Factorizing Hidden Particle Production Rates

A method is proposed to streamline the computation of hidden particle production rates by factorizing them into i) a model-independent SM contribution, and ii) a observable-independent hidden sector contribution. The Standard Model (SM) contribution can be computed once for each observable and re-used for a wide array of hidden sector models, while the hidden sector contribution can be computed once for each model, and re-used for a wide array of observables. The SM contribution also facilitates extracting model independent constraints on hidden particle production. The method is compatible with effective field theory (EFT) and simplified model approaches. It is illustrated by factorizing the rate of charged kaon decays into a charged lepton and a number of hidden particles, and a single form factor $F_{\ell}$ is found to parametrize the impact of general hidden sectors. We derive model-independent constraints on the form factor $F_e$ that governs decays into positrons and hidden particles.


Introduction
Since the Standard Model (SM) of particle physics is known to be incomplete, the search for physics beyond the Standard Model, and in particular for new particles, is an integral part of modern high energy physics. Focusing on searches at collider and fixed target experiments, the phenomenology of candidate particles crucially depends on whether or not they are light enough to be produced at current generation experiments. Light new particles are more strongly constrained, and they can only couple very feebly to the SM.
In recent years, a significant program of searches for such feebly interacting new particles has begun to emerge [ -]. Relevant constraints can be obtained from high intensity data sets of CMS [ -] and ATLAS [ ], from flavour physics experiments such as LHCb [ -] or Belle-II [ -], and from high luminosity runs of the Large Hadron Collider [ ]. There is also a number of ongoing and proposed searches at low-energy fixed target experiments such NA [ -], KOTO [ -], SeaQuest [ , ], or SHiP [ ], and further complimentary constraints can be obtained from searches at dedicated long-lived particle experiments [ ] such as MATHUSLA [ , ], FASER [ , ] and CODEX-b [ ].
In spite of these searches, numerous viable hidden sector models still predict a large variety of hidden particle candidates. Examples include axion-like particles (ALPs) [ -], heavy neutral leptons (HNLs) [ , , ], and new vector Bosons [ , , ]. Since standard perturbative methods of computing predictions for the relevant observables (i.e. hidden particle production, scattering, and decay rates) depend on detailed knowledge of the number and the properties of the hidden fields that are presumed to exist as well as their interactions, this poses a challenge for the model independent interpretation of the experimental constraints.
In recent years, effective field theories (EFTs) such as Standard Model effective field theory [ , ] and Higgs effective field theory [ -] have become a standard tool for extracting model independent constraints on the SM coupling to heavy new particles [ -]. Due to this success, some effort has been put towards constructing EFTs that also account for light new particles. For instance, there are a number of EFTs that couple the SM to specific candidate particles such as ALPs [ ], HNLs [ , ], or dark photons [ ], as well as generic dark matter candidates [ -]. EFTs are also commonly used to describe non-relativistic interactions between the SM and dark matter candidates [ -]. Finally, there has been significant work to create a comprehensive framework for constructing portal effective theories (PETs) that systematically extend EFTs of the SM by coupling them to generic hidden particles while making only a minimal number of assumptions [ ]. Another approach for extracting model independent bounds consists in constructing so-called "simplified models", which are designed to capture certain features of realistic SM extensions in a minimalist and therefore more generic setup. Simplified models have become popular e.g. within the context of searches for particles that are on the cusp of being collider accessible [ , ] and dark matter candidates [ , ]. However, while both EFTs and simplified models are useful for studying generic features of hidden sectors, it is not always straightforward to translate the resulting constraints into hard constraints that are applicable to realistic SM extensions.
In this work, we focus on the computation of hidden particle production rates. We argue that some of the challenges associated with establishing model independent constraints can be ameliorated by factorizing these production rates into a product of i) model-independent "reduced matrix elements" that depend only on SM interactions and ii) observable-independent "hidden current correlation matrices" that capture the impact of general hidden sectors. As we will show, this factorization is completely generic and relies only on a minimal set of assumptions.
Using the reduced matrix elements, it is possible to derive model-independent "master formulae" for a wide range of observables that parametrize the impact of general hidden sectors via a number of generic form factors. The master formulae can be fitted to experimental data in order to extract model-independent constraints on the form factors. In addition, it is possible to use the hidden current correlation matrices in order to translate the form-factor constraints into more specific constraints on the individual parameters of a given hidden sector model. One major advantage of this two-step procedure is that it divides the workload required for computing hidden particle production rates into two largely independent packages: On the one hand, the hidden current correlation matrices depend only on hidden sector physics and can be computed without having to account for the intricacies of SM model physics or collider phenomenology. On the other hand, the master-formulae depend only on SM physics, and can be computed without having to specify the feature of the hidden sector. This division also makes it possible to mix and match results in a more transparent and systematic way while minimizing the need for re-doing computations in order to adapt a given result to a new model.
In order to make use of the factorization procedure, it is necessary to supply a comprehensive list of relevant portal operators that can mediate a given production process. In principle, any EFT that accounts for the presence of light hidden particles can be used to provide such a list. However, the PET framework is particularly well suited for this task, since it makes only minimal assumptions about the symmetries obeyed by the portal interactions and about the internal structure of the hidden sector, while also accounting for the presence of any additional heavy new particles. When combined with an approriately constructed list of portal operators, the factorization approach is a powerful tool for establishing model independent constraints on hidden sectors. It effectively extends the EFT approach by providing a description that accounts for both light and heavy new particles.
The remainder of this paper is structured as follows: In section , we provide a short proof of the factorization rule, and derive the general recipes for computing the reduced matrix elements and correlation matrices. In section , we illustrate the procedure by considering the example of charged Kaon decays K + → + X into a charged lepton and a number of hidden sector particles. To do so, we first compute a model-independent master amplitude that encodes the impact of generic hidden sectors using a single form factor F , and then show how to compute this form factor for an example model that couples the SM to a number of hidden Fermions. We also extract model-independent upper bounds on the form factor F e that governs charged kaon decays K + → e + X into a positron and a number of hidden particles. Section concludes the paper.

