SIDIS

The differential cross section is given by

\[\begin{align} \frac{d\sigma}{dx ~ dy ~ d\phi_S ~ dz ~ d\phi_h ~ dP_{\rm T}^2} = \frac{\alpha^2}{xyQ^2}\frac{y^2}{2(1-\varepsilon)}\left(1+\frac{\gamma^2}{2x} \right) \sum_{i=1}^{18} F_i(x,z,Q^2,P_{\rm T}) \beta_i \end{align}\]

where

SIDIS structure functions
\(F_i\) Standard Label \(\beta_i\)
\(F_1\) \(F_{UU,T}\) \(1\)
\(F_2\) \(F_{UU,L}\) \(\varepsilon\)
\(F_3\) \(F_{LL}\) \(S_{||}\lambda_e\sqrt{1-\varepsilon^2}\)
\(F_4\) \(F_{UT}^{\sin(\phi_h+\phi_S)}\) \(|\vec{S}_\perp|\varepsilon\sin(\phi_h+\phi_S)\)
\(F_5\) \(F_{UT,T}^{\sin(\phi_h-\phi_S)}\) \(|\vec{S}_\perp|\sin(\phi_h-\phi_S)\)
\(F_6\) \(F_{UT,L}^{\sin(\phi_h-\phi_S)}\) \(|\vec{S}_\perp|\varepsilon\sin(\phi_h-\phi_S)\)
\(F_7\) \(F_{UU}^{\cos2\phi_h}\) \(\varepsilon~\cos(2\phi_h)\)
\(F_8\) \(F_{UT}^{\sin(3\phi_h-\phi_S)}\) \(|\vec{S}_\perp|\varepsilon\sin(3\phi_h-\phi_S)\)
\(F_9\) \(F_{LT}^{\cos(\phi_h-\phi_S)}\) \(|\vec{S}_\perp|\lambda_e\sqrt{1-\varepsilon^2}\cos(\phi_h-\phi_S)\)
\(F_{10}\) \(F_{UL}^{\sin~2\phi_h}\) \(S_{||}\varepsilon\sin(2\phi_h)\)
\(F_{11}\) \(F_{LT}^{\cos\phi_S}\) \(|\vec{S}_\perp|\lambda_e\sqrt{2\varepsilon(1-\varepsilon)}\cos\phi_S\)
\(F_{12}\) \(F_{LL}^{\cos\phi_h}\) \(S_{||}\lambda_e\sqrt{2\varepsilon(1-\varepsilon)}\cos\phi_h\)
\(F_{13}\) \(F_{LT}^{\cos(2\phi_h-\phi_S)}\) \(|\vec{S}_\perp|\lambda_e\sqrt{2\varepsilon(1-\varepsilon)}\cos(2\phi_h-\phi_S)\)
\(F_{14}\) \(F_{UL}^{\sin\phi_h}\) \(S_{||}\sqrt{2\varepsilon(1+\varepsilon)}\sin\phi_h\)
\(F_{15}\) \(F_{LU}^{\sin\phi_h}\) \(\lambda_e\sqrt{2\varepsilon(1-\varepsilon)}\sin\phi_h\)
\(F_{16}\) \(F_{UU}^{\cos\phi_h}\) \(\sqrt{2\varepsilon(1+\varepsilon)}~\cos\phi_h\)
\(F_{17}\) \(F_{UT}^{\sin\phi_S}\) \(|\vec{S}_\perp|\sqrt{2\varepsilon(1+\varepsilon)}\sin\phi_S\)
\(F_{18}\) \(F_{UT}^{\sin(2\phi_h-\phi_S)}\) \(|\vec{S}_\perp|\sqrt{2\varepsilon(1+\varepsilon)}\sin(2\phi_h-\phi_S)\)

were

\[\begin{split}\begin{align} \varepsilon &= \frac{1-y-\frac{1}{4}\gamma^2 y^2}{1-y+\frac{1}{2} y^2 + \frac{1}{4}\gamma^2 y^2}\, , \\ \gamma &= \frac{2 M x}{Q} \, , \end{align}\end{split}\]

and in the nucleon rest frame the polarization vector is given by \(S=(0,\vec{S}_\perp,S_L)\) with \(\vec{S}_\perp^2+S_L^2=1\).

The 18 structure function in SIDIS at leading-order will be expressed in the context of WW-type approximation in terms of a minimal TMD basis using the gaussian ansatz:

\[\begin{split}\begin{align} {\cal F}_q(\xi,p_{\perp})&={\cal K}_q ~ {\cal C}_q(\xi) \frac{\exp\left(-k_{\perp}^2/\omega_q\right)}{\pi \omega_q}\\ {\cal D}_q(\xi,p_{\perp})&={\cal K}_q ~ {\cal C}_q(\xi) \frac{\exp\left(-P_{\perp}^2/\omega_q\right)}{\pi \omega_q}. \end{align}\end{split}\]

We denote the transverse momentum of the quark inside a fast moving proton by \(k\). We use the notation \(p_{\perp}\) for the transverse momentum of the quark relative to the fragmenting parton motion. The structure functions are expressed as

\[\begin{split}\begin{align} F&=\sum_q e_q^2 ~ {\cal K}_q ~ {\cal F}_q(x) ~{\cal D}_q(z) \frac{\exp\left(-P_{\perp}^2/\Omega_q\right)}{\pi \Omega_q}\\ \Omega_q&=z^2\left<k\right> + \left<p_{\perp}\right> \end{align}\end{split}\]
Nucleon distributions and fragmentation functions
type Name \({\cal K}_q\) \({\cal C}_q\)
\({\cal F}_q\) upol.PDF \(1\) \(f_1\)
\({\cal F}_q\) pol.PDF \(1\) \(g_1^q\)
\({\cal F}_q\) Transversity \(1\) \({h_1^q}\)
\({\cal F}_q\) Sivers \(\frac{2M^2}{\omega_q}\) \({{f^{\perp(1)q}_{1T}}}\)
\({\cal F}_q\) Boer-Mulders \(\frac{2M^2}{\omega_q}\) \({{h^{\perp(1)q}_1 }}\)
\({\cal F}_q\) Pretzelosity \(\frac{2M^4}{\omega_q^2}\) \({{h^{\perp(2)q}_{1T} }}\)
\({\cal F}_q\) WormGear \(\frac{2M^2}{\omega_q}\) \({{g^{\perp (1) q}_{1T}}}\)
\({\cal F}_q\) WormGear \(\frac{2M^2}{\omega_q}\) \({{h^{\perp (1) q}_{1L}}}\)
\({\cal F}_q\) twist-3 \(1\) \({{g^{q}_{T}}}\)
\({\cal F}_q\) twist-3 \(\frac{2M^4}{\omega_q^2}\) \({ {g^{\perp(2)q}_{T} }}\)
\({\cal F}_q\) twist-3 \(\frac{2M^2}{\omega_q}\) \({{f^{\perp (1) q}}}\)
\({\cal F}_q\) twist-3 \(\frac{2M^2}{\omega_q}\) \({{h^{\perp (1) q}_T}}\)
\({\cal F}_q\) twist-3 \(\frac{2M^2}{\omega_q}\) \({{h^{(1) q}_T}}\)
\({\cal F}_q\) twist-3 \(\frac{2M^4}{\omega_q^2}\) \({{f^{\perp(2)q}_{T} }}\)
\({\cal C}_q\) FF \(1\) \({{d_1^{q}}}\)
\({\cal C}_q\) Collins \(\frac{2z^2m_h^2}{\omega_q}\) \({{H^{\perp(1)q}_{1} }}\)

twist-3 functions can be related to twist-2 functions using the following formulas:

\[\begin{split}\begin{align} g^{q}_{T}(x) = \int_x^1\frac{ d y}{y}\,g_1^q(y) \\ \end{align}\end{split}\]
\[\begin{split}\begin{align} xg^{\perp(2)q}_{T}(x) = \frac{\left<k_{\perp}^2\right>_{g^{\perp (1) q}_{1T}}}{M^2}\; g^{\perp (1) q}_{1T}(x)\\ \end{align}\end{split}\]
\[\begin{split}\begin{align} x\,f^{\perp (1) q}(x) = \frac{\left<k_{\perp}^2\right>_{f_1}}{2M^2}\,f_1(x)\\ \end{align}\end{split}\]
\[\begin{align} -\frac{1}{2}x(h^{\perp (1) q}_T(x)-h^{(1) q}_T(x)) = \frac{\left<k_{\perp}^2\right>_{h_1^q}}{2M^2}\;h_1^q(x) \end{align}\]
\[\begin{equation} -\frac{1}{2}x(h^{\perp (1) q}_T(x)+h^{(1) q}_T(x)) = h^{\perp(2)q}_{1T}(x) \end{equation}\]
Structure functions in terms of partonic d.o.f
\(F_i\) standard notation \({\cal K}_q\) \({\cal F}_q(x)\) \({\cal D}_q(z)\)
\(F_1\) \(F_{UU,T}\) \(x\) \(f_1\) \(d_1\)
\(F_2\) \(F_{UU,L}\) \(0\) \(0\) \(0\)
\(F_3\) \(F_{LL}\) \(x\) \({g_1^q}\) \({d_1^q}\)
\(F_4\) \(F_{UT}^{\sin(\phi_h+\phi_S)}\) \(\frac{2xz{P^h_{\perp}} m_h}{w_q}\) \({h_1^q}\) \({H^{\perp(1)q}_{1}}\)
\(F_5\) \(F_{UT,T}^{\sin(\phi_h-\phi_S)}\) \(-\frac{2xzM{P^h_{\perp}}}{w_q}\) \({f^{\perp(1)q}_{1T}}\) \({d_1^q}\)
\(F_6\) \(F_{UT,L}^{\sin(\phi_h-\phi_S)}\) \(0\) \(0\) \(0\)
\(F_7\) \(F_{UU}^{\cos(2\phi_h)}\) \(\frac{4xz^2M{P^h_{\perp}}^2m_h}{w_q^2}\) \({h^{\perp(1)q}_1}\) \({H^{\perp(1)q}_{1}}\)
\(F_8\) \(F_{UT}^{\sin(3\phi_h-\phi_S)}\) \(\frac{2xz^3{P^h_{\perp}}^3m_hM^2}{w_q^3}\) \({h^{\perp(2)q}_{1T}}\) \({H^{\perp(1)q}_{1}}\)
\(F_9\) \(F_{LT}^{\cos(\phi_h-\phi_S)}\) \(\frac{2xzM{P^h_{\perp}}}{w_q}\) \({g^{\perp (1) q}_{1T}}\) \({d_1^q}\)
\(F_{10}\) \(F_{UL}^{\sin(2\phi_h)}\) \(\frac{4xz^2M{P^h_{\perp}}^2m_h}{w_q^2}\) \({h^{\perp (1) q}_{1L}}\) \({H^{\perp(1)q}_{1}}\)
\(F_{11}\) \(F_{LT}^{\cos\phi_S}\) \(-\frac{2M}{Q}x\) \({g^{q}_{T}}\) \({d_1^q}\)
\(F_{12}\) \(F_{LL}^{\cos\phi_h}\) \(-\frac{2xz{P^h_{\perp}}}{Q}\frac{{\left<p_{\perp}\right>}}{w_q}\) \({g_1^q}\) \({d_1^q}\)
\(F_{13}\) \(F_{LT}^{\cos(2\phi_h-\phi_S)}\) \(-\frac{2xz^2M^3{P^h_{\perp}}^2}{Q}\frac{1}{w_q^2}\) \(x{g^{q}_{T}}_{\perp}\) \({d_1^q}\)
\(F_{14}\) \(F_{UL}^{\sin\phi_h}\) \(\frac{4Mm_h}{Q}\frac{{P^h_{\perp}}z}{w_q}\) \({h^{\perp (1) q}_{1L}}\) \({H^{\perp(1)q}_{1}}\)
\(F_{15}\) \(F_{LU}^{\sin\phi_h}\) \(0\) \(0\) \(0\)
\(F_{16}\) \(F_{UU}^{\cos\phi_h}(i)\) \(0\) \({h^{\perp(1)q}_1 }\) \({H^{\perp(1)q}_{1}}\)
\(F_{16}\) \(F_{UU}^{\cos\phi_h}(ii)\) \(-\frac{2M}{Q}\frac{2xz{P^h_{\perp}}M}{w_q}\) \(xf^{\perp (1) q}\) \({d_1^q}\)
\(F_{17}\) \(F_{UT}^{\sin\phi_S}(i)\) \(0\) \({f^{\perp(1)q}_{1T}}\) \({d_1^q}\)
\(F_{17}\) \(F_{UT}^{\sin\phi_S}(ii)\) \(\frac{2M}{Q}\frac{4xz^2m_hM}{w_q}(1-\frac{{P^h_{\perp}}^2}{w_q})\) \(-\frac{1}{2}x({h^{\perp (1) q}_T}-{f^{\perp(2)q}_{T} })\) \({H^{\perp(1)q}_{1}}\)
\(F_{18}\) \(F_{UT}^{\sin(2\phi_h-\phi_S)}(i)\) \(\frac{2M}{Q}x\frac{M^2z^2{P^h_{\perp}}^2}{w_q^2}\) \(x{f^{\perp (1) q}}\) \({d_1^q}\)
\(F_{18}\) \(F_{UT}^{\sin(2\phi_h-\phi_S)}(ii)\) \(-\frac{2M^2}{Q}x\frac{4z^2{P^h_{\perp}}^2Mm_h}{w_q^2}\) \(-\frac{1}{2}x({h^{\perp (1) q}_T}+{f^{\perp(2)q}_{T} })\) \({H^{\perp(1)q}_{1}}\)
\(F_{19}\) \(F_{\rm CAHN}^{\cos(2\phi_h)}\) \(\frac{1}{Q^2}\frac{2xz^2{P^h_{\perp}}^2{\left<p_{\perp}\right>}^2}{w_q^2}\) \({f_1^q}\) \({d_1^q}\)