Fourier transforming the virtual photon transverse momentum in \gamma*(q)+N
\to f processes allows new insight into hadron dynamics as a function of impact
parameter b. I discuss how previous analyses of charge density based on elastic
and transition form factors (f=N, N*) can be generalized to any multi-hadron
final state (f=\pi N, \pi\pi N, \bar D \Lambda_c,...). The b-distribution
determines the transverse positions of the quarks that the photon couples to,
and can be studied as a function of multiplicity, the relative transverse
momenta, quark masses and polarization. The method requires no factorization
nor leading twist approximation. Data with spacelike photon virtualities in the
range 0  1/Q_{max} in impact
parameter.Comment: Talk at the Third International Workshop on Transverse Polarization
  Phenomena in Hard Scattering (Transversity 2011), in Veli Losinj, Croatia, 29
  August - 2 September 2011. 7 page

Hoyer, Paul

English

arXiv

Fourier transforming the virtual photon transverse momentum in

γ∗(q⊥) + N → f processes allows new insight into hadron dynamics as a function

of impact parameter b. I discuss how previous analyses of charge density based on

elastic and transition form factors (f = N,N∗) can be generalized to any multihadron final state (f = πN, ππN, D¯Λc,...). The b-distribution determines the

transverse positions of the quarks that the photon couples to, and can be studied

as a function of multiplicity, the relative transverse momenta, quark masses and

polarization. The method requires no factorization nor leading twist approximation.

