Relativistic Form Factors and Bound States

Transcription

1 1 Relativistic Form Factors and Bound States Johannes Gutenberg Universität Mainz 10 December 2009 Paul Hoyer University of Helsinki

2 Plan of Seminar 2 I. Density and spin distributions in transverse distance space Complementary to distributions in longitudinal momentum x Parton distributions in b-space determined by elastic form factors Gives some insight into g 2 of electron (Feynman s challenge) Relation to Color Transparency: Transverse size of Fock states Work done with Samu Kurki (Helsinki) arxiv: II. Relativistic bound states (outline) What we do (not) know about field theory bound states hbar (loop) expansion: A Born term for bound states Boost covariance of equal-time wave function Many-body Fock states with simple wave functions arxiv:

3 Light-Front Fock Expansion of Hadrons 3 A hadron state of momentum P + = P 0 + P 3 defined at given x + = x 0 + x 3 of 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 k i ) ψ n (x i, k i, λ i ) n; x i P +,x i P + k i, λ i x+ =0 where ψn(xi, ki,λi) are its LF wave functions. Note that they are independent of the hadron momentum P +, P. Both inclusive Deep Inelastic Scattering e + N e + X (at high Q 2 ) and exclusive elastic scattering e + N e + N (at any Q 2 ) are given by a sum of squares of LF wave functions. This allows a physical interpretation of the measurements

4 Inclusive Deep Inelastic Scattering (DIS) e' 4 e γ* q P N X Through factorization (for Q 2 ), measures parton longitudinal momentum fractions x, e.g.: f q/n (x) = n,λ i,k n [ dx i d 2 k ] i 16π 3 16π 3 δ(1 i i=1 x i ) δ (2) ( i k i ) δ(x x k ) ψ n (x i, k i, λ i ) 2

5 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 F 1 (Q 2 )γ µ + F 2 (Q 2 ) i 2m σµν q ν e γ* q F1, F2 e p+q/2 N ] u(p 1 2q, λ) e p e p is like electron microscopy of the proton.

6 Electron microscopy 6 When the target is static (mq, mn >> Q), the 3-dim. Fourier transform of the form factors gives the spatial distribution of electric charge and magnetization. p e γ quarks e p+q However, this electron microscopy interpretation is not correct for relativistic quark constituents and due to quark and nucleon recoil.

7 Form factors give transverse charge densities (I) 7 The Dirac and Pauli form factors have an exact representation in terms of LF charge densities ρ(b) in transverse position space, or impact parameter b. Choose a frame where the momentum transfer is purely transverse, q = (0 +, 0, q) and Fourier q b For λ = λ = ± 1/2 find ρ 0 (b) = 0 ρ 0 (b) = n,λ i,k dq 2π QJ 0(bQ)F 1 (Q 2 ) [ n e k i=1 dx i p q/2 4πd 2 b i ] δ(1 i N,λ e γ* F1, F2 x i ) 1 4π δ(2) ( i δ (2) (b b k ) ψ λ n(x i, b i, λ i ) 2 Soper (1977) Burkardt (2000) Diehl (2002) e q = (0 +, 0, q) x i b i ) p+q/2 N,λ Center of momentum at the origin Fouriertransformed wave function

8 Form factors give transverse charge densities (II) 8 For a transversely polarized state, Sx = Sx = + 1/2 one gets analogously that ρ x (b) =ρ 0 (b) + sin(φ b ) 0 dq 2π Q 2 2m J 1(bQ)F 2 (Q 2 ) can be similarly expressed as a sum of squares of wave functions, with ψ λ n (ψ λ=1/2 n Note: + ψ λ= 1/2 n )/ 2 The densities are given by Fourier transforming F1, F2 all Q 2 contribute, and b gets sharply defined. The momentum fractions xi are integrated over. Density is analogous to the parton distribution, with xi bi

