ons
• Deep-inelas0csca9ering(DIS)
• Semi-inclusiveDIS(SIDIS)
• Single-inclusiveannihila0on(SIA)
d� =X
f
Zd⇠f(⇠)d�
d� =X
f
Zd⇠d⇣f(⇠)D(⇣)d�
d� =X
f
Zd⇣D(⇣)d�
Fragmenta0onfunc0ons(FFs)
QCDObservables
e+e� ! hX
e�N ! e�X
e�N ! e�hX
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• Collinearfactoriza0on!QCDobservablessplitintohardsca9eringcrosssec0onanduniversal,non-perturba0vefunc0ons
• Deep-inelas0csca9ering(DIS)
• Semi-inclusiveDIS(SIDIS)
• Single-inclusiveannihila0on(SIA)
d� =X
f
Zd⇠f(⇠)d�
d� =X
f
Zd⇠d⇣f(⇠)D(⇣)d�
d� =X
f
Zd⇣D(⇣)d�
PDFsandFFsmustbedeterminedthroughanalysesofexperimentaldata
QCDObservables
e+e� ! hX
e�N ! e�X
e�N ! e�hX
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Proton spin structure from DIS • Spin sum rule:
�⌃(Q2) =
Z 1
0dx
��u
+(x,Q2) +�d
+(x,Q2) +�s
+(x,Q2)�
�q+ = �q +�q1
2=
1
2�⌃+�g + L
! Quark contribution:
! Gluon contribution:
! Orbital angular momentum: requires information from GPDs
⇡ 0.3[10�3,1]
�g(Q2) =
Z 1
0dx�g(x,Q2) ⇡ 0.1[0.05,0.2]
• Spin PDFs:
• Polarized DIS experiments extract information on the polarized structure function g1
! Determined from weak baryon decays under exact SU(2) and SU(3) flavor symmetries
Z 1
0dxg
p1(x,Q
2) =1
36[8�⌃+ 3gA + a8]
⇣1� ↵s
⇡
+O(↵2s)⌘+O(
1
Q
2)
• First moment requires additional knowledge of gA and a8 quantities to extract ΔΣ
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Strange polarization
10�3 10�2�0.06
�0.04
�0.02
0.00
0.02
0.04
x�
s+
0.1 0.3 0.5 0.7 x 10�3 10�20.0
0.1
0.2
0.3
0.4
0.5
x�
u+
JAM15
JAM13
DSSV09
AAC09
BB10
LSS10
NNPDF14
0.1 0.3 0.5 0.7 x
�s+(Q2) =
Z 1
0dx�s
+(x,Q2)
JAM15: DSSV09: Q2 = 1GeV2�s+ = �0.1± 0.01 �s+ = �0.11
• Discrepancy in shape of strange polarization (DSSV, LSS)! ! How does semi-inclusive DIS affect the shape of Δs+?
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Strange polarization
10�3 10�2�0.06
�0.04
�0.02
0.00
0.02
0.04
x�
s+
0.1 0.3 0.5 0.7 x 10�3 10�20.0
0.1
0.2
0.3
0.4
0.5
x�
u+
JAM15
JAM13
DSSV09
AAC09
BB10
LSS10
NNPDF14
0.1 0.3 0.5 0.7 x
�s+(Q2) =
Z 1
0dx�s
+(x,Q2)
JAM15: DSSV09: Q2 = 1GeV2�s+ = �0.1± 0.01 �s+ = �0.11
Extracted under SU(3) constraint
• Discrepancy in shape of strange polarization (DSSV, LSS)! ! How does semi-inclusive DIS affect the shape of Δs+?
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Semi-inclusive DIS • SIDIS observables require information on FFs ! contains information
about quark to hadron fragmentation.
d�SIDIS =X
f
Zd⇠d⇣�f(⇠)Df (⇣)d�
• Which SIDIS observable is most sensitive to strange PDF?
