Running beyond ALPs: shift-breaking and CP-violating effects

We compute the renormalization group equations (RGEs) of the Standard Model effective field theory (EFT) extended with a real scalar singlet, up to dimension-five and one-loop accuracy. We compare our renormalization results with those found in the shift-symmetry preserving limit, which characterizes axion-like particles (ALPs). The matching and running equations below the electroweak scale are also obtained, including the mixing effects in the scalar sector. Such mixing leads to interesting phenomenological consequences that are absent in the EFT at the renormalizable level, namely new correlations among the triplet and quartic Higgs couplings are predicted. All RGEs obtained in this work are implemented in a new Mathematica package — ALPRunner, together with functions to solve the running numerically for an arbitrary set of UV parameters. As an application, we obtain electric dipole moment constraints on particular regions of the singlet parameter space, and quantify the level of shift-breaking in these regions.


Introduction
Singlet scalars are one of the most promising candidates of beyond the Standard Model (BSM) physics.Not only can they provide solutions to long-standing puzzles of the SM, but their elusive nature can also explain their challenging detection, in spite of the increasing experimental efforts which span a wide range of energies and sophisticated techniques [1].
If a new scalar particle is ever discovered, compelling questions will be addressed which could impact significantly our knowledge of Nature.One of these questions is how much the exotic scalar can interact with the only other we know, the Higgs boson.If the Higgs is afterall a composite particle, a plausible solution to the electroweak (EW) hierarchy problem [2,3], it should be accompanied by several other scalar "siblings", most predicted to transform as singlets under the SM gauge group [4][5][6][7][8].The size of the scalar interactions, being UV dependent, could therefore provide valuable information about the underlying dynamics responsible for EW symmetry breaking (EWSB) [4].Moreover, the Higgs-singlet portal is one of the simplest avenues to generate the dark matter relic density we observe [9][10][11], or even a first order phase transition, in turn connected with the possibility of explaining baryogenesis [12][13][14][15].The amount of CP violation the singlet carries would be then a key property to determine the viability of this scenario.
Another prominent question to face will be how much the new singlet looks like an axion-like particle (ALP).The existence of ALPs is a common prediction of more fundamental constructions, like string theory [16,17].They are furthermore motivated by another fine-tuning problem in the SM, the strong CP problem, which is the fine adjustment one is required to make (of more than ten orders of magnitude) to explain the absence of CP-violation in the QCD vacuum.To successfully solve this problem, the singlet interactions would have to preserve a shift-symmetry of very high quality [18][19][20][21].
At the renormalizable level, the answers to these questions lie on measuring the strength of only a few singlet interactions with the SM particles.For instance, in the absence of mixing, the communication between a scalar singlet and the SM is entirely dictated by triplet and quartic interactions which are in turn related by the Higgs vacuum expectation value (VEV).Instead, if the two scalars mix, the singlet can also develop universal couplings to the SM fermions which are Yukawa suppressed, and moreover incompatible with a shift-symmetry.
Effective corrections can however change the answer to these questions, while renormalization group (RG) running can affect their stability.Therefore, in this work, we present the one-loop renormalization group equations (RGEs) characterizing the evolution across energy scales of all singlet interactions with the SM, up to mass dimension five.The RGEs of the singlet effective field theory (EFT) were previously obtained in two particular limits: ignoring CP-odd couplings (under the assumption that the singlet is a pseudoscalar) [22]; or including only shift-preserving interactions, that is assuming the singlet is an ALP [23,24].Our work is therefore the most complete study of the singlet EFT presented in the literature so far, not only at the level of RGEs, but also in what concerns all effective contributions to the scalar mass mixing.We obtain the running of the Wilson coefficients both in the high-energy (HE) and in the low-energy (LE) EFTs, the latter resulting from integrating out the heavy top quark, the Higgs and the Z and W gauge bosons.The matching between the two theories is also obtained at tree level, including the effects which arise from scalar mixing.The shift-symmetric limit of our results is commented throughout the work.
As an application, we obtain the electric dipole moment (EDM) constraints on particular UV scenarios taking the full running of couplings into account.In partic-ular, we set constraints on the CP violation arising from a CP-even BSM scenario, due to the mixing with the Yukawa couplings of the SM.We also exemplify how the experimental constraints of a generic singlet can be directly compared with those of an ALP.This comparison is based on the analysis presented in Ref. [25], where the shift-symmetry flavour invariants induced by the singlet-fermionic couplings were constructed.
All the results obtained in this work are provided in a new Mathematica package, ALPRunner [26], that can be used to study in detail the phenomenology of the singlet EFT.The main functionalities are described in App. A. Using this tool, it is straightforward for the user to find the whole set of EFT parameters generated at an IR scale given the inputs defined in the UV.A procedure is also implemented such that, given low-energy bounds, constraints can be placed on the UV parameters of the theory.
The structure of this work can be easily understood from the table of contents.

