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The Gauge-Invariant Singlet Scalar Field and Its Higgs-Portal Coupling John Edward Holland 1 1 Independent Researcher; Chartered Engineer (CEng MIMechE); Six-Sigma Master Black Belt; United Kingdom Email: john.holland@expansetension.org ORCID: 0009-0001-5120-8712 © Copyright. 2025. John Holland. All rights reserved. Abstract We examine the theoretical consistency of a minimal extension of the Standard Model in which a real scalar field Φ, singlet under all gauge interactions, couples to the Higgs doublet through a density-modulated Higgs- portal interaction. The scalar sector is defined by the most general renormalisable potential consistent with gauge invariance and a discrete ℤ 2 symmetry, with the portal coupling rescaled by a dimensionless function 푆(Σ) of a coarse-grained macroscopic energy density Σ. Treating 푆(Σ) as an external macroscopic parameter, we analyse the vacuum structure, boundedness-from-below conditions, perturbative unitarity, and loop stability of the model. For any fixed value of the modulation factor the theory reduces to the conventional singlet-Higgs extension with an effective portal coupling 휆 ̃ Φ퐻 =푆(Σ)휆 Φ퐻 , and all stability and unitarity constraints follow the usual forms with this rescaling. The presence of 푆(Σ) introduces no new divergences, no non-renormalisable operators, and no modifications to the counterterm structure, ensuring that the model remains local, polynomial, and renormalisable at fixed Σ. We conclude that a gauge-invariant singlet scalar with density-modulated Higgs- portal strength constitutes a theoretically coherent and perturbatively consistent extension of the Standard Model. Phenomenological implications of density variation are left to future work. Keywords: methods: analytical – elementary particles – particle physics – theory

1. Introduction Scalar extensions of the Standard Model (SM) remain a widely explored and theoretically well-motivated direction in particle physics. Among the simplest possibilities is the addition of a real scalar field that is a complete singlet under the SM gauge group. Such singlet fields appear naturally in numerous contexts, including studies of vacuum stability, electroweak symmetry breaking, hidden-sector interactions, and minimal Higgs-portal scenarios [1-10]. A key virtue of singlet-scalar models is their theoretical economy: they preserve gauge invariance, require no additional gauge interactions, and allow renormalisable interactions with the Higgs doublet through the scalar bilinear 퐻 † 퐻. These features make them suitable as a baseline for exploring structurally modest variations in the scalar sector. In the conventional Higgs-portal framework, the interaction between the singlet and Higgs fields is mediated by a quartic coupling with constant strength [3,4,6,10]. This assumption is adequate for many applications; however, it is not the only theoretically consistent possibility. A broader class of effective descriptions becomes available when one recognises that the low-energy parameters appearing in renormalisable Lagrangians may depend on environmental inputs, provided such dependence respects locality, symmetry, and the operator structure of the theory. Examples of such behaviour are familiar across physics: electromagnetic propagation in materials depends on macroscopic properties encoded in the dielectric constant [17,19]; photon dispersion in a plasma depends on the ambient electron density; and renormalised couplings in quantum field theory vary with the probing energy scale [20,21]. These analogies suggest that, under appropriate conditions, it may be meaningful to consider a scalar-portal framework in which the portal strength varies across environments characterised by different coarse-grained macroscopic energy densities - a variable that naturally arises in inhomogeneous environments like cosmic structures.

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The purpose of this paper is to examine the theoretical viability of such a construction. We consider a gauge- invariant real scalar singlet Φ interacting with the Standard Model only through a Higgs-portal term, where the portal coupling is modulated by a dimensionless function 푆(Σ) of a coarse-grained mass–energy density Σ. The scalar sector remains renormalisable at fixed density, and the density dependence is treated as an external macroscopic input, not as the result of new dynamical degrees of freedom. The goal of this paper is intentionally limited: we do not propose a specific ultraviolet completion, introduce new dynamical mechanisms, or explore detailed phenomenological consequences. The viability of any future phenomenological applications depends entirely on the specific form of f(ρ), which is not assumed here. Instead, we focus strictly on the field-theoretic questions:

1. Does a density-modulated Higgs-portal interaction remain renormalisable?
2. Does it preserve the boundedness and stability of the scalar potential?
3. Does it satisfy perturbative unitarity?
4. Does it introduce any new divergences or counterterms through loop corrections?
5. How does the vacuum structure respond to the modulation of the portal coupling? These questions can be addressed without making assumptions about the microscopic origin or phenomenological role of 푆(Σ). By separating the consistency analysis from potential applications, we ensure that the discussion remains conservative and grounded in well-defined quantum field theory. The structure of the paper is as follows. In Section 2, we define the model, including field content, symmetries, Lagrangian structure, and the role of the density-dependent modulation. We emphasise that the operator content is identical to that of a conventional singlet-scalar extension and that 푆(Σ) enters only as a multiplicative factor modifying the effective portal strength. Section 3 examines the vacuum structure of the theory in detail. Depending on parameters, the model admits either a vacuum in which the singlet does not condense or a mixed vacuum in which both fields acquire expectation values. We derive the stationary conditions, classify the vacuum branches, and summarise their physical properties. In Section 4, we analyse boundedness-from-below conditions, showing that the stability constraints reduce to the usual positivity requirements on the scalar quartics, with the portal term rescaled by 푆(Σ). The modulation does not introduce new forms of instability; it simply shifts the numerical range of admissible parameters. Section 5 addresses perturbative unitarity, running couplings, and loop corrections. We show that unitarity bounds place constraints on the effective portal coupling, and that the loop structure and counterterm content remain unchanged because 푆(Σ)is not a field. The renormalisation-group behaviour of the quartics follows the standard singlet-portal pattern. In Section 6, we discuss the interpretation of the model as an effective field theory in a medium. We provide several analogies from well-established physics where environmental parameters influence effective couplings, clarifying how density-dependent modulation can be incorporated without violating locality or renormalisability. Section 7 briefly compares the present construction with existing scalar-field models, including screened modified-gravity scenarios and singlet-Higgs extensions used in cosmology or dark-sector phenomenology. This highlights the conceptual differences between density-modulated and dynamical screening mechanisms. Finally, Section 8 summarises the findings and outlines future directions. The analysis presented here establishes that a gauge-invariant singlet scalar with a density-dependent Higgs-portal interaction forms a theoretically coherent extension of the Standard Model [11-16]. The model is minimal, renormalisable, and structurally conservative, providing a well-defined foundation for potential future studies in more specific phenomenological settings.

