Generalized Parton Distributions in PT

Transcription

1 Generalized Parton Distributions in PT Nikolai Kivel in collaboration with M. Polyakov & A. Vladimirov

2 Deeply Virtual Compton Scattering e e * A DVCS Q 2 large >> 1/R N Q 2 /s =x Bj fixed Δ 2 ~ 1/R N <<Q 2 Highly virtual photon interacts with the quarks and gluons inside nucleon

3 DVCS Amplitude Breit Frame q=(0,0,0,q) P + =E+P z ~Q large * q ~Q 2 xp + + xp ) Δ(Δ +, Δ, Δ - ) Δ + ξ P + Δ ~ 1/R N <<Q A = dx 1 x ξ + i0 H(x, ξ, 2 )

4 GPDs n=(1,0,0,-1) Δ(Δ +, Δ ) N q(λn) γ + q( λn) N xp + + xp + + Δ + ξ P + 2 ) N q(λn) γ + q( λn) N FT [dλ] H(x, ξ, 2 ) GPDs to PDF: H q (x, 0, 0) = f q (x) GPDs to FF: dxh(x, ξ, 2 ) = F em ( 2 )

5 x=0.05 x= 1 Parton Distribution in Transverse Plane 0.5! fm in transverse -0.5 x=0.3 x= fm fm uv-quark x=0.3 Kroll et al, x= fm fm 0.5 Density 2 partons of 0 plane (b ) with given -1 x= fm momentum fraction x fm d i( b ) e H(x, ξ = 0, 2 ) = q(x, b ) 2π -0.5 x= fm

6 Nucleon GPD at small momentum transfer χpt : At large distances pions are the main dynamical degrees of freedom Nucleon has large mass m N Δ, m π static source of pions (pion cloud) dominant contribution at small-t due to the pion cloud skewedness mom fraction large small x m π m N ξ = + P + m π /m N light cone distance b y λ 1/m N b x ~ b y ~1/m π N b x λ ~1/m π x m π m N λ 1/m π

7 Breakdown of χpt expansion at small-x Kivel, Polyakov 07 Structure of chiral expansion a χ m π = 0 2 (4πF π ) 2 1 SU(2) L xsu(2) R SU(2) V H(x, ξ = 0, ) q(x) + n>0 ω n δ (n 1) (x)a n χ ln n [µ 2 χ/ 2 ] +... If x 2 (4πF π ) 2 1 small then the small parameter is compensated by δ-function (a χ ) n δ (n 1) (x) = δ (n 1) (x/a χ ) O(1)

8 Breakdown of χpt expansion at small-x Kivel, Polyakov 07 Structure of chiral expansion a χ m π = 0 2 (4πF π ) 2 1 SU(2) L xsu(2) R SU(2) V H(x, ξ = 0, ) q(x) + n>0 ω n δ (n 1) (x)a n χ ln n [µ 2 χ/ 2 ] +... Resummation If n>0 x 2 (4πF π ) 2 1 ω n δ (n 1) (x)ɛ n = q sing ( x ɛ small ) LLog s accuracy: ɛ = a χ ln[1/a χ ]

9 the sum of s- and t- channel resonances π O(λ) ππ R j 0 O(λ) R J 1 M 2 R t πr J π = R j R J t-channel H(x, ξ, t) R(l) c l G Rππ(M R ) l M 2 R t s-channel ( ) ( 1 x P l θ( x ξ)f R ϕ R ξ ξ ) ξ 0 limit: ξ 0 δ (l 1) (x) H(x, ) R j c j f R G Rππ M l R M 2 R + 2 δ (l 1) (x) +...

10 QCD operator as a pion probe LO chiral Lagrangian L 2 = 1 2 πa 2 π σ 2 σ + m 2 F π σ SU(2) L xsu(2) R SU(2) V π 2 + σ 2 = F 2 π Light-cone operator O c (λ) = q ( ) 1 2 λn τ c γ + q ( 12 λ n ) O(p), λ O(p 1 ), x ξ O(p 0 ) O c (λ) ε abc F (α, β) π a (α λn) + π b (β λn)

11 LLog s accuracy χpt case: dim[g n ] ~(F π ) -2n Lessons from 3-loop calc: p k p massless pions only 2-particle vertex a n χ ln n [1/a χ ]δ (n 1) (x) maximal spin for fixed dimension RG in renormalizable case: power divergencies to LLog accuracy considerable simplifications LLog s one loop calc of RG-functions