• From proton target:
d�K+
⇠ 4�uDK+
u +�sDK+
s
! Kaon valence structure contains strange flavor
d�K�⇠ 4�uDK�
u +�sDK�
s + 4�uDK�
u
K+(us) K�(us)
• From deuteron target:
d�K�⇠ 4(�u+�d)DK�
u + 2�sDK�
s + 4(�u+�d)DK�
u
d�K+
⇠ 4(�u+�d)DK+
u + 2�sDK+
s
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Semi-inclusive DIS • SIDIS observables require information on FFs ! contains information
about quark to hadron fragmentation.
d�SIDIS =X
f
Zd⇠d⇣�f(⇠)Df (⇣)d�
• Which SIDIS observable is most sensitive to strange PDF?
• From proton target:
d�K+
⇠ 4�uDK+
u +�sDK+
s
! Kaon valence structure contains strange flavor
d�K�⇠ 4�uDK�
u +�sDK�
s + 4�uDK�
u
K+(us) K�(us)
• From deuteron target:
d�K�⇠ 4(�u+�d)DK�
u + 2�sDK�
s + 4(�u+�d)DK�
u
small small
d�K+
⇠ 4(�u+�d)DK+
u + 2�sDK+
s
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0z
�0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
(a) zDK�s(a) zDK�s
HKNSDSS
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0z
�0.05
0.00
0.05
0.10
0.15
0.20
(b) zDK�u(b) zDK�u
HKNSDSS
! Recent JAM analysis extracted FFs from single-inclusive e+e- annihilation using the iterative Monte Carlo technique
Fragmentation Functions
! Choice of kaon FF parameterization influences shape of strange polarization density in SIDIS analysis (Leader, et al)
• SIDIS observables require information on FFs ! contains information about quark to hadron fragmentation.
(arXiv:1609:00899)
d�SIDIS =X
f
Zd⇠d⇣�f(⇠)Df (⇣)d�
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JAM17 Combined Analysis
• We perform the first ever combined Monte Carlo analysis of polarized DIS, polarized SIDIS, and SIA data (at NLO)
• Spin PDFs and FFs are fitted simultaneously
d�DIS =X
f
Zd⇠�f(⇠)d� d�SIA =
X
f
Zd⇣Df (⇣)d�
d�SIDIS =X
f
Zd⇠d⇣�f(⇠)Df (⇣)d�
• SU(2) and SU(3) constraints used in DIS only analyses are released Z 1
0dx
��u
+ ��d
+� ?= gA
Z 1
0dx
��u
+ +�d
+ � 2�s
+� ?= a8
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Traditional Fitting Method
• Functional form for PDF and FF, e.g.
• Single fit of parameters
Typically fix parameters that are difficult to constrain
• Uncertainties determined by Hessian or Lagrange multiplier methods
Introduces tolerance criteria
• Since is a highly non-linear function of the fit parameters, there can be various local minima
�2
• We can efficiently explore the parameter space using an iterative Monte Carlo procedure
xf(x) = Nx
a(1� x)b(1 + c
px+ dx)
�2
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Iterative Monte Carlo (IMC) Fitting Methodology
Initial iteration: flat sample priors
! Set of parameters used as initial guess for least-squares fits
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Iterative Monte Carlo (IMC) Fitting Methodology
Perform thousands of fits
! Pseudo-data constructed by bootstrap method
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Iterative Monte Carlo (IMC) Fitting Methodology
