Moments of generalized parton distribution functions viewed from chiral effective field theory.

Transcription

1 Moments of generalized parton distribution functions viewed from chiral effective field theory. Marina Dorati Universita di Pavia, Dipartimento di Fisica Nucleare e Teorica Technische Universität München, Physik T39 Marina Dorati (UniPV) Moments of GPDs in HBChPT / 9

2 Outline Form Factors of the nucleon to O(p 3 ) Definition HBChPT Chiral ChPT results up to order O(p 3 ) Generalized Parton Distributions Generalized Form Factors to O(p 4 ) Definition ChPT results up to order O(p 4 ) Forward limit Conclusions and Outlook Marina Dorati (UniPV) Moments of GPDs in HBChPT / 9

3 Form Factors of the nucleon to O(p 3 ) Isovector Form Factors Definition Nucleon matri element of the isovector component of the quark vector current Vµ a = qγµ( τ a )q : [ N(p ) Vµ N(p a ) = u(p ) F υ (q )γ µ + i ] F υ (q )σ µνq ν u(p ) η τ a m N η isovector F υ i Sachs Form Factors = F p F n Form factor y GE υ (q ) = F υ (q )+ q 4m F υ (q ) N GM υ (q ) = F υ (q ) + F υ (q ) z b ρ( b ) [ Fi υ (q ) = Fı υ (0) + ] 6 (r i υ ) q + O(q 4 ) (ri υ ) = 6 df υ (q i ) F i (0) dq q =0 0 b Marina Dorati (UniPV) Moments of GPDs in HBChPT 3 / 9

4 Form Factors of the nucleon to O(p 3 ) Non-relativistic reduction Definition Nucleons treated non relativistically = Heavy Baryon approimation non-relativistic reduction in the Breit frame Breit frame for ep ep p = + q / P = q / p = q / P = + q / q = (0, q ) = Q = q N(p ) V a µ N(p ) = p µ mυ µ + r µ [ u υ(r ) G E (q )υ µ + ] GM (q )[S µ, S ν]q ν u υ(r ) η τ a N N m N η u υ(r) = P υ + u(p) = ( + /υ)u(p) E N i = i + m N m N with the choice υ µ = (, 0, 0, 0) G E (q ) G E (q ) G M (q ) G M (q ) Marina Dorati (UniPV) Moments of GPDs in HBChPT 4 / 9

5 Form Factors of the nucleon to O(p 3 ) HBChPT Heavy Baryon Chiral Perturbation Theory L ππ () = { } 4 F π Tr[ µu µ U + χ U + χu ] [ i τ ] π U = ep F π µu = µu i(v µ + a µ)u + iu(v µ a µ) χ = B(s + ip) L () πn = N(iυ D + g AS u)n N υ e imυ P υ+ Ψ, D µψ = µψ + Γ µψ Γ µ = [u, µu] i (vµ + aµ)u i u(vµ aµ)u u µ i(u µu u µu ) S µ = iγ5σµνυµ Marina Dorati (UniPV) Moments of GPDs in HBChPT 5 / 9

6 Form Factors of the nucleon to O(p 3 ) Chiral ChPT results up to order O(p 3 ) Non-zero loop diagrams at order O(p 3 ) a) b) c) V.Bernard,N.Kaiser,J.Kambor,U.Meissner Nucl.Phys.B388(99) where N L number of independent loop momenta I M number of internal pion lines Power Counting Scheme of HBChPT: D = 4N L I M I B + n Nn M + n Nn B Nn M number of pion vertices originating from L n N M = n= nnn M total number of pion vertices I B number of internal nucleon lines Nn B number of baryonic vertices originating from L(n) πn N B = n= number of baryonic vertices n= Marina Dorati (UniPV) Moments of GPDs in HBChPT 6 / 9 n=

