arXiv:1811.07898v1 [hep-ph] 19 Nov 2018

Higgs stability-bound and fermionic dark matter

Aaron Held1, ∗ and René Sondenheimer2, †

1Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany2Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena, Germany

Higgs-portal interactions of fermionic dark matter – in contrast to fermions coupled via Yukawa in-teractions – can have a stabilizing effect on the standard-model Higgs potential. A non-perturbativerenormalization-group analysis reveals that, similar to higher-order operators in the Higgs potentialitself, the fermionic portal coupling can increase the metastability scale by only about one orderof magnitude. Furthermore, this regime of very weakly coupled dark matter is in conflict withrelic-density constraints. Conversely, fermionic dark matter with the right relic abundance requireseither a low cutoff scale of the effective field theory or a strongly interacting scalar sector. Thisresults in a triviality problem in the scalar sector which persists at the non-perturbative level. Thecorresponding breakdown of the effective field theory suggests a larger dark sector to be present nottoo far above the dark-fermion mass-scale.

I. THE HIGGS POTENTIAL AND NEWPHYSICS

The most surprising result obtained at the LargeHadron Collider might be, that the masses and couplingsconspire to render the standard model consistent up tovery high energy scales, possibly even all the way up tothe Planck scale [1–5]. It is a delicate balance betweenthe running of gauge couplings, Yukawa couplings andthe Higgs potential that results in this unexpected exper-imental result. Although this so-called “desert” disfavorsnew degrees of freedom to be discovered any time soon,it should maybe not be taken as discouragement. Ex-perimental evidence suggests that our knowledge of theUniverse is more complete than we might have hoped for.The vast other parts of parameter space, in which thestandard model as an effective field theory (EFT) couldin principle be realized, results either in an instability ofthe scalar potential or in a sub-Planckian triviality prob-lem. Both of these could be interpreted as a clear signfor a breakdown of the standard model and an associatednew physics scale.

Focusing on the central measurement values [6], thestandard model as an EFT is potentially valid up toΛ(SM)

EFT ≈ 1041 GeV. At this scale, the theory reaches aU(1) Landau-pole in perturbation theory [7], reflected ina triviality problem at the non-perturbative level [8–10].This would suggest that the next energy scale that the-orists should be concerned about is the Planck scale, atwhich a joint theory of gravity and matter becomes nec-essary, see e.g., [11–17]. On the other hand, the standardmodel Higgs-potential, assuming that the current cen-tral values of measured mass and coupling parameterswere exact, develops a metastability. The energy scaleassociated with this metastability is Λ(SM)

meta ≈ 1010− 1011

GeV [1–5, 18]. Much interesting physics can be associ-ated with this metastability [19–23], and the metastable

∗ [email protected]† [email protected]

electroweak vacuum could be long-lived enough to al-low for the current age of the universe [24–29]. Here,we will take the most conservative viewpoint and inter-pret the metastability as another sign of a breakdownof the effective theory and as a demand for new de-grees of freedom below Λ(SM)

meta . We will therefore com-bine the minimum of both of these scales into the scaleat which new degrees of freedom are to be expected, i.e.,Λ(SM)

new-phys = Min(Λ(SM)meta ,Λ

(SM)EFT ).

Since the standard model looks surprisingly complete,it is desirable that necessary new physics, such as darkmatter, will not destroy this delicate balance. It iswell-known that an additional weakly-interacting scalarfield, coupled via a Higgs portal, could act as dark mat-ter and at the same time might fully remove a sub-Planckian metastability scale [30–35]. Such scalar dark-matter models could account for the total relic density ifthe perturbatively renormalizable Higgs-portal couplingis sufficiently weak [31, 33, 36–38].In this paper, we analyze whether fermionic dark mat-

ter coupled through a Higgs portal could – despite itsperturbative non-renormalizability – delay the metasta-bility scale in a similar manner. Naïvely, one might objectthat top-quark fluctuations are known to be the causeof the metastability in the first place: consequently onesometimes reads that fermionic fluctuations destabilizethe Higgs potential. We will argue that the declining ofthe Higgs quartic coupling towards the UV is only truefor fermions coupled via a Yukawa interaction yHtt. Ithas recently been suggested [39], that the fundamentallydifferent vertex structure of fermions coupled via a Higgs-portal coupling of the form

LHiggs-portal ∼ λhψH†Hψψ (1)

could lead to a regime in which dark fermions, with suf-ficiently light mass mψ, in fact delay the onset of theHiggs-instability scale. This approach is different fromother scenarios discussed in the literature with additionalright-handed neutrinos where the stabilizing effect wascaused by UV boundary conditions governed by high-scale supersymmetry [40].

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We will address these questions in the framework ofEFT by use of the functional renormalization group (RG)[41], for reviews see [42–49]. Validity of the standardEFT requires that all dimensionless combinations, i.e.,λhψ mψ and λ2, the quartic Higgs self-coupling, enteringcross sections remain perturbative at all energy scalesk below the EFT cutoff scale ΛEFT. We will find thatthe following intuition persists in the non-perturbativeanalysis:

• Decoupling regime: mψ � Λ(SM)meta . It is obvious

that very heavy dark fermions will decouple andnot influence the metastability scale.

• Destabilizing regime: mψ ≈ Λ(SM)meta . Dark fermions

of a mass comparable with the metastability scalewill further destabilize the Higgs sector.

• Stabilizing regime: mψ � Λ(SM)meta . Comparatively

light dark matter can stabilize the Higgs-potentialand increase the metastability scale. However, thenon-perturbative analysis shows that portal cou-plings λhψ > 1/Λ(SM)

meta will introduce novel Landau-pole like instabilities requiring new physics even be-low the original metastability scale.

In principle, the functional RG allows the analysis tobe extended into the non-perturbative regime, which wedefine as λhψ mψ > 4π. To remain as conservative aspossible, we refrain from such an extension and interpretthe onset of non-perturbativity as a breakdown of theconventional EFT description. This is corroborated bythe non-perturbative analysis which shows that the flowequations in our approximation become singular in thisregime.