9 Using measured form factors, find the 9 by + ρ0(b) empirical quark transverse densities in neutron ρ x bx ρx(b) by + ρ 0 induced EDM : d y = - F 2n (0). e / (2 M N ) data : Bradford, Bodek, Budd, Arrington (2006) bx densities : Miller (2007); Carlson, Vdh (2007) M. Vanderhaeghen, ECT* May 2008

10 Qualitative change in central neutron charge density 10 ρ ch (r) 3-dimensional Fourier transform with phenomenological factors (2001) ρ ch (r = 0) > 0 J. J. Kelly, hep-ph/ ρ 0 (b) ρ 0 (b = 0) < 0 Transverse Fourier transform (2007) G. Miller, PRL 99 (2007)

11 Form factors from charge densities 11 Conversely, the form factors are given by the charge densities as F 1 (Q 2 ) = 2π F 2 (Q 2 ) = 2πm Q 0 db b J 0 (bq)ρ 0 (b) 0 db b J 1 (bq) ( ρ φ b=π/2 x ) (b) ρ φ b=3π/2 x (b) The latter relation allows to express the anomalous magnetic moment F2(0) as F 2 (0) = πm 0 db b 2 ( ρ φ b=π/2 x (b) ρ φ b=3π/2 x (b) Hence intuition about the densities may give an idea about the magnetic moment. Consider the remark on the electron g 2 made nearly 50 years ago by Feynman, at the 12th Solvay Congress: )

12 12 douzième Conseil de Physique Bruxelles, 9-14 octobre 1961 S. MANDELSTAM G. CHEW M.L. GOLDBERGER G.C. WICK M. GELL-MANN G. KÄLLEN E.F. WIGNER E. WENTZEL J. SCHWINGER M. CINI A. S. WIGHTMAN(?) I. PRIGOGINE A. PAIS A. SALAM W. HEISENBERG F.J. DYSON R.P. FEYNMAN L. ROSENFELD P.A.M. DIRAC L. VAN HOVE O. KLEIN S. TOMONAGA W. HEITLER Y. NAMBU N. BOHR F. PERRIN J.R. OPPENHEIMER E.W. LAWRENCE BRAGG C. MØLLER C.J. GORTER H. YUKAWA R.F. PEIERLS H.A. BETHE

13 Feynman s Challenge (1961) 13 In his report to the 12th Solvay Congress (Brussels, 1961) on The Present Status of Quantum Electrodynamics (QED), Feynman called for more insight and physical intuition in QED calculations. It seems that very little physical intuition has yet been developed in this subject. In nearly every case we are reduced to computing exactly a coefficient of some specific term. We have no way to get a general idea of the result to be expected. To make my view clearer, consider, for example, the anomalous electron moment, (g 2)/2 = α/2π 0.328α 2 /π 2. We have no physical picture by which we can easily see that the correction is roughly α/2π, in fact, we do not even know why the sign is positive (other than by computing it). In another field we would not be content with the calculation of the second order term to three significant figures without enough understanding to get a rational estimate of the order of magnitude of the third. We have been computing terms like a blind man exploring a new room, but soon we must develop some concept of this room as a whole, and to have some idea of what is contained in it. As a specific challenge, is there any method of computing the anomalous moment of the electron which, on first rough approximation, gives a fair approximation to the α term and a crude one to α 2 ; and when improved, increases the accuracy of the α 2 term, yielding a rough estimate of α 3 and beyond? Drell and Pagels, Physical Review 140 (1965) B397

14 Current of classical spinning particle 14 Recall that ρx(b) is the matrix element of j + = j 0 + j 3 ρ x (b) d 2 q e iq b 1 (2π) 2 2P + P +, 1 2 q,sx =+ 1 2 j+ (0) P +, 1 2 q,sx =+ 1 2 x + =0 ( Classical picture of spinning particle: p γ j z > 0 ŷ ẑ Burkardt j 0 (y>0) = j 0 (y<0) j z < 0 j 3 (y>0) = j 3 (y<0) > 0 ) 0 in (3.1). ρ φ b=π/2 x (b) ρ φ b=3π/2 x (b) > 0 e F2(0) > 0 Similar arguments apply to single spin asymmetries