Inclusive production rates
In this section, we demonstrate the factorization of inclusive hidden sector production rates into a product of reduced matrix elements M d and hidden sector correlation matrices J de . Although the proof is not complicated, it also serves as a derivation of the diagrammatic expressions for both quantities. In the interest of full generality, we consider a generic theory that is composed of a visible sector A and a hidden sector B, The two sectors are linked by a number of weak portal interactions whose strength is measured by a small parameter . Each portal operator is the product of a local operator constructed from a collection of visible fields {φ a } and a conjugated local operator constructed from a collection of hidden fields ξ b . We are interested in computing inclusive rates for the production of hidden particles in experimental setups where the hidden particles cannot be observed directly. In this case, only the number and the properties of the visible particles in the final state are known, and the resulting transition rates involve a sum over an infinite number of viable final states, that consists of m visible particles t j with four-momenta p j = (π j , p j ) and k hidden particles r l with four-momenta q l = (ω l , q l ). Here and in the following, we use the multi-indices s = (s 1 , . . . , s n ), t = (t 1 , . . . , t m ), and r = (r 1 , . . . , r k ) to collectively denote the species and the helicity of each particle. The integration measure includes a sum over both the species and the helicity of each hidden particle as well as an integral over its associated phase space. Finally, the sums denote the total momentum of the visible particles in the initial and final states as well as the corresponding missing momentum.
Most of the relevant production modes for hidden particles involve either decays or scatterings of visible particles. The general sum ( ) determines the overall rates for hidden particle production in both of these cases, yielding where v 12 = |v 1 − v 2 | is the relative velocity of the two initial state particles in the scattering process. Our aim is to show that the inclusive rate M (K → P) factorizes according to where the reduced matrix elements M d encode the dynamics of the visible sector while the hidden current correlation matrix J de encodes the impact of hidden sectors. We note that, although equation ( ) is completely general, its usefulness crucially depends on the size of the small parameter . However, this is of no concern within the context of hidden sector searches, since higher order corrections are almost always negligible due to the required smallness of the portal coupling. In this case, the factorization equation ( ) becomes a very good approximation.