Perform thousands of fits
! Pseudo-data constructed by bootstrap method
! Data is partitioned for cross-validation – training set is fitted via chi-square minimization
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Iterative Monte Carlo (IMC) Fitting Methodology
Obtain a set of posteriors
! Set of parameters that minimize validation chi-square are chosen as posteriors
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Iterative Monte Carlo (IMC) Fitting Methodology
Posteriors are sent through a sampler
! Kernel density estimation (KDE): estimates the multi-dimensional probability density function of the parameters
! A sample of parameters is chosen
from the KDE and used as starting priors for the next iteration
! Iterated until distributions are converged
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Iterative Monte Carlo (IMC) Fitting Methodology
Obtain final set of parameters
E[O] =1
n
nX
k=1
O(ak)
V[O] =1
n
nX
k=1
(O(ak)� E[O])2
! Compute mean and standard deviation of observables
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Parameterizations and Chi-square
• Chi-squared definition:
�2(a) =X
e
2
4X
i
D(e)
i N (e)i � T (e)
i (a)
↵(e)i N (e)
i
!2
+X
k
⇣r(e)k
⌘23
5+X
`
✓a(`) � µ(`)
�(`)
◆2
• PDFs: n = 1
• FFs: n = 2, c = 0 Favored: Dhq+ = T(z;a) + T(z;a0
)
Unfavored: Dhq+,g = T(z;a)
Pions: Kaons:
DK+
s = T(z;a)
DK+
u = DK+
d =1
2DK+
d+
D⇡+
s = D⇡+
s =1
2D⇡+
s+
D⇡+
u = D⇡+d = T(z;a)
T(x;a) =M x
a(1� x)b(1 + c
px)
B(n+ a, 1 + b) + cB(n+ 12 + a, 1 + b)
Template function:
�q
+,�q,�g = T(x;a)
�2(a) =X
e
2
4X
i
D(e)
i N (e)i � T (e)
i (a)
↵(e)i N (e)
i
!2
+X
k
⇣r(e)k
⌘23
5+X
`
✓a(`) � µ(`)
�(`)
◆2
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Parameterizations and Chi-square
Penalty for fitting normalizations
• Chi-squared definition:
• PDFs: n = 1
• FFs: n = 2, c = 0 Favored: Dhq+ = T(z;a) + T(z;a0
)
Unfavored: Dhq+,g = T(z;a)
Pions: Kaons:
DK+
s = T(z;a)
DK+
u = DK+
d =1
2DK+
d+
D⇡+
s = D⇡+
s =1
2D⇡+
s+
D⇡+
u = D⇡+d = T(z;a)
T(x;a) =M x
a(1� x)b(1 + c
px)
B(n+ a, 1 + b) + cB(n+ 12 + a, 1 + b)
Template function:
�q
+,�q,�g = T(x;a)
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Parameterizations and Chi-square
Modified likelihood to include prior information
• Chi-squared definition:
�2(a) =X
e
2
4X
i
D(e)
i N (e)i � T (e)
i (a)
↵(e)i N (e)
i
!2
+X
k
⇣r(e)k
⌘23
5+X
`
✓a(`) � µ(`)
�(`)
◆2
• PDFs: n = 1
• FFs: n = 2, c = 0 Favored: Dhq+ = T(z;a) + T(z;a0
)
Unfavored: Dhq+,g = T(z;a)
Pions: Kaons:
DK+
s = T(z;a)
DK+
u = DK+
d =1
2DK+
d+
D⇡+
s = D⇡+
s =1
2D⇡+
s+
D⇡+
u = D⇡+d = T(z;a)
T(x;a) =M x
a(1� x)b(1 + c
px)
B(n+ a, 1 + b) + cB(n+ 12 + a, 1 + b)
Template function:
�q
+,�q,�g = T(x;a)
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Data vs Theory – SIDIS
10�10.0
0.2
0.4
0.6
0.8
1.0
1.2
A⇡+
1,p HERMES
10�10.00.10.20.30.40.50.60.70.80.9
A⇡�1,p
10�1�0.1
0.0
0.1
0.2
0.3
0.4
0.5
A⇡+
1,d
10�1�0.4�0.3�0.2�0.1
0.00.10.20.30.40.5
A⇡�1,d
10�2 10�1