The complete singlet EFT
The renormalisable Lagrangian of the SM extended with a real scalar singlet (s) reads [27][28][29]: where we have used e R , u R and d R to denote the right-handed (RH) leptons and quarks; while l L and q L are the left-handed (LH) counterparts.In turn, y u , y d , and y e are the SM Yukawa matrices.We represent the Higgs doublet by ϕ = (ϕ + , ϕ 0 ) T and its charge conjugate by ϕ = i σ 2 ϕ * .G A µν , W a µν , and B µν are the field strength tensors of the SM gauge groups SU (3) C , SU (2) L , and U (1) Y , respectively.The corresponding gauge couplings are denoted by g i , with i = 1, 2, 3.The dual strength tensor of the gluon field is defined as G A,µν ≡ 1 2 ϵ µνρσ G A,ρσ and similarly for the other gauge fields.We use the minus sign convention for the covariant derivative.
At dimension-five and assuming lepton number conservation, a minimal basis for the SM+s EFT is given by the following operators [22]: where a sψϕ are complex matrices in flavour space to account for both CP-conserving and CP-violating interactions.In our notation, all a i ≡ a 0 i /Λ are dimension-full couplings, with Λ denoting the cutoff scale up until which the EFT is a valid expansion.In what follows, we assume s is a pseudoscalar.The previous EFT and the results to obtain next are however valid beyond this assumption.
To order v/Λ, the dimension-five Wilson coefficients are only renormalized by other dimension-five couplings: The anomalous dimensions γ (1) were computed in Ref. [22] for the CP-even sector in isolation.We generalize Eq. 3 by including the contributions from CP-odd couplings.
In this more general scenario, dimension-five terms renormalize lower dimensional interactions as well, due to the presence of light mass scales in the theory.This effect is absent assuming only CP-even interactions as, in this case, there is a Z 2 symmetry acting on s → −s, under which the (non-) renormalizable sector of the EFT is (odd) even.
Considering renormalizable couplings, we therefore expect, schematically, the following structure: where m = m s , µ.The RGEs of the CP-odd couplings in Eq. 4 are presented for the first time in this work, as well as the contributions stemming from γ (5) .

A comparison with the ALP basis
In order to account for the most generic interactions, we did not impose shiftsymmetry in the EFT defined in Eq. 2, as commonly done in studies of axions and ALPs, which arise as the Goldstone bosons of a spontaneously broken global symmetry.In particular, the axion is a prediction of the Peccei-Quinn mechanism [18][19][20][21] to solve the strong CP problem, which requires an exact shift symmetry in the axion Lagrangian, broken at the quantum level by the anomalous interaction with gluons.Upon QCD confinement, such interaction generates a periodic potential for the axion, V (s) = V (s + 2πf s ), with f s denoting the field decay constant.Such potential also predicts a strict relation between the mass and decay constant for the axion field, that ALPs are not required to abide.
Up to a bare mass term, the minimal set of ALP interactions can be described by the following EFT [30]: where ψ runs over q L , l L , u R , d R , e R , c ψ are hermitian matrices in flavour space and X = G, W, B. When the gauge boson sector of this theory is considered in isolation, the 2π periodicity imposes that c X ∈ Z [31].While the shift s → s + 2πf s modifies the interaction Lagrangian, the action is left invariant whenever this quantization condition is fulfilled.It is well known that a chiral rotation produces, however, additional interactions which modify this condition [32].In other words, the requirement c X ∈ Z follows from the ALP periodicity only in the basis of Eq. 6.
Consequently, in this basis, the running of c X couplings is expected to vanish1 , as found in previous works [22][23][24] (up to two-loop accuracy in the gauge couplings).On the other hand, there is no constraint on the running of the derivative couplings.
In the interaction basis considered in this work (Eq.2), the singlet periodicity is explicitly broken.Therefore, no quantization constraint applies2 .The previous discussion still connects to our work in two main ways: • The more restrictive running in the SM+s EFT invariant under discrete ALP shifts constitutes an important case limit of our study, which should be recovered from the complete RGEs to be derived next.
• Using the freedom to write the derivative interactions between the ALP and fermions in terms of chirality-flipping operators, it is straightforward to find the conditions to preserve shift-symmetry in the basis of Eq. 2. Hence, the ALP phenomenology can be studied using the EFT basis of this work, even though it is not possible in general to map the couplings back to the chirality-preserving basis in Eq. 6.
Indeed, by integrating by parts the second term in Eq. 6 and using the SM equations of motion (EOMs), we obtain the fermionic operators in Eq. 2 with the following coefficients [22]: These three conditions are sufficient to preserve a continuous shift-symmetry at the perturbative level in our EFT, up to terms in the scalar potential.However these conditions are implicit and do not, in general, allow to identify the c-matrices given a set of a sψϕ couplings.
Only under specific assumptions, such as Minimal Flavour Violation or in the presence of just one fermion generation coupled to the ALP, it is possible to trade completely c l , c q by their RH counterparts and find one-to-one correspondence between the chirality-preserving and chirality-flipping couplings [22][23][24].In those cases, to match the results derived using the basis in Eq. 6 and ours, shifts on the anomalous couplings also need to be taken into account, that arise from the axial anomaly equation [33]: Assuming this set of boundary conditions (Eqs.7-10), the results obtained with any of the two bases must agree within an accuracy of O(α/Λ).
Finally, we remark that the general basis adopted in this work can be used to study the fate of shift-symmetric scenarios under running, without having to refer implicitly to the matrices c ψ .This has become possible with the work presented in Ref. [25], where the complete set of flavour invariants associated to the singlet shift-symmetry was constructed, and shown to form a closed set under RG evolution.

