Beyond the Standard Model Effective Field Theory: The Singlet Extended Standard Model

One of the assumptions of simplified models is that there are a few new particles and interactions accessible at the LHC and all other new particles are heavy and decoupled. Effective field theory (EFT) methods provide a consistent method to test this assumption. Simplified models can be augmented with higher order operators involving the new particles accessible at the LHC. Any UV completion of the simplified model will be able to match onto these beyond the Standard Model EFTs (BSM-EFT). In this paper we study the simplest simplified model: the Standard Model extended by a real gauge singlet scalar. In addition to the usual renormalizable interactions, we include dimension-5 interactions of the singlet scalar with Standard Model particles. As we will show, even when the cutoff scale is 3 TeV, these new effective interactions can drastically change the interpretation of Higgs precision measurements and scalar searches. In addition, we discuss how power counting in a BSM-EFT depends strongly on the processes and parameter space under consideration. Finally, we propose a $\chi^2$ method to consistently combine the limits from new particle searches with measurements of the Standard Model. Unlike imposing a hard cut off on heavy resonance rates, our method allows fluctuations in individual channels that are consistent with global fits.


I. INTRODUCTION
The Large Hadron Collider (LHC) has had two very successful runs. While no new physics beyond the Standard Model (BSM) has been discovered, we may yet expect it to show up in currently unanalyzed data or in future runs at the LHC. In the absence of discoveries of more complete models such as Supersymmetry, extra dimensions, or composite Higgs models, it is useful to study simplified models [1]. A frequent assumption of simplified models is that there are at most a handful of new particles accessible at LHC energies, while all additional new particles are too heavy to be produced. However, this raises the question: can the effects of the inaccessible new particle be truly neglected? For example, consider a simplified model with a new up-type vector like quark (VLQ). If there is a new scalar in the theory, even if the scalar cannot be directly produced, it can mediate new loop level decays of the VLQ into photons and gluons [2]. Indeed, in certain regions of parameter space, these decay modes can be dominant [2][3][4], fundamentally changing the phenomenology of the simplified VLQ model. The most "model independent" method to determine the effects of new, heavy particles is an effective field theory (EFT). An EFT is a power expansion in inverse powers of some new physics scale Λ: where f k,n are Wilson coefficients, L ren is the renormalizable Lagrangian, and O k,n are dimension-n operators. In the Standard Model EFT (SMEFT) [5][6][7], L ren and O k,n con- This approach is agnostic about the high scale new physics since any UV completion of a simplified model will match onto the BSM-EFT.
In this paper we study the BSM-EFT of the simplest possible extension of the SM, the addition of a real scalar singlet S [16][17][18]. Beyond being the simplest extension of the SM, the singlet model can help provide a strong first order electroweak phase transition necessary of electroweak baryogenesis [19][20][21][22][23]. At the renormalizable level, the new singlet only enters the scalar potential and its interactions with fermions and gauge bosons are inherited by its mixing with the SM Higgs boson. However, it is highly unlikely that a singlet scalar singlet would appear without any new physics. For example, even if it can give rise a strong first order electroweak phase transition, in order to successfully have electroweak baryogenesis new sources of CP violation are needed [24][25][26][27]. In fact, it has been shown [28][29][30] that the BSM-EFT for the real scalar singlet can provide the CP violation necessary for electroweak baryogenesis.
Our analysis will consist of two major portions: re-interpreting Higgs precision measurements in the singlet extended SM and re-interpreting searches for new heavy scalars. After EWSB, the new singlet scalar and Higgs boson will mix. Without the new EFT interactions, this mixing results in a universal suppression of Higgs boson production rates. Hence, Higgs precision measurements have a very simple interpretation [31][32][33][34][35]. However, the BSM-EFT will introduce new interactions between the Higgs boson and fermions/gauge bosons. As we will show, these can significantly alter the interpretation of Higgs measurements. A similar argument can be made for constraints coming from heavy scalar searches. At the renormalizable level, the new scalar inherits all of its interactions with fermions and gauge bosons from the SM Higgs boson. Hence, its production rates are the same as a heavy Higgs boson but suppressed by a mixing angle. Similarly, its decay rates are the same as a heavy Higgs boson suppressed by a mixing angle, except when a di-Higgs resonance is kinematically available.
