Pion Electromagnetic Form Factor in Virtuality Distribution Formalism

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1 & s Using s Pion Electromagnetic Form Factor in Distribution Formalism A. Old Dominion University and Jefferson Lab QCD 215 Workshop Jefferson Labs, May 26, 215

2 Pion Distribution Amplitude & s ϕ π (x): momentum sharing for pion in valence qq configuration (e.g. in γ (q 1 )γ (q 2 ) π process) q 1 (1 x)p p Using s q 2 xp Chernyak; A.R. (1977): DA is function whose x n moments f n = 1 x n ϕ π (x) dx are given by matrix elements of twist-2 local operators i n+1 d()γ 5 {γ ν D ν1... D νn } u() π +, P = {P ν P ν1... P νn } f n

3 Pion Light-Front Wave Function & s Using s Jackson (1977); Lepage & Brodsky (1979): k -integral of light-front wave function Ψ(x, k ) 6 ϕ π (x, µ) = (2π) 3 Ψ(x, k ) d 2 k k 2 µ2 Zeroth moment of ϕ π (x): matrix element of axial current 1 ϕ π (x, µ) dx = pion decay constant f π 13 MeV Automatic in OPE definition Violated (µ-dependent) in LF definition, e.g. 2 Λ 2 e k /Λ2 dk 2 = 1 e µ2 /Λ 2 k 2 µ2 1

4 and of & s Using s Integral under ϕ π (x) curve (in OPE definition) is fixed, but not its shape depends on renormalization scale µ: ϕ π (x) ϕ π (x, µ). equation for pion DA may be written in matrix form A.R. (1977) Or in kernel form µ d dµ f n(µ) = µ d dµ ϕ π(x, µ) = Lepage&Brodsky (1979) 1 n Z nk f k (µ) k= V (x, y) ϕ π (y, µ) dy

5 Solution of Equation & s Using s Expansion over Gegenbauer polynomials { } a 2n C 3/2 2n (2x 1) ϕ π (x, µ) = 6f π x(1 x) 1 + [ln(µ 2 /Λ 2 )] γ2n/β n=1 Efremov & A.R. (1978) Lepage & Brodsky (1979) γ 2n > is anomalous dimension of composite operator β > is the lowest coefficient of QCD β-function When µ, pion DA acquires simple form Efremov & A.R. (1978) ϕ π (x, µ ) = 6f π x(1 x) It is known as asymptotic DA

6 Width of & s Using s Quantitative measure of DA width: moments ξ 2 and ξ 4 in relative variable ξ x (1 x) ξ 2 = for infinitely narrow DA ϕ π (x) = f π δ(x 1/2) ξ 2 = 1/5 for asymptotic DA ϕ π (x) = 6f π x(1 x) ξ 2 = 1/4 for root DA ϕ π (x) = 8 π f π x(1 x) ξ 2 = 1/3 for flat DA ϕ π (x) = f π

7 CZ Estimates of Width of & s Using s QCD sum rules were used by Chernyak & Zhitnitsky (1981) CZ obtained ξ 2 =.4 for µ 2 = 1.5 GeV 2 Translates into ξ 2 =.46 for low scale µ 2 =.25 GeV 2 CZ result is larger than 1/3 of flat DA Flat DA φ π (ξ) = a + 3(1 a)ξ 2 could be used for fit CZ fitting model (CZ DA) φ CZ π (ξ) = 15 ξ 2 (1 ξ 2 )/4 is sum of two first terms of Gegenbauer expansion

8 Structure of QCD Sum Rule for & s Using s When written for DA rather than its moments: f πϕ π(x) = 3M 2 2π 2 (1 e s /M 2 )x(1 x) + αs GG [δ(x) + δ(1 x)] 24πM πα s qq 2 { } 81 M 4 11[δ(x) + δ(1 x)] + 2[δ (x) + δ (1 x)] Non-locality of condensates changes δ(x) 2x θ(x < ) 2 with = λ 2 q /2M 2 and λ 2 q = qd2 q / qq.4 GeV 2 Decreases resulting ξ 2 to.25 Mikhailov & A.R. (1986) Most recent lattice result:.24 at µ = 2 GeV Braun et al. (215) Corresponds to ϕ MR π (x) = 8 π x(1 x) NB: Value in the middle ϕ MR π (1/2) = 4/π 1.27 is close to 1.2 found by Braun & Filyanov (1989)

9 Observables: γ γπ Transition Form Factor & s q1 2 = Q2, q2 2 = q 1 (1 x)p xp q 2 p Using s Leading-order prediction F 2fπ 1 γ γπ (Q2 ) = 3 Q 2 J with J ϕ π(x) dx x Factor J is sensitive to shape of pion DA J = 2 for infinitely narrow δ(x 1/2) DA J = 3 for asymptotic 6x(1 x) DA J = 4 for 8 π x(1 x) DA J = 5 for CZ 3x(1 x)(1 2x) 2 DA J = for flat ϕ π(x) = f π DA

