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\title[UPC at LHC]{Ultraperipheral collisions at the Large Hadron Collider}

\institute{
  Lebedev Physical Institute \\
  Moscow, Russia
}

\author[E.~V.~Zhemchugov]{
  S.~I.~Godunov, E.~K.~Karkaryan, V.~A.~Novikov, A.~N.~Rozanov, M.~I.~Vysotsky,
  \underline{E.~V.~Zhemchugov}
}

\date{
  JINR, Dubna \\
  June 17, 2022
}

\begin{document}

\begin{frame}[plain, noframenumbering]
  \maketitle
\end{frame}

\begin{frame}{Contents}
  \begin{itemize}
    \item Ultraperipheral collisions [1806.07238].
    \item Survival factor [2106.14842].
    \item Long-lived charged particles [1906.08568].
    \item Semi-inclusive processes [to be published].
  \end{itemize}
\end{frame}

\section{Ultraperipheral collisions}
\renewcommand{\insertarxiv}{1806.07238}

\begin{frame}[plain, noframenumbering]
  \centering { \LARGE Ultraperipheral collisions } \\[2ex]
  {\large [1806.07238]}
\end{frame}

\begin{frame}{Ultraperipheral collisions at the LHC}
  \centering
  \hfill
  \begin{tikzpicture}[>=stealth]
    \coordinate (A) at (-1,  1);
    \coordinate (B) at ( 1, -1);

    \path [shade, inner color=black, outer color=white]
       (A) ellipse (0.4 and 1.2);
    \path [shade, inner color=black, outer color=white]
       (B) ellipse (0.4 and 1.2);
    \draw [fill] (A) ellipse (0.1 and 0.3);
    \draw [fill] (B) ellipse (0.1 and 0.3);

    \draw [->] (A) -- ++( 0.8, 0);
    \draw [->] (B) -- ++(-0.8, 0);
  \end{tikzpicture}
  \hfill
  \begin{tikzpicture}
    \coordinate (P1in)  at (-1.7,  0.7);
    \coordinate (V1)    at (-0.7,  0.7);
    \coordinate (P1out) at ( 1.0,  1.7);
    \coordinate (V)     at (   0,    0);
    \coordinate (P2in)  at (-1.7, -0.7);
    \coordinate (V2)    at (-0.7, -0.7);
    \coordinate (P2out) at ( 1.0, -1.7);

    \draw [fermion] (P1in) node [left] {$p~\text{ (Pb)}$} -- (V1);
    \draw [fermion] (V1)   -- (P1out) node [right] {$p~\text{ (Pb)}$};
    \draw [fermion] (P2in) node [left] {$p~\text{ (Pb)}$} -- (V2);
    \draw [fermion] (V2)   -- (P2out) node [right] {$p~\text{ (Pb)}$};
    \draw [photon]  (V1)   -- node [above right] {$\gamma$} (V);
    \draw [photon]  (V2)   -- node [below right] {$\gamma$} (V);

    \draw (V) -- +( 15:1);
    \draw (V) -- +(  5:1);
    \draw (V) -- +( -5:1);
    \draw (V) -- +(-15:1);

    \draw [fill=gray] (V) circle (0.2);
  \end{tikzpicture}
  \hfill
  \phantom{.}
  \\
  \vfill
  \begin{tikzpicture}
    \coordinate (O1) at (-3, 0);
    \coordinate (O2) at ( 3, 0);

    \path [shade, inner color=black, outer color=white] (O1) circle (1);
    \draw [fill] (O1) circle (0.25);

    \path [shade, inner color=black, outer color=white]
      (O2) ellipse (0.4 and 1.2);
    \draw [fill]
      (O2) ellipse (0.1 and 0.3);

    \draw [->, >=stealth] (-1, 0) -- node [below] {$\gamma \gg 1$} (2, 0);

    \draw [photon]
      (O1) ++(0.4, 0.4) -- ++(25:1) node [right] {$(0, q_x, q_y, q_z)$};
    \draw [photon]
      (O2) ++(0.1, 0.4)
      -- ++(0:1) node [right] {$(\sqrt{\gamma^2 - 1} q_z, q_x, q_y, \gamma q_z)$};
  \end{tikzpicture}
  Photon virtuality:
  $Q^2 \equiv -q^2 = q_x^2 + q_y^2 + q_z^2 \ll (\gamma q_z)^2 \equiv \omega^2$
\end{frame}  

\begin{frame}{Photon-photon processes studied at LHC}
  \centering
  \begin{tabular}[t]{cl}
    $\gamma \gamma \to \gamma \gamma$ & \\
    \begin{tikzpicture}[scale=0.5, baseline=1]
      \draw [photon] (-3,  1) node [left] {$\gamma$} -- (-1,  1);
      \draw [photon] (-3, -1) node [left] {$\gamma$} -- (-1, -1);
      \draw [photon] (1,  1) -- (3,  1) node [right] {$\gamma$};
      \draw [photon] (1, -1) -- (3, -1) node [right] {$\gamma$};
      \draw [fermion] (-1,  1) -- (-1, -1);
      \draw [fermion] (-1, -1) -- ( 1, -1);
      \draw [fermion] ( 1, -1) -- ( 1,  1);
      \draw [fermion] ( 1,  1) -- (-1,  1);
    \end{tikzpicture}
    \vspace{5ex}
    & \parbox{0.5\textwidth}{
        Nature Physics 13, 852 (2017) \\
        Phys.Lett.~B797, 134826 (2019)
      }
    \\
    $\gamma \gamma \to \mu^+ \mu^-$ & \\
    \begin{tikzpicture}[scale=0.5, baseline=1]
      \draw [photon] (-2,  1) node [left] {$\gamma$} -- (0,  1);
      \draw [photon] (-2, -1) node [left] {$\gamma$} -- (0, -1);
      \draw [fermion] (2, 1) node [right] {$\mu^+$} -- (0, 1);
      \draw [fermion] (0, 1) -- (0, -1);
      \draw [fermion] (0, -1) -- (2, -1) node [right] {$\mu^-$};
    \end{tikzpicture}
    \vspace{5ex}
    & \parbox{0.5\textwidth}{
        Phys.Lett.~B777, 303 (2018) \\
        JHEP 1201, 052 (2012)
      }
    \\
    $\gamma \gamma \to W^+ W^-$ & \\
    \begin{tikzpicture}[scale=0.5, baseline=1]
      \draw [photon] (-1, -1) node [below left] {$\gamma$} -- (0, 0);
      \draw [photon] (-1,  1) node [above left] {$\gamma$} -- (0, 0);
      \draw [boson]  (0, 0) -- (1, -1) node [below right] {$W^-$};
      \draw [boson]  (0, 0) -- (1,  1) node [above right] {$W^+$};
    \end{tikzpicture}
    \vspace{5ex}
    & \parbox{0.5\textwidth}{
        Phys.Rev.~D94, 032011 (2016) \\
        JHEP 1608, 119 (2016)
      }
  \end{tabular}
\end{frame}

\begin{frame}{$p$ vs Pb}
  \begin{center}
    \begin{tikzpicture}[scale=0.7]
      \coordinate (P1in)  at (-1.7,  0.7);
      \coordinate (V1)    at (-0.7,  0.7);
      \coordinate (P1out) at ( 1.0,  1.7);
      \coordinate (V)     at (   0,    0);
      \coordinate (P2in)  at (-1.7, -0.7);
      \coordinate (V2)    at (-0.7, -0.7);
      \coordinate (P2out) at ( 1.0, -1.7);

