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Hadron Structure Theory II
Alexei Prokudin
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l Lecture I:Structure of the nucleon
l Lecture IITransverse Momentum Dependent distributions (TMDs)Semi Inclusive Deep Inelastic Scattering (SIDIS)
l TutorialCalculations of SIDIS structure functions using Mathematica
• Lecture IIIAdvanced topics. Evolution of TMDs
The plan:
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6
How do we study the structure of the nucleon?
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Deep Inelastic Scattering (DIS)
In order to access distributions we could usedeep inelastic scattering
The energy is big enough totransform the proton ina lot of final states
Bjorken limit is
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Distributions measured in deep inelastic scattering
This sum makes it sensitive to parton structure!
Deep Inelastic Scattering (DIS)
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Distributions and parton model
Parton model is a logical step, partons are pointlike and dilute, so the photon interactswith them incoherently
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Distributions and parton model
CONSTANT!
Parton model is a logical step, partons are pointlike and dilute, so photon interactswith them incoherently
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Factorization
3
W
µ⌫
DY
=
X
f
|Hf
(Q;µ/Q)|µ⌫
⇥Z
d
2k1T
d
2k2T
F
f/P1(x
1
,k1T
;µ; ⇣
1
) F
f/P2(x
2
,k2T
;µ; ⇣
2
) �
(2)
(k1T
+ k2T
� qT
)
+ Y (q
T
, Q)
+O✓✓
⇤
Q
◆a
◆
˜
F
f/P1(x
1
,bT
;µ, ⇣
1
) =
Zd
2kT
e
�ikT ·bTF
f/P1(x
1
,kT
;µ, ⇣
F
)
˜
K(bT
;µ) =
Zd
2kT
e
�ikT ·bTK(k
T
;µ)
@
@ ln
p⇣
F
F
f/P1(x
1
,kT
;µ, ⇣
F
) =
Zd
2qT
K(qT
;µ)F
f/P1(x
1
,kT
� qT
;µ, ⇣
F
)
d
d lnµ
K(k
T
;µ) = ��
K
(g(µ)) �(kT
)
d
d lnµ
F
f/P1(x
1
,kT
;µ, ⇣
F
) = �
F
(g(µ); ⇣
F
/µ
2
)F
f/P1(x
1
,kT
;µ, ⇣
F
)
k⇤(kT
) ⌘ k̂T
qk
2
min
+ k
2
T
µ⇤(kT ) ⌘ C
1
k⇤
↵
s
(µ⇤(kT ))kT!0
= ↵
s
(C
1
k
min
)
b⇤(bT
) ⌘ bTp
1 + b
2
T
/b
2
max
µ⇤(bT ) = C
1
/b⇤
↵
s
(µ⇤(bT ))bT!1= ↵
s
(C
1
/b
max
)
d�
dqT
· · ·
P
1
P
2
k
1
⌘ k k
2
⌘ q � k
q + k (34)
3
W
µ⌫
DY
=
X
f
|Hf
(Q;µ/Q)|µ⌫
⇥Z
d
2k1T
d
2k2T
F
f/P1(x
1
,k1T
;µ; ⇣
1
) F
f/P2(x
2
,k2T
;µ; ⇣
2
) �
(2)
(k1T
+ k2T
� qT
)
+ Y (q
T
, Q)
+O✓✓
⇤
Q
◆a
◆
˜
F
f/P1(x
1
,bT
;µ, ⇣
1
) =
Zd
2kT
e
�ikT ·bTF
f/P1(x
1
,kT
;µ, ⇣
F
)
˜
K(bT
;µ) =
Zd
2kT
e
�ikT ·bTK(k
T
;µ)
@
@ ln
p⇣
F
F
f/P1(x
1
,kT
;µ, ⇣
F
) =
Zd
2qT
K(qT
;µ)F
f/P1(x
1
,kT
� qT
;µ, ⇣
F
)
d
d lnµ
K(k
T
;µ) = ��
K
(g(µ)) �(kT
)
d
d lnµ
F
f/P1(x
1
,kT
;µ, ⇣
F
) = �
F
(g(µ); ⇣
F
/µ
2
)F
f/P1(x
1
,kT
;µ, ⇣
F
)
k⇤(kT
) ⌘ k̂T
qk
2
min
+ k
2
T
µ⇤(kT ) ⌘ C
1
k⇤
↵
s
(µ⇤(kT ))kT!0
= ↵
s
(C
1
k
min
)
b⇤(bT
) ⌘ bTp
1 + b
2
T
/b
2
max
µ⇤(bT ) = C
1
/b⇤
↵
s
(µ⇤(bT ))bT!1= ↵
s
(C
1
/b
max
)
d�
dqT
· · ·
P
1
P
2
k
1
⌘ k k
2
⌘ q � k
q + k (34)
2
ps = 1.8 TeV / 1.97 TeV (13)
�
D
e↵
= 11mb (14)
ps = 7 TeV (15)
Zd
2b�
2n(s, b; pct) = �
inc
2n (s, pct) (16)
�
D
e↵
(17)
⇡ 34 mb (18)
p
ct = 3.5 GeV (19)
�
1 di↵
(s, b; p
ct) = �
diff(s, b; p
ct)� �
diff(s, b; p
ct)
1X
n=1
(�1)
n�1
�
2n(s, b; pct) (20)
�
1 di↵
(s, b; p
ct) = �
diff(s, b; p
ct) exp {��
2
(s, b; p
ct)} (21)
�
1 di↵
(s, b; p
ct) = �
diff(s, b; p
ct) exp {��
2
(s, b; p
ct)} (22)
�
n di↵
(s, b; p
ct) =
1
n!
