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The philosophy of our ``model-exhaustive'' analysis. Traditional model-independent analyses express the new physics contribution to $\g$ as a non-renormalizable operator, either in the low-energy theory after EW symmetry breaking (left) or in the full SM gauge invariant formulation (middle). This makes no assumptions about the new physics but is limited to indirect signatures of the new physics produced by the same operator. Since we want to probe direct signatures of the BSM physics which solves the $\g$ anomaly, we add the single assumption of perturbativity to the traditional model-independent analysis, which resolves the new $\deltaa$ contributions into explicit loop diagrams of new states $\{\psi_i \}$ carrying specific SM quantum numbers (right). If the Higgs insertion lies on the external muon, $\Delta a_\mu$ is suppressed by $y_\mu$, while $\deltaa$ can be significantly enhanced if the Higgs couples to new particles in the loop. By exhaustively analyzing all possible choices of new states, we can derive predictions for direct signatures that are as universal as the traditional model-independent predictions for indirect signatures.

Schematic representation of the model-exhaustive space of BSM theories that can solve the $\g$ anomaly, and our mutually exclusive and collectively exhaustive categorization into Singlet Scenarios and Electroweak Scenarios. For these two classes of theories, the phenomenological questions are distinct. To understand how to discover Singlet Scenarios, we have to not only find the heaviest possible mass of the singlet(s), but also how to discover this singlet for all possible masses, since its phenomenology depends on its stability and decay mode, and lighter singlets have weaker coupling. Electroweak Scenarios predict new charged states, and since those have to produce visible final states in a collider and are efficiently produced at lepton colliders for $m \lesssim \sqrt{s}/2$, we only have to find the maximum mass the lightest new charged state in the BSM theory can have. (We limit ourselves to scenarios that generate $\deltaaexp$ at one-loop, since higher-loop solutions have lower BSM mass scales.)

Representative 1-loop contributions to $\g$ in the simplified models we consider. Top row: Singlet Scenarios with a SM neutral vector $V$ or scalar $S$ that couple to the muon. Note that the Higgs VEV on the muon line gives both the chirality flip and the EW breaking insertions in these models. Bottom left: EW Scenario of SSF type, with one BSM fermion and two BSM scalars that mix via a Higgs insertion. Bottom right: EW Scenario of FFS type, with one BSM scalar and two BSM fermions that mix via a Higgs insertion.

The coupling of the singlet scalar ($g_S$) and vector ($g_V$) required to account for the $\g$ anomaly as a function of its mass $m_{S, V}$ and multiplicity. For $N_{\rm BSM}=1$, perturbative unitarity imposes $g_S\leq3.5$ and $g_V\leq6.1$, which implies an upper bound on the masses needed for $\g$ of $m_s\leq2.7\tev$ and $m_V\leq1.1\tev$, respectively. If one imposes MFV in the scalar couplings, the upper bounds for scalars become $(g_S,m_s) \leq (0.2,155\gev)$. Note that the $N_{\rm BSM}$-dependence of the singlet mass drops out by requiring $\deltaa = \deltaaexp$.

Single production of the singlet in association with a photon at a muon collider. The singlets can be stable and constitute missing energy, or decay to any SM final states. The search is defined by the search for the recoiling photon, as well as any possible SM final states (including missing energy) inside the singlet decay cone.

Luminosity needed for $5\sigma$ discovery significance of inclusive Singlet Scenario searches at a 215 GeV and 3 TeV muon collider for singlet scalars (green) and singlet vectors (orange). This is shown for singlet masses up to the perturbativity limit calculated in Section \ref{singlet-unitarity}. Dashed lines (solid lines) show the results from the inclusive direct $\gamma+X$ analysis (Bhabha scattering analysis). Note that these sensitivities do not depend on $N_{\rm BSM}$.

