arXiv subjects

Brandon Manley

Publications and source records attributed to Brandon Manley.

5 recordsLinked to original sources

arXiv2026

Orbital angular momentum at small $x$ in the large $N_c\&N_f$ limit

We extend the small-$x$ analysis of the quark and gluon orbital angular momentum (OAM) distributions in the proton from the large-$N_c$ limit considered in our earlier works to the large-$N_c\&N_f$ limit, in which the numbers of quark colors $N_c$ and flavors $N_f$ are large with their ratio held fixed. Working in the double-logarithmic approximation (DLA), summing powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ the strong coupling and $x$ the proton's momentum fraction carried by a parton, we correct the small-$x$ operator expression for the quark OAM distribution suggested earlier in \cite{Kovchegov:2024wjs} and relate both the quark and gluon OAM distributions to the impact-parameter moments of the polarized dipole amplitudes. We derive the large-$N_c\&N_f$ evolution equations for these moment amplitudes in the DLA; these include a new equation for the moment amplitude governing the quark OAM. We then solve these evolution equations numerically together with the helicity evolution for $N_f = 2,3,4,5,6$ and $N_c =3$. We find that, similar to the large-$N_c$ case, the OAM distributions share a common small-$x$ intercept with the helicity distributions, $L_{q+\bar{q}}(x,Q^2) \sim L_G(x,Q^2) \sim \Delta \Sigma (x,Q^2) \sim \Delta G(x,Q^2) \sim \left(\frac{1}{x} \right)^{\alpha_h}$ with the intercept $\alpha_h \approx {3.48 \sqrt{\alpha_s N_c/2\pi}}$ for $N_f = N_c = 3$ (cf.~\cite{Borden:2025ehe}): this result, along with $\alpha_h$ for other values of $N_f \neq 0$ that we studied, is smaller than the intercept of $3.66\sqrt{\alpha_s N_c/2\pi}$ found in the large-$N_c$ limit. We also compute the ratios of the OAM distributions to the helicity parton distribution functions as $x\to 0$, obtaining $L_{q+\bar{q}}(x,Q^2)/\Delta\Sigma(x,Q^2) \approx -1.01$ and $L_G(x,Q^2)/\Delta G(x,Q^2) \approx -1.94$ at $Q^2=10\, \mathrm{GeV}^2$, with both ratios being nearly independent of $N_f$.

hep-ph↗

arXiv2024

Elastic Dijet Production in Electron Scattering on a Longitudinally Polarized Proton at Small $x$: A Portal to Orbital Angular Momentum Distributions

We calculate the elastic production of dijets from electron collisions with a longitudinally polarized proton target at small values of the Bjorken $x$ variable. Building on the pioneering proposals of \cite{Hatta:2016aoc,Bhattacharya:2022vvo, Bhattacharya:2023hbq, Bhattacharya:2024sck} for measuring the quark and gluon orbital angular momentum (OAM) distributions, our focus is on both the longitudinal double spin asymmetry (DSA) and longitudinal single spin asymmetry (SSA). We compute the numerators of these asymmetries in the small-$x$ formalism of the light-cone operator treatment. Utilizing the small-$x$ expressions for the OAM distributions derived in our earlier paper, we demonstrate that the DSA provides a robust probe for both the quark and gluon OAM distributions within the proton. In contrast, we find that while the SSA is also sensitive to the OAM distributions, extraction of the latter from the SSA would require new developments in small-$x$ theory and phenomenology, and is probably not feasible at this point in time. These findings highlight the potential of DSA measurements in elastic dijet production at the future Electron-Ion Collider to provide the first-ever direct access to the quark and gluon OAM distributions at small $x$, paving the way for new insights into the proton spin puzzle.

hep-ph↗

arXiv2024

Orbital Angular Momentum Small-$x$ Evolution: Exact Results in the Large-$N_c$ Limit

