Nucleon Electromagnetic Form Factors on GPDs

Presented by

Negin Sattary Nikkhoo

1

üMost of what we know on the structure of the nucleon has comefrom the scattering of high energy leptons. This is due to theirstructureless nature, their well-known and quantifiedelectromagnetic interaction with matter.

1. Inclusive scattering Only the scattered lepton is detected and the target nucleon is unobserved.

2. Exclusive scatteringThe final hadronic state of the reaction is fully determined.

2

3

Deep-Inelastic scatteringIn DIS, 𝑓" 𝑥 is related to the unpolarized 𝑥 -momentum distributions of the partons 𝑞(𝑥).

𝑓" 𝑥 =12*𝑒,-𝑞(𝑥),

The non-perturbative parton distribution functions (PDFs) is defined viamatrix elements of parton operators between nucleon states with equalmomenta:

𝑞 𝑥 =𝑝/

4𝜋2𝑑𝑦5𝑒6789:; 𝑝 𝜓=, 0 𝛾/𝜓, 𝑦 𝑝 @

:9A:BAC

4

The FFs are related to the following QCD matrix element in space–time coordinates

Elastic scattering

𝑝D   𝜓=, 0 𝛾/𝜓, 0 𝑝

= 𝐹", 𝑡 𝑁I 𝑝D   𝛾/𝑁 𝑝 + 𝐹-

, 𝑡 𝑁I 𝑝D   𝑖𝜎/MΔM2𝑚P

𝑁 𝑝 ,

𝐹", 𝑡

𝐹-, 𝑡

Dirac form factor

Pauli form factor

𝐺S 𝑡 =𝐹" 𝑡 + 5TUVW

X𝐹- 𝑡

𝐺Y 𝑡 =𝐹" 𝑡 + 𝐹- 𝑡

Sachs form factors

Generalized Parton Distributions

Like usual PDFs, GPDs are non-perturbative functions defined via thematrix elements between nucleon states with different momenta [1-3]:

𝑃/

2𝜋2 𝑑𝑦5𝑒67[

9:; 𝑝D 𝜓=, 0 𝛾/𝜓, 𝑦 𝑝 @:9A:BAC

= 𝐻, 𝑥, 𝜉, 𝑡 𝑁I 𝑝D   𝛾/𝑁 𝑝 +𝐸, 𝑥,𝜉, 𝑡 𝑁I 𝑝D   𝑖𝜎/M𝛥M2𝑚P

𝑁 𝑝

5

[1] D. Muller, et.al Phys. 42, 101 (1994).[2] X. D. Ji, Phys. Rev. Lett. 78, 610 (1997).[3] A. V. Radyushkin, Phys. Rev. D 56, 5524 (1997).

𝑃/

2𝜋2 𝑑𝑦5𝑒67[

9:; 𝑝D 𝜓=,(0)𝛾/𝛾 𝜓,(𝑦) 𝑝 @:9A:BAC

=𝐻a, 𝑥, 𝜉, 𝑡 𝑁I 𝑝D   𝛾/𝛾 𝑁 𝑝 + 𝐸b, 𝑥,𝜉, 𝑡 𝑁I 𝑝D   𝛾 cd-VX

𝑁 𝑝

GPDs depend on the squared 4-momentumtransfer t, on the average momentum fraction𝑥 of the active quark, and on the skewnessparameter ξ, which is the longitudinalmomentum transfer.

𝐹-(𝑡) =*𝑒,𝐹-, 𝑡

,

𝐹"(𝑡) =*𝑒,𝐹", 𝑡

,

6

• Traditionally, elastic form factors and parton distribution functions(PDFs) were considered totally unrelated;

• Elastic form factors give information on the charge and magnetizationdistributions in transverse plane.

• PDFs describe the distribution of partons in the longitudinal direction.