Luminosity needed for $5\sigma$ discovery significance of inclusive Singlet Scenario searches at a 215 GeV and 3 TeV muon collider for singlet scalars (green) and singlet vectors (orange). This is shown for singlet masses up to the perturbativity limit calculated in Section \ref{singlet-unitarity}. Dashed lines (solid lines) show the results from the inclusive direct $\gamma+X$ analysis (Bhabha scattering analysis). Note that these sensitivities do not depend on $N_{\rm BSM}$.

Feynman diagrams for Bhabha scattering in the SM (top) and contributions from singlet scalars or vectors (bottom). (Note that the arrows in this diagram represent charge flow, not helicity.)

Feynman diagrams for Bhabha scattering in the SM (top) and contributions from singlet scalars or vectors (bottom). (Note that the arrows in this diagram represent charge flow, not helicity.)

Feynman diagrams for Bhabha scattering in the SM (top) and contributions from singlet scalars or vectors (bottom). (Note that the arrows in this diagram represent charge flow, not helicity.)

Feynman diagrams for Bhabha scattering in the SM (top) and contributions from singlet scalars or vectors (bottom). (Note that the arrows in this diagram represent charge flow, not helicity.)

Prediction for the forward-backward asymmetry variable $r_{\rm FB}$ in Bhabha scattering for Singlet Scenarios at a 215 GeV and 3 TeV MuC. This is independent of $N_{\rm BSM}$.

Prediction for the forward-backward asymmetry variable $r_{\rm FB}$ in Bhabha scattering for Singlet Scenarios at a 215 GeV and 3 TeV MuC. This is independent of $N_{\rm BSM}$.

Contours show mass in TeV of lightest charged state in two representative SSF models with $N_{\rm BSM} = 1$ as a function of scalar masses $m_A, m_B$. The largest possible fermion mass $m_F$ was determined by $\Delta a^\mathrm{BSM} = \deltaaexp$, with the couplings $y_1, y_2, \kappa$ chosen to maximize $\g$ while obeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity + naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, $(R, R^A, R^B) = (1_{-2}, 2_{3/2}, 1_1)$, and all fields contributing to $\g$ are charged. On the right, $(R, R^A, R^B) = (1_{-1}, 2_{1/2}, 1_0)$, and the scalars in the $\g$ loop are neutral but since $\Phi_A$ is an EW doublet, there is a charged scalar with mass $m_A$.

Contours show mass in TeV of lightest charged state in two representative SSF models with $N_{\rm BSM} = 1$ as a function of scalar masses $m_A, m_B$. The largest possible fermion mass $m_F$ was determined by $\Delta a^\mathrm{BSM} = \deltaaexp$, with the couplings $y_1, y_2, \kappa$ chosen to maximize $\g$ while obeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity + naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, $(R, R^A, R^B) = (1_{-2}, 2_{3/2}, 1_1)$, and all fields contributing to $\g$ are charged. On the right, $(R, R^A, R^B) = (1_{-1}, 2_{1/2}, 1_0)$, and the scalars in the $\g$ loop are neutral but since $\Phi_A$ is an EW doublet, there is a charged scalar with mass $m_A$.

Contours show mass in TeV of lightest charged state in two representative SSF models with $N_{\rm BSM} = 1$ as a function of scalar masses $m_A, m_B$. The largest possible fermion mass $m_F$ was determined by $\Delta a^\mathrm{BSM} = \deltaaexp$, with the couplings $y_1, y_2, \kappa$ chosen to maximize $\g$ while obeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity + naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, $(R, R^A, R^B) = (1_{-2}, 2_{3/2}, 1_1)$, and all fields contributing to $\g$ are charged. On the right, $(R, R^A, R^B) = (1_{-1}, 2_{1/2}, 1_0)$, and the scalars in the $\g$ loop are neutral but since $\Phi_A$ is an EW doublet, there is a charged scalar with mass $m_A$.