In Sec. II we analyze the three regimes in functionalRG flows in a simplified Higgs-Yukawa toy model as in[50]. We compute the effective potential in different ap-proximations and validate the usefulness of the pertur-bative beta functions to obtain a first estimate for a pos-sible scale of new physics. In Sec. III, we improve ouranalysis to obtain reliable results for the actual metasta-bility scale of the standard model and investigate thephenomenological implications of fermionic dark matter.In particular, we discuss our results in the context of thedark-matter relic density. Assuming that the fermionicdark matter is thermally produced, the relic-density con-straint will rule out a shift of the EFT cutoff scale beyondthe standard-model instability scale. Thus, a larger darksector is required, if fermionic dark matter coupled via aHiggs portal is realized in Nature.

II. RG FLOW OF A SIMPLE HIGGS-YUKAWAMODEL WITH FERMIONIC DARK MATTER

To obtain an understanding of the impact of thehigher-dimensional portal coupling on the RG running of

the Higgs potential, we restrict the discussion to the de-grees of freedom relevant for the lower Higgs-mass bound.Therefore, we consider a real scalar field φ coupled to twodifferent species of fermions. The first fermion species tis coupled via an ordinary Yukawa coupling to the scalarfield yt φ t t and thus represents the top quark of the stan-dard model. The second species denoted by ψ describesa fermionic dark matter particle which is a singlet withrespect to the standard model gauge group and couplesthrough a dimension-5 operator to the scalar field. Thus,the classical action for this toy model reads,

S =∫x

[12∂µφ∂

µφ+m2φ

2 φ2 + λ2

8 φ4 + t i/∂ t+ iyt√2φ t t

+ ψ i/∂ ψ + imψψ ψ + iλhψ2 φ2ψ ψ

], (2)

where we work in Euclidean signature. The couplingsand mass parameters with an overbar indicate that theseare dimensionful quantities. This model strongly sim-plifies the electroweak sector. Nevertheless, it containsthe main ingredients to discuss the generic properties ofthe lower Higgs-mass bound and the stability propertiesof the effective Higgs potential [51–54]. Note that themodel under consideration is invariant under a discretechiral symmetry φ→ −φ, t→ eiπ2 γ5t, t→ t eiπ2 γ5 as wellas a continuous U(1) symmetry ψ → eiαψ, ψ → e−iαψ.γ5 is the product of the Dirac matrices as usual. The dis-crete chiral symmetry mimics the electroweak symmetrygroup of the standard model and forbids the occurrenceof a mass term for the top quark.Investigating the RG flow of the model, we are con-

fronted with the perturbative nonrenormalizability of thedimension-5 coupling λhψ. It was pointed out in Ref. [39]that also the running and threshold effects of the darkfermion mass term mψ might play an important role inthe running of the Higgs sector. Hence, our investiga-tion has to go beyond conventional analyses in the deepEuclidean region of the RG flow.A well-suited tool capable of handling all these tasks

is the functional RG. In its modern form, the runningof a scale-dependent effective average action Γk is stud-ied. The effective average action interpolates smoothlybetween the classical action at a high UV cutoff scaleΓΛ = S and the full quantum effective action Γk=0 = Γ,the generating functional of 1PI Greens functions. TheRG-scale dependence of Γk is governed by the functionalRG equation [41], cf. also [55]

∂tΓk = 12STr

[(Γ(2)k +Rk

)−1∂tRk

](3)

which has a technically simple one-loop structure butcontains the full propagator at each RG step. The reg-ulator function Rk ensures IR and UV finiteness. Itspresence in the denominator, where it acts like a scale-dependent mass term, regulates IR divergences. Its scale-derivative in the numerator regulates UV divergences byimplementing Wilson’s idea to integrate out fluctuations

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in a momentum-shell wise fashion. We refer to the liter-ature for more details on the functional RG [42, 44–48].

In order to investigate the running of the Higgs po-tential, the Yukawa coupling, and the portal coupling,we choose the following ansatz for the effective averageaction based on a systematic derivative expansion,

Γk =∫x

[Zφ2 ∂µφ∂

µφ+ U(ρ) + Zt ti/∂t+ i√2Ht(ρ)φ tt

+ Zψ ψi/∂ψ + iHψ(ρ)ψψ]. (4)

We have introduced a generic potential U(ρ) for thescalar self-interactions. Further we promote the Yukawacoupling as well as the dark-matter coupling to functionsof the symmetry invariant ρ = φ2/2, Ht(ρ) and Hψ(ρ),respectively. Another advantage of the functional RGsetup is that the running of full coupling functions de-pending on various mass scales can be studied. In partic-ular, these beta-functionals serve as closed-form expres-sions to extract the flow equations for any higher-ordercoupling in terms of the scalar field by suitable expan-sions of the various potentials in polynomials.

It is convenient and useful to introduce renormalizedfields to fix the RG invariance of field rescalings and tointroduce dimensionless quantities, i.e.,

ρ = Zφkd−2ρ, u = k−dU,

ht = Z− 1

2φ Z−1

ψ k4−d

2 Ht, hψ = Z−1ψ k−1Hψ. (5)

This allows to track the flow over many orders of mag-nitude of the RG scale. Finally, the flow equations forthe potentials can be derived from Eq. (3). In terms ofdimensionless, renormalized variables they read

∂tu(ρ) = −d u+ (d− 2 + ηφ) ρu′ + 2vd[l(B)d0 (ωφ; ηφ)

− dγ l(F)d0 (ωt; ηt)− dγNf l

(F)d0 (ωψ; ηψ)

], (6)

∂tht(ρ) = 12(d− 4 + ηφ + 2ηt)ht + (d− 2 + ηφ)ρh′t

+ 2vd ht(ht + 2ρh′t)2 l(BF)d1,1 (ωφ, ωt; ηφ, ηt)

