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$π^+ π^-$ and $π^0 π^0$ pair production in photon-photon and in ultraperipheral ultrarelativistic heavy ion collisions

We discuss $γγ\to ππ$ reactions, starting from the two-pion threshold up to about $\sqrt{s_{γγ}}$ = 6 GeV. We include the dipion continuum due to pion exchange (for $π^+π^-$) and $ρ^\pm$ exchange (for $π^0π^0$) as well as the dipion $σ(600)$, $f_0(980)$ and $f_2(1270)$ s-channel resonances. We discuss also a possible contribution of less pronounced scalar resonances $f_0(1500)$ and $f_0(1710)$ as well as the tensor resonances $f'_2(1525)$, $f_2(1565)$ and $f_2(1950)$. We find that the inclusion of the spin-four $f_4(2050)$ resonant state improves the situation at $\sqrt{s_{γγ}} \approx$ 2 GeV. We estimate the relevant diphoton partial decay width to be $Γ_{f_4 \to γγ} \approx$ 0.7 keV. At higher energies, Brodsky-Lepage and hand-bag mechanisms are included in addition. We nicely describe the world data for $γγ\to ππ$ for the first time both for the total cross section and angular distributions. The cross section for the production of two-pions in ultraperipheral ultrarelativistic heavy ion collisions is calculated in the impact parameter space Equivalent Photon Approximation (EPA). We obtain the nuclear cross section of 46.7 mb for $π^+π^-$ and 8.7 mb for $π^0π^0$ at the LHC energy $\sqrt{s_{NN}} =$ 3.5 TeV and minimal cut on pion transverse momentum $p_t>$ 0.2 GeV. Differential distributions in impact parameter, dipion invariant mass, single pion and dipion rapidity, pion transverse momentum as well as pion pseudorapidity are shown. The $γγ\to π^+ π^-$ subprocess constitutes a background to $A A \to A ρ^0 (\to π^+ π^-) A$ process.