Why can’t BSM particles enter at tree level and why can they only exist in loops?

The reason why Beyond the Standard Model (BSM) particles typically do not appear at tree level (directly in the initial or final states of a scattering process) and instead manifest primarily in loops (appearing virtually in intermediate states) is closely tied to the structure of the Standard Model (SM) and the nature of quantum field theory.

1. Conservation Laws and Symmetries: In the SM, certain conservation laws and symmetries (like gauge symmetries) constrain the possible interactions and particle content at tree level. For example, tree-level processes involving the exchange of known gauge bosons (like photons, W, Z bosons) must respect the underlying gauge symmetry of the SM. Introducing new BSM particles directly into these tree-level processes can potentially violate these symmetries or conservation laws.
2. Mass and Energy Constraints: BSM particles often have masses and properties that differ significantly from the SM particles. This can affect the kinematics and energy-momentum conservation in tree-level processes. If a BSM particle is much heavier than the particles involved in the interaction, direct production or decay at tree level may be kinematically forbidden or highly suppressed.
3. Loop Corrections and Quantum Fluctuations: BSM particles typically interact with SM particles through quantum loop processes. In these loops, BSM particles can be temporarily created as virtual particles, influencing the behavior of SM particles indirectly. The existence of BSM particles as virtual states in loops can lead to measurable effects through radiative corrections or higher-order processes.
4. Gauge Boson Propagators and Off-Shell Particles: When a gauge boson (like the W boson) participates in a loop, it can indeed be off-shell (not on its mass shell, i.e., not satisfying the energy-momentum relation of an on-shell particle). However, the context of off-shellness in loops versus tree-level processes is different:
   • In tree-level processes, the conservation laws and interactions are more directly dictated by the on-shell properties of particles involved.
   • In loop processes, the virtual nature of particles allows for more flexibility in energy-momentum conservation, permitting the appearance of BSM particles as fluctuations in the quantum vacuum.
5. Quantum Corrections and Precision Measurements: The presence of BSM particles in loops can often manifest as subtle quantum corrections to SM predictions, such as modifications to particle masses, interaction strengths, or decay rates. Observing these loop effects through precision measurements can provide indirect evidence for the existence of BSM physics beyond what the SM predicts at tree level.

In summary, while tree-level processes in the SM are governed by strict conservation laws and symmetries that typically exclude direct participation of BSM particles, quantum loop processes offer a more nuanced and flexible framework where BSM effects can manifest indirectly, affecting measurable observables through quantum corrections and fluctuations in the quantum field theory framework.