− 2vd (3h′t + 2ρh′′t ) l(B)d1 (ωφ; ηφ), (7)

∂thψ(ρ) = (−1 + ηψ)hψ + (d− 2 + ηφ)ρh′ψ+ 8vd ρ hψh′

2ψ l

(BF)d1,1 (ωφ, ωψ; ηφ, ηψ)

− 2vd (h′ψ + 2ρ h′′ψ) l(B)d1 (ωφ; ηφ). (8)

Here, primes denote derivatives with respect to ρ andωφ = u′ + 2ρu′′, ωt = ρh2

t , and ωψ = h2ψ are RG-

scale and field-amplitude dependent mass parameters forthe different fields. Further, d denotes the spacetime di-mension, dγ is the dimension of the Clifford algebra andv−1d = 2d+1π

d2 Γ(d/2). The running of the wave function

renormalizations Zφ/t/ψ is encoded in the anomalous di-mensions of the fields, ηφ/t/ψ = −∂t lnZφ/t/ψ. Their flow

is governed by

ηφ = 4vdd

{2κ(3u′′ + 2κu′′′

)2m

(B)d2 (ωφ; ηφ)

+ dγ(ht + 2κh′t)2[m

(F)d4 (ωt; ηt)− κh2

tm(F)d2 (ωt; ηt)

]+ 4dγκh′ψ

[m

(F)d4 (ωψ; ηψ)− h2

ψm(F)d2 (ωψ; ηψ)

]}ρ=κ

,

(9)

ηt = 4vdd

(ht + 2κh′t)2m(FB)d1,2 (ωt, ωφ; ηt, ηφ)

∣∣∣ρ=κ

, (10)

ηψ = 16vdd

κh′2ψ m(FB)d1,2 (ωψ, ωφ; ηψ, ηφ)

∣∣∣ρ=κ

, (11)

where the right-hand sides are evaluated at ρ = κ.The latter is the actual scale-dependent minimum of thescalar potential. It lies at vanishing field amplitude inthe symmetric (SYM) regime and at nonvanishing fieldvalues in case of spontaneous symmetry breaking (SSB).The so-called threshold functions l and m on the right-hand sides of the flow equations arise from the integrationover loop momenta and contain the nonuniversal regula-tor dependence. They can be evaluated analytically fora piece-wise linear regulator function which is optimizedwith respect to the present truncation scheme [56, 57]and can be found, e.g., in [58].The non-perturbative RG evolution of Yukawa mod-

els has been investigated extensively in many contexts inthe literature. This includes effective quark-meson mod-els [59–64], statistical systems [65–69], as well as asymp-totic freedom and safety [39, 70, 71]. In particular, asimple Higgs-Yukawa toy model has proven useful to in-vestigate the generic mechanisms and properties for theoccurrence of lower Higgs-mass bounds within the setof power-counting renormalizable operators and beyond[18, 50–53, 58, 72–74].

A. Preliminary analysis of the quartic polynomialpotential

The basic mechanisms at work can already be analyzedin polynomial approximations of the potentials. In thesymmetric regime, we expand the three different poten-tials in power series around vanishing field amplitude,

u(ρ) =Nu∑n=1

λnn! ρ

n, ht(ρ) =Nh∑n=0

ynn! ρ

n,

hψ(ρ) =Nψ∑n=0

λhψ,nn! ρn. (12)

In this parametrization, the coefficients have the natu-ral interpretation of conventional coupling constants andmass terms. For instance, the dimensionless scalar mass-parameter is given by the linear term of the scalar po-tential u, i.e., λ1 ≡ m2

φ. The quartic coupling is given byλ2. The higher-order coefficients describe higher-order

4

interaction vertices which will be generated during theRG flow even if they are not present at the UV scale.Further, we identify y0 ≡ yt as the usual top Yukawacoupling as well as λhψ,0 ≡ mψ and λhψ,1 ≡ λhψ as thedimensionless dark fermion mass term and Higgs-portalcoupling, respectively. We focus on the basic stabiliza-tion mechanisms of the Higgs potential, which take placein the SYM regime, i.e., for energy scales much higherthan the electroweak scale. In this regime, the simpleapproximation of the potentials in terms of polynomialsis sufficient to analyze the system [50].

The perturbative RG evolution of the quartic Higgsself-coupling already provides a good indicator for theoccurrence of an instability if the impact of higher-dimensional couplings is neglected at UV scales. In thiscase, the zero crossing of the coupling determines the in-stability scale. This is corroborated by investigations ofthe full non-perturbative functional flow of the potentialas well as polynomial approximations thereof [50, 58].The latter serve as the simplest possible extension of theperturbative flow equations.

Permitting the existence of higher-dimensional opera-tors at the UV scale, the instability scale can be shiftedtowards larger scales. In this case, the metastability ofthe potential arises as an intricate interplay of the non-trivial bare structure of the Higgs potential at the UVcutoff scale and the fluctuating fields. A second or evenseveral competing minima can occur. Then, polynomialtruncations might no longer be an optimal choice for theproper tracking of the RG evolution of the Higgs poten-tial. Instead, global information in field space is requiredto investigate all implications properly.

Nonetheless, polynomial approximations of the poten-tial are still able to capture the relevant information ofthe occurrence of an instability for bare interactions ofpolynomial type [50]. For this particular class, the radiusof convergence of the potential in field space is finite andshrinks during the RG flow. Due to the fact that the sec-ond minimum usually occurs close to the cutoff scale, thestability issue can be addressed within these polynomialtruncations.