      \draw [fermion] (P1in) node [left] {$p~\text{ (Pb)}$} -- (V1);
      \draw [fermion] (V1)   -- (P1out) node [right] {$p~\text{ (Pb)}$};
      \draw [fermion] (P2in) node [left] {$p~\text{ (Pb)}$} -- (V2);
      \draw [fermion] (V2)   -- (P2out) node [right] {$p~\text{ (Pb)}$};
      \draw [photon]  (V1)   -- node [above right] {$\gamma$} (V);
      \draw [photon]  (V2)   -- node [below right] {$\gamma$} (V);

      \draw (V) -- +( 15:1);
      \draw (V) -- +(  5:1);
      \draw (V) -- +( -5:1);
      \draw (V) -- +(-15:1);

      \draw [fill=gray] (V) circle (0.2);
    \end{tikzpicture}
    \\
    $\sigma \sim Z^4$
    \vfill
    \begin{tabular}{r|cc}
      & $pp$ & Pb~Pb \\ \hline
      Energy & $13$ TeV & $5.02$ TeV/(nucleon pair) \\ \hline
      $Z$   & $1$ & $82$ \\ \hline
      $Z^4$ & $1$ & $4.5 \cdot 10^7$ \rule{0pt}{1em} \\ \hline
      Luminosity & $159~\text{fb}^{-1}$ & $2.4~\text{nb}^{-1}$ \rule{0pt}{1em} \\
      & \multicolumn{1}{r}{ratio:} & \multicolumn{1}{l}{$6.6 \cdot 10^7$} \\ \hline
      Duration & 21 months & 2 months \\
      & (Run 2)   & (2015, 2018) \\ \hline
      $\sqrt{s_{\gamma \gamma}}$ & $\lesssim 2.6$~TeV & $\lesssim 160$~GeV \\
    \end{tabular}
  \end{center}
\end{frame}

\begin{frame}{$p$ vs Pb: forward detectors}
  \centering
  \begin{flushright}
    \tiny [1909.10827]
  \end{flushright}
  \includegraphics[scale=0.3]{FPD.png} \\
  \begin{tabular}{lr@{--}lr@{--}l}
    \hline \hline
    Distance from the IP, m
    & \multicolumn{2}{c}{$200$}
    & \multicolumn{2}{c}{$420$} \\
    $\xi$ range & $0.015$ & $0.15$ & $0.002$ & $0.02$ \\
    $6.5$~TeV $p$ energy loss, GeV & $97.5$ & $975$ & $13$ & $130$ \\
    \hspace{2ex} in the center-of-mass frame, MeV & $14$ & $141$ & $1.9$ & $19$ \\
    $0.5$~PeV \nuc{Pb}{208} energy loss, TeV & $7.8$ & $78$ & $1.0$ & $10$ \\
    \hspace{2ex} in the center-of-mass frame, GeV & $2.9$ & $29$ & $0.37$ & $3.7$ \\
    \hline \hline
  \end{tabular}
\end{frame}

\begin{frame}{Equivalent photons approximation}
  \begin{center}
    \begin{tikzpicture}
      \coordinate (P1in)  at (-1.7,  0.7);
      \coordinate (V1)    at (-0.7,  0.7);
      \coordinate (P1out) at ( 1.0,  1.7);
      \coordinate (V)     at (   0,    0);
      \coordinate (P2in)  at (-1.7, -0.7);
      \coordinate (V2)    at (-0.7, -0.7);
      \coordinate (P2out) at ( 1.0, -1.7);

      \draw [fermion] (P1in) node [left] {$p$} -- (V1);
      \draw [fermion] (V1)   -- (P1out) node [right] {$p$};
      \draw [fermion] (P2in) node [left] {$p$} -- (V2);
      \draw [fermion] (V2)   -- (P2out) node [right] {$p$};
      \draw [photon]  (V1)   -- node [above right] {$\gamma$} (V);
      \draw [photon]  (V2)   -- node [below right] {$\gamma$} (V);

      \draw (V) -- +( 15:1);
      \draw (V) -- +(  5:1);
      \draw (V) -- +( -5:1);
      \draw (V) -- +(-15:1);

      \draw [fill=gray] (V) circle (0.2);
    \end{tikzpicture}
  \end{center}

  \begin{gather*}
    \sigma(pp \to pp X)
    = \int\limits_0^\infty \mathrm{d} \omega_1
      \int\limits_0^\infty \mathrm{d} \omega_2
      \, \sigma(\gamma \gamma \to X)
      \, n_p(\omega_1)
      \, n_p(\omega_2)
    \\
    n_p(\omega)
    = \frac{2 Z^2 \alpha}{\pi \omega}
      \int\limits_0^\infty
        \left[
          \frac{F_p(q_\perp^2 + (\omega / \gamma)^2)}
               {q_\perp^2 + (\omega / \gamma)^2}
        \right]^2
%        \frac{D_p(q_\perp^2 + (\omega / \gamma)^2)}
%             {(q_\perp^2 + (\omega / \gamma)^2)^2}
        q_\perp^3
      \mathrm{d} q_\perp
    \\
    Q^2 \equiv -q^2 = q_\perp^2 + \left( \frac{\omega}{\gamma} \right)^2
  \end{gather*}
\end{frame}

\begin{frame}{Proton form factor}
  \vspace{-3ex}
  \begin{minipage}{0.5\textwidth}
    \begin{gather*}
%      D(Q^2) = \frac{G_E^2(Q^2) + \tau G_M^2(Q^2)}{1 + \tau}
      F_p(Q^2) = \frac{G_E(Q^2) + \tau G_M(Q^2)}{1 + \tau}
      \\
      G_E(Q^2) = \frac{1}{\left( 1 + \frac{Q^2}{\Lambda^2} \right)^2}
      \\
      G_M(Q^2) = \frac{\mu_p}{\left(1 + \frac{Q^2}{\Lambda^2} \right)^2}
      \\
      \tau = \frac{Q^2}{4 m_p^2} \lesssim 0.01
      \\
      \mu_p = 2.79
      \\ 
      \Lambda_\text{std.}^2 = 0.71~\text{GeV}^2
    \end{gather*}
    Proton radius \\
    $r_p = 0.8414~\text{fm} = \sqrt{12} / \Lambda$ \\
    $\Rightarrow \Lambda^2 = 0.66~\text{GeV}^2$.
  \end{minipage}
  \begin{minipage}{0.45\textwidth}
    \includegraphics[width=\textwidth]{1307.6227-3.pdf} \\
  \end{minipage}
  \begin{flushright}
    { \tiny [1307.6227]}
  \end{flushright}
\end{frame}

