On the consistent disformal couplings to fermions

arXiv:2510.07419 · hep-th, astro-ph.CO, gr-qc · Submitted 2025-10-08 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: We're looking at the paper: "On the consistent disformal couplings to fermions".

Jocelyn: The gist: Disformal couplings to fermions lead to a unique derivative coupling to the axial fermionic current, which contains higher derivatives in general.

Vera: First, who's behind it and why it matters.

Paper summary: Vera: So we're looking at this paper called "On the consistent disformal couplings to fermions," and what it claims is that these couplings lead to a specific kind of derivative coupling between the disformal fields and the axial fermionic current, which generally has higher derivatives in it.

Jocelyn: Right, so they are looking at how these couplings work with fermions and finding this unique derivative coupling that brings in those higher derivatives. It seems like they're setting up some conditions to keep things consistent without introducing ghosts.

Subrahmanyan: That sounds like a tricky area because higher derivatives often bring in those pesky ghost degrees of freedom, so the focus on consistency is important for cosmology as well, considering how we model dark energy and dark matter one <ref:2510.07419#pg1>.

Vera: Exactly. The core thesis they push for is that to avoid these ghosts, you end up needing some specific conditions on the field space metric if you're using two scalar fields.

Jocelyn: And what they found was that for those two-scalar field disformal transformations, a consistent coupling must have a degenerate field space metric. That opens up avenues for new kind of gravity models where you can still explore modified gravity but keep it ghost-free twelve.

Subrahmanyan: That's interesting because it connects the structure of the transformation directly to the stability of the resulting theory, which is crucial when you're trying to build a model that describes our universe on large scales.

Vera: They then show that this degeneracy condition means that for those two scalar fields, h2 must equal h3, which is the same as b2 equaling b3. This makes the coupling term, called M alpha, depend only on derivatives of the scalar fields X, Q and Z.

Jocelyn: So they're saying that these integrability conditions for the total differential dH lead precisely to the absence of higher time derivatives in those cases forty-three <ref:2510.07419#pg2>.

Subrahmanyan: That really tightens the constraint on how these couplings can behave dynamically; it means you can't just have arbitrary higher derivatives without running into ghosts unless you impose that degeneracy condition.

Paper summary: Vera: Moving on, they also explored a case with a single scalar field but with higher order spatial derivatives, where they consider a disformal transformation like mu nu = f zero g mu nu + f one phi mu phi nu + 2f two phi(mu not equal to nu) + f three delta mu nu.

Jocelyn: And for a consistent matter coupling in that single scalar case, they found that the condition c twenty = -c twenty-one d X implies f3 has to be equal to 2C 2c eleven. This shows that there are still disformal couplings to matter where f3 is not zero and it's consistent.

Subrahmanyan: That’s a bit encouraging because it means the restriction on the transformation form isn't as severe as in the two-scalar case, opening up a larger class of disformal Horndeski theories that we can look at thirty-eight forty-three <ref:2510.07419#pg2>.

Vera: And they extend this idea to single vector fields with higher derivatives in section V, and they demonstrate that these transformations including the field strength are highly restricted.

Jocelyn: Specifically, for a U(one) gauge symmetry, the axial coupling expression M alpha ends up involving second time derivatives multiplied by first derivatives of the vector field <ref:2510.07419#pg1>.

Subrahmanyan: That is significant because from that term in equation ninety-six the equations of motion then contain third derivatives of the vector field, which means they conclude that this disformal coupling with U(one) gauge symmetry is inconsistent with fermions ninety-two.

Vera: So what we have here is a lot about how these couplings interact with different field setups—scalar fields versus vector fields.

Jocelyn: And to wrap up the structure, for two scalar fields, you need that degeneracy condition on the metric space, while for single scalar fields with higher spatial derivatives, the transformation form is much more restricted.

Subrahmanyan: The paper highlights that the formalism can also be applied to single field disformal transformations with higher derivatives and extends conditions from other work thirty-nine <ref:2510.07419#pg2>.

Vera: In essence, they’ve provided a larger class of disformal Horndeski theories by showing how these constraints apply across different setups.

Jocelyn: And for vector fields, they managed to propose a new class of transformations for higher derivative Proca theories that are free of ghost degrees of freedom fifty-eight fifty-nine <ref:2510.07419#pg3,propose a new class of transformations for higher derivative Proca theories>.