12 LLog s accuracy χpt case: dim[g n ] ~(F π ) -2n Lessons from 3-loop calc: p k p a n χ ln n [1/a χ ]δ (n 1) (x) L = 1 2 πa 2 π a 1 8F 2 π power divergencies to LLog accuracy considerable simplifications RG works like in renormalizable case: π 2 2 π 2 + g 2 8F 4 π π 2 4 π 2 + g 3 8F 6 π 1-loop β-functions of the couplings g 2,3 fix LLog coefficients π 2 6 π

13 Chiral Leading Log resummation χpt expansion H sing (x, ) = n>0 ω n n! xn 1 δ (n 1) (x)ɛ n +... small chiral parameter ɛ = 2 (4πF π ) 2 ln 2 (4πF π ) 2 1 PDF moments x n 1 = 1 1 dβ β n 1 q(β) Assume that then ω I n n = 1 q sing (x) = θ( x < ɛ) 0 dz z n 1 W (z) 1 x ɛ dβ β q(β) W ( ) x βɛ

14 L = L 0 + V i = 1 8 i=1 [i/2] k=0 V i LLog Lagrangian g 10 = 1 F 2 π g ik π a µ1...µ k π a 2(i 2k) π b µ1...µ k π b renormalizable for massless pions (no tadpoles) 1-loop β-functions of the couplings g ik fix the LLogs coefficients µ d dµ g nr(µ) = β nr (g ik ), i < n

15 Results V i = 1 8 [i/2] g ik π a µ1...µ k π a 2(i 2k) π b µ1...µ k π b k=0 Text g i g ij g n i,m n-i g nr H sing (x, ) = n>0 ω n n! xn 1 δ (n 1) (x)ɛ n +... ω n = [n/2] ω nr ω n0 ω 10 = 1 r=0 ω nr = 1 (n 1) n 1 i=1 (jm) β nr [ij, (n i)m] ω ij ω (n i)m.

16 ω nr = 1 (n 1) β-functions: n 1 i=1 (jm) β nr [ij, (n i)m] ω ij ω (n i)m. β nr [ji, (n i)m] = 3 δ rj δ rm 2 4r δ rmω i(2j) 2r n +8(4r + 1) k=0 (2k + 1)Ω i(2j) k Ω (n i)(2m) k Ω nk (2r) Ω nm j = 2 n [ ] dx P j (x) (x 1) n (x + 3) P m (x 1) P i (x) Legendre polynomials

17 Checks: reproduce correctly LLogs in all higher order calculations existing in literature 2-loop LLogs for pion-pion scattering 2-loop LLogs for em. form factor 4-loop LLogs for the scalar form factor

18 Properties ω n ω n = ω n(r+1) /ω nr 1/10 ω n = Large-n: ω n ( ) n 3a, a < 1 2 I=1 a = 0.766, n=1,3,... I=0 a = 0.750, n=2,4,... [n/2] r=0 ω nr ω n0 H sing (x, ) = n>0 ln Ω n I [n/2] r=0 ω nr ω n0 15 I 0 10 I n ω n n! xn 1 δ (n 1) (x)ɛ n +

19 Approximate solutions ω I=1 n a n ( e (n 1)/b), a = , b = 8, Ratio Ω n I model Ωn I exact model asymptotic n

20 Approximate solution Model q sing (x) = θ( x < ɛ) 1 x ɛ dβ β q(β) W ( ) x βɛ ω I=1 I=1 n a ( n 1 e, a b 8, n a n e (n 1)/b), a = , b = 8, q sing (x) 2 3 θ( x < a ɛ) 1 a = 0.766, a = θ( x < a ɛ) x /aɛ 1 dβ β q(β) x /a ɛ dβ β q(β)

21 Properties of the solution Small-x: q(x) 1/x ω, ω 1/2 Small-t and small-x asymptotic (m π =0) H sing (x, ) 1 x ω [ 2 [ (4πF π ) 2 ln (4πFπ ) 2 ]] ω 2 H reg (x, ) 1 x ω 4 [ (4πF π ) 4 ln (4πFπ ) 2 ] 2 Phenomenology Regge motivated model H(x, ) q(x)x α 2 x ω α 2

22 Properties of the solution Small-x: q(x) 1/x ω, ω 1/2 Distribution of partons in transverse plane (m π =0) d 2π ei( b ) H(x, ξ = 0, 2 ) = q(x, b ) b q(x, b ) 1 b 2 1 x ω [ ln[b 2 F 2 π] b 2 F 2 π ] ω naive NLO result: q(x, b ) 1 b 4 δ(x).

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