Flavour constraints on light spin-1 bosons within a chiral Lagrangian approach

We discuss the construction of the chiral Lagrangian for a light spin-1 boson, here denoted as X, featuring both vector and axial-vector couplings to light u, d, s quarks. Focusing on ∆S = 1 transitions, we show that there are model-independent tree-level contributions to K±→ π±X, sourced by Standard Model charged currents, which receive an mK2/mX2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {m}_K^2/{m}_X^2 $$\end{document} enhancement from the emission of a longitudinally polarized X. This flavour observable sets the strongest to date model-independent bound on the diagonal axial-vector couplings of X to u, d, s quarks for mX< mK– mπ, superseding the bounds arising from beam-dump and collider searches.


Introduction
The lack of new particles at the LHC may be suggestive of the fact that they are either too heavy to be directly produced or too weakly coupled to Standard Model (SM) particles.New Physics (NP) models containing new feebly interacting massive particles with sub-GeV masses are currently among the most studied NP scenarios both theoretically and experimentally.Many of these studies were dedicated to the dark photon [1,2], a new massive spin-1 particle which is kinetically mixed with the ordinary photon and that could act as a portal to a dark sector.Dark photon searches have been conducted by a number of experiments, including beam-dump [3], fixed-target [4,5], collider [6][7][8][9][10][11][12], as well as meson decay [13][14][15][16][17][18][19] experiments.
Moreover, comprehensive analyses aiming at probing a light spin-1 boson X with general couplings to quarks and leptons have been also carried out (see e.g.[20][21][22][23]).If X is coupled to SM particles through a non-conserved current, such as the axial-vector current, processes which are enhanced by the ratio (energy/m X ) 2 involving the longitudinal mode of X are generally induced.The same happens if X is coupled to a tree-level conserved current which is broken by the chiral anomaly [24,25].These energy-enhanced contributions generally provide the dominant effects both to the production mechanisms of X in high-energy experiments, as well as to flavor-changing neutral current processes such as K ± → π ± X.
The aim of this paper is to revisit the sensitivity to the above NP scenarios of the rare decay K ± → π ± X induced by an underlying s → d quark transition.In order to accomplish this task, we will extend previous studies by constructing the most general ∆S = 1 chiral Lagrangian up to the order O(p 4 ), which will enable us to account for all of the dominant effects stemming from weak interactions.Indeed, the lowest-order O(p 2 ) terms of chiral perturbation theory (χPT) will capture the weak effects to the s → dX transition discussed in [24,25] and arising from the one-loop exchange of the W -boson and up-quarks.Instead, as we will see, weak effects stemming from the ∆S = 1 four-quark Lagrangian [26], can be included only by keeping O(p 4 ) terms in χPT.Although subleading in the chiral expansion, the latter contributions to K ± → π ± X arise already at tree level (they can be thought as arising from initial or final state radiation of X from the external quark legs of the ∆S = 1 effective Lagrangian) and therefore their inclusion appears to be mandatory.
Moreover, the tree-level weak contributions discussed in this work are model-independent and therefore they represent a general and robust prediction of any ultraviolet (UV) complete NP model entailing a light spin-1 boson.Instead, the loop-induced effects discussed in [24,25] are sensitive to the specific UV completion responsible for the mass generation of X (see e.g.[27]).
The paper is organised as follows.In Sect.2, we will present the general derivation of the ∆S = 1 chiral Lagrangian, as well as the related Feynman rules for spin-1 bosons up to the O(p 4 ) order.In Sect.3, we will compute the K ± → π ± X decay rate in χPT exploiting the Feynman rules derived in Sect.2, comparing our tree-level effects with the results obtained at one-loop level in [24,25].In Sect.4, we will discuss our flavour bounds vs. beam-dump and collider searches as reported in [21,22].Sect. 5 is dedicated to our conclusions, while more technical details about the construction of the chiral Lagrangian are deferred to App. A.

