Hadron Masses and Factorization (in DIS) Ted Rogers Jefferson - - PowerPoint PPT Presentation

Hadron Masses and Factorization (in DIS)

Jefferson Lab/Old Dominion University

Ted Rogers

Quark-Hadron Duality 2018, James Madison University, Sept 24 2018

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2

Hadronic vs. Partonic Degrees of Freedom

• Q ≈ 1 GeV, large x.
• Approach kinematical issues in terms of what

they reveal about underlying degrees of freedom.

Two Questions

• What are kinematical target mass

approximations?

• When/how do they matter?

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4

Target Mass Corrections

• Large number of TMC formalisms:

– OPE based – Feynman graph based – Higher twist – Standard factorization (Aivazis, Olness, Tung)

Brady, Accardi, Hobbs, Melnitchouk PHYSICAL REVIEW D 84, 074008 (2011)

Standard setup

5

• Definition of a cross section

d = |M e,P !N|2 2(s, m2

e, M 2)1/2

d3p1 (2⇡)32E1 d3p2 (2⇡)32E2 · · · × d3pN (2⇡)32EN (2⇡)4(4) P + l −

N

X

i=1

pi !

− E0 d d3l0 = 2 ↵2

em

(s − M 2) Q4 Lµ⌫W µ⌫

W µ⌫ = ✓ −gµ⌫ + qµq⌫ q2 ◆ F1 + (P µ − qµP · q/q2)(P ⌫ − q⌫P · q/q2) P · q F2

Single photon exchange

6

Massless Target Approximation (MTA)

• Exact:
• The approximation:
• Usually taken for granted at large Q and small x

P = ⇣p M 2 + P 2

z , 0, 0, Pz

⌘ = ✓ P +, M 2 2P + , 0T ◆

P → ˜ P = (Pz, 0, 0, Pz) =

• P +, 0, 0T
• 2P · q → 2 ˜

P · q M 2/Q2 → 0

7

MTA in Light-Cone Fractions

• Light-cone ratios:

– No MTA: – MTA:

− q+ P + = xN ≡ 2xBj 1 + q 1 +

4x2

BjM 2

Q2

− q+ P + = xBj + O x2

BjM 2

Q2 !

Structure Functions

8

W µ⌫ = ✓ −gµ⌫ + qµq⌫ q2 ◆ F1 (xN, Q) + ✓ P µ − P · q q2 qµ ◆ ✓ P ⌫ − P · q q2 q⌫ ◆ F2 (xN, Q) P · q

Structure Functions

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P1(xN, Q2, M 2)µν ≡ −1 2gµν + 2Q2x2

N

(M 2x2

N + Q2)2 P µP ν

P2(xN, Q2, M 2)µν ≡ 12Q4x3

N

• Q2 − M 2x2

N

• (Q2 + M 2x2

N)4

P µP ν −

• M 2x2

N + Q22

12Q2x2

N

gµν !

W µ⌫ = ✓ −gµ⌫ + qµq⌫ q2 ◆ F1 (xN, Q) + ✓ P µ − P · q q2 qµ ◆ ✓ P ⌫ − P · q q2 q⌫ ◆ F2 (xN, Q) P · q

F1(xN, Q2) ≡ P1(xN, Q2, M 2)µνWµν F2(xN, Q2) ≡ P2(xN, Q2, M 2)µνWµν

MTA

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2P · q ! 2 ˜ P · q M 2/Q2 ! 0

W µ⌫ → ✓ −gµ⌫ + qµq⌫ q2 ◆ F1(xBj, Q2) + ( ˜ P µ − qµ ˜ P · q/q2)( ˜ P ⌫ − q⌫ ˜ P · q/q2) ˜ P · q F2(xBj, Q2)

F1(xBj, Q2) ≡ P1(xBj, Q2, 0)µνWµν F2(xBj, Q2) ≡ P2(xBj, Q2, 0)µνWµν

• Power expansion
• m2 = parton virtuality, transverse momentum,

mass…

• What about hadron masses?

