Measuring transverse size with virtual photons

Transcription

1 Measuring transverse size with virtual photons 1 Paul Hoyer University of Helsinki Work done with Samu Kurki arxiv: arxiv: How to determine the size of the interaction region in electroproduction from the dependence on the photon virtuality.

2 Photons are useful probes of strong dynamics at any Q 2 2 Scattering from pointlike sources at LT in the Bj limit σ(q 2 ) 1/q 2 The q 2 dependence reflects the effective size of the interaction region e e γ*(q) A Drell-Yan

3 Example: Deuteron photodisintegration 3 q γ * E 22 CM D d" (!d pn) / kb GeV dt # 20 p n The 90 break-up cross section at q 2 =0 agrees with dimensional scaling for Eγ > 1 GeV. Does this mean that only compact configurations of the deuteron, with R < 0.2 fm, contribute to this process? If so, expect no q 2 -dependence for q 2 < 1 GeV 2. This can be formulated more precisely through a Fourier transform to coordinate space. E (GeV)! C. Bochna et al, PRL 81 (1998) 4576

4 Relativistic effects on photon resolution 4 From DIS, recall that resolution of photon with q 2 = Q 2 is different in the longitudinal and transverse directions: r 1 Q ν Q = 1 2mx Bj r 1 Q Hence we can accurately measure only the transverse size (impact parameter b): d 2 q (2π) 2 e iq b p(p 1 ) n(p 2 ) J µ (q) D(p) q is frame dependent: Is there a preferred frame? p1+p2 = p+q depends on q: What is p1(q), p2(q)? Does the result depend on µ?

5 Recall: Nucleon Form Factors 5 Using Lorentz and gauge invariance, the scattering amplitude is expressed in terms of the Dirac F1 and Pauli F2 form factors, which depend on Q 2 = q 2 A µ λλ = p q, λ J µ (0) p 1 2q, λ [ =ū(p q, λ ) p q/2 N γ* F 1 (Q 2 )γ µ + F 2 (Q 2 ) i 2m σµν q ν e q F1, F2 e p+q/2 N ] u(p 1 2q, λ) The photon scatters from quarks, which are ultrarelativistic. How can we measure the quark postions with resolution Δb ~ 1/Q, when the photon itself is not moving faster than the quarks?

6 Boosting to the Infinite Momentum Frame The photon probes a hadron at an instant of x + = t+z, not at an instant in t 6 The Light Front Infinite Momentum Frame Quark motion in the transverse direction slows down in the IMF: A hadron state of momentum P + = P 0 + P 3 defined at given x + = x 0 + x 3 can be expanded in terms its quark and gluon Fock states as P +, P, λ x + =0 = n [ 1 n,λ i i=1 0 dx i xi d 2 k ] i 16π 3 16π 3 δ(1 i x i ) δ (2) ( i ψ n (x i, k i, λ i ) n; x i P +,x i P + k i, λ i x+ =0 where the LF wave functions ψn(xi, ki,λi) are independent of the hadron momentum P +, P. v = p xe h k i ) Note: The partons carry fractions xi of the hadron momentum P, like in non-relativistic physics with mi /M xi.

7 f q/n (x) = Inclusive Deep Inelastic Scattering (DIS) In the DIS cross section the photon vertices of the amplitude and amplitude * are separated by a light-like distance z: z +, z 0; z ~ 1/(2mxBj). The parton distributions can be expressed in terms of LF wave functions: n,λ i,k n [ dx i d 2 k ] i 16π 3 16π 3 δ(1 i i=1 * N P x i ) δ (2) ( i k i ) δ(x x k ) ψ n (x i, k i, λ i ) 2 x 0 z... f q/n (x) x * N P 7 Notes: The parton distribution is defined in the Bj limit (Q 2 ) The above expression is approximate, since rescattering of the struck parton (described by the Wilson line) is neglected.

8 The Generalized Parton Distributions: GPD s 8 1 8π The GPD s are non-forward matrix elements of the PDF operator: dr e imxr /2 P q( 1 2 r)γ+ W [ 1 2 r, 1 2 r ]q( 1 2 r) P 1 2 r + =r =0 = 1 [ 2P + ū(p ) H(x, ξ,t)γ + + E(x, ξ,t)iσ +ν ] ν u(p 1 2m 2 ) The GPD amplitudes can be accessed experimentally through the Deeply Virtual Compton Scattering cross section at leading twist: Q 2. DVCS: e N e + γ + N Through Δ, the GPD s contain information about the parton distributions in transverse space. dx N * x + ξ t = Δ 2... x - ξ ξ = N x B 2 x B P - Δ/2 GPD (x, ξ,t) P + Δ/2

9 Impact parameter distributions via the GPD s 9 Extrapolating the GPD to ξ = 0 and Fourier transforming it wrt. q f q/n (x, b) = d 2 q (2π) 2 e iq b dz + 8π eixp z /2 Soper (1977) Burkardt (2000) Diehl (2002) P +, 1 2 q, λ q(0+, 1 2 z, 0 )γ + q(0 +, 1 2 z, 0 ) P +, 1 2q, λ the GPD can be expressed in terms of LF wf s with the struck quark at transverse position b (still ignoring the Wilson line): f q/n (x, b) = n,λ i,k i=1 n [ dx i N (P +, 1 2 q ) ] ( 4πd 2 b i δ 1 i x x GPD (x, ξ=0, q ) ) ( 1 x i 4π δ2 i x i b i ) δ (2) (b b k )δ(x x k ) ψn(x λ i, b i, λ i ) 2 momentum at the origin Center of N (P +, 1 2 q ) Fourier-transformed wave function

10 Relation of GPD s to Form Factors 10 When GPD s are integrated over x the GPD reduces to a form factor, since dx exp(ixp + z /2) δ(z ) z = 0 ensures that the photon vertices coalesce. The GPD s vanish for x > 1, hence the relations reduce to dxh q (x, ξ,t)=f q 1 (t) dxe q (x, ξ,t)=f q 2 (t) Dirac Pauli This gives constraints on GPD models and a great experimental simplification: Form factors are easy to measure (compared to GPD s!).

11 Impact parameter picture of GPD s inherited by FF s 11 Fourier transforms of form factors give charge densities in impact parameter space: q ρ 0 (b) = 0 ρ 0 (b) = n,λ i,k dq 2π QJ 0(bQ)F 1 (Q 2 ) [ n ] e k dx i 4πd 2 b i δ(1 i=1 i δ (2) (b b k ) ψ λ n(x i, b i, λ i ) 2 Δ = q x i ) 1 4π δ(2) ( i x i b i ) No more Wilson line: Fock expansion is exact No more leading twist : Resolution in b 1/qmax

12 Using measured form factors, find the 12 by + ρ0(b) empirical quark transverse densities in neutron ρ x bx ρx(b) by + ρ 0 J x induced EDM : d y = - F 2n (0). e / (2 M N ) bx Miller (2007) Carlson and Vanderhaeghen (2008) data : Bradford, Bodek, Budd, Arrington (2006)

13 Generalization to transition Form Factors 13 In the case of transition form factors, the density is no longer positive definite but the charge distribution is still interesting: It is found that the transition from the proton to its first radially excited state is dominated by up quarks in a central region of around 0.5 fm and by down quarks in an outer band which extends up to about 1 fm. Tiator and Vanderhaeghen (2009) γ* N P 11 (1440) te is dominated b

14 Generalization to any γ* transition M(lN l f)= e 2 ū(l )γ µ u(l) 1 q 2 Need to identify J + current contribution for LF Fock expansion: E.g.: l at fixed q d 4 xe iq x f J µ (x) N(p) N e l q e π N 14 PH and S. Kurki arxiv: J + (x) =e q q(x)γ + q(x) = 2e q q +(x)q + (x) q+ (x) = ¼ γ γ + q(x) q + (0 +,x, x) = dk + k + θ(k+ ) [b(k +, x)u + (k + )e [ i 1 2 k+ x + d (k +, x)v + (k + )e i 1 2 k+ x ] where the LF spinors satisfy: u +(k +, λ )u + (k +, λ) =k + δ λ λ Fourier transform to impact parameter: p +, p =4π d 2 b e ip b p +, b

15 Expand into LF Fock states: p +, b = 1 4π [ n n i=1 1 0 dx ] i 4πd 2 b i δ(1 x i )δ 2 (b x i b i ) xi i i n ψ n (x i, b i b) b (x i p +, b i )d ()a () 0 1 2p + f(p+, b f ) J + (0) N(p +, b N ) 1 (4π) 2 δ2 (b f b N )A fn ( b N ) where A fn (b) = 1 4π [ n n i=1 1 0 dx i 4πd 2 b i ] δ(1 i x i )δ 2 ( i x i b i )ψ 15 )ψn f (xi, b i )ψn N (x i, b i ) k e k δ 2 (b k b) is diagonal in Fock states n in frames where q + = 0 ( no pair production)

16 FT of γ* matrix element in momentum space 16 In the frame: p = (p +,p, 1 2 q) q = (0 +,q, q) p f = (p +,p + q, 1 2 q) we have N d 2 q e iq b 1 (2π) 2 2p + f(p f ) J + (0) N(p) = A fn (b) p e l + q e f pf where A fn (b) is given by the previous overlap of Fock amplitudes, which are universal features of N and f. The b-distribution may be studied as a function of the final state f, providing information about the transverse size of the contributing Fock states. When f consists of several hadrons their relative momenta must be consistent with the LF Fock expansion at all pf = q + p

17 Example: f = π (p 1 ) N(p 2 ) ( ) In order to conform with the Lorentz covariance of LF states, at any pf : 1 πn(p + f, p f ; Ψ f dx d 2 k ) 0 x(1 x) 16π 3 Ψf (x, k) π(p 1 )N(p 2 ) [ ] ( where Ψ f (x,k) is a freely chosen function of the relative variables x, k : p + 1 = xp+ f p 1 = xp f + k p + 2 = (1 x)p+ f p 2 = (1 x)p f k With x, k being independent of pf, this defines the pion and nucleon momenta p1, p2 at all photon momenta q. The πn(p + f, p f ; Ψ f ) state has an LF Fock expansion of standard form, in terms of the pion and nucleon Fock amplitudes. 17

18 Illustration (1): γ*+ µ µ + γ 18 The QED matrix element A µγ λ 1,λ 2 = 1 2p + µ(p 1, λ 1 )γ(p 2, λ 2 ) J + (0) µ(p, λ = 1 2 ) expressed in terms of the relative variables x, k is: A µγ (q; x, k) =2e x where e λ k = λe iλφ k k / 2. ndix. The Fourier transform A µγ (b; x, k) = 2e x [ [ [ ] e k (1 x) 2 m 2 + k 2 e (k (1 x)q) (1 x) 2 m 2 +(k (1 x)q) 2 e k (1 x) 2 m 2 + k 2 δ2 (b) i 2 2π The Fourier transform gives: me iφ ( b 1 x K 1(mb) exp i k b )] 1 x In the first term the γ* interacts with the initial muon, which by definition is at b = 0. The second term reflects the distribution of the final muon in transverse space. This expression agrees exactly with the wave function overlap formula. ]

19 ( ) Illustration (2): γ*+ µ µ + γ 19 Choosing Ψ(x, k) =δ(x x) ( x(1 x) exp i k b µ ( ) [ 1 x ) corresponds to fixing the impact parameter bµ of the final muon. Then ( ) A µγ (b; x, b µ)= [ ] x(1 x) ψ (x, b µ) δ (2) (b)+δ (2) (b b µ) 2 which again conforms with the general overlap expression of LF Fock state wave functions.

20 Fourier transform of the cross section 20 The γ*+n f amplitudes [ have dynamical ] phases (resonances,...). Calculating their Fourier transforms requires a partial wave analysis. However, one can Fourier transform the measured cross section itself. Then the b-distribution reflects the difference between the impact parameters of the photon vertex in the amplitude and its complex conjugate: For d 2 q (2π) 2 e iq b efers to the internal f = π(p 1 )N(p 2 ), S fn (b; x, k) = 1 2p + f(p f ) J + (0) N(p) 2 = d 2 q e iq b (2π) 2 q 4 dσ(ln l πn) d 2 q dx d 2 k = α2 4π 3 1 x(1 x) d 2 b q A fn (b q ) A fn(b q b), parametrized with the relative variables x and k, d 2 b q A fn (b q ; x, k) A fn(b q b; x, k)

21 Illustration (3): σ(γ*+ µ µ + γ) 21 For the QED example considered above the Fourier transform of the cross section can be done analytically: { S µγ (b; x, k) =4e 2 k 2 /2 x [(1 x) 2 m 2 + k 2 ] 2 δ(2) (b) k cos(φ b φ k ) im (1 x) 2 m 2 + k 2 2π + 1 4π exp ( ) i k b 1 x K 1 (mb) 1 x exp ( ) i k b [ ] } 1 x (1 x) 2 K 0 (mb) 1 2 mb K 1(mb) The 3 terms correspond to 2, 1 and 0 of the γ* interactions occurring on the initial muon. The imaginary part arises from an angular correlation between b and k.

22 Remarks In γ*n πn, expect the b-distribution to narrow with the relative transverse momentum k between the π and the N. σ(γd pn) E 22 at large angles, suggesting compact states. A measurement of the q 2 - dependence would allow a direct measurement of the transverse size. states. Howe γ N KΛ In heavy quark production: γ N DΛ c, light quarks, directly to the heavy quarks. 22 E d" CM (!d pn) / kb GeV dt E (GeV)! the b-distribution should narrow with the quark mass if the photon couples # One may compare the b-distribution in ordinary and diffractive events.

23 Summary (1) 23 Intuitively, the q -dependence of a virtual photon interaction gives information about the charge distribution in space. The target is illuminated instantaneously only when the charge carriers are non-relativistic. This is the case in electron microscopy. Quarks move inside hadrons with velocity of light. The photon phase is constant at fixed Light-Front time x + = t + z In the IMF LF formulation, transverse quark velocities are non-relativistic 2-dim. FT s of form factors describe charge densities in transverse space Unlike pdf s, no leading twist limit is implied. The resolution in impact parameter is expected to be Δb 1/Qmax

24 Summary (2) 24 The formulation can be generalized to transition form factors γ*n N* and to any (multi-hadron) final (and initial) state: γ*a f FT of the cross section σ(γ*n f) gives the distribution in the transverse distance b between the photon vertex in T(γ*N f) and [T(γ*N f)]*