\begin{frame}{Proton EPA spectrum}
  \small
  Proton form factor with magnetic contribution taken into account:
  \begin{gather*}
    F_p(Q^2) = \frac{G_E(Q^2) + \tau \textcolor{blue}{G_M(Q^2)}}{1 + \tau},
    \\
    G_E(Q^2) = \frac{1}{\left( 1 + \frac{Q^2}{\Lambda^2} \right)^2},
    \qquad
    G_M(Q^2) = \frac{\mu_p}{\left(1 + \frac{Q^2}{\Lambda^2} \right)^2}
    \\
    \tau = \frac{Q^2}{4 m_p^2} \lesssim 0.01, 
    \ 
    \Lambda^2 = 0.61~\text{GeV}^2,
    \ 
    \mu_p = 2.79.
  \end{gather*}
  EPA spectrum:
  \begin{gather*}
    \begin{split}
      n_p(\omega)
      &= \frac{\alpha}{\pi \omega}
         \left\{
             \left(
               1 + 4 u \textcolor{blue}{{} - 2 (\mu_p - 1) \frac{u}{v}}
             \right)
             \ln \left( 1 + \frac{1}{u} \right)
           - \frac{24 u^2 + 42 u + 17}{6 (u + 1)^2}
      \right. \\ &\qquad \left. {}
           \textcolor{blue}{
           + \frac{\mu_p - 1}{(v - 1)^4} \left[
                 \frac{\mu_p - 1}{v - 1} (1 + 4 u + 3 v)
               - 2 \left( 1 + \frac{u}{v} \right)
             \right]
             \ln \frac{u + v}{u + 1}
          }
      \right. \\ &\qquad \left. {}
           \textcolor{blue}{
           + (\mu_p - 1) \,
             \frac{
               6 u^2 (v^2 - 3 v + 3) + 3 u (3 v^2 - 9 v + 10) + 2 v^2 - 7 v + 11
             }{
               3 (u + 1)^2 (v - 1)^3
             }
           }
      \right. \\ &\qquad \left. {}
           \textcolor{blue}{
           - (\mu_p - 1)^2 \,
             \frac{
               24 u^2 + 6 u (v + 7) - v^2 + 8 v + 17
             }{
               6 (u + 1)^2 (v - 1)^4
             }
           }
         \right\},
    \end{split}
    \\
    u = (\omega / \Lambda \gamma)^2, \ v = (2 m_p / \Lambda)^2.
  \end{gather*}
\end{frame}

%\begin{frame}{Proton EPA spectrum}
%  \small
%  Proton form factor with magnetic contribution taken into account:
%  \begin{gather*}
%    D_p(Q^2) = \frac{G_E^2(Q^2) + \tau \textcolor{blue}{G_M^2(Q^2)}}{1 + \tau},
%    \\
%    G_E(Q^2) = \frac{1}{\left( 1 + \frac{Q^2}{\Lambda^2} \right)^2},
%    \qquad
%    G_M(Q^2) = \frac{\mu_p}{\left(1 + \frac{Q^2}{\Lambda^2} \right)^2}
%    \\
%    \tau = \frac{Q^2}{4 m_p^2} \lesssim 0.01, 
%    \ 
%    \Lambda^2 = 0.61~\text{GeV}^2,
%    \ 
%    \mu_p = 2.79.
%  \end{gather*}
%  EPA spectrum:
%  \begin{gather*}
%    \begin{split}
%      n_p(\omega)
%      &= \frac{\alpha}{\pi \omega}
%         \left\{
%           \left(
%             1 + 4 u \textcolor{blue}{{} - (\mu_p^2 - 1) \frac{u}{v}}
%           \right)
%           \ln \left( 1 + \frac{1}{u} \right)
%           - \frac{24 u^2 + 42 u + 17}{6 (u + 1)^2}
%      \right. \\ & \quad
%           \textcolor{blue}{
%             {}
%             + \frac{\mu_p^2 - 1}{(v - 1)^3}
%               \left[
%                 - \frac{1 + u / v}{v + 1} \ln \frac{u + v}{u + 1}
%      \right. } \\ & \left. \textcolor{blue}{ \left. \qquad {}
%                 + \frac{
%                       6 u^2 (v^2 - 3 v + 3)
%                     + 3 u (3 v^2 - 9 v + 10)
%                     + 2 v^2 - 7 v + 11
%                   }{6 (u + 1)^2}
%               \right]
%           }
%         \right\}
%    \end{split}
%    \\
%    u = (\omega / \Lambda \gamma)^2, \ v = (2 m_p / \Lambda)^2.
%  \end{gather*}
%\end{frame}

\begin{frame}{\nuc{Pb}{208} form factor}
  Charge distribution is provided in form of Bessel spherical functions:
  \begin{equation*}
    \rho(r) = \sum\limits_{k=1}^N a_k j_0(k \pi r / R) \Theta(R - r);
    \ j_0(x) = \sin(x) / x.
  \end{equation*}
  $a_k$, $R$ are numerical parameters. \\
  Form factor is the Fourier transform of charge:
  \begin{equation*}
    F_\text{Pb}(Q^2)
    = \frac{\int \rho(r) \mathrm{e}^{i \vec q \vec r} \mathrm{d}^3 r}
           {\int \rho(r) \mathrm{d}^3 r}
    = \frac{\sin Q R}{Q R}
      \frac{\sum\limits_{k=1}^N \frac{(-1)^k a_k}{(k \pi)^2 - (Q R)^2}}
           {\sum\limits_{k=1}^N \frac{(-1)^k a_k}{(k \pi)^2}}.
  \end{equation*}
  EPA spectrum:
  \begin{equation*}
    n_\text{Pb}(\omega)
    = \frac{2 Z^2 \alpha}{\pi \omega}
      \int\limits_0^\infty
        \left[
          \frac{F(q_\perp^2 + (\omega / \gamma)^2)}
               {q_\perp^2 + (\omega / \gamma)^2}
        \right]^2
        q_\perp^3
      \, \mathrm{d} q_\perp
  \end{equation*}
  Monopole approximation:
  \begin{equation*}
    F_1(Q^2) = \frac{1}{1 + Q^2 / \Lambda^2},
    \ 
    n_1(\omega)
    = \frac{Z^2 \alpha}{\pi \omega}
      \left[ (1 + 2 u) \ln \left(1 + \frac{1}{u} \right) - 2 \right],
    \ 
    u = \left( \frac{\omega}{\Lambda \gamma} \right)^2
  \end{equation*}
\end{frame}

\begin{frame}{\nuc{Pb}{208} form factor} \relax
  \begin{center}
    \includegraphics[width=0.95\textwidth]{form-factor.pdf}
  \end{center}
  \vspace{-2ex}
  {[1987]}: At.Data and Nucl.Data Tabl. 36, 495 (1987) \\
  {[1995]}: At.Data and Nucl.Data Tabl. 60, 177 (1995)
\end{frame}

\begin{frame}{\nuc{Pb}{208} charge density}
  \small
  \vspace{-2.5ex}
  \begin{equation*}
    \rho(r) = \sum\limits_{k=1}^N a_k \, j_0(k \pi r/R) \, \Theta(R - r),
    \quad
    F(Q)
    = \frac{\int \rho(r) \mathrm{e}^{i \vec q \vec r} \mathrm{d}^3 r}
           {\int \rho(r) \mathrm{d}^3 r}
  \end{equation*}
  \begin{center}
    \begin{tabular}{c|cc|cc}
      & \multicolumn{2}{c|}{[1987]}
      & \multicolumn{2}{c}{[1995]}
      \\
      & Quoted & Actual  & Quoted & Actual
      \\
      \hline
      $Z = \int \rho(r) \mathrm{d}^3 r$ \rule{0pt}{2.5ex} &
      $82$ & $82.03$ & $82$ & $3806$ \\
      $
       \left< r^2 \right>^{1/2}
       = \left(
           \frac{\int \rho(r) r^2 \mathrm{d}^3 r}{\int \rho(r) \mathrm{d}^3 r}
         \right)^{\frac{1}{2}}
      $, fm
      & $5.499(1)$ & $5.501$ & $5.4785$ & $8.0850$ \\
    \end{tabular}
  \end{center}
  \vspace{-3.5ex}
  \begin{center}
    \includegraphics[width=0.95\textwidth]{pb-charge.pdf}
  \end{center}
  \vspace{-2ex}
  {[1987]}: At.Data and Nucl.Data Tabl. 36, 495 (1987) \\
  {[1995]}: At.Data and Nucl.Data Tabl. 60, 177 (1995) 
  \textcolor{red}{?}
\end{frame}