Data with spacelike photon virtualities in the range 0 ≤ Q ≤ Qmax provides a

resolution Δb  1/Qmax in impact parameter

Hoyer, P.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2012-11175-yColloquia: Transversity 2011IL NUOVO CIMENTO Vol. 35 C, N. 2 Marzo-Aprile 2012Measuring transverse size with virtual photonsP. HoyerDepartment of Physics and Helsinki Institute of PhysicsPOB 64, FIN-00014 University of Helsinki, Finlandricevuto il 15 Ottobre 2011; approvato il 25 Novembre 2011pubblicato online l’1 Marzo 2012Summary. — Fourier transforming the virtual photon transverse momentum inγ∗(q⊥) + N → f processes allows new insight into hadron dynamics as a functionof impact parameter b. I discuss how previous analyses of charge density based onelastic and transition form factors (f = N,N∗) can be generalized to any multi-hadron final state (f = πN, ππN, D¯Λc, . . .). The b-distribution determines thetransverse positions of the quarks that the photon couples to, and can be studiedas a function of multiplicity, the relative transverse momenta, quark masses andpolarization. The method requires no factorization nor leading twist approximation.Data with spacelike photon virtualities in the range 0 ≤ Q ≤ Qmax provides aresolution Δb  1/Qmax in impact parameter.PACS 13.60.-r – Photon and charged-lepton interactions with hadrons.1. – Charge density from elastic form factorsThe quark density of a hadron in transverse (impact parameter) space b is given by atwo-dimensional Fourier transform of the electromagnetic form factor of the hadron [1-4].For a proton with helicity λ the density distribution isρq/N (b) ≡∫d2q⊥(2π)2e−i q⊥·b12P+〈P+,12q⊥, λ∣∣∣∣ j+(0)∣∣∣∣P+,−12q⊥, λ〉(1)=∫ ∞0dQ2πQJ0(bQ)F1(Q2),where F1(Q2) is the Dirac form factor. It is not immediately obvious why precisely thisdefinition corresponds to a density. Until recently it was in fact common to define thecharge density in terms of a three-dimensional Fourier transform. Such a definition is,however, not compatible with relativistic effects. Quarks in the proton move with nearlythe speed of light, vq  c, so a photon cannot give a sharp picture of the charge distri-bution at an instant of time. However, the transverse velocity v⊥q = p⊥q /Eq decreasesc© Societa` Italiana di Fisica 277278 P. HOYERwith the energy Eq of the quark. In the Infinite Momentum Frame (IMF) v⊥q = 0 anda high-resolution picture of the charge distribution can be obtained in the transverseplane, i.e., as a function of impact parameter as in (1). Formally, the IMF is equivalentto quantization at equal Light-Front (LF) time, x+ = t + z. More intuitively, a photonmoving along the negative z-axis interacts at fixed x+. The Fourier transform in (1) isin fact defined in a frame with photon momentum q+ = 0. It is also important that thematrix element in (1) involves only the j+ component of the quark current, and that theinitial and final states have opposite transverse momenta.To see why ρ(b) merits being viewed as a charge density one needs to expand thehadron h state in terms of its quark and gluon Fock components taken at equal x+ [5],∣∣P+,P⊥, λ〉hx+=0 = ∑n,λin∏i=1[∫ 10dxi√xi∫d2k⊥i16π3]16π3δ(1−∑ixi)δ(2)(∑ik⊥i)(2)×ψhn(xi,k⊥i, λi)∣∣n; xiP+, xiP⊥ + k⊥i, λi〉 .For a proton the Fock states n would include |uud〉 , |uudg〉 , |uuduu¯〉 , . . ., the infinite andcomplete sum of all quark and gluon states, integrated over the longitudinal momentumfraction xi and the relative transverse momentum k⊥i of each parton(1). The uniqueproperty of this LF expansion is that the wave functions ψhn(xi,k⊥i, λi) do not dependon the hadron momentum P+,P⊥. Hence the same wave functions ψhn describe theinitial and final states in (1). Each parton i carries a share xi of the parent hadron’slongitudinal and transverse momentum, and a relative transverse momentum k⊥i.When the initial and final states in (1) are expanded in their Fock states (2), theimpact parameter distribution of a quark q is found to be [3]ρq/h(b) =∑n,λi[n∏i=1∫dxi∫4πd2bi]δ(1−∑ixi)14πδ(2)(∑ixibi)(3)×|ψhn(xi, bi, λi)|2∑kek δ(2)(b− bk),where the wave functions ψhn(xi, bi, λi) are related to the momentum space wave functionsψhn(xi,k⊥i, λi) of (2) by standard (two-dimensional) Fourier transforms of the k⊥i. Thusρq/h(b) indeed is a charge density: the probability that there is a quark k in the hadronat impact parameter bk = b (relative to the parent hadron), weighted by its charge ek.The usual parton distributions fq/h(x,Q2) measured in hard inclusive processes maylikewise be expressed in terms of the LF wave functions,fq/h(x, μ2) =∑n,λi[n∏i=1∫ 10dxi∫ k⊥<μ d2k⊥i16π3]16π3δ(1−∑ixi)δ(2)(∑ik⊥i)(4)×|ψhn(xi,k⊥i, λi)|2∑kδ(xk − x).(1) This is a formally exact expansion, but possible contributions from partons with xi = 0(zero-modes) will be neglected.MEASURING TRANSVERSE SIZE WITH VIRTUAL PHOTONS 279qp-q/2q/N(b)*x xpfq/N(x)...Q2 Q2p* *(a) (b)p +q/2Fig. 1. – (a) The elastic proton form factor measured by the ep → ep scattering amplitude. Inthe frame q+ = 0 its Fourier transform (1) over q⊥ gives the charge density in impact parameter.