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

Hadron Masses and Factorization (in DIS) Ted Rogers Jefferson Lab/Old Dominion University Quark-Hadron Duality 2018, James Madison University, Sept 24 2018 1

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. 2

Two Questions • What are kinematical target mass approximations? • When/how do they matter? 3

Target Mass Corrections • Large number of TMC formalisms: Brady, Accardi, Hobbs, Melnitchouk PHYSICAL REVIEW D 84, 074008 (2011 ) – OPE based – Feynman graph based – Higher twist – Standard factorization (Aivazis, Olness, Tung) 4

Standard setup • Definition of a cross section N ! | M e,P ! N | 2 d 3 p 1 d 3 p 2 d 3 p N X (2 ⇡ ) 4 � (4) d � = · · · × P + l − p i 2 � ( s, m 2 e , M 2 ) 1 / 2 (2 ⇡ ) 3 2 E 1 (2 ⇡ ) 3 2 E 2 (2 ⇡ ) 3 2 E N i =1 − 2 ↵ 2 E 0 d � Single photon exchange ( s − M 2 ) Q 4 L µ⌫ W µ⌫ em d 3 l 0 = F 1 + ( P µ − q µ P · q/q 2 )( P ⌫ − q ⌫ P · q/q 2 ) − g µ⌫ + q µ q ⌫ ✓ ◆ W µ⌫ = F 2 q 2 P · q 5

Massless Target Approximation (MTA) • Exact: P + , M 2 ✓ ◆ ⇣p ⌘ M 2 + P 2 P = z , 0 , 0 , P z = 2 P + , 0 T • The approximation: P → ˜ P + , 0 , 0 T � � P = ( P z , 0 , 0 , P z ) = M 2 /Q 2 → 0 2 P · q → 2 ˜ P · q • Usually taken for granted at large Q and small x 6

MTA in Light-Cone Fractions • Light-cone ratios: − q + 2 x Bj – No MTA: P + = x N ≡ q 4 x 2 Bj M 2 1 + 1 + Q 2 ! x 2 Bj M 2 − q + – MTA: P + = x Bj + O Q 2 7

Structure Functions − g µ ⌫ + q µ q ⌫ ✓ ◆ W µ ⌫ = F 1 ( x N , Q ) q 2 ◆ F 2 ( x N , Q ) ✓ ◆ ✓ P µ − P · q P ⌫ − P · q q 2 q µ q 2 q ⌫ + P · q 8

Structure Functions − g µ ⌫ + q µ q ⌫ ✓ ◆ W µ ⌫ = F 1 ( x N , Q ) q 2 ◆ F 2 ( x N , Q ) ✓ ◆ ✓ P µ − P · q P ⌫ − P · q q 2 q µ q 2 q ⌫ + P · q F 1 ( x N , Q 2 ) ≡ P 1 ( x N , Q 2 , M 2 ) µν W µν F 2 ( x N , Q 2 ) ≡ P 2 ( x N , Q 2 , M 2 ) µν W µν 2 Q 2 x 2 P 1 ( x N , Q 2 , M 2 ) µ ν ≡ − 1 2 g µ ν + N N + Q 2 ) 2 P µ P ν ( M 2 x 2 N + Q 2 � 2 ! P 2 ( x N , Q 2 , M 2 ) µ ν ≡ 12 Q 4 x 3 � Q 2 − M 2 x 2 � � M 2 x 2 N N P µ P ν − g µ ν N ) 4 12 Q 2 x 2 ( Q 2 + M 2 x 2 N 9

MTA M 2 /Q 2 ! 0 2 P · q ! 2 ˜ P · q F 1 ( x Bj , Q 2 ) ≡ P 1 ( x Bj , Q 2 , 0) µν W µν F 2 ( x Bj , Q 2 ) ≡ P 2 ( x Bj , Q 2 , 0) µν W µν − g µ ⌫ + q µ q ⌫ ✓ ◆ W µ ⌫ → F 1 ( x Bj , Q 2 ) q 2 P µ − q µ ˜ P ⌫ − q ⌫ ˜ + ( ˜ P · q/q 2 )( ˜ P · q/q 2 ) F 2 ( x Bj , Q 2 ) ˜ P · q 10