That is, at the renormalizable model, the phenomenology is well defined. As we will show, with the introduction of new interactions between the scalar and fermions/gauge bosons the phenomenology can significantly change. Even though it is typically assumed that heavy new physics can be neglected, we will show that even in the simplest of all simplified models this assumption must be called into question.
This paper is an extension of work in Ref. [8], where only effective interactions between the scalar singlet and gauge bosons were considered. We should note that the full BSM-EFT was considered in Ref. [9]. However, they also considered dimension-6 SMEFT operators.
While these effects can be important, we are interested in the question of how the EFT including new particles can change the phenomenology of the simplified models. Hence, we will focus on dimension-5 operators involving SM gauge bosons, SM fermions, and the new scalar singlet. In addition, we will include the most up-to-date Higgs precision data and searches for scalar singlets. Also, we give a robust discussion of power counting the BSM-EFT and propose a new χ 2 analysis to combine heavy resonance searches with precision measurements.
In Section II we develop the BSM-EFT for the real scalar singlet.

II. MODEL
We consider the SM extended by a real gauge singlet scalar, S, and will not impose an additional Z 2 upon S. In order to focus on the effects of new physics on the scalar singlet properties, we will consider only dimension-5 EFT operators. For simplicity, we will also only focus on CP even operators. These are the lowest order effective operators that include a scalar singlet [8,9]. At dimension-5, the only SMEFT operators are those that contribute to Majorana neutrino masses [5,6,36], which are not relevant for LHC analyses. Hence, these will be neglected and the BSM-EFT will only consist of operators including the new singlet scalar.
Adapting the notation of Refs. [37], to or Λ −1 the scalar potential is: where Φ = (0, φ 0 / √ 2) T is the SM Higgs doublet in the Unitary gauge, φ 0 = h + v is the neutral scalar component of Φ, h is the Higgs boson, and φ 0 = v is the SM Higgs vacuum expectation value (vev). Since S is not charged under any symmetry, its vev does not break any symmetry and results in an unphysical redefinition of parameters [34,37]. Hence, without loss of generality we can impose S = 0.
After electroweak symmetry breaking (EWSB), the Higgs boson h and scalar singlet have the same quantum numbers and can mix: where h 1,2 are mass eigenstates with masses m 1,2 . We will assume m 1 = 125 GeV < m 2 , since the other mass hierarchy is strongly constrained by LEP [31]. With the masses, mixing, and vevs, we can now solve for five parameters in the potential These are O(v/Λ) corrections on the relationships founds in Refs. [34,37]. The free parameters of the scalar potential are then The scalar potential gives rise to important trilinear scalar couplings after EWSB: where When kinematically allowed, the coupling λ 211 gives rise to resonant double Higgs production via the decay h 2 → h 1 h 1 . The Higgs trilinear coupling λ 111 can alter the non-resonant di-Higgs rate away from SM predictions.
There are important theoretical constraints on the scalar potential, Eq. (2), to consider.
Limits from the potential effect the allowed values of λ 211 and can have a significant impact on the h 2 → h 1 h 1 branching ratio [37]. First, there are quintic terms S(Φ † Φ) 2 , S 3 Φ † Φ and S 5 that dominate at large field values and can be negative, indicating an unstable potential.
