Moments of leading and next-to-leading twist Nucleon Distribution Amplitudes. Institut für Theoretische Physik Universität Regensburg

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1 Moments of leading and next-to-leading twist Nucleon Distribution Amplitudes Nikolaus Warkentin for QCDSF Institut für Theoretische Physik Universität Regensburg

2 Theoretical framework for hard exclusive processes QCD factorization e.g. magnetic form factor of the proton can be written as: G M (Q 2 ) = 1 [dx] 1 [dy]φ (y i, Q y )T H (x i, y i, Q)Φ(x i, Q x ) [ 1 + O(m 2 /Q 2 ) ] 0 0 [dx] = dx 1 dx 2 dx 3 δ(1 i x i ) and Q x min i (x i Q) x i, y i are longitudinal momentum fractions carried by the i-th quark. Φ as (x i ) = 120x 1 x 2 x 3 Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

3 Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

4 Starting Point In the light-cone gauge with lightlike z 1, z 2, z 3 -separation (z 1 z 2 ) 2 (z 2 z 3 ) 2 (z 3 z 1 ) 2 1/µ 2 0 uα(z a 1 )uβ(z b 2 )dγ(z c 3 ) P ɛ abc = 1 4 f [ N ( /pc) αβ (γ 5 N) γ V (z i p) + (/pγ 5 C) αβ N γ A(z i p) +(iσ µν p ν C) αβ (γ µ γ 5 N) γ T (z i p)] φ(x 1, x 2, x 3 ) :=V (x 1, x 2, x 3 ) A(x 1, x 2, x 3 ) + 2T (x 1, x 3, x 2 ) P = [dx] 12 6 (2φ(x 1, x 2, x 3 ) φ(x 3, x 2, x 1 )) u (x 1 ) ( u (x 2 )d (x 3 ) d (x 2 )u (x 3 ) ) Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

5 The Wish List ) l u(0)] a Cγ λ [(id µ j ) n γ 5d τ (0)] c P = f N V lmn (p ρ i ) l u(0)] a Cγ λ γ 5[(iD µ j ) n d τ (0)] c P = f N A lmn (p ρ i ) l u(0)] a Cγ λ ( iσ µν)[(id µ j ) n γ µ γ 5d τ (0)] c P = 2f N T lmn (p ρ i ) l u(0)] a Cγ µ [(id µ j ) n γ 5γ µd τ (0)] c P = λ 1f lmn 1 (p ρ i ) n m N N τ ) l u(0)] a Cσ µν [(id µ j ) n γ 5σ µνd τ (0)] c P = λ 2f lmn 2 (p ρ i ) n m N N τ Symmetries Knowledge of φ lmn sufficient Momentum Conservation φ lmn = φ (l+1)mn + φ l(m+1)n + φ lm(n+1) Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

6 The Wish List ) l u(0)] a Cγ λ [(id µ j ) n γ 5d τ (0)] c P = f N V lmn (p ρ i ) l u(0)] a Cγ λ γ 5[(iD µ j ) n d τ (0)] c P = f N A lmn (p ρ i ) l u(0)] a Cγ λ ( iσ µν)[(id µ j ) n γ µ γ 5d τ (0)] c P = 2f N T lmn (p ρ i ) l u(0)] a Cγ µ [(id µ j ) n γ 5γ µd τ (0)] c P = λ 1f lmn 1 (p ρ i ) n m N N τ ) l u(0)] a Cσ µν [(id µ j ) n γ 5σ µνd τ (0)] c P = λ 2f lmn 2 (p ρ i ) n m N N τ Symmetries Knowledge of φ lmn sufficient Momentum Conservation φ lmn = φ (l+1)mn + φ l(m+1)n + φ lm(n+1) Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

7 The Wish List ) l u(0)] a Cγ λ [(id µ j ) n γ 5d τ (0)] c P = f N V lmn (p ρ i ) l u(0)] a Cγ λ γ 5[(iD µ j ) n d τ (0)] c P = f N A lmn (p ρ i ) l u(0)] a Cγ λ ( iσ µν)[(id µ j ) n γ µ γ 5d τ (0)] c P = 2f N T lmn (p ρ i ) l u(0)] a Cγ µ [(id µ j ) n γ 5γ µd τ (0)] c P = λ 1f lmn 1 (p ρ i ) n m N N τ ) l u(0)] a Cσ µν [(id µ j ) n γ 5σ µνd τ (0)] c P = λ 2f lmn 2 (p ρ i ) n m N N τ Symmetries Knowledge of φ lmn sufficient Momentum Conservation φ lmn = φ (l+1)mn + φ l(m+1)n + φ lm(n+1) Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

8 Leading Twist Projection Weyl representation: γ 5 = +1 spinor: χ α (χ ) α γ 5 = 1 spinor: χ α (χ ) α σ µ matrices: (σ µ ) α α ( σ µ ) α α Converts vector-like objects to spinor-like objects Different classes of 3-quark operators: Quarks with same (ɛ abc q a αq b β qc γ) and different (ɛ abc q a α q b β qc γ) chirality Leading Twist projection (for nucleon) 1 : O α βγ = ɛ abc q a α q b {β qc γ} Easy to generalize to composite operators with derivatives 1 M.E.Peskin (1979) Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

