We construct a numerical solution of the recently-derived large-N c &N f smallx helicity evolution equations [1] with the aim to establish the small-x asymptotics of the quark helicity distribution beyond the large-N c limit explored previously in the same framework. (Here N c and N f are the numbers of quark colors and flavors.) While the large-N c helicity evolution involves gluons only, the large-N c &N f evolution includes contributions from quarks as well. We find that adding quarks to the evolution makes quark helicity distribution oscillate as a function of x. Our numerical results in the large-N c &N f limit lead to the x-dependence of the flavor-singlet quark helicity distribution which is wellapproximated byThe power α q h exhibits a weak N f -dependence, and, for all N f values considered, remains very close to α q h (N f = 0) = (4/ √ 3) α s N c /(2π) obtained earlier in the large-N c limit [2, 3]. The novel oscillation frequency ω q and phase shift ϕ q depend more strongly on the number of flavors N f (with ω q = 0 in the pure-glue large-N c limit). The typical period of oscillations for ∆Σ is rather long, spanning many units of rapidity. We speculate whether the oscillations we find are related to the sign variation with x seen in the strange quark helicity distribution extracted from the data [4][5][6][7].

We calculate single-logarithmic corrections to the small-x flavor-singlet helicity evolution equations derived recently [1–3] in the double-logarithmic approximation. The new single-logarithmic part of the evolution kernel sums up powers of αs ln(1/x), which are an important correction to the dominant powers of αs ln2(1/x) summed up by the double-logarithmic kernel from [1–3] at small values of Bjorken x and with αs the strong coupling constant. The single-logarithmic terms arise separately from either the longitudinal or transverse momentum integrals. Consequently, the evolution equations we derive employing the light-cone perturbation theory simultaneously include the small-x evolution kernel and the leading-order polarized DGLAP splitting functions. We further enhance the equations by calculating the running coupling corrections to the kernel.

We revisit the problem of small Bjorken-x evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave (s-channel) formalism. Earlier works on the subject in the same framework [1–3] resulted in an evolution equation for the gluon field-strength F12 and quark “axial current” $$ \overline{\psi}\gamma $$ ψ ¯ γ +γ5ψ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (summing powers of αs ln2(1/x) with αs the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, $$ {}_D{}^{\leftarrow i} $$ D ← i Di, also sandwiched between the light-cone Wilson lines (with the repeated transverse index i = 1, 2 summed over), was missing in the previous works [1–3]. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, Di − $$ {}_D{}^{\leftarrow i} $$ D ← i , which is related to the Jaffe-Manohar polarized gluon distribution [4]. In this work we include this new operator into small-x helicity evolution, and construct novel evolution equations mixing all three operators (Di − $$ {}_D{}^{\leftarrow i} $$ D ← i , F12, and $$ \overline{\psi}\gamma $$ ψ ¯ γ +γ5ψ), generalizing the results of [1–3]. We also construct closed double-logarithmic evolution equations in the large-Nc and large-Nc&Nf limits, with Nc and Nf the numbers of quark colors and flavors, respectively. Solving the large-Nc equations numerically we obtain the following small-x asymptotics of the quark and gluon helicity distributions ∆Σ and ∆G, along with the g1 structure function,$$ \Delta \Sigma \left(x,{Q}^2\right)\sim \Delta G\left(x,{Q}^2\right)\sim {g}_1\left(x,{Q}^2\right)\sim {\left(\frac{1}{x}\right)}^{3.66\sqrt{\frac{\alpha_s{N}_c}{2\pi }}} $$ ∆ Σ x Q 2 ∼ ∆ G x Q 2 ∼ g 1 x Q 2 ∼ 1 x 3.66 α s N c 2 π in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin [5].

In the context of the Bank-Fishler-Shenker-Susskind Matrix theory, we analyze a spherical membrane in light-cone M theory along with two asymptotically distant probes. In the appropriate energy regime, we find that the membrane behaves like a smeared Matrix black hole; and the spacetime geometry seen by the probes can become non-commutative even far away from regions of Planckian curvature. This arises from nonlinear Matrix interactions where fast matrix modes lift a flat direction in the potentialakin to the Paul trap phenomenon in atomic physics. In the regime where we do have a notion of emergent spacetime, we show that there is non-zero entanglement entropy between supergravity modes on the membrane and the probes. The computation can easily be generalized to other settings, and this can help develop a dictionary between entanglement entropy and local geometry -similar to Ryu-Takayanagi but instead for asymptotically flat backgrounds.