We only consider parameter space where the global minimum is inside the field value region |S| < Λ and |φ 0 | < Λ and not along the boundaries. Above the cut-off scale, it assumed new physics comes in and stabilizes the potential. Second, the potential is much more complicated than the SM and has many different minimum even inside the allowed field value regions. The singlet vev cannot contribute to the W and Z masses. Hence, the Higgs vev must give the correct masses and we only consider parameter space where the global minimum is φ 0 = v = 246 GeV and S = 0. Finally, in the scalar potential we require all dimensionless parameters to be bounded by 4π and all dimensionful parameters to be bounded by Λ.
In addition to the scalar potential, the scalar singlet obtains new interactions with SM fermions and gauge bosons [8,9]. Current measurements of the observed Higgs boson are only sensitive to third generation quarks, and second and third generation leptons. Hence, we will only consider those interactions in addition to the gauge bosons. The relevant effective interactions in the fermion mass eigenbasis for our study are then: where L 2,3 are second and third generation lepton SU (2) L doublets, Q 3 is the third generation quark SU (2) L doublet, µ R , τ R , b R , t R are SU (2) L singlets, and m µ , m τ , m b , m t are the masses of the relevant fermions. All Wilson coefficients are assumed to be real. The Feynman rules from Eqs. (2,9) can be found in Appendix A.

III. POWER COUNTING
In traditional SMEFT counting, the amplitude squared terms should be truncated to the same order as the Lagrangian. As an example, consider a baryon and lepton number conserving SMEFT amplitude to dimension-8 1 where A ren is the dimension-4 renormalizable amplitude, and A n,SM EF T are SMEFT amplitudes originating from operators at dimension-n. The amplitude squared is then As can be clearly seen, at the amplitude squared level the dimension-8 term is of the same order as the dimension-6 squared term. Hence, for self-consistency, if only the dimension-6 term is included in the amplitude, then the amplitude squared should also be truncated at According to this argument, since the interactions in Eqs. (2,9) are truncated at dimension-5 the amplitude squareds should be truncated at O(Λ −1 ). Here we note that while this is the SMEFT procedure, in the model presented the counting is more complicated due to the unknown scalar mixing angle. First, consider h 1 single production and decay. The relevant singlet scalar interactions are all dimension-5 or higher. Hence, to order Λ −2 , amplitudes for h 1 production and decay are schematically where A 5,S and A 6,S are, respectively, dimension-5 and dimension-6 operators involving the scalar singlet S. Note that due to mixing among the scalars after EWSB, in the production and decay of the mass eigenstate h 1 the SMEFT and renormalizable terms are proportional to cos θ and the singlet scalar EFT terms are proportional to sin θ. The amplitude squared is then In the small mixing angle limit the SM and SMEFT contributions dominate, and the usual power counting is valid. In the large mixing angle limit, sin θ → ±1, the cos θ terms go to zero and the amplitude squared is Hence, the dimension-5 squared piece dominates the dimension-6 terms. That is, in the large mixing angle limit we can take the full dimension-5 amplitude squared and not violate power counting rules.
For h 2 single production and decay the relevant singlet scalar renormalizable interaction comes from the potential and induces h 2 → h 1 h 1 when kinematically allowed. This process depends λ 211 . From Eq. (8) it is clear that the renormalizable piece of λ 211 is proportional to sin θ. Hence, all renormalizable contributions to h 2 single production and decay are proportional to sin θ and the amplitude is schematically Now, in the large mixing angle limit sin θ → ±1, the amplitude becomes SM-like and the SMEFT power counting is correct. While in the small mixing angle limit the dimension-5 term is the leading term, similar to Eq. (12). Hence, the leading term in the amplitude squared is the dimension-5 squared piece and the full dimension-5 amplitude squared does not violate power counting. Note that h 2 h 2 production depend on λ 221 which is not sin θ mixing angle suppressed as shown in Eq. (A2) and studied in Ref. [23]. That is h 2 h 2 production the power counting changes again.
As this discussion makes clear, the power counting in BSM-EFT depends intimately on the what parameter space is being considered and exactly what processes are under consideration. We expect that LHC limits will force this model into the small mixing angle limit. Hence, to test the validity of the EFT, for Higgs precision measurements we will compare O(Λ −1 ) rates to O(Λ −2 ) rates. For scalar singlet searches we will always keep rates at O(Λ −2 ).