15 Generalized parton distributions give ρ(x,b) 15 The pointlike current measured in form factors may be generalized to the LF operator relevant for Generalized Parton Distributions (GPD s): j + (0) dz 8π eixp + z /2 q(0 +, 1 2 z, 0 )γ + q(0 +, 1 2 z, 0 ) he transverse densities to be measured as a function of the lon The GPD s allow to measure densities simultaneously in x and b ρ 0 (x, b) = n,λ i,k n [ dx i i=1 4πd 2 b i ] δ(1 i x i ) 1 4π δ(2) ( i δ (2) (b b k )δ(x x k ) ψ λ n(x i, b i, λ i ) 2 The GPD s are not as well determined experimentally as are the form factors. x i b i )

16 Transverse size of Fock states contributing to form factors at high Q 2 16 Brodsky Lepage Feynman φ x ~1/Q 1-x γ*(q 2 ) φ γ*(q2 ) ~ (1-x)Q 2 x ~1 ~1 fm 1-x ~ 0 Constituents carry 0<x<1 Hard ~ Q gluon exchange Fock states have R ~ 1/Q Constituents carry x ~ 0, 1 Soft overlap of x ~ 0 quark Fock states have R ~ 1 fm ρ(x,b) determines the transverse size of Fock states contributing to form factors F 1 (Q 2 ) = 2π 0 db b J 0 (bq) 1 0 dx ρ 0 (x, b) Diehl et al (2004) Guidal et al (2004)

17 The eγ Fock state of the electron 17 The LF wave functions ψ λ λ e λ γ (x, b) which describe e(λ) e(x, b, λ e ) γ(1 x, b γ, λ γ ) e(x,b,λ) with b γ = xb/(1 x) and b = b(cosφb, ± ± sinφb): e(λ) ψ (x, b) =ψ 1 2 1(x, b) = i em 1 x 4 2 π 2 e iφ b K 1 (mb) γ(1-x,b γ,λ) ψ (x, b) =ψ (x, b) = iem 1 x 4 2 π 2 xe +iφ b K 1 (mb) ψ (x, b) =ψ (x, b) = em 1 x 4 2 π 2 (1 x)k 0 (mb) ψ + 1 1(x, b) =ψ (x, b) = where KL(mb) are Bessel functions and L the orbital angular momentum. Wave functions in p- space by Brodsky & Drell (1980)

18 b-dependence of the Pauli form factor 18 The wave functions give the densities of the eγ Fock state of the electron: ρ 0 (x, b) = αm2 [ 1+x 2 ] 2π 2 1 x K2 1(mb) + (1 x) K0(mb) 2 ρ x (x, b) =ρ 0 (x, b)+ αm2 π 2 x sin(φ b) K 0 (mb)k 1 (mb) from which the Pauli form factor is Oobtained (exact at order α) F 2 (Q 2 )= 4αm3 πq = 2αm2 π 1 0 dx x 1 Q Q 2 +4m 2 log 0 db b J 1 (bq)k 0 (mb)k 1 (mb) [ 1 2m Recall: be b γ = b/(1 x) grows as x 1 x- and b-dependence factorizes ( ) ] Q 2 +4m 2 Exact expression + Q from loop integral

19 Impact parameter structure of electron F 2 (Q 2 ) 19 For b > 1/Q, the Bessel function J 1 (bq) cos(bq)/ bq Oscillations arise from photon of wavelength ~ 1/Q probing transverse distances ~ b Using b 0 in the KL (mb): F 2 (Q 2,b 0) 2αm2 πq 2 0 dtj 1 (t) log ( Q m F 2 (Q 2, b max ) F 2 (Q 2 ) ) ( ) 1 = αm2 Q 2 t πq 2 log m Q 2 = 100 m 2 m b max Dependence on upper limit of b - integral in F2 of electron Correct Q 2 limit Stability wrt. rescattering: Fock state size b/(1-x) Weighted by b/(1-x), the b-integral diverges. Hence dominance of small b (and color transparency) is marginal

