Parton Energy Loss in Generalized High-twist Approach Yuan-Yuan - - PowerPoint PPT Presentation

Parton Energy Loss in Generalized High-twist Approach

Yuan-Yuan Zhang Central China Normal University (CCNU) Collaborators: Guang-You Qin, Xin-Nian Wang

ATHIC 2018 Hefei, China

Introduction Generalized High Twist approach Results and approximations Summary

ATHIC 2018

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Outline

High energy partons travel through hot QGP or cold nuclei (eA DIS process) Energy loss mechanisms reveal medium properties

Parton energy loss in medium

Parton energy loss mechanisms

collisional energy loss radiative energy loss

vacuum medium induced

l⊥

k⊥

A e

l⊥

{

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Introduction ATHIC 2018

Radiative energy loss:assumptions

Scattering Center : Static or Dynamic?

Extension of GLV to dynamic S.C. Djordjevic,Heinz PRL 101,022302 Discussion on soft appr. of GLV Blagojevic et al. arXiv:1804.07593

k⊥ ⌧ l⊥

Radiated Gluon : Soft or hard ? Transverse momentum transfer : smaller or same order ?

k⊥

l⊥

Dynamic: both momentum and energy transfer Static: no energy transfer (BDMPS-Z, GLV) z 0 (BDMPS-Z, GLV, SCET) z finite

k⊥ ∼ l⊥

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Introduction

Approaches to radiative energy loss : BDMPS-Z, GLV, AMY, SCET, High Twist

l⊥

(High Twist)

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lepton-nucleus scattering identify one hadron in final state cross section and hadronic tensor dσ = e4 2s P

q e2 q

q4 Z d4l2 (2π)4 2πδ(l2

2)1

2LµνW µν

Semi-inclusive Deeply Inelastic Scattering

l1 l2 lh Z

q

Ap

e(l1) + A(Ap) → e(l2) + h(lh) + Z

dσ

W µν

Introduction

4/20

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Factorization Separate non-perturbative part (pdf, fragmentation function)

from perturbative part (hard scattering)

SIDIS process

Collinear factorization when final hadron integrated

Collinear Factorization for SIDIS

xp x1p q q y lh S µ ν lq X Ap Ap

dW µν

S(0)

dzh = Z dxf A

q (x)Hµν (0)Dq→h(zh)

f A

q (x)

Dq→h(zh) quark fragmentation function

lh⊥

handbag diagram hadronic tensor

quark distribution function

Introduction

5/20

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Factorization of Medium Induced Radiation

xp q q x1p x3p + k⊥ x2p + k⊥ l lq y2 y1 y lh S

X

Collinear expansion of hard part

dW µ⌫

D(1)q

dzh = Z 1

zh

dz z Dq→h(zh/z) Z dy− 2⇡ dy−

1 dy− 2

d2y1⊥ (2⇡)2 d2y2⊥d2k⊥e−i~

k⊥·(~ y1⊥−~ y2⊥)

1 2 < A| ¯ q(0)+A+(y−

2 , ~

y2⊥)A+(y−

1 , ~

y1⊥) q(y−)|A > Hµ⌫

D (k⊥, y−, y− 1 , y− 2 , p, q, z)

Hµν

D (k⊥, y−, y− 1 , y− 2 , p, q, z) =Hµν D (k⊥ = 0) + ∂Hµν D

∂kα

⊥

• k⊥=0

kα

⊥

+ 1 2 ∂2Hµν

D

∂kα

⊥kβ ⊥

• k⊥=0

kα

⊥kβ ⊥ + · · ·

contribute to gauge link of initial quark PDF contribute zero for unpolarized beam

XF Guo, XN Wang (2000) PRL 85(17), 3591 XN Wang and XF Guo (2001) Nucl. Phys. A 696, 788

approximation

k⊥ = 0

kα

⊥

k⊥ ⌧ l⊥

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Generalized HT approach

High Twist approach

Collinear factorization

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Factorization of Medium Induced Radiation

xp q q x1p x3p + k⊥ x2p + k⊥ l lq y2 y1 y lh S

X

Generalized High Twist approach

without collinear expansion

YY Z, GY Qin, XN Wang (to be published)

relax k⊥ ⌧ l⊥ factorize quark PDF and gluon PDF directly

Generalized HT approach

7/20

splitting function

One example diagram

quark pdf TMD gluon pdf nucleon density

dW µν

D(1)q

dzh ∼↵s 2⇡ CF 2⇡↵s Nc 1 + z2 1 − z Hµν

(0)(x) ⊗ ⇢(y− 1 , ~

y1⊥) ⊗ Dq→h(zh/z) z ⊗ f A

q (x)