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1. Theory and Model Definition This work considers one of the simplest possible extensions of the Standard Model (SM): the addition of a single real scalar field, denoted Φ, which transforms as a gauge-invariant singlet under the full SM symmetry group

푆푈(3) 푐 ×푆푈(2) 퐿 ×푈(1) 푌

[Eq. 2.1] The objective of this section is to define the field content, Lagrangian structure, symmetry constraints, and the theoretical assumptions underpinning the model. No phenomenological claims are made here; the focus is on establishing theoretical self-consistency. 2.1 Scalar content and symmetry structure The new scalar field Φ is taken to be a Lorentz scalar and a complete singlet under all Standard Model gauge transformations. It therefore carries no colour, weak isospin, or hypercharge. This choice ensures that the introduction of Φ does not alter the gauge sector of the SM, does not require additional gauge fields, and preserves the renormalisability of the extended scalar sector. Because Φ is a gauge singlet, the most general renormalisable scalar potential consistent with gauge invariance may include [1-4]: • a quadratic mass term in Φ, • a quartic self-interaction Φ 4 , • and a mixed “portal” interaction connecting Φ to the gauge-invariant Higgs bilinear 퐻 † 퐻. To restrict the form of the potential and avoid unnecessary complications such as linear or cubic terms in Φ, we adopt a ℤ 2 symmetry, [3,4,6,10] defined by:

Φ→−Φ,퐻→퐻 [Eq.2.2] This discrete symmetry forbids odd powers of Φ in the potential. Z₂ is optional and not required for renormalisability. While such a symmetry is not strictly required, it is commonly used in singlet-scalar extensions for simplicity and to stabilise the scalar in dark-matter applications [1,2,6,9,10]. In this paper the symmetry is simply a structural assumption that keeps the scalar potential minimal and transparent. 2.2 Lagrangian structure With these ingredients, the scalar sector of the theory is defined by the Lagrangian. This consists of a canonical kinetic terms for 퐻and Φ, the Standard Model Higgs potential, the singlet mass and self-interaction terms, and a Higgs-portal interaction whose strength is modulated by an environmental parameter. The kinetic terms are standard:

ℒ kin =(퐷 휇 퐻) † (퐷 휇 퐻)+ 1 2 ∂ 휇 Φ ∂ 휇 Φ. [Eq.2.3] The scalar potential is:

푉(퐻,Φ)=−휇 퐻 2 퐻 † 퐻+휆 퐻 (퐻 † 퐻) 2 + 1 2 휇 Φ 2 Φ 2 + 1 4 휆 Φ Φ 4 +푆(Σ) 휆 Φ퐻 Φ 2 (퐻 † 퐻).

[Eq.2.4]

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This potential has the standard structure of a renormalisable two-scalar system, with one modification: the Higgs-portal coupling is multiplied by a dimensionless modulation function 푆(Σ), which depends on a coarse-grained macroscopic energy density Σ [3,4,6,10]. The function 푆 ( Σ ) is treated as an external classical parameter, not a dynamical field. It does not break any gauge symmetry. It does not introduce operators beyond dimension four. It preserves renormalisability at fixed Σ. Reduces the model to the familiar singlet-scalar extension when 푆(Σ)=1. No new counterterms arise beyond those of the standard singlet model at fixed Σ. We require only that 푆(Σ) is real and non-negative in the environments of interest. As we will show later, the effective portal coupling 휆 ̃ Φ퐻 =푆(Σ)휆 Φ퐻 is the parameter that enters all stability and unitarity constraints. The function 푆(Σ) is assumed to be dimensionless, real, and non-negative in the environments of interest. In addition, we assume that its dependence on the coarse-grained density Σ is smooth on macroscopic scales, varying only across regions where the density itself changes appreciably. This ensures that the modulation of the portal coupling does not introduce rapid spatial or temporal fluctuations beyond the resolution of the effective field theory and preserves the locality of the underlying Lagrangian description. For any fixed value of Σ, the theory reduces to a conventional renormalisable singlet extension of the Standard Model. 2.3 Electroweak symmetry breaking with a singlet scalar To examine the vacuum structure of the theory we work in unitary gauge, where unphysical Goldstone modes are absorbed, writing:

퐻= 1 √ 2 ( 0 푣+ℎ ),Φ=푥+휑 [Eq.2.5]

where: • 푣 is the Higgs Vacuum Expectation Value (VEV), • 푥 is the possible singlet VEV, • ℎand 휑 are the fluctuations around these VEVs. Substituting these into the potential yields the tree-level vacuum potential:

푉(푣,푥)=− 1 2 휇 퐻 2 푣 2 + 1 4 휆 퐻 푣 4 + 1 2 휇 Φ 2 푥 2 + 1 4 휆 Φ 푥 4 + 1 2 푆(Σ)휆 Φ퐻 푣 2 푥 2

[Eq.2.6] Stationarity conditions give the coupled equations:

−휇 퐻 2 푣+휆 퐻 푣 3 +푆(Σ)휆 Φ퐻 푣푥 2 =0 [Eq.2.7]

휇 Φ 2 푥+휆 Φ 푥 3 +푆(Σ)휆 Φ퐻 푣 2 푥=0

[Eq.2.8]

These equations determine whether the singlet acquires a VEV in addition to the Higgs VEV. Two important regimes arise: These stationary conditions admit two distinct vacuum configurations:

1. The “Φ-symmetric vacuum”, in which the singlet does not acquire a VEV ( 푥=0 ) . This preserves the discrete ℤ 2 symmetry and yields a scalar sector with no Higgs–singlet mixing.