10 & s Using s Q 2 F(Q 2 ) (GeV) CELLO CLEO BABAR Q 2 (GeV 2 ) BABAR data may be fitted by Q 2 F γ γπ (Q2 ) = with M 2 =.6 GeV 2 2fπ 3 J J exp NB: J L (Q 2 ) does not flatten to a particular value! J L J as Q 2 (GeV 2 ) ln ( Q 2 /M ) 2fπ J L (Q 2 ) 3 J L (Q 2 ) is obtained if ϕ π(x) = f π and xq 2 xq 2 + M 2 J L (Q 2 ) = Q 2 1 dx xq 2 + M 2 M is usually treated as average intrinsic transverse momentum

11 & s Using s Use coordinate representation q z q xp xp p γ(q )γ (q) π (p) q 2 =, q 2 = Q 2 Introduce virtuality distribution amplitude () Φ(x, σ) (scalar case for illustration) 1 p φ()φ(z) = dσ dx Φ(x, σ) e i x(pz) iσ(z2 iɛ)/4 Handbag diagram in representation 1 T (Q 2 dx ) = xq 2 dσ Φ(x, σ) {1 } e [ixq2 +ɛ]/σ First term: twist-2 approximation, dσ Φ(x, σ) = ϕπ(x)

12 Transverse momentum dependent DAs & s Using s Pion momentum is defined to have no transverse components, p p + Projection on z + = interval z = (z, z ) 1 p φ()φ(z) z + =,p = = dx ϕ(x, z ) e i x(pz ) Impact parameter distribution amplitude (IDA) ϕ(x, z ) Transverse momentum dependent distribution amplitude ϕ(x, z ) = Ψ(x, k ) e i(k z ) d 2 k = dσ Φ(x, σ) e iσ(z2 +iɛ)/4 can be written in terms of Ψ(x, k ) = i dσ π σ Φ(x, σ) e i(k2 iɛ)/σ Handbag term may be written in terms of (spinor case formula) 1 [ ] T (Q 2 dx ) = xq 2 Ψ(x, k ) 1 k2 k 2 xq2 xq 2 d 2 k

13 & s Using s Generic representation treats (pz) and z 2 as independent variables 1 p φ()φ(z) F ((pz), z 2 ) = dσ dx Φ(x, σ) e i x(pz) iσ(z2 iɛ)/4 Lorentz invariance is fully incorporated already no a priori correlation of x and σ dependence in is expected Simplest example: ized models for Factorized models for Φ(x, σ) = ϕ(x) Φ(σ) Ψ(x, k ) = ϕ(x) ψ(k 2 )/π

14 Modeling soft s & s Using s Gaussian dependence on k Impact parameter Gaussian DA Ψ G (x, k ) = ϕ(x) πλ 2 e k2 /Λ2 ϕ G (x, z ) = ϕ(x) e z2 Λ2 /4 Faster fall-off at large z compared to e z m of massive propagator D c (z, m) = 1 16π 2 e iσz2 /4 i(m 2 iɛ)/σ dσ But we need p φ()φ(z) finite at z 2 = Add a constant term ( 4/Λ 2 ) to z 2 in the representation, i.e. take Φ m(x, σ; Λ) = ϕ(x) eiσ/λ2 im 2 /σ ɛσ 2imΛK 1 (2m/Λ) Concentrating on finite-size effects: take m = model Φ m= (x, σ; Λ) = ϕ(x) eiσ/λ2 ɛσ ) K (2 k 2 + m2 /Λ ; Ψ m(x, k ) = ϕ(x) πmλk 1 (2m/Λ) iλ 2 ; ϕ m= (x, z ) = ϕ(x) 1 + z 2 Λ2 /4

15 Modeling transition form & s Using s Gaussian model (spin-1/2 quarks) 1 F G (Q 2 dx ) = [1 xq 2 ϕ(x) Λ2 ( xq 2 1 e xq2 /Λ 2)] Power-like (under x-integral) twist-4 contribution Formal Q 2 limit is finite: F G (Q 2 = ) = fπ 1 2Λ 2 ; fπ ϕ(x) dx Note: F (Q 2 ) is finite for Q 2 = in any model with finite Ψ(x, k = ) Non-Gaussian m = model 1 F (Q 2 ) = F (Q 2 = ) = π 1 Ψ(x, k = ) dx 2 dx xq 2 ϕ(x) [1 Λ2 xq 2 + 2K 2(2 ] xq/λ) Size of twist-4 term is governed by confinement scale Λ