IV. h 1 PRODUCTION AND DECAY
After mixing with the singlet scalar, the observed Higgs boson h 1 obtains additional, BSM-EFT couplings to gauge bosons and fermions via Eq. (9). These additional couplings will change the partial widths of h 1 . In this section, we show the numerical dependence of the relevant branching ratios on the various Wilson coefficients.
In addition to the Higgs boson mass, our input parameters are the same as the LHC Higgs cross section working group [54]:    O(Λ −2 ) results agree well. This indicates that the BSM-EFT is valid in these regions of parameter space. The only exception is the dependence of h 1 → gg on f GG . However, as we will show in the next section, the fits to the Higgs precision data also indicate the BSM-EFT is valid in the allowed parameter regions.
While we do not explicitly show the variation of the Higgs production cross section, it should be noted that gluon fusion is the main production mode. For on-shell h 1 decay, the LO gluon fusion production rate is where the parton luminosity is where √ S is the hadronic center-of-momentum energy and τ 0 = m 2 1 /S. As show in Fig. 1(c) and 2(c), this production rate will have a strong dependence on f GG and f t . Other subdominant but important production modes are Higgs production in association with W/Z (W h 1 /Zh 1 ) and vector-boson-fusion (VBF). The relevant Wilson coefficients for these production modes are f BB and f W W . However, as evidenced in Figs. 1(a,b) W h 1 , Zh 1 , and VBF production will have little dependence on f BB and f W W .

V. HIGGS SIGNAL STRENGTHS
Now we perform a fit to the Higgs precision data. The effects of the additional interactions on the Higgs measurements are parameterized using Higgs signal strengths: where i is the initial state, f is the final state, and the subscript SM indicates SM values.
We combine the signal strengths into a chi-square: where µ f i is a calculated signal strength,μ f i is a signal strength measured at the LHC, and δ f i is the one standard deviation uncertainty onμ f i . We combine measurements from both ATLAS and CMS at the 13 TeV LHC. The set of signal strengths we use can be found in Tables I and II in Appendix B. For the gluon fusion (ggF) production rate we use Eq. (17) to calculate where the partial widths into gluons are calculated as discussed in Sec. (IV). We also include W h 1 , Zh 1 , Higgs production in association with a tt pair (tth 1 ), Higgs production in association with a top plus jet or top plus W (collectively th 1 ), and VBF. For these production modes the model is implemented in MadGraph5 aMC@NLO [55] via FeynRules [56]. The default NNPDF2.3LO pdf sets [57] are used and the renormalization and factorization scales are set to the sum of the final state particle masses. For the VBF mode, we apply the cuts [58] p j T > 20 GeV, |η j | < 5, |∆η jj | > 3, and m jj > 130 GeV, where p j T are jet transverse momenta, η j are jet pseudo-rapidity, ∆η jj is the difference in the jet pseudo-rapidity, and m jj is the di-jet invariant mass. These production and decay modes are calculated at LO in QCD, and it is hoped that most of the QCD corrections cancel in the ratio of the cross sections used for the signal strengths. However, it should be pointed out that in the SMEFT, for some observables the QCD corrections can be strongly dependent upon the EFT operators [59][60][61][62].