The running of the various polynomial couplings can beextracted by suitable projections of the beta functionalsof the potentials. The flow equation for the quartic scalarcoupling, ∂tλ2 = ∂tu

′′(0) in the SYM regime, reads

∂tλ2 = 116π2

[9λ2

2(1 +m2

φ)3 + 4λ2y2t − 4y4

t

+4λ2

hψ

(1 +m2ψ)2 −

16m2ψλ

2hψ

(1 +m2ψ)3

]. (13)

Here, we have neglected any term coming from higher-dimensional operators except for the Higgs-portal cou-pling, i.e., we have truncated the potentials at the sim-plest nontrivial order (Nu, Nh, Nψ) = (2, 0, 1). Further,we have neglected contributions from the anomalous di-mensions within the threshold functions. These corre-

spond to various resummed diagrams and thus higher-loop orders. As we have derived the flow equationfrom the Wetterich equation, threshold effects of mas-sive modes are automatically incorporated. Due to thefact that the top quark is massless in the SYM regime,no threshold corrections appear for top fluctuations. Bycontrast, a mass term for the dark fermion ψ is not forbid-den by the symmetries of the action (2), thus modifyingthe running for nonvanishing masses.First, we analyze the flow equation for the quartic cou-

pling in the massless limit, mψ → 0 and mφ → 0, i.e.,we set the threshold corrections to zero and obtain theone-loop coefficients in the deep Euclidean region. Cru-cially, the impact of the portal coupling λhψ comes withthe opposite sign than the pure top quark loop withinthis limit. This might be rather surprising at first sight.It is commonly known that fermionic fluctuations causethe diminishing of the quartic coupling towards the UV,resulting in the stability constraints on the Higgs mass.Thus, the inclusion of further fermion species usually low-ers the instability scale. However, the dark fermion cou-ples to the scalar field in a different manner than theusual Yukawa-type structure as dictated by symmetry.Of course, a dark-fermion loop generally comes with anegative sign, just as the top loop does. This is due totheir fermionic nature which can be seen in the flow equa-tion for the scalar potential, cf. Eq. (6). But, we obtaina relative sign between the pure top-quark loop and thedark-fermion loop in the flow equation of the quartic cou-pling due to the different vertex structure of the differentscalar-fermion interactions. Hence, the dimension-5 op-erator can effectively reduce the Higgs mass for a fixedcutoff scale. From another perspective, this implies thatthe instability scale is shifted towards larger scales.However, this lowering mechanism is only present

in a certain range of the dimensionless dark-fermionmass-parameter. Including threshold effects of the darkfermion, we can directly infer from the second linein Eq. (13) that the contribution of the dark fermionchanges sign at mψ = 1/3, cf. [39]. Thus, as long asmψ < 1/3, the quartic coupling λ2 is effectively reducedduring the flow from UV to IR. In case mψ > 1/3, thedark fermion loops contribute to the growth of the Higgsmass in the IR, similar to the top quark. As the dimen-sionless parameter mψ generically grows towards the IR,both effects will contribute to the running of the scalarsector if the initial parameter at the UV scale is below thecritical value of 1/3. Nonetheless, the latter effect con-tributes only over a comparatively short amount of RG“time” because the dark fermion decouples from the flowas soon asmψ � 1 due to threshold effects. Thus, we candivide the impact of the dark fermion on the running ofthe Higgs sector into three different regimes; stabilizing,destabilizing, and decoupling.It depends on the precise initial parameters whether

the stabilizing or destabilizing mechanism dominates theRG running of the scalar sector. In case the initial massparameter is larger than one-third of the cutoff scale,

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i.e., mψ(k = Λ) > 1/3, the Higgs mass increases dueto the dark fermion. For mψ(k = Λ) smaller but suffi-ciently close to 1/3, the diminishing effect is present fora few RG steps but erased by the destabilizing effect formψ > 1/3 before the decoupling limit governs the flow(depending on the strength of the portal coupling as wellas the scalar sector itself). Therefore, the Higgs mass isonly reduced for sufficiently small bare mψ(k = Λ) suchthat the contribution from the integrated running in thestabilizing regime overwhelms the contribution from thedestabilizing regime.

In order to check in which parameter regions the darkfermion diminishes the lower mass bound or compoundsthe stability issue, we investigate in detail the RG evo-lution of the proposed toy model and perform varioustruncation tests to substantiate the picture arising fromthis simple analysis in the following.

B. Functional analysis of the Higgs potential

Solving the system of flow equations for the scalar,Yukawa, and portal potential as well as for the anoma-lous dimensions of the fields allows us to investigate theHiggs-stability problem on a global level in field space.However, this is a rather computing-time consuming en-deavor as a system of coupled nonlinear partial differen-tial equations has to be solved over many orders of mag-nitude with a sufficiently high precision to separate theUV cutoff scale ΛEFT from the electroweak scale. There-fore, we begin the investigation with the simplest possibleextension of the perturbatively renormalizable flow equa-tions by expanding the potentials in polynomials up to afinite order.

The beta functionals of the potentials can be viewedas closed-form expressions for the flow equations for ar-bitrary higher-order operators. The main advantage ofthe expansion of the potentials in Taylor series aroundthe minimum of the scalar potential is that the sys-tem of partial differential equations is reduced to a sys-tem of ordinary differential equations. A typical flowfor Higgs masses close to the lower bound starts in theSYM regime. We fine-tune the initial mass parameterfor the scalar field such that fermionic fluctuations drivethe system into the SSB regime as soon as the RG scaleapproaches the electroweak scale and the Higgs poten-tial develops a nonvanishing minimum. At this point, weswitch to expansions of the potentials which are givenby Taylor series around the flowing minimum. The lat-ter is the main point of interest in field space because itdetermines the mass and couplings of the particles. Thenonvanishing vacuum expectation value of the scalar fieldinduces a mass for the top quark as well as for the Higgsexcitation. Therefore, all particles involved in the RGflow become massive and the flow freezes out completely.Further, we vary the bare Yukawa coupling y0,Λ to ob-tain a top mass of 173 GeV. For the moment, we leavethe Higgs mass as a free parameter which depends on the

