Ultrarelativistic quark-nucleus scattering in a light-front Hamiltonian approach

We investigate the scattering of a quark on a heavy nucleus at high energies using the time-dependent basis light-front quantization (tBLFQ) formalism, which is the first application of the tBLFQ formalism in QCD. We present the real-time evolution of the quark wave function in a strong classical color field of the relativistic nucleus, described as the color glass condensate. The quark and the nucleus color field are simulated in the QCD SU(3) color space. We calculate the total and the differential cross sections, and the quark distribution in coordinate and color spaces using the tBLFQ approach. We recover the eikonal cross sections in the eikonal limit. We find that the differential cross section from the tBLFQ simulation is in agreement with a perturbative calculation at large $p_{\perp }$, and it deviates from the perturbative calculation at small $p_{\perp }$ due to higher-order contributions. In particular, we relax the eikonal limit by letting the quark carry realistic finite longitudinal momenta. We study the sub-eikonal effect on the quark through the transverse coordinate distribution of the quark with different longitudinal momentum, and we find the sub-eikonal effect to be sizable. Our results can significantly reduce the theoretical uncertainties in small $p_{\perp }$ region which has important implications to the phenomenology of the hadron-nucleus and deep inelastic scattering at high energies.

Figure 1

The quark is moving along the positive-z direction scatters on the nucleus along the negative-z direction. The dashed line is the worldline of the quark, $z={\beta }_{q}t$ with ${\beta }_q$ the speed of the quark. The band represents worldlines of the nucleus, $z=-{\beta }_{A}t$ for one end and $z=-{\beta }_{A}t+d^{\prime }$ for the other end. ${\beta }_A$ is the speed of the nucleus and $d^{\prime }=d\sqrt{1-{\beta }_A^2}$ with $d$ the width of the nucleus in its rest frame. In the ultra-relativistic limit of ${\beta }_A\to 1$, the red band in the diagram shrinks to a single line aligned with $x^{+}=0$.

Figure 2

The dependence on the transverse grid number $N$ of (a) the total and (b) the elastic cross sections at $L=50{\mathrm{GeV}}^{-1}$. The cross sections are calculated as functions of $g^2\mu$ with $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$ and $p^{+}=\infty$. The solid lines are the eikonal predictions as calculated from Eq. (20). Each data point results from an average over 100 configurations, and the standard deviation is taken as the uncertainty bar.

Figure 3

The dependence on the transverse grid length $L$ of (a) the total and (b) the elastic cross sections. The lattice spacing is fixed as $a=L/N=6.25{\mathrm{GeV}}^{-1}$ for these results. The cross sections of the quark are plotted as functions of $g^2\mu$ at $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$ and $p^{+}=\infty$. The solid lines are the eikonal predictions as calculated from Eq. (20). Each data point is averaged over 100 configurations, and the standard deviation is taken as the uncertainty bar.

Figure 4

The dependence on $N_{\eta }$ of (a) the total and (b) the elastic cross sections. Parameters for those results: $L=50{\mathrm{GeV}}^{-1}$, $N=8$, $L_{\eta }=50{\mathrm{GeV}}^{-1}$ and $p^{+}=\infty$. The solid lines are the eikonal predictions as calculated from Eq. (20). Each data point is averaged over 100 configurations, and the standard deviation is taken as the uncertainty bar.

Figure 5

The dependence of (a) the total and (b) the elastic cross sections on $m_g$. Parameters for those panels, $N=8$, $L=50{\mathrm{GeV}}^{-1}$, $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$ and $p^{+}=\infty$. The transverse grid parameters introduce numerical IR cutoff ${\lambda }_{\mathrm{IR}}=\pi /L\approx 0.06\mathrm{GeV}$ and UV cutoff ${\lambda }_{\mathrm{UV}}=N\pi /L\approx 0.5\mathrm{GeV}$ to the momentum space. The physical IR cutoff $m_g$ should be inside the numerical range to obtain a valid result. The solid lines are the eikonal predictions as calculated from Eq. (20). Each data point is averaged over 100 configurations, and the standard deviation is taken as the uncertainty bar.

Figure 6

The dependence on $p^{+}$ of (a) the total and (b) the elastic cross sections at $L=50{\mathrm{GeV}}^{-1}$ and $N=18$. The cross sections of the quark as functions of $g^2\mu$ for $L_{\eta }=50{\mathrm{GeV}}^{-1}$ with $N_{\eta }=4$. The solid lines are the eikonal predictions ($p^{+}=\infty$). Each data point is averaged over 100 configurations, and the standard deviation is taken as the uncertainty bar.

