High energy spectrum of internal positrons from radiative muon capture on nuclei

The Mu2e and COMET collaborations will search for nucleus-catalyzed muon conversion to positrons (${\mu }^{-}\to e^{+}$) as a signal of lepton number violation. A key background for this search is radiative muon capture where either (1) a real photon converts to an $e^{+}{e}^{-}$ pair “externally” in surrounding material, or (2) a virtual photon mediates the production of an $e^{+}{e}^{-}$ pair “internally.” If the $e^{+}$ has an energy approaching the signal region then it can serve as an irreducible background. In this work we describe how the near end point internal positron spectrum can be related to the real photon spectrum from the same nucleus, which encodes all nontrivial nuclear physics.

Figure 1

Radiative muon capture on a nucleus (double lines) resulting in (a) a real photon and (b) a virtual photon that mediates the production of an electron positron pair. In this work we study how the $e^{+}$ (or $e^{-}$) spectrum in (b) is related to the photon spectrum in (a).

Figure 2

Spectrum of positrons/electrons as compared to real photons near the end point assuming that the photon spectrum is given by $\mathrm{d}\mathrm{\Gamma }/\mathrm{d}{E}_{\gamma }\propto (1-2x+2x^2)x(1-x{)}^2$ with $x=E_{\gamma }/q_{\max }$ and $q_{\max }=100\mathrm{MeV}$. This corresponds to the closure approximation spectrum used in the analysis of RMC data on various nuclear isotopes in [20]. The electron/positron spectrum is computed using Eqs. (36) and (44) below. The spectrum is softened relative to photons because (1) the end point is shifted by $m_e=0.511\mathrm{MeV}$, and (2) the probability of producing an $e^{+}{e}^{-}$ pair rises sharply as one moves further from the end point as shown in Fig. 4. Errors due to virtual-photon nuclear matrix elements (gray band) are estimated by treating $C_L$ and $C_T$ (see Appendix pp1) as independent random variables drawn from a Gaussian distribution with unit variance.

Figure 3

Phase space for radiative muon capture. Varying $\cos \vartheta$ from $-1$ to 1 at fixed $m_{*}$ moves from the bottom of the hull to the top of the hull. Changing $E_{\gamma }=q_0$ changes the size of the hull to be integrated over (i.e., orange dotted curve [$E_{\gamma }=85\mathrm{MeV}$] $\to$ blue dashed curve [$E_{\gamma }=100\mathrm{MeV}$]). If $E_{+}\to q_0$ then only the top tip of the hull contributes to the phase space integration. Units of $E_{+}$ and $m_{*}$ are in MeV.

Figure 4

Probability per differential bin (i.e., of width $\mathrm{d}{E}_{\gamma }$) of producing a positron with energy $E_{+}$ given a photon energy of $E_{\gamma }=100\mathrm{MeV}$, Eqs. (39) and (44). We plot $P_{\mathrm{int}}^{(0)}$ of Eq. (43) as well as error estimates (gray band) obtained by considering an ensemble of the two parameters $C_L$, and $C_T$, that capture nuclear structure dependence (see discussion in Appendix pp1). We treat $C_L$ and $C_T$ as random variables drawn from independent Gaussian distributions such that $⟨C_L^2⟩=⟨C_T^2⟩=1$.