 #1
 2,425
 6
Main Question or Discussion Point
In some of my previous posts, I refered to GPDs, a recently developped formalism describing hadronic states. I am afraid that this formalism remains largely ignored partly because few introduction review are only recently available.
See for instance :
Markus Diehl, Phys.Rept. 388 (2003) 41277, hepph/0307382
Link to Markus' thesis
and references therein. I especially also like Ji's papers.
Since this a lengthy paper, I feel it is not useless that I write here some infos I think give motivations for it. The following is schematic. It only intends to lead as fast as possible to the formalism of GPDs. I hope this is not going to be considered as spamming
The early data on the nucleon (hadronic) structure came mainly from Elastic Lepton Scattering (ELS) [tex]lN\rightarrow l'N'[/tex] or from Deep Inelastic Scattering (DIS) [tex]lN\rightarrow l'X[/tex] with the final hadronic state [tex]X[/tex] undetermined. ELS and DIS seem at first rather difficult to unify, although a satisfactory understanding of the nucleon structure would require such a full picture. ELS provides access to the usual Elastic Form Factors (EFF) : Dirac [tex]F_1(t)[/tex], Pauli [tex]F_2 (t)[/tex], axial [tex]g_A(t)[/tex] and pseudoscalar [tex]g_P(t)[/tex]. For each of the quark species, they parameterize the electroweak vertex :
[tex]\langle p'\bar{q}(0)\gamma^\mu q(0)p\rangle =
\bar{u}(p')\left[ F^q_1(t)\gamma^\mu + F^q_2(t)\frac{\imath \sigma^{\mu \alpha}\Delta_\alpha}{2m} \right] u(p) [/tex]
[tex]\langle p'\bar{q}(0)\gamma^\mu\gamma_5 q(0)p\rangle =
\bar{u}(p')\left[ g^q_A(t)\gamma^\mu\gamma_5 + g^q_P(t)\frac{\gamma_5\Delta^\mu}{2m} \right] u(p) [/tex]
with [tex]\Delta = p'p[/tex] and [tex]t=\Delta^2[/tex] as usual. On the other hand, DIS provides the quark distributions [tex]q(x)[/tex] and helicity distributions [tex]\Delta q(x)[/tex] with respect to Bjorken's [tex]x[/tex] scaling variable, interpreted as the momentum fraction carried by the struck quark in the socalled infinite momentum frame : [tex]x\in[0,1][/tex] In DIS restricted to the EM probe (not the full electroweak interaction involving the Z in neutral currents and the W in charged currents as well), the optical theorem relates the crosssection to the imaginary part of the forward Compton amplitude : ( cf theoroptq.jpg : The Feynmam graph on the LHS omits the leptons [tex]l[/tex] and [tex]l'[/tex].) The optical theorem here is only a mean of calculation to avoid the sum over [tex]X[/tex] : the forward Compton process is not kinematically allowed !
The introduction of GPDs relies on physical process corresponding not merely to quark distributions, but to quark correlations. Such processes are the Deeply Virtual Compton Scattering (DVCS), or the Deeply Virtual Meson Production (DVMP) : (cf GPDs.jpg).
The important property of Quantum ChromoDynamics (QCD) called factorization allows one to separate a process into a hard subprocess involving only one parton and calculated with perturbative techniques in the standard Quantum ElectroDynamics (QED) or perturbative QCD (pQCD) framework, and a soft subprocess encoding the nonperturbative physics of the proton structure and in our case parameterized by the GPDs. The validity of factorization relies on theorems specifying a large scale limit, such as a large momentum transfer, or a large virtuality of a probing particle for instance. The GPDs are nonforward (offdiagonal or skewed) matrix elements because they appear in an amplitude relating twoparton states of the form [tex]\Psi^*(x\xi)\Psi(x+\xi)[/tex], as compared to ordinary parton distributions [tex]q(x)[/tex] and [tex]\Delta q(x)[/tex] which are diagonal operators [tex]\Psi^*(x)\Psi(x)[/tex].
