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Electroweak Stability and Discovery Luminosities for New Physics

What is the luminosity needed for discovering new physics if the electroweak scale is to remain stable? In this work we study this question, with the example of a real singlet scalar which couples to the Higgs field already at the renormalizable level. Observing that the electroweak scale remains stable if the two scalars couple in a seesawic fashion, we show that the HL-LHC, expected to deliver an integrated luminosity around 3/ab, can discover scalars weighing up to 800 GeV. The FCC-hh, on the other hand, can discover scalars as heavy as 2.3 TeV at 100/ab luminosity. It thus follows that the new physics that does not destabilize the electroweak scale can be accessed only at high luminosities, and is not possible exclude by the current LHC results.


I. INTRODUCTION
The standard model of elementary particles (SM), experimentally completed by the discovery of the Higgs boson at the ATLAS and CMS [1], has shown excellent agreement with all the available data so far [2]. The TeV domain seems to be devoid of any new particles beyond the SM spectrum [3]. There are, however, astrophysical (dark matter, dark photon), cosmological (dark energy, inflation) and structural (neutrino masses, flavor, unification · · · ) phenomena which require the SM to be extended. Each extension comes with its own scale and mechanisms, and tends to pull up the SM towards its own scale. In fact, if Λ is a UV cutoff lying above all the aforementioned extensions then loops of matter lead to the masses for the photon γ and gluon g, and for the Higgs boson h [4] such that c i and c i are O(10 −2 ) loop factors, and λ hψ is a coupling between the Higgs boson h and the non-SM field ψ .
In the gauge sector, the correction (1) completely destructs the SM by breaking the color and the electric charge [6,7]. The destruction can be prevented only if the quadratic corrections c g Λ 2 and c γ Λ 2 are eradicated.
In the Higgs sector, with similar corrections for W and Z masses, even if c h Λ 2 is alleviated along with the gluon and photon masses, the logarithmic part remains to destabilize the SM with its quadratic sensitivity to m ψ [4].
If Λ does not correspond to a physical scale then the quadratic c i Λ 2 terms in (1) and (2) can all be ignored but the logarithmic part of (2) remains as a physical contribution.
The simplest way to see this is to switch from cutoff to dimensional regularization (it is a regularization because Λ is unphysical) to dimensional regularization in which color and charge conservation is automatic [5].
In reality Λ is physical. The reason is that gravity exists and need be incorporated into the SM. Indeed, incorporation of gravity into the SM rests on a Poincare-breaking scale (as a feature of curved geometry) and such a scale does necessarily form a momentum cutoff (as it breaks Poincare symmetry) [8]. This means that gravity requires a physical UV cutoff Λ to be implemented in the flat spacetime theory (the SM), and this cutoff renders the QFT effective with corrections like (1) and (2). To this end, the mechanism of [9] (see also [8] and [6]), the so-called symmergence, incorporates gravity into the SM such that (i) it predicts a BSM sector (containing the ψ fields in (2)), (ii) it restores color and electric charge by curving away (1) allow the Higgs mass correction in (2) to remain within the bounds, and symmergence allows this bound while the others can't. This coupling scheme, which implies that heavier the BSM smaller its couplings to the SM, gives way to a novel approach to collider and other searches for the BSM physics. It is true that the seesawic relation (5) is imposed empirically but it is a natural requirement since λ hψ renormalizes multiplicatively and maintains its size. In this sense, the relationship in (5) can be viewed as a natural "electroweak stability" criterion.
In the present work, our goal is to analyze collider (LHC and FCC, in particular) searches for BSM sectors coupling to the SM as in (5). For definiteness and simplicity, we focus on a BSM sector made up only of a single real SM-singlet scalar S, which couples to the Higgs field at the renormalizable level with a coupling like (5). We then raise the question: What energy and luminosity does it take to discover of a singlet scalar S of mass m S if the electroweak scale is to remain stable? (6) and investigate it in detail by first modeling (Sec. II) then computing one-loop corrections like (2) (Sec. III), and finally performing a collider study at the LHC and FCC energies (Sec. IV). Our analysis is expected to put an electroweak stability bound on different discovery limits at colliders. In Sec. V we conclude.

II. THE MODEL
In view of the question (6), the most general, renormalizable, symmetric Lagrangian density extending the SM with a real singlet scalar field S is given by [11] where is the potential, and H is the usual SM Higgs doublet with the Higgs boson h remaining as a CP-even scalar after the Goldstone bosons φ i are swallowed as longitudinal components of the W and Z bosons. Indeed, for λ H > 0 and λ S > 0, the potential gets bounded from below and the minimum of the potential breaks the electroweak symmetry spontaneously via the Higgs vacuum expectation value (VEV) υ H = 0. If the scalar S is not inert (see for instance [12]), that is, if it gets a VEV υ S = 0 then the minimum of the potential (8) occurs at with the singlet boson s defined as S = υ S + s in parallel to (9).
In the vicinity of the vacuum (10), the mass-squared matrix of the h and s bosons assume two eigenvalues corresponding to the two physical eigenstates h 1 (which should be identified with the scalar boson observed at the LHC [1]) and h 2 (the extra scalar boson under search at the LHC and to be searched for at future colliders like the FCC). The key parameter is their mixing which is proportional to λ HS -the strength of the SM-BSM coupling.