∆S = 1 chiral Lagrangian for spin-1 bosons
The most general Lagrangian describing the lowest-order interactions of a new spin-1 particle X with SM fermions includes both vectorial and axial couplings and, focusing on the interactions with the lightest quark flavours q = (u, d, s) T , it can be written as where g x measures the strength of the universal coupling of X to quarks.The vectorial and axial charges, x V,A , are defined in flavour space and include off-diagonal entries in the 2-3 sector.

Lowest-order chiral Lagrangian
At energies above few GeV, the Lagrangian of Eq. (2.1) can be directly employed to analyse the interactions of X µ with quarks.Here, instead we focus on the low-energy range below the GeV scale, where we can resort to χPT -see e.g.[28,29].In order to construct our χPT in the presence of X µ , we proceed as follows.Let us consider the massless QCD Lagrangian with chiral symmetry group where q = (u, d, s) T and λ a are the Gell-Mann matrices.Chiral symmetry-breaking terms (like mass terms or interactions with external gauge fields other than gluons) can be implemented by introducing appropriate spurions (a µ , v µ , s, p) as external source fields [28].Therefore, the resulting Lagrangian L ext QCD reads (2.3) where 2r µ = v µ + a µ and 2ℓ µ = v µ − a µ .Its chiral counterpart is then found to be where V spontaneous breaking.Moreover, we have defined In our model, described by the Lagrangian of Eq. (2.1), the covariant derivative D µ U reads where . Expanding the Lagrangian in (2.4) and keeping only the lowest order terms in the NP coupling, we find with the corresponding Feynman rules given in Fig. 1 (all momenta flow from left to right).Note that all couplings in Eq. (2.7) are of vector type.This is due to the fact that the matrix element of the axial-vector quark operators in Eq. (2.1) vanishes between external pseudo-scalar meson states.Moreover, in the limit of universal vector couplings, i.e. x u V = x d V = x s V , the K + K − X and π + π − X interaction terms vanish as well, as a result of the underlying SU (3) V chiral symmetry, while the K + π − X vector coupling still survives as the flavour-changing current is not conserved.
Moreover, tree-level contributions to ∆S = 1 processes, such as K ± → π ± X, are generated only if the couplings x V are flavor off-diagonal.Yet, even for flavour-diagonal couplings, irreducible flavour-violating effects to x V are loop-induced by the exchange of the W boson and up-quarks (see e.g.[25]).In the following, we will show that weak interactions provide additional sources of flavour-violation to the ∆S = 1 chiral Lagrangian, already at tree level, when we include higher-order terms in the momentum expansion corresponding to four-quark operators.