\begin{frame}{EPA spectra}
  \centering
  $E_p = 6.5$~TeV, $E_\text{Pb} = 522$~TeV ($2.5~\text{TeV}/\text{nucleon}$) \\
  \includegraphics[width=0.93\textwidth]{epa-spectrum.pdf}
  $\hat q_p \gamma_p = 1.3$~TeV, $\hat q_\text{Pb} \gamma_\text{Pb} = 79$~GeV
\end{frame}

\begin{frame}{CMS detector}
  \small
  \centering
  \includegraphics[scale=0.3]{Schematic-view-of-the-CMS-detector.png}
  \vfill
  \begin{minipage}{0.7\textwidth}
    Typical experimental requirements: $\abs{\eta} < \hat \eta$, $p_T > \hat
    p_T$.
    \\
    $\eta = -\ln \tan \frac{\theta}{2}$, \\
    $y
     = \frac12
       \ln \frac{E_\chi + p_\chi \cos \theta}
                {E_\chi - p_\chi \cos \theta}
     \overset{m_\chi \ll E_\chi}{=} \eta$ \\
     $p_T = p_\chi \sin \theta$
  \end{minipage}
  \hspace{3ex}
  \begin{minipage}{0.17\textwidth}
    \vspace{-6ex}
    \begin{tikzpicture}[scale=0.6]
      \draw [->] (-1, 0) node [left]  {$p$} -- (-0.1, 0);
      \draw [->] ( 1, 0) node [right] {$p$} -- ( 0.1, 0);
      \draw [->] ($(0,0) + (55:0.1)$)   -- node [above left] {$\vec p_\chi$} (55:1)   node [above right] {$\chi^+$};
      \draw [->] ($(0,0) + (-125:0.1)$) -- (-125:1) node [below]  {$\chi^-$};
      \draw      (0.4, 0) node [above right] {$\theta$} arc (0:55:0.4);
    \end{tikzpicture}
  \end{minipage}
\end{frame}

\begin{frame}{Fiducial cross section}
  \begin{equation*}
    \frac{\mathrm{d} \sigma_\text{fid}(pp \to pp \chi^+ \chi^-)}{\mathrm{d} s}
    = \int\limits_{
        \max \left(
          \hat p_T,
          \frac{\sqrt{s}}{2 \cosh \hat \eta} \sqrt{1 - \frac{4 m_\chi^2}{s}}
        \right)
      }^{\frac{\sqrt{s}}{2} \sqrt{1 - \frac{4 m_\chi^2}{s}}}
      \mathrm{d} p_T
      \, \frac{\mathrm{d} \sigma(\gamma \gamma \to \chi^+ \chi^-)}
              {\mathrm{d} p_T}
      \, \frac{\mathrm{d} \hat L}{\mathrm{d} s}
  \end{equation*}
  Photon-photon luminosity:
  \begin{equation*}
    \frac{\mathrm{d} \hat L}{\mathrm{d} s}
    = \frac14
      \int\limits_{-\hat y}^{\hat y} \mathrm{d} y
      \, n \left( \frac{\sqrt{s}}{2} \mathrm{e}^y \right)
      \, n \left( \frac{\sqrt{s}}{2} \mathrm{e}^{-y} \right),
  \end{equation*}
  $\sqrt{s}$~--- invariant mass of $\chi^+ \chi^-$ pair, $y = \tfrac12 \ln
  \tfrac{\omega_1}{\omega_2}$~--- its rapidity,
  \begin{equation*}
    \hat y = \asinh \left[
      \frac{\sqrt{s} p_T}{2 (p_T^2 + m_\chi^2)}
      \left(
        \sinh \hat \eta
        - \sqrt{\cosh^2 \hat \eta + \frac{m_\chi^2}{p_T^2}}
    \cdot \sqrt{1 - \frac{4 (p_T^2 + m_\chi^2)}{s}}
      \right)
    \right]
  \end{equation*}
\end{frame}

\begin{frame}{ATLAS experiment: $pp \to pp \mu^+ \mu^-$}
  \vspace{-2ex}
  \begin{center}
    \includegraphics[scale=0.65]{atlas-pp.pdf}
  \end{center}
  \vspace{-4ex}
  Integrated cross section: 
  \begin{itemize}
    \item Experiment: $3.12 \pm 0.07~\text{(stat.)} \pm 0.10~\text{(syst.)}$ pb.
    \item Theory: $3.39$~pb.
  \end{itemize}
\end{frame}

\begin{frame}{
  ATLAS experiment: $\text{Pb}~\text{Pb} \to \text{Pb}~\text{Pb}~\mu^+ \mu^-$
  (preliminary)
}
  \begin{center}
    \includegraphics[scale=0.8]{atlas-pb-pb.pdf}
  \end{center}
  \vspace{-4ex}
  Integrated cross section:
  \begin{itemize}
    \item Experiment: 
      $32.3 \pm 0.3~\text{(stat.)}^{+4.0}_{-3.4}~\text{(syst.)}~\mu$b.
    \item Theory: $34.4~\mu$b ($\Lambda = 50$~MeV) or $57.4~\mu$b ($\Lambda =
    80$~MeV).
  \end{itemize}
\end{frame}

\section{Survival factor}
\renewcommand{\insertarxiv}{2106.14842}

\begin{frame}[plain, noframenumbering]
  \centering
  { \LARGE Survival factor } \\[2ex]
  { \large [2106.14842] }
\end{frame}

\begin{frame}{Correction from strong interactions}
  \vspace{5pt}
  \begin{minipage}{0.6\textwidth}
    Assuming only electromagnetic interactions:
    \begin{equation*}
      n(\omega)
      = \frac{2 Z^2 \alpha}{\pi \omega}
        \int\limits_0^\infty
          \left[
            \frac{F(q_\perp^2 + (\omega / \gamma)^2)}
                 {q_\perp^2 + (\omega / \gamma)^2}
          \right]^2
          q_\perp^3
          \mathrm{d} q_\perp,
     \end{equation*}
   \end{minipage}
   \hfill
   \begin{minipage}{0.2\textwidth}
     \begin{tikzpicture}[>=Stealth]
       \tikzmath{\R = 0.5; \l = 0.07;}

       \coordinate (Cl) at (-1.5 * \R, 0);
       \coordinate (Cr) at ( 1.5 * \R, 0);

       \filldraw [black!30!white] (Cl) circle [radius=\R];
       \filldraw [black!30!white] (Cr) circle [radius=\R];
       \draw     ($(Cr) + (\l, \l)$) -- ($(Cr) - (\l, \l)$);
       \draw     ($(Cr) + (\l, -\l)$) -- ($(Cr) + (-\l, \l)$);
       \filldraw (Cl) circle [radius=\l];