We compare the diagram for Deeply Virtual Compton Scattering (DVCS) (called "handbag" diagram) and the diagram for DIS, related to a diagonal matrix element through the optical theorem. The DVCS process requires a nonzero momentum transfer between the initial and final nucleons. The matrix element [tex]\Psi^*(x\xi)\Psi(x+\xi)[/tex] must be properly defined in a convenient gauge where the Wilson gauge link along the lightcone is the identity, and using the nonlocal operator product expansion with respect to the order [tex]\tau[/tex] called twist and defined as [tex]\tau = d_{ O}  s_{ O}[/tex] where [tex]d_{ O}[/tex] is the dimension of the operator [tex]\Psi^*(x\xi)\Psi(x+\xi)[/tex] and [tex]s_{ O}[/tex] is its spin. At lowest twist (i.e. 2), each parton species (quarks and gluons) has two quark helicity dependent GPDs denoted [tex]\tilde{H}(x,\xi,t)[/tex] and [tex]\tilde{E}(x,\xi,t)[/tex], as well as two quark helicity averaged GPDs denoted [tex]H(x,\xi,t)[/tex] and [tex]E(x,\xi,t)[/tex]. They depend on the average longitudinal momentum fraction [tex]x[/tex] of the parton involved in the hard subprocess, the longitudinal momentum transfer [tex]\xi[/tex] between the initial and final parton, and the squared fourmomentum transfer [tex]t[/tex]. In this context, a negative value of the longitudial momentum fraction [tex]x[/tex] corresponds to an antiquark. Two more GPDs [tex]H_T[/tex] and [tex]\tilde{H}_T[/tex] are needed when one includes parton helicity flip. The GPDs [tex]q^2[/tex] evolution interpolates between the DGLAP regime of parton density and the ERBL sector of meson amplitudes. The first [tex]x[/tex] moment of the GPDs allows one to recover the usual proton's form factors (Dirac, Pauli, axial, and inducedpseudoscalar). At [tex]\xi=t=0[/tex], the GPDs [tex]H[/tex] and [tex]\tilde{H}[/tex] reduce to the quark longitudinal momentum and helicity distributions [tex]q(x)[/tex] and [tex]\Delta q(x)[/tex]. New physics is available in the kinematical region [tex]\xi<x<\xi[/tex], a regime in which GPDs probe [tex]q\bar{q}[/tex] and gluons pairs, and are thus sensitive to the dynamics of sea quarks and meson degrees of freedom. The [tex]t[/tex] dependence is related to the transverse spatial distributions of partons in the nucleon. Thus, mapping out the GPDs will allow for the first time a full 3dimensional picture of the proton. Finally, Ji's sum rule (the second moment relative to [tex]x[/tex] of the sum [tex]H+E[/tex] equals the total angular momentum carried by the quarks) will settle the issue of the spin of the proton, by providing a measure of the total contribution of quarks to the spin of the proton, including orbital angular momentum. Even more, GPDs furnish access to the Energymomentum tensor of each parton species ! The only other known way to obtain this information is graviton scattering. One experimental evidence for this GPD formalism to hold is to look for the [tex]q^2[/tex] behavior of the cross sections and other observables, which should follow definite scaling laws. DVCS is perceived as the cleanest process to experimentally access the GPDs, because the hard part of the process involves only electromagnetic couplings.
See for instance :
Markus Diehl, Phys.Rept. 388 (2003) 41277, hepph/0307382
Link to Markus' thesis
and references therein. I especially also like Ji's papers.
Since this a lengthy paper, I feel it is not useless that I write here some infos I think give motivations for it. The following is schematic. It only intends to lead as fast as possible to the formalism of GPDs. I hope this is not going to be considered as spamming
The early data on the nucleon (hadronic) structure came mainly from Elastic Lepton Scattering (ELS) [tex]lN\rightarrow l'N'[/tex] or from Deep Inelastic Scattering (DIS) [tex]lN\rightarrow l'X[/tex] with the final hadronic state [tex]X[/tex] undetermined. ELS and DIS seem at first rather difficult to unify, although a satisfactory understanding of the nucleon structure would require such a full picture. ELS provides access to the usual Elastic Form Factors (EFF) : Dirac [tex]F_1(t)[/tex], Pauli [tex]F_2 (t)[/tex], axial [tex]g_A(t)[/tex] and pseudoscalar [tex]g_P(t)[/tex]. For each of the quark species, they parameterize the electroweak vertex :
[tex]\langle p'\bar{q}(0)\gamma^\mu q(0)p\rangle =
\bar{u}(p')\left[ F^q_1(t)\gamma^\mu + F^q_2(t)\frac{\imath \sigma^{\mu \alpha}\Delta_\alpha}{2m} \right] u(p) [/tex]
[tex]\langle p'\bar{q}(0)\gamma^\mu\gamma_5 q(0)p\rangle =
\bar{u}(p')\left[ g^q_A(t)\gamma^\mu\gamma_5 + g^q_P(t)\frac{\gamma_5\Delta^\mu}{2m} \right] u(p) [/tex]
The introduction of GPDs relies on physical process corresponding not merely to quark distributions, but to quark correlations. Such processes are the Deeply Virtual Compton Scattering (DVCS), or the Deeply Virtual Meson Production (DVMP) : (cf GPDs.jpg).