Chiral Lagrangian for weak interactions
In the SM, at energies above the scale of chiral symmetry breaking, ∆S = 1 transitions are induced by the effective four-fermion Lagrangian [26] where q = u, d, s, e u = 2/3 and e d = e s = −1/3; α and β are colour indices which, if unspecified, are assumed to be contracted between the two quarks in the same current.
The construction of the chiral counterpart to Eq. (2.9) proceeds in two steps: • In the first step, one constructs the chiral structures describing the product of two fermionic currents.These structures must possess the same chiral transformation properties of the corresponding quark currents and are obtained by exploiting the quark-hadron duality between the Lagrangians of Eqs.(2.3) and (2.4).At low energies, one has and taking the functional derivatives of the QCD and the χPT actions with respect to the external sources one can readily find the chiral counterparts to the various Dirac structures.For instance, up to order O(p 2 ) one finds where in the previous expressions we have defined the chiral currents L µ and R µ (2.12) • In the second step, one decomposes the product of quark currents into irreducible representations of the flavour algebra by defining appropriate projectors.These are to be applied as well to the chiral realisation of the quark currents in order to obtain the desired operators in the chiral Lagrangian, classified according to the irreducible representation of the flavour algebra they belong to (see e.g.[30][31][32]).Further details are discussed in App. A.
After carrying out the program outlined above, we finally reproduce the ∆S = 1 chiral Lagrangian of Ref. [30], which takes the following simple form where λ ≡ 1 2 (λ 6 − iλ 7 ) is responsible for the s → d flavour transition and we have specialised Q = 1  3 diag(2, −1, −1) to be the charge matrix for quarks.Out of the pieces making up Eq. (2.13), the first one transforms in the (27 L , 1 R ) representation of the flavour group, while the second and the third ones transform in the (8 L , 1 R ) and (8 L , 8 R ) representation, respectively.Clearly, no singlet term can have any effect on ∆S = 1 transitions.The coefficients g 27 , g 8 , g S 8 and g ew are functions of non-perturbative effective parameters, as well as of the Wilson coefficients of the weak operators, see Eq. (2.8).Their values are found to be g 8 = 3.07 ± 0.14 [30] , ( g S 8 = −1.17± 0.37 [33] , (2.15) ) (2.17) Expanding the Lagrangian of Eq. (2.13) and keeping only the contributions relevant for our analysis, we find which includes both a Kπ mixing term as well as a flavour-violating K ± → π ± X interaction, as depicted in the Feynman rules of Fig. 2. Note that, if Eq. (2.1) contains an explicit source of flavour violation, the latter Feynman rule has to be supplemented by the last contribution displayed in Fig. 1.
Differently from the leading-order chiral Lagrangian of Eq. (2.7), we are now sensitive both to x f V and x f A couplings.Indeed, the hadronic matrix element ⟨K|O i |π⟩, where O i are the weak operators defined in Eq. (2.9), receives contributions from both vector and axial-vector currents, as there are not symmetry arguments to forbid them.Again, for universal vector couplings, the K + π − X interaction vanishes because of the underlying SU (3) V chiral symmetry.
In this section we will compute the decay rate of the process K ± → π ± X in χPT.Exploiting the Feynman rules derived in the previous section, we will first analyse the tree-level contributions and then we will compare them with the results obtained at one-loop level in [24].

Tree-level contribution
At the tree level, the process K ± → π ± X is generated by the diagrams in Fig. 3.The X boson can be emitted either at the same vertex where the flavour transition takes place (first diagram) or at a different one.In the latter case (second and third diagrams) weak interactions prompt a flavour transition while the leg emission of an X boson occurs at a different interaction point.The total amplitude M = M 8 + M S 8 + M 27 + M ew receives four independent contributions proportional to g 8 , g S 8 , g 27 and g ew which are given by On the other hand, the decay rate can be written as where the total unpolarised amplitude squared is given by Assuming generation universality of the couplings, i.e. x u V,A = x d V,A = x s V,A , and taking the limit m K ≫ m X , m π , one can find the simple expression . (3.7) A few important comments are in order: • In the limit of universal vector couplings, the decay rate of K ± → π ± X becomes independent of these couplings as a result of the underlying SU (3) V chiral symmetry.
• The enhancement factor (m K /m X ) 2 in Eq. (3.7) for small m X is conceptually similar to the enhancement obtained in [24,25].This enhancement here is produced by the longitudinal component of the polarization vector: .
In order to see where we stand, we write the branching ratio of K ± → π ± X as where F /4π and B(K → µν) ≈ 64%.Moreover, we assume the equality f K = f π which holds in the SU (3) chiral limit.Finally, exploiting the E949 measurement B(K + → π + νν) = (1.73 +1. 15  −1.05 ) × 10 −10 [35], we obtain the 2σ level constraint BR(K + → π + X) ≲ 4 × 10 −10 .As a result, we find the following bound where the charges x u A and x d A are typically expected to be of order one.The above result will be fully confirmed by the numerical analysis of Sect. 4.

