Virtuality Distributions and γγ π 0 Transition at Handbag Level

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1 and γγ π Transition at Handbag Level A.V. Radyushkin form hard Physics Department, Old Dominion University & Theory Center, Jefferson Lab May 16, 214, QCD Evolution 214, Santa Fe

2 Transverse Momentum form hard Longitudinal Momentum Distribution Transverse Momentum Distribution f(x) x N dx = p φ() + N φ() p f(x, k T ) x N (kt 2 )l dx = p φ() + N ( 2 )l φ() p Operators with ( 2 )l : higher twists Usual twist of φ() µ 1... µn φ() involves matrix elements containing 2 rather than 2 2 is related to parton virtuality p φ() N + ( 2 ) l φ() p Relate virtuality distributions with TMDs

3 γ γ π transition amplitude at twist 2 form hard q z q xp xp -2 distribution amplitude: p γ(q )γ (q) π (p) q 2 =, q 2 = Q 2 p φ()φ(z) = ϕ(x) e i x(pz) dx + O(z 2 ) -2 transition amplitude (for p 2 = and (q p) 2 = Q 2 ) T (p, q) = = dx d 4 z e i(qz) i x(pz) D c (z) ϕ(x) 1 (q xp) 2 dx = ϕ(x) xq 2 dx

4 of bilocal operator form hard Taylor expansion in bilocal operator φ()φ(z) φ(z) = n= 1 n! zµ 1... zµn µ 1... µn φ() expansion (with {z } n {z µ1... z µn } µ 1... µn ) φ(z) = ( z 2 ) l l= 4 N= -dependent matrix elements (p 2 = ) N + 1 l!(n + l + 1)! {z }N ( 2 ) l φ() p φ(){z } k ( 2) l φ() [i(zp)] k Λ 2l A kl Treating A kl as x k moments of higher-twist DAs ϕ l (x) ( Λ 2 z 2 ) l p φ()φ(z) = ϕ l (x) e i x(pz) dx 4 l= B(x, z 2 /4) e i x(pz) dx Bilocal function B(x, z 2 /4) accumulates information about parton virtuality

5 distributions form hard In fact, no assumptions about finiteness of matrix elements p φ(){z } k ( 2) l φ() are necessary! Schwinger alpha-representation for any contributing diagram l p φ()φ(z) = const dα j [A(α) + B(α)] d/2 j=1 { z 2 } /4 exp i A(α) + B(α) + i(pz) B(α) A(α) + B(α) exp ip2 C(α) i α j (m 2 j iɛ) j with positive A(α), B(α), C(α) Representation through virtuality distribution amplitude () Φ(x, σ) p φ()φ(z) = dσ dx Φ(x, σ) e i x(pz) iσ(z2 iɛ)/4 Note: dependence on z 2 iɛ, support σ, and in general p 2

6 Transverse momentum dependent DAs form hard Pion momentum is defined to have no transverse components Projection on z + = interval z = (z, z ) p φ()φ(z) z + =,p = = dx ϕ(x, z ) e i x(pz ) Impact parameter distribution amplitude (IDA) ϕ(x, z ) B(x, z 2 /4) Transverse momentum dependent distribution amplitude ϕ(x, z ) = Ψ(x, k ) e i(k z ) d 2 k = dσφ(x, σ) e iσ(z2 +iɛ)/4 can be written in terms of (valid always ) Ψ(x, k ) = i π Relation for moments (valid for soft functions) Ψ(x, k ) k 2n d2 k = n! i n dσ σ Φ(x, σ) e i(k2 iɛ)/σ σ n Φ(x, σ) dσ

7 Handbag diagram in representation q z q xp xp p form hard Starting expression T (p, q) = dx Using representation T (Q 2 dx ) = xq 2 d 4 z e i(q z) ix(pz) D c (z) B(x, z 2 /4) dσ Φ(x, σ) {1 } e [ixq2 +ɛ]/σ First term: twist-2 approximation Integral of over σ may be written as integral of over k : T (Q 2 dx ) = xq 2 Ψ(x, k ) d 2 k k 2 xq2

8 form hard q z q xp xp Handbag contribution d 4 z e i(qz) p ψ()γ ν S c ( z) γ µ ψ(z) = iɛ µναβ p αq β F (Q 2 ) Antisymmetric part of γ ν /z γ µ is iz β ɛ µναβ γ 5 γ α, and we need p ψ() γ 5 γ α ψ(z) = ip α dσ dx Φ(x, σ) e i x(pz) iσ(z2 iɛ)/4 Result in terms of (based on S c ( z) /z/(z 2 ) 2 F (Q 2 ) = dσ Φ(x, σ) dx { xq iσ [ ]} xq 2 1 e [ixq2 +ɛ]/σ Result in terms of F (Q 2 dx xq 2 dk 2 ) = xq 2 xq 2 Ψ(x, k k 2 k 2 ) d2 k p

9 q q z p form hard In a covariant gauge handbag should be complemented by ψ()... /A(z i )... ψ(z) insertions of twist- gluonic field A µi (z i ) Can be organized into path-ordered exponential of zero-twist field A µ ] E(, z; A) P exp [ig z µ dt A µ (tz) and insertions of non-zero twist gluon field A µ (z) = z ν G µν (sz) s ds, which is the vector potential in the Fock-Schwinger gauge