The important property of Quantum ChromoDynamics (QCD) called factorization allows one to separate a process into a hard subprocess involving only one parton and calculated with perturbative techniques in the standard Quantum ElectroDynamics (QED) or perturbative QCD (pQCD) framework, and a soft subprocess encoding the nonperturbative physics of the proton structure and in our case parameterized by the GPDs. The validity of factorization relies on theorems specifying a large scale limit, such as a large momentum transfer, or a large virtuality of a probing particle for instance. The GPDs are nonforward (offdiagonal or skewed) matrix elements because they appear in an amplitude relating twoparton states of the form [tex]\Psi^*(x\xi)\Psi(x+\xi)[/tex], as compared to ordinary parton distributions [tex]q(x)[/tex] and [tex]\Delta q(x)[/tex] which are diagonal operators [tex]\Psi^*(x)\Psi(x)[/tex].
We compare the diagram for Deeply Virtual Compton Scattering (DVCS) (called "handbag" diagram) and the diagram for DIS, related to a diagonal matrix element through the optical theorem. The DVCS process requires a nonzero momentum transfer between the initial and final nucleons. The matrix element [tex]\Psi^*(x\xi)\Psi(x+\xi)[/tex] must be properly defined in a convenient gauge where the Wilson gauge link along the lightcone is the identity, and using the nonlocal operator product expansion with respect to the order [tex]\tau[/tex] called twist and defined as [tex]\tau = d_{ O}  s_{ O}[/tex] where [tex]d_{ O}[/tex] is the dimension of the operator [tex]\Psi^*(x\xi)\Psi(x+\xi)[/tex] and [tex]s_{ O}[/tex] is its spin. At lowest twist (i.e. 2), each parton species (quarks and gluons) has two quark helicity dependent GPDs denoted [tex]\tilde{H}(x,\xi,t)[/tex] and [tex]\tilde{E}(x,\xi,t)[/tex], as well as two quark helicity averaged GPDs denoted [tex]H(x,\xi,t)[/tex] and [tex]E(x,\xi,t)[/tex]. They depend on the average longitudinal momentum fraction [tex]x[/tex] of the parton involved in the hard subprocess, the longitudinal momentum transfer [tex]\xi[/tex] between the initial and final parton, and the squared fourmomentum transfer [tex]t[/tex]. In this context, a negative value of the longitudial momentum fraction [tex]x[/tex] corresponds to an antiquark. Two more GPDs [tex]H_T[/tex] and [tex]\tilde{H}_T[/tex] are needed when one includes parton helicity flip. The GPDs [tex]q^2[/tex] evolution interpolates between the DGLAP regime of parton density and the ERBL sector of meson amplitudes. The first [tex]x[/tex] moment of the GPDs allows one to recover the usual proton's form factors (Dirac, Pauli, axial, and inducedpseudoscalar). At [tex]\xi=t=0[/tex], the GPDs [tex]H[/tex] and [tex]\tilde{H}[/tex] reduce to the quark longitudinal momentum and helicity distributions [tex]q(x)[/tex] and [tex]\Delta q(x)[/tex]. New physics is available in the kinematical region [tex]\xi<x<\xi[/tex], a regime in which GPDs probe [tex]q\bar{q}[/tex] and gluons pairs, and are thus sensitive to the dynamics of sea quarks and meson degrees of freedom. The [tex]t[/tex] dependence is related to the transverse spatial distributions of partons in the nucleon. Thus, mapping out the GPDs will allow for the first time a full 3dimensional picture of the proton. Finally, Ji's sum rule (the second moment relative to [tex]x[/tex] of the sum [tex]H+E[/tex] equals the total angular momentum carried by the quarks) will settle the issue of the spin of the proton, by providing a measure of the total contribution of quarks to the spin of the proton, including orbital angular momentum. Even more, GPDs furnish access to the Energymomentum tensor of each parton species ! The only other known way to obtain this information is graviton scattering. One experimental evidence for this GPD formalism to hold is to look for the [tex]q^2[/tex] behavior of the cross sections and other observables, which should follow definite scaling laws. DVCS is perceived as the cleanest process to experimentally access the GPDs, because the hard part of the process involves only electromagnetic couplings.
Attachments

5.7 KB Views: 263

9 KB Views: 363