---
abstract: 'A study is presented to extend the measurements of photon-photon scattering in ultra-peripheral Pb-Pb collisions at the LHC into the mass region of the pseudoscalar resonances $\eta$ and $\eta''$. The elementary photon-photon scattering cross section is presented. The cross section for photon-photon scattering in Pb-Pb is derived by convoluting the elementary photon-photon cross section with the Pb-Pb photon luminosity. The main background to two-photon final states, arising from double $\pi^{0}$ production with two of the four decay photons escaping detection, is examined, and possible kinematical conditions are discussed to optimize the signal-to-background ratio for such measurements at mid-rapidity.'
address:
- 'Phys. Inst., Heidelberg'
- 'Inst. Nucl. Phys., PAN Krak$\acute{o}$w, Univ. of Rzesz$\acute{o}$w'
author:
- |
    Rainer Schicker\
    [Mariola Kłusek-Gawenda, Antoni Szczurek]{}
title: 'Photon-photon scattering in the resonance region at midrapidity at the LHC. [^1]'
---

Introduction
============

Classical electrodynamics is epitomised by Maxwell’s equations $$\partial_{\alpha}F^{\alpha\beta} = \frac{4\pi}{c}J^{\beta},\hspace{1.cm}
  \partial_{\alpha}{\cal F}^{\alpha\beta} = 0 \, .
\label{Eq:Maxwell}$$

The Maxwell equations in vacuum are linear in the electromagnetic fields [**E**]{} and [**B**]{}. Two electromagnetic waves will pass through each other without scattering. The superposition principle conveniently expresses the non-interaction of electromagnetic fields at the classical level. The electromagnetic field energy carried by the electron, however, poses a conceptual challenge at the classical level. Attempts to circumvent the infinite Coulomb energy of a point charge resulted in Born-Infeld electrodynamics which affirms the linearity of Maxwell’s equations down to some length scale intrinsic to the electron, and introduces non-linear equations for smaller lengths scales [@Born]. The polarisation of the vacuum in view of Dirac’s positron theory led to the Euler-Kockel-Heisenberg Lagrangian which modifies the classical Maxwell’s equation in vacuum by leading non-linear terms [@Heisenberg]. The advent of accelerating heavy-ions at the LHC has opened up the possibility of measuring the photon-photon scattering cross section due to the large associated photon luminosity of the heavy-ion beams. Evidence of such events have been reported by the ATLAS and CMS collaborations at the LHC [@ATLAS; @CMS]. These measurements are, however, restricted to photon-photon invariant masses W$_{\gamma\gamma} >$ 5 and 6 GeV for the CMS and ATLAS data, respectively. The purpose of the analysis presented here is to study the feasibility of measuring photon-photon scattering in the range 0.4$<$W$_{\gamma\gamma}<$5 GeV. As a first step, we analyse the corresponding cross section in this range, and examine the background which is dominated by $\pi^{0}$ pair production with only two of the four decay photons being within the detector acceptance.

The elementary photon-photon scattering cross section
=====================================================

Different mechanisms contribute to the elementary $\gamma\gamma\rightarrow\gamma\gamma$ scattering.

![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_box.pdf "fig:"){width="3.2cm"} ![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_gluon.pdf "fig:"){width="4.4cm"} ![Mechanisms of $\gamma\gamma\rightarrow\gamma\gamma$ scattering (Fig. taken from Ref. [@Sz1]).[]{data-label="Fig:gg_elem"}](gg_gg_IPIR.pdf "fig:"){width="4.4cm"}

In Fig. \[Fig:gg\_elem\], the different mechanisms of $\gamma\gamma$ scattering are presented. On the left, the loop diagram for fermions, leptons and quarks, is shown. In the center, a QCD correction is displayed corresponding to a three-loop mechanism. On the right, the analogous process as expressed in the vector dominance approach is shown [@Sz1].