       \draw [->] (Cl) -- node [above] {$\vec b$} (1.5 * \R, 0);

       \coordinate (A) at ($(Cl) + (45:3 * \R)$);

       \draw [->] (Cl) -- node [above left]  {$\vec b_1$} (A);
       \draw [->] (Cr) -- node [above right] {$\vec b_2$} (A);
    \end{tikzpicture}
   \end{minipage}
   \\
   \begin{equation*}
    \hspace{-18ex}
    \sigma(pp \to pp X)
    = \int\limits_0^\infty
      \mathrm{d} \omega_1
      \int\limits_0^\infty
      \mathrm{d} \omega_2
      \, \sigma(\gamma \gamma \to X)
      \, n_p(\omega_1)
      \, n_p(\omega_2).
   \end{equation*}
   \\
   Including strong interactions:
   \begin{gather*}
     n(\omega) = \int n(b, \omega) \, \mathrm{d}^2 b,
     \quad
     n(b, \omega)
     = \frac{Z^2 \alpha}{\pi^2 \omega}
       \left[
         \int\limits_0^\infty
           \frac{F(q_\perp^2 + (\omega / \gamma)^2)}
                {q_\perp^2 + (\omega / \gamma)^2}
           J_1(b q_\perp)
           \, q_\perp^2
           \mathrm{d} q_\perp
       \right]^2,
     \\
     \begin{multlined}
       \sigma(pp \to pp X) \\
       = \int\limits_0^\infty \mathrm{d} \omega_1
         \int\limits_0^\infty \mathrm{d} \omega_2
         \int \mathrm{d}^2 b_1
         \int \mathrm{d}^2 b_2
         \, \sigma(\gamma \gamma \to X)
         \, n_p(b_1, \omega_1)
         \, n_p(b_2, \omega_2)
         \, P(\abs{\vec b_1 - \vec b_2}).
     \end{multlined}
   \end{gather*}
   $P(b)$ is the probability for the protons to survive after the collision with
   the impact parameter $b$. 
\end{frame}

\begin{frame}{Polarization}
  \vspace{-5ex}
  \begin{center}
    \begin{tikzpicture}[>=Stealth, scale=0.6]
      \tikzmath{\R = 1; \l = 0.07;}

      \coordinate (Cl) at (-1.5 * \R, 0);
      \coordinate (Cr) at ( 1.5 * \R, 0);

      \filldraw [black!30!white] (Cl) circle [radius=\R];
      \filldraw (Cl) circle [radius=\l];
      \filldraw [black!30!white] (Cr) circle [radius=\R];
      \draw     ($(Cr) + (\l, \l)$) -- ($(Cr) - (\l, \l)$);
      \draw     ($(Cr) + (\l, -\l)$) -- ($(Cr) + (-\l, \l)$);

      \draw [->] (Cl) -- node [above] {$\vec b$} (Cr);

      \coordinate (A) at ($(Cl) + (45:3 * \R)$);

      \draw [->] (Cl) -- node [above left]  {$\vec b_1$} (A);
      \draw [->] (Cr) -- node [above right] {$\vec b_2$} (A);

      \draw [->] (A) -- +(45:1)    node [below right] {$\vec \varepsilon_1$};
      \draw [->] (A) -- +(112.5:1) node [below left]  {$\vec \varepsilon_2$};
      \draw      ($(A) + (45:0.3)$) arc (45:112.5:0.3);
      \node at ($(A) + (78.75:0.5)$) {$\varphi$};
    \end{tikzpicture}
  \end{center}
  \begin{equation*}
    \sigma(pp \to pp X)
    = \int\limits_0^\infty \mathrm{d} s
      \left[
       \, \sigma_\parallel(\gamma \gamma \to X)
       \, \frac{\mathrm{d} L_\parallel}{\mathrm{d} s}
       +  \sigma_\perp(\gamma \gamma \to X)
       \, \frac{\mathrm{d} L_\perp}{\mathrm{d} s}
      \right],
  \end{equation*}
  where
  \begin{equation*}
    \begin{aligned}
      \frac{\mathrm{d} L_\parallel}{\mathrm{d} s}
      &= \frac14
         \int\limits_{-\infty}^\infty \mathrm{d} y
         \int \mathrm{d}^2 b_1
         \int \mathrm{d}^2 b_2
         \, n_p \left( b_1, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^y \right)
         \, n_p \left( b_2, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^{-y} \right)
         \, P(b)
         \, \textcolor{red}{\cos^2 \varphi},
      \\
      \frac{\mathrm{d} L_\perp}{\mathrm{d} s}
      &= \frac14
         \int\limits_{-\infty}^\infty \mathrm{d} y
         \int \mathrm{d}^2 b_1
         \int \mathrm{d}^2 b_2
         \, n_p \left( b_1, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^y \right)
         \, n_p \left( b_2, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^{-y} \right)
         \, P(b)
         \, \textcolor{red}{\sin^2 \varphi}
    \end{aligned}
  \end{equation*}
  are photon-photon luminosities.
\end{frame}
\begin{frame}{Survival factor}
  \begin{equation*}
    \begin{multlined}
      \sigma(pp \to pp X) \\
      = \int\limits_0^\infty \mathrm{d} \omega_1
        \int\limits_0^\infty \mathrm{d} \omega_2
        \int \mathrm{d}^2 b_1
        \int \mathrm{d}^2 b_2
        \, \sigma(\gamma \gamma \to X)
        \, n_p(b_1, \omega_1)
        \, n_p(b_2, \omega_2)
        \, P(\abs{\vec b_1 - \vec b_2})
    \end{multlined}
  \end{equation*}
  vs
  \begin{equation*}
    \sigma(pp \to pp X)
    = \int\limits_0^\infty \mathrm{d} \omega_1
      \int\limits_0^\infty \mathrm{d} \omega_2
      \, \sigma(\gamma \gamma \to X)
      \, n_p(\omega_1)
      \, n_p(\omega_2)
  \end{equation*}
  Neglecting the polarization,
  \begin{gather*}
    S(s)
    = \frac{\mathrm{d} L / \mathrm{d} s}
           {\mathrm{d} L / \mathrm{d} s \rvert_{P=1}},
  \end{gather*}
  where $L = L_\parallel + L_\perp$.
  \vfill
  \phantom{.}
\end{frame}

\begin{frame}{Survival factor in $pp$ collisions}
  \centering
  \includegraphics[width=\textwidth]{survival.pdf}
\end{frame}