(b) The quark distribution in longitudinal momentum fq/h(x,Q2) (4) may be obtained via QCDfactorization in the Q2 →∞ limit from Deep Inelastic Scattering, ep→ eX.It is apparent that the quark distributions in impact parameter (3) (fig. 1a) and inlongitudinal momentum fraction (4) (fig. 1b) give similar and complementary informationon hadron structure. However, there are also important differences. First of all, theexpression (4) is not exact due to the neglect of the “Wilson line”, indicated by the verticalCoulomb gluon exchanges in fig. 1b. This contribution arises from the rescattering ofthe struck quark in the color field of the hadron and adds coherently to the bound statewave function. There is no Wilson line in the form factor since it has only one photonvertex. Hence the expression (3) of the impact parameter distribution ρq/h(b) in termsof the LF target wave functions is formally exact.Parton distributions like fq/h(x, μ2) are obtained from hard inclusive scatteringthrough QCD factorization at leading power (“twist”) in the Q2 → ∞ limit, and de-pend on the factorization scale μ. Corrections to the hard subprocess of higher order inαs must be taken into account in the extraction of fq/h(x, μ2) from data. Conversely,the expression (3) for the impact parameter distribution only has corrections from higherorders in the QED coupling α. Gluon corrections to the photon vertex are included inthe sum over Fock states n(2). Data in the whole range of Q2 is used, in fact the integralin (1) is over all spacelike q2 = −q⊥2 = −Q2 ≤ 0. A finite range 0 ≤ Q ≤ Qmax = q⊥maxlimits the resolution in impact parameter to Δb  1/Qmax.2. – Charge distribution of inelastic processesThe analysis described above has been applied to data on elastic (γ∗N → N) andtransition (γ∗N → N∗) form factors [6, 7]. The sum over initial and final Fock statesin (3) remains diagonal for transition form factors, but the product of wave functionsψN∗n (xi, bi, λi)∗ψNn (xi, bi, λi) is no longer positive definite. The impact parameter dis-tribution nevertheless reflects the transverse positions of the quarks which couple to thevirtual photon.Here I shall describe the generalization of the impact parameter analysis to any processγ∗i → f , where i and f are arbitrary (multi-)hadron states [8]. The possibility to study(2) The renormalization of vertex corrections would introduce a scale if the charge density werefactorized into a photon vertex and wave functions. However, such scale dependence is absentin the measurable density (3). I am grateful for a discussion with Jian-Wei Qiu on this point.280 P. HOYERhow the impact parameter distribution depends on the type and relative momenta ofthe produced hadrons allows new insight into hadron dynamics. I shall use the processγ∗N(p) → π(p1)N(p2) to illustrate the procedure.The Fock expansion (2) is valid for any hadronic state, and thus also for |π(p1)N(p2)〉.However, we need to make sure that πN states with different total momenta pf = p+q =p1+p2 are described by the same LF wave functions. The obvious guess is to parametrizethe hadron momenta in the same way as the parton momenta in (2):p+1 = xp+f , p1⊥ = xpf⊥ + k⊥,(5)p+2 = (1− x)p+f , p2⊥ = (1− x)pf⊥ − k⊥.We may specify the πN state using a wave function Ψf (x,k⊥) of our choice,(6)∣∣∣πN(p+f ,pf⊥; Ψf )〉 ≡∫ 10dx√x(1− x)∫d2k⊥16π3Ψf (x,k⊥) |π(p1)N(p2)〉 ,where x and k⊥ are independent of the total πN momentum pf . The asymptotic |πN〉state (6) then has an LF expansion of standard form, with quark and gluon wave functionsgiven by Ψf and the Fock state wave functions ψπn and ψNn of (2). It should be noted thatthe wave functions which determine the density distribution are the ones at the time ofthe photon interaction (x+ = 0), not those of the asymptotic (x+ →∞) πN state. The|πN〉 state evolves with x+: the hadrons fly apart at large times and converge towardthe photon vertex as x+ → 0. As the pion and the nucleon get close to each other theystart to interact and can form resonances. Hence the A(γ∗N → πN) amplitude has adynamical phase, unlike the spacelike form factor A(γ∗N → N) which is real. A singlehadron in the final state has a stationary time development, i.e., its Fock state wavefunctions are independent of x+.The impact parameter analysis of the γ∗N → πN transition amplitude is similar tothat of form factors. In the frame wherep =(p+, p−,−12q⊥), q = (0+, q−, q⊥) , pf =(p+, p− + q−,12q⊥),(7)the Fourier transform of the j+ current matrix element can be expressed as a diagonalsum over Fock states,AfN (b) ≡∫d2q⊥(2π)2e−iq⊥·b12p+〈f(pf )|j+(0) |N(p)〉 = 14π∑n[n∏i=1∫ 10dxi∫4πd2bi](8)×δ(1−∑ixi)δ2(∑ixibi)ψfn∗(xi, bi)ψNn (xi, bi)∑kekδ2(bk − b),where ψfn∗ are the LF wave functions of the state (6) at the photon vertex (x+ = 0).AfN gives the impact parameter distribution of the quarks to which the photon couples,when the center-of-momentum of the initial (N) and final (f) state is at∑i xibi = 0.Since the amplitude 〈f(pf )|j+(0) |N(p)〉 has a dynamical phase due to final-stateinteractions the Fourier transform in (8) generally requires a partial wave analysis.MEASURING TRANSVERSE SIZE WITH VIRTUAL PHOTONS 281Alternatively, the Fourier transform of the square of the amplitude,SfN (b) ≡∫d2q⊥(2π)2e−iq⊥·b∣∣∣∣ 12p+ 〈f(pf )|J+(0) |N(p)〉∣∣∣∣2(9)=∫d2bq AfN (bq)A∗fN (bq − b)gives a convolution of the impact parameter distributions (8). Now b is the differencebetween the impact parameter of the quark struck in the amplitude and in its complexconjugate. This also gives a measure of the transverse distribution of the active quarks.The quantity SfN (b) has an imaginary part if the squared amplitude in (9) is asymmetricfor q⊥ → −q⊥. The imaginary part thus measures the azimuthal correlation betweenq⊥ and a transverse direction defined, e.g., by a relative transverse momentum betweenthe particles in the final state (such as k⊥ in (5)).The square of the amplitude is obtained from the measured cross section—with thecaveat that the analysis concerns only the matrix element of the j+ component of thecurrent. This component dominates at high lepton energies, or may be identified via aRosenbluth separation.The above method can be illustrated using the Born level QED amplitude for scat-tering on a muon, γ∗(q) + μ(p) → μ(p1) + γ(p2),(10) Aμγ,+ 12+ 12+1(q⊥) = 2e√x{e− · k⊥(1− x)2m2 + k⊥2− e− · [k⊥ − (1− x)q⊥](1− x)2m2 + [k⊥ − (1− x)q⊥]2},where e− · k⊥ = e−iφk |k⊥|/√2. The muons and the photon are taken to have positivehelicities, the final state momenta are parametrized as in (5) and the wave functionΨf (x,k⊥) of (6) is a δ-function in x and k⊥. The Fourier transform (8) givesAμγ,+ 12+ 12+1(b) =(11)2e√x[e− · k⊥(1− x)2m2 + k⊥2δ2(b)− i2√2πm e−iφb1− x K1(mb)]exp(−ik⊥ · b1− x).In the first term of (10) the virtual photon interacts with the initial muon, and this termcontributes to (11) at the impact parameter b = 0 of the target. In the second term thevirtual photon interacts with the muon after the emission of the real photon, and itsb-dependence agrees with the known μ → μ + γ QED wave function.If the Fourier transform is taken of the square of the QED amplitude (10) as in (9)the result isSμγ,+ 12+ 12+1(b;x,k⊥) = 4e2x{k⊥2/2[(1− x)2m2 + k⊥2]2δ(2)(b)(12)−im |k⊥| cos(φb − φk)(1− x)2m2 + k⊥2K1(mb)2π(1− x)+K0(mb)− 12mb K1(mb)4π(1− x)2}exp(−ik⊥ · b1− x).The three terms within { } correspond, respectively, to the virtual photon interacting282 P. HOYERi) with the initial muon in both Aμγ and (Aμγ)∗, ii) once with the intial and once withthe final muon, and iii) twice with the final muon. The imaginary part can be seen toarise from the angular correlation between the lepton scattering plane (defined by b) andthe relative transverse momentum k⊥ in the final state.3. – DiscussionThe analysis presented here generalizes previous work on charge densities of (tran-sition) form factors. Being applicable to any final state in γ∗N → f it opens up anew window on the dynamics of lepton-nucleon scattering. Data at all spacelike pho-ton virtualities 0 ≤ Q ≤ Qmax are used, with an expected impact parameter resolutionΔb  1/Qmax. The absence of a QCD factorization removes uncertainties related to theleading twist approximation and the factorization scale. The possibilities to study howthe impact parameter distribution depends on properties of the final state (multiplicity,relative momenta, quark masses, . . . ) can give new insight into hadron dynamics. Forexample, the dimensional scaling observed [9] in deuteron photodisintegration, γd → pnat θCM = 90◦, suggests transversally compact configurations of the deuteron and nu-cleons. A Fourier transform of the electroproduction process, γ∗d → pn, can reveal thetransverse distribution of the active quarks.The absence of QCD factorization also implies less predictions. The Q2-dependenceof the quark distributions fq/N (x,Q2) measured in DIS can be calculated, and the uni-versality of the distributions tested in other hard processes. The LF wave functions ψnwhich determine the impact parameter distribution in (8) are also universal, but are moredifficult to reconstruct from measured data. The impact parameter analysis discussedhere thus is complementary to the traditional analyses of hard inclusive (and exclusive)scattering processes.∗ ∗ ∗I wish to thank the organizers of the Transversity 2011 workshop for their kind invi-tation. The Magnus Ehrnrooth foundation provided travel support.REFERENCES[1] Soper D. E., Phys. Rev. D, 15 (1977) 1141.[2] Burkardt M., Phys. Rev. D, 62 (2000) 071503; 66 (2002) 119903(E); Burkardt M., Int.J. Mod. Phys. A, 18 (2003) 173.[3] Diehl M., Eur. Phys. J. C, 25 (2002) 223; 31 (2003) 277(E).[4] Ralston J. P. and Pire B., Phys. Rev. D, 66 (2002) 111501.[5] Brodsky S. J. and Lepage G. P., SLAC-PUB-4947, published in Adv. Ser. Direct. HighEnergy Phys., 5 (1989) 93; Brodsky S. J. and Pauli H. C., SLAC-PUB-5558, publishedin Lect. Notes Phys., 396 (1991) 51. Brodsky S. J., Pauli H. C. and Pinsky S. S., Phys.Rep., 301 (1998) 299.[6] Miller G. A., Phys. Rev. Lett., 99 (2007) 112001; Miller G. A., Phys. Rev. C, 80 (2009)045210.[7] Carlson C. E. and Vanderhaeghen M., Phys. Rev. Lett., 100 (2008) 032004; CarlsonC. E. and Vanderhaeghen M., Eur. Phys. J. A, 41 (2009) 1; Tiator L. andVanderhaeghen M., Phys. Lett. B, 672 (2009) 344; Tiator L., Drechsel D., KamalovS. S. and Vanderhaeghen M., arXiv:1109.6745.[8] Hoyer P. and Kurki S., Phys. Rev. D, 83 (2011) 114012.[9] Bochna C. et al. (E89-012 Collaboration), Phys. Rev. Lett., 81 (1998) 4576; SchulteE. C. et al., Phys. Rev. Lett., 87 (2001) 102302.

Measuring transverse size with virtual photons

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