20 Impact parameter structure of electron F 1 (Q 2 ) 20 F 1 (Q 2 )= αm2 π 1 0 dx Log divergencies for x 1 (IR) and b 0 (UV) 0 [ ] 1+x 2 db b J 0 (bq) 1 x K2 1(mb) + (1 x)k0(mb) 2 If integrand is weighted by the Fock state size b/(1-x) the integral diverges linearly at the endpoint x = 1. e p e γ*(q 2 ) A p Searches for color transparency in form factors and large angle scattering have not been successful. Garrow et al (Jlab) Combining the A and Q 2 dependence analysis results, we find no evidence for the onset of CT within our range of Q 2.

21 Color Transparency in meson electroproduction 21 In γ* + A π + A the photon has large longitudinal momentum and its quark-antiquark Fock states have transverse size distribution ~ exp( bq) b - distribution is limited in size except at end-points, which are suppressed for longitudinally polarized photons γ*(q 2 ) A ψ γ (b) ~ K0(bQ) x ~ exp( bq) 1 x π CT of compact photon states is required for scaling in DIS and factorization in γ* + p π + p. Positive evidence for CT seen in meson electroproduction (Fermilab, DESY, Jlab) γ* + A π + A Clasie et al (Jlab)

22 Spin carried by constituents of eγ Fock state 22 The LF wf s show how the helicity λ of the parent electron is carried by the constituents of its eγ Fock state, as a function of x and b. N E.g., the expectation value of the electron helicity: λ e λ 1 N λ e,λ γ λ; λ e, λ γ S z e λ; λ e, λ γ = 1 2N [ ψ+,λ λ γ 2 ψ,λ λ γ 2] λ γ λ e,λ γ ψ + λ e λ γ 2 Integrated over b, this is proportional to the spin dependent g1(x) distribution. λ e + + λ γ + + L z eγ + = 1 2 for all (x,b) Spin is locally conserved ! e ! e x x L b = 0.1 / m L b = 2 / m

23 Transversity distribution of the electron 23 The expectation value of Se x in a transversely polarized electron Se x x 1 S x = 1 N 2 ; λ e, λ γ Se x S x = 1 2 ; λ e, λ γ λ e,λ γ [ ] integrated over b, gives the transversity distribution h1(x). The shape of the distribution is weakly dependent on b: ! S x " e x b = 1/m x

24 Conclusions on density distributions 24 Remarkable that form factors can be related to parton densities. Densities give an intuitive understanding of spin dependence, e.g., (g 2)e Transverse size of Fock states makes form factors sensitive to rescattering

25 II. Relativistic Bound States 25 Study of hadrons as QCD bound states is important: Fascinating: Only truly relativistic bound states in Nature Central for our understanding of QCD Needed for interpreting hadron scattering data The quark model indicates a simple first approximation to hadrons Then why do we pay so little attention to bound states? Field theory bound states are often omitted from textbooks & courses. We do not consider the NR Hydrogen atom in flight. M. Järvinen (2004) We do not teach the derivation of the Dirac equation. Do we understand the Fock structure of the Dirac bound states?

26 Wave functions at equal LF or real time? 26 Equal-time wf s turn into LF wf s when boosted to the infinite momentum frame (IMF) P x1 Ψ + = Ψ 0 (P z ) Ψ 0 t1 = t2 Wee partons whose momenta remain finite in the IMF are zero modes on LF (p + = 0). Such modes are generally ignored on the LF. x1 x2 The LF vacuum is trivial (empty), apart from zero-modes. Confinement effects (condensates) may be linked to zero-modes. P Ψ + x2 The Quark Model is formulated in the CM, while the Parton Model is valid in the IMF. (t+z)1 = (t+z)2 Use of equal ordinary time wave functions is safer, and allows to consider both the CM and IMF frames. But: Boosts of equal-time wave functions are dynamical!