. Factorization
The factorization of the inclusive rate ( ) is a corollary of an equivalent factorization of the time-ordered, connected correlation functionŝ that capture the dynamics of hidden particle production. Here and in the following, the average 0| • |0 conn. is defined to include only connected diagrams. In order to avoid superfluous notational clutter, we have defined the multi-field operatorŝ We also define the corresponding momentum-space operators Expanding the path-integral of the full theory to leading order in , one immediately finds that the above correlation functions factorize according tô where the reduced correlation functionĜ ab;d only depends on physics of the visible sector, while the form factorF conn.
( ) encodes the impact of hidden sectors. Moving on to momentum space, this gives where the momentum space versions of the reduced Greens functions and the hidden sector form factors areδ In order to translate the factorization equation ( ) into a statement about the inclusive transition rate ( ), we apply the Lehmann-Symanzik-Zimmermann (LSZ) reduction formula to ( ). Using the multi-field notation, this gives

the initial and final state polarization vectors
The external current correlation matrix J de fully encodes the impact of hidden sectors. Since the integration k D k Q in the definition of J de includes a sum over the species and the helicities of the hidden particles in the final state, the correlation matrix can be written as an infinite sum of terms J r de = J r de (Q) that capture the contribution associated with each individual final state. Notice also that M d and J d (and with them J de ) can, depending on the precise structure of the corresponding portal operator, carry free Lorentz and spinor indices. For example, if the theory contains a portal operator A µ B µ = ψσ µ ψ † v µ that couples a pair of visible Fermions A µ = ψσ µ ψ † to a hidden vector particle B µ = v µ , then the corresponding reduced matrix element M µ and hidden currents J µ carry free Lorentz indices that are to be contracted with each other. .

Feynman rules
The expressions ( ) and ( ) define series of Feynman diagrams that can be used to compute the reduced matrix elements M d as well as the hidden currents J d (r, Q) associated with each viable hidden sector final state. Figure shows the diagrammatic expansions for both M d and J d . As in the case standard Smatrix element computations, they are given as a sum of all available connected and amputated Feynman diagrams with the appropriate number and kind of particles in the initial and final states, where M d diagrams only contain vertices and propagators associated with visible fields and interactions while J d diagrams only contain vertices and propagators associated with hidden fields and interactions. The main difference compared to the recipe for standard Smatrix elements is that all relevant diagrams have to contain exactly one portal vertex that is constructed from either the "visible" part A d (in the case of M d ) or the "hidden" part B d (in the case of J d ) of the corresponding portal operator.
The rules for constructing these portal vertices are largely the same as those for constructing standard vertices. (i.e. symmetrize the operator under exchange of identical fields, go to momentum space, strip away all fields, and multiply by an overall factor of i.) However, there are two differences: First, and in complete analogy to the composite operators A d and B d , the portal vertices may carry free Lorentz and spinor indices, which result in M d and J d likewise carrying such free indices. Second, both portal vertices do not conserve four-momentum in the strict sense. Rather, the appropriate sum of all ingoing and outgoing momenta has to be equal to some missing momentum q. This is depicted symbolically using a dashed line (cf. figure ). The missing momentum is outflowing for M d , so that q = K − P , and it is inflowing for J d , so that q = Q.

