Supersymmetry in a Multiverse Originating from a Wormhole Collapse

Developing new equations for supersymmetry (SUSY) in a universe with the unique conditions you've described—where the universe is one of an infinite number, originating from the collapse of a wormhole that injected baryonic matter into a dark matter universe—requires us to integrate several advanced concepts from theoretical physics. This scenario introduces novel dynamics involving both baryonic and dark matter, the geometry of wormholes, and the multiverse hypothesis.

Factors to Consider in Developing New Equations

To develop these new equations, we'll need to consider the following factors:

1. Supersymmetry Basics: SUSY posits that every particle in the Standard Model has a superpartner with different spin properties. These equations typically involve relations between bosons (integer-spin particles) and fermions (half-integer-spin particles).
2. Baryonic and Dark Matter Interaction: In your scenario, the injection of baryonic matter into a predominantly dark matter universe suggests a new kind of interaction. Traditionally, baryonic and dark matter have different properties and interact only through gravity. However, in this model, the dynamics of such an injection must be accounted for, potentially altering the dark matter distribution and behavior.
3. Wormhole Dynamics: A wormhole's collapse might have unique effects on the conservation laws (like energy, momentum, and angular momentum) and on the structure of spacetime itself. This collapse could affect the equations governing SUSY by altering the background spacetime metric.
4. Multiverse Considerations: Assuming the multiverse concept, the equations may need to consider transitions or connections between different universes, possibly through these wormhole dynamics. This could imply new terms or conditions in the supersymmetry equations to account for cross-universe effects.

Supersymmetric Action with Dark Matter and Wormhole Terms

Let's begin by modifying the standard supersymmetric action \(S\):

$$ S = \int d^4x \, \mathcal{L}_{\text{SUSY}} + \mathcal{L}_{\text{DM}} + \mathcal{L}_{\text{WH}} + \mathcal{L}_{\text{Int}} $$

where:

• \(\mathcal{L}_{\text{SUSY}}\) is the standard supersymmetric Lagrangian,
• \(\mathcal{L}_{\text{DM}}\) is the Lagrangian for dark matter,
• \(\mathcal{L}_{\text{WH}}\) represents the effects of the wormhole collapse,
• \(\mathcal{L}_{\text{Int}}\) describes interactions between baryonic matter, dark matter, and the wormhole effects.

1. Standard Supersymmetric Lagrangian (\(\mathcal{L}_{\text{SUSY}}\))

The standard SUSY Lagrangian typically includes kinetic terms for the fields, potential terms, and interaction terms between the fields and their superpartners. For a chiral superfield \(\Phi\), it is given by:

$$ \mathcal{L}_{\text{SUSY}} = \int d^2 \theta \, d^2 \bar{\theta} \, \Phi^\dagger \Phi + \left( \int d^2 \theta \, W(\Phi) + \text{h.c.} \right) $$

where \(W(\Phi)\) is the superpotential.

2. Dark Matter Lagrangian (\(\mathcal{L}_{\text{DM}}\))

We need to model dark matter as a supersymmetric field, possibly as a Majorana fermion or another SUSY partner. The simplest dark matter Lagrangian in SUSY might involve a neutralino (\(\tilde{\chi}^0\)):

$$ \mathcal{L}_{\text{DM}} = \frac{1}{2} \bar{\tilde{\chi}^0} (i \gamma^\mu \partial_\mu - m_{\tilde{\chi}^0}) \tilde{\chi}^0 $$

where \(\tilde{\chi}^0\) is the neutralino field.

3. Wormhole Collapse Lagrangian (\(\mathcal{L}_{\text{WH}}\))

The effects of a wormhole collapse can be modeled by a non-trivial spacetime metric \(g_{\mu\nu}\) and possibly additional fields (like a scalar field \(\phi\) associated with the wormhole's throat):

$$ \mathcal{L}_{\text{WH}} = \frac{1}{2} \sqrt{-g} \left( R - \frac{1}{2} g^{\mu\nu} \partial_\mu \phi \partial_\nu \phi - V(\phi) \right) $$

where \(R\) is the Ricci scalar of the curved spacetime and \(V(\phi)\) represents the potential energy of the scalar field.

4. Interaction Lagrangian (\(\mathcal{L}_{\text{Int}}\))

The interaction between baryonic matter, dark matter, and the wormhole collapse dynamics can be expressed as:

$$ \mathcal{L}_{\text{Int}} = \lambda_{\text{int}} \bar{\psi} \gamma^\mu A_\mu \tilde{\chi}^0 + g_{\phi \psi} \phi \bar{\psi} \psi $$

where \(\psi\) represents the baryonic matter field, \(A_\mu\) is a gauge field representing possible mediator particles, \(\lambda_{\text{int}}\) is the interaction strength between dark matter and baryonic matter via this mediator, and \(g_{\phi \psi}\) is the coupling constant for the interaction with the wormhole scalar field \(\phi\).

New Supersymmetry Equations

From these Lagrangian components, the new equations for supersymmetry under the conditions described can be derived by varying the action with respect to the fields involved:

1. Modified Supersymmetry Transformation Rules: These will include terms that account for the interactions with the dark matter fields and the effects due to the altered spacetime metric from the wormhole collapse.
2. Field Equations: Derived from the Euler-Lagrange equations for each field, incorporating the new interaction terms.
3. Constraints from Wormhole Collapse: Any topological constraints or boundary conditions imposed by the wormhole collapse could introduce additional conditions on the SUSY field equations.

Example: Modified Field Equation for a Fermion Field

A typical fermion field equation modified by dark matter and wormhole effects could look like:

$$ (i \gamma^\mu D_\mu - m_\psi) \psi = \lambda_{\text{int}} \gamma^\mu A_\mu \tilde{\chi}^0 + g_{\phi \psi} \phi \psi $$

where \(D_\mu\) is a covariant derivative that includes corrections from the curved spacetime metric due to the wormhole.

Conclusion

Developing these new equations involves merging concepts from supersymmetry, dark matter physics, and general relativity. Further refinement would require detailed calculations and possibly introducing new fields or mechanisms to account for the unique dynamics at play. This is a highly speculative and complex endeavor that could lead to novel insights into the nature of the universe and the interplay between different forms of matter and spacetime.