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