Factorization • Power expansion ✓ m 2 ◆ d � Z dˆ � d x Bj d Q 2 = d ⇠ x Bj d Q 2 f ( ⇠ ) + O Q 2 dˆ • m 2 = parton virtuality, transverse momentum, mass … • What about hadron masses? For now assume M 2 ≠ O(m 2 ) 11

Factorization and partonic light-cone fractions Q 2 ✓ ◆ � x N P + , q = 2 x N P + , 0 T q k + = O ( Q ) k + q k 2 = O m 2 � � k ( k + q ) 2 = O m 2 � � 12

Factorization and partonic light-cone fractions Q 2 ✓ ◆ � x N P + , q = 2 x N P + , 0 T q k + = O ( Q ) k + q k 2 = O m 2 � � k ( k + q ) 2 = O m 2 � � � � � � 2 k + q − + 2 k − q + � Q 2 + k 2 = O m 2 � � 2 k + q − = Q 2 + O m 2 � � 13

Factorization and partonic light-cone fractions Q 2 ✓ ◆ � x N P + , q = 2 x N P + , 0 T q k + = O ( Q ) k + q k 2 = O m 2 � � k ( k + q ) 2 = O m 2 � � � � � � 2 k + q − + 2 k − q + � Q 2 + k 2 = O m 2 � � 2 k + q − = Q 2 + O m 2 � � ⇠ ⌘ k + ✓ m 2 ◆ P + = x N + O Q 2 14

Factorization and partonic light-cone fractions Q 2 ✓ ◆ � x N P + , q = 2 x N P + , 0 T q k + = O ( Q ) k + q k 2 = O m 2 � � k ( k + q ) 2 = O m 2 � � � � � � 2 k + q − + 2 k − q + � Q 2 + k 2 = O m 2 � � 2 k + q − = Q 2 + O m 2 � � ⇠ ⌘ k + ✓ m 2 ◆ P + = x N + O Q 2 ! x 2 Bj M 2 ✓ m 2 ◆ = x Bj + O + O Q 2 Q 2 15

Factorization Power Series • Drop O(m 2 /Q 2 ) ?: Necessary for factorization. • Drop O(x 2 M 2 /Q 2 ) ?: Not necessary for Bj factorization. 16

MTA with factorization • Make approximations with exact target momentum: W µ ν ! W µ ν fact Introduce O(m 2 /Q 2 ) errors • Then do MTA: W µν fact ! W µν fact , TMC 2 M 2 /Q 2 ) errors Introduce O(x Bj 17

Aivazis, Olness, Tung (AOT) Phys. Rev. D 50, 3085 (1994) • Normal factorization, just keeping exact mass. – MTA Z 1 d ⇠ W µ ν = ˆ W µ ν ( x Bj / ⇠ , q ) f ( ⇠ ) + O m 2 /Q 2 � M 2 /Q 2 � � � + O ⇠ x Bj – TMC Z 1 d ⇠ W µν = ˆ W µν ( x N /⇠, q ) f ( ⇠ ) + O m 2 /Q 2 � � ⇠ x N • The only “pure” kinematical correction. Others involve assumptions about dynamics. 18

What if the target mass is important? • How to test? – Scaling with Nachtmann rather than Bjorken variable? – Improved universality. Extend range of pQCD? See N. Sato talk • Why does it give improvement? Something about nucleon structure? 19

Partonic interpretation of target mass effects • Small scales q k + q • Exact target k mass useful if M 2 J k 2 Q 2 Q 2 k 2 suppression by T Q 2 partonic scales is greater than x 2 Bj M 2 target mass x 2 Bj M 2 Q 2 X Q 2 20

Partonic interpretation of target mass effects • Parton virtuality q vs. hadron mass k + q k k 2 Q 2 x 2 Bj M 2 + B k 2 ⇠ A Q 2 Q 2 ?? ?? 21

Partonic interpretation of target mass effects • Two scales? See E. Moffat talk | k 2 | ⌧ x 2 Bj M 2 q k + q k P 22

Operator product expansion versus AOT • 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 (F 1 (0) ) – Invert series of M/Q (F 1 TMC ) – Relate F 1 (0) and F 1 TMC 23

Summary • Normal factorization derivation naturally leads to x N 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 24