\begin{frame}{Survival factor in Pb Pb collisions --- no result}
  \begin{gather*}
    F_\text{Pb}(Q^2)
    = \frac{\sin QR}{QR}
      \frac{\sum\limits_{k=1}^N \frac{(-1)^k a_k}{(k \pi)^2 - (QR)^2}}
           {\sum\limits_{k=1}^N \frac{(-1)^k a_k}{(k \pi)^2}}
    \\
    \begin{split}
      n(b, \omega)
      &= \frac{Z^2 \alpha}{\pi^2 \omega}
        \left[
          \int\limits_0^\infty
            \frac{F(q_\perp^2 + (\omega / \gamma)^2)}
                 {q_\perp^2 + (\omega / \gamma)^2}
            J_1(b q_\perp) q_\perp^2
          \mathrm{d} q_\perp
        \right]^2
     \\
     &= \frac{Z^2 \alpha}{\pi^2 \omega R^4}
        \left[ \sum\limits_{k=1}^K \frac{(-1)^k a_k}{(k \pi)^2} \right]^{-2}
     \times \\ & \qquad
        \left[
          \sum\limits_{k=1}^K (-1)^k a_k
          \int\limits_0^\infty
            \frac{x^2 \sin \sqrt{x^2 + y^2} \, J_1(cx)}
                 {(x^2 + y^2 - (k \pi)^2) (x^2 + y^2)^{\frac32}}
            \, \mathrm{d} x
       \right]^2
    \end{split}
    \\ x = q_\perp R, ~ y = R \omega / \gamma, ~ c = b / R.
  \end{gather*}
  % -y/(kπ)^2 K_1(c y) (c >= 1)
\end{frame}

\begin{frame}{ATLAS experiment: $pp \to pp \mu^+ \mu^-$}
  \vspace{-2ex}
  \begin{center}
    \includegraphics[scale=0.65]{atlas-pp-s.pdf}
  \end{center}
  \vspace{-4ex}
  Integrated cross section: 
  \begin{itemize}
    \item Experiment: $3.12 \pm 0.07~\text{(stat.)} \pm 0.10~\text{(syst.)}$ pb.
    \item No strong interactions: $3.39$~pb.
    \item With strong interactions: $3.26$~pb.
  \end{itemize}
\end{frame}

\begin{frame}{
  ATLAS experiment: $\text{Pb}~\text{Pb} \to \text{Pb}~\text{Pb}~\mu^+ \mu^-$
  (preliminary)
}
  \begin{center}
    \includegraphics[scale=0.7]{atlas-pb-pb-s.pdf}
  \end{center}
  \vspace{-4ex}
  Integrated cross section:
  \begin{center}
    \begin{tabular}{rll}
      $\Lambda$ & 50~MeV & 80~MeV \\
      No strong interactions   & $34.4~\mu$b & $57.4~\mu$b \\
      With strong interactions & $32.3~\mu$b & $50.8~\mu$b \\
      Experiment & \multicolumn{2}{l}{
        $32.3 \pm 0.3~\text{(stat.)}^{+4.0}_{-3.4}~\text{(syst.)}~\mu$b.
      } 
   \end{tabular}
 \end{center}
%
%  \begin{itemize}
%    \item Experiment: 
%      $32.3 \pm 0.3~\text{(stat.)}^{+4.0}_{-3.4}~\text{(syst.)}~\mu$b.
%    \item No strong interactions: $34.4~\mu$b ($\Lambda = 50$~MeV) or
%    $57.4~\mu$b ($\Lambda = 80$~MeV).
%    \item With strong interactions: $32.3~\mu$b ($\Lambda = 50$~MeV) or $50.8~\mu$b ($\Lambda =
%    80$~MeV).
%  \end{itemize}
\end{frame}

\section{Long-lived charged particles}
\renewcommand{\insertarxiv}{1906.08568}

\begin{frame}[plain, noframenumbering]
  \centering
  { \LARGE Long-lived charged particles } \\[2ex]
  { \large [1906.08568] }
\end{frame}

\begin{frame}
  \begin{flushright}
    \begin{tikzpicture}[scale=0.4, baseline=1]
      \draw [photon] (-2,  1) node [left] {$\gamma$} -- (0,  1);
      \draw [photon] (-2, -1) node [left] {$\gamma$} -- (0, -1);
      \draw [fermion] (2, 1)  node [right] {$\chi$}-- (0, 1);
      \draw [fermion] (0, 1)  -- (0, -1);
      \draw [fermion] (0, -1) -- (2, -1) node [right] {$\chi$};
    \end{tikzpicture}
  \end{flushright}

  \vspace*{-7ex}
  \begin{itemize}
    \item Long-lived charged particles
    \begin{itemize}
      \item Live long enough to escape the detector (like muons).
      \item Usual search techniques: $\mathrm{d} E / \mathrm{d} x$, time of
      flight.
      \item Current bounds [1506.09173, 1609.08382, 1902.01636] are
      model-dependent.
      \item Example: SUSY chargino nearly degenerated with neutralino, $m_\chi
      \gtrsim 100$~GeV.
    \end{itemize}
    \item UPC approach:
    \begin{itemize}
      \item The particles leave tracks in the central detector allowing for
      reconstruction of their momenta $\vec p_1$, $\vec p_2$.
      \item Forward detectors provide the proton energies after the collision
      $E_1$, $E_2$.
      \item Collision kinematics is reconstructed. The mass of the particle
      \begin{equation*}
        m = \sqrt{
          \frac{
            (2 E_1 E_2 + \vec p_1 \vec p_2)^2 - \vec p_1^{\, 2} \vec p_2^{\, 2}
          }{4 E_1 E_2 + (\vec p_1 + \vec p_2)^2}
        }.
      \end{equation*}
    \end{itemize}
    \item Complementary to $\mathrm{d} E / \mathrm{d} x$ or time of flight
    measurements.
    \item Background:
    \begin{itemize}
      \item $pp \to pp \mu^+ \mu^-$ (and other processes producing muons).
      \item Pileup and diffractive scattering.
    \end{itemize}
  \end{itemize}
\end{frame}

\begin{frame}{Fermion pair production}
  \centering
  \includegraphics[scale=0.9]{xsection-m.pdf} \\
  Available integrated luminosity: $pp$ --- $159~\text{fb}^{-1}$, Pb~Pb ---
  $2.4~\text{nb}^{-1}$.
\end{frame}

\begin{frame}{Fermion pair production (fiducial)}
  \centering
  \includegraphics[scale=0.9]{xsection-m-fid.pdf} \\
  $p_T > 20$~GeV, $\abs{\eta} < 2.5$, both protons hit the forward detectors.
\end{frame}

\begin{frame}{Expected signal (no pileup)}
  \centering
  \includegraphics[width=\textwidth]{chargino-clean.pdf}
  Assuming $m_\chi = 100$~GeV and integrated luminosity 150/fb.
\end{frame}

\begin{frame}{Expected signal (with pileup)}
  \centering
  \includegraphics[width=\textwidth]{chargino-pileup.pdf}
  Assuming pileup of 50 collisions at once.
\end{frame}

\begin{frame}{Expected signal (with pileup and longitudinal selection)}
  \centering
  \includegraphics[width=\textwidth]{chargino.pdf}
  Longitudinal selection: $\abs{p_{\parallel, 1} + p_{\parallel, 2} - (E_1 -
  E_2)} < 20$~GeV.
\end{frame}

\begin{frame}{Expected signal (with pileup and longitudinal selection)}
  \centering
  \includegraphics[width=\textwidth]{chargino-2.pdf}
  $m_\chi = 190$~GeV.
\end{frame}

\begin{frame}
  \centering
  \includegraphics[scale=0.8]{lifetimes.pdf} \\
  Assuming integrated luminosity 150/fb.
\end{frame}