Subrahmanyan: So the implication is that we can construct viable disformal transformations even in more complex setups like single vector fields, provided we stick to these restrictions.

Paper summary: Vera: It seems like the main thing they're pointing toward is how these couplings can be used to construct new modified gravity models by performing the inverse transformation.

Jocelyn: And they emphasize that if the metric transformation is invertible, the number of degrees of freedom doesn't change, which means the resulting model can be free of Ostrogradsky ghosts as long as there were no ghosts to begin with thirty-four thirty-five <ref:2510.07419#pg2,free of Ostrogradsky ghosts as long as there were no ghosts to>.

Subrahmanyan: It really boils down to how you treat those higher-derivative interactions in the gravitational sector when you introduce this disformal coupling.

Vera: So, if we look at the title of "On the consistent disformal couplings to fermions," it suggests a deep dive into making sure these couplings are physically sound with matter fields.

Jocelyn: And for someone who only listens to this, it means that constructing viable modified gravity models using disformal transformations depends heavily on imposing specific degeneracy conditions based on the number of scalar fields you're using.

Subrahmanyan: This work gives us a concrete set of rules about when these couplings lead to higher derivatives and when they stay within the bounds of what we consider consistent theories.

Vera: So, to bring it all together, this paper shows that for two scalar fields you need to impose degeneracy conditions on the form of the field space metric.

Jocelyn: And in contrast, for single scalar field disformal transformation with higher derivatives, the form of that transformation is highly restricted.

Subrahmanyan: The formalism extends to single field disformal transformations with higher derivatives and leads to a larger class of disformal Horndeski theories based on conditions derived in work like thirty-nine <ref:2510.07419#pg2>.

Vera: And they also managed to construct a viable disformal transformation of a single vector field depending directly on the field strength, which can be used to explore new classes of Proca theories beyond current approaches.

Jocelyn: It seems like the main point is that you can build consistent disformal transformations even in more complex setups, but you have to follow these specific rules about degeneracy or restriction.

Conclusion: Vera: So we've seen how these disformal couplings lead to a specific derivative coupling for fermions, and now we're at the end of this paper, "On the consistent disformal couplings to fermions."

Jocelyn: Right, so what does that title actually mean when you read it? It sounds pretty technical.

Subrahmanyan: It means they’re figuring out how these couplings have to behave consistently with matter fields, which is a big deal because we need models that don't have ghosts.

Vera: Exactly. The main point here is about the constraints they put on these couplings, especially when you look at different setups, like two scalar fields versus one.

Jocelyn: And what are those constraints? I'm guessing it involves some kind of metric structure or something that prevents instability.

Subrahmanyan: They show that for two scalar fields, you have to impose a degeneracy condition on the field space metric for the coupling to be consistent and avoid those higher-derivative ghosts.

Vera: That’s right, they found that if you're using two scalars, h2 has to equal h3, which simplifies the coupling term significantly.

Jocelyn: So it boils down to a specific relationship between those field components, rather than letting you pick anything arbitrarily.

Subrahmanyan: And for single scalar fields with higher spatial derivatives, the paper shows that the form of that transformation is actually much more restricted than in the two-scalar case.

Vera: That's a key difference they point out; the restrictions are different depending on how many fields you start with or if you're looking at spatial versus temporal derivatives.

Jocelyn: So, what does this mean for us when we think about building modified gravity models?

Subrahmanyan: It gives us a roadmap. You can build viable theories like disformal Horndeski theories by following those specific conditions they derived from the consistency checks.

Vera: And for vector fields, they found a way to construct a viable transformation that depends directly on the field strength, which is useful for exploring Proca theory ideas.

Jocelyn: So it sounds like this paper is really about showing that even in these complex setups, we can still find physically sound ways to couple gravity and matter.

Subrahmanyan: It’s about setting the rules for consistency so that the resulting theories are actually workable models for cosmology and structure formation.

GUILLEM DOMÈNECH, ALEXANDER GANZ, APOSTOLOS TSABODIMOS

Institute for Theoretical Physics, Leibniz University Hannover · Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Hannover · Center for Physical Sciences and Technology, Vilnius University

hep-th, astro-ph.CO, gr-qc

Submitted: 2025-10-08

Updated: 2025-10-08

The gist: The gist: Disformal couplings to fermions lead to a unique derivative coupling to the axial fermionic current, which contains higher derivatives in general.

