RECONSTRUCTING PARTON DISTRIBUTION FUNCTIONS FROM THEIR COORDINATE SPACE BEHAVIOR

(1)

RECONSTRUCTING PARTON

DISTRIBUTION FUNCTIONS FROM

THEIR COORDINATE SPACE

BEHAVIOR

Light Cone Meeting,

Jefferson Lab, May 14-19, 2018

Simonetta Liuti

(2)

(3)

The aim of our work is to pose (and possibly answer) the

following questions:

1.  What information from PDF fits is relevant to constrain/test/validate

lattice calculations?

2.  What PDF-related quantities we would like lattice QCD to compute?

5/15/18 3

1.  Can we quantify the accuracy that lattice calculations should

have in order to have a direct impact on PDF fits

2.  To which extent available lattice results agree with the results of

global PDF fits?

….extend to Generalized Parton Distributions!

The Review poses (and possibly answers some of) the

following questions:

(4)

(5)

F. Yndurain, PLB 1978 Rep. Prog. In Physics 1983

Early attempts: Reconstructing DIS structure functions from their Mellin moments, using the orthogonal polynomials technique

N=3 N=8 Mn(Q2) = Z 1 0 dx xn 2F2(x, Q2) Mellin Moment

(6)

Z 1 0 dx B_km(x)F2(x, Q2) = (m+ 1)! k! m k_X l=0 ( 1)l l!(m k l)! Mk+l+2(Q 2₎

In a nutshell….

… replace Mellin Moment with Orhtogonal Polynomial Moment

Mn(Q2) =

Z 1

0

(7)

0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 Polynomials x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 11th Bernstein Polynomial x 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x F2P 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x F2P Structure Function

Integrand of Bernstein Moment Bernstein Polynomial -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x Q2 = 1.8 GeV2 F₂P

(8)

GPD/PDF CORRELATION FUNCTION

W_⇤0_⇤(x,⇠, ) = 1 2 Z dz (2⇡)e ixP+z _h_P _,_⇤0 _| _q_¯₍₀₎ _W ₀_,_z _q _z _| _P,_⇤_i zT=0,z+=0 .

q

,

Λ

_γ

k’= k –

Δ

.

λ

’

q'=q+

Δ

,

Λ

_γ

’

P’= P-

Δ

,

Λ

’

P

,

Λ

k

,

λ

t Q2 x, ξ h

(9)

GPDs involve two types of distance

5/15/18 9

H

q

(x

,

0 ,

) =

Z

dz

2 ⇡

e

ixP+z

h

P

,

⇤

0

_|

q

¯

(0)

+

q

(

z

)

_|

P,

⇤

_i

zT=0

¯

q

₊†

(0

,

b

)

q

₊

(

z

,

b

)

_!

⇢

(0

,

b

;

z

,

b

)

x distribution Fourier transform of non-diagonal density distribution in z

(10)

Hu_(x,0,_Δ₎ Fq(t _⌘ 2) = Z dxHq(x,0, ) _! ⇢q(b) O. Gonzalez-Hernandez et al., PRC88 (2013) x dependence t large t small

(11)

5/15/18 11

h

P

,

⇤

0

_|

q

¯

(0)

+

_W

(0

, z

)

q

(

z

)

_|

P,

⇤

_i

_z_T ₌₀ P’ 0 z -b P P-P’=Δ b

(12)

Reconstructing GPDs from a finite number of moments

0.0 1.0 2.0 3.0 4.0 H_u-dat =0.0 t=-0.0 N=2 N=7 0.0 1.0 2.0 3.0 4.0 t=-0.1 0.0 0.5 1.0 1.5 2.0 2.5 t=-0.2 0.0 0.5 1.0 1.5 2.0 2.5 t=-0.3 0.0 0.2 0.4 0.6 0.8 x 0.0 0.4 0.8 1.2 1.6 t=-0.5 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.4 0.8 1.2 1.6 t=-1.0

Bernstein polynomials based analysis

H. Honkanen et al., EPJC(2007)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 Hu-dat Q 2 =4 GeV2 =0.18 t=-0.035 0.0 0.5 1.0 1.5 2.0 2.5 3.0 =0.18 t=-1.00 0.0 0.5 1.0 1.5 2.0 2.5 =0.25 t=-0.073 0.0 0.5 1.0 1.5 2.0 2.5 =0.25 t=-1.00 0.0 0.5 1.0 1.5 2.0 2.5 =0.36 t=-0.18 0.0 0.5 1.0 1.5 2.0 2.5 =0.36 t=-1.00 0.0 0.2 0.4 0.6 0.8 x 0.0 0.5 1.0 1.5 2.0 2.5 =0.53 t=-0.53 0.0 0.2 0.4 0.6 0.8 1.0 x 0.0 0.5 1.0 1.5 2.0 2.5 =0.53 t=-1.00

(Using lattice moments from LHPC, Ph. Hägler et al., 2005, 2006, 2008)

3 moments from LHPC 8 moments from param.