⇡ [~ l⊥ − (1 − z)~ k⊥]2 (xL + xD,~ k⊥) k2

⊥

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depend on the parton energy and medium color source

TMD gluon distribution function emerge from TMD transport parameter

TMD gluon distribution function

ˆ q(~ k⊥)

k′ k′ l l′ l′ l X xp + k⊥ xp + k⊥ y

color source density*average kT broadening per scattering

ˆ q(~ k⊥) (x,~

k⊥)

emerge naturally

energy transfer via x

Generalized HT approach

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ˆ q = ρ Z dk2

⊥

hdσi dk2

⊥

k2

⊥

(x,~ k⊥)

ˆ q ≡ Z d2k⊥ (2⇡)2 Z dx(x − k2

⊥

2p+l− )4⇡↵sC2(R) N 2

c − 1

⇢(y)(x,~ k⊥) ≡ Z d2k⊥ (2⇡)2 ˆ q(~ k⊥)

energy via x

l−

p+

(x,~ k⊥) ⌘ Z dy− 2⇡p+ Z d2y⊥eixp+y−−i~

k⊥·~ y⊥hp|F↵ +(0)F +↵(y−, y⊥)|pi

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Gluon Spectrum Result

summing up all the diagrams

xp q q x1p x3p + k⊥ x2p + k⊥ l lq y2 y1 y lh S X Ap Ap

Results and Approximations

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xp q q x1p x3p + k⊥ x2p + k⊥ l lq y2 y1 y lh S X Ap Ap q q l lq x3p + k⊥ x2p + k⊥ y2 y1 y lh xp x1p Ap Ap S X xp x1p x3p + k⊥ x2p + k⊥ lq l q q y2 y1 y lh Ap Ap X S xp x1p x3p + k⊥ x2p + k⊥ lq l q q y1 y2 y lh Ap Ap S X q q l lq x3p + k⊥ x2p + k⊥ y2 y1 y lh xp x1p Ap Ap S X

q q l lq x3p + k⊥ x2p + k⊥ y2 y1 y lh xp x1p Ap Ap S X q q l lq x3p + k⊥ x2p + k⊥ y2 y1 y lh xp x1p Ap Ap S X q q l lq x3p + k⊥ x2p + k⊥ y2 y1 y lh xp x1p Ap Ap S X

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Gluon Spectrum Result

vacuum + medium induced radiation interference left cut diagrams symmetric to right cut diagrams

xp x3p + k⊥ x2p + k⊥ q x1p q y y1 y2 lq l

S

X Ap Ap xp x3p + k⊥ x2p + k⊥ q x1p q y y1 y2 lq l lh Ap Ap

S

X xp x3p + k⊥ x2p + k⊥ q x1p q y y1 y2 lq l lh Ap Ap

S

X

xp x3p + k⊥ x2p + k⊥ q x1p q y y1 y2 lq l lh Ap Ap

S

X xp x3p + k⊥ x2p + k⊥ q x1p q y y1 y2 lq l lh Ap Ap X

S

xp x3p − k⊥ x2p − k⊥ q x1p q y y1 y2 lq l lh Ap Ap

S

X xp x3p − k⊥ x2p − k⊥ q x1p q y y1 y2 lq l lh Ap Ap X

S

Results and Approximations

10/20

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Gluon Spectrum Result: Full Result

We get gluon spectrum from the hadronic tensor Gluon spectrum : probalibility of quark to radiate one gluon with longitudinal momentum transverse momentum

zq−

Results and Approximations

11/20

Quark-gluon correlation function

dW µν

D(1)g

dzh = Z dxHµν

(0)(x)

Z dz z Dg→h(zh/z) Z dl2

⊥Tqg(x, z, l2 ⊥)

Tqg(x, z, l2

⊥)

Tqg =⇡ ↵s 2⇡ 1 + (1 − z)2 z 2⇡↵s Nc Z d2k⊥ (2⇡)2 Z dy−

1

Z d2~ y1⊥⇢(y−

1 , ~

y1⊥) ⇥ HD

C ✓(y− − y− 1 )✓(−y− 2 ) + HD L ✓(y− 1 − y− 2 )✓(y− − y− 1 ) + HD R ✓(y− 2 − y− 1 )✓(−y− 2 )

⇤

~ l⊥ dN dl2

⊥dz ∼

Tqg f A

q (x)

nuclear enhancement

xp q q x1p x3p + k⊥ x2p + k⊥ l lq y2 y1 y lh S

X

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Gluon Spectrum Result: Full Result