16 Comparison with data & s Using s Recent experimental data from BaBar and Belle do not show flattening yet I I 5 5 BaBar BELLE Q 2 (GeV 2 ) Q 2 (GeV 2 ) Curves for BaBar data with flat DA ϕ(x) = f π and Λ 2 G =.35 GeV2 or Λ 2 m= =.6 GeV2 Curves for Belle data with ϕ(x) f π(x x).4 and Λ 2 G =.3 GeV2 or Λ 2 m= =.4 GeV2 NB: J = 4.5 for ϕ(x) f π(x x).4, i.e. curve I(Q 2 ) for Q 2 1 GeV 2 is well below asymptotic value For Q 2 2 GeV 2 (where BaBar and Belle agree) the data may be described by models with both flat and (x x).4 behavior

17 Pion Electromagnetic Form Factor & s q Using s p 1 p 2 xp 1 yp 2 Perturbative QCD expression for one-gluon-exchange term in pion EM form (Jackson; A.R.; CZS 1977) as () Fπ (Q 2 ) = 8παs 1 1 dx dy 9 where s = 4π 2 fπ 2.7 GeV2 ϕπ(x) ϕπ(y) xyq 2 = 2 s Q 2 α s π (J/3)2 Diverges for flat DA ϕ π(x) = f π xyq 2 is virtuality of exchanged gluon: expect xyq 2 xyq 2 + M 2 when transverse momentum is included

18 Exchange diagram with virtualities & s Using s p 1 p 2 Take simplified diagram that has the same asymptotic structure 1 1 F as (Q 2 1 ) =a ϕ(x) dx ϕ(y) dy xyq 2 In coordinate representation 1 1 T (p 1, p 2 ) = dx dy d 4 z e ix(p 1z)+iy(p 2 z) D c(z) B(x, z 2 /4) B(y, z 2 /4) In representation 1 1 F (Q 2 ) =a dσ 1 dσ 2 Φ(x, σ 1 ) Φ(y, σ 2 ) dxdy [ ] xyq 2 1 e [ixyq2 +ɛ]/(σ 1 +σ 2 ) z

19 Impact parameter representation & s Using s In terms of impact parameter DAs or, taking integral over k p 1 p 2 z F (Q 2 ) = a 1 1 d 2 k (2π) 2 dx dy k 2 xyq2 xyq 2 e i(k b ) ϕ(x, b ) ϕ(y, b ) d 2 b 1 1 F (Q 2 db ) =a dx dy xyq 2 J 1 (b xyq 2 ) ϕ(x, b) ϕ(y, b)

20 IDA modeling of exchange diagram & s Using s Taking ized Gaussian or Power-law Ansatz ϕ G (x, b ) = ϕ(x)e b2 Λ2 /4, ϕ P (x, b ) = ϕ(x)/(1 + b 2 Λ2 /4) gives expressions 1 F G (Q 2 dx dy [ ) =a xyq 2 ϕ(x)ϕ(y) 1 e xyq2 /2Λ 2] a f π 2 Q 2 I G(Q 2 ) 1 F P (Q 2 ) =a dx dy xyq 2 ϕ(x)ϕ(y) finite for Q 2 =, with the values F G (Q 2 = ) =a f 2 π 2Λ 2 [1 2 xyq2 Λ 2 K 2 (2 ] xyq/λ), F P (Q 2 = ) = a f 2 π Λ 2 a f 2 π Q 2 I P (Q 2 )

21 Sensitivity to DA shape & s Using s Taking Λ 2 G =.35 GeV2 and Λ 2 P =.6 GeV2 we plot ratio I(Q 2 )/9 ( 1 corresponds to with asymptotic DA) I G (Q 2 )/9 FLAT I P (Q 2 )/9 FLAT ROOT 1. ROOT AS 1. AS Q 2 (GeV 2 ) Q 2 (GeV 2 ) Q 2 (GeV 2 ) Asymptotic behavior sets very slowly I G (Q 2 )/9 I P (Q 2 )/9 FLAT FLAT.6.6 ROOT ROOT.5.5 AS AS Q 2 (GeV 2 ) Very small sensitivity to DA shape in accessible Q 2 region Even for Q 2 = 1 GeV 2, gluon exchange contribution reaches just about half of naive value

22 Soft overlap & s Q: If gluon exchange contribution is small, what is the mechanism explaining observed pion FF behavior? A: Soft overlap = Feynman mechanism, starting with Drell-Yan term q q q p 1 p 2 + p 1 p 2 = p 1 p 2 Using s Overlap contribution may be written in terms of pion GPDs F π(q 2 ) = 1 e q dx H q(x, Q 2 ) q restricted by H q(x, ) = f q(x), Regge behavior for small x and Q 2 : H q(x, Q 2 ) x α Q 2, etc. How GPDs work here, is a subject of a different talk!

23 & s We discussed a kind of transverse momentum effects invisible in OPE They are very essential in pion form s at accessible momentum transfers Sensitivity to the shape of DA is very small in accessible region: widening of DA does not increase the size of hard gluon exchange contribution is dominated by overlap/ GPD term Studying F π(q 2 ) we investigate primordial transverse structure of the pion Using s