We rederive the small-$x$ evolution equations governing quark helicity distribution in a proton using solely an operator-based approach. In our previous works on the subject, the evolution equations were derived using a mix of diagrammatic and operator-based methods. In this work, we re-derive the double-logarithmic small-$x$ evolution equations for quark helicity in terms of the "polarized Wilson lines", the operators consisting of light-cone Wilson lines with one or two non-eikonal local operator insertions which bring in helicity dependence. For the first time we give explicit and complete expressions for the quark and gluon polarized Wilson line operators, including insertions of both the gluon and quark sub-eikonal operators. We show that the double-logarithmic small-$x$ evolution of the "polarized dipole amplitude" operators, made out of regular light-cone Wilson lines along with the polarized ones constructed here, reproduces the equations derived in our earlier works. The method we present here can be used as a template for determining the small-$x$ asymptotics of any transverse momentum-dependent (TMD) quark (or gluon) parton distribution functions (PDFs), and is not limited to helicity.
We generalize the JIMWLK evolution equation to include the small-$x$ evolution of helicity distributions. To derive helicity JIMWLK, we use the method devised by A.H. Mueller for the usual unpolarized JIMWLK, and apply it to the helicity evolution equations derived earlier. We obtain a functional evolution equation for the generalized JIMWLK weight functional. The functional now is a sum of a helicity-independent term and a helicity-dependent term. The kernel of the new evolution equation consists of the standard eikonal leading-order JIMWLK kernel plus new terms containing derivatives with re…
We extend our earlier results for the quark helicity evolution at small $x$ to derive the small-$x$ asymptotics of the flavor singlet and flavor non-singlet quark helicity TMDs and PDFs and of the $g_1$ structure function. In the flavor singlet case we re-derive the evolution equations obtained in our previous paper on the subject, performing additional cross-checks of our results. In the flavor non-singlet case we construct new small-$x$ evolution equations by employing the large-$N_c$ limit. All evolution equations resum double-logarithmic powers of $α_s \, \ln^2 (1/x)$ in the polarization-d…
We construct small-$x$ evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $α_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $α_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in th…
The small-$x$ quark helicity evolution equations at double-logarithmic order, with the kernel $\simα_s\ln^2(1/x)$, have been derived previously. In this work, we derive the single-logarithmic corrections to the equations, to order $α_s\ln(1/x)$ of the evolution kernel. The new equations include the effects of the running coupling and the unpolarized small-$x$ evolution, both of which are parametrically significant at single-logarithmic order. The large-$N_c$ and large-$N_c\& N_f$ approximations to the equation are computed. (Here, $N_c$ and $N_f$ are the numbers of quark colors and flavors, re…
We calculate single-logarithmic corrections to the small-$x$ flavor-singlet helicity evolution equations derived recently in the double-logarithmic approximation. The new single-logarithmic part of the evolution kernel sums up powers of $α_s \ln(1/x)$, which are an important correction to the dominant powers of $α_s \ln^2(1/x)$ summed up by the double-logarithmic kernel at small values of Bjorken $x$ and with $α_s$ the strong coupling constant. The single-logarithmic terms arise separately from either the longitudinal or transverse momentum integrals. Consequently, the evolution equations we d…
In this Letter, we analytically solve the evolution equations for the small-$x$ asymptotic behavior of the (flavor singlet) quark helicity distribution in the large-$N_c$ limit. These evolution equations form a set of coupled integro-differential equations, which previously could only be solved numerically. This approximate numerical solution, however, revealed simplifying properties of the small-$x$ asymptotics, which we exploit here to obtain an analytic solution. We find that the small-$x$ power-law tail of the quark helicity distribution scales as $Δq^S (x, Q^2) \sim \left(\tfrac{1}{x} \ri…
We present a Monte Carlo based analysis of the combined world data on polarized lepton-nucleon deep-inelastic scattering at small Bjorken $x$ within the polarized quark dipole formalism. We show for the first time that double-spin asymmetries at $x<0.1$ can be successfully described using only small-$x$ evolution derived from first-principles QCD, allowing predictions to be made for the $g_1$ structure function at much smaller $x$. Anticipating future data from the Electron-Ion Collider, we assess the impact of electromagnetic and parity-violating polarization asymmetries on $g_1$ and demonstr…
We construct small-x evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $α_s \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $α_s \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the …
We present the first-ever description the world data on the $g_1^{p,n}$ structure function at small Bjorken $x$ using evolution equations in $x$ derived from first principles QCD. This is a Monte-Carlo analysis within the JAM global framework that allows us to fit all existing polarized DIS data below $x<0.1$ as well as predict future measurements of small $x$ $g_1^{p,n}$ at the EIC. This is a necessary step in determining the quark helicity PDFs and, ultimately, the quark contribution to the proton spin.
Magnetic field is unstable in a medium with time-independent chiral conductivity. Owing to the chiral anomaly, the electromagnetic field and the medium exchange helicity which results in time-evolution of the chiral conductivity. Using the fastest growing momentum and helicity state of the vector potential as an ansatz, the time-evolution of the chiral conductivity and magnetic field is solved analytically. The solution for the hot and cold equations of state shows that the magnetic field does not develop an instability due to he helicity conservation. Moreover, it develops a peak only if a si…
We construct a numerical solution of the recently-derived large-$N_c \& N_f$ small-$x$ helicity evolution equations 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 num…
Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept
Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept
In the quantum statistical parton distributions approach proposed more than one decade ago to describe the parton structure, new properties are now understood, in particular, the relation between quarks and antiquarks which leads to very specific properties. The simultaneous treatment of unpolarized and polarized Parton Distribution Functions (PDFs) allows a determination of thermodynamical potentials (the master parameters of the model) which drive their behavior and as a consequence those of the structure functions. The existence of a possible relation between the gluon and a $q~\bar q$ pair…
We analyze the role of the surface terms in the conservation law for the sum of the magnetic helicity density and the chiral imbalance of the charged particle densities. These terms are neglected in the Anomalous MagnetoHydroDynamics (AMHD), where infinite volume is considered typically. We discuss a finite volume system, such as a magnetized neutron star (NS), and study the contribution of the surface terms to the evolution of the magnetic helicity. Accounting for the fast washing out of the chiral imbalance in a nascent NS, we demonstrate that the surface terms contribution can potentially l…
Since decoupling in the early universe in helicity states, primordial neutrinos propagating in astrophysical magnetic fields precess and undergo helicity changes. In view of the XENON1T experiment possibly finding a large magnetic moment of solar neutrinos, we estimate the helicity flipping for relic neutrinos in both cosmic and galactic magnetic fields. The flipping probability is sensitive both to the neutrino magnetic moment and the structure of the magnetic fields, thus potentially a probe of the fields. As we find, even a magnetic moment well below that suggested by XENON1T could signific…
Relic neutrinos from the Big Bang decoupled from the hot plasma predominantly in helicity eigenstates. Their subsequent propagation through gravitational inhomogeneities of the Universe alters the helicities of both Dirac and Majorana neutrinos, thus providing an independent probe of the evolving universe. We determine here the probability that relic neutrinos flip their helicity, in terms of the spectrum of density inhomogeneities measured in the Cosmic Microwave Background. As we find, for Dirac neutrinos the gravitational helicity modifications are intermediate between the effects of magnet…
We study hypermagnetic helicity and lepton asymmetry evolution in plasma of the early Universe before the electroweak phase transition (EWPT) accounting for chirality flip processes via inverse Higgs decays and sphaleron transitions which violate the left lepton number and wash out the baryon asymmetry of the Universe (BAU). In the scenario where the right electron asymmetry supports the BAU alone through the conservation law $B/3 - L_{eR}=const$ at temperatures $T>T_{RL}\simeq 10~TeV$ the following universe cooling leads to the production of a non-zero left lepton (electrons and neutrinos) as…
We study the small-$x$ evolution equation for the gluon generalized parton distribution (GPD) $E_g$ of the nucleon. It is shown that $E_g$ at vanishing skewness exhibits the Regge behavior identical to the BFKL Pomeron, despite its association with nucleon helicity-flip processes. We also consider the effect of gluon saturation and demonstrate that $E_g$ gets saturated in the same way as its helicity-nonflip counterpart $H_g$. Our result has a direct impact on the modeling of $E_g$ as well as the small-$x$ contribution to nucleon spin sum rules.
We construct a numerical solution of the small-$x$ evolution equations recently derived in \cite{Kovchegov:2015pbl} for the (anti)quark helicity TMDs and PDFs as well as the $g_1$ structure function. We focus on the case of large $N_c$ where one finds a closed set of equations. Employing the extracted intercept, we are able to predict directly from theory the behavior of the helicity PDFs at small $x$, which should have important phenomenological consequences. We also give an estimate of how much of the proton's spin may be at small $x$ and what impact this has on the so-called "spin crisis."
Starting from operator equations of motion and making arguments based on a separation of time scales, a set of equations is derived which govern the non-equilibrium time evolution of a GeV-scale sterile neutrino density matrix and active lepton number densities at temperatures T > 130 GeV. The density matrix possesses generation and helicity indices; we demonstrate how helicity permits for a classification of various sources for leptogenesis. The coefficients parametrizing the equations are determined to leading order in Standard Model couplings, accounting for the LPM resummation of 1+n <-> 2…
We determine the small-$x$ asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large $N_c$. To achieve this, we begin by evaluating the dipole gluon helicity TMD at small $x$. In the process we obtain an interesting new result: in contrast to the unpolarized dipole gluon TMD case, the operator governing the small-$x$ behavior of the dipole gluon helicity TMD is different from the operator corresponding to the polarized dipole scattering amplitude (used in our previous work to determine the small-$x$ asymptotics of the quark helicity distribution).…