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1. The “mixed vacuum”, in which both 퐻 and Φ acquire non-zero expectation values ( 푣≠0, 푥≠0 ) . This spontaneously breaks the ℤ 2 symmetry and leads to a two-scalar mass matrix with non-zero mixing. Case 1: No singlet condensate (푥=0) This is the simplest and most commonly realised vacuum configuration in singlet-scalar extensions. In this regime electroweak symmetry breaking proceeds exactly as in the Standard Model, with the Higgs vacuum expectation value given by

푣 2

휇 퐻 2 휆 퐻

[Eq.2.9] The singlet field does not acquire a vacuum expectation value, and the physical singlet excitation has mass

푚 휑 2 =휇 Φ 2 + 1 2 푆(Σ)휆 Φ퐻 푣 2

[Eq.2.10] For sufficiently positive values of 휇 Φ 2 + 1 2 푆(Σ)휆 Φ퐻 푣 2 , the singlet remains massive and does not condense. Electroweak symmetry breaking is identical to the SM at tree level. The singlet behaves as an additional massive scalar, whose mass depends on the effective portal combination 푆(Σ)휆 Φ퐻 , but it does not modify the Higgs vacuum or couplings in this branch. This ensures that the scalar sector stays in the 푥=0 branch of the vacuum manifold and that no Higgs–singlet mixing occurs in this configuration. The portal interaction therefore influences only the singlet’s effective mass, without altering the Higgs sector at tree level. Case 2: Non-zero singlet VEV (풙≠ퟎ) If the parameters allow 푥≠0, the physical fields ℎand 휑mix, giving a two-scalar mass matrix: Diagonalising this matrix yields two physical scalars, whose masses and mixing angles depend explicitly on 푆(Σ).
The physical “Higgs-like” state is a mixture of the original Higgs and singlet, modifying its couplings to SM particles. The degree of mixing and the scalar spectrum can vary with density through 푆(Σ), potentially leading to environment-dependent scalar properties. This regime is more tightly constrained experimentally by precision Higgs measurements at the LHC and is not analysed in detail in this work. This provides a pathway for the model to have environment-dependent phenomenological signatures, however,
such applications are not pursued in this paper. 2.4 Renormalisability and operator structure A crucial point is that, because 푆(Σ) is treated as an external parameter, the full Lagrangian contains only dimension-four or lower operators in the fields 퐻and Φ. Power-counting renormalisability is preserved,
there are no non-renormalisable interactions are introduced, and perturbative loop corrections proceed as in the standard singlet-scalar model.

ℳ 2

( −휇 퐻 2 +3휆 퐻 푣 2 +푆(Σ)휆 Φ퐻 푥 2 2푆(Σ)휆 Φ퐻 푣푥 2푆(Σ)휆 Φ퐻 푣푥휇 Φ 2 +3휆 Φ 푥 2 +푆(Σ)휆 Φ퐻 푣 2 )

[Eq.2.11]

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All vertices remain polynomial in the dynamical fields, and standard counterterm structures apply. The only modification is that the numerical value of the portal coupling is rescaled in different environments. Because 푆(Σ)is an external parameter rather than a field, it does not generate additional loop diagrams, counterterms, or operator mixing beyond those present in the constant portal model. We therefore propose this to be a clean, theoretically conservative extension of the SM. 2.5 Interpretation of 푺(횺) From an effective field-theory viewpoint, the density-dependent modulation function 푆(Σ) encapsulates how the parameters of a low-energy scalar theory may vary across environments with different coarse-grained macroscopic energy density (units J m⁻³, or equivalently kg m⁻³ via Σ=휌푐 2 ). It is treated as an external classical quantity that varies only on scales large compared with particle-physics interactions. The microscopic origin of this dependence is not specified here; instead, 푆(Σ) is treated as an external classical parameter that varies only on macroscopic scales. For any fixed value of Σ, the scalar sector reduces to a conventional renormalisable singlet–Higgs model with an effective portal coupling 휆 ̃ Φ퐻 =푆 ( Σ ) 휆 Φ퐻 . The modulation is macroscopic and does not modify the micro-local quantum field interactions; it simply changes the effective parameters describing them. This approach is directly analogous to familiar situations in physics where effective parameters depend on environmental conditions. A common example is the dielectric constant in electromagnetism, where the response of a medium modifies the propagation of electric and magnetic fields. Similar ideas appear in plasma physics, where photon propagation is influenced by the plasma frequency and effective refractive index of the medium. Even within quantum field theory itself, coupling constants “run” with energy scale, reflecting the fact that effective interactions depend on the resolution at which the system is probed. These analogies provide useful intuition: just as the electromagnetic, strong, and weak couplings acquire scale dependence through renormalisation, and just as material media modify the effective equations governing photons, a density-dependent modulation of a Higgs-portal interaction is entirely compatible with established EFT principles. Crucially, the present framework preserves locality, gauge symmetry, and renormalisability at fixed density. The role of 푆(Σ) is simply to parameterise how the effective portal strength may shift in different macroscopic environments. 2.6 Summary of the model The Theory & Model Definition section establishes that:

1. The scalar Φ is a gauge-invariant singlet.
2. The Lagrangian includes only renormalisable operators.
3. A novel density-modulated Higgs-portal term is introduced, but it does not break any symmetry or renormalisation structure.
4. Electroweak symmetry breaking proceeds as usual, with optional singlet mixing.
5. All theoretical constraints (renormalisability, stability, unitarity) reduce to standard singlet-scalar conditions with the replacement This is a mathematically clean, conservative extension of the Standard Model, designed to serve as a stable foundation for later phenomenological development. Section 3. Potential Analysis & Vacuum Structure The vacuum properties of the scalar sector follow from the structure of the potential

휆 Φ퐻 →푆(Σ)휆 Φ퐻

[Eq.2.12]

푉(퐻,Φ)=− 휇 퐻 2 퐻 † 퐻+휆 퐻 (퐻 † 퐻) 2 + 1 2 휇 Φ 2 Φ 2 + 1 4 휆 Φ Φ 4 +푆(Σ) 휆 Φ퐻 Φ 2 (퐻 † 퐻) [Eq.3.1]

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evaluated after spontaneous electroweak symmetry breaking. To characterise the vacuum configurations in a systematic way, we analyse the potential in both field directions, identify the stationary points, and determine their physical properties. The results clarify when the singlet does or does not acquire a vacuum expectation value, how mixing arises, and how the density-modulated portal coupling influences the vacuum landscape.