Photon-photon scattering in ultra-peripheral heavy-ion reactions
================================================================

The cross section for photon-photon scattering in ultra-peripheral heavy-ion collisions can be calculated by folding the cross section of the elementary mechanisms shown in Fig. \[Fig:gg\_elem\] with the equivalent photon flux [@Baur], $$N(\omega,b) = \frac{Z^{2}\alpha}{\pi^{2}}
\frac{1}{\beta^{2}b^{2}}
\bigg| \int_0^{\infty} dv\; v^{2} J_{1}(v)\frac{F_{el}(-\frac{u^{2}+v^{2}}{b^{2}})}{u^{2}+v^{2}}\bigg|^{2} \, .
\label{Eq:EPA}$$

The equivalent photon flux is defined in Eq. \[Eq:EPA\]. Here, $\omega$ denotes the energy of the photon, and b represents the transverse distance from the center of the nucleus where the photon density is evaluated. The integral in Eq. \[Eq:EPA\] represents the form factor of the charge distribution of the source.

![Differential cross section d$\sigma$/dM for photon scattering in PbPb-collisions.[]{data-label="Fig:dsigma1"}](dsigmadM_signal.png){width="6.5cm"}

The cross section for $\gamma\gamma\rightarrow\gamma\gamma$ in PbPb-collisions is calculated by convoluting the elementary cross section with the photon flux ,

$$\sigma^{EPA}_{PbPb \rightarrow PbPb\gamma\gamma} = \int\!\!\int dn^{1}_{\gamma} 
\; dn^{2}_{\gamma} \; \sigma_{\gamma\gamma \rightarrow \gamma\gamma}(\omega_1,\omega_2) \, .
\label{Eq:dsigmadM_PbPb}$$

The differential cross section for photon-photon scattering in PbPb-collisions at $\sqrt{s}$=5.02 TeV resulting from the convolution of Eq. \[Eq:dsigmadM\_PbPb\] is shown in Fig. \[Fig:dsigma1\]. This cross section is derived with the box diagrams of Fig. \[Fig:gg\_elem\], and with conditions of the two final state photons being within the pseudorapidity range $|\eta|<$0.9, and having an energy E$_{phot} >$200 MeV.

Resonance signal from $\eta,\eta'$ decays
=========================================

The cross section for photoproduction of $\eta,\eta'$ is taken according to [@Budnev] $$\sigma_{\gamma\gamma\rightarrow R} = 8\pi(2J+1)\frac{\Gamma_{\gamma\gamma}\Gamma_{tot}}{(W^{2}-M^{2}_{R})^{2}+M^{2}_{R}\Gamma^{2}_{tot}} \, .
  \label{Eq:sigma_res}$$

The cross section for photoproduction of $\eta,\eta'$ in PbPb-collisions is calculated according to the convolution defined in Eq. \[Eq:dsigmadM\_PbPb\].

![Cross section for photoproduction of $\eta,\eta'.$[]{data-label="Fig:dsigmadM_eta"}](dsigmadM_signaleta.png){width="6.5cm"}

The $\eta,\eta'$ cross section in PbPb-collisions at $\sqrt{s}$= 5.02 TeV multiplied by the branching ratio $\eta,\eta'\rightarrow\gamma\gamma$ is shown in Fig. \[Fig:dsigmadM\_eta\]. The values shown are derived with conditions of the two decay photons being within the pseudorapidity range $|\eta|<$0.9, and having an energy E$_{phot} >$200 MeV. The finite width of these two resonances results from the detector resolution of the photon measurements which is taken here to be $\sigma_{E_{phot}}/E$ = 0.02.

Photoproduction of $\pi^{0}\pi^{0}$ pairs
=========================================

The main background to the signal of two photons in the final state results from $\pi^{0}\pi^{0}$ production with two of the four decay photons escaping detection. The two measured photons from this $\pi^{0}\pi^{0}$ decay cannot be distinguished from the signal of photon-photon scattering.

![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](pires.pdf "fig:"){width="4.cm"} ![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](picoupl1.pdf "fig:"){width="4.cm"} ![Mechanisms of $\pi^{0}\pi^{0}$ photoproduction (Fig. taken from [@Sz2]).[]{data-label="Fig:pion_pairs"}](picoupl2.pdf "fig:"){width="4.cm"}

The contribution of resonance decays to the $\pi^{0}\pi^{0}$ final state is shown in Fig. \[Fig:pion\_pairs\] on the left. Here, the resonances $\sigma$(600), $f_{0}$(980), $f_{0}$(1500), $f_{0}$(1710), $f_{2}$(1270), $f^{`}_{2}$(1525), $f_{2}$(1565), $f_{2}$(1950) and $f_{4}$(2050) are considered [@Sz2]. The cross section $\gamma\gamma\rightarrow\pi^{+}\pi^{-}$ is much larger than for $\gamma\gamma\rightarrow\pi^{0}\pi^{0}$, hence even a small coupling between the charged and neutral pion channel might have an influence on the $\gamma\gamma\rightarrow\pi^{0}\pi^{0}$ cross section. Examples of processes leading to such channel couplings are shown in Fig. \[Fig:pion\_pairs\] in the middle and on the right.