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Factorization

d dxBj dQ2 = Z d⇠ dˆ

• dˆ

xBj dQ2 f(⇠) + O ✓m2 Q2 ◆

For now assume M2 ≠ O(m2)

12

Factorization and partonic light-cone fractions

k k + q q

q = ✓ xNP +, Q2 2xNP + , 0T ◆

(k + q)2 = O

• m2

k2 = O

• m2

k+ = O (Q)

13

Factorization and partonic light-cone fractions

k k + q q

q = ✓ xNP +, Q2 2xNP + , 0T ◆

(k + q)2 = O

• m2

k2 = O

• m2

k+ = O (Q)

• 2k+q− + 2k−q+ Q2 + k2 = O
• m2

2k+q− = Q2 + O

• m2

14

Factorization and partonic light-cone fractions

k k + q q

q = ✓ xNP +, Q2 2xNP + , 0T ◆

(k + q)2 = O

• m2

k2 = O

• m2

k+ = O (Q)

• 2k+q− + 2k−q+ Q2 + k2 = O
• m2

2k+q− = Q2 + O

• m2

⇠ ⌘ k+ P + = xN + O ✓m2 Q2 ◆

15

Factorization and partonic light-cone fractions

k k + q q

q = ✓ xNP +, Q2 2xNP + , 0T ◆

(k + q)2 = O

• m2

k2 = O

• m2

k+ = O (Q)

• 2k+q− + 2k−q+ Q2 + k2 = O
• m2

2k+q− = Q2 + O

• m2

⇠ ⌘ k+ P + = xN + O ✓m2 Q2 ◆

= xBj + O x2

BjM 2

Q2 ! + O ✓m2 Q2 ◆

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Factorization Power Series

• Drop O(m2/Q2) ?: Necessary for factorization.
• Drop O(x2

Bj M2/Q2) ?: Not necessary for

factorization.

• Make approximations with exact target

momentum:

• Then do MTA:

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MTA with factorization

Introduce O(m2/Q2) errors Introduce O(xBj

2M2/Q2) errors

W µν ! W µν

fact

W µν

fact ! W µν fact,TMC

Aivazis, Olness, Tung (AOT)

• Normal factorization, just keeping exact mass.

– MTA – TMC

• The only “pure” kinematical correction. Others

involve assumptions about dynamics.

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• Phys. Rev. D 50, 3085 (1994)

W µν = Z 1

xBj

d⇠ ⇠ ˆ W µν(xBj/⇠, q)f(⇠) + O

• m2/Q2

+ O

• M 2/Q2

W µν = Z 1

xN

d⇠ ⇠ ˆ W µν(xN/⇠, q)f(⇠) + O

• m2/Q2

What if the target mass is important?

• How to test?

– Scaling with Nachtmann rather than Bjorken variable? – Improved universality. Extend range of pQCD?

• Why does it give improvement? Something

about nucleon structure?

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See N. Sato talk

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Partonic interpretation of target mass effects

k k + q q

• Small scales
• Exact target

mass useful if suppression by partonic scales is greater than target mass

x2

BjM 2

Q2 x2

BjM 2 X

Q2

k2 Q2 k2

T

Q2

M 2

J

Q2

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Partonic interpretation of target mass effects

• Parton virtuality
• vs. hadron mass

k k + q q

?? ??

k2 Q2

⇠ A x2

BjM 2

Q2 + B k2 Q2

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Partonic interpretation of target mass effects

k k + q q

• Two scales?

P

|k2| ⌧ x2

BjM 2

See E. Moffat talk

• AOT (direct factorization):

– Direct power expansion in small partonic mass scales – Keep exact momentum expressions

• OPE:

– Transform to Mellin moment space – Expand in 1/Q (both twist and target momentum) – Truncate twist

• Identify leading M/Q part
• Identify remaining series of M/Q

– Invert leading M/Q (F1

(0) )

– Invert series of M/Q (F1

TMC )

– Relate F1

(0) and F1 TMC

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Operator product expansion versus AOT

Summary

• Normal factorization derivation naturally leads

to xN as scaling variable/independent variable.

• These are easy to retain (AOT).
• Sensitivity to a target mass might say

something about nucleon structure.

– Compare Proton, Kaon, Pion, Nucleus targets

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Backup

25

• AOT (direct factorization):

– Direct power expansion in small partonic mass scales – Keep exact momentum expressions

• OPE:

– Transform to Mellin moment space – Expand in 1/Q (both twist and target momentum) – Truncate twist

• Identify leading M/Q part
• Identify remaining series of M/Q

– Invert leading M/Q (F1

(0) )

– Invert series of M/Q (F1

TMC )

– Relate F1

(0) and F1 TMC

26

Operator product expansion versus AOT

OPE-based

• Georgi-Politzer (1976)
• What is F1,2

(0) ?

27

ρ2 ≡ 1 + 4x2

BjM 2

Q2

F TMC

1

(xBj, Q2) = 1 + ρ 2ρ F (0)

1

(xN, Q2) +

) + ρ2 1 4ρ2 Z 1

xN

du u2 F (0)

1

(u, Q2) + ) + (ρ2 1)2 8xBjρ3 Z 1

xN

du Z 1

u

dv v2 F (0)

1

(v, Q2)

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As exact structure function

• Power series:
• Mellin moments:
• Leading twist
• F2 =

2 xBj

∞

X

l=0

1 x2l

Bj ∞

X

j=0

✓M 2 Q2 ◆j Nj,lC2l+2jA2j+2l+2 Z 1 xn−2

Bj F2 = ∞

X

j=0

✓M 2 Q2 ◆j ¯ Nn,jCn+2jA2j+n

Z 1 dy yn−2F (0)

2

(y, Q2) ⌘ CnAn

F2 = 1 2πi Z i∞

−i∞

dn x1−n

Bj

✓M 2 Q2 ◆j ¯ Nn,jCn+2jA2j+n

Drop Higher Twist

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As exact structure function

• Power series:
• Integer Mellin moments:
• Leading twist
• F2 =

2 xBj

∞

X

l=0

1 x2l

Bj ∞

X

j=0

✓M 2 Q2 ◆j Nj,lC2l+2jA2j+2l+2 Z 1 xn−2

Bj F2 = ∞

X

j=0

✓M 2 Q2 ◆j ¯ Nn,jCn+2jA2j+n

Z 1 dy yn−2F (0)

2

(y, Q2) ⌘ CnAn

F2 = 1 2πi Z i∞

−i∞

dn x1−n

Bj

✓M 2 Q2 ◆j ¯ Nn,jCn+2jA2j+n

Series in αs OPE provides info about integer values

30

As exact structure function

• Power series:
• Integer Mellin moments:
• Leading twist, zero mass

F2 = 2 xBj

∞

X

l=0

1 x2l

Bj ∞

X

j=0

✓M 2 Q2 ◆j Nj,lC2l+2jA2j+2l+2 Z 1 xn−2

Bj F2 = ∞

X

j=0

✓M 2 Q2 ◆j ¯ Nn,jCn+2jA2j+n

Z 1 dy yn−2F (0)

2

(y, Q2) ⌘ CnAn

F2 = 1 2πi Z i∞

−i∞

dn x1−n

Bj

✓M 2 Q2 ◆j ¯ Nn,jCn+2jA2j+n

Extend to non-integer values

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As exact structure function

• Invert:

F2 = 1 2πi Z i∞

−i∞

dn x1−n

Bj

✓M 2 Q2 ◆j ¯ Nn,jCn+2jA2j+n F TMC

1

(xBj, Q2) = 1 + ρ 2ρ F (0)

1

(xN, Q2) +

) + ρ2 1 4ρ2 Z 1

xN

du u2 F (0)

1

(u, Q2) + ) + (ρ2 1)2 8xBjρ3 Z 1

xN

du Z 1

u

dv v2 F (0)

1

(v, Q2)

32

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x 0.010 0.050 0.100 0.500 1 5 F(x)

Structure Functions

• Functions with equal moments up to N = 13

33

As fit to phenomenological structure function

• Finite order hard part
• Parametrization of pdf
• Fit needed all the way to x = 1
• Theoretical leading twist ≠ pheno fit near x = 1

F2 = X

j

Cαn

s

j/i(x/ξ, Q) ⌦ fi/P (ξ; Q)

As a parton density

• Ellis-Furmanski-Petronzio (Intro - 1982)
• Impose k2 = 0 but allow kT > 0

34

P

k

• Z

dk−d2kt

35

As a parton density

• In low order Yukawa theory

P

k

Z dw− (2⇡) e−i⇠P +w− hP| ¯ 0(0, w−, 0t)+ 2 0(0, 0, 0t) |Pi

36

0.0 0.2 0.4 0.6 0.8 1.0

kT (GeV)

0.2 0.6 1.0 1.4

v (GeV)

Q = 2 GeV

As a parton density

• v = √-k2
• Blue = in pdf
• Red = in

unapproximated graph

Moffat et al., (2017)

• In low order Yukawa theory