High-energy anomalous dimensions
We renormalize the SM+s EFT by computing the divergences in the basis of independent Green's functions of Ref. [22]; see App.B. We use the background field method (BFM) and work in the Feynman gauge with d = 4 − 2ϵ spacetime dimensions.The computations are based on dedicated routines that rely on FeynRules [34], FeynArts [35], and FormCalc [36].All of our results have been cross-checked with matchmakereft [37], from which we have obtained the µ 2 contributions.
The one-loop divergences of the renormalizable couplings read: As we work in BFM approach, the divergences of the gauge couplings are automatically fixed by the wave function renormalization (WFR) factors of the gauge bosons [38].The latter are not modified by the presence of CP-odd terms and can therefore be read from Ref. [22].
We obtain the following divergences for the effective couplings3 : where the r-couplings are associated to redundant operators which will be redefined away; see App.B. All other counterterms vanish up to the EFT and loop order we are interested in.
The divergences obtained above can be projected onto the minimal basis of Eq. 2 using suitable field redefinitions, which can be implemented via the EFT EOMs within O(1/Λ) accuracy.In this procedure, we neglect O(α/4π) corrections that arise from the anomaly equations which relate chirality-preserving operators with the chiralityflipping ones in our basis; see Eqs. 8-10.This would lead to O(α/4π) 2 corrections which are beyond the scope of our analysis.The on-shell results are obtained after the following replacements4 : a seϕ y † e y e + y e y † e a seϕ − Table 1: Structure of the dimension-five anomalous dimension matrix.Only BSM and the leading SM contributions are kept.For instance, y t (the top-quark Yukawa coupling) and g 2,3 are the largest contributions in their respective sub-classes.Terms in blue show the contributions that deviate significantly from naive dimensional analysis; see the text for details.
We can now obtain the anomalous dimensions of all couplings in the singlet EFT.Taking the CP-even limit of our results, we find exact agreement with those presented in Ref. [22].
The complete expressions for the RGEs are provided in the ALPRunner package [26] published along this work.Instead, a more graphic picture is presented in Tabs. 1 and 2, where we show the structure of the anomalous dimension matrices, resulting from one-loop insertions of the SM and new physics (NP) couplings up to O (1/Λ).Each Wilson coefficient in the rows is renormalized by the non-zero coefficients in the columns.We have highlighted in blue the contributions which deviate from naive dimensional analysis [39] by at least one order of magnitude, and which are therefore expected to play an important role in the singlet phenomenology.Most of the zeros in these tables are trivial.For example, it is not possible to insert an s 5 operator in a one-loop diagram without leaving at least 3 external singlet legs, which explains the non-renormalization results in the first column of Tab. 1.On the other hand, operators with n singlet fields can renormalize others with m > n singlet fields via the insertion of renormalizable NP interactions.Interestingly, both sXX and sX X operators renormalize the fermionic interactions of the singlet, while sXX renormalizes also the mixed interactions in the scalar sector.
At one-loop, in the most generic basis, we find that the pseudoscalar interactions with gauge bosons run proportionally to the gauge coupling, In fact, if we redefine a sX, s X ≡ g 2 X c X, X , we find that the c X, X couplings do not run at all [22][23][24].This follows from a non-trivial cancellation of the diagrams shown in Fig. 2 and agrees with the considerations presented in the previous section.Indeed, considering the gauge sector in isolation, the CP-even couplings c X cannot be multiplicatively renormalized since they are multiplied by an effective angle that is 2π periodic.The additional pseudoscalar interactions in the EFT could in principle break the scale invariance of these couplings, as they explicitly break this periodicity.

