Wally Melnitchouk

Novel Probes of Nucleon Structure in SIDIS, e+e- & pp (FF2019) Duke University, March 14, 2019

Simultaneous analysis ofPDFs and fragmentation functions

— JAM19 —

JLab Angular Momentum collaboration

https://www.jlab.org/jam

OverviewJAM (Jefferson Lab Angular Momentum) collaboration aimsto study the quark-gluon structure of hadrons throughextraction of quantum probability distributions (PDFs, FFs, TMDs) via global QCD analysis using Monte Carlo based methods

Methodology based on Bayesian statistics and Monte Carlosampling of the parameter space

Inter-dependence of observables on distributions requiressimultaneous extraction of PDFs & fragmentation functions

�AB!CX(pA, pB) =X

a,b

Zdxa dxb fa/A(xa, µ) fb/B(xb, µ)

⇥X

n

�ns (µ) ⇥

(n)ab!CX (xapA, xbpB , Q/µ)

A B

C

a b�

X

extraction of functions characterizing structureof bound state A is a typical “inverse problem”

fa/A

xa xb

Parton distributions in hadrons

Generic process: inclusive particle production AB ! C X

Bayesian approach to global analysisAnalysis of data requires estimating expectation values Eand variances V of “observables” (functions of PDFs) which are functions of parameters

O

E[O] =

ZdnaP(~a|data)O(~a)

V [O] =

ZdnaP(~a|data) ⇥O(~a)� E[O]

⇤2

P(~a|data) = 1

ZL(data|~a)⇡(~a)

in terms of the likelihood function L

Using Bayes’ theorem, probability distribution given byP

“Bayesian master formulas"

Bayesian approach to global analysis

�2(~a) =

X

i

✓data i � theoryi(~a)

�(data)

◆2

Likelihood function

L(data|~a) = exp

✓�1

2

�2(~a)

◆

is a Gaussian form in the data, with function�2

with priors and evidence⇡(~a) Z

Z =

ZdnaL(data|~a)⇡(~a)

Z tests if e.g. an n-parameter fit is statistically differentfrom (n+1)-parameter fit

maximize probability distributionP(~a|data) ! ~a0

E[O(~a)] = O(~a0) V [O(~a)] ! Hessian

if is linear in parameters, and if probability issymmetric in all parametersO

need more robust (Monte Carlo) approach

,

E[O] ⇡ 1

N

X

k

O(~ak) V [O] ⇡ 1

N

X

k

⇥O(~ak)� E[O]⇤2,

Hij =1

2

@�2(~a)

@ai@aj

���~a=~a0

Standard method for evaluating E, V via maximum likelihood

In practice, since in general ,maximum likelihood method sometimes fails

E[f(~a)] = f(E[~a])

Bayesian approach to global analysis

Monte Carlo methods

f(x) = N x

↵(1� x)� P (x)

Gbedo, Mangin-Brinet (2017)

Skilling (2004)

Forte et al. (2002)P (x)

First group to use MC for global PDF analysis was NNPDF,using neural network to parametrize in

Iterative Monte Carlo (IMC), developed by JAM Collaboration,variant of NNPDF, tailored to non-neutral net parametrizations

Markov Chain MC (MCMC) / Hybid MC (HMC)— recent “proof of principle” analysis, ideas from lattice QCD

Nested sampling (NS) — computes integrals in Bayesian masterformulas (for E, V, Z) explicitly

N. Sato et al. (2016)

E. Nocera talk

10�2

0.1

0.2

0.3

0.4

0.5JAM15

no JLab

0.1 0.3 0.5 0.7

x�u+

10�2

�0.15

�0.10

�0.05

x�d+

0.1 0.3 0.5 0.7

10�2

�0.04

�0.02

0.00

0.02

0.04

0.1 0.3 0.5 0.7

x�s+

10�2�0.1

0.0

0.1

0.2

0.1 0.3 0.5 0.7 x

x�g

10�2

�0.05

0.00

0.05

0.10

0.15

0.1 0.3 0.5 0.7

xDu

10�2

�0.10�0.05

0.000.050.100.15 xDd

0.1 0.3 0.5 0.7

10�2

�0.010

�0.005

0.000

0.005

0.010

xHp

0.1 0.3 0.5 0.7

10�2�0.04

�0.02

0.00

0.02

0.04

0.06

xHn

0.1 0.3 0.5 0.7 x

First application of IMC — proton spin structure

s-quark polarization negativefrom inclusive DIS data(assuming SU(3) symmetry)