Background from $\pi^{0}\pi^{0}$ decays
=======================================

The $\pi^{0}\pi^{0}$ cross section in PbPb-collisions is calculated the elementary $\pi^{0}\pi^{0}$ cross section with the photon flux according to Eq. \[Eq:EPA\]. The background from $\pi^{0}\pi^{0}$ decays is shown in Fig. \[Fig:bck\_pi0\] by the solid red line. This background cross section is derived by the conditions that exactly one decay photon from each of the two $\pi^{0}$’s is within the pseudorapidity range $|\eta| <$0.9, and that these photons have an energy E$_{phot} >$200 MeV.

![Signal and background from $\pi^{0}\pi^{0}$ decays.[]{data-label="Fig:bck_pi0"}](dsigmadM_signaletabck.png){width="6.5cm"}

Background suppression by asymmetry cuts
----------------------------------------

The two photons of the signal are of equal transverse momentum and are back-to-back in azimuth. These correlations are smeared out due to finite resolution in the measurement of photon energy and azimuthal angle. The two photons from $\pi^{0}\pi^{0}$ decay do not show these correlations. and vector ($A_{V}$) asymmetry can be defined for background suppression, $$A_S=\left|\frac{|\vec{p}_T(1)|-|\vec{p}_T(2)|}
  {|\vec{p}_T(1)|+|\vec{p}_T(2)|}\right|,\hspace{1.cm}
A_V=\frac{|\vec{p}_T(1)-\vec{p}_T(2)|}
{|\vec{p}_T(1)+\vec{p}_T(2)|} \, .$$

![Background suppression by asymmetry cut A$_{S}$.[]{data-label="Fig:bck_pi0_asym"}](dsigmadM_sigbck.png){width="6.5cm"}

The background reduction for condition A$_{S}\!<$0.1 and A$_{S}\!<$0.02 is shown in Fig. \[Fig:bck\_pi0\_asym\] by the blue and red line, respectively. The condition A$_{S}<$0.02 reduces the background by about a factor $\sim$10 while keeping 98%

Background correction by sideband subtraction
---------------------------------------------

The background remaining after the cut A$_{S}\!<\!0.02$ can be subtracted by sideband correction. The signal band is given by with sideband 1 and 2 defined by $0.02\!<\!A_{s}\!<\!0.04$ and $0.04\!<\!A_{s}\!<\!0.06$, respectively. An estimator for the background in the signal region can be defined by linear extrapolation of the sidebands 1 and 2 into the signal region.

![Sideband corrected background.[]{data-label="Fig:bck_pi0_asym_corr"}](dsigmadM_sidecorr.png){width="6.5cm"}

The background remaining after sideband correction is shown by the red line. The sideband correction could be further improved by defining more than two sidebands, and by a non-linear extrapolation of the background into the signal region.

ACKNOWLEDGMENTS
===============

This work is supported by the German Federal Ministry of Education and Research under promotional reference 05P15VHCA1.

[99]{} M.Born, Nature 132:3329 (1933) 282. W.Heisenberg, H.Euler, Zeitschrift für Physik 98: 11-12 (1936) 714. ATLAS Collaboration, Nature Phys. 13 (2017) no.9, 852. CMS Collaboration, arXiv:1810.04602. M.Kłusek-Gawenda et al., Phys.Rev. C93 (2016), no.4, 044907. G.Baur et al., Phys.Rept. 364 (2002) 359. V.M.Budnev, $\!$I.F.Ginzburg, $\!$G.V.Meledin, $\!$V.G.Serbo, $\!$Phys.Rept.15, $\!$(1975) $\!$181. M.Kłusek-Gawenda, A.Szczurek, Phys.Rev. C87 (2013), no.5, 054908.

[^1]: Presented at Diffraction and Low-x Conference 2018, Reggio Calabria,