< l a t e x i t s h a 1 _ b a s e 6 4 = " 2 4 t M v W S t P B o n B k 2 I S b V x G T a m 5 l A = " >
r t G y x D r s K v + q 9 2 y 2 / k / M m 2 5 5 I + 8 f d 9 8 N z 9 T g / F g 8 a h 3 + 3 5 B y g e r 1 r b z n y 9 O 9 / a + G c Y p i O M W p S c 5 3 q j 6 v L 2 v t 7 n U q q r e s Z 4 o 4 t r g 2 8 b Z 7 5 L u u 7 + X / v b r 0 7 m d / v A + d n 5 x W k 5 x D l 0 X j m / O + + c U 4 c 1 n j X 6 j Z P G 6 3 v / N R 8 1 n z a f 1 9 A 7 G / O Y Z 8 7 K a r 7 4 H 1 p L 8 a c = < / l a t e x i t > s V V Figure 2: One-loop diagrams that contribute to the s → V V divergent amplitude, V denoting any SM vector, at order O (1/Λ).
However, no one-loop diagram can be constructed with a single insertion of sψψϕ (or other) terms.On the other hand, the CP-odd couplings c X are not renormalized because X 2 is the trace of the conserved energy momentum tensor [40].
Regarding the singlet-fermion interactions, our results show that they produce at low energies interactions between the pseudoscalar and the Higgs, both at the renormalizable and non-renormalizable levels.These effects are absent in shift-symmetric scenarios, that is for an ALP, as in this case the fermionic interactions grow with momentum and hence can only renormalize themselves [23].Furthermore, the presence of the SM Yukawas in the RGEs of the singlet fermionic couplings implies that CP-even UV setups are not, in general, stable upon running.

Matching at the electroweak scale
After the Higgs field develops a VEV, at energies v ∼ 246 GeV, the SM particles develop masses.Given the large hierarchy between the masses of the EW bosons and the top quark, in comparison to that of the remaining particles in the SM, the former can be integrated out rendering a simpler low-energy EFT (LEFT).While this theory can be fully general and matched to different models above the EW scale, we present in Sec.4.2 the tree-level matching results assuming that the singlet s is the only BSM d.o.f. in the UV.With this aim, we start by diagonalizing the mass matrix of the scalar sector.