27 Synopsis: Bound states at lowest order in hbar 27 PH arxiv: Recall how the Schrödinger and Dirac equations emerge from QED Bound states are due to a divergence of the perturbative expansion Recall cause of divergence: Classical instantaneous A 0 potential Schrödinger and Dirac states are analogous to Born amplitudes Represent binding at lowest order in hbar (no loops) Derive analogous relativistic f-fbar bound states at lowest order in hbar Linear potential allowed as a homogenous solution of the EOM Poincare invariance preserved, despite equal time in all frames

28 The Dirac equation from Feynman diagrams 28 e µ For m µ the above sum of diagrams gives the Dirac eq. for the electron (in the rest frame of the muon): Tree-level scattering from external potential Note: All crossed photon ladders must be included for a relativistic electron! In a Bethe-Salpeter framework this implies that the kernel K must be of infinite order in α. P p P Ψ P = Ψ P K S p k p A finite order kernel will break boost covariance.

29 Basic ideas (I) At tree-level (no loops, lowest order in hbar) the iε prescription of the fermion propagator is irrelevant for scattering amplitudes. Using retarded propagators avoids the Z-diagrams of equal-time wave functions and allows an equal-time hamiltonian formulation. The one-particle Dirac wave function is obtained using retarded boundary conditions. The physical (Feynman) Dirac Fock states contain an infinite number of particle pairs, due to the Z-diagrams. 29 Using SF: E i > 0 E i < 0 p 0 p 0, p 0 p 0 p 0 E ki = + k1 k2 k1 k2 t1 t2 t1 t2 E Using SR: Ei > 0 Ei < 0 E E E E E k1 k2 = k1 k2 t1 t2 k2 k1

30 Basic ideas (II) 30 The same equal-time hamiltonian method may be applied to fermion-antifermion states, to obtain Born-level, 2-body bound states. Instead of an external potential, there is the instantaneous A 0 interaction, determined using the EOM for each position of the fermions. (It is crucial not to introduce a 0 A 0 term, i.e., one must use Coulomb gauge.) A linear potential is allowed as a homogeneous solution of the EOM, with direction along the fermion separation determined by minimizing the action. It generates a potential of order g. One-gluon exchange gives both A 0 and transverse A fields for relativistic sources, but contributes only at order g 2. At order g there is only a linear potential and the derivation is accurate to lowest order in hbar. There is an accidental relation between bound state solutions in different frames, with E = k 2 CM + M 2 This is only true for a linear potential.

31 QED: relativistic e µ + bound state equation 31 γ 0 ( i 1 γ + m e )χ(x 1, x 2 ) χ(x 1, x 2 )γ 0 (i 2 γ + m µ ) = [E V (x 1, x 2 )]χ(x 1, x 2 ) where χ(x1,x2) is the 4x4 wave function of the e (x1) µ + (x2) Fock state, and the potential V (x 1, x 2 )= 2 3 eλ2 x 1 x 2 is purely linear at O(e). The equation was derived at O ( 0) from first principles, hence gives a Born term for bound state calculations. It is a natural extension of the Dirac equation and as such was proposed by Breit already in It has been studied phenomenologically for a linear + 1/r potential Suura et al (1977) Krolikowski et al

32 Interesting properties of the solutions 32 Lorentz covariance: E = k 2 CM + M 2 χ transforms in a novel way Rotational invariance: J = S + L commutes with the Hamiltonian. Allows separation of angular dependence in CM. Linear Regge trajectories: α = 1/8gΛ 2 High relative momentum components with oscillating phase ( Klein paradox ). Related to Z-diagrams, i.e., to multi-particle Fock states.