10 representation in gauge form hard Two-body qq Fock component is given by gauge-invariant bilocal operator O α (, z; A) ψ() γ 5 γ α E(, z; A)ψ(z) Taylor expansion involves covariant derivatives D µ = µ iga µ E(, z; A) ψ(z) = n= We can introduce parametrization 1 n! zµ 1... zµn Dµ 1... D µn ψ() p O α (, z; A) =ip α dσ dx Φ(x, σ) e i x(pz) iσ(z2 iɛ)/4 and proceed as in non-gauge case

11 form hard Generic representation treats (pz) and z 2 as independent variables 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

12 soft form hard 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

13 transition form form hard Gaussian model 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 = ) F (Q 2 = ) = π Ψ(x, k = ) dx 2 Non-Gaussian m = model F (Q 2 dx ) = [1 xq 2 ϕ(x) Λ2 xq 2 + 2K 2(2 ] xq/λ) Size of twist-4 term is governed by confinement scale Λ

14 Comparison with data form hard In leading-order perturbative QCD F LOpQCD (Q 2 dx ) = xq 2 ϕ(x) I(Q2 )f π/q 2 For DAs ϕ r(x) (x x) r, one has I LOpQCD r (Q 2 ) = 1 + 2/r I as (Q 2 ) = 3 for asymptotic wave function ϕ as (x) = 6f πx x Recent experimental data from BaBar and Belle do not show flattening yet I I 5 5 BaBar BELLE 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 GeV Q 2 (GeV 2 )

15 hard, scalar case form hard Quarks generated from local current z z 1 p Φ point (x, σ) = 1 σ ei(x xp2 m 2 )/σ Ψ point (x, k ) = 1 π 1 k 2 + m2 x xp 2 ϕ point (x, z ) = 2K (z m 2 x xp 2 ) logarithmic singularity for z = Hard exchange model z z 2 ȳp Φ exch (x, σ; y) = ig 2 ei(x xp2 m 2 )/σ 16π 2 σ 2 { } min xy, ȳ x e yp iyȳβ p2 /σ dβ z 1 For p 2 =, β-integral gives part of ERBL evolution kernel V (x, y) = x y θ(x < y) + x θ(x > y) ȳ generated in p 2 = limit (using α g g 2 /16π 2 ) Ψ exch (x, k ; y) = αg π V (x, y) (k 2 + m2 ) 2

16 Convolution model, scalar case form hard Superpose yp, ȳp states the weight ϕ (y) = primordial DA is given by a convolution Ψ conv (x, k ) = αg π 1 (k 2 + m2 ) 2 V (x, y) ϕ (y) dy Use primordial soft Ψ (y, k ) ψ (x, k 2 )/π (and m = ) z z 2 ȳp Ψ B (x, k ) = αg 1 πk 2 dy p [ )] 1 ξk dξ ψ (y, 2 yp V (x, y) z 1 Term in square brackets may be written as [ ] = V (x, y) k 2 { } ϕ (y) ψ (y, k 2 k 2 /V (x,y) ) dk 2 For large k, leading 1/k 4 term is determined by DA ϕ (y) only

17 Convolution model, spinor form hard For spin-1/2 interacting via (pseudo)scalar gluon field extra k 2 from numerator trace k limit is finite Ψ B (x, k ) = αg π Ψ B Using Gaussian model for B ) 1 ξk dy dξ ψ (y, 2 V (x, y) = αg π 1 k 2 [V ϕ ](x) +... Y (x, k = ) =α g dy Ψ (y, k = ) Ψ B,G Y (x, k = ) = α g f π Λ 2

18 Evolution in impact parameter space form hard In impact parameter space: ϕ B Y (x, z2 ) =αg dν ) dy V (x, y) 1 ν ϕ (y, ν z 2 V (x, y) Integral over ν is cut at ν 4/z 2 Λ2 ln(z 2 Λ2 /4) We can keep hard massless Illustration for Gaussian model with flat DA ϕ (x, z ) = exp[ z 2 Λ2 /4] '(x =.5,z? ) (ln 2) ln(4/z 2? 2 ) ' B Y (x =.5,z?) ' (x =.5,z?) z? 2 2 /4 '(x =.5,z? ) ' +.2 ' B Y ' z 2? 2 /4

19 Adding UV divergent correction form hard Adding self-energy part (with µ = Λ/2 and Bessel form for log singularity) [ δϕ Y (x, z 2 ) = dν ) αg dy V (x, y) 1 ν ϕ (y, ν z 2 V (x, y) ] K (z Λ/2) ϕ (x, z 2 ) Total IDA ϕ(x, z 2 ) = ϕ (x, z 2 ) + δϕ Y (x, z 2 ) Illustration for Gaussian model with flat DA ϕ (x) = 1 and α g =.2 (x, z 2?)/ (x, z 2?) 1.2 z? 2 2 /4= x

20 form Outlined a new approach to transverse momentum dependence Introduced virtuality distribution Φ(x, σ) Introduced transverse momentum distribution Ψ(x, k ) and wrote it in terms of Φ(x, σ) Results of covariant calculations in terms of Φ(x, σ) converted into expressions involving Ψ(x, k ) Proposed simple models for soft s/, and used them for comparison with experimental data on the pion transition form Described generation of hard s from soft primordial for scalar gluons hard

21 Outlook Future directions: building hard for QCD case Extension of approach onto inclusive reactions, such as Drell-Yan and SIDIS processes Building -based models for soft parts of TMDs that would have a non-gaussian behavior at large k Generating hard s from these soft TMDs form hard