Property 횽−퐬퐲퐦퐦퐞퐭퐫퐢퐜 퐯퐚퐜퐮퐮퐦 (풙=ퟎ) 퐌퐢퐱퐞퐝 퐯퐚퐜퐮퐮퐦 (풙≠ퟎ) ℤ 2 symmetry Preserved Spontaneously broken Higgs VEV v=246 GeV v≈246 GeV (slightly shifted) Singlet VEV 푥=0 푥≠0 Higgs–singlet mixing None (θ=0) Yes (휃≠0); depends on 푆(Σ)휆 Φ퐻

Singlet mass mφ2=μΦ2+21S(Σ)λΦHv2 Determined by diagonalising the 2×2 mass matrix LHC constraints Minimal Strong (Higgs coupling deviations) Table 1 The summarised physical implications of each configuration. Table 1 summarises the physical implications of the two vacuum configurations of the Higgs–singlet system. 3.1 Stationary conditions Working in unitary gauge and expressing the fields as the tree-level potential becomes

Stationary points satisfy yielding the coupled equations

퐻= 1 √ 2 (0, 푣+ℎ) 푇 ,Φ=푥+휑

[Eq.3.2]

푉(푣,푥)=− 1 2 휇 퐻 2 푣 2 + 1 4 휆 퐻 푣 4 + 1 2 휇 Φ 2 푥 2 + 1 4 휆 Φ 푥 4 + 1 2 푆(Σ)휆 Φ퐻 푣 2 푥 2

[Eq.3.3]

∂푉 ∂푣 =0, ∂푉 ∂푥 =0

[Eq.3.4]

−휇 퐻 2 푣+휆 퐻 푣 3 +푆(Σ)휆 Φ퐻 푣푥 2 =0

[Eq.3.5]

휇 Φ 2 푥+휆 Φ 푥 3 +푆(Σ)휆 Φ퐻 푣 2 푥=0

[Eq.3.6]

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These admit two distinct classes of solutions, corresponding to whether 푥=0or 푥≠0, which we label:

1. The Φ-symmetric vacuum (푥=0)
2. The mixed vacuum (푥≠0) Case Vacuum expectation values Physical mass eigenstates Φ- symmetric vacuum ⟨퐻⟩=(0,푣/ √ 2),⟨Φ⟩=0 Higgs: 푚 ℎ 2 =2휆 퐻 푣 2 . Singlet: 푚 휑 2 =휇 Φ 2

1 2 푆(Σ)휆 Φ퐻 푣 2

Mixed vacuum

⟨퐻⟩=(0,푣/ √ 2) 푇 ,⟨Φ⟩=푥≠0 Two scalar eigenstates ℎ 1 ,ℎ 2 , given by diagonalising the 2×2 mass matrix in the ( ℎ , 휑 ) basis. Masses depend on 휆 퐻 ,휆 Φ ,휆 Φ퐻 , the VEVs 푣,푥, and the modulation factor 푆(Σ). Table 2. VEV and Physical Mass Eigenstates Table 2 shows the vacuum configurations of the Higgs–singlet system, highlighting the corresponding VEVs, scalar mass eigenstates and mixing properties. Φ-symmetric vacuum: No mixing between ℎ and 휑; the mass matrix is diagonal in the ( ℎ , 휑 ) basis. Mixed vacuum: Non-zero mixing angle 휃, with ℎ 1 =cos 휃 ℎ+sin 휃 휑, ℎ 2 =−sin 휃 ℎ+cos 휃 휑. The off- diagonal terms in the mass matrix are proportional to 푆(Σ)휆 Φ퐻 푣푥. 3.2 The 횽-symmetric vacuum In the first class of solutions, the singlet does not acquire a vacuum expectation value. Setting 푥=0 identically reduces the Higgs stationary condition to its Standard Model form so electroweak symmetry breaking proceeds without modification, with no Higgs–singlet mixing; singlet mass depends on 푆(Σ). The singlet’s physical mass in this vacuum is Thus, the effective mass receives a density-dependent contribution via the factor 푆(Σ). As noted previously, for sufficiently positive 푚 휑 2 , the scalar remains massive and does not condense, stabilising the Φ-symmetric solution. There is no mixing between the scalar excitations: the Higgs and singlet mass matrix is diagonal in the ( ℎ , 휑 ) basis. Consequently, the observed Higgs boson is unaffected at tree level, the singlet behaves as a spectator field with environment-modulated mass, and Higgs couplings to Standard Model fields remain unchanged. This vacuum is phenomenologically safe and the one most often considered in minimal singlet- scalar models [6,8,10]. 3.3 The mixed vacuum If the parameters permit 푥≠0, the discrete ℤ 2 symmetry is spontaneously broken, generating a second class of stationary points. Substituting 푣≠0and 푥≠0into the minimisation equations yields a coupled system whose solutions must satisfy

푣 2

휇 퐻 2 휆 퐻

[Eq.3.7]

푚 휑 2 =휇 Φ 2 + 1 2 푆(Σ)휆 Φ퐻 푣 2

[Eq.3.8]

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Expanding the potential around this vacuum leads to a mass matrix with off-diagonal mixing terms proportional to Diagonalising the matrix yields mass eigenstates

with a mixing angle determined by both the portal strength and the density modulation. The mixing angle satisfies tan(2θ) = 2λ_p f(ρ) v_h v_s / (m²_11 - m²_22), making it explicitly density-dependent in this vacuum. The mixed vacuum carries two important implications:

1. The Higgs-like scalar is no longer purely the SM Higgs, modifying its couplings to SM particles.
2. The degree of mixing may depend on ambient density through 푆(Σ), a feature not present in constant-portal models. Higgs couplings modified; degree of mixing may depend on density; phenomenology more constrained. Although such vacua can be studied, they are experimentally constrained because Higgs mixing angles are tightly bounded by LHC data. For this reason, and to keep the focus on theoretical consistency, we do not explore the mixed vacuum phenomenology in detail here. The mixed vacuum necessarily produces Higgs– singlet mixing, leading to a physical Higgs-like state with couplings uniformly suppressed by cos휃. Current LHC Higgs signal–strength measurements place significant limits on such mixing. Global fits to the combined ATLAS and CMS Run-2 data, as well as dedicated analyses within singlet-extended Higgs sectors, typically constrain the mixing angle to ∣sin 휃∣≲0.3at 95% CL [6,10]. Consequently, viable mixed-vacuum scenarios require parameter choices that keep the mixing sufficiently small to remain consistent with collider bounds. 3.4 Vacuum stability within each branch After identifying stationary points, the next step is selecting the true vacuum. Each candidate must satisfy: • positivity of second derivatives (mass squared eigenvalues), • boundedness-from-below conditions for quartic terms, • and global minimisation of the scalar potential. Section 4 will examine these constraints in more detail, showing that stability conditions reduce to forms identical to the constant-portal model with the replacement Specifically, boundedness from below requires λ_s > 0, λ_h > 0, and λ_h λ_s > [λ_p f(ρ)]²/4, which are the standard constraints with f(ρ)-rescaled portal coupling.

