Moments of leading and next-to-leading twist Nucleon Distribution Amplitudes Nikolaus Warkentin for QCDSF Institut für Theoretische Physik Universität Regensburg
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 2007 2 / 14
Nikolaus Warkentin (Regensburg) NDA moments 2nd August 2007 3 / 14
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 2007 4 / 14
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 2007 5 / 14
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 2007 5 / 14
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 2007 5 / 14
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 2007 6 / 14
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 1234 0 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 1 0 0 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 1234 1 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 3 0 0 O 1234 2 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 2007 7 / 14
Example for the first moment Rewrite the irreducible operator as sum of DA s operators B D+ + 42 = 1 2 1 8 ( 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 13 2 + ia 14 2 + i A 23 2 A 24 2 + A 31 2 + i A 32 2 + i A 41 2 A 42 iv2 14 i V2 23 + V2 24 V2 31 i V2 32 iv2 41 + V2 42 Use different copies of the representation for isospin 1/2 operators Calculate the matrix element 4 0 B D+ + 42 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 2007 8 / 14
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] 5.29 0.1340 (1.411), 0.1350 (1.029), 0.1359 (0.587) 16 3 32 0.08 1.28 5.29 0.1355 (0.800), 0.1359 ( 0.587), 0.1362 (0.383) 24 3 48 0.08 1.92 5.40 13560 (0.856), 13640 ( 0.421) 24 3 48 0.07 1.68 general 3-quark operators calculated on APE-machine on 128 nodes (24 3 48) and 32 nodes (16 3 32) 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 2007 9 / 14
LTW normalization constant f N 0,03 f N f N [GeV 2 ] 0,025 0,02 0,015 f N = 0.00746 +/- 0.00079 GeV 2 β=5.29 (16 3 32) 0,01 f N = 0.005 +/- 3.39e-05 GeV 2 β=5.29 (24 3 48) 0,005 0 0 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 2007 10 / 14
LTW (100)-moment 1,2 φ 100 1,1 1 φ 100 = 1.05 +/ 0.00127 β = 5.29 (16 3 32) φ 100 = 0.873 +/ 0.026 β = 5.29 (24 3 48) 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 2007 11 / 14
LTW (020)-moment φ 002 0,5 φ 002 = 0.31 +/ 0.0299 β=5.29 (16 3 32) φ 002 = 0.365 +/ 0.0296 β=5.29 (24 3 48) 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 2007 12 / 14
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 = 0.0443 +/- 0.000422 GeV 2 λ 1 (β=5.29, 16 3 x 32) λ 1 = 0.033 +/- 0.00218 GeV 2 λ 1 (β=5.29, 24 3 x 48) λ 2 = 0.0942 +/- 7.2e-05 GeV 2 λ 2 (β=5.29, 16 3 x 32) λ 2 = 0.0678 +/- 0.0044 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 2007 13 / 14
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 5 10 3 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 2007 14 / 14