7 Form Factors of the nucleon to O(p 3 ) Chiral ChPT results up to order O(p 3 ) Eample of Loop Diagram calculation defining m = m π q ( ). [ L = λd 4 6π d 4 + ] (γ E ln4π) A a = dl 4 (π) 4 u(r ) eɛ c3b ɛ (l + l + q) g as (l + q)τ a F π iδ dc l mπ + i0+ { u(r ) ɛ υ u(r ) i υ (r l) + i0 + g as lτ d iδ ab (l + q) m π + i0+ u(r ) F π [(6m π 53 ) ( q 6π L + log mπ λ ga = i τ i (4πF π) η η + mπ [ m 3 q + d (3mπ ]] 5q ( ))log 0 mπ } u(r )[S µ, S ν]ɛ µ ν q ν u(r ) d 4π m 0 ) Marina Dorati (UniPV) Moments of GPDs in HBChPT 7 / 9

8 Form Factors of the nucleon to O(p 3 ) Chiral ChPT results up to order O(p 3 ) Dirac Form Factor F υ (q ) = + ( {q (4πF π) ) ( 3 g A B(r) 0 + q 5 3 g A ) log 3 F v q [ d mπ (3g A + ) q ( ) q GeV [ ] mπ λ )] [ m (5g A ]} + log mπ Marina Dorati (UniPV) Moments of GPDs in HBChPT 8 / 9

9 Form Factors of the nucleon to O(p 3 ) Chiral ChPT results up to order O(p 3 ) Pauli Form Factor F υ (q ) = κ υ { ga 4πM [ ]} N (4πF π) d m m π 0 F v q q GeV Marina Dorati (UniPV) Moments of GPDs in HBChPT 9 / 9

10 Form Factors of the nucleon to O(p 3 ) Generalized Parton Distributions Generalized Parton Distributions GPDs as generalization of Parton Distributions. Parton density DIS ep ex (a) γ (q) γ (q) δ z p y FACTORIZATION: hard partonic subprocess + parton distributions p p z f ( ) Case finite momentum transfer to the target Handbag for CS PDs (a) (b) (c) γ (q) γ (q) γ (q) γ (q ) 0 +ξ ξ ξ+ ξ p p p p Handbag for DVCS GPDs Marina Dorati (UniPV) Moments of GPDs in HBChPT 0 / 9

11 Form Factors of the nucleon to O(p 3 ) Generalized Parton Distributions Notation: Working in light-cone coordinates v ± = (v 0 ± v 3 ), v = (v, v ) Definition: p = p + p, = p p, t = ξ = p+ p + p + + p + F q = = p + dz [ + z π eip p q( z)γ+ q( z+=0,z=0 z) p ] H q (, ξ, t)u(p )γ + u(p) + E q (, ξ, t)u(p ) iσ+α α m u(p) [, ], ξ 0 ξ ξ +ξ ξ +ξ ξ ξ 0 ξ Marina Dorati (UniPV) Moments of GPDs in HBChPT / 9

12 Form Factors of the nucleon to O(p 3 ) Generalized Parton Distributions GPDs: properties and relation to Form Factors caso ξ = t = 0 [forward limit] = GPDs reduce to PDs H q (, 0, 0) = q() no corrisponding relation for E! case ξ = 0, t 0 [purely transverse momentum transfer] H q (, 0, ) = d r f (, b )e b = info about transverse structure of the target Generalized parton distribution at =0 δ z p z b y O th moments of GPDs f (, b ) d H q (, ξ, ) = F q ( ), d E q (, ξ, ) = F q ( ) 0 b Marina Dorati (UniPV) Moments of GPDs in HBChPT / 9

13 Generalized Form Factors to O(p 4 ) Generalized Form Factors Definition higher -moment of GPDs are form factors of the local twist-two operator O (n),q µ,µ...µn = n q γ {µ (i ) D µ... (i ) D µn } q traccia n = th moments of GPDs d H q (, ξ, ) = A q,0 ( ) + ξ C q,0 ( ) d E q (, ξ, ) = B q,0 ( ) ξ C q,0 ( ) Matri element of singlet twist- operator: p O q {µν} p i p qγ {µ D ν} q p = A q,0 ( ) u(p )γ {µ p ν} u(p) B q,0 ( ) i m N u(p ) α σ α{µ p ν} u(p) + C q,0 ( ) m N u(p )u(p) {µ ν} p = (p + p), where D µ = ( D µ D µ) Marina Dorati (UniPV) Moments of GPDs in HBChPT 3 / 9

