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

1

arXiv:1705.05889 accepted to PRL

2

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

3





d� =X

f

Zd⇠f(⇠)d�

d� =X

f


d� =X

f

Zd⇣D(⇣)d�

Partondistribu0onfunc0ons(PDFs)

QCDObservables

e+e� ! hX

e�N ! e�X

e�N ! e�hX

4





d� =X

f

Zd⇠f(⇠)d�

d� =X

f


d� =X

f

Zd⇣D(⇣)d�

Fragmenta0onfunc0ons(FFs)

QCDObservables

e+e� ! hX

e�N ! e�X

e�N ! e�hX

5





d� =X

f

Zd⇠f(⇠)d�

d� =X

f


d� =X

f

Zd⇣D(⇣)d�

PDFsandFFsmustbedeterminedthroughanalysesofexperimentaldata

QCDObservables

e+e� ! hX

e�N ! e�X

e�N ! e�hX

6

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

7

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+?

8


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+?

9

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

10

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

11

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�

12

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

13

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

14

Iterative Monte Carlo (IMC) Fitting Methodology

15


Initial iteration: flat sample priors

! Set of parameters used as initial guess for least-squares fits

16


Perform thousands of fits

17



! Pseudo-data constructed by bootstrap method

18



! Pseudo-data constructed by bootstrap method

! Data is partitioned for cross-validation – training set is fitted via chi-square minimization

19


Obtain a set of posteriors

! Set of parameters that minimize validation chi-square are chosen as posteriors

20


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

21


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

22

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

23


Penalty for fitting normalizations


•  PDFs: n = 1


)


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)

24


Modified likelihood to include prior information


�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


)


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)

25

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!

26

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

27


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

28


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)

32

BACKUP SLIDES

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


d�h

dz=

d�hL

dz+

d�hT

dz


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


d�h

dz=

d�hL

dz+

d�hT

dz


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

first simultaneous extraction of spin pdfs and ffs from a

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