Lensing function relation in Hadrons Simone Rodini In collaboration with Barbara Pasquini and Alessandro Bacchetta Based on arXiv:1907:06960
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