�
diff(s, b; p
ct)
nexp {��
2
(s, b; p
ct)} (23)
�
diff(s, b; p
ct) =
�1� exp
���
diff(s, b; p
ct) �
exp {��
2
(s, b; p
ct)} (24)
�(s, b) = 1� exp [��h(s, b; pct)� �s(s, b; p
ct) + · · · ] (25)
�h(s, b; pct) (26)
d�
dq
2
T
(27)
qT (28)
q
2 ⇠ Q
2 � ⇤
2
QCD
(29)
qT ⇠ ⇤
QCD
(30)
⇤QCD ⌧ k
1T ⌧ Q (31)
qT ⌧ Q (32)
P (33)
II. DISCUSSION
...................
Acknowledgments
This work was supported by...
2
ps = 1.8 TeV / 1.97 TeV (13)
�
D
e↵
= 11mb (14)
ps = 7 TeV (15)
Zd
2b�
2n(s, b; pct) = �
inc
2n (s, pct) (16)
�
D
e↵
(17)
⇡ 34 mb (18)
p
ct = 3.5 GeV (19)
�
1 di↵
(s, b; p
ct) = �
diff(s, b; p
ct)� �
diff(s, b; p
ct)
1X
n=1
(�1)
n�1
�
2n(s, b; pct) (20)
�
1 di↵
(s, b; p
ct) = �
diff(s, b; p
ct) exp {��
2
(s, b; p
ct)} (21)
�
1 di↵
(s, b; p
ct) = �
diff(s, b; p
ct) exp {��
2
(s, b; p
ct)} (22)
�
n di↵
(s, b; p
ct) =
1
n!
�
diff(s, b; p
ct)
nexp {��
2
(s, b; p
ct)} (23)
�
diff(s, b; p
ct) =
�1� exp
���
diff(s, b; p
ct) �
exp {��
2
(s, b; p
ct)} (24)
�(s, b) = 1� exp [��h(s, b; pct)� �s(s, b; p
ct) + · · · ] (25)
�h(s, b; pct) (26)
d�
dq
2
T
(27)
qT (28)
q
2 ⇠ Q
2 � ⇤
2
QCD
(29)
qT ⇠ ⇤
QCD
(30)
⇤QCD ⌧ k
1T ⌧ Q (31)
qT ⌧ Q (32)
P (33)
k + q (34)
II. DISCUSSION
...................
Acknowledgments
This work was supported by...
3
P (33)
ˆ
k + q (34)
ˆ
k (35)
d�̂ (36)
electron quark ! electron quark (37)
d� (38)
electron proton ! electron proton (39)
d� =
Zd⇠f(⇠) d�̂ (40)
L
µ⌫W
µ⌫(41)
II. DISCUSSION
...................
Acknowledgments
This work was supported by...
3
P (33)
ˆ
k + q (34)
ˆ
k (35)
d�̂ (36)
electron quark ! electron quark (37)
d� (38)
electron proton ! electron proton (39)
d� =
Zd⇠f(⇠) d�̂ (40)
Lµ⌫ W
µ⌫(41)
II. DISCUSSION
...................
Acknowledgments
This work was supported by...
3
P (33)
ˆ
k + q (34)
ˆ
k (35)
d�̂ (36)
electron quark ! electron quark (37)
d� (38)
electron proton ! electron proton (39)
d� =
Zd⇠f(⇠) d�̂ (40)
Lµ⌫ W
µ⌫(41)
E
0 d�
d
3l0=
2↵em
sQ
4
Lµ⌫Wµ⌫
(42)
II. DISCUSSION
...................
Acknowledgments
This work was supported by...
p p
Leptonic tensor
Hadronic tensor
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Distributions and parton model
This diagram is called “handbag diagram”
- parton distribution
![Page 10: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/10.jpg)
Distributions and parton model
Why quarks are on mass-shell?