33 Frame dependence (t 1 = t 2 in all frames!) 33 The wave function of a bound state with CM momentum k has χ(x 1, x 2 ) = exp [ ik (x 1 + x 2 )/2 ] φ(x 1 x 2 ) The equation for φ(x) becomes (for m1 = m2 = m): i [α, φ]+ 1 2 k {α, φ} + m [ γ 0, φ ] =(E V )φ where the solutions φ(x) and E depend on the CM momentum k. E = k 2 CM + M 2 holds only for a purely linear potential V( x )! PH (1986)

34 Boost covariance of the wave function 34 How should relativistic, equal-time wave functions transform under Lorentz boosts? The above bound state equation gives, for k = (0,0,k): γ 0 φ k (s) =e ζα 3/2 γ 0 φ k=0 (s)e ζα 3/2 for φk(s) φk(x1=0, x2=0, x3(s)) (and its transverse derivative) on the z-axis and with the invariant distance s defined by s(x 3 )= 1 2 x 3[ E 1 2 V (x 3) ] and tanh ζ(s) = k E V Note: For V << E this reduces to standard Lorentz contraction, but in general the boost depends on the canonical energy p 0 ea 0. The present field theoretic derivation may allow to better understand the above Lorentz transformation properties.

35 Rotational covariance in the CM 35 For k = 0 and m1 = m2 the bound state equation becomes i [α, φ(x)] + m [ γ 0, φ(x) ] =(M V )φ(x) Geffen and Suura (1977) Use a direct product of Pauli matrices {ρi}, {σi}: αi = ρ1 σi, γ 0 = ρ3 1. to express the 4 4 wave function φ(x) in terms of four 2 2 χ µ : ( ) χ4 + χ 3 χ 1 iχ 2 φ(x) = Σi ρi χi = χ 1 + iχ 2 χ 4 χ 3 The angular momentum operator J: Jφ(x) = 1 2 [1 σ, φ(x)] + Lφ(x) where satisfies L ix [J i,j j ]=iɛ ijk J k and commutes with the Hamiltonian.

36 Separation of variables in the CM 36 There are four independent eigenfunctions of J 2 and J z : χ (1) jm = Y jm(θ, ϕ) F 1 (r) χ (2) jm = σ Y jm(θ, ϕ) F 2 (r) χ (3) jm = σ x Y jm(θ, ϕ) F 3 (r) χ (4) jm = σ L Y jm(θ, ϕ) F 4 (r) Considering parity and charge conjugation, the 2 2 wave functions χ µ may be expressed as linear combinations of (some of) the above structures. This way one can identify states on the π, A1, and ρ trajectories, and obtain the corresponding radial equations for the Fi(r). The radial functions are potentially singular at x = 0 and at M = V. The regular solutions have a discrete mass (M) spectrum.

37 Comparison with the Quark Model 37 The quark model uses a potential where the Coulomb term (one gluon exchange) is perturbative. In the present approach the linear (non-perturbative) term emerges as a homogenous solution of the equations of motion. Both Coulomb and transverse gluon exchange contribute at order α s, hence this is more difficult to treat than the linear (purely Coulomb) term (cf. Hydrogen atom in flight). However, gluon exchange (order g 2 ) is subdominant to the (order g) linear term in a perturbative sense, hence at lowest order in α s (and in hbar) we may retain only the linear term. This is the why boost covariance is expected to, and in fact does, hold only for a purely linear potential: Terms of order g 2 were dropped in the derivation of the bound state equation. V (r) =gλ 2 α s r C F r

38 Some open issues 38 Boosting bound states: Determine dynamical boost generators which satisfy the Lie algebra. Generalization of Dirac covariance in an external potential. Understand the Fock state structure: Contributions of Z diagrams. Concerns also the Dirac equation. Extension to QCD baryons Including separation of variables. Chiral symmetry breaking: Consider vacua with condensates?