Light-by-light scattering: asymptotic expansions, Coulomb resummation and NLO corrections

Abstract

Light-by-light (LbL) scattering is one of the earliest predictions of quantum electrodynamics (QED). Interest in this process has been renewed following its experimental observation at the LHC and the prospects of future measurements at free-electron laser facilities. In this paper, we refine theoretical predictions for LbL scattering by improving the full fermion-mass-dependent two-loop QCD and QED helicity amplitudes using high- and low-energy asymptotic expansions, and by performing Coulomb resummation in the threshold region. We present state-of-the-art predictions for LbL cross sections in the Standard Model and provide a new event generator, LbLatNLO, for Monte Carlo simulations of LbL scattering.

Figures (12)

• Figure 1: The real parts of all-plus (upper left), single-minus (upper right), and two-minus-two-plus (lower) two-loop QCD amplitudes $i\mathcal{M}_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(1,0,f)}$ as functions of $s/m_f^2$ in the LE region. Exact results obtained with double precision (open diamonds) and quadruple precision (filled circles) are compared with the LE expansion truncated at $\mathcal{O}(m_f^{-4})$ (black curves), $\mathcal{O}(m_f^{-6})$ (red curves), $\mathcal{O}(m_f^{-8})$ (green curves), and $\mathcal{O}(m_f^{-10})$ (blue curves). The scattering angle is fixed at $t/s = -0.3$. The lower panels show the ratios with respect to the LE result truncated at $\mathcal{O}(m_f^{-10})$.
• Figure 2: All-plus (left) and single-minus (right) two-loop QCD amplitudes $i\mathcal{M}_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(1,0,f)}$ as functions of $s/m_f^2$ in the range $1<s/m_f^2<10^5$. The real (upper panels) and imaginary (lower panels) parts are shown. Exact results obtained in double precision (open diamonds) and quadruple precision (filled circles) are compared with the HE expansion truncated at $\mathcal{O}(m_f^0)$ (black curves), $\mathcal{O}(m_f^2)$ (red curves), $\mathcal{O}(m_f^4)$ (green curves), and $\mathcal{O}(m_f^{14})$ (blue curves). The scattering angle is fixed at $t/s=-0.3$. The bottom panels display the ratios with respect to the exact quadruple-precision result.
• Figure 3: Two-minus-two-plus two-loop QCD amplitudes $i\mathcal{M}_{--++}^{(1,0,f)}$ (left), $i\mathcal{M}_{+-+-}^{(1,0,f)}$ (middle), and $i\mathcal{M}_{+--+}^{(1,0,f)}$ (right) as functions of $s/m_f^2$ in the range $1<s/m_f^2<10^5$. The real (upper panels) and imaginary (lower panels) parts are shown. Exact results obtained in double precision (open diamonds) and quadruple precision (filled circles) are compared with the HE expansion truncated at $\mathcal{O}(m_f^0)$ (black curves), $\mathcal{O}(m_f^2)$ (red curves), $\mathcal{O}(m_f^4)$ (green curves), and $\mathcal{O}(m_f^{14})$ (blue curves). The scattering angle is fixed at $t/s=-0.3$. The bottom panels display the ratios with respect to the exact quadruple-precision result.
• Figure 4: All-plus (black) and two-minus–two-plus (blue) two-loop QCD amplitudes $i\mathcal{M}_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(1,0,f)}$ (points), their Coulomb-approximated counterparts $i\mathcal{M}_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(1,0,f),\mathrm{Coul~approx}}$ (long dashed), and the LP Coulomb-resummation-improved amplitudes $i\mathcal{M}_{\lambda_1\lambda_2\lambda_3\lambda_4}^{(1,0,f),\mathrm{LP}}$, shown as functions of $s/m_f^2$. From left to right, the panels display the real part of the amplitudes for $s<4m_f^2$ (left) and $s>4m_f^2$ (middle), and the imaginary part for $s>4m_f^2$ (right). The scattering angle is fixed at $t/s=-0.3$. The lower panels show the ratio of the Coulomb approximation to the full two-loop amplitude.
• Figure 5: The partonic LbL cross sections induced by a top-quark loop at LO (black), NLO QCD (red), NLO$^\prime$ QCD (green), NLO+LP QCD (blue), and (NLO+LP)$^\prime$ QCD (magenta). Both the below-threshold region ($\sqrt{s}<2m_f$, left) and the above-threshold region ($\sqrt{s}>2m_f$, right) are shown. The lower panels display the corresponding $K$ factors.