x�0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
A⇡+
1,p COMPASS
10�2 10�1
x�0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
A⇡�1,p
10�2 10�1
x�0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
A⇡+
1,d
10�2 10�1
x�0.05
0.00
0.05
0.10
0.15
0.20
0.25
A⇡�1,d
10�1
0.0
0.2
0.4
0.6
0.8 AK+
1,d HERMES
10�1�1.0
�0.5
0.0
0.5
1.0
1.5
2.0
2.5
AK�1,d
10�1�0.2
0.0
0.2
0.4
0.6
0.8
1.0
AK++K�
1,d
10�2 10�1
x�0.2
0.0
0.2
0.4
0.6
0.8
1.0
AK+
1,p COMPASS
10�2 10�1
x�0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
AK�1,p
10�2 10�1
x�0.10
�0.05
0.00
0.05
0.10
0.15
0.20
0.25
0.30
AK+
1,d
10�2 10�1
x�0.20�0.15�0.10�0.05
0.000.050.100.150.200.25
AK�1,d
process target Ndat �2
DIS p, d, 3He 854 854.8SIA (⇡±,K±) 850 997.1SIDIS (⇡±)HERMES d 18 28.1HERMES p 18 14.2COMPASS d 20 8.0COMPASS p 24 18.2
SIDIS (K±)HERMES d 27 18.3COMPASS d 20 18.7COMPASS p 24 12.3
Total: 1855 1969.7
• Good agreement overall with all data!
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Polarized PDF Distributions
• Δu+ consistent with previous analysis
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4 x�u+
JAM17
JAM15
0 0.2 0.4 0.6 0.8 1
�0.15
�0.10
�0.05
0 x�d+
0.4 0.810�3 10�2 10�1
�0.04
�0.02
0
0.02
0.04 x(�u + �d)
DSSV09
0.4 0.810�3 10�2 10�1
�0.04
�0.02
0
0.02
0.04
x(�u � �d)
0.01 0.1 0.5x0
0.2
0.4
0.6
A⇡�1p
JAM17
�q = 0
COMPASS
Q2 = 1GeV2 • Δd+ slightly larger in magnitude
! Anti-correlation with ∆s+, which is less negative than JAM15 at x ~ 0.2
• Isoscalar sea distribution consistent with zero
• Isovector sea slightly prefers positive shape at low x
! Non-zero asymmetry given by small contributions from SIDIS asymmetries
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Strange polarization
0.4 0.8x
10�3 10�2 10�1
�0.04
�0.02
0
0.02
0.04x�s+ JAM17 + SU(3)
DSSV09
JAM15
0.03 0.1 0.2 0.5x�0.2
0
0.2
0.4
0.6
0.8
AK�1d
JAM17
�s+ < 0
HERMES
• Δs+ distribution consistent with zero, slightly positive in intermediate x range
• Primarily influenced by HERMES K- data from deuterium target
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Strange polarization
0.4 0.8x
10�3 10�2 10�1
�0.04
�0.02
0
0.02
0.04x�s+ JAM17 + SU(3)
DSSV09
JAM15
0.03 0.1 0.2 0.5x�0.2
0
0.2
0.4
0.6
0.8
AK�1d
JAM17
�s+ < 0
HERMES
Why does DIS+SU(3) give large negative ∆s+?
• Low x DIS deuterium data from COMPASS prefers small negative Δs+
• Needs to be more negative in intermediate region to satisfy SU(3) constraint
• Δs+ distribution consistent with zero, slightly positive in intermediate x range
• Primarily influenced by HERMES K- data from deuterium target
• b parameter for ∆s+ typically fixed to values ~6-10, producing a peak at x~0.1
29
Moments
1.1 1.2 1.3
Nor
mal
ized
yiel
d a3 SU(2)
0 0.5 1
a8 SU(3)
0.2 0.3 0.4 0.5
Nor
mal
ized
yiel
d
�⌃
�0.2 0 0.2
�u �d�
Confirmation of SU(2) symmetry to ~2%
~20% SU(3) breaking ± ~20%; large uncertainty
• Need better determination of Δs+ moment to reduce a8 uncertainty!