Figure 7

The differential cross section of the $qA$ scattering at different $g^2\mu$, (a) $g^2\mu =0.05{\mathrm{GeV}}^{3/2}$, (b) $g^2\mu =0.14{\mathrm{GeV}}^{3/2}$, and ($c,d$) $g^2\mu =0.49{\mathrm{GeV}}^{3/2}$. The top panels are plotted on a linear scale, and the bottom panels are on a log-log scale. The tBLFQ results are plotted as empty diamonds (or black stars), and each data point is averaged over 50 events. The vertical dashed line is at the saturation scale $Q_s^2=(g^2\mu {)}^2{L}_{\eta }/(2{\pi }^2)$. LO (NLO) is the leading (next-to leading) order perturbative approximation (see Appendix pp5). The tBLFQ results in panels (a-c) are calculated on the transverse lattice of $L=50{\mathrm{GeV}}^{-1}$, and that in panel (d) is evaluated on the lattice of $L=5{\mathrm{GeV}}^{-1}$ to reveal the large $p_{\perp }^2$ range at the same $g^2\mu =0.49{\mathrm{GeV}}^{3/2}$ as (c). Other parameters for those tBLFQ results: $N=18$, $m_g=0.1\mathrm{GeV}$, $L_{\eta }=50{\mathrm{GeV}}^{-1}$ and $N_{\eta }=4$.

Figure 8

The differential cross section of the $qA$ scattering with different $N_{\eta }$ plotted on (a) linear scale and (b) log-log scale. The tBLFQ results are plotted as empty diamonds, and each data point is averaged over 50 events. Parameters for those results: $N=18$, $L=50{\mathrm{GeV}}^{-1}$, $m_g=0.1\mathrm{GeV}$, $L_{\eta }=50{\mathrm{GeV}}^{-1}$ and $g^2\mu =0.05{\mathrm{GeV}}^{3/2}$. The vertical dashed line is at $Q_s^2=(g^2\mu {)}^2{L}_{\eta }/(2{\pi }^2)\approx 0.006{\mathrm{GeV}}^2$. LO (NLO) is the leading (next-to leading) order perturbative approximation (see Appendix pp5).

Figure 9

The evolution of the quark’s transverse coordinate distribution at different $p^{+}$, (a) $p^{+}=\infty$, ($b,c$) $p^{+}=10\mathrm{GeV}$. The initial state of the quark is distributed as $C{e}^{-|{\overrightarrow{r}}_{\perp }{|}^2/(0.2*50{\mathrm{GeV}}^{-1}{)}^2}$, where $C$ is the normalization coefficient. From left to right, the transverse coordinate distributions of the quark are shown at a sequential interaction time calculated by tBLFQ. Parameters in those panels: $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$, $m_g=0.1\mathrm{GeV}$, $N=18$, $L=50{\mathrm{GeV}}^{-1}$, $g^2\mu =0.486{\mathrm{GeV}}^{-3/2}$. The result in (a) is identical for each event with the same parameters. The result in (b) is an single event and could be different for a different event with the same parameters. The result in (c) is an average of 50 events.

Figure 10

The evolution of the quark’s transverse coordinate distribution when no source exists—that is, the light-front time evolution of the wave packet without interactions. The initial state of the quark is distributed as $C{e}^{-|{\overrightarrow{r}}_{\perp }{|}^2/(0.2*50{\mathrm{GeV}}^{-1}{)}^2}$, where $C$ is the normalization coefficient. From left to right, the transverse coordinate distributions of the quark are shown at a sequential interaction time calculated by tBLFQ. Parameters in those panels: $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$, $m_g=0.1\mathrm{GeV}$, $N=18$, $L=50{\mathrm{GeV}}^{-1}$, $p^{+}=10\mathrm{GeV}$.

Figure 11

The evolution of the expectation value of the quark’s transverse coordinate at different $p^{+}$ and at different color charge densities. The initial state of the quark is distributed as $C{e}^{-|{\overrightarrow{r}}_{\perp }{|}^2/(0.2*50{\mathrm{GeV}}^{-1}{)}^2}$, where $C$ is the normalization coefficient. From left to right, the first panel is calculated without an external field, the following three panels are calculated with increasing color charge density $g^2\mu$. The results are averaged over 10 events. Parameters in those panels: $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$, $m_g=0.1\mathrm{GeV}$, $N=18$, $L=50{\mathrm{GeV}}^{-1}$.