휇 퐻 2 =휆 퐻 푣 2 +푆(Σ)휆 Φ퐻 푥 2

[Eq.3.9]

휇 Φ 2 =− 휆 Φ 푥 2 −푆(Σ)휆 Φ퐻 푣 2

[Eq.3.10]

2푆(Σ)휆 Φ퐻 푣푥

[Eq.3.11]

ℎ 1 =cos 휃 ℎ+sin 휃 휑,ℎ 2 =−sin 휃 ℎ+cos 휃 휑

[Eq.3.12]

휆 Φ퐻 → 푆(Σ)휆 Φ퐻

[Eq.3.13]

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3.5. Potential Contour Plots for Symmetric vs. Mixed Vacua To complement the analytic discussion above, it is useful to visualise the tree-level potential in the ( 푣 , 푥 ) field space. Contour plots provide an intuitive illustration of how the Φ-symmetric and mixed vacua arise from different choices of parameters, and how the location of the minimum shifts from the 푥=0axis to an off-axis point when the singlet acquires a non-zero expectation value.

Figure 1. Potential Contour Plots for Symmetric vs. Mixed Vacua • Panel (a): “Φ-symmetric vacuum” o Contours of 푉(ℎ,푥) in the ( ℎ , 푥 ) plane. o The minimum (marked with a dot) lies on the Higgs axis (푥≈0), illustrating the branch with ⟨퐻⟩≠0, ⟨Φ⟩=0 (Z 2 preserved, no mixing). • Panel (b): “Mixed vacuum” o Contours of the same potential but with parameters chosen so the minimum moves off-axis. o The minimum (dot) is at 푥≠0, illustrating the branch with ⟨퐻⟩≠0, ⟨Φ⟩≠0 (Z 2 broken, mixing allowed). Figure 1. Schematic contour plots of the tree-level scalar potential 푉(ℎ,푥)in the Higgs–singlet field space. Panel (a) shows a Φ-symmetric vacuum with ⟨Φ⟩=0, while panel (b) shows a mixed vacuum with ⟨Φ⟩=푥≠ 0. Parameter values are chosen for illustration only, to highlight the qualitative difference between the two vacuum configurations. 4. Stability & Boundedness The scalar potential must remain positive at large field values to ensure that the theory possesses a stable ground state. In models with two interacting scalar fields, stability is governed by the behaviour of the quartic terms, which dominate the potential in the limit of large field amplitudes. The potential of interest is and the relevant part for stability is the quartic expression

푉(퐻,Φ)=− 휇 퐻 2 퐻 † 퐻+휆 퐻 (퐻 † 퐻) 2 + 1 2 휇 Φ 2 Φ 2 + 1 4 휆 Φ Φ 4 +푆(Σ) 휆 Φ퐻 Φ 2 (퐻 † 퐻)

[Eq.4.1]

푉 4 (퐻,Φ)=휆 퐻 (퐻 † 퐻) 2 + 1 4 휆 Φ Φ 4 +푆(Σ) 휆 Φ퐻 Φ 2 (퐻 † 퐻)

[Eq.4.2]

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Since 퐻 † 퐻and Φ 2 are non-negative, the quartic form must be positive for all field directions. A standard parameterisation introduces polar coordinates in the two-dimensional field space: Then

with

The potential is bounded from below if and only if 푓(휃)≥0for all 휃. This requirement leads to well-known positivity conditions for interacting quartics:

and the additional constraint which match the familiar boundedness-from-below constraints in singlet–Higgs portal models [6,7,8,10]. Thus, the density modulation factor 푆(Σ) plays a direct role in determining the range of portal couplings compatible with vacuum stability. A positive 푆(Σ) increases the effective portal strength and can tighten the upper bound on 휆 Φ퐻 if 휆 Φ퐻 <0. Conversely, if 푆(Σ)is small in a given environment, the portal term becomes less influential, and the stability region closely resembles the standard constant-portal model. The boundedness conditions apply independently within each vacuum branch. For the Φ-symmetric vacuum (푥=0), stability requires that the curvature in the Φ direction, is positive. This ensures that 푥=0 is a local minimum. For the mixed vacuum (푥≠0), positivity of both eigenvalues of the mass matrix is required, which again depends on the effective portal strength 푆(Σ)휆 Φ퐻 .
Importantly, the density-dependent modulation does not introduce new types of instabilities: the stability constraints are identical to those of the constant portal model, with a simple replacement The structure of the scalar potential remains conventional, and all qualitative behaviour follows directly from this rescaling. In summary, the model admits stable vacua across a broad parameter space, provided the quartic couplings are positive and the effective portal interaction satisfies the usual boundedness requirement. The density modulation does not destabilise the theory; it simply shifts the allowed range of couplings in a predictable and transparent way.