9 Complete set of operators for Leading Twist Use only operators from one "good" irreducible representation See next talk by Thomas Kaltenbrunner 0 Derivatives (τ 12 1 ) 0 O 12 0 p = f N(p 1 γ 1 p 2 γ 2 )N(p) Bad choice 0 O 34 0 p = f N(p 3 γ 3 p 4 γ 4 )N(p) 0 O p = f N (p 1 γ 1 + p 2 γ 2 p 3 γ 3 p 4 γ 4 )N(p) Notation N(p) Nucleon spinor f N Wave function normalisation φ lmn NDA moment 1 Derivative (τ 12 2 ), l + m + n = 1 p O 12 1 p = f Nφ lmn [(γ 1 p 1 γ 2 p 2 )(γ 3 p 3 + γ 4 p 4 ) 2p 1 p 2 γ 1 γ 2 ] N(p) 0 O 34 1 p = f Nφ lmn [(γ 1 p 1 + γ 2 p 2 )(γ 3 p 3 γ 4 p 4 ) 2p 3 p 4 γ 3 γ 4 ] N(p) 0 O p = f N φ lmn (γ 1 p 1 γ 2 p 2 )(γ 3 p 3 γ 4 p 4 )N(p) 2 Derivatives (τ 4 2 ), l + m + n = 2 p 2 p O p = f N φ lmn [p 1 p 2 γ 1 γ 2 (p 3 γ 3 p 4 γ 4 ) + p 3 p 4 γ 3 γ 4 (p 1 γ 1 p 2 γ 2 )] N(p) Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

10 Example for the first moment Rewrite the irreducible operator as sum of DA s operators B D = ( D1 u2u a 3 b d2 c id 2 u2 a u3 b d2 c D 3 u2u a 4 b d2 c id 4 u2u a 4 b d2 c ) = ( A ia i A 23 2 A A i A i A 41 2 A 42 iv2 14 i V V2 24 V2 31 i V2 32 iv V2 42 Use different copies of the representation for isospin 1/2 operators Calculate the matrix element 4 0 B D B D++ 66 p = N 2 (p 1 + ip 2 ) (p 3 + ip 4 ) f N φ lmn ) 2 V 13 2 Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

11 Lattice Setup valence & sea quark Clover-Wilson action plaquette gluon action QCDSF/UKQCD configurations Used configurations β κ (m π[gev ]) volume a[fm] L[fm] (1.411), (1.029), (0.587) (0.800), ( 0.587), (0.383) (0.856), ( 0.421) general 3-quark operators calculated on APE-machine on 128 nodes ( ) and 32 nodes ( ) irreducible combinations calculated on PC and normalisation extracted from nucleon-nucleon corellator 0 T [O(t) N(0) 0 and 0 T [N(t) N(0) 0 Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

12 LTW normalization constant f N 0,03 f N f N [GeV 2 ] 0,025 0,02 0,015 f N = / GeV 2 β=5.29 ( ) 0,01 f N = /- 3.39e-05 GeV 2 β=5.29 ( ) 0, ,3 0,6 0,9 1,2 1,5 1,8 2,1 2 2 m π [GeV ] Lattice (unrenormalized) Here (Z 0.8) f N = (5 ± 0.04) 10 3 GeV 2 Lattice 1988 (renormalized) G.Martinelli, C.T.Sachrajda f N = (2.9 ± 0.6) 10 3 GeV 2 LCSR V.Braun et al. f N = (5.3 ± 0.5) 10 3 GeV 2 Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

13 LTW (100)-moment 1,2 φ 100 1,1 1 φ 100 = / β = 5.29 ( ) φ 100 = / β = 5.29 ( ) 0,9 0,8 0,7 0,6 0 0,3 0,6 0,9 1,2 1,5 1,8 2,1 m π 2 [GeV 2 ] Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

14 LTW (020)-moment φ 002 0,5 φ 002 = / β=5.29 ( ) φ 002 = / β=5.29 ( ) 0,4 0,3 0,2 0,1 0 0,3 0,6 0,9 1,2 1,5 1,8 2,1 m π 2 [GeV 2 ] Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

15 NLTW constants [GeV 2 ] 0,14 0,13 0,12 0,11 0,1 0,09 0,08 0,07 0,06 Next-leading-twist constants λ 1 = / GeV 2 λ 1 (β=5.29, 16 3 x 32) λ 1 = / GeV 2 λ 1 (β=5.29, 24 3 x 48) λ 2 = /- 7.2e-05 GeV 2 λ 2 (β=5.29, 16 3 x 32) λ 2 = / GeV 2 λ 2 (β=5.29, 24 3 x 48) 0,05 0,04 0,03 0,02 0,01 0 0,3 0,6 0,9 1,2 1,5 1,8 2,1 2 2 m π [GeV ] Lattice (unrenormalized) Here λ 1 = ( 33 ± 2) 10 3 GeV 2 λ 2 = (68 ± 4) 10 3 GeV 2 LCSR V.Braun et. al. λ 1 = ( 27 ± 5) 10 3 GeV 2 λ 2 = (54 ± 19) 10 3 GeV 2 Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14

16 Done To do General 3-quark local operator calculated. Using the irreducible operators for the 3-quark distribution amplitude operators DA s moments are calculated. f N GeV 2 (not renormalized value) is in good agreement with LCSR. λ 1, λ 2 have different signs. Statistics improved by using different momentum combinations for nucleon corellator (too expensive for the DA s ;-( ). Increase statistics (important for higher moments). Extend calculation to the parity partner of the nucleon. Extend to higher moments of higher twists. Nikolaus Warkentin (Regensburg) NDA moments 2nd August / 14