Figure 12

The evolution of the expectation value of the quark’s transverse coordinate at different lattice sizes and at different color charge densities with a fixed lattice spacing of $a=L/N=5{\mathrm{GeV}}^{-1}$. Parameters in those panels: $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$, $m_g=0.1\mathrm{GeV}$, $p^{+}=10\mathrm{GeV}$. The initial state of the quark is distributed as $C{e}^{-|{\overrightarrow{r}}_{\perp }{|}^2/(0.2*50{\mathrm{GeV}}^{-1}{)}^2}$, where $C$ is the normalization coefficient. From left to right, the first panel is calculated without an external field, the following three panels are calculated with increasing color charge density $g^2\mu$. The results are averaged over 10 events.

Figure 13

The evolution of the expectation value of the quark’s transverse coordinate at four different values of the grid number $N$ and at different color charge densities ($g^2\mu$) with a fixed lattice size of $L=50{\mathrm{GeV}}^{-1}$. The initial state of the quark is $C{e}^{-|{\overrightarrow{r}}_{\perp }{|}^2/(0.2L{)}^2}$, where $C$ is the normalization coefficient. For both (a) and (b), from left to right, the first panel is calculated without an external field, the following three panels are calculated with increasing color charge density $g^2\mu$. In (b), we impose a UV cutoff when solving the gluon field so that $\tilde{A}({\overrightarrow{k}}_{\perp })=0$ for $|{\overrightarrow{k}}_{\perp }|\ge {\mathrm{\Lambda }}_{\mathrm{UV}}=0.2\mathrm{GeV}$. The results are averaged over 10 events. Parameters in those panels: $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$, $m_g=0.1\mathrm{GeV}$, $p^{+}=10\mathrm{GeV}$.

Figure 14

The evolution of the expectation value of the quark’s transverse coordinate with different quark masses and at different color charge densities. Parameters in those panels: $L=50{\mathrm{GeV}}^{-1}$, $N=18$, $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$, $m_g=0.1\mathrm{GeV}$, $p^{+}=10\mathrm{GeV}$. The initial state of the quark is distributed as $C{e}^{-|{\overrightarrow{r}}_{\perp }{|}^2/(0.2*50{\mathrm{GeV}}^{-1}{)}^2}$, where $C$ is the normalization coefficient. From left to right, the first panel is calculated without an external field, the following three panels are calculated with increasing color charge density $g^2\mu$. The results are averaged over 10 events.

Figure 15

The evolution of the quark’s distribution in the color space at a selection of color charge densities with different $p^{+}$, (a) $p^{+}=\infty$ and (b) $p^{+}=10\mathrm{GeV}$. The results are averaged over 50 events. Parameters in those panels: $N=18$, $L=50{\mathrm{GeV}}^{-1}$, $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$, $m_g=0.1\mathrm{GeV}$. For both (a) and (b), from left to right, the first panel is calculated without an external field, the following three panels are calculated with increasing color charge density $g^2\mu$. The initial state of the quark is a single color state ($c=1$) with space distribution as $C{e}^{-|{\overrightarrow{r}}_{\perp }{|}^2/(0.2L{)}^2}$, where $C$ is the normalization coefficient. The dashed line marks the average probability of the three colors: 0.33.

Figure 16

The transverse profiles of the CGC field and related quantities. (a–c): single-event source charges scaled by profile functions plotted on the transverse plane ${\overrightarrow{r}}_{\perp }$. The shapes of the adopted profiles are sketched above each panel for reference. (d): the total cross sections using the three different profiles. The initial state of the quark is set as ${\overrightarrow{p}}_{\perp }={\overrightarrow{0}}_{\perp }$. The solid line is the eikonal prediction with the uniform profile according to Eq. (20). (e): the differential cross sections using the three different profiles. The solid lines are the perturbative approximations with the uniform profile, see the caption of Fig. 7. For this result, we choose the initial state of the quark as ${\overrightarrow{p}}_{\perp }={\overrightarrow{0}}_{\perp }$ and the color charge density of value $g^2\mu =0.14{\mathrm{GeV}}^{3/2}$. (f): the evolution of the expectation value of the quark’s transverse coordinate with different profiles and at a selection of color charge densities. The initial state of the quark is $C{e}^{-|{\overrightarrow{r}}_{\perp }{|}^2/(0.2L{)}^2}$, where $C$ is the normalization coefficient. The results are averaged over 100 events. Parameters for those panels: $N=18$, $L=50{\mathrm{GeV}}^{-1}$, $L_{\eta }=50{\mathrm{GeV}}^{-1}$, $N_{\eta }=4$, $m_g=0.1\mathrm{GeV}$, $p^{+}=10\mathrm{GeV}$.