Hadron optics in three-dimensional invariant coordinate space from deeply virtual Compton scattering

The Fourier transform of the deeply virtual Compton scattering amplitude (DVCS) with respect to the skewness parameter $\zeta =Q^2/2p·q$ can be used to provide an image of the target hadron in the boost-invariant variable $\sigma$, the coordinate conjugate to light-front time $\tau =t+z/c$. As an illustration, we construct a consistent covariant model of the DVCS amplitude and its associated generalized parton distributions using the quantum fluctuations of a fermion state at one loop in QED, thus providing a representation of the light-front wave functions (LFWFs) of a lepton in $\sigma$ space. A consistent model for hadronic amplitudes can then be obtained by differentiating the light-front wave functions with respect to the bound-state mass. The resulting DVCS helicity amplitudes are evaluated as a function of $\sigma$ and the impact parameter ${\overrightarrow{b}}_{\perp }$, thus providing a light-front image of the target hadron in a frame-independent three-dimensional light-front coordinate space. Models for the LFWFs of hadrons in $(3+1)$ dimensions displaying confinement at large distances and conformal symmetry at short distances have been obtained using the AdS/CFT method. We also compute the LFWFs in this model in invariant three-dimensional coordinate space. We find that, in the models studied, the Fourier transform of the DVCS amplitudes exhibit diffraction patterns. The results are analogous to the diffractive scattering of a wave in optics where the distribution in $\sigma$ measures the physical size of the scattering center in a one-dimensional system.

Figure 1

The handbag diagram for the DVCS amplitude viewed in coordinate space. The position of the struck quark differs by $x^{-}$ in the two wave functions (whereas $x^{+}=x^{\perp }=0$).

Figure 2

Two-particle LFWFs of the electron vs $x$ for $M=0.51\mathrm{MeV}$, $m=0.5\mathrm{MeV}$, $\lambda =0.02\mathrm{MeV}$ and fixed values of $|k^{\perp }|=k$ in units of MeV. In (b) and (c) we have divided the LFWFs by the factors $(k^1+i{k}^2)$ and $(-k^1+i{k}^2)$, respectively.

Figure 3

Fourier spectrum of the two-particle LFWFs of the electron vs $\sigma$ for $M=0.51\mathrm{MeV}$, $m=0.5\mathrm{MeV}$, $\lambda =0.02\mathrm{MeV}$ and fixed values of $|k^{\perp }|=k$ in MeV. In (b) and (c) we have divided the LFWFs by the factors $(k^1+i{k}^2)$ and $(-k^1+i{k}^2)$, respectively.

Figure 4

FT of the two-particle LFWFs of the electron vs $\sigma$ for $M=0.51\mathrm{MeV}$, $m=0.5\mathrm{MeV}$, $\lambda =0.02\mathrm{MeV}$ and fixed values of $|k^{\perp }|=k$ in MeV. Re and Im denote the real and imaginary parts of the FT, and FS denotes the Fourier spectrum presented in Figs. 3b, 3c.

Figure 5

Imaginary part of the DVCS amplitude for an electron vs $\zeta$ for different values of $t$: (a) helicity-flip part, (b) helicity-nonflip part. We have taken $M=0.51\mathrm{MeV}$, $m=0.5\mathrm{MeV}$, $\lambda =0.02\mathrm{MeV}$. The parameter $t$ is given in MeV.

Figure 6

Real part of the DVCS amplitude for an electron vs $\zeta$ for different values of $t$: (a) helicity-flip part, (b) helicity-nonflip part. We have taken $M=0.51\mathrm{MeV}$, $m=0.5\mathrm{MeV}$, $\lambda =0.02\mathrm{MeV}$. The parameter $t$ is given in MeV.