Backup 25

Operator product expansion versus AOT • 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 (F 1 (0) ) – Invert series of M/Q (F 1 TMC ) – Relate F 1 (0) and F 1 TMC 26

ρ 2 ≡ 1 + 4 x 2 Bj M 2 OPE-based Q 2 • Georgi-Politzer (1976) ( x Bj , Q 2 ) = 1 + ρ 2 ρ F (0) F TMC ( x N , Q 2 ) + 1 1 Z 1 ) + ρ 2 � 1 d u u 2 F (0) ( u, Q 2 ) + 1 4 ρ 2 x N Z 1 Z 1 ) + ( ρ 2 � 1) 2 d v v 2 F (0) ( v, Q 2 ) d u 1 8 x Bj ρ 3 x N u • What is F 1,2 (0) ? 27

As exact structure function • Power series: ◆ j ✓ M 2 ∞ ∞ 2 1 X X N j,l C 2 l +2 j A 2 j +2 l +2 F 2 = x 2 l Q 2 x Bj Bj j =0 l =0 Drop Higher Twist • Mellin moments: Z 1 ◆ j ∞ ✓ M 2 ¯ X x n − 2 N n,j C n +2 j A 2 j + n Bj F 2 = Q 2 0 j =0 • Leading twist Z 1 d y y n − 2 F (0) ( y, Q 2 ) ⌘ C n A n 2 0 Z i ∞ ◆ j ✓ M 2 1 • ¯ d n x 1 − n N n,j C n +2 j A 2 j + n F 2 = Bj Q 2 2 π i 28 − i ∞

As exact structure function • Power series: ◆ j ✓ M 2 ∞ ∞ 2 1 X X N j,l C 2 l +2 j A 2 j +2 l +2 F 2 = x 2 l Q 2 x Bj Bj j =0 l =0 • Integer Mellin moments: Z 1 ◆ j ∞ ✓ M 2 ¯ X x n − 2 N n,j C n +2 j A 2 j + n Bj F 2 = Q 2 0 j =0 • Leading twist Z 1 Series in α s d y y n − 2 F (0) ( y, Q 2 ) ⌘ C n A n OPE provides info about 2 0 integer values Z i ∞ ◆ j ✓ M 2 1 • ¯ d n x 1 − n N n,j C n +2 j A 2 j + n F 2 = Bj Q 2 2 π i 29 − i ∞

As exact structure function • Power series: ◆ j ✓ M 2 ∞ ∞ 2 1 X X N j,l C 2 l +2 j A 2 j +2 l +2 F 2 = x 2 l Q 2 x Bj Bj j =0 l =0 • Integer Mellin moments: Z 1 ◆ j ∞ ✓ M 2 ¯ X x n − 2 N n,j C n +2 j A 2 j + n Bj F 2 = Q 2 0 j =0 • Leading twist, zero mass Z 1 d y y n − 2 F (0) ( y, Q 2 ) ⌘ C n A n 2 0 Z i ∞ ◆ j ✓ M 2 1 ¯ Extend to non-integer values d n x 1 − n N n,j C n +2 j A 2 j + n F 2 = Bj Q 2 2 π i 30 − i ∞

As exact structure function • Invert: Z i ∞ ◆ j ✓ M 2 1 ¯ d n x 1 − n N n,j C n +2 j A 2 j + n F 2 = Bj Q 2 2 π i − i ∞ ( x Bj , Q 2 ) = 1 + ρ 2 ρ F (0) F TMC ( x N , Q 2 ) + 1 1 Z 1 ) + ρ 2 � 1 d u u 2 F (0) ( u, Q 2 ) + 1 4 ρ 2 x N Z 1 Z 1 ) + ( ρ 2 � 1) 2 d v v 2 F (0) ( v, Q 2 ) d u 1 8 x Bj ρ 3 x N u 31

• Functions with equal moments up to N = 13 Structure Functions F ( x ) 5 1 0.500 0.100 0.050 0.010 x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 32

As fit to phenomenological structure function C α n X F 2 = j/i ( x/ ξ , Q ) ⌦ f i/P ( ξ ; Q ) s j • Finite order hard part • Parametrization of pdf • Fit needed all the way to x = 1 • Theoretical leading twist ≠ pheno fit near x = 1 33