0.01

Q2

(GeV

2 )

1

10

100EMC

SMC

COMPASS

HERMES

SLAC

JLab

W 2 = 10 GeV2

W 2 = 4 GeV2

0.1 0.3 0.5 0.7 0.9x

reduced uncertainty in , through Q evolution

�s+ �g2

inclusion of JLab data increases# data points by factor ~ 2

Previous JAM analyses

Sato, WM, Kuhn, Ethier, Accardi (2016)

γ∗

q

N*

M

X

N

=N*,N’* q, X

Σ

γ∗M

N’*N

Σ

Inclusive DIS data cannot distinguish between q and q_

semi-inclusive DIS sensitive to �q & �q

⇠X

q

e

2q

⇥�q(x)Dh

q (z) +�q(x)Dhq (z)

⇤

but need fragmentation functions…

Global analysis of DIS + SIDIS data gives different sign forstrange quark polarization for different fragmentation functions!

for “DSS” FFs�s < 0

�s > 0

need to understand origin of differences in fragmentationfunctions

for “HKNS” FFs Hirai et al. (2007)

de Florian et al. (2007)

Previous JAM analyses

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

init

ial

u+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

d+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

s+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

c+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

b+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

g

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

inte

rmed

iate ⇡+

K+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

⇡+

K+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

inte

rmed

iate

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

final

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

convergence after ~ 20 iterations

Sato, Ethier, WM, Hirai,Kumano, Accardi (2016)

First MC analysis of fragmentation functions

e+e� ! hXsingle-inclusiveannihilation (SIA)

Previous JAM analyses

0.2 0.4 0.6 0.8

0.2

0.6

1.0

1.4

zD(z

) ⇡+

K+

u+

0.2 0.4 0.6 0.8

0.2

0.6

1.0

1.4

⇡+

K+

d+

0.2 0.4 0.6 0.8

0.2

0.6

1.0

1.4

⇡+

K+

s+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

zD(z

) ⇡+

K+

c+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

⇡+

K+

b+

0.2 0.4 0.6 0.8 z

0.2

0.6

1.0

1.4

⇡+

K+

g

favored FFs well constrained; unfavored not as well…

hard g K fragmentation?nontrivial shape of s K fragmentation

Sato, Ethier, WM, Hirai,Kumano, Accardi (2016)

Previous JAM analysesFirst MC analysis of fragmentation functions

0 0.2 0.4 0.6 0.8 10

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)

0.4 0.8x

10�3 10�2 10�1

�0.04

�0.02

0

0.02

0.04 x�s+

JAM17 + SU(3)0.4 0.8

x10�3 10�2 10�1

�0.1

�0.05

0

0.05

0.1 x�s�

no assumption ofSU(3) symmetry

First simultaneous extraction of spin PDFs and FFs,fitting polarized DIS + SIDIS (HERMES, COMPASS) and SIA data

slightly > 0 at high x, consistent with zero�s

consistent with zero�s��s & �u��d

Previous JAM analyses

Ethier, Sato, WMPRL 119, 132001 (2017)

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+

HKNS

DSS

new result more consistent with DSS at moderate z

some constraint from SIDIS on unfavored FFs , but uncertainties still large(e.g. s ! K+)

Ethier, Sato, WM (2017)

Previous JAM analysesFirst simultaneous extraction of spin PDFs and FFs,fitting polarized DIS + SIDIS (HERMES, COMPASS) and SIA data

0 0.2 0.4 0.6 0.8 10

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)

0.4 0.8x

10�3 10�2 10�1

�0.04

�0.02

0

0.02

0.04 x�s+

JAM17 + SU(3)0.4 0.8

x10�3 10�2 10�1

�0.1

�0.05

0

0.05

0.1 x�s�

DIS only,with SU(3)

DIS+SIDIS,no SU(3)

Polarized strangeness in previous, DIS-only analyses wasnegative at x ~ 0.1, induced by SU(3) and parametrization bias

weak sensitivity to from DIS data & evolution— SU(3) pulls to generate moment ~ -0.1

�s+

�s+

— negative peak at x ~ 0.1 induced by fixing b ~ 6 - 8

Ethier, Sato, WM (2017)

�s = �0.03(10)less negative gives larger total helicity �⌃ = 0.36(9)

Previous JAM analyses

What impact does unpolarized strange PDF have on theextraction of polarized strange?

only systematic way is to fit unpolarized PDFs, polarized PDFsand fragmentation functions simultaneously…

Shape of unpolarized strange PDF is interesting(and controversial) in its own right!