퐻 † 퐻=푟 2 cos 2 휃,Φ 2 =푟 2 sin 2 휃,0≤휃≤ 휋 2 ,푟→∞

[Eq.4.3]

푉 4 =푟 4 푓(휃) [Eq.4.4]

푓(휃)=휆 퐻 cos 4 휃+ 1 4 휆 Φ sin 4 휃+푆(Σ) 휆 Φ퐻 cos 2 휃sin 2 휃

[Eq.4.5]

휆 퐻

0,휆 Φ 0

[Eq.4.6]

푆(Σ) 휆 Φ퐻

− 2 √ 휆 퐻 휆 Φ

[Eq.4.7]

푚 휑 2 =휇 Φ 2 + 1 2 푆(Σ)휆 Φ퐻 푣 2

[Eq.4.8]

휆 Φ퐻 → 푆(Σ) 휆 Φ퐻

[Eq.4.9]

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1. Unitarity, Perturbativity, Loop Corrections The consistency of any extended scalar sector requires that scattering amplitudes remain perturbative at high energies. In renormalisable models with multiple scalar fields, the strongest constraints arise from tree-level unitarity bounds on 2→2 scattering amplitudes and from the behaviour of the quartic couplings under radiative corrections. In the present model, these considerations apply in a particularly transparent way because the density modulation enters only as a multiplicative factor rescaling the Higgs–portal interaction. 5.1 Tree-level unitarity constraints At energies well above the scalar masses but below any potential ultraviolet completion, longitudinal gauge- boson scattering and scalar–scalar scattering impose bounds on combinations of the quartic couplings. These constraints are obtained by constructing the partial-wave expansion of the scattering matrix and requiring that the 푠-wave eigenvalues satisfy ...[7,8,10] are examples where similar unitarity/perturbativity bounds are applied. In the Standard Model extended by a real singlet scalar, the channels most relevant for the scalar sector involve the fields ℎ, 휑, and their mixed quartic interactions. For any fixed value of the density Σ, the quartic coefficients entering these amplitudes are • 휆 퐻 (Higgs self-coupling), • 휆 Φ (singlet self-coupling), • 휆 ̃ Φ퐻 =푆(Σ) 휆 Φ퐻 (effective portal coupling). The unitarity bounds take the schematic form where the precise numerical coefficients depend on the basis of scattering channels. Importantly, the presence of 푆(Σ) does not alter the structure of these constraints; it simply changes the effective coefficient appearing in the mixed scalar scattering amplitude. As a result, perturbative unitarity is maintained provided that the combination 푆(Σ)휆 Φ퐻 remains within the usual perturbative bounds for the environments under consideration. 5.2 Perturbativity and running couplings Even if unitarity holds at tree level, the quartic couplings must remain perturbative under renormalisation-group evolution. The beta functions for a singlet-extended Higgs model are well known and depend only on the couplings of the theory. Because 푆(Σ) is treated as an external parameter rather than a dynamical field, it does not enter the loop diagrams or the counterterm structure. Therefore, the one-loop running of the quartic couplings is identical to that of a standard singlet-portal model with a constant portal coupling replaced by its effective value [7,8,17,18,20,21], where the beta functions for a singlet-extended Higgs model are well known. The requirement of perturbativity then reduces to the standard conditions, including, quartic couplings remain small enough to avoid Landau poles below the cutoff scale, the portal coupling does not drive either scalar quartic negative through radiative effects, and the loop-corrected potential remains bounded from below. As in any effective theory, these conditions restrict the viable parameter space but do not introduce new constraints beyond those already known for constant-portal singlet models.

∣푎 0 ∣≤ 1 2

[Eq.5.1]

∣휆 퐻 ∣≲풪(1),∣휆 Φ ∣≲풪(1),∣휆 ̃ Φ퐻 ∣≲풪(1) [Eq.5.2]

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5.3 Loop stability of the vacuum structure Loop corrections can shift the minima of the potential and modify the mass matrix. In models with a gauge- singlet scalar, the dominant one-loop contributions arise from; • the Higgs quartic, • the singlet quartic, • the portal interaction, • and the Standard Model gauge and Yukawa sectors. Because the density modulation 푆(Σ) acts as a fixed multiplicative parameter in a given environment, it changes the numerical value of the portal coupling entering the loop integrals but does not generate new loop diagrams. The Coleman–Weinberg effective potential therefore retains the same structural form. In particular, the dominant one-loop Coleman–Weinberg corrections scale as ∼휆 2 /(16휋 2 ), ensuring that radiative effects remain perturbative for quartic couplings in the parameter ranges considered. Vacuum configurations that are stable at tree level remain stable under loop corrections provided the couplings remain in the perturbative regime defined above. If the singlet remains in the Φ-symmetric vacuum (푥=0), the one-loop corrections simply shift the mass of the singlet according to the usual renormalisation-group flow. If the mixed vacuum appears, loop corrections modify the scalar mixing angle and eigenvalues in exactly the same way as in constant-portal models. 5.4 Summary of consistency conditions The theory maintains perturbative consistency under the following conditions:

1. Quartic couplings remain within perturbative bounds
2. Running couplings avoid Landau poles below the cutoff scale of the effective theory.
3. Loop-corrected scalar masses remain positive, ensuring that the chosen vacuum branch is stable.
4. No new divergences or counterterms arise from the density modulation, since 푆(Σ)is not a field but an external parameter. Together, these results show that the density-modulated singlet–Higgs model is theoretically well-behaved at the perturbative level. All consistency constraints reduce to those of the standard singlet-scalar extension, with the replacement The environmental modulation does not compromise unitarity, renormalisability, or loop stability.
5. Effective Field-Theory Interpretation with Density Modulation The introduction of a density-dependent modulation function 푆(Σ) raises important conceptual questions regarding interpretation, consistency, and scope. In this section we clarify how the model operates as an effective field theory (EFT), how density dependence can arise without violating locality or renormalisability, and how the framework relates to familiar examples of environment-dependent effective parameters in physics. Throughout, we emphasise that Σ is treated as an external, coarse-grained macroscopic variable, not a dynamical quantum field. This distinction is central: it ensures that the model maintains the operator structure and renormalisability of a standard scalar extension of the Standard Model (SM), while allowing its parameters to vary across regions of differing density.

