Light front approach to axial meson photon transition form factors: probing the structure of χ c 1 (3872)

Abstract. We propose to study the structure of the enigmatic χc1(3872) axial vector meson through its γL*γ → χc1 (3872) transition form factor. We use our recently derived light-front wave function representation of the form factor for the lowest cc¯ Fock-state. We found that the reduced width of the state is well within the current experimental bound recently published by the Belle collaboration. This strongly suggests a crucial role of the cc¯ Fock-state in the photon-induced production. Our predictions for the Q2 dependence can be tested by future single tagged e+e− experiments, giving further insights into the short-distance structure of this meson.

gives rise to so-called "reduced width" Γ.

Accessing transition form factors
e + /p/A We need at least one virtual photon to produce an axial vector in photon photon collisions.This excludes ultraperipheral heavy ion collisions, where photons are quasi-real.
Electron scattering gives us access to finite Q 2 and a whole polarization density matrix of virtual photons.Feasible options are: 1 single tag e + e − collisions.Here the tagged lepton couples to the virtual photon, while photons from the lepton "lost in the beampipe" are quasireal. 2electron-proton or electron-ion scattering.Here especially heavy ions such as Gold which large charge Z = 79 give rise to a large quasireal photon flux enhanced by Z 2 .
Transition amplitude in the Drell-Yan frame LF-Fock state expansion We evaluate the γ * γ * → χ c1 amplitude in the Drell-Yan frame where q 1µ = q 1+ n + µ + q 1− n − µ and q 2µ = q 2− n − µ + q ⊥ 2µ , using the light front plus-component of the current: for spacelike photons, the plus component of the current is free from parton number changing or instantaneous fermion exchange contributions.

Quarkonium light front wave functions
We adopt two different approaches to LFWFs:

Terentev substitution -LFWF from potential model
Quark three-momentum in bound state rest frame canonical spin is substituted by LF helicity via Melosh transform We use a variety of interquark potentials summarized in J.

Light front wave functions from potential models
For the weakly bound systems a procedure to obtain the LFWF from Schrödinger WFs has been proposed by Terentev.In this case the helicity dependent WF Ψ (λ A ) λ λ (z, k) factorizes into a "radial" part, and a spin-orbit part obtained by a Melosh-rotation R(z, k).
Rest frame: Light front: with:

Transition form factor from light front wave functions
We use the well known perturbative LFWF of the longitudinal photon Only the Sz = 0 component with antiparallel quark helicities and one unit of orbital angular momentum contributes.
In the Melosh transform formalism for the n 3 P 1 state, we have γ * γ * cross sections and the reduced width photon-photon cross sections: where {i, j} ∈ {T, L}, and N T = 2, N L = 1 are the numbers of polarization states of photons.
In terms of our helicity form factor, we obtain for the LT configuration, putting at the resonance pole ŝ → M 2 , and J = 1 for the axial-vector meson: provides a useful measure of size of the relevant e + e − cross section in the γγ mode.For a cc state: The square of the effective form factor as a function of photon virtuality within LFWF approach (on the l.h.s.) and in the nonrelativistic limit (on the r.h.s.).
. A measurement of the reduced width would therefore be very valuable.We use LFWFs for n = 1 radial excitation of the p-wave charmonium.
We trace the different Q 2 -dependences to differences of the z-dependence and constituent c-quark mass used in different models.
error band for BLFQ reflects dependence on basis-size as proposed by its authors.using a corrected value for the branching ratio Br(χ c1 (3872) → π + π − J/ψ) and reads 0.024 keV < Γγγ (χ c1 (3872)) < 0.615 keV all our results, including the BLFQ approach, lie well within the experimentally allowed range.Therefore, γγ data do not exclude the cc option, although there is certainly some room for a contribution from an additional meson-meson component.
Possible molecule contribution to Γ? apparently nothing (?) is known about the molecular contribution to the reduced width.
What about the analogous contribution to the one we adopted in the hadronic case?Say γ * γ → cc → DD * , and FSI of D D * generates the X (3872).
Spins of heavy quarks in χ c1 (3872) are entangled to be in the spin-triplet state (M.Voloshin, 2004).But near threshold the cc state produced via γγ-fusion is in the 1 S 0 state.(It's different for gluons, where color octet populates 3 S 1 !) → "handbag mechanism" suppressed in heavy quark limit.

Table :
The reduced width of the χc1(2P) state for several models of the charmonium wave functions with specific c-quark mass.