HD

C =

(" CA (l⊥ − k⊥)2 (− 1−z

z xT ,~

k⊥) k2

⊥

f A

q (x + xL + xT

z ) − CA l2

⊥

( 1−z

z xT , −~

k⊥) k2

⊥

f A

q (x + xL)

# + " CF l2

⊥

+ CA ~ k⊥ ·~ l⊥ l2

⊥(~

l⊥ − ~ k⊥)2 + CF [~ l⊥ − z~ k⊥]2 + 1 N ~ l⊥ · [~ l⊥ − z~ k⊥] l2

⊥[~

l⊥ − z~ k⊥]2 −CA (~ l⊥ − ~ k⊥) · [~ l⊥ − z~ k⊥] (l⊥ − k⊥)2[~ l⊥ − z~ k⊥]2 ! (xL + xT ,~ k⊥) k2

⊥

f A

q (x)

# " − CA (~ l⊥ − ~ k⊥)2 + CA 2 ~ l⊥ · (~ l⊥ − ~ k⊥) l2

⊥(~

l⊥ − ~ k⊥)2 + CA 2 (~ l⊥ − ~ k⊥) · [~ l⊥ − z~ k⊥] (l⊥ − k⊥)2[~ l⊥ − z~ k⊥]2 ! (xL + xT ,~ k⊥) k2

⊥

e−i(xL+ xT

z )p+y− 1 f A

q (x + xL + xT

z ) # " − CA (~ l⊥ − ~ k⊥)2 + CA 2 ~ l⊥ · (~ l⊥ − ~ k⊥) l2

⊥(~

l⊥ − ~ k⊥)2 + CA 2 (~ l⊥ − ~ k⊥) · [~ l⊥ − z~ k⊥] (l⊥ − k⊥)2[~ l⊥ − z~ k⊥]2 ! (− 1−z

z xT ,~

k⊥) k2

⊥

ei(xL+ xT

z )p+y− 1 f A

q (x)

#)

HD

L = · · ·

HD

R = · · ·

Results and Approximations

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xp q q x1p x3p + k⊥ x2p + k⊥ l lq y2 y1 y lh S

X

x : energy transfer

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Gluon Spectrum Result: Contact Terms

There are contact terms, which are negligible

dNcontact dl2

⊥dz

= ⇡ f A

q (x)

↵s 2⇡ 1 + (1 − z)2 z 2⇡↵s Nc Z d2k⊥ (2⇡)2 Z dy−

1

Z d2~ y1⊥⇢(y−

1 , ~

y1⊥) Hcontact ⇥ ✓(y− − y−

1 )✓(−y− 2 ) − ✓(y− 1 − y− 2 )✓(y− − y− 1 ) − ✓(y− 2 − y− 1 )✓(−y− 2 )

⇤

The functions constrain the integration region

θ

Z dy−

1 dy− 2

⇥ θ(y− − y−

1 )θ(−y− 2 ) − θ(y− 1 − y− 2 )θ(y− − y− 1 ) − θ(y− 2 − y− 1 )θ(−y− 2 )

⇤ f(y−

1 , y− 2 )

= Z y− dy−

1

Z y−

1

dy−

2 f(y− 1 , y− 2 )

gives , quark and gluon comes from same nucleon, no nuclear enhancement.

0 < y−

2 < y− 1 < y− Results and Approximations

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Gluon Spectrum Approximations

no energy transfer for medium scattering gluon spectrum reduces to

dN dl2

⊥dz =⇡ ↵s

2⇡ 1 + (1 − z)2 z 2⇡↵s Nc Z d2k⊥ (2⇡)2 Z dy−

1

Z d2~ y1⊥⇢(y−

1 , ~

y1⊥)  H

D C + 1

2H

D L + 1

2H

D R

(0,~ k⊥) k2

⊥

• 1. Static scattering center approximation:

Results and Approximations

14/20 H

D C =

⇢ CA (l⊥ − k⊥)2 − CA l2

⊥

• + · · ·

H

D R = · · ·

H

D L = · · ·

Q2

l2

⊥

z(1−z), k2

⊥

z(1−z) or

(xL + 1 − z z xT ,~ k⊥) ≈ (xT ,~ k⊥) ≈ · · · ≈ (0,~ k⊥)

no x dependence

f A

q (xB + xL + xT

z ) ≈ f(xB + xL) ≈ f A

q (xB)

xB xL, xT z

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Gluon Spectrum Approximations