(13)

2. DETERMINATION OF PDF

FROM COORDINATE SPACE

BEHAVIOR

(14)

Example: Unpolarized PDFs

5/15/18 14

Parton x-momentum distribution is the Fourier transform of a non-diagonal one body density distribution in coordinate z-space

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0.0001 0.001 0.01 0.1 1 PDFs

U_v + D_v Q2 = 150 GeV2 Collaboration and SOM PDFs

x ABM CT10 CJMid 6x6

¯

q

₊†

(0)

q

₊

(

z

)

_!

⇢

(0

,

z

)

E. Askanazi, K. Holcomb, S. L., J. Phys. G 42, no. 3, 034030 (2015) [arXiv:1411.2487 [hep-ph]].

F

₂

(

x

) =

X

q

xf

₁q

(

x

)

f

₁q

(

x

) =

Z

dz

2 ⇡

e

ixP+z

h

P,

⇤

_|

q

¯

(0)

+

q

(

z

)

_|

P,

⇤

_i

_z_T ₌₀

(15)

Symmetry: C-parity

¯

q

(

x

) =

q

(

x

)

q

(

x

) + ¯

q

(

x

)

q

(

x

)

q

¯

(

x

)

Collins and Soper, ‘80s

C-odd C-even

Symmetric with respect to x=0 Anti-symmetric with respect to x=0

(16)

hP,⇤0 _| q¯(0) +q z _| P,⇤_i + _hP,⇤0 _| q¯(0) +q z _| P,⇤_i = i 2P+ Z 1 0 dxqs(x) sin(xz) hP,⇤0 _| q¯(0) +q z _| P,⇤_i _hP,⇤0 _| q¯(0) +q z _| P,⇤_i = 1 2P+ Z 1 0 dxqv(x) cos(xz) T_c T_s

(17)

T

_s

(

z

)

=

i

2 M

Z

1 0

dxq

(

x

) sin(

xz

) =

i

2 M



M

₂

z

+

1 3!

M

4

z

3

₊

_...

M

₂

=

Z

1 0

dxx q

(

x

)

M

4

=

Z

1 0

dxx

3

q

(

x

)

T

_c

(

z

)

=

1

2 M

Z

1 0

dxq

_s

(

x

) cos(

xz

) =

i

2 M



M

₁

+

1

2 M

3

z

2

₊

_...

M

1

=

Z

1 0

dx q

(

x

)

M

₃

=

Z

1 0

dxx

2

q

(

x

)

(18)

T

_s

T

_c

Regge tail z

-α+1

proton size

(19)

(20)

0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 25 30 Q(z) z CT10 u CT10 xmin=0.1

(21)

Space-time picture of the correlation function

!  At small distances - z- < R - the virtual photon scatters

incoherently from the individual partons. This region is dominated by bulk properties such as the average

momentum.

!  At larger distances - z- >> R - the interaction is given by

coherent scattering over several partons. The virtual photon converts into a q-qbar pair covering a distance z

-z- zo

z3

z+

(22)

2), 3) 4)

3. HOW WELL CAN WE RECONSTRUCT

THE X DEPENDENT FUNCTION (PDF/

(23)

Information from lattice: Mellin Moments

W. Detmold et al., Moment u-d µ2_{= 4 GeV}2 Linear extrapolation Chiral extrapolation Phenomenology CT10 M₁ 1 1 1 M₂ 0.262 0.18(3) 0.169 M₃ 0.0843 0.05(2) 0.0536 M₄ 0.0340 0.02(1) 0.0221 Mod.Phys.Lett. A18 (2003) Eur.Phys.J.3 (2001),

u-d

Mn(Q2) = Z 1 0 dx xn 2F2(x, Q2)

(24)