Forward limit (𝑝 = �́�, 𝑡 = 0) 𝐻, 𝑥, 0,0 = f𝑞 𝑥                      𝑥 > 0−𝑞= −𝑥          𝑥 < 0

ü At finite momentum transfer, there are model independent sum ruleswhich relate the first moments of the GPDs to the elastic form factors

2 𝑑𝑥  𝐻, 𝑥, ξ, 𝑡/"

5"= 𝐹"

, 𝑡

𝐹"j 0 = 2 𝐹"k 0 = 1

2 𝑑𝑥  𝐸, 𝑥, ξ, 𝑡/"

5"= 𝐹-

, 𝑡

𝐹-j 0 = 𝜅j 𝐹-k 0 = 𝜅k

𝐹"8 0 = 1 𝐹"m 0 = 0 𝐹-

8 0 = 1.79 𝐹-m 0 = −1.91

𝐹-(𝑡) = *𝑒,,

2 𝑑𝑥ℇ, 𝑥, 𝑡"

C𝐹"(𝑡) =*𝑒,

,

2 𝑑𝑥ℋ, 𝑥, 𝑡"

C

In some studies [4-6], the dependence on 𝑡 and 𝑥 for ℋ, 𝑥, 𝑡 has beenconsidered separately

ℋ, 𝑥, 𝑡 = 𝑞s 𝑥  𝐺 𝑡

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ü Our goal is to calculate electromagnetic form factors in the region of highmomentum transfers.

ü As yet, the most common ansatzes used in literature are Gaussian andRegge ansatzes for calculating the electromagnetic form factors.

ü These ansatzes can reproduce experimental electromagnetic form factors inthe region of low and medium momentum transfers.

ℋ, 𝑥, 𝑡 = 𝑞s 𝑥 𝑒(t;u)vwuxW

 

ℋ, 𝑥, 𝑡 = 𝑞s 𝑥 𝑥5  yzT

Gaussian ansatz

Regge ansatz

If the momentum transfer 𝑡 = ∆|- is transverse, then ξ=0. The integration regioncan be reduced to the interval 0   <  𝑥   <  1. So, We can rewrite 𝐹" 𝑡   and 𝐹- 𝑡  asfollow:

[4] M. Guidal et al., Phys. Rev. D 72, 054013 (2005).[5] A. V. Radyushkin, Phys. Rev. D 58, 114008 (1998).[6] O. V. Selyugin, Phys. Rev. D 89, 093007 (2014).

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ü However, by considering Modified Gaussian and Extended Regge ansatzeswe can reproduce experimental form factors also in the region of highmomentum transfers.

ℋ 𝑥, 𝑡 = 𝑞 𝑥 𝑒𝑥𝑝 𝛼𝑡1 − 𝑥 -

𝑥V

ℋ 𝑥, 𝑡 = 𝑞 𝑥 𝑥5yzT("57)

q In the case of the Pauli form factor, 𝐹- 𝑡 , we take the same

parameterization for ℇ, 𝑥, 𝑡 , which is given by:

ℇ, 𝑥, 𝑡 = ℇ, 𝑥 𝑥5  yz("57)T and ℇ, 𝑥, 𝑡 = ℇ, 𝑥 exp  [𝛼 "57 W

7�𝑡]

for extended Regge and modified Gaussian ansatzes, respectively.

Modified Gaussian ansatz

Extended Regge ansatz

The experimental proton Pauli form factor at large t exhibits a fasterreduction than Dirac form factor. In order to yield a faster reduction with t,one has to multiply an additional power of 1 − 𝑥 to ℇ   𝑥 . Thus, we have

ℇj 𝑥 = ��P�

1 − 𝑥 ��𝑢s 𝑥 and

ℇk 𝑥 =𝜅k𝑁k

1 − 𝑥 ��𝑑s 𝑥

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ü In some previous studies [4, 7], Dirac and Pauli form factors were extracted by choosing MSRT 2002 PDFs.

ü In Ref. [8], we used CJ15 , JR09 , and GJR07 PDFs to extract the Dirac and Pauli form factors.

[7] O. V. Selyugin and O. V. Teryaev, Phys. Rev. D 79, 033003 (2009).[8] Negin Sattary Nikkhoo and M. R. Shojaei, Phys. Rev. C 97, 055211 (2018).

ModifiedGaussian (MG)

𝜶 𝜼𝒖 𝜼𝒅 m

MSRT2002 1.1 𝐺𝑒𝑉5- 1.71 0.56 0.4CJ15 1.09𝐺𝑒𝑉5- 1.41 0.61 0.39JR09 1.25𝐺𝑒𝑉5- 1.89 -0.11 0.42GRJ07 1.345  𝐺𝑒𝑉5- 1.7 0.35 0.36

Extended Regge(ER)

𝜶’ 𝜼𝒖 𝜼𝒅

MSRT2002 1.105  𝐺𝑒𝑉5- 1.713 0.556

CJ15 1.05  𝐺𝑒𝑉5- 1.56 0.19

JR09 1.22  𝐺𝑒𝑉5- 1.51 0.31

GRJ07 1.29  𝐺𝑒𝑉5- 1.84 −0.05

10

0 5 10 15 20 25 300.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1(a)