Scalar mixing
The scalar mass matrix induced by the interactions in our EFT reads where and In the expressions above, v is the observed Higgs VEV, determined from muon decay measurements, while v s denotes the singlet VEV which solves the tadpole equations Moreover, (v, v s ) is guaranteed to be a minimum of the scalar potential if it satisfies the following conditions: The mass matrix in Eq.48 can be diagonalized by the rotation with .
(54) To obtain this expression, we have used Eqs.50 and 51 to rewrite (µ 2 , κ sϕ ) in terms of the remaining parameters in the EFT.In the absence of CP-odd interactions, v s = 0 is a solution of the tadpole equations consistent with the SM assignment of the Higgs parameters, i.e. µ 2 = λv 2 .In this limit, the weak and the physical bases of the scalar sector automatically coincide.From the expression above, it is also clear that at the renormalizable level, if the UV parameters are chosen so that v s vanishes, no mixing with the Higgs boson can arise.Neglecting the effective interactions, our expressions agree exactly with the results obtained in previous works [12,15,41].Nonetheless and in spite of the popularity of the singlet framework, there are several studies in the literature exploring the consequences of scalar mixing that neglect the implications of v s ̸ = 0.
The correlation between the singlet VEV and the scalar mixing is broken by the dimension-five interactions.Indeed, if the coupling a s is generated by the UV model, h − s mixing can arise even for v s = 0.This scenario is very particular as it implies an alignment between renormalizable and effective interactions, κ sϕ ∼ a s v 2 , leading to Therefore, in full generality, we keep track of the v s dependence in the physical couplings arising from this framework: where for instance the quartic couplings read as λh = 6 (λ − a s v s + 4θva s ) , (57) λs = λ s − 24 (θva in the small mixing limit.They are related to the physical scalar masses via the relations: It is also interesting to point out the relation between the triple and quartic Higgs couplings in the presence of the singlet, taking into account all effective contributions up to O(1/Λ): or, in terms of physical parameters, From this expression, it follows that the SM relation between the two physical Higgs couplings can only be broken by non-renormalizable singlet interactions.
A similar effect occurs in the singlet-Higgs portal couplings.To give an example, in the particular case of v s → 0, we find that: where θ is now determined from Eq. 55.The parameters on the right-hand-side of this equation can be further traded by the physical ones, using the results in App.C: The gauge sector receives also corrections from a non-vanishing v s .After redefining the gauge couplings, as well as the gauge bosons such that the product g i V i remains invariant, the physical vector masses read the same as in the SM 5 .Furthermore, the singlet develops renormalizable interactions with the EW gauge bosons due to the mixing with the Higgs particle: where In turn, by mixing with the singlet, the Higgs particle can also couple to all gauge bosons at the non-renormalizable level: where the ellipsis represent additional 4-body interactions, with at least 2 heavy fields, which will not be important in the discussion that follows.The CP-even couplings above are given by: âsAA =c 2 w a sB + s 2 w a sW , âsZZ =s 2 w a sB + c 2 w a sW , (71) âsAZ =2s w c w (a sW − a sB ) , âsWW =2a sW , âsGG =a sG , âhV V = − θâ sV V , while the CP-odd ones are obtained upon the replacement V → V and are in agreement with the expressions in the literature [43].Note that, in the limit a sW = a sB , the singlet coupling to a mixed state of gauge bosons vanishes.
The QCD vacuum also receives contributions from the singlet VEV, as expected from the axion mechanism, Turning to the fermionic sector, we perform the usual unitary rotations into the mass basis [44]: with the usual relation to the fermion masses: The corresponding Lagrangian, including the interactions with the scalar sector, reads:

The low-energy singlet EFT
We can now integrate out the heavy d.o.f. to obtain the LEFT description of the interactions among light particles (including the singlet).To dimension-five, the most general and minimal LEFT Lagrangian can be written as where the non-renormalizable couplings are dimensionfull, ã ≡ ã0 /v.The parameters in this tilde basis can be fully fixed at the scale µ = v by requiring that both theories, before and after EWSB, describe the same physics.While the RGEs we will derive next include all the couplings in Eq. 84, below we present the matching equations assuming that the UV model is well described by the SM+s EFT, presented in Sec. 2. At tree level, we obtain: ãsV = âsV , and ãs with all other Wilson coefficients vanishing.Special care needs to be taken with respect to contributions of O(κ s 2 h / m2 h ).While naively they seem to be of the same order as that of operators we are neglecting in the LEFT, in fact κs 2 h / m2 h ∼ O(1/v) in the small mixing limit.On the contrary, since ĉhψ ∼ O( mψ /v), the second term in Eq. 89 should be neglected in our study.Large mixing angles can however spoil this power counting; see Eq. 59 and App.C.
Finally, we remark that with the results of the previous section, where we have obtained the full matching to the physical basis including operators with more than one heavy particle, the accuracy of the previous relations could be easily improved beyond tree level.and similarly for β ãsG upon the replacement ãψG → iã ψG .The first contribution in Eq. 92 stems entirely from the running of the gauge coupling, such that if all other contributions vanish, c G ≡ ãs G /g 2 3 remains scale invariant.This occurs if the UV physics above EWSB can be described by the SM+s EFT with θ ≪ 1, as in this case cψ couplings are O(1/Λ).We might still wonder about the additive contributions to this RGE due to the presence of dipole operators, in view of the arguments presented in Sec.2.1, as these operators could be generated by additional field content in the UV without spoiling shift-symmetry.Nonetheless, if this symmetry is assumed, that is in the ALP scenario, ãψG cψ terms are at most O(Λ −2 ).
Therefore, up to the EFT expansion order considered in this work, the running of ãs X couplings is consistent with the expectations discussed in Sec.2.1.To test their robustness at one-loop level, the inclusion of dimension-six operators in the ALP EFT is required.
Moreover, we find that no contributions to the mass are generated by the LEFT running assuming the singlet is an ALP.For a generic pseudoscalar however, apart from multiplicative contributions, the mass can be renormalized by the trilinear coupling κ2 s as well as by m2 ψ c2 ψ and m3 ψ ãψ terms.In turn, the singlet mass renormalizes the mass of fermions in the LEFT.This contribution is again absent once matching this theory to the ALP EFT in the UV, up to the expansion order we are considering.
Regarding the running of fermionic couplings, we obtain: simplifying into the first three terms upon the use of Eqs.85-91 in the absence of scalar mixing.Similarly to this case, ãsV cψ are the only new terms in the RGEs of both ãψ and ãψV couplings relatively to the CP-even scenario explored in Ref. [22].The RGE of the singlet self-coupling is also modified in the presence of CP-odd interactions: where the large deviations from naive dimensional analysis are apparent.
Besides, the running of the pure CP-odd couplings in the theory is given by: 6 ALPRunner usage: phenomenological application In this section, we show the non-trivial impact of running in phenomenological studies of the singlet by computing the strongest EDM constraints on particular UV scenarios (where only two BSM couplings are assumed non-zero).We use the EDM expressions obtained in Ref. [45] 6 .To include all the running effects computed in this work, namely contributions from scalar mixing, we take into account the energy evolution of the couplings involved in such expressions 7 .
Before presenting the results, we illustrate below the main features of the new ALPRunner package.Our EDM analyses are based on dedicated routines which can be generalized to scan the HE parameter space and impose bounds on more general UV scenarios than those considered in this section.The functions available are enumerated in Tab. 3.
Users can download the package from the link in Ref. [26] and initialize it in the following way: Out [1]= SDB, JMR, MR. arXiv:2306.08036 The complete RGE of a given Wilson coefficient can be obtained via the function 'rge$variable name' (e.g.rge$asB).Additional functions have been included to solve numerically the running equations for user-given boundary conditions (BCs).In the code line below, the SM BCs (defined at the scale Λ ′ = 172 GeV) are stored in the list 'smLOWBC'.The following function finds then the values of the SM couplings at a user-set UV scale (Λ = 10 3 GeV in the example below), which are compatible with the given low-scale BCs.
In [2]:= rgeSMtoΛreset[smLOWBC,10 3 ,172]; The BSM BCs are stored in a different list, 'bsmBC', defined at the Λ scale.The nomenclature used for the Wilson coefficients in the package is the same as in Eq. 2.Moreover, we have included a function 'basisChange' to translate operators from the derivative basis in Eq. 6 to the EFT basis used in this work.After defining the BCs, the function 'rgePSEFTparamsolvereset' solves the HE RGEs parametrically 8 .For instance, in the example below, two UV parameters remain free, a s B ≡ asBt and a sB ≡ asB: In [3]:= rgePSEFTparamsolvereset[bsmBC,10 3 ,172, {{asBt,asBtparam},{asB,asBparam}}]; In the next step, we show how to match automatically the HE and the LE EFT couplings, as described in Sec. 4. In this procedure, the UV values of the Higgs coupling and scale, µ and λ, are fixed in order to comply with the experimental constraints9 v = 246 GeV and mh (v) = 125 GeV [47].The singlet VEV is then directly obtained from Eqs. 50 and 51, as well as the mixing angle defined in Eq. 54.Points (v, v s ) which are not minima of the scalar potential are discarded.
Under these conditions and with fixed v s , the LE BCs are set by the function 'bcLE', as written below.
In [4] rgePSEFTparamsolve,{2*10 -6 ,3*10 -6 }]; The first argument in this function requires the matching conditions (defined internally), which are defined in terms of the UV couplings that result from solving parametrically the HE RGEs, given in the second argument.The latter have been evaluated for specific values of (a s B , a sB ) = (2 × 10 −6 , 3 × 10 −6 ).
Given the LE BCs, the following function solves numerically the LE RGEs, outputing the EFT parameters at a given IR scale (chosen below to be 5 GeV): In [5]:= rgeLEreset[boundaryConditionLE,172,5]; After these steps, any LE Wilson coefficients can be determined at a scale < v, e.g.If the user is interested in generating several UV points, to understand namely which of those are bounded by IR constraints, the following scan function can be used: In[100]:= scanHEspace[bsmBC,1000,172,5, {{asBt,asBtparam},{asB,asBparam}}, {10000,{-5*10 -6 ,+5*10 -6 },{-5*10 -6 ,+5*10 -6 }}, Observable, "File.m"]; where the first argument is the list of HE BCs; the second, third and fourth arguments are, respectively, the UV, matching and IR scales; the fifth is the list of UV parameters that the user chooses to let free (and which are not defined in bsmBC); whereas the sixth entry corresponds to the number and interval of random points to be generated.Finally, the two last arguments encode the expression -defined by the user -that will be evaluated at the IR scale, for each point that was generated, and the file where the information will be stored.In the analyses presented below, 'Observable' is a list of expressions of the different contributions to EDMs induced by the singlet interactions -defined in Eq. 9 of Ref. [45] -that we have subsequently compared with the relevant experimental bounds, in order to constrain a given UV scenario.Some results are shown in Figs. 3 and 4.
UV gauge scenarios.Assuming that only (a s B , a sB ) are generated by the UV, the constraints presented in the left panel of Fig. 3  for m s = 5 GeV.This bound is set by measurements of the electron EDM using the molecule ThO [48] and agrees with the results reported in Ref. [45].The results are largely insensitive to m s .Tree level mixing effects can have a non-negligible contribution to the fermion coupling and therefore modify the bound obtained above.For instance, assuming κ sϕ ∼ −0.1, which leads to θ ∼ O(10 −3 ), we find a stronger constraint on the gauge couplings as represented by the blue points in Fig. 3, which turns out to be largely independent of a sB .This is because, for a (red) point in the positive quadrant that was allowed by the previous analysis, the θ contribution in Eq. 79 dominates over the others induced by running.As such, the EDM contribution, which essentially goes as Re[c e ]a sB + Im[c e ]a s B becomes a function of a s B only.However, the width of the new allowed region is not exactly constant and more points are allowed in the a sB < 0 area.This is due to a possible cancellation between the imaginary terms in Eq. 79, that is, those induced by running vs. those induced by scalar mixing.
In the right panel of Fig. 3, we have explored another UV scenario where only the singlet-gluon couplings are assumed non-zero.The leading observable setting the bounds in this case is the neutron EDM of the 199 Hg atom [49], which requires for m s = 5 GeV.This bound is within the same order of magnitude than the one obtained in Ref. [45] 10 and is insensitive to the value of κ sϕ .UV fermionic scenarios.In Fig. 4, we instead explored a UV fermionic scenario where the only non-vanishing couplings are (a sdϕ ) 11 and (a sdϕ ) 13 , assumed to take real values 11 .This setup aims to illustrate an important point: strong EDM bounds can be applied to some CP-even BSM scenarios.The CP violation induced in this case stems from the mixing of the CP-even physics with the SM Yukawa couplings 12 , induced by the fermion mass diagonalization and the RGEs (see Tab. 1).As in the previous case, the leading observable setting the bounds is the neutron EDM.
In Fig. 4, we have also represented the size of the shift-breaking invariants 13 , i.e.
for each point represented in the plot.We see that the most shift-symmetric points lie close to the horizontal axis.In particular, ALPs lie exactly in the blind direction of the plot, since the UV texture is not compatible with an exact shift-symmetry.Using ALPRunner, it is straightforward to generalize this flavour matrix and obtain meaningful bounds in the ALP parameter space.

Conclusions
In this work, we have computed the energy evolution of all couplings in the most general EFT obtained by extending the SM with a real (pseudo) scalar singlet.Our results are valid up to one-loop accuracy and order 1/Λ in the EFT expansion.The renormalization group equations have been furthermore included in a new Mathematica package -ALPRunner -with functions to solve the running numerically, as well as to match directly the high-energy with the low-energy EFT couplings, assuming that the singlet is the only BSM d.o.f. at energies above the EW scale.
We have found that CP-odd couplings in the singlet EFT are, in general, renormalized by CP-even ones due to the mixing with the Yukawa couplings of the SM.Namely, CP-even fermiophilic scenarios can generate, at low energies, CP-odd couplings between the singlet and the Higgs boson, both at the renormalizable and effective levels.This is possible due to the RGE mixing of operators of different mass dimensions, which is a common feature of both the high-energy and low-energy EFTs.Moreover, BSM operators can renormalize pure SM ones, such as the Higgs quartic coupling.At the matching level, we have computed all effective contributions to scalar mixing, which modify important relations among the physical couplings.In particular, we found that via this mixing and only at dimension-five, the SM relation between the triplet and quartic Higgs couplings can be distorted.Furthermore, it is possible for the singlet to have a vanishing VEV and still mix with the Higgs boson, an effect that is absent in the renormalizable framework, and which could be of phenomenological relevance.(While mixing effects are expected to be small from experimental searches [51], the current interpretations assume correlations that do not necessarily hold at the effective level.A re-interpretation of the experimental analyses is therefore necessary to quantify the actual size of these effects 14 .)Throughout this work, we have also discussed the shift-symmetric limit of our results and showed that they are in perfect agreement with the quantization rules imposed by the ALP periodicity.Conversely, for a generic singlet, the couplings to gauge bosons can in general run and the interpretation of the experimental bounds must take this running effect into account.
Finally, as an application of our results, we obtained EDM bounds on particular UV scenarios.We have shown that the constraints on the singlet couplings to the abelian gauge boson can be significantly impacted by the presence of scalar mixing.Generic constraints on the singlet parameter space can be obtained for more general UV setups using the package presented along this work, as we have explained in Sec. 6.The diagonalization of the scalar and fermion systems has been implemented in the package, as well as the full dependence of the EFT coefficients on the scalar VEVs.An algorithm was included to make sure the UV inputs comply with EW constraints.Finally, the shift-breaking invariants [25] of the singlet EFT have also been defined in the package, such that the phenomenology of a more shift-symmetric singlet, or that of an ALP, can be studied in full detail with ALPRunner.

Scalar
Yukawa Gauge Derivative Green's basis of the SM+s EFT at dimension-five [22].The operators labelled with R are redundant when evaluating an amplitude on-shell.

Scalar
Table 5: Green's basis of the SM+s LEFT at dimension-five [22].The operators labeled with R are redundant when evaluating an amplitude on-shell.

C Scalar couplings in the physical basis
The scalar interactions defined in the physical basis of Eq. 56 read as follows, in the small mixing angle limit: κs =κ s + v s λ s + 3θvλ sϕ − 3 v (v + 6θv s ) a s 3 + 20v L r P a 0 d d 1 R r j j a 6 u D d 4 8 O f O 7 Z N A 9 / O l w 9 9 F J c 7 e 3 n a + d b 5 y O Q 5 w j 5 5 H z v j G p e P s f G A 0 I x m Y d C T S G h L E b 5 6 f l 6 L y x N M m p R c p 7 r T a r L O / h 0 n y u p b u o Z 4 4 0 u r w 0 + b 1 z 4 L u m 7 h 3 8 c 7 r 4 8 a + 7 2 s f O D 8 5 P T c Y h z 7 L x 0 f n X e O u c O a 7 1 o D V t n r V e P / m s / b T 9 v f 1 9 D H 2 w 1 M S + c t d X + 8 X 9 d s / G p < / l a t e x i t > s V V < l a t e x i t s h a 1 _ b a s e 6 4 = " 5 j

Figure 3 :
Figure 3: EDM constraints on the UV gauge scenarios discussed in the text for Λ = 1 TeV and m s = 5 GeV.The red points are allowed in the scenarios with vanishing κ sϕ , but are further constrained to the blue region for κ sϕ = −0.1.The running of all couplings in the EFT is considered, from the scale Λ down to µ = 5 GeV.The leading observable setting the bounds is identified in the plot legend.

Figure 4 :
Figure 4: EDM constraints on the fermionic UV scenario discussed in the text for Λ = 1 TeV and m s = 5 GeV.The leading observable setting the bounds is identified in the plot legend.The colour bar identifies the size (in log scale) of the shift-breaking invariants; see Eq. 99.

s 5
Õu = s 2 u L u R Õs A = sA µν A µν ÕuA = u L σ µν u R A µν Rs□ = s 2 ∂ 2 s Õd = s 2 d L d R Õs G = sG µν G µν ÕdA = d L σ µν d R A µν Ru□ = u L D 2 u R Õe = s 2 e L e R ÕsA = sA µν A µν ÕeA = e L σ µν e R A µν Rd□ = d L D 2 d R ÕsG = sG µν G µν ÕuG = u L σ µν T A u R G A µν Re□ = e L D 2 e R ÕdG = d L σ µν T A d R G A µν Rsu L = isu L / Du L ÕeG = e L σ µν T A e R G A µν

Table 2 :
Structure of the renormalizable anomalous dimension matrix induced by effective interactions.Terms in blue show the contributions that deviate significantly from naive dimensional analysis; see the text for details.