Strange quarks

x

xS(x

)

0.02 0.1 0.6

CTEQ6L

CTEQ6.5S-0NNPDF2.3

Fit

x(u–(x)+d

–(x))

HERMES with ∫DSK(z,Q2)dz=1.27⟨Q2⟩=2.5 GeV2

0

0.2

0.4

HERMES, PRD 89 (2014) 097101

=s+ s

u+ d⇠ 0.2� 0.5

~ 0.2 - 0.5historically, strange to nonstrange ratio

(2012)pp ! W (Z) +X

ATLASPRL 109 (2012) 012001

rs = (s+ s)/2d

= 1.00+0.25�0.28

+DIS (no HERA)

+DIS (HERA)

+DY

+SIA

+SIDIS

+�DIS

+�SIDIS

w1

w2

w3

w4

w5

w6

w7

trial PDFs

PDFs 1

PDFs 2

PDFs 3

PDFs 5

PDFs 7

final PDFs

trial FFs

FFs 4

FFs 5

FFs 7

final FFs

trial �PDFs

�PDFs 6

�PDFs 7

final �PDFs

Visualization

Parametrization

lNNN

ll

Multi step

MCsampling

JAM 2019 analysis

Multistep MC sampling

JAM 2019 analysis

Use k-means clustering algorithm to identify regionsin parameter space where solutions cluster

“Inverse problem” leads to multiple solutionsexplore possible correlations between solutionsby identifying clusters of solutions

e.g.

JAM 2019 analysisfixed-target DIS only

PRELIMINARY

red cyan green yellow

g uv dv

d

u

s s s s-

-u

�2 "�2 #

d

JAM 2019 analysisfixed-target DIS + HERA

PRELIMINARYg uv dvu

d s s -s s

d-u

JAM 2019 analysisfixed-target DIS + HERA + DY

PRELIMINARYg

d

uv dvu

ss

s-s

d u-

JAM 2019 analysisSIA (pion)

D⇡g D⇡

u D⇡d

D⇡s D⇡

c

PRELIMINARY

JAM 2019 analysisSIA (pion) + SIDIS (pion)

D⇡g D⇡

u D⇡d

D⇡s D⇡

c

PRELIMINARY

JAM 2019 analysisfixed-target DIS + HERA + DY + SIA (pion) + SIDIS (pion)

g

PRELIMINARYuv dvu

d s s s-s

d-u

JAM 2019 analysisSIA (kaon)

PRELIMINARYDKg DK

uDK

d

DKs

DKs DK

c

JAM 2019 analysisSIA (kaon) + SIDIS (kaon) + …

PRELIMINARYDKg

DKs

DKs

DKdDK

u

DKc

JAM 2019 analysis

need more precise SIDIS data (JLab12, EIC)

fixed-target DIS + HERA + DY + SIA (pion) + SIDIS (pion)

PRELIMINARYg uv

existing SIDIS data do notstrongly discriminate between different strange PDFs

dvu

ds s

s s-

d-u

Strange quarks from PVDIS

Parity-violating DIS allows strange contribution to be isolated, when combined with e.m. p and n DIS data at low/intermediate x

e (k) e (k’)

N (p) N (p’)

Z (q 2)γ

∗ (q 1)

e e

N N

Z�⇤F

�p2 =

4

9x(u+ u) +

1

9x(d+ d+ s+ s) + · · ·

at leading order

F

�n2 =

4

9x(d+ d) +

1

9x(u+ u+ s+ s) + · · ·

F

�Z,p2 =

✓1

3� 8

9sin2 ✓W

◆x(u+ u) +

✓1

6� 2

9sin2 ✓W

◆(d+ d+ s+ s) + · · ·

⇡ 1

9x(u+ u+ d+ d+ s+ s) + · · · for sin

2 ✓W ⇡ 1/4

3 equations with 3 unknowns;

V x A term also sensitive to s� s

F �Z,p3 =

2

3(u� u) +

1

3(d� d+ s� s) + · · ·

order of magnitude greater sensitivity of to strange PDF�Z

27

OutlookNew paradigm in global analysis — simultaneous determinationof collinear distributions using MC sampling of parameter space

Short-term: “universal” QCD analysis of all observables sensitive to collinear (unpolarized & polarized) PDFs and FFs

Longer-term: technology developed will be applied to global analysis of transverse momentum dependent (TMD) distributionsto map out full 3-d image of hadrons