 

 

t2 FP 1

-t [GeV2]

0 1 2 3 4 5 6 7 8 9 100.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

 

 

-tFp 2/κ

p Fp 1

-t [GeV2]

(b)

The ratio of Pauli to Dirac form factor

The proton Dirac form factor 𝑭𝟏multiplied by 𝒕𝟐

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

 

 -tF

u 1

-t [GeV2]

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20 (b)

 

 

-tFd 1

-t [GeV2]

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.50.00

0.05

0.10

0.15

0.20

0.25

0.30 (c)

 

 

-tFu 2

-t [GeV2]0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35(d)

 

 

|-tFd 2|

-t [GeV2]

F1 and F2 form factors of u and d quarks

12

0.01 0.1 1 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2(a)

 

 

µp GP E/G

P M

-t [GeV2]

0.01 0.1 1 100.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20(b)

 

 

Gp M

/µp G

D

-t [GeV2]

The ratio of the proton electric and magnetic form factors

The ratio of the proton magnetic and dipole form factors

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0.01 0.1 1 100.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09 (a)

 

 Gn E

-t [GeV2]

0.01 0.1 1 100.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2(b)

 

 

Gn M

/µn G

D

-t [GeV2]

The neutron electric form factor

The ratio of the neutron magnetic and dipole form factors

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The Dirac mean radii squared is calculated both with the Extended Reggeansatz:

𝑟",8- = −6𝛼’ 2 𝑑𝑥 𝑒j𝑢s 𝑥 + 𝑒k𝑑s 𝑥 1− 𝑥 ln 𝑥"

C,

𝑟",m- = −6𝛼’ 2 𝑑𝑥 𝑒j𝑑s 𝑥 + 𝑒k𝑢s 𝑥 1− 𝑥 ln 𝑥"

C,

and with the Modified Gaussian ansatz:

𝑟",8- = −6𝛼2 𝑑𝑥 𝑒j𝑢s 𝑥 + 𝑒k𝑑s 𝑥1 − 𝑥 -

𝑥V"

C,

𝑟",m- = −6𝛼2 𝑑𝑥 𝑒j𝑑s 𝑥 + 𝑒k𝑢s 𝑥1 − 𝑥 -

𝑥V"

C.

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The electric mean squared radius of proton and neutron are determined as

𝑟S,P- = 𝑟",P- +32𝜅P𝑀P-

Nucleon electric mean squared radius

6  𝑑𝐹"P

𝑑𝑡�TAC

Foldy term6  𝑑𝐺SP

𝑑𝑡�TAC

𝒓𝑬,𝒑 𝒓𝑬,𝒏𝟐

Experimental data 𝟎. 𝟖𝟕𝟕  𝐟𝐦 −𝟎. 𝟏𝟏𝟔𝟏  𝒇𝒎𝟐

ER+MSRT2002 𝟎. 𝟖𝟏𝟖  𝐟𝐦 −𝟎.𝟏𝟎𝟏  𝒇𝒎𝟐

ER+CJ15 𝟎. 𝟖𝟑𝟗  𝐟𝐦 −𝟎. 𝟏𝟎𝟓𝟓  𝒇𝒎𝟐

ER+JR09 𝟎. 𝟖𝟓𝟕  𝐟𝐦 −𝟎. 𝟏𝟒𝟎𝟏  𝒇𝒎𝟐

ER+GRJ07 𝟎. 𝟖𝟕𝟏  𝐟𝐦 −𝟎. 𝟏𝟏𝟏𝟖  𝒇𝒎𝟐

MG+MSRT2002 𝟎. 𝟖𝟔𝟔  𝐟𝐦 −𝟎. 𝟎𝟖𝟗𝟔  𝒇𝒎𝟐

MG+CJ15 𝟎. 𝟖𝟗𝟗  𝐟𝐦 −𝟎. 𝟏𝟎𝟎𝟑  𝒇𝒎𝟐

MG+JR09 𝟎. 𝟗𝟒𝟑  𝐟𝐦 −𝟎. 𝟏𝟓𝟓𝟗  𝒇𝒎𝟐

MG+GRJ07 𝟎. 𝟖𝟗𝟕  𝐟𝐦 −𝟎. 𝟏𝟎𝟕𝟎  𝒇𝒎𝟐

The proton and neutron electric radius

16

Thanks for your attention