\section{Semi-inclusive processes}
\renewcommand{\insertarxiv}{to be published}

\begin{frame}[plain, noframenumbering]
  \centering
  { \LARGE Semi-inclusive processes } \\[2ex]
  { \large [to be published] }
\end{frame}

\begin{frame}{[2009.14537] measurement}
  \begin{center}
    \begin{minipage}[t]{0.45\textwidth}
      \strut\vspace*{-\baselineskip}\newline
      \begin{tikzpicture}[scale=1.3]
        % P1i -- A ---- P1o
        %         \
        %          B -- M1
        %          |
        %          C -- M2
        %         /
        % P2i -- D ---- P2o

        \coordinate (A)   at (-0.7,  1.2);
        \coordinate (B)   at (   0,  0.5);
        \coordinate (C)   at (   0, -0.5);
        \coordinate (D)   at (-0.7, -1.2);
        \coordinate (P1i) at (-1.2,  1.2);
        \coordinate (P1o) at (   1,  1.2);
        \coordinate (P2i) at (-1.2, -1.2);
        \coordinate (P2o) at (   1, -1.2);
        \coordinate (M1)  at (   1,  0.5);
        \coordinate (M2)  at (   1, -0.5);

        \draw[fermion] (P1i) node [left] {$p$} -- (A);
        \draw[fermion] (A)   -- (P1o) node [right] {$p$};
        \draw[fermion] (P2i) node [left] {$p$} -- (D);
        \draw[fermion] (D)   -- (P2o) node [right] {$X'$};
        \draw[photon]  (A)   -- node [below left] {$\gamma$} (B);
        \draw[photon]  (D)   -- node [above left] {$\gamma$} (C);
        \draw[fermion] (M1)  node [right] {$\mu^+$} -- (B);
        \draw[fermion] (B)   -- (C);
        \draw[fermion] (C)   -- (M2) node [right] {$\mu^-$};
        \draw (D) -- +( 5:1.7);
        \draw (D) -- +(-5:1.7);
      \end{tikzpicture}
    \end{minipage}
    ~
    \begin{minipage}[t]{0.45\textwidth}
      Experimental selections:
      \begin{itemize}
        \item $p_T > 15$~GeV.
        \item $\abs{\eta} < 2.4$.
        \item $p_T^{\mu \mu} < 5$~GeV.
        \item $20~\text{GeV} < m_{\mu \mu} < 70$~GeV or $m_{\mu \mu} > 105$~GeV.
        \item At least one proton hits a forward detector.
      \end{itemize}
    \end{minipage}
  \end{center}
  \begin{gather*}
    \sigma(pp \to pX \mu^+ \mu^-) = \sum\limits_q \sigma(pq \to pq \mu^+ \mu^-),
    \\
    \text{For } \left( \frac{p_T^{\mu \mu}}{m_{\mu \mu}} \right)^2 \ll 1,
    \sigma(pq \to pq \mu^+ \mu^-)
    = \int\limits_0^\infty \mathrm{d} \omega_1
      \int\limits_0^\infty \mathrm{d} \omega_2
      \, \sigma(\gamma \gamma \to \mu^+ \mu^-)
      \, n_p(\omega_1)
      \, n_q(\omega_2),
    \\
    n_q(\omega)
    = \frac{\alpha}{\pi \omega}
    \int\limits_{\omega / E}^1 \mathrm{d} x
    \int\limits_{(\omega / x \gamma)^2}^{(p_T^{\mu \mu})^2} \mathrm{d} Q^2
    \, \frac{q_\perp^2}{Q^2} f_q(x, Q^2)
  \end{gather*}
\end{frame}

\begin{frame}{Preliminary results}
  \begin{itemize}
    \item Experiment: $7.2 \pm 1.6~\text{(stat.)} \pm 0.9~\text{(syst.)} \pm
    0.2~\text{(lumi.)}$~fb.
    \item Exclusive process ($pp \to pp \mu^+ \mu^-$): $8.0$~fb.
    \item Inclusive process ($pp \to pX \mu^+ \mu^-$): $8.7$~fb.
  \end{itemize}
  Survival factor should reduce the calculated values by $\sim 10$\%. Errors are
  to be estimated yet.
\end{frame}

\begin{frame}{Conclusions}
  \begin{itemize}
    \item Ultraperipheral collisions are a clean source of photon-photon
    collisions.
    \item Proton dipole form factor yields accurate results. Lead form factor
    is weird.
    \item Survival factor gives corrections of a few percents in the
    measurements considered.
    \item A new technique for detecting long-lived charged particles with the
    help of forward detectors. With 150/fb of integrated luminosity, particles
    with lifetimes down to $\sim 10^2$ ns and masses up to $\sim 150$~GeV can be
    detected.
    \item Our calculation of semi-inclusive muon pair production cross section
    overshoots the measurement by a factor of 2.
    \item \texttt{libepa}
    (\href{https://github.com/jini-zh/libepa}{https://github.com/jini-zh/libepa})
    --- A library for calculations of cross sections of ultraperipheral
    collisions under the equivalent photons approximation.
  \end{itemize}
\end{frame}

\appendix

\section{Backup}
\renewcommand{\insertarxiv}{2106.14842}

\begin{frame}[plain, noframenumbering]
  \centering \LARGE Backup
\end{frame}

\begin{frame}{Proton EPA spectrum}
  \small
  \begin{gather*}
    F_p(Q^2) = \frac{G_E(Q^2) + \tau \textcolor{blue}{G_M(Q^2)}}{1 + \tau},
    \\
    G_E(Q^2) = \frac{1}{\left( 1 + \frac{Q^2}{\Lambda^2} \right)^2},
    \quad
    G_M(Q^2) = \frac{\mu_p}{\left(1 + \frac{Q^2}{\Lambda^2} \right)^2},
    \quad
    \tau = -\frac{Q^2}{4 m_p^2},
    \\
    \begin{split}
      n_p(b, \omega)
      &= \frac{\alpha}{\pi^2 \omega} \left[
           \int\limits_0^\infty
             \frac{F_p(q_\perp^2 + (\omega / \gamma)^2)}
                  {q_\perp^2 + (\omega / \gamma)^2}
             J_1(b q_\perp)
             q_\perp^2
           \mathrm{d} q_\perp
         \right]^2
      \\
      &= \frac{\alpha}{\pi^2 \omega}
         \left[
             \frac{\omega}{\gamma} K_1 \left( \frac{b \omega}{\gamma} \right)
           - \textcolor{blue}{
               \left(
                   1
                 + \frac{(\mu_p - 1) \frac{\Lambda^4}{16 m_p^4}}{
                     \left( 1 - \frac{\Lambda^2}{4 m_p^2} \right)^2
                   }
               \right)
             }
             \sqrt{\Lambda^2 + \frac{\omega^2}{\gamma^2}}
             \, K_1 \left( b \sqrt{\Lambda^2 + \frac{\omega^2}{\gamma^2}} \right)
      \right. \\ &\qquad \left. \textcolor{blue}{ {}
           + \frac{(\mu_p - 1) \frac{\Lambda^4}{16 m_p^4}}{
               \left( 1 - \frac{\Lambda^2}{4 m_p^2}  \right)^2
             }
             \sqrt{4 m_p^2 + \frac{\omega^2}{\gamma^2}}
             \, K_1 \left( b \sqrt{4 m_p^2 + \frac{\omega^2}{\gamma^2}} \right)
      }
      \right. \\ &\qquad \left. {}
           - \textcolor{blue}{
               \frac{1 - \frac{\mu_p \Lambda^2}{4 m_p^2}}
                    {1 - \frac{\Lambda^2}{4 m_p^2}}
             }
           \cdot
             \frac{b \Lambda^2}{2}
             \, K_0 \left( b \sqrt{\Lambda^2 + \frac{\omega^2}{\gamma^2}} \right)
         \right]^2,
    \end{split}
  \end{gather*}
\end{frame}