Once all the signal strengths are known, we perform a fit to the Wilson coefficients and scalar mixing angle. As shown in Appendix A, the h 1 → γγ decay depends on the Fig. 1 shows that processes with external W and Z bosons do not depend strongly on f W W and f BB . Hence, we define Now h 1 → γγ will constrain only f + , and W h 1 , Zh 1 , VBF, h 1 → W W * , and h 1 → ZZ * have negligible dependence on both f ± . Hence, we set f − = 0. Also, the constraints on are not yet strong and we set f µ = 0. Hence, the following parameters are fit using just Higgs signal strengths: All µ, rates to O(Λ -2 ) and |f i | < 4 π 95% CL Higgs Fits All µ, rates to O(Λ -2 ) and |f i | < 4π 95% CL Higgs Fits  In Fig. 3 we show the results of the χ 2 fits to Higgs data at 95% CL. As can be seen from Eq. (13) allowed. This is because ggF, tth 1 , and th 1 depend relatively strongly on f t and f GG . Hence, deviations in sin θ can be compensated for by changes in f t and f GG . The major effect of vector boson fusion and Higgs production in association with W ± or Z is to eliminate the largest sin θ regions. As discussed above in Sec. IV, VBF, W h 1 , and Zh 1 do not depend strongly on the Wilson coefficients 2 . Hence, the production rates for these modes are approximately the SM rate suppressed by the mixing angle cos 2 θ: where V h 1 = Zh 1 , W h 1 . Limits on these production rates then essentially place limits on the scalar mixing angle. Additionally, at large Wilson coefficient values, fits including only ggF, tth 1 , and th 1 agree well with the full fit, except for f b and f τ . This is can be understood by noting that the strongest constraints on h 1 → τ + τ − and h 1 → bb come from VBF, W h 1 , and Zh 1 . From general perturbativity arguments, it is expected that the Wilson coefficients are bounded |f i | < 4π. The red dotted contours in Fig. 3 shows the results of requiring |f i | < 4π First, we describe how heavy resonance searches are incorporated into our χ 2 fits, then we give the results.

A. χ 2 for Heavy Resonance Searches
Similar to the Higgs signal strengths, it is assumed that scalar resonance searches are Gaussian and χ 2 fit is performed: where χ f,2 i,h 2 is a chi-square of a single h 2 process, σ f i is the calculated cross section for initial state i into final state f ,σ i f is the measured cross section at the LHC, and δσ f i is the one standard deviation uncertainty onσ f i . To calculate the cross section, both the SM rate as well as the new physics contribution must be included. Using the narrow width approximation, we have Typically, the observed and expected 95% CL upper limits on resonance production are reported. Assuming that there are no large fluctuations away from the SM predictions, a SM cross section is measured in all new physics searches. The allowed fluctuations away from the SM cross section at 95% CL are then the expected 95% CL upper limits on new resonance cross sections. That is, the uncertainty on the cross section is approximated as whereσ i,Exp is the expected 95% CL upper limit on the resonance cross section. Again assuming there are no large excesses, the measured cross section is mostly SM-like with a small deviation given by the difference in the observed and expected bounds: whereσ f i,Obs is the observed 95% CL upper limit on the resonance cross section. With these approximations, we finally have One final complication is ifσ f i,Obs <σ f i,Exp then according to Eq. (30) the best fit signal cross section σ(i → h 2 )BR(h 2 → f ) will be negative, which is nonsensical. We propose to alter the definition in Eq. (30) to The second line forces the best fit value of σ i (pp → h 2 )BR(h 2 → f ) to be bounded from below by zero. Also in the second line, the uncertainty has been changed from the expected to observed signal rate. If the best fit value of the signal cross section is at zero, thenσ i,Obs is how far away it can fluctuate from zero at 95% CL. Hence, this form of the χ 2 allows for upward fluctuations with a best fit value of the signal cross section away from zero as well as bounding the best fit value of the cross section to be positive.
The usual use of the reported 95% CL upper bounds is to put a strict upper bound on resonance cross sections: To check that our proposal is consistent, it must be checked that this interpretation can be derived from Eq. (31).
Assuming a 1-parameter fit, the value of the resonance cross section at the minimum χ 2 is Then, the 1-parameter fit limit is found by requiring ∆χ 2 = χ 2 − χ 2 min < 3.84, where χ 2 min is the minimum χ 2 . It can then be shown that Eq. (31) gives the limit which is consistent with the usual interpretation of these bounds.