gA = 1.24± 0.04
a8 = 0.46± 0.21
�s+ = �0.03± 0.09
30
Moments
Slightly larger central value than previous analyses, but consistent within uncertainty �⌃ = 0.36± 0.09
�u��d = 0.05± 0.08Preference for slightly positive sea asymmetry; not very well constrained by SIDIS
1.1 1.2 1.3
Nor
mal
ized
yiel
d a3 SU(2)
0 0.5 1
a8 SU(3)
0.2 0.3 0.4 0.5
Nor
mal
ized
yiel
d
�⌃
�0.2 0 0.2
�u �d�
31
Summary and Outlook
• QCD observables yet to be implemented: • W asymmetries for constraints on up and down sea polarization • Unpolarized SIDIS and single-inclusive pp collision for FFs
• Working towards a universal fit of quark helicity distributions q", q#
• Global analyses of combined unpolarized and polarized data
• Difficult to determine a8 with DIS+SIDIS, but results confirm SU(2) symmetry to ~2%
• Data sensitive to Δs+ distribution give result consistent with zero with large uncertainties
! Need higher precision polarized SIDIS kaon data
• Analysis suggests the resolution of the “strange polarization puzzle” !Shape of Δs+ in DIS+SU(3) analyses is artificial ( caused by SU(3) constraint + large-x shape parameter)
33
Fragmentation Functions
0.2 0.4 0.6 0.8 1z
0.2
0.4
0.6
0.8
zDh q(z
)
u+
u
⇡+
Q2 = 5 GeV2
0.2 0.4 0.6 0.8 1z
0.1
0.2
0.3
0.4
s+
u+
s
K+
• Little change in ‘plus’ distributions from JAM16
• Agreement with DSS’s strange FF (dashed red line)
• Uncertainty for unfavored ū to π distribution smaller than s to K
!s+ to K+ FF marginally smaller at low-z compared to JAM16
!Due to lower precision kaon production data
34
PolarizedDIS
Ak =�"+ � �"*
�"+ + �"* = D (A1 + ⌘A2)
• Wefitmeasurementsofparallelandperpendicularspinasymmetries
A? =�") � �"(
�") + �"( = d (A2 + ⇣A1)
A1 =(g1 � �2g2)
F1A2 = �
(g1 + g2)
F1�
2 =4M2
x
2
Q
2
• Defineourpolarizedstructurefunc0ons:
g1,2(x,Q2) = g
LT+TMC1,2 (�u
+,�d
+,�g, ...)
35
Leading Twist Structure Functions
⇠ =2x
1 + ⇢
, ⇢
2 = 1 + �
2
g
(⌧2)1 (x,Q2) =
1
2
X
q
e
2q
⇥(�Cq ⌦�q
+)(x,Q2) + (�Cg ⌦�g)(x,Q2)⇤
g
(⌧2+TMC)1 (x,Q2) =
x
⇠⇢
3g
(⌧2)1 (⇠, Q2) +
(⇢2 � 1)
⇢
4
Z 1
⇠
dz
z
(x+ ⇠)
⇠
� (3� ⇢
2)
2⇢ln
z
⇠
�g
(⌧2)1 (z,Q2)
g