UV cutoff scale Λ, the bare quartic coupling λ2,Λ, as wellas other details in the bare action encoded in the higher-dimensional couplings λ3,Λ, λ4,Λ, . . . , y1,Λ, y2,Λ, . . . andthe dark sector mψ,Λ, λhψ,1,Λ, λhψ,2,Λ, . . . .In order to check the convergence properties of the

polynomial approximation of the full potential, we per-form calculations for different truncation orders Nu, Nh,and Nψ and compare the extracted masses for the scalarfield. Here we follow the same strategy as in [50, 53]for the model without dark fermions. We observe thefollowing behavior. As most of the important physics isstored in the shape of the potentials at the Fermi scale,the expansion around the flowing minimum of the scalarpotential u is well suited to extract the masses of theparticles and we obtain a rather fast convergence. Withrespect to the truncation order of the scalar potential,Nu = 2 is the simplest nontrivial order. Increasing Nu,i.e., including more and more higher-dimensional scalarself-interactions, we find Nu = 4 as an optimal trunca-tion parameter for fixed Nh and Nψ. From Nu = 2 toNu = 4 we find a slight deviation in the computed Higgsmass by a few percent in small as well as large couplingregimes. For Nu > 4, we find no deviations from theresults obtained for Nu = 4 within our numerical ac-curacy. Performing the same investigation for fixed Nuand Nψ and varying Nh, we find that the resulting Higgsmasses are converged for Nh = 2. Similarly, we obtainNψ = 1 as an optimal truncation order regarding con-vergence and computing time, thus demonstrating theremarkable convergence of the polynomial truncation forthe present purpose.In order to test the convergence regarding the deriva-

tive expansion of the effective average action, we com-pare leading order to next-to-leading order results. Wedrop the running of the kinetic terms at leading orderby setting the anomalous dimensions to zero in the flowequations of the potentials (local potential approxima-tion). We find mild differences about 5 % for the lowermass bounds. This difference is mainly caused by thefact that the local potential approximation does not in-clude all one-loop contributions given by one-particle re-ducible diagrams. These are stored in the anomalousdimensions which are not present inside a threshold func-tion, i.e., the contribution of the anomalous dimensionswhich are separated in the first lines on the right-handsides of Eqs. (6)-(8) which modify the canonical scalingdimensions. Including these contributions but setting theanomalous dimensions inside the threshold functions tozero, we obtain changes of merely a few 100 MeV of theresulting Higgs masses compared to the leading-order re-sult of the derivative expansion.

C. Higgs mass bounds and phenomenologicalimplications

In Fig. 1 we plot the shift of the low energy Higgs massas a function of the dimensionless dark fermion mass pa-

6

● ● ● ● ● ● ● ● ●

▲ ▲ ▲▲

▲ ▲ ▲ ▲ ▲

○○

○

○

○

○ ○ ○ ○

0.01 0.10 1 10 100-40

-30

-20

-10

0

10

mψ,Λ

ΔmH[GeV]

Figure 1. Relative shift of the Higgs mass from the conven-tional lower mass bound as a function of the dimensionlessdark fermion mass parameter at the cutoff scale for Λ = 105

GeV. The black curve with black filled circles denotes val-ues for λhψ,Λ = 0.1. The blue curve with triangles and thered curve with empty circles show results for λhψ,Λ = 1 andλhψ,Λ = 3, respectively. Solid lines indicate that the shiftcan be obtained without introducing a metastability in theeffective Higgs potential. A second minimum is present inthe effective potential for dashed curves. The dotted branchdepicts a shift in the Higgs mass with stable potential whichis stabilized by a nonvanishing scalar sector in the UV.

rameter at the cutoff scale Λ for different values of the di-mensionless portal coupling λhψ,Λ.1 In this example, thecutoff scale is set to Λ = 105 GeV but similar conclusionshold also for other scales. The mass shift measures thedeparture from the conventional lower bound. For theconsidered toy model, the equivalent of the conventionallower bound is given by mH,min = 94.9 GeV for a cutoffscale Λ = 105 GeV. As in our preliminary analysis of thequartic coupling, we find that a diminishing of the lowermass bound is achieved for small bare mass parametersmψ,Λ. However, this effect is only measurable for suffi-cient strong portal couplings of O(1) or even larger. Forλhψ,Λ = 0.1, depicted as a black line with filled circles,the deviation from the conventional lower mass boundwithout dark sector and vanishing higher-order opera-tors at the cutoff scale is less than 0.1 GeV. The effectof the dark fermions diminishing the lower Higgs-massbound grows rapidly for larger values of λhψ,Λ. We de-pict the results for λhψ,Λ = 1 and λhψ,Λ = 3 as a bluecurve with triangles and a red curve with empty circles,respectively. The largest mass shift is achieved for a van-

1 The fact that we use mψ,Λ instead of the observable IR mass ofthe dark fermion is merely for reasons of convenience. As a ruleof thumb, the dark fermion mass can be computed by the follow-ing simple relation mψ,IR = (mψ,Λ + λhψ,Λ

64π2 )Λ which deviatesfrom the actual dark fermion mass by at most 3% for all testedparameter combinations. This approximation is based on thefact that the dark fermion sector renormalizes multiplicativelyand is mainly dominated by its power counting behavior.

ishing dark fermion mass parameter at the cutoff scale.Already for mψ,Λ < 0.01 we observe a saturation of thecurve towards the limit mψ,Λ → 0.By contrast, neither in the small nor large coupling

regime a deviation from the conventional lower massbound can be observed for large bare mass parameters.Of course, this is expected as the threshold effects ofthe dark fermion dominate the impact on the RG flowof the scalar sector already in the UV in this limit. Thiscan be circumvented for extremely large portal couplings,λhψ & m2