With these results, all heavy scalar searches can be combined into one χ 2 : Unlike imposing a hard cut off on the heavy resonance rates, this method will allow for fluctuations in some channels that are consistent with a global fit at 95% CL. The combined limits from scalar searches and Higgs measurements are found by combing the χ 2 in Eqs. (20) and (34):

B. Results for Heavy Resonance Searches and Higgs Precision
As with the Higgs boson, the main production channel of h 2 is gluon fusion due to the large gluon parton luminosities. Hence, we will only consider the ggF initial state. To calculate this we reweight partial widths with the NNLO+NNLL SM-like Higgs predictions provided by the LHC Higgs Cross Section Working Group [54]: where the subscript SM indicates the prediction for a SM-like Higgs boson at a mass m 2 .
The values ofσ f ggF,Obs andσ f ggF,Exp for the final states under consideration are given in Tables III and IV  As discussed in Sec. III, this is valid power counting for h 2 processes in the small mixing limit. Finally, we always require that Wilson coefficients are bounded by |f i | < 4π and fit to all relevant Wilson coefficients and scalar trilinear couplings:   Many of the scalar searches that are included require that h 2 be a narrow resonance.
Hence, the results from requiring that the h 2 total width, Γ 2 , be less than 10% of the h 2 mass are also shown in Fig. 5. For the scalar searches, the width constraint limits both sin θ and λ 221 . When sin θ is large, the decays h 2 → W W and h 2 → ZZ are SM like and large for large m 2 [54]. Hence, Γ 2 /m 2 < 0.1 places stronger constraints on sin θ than just blindly applying the scalar searches. Also, if λ 211 is too large the partial width of the decay h 2 → h 1 h 1 becomes large. As a result, requiring a narrow resonance puts strong constraints  As can be seen in Fig. 6, Higgs measurements and scalar searches are complementary.
That is, the allowed regions for scalar searches and Higgs measurements do not fully overlap.
Indeed, the combined allowed region is smaller than the individual allowed regions. This is particularly striking for m 2 = 600 GeV.
Finally, in Fig. 7 we show the ∆χ 2 distributions as a function sin θ for the BSM-EFT and renormalizable model with all other parameters profiled over. In the BSM-EFT, the shape of ∆χ 2 changes dramatically. The 95% CL and 68% CL allowed regions also change drastically and exactly how they change depends strongly on the h 2 mass. It is clear that even 3 TeV new physics effects can make a significant impact on the interpretation of current measurements.

VII. CONCLUSION
A common assumption of simplified models at the LHC is that there are a few new BSM particles that can be produced, while all other new particles are heavy and decoupled.
Under these assumptions, most studies of simplified models are renormalizable. However, using EFT techniques it is possible to test the basic assumption that all other new particles are indeed decoupled.
In this paper, we studied a popular simplified model, the real singlet extended SM, and supplemented it with all possible dimension-5 operators involving the scalar singlet. We studied the effects of the effective operators on the interpretation of Higgs signal strengths as well as searches for heavy new resonances. As we showed, even if the new physics occurs at 3 TeV, the interpretation of these measurements and searches are changed drastically.
This study shows that even in the simplest of simplified model, the heavy new physics is not "decoupled" even when the BSM-EFT expansion is valid. That is, it cannot be neglected and the BSM-EFT should generically be considered.
In addition to the numerical results, we also gave a comprehensive discussion of the counting in BSM-EFT for production and decay rates. We showed that while in the linear SMEFT power counting is relatively straightforward, power counting in a BSM-EFT is strongly process and parameter space dependent. We also developed a new proposal to consistently combine the limits from new resonance searches and precision measurements via a χ 2 . This method allows for fluctuations in individual channels, while keeping the global EPSCoR grant program. The data to reproduce the plots has been uploaded with the arXiv submission or is available upon request.
Appendix A: Feynman Rules