(⌧2+TMC)2 (x,Q2) = � x
⇠⇢
3g
(⌧2)1 (⇠, Q2) +
1
⇢
4
Z 1
⇠
dz
z
x
⇠
� (⇢2 � 1) +3(⇢2 � 1)
2⇢ln
z
⇠
�g
(⌧2)1 (z,Q2)
• Leading twist structure function defined in collinear factorization as
• Leading twist + target mass corrections
• In Bjorken limit :
J. Blümlein and A. Tkabladze Nucl. Phys. B553, 427 (1999)
�Q2 ! 1
�
g(⌧2+TMC)1 = g(⌧2)1 g
(⌧2)2 (x,Q2) = �g
(⌧2)1 (x,Q2) +
Z 1
x
dz
z
g
(⌧2)1 (z,Q2)
36
Semi-inclusive DIS • In SIDIS, we fit the photon-nucleon spin asymmetries (assuming Bjorken limit)
• The (un)polarized structure functions are defined
A
h1
�x, z,Q
2�=
g
h1 (x, z,Q
2)
F1(x, z,Q2)
g
h1 (x, z,Q
2) =1
2
X
q
e
2q
⇢�q(x, µF )D
hq (z, µFF ) +
↵s(µR)
2⇡
⇥✓�q ⌦�Cqq ⌦D
hq +�q ⌦�Cgq ⌦D
hg +�g ⌦�Cqg ⌦D
hq
◆�
F
h1 (x, z,Q
2) =1
2
X
q
e
2q
⇢q(x, µF )D
hq (z, µFF ) +
↵s(µR)
2⇡
⇥✓q ⌦ Cqq ⌦D
hq + q ⌦ Cgq ⌦D
hg + g ⌦ Cqg ⌦D
hq
◆�
37
Single-inclusive Annihilation (SIA)
• Cross sections for
d�h
dz=
d�hL
dz+
d�hT
dz
• Transverse and longitudinal cross sections
d�hT
dz=
X
i
�i
hDi(z,Q
2) +↵s
2⇡
�CT
i ⌦Di
�(z,Q2)
i+ �0
↵s
2⇡
�CT
g ⌦Dg
�(z,Q2)
d�hL
dz=
X
i
�i↵s
2⇡
�CL
i ⌦Di
�(z,Q2) + �0
↵s
2⇡
�CL
g ⌦Dg
�(z,Q2)
e+e� ! hX
z =2Ehp
s
38
Single-inclusive Annihilation (SIA)
d�h
dz=
d�hL
dz+
d�hT
dz
• Transverse and longitudinal cross sections
d�hT
dz=
X
i
�i
hDi(z,Q
2) +↵s
2⇡
�CT
i ⌦Di
�(z,Q2)
i+ �0
↵s
2⇡
�CT
g ⌦Dg
�(z,Q2)
d�hL
dz=
X
i
�i↵s
2⇡
�CL
i ⌦Di
�(z,Q2) + �0
↵s
2⇡
�CL
g ⌦Dg
�(z,Q2)
• Electroweak cross section :
�i(s) = �0
he2i + 2ei g
iV g
eV ⇢1(s) +
�ge 2A + ge 2V
� �gi 2A + gi 2V
�⇢2(s)
ie+e� ! �, Z ! qq
• Cross sections for e+e� ! hX
z =2Ehp
s
39
Single-inclusive Annihilation (SIA)
d�h
dz=
d�hL
dz+
d�hT
dz
• Transverse and longitudinal cross sections
d�hT
dz=
X
i
�i
hDi(z,Q
2) +↵s
2⇡
�CT
i ⌦Di
�(z,Q2)
i+ �0
↵s
2⇡
�CT
g ⌦Dg
�(z,Q2)
d�hL
dz=
X
i
�i↵s
2⇡
�CL
i ⌦Di
�(z,Q2) + �0
↵s
2⇡
�CL
g ⌦Dg
�(z,Q2)
• Electroweak cross section :
�i(s) = �0
he2i + 2ei g
iV g
eV ⇢1(s) +
�ge 2A + ge 2V
� �gi 2A + gi 2V
�⇢2(s)
ie+e� ! �, Z ! qq
• Cross sections for e+e� ! hX
• Typically, observables are normalized by total hadronic cross section
�tot
(s) =X
i
�i
⇣1 +
↵s
⇡
⌘
z =2Ehp
s