Key concepts

Disformal Coupling: This refers to a specific way of coupling gravity (spacetime metrics) to matter fields, like fermions. It involves transforming the metric itself, which is a non-standard way of modifying gravity that needs careful checking to ensure physical consistency.
Axial Fermionic Current Coupling: The analysis shows that disformal couplings uniquely generate a derivative coupling between the disformal fields and the fermionic axial current. This current is crucial in describing how fermions interact with these modified gravitational theories, and its structure often involves higher derivatives.
Ostrogradsky Instabilities (Ghosts): These are unphysical modes that appear in certain higher-derivative theories when trying to describe physical systems. The paper shows that disformal couplings can introduce these ghosts unless specific constraints, like degeneracy conditions on the field space metric, are imposed to keep the resulting equations of motion stable.
Degeneracy Conditions: These are mathematical requirements placed on the structure of the transformation (the field space metric) when dealing with multiple scalar fields. Imposing these conditions is necessary to eliminate higher time derivatives and avoid unphysical ghosts, allowing for consistent modified gravity models.

Terminology

Degeneracy — 27× in this paper · explained in 4 episodes
Consistency — 6× in this paper · explained in 3 episodes
Generalization — 4× in this paper · explained in 13 episodes
Degenerate Higher Order Scalar-Tensor — 2× in this paper
While General Relativity — 2× in this paper
Core — 1× in this paper · explained in 2 episodes
Inflation — 1× in this paper · explained in 2 episodes
Trace — 1× in this paper · explained in 2 episodes

Summary

The gist: Disformal couplings to fermions lead to a unique derivative coupling to the axial fermionic current, which contains higher derivatives in general.

Introduction and Motivation

Disformal couplings to fermions lead to a unique derivative coupling between the disformal fields and the fermionic axial current, which contains higher derivatives in general While General Relativity (GR) is successful in describing gravity on local scales, on cosmological scales, there remain open questions, most notably the nature of dark energy and dark matter. Among a wide variety of modified gravity models [2], the most popular approaches are the inclusion of a scalar or vector field. In order to study these models in a unified manner, there has been a focus on constructing general scalar- or vector-tensor theories with second-order equations of motion. Since the equations of motion are second-order, there are no Ostrogradsky instabilities [12], commonly present in higher derivative theories. However, it was later realized that the restriction on the second order equation of motions can be relaxed by requiring that the kinetic matrix is degenerate, leading to Degenerate Higher Order Scalar-Tensor (DHOST) theories [15–20] and beyond generalized Proca theories [21, 22].

Transformation and Consistency Conditions

The action under consideration can be schematically expressed as S = Sgrav[gμν, φI, B J] + Sm[̃gμν, ΨM, A N, φ] (1) where ΨM denotes “M” number of spinors, A N are “N” number of gauge vector fields and φ the Higgs field of the standard model. Disformal coupling is consistent if it does not introduce new Ostrogradsky ghosts and leaves the original number of degrees of freedom unchanged. The analysis shows that disformal couplings generically lead to derivative coupling between the disformal fields and the fermionic axial current via Mμ. This coupling can prevent a consistent matter coupling and revive the Ostrogradsky ghost as discussed in [39].

Two Scalar Fields Analysis

For a two-scalar field disformal transformation, we show that the consistent disformal coupling must have a degenerate field space metric. This allows us to explore consistent, new twoscalar field modified gravity models. The analysis of the axial coupling shows that Mα contains in general second time derivatives of the scalar fields, which leads to higher order equation of motions and Ostrogradsky ghosts unless degeneracy conditions are imposed To avoid the presence of Ostrogradsky ghosts in Eq. (15), we need to impose degeneracy conditions, specifically h2 = h3, which is equivalent to b2 = b3. This condition leads to Mα being dependent only on derivatives of X, Q and Z. The integrability conditions for the total differential dH (43) are precisely those that also lead to the absence of higher time derivatives.

Single Scalar Field with Higher Derivatives

In exploring the case with a single scalar field but with higher order spatial derivatives, we consider a disformal transformation of the form ̃gmunu = f0 gmunu + f1φmuφnu + 2f2φ(μν) + f3muν. The conditions for invertibility of the metric are given in Ref. [36]. For a consistent matter coupling, we find that the condition c20 = -c21X implies f3 = 2C2c11. This demonstrates that there are disformal couplings to matter with f3 ≠ 0 which are consistent.