ψ, such that the decoupling in the flow equa-tion of the Higgs potential is compensated by the largecoupling. However, the dark fermion fluctuations in-duce larger Higgs masses, i.e., lower the instability scale,in this limit as can be directly inferred from Eq. (13).Therefore, we do not further explore the parameter spacein this regime.In the intermediate mass regime, we observe the pre-

viously discussed change in the effect of dark-fermionfluctuations on the lower Higgs-mass bound. Instead ofdiminishing the lower mass bound, the fermionic fluc-tuations start raising it close to the critical value ofmψ = 1/3. At this critical value, the dark-fermion con-tribution to the running of the quartic coupling changessign. The raising of the lower Higgs-mass bound ap-proaches a maximum around mψ,Λ ≈ 1, cf. Fig. 1. Weobserved that a large cutoff portal-coupling λψ,Λ quicklyenhances the lowering effect for mψ,Λ < 1/3. Similarly,the effect of raising the Higgs mass bound for mψ,Λ ≈ 1quickly increases with growing λψ,Λ.Depending on the specific values of the dark-fermion

mass and the portal coupling, dark fermions have the po-tential to lower the Higgs mass by an arbitrary amount ascan be seen in Fig. 1. Nevertheless, this does not meanthat the Higgs potential remains stable during the RGflow. We observe that the dark fermions can indeed sta-bilize the Higgs potential and thus raise the instabilityscale but only for a certain finite amount. Once a criticalupper value of the shift in the Higgs mass is passed, theeffective average potential starts to develop a second non-trivial minimum at large field values usually below butclose to the cutoff scale Λ, rendering the Higgs potentialmetastable.This is similar to investigations on the impact of

higher-dimensional operators of just standard-model de-grees of freedom. Within these examples, the instabilityscale can also be shifted towards larger scales. As the run-ning of higher-dimensional couplings is mainly governedby their power-counting dimension, they quickly die out.This implies that they can only stabilize the Higgs po-tential by 1-2 orders of magnitude if the dimensionlesshigher-order couplings are of order one. We observe asimilar effect in the model with dark fermions wherethe stabilizing effect is also caused by a power-countingnonrenormalizable operator. By contrast, the instabilityscale can easily be shifted towards larger scales for scalardark matter, where the portal coupling is marginal [30–35].

7

In Fig. 1 we indicate this fact by different codings ofthe interpolating lines. Solid lines depict shifts in theHiggs mass for which the scalar potential u is stable dur-ing the entire RG flow. Dashed lines depict a possiblediminishing of the lower mass bound accompanied by asecond minimum of the effective potential beyond theFermi scale. In almost all cases this is the global one,thereby again rendering the potential metastable. Thisoccurs only for sufficiently strong portal couplings andsmall dark-fermion masses. Then the RG flow of thequartic coupling is driven towards negative values. Theseare not compensated for by positive higher-dimensionalcouplings of the scalar sector to render the potential sta-ble. However, this effect can be mitigated if a positiveand sufficient large quartic or higher-dimensional cou-pling in the scalar sector is present in the UV. The low-est possible value of the Higgs mass with stable Higgspotential is indicated as a dotted line for λhψ,Λ = 3. Forthis, we scanned the parameter region by allowing forsuch positive bare couplings and examined the transitionfrom a stable to an unstable potential. Thus, any shiftof the Higgs mass above the dotted or solid curve canbe reached for λhψ,Λ = 3 without introducing a secondminimum in the effective potential.

In Fig. 2, we plot the conventional lower Higgs massbound for this toy model, i.e., all higher-dimensional cou-plings are zero at the cutoff scale, as a black dashed linefor various cutoff scales Λ. In addition, the red curve de-notes the diminishing of the lower bound due to the por-tal coupling to dark fermions for fully stable Higgs poten-tials during the entire RG flow for bare couplings of O(1).A further diminishing would result in a metastable poten-tial due to the complex interplay between the bare struc-ture of the potentials and the quantum fluctuations. Sim-ilarly to investigations in models without a dark fermion,we find that the stability scale can be shifted by roughlyone order of magnitude towards larger cutoff scales byhigher-dimensional operators.

Finally, we address the question of the validity of thepolynomial approximation of the potential to investigatethe stability problem globally in field space. To that end,we solve the full partial differential equation for the scalarpotential u in leading order in the derivative expansion.In order to minimize the computational effort, we treatthe couplings of the other sectors as external sources, i.e.,we neglect their RG running forced by quantum fluctua-tions. Of course, this does not capture the mutual backreactions between the different sectors but is sufficient toinvestigate the occurrence of a possible second minimumon a qualitative level. We find that the polynomial trun-cation of the potential is able to describe the occurrenceof a second minimum during the flow as in the previ-ous studies without a dark fermion [50]. The radius ofconvergence is large enough to appropriately track thedevelopment of a second minimum at large scales duringthe RG flow. For instance, for Λ = 105 GeV, we haveexamined that both, the full flow as well as the polyno-mial approximation of the potential, allow for the same

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4 6 8 10 12 14

50

100

150

200

Log10 Λ[GeV]

mH[GeV]

Figure 2. Lower Higgs-mass bounds with a single minimumin the effective Higgs potential. The black-dotted line depictsthe conventional lower mass bound for the considered toymodel without dark matter or the influence of any higher-dimensional operator in the UV. The red solid line demon-strates how fermionic dark matter, as well as other higher-dimensional operators, can diminish this bound without in-troducing a metastability in the effective potential. Thus thecutoff scale of new physics Λ is increased.

shift of ≈24 GeV of the lower mass bound due to the por-tal coupling with a stable electroweak minimum. In bothapproximations, increasing the portal coupling, which re-sults in a further decrease of the Higgs mass, causes theoccurrence of a second minimum.The full flow of the potential shows the same behavior

as investigated in previous toy models without fermionicdark matter. Thus, the zero crossing of the quartic cou-pling provides a good estimate for the instability scaleas long as a further stabilization by higher-dimensionaloperators is absent. However, the latter does not al-ter the order of magnitude of the shift of the instabilityscale but causes small quantitative modifications towardslarger scales. In the following, we will use these results ofthe simple toy model to address the implications for thestandard-model Higgs potential. In particular, we willconfront our results with experimental constraints on therelic abundance which further constrains the parametersof the dark sector.