Illustrative example: Inclusive
To illustrate the factorization procedure as well as the computation of the reduced matrix elements and hidden currents, we consider the production of hidden particles in inclusive K + → + X decays, where X denotes any collection of hidden particles. There already exists a model-independent master-formula for the production rate of generic spin / particles in this type of charged kaon decay [ ], and the present computation improves this result by accounting for the production of hidden particles with different spins as well as the production of multiple hidden particles via the same decay process. .

Reduced matrix elements
The primary input required to factorize inclusive matrix elements according to ( ) is an exhaustive list of all relevant portal operators. Here, we use the list compiled in [ ], which encompasses all leading operators that couple the SM to a single hidden particle of spin 0, 1 /2, or 1 at the strong scale. The sector of the portal Lagrangian that is relevant for K + → + X decays is where u and s are the up-and strange quark fields, = e, µ is a charged lepton field, and ν is the corresponding SM neutrino field. Following two-component notation of [ ], the SM Fermion fields are taken to be left-handed Weyl spinors. v = 174.10358 ± 0.00004 GeV is the vacuum expectation value of the Higgs Boson and |V us | = 0.2252 ± 0.0008 is the u-s element of the Cabibbo-Kobayashi-Maskawa matrix [ ]. The small parameter has been reabsorbed into the operators Ξ ν , Ξ , which now take on the role of the generic local operators B d . It is not necessary to specify their detailed shape in order to compute the reduced matrix elements. However, we note that, although the master formula that results from using the portal Lagrangian ( ) is largely model-independent, the factorization procedure outlined here can also be used to obtain an even more general master formula by including additional sub-leading portal operators. A collection of the relevant interactions is given e.g. in [ , ].
Given the portal Lagrangian ( ), the resulting coupling of hidden sectors to the pseudoscalar mesons is captured by the portal chiral perturbation theory (χPT) action constructed in [ ]. We work at leading order in the χPT power counting, at tree-level, and neglect electromagnetic corrections. At this level of accuracy, the coupling of the operator Ξ to charged kaons can be obtained by replacing the quark bilinears according to [ , , ] where f 0 is the kaon decay constant. This replacement yields the "visible" portal vertex depicted in figure a, while neutrino portal operator in ( ) yields the additional vertex depicted in figure b.
In general, there is one reduced matrix element for each portal operator. Since there are two relevant portal operators, there are also two reduced matrix elements M and M ν . Using the Feynman rules for two-component spinors given in [ , ], one finds the leading order expressions where y is the two-component polarization vector of the final state lepton, k = (κ, k) is the kaon four-momentum, p = (π, p) is the lepton four-momentum, and q = k − p is the missing momentum. The free spinor indices α andα are to be contracted with the external current correlation matrices. Summing over the available + spin polarizations, the resulting inclusive rate for The impact of hidden sector contributions is encoded by the form factor where tr D is the trace taken with respect to the (un)dotted spinor indices. Since the Levi-Civita tensor is used to raise and lower indices associated with the same chirality, V α = αβ V β , one has tr D J a = αβ J a βα . This gives the partial decay width The leading contribution to M is generated by the tree-level diagram that contains only the portal vertex in figure a, and the leading contribution to M ν is generated by the tree-level diagram that contains the SM three-point vertex that mediates K + → + ν decays as well as the portal vertex in figure b is a phase-space factor, m K = 493.636 ± 0.007 MeV is the charged kaon mass, and m is the mass of the relevant charged lepton [ ]. Comparing equation ( ) with the known partial width for K + → + ν decays [ ], one finally obtains the ratio of branching ratios Since the master formula ( ) only depends on the overall shape of the portal Lagrangian ( ), but not on the specifics of the portal interactions or on the internal structure of the hidden sector, formula ( ) is almost completely model independent. In particular, it captures the production of an arbitrary number of hidden particles with arbitrary masses, spins, and interactions. Since we have also allowed for the possibility of higher dimensional SM and portal operators, formula ( ) even accounts for the presence of heavy new particles. Formula ( ) can be used to extract model-independent constraints on the size of the form factors F (x q ). To illustrate how this can be done in practice, we re-interpret the analysis in [ ], which uses a missing mass search to establish upper bounds for the branching ratio of charged kaon decays into a positron and a HNL from NA data. The HNLs were assumed to decay outside the detector, and the search was hence directed at finding events with a single visible positron in the final state and some finite missing mass q 2 = m 2 miss = (k − p) 2 . This setup is a special case of the generic setup that we have considered in this work, where the number and the type of visible particles in the final state is known, while any produced hidden particles are not observed directly.
The analysis in [ ] sampled a large number of missing masses in the range 122 MeV < m miss < 465 MeV, searching for candidate events in a bin of width 2∆m = 0.3 MeV centered around the sampled missing mass. For each bin, they extracted an upper bound on the corresponding branching ratio. This upper bound is roughly constant over the entire mass range, and comes out to B(K + → e + N ) 4·10 −9 . In the following, we interpret this number as a bound on the fractional branching ratio ∆B e = ∆x q · dB e /dx q , which captures the likelihood of producing a positron and a collection of generic hidden particles with an aggregate missing mass in the range [x q − ∆x q , x q + ∆x q ], where ∆x q = ( ∆m/m K ) 2 . Using the known values of B(K + → e + ν e ) = (1.582 ± 0.007) · 10 −5 and x e = m 2 e m 2 K ≈ 1.07 · 10 −6 [ ], this gives the constraint where F e is the average of F e (x q ) taken over the range [x q − ∆x q , x q + ∆x q ]. To understand how to interpret this constraint, recall that the form factor captures the production of an arbitrary number of hidden particles. Generically, a contribution associated the production of a single hidden particle will include the delta distribution δ(q 2 − m 2 i ), where m i is the mass of the relevant hidden particle, to ensure that only on-shell particles are produced. In contrast, the phase-space integral d k Q in equation ( ) is sufficient to smooth out the contributions associated with the production of two or more hidden particles. Hence, the form factor is of the general shape where the A i are amplitudes for single particle production and B(X q ) is a smooth function that captures the production of multiple hidden particles. Averaging ( ), one finds where we have used that B ≈ B for sufficiently small bins. Combining this decomposition with the general expression ( ), one now obtains separate constraints on the single-particle amplitudes A i and the multi-particle amplitude B(x q ). The single-particle amplitudes have to obey the model-independent constraint while the constraint on the multi-particle amplitude is much less stringent, This is to be expected, since a signal for the production of a single hidden particle would appear as a sharp peak that is concentrated into a single bin, while a signal for the production of multiple particles would be spread over a whole range of viable missing masses. While the peak is relatively easy to observe, and therefore constrain, it is more difficult to constrain the spread-out signal. .

Hidden current matrices
In order to translate constraints on the form factors F (x q ) into more specific constraints, it is necessary to plug in appropriate expressions for the hidden current correlation matrices J de . In this section we demonstrate how to compute general correlation matrices by considering an example case in which the SM couples to a number of left-handed Weyl Fermions ξ i . Since both Dirac and Majorana Fermions can always be written as a combination of Weyl Fermions, this setup remains quite general. The leading contributions to the portal operators are where the constants c ν, are model-dependent coupling constants. The corresponding "hidden" portal vertices are depicted in figure c. In general, there is one collection of hidden currents J d = J d (Q, r) for each viable hidden sector final state r = (r 1 , . . . , r k ), and each of these collections contains one current for each available portal operator. In the concrete case of K + → + X decays this means that there are two hidden currents J ν (Q, r) and J (Q, r) for each final state. Assuming that the hidden sector interactions are perturbatively small, the leading contributions are generated by diagrams without any hidden sector vertices. Since the portal operators in equation ( ) contain only a single field operator, the only final states that are viable at this order of accuracy (i.e. neglecting hidden sector interactions) are those with a single hidden Fermion, and no additional hidden particles. The two hidden currents that are associated with this type of final state are given as where index i = i(r 1 ) denotes ξ i that creates the single-particle final state associated with r 1 . Evaluating the spin sums and the phase-space integration, one obtains the correlation matrices where m i is the mass of the hidden Fermion ξ i . Inserting these matrices into the form factor equation ( ), one finds ( )