Vector Field Analysis

For vector disformal transformations, the analysis of a U(1) gauge symmetry shows that the axial coupling (17) is expressed in tensor components as Mα = 1/4 εβγδα (δνδ + Pνδ) Pμγ∇νLβμ = -D3/4(1 − a)2 εβγδα ((1 + 1/4 Tr F2) δνδ - FνσFσ δ) FμκFκ γ∇ν(FρβFρμ). From the last term in Eq. (96), we see that the axial coupling contains second time derivatives multiplied by first derivatives of the vector field, which means the equations of motion contain third derivatives of the vector field. Thus, we conclude that the disformal coupling with U(1) gauge symmetry (92) is inconsistent with fermions.

Conclusion

We have demonstrated that for two scalar fields, one needs to impose degeneracy conditions on the form of the field space metric of the disformal transformation. This demonstrates that in contrast to single scalar field disformal transformation the form of the transformation is highly restricted. The formalism can also be applied to the single field disformal transformation with higher derivatives, extending conditions derived in Ref. [39] leading to a larger class of disformal Horndeski theories. Similarly, one can construct a viable disformal transformation of a single vector field depending directly on the field strength, which can be used to explore new classes of Proca theories beyond current approaches.

Acknowledgments

We would like to thank Kazufumi Takahashi for the helpful comments and discussions. This research is supported by the DFG under the Emmy-Noether program, project number 496592360, and by the JSPS KAKENHI grant No. JP24K00624.

Appendix Summary

Appendix A reviews the disformal transformation of the spin connection and Dirac Lagrangian using tensor components. Appendix B provides details on how to remove any antisymmetric component of the tetrad disformal transformation via a spinor field redefinition. In Appendix B, we obtain that up to quadratic order in L2 using Tμν = L[μν] we recover the result in Eq. (81) up to quadratic order in (L2).

References

[1] E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys. D 15, 1753 (2006), arXiv:hep-th/0603057.

[8] G. W. Horndeski, Int. J. Theor. Phys. 10, 363 (1974).

[9] C. Charmousis, E. J. Copeland, A Padilla, and P M Saffin, Phys Rev Lett 108 051101 (2012).

[36] K Takahashi, H Motohashi, and M Minamitsuji Phys Rev D 105 024015 (2022).

[39] K Takahashi, R Kimura, and H Motohashi Phys Rev D 107 044018 (2023).

[57] G Domènech, A Ganz, F Kirk, N Sherrill, and A Tsabodimos arXiv:2509.XXXXX [in preparation].

[61] A De Felice, S Mukohyama, and K Takahashi JCAP 12 (12) 020 arXiv:2110.03194 [gr-qc].

[88] We can later expand the resulting action in terms of φ and χ.

[97] After imposing all degeneracy conditions, we conclude that Mα (42) must be Mα = 1/4 εβγδα φβχ∇δH, namely only a function of ∇δH.

Improvements for AI systems

Bold header: Enhanced Ghost-Free Theory Construction

The improved system can now rigorously classify consistent disformal couplings to fermions by requiring the absence of higher time derivatives, as they typically lead to ghost degrees of freedom, allowing it to derive general conditions on consistent disformal couplings.

Bold header: Degeneracy Condition Identification

The AI will be capable of automatically identifying the necessary degeneracy conditions for consistency, such as the requirement that the kinetic matrix is degenerate for two-scalar field transformations, which leads to a degenerate field space metric.

Bold header: Mapping to Specific Gravity Models

The system can perform direct mapping from consistent disformal couplings to specific modified gravity theories by showing that the transformation of the Einstein-Hilbert action leads to two-field Horndeski or two-field DHOST theories.

Bold header: Analysis of Higher Derivative Constraints

The AI can analyze how disformal couplings with higher derivatives affect consistency, noting that the absence of ghosts restricts the form of the disformal transformation, and specifically identifying when a coupling leads to Ostrogradsky ghost instabilities.

Bold header: Vector Field Consistency Assessment

For vector fields, the system can determine consistency by checking if a transformation is consistent with gauge symmetry; for instance, it can show that the disformal coupling with U(1) gauge symmetry (92) is inconsistent with fermions because of terms like those in Eq. (97).

Bold header: Extension to Multi-Field Systems

The system can explore generalizations to multi-field extensions by utilizing the derived constraints, such as the most general equation of motion for two scalar fields with a second-order equation of motion, linking this to multifield disformal transformations.