III. FERMIONIC DARK MATTER AND THESTANDARD-MODEL HIGGS-POTENTIAL

To compare our analysis with experimental bounds onthe fermionic dark matter couplings, we supplement thesimple model with the standard model gauge structure,cf. [18], without explicitly gauging the Higgs. In par-ticular, we add an SU(3) gauge group under which wecharge the nf Yukawa-type fermions, but neither thedark-matter fermions nor the Higgs, modifying the cor-

8

responding effective action, cf. Eq. (2), to read

Γk =∫x

[ZG4 F aµν F

aµν + Zφ2 ∂µφ∂

µφ+nf∑i

Zi qii /Dqi

+ Zψ ψi/∂ψ + U(ρ) + i√2Ht(ρ)φ qtqt

+ iHψ(ρ)ψψ]. (14)

We also keep neglecting all Yukawa couplings from thenon-heavy fermions qi6=t because of their negligible im-pact on the running of the Higgs potential. This capturesthe influence of the strong interaction in the standardmodel above ΛQCD.Gauging the scalar under the electroweak gauge-group,

i.e., transitioning to a Higgs-doublet, would require ourmodel to capture the full electroweak sector of the stan-dard model to observe the freeze-out of all massive modescorrectly. Instead, we add a fiducial gauge-contributionto the running of the dimensionless effective scalar po-tential u(ρ) and the Yukawa potential ht(ρ). It has beenchecked that such a model captures all of the standard-model structure required for a quantitative discussion ofthe lower Higgs-mass bound [18], but avoids intricatequestions arising from the involved Higgs-gauge bosoninterplay [75–82]. The dark fermions are left unchargedunder both the strong and the fiducial electroweak gaugesector.

The running of this toy standard model is captured byadding the following additional contributions from fidu-cial electroweak gauge loops as well as the strong sectorto Eqs (6)-(8):

∂tu(ρ)∣∣∣toy-SM

= ∂tu(ρ) + 4vdd

cu

1 + ρ g2F

2

, (15)

∂tht(ρ)∣∣∣toy-SM

= ∂tht(ρ) + 2 vd ct g2Fht (16)

− 4 vdN2c − 12Nc

g2s ht(3− ξ) l(BF)d

1,1 (0, ωt; ηG, ηt).

The strong gauge coupling is denoted by gs. The termscontaining the fiducial gauge coupling gF model elec-troweak contributions. In all following numerical compu-tations we will employ Landau gauge, i.e., ξ = 0, which isalso a fixed point of the RG flow [83]. Further, one has toaccount for closed fermionic loops in the scalar potentialand anomalous dimension, that now contribute with colormultiplicity, i.e., dγ → dγ Nc in the second line of Eqs (6)and (9). Finally, there are also additional gauge contribu-tions to the fermionic anomalous dimension, cf. Eq. (C9)in [18], but we refrain from showing them here since theycancel out in Landau gauge (ξ = 0). With gF = 0.57,ct = 97

30 , and cu = 9 the running of the conventional top-Yukawa coupling yt = ht(0) and the quartic Higgs cou-pling λ2 = u′′(0) falls back to the known standard-model1-loop running when threshold effects are neglected [18].

A. Parameter space of fermionic dark matter

As we have carefully analyzed in Sec. II, the lowestnon-trivial truncation of the polynomial approximation,i.e., (Nu, Nh, Nψ) = (2, 0, 1), already captures the quali-tative features of the effects of fermionic dark matter onthe lower Higgs-mass bound. We will therefore presentresults in the (2, 0, 1)-scheme and confirm the validity ofthis approximation at specific points, cf. App. A. We fixthe coupling values of the strong gauge, the top Yukawaand the scalar quartic coupling to match the standardmodel values gs(1TeV) = 1.060, yt(1TeV) = 0.867 andλ2(1TeV) = 0.202 at 1 TeV at one loop. Then we mapout the possible parameter space of fermionic dark mat-ter mass mψ(1TeV) and portal coupling λhψ(1TeV) atthe matching scale. Since we only consider dark-mattermasses above mψ > 1 TeV, the dark matter couplings willalways freeze out above 1 TeV. Below 1 TeV they scalenear-canonically. Hence, the dimensionful values of mψ

and λhψ essentially do not run below 1TeV. We confirmthe validity of this approximation in App. A.Fig. 3 shows the difference of the metastability scale

with fermionic dark matter and the standard-modelmetastability scale, i.e., Λ(fdmSM)

meta −Λ(SM)meta . In the simple

(2, 0, 1)-scheme, the metastability scale is estimated bythe respective scale at which the quartic coupling turnsnegative. For our toy standard-model (without fermionicdark matter) this amounts to Λ(SM)

meta = 2.54 × 109 GeV.The contours within the stabilizing (green) regime in-dicate by how much the metastability scale can be in-creased. In accordance with the analysis of the simpleHiggs-Yukawa model in Sec. II, we again find a desta-bilizing regime whenever mψ ≈ Λ(SM)

meta , cf. red region inFig. 3.