\begin{frame}{$\gamma \gamma$ luminosities in $pp$ collisions}
  \small
  The probability for both protons to survive [hep-ph/0608271],
  \begin{equation*}
    P(b)
    = \left( 1 - \mathrm{e}^{-\frac{b^2}{2 B}} \right)^2,
    \ B = 21~\text{GeV}^2.
  \end{equation*}
  Photon-photon luminosities:
  \begin{equation*}
    \begin{gathered}
      \begin{multlined}
        \frac{\mathrm{d} L_{pp}^\parallel}{\mathrm{d} s}
        = \frac{\pi^2}{2}
          \int\limits_{\hat r}^\infty b_1 \, \mathrm{d} b_1
          \int\limits_{\hat r}^\infty b_2 \, \mathrm{d} b_2
          \int\limits_{-\infty}^\infty \mathrm{d} y
          \, n_p \left( b_1, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^y \right)
          \, n_p \left( b_2, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^{-y} \right)
          \\  \times
          \left\{
              1
            - 2 \mathrm{e}^{-\frac{b_1^2 + b_2^2}{2 B}}
              \left[
                I_0 \left( \frac{b_1 b_2}{B} \right)
                \textcolor{red}{{}+{}}
                I_2 \left( \frac{b_1 b_2}{B} \right)
              \right]
            + \mathrm{e}^{-\frac{b_1^2 + b_2^2}{B}}
              \left[
                I_0 \left( \frac{2 b_1 b_2}{B} \right)
                \textcolor{red}{{}+{}}
                I_2 \left( \frac{2 b_1 b_2}{B} \right)
              \right]
          \right\},
      \end{multlined}
      \\
      \begin{multlined}
        \frac{\mathrm{d} L_{pp}^\perp}{\mathrm{d} s}
        = \frac{\pi^2}{2}
          \int\limits_{\hat r}^\infty b_1 \, \mathrm{d} b_1
          \int\limits_{\hat r}^\infty b_2 \, \mathrm{d} b_2
          \int\limits_{-\infty}^\infty \mathrm{d} y
          \, n_p \left( b_1, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^y \right)
          \, n_p \left( b_2, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^{-y} \right)
          \\  \times
          \left\{
              1
            - 2 \mathrm{e}^{-\frac{b_1^2 + b_2^2}{2 B}}
              \left[
                I_0 \left( \frac{b_1 b_2}{B} \right)
                \textcolor{red}{{}-{}}
                I_2 \left( \frac{b_1 b_2}{B} \right)
              \right]
            + \mathrm{e}^{-\frac{b_1^2 + b_2^2}{B}}
              \left[
                I_0 \left( \frac{2 b_1 b_2}{B} \right)
                \textcolor{red}{{}-{}}
                I_2 \left( \frac{2 b_1 b_2}{B} \right)
              \right]
          \right\}.
      \end{multlined}
    \end{gathered}
  \end{equation*}
\end{frame}

\begin{frame}{Fiducial cross section}
  \vspace{-3ex}
  \begin{multline*}
    \frac{\mathrm{d} \sigma_\text{fid.}(pp \to pp \chi^+ \chi^-)}
         {\mathrm{d} s} 
    \\
    = \int\limits_{
        \max \left(
          \hat p_T,
          \frac{\sqrt{s}}{2 \cosh \hat \eta}
          \sqrt{1 - \frac{4 m_\chi^2}{s}}
        \right)
      }^{\frac{\sqrt{s}}{2} \sqrt{1 - \frac{4 m_\chi^2}{s}}}
        \mathrm{d} p_T
        \left[
          \frac{\mathrm{d} \sigma_\parallel(\gamma \gamma \to \chi^+ \chi^-)}
               {\mathrm{d} p_T}
          \frac{\mathrm{d} \hat L^\parallel}{\mathrm{d} s}
          + 
          \frac{\mathrm{d} \sigma_\perp(\gamma \gamma \to \chi^+ \chi^-)}
               {\mathrm{d} p_T}
          \frac{\mathrm{d} \hat L^\perp}{\mathrm{d} s}
        \right]
  \end{multline*}
  Differential photon-photon cross sections:
  \begin{equation*}
    \begin{aligned}
      \frac{\mathrm{d} \sigma_\parallel(\gamma \gamma \to \chi^+ \chi^-)}{\mathrm{d} p_T}
    & = \frac{8 \pi \alpha^2 p_T}{s (p_T^2 + m_\chi^2)}
        \cdot
        \frac{1 - \dfrac{2 (p_T^4 \textcolor{red}{{} + 2 m_\chi^4})}{s (p_T^2 + m_\chi^2)}}
             {\sqrt{1 - \dfrac{4 (p_T^2 + m_\chi^2)}{s}}}
    \\
      \frac{\mathrm{d} \sigma_\perp(\gamma \gamma \to \chi^+ \chi^-)}{\mathrm{d} p_T}
    & = \frac{8 \pi \alpha^2 p_T}{s (p_T^2 + m_\chi^2)}
        \cdot
        \frac{1 - \dfrac{2 p_T^4}{s (p_T^2 + m_\chi^2)}}
             {\sqrt{1 - \dfrac{4 (p_T^2 + m_\chi^2)}{s}}}
    \end{aligned}
  \end{equation*}
\end{frame}

\begin{frame}{Fiducial cross section}
  Fiducial luminosity:
  \begin{equation*}
    \begin{aligned}
      \frac{\mathrm{d} \hat L_\parallel}{\mathrm{d} s}
    & = \frac14
        \iint \mathrm{d}^2 b_1 \, \mathrm{d}^2 b_2
        \, P(b)
        \int\limits_{\textcolor{magenta}{-\hat y}}^{\textcolor{magenta}{\hat y}}
        \mathrm{d} y
        \, n \left( b_1, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^y \right)
        \, n \left( b_2, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^{-y} \right)
        \, \cos^2 \varphi
    \\
      \frac{\mathrm{d} \hat L_\perp}{\mathrm{d} s}
    & = \frac14
        \iint \mathrm{d}^2 b_1 \, \mathrm{d}^2 b_2
        \, P(b)
        \int\limits_{\textcolor{magenta}{-\hat y}}^{\textcolor{magenta}{\hat y}}
        \mathrm{d} y
        \, n \left( b_1, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^y \right)
        \, n \left( b_2, \tfrac{\sqrt{s}}{2} \, \mathrm{e}^{-y} \right)
        \, \sin^2 \varphi
    \end{aligned}
  \end{equation*}
  Rapidity cut:
  \begin{equation*}
    \hat y = \asinh \left[
      \frac{\sqrt{s} p_T}{2 (p_T^2 + m_\chi^2)}
      \left(
        \sinh \hat \eta
        - \sqrt{\cosh^2 \hat \eta + \frac{m_\chi^2}{p_T^2}}
    \cdot \sqrt{1 - \frac{4 (p_T^2 + m_\chi^2)}{s}}
      \right)
    \right]
  \end{equation*}
\end{frame}

\end{document}