14 Generalized Form Factors to O(p 4 ) Connection to phenomenology Definition caso ξ = 0 [limite forward] H q (, 0, 0) = q() A q (0) = q = d (q () + q ()) 0 as for Form Factors A q,0 = Aq,0 [ (0) + ] 6 (r )q + O(q 4 ) (r υ A,) = 6 A,0 (0) da υ,0(q ) dq q =0 Marina Dorati (UniPV) Moments of GPDs in HBChPT 4 / 9

15 Generalized Form Factors to O(p 4 ) Non-relativistic reduction Definition By performing the non-relativistic reduction we define three new structures: p O q {µν} p = { u υr G (t)(υ µυ ν + υ νυ µ N N gµν) + G (t)([s µ, S α] α υ ν + υ µ[s ν, S α] α ) + G 3 (t)( µ ν + ν µ gµν ) }u υ(r ) η τ a η. related to the Generalized Form Factors through the following epressions: G (t) = uu mn t [ A v,0 4 (t) + t ] 4m B v,0 (t) N G (t) = uu mn t [ ] (A v,0 4 m (t) + Bv,0 (t)) N [ ] G 3 (t) = uu C v,0 m (t) N Marina Dorati (UniPV) Moments of GPDs in HBChPT 5 / 9

16 Generalized Form Factors to O(p 4 ) ChPT results up to order O(p 4 ) Loop diagrams up to order O(p 4 ) To this order the three structures take the form G (t) = uu mn t [ A v,0 4 (t)+ t ] B v 4mN,0 (0)+O(p5 ) G (t) = uu mn t [ ] (A v,0 4 m (t)+bv,0 (0))+O(p5 ) N [ ] G 3 (t) = uu C v,0 m (0) + O(p5 ) N Introduction of new mathematical tools: V µν = ṽ µν (u τ a u + uτ a u ) L () πn, L(3) πn, L(4) πn Marina Dorati (UniPV) Moments of GPDs in HBChPT 6 / 9

17 ξ = t = 0 Generalized Form Factors to O(p 4 ) Forward limit The first structure G (t) reduces to and G (0) = uu m N [ A v,0 (0) + O(p5 ) ] = uu m N u d u d = c 8 { (3g A + ) (4πF π) m π log [ m π λ ] } (4πF π) g A m π + 4m π S(r) 40 Marina Dorati (UniPV) Moments of GPDs in HBChPT 7 / 9

18 Results Generalized Form Factors to O(p 4 ) Forward limit The forward limit HBChPT calculation, performed for the Generalized Form Factors n = up to order O(p 4 ), give the following results: A v,0 (t) = c 8 { (3g A + ) (4πF π) m π log [ m π λ + 4m πs (r) 40 + S 44t + O(p 5 ). ] } (4πF π) g A m π B v,0 (0) = B 40 m N + O(p 5 ) C v,0 (0) = S 4 m N + O(p 5 ) Marina Dorati (UniPV) Moments of GPDs in HBChPT 8 / 9

19 Generalized Form Factors to O(p 4 ) Conclusions and Outlook Conclusions and Outlook Reproduction of HBChPT results for Dirac (F v (q )) and Pauli (F v (q )) Form Factors; Introduction of GPDs and definiton of the Generalized Isovector Form Factors (A v,0 (t), Bv,0 (t), Cv,0 (t)); Derivation of the non-relativistic Form Factors G i (t); Construction of new Lagrangians and introduction of new counterterms; Forward limit up to order O(p 4 ), derivation of < > u d and comparison with phenomenological and lattice data. NEXT: OFF-FORWARD LIMIT UP TO ORDER O(p 5 ) AND MAPPING OF LOOP DIAGRAMS INTO A v,0 (t), Bv,0 (t), Cv,0 (t) Ph.Hägler et al.,hep lat/04096 Marina Dorati (UniPV) Moments of GPDs in HBChPT 9 / 9