∣휆 퐻 ∣, ∣휆 Φ ∣, ∣푆(Σ)휆 Φ퐻 ∣≲풪(1) [Eq.5.3]

휆 Φ퐻 →푆(Σ) 휆 Φ퐻

[Eq.5.4]

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6.1 Density as a classical, coarse-grained parameter In any medium, one may define a coarse-grained mass–energy density by averaging over scales large compared with the microscopic structure but small compared with the scale on which macroscopic gradients vary. This is the standard coarse-graining procedure used in field theories in media and cosmology [17–19]. This is standard in both condensed-matter physics and astrophysics. We adopt the same viewpoint: • Σ is defined over macroscopic spacetime regions; • spatial/temporal variations in 푆(Σ) are assumed to be smooth on scales larger than those probed by local QFT interactions; • no quantum fluctuations of Σ are introduced; • the local Lagrangian at each point is simply the renormalisable Lagrangian with effective couplings determined by the value of Σ. This ensures that the theory remains local, Lorentz covariant, and operator-renormalisable within each region. Under these assumptions, the model behaves as a standard singlet–Higgs theory with where all field-theoretic computations proceed with 휆 ̃ Φ퐻 treated as a constant. 6.2 Analogies with well-understood EFT contexts Environment-dependent effective parameters are common across physics. The present model belongs to this tradition, and it is helpful to place it alongside familiar examples: (1) Dielectric and magnetic response in matter The propagation of electromagnetic fields through a material is governed not by the bare vacuum constants but by the medium’s permittivity and permeability. These effective parameters depend on the material’s density, composition, and polarisation response. Crucially, quantum electrodynamics remains unchanged at the microphysical level; the material only modifies the effective couplings seen by long-wavelength modes [17,19]. (2) Photon propagation in plasma physics In a plasma, photons acquire an effective mass through the plasma frequency. Their dispersion relation changes, even though the fundamental Maxwell Lagrangian is unaltered. This is a textbook example of how ambient density alters the effective dynamics of a field [19]. (3) Running coupling constants in quantum field theory In QFT, couplings such as 푔 푠 , 푒, and 푔 depend on the energy scale according to renormalisation-group equations. Although the coupling constants are “fundamental” in the Lagrangian, they run in response to environmental inputs - the probing scale - illustrating that ‘environmental dependence’ of parameters is standard practice in theory and experiment. These analogies underscore that effective parameters depending on environment or scale are ubiquitous. The function 푆(Σ) plays a similar conceptual role for the effective portal coupling [20,21]. 6.3 No violation of renormalisability or locality A crucial question is whether the modulation 푆(Σ) spoils renormalisability. It does not. Because 푆(Σ) is not a field:

휆 ̃ Φ퐻 =푆(Σ) 휆 Φ퐻

[Eq.5.5]

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• it does not appear in loop diagrams; • it does not generate new counterterms; • it does not introduce operators beyond dimension four; • it does not alter the structure of the beta functions; • it does not produce new non-local interactions. For fixed Σ, the theory remains a fully renormalisable, polynomial scalar extension of the SM. Varying Σ across macroscopic regions simply produces a patchwork of EFTs, each with its own effective portal coupling. This is analogous to computing the dielectric constant in different materials or evaluating QED in different media. The underlying field theory remains unchanged; the environment selects the effective parameters [18,18,20]. 6.4 Relation to screened scalar-field models It is helpful to contrast the present construction with scalar-field theories in which the environment dependence is dynamical, such as chameleon scalars, symmetron fields, dilaton-like models, and f(R)-inspired scalar– tensor theories. In these models: • the scalar field’s effective potential depends non-linearly on the local density; • the scalar’s mass and couplings are themselves functions of the field value; • screening mechanisms emerge dynamically through minimisation of the effective potential. By contrast: • in the current model, no new dynamics are introduced; • 푆(Σ)is not derived from additional field degrees of freedom; • the density modulation acts purely as an external parameter, not a self-consistency condition. This distinction makes the model simpler, more conservative, and easier to analyse than screened modified- gravity frameworks. 6.5 Interpretation as an EFT in a medium The structure of the model corresponds directly to a well-defined EFT: • At each point in spacetime, the theory is the Standard Model plus a singlet scalar with couplings evaluated at the local density. • Provided gradients of Σ are small, the EFT is valid in any macroscopic region. • Discontinuities or sharp boundaries in Σ may require interface conditions, analogous to boundary conditions on electromagnetism at dielectric interfaces, however, do not invalidate the EFT. Importantly, no new high-energy behaviour is implied. The modulation by 푆(Σ) does not indicate a breakdown of field theory at short distances; rather, it captures environmental sensitivity at long wavelengths. 6.6 Summary of EFT interpretation The density-modulated Higgs–portal model:

1. Remains a local, polynomial, renormalisable QFT at fixed density.
2. Interprets 푆(Σ) as a coarse-grained effective parameter varying only on macroscopic scales.
3. Fits naturally into the tradition of environment-dependent EFTs, similar to dielectric response, plasma dispersion relations, and running couplings in QFT.
4. Avoids any dynamical screening or additional fields, making it simpler than chameleon/symmetron- like models.
5. Provides a well-posed theoretical foundation upon which more detailed phenomenological applications can later be built.