• 2. Soft radiated gluon approximation:

the radiated gluon momentum fraction z 0 gluon spectrum reduces to

dNsoft dl2

⊥dz =

⇡ f A

q (x)

↵s 2⇡ 1 + (1 − z)2 z 2⇡↵s Nc Z d2k⊥ (2⇡)2 Z dy−

1

Z d2~ y1⊥⇢(y−

1 , ~

y1⊥) ⇥ (HD

C )soft✓(y− − y− 1 )✓(−y− 2 )

+(HD

L )soft✓(y− 1 − y− 2 )✓(y− − y− 1 ) + (HD R )soft✓(y− 2 − y− 1 )✓(−y− 2 )

⇤

no dependence on z except splitting function, and

f A

q

φ

Ç

Results and Approximations

15/20

(HD

C )soft =

(" CA (l⊥ − k⊥)2 (− 1−z

z xT ,~

k⊥) k2

⊥

f A

q (x + xL + xT

z ) − CA l2

⊥

( 1−z

z xT , −~

k⊥) k2

⊥

f A

q (x + xL)

# + · · ·

(HD

R )soft = · · ·

(HD

L )soft = · · ·

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• 3. Static scattering + Soft radiated gluon approximation:

Gluon Spectrum Approximations

and z 0 give notice here Q2

l2

⊥

z(1−z), k2

⊥

z(1−z)

as z 0 , or xB xL, xT z

the gluon spectrum is

d e N dl2

⊥dz =⇡ ↵s

2⇡ 1 + (1 − z)2 z 2⇡↵s Nc Z d2k⊥ (2⇡)2 Z dy−

1

Z d2~ y1⊥⇢(y−

1 , ~

y1⊥) CA 2~ k⊥ ·~ l⊥ l2

⊥(~

l⊥ − ~ k⊥)2 ⇣ 1 − cos[(xL + xT z )p+y−

1 ]

⌘ (0,~ k⊥) k2

⊥

Ç

Results and Approximations

16/20

(xL + 1 − z z xT ,~ k⊥) ≈ (xT ,~ k⊥) ≈ · · · ≈ (0,~ k⊥)

f A

q (xB + xL + xT

z ) ≈ f(xB + xL) ≈ f A

q (xB)

{

(x,~ k⊥)

non-perturbative

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Gluon Spectrum Approximations

• 3. Static scattering + Soft radiated gluon approximation:

TMD gluon pdf relation to transport parameter ˆ

q

ˆ q = 4⇡↵sC2(R) N 2

c − 1

⇢(y) Z d2k⊥ (2⇡)2 (0,~ k⊥)

Ç

Results and Approximations

17/20

One method : Static potential model to calculate ˆ

q

< dσ >= C2(R)C2(T) dA 4πα2

s

t2 dt Mandelstam variable t

t = k2

⊥ + µ2

a(p) b(k′) c(p′) d(k′) a(p) b(k) c(p′) d(k′)

ˆ q = ρ Z dk2

⊥

C2(R)C2(T) dA 4πα2

s

k2

⊥ + µ2

ˆ q

(0,~ k⊥) = C2(T) 4↵s k2

⊥ + µ2

compare two

Casimir C2(R)

C2(T)

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• 3. Static scattering + Soft radiated gluon approximation:

Gluon Spectrum Approximations

(0,~ k⊥) substitute into gluon spectrum, one get

d e N dl2

⊥dz =8⇡↵3 s

C2(T)CA Nc 1 + (1 − z)2 z Z d2k⊥ (2⇡)2 Z dy−

1 ⇢(y− 1 )

~ k⊥ ·~ l⊥ l2

⊥(~

l⊥ − ~ k⊥)2 ⇣ 1 − cos[(xL + xT z )p+y−

1 ]

⌘ 1 (k2

⊥ + µ2 0)2

agrees with of GLV formalism at first order in opacity arguments in Cosine function

ω1 ≈ √ 2(xL + 1 z xT )p+ y10 = y1 − y0 ≈ y−

1

√ 2

{

Ç

Results and Approximations

18/20

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Summary

Radiative energy loss in semi-inclusive DIS process Generalized High Twist Approach

• no soft radiated gluon approximation
• dynamic scattering center (energy transfer)
• relax approximation,
• transverse momentum dependent (TMD) gluon pdf

TMD transport parameter Relation of Generalized High Twist approach result with GLV result

Further work

gauge link of TMD gluon distribution implementation into CoLBT-Hydro Model

Ç

Results and Approximations

19/20

k⊥ ⌧ l⊥

k⊥ ∼ l⊥

ATHIC 2018

Thanks!