One-loop contribution
In this section, we will calculate the one-loop contributions to the flavour-violating process K ± → π ± X.At the quark level, the Feynman diagrams generating the underlying s → d transition are displayed in Fig. 4. Notice that these diagrams are sensitive to different couplings Figure 4: Feynman diagrams contributing to the process K ± → π ± X at the one-loop level.
of the X boson to quarks.Summing up all contributions, the full amplitude reads where W with m i being the mass of the up-type quark running in the loop and we have defined the chiral charges as The divergences, originating from the nonrenormalizability of our model, can be interpreted within a hard cutoff regularization scheme as 2/ϵ + log(µ 2 /m 2 i ) = log(Λ 2 /m 2 i ) where Λ is the UV cutoff.In specific renormalizable models, Λ will be identified with the mass scale of particles belonging to the NP sector which will provide a UV completion of our model.
In the limit of universal couplings, i.e.
, and keeping only the dominant loop effects stemming from the exchange of the top quark, we obtain where The inclusion of the above loop effects in the decay rate of K ± → π ± X can be implemented by the following replacement in Eq. (3.6): As a result, we can estimate the relative size of loop effects and tree-level ones as where f (x t ) is a model-dependent loop function which depends on the specific UV completion of our effective theory.Therefore, we have learned that loop-effects have a similar size of treelevel contributions.However, while the former suffer from sizeable uncertainties, the latter provide a robust model-independent result.Moreover, we also remark that loop and tree-level contributions generally depend on different couplings and therefore the comparison in Eq. (3.14) is valid only in the universal scenario 4 Flavour bounds vs. beam-dump and collider searches We are ready now to exploit the results derived in the previous section, in order to explore the capability of the process K ± → π ± X to unveil new light vector bosons.We are going to use the DarkCast package [21,22], which enables us to derive bounds on vector and axial couplings of models entailing new spin-1 states by imposing current and future experimental constraints on several processes.In Fig. 5, we show the bounds in the (m X , g x ) plane arising from a variety of beam-dump and collider searches [22] as well as from the flavour changing process K ± → π ± X discussed in this paper.The three plots refer to the benchmark models dubbed as axial, chiral and 2HDM [22] which differ for the values of the x V,A charges, see Table 1.
Table 1: Charges of the SM fermions under X boson interactions for the models considered in Ref. [22] where, for simplicity, flavor universal couplings have been assumed.

Conclusions
Extensions of the SM entailing new feebly interacting massive particles with sub-GeV masses are currently among the most studied scenarios of NP.In particular, comprehensive analyses aiming to probe a light spin-1 boson X featuring general couplings to quarks and leptons have been carried out in the literature exploiting beam-dump and collider searches.
In this work, we revisited the flavour constraints to this scenario by means of the rare decay K ± → π ± X.In particular, we extended previous studies by constructing the most general ∆S = 1 chiral Lagrangian as induced by weak interactions up to the order O(p 4 ).
The lowest-order O(p 2 ) terms of our χPT capture the effects to K ± → π ± X discussed in [25], which are loop-induced and suppressed by the fifth power of the Cabibbo angle.Instead, the inclusion of subleading O(p 4 ) terms in the chiral expansion generates K ± → π ± X already at the tree-level and the related amplitude is only singly Cabibbo suppressed.As a result, rather surprisingly, the two contributions turn out to be of comparable size, see Eq. (3.14).
However, while the tree-level weak effects discussed in this work are model-independent, the loop-induced contributions of Ref. [25] are instead sensitive to the specific UV completion accounting for the mass of the spin-1 boson.
In conclusion, we have shown that the process K ± → π ± X sets the strongest to date modelindependent bound on the diagonal axial-vector couplings to u, d, s quarks of a light X with m X < m K − m π , superseding the bounds arising from beam-dump and collider searches.The dark shaded area represents the tree-level K ± → π ± X bound obtained in this work.Limits from beam-dump and collider searches are obtained with DarkCast [22] and are shown for the purpose of comparison for the three benchmark models given in Table 1.