Figure 7

Fourier spectrum of the imaginary part of the DVCS amplitude for an electron vs $\sigma$ for different values of $t$: (a) when the electron helicity is flipped; (b) and (c) when the helicity is not flipped. In (b) $Q=10\mathrm{MeV}$; in (c) $Q=50\mathrm{MeV}$. The mass parameters are $M=0.51\mathrm{MeV}$, $m=0.5\mathrm{MeV}$, $\lambda =0.02\mathrm{MeV}$. The parameter $t$ is given in MeV.

Figure 8

FT of the imaginary part of the helicity-flip DVCS amplitude for an electron vs $\sigma$ for $t=-5.0$. Re and Im denote the real and imaginary parts of the FT. The mass parameters are $M=0.51\mathrm{MeV}$, $m=0.5\mathrm{MeV}$, $\lambda =0.02\mathrm{MeV}$.

Figure 9

Fourier spectrum of the real part of the DVCS amplitude for an electron vs $\sigma$ for different values of $t$: (a) when the electron helicity is flipped; (b) and (c) when the helicity is not flipped. In (b) $Q=10\mathrm{MeV}$; in (c) $Q=50\mathrm{MeV}$. The mass parameters are $M=0.51\mathrm{MeV}$, $m=0.5\mathrm{MeV}$, $\lambda =0.02\mathrm{MeV}$. The parameter $t$ is given in MeV.

Figure 10

Imaginary part of the helicity-nonflip DVCS amplitude for an electron vs $\sigma$ for fixed $T$ in MeV.

Figure 11

Fourier spectrum of the LFWFs of the simulated hadron model vs $\sigma$ for $M=0.5\mathrm{MeV}$, $m=\lambda =1.0\mathrm{MeV}$ and fixed values of $|k^{\perp }|=k$ in MeV. We have divided the LFWFs by the normalization constant. In (b) and (c) we have divided by the factors $(k^1+i{k}^2)$ and $(-k^1+i{k}^2)$, respectively, as well.

Figure 12

Fourier spectrum of the LFWFs of the simulated hadron model plotted in $\sigma ,|b_{\perp }|$ space for $M=150\mathrm{MeV}$, $m=\lambda =300\mathrm{MeV}$. ${\chi }_1$, ${\chi }_2$, and ${\chi }_3$ are $|{\chi }_{1/2+1}|$, $|{\chi }_{1/2-1}|$, and $|{\chi }_{-1/2+1}|$, respectively. In the plots $|b_{\perp }|$ runs from 0.001 to $0.01{\mathrm{MeV}}^{-1}$ and $\sigma$ runs from $-25$ to $+25$.

Figure 13

Real part of the DVCS amplitude for the simulated hadron state. The parameters are $M=150$, $m=\lambda =300\mathrm{MeV}$. (a) Helicity-flip amplitude vs $\zeta$; (b) Fourier spectrum of the same vs $\sigma$. The parameter $t$ is in ${\mathrm{MeV}}^2$.

Figure 14

Real part of the DVCS amplitude for the simulated hadron bound state. The parameters are $M=150$, $m=\lambda =300\mathrm{MeV}$. (a) Helicity-nonflip amplitude vs $\zeta$; (b) Fourier spectrum of the same vs $\sigma$; (c) structure function vs $x$. The parameter $t$ is in ${\mathrm{MeV}}^2$.

Figure 15

The ground state $(L=0,k=1)$ of the two-parton holographic light-front wave function in 3D space. We have taken ${\Lambda }_{\mathrm{QCD}}=0.32\mathrm{GeV}$. Here $|b_{\perp }|$ runs from 0.001 to $6.0{\mathrm{GeV}}^{-1}$ and $\sigma$ from $-25$ to 25.

Figure 16

(a) The DVCS amplitude vs $\zeta$ and (b) the Fourier spectrum of the DVCS amplitude in the $\sigma$ space using the light-front wave function for mesons obtained from holographic QCD [24]. We have taken ${\Lambda }_{\mathrm{QCD}}=0.32\mathrm{GeV}$. Plots are in units of $e_q^2$. $b^{\perp }$ is given in units of ${\mathrm{GeV}}^{-1}$.