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This establishes that the model is theoretically conservative, internally consistent, and conceptually well aligned with effective-field-theoretic principles. 7. Comparison with Existing Scalar-Field Models Extensions of the Standard Model that introduce additional scalar degrees of freedom are common in particle physics and cosmology, and it is therefore important to situate the present model within that broader landscape. The simplest and most widely studied example is the real singlet scalar with a constant Higgs-portal coupling, originally introduced in dark-matter contexts and subsequently explored in models addressing vacuum stability, electroweak phase transitions, hidden sectors, and light scalar mediators [1-10]. In these frameworks, the scalar is gauge-invariant and renormalisable, and its interaction with the Standard Model is governed by a quartic portal term of the form 휆 Φ퐻 Φ 2 퐻 † 퐻. This forms the baseline theoretical structure against which the present model can be compared. A parallel line of research investigates environmentally dependent scalar interactions, most notably in the context of “screened” scalar fields such as chameleons, symmetrons, and dilaton-like models[11-16]. In these theories, effective scalar masses or couplings depend on local macroscopic energy density through non-linear self-interactions or curvature-dependent terms. Unlike chameleon models, which rely on field-dependent screening where the effective mass satisfies 푉 ′ (휙)∝휌 and the scalar dynamically adjusts to local density, the present framework introduces no such dynamical mechanism: 푆(Σ) is an external macroscopic parameter, and no field-dependent screening or density-driven self-consistency condition arises. These mechanisms allow a scalar field to mediate large-scale effects while remaining consistent with laboratory and Solar System constraints. Although such models introduce density dependence, they typically do so through additional dynamical fields or non-linear potentials, not through a direct modulation of a Higgs-portal interaction. Between these two categories - constant-portal singlet models and screened modified-gravity scalars - there is relatively little work exploring Higgs-portal interactions whose strength is directly modulated by ambient coarse-grained macroscopic energy density. Some studies of scalar–Higgs mixing in media note that effective parameters can shift in the presence of dense environments, but these shifts arise indirectly from field mixing or thermal corrections rather than from an explicit density-modulation function. The model proposed here occupies a distinct conceptual space. It retains the simplicity and renormalisability of the conventional Higgs-portal singlet extension, while introducing a density-dependent modulation factor 푆(Σ) that scales the portal coupling without altering the underlying symmetry structure. This approach can be viewed as a simplified parameterisation of more complex environmental effects, compatible with an effective field-theory treatment in which density acts as an external parameter. To our knowledge, this precise combination - a gauge-singlet scalar with a density-modulated Higgs-portal interaction - has not been developed in existing literature, and therefore provides a novel variation within the broader class of scalar extensions of the Standard Model. 8. Discussion and Conclusions In this work we have examined the theoretical consistency of a gauge-invariant real scalar singlet that interacts with the Standard Model exclusively through a Higgs-portal coupling modulated by a coarse-grained density- dependent function 푆(Σ). The construction is deliberately conservative: the field content is minimal, the interaction terms are renormalisable, and the dependence on ambient density is treated as a parametric modulation rather than a dynamical effect. Within this framework, all aspects of the scalar sector, including vacuum structure, stability, perturbativity, and radiative corrections, all reduce to those of a standard singlet- extension with the effective portal coupling 휆 ̃ Φ퐻 =푆(Σ)휆 Φ퐻 . Our analysis shows that this model is theoretically well defined. The potential remains bounded from below provided the usual positivity conditions on the quartic terms are satisfied, with the density modulation entering only through a straightforward rescaling of the mixed quartic coefficient. Perturbative unitarity places constraints on the effective portal coupling that are identical in structure to those in conventional singlet–Higgs models. Loop corrections do not introduce any new divergences or counterterms beyond those already known for the constant-portal case, because 푆(Σ) acts as an external macroscopic parameter rather than a dynamical

17

field. These results confirm that the model is renormalisable and perturbatively controlled across a broad parameter region. The vacuum structure is likewise governed by familiar considerations. Depending on the parameters, the theory admits a symmetric vacuum in which the singlet does not condense, and a mixed vacuum where both the Higgs and singlet acquire expectation values. The properties of the physical scalar eigenstates depend on the density through 푆(Σ), influencing their masses and mixing angles. Although a detailed phenomenological analysis lies beyond the scope of this paper, the framework permits the possibility that scalar-sector properties may vary across environments with different densities, a feature not present in constant-portal models. Conceptually, the interpretation of the model as an effective field theory clarifies its naturalness. Environment- dependent effective parameters are ubiquitous across physics, for example in dielectric media, plasma-modified photon propagation, and the running of gauge couplings with energy [17-21]. The present construction fits naturally within this paradigm: it introduces no new non-locality, no new high-dimensional operators, and no alteration to the Standard Model gauge structure. The theory therefore offers a structurally simple and well- grounded extension of the Standard Model scalar sector that remains consistent across a wide range of densities. The results presented here establish the theoretical foundations necessary for further work. Potential future directions include: analysing the phenomenology of the density-modulated portal interaction in the subatomic, laboratory, astrophysical, or cosmological environments; exploring whether the modulation may arise from a more fundamental UV completion; assessing constraints from Higgs mixing and precision electroweak observables; and investigating whether density-dependent scalar sectors could play a role in settings where coarse-grained macroscopic energy density varies strongly, such as stellar interiors, neutron stars, or the early universe. These applications require dedicated study and lie beyond the strictly field-theoretic scope of the present paper. In conclusion, a gauge-invariant singlet scalar with a density-modulated Higgs-portal interaction constitutes a theoretically coherent, renormalisable, and perturbatively valid extension of the Standard Model. By isolating the field-theoretic structure from any phenomenological claims, this work provides a clear foundation on which more detailed analyses of the scalar sector in varying-density environments can be built. References Singlet–Higgs portal & scalar extensions [1] Silveira V., Zee A., 1985, Phys. Lett. B, 161, 136 [2] McDonald J., 1994, Phys. Rev. D, 50, 3637 [3] Patt B., Wilczek F., 2006, arXiv:hep-ph/0605188 [4] Schabinger R. M., Wells J. D., 2005, Phys. Rev. D, 72, 093007 [5] O’Connell D., Ramsey-Musolf M. J., Winslow P., 2007, Phys. Rev. D, 75, 037701 [6] Barger V., Langacker P., McCaskey M., Ramsey-Musolf M. J., Shaughnessy G., 2008, Phys. Rev. D, 77, 035005 [7] Gonderinger M., Li H., Patel S., Ramsey-Musolf M. J., 2010, J. High Energy Phys., 01, 053 [8] Lebedev O., 2012, Eur. Phys. J. C, 72, 2058 [9] Cline J. M., Kainulainen K., Scott P., Wen C., 2013, Phys. Rev. D, 88, 055025 [10] Robens T., Stefaniak T., 2015, Eur. Phys. J. C, 75, 104 Screened / environment-dependent scalar fields [11] Khoury J., Weltman A., 2004, Phys. Rev. D, 69, 044026 [12] Hinterbichler K., Khoury J., 2010, Phys. Rev. Lett., 104, 231301 [13] Olive K. A., Pospelov M., 2008, Phys. Rev. D, 77, 043524 [14] Brax P., van de Bruck C., Davis A.-C., Shaw D. J., 2012, Phys. Rev. D, 86, 044015 [15] De Felice A., Tsujikawa S., 2010, Living Rev. Relativ., 13, 3 Modified gravity / scalar–tensor context

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[16] Clifton T., Ferreira P. G., Padilla A., Skordis C., 2012, Phys. Rep., 513, 1 EFT, running couplings, fields in media [17] Burgess C. P., 2004, Ann. Rev. Nucl. Part. Sci., 57, 329 [18] Burgess C. P., 2020, Ann. Rev. Nucl. Part. Sci., 70, 329 [19] Weldon H. A., 1982, Phys. Rev. D, 26, 2789 [20] Peskin M. E., Schroeder D. V., 1995, An Introduction to Quantum Field Theory, Addison-Wesley [21] Weinberg S., 1995, The Quantum Theory of Fields, Vol. I, Cambridge Univ. Press