The functional RG is a non-perturbative tool. Henceour analysis might still capture the correct dynamics inthe non-EFT regime. Nevertheless, we also determinethe scale, Λ( fdmSM)

EFT , at which the perturbative EFT con-ditions λhψmψ < 4π and λ2 < 4π break down. Whilefor the standard model on its own this trans-Planckianscale is as high as Λ(SM)

EFT ≈ 1041 GeV, the influence ofthe fermionic dark sector can significantly lower thisbound. We mark these regions in Fig. 3 as dark-grayshaded. The upper right triangle in parameter spaceshows where the EFT criterion λhψmψ < 4π is alreadyviolated at 1 TeV. We do not perform further computa-tions in this regime. The region on the upper left-handside λ2 > 4π shows where the Higgs-quartic couplingleaves the EFT-regime below the standard metastabilityscale, i.e., Λ( fdmSM)

EFT < Λ(SM)meta . Shortly thereafter, it also

runs into a perturbative Landau pole. This occurs due totoo large screening contributions from the fermionic darkmatter. While the fermionic dark sector remains of sta-bilizing nature, it nevertheless fails to increase the scaleof new physics Λ( fdmSM)

new-phys = Min(Λ( fdmSM)meta ,Λ(fdmSM)

EFT ).Finally, we check the results obtained in

(Nu, Nh, Nψ) = (2, 0, 1) against the ones obtained

9

Figure 3. Regions in the parameter space of dark mattermass mψ and portal coupling λhψ, in which the Higgs po-tential is (de)stabilized. In the green (red) region the Higgsstability-bound is shifted to larger (smaller) values. The thincontours show the relative increase of the metastability scale,i.e., (Λ(fdmSM)

meta − Λ(SM)meta)/Λ(SM)

meta . The shaded regions indicatewhere a perturbative EFT description, either in the fermionicsector or in the scalar sector breaks down.

in the optimal truncation order (Nu, Nh, Nψ) = (4, 2, 1),cf. Sec. II. This requires a tuning of the bare massparameter for each point individually. For all pointsthat we tested the qualitative behavior agrees with thelower order truncation, with quantitative shifts of thecontours up to 9% at log-log scales. Details on thisanalysis can be found in App. A.

B. Cosmologically viable fermionic dark matterrequires additional new physics

The standard-model analysis confirms the findings ofthe simple Higgs-Yukawa model. Fermionic dark mattercan increase the new physics scale Λ(fdmSM)

new-phys > Λ(SM)new-phys

to which the standard model can be extrapolated as astable EFT. For this, the dark-fermion mass has to belight in comparison to the metastability scale, in or-der to avoid a complete decoupling of the dark sectoror a Yukawa-like influence on the running of the Higgs-potential. Secondly, the portal coupling has to be veryweak, i.e., λhψ < 1/Λ(SM)

meta . Otherwise, it induces a break-down of the EFT in the form of a Landau-pole-like in-stability below Λ(SM)

meta . Even if both requirements aremet, the Higgs-portal coupling is only able to increaseΛnew-phys by about one order of magnitude, cf. Fig. 2.Therefore, in contrast to scalar dark matter, fermionicdark matter cannot extend the standard model as a sta-ble EFT up to the Planck scale, even within the presentnon-perturbative analysis.

Furthermore, the regime in which fermionic dark mat-ter has a slight stabilizing effect, i.e., λhψ < 10−8 GeV−1,is in direct conflict with relic density constraints. Per-

turbative studies [84] suggest that any fermionic por-tal coupling λhψ < 10−3 GeV−1 would oversaturate therelic-density bound and therefore overclose the universe.Under the assumptions that fermionic dark matter isproduced via a post-inflationary thermal freeze-out andthe relic-density analysis is not drastically modified bynon-perturbative effects, this observational constraint ex-cludes the possibility of fermionic dark matter to stabilizethe Higgs potential without introducing a Landau-pole-like instability below Λ(SM)

meta .Therefore, if such fermionic dark matter was to be

discovered experimentally, it would significantly impactthe Higgs potential. The stabilizing fermionic fluctua-tions would drive the quartic coupling towards a pertur-bative Landau pole far below the current metastabilityscale, i.e., Λ( fdmSM)

new-phys ≈ 1/λhψ. In the present functionalRG study, we find this behavior to persist at the non-perturbative level. Our study therefore tentatively sug-gests that fermionic dark matter, in agreement with relic-density constraints, either requires the transition into aregime of fully non-perturbative dynamics or suggests alarger dark sector close to Λ( fdmSM)

new-phys ≈ 1/λhψ in whichthe Higgs-portal coupling appears as a higher-order in-teraction.One possibility for a strongly coupled regime with-

out the requirement of additional new degrees of free-dom could be offered by a non-perturbative fixed point,as suggested by the analysis in [39]. Whether tentativehints for the fixed point can be confirmed in extendedstudies and it might then be used to define a stronglycoupled Higgs-portal sector remains a promising futurestudy. This presumably requires the inclusion of higher-order operators in the fermionic sector.

IV. CONCLUSIONS

We have investigated the global stability properties ofthe standard-model Higgs-potential with the functionalrenormalization group under the impact of fermionicdark matter coupled via a Higgs portal. The motivationfor this study stems from the low-truncation (perturba-tive) indications that the fermionic portal coupling, con-trary to Yukawa couplings, can reduce the lower Higgs-mass bound, i.e., delay the onset of the standard-modelmetastability. The reason for this distinct behavior is en-coded in the different symmetry structure of the fermionscoupled to the Higgs via Yukawa or portal interactions.While the non-perturbative analysis confirms that the

metastability scale can be shifted towards higher scales,the dimensionful nature of the portal coupling onlyallows stabilizing the Higgs-potential for about one orderof magnitude. This already follows from a dimensionalargument, as the Higgs portal to fermionic dark matter isa dimension-5 operator. We find that this perturbativelyintuitive conclusion also holds at the non-perturbativelevel. We numerically solve the flow equation for thescalar potential, as obtained from the functional RG

10

equation at lowest order in the derivative expansion andconfirm that the polynomial truncation sufficiently cap-tures the structure of a possibly arising minimum. Everyattempt to stabilize the quartic coupling by significantlymore than one order of magnitude leads to the emergenceof a novel minimum for polynomial interactions. Thisresembles a low-scale Landau-pole like instability foundin perturbative approximations. In non-perturbativetruncations, the running couplings become singular ina similar fