Evolution of the electroweak structure functions of nucleons

Abstract

Electroweak scattering of neutrinos with hadronic targets reveal the underlying structure of protons and neutrons. Electroweak structure functions for nucleon targets are reviewed in the context of muon and tau neutrino and antineutrino scattering. Quantum chromodynamic, target mass and quark mass corrections and the regimes where they are relevant are discussed. In future experiments in the Large Hadron Collider forward region at CERN, there are prospects for future measurements for neutrino and antineutrino energies in the 100’s of GeV to TeV energy regime with a large flux of $\nu _\tau +\overline{\nu }_\tau$ as well as the electron and muon flavors. At lower energies, DUNE will probe structure functions at the boundary of perturbative and nonperturbative regimes.

Appendices

NLO coefficient functions

The coefficient functions that multiply quark PDFs in the massless quark limit with $\overline{\mathrm{MS}}$ renormalization are [33,34,35]

$$\begin{aligned} f_{2,q}(z)= & {} C_F\Bigl [ 2\Bigl (\frac{\ln (1-z)}{1-z}\Bigr )_+\nonumber \\&-\frac{3}{2}\Bigl (\frac{1}{1-z}\Bigr )_+ - (1+z)\ln (1-z)-\frac{1+z^2}{1-z}\ln z\nonumber \\+ & {} 3+2z-\Bigl (\frac{\pi ^2}{3}+\frac{9}{2}\Bigr )\delta (1-z)]\Bigr ] \end{aligned}$$

(33)

$$\begin{aligned} f_{1,q}(z)= & {} f_{2,q}(z) - C_F\bigl ( 2z\bigr ) \end{aligned}$$

(34)

$$\begin{aligned} f_{3,q}(z)= & {} f_{2,q}(z) - C_F \bigl ( 1+z\bigr ) \end{aligned}$$

(35)

$$\begin{aligned} f_{4,q} (z)= & {} C_F \bigl ( z\bigr ) \end{aligned}$$

(36)

$$\begin{aligned} f_{5,q} (z)= & {} f_{2,q}(z)\,, \end{aligned}$$

(37)

for $C_F=4/3$. The coefficient functions that multiply the gluon PDF at NLO in the massless quark limit are

$$\begin{aligned} f_{2,G}(z)= & {} T_R\Bigl ( \bigl ( z^2+(1-z)^2 \bigr )\ln \frac{1-z}{z} + -8z^2+8z-1\Bigr ) \nonumber \\ \end{aligned}$$

(38)

$$\begin{aligned} f_{1,G}(z)= & {} f_{2,G}(z) - T_R \Bigl (4z(1-z)\Bigr ) \end{aligned}$$

(39)

$$\begin{aligned} f_{3,G}(z)= & {} 0 \end{aligned}$$

(40)

$$\begin{aligned} f_{4,G}(z)= & {} 2z(1-z) \end{aligned}$$

(41)

$$\begin{aligned} f_{5,G} (z)= & {} f_{2,G}(z)\,, \end{aligned}$$

(42)

where $T_R=1/2$.

Target mass corrections to $F_4$, $F_5$ and $F_L$

The target mass corrections to $F_4$ and $F_5$ can be written [58, 59]

$$\begin{aligned} F_{4}^{\mathrm{TMC}}(x,Q^{2})= & {} \frac{1}{1+M^2\xi ^{2}/Q^2}F_{4}^{(0)}(\xi ,Q^2) \nonumber \\&+ \frac{x^3 M^4}{Q^4 r^3} F_2^{(0)}(\xi ,Q^2) -\frac{2 M^2 x^2}{Q^2 r^2}F_5^{(0)}(\xi ,Q^2)\nonumber \\- & {} \frac{2M^{4}x^{4}}{Q^{4}r^{4}}(2-\frac{M^2 \xi ^2}{Q^2 })h_{2}(\xi ,Q^2)+\frac{x^2 M^2}{Q^2 r^3} h_5(\xi ,Q^2) \nonumber \\+ & {} \frac{2M^{4}x^{3}}{Q^{4}r^{5}}(1-\frac{2M^2x^2}{Q^2})g_{2}(\xi ,Q^2)\, , \end{aligned}$$

(43)

$$\begin{aligned} F_{5}^{\mathrm{TMC}}(x,Q^{2})= & {} \frac{x}{\xi r^{2}}F_{5}^{(0)}(\xi ,Q^2) - \frac{M^2 x^2}{Q^2 \xi r^3}F_{2}^{(0)}(\xi ,Q^2)\nonumber \\+ & {} \frac{M^{2}x^{2}}{Q^{2}r^{3}}h_{5}(\xi ,Q^2) +\frac{2M^2 x^2}{Q^2 r^4}(1-\frac{M^2 x\xi }{Q^2}) h_2(\xi ,Q^2)\nonumber \\+ & {} \frac{6M^{4}x^{3}}{Q^{4}r^{5}}g_{2}(\xi ,Q^2)\,, \end{aligned}$$

(44)

where

$$\begin{aligned} h_{5}(\xi ,Q^{2})= & {} \int _{\xi }^{1}du\ \frac{2F_{5}^{(0)}(u,Q^{2})}{u}\,. \end{aligned}$$

(45)

For reference, the longitudinal structure function is:

$$\begin{aligned} F_{L}^{\mathrm{TMC}}(x,Q^{2})= & {} r^{2}F_{2}^{\mathrm{TMC}}(x,Q^{2}) \nonumber \\&-2xF_{1}^{\mathrm{TMC}}(x,Q^{2}) \nonumber \\= & {} \frac{x^{2}}{\xi ^{2}r} \left[ F_{2}^{(0)}(\xi ,Q^2)-2\xi F_{1}^{(0)}(\xi ,Q^2)\right] \nonumber \\&+\frac{4M^{2}x^{3}}{Q^{2}r^{2}}h_{2}(\xi ,Q^2) +\frac{8M^{4}x^{4}}{Q^{4}r^{3}}g_{2}(\xi ,Q^2) \nonumber \\\simeq & {} \frac{x^{2}}{\xi ^{2}r}F_{L}^{(0)}(\xi ,Q^2)\nonumber \\&+\frac{4M^{2}x^{3}(1-\xi )^2}{Q^{2}\xi r^{2}}F_2^{(0)}(\xi ,Q^2)\,, \end{aligned}$$

(46)

where $F_L{(0)}(\xi ,Q^2)=F_2^{(0)}(\xi ,Q^2)-2\xi F_1^{(0)}(\xi ,Q^2)$. Care must be taken not to mix the M dependent relation in the first line of eq. (46) and the relation between $F_1^{(0)}$, $F_2^{(0)}$ and $F_L^{(0)}$ when $F_2$ and $F_L$ are calculated directly rather than $F_1$ and $F_2$.