Lensing function relation in Hadrons

Simone RodiniIn collaboration with

Barbara Pasquini and Alessandro Bacchetta

Based on arXiv:1907:06960

Photon Final State Interactions

Detected hadron

Proton Polarization

x

zy

Generalized Parton DistributionInformation

Transverse Momentum PDFInformationsBurkardt PRD66 (2002)

TMD

FSI

Detected hadron

Proton Polarization

Unpolarized quark

GPD

Proton Polarization

Zd�?(2⇡)2

e�ib?·�?

�[�] (x,k?, S) =1

2

Zdz�dz?(2⇡)3

eixp+z��ik?·z?

⇥ hp, S| ⇣�z

2

⌘�W

⇣�z

2,z

2

⌘ ⇣z2

⌘|p, Si |z+=0

+ h.c.

hki?(x)iUT =

Zdk?k

i?�

[�+](x,k?,S?)

Meissner, Metz, Goeke PRD76 (2007)

Zd�?(2⇡)2

e�ib?·�?

In light cone gauge

z1,2 =

✓0+,⌥z�

2, b?

◆

F [�](x, b?, S) =1

2

Zdz�

2⇡eixp

+z�

⇥ hp+,R? = 0?, S| (z1)� (z2) |p+,R? = 0?, Si

hki?(x)iUT ⇡Z

db?Li�b?/(1� x)

�F [�+](x, b?,S?)

Soper PRD15 (1977)

Burkardt IJMPA18 (2003)

+ h.c.

In light cone gauge

Burkardt PRD69 (2004)

Ii(z2) =gs2Ai

?(1�, 0+, b?)

hki?(x)iUT =1

2

Zdb?

Zdz�

2⇡eixp

+z�

⇥ hp+,R? = 0?,S?| ̄(z1)W(z1; z2)Ii(z2)�+ (z2)|p+,R? = 0?,S?i

hki?(x)iUT =1

2

Z{dk1}{dk2}{dl}

Zdz�

2⇡eixp

+z�e�i z�

2 (k+1 +k+

2 +l+)

X

n,m

X

�,�0

Z nY

i=1

dq+i dq?,i

(2⇡)32q+i

mY

i=1

dw+i dw?,i

(2⇡)32w+i

⇥ hp+,p? = 0?,S?|�(k1)�+|{q+i , q?,i}n,�0i⇥ h{q+i , q?,i}n,�0|Ii(l)|{w+

i ,w?,i}m,�i⇥ h{w+

i ,w?,i}m,�| (k2)|p+,p? = 0?,S?i

h{q+i , q?,i}n,�0|Ii(l)|{w+i ,w?,i}m,�i

= 2⇡Li

✓l?

1� x

◆�n,m���0�(l+)

⇥nY

i=1

(2⇡)32q+i �(q+i � w+

i )�

✓q?,i �w?,i � xi

l?1� x

◆

1) connect Fock states with the same number of constituents and the same parton, helicity and color content; 2) transfer the total transverse momentum ! to the whole spectator system;3) NOT transfer momentum in the light-cone direction to the spectator system;4) transfer a fraction ! of the total transverse momentum to each constituent of the spectator system.

l⊥/(1 − x)

xi = w+i /p+

h{q+i , q?,i}n,�0|Ii(l)|{w+i ,w?,i}m,�i

= 2⇡Li

✓l?

1� x

◆�n,m���0�(l+)

⇥nY

i=1

(2⇡)32q+i �(q+i � w+

i )�

✓q?,i �w?,i � xi

l?1� x

◆

The Final State Interactions should:

When does it work?

Two body system: Pion (q-antiq pair)Scalar diquark model for the protonMassive remnant with spin ≤ 1/2

When does it NOT work?

Many body system: Three-quark model for the protonMassive remnant with spin > 1/2:Axial-vector di quark model for the proton

Bacchetta, Conti, Radici PRD78 (2008)

Pasquini, Yuan PRD81 (2010)

Gamberg, Schlegel PLB685 (2010)Burkardt NPA735 (2004)

�k?

2M⇡

eHT,⇡(x, 0,��2?) =

T2⇡

2(2⇡)3

Zdk?G

k (x,k?| |x,k? + (1� x)�?)

kk?h?1,⇡

�x,k2

?�=

2↵s

(2⇡)44

3T 2⇡M⇡

Zdq?q2?

Gk (x,k?| |x,k? � q?)

�q? = (1� x)�?

Two body system

i✏ij?�j?S

i?

ME(x,��2

?)

=

Zdk?

2(2⇡)3

Z x

0dy

Zdt?

2(2⇡)3GT (x,k?; y, t?| |x,k? + (1� x)�?; y, t? � y�?) ,

✏ij?kj?S

i?

Mf?1T

�x,k2

?�

= � ↵s

3(2⇡)7

Zdq?q2?

Z x

0dy

Zdt?GT (x,k?; y, t?| |x,k? � q?; y, t? + q?)

Three body system

�k? � t?

+ h.c.q?

t?

k?

PossibleSpin Flip

No Spin Flip

+ h.c.

Bacchetta, Conti, Radici PRD78 (2008)

With longitudinal polarisation allowed

Conclusions

Model-dependent relations between distributions can be useful

But they should not be extrapolated to different models,nor assumed as a general feature of the theory

Model studies are useful to get insight on complex physics phenomena