LHPC (Ph. Hägler et al.) Moment u-d µ2_{= 4 GeV}2 Linear extrapolation Chiral extrapolation Phenomenology CT10 M₁ 1 1 1 M₂ / 0.200(-7/+9) 0.169 M₃ / / 0.0536 M₄ / / / Phys.Rev. D77 (2008) Moment u-d µ2_{= 4 GeV}2 m_π=352 MeV Chiral extrapolation Phenomenology CT10 M₁ 1 1 1 M₂ 0.206(14) 0.157(10) 0.169 M₃ 0.078(16) / 0.0536 M₄ / / 0.0221

G. Bali et al. PoS LATTICE2015 (2016)

(25)

Moment u-d µ2_{= 4 GeV}2

m_π=physical Hägler: Chiral extrapolation Phenomenology CT10 M₁ 1 1 1 M₂ 0.140(21) 0.157(10) 0.169 M₃ / / 0.0536 M₄ / / 0.0221

ETMC (C. Alexandrou et al., Phys Rev D93, 2016)

_u-d

LHPC (J. Green et al., Phys. Lett. B, 2014 )

Moment u-d µ2_{= 4 GeV}2 m_π=physical (largest source sink) Hägler: Chiral extrapolation Phenomenology CT10 M₁ 1 1 1 M₂ 0.208(24) 0.157(10) 0.169 M₃ / / 0.0536 M₄ / / 0.0221

(26)

Reconstructed pdf: using its own moments up

to x

5 no error shown

(27)

Using Detmold et al.

Weigl and Mankiewicz, PLB 1996

(28)

(29)

(30)

Error analysis in progress

ii) What PDF-related quantities would we like lattice QCD to

compute?

Sources of error: •  lattice moments

•  Regge tail (CT10)

•  unknown region

A careful analysis needs

to be done and it’s on its

way!

(31)

First stab at GPDs…

0.4 ₀_.₂ 0 0.2 ₀_.₄ 0.6 ₀_.₈ 1 0.1 0 0.1 0 0.5 x ⇠ Hu ( x, ⇠ ,t) t= 0.1GeV2_{reconstructed} 0.2 0 0.2 0.4 0.6 1 0.4 _0.2 0 0.2 _0.4 0.6 _0.8 1 0.1 0 0.1 0 1 x ⇠ Hu ( x, ⇠ ,t) t= 0.1GeV2 0 0.2 0.4 0.6 0.8 1 1.2 1 Reconstructed _Model x x ξ ξ H(x,ξ,t) H(x,ξ,t)

(32)

0.4 ₀_.₂ 0 0.2 ₀_.₄ 0.6 ₀_.₈ 1 0.1 0 0.1 0 0.5 x ⇠ Hu ( x, ⇠ ,t) t= 0.1GeV2_{reconstructed} 0.2 0 0.2 0.4 0.6 1 0.4 ₀_.₂ 0 0.2 ₀_.₄ 0.6 ₀_.₈ 1 0.1 0 0.1 0 1 x ⇠ Hu ( x, ⇠ ,t) t= 0.1GeV2 0 0.2 0.4 0.6 0.8 1 1.2 1 Perturbative evolution x ξ H(x,ξ,t)

(33)

(34)

"  Pseudo-PDFs interpolate between PDFs evaluated on the LF and lattice

QCD space-like correlations

"  Pseudo-PDFs are the Fourier transforms of z_T(z₃)-dependent generalized

Ioffe time distributions,

"  For the ratio reduces to

"  is the Ioffe time distribution conjugate to

M(

ν

,

z

₃2

)

M(0,

z

₃2

)

z

₃

→

0 M

(

ν

,

z

₃2

)

M

(

ν

, 0)

f

(

x

)

M

(

ν

, 0)

(35)

(36)

-0.2 0 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 Im M(z - ,z 3 ) z- (fm) Experimental data (CT10) Orginos et al. (2017) Haegler et al., LHPC 2007

(37)

0 2 4 6 8 10 12 0.1 1 10 Im M( ,z 3 ) z₃2 (fm2) 0 2 4 6 8 10 12 1 2 3 4 5 6 7 8 9 Im M( ,z3 ) z₃2 (fm2)

Evolution in z

_T Non-perturbative region ν=5 ν=5 transition

(38)

Conclusions and Outlook

"  We presented results on the reconstruction of PDFs from their Mellin

moments which are available in lattice QCD

"  Lattice QCD moments evaluation complements pseudo-PDFs

analysis

"  Both methods need assumptions on small x behavior

"  Lattice QCD moments evaluation can be extended to GPDs

"  Error analysis is crucial