A Details of the chiral Lagrangian construction
In this appendix we will provide an extensive derivation of the chiral Lagrangian describing ∆S = 1 transitions.The operators of Eq. (2.9) have the form Let us first discuss how to identify those combinations of four-quark operators belonging to irreducible representations of the chiral group [30][31][32].From the invariance under the flavour group U (3) F ≡ U (3) L × U (3) R , we know that the LL currents transform as the 81-dimensional representation of (3 ⊗ 3) ⊗ (3 ⊗ 3) of U (3) L which can be further decomposed into irreducible symmetric or antisymmetric representations having dimension 1, 8, 10, 27.
In particular, one has symmetric-symmetric combinations SS transforming in the 1 ⊕ 8 ⊕ 27 representations and antisymmetric-antisymmetric combinations AA transforming as 1 ⊕ 8 representations.We can disregard the symmetric-antisymmetric and antisymmetric-symmetric combinations transforming in the 8 ⊕ 10 representations because they cannot be generated by the operators in (A.1) which are symmetric under the simultaneous exchange of upper indices with the lower ones, (i, k) ↔ (j, l).In any case, the particular combinations of quark currents transforming in each one of these representations can be obtained by projecting the fundamental structure in (A.1) on the orthonormal basis of irreducible representations of (3 ⊗ 3) ⊗ (3 ⊗ 3): {(e a 27 ) S S , (e a 8 ) S S , (e 1 ) S S , (e a 10 ) A S , (e a 8 ) A S , (e a 10 ) S A , (e a 8 ) S A , (e a 8 ) A A , (e 1 ) A A } .
This task is accomplished by making use of the tensorial product defined as where we exploited the decomposition T = T a M r N e a M r N .Now, the fully symmetric singlet and octet basis elements are whereas the fully antisymmetric singlet and octet basis elements are The symmetric-symmetric 27-plet basis element is harder to construct and a better strategy is to extract the corresponding component by subtracting from a fully symmetric tensor its octet and singlet parts, namely = a In principle, one should consider also the structure Tr (λ a (U † χ + χ † U )) along with Tr (λ a L µ L µ ).However, these additional structures induce vacuum misalignment effects through Goldstone tadpoles and can be rotated away by properly redefining the Goldstone fields [30].The 27-plet Lagrangian term is finally given by In the expressions (A.8) and (A.9), the parameters a S 8 , a A 8 and a 27 parametrize our ignorance about the hadronization dynamics and are to be determined experimentally.
LR currents: Since left-handed and right-handed currents transform under different U (3) groups, these combinations correspond to the product of a 3 ⊕ 3 = 1 ⊕ 8 representation in each chiral sector, resulting in four possible different structures transforming as (1 L , 1 R ), (8 L , 1 R ), (1 L , 8 R ) and (8 L , 8 R ).We first identify the associated orthonormal basis , (e a 1 L ,8 R ) kl ij = which will be then used in order to project the appropriate structure onto the low-energy operators possessing definite chiral transformation properties.Exploiting the completeness relation we can recast Eq. (A.2) in the following form L EW, LR eff = [t δδ LR ] jl ik (q i L γ µ q Lj )(q k R γ µ q Rl ) −

2N C
[t λλ LR ] jl ik (q i L γ µ q Lj )(q k R γ µ q Rl ) .(A.12) At this point the Fierz identity (q i L γ µ q Lj )(q k R γ µ q Rl ) = −2(q i L q Rl )(q k R q Lj ) (A. 13) can be used in order to identify the various operators.The leading order chiral structure that is compatible with an (8

Figure 1 :
Figure 1: Feynman rules for the lowest-order chiral Lagrangian

Figure 2 :
Figure 2: Feynman rules for the chiral Lagrangian of weak interactions.

Figure 5 :
Figure5: The dark shaded area represents the tree-level K ± → π ± X bound obtained in this work.Limits from beam-dump and collider searches are obtained with DarkCast[22] and are shown for the purpose of comparison for the three benchmark models given in Table1.