This one is virtual! However the main contribution comes from
![Page 11: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/11.jpg)
Distributions and parton model
Definition of parton distribution
![Page 12: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/12.jpg)
Distributions and parton model
Definition of parton distribution
Fourier transform from coordinate to momentum space
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Distributions and parton model
Definition of parton distribution
Quark field operator
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Definition of parton distribution
The proton state vector
Distributions and parton model
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Definition of parton distribution
Position of the field in coordinate space
Distributions and parton model
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Distributions and parton model Definition of parton distribution
This matrix element is called “bilocal”
![Page 17: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/17.jpg)
Distributions and parton model
What do we know about quark momentum? Suppose that protonis moving along Z direction with a high momentum, then
“Big”component
is a new variable called lightcone momentumfraction
![Page 18: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/18.jpg)
Distributions and parton model
What do we know about quark momentum?
“Big”component
“Small” component
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“Small” component
Distributions and parton model
What do we know about quark momentum?
“Big”component
“Transverse” component
![Page 20: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/20.jpg)
Distributions and parton model
What do we know about hadronic tensor?
Quarks are “probed” at value of
![Page 21: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/21.jpg)
Gauge invariance
The quark and the remnant are colored thus they interactvia gluon exchanges! If “–” and perpendicular component of parton momentum areneclected, than in configuration space only “–” component survives,
This object is called Wilson line
For DIS:
ip · ⇠ = ixP
+⇠
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Factorization
Distribution
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Success of QCD factorization§ Universality of PDFs: mapped in one process (say DIS),
used in other processes
3
What Do We Know About Glue in Matter?
• Scaling violation: dF2/dlnQ2 and
linear DGLAP Evolution !
G(x,Q2)!
Deep Inelastic Scattering : d
2" ep#eX
dxdQ2
=4$%e.m.
2
xQ4
1& y +y
2
2
'
( )
*
+ , F2(x,Q2) &
y2
2FL (x,Q2)
-
. /
0
1 2
Gluons dominate low-x wave function
)20
1( !xG
)20
1( !xS
vxu
vxd
!
![Page 24: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/24.jpg)
Success of QCD factorization§ Universality of PDFs: mapped in one process (say DIS),
used in other process
![Page 25: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/25.jpg)
Success of QCD factorization§ Universality of PDFs: mapped in one process (say DIS),
used in other process
![Page 26: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/26.jpg)
Success of QCD factorization§ Universality of PDFs: mapped in one process (say DIS),
used in other process
3
What Do We Know About Glue in Matter?
• Scaling violation: dF2/dlnQ2 and
linear DGLAP Evolution !
G(x,Q2)!
Deep Inelastic Scattering : d
2" ep#eX
dxdQ2
=4$%e.m.
2
xQ4
1& y +y
2
2
'
( )
*
+ , F2(x,Q2) &
y2
2FL (x,Q2)
-
. /
0
1 2
Gluons dominate low-x wave function
)20
1( !xG
)20
1( !xS
vxu
vxd
!
![Page 27: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/27.jpg)
Transverse structure: Momentum vs Position
Variables are related by 2 dimensional Fourier transform
At the level of squared amplitudes one has
The ‘average’ transverse momentum is Fourier conjugate to position difference (TMD)
¯̃ (k?, z�) =
Zd2z?e
�iz?k? ̄(z?, z�)
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z?k? � y?l? =1
2(z? � y?)(k? + l?) +
1
2(z? + y?)(k? � l?)
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The momentum transfer is Fourier conjugate to ‘average’ position (GPD)
¯̃ (k?) ̃(l?) =
Zd2z?d
2y?e�i(z?k?�y?l?) ̄(z?) (y?)
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32
GPDs TMDs
DVCS SIDIS
ensures hard scale, pointlike interactionmomentum transfer can be variedindependently
Connection to 3D structure Burkardt (2000)Burkardt (2003)
Drell-Yan frame Weiss (2009)
ensures hard scale, pointlike interactionfinal hadron transverse momentumcan be varied independently
Connection to 3D structure
is the transverse separation of parton fieldsin configuration space
Ji, Ma, Yuan (2004)Collins (2011)
AP (2012)
Kotzinian (1995),
Mulders,
Tangerman (1995),
Boer, Mulders (1998)
Ji (1997)
Radyushkin (1997)
Imaginary part, momentum transfer is zero
![Page 29: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/29.jpg)
Transverse Momentum Dependent distributions
Gauge link
Ensures gauge invariance ofthe distribution, cannot be canceled by gauge choice
SIDIS in IMF:
Struckquark
![Page 30: Hadron Structure Theory II - Institute for Nuclear Theory · Hadron Structure Theory II Alexei Prokudin . l Lecture I: Structureof the nucleon l Lecture II ... xQ4 1&y+ y2 2 ' ( )](https://reader034.vdocuments.mx/reader034/viewer/2022042123/5e9e19e9b98669093d52cfa7/html5/thumbnails/30.jpg)
Transverse Momentum Dependent distributionsIndividual TMDs can be projected out of the correlator
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1
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Unpolarized quarks
Longitudinally polarized quarks
Transversely polarized quarks
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