III. ONE-LOOP CORRECTIONS AND MODEL SPACE
In this section, we give a detailed analysis of the logarithmic corrections mentioned in (2). The Feynman diagrams which contributes the logarithmic corrections are depicted in where the various couplings (like the quartic coupling λ h i h j h k h l ) are listed explicitly in the Appendix as functions of λ H , λ S , λ HS and the mixing angle θ.
The h 1 mass receives non-trivial corrections from the h 2 loops. This feature, explicated in (14), requires λ HS to be bounded appropriately. The vacuum stability already gives a bound λ 2 which means that |λ HS | is typically at the 30% level depending on precise values of λ H and λ S . We will consider different parameter ranges during the analysis.
The bound above is however not sufficient to ensure electroweak stability. The reason is that h 2 can be too heavy to keep h 1 mass within the LHC bound. To this end, one comes back to the see-sawic bound in (5). In what follows thus we require thus λ HS to have the value after expressing and analyze the model in terms of the remaining two free parameters: the S quartic coupling λ S and the S VEV υ S .
The allowed ranges of the model parameters can be determined numerically. In doing so we consider υ S values as large as 20 TeV in view of the sensitivity of the exotica searches at the LHC [3]. To this end, we plot in Fig. 2 variation of λ HS with υ S in the small λ S regime of 0.01 ≤ λ S ≤ 0.1. It is seen that λ HS , which decreases with m 2 S due to its see-sawic structure in (16), in magnitude, remains below λ S at least by two orders of magnitude. For larger λ S , from 0.5 to 0.9, we find that λ HS takes unacceptably large values (a thousand), we do not consider therefore λ S values above 0.5. In fact, hereon we set λ S = 0.1 as a nominal value revealing the physics implications of the heavy scalar.
To see the difference between setting λ HS to a fixed (albeit small) value as in most phenomenological analyses [12] and requiring λ HS to obey the see-sawic bound in (16)   To see further how λ HS varies with υ S we list in Table I λ HS values as υ S ranges from 2 TeV to 20 TeV. In agreement with Figs. 2 and 3, λ HS remains negative throughout and well satisfies the vacuum stability bound (15). It is clear that larger the m S of scalar field, the weaker its interaction with Higgs. This decrease could explain why we have not observed any fingerprint of BSM physics (the scalar S here) at LHC experiments.

IV. COLLIDER PHENOMENOLOGY
Now, we come to the question of investigation (6). The production cross section of real singlet scalar depends on its mass and its coupling to the SM Higgs field. In view of the see-sawic coupling (16) the production cross section is directly set by m h 2 (or v S ). It sets also branching fractions of h 2 decays.
For analysis purposes, we have modified the SM package in LanHEP-3.2.0 [15] by including the real singlet S, and exported the extended model to CalcHEP-3.7.5 [14]. The parton distribution functions are evaluated by using LHAPDF6 [16], and simulations are performed with cteq6l1 PDF set [17].
As revealed by Fig.(5), the most dominant decay channels of singlet scalar are h 2 → W W This constancy of the branching fractions proves useful for putting discovery limits (as in simplified models [18]. In Fig.6, h 2 production cross sections as a function of the h 2 mass are shown for the LHC with a center-of-mass energy of 13 TeV (solid lines) and for the FCC with a center-of-mass energy of 100 TeV (dashed lines). In both cases, gluon-gluon fusion dominates the cross section, as expected. In the figure, the gray lines indicate the cross sections that would produce 10 events assuming integrated luminosities of 150 fb −1 (LHC), 3 ab −1 (HC-LHC), and 100 ab −1 (FCC) [19][20][21]. With ∼ 150 fb −1 , the total integrated luminosity recorded at 13 TeV during Run 2, the LHC does not seem to be able to produce a sufficient amount of h 2 's above ∼ 1 TeV. The amount of data that experiments can collect during high luminosity LHC (HL-LHC), 3 ab −1 , may not reveal any h 2 signature in multi-TeV region. Here it should be noted that proton-proton collisions at the HL-LHC will actually occur at 14 TeV, however, the conclusion remains the same.
The prospects are more promising for a 100 TeV proton-proton collider, for example the FCC-hh. h 2 events can already be produced with 3 ab −1 data to be collected at the FCC-hh, and moreover, with an integrated luminosity of 100 ab −1 , more than 10 h 2 events production mechanisms, such as qq → qqh 2 and qq → qth 2 , are also capable of producing h 2 events with m h 2 > 1 TeV.
In Table II

V. CONCLUSION
In this work we have studied impact of the electroweak stability on the collider discovery of the BSM physics, with the example of a single SM-singlet scalar. Our general discussion in the Introduction and more specific analysis in Sec. III have revealed that bounding the SM-BSM coupling λ SM as in (16), admissible only for symmergence, has important implications for new particle searches. It tells us that there can exist heavy particles like h 2 and they can directly couple to the SM Higgs boson but they do not destabilize the electroweak scale thanks to see-sawic structure in (16). This empirical structure has a room only in symmergence.
Our phenomenological analysis in Sec. III and simulations in Sec. IV show that elec-  It must be emphasized that the limits we report here are optimistic in that we did not perform a background analysis. It is after eliminating the background that discovery limits can be more realistic. We nevertheless expect that discovery limits for 10 events give a satisfactory account of the collider discovery limits. (We will give complete background analysis that in our follow-up work [22]). It must also be emphasized that the scalar field we studied is not linked to dark matter. It would be a dark matter candidate if its VEV vanishes and if it possesses correct relic density. In that case bounds on the model parameter space would increase, and its phenomenology would be affected accordingly.