Contours show mass in TeV of lightest charged state in two representative SSF models with $N_{\rm BSM} = 1$ as a function of scalar masses $m_A, m_B$. The largest possible fermion mass $m_F$ was determined by $\Delta a^\mathrm{BSM} = \deltaaexp$, with the couplings $y_1, y_2, \kappa$ chosen to maximize $\g$ while obeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity + naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, $(R, R^A, R^B) = (1_{-2}, 2_{3/2}, 1_1)$, and all fields contributing to $\g$ are charged. On the right, $(R, R^A, R^B) = (1_{-1}, 2_{1/2}, 1_0)$, and the scalars in the $\g$ loop are neutral but since $\Phi_A$ is an EW doublet, there is a charged scalar with mass $m_A$.

Contours show mass in TeV of lightest charged state in two representative SSF models with $N_{\rm BSM} = 1$ as a function of scalar masses $m_A, m_B$. The largest possible fermion mass $m_F$ was determined by $\Delta a^\mathrm{BSM} = \deltaaexp$, with the couplings $y_1, y_2, \kappa$ chosen to maximize $\g$ while obeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity + naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, $(R, R^A, R^B) = (1_{-2}, 2_{3/2}, 1_1)$, and all fields contributing to $\g$ are charged. On the right, $(R, R^A, R^B) = (1_{-1}, 2_{1/2}, 1_0)$, and the scalars in the $\g$ loop are neutral but since $\Phi_A$ is an EW doublet, there is a charged scalar with mass $m_A$.

Contours show mass in TeV of lightest charged state in two representative SSF models with $N_{\rm BSM} = 1$ as a function of scalar masses $m_A, m_B$. The largest possible fermion mass $m_F$ was determined by $\Delta a^\mathrm{BSM} = \deltaaexp$, with the couplings $y_1, y_2, \kappa$ chosen to maximize $\g$ while obeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity + naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, $(R, R^A, R^B) = (1_{-2}, 2_{3/2}, 1_1)$, and all fields contributing to $\g$ are charged. On the right, $(R, R^A, R^B) = (1_{-1}, 2_{1/2}, 1_0)$, and the scalars in the $\g$ loop are neutral but since $\Phi_A$ is an EW doublet, there is a charged scalar with mass $m_A$.

Contours show mass in TeV of lightest charged state in two representative SSF models with $N_{\rm BSM} = 1$ as a function of scalar masses $m_A, m_B$. The largest possible fermion mass $m_F$ was determined by $\Delta a^\mathrm{BSM} = \deltaaexp$, with the couplings $y_1, y_2, \kappa$ chosen to maximize $\g$ while obeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity + naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, $(R, R^A, R^B) = (1_{-2}, 2_{3/2}, 1_1)$, and all fields contributing to $\g$ are charged. On the right, $(R, R^A, R^B) = (1_{-1}, 2_{1/2}, 1_0)$, and the scalars in the $\g$ loop are neutral but since $\Phi_A$ is an EW doublet, there is a charged scalar with mass $m_A$.

Contours show mass in TeV of lightest charged state in two representative SSF models with $N_{\rm BSM} = 1$ as a function of scalar masses $m_A, m_B$. The largest possible fermion mass $m_F$ was determined by $\Delta a^\mathrm{BSM} = \deltaaexp$, with the couplings $y_1, y_2, \kappa$ chosen to maximize $\g$ while obeying the constraint from perturbative unitarity (1st row), unitarity + MFV (2nd row), unitarity + naturalness (3rd row) or unitarity + naturalness + MFV (4th row) On the left, $(R, R^A, R^B) = (1_{-2}, 2_{3/2}, 1_1)$, and all fields contributing to $\g$ are charged. On the right, $(R, R^A, R^B) = (1_{-1}, 2_{1/2}, 1_0)$, and the scalars in the $\g$ loop are neutral but since $\Phi_A$ is an EW doublet, there is a charged scalar with mass $m_A$.