When combined with the master formula ( b), this result is consistent with the modelindependent formula given in [section of ].
This computation exemplifies a second use of the factorization procedure: In addition to facilitating model-independent constraints on the coupling to hidden sectors, it can also help streamline the derivation of more specific model-dependent constrains. Once the masterformula associated with a given observable has been computed, it does not need to be adjusted anymore, and in order to adapt the result for a new model, it is sufficient to recompute the form-factors F (x q ). As a final remark, we note that while the form factors F (x q ) depends on the observable in question, the hidden current correlation matrices in equation ( ) do not. They can be provided once and for all, and re-used to compute a wide array of form factors associated with different observables, further reducing the need for re-computing ingredients in order to adapt a known result for a new model or observable.

Conclusion and outlook
In this work, we have shown that inclusive hidden particle production rates approximately factorize according to relation ( ), and derived recipes for computing the reduced matrix elements M d and the hidden currents J d as series of Feynman diagrams. We illustrated the factorization procedure by considering decays K + → + X of charged koans into charged leptons and a number of hidden particles. The resulting model-independent master formula ( ) improves the model-independent formula given in [ ] and parametrizes the impact of hidden sectors in terms of a single form factor F (x q ) that can be constrained in a largely modelindependent fashion. To illustrate how to constrain the form factor in practice, we have re-interpreted the analysis of [ ] and derived the model-independent bounds ( ) and ( ).
If the factorization approach is combined with an appropriately general list of portal operators, which can be constructed in a systematic and consistent fashion using e.g. the PET framework [ ], it correctly accounts for both light and heavy new particles. In this case, the factorization approach is strictly more general than the EFT approach, and we have argued that it provides a powerful tool for the model-independent interpretation of hidden sector searches.
Using the present work as a foundation, there are many potentially interesting avenues for future investigation.
Since the factorization approach relies on lists of portal operators being provided as a necessary input, it will profit greatly from further efforts of extending the EFT approach to also account for light new particles. In particular, the PET framework is well suited for providing the needed lists of portal operators, and constructing further PETs will allow the factorization approach to be applied to a significantly larger variety of observables. At this time, there are PETs that extend the full SM, light effective field theory, which describes the physics of the light SM fields [ -], and χPT, which describes the physics of the light pseudoscalar mesons [ -], by coupling them to a single light new particle of spin , / , or [ ]. These PETs also account for the possibility of SM fields coupling to multiple hidden fields with the same spin, but they are not sufficient for describing a situation in which the SM couples to multiple hidden particles with different spins. Hence, it would be useful to costruct a corresponding set of PETs that include the relevant additional portal operators.
Likewise, it would be interesting to construct PETs that extend EFTs that capture different regimes of the SM. For instance, constructing PETs that extend heavy quark effective theory, which captures the physics of heavy non-relativistic quarks [ -], and soft-collinear effective theory, which captures the physics of light but highly energetic particles [ -], would make it possible to apply the factorization approach to hidden particle production in B and D meson decays, where B meson decays are of particular interest due to the persistent B anomalies [ -].
On the hidden current side, it would be of great use to tabulate hidden current correlation matrices for a number of popular hidden sector models that include particles such as ALPs, HNLs, and dark photons. Since the expressions for the hidden current correlation matrices are observable independent, these tabulated expressions can be used as input for a wide array of master formulae, and could help facilitate e.g. global parameter scans that combine constraints from a wide range of observables. Finally, it would be interesting the study the renormalization scale dependence of the reduced matrix elements M d and the hidden currents J d , since a robust understanding of this running is necessary in order to combine constraints from observations at multiple characteristic energy scales.