We construct an exact solution to the revised small-$x$ orbital angular momentum (OAM) evolution equations derived recently, based on an earlier work. These equations are derived in the double logarithmic approximation (summing powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ the strong coupling constant and $x$ the Bjorken $x$ variable) and the large-$N_c$ limit, with $N_c$ the number of quark colors. From our solution, we extract the small-$x$, large-$N_c$ expressions of the quark and gluon OAM distributions. Additionally, we determine the large-$N_c$ small-$x$ asymptotics of the OAM distributions to be \begin{align} \notag L_{q+\bar{q}}(x,Q^2) \sim L_G(x,Q^2) \sim \Delta \Sigma (x,Q^2) \sim \Delta G(x,Q^2) \sim \left(\frac{1}{x} \right)^{\alpha_h}, \end{align} with the intercept $\alpha_h$ the same as obtained in the small-$x$ helicity evolution, which can be approximated as $\alpha_h \approx 3.66074 \sqrt{\frac{\alpha_s N_c}{2\pi}}$. This result is in complete agreement with the literature. Additionally, we calculate the ratio of the quark and gluon OAM distributions to the flavor-singlet quark and gluon helicity parton distribution functions respectively in the small-$x$ region.

hep-ph↗

arXiv2023

Orbital Angular Momentum at Small $x$ Revisited

We revisit the problem of the small Bjorken-$x$ asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton utilizing the revised small-$x$ helicity evolution derived recently. We relate the quark and gluon OAM distributions at small $x$ to the polarized dipole amplitudes and their (first) impact-parameter moments. To obtain the OAM distributions, we derive novel small-$x$ evolution equations for the impact-parameter moments of the polarized dipole amplitudes in the double-logarithmic approximation (summing powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ the strong coupling constant). We solve these evolution equations numerically and extract the leading large-$N_c$, small-$x$ asymptotics of the quark and gluon OAM distributions, which we determine to be \begin{align} L_{q+\bar{q}}(x, Q^2) \sim L_{G}(x,Q^2) \sim \Delta \Sigma(x, Q^2) \sim \Delta G(x,Q^2) \sim \left(\frac{1}{x}\right)^{3.66 \, \sqrt{\frac{\alpha_s N_c}{2\pi}}}, \notag \end{align} in agreement with the existing results in the literature within the precision of our numerical evaluation. (Here $N_c$ is the number of quark colors.) We also investigate the ratios of the quark and gluon OAM distributions to their helicity distribution counterparts in the small-$x$ region.

hep-ph↗

arXiv2023

Orbital Angular Momentum at Small $x$

We revisit the problem of the small Bjorken-$x$ asymptotics of the quark and gluon orbital angular momentum (OAM) distributions in the proton utilizing the revised formalism for small-$x$ helicity evolution derived recently in a paper by Cougoulic, Kovchegov, Tarasov, and Tawabutr. We relate the quark and gluon OAM distributions at small $x$ to the polarized dipole amplitudes and their (first) impact-parameter moments. To obtain the $x$-dependence of the OAM distributions, we derive novel small-$x$ evolution equations for the impact-parameter moments of the polarized dipole amplitudes in the double-logarithmic approximation (summing powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ the strong coupling constant). We solve these evolution equations numerically and extract the large-$N_c$, small-$x$ asymptotics of the quark and gluon OAM distributions, which we determine to be \[ L_{q+\bar{q}}(x, Q^2) \sim L_{G}(x,Q^2) \sim \Delta \Sigma(x, Q^2) \sim \Delta G(x,Q^2) \sim \left(\frac{1}{x}\right)^{3.66 \, \sqrt{\frac{\alpha_s N_c}{2\pi}}},\] in agreement with an earlier work by Boussarie, Hatta, and Yuan within the precision of our numerical evaluation (here $N_c$ is the number of quark colors). We also investigate the ratios of the quark and gluon OAM distributions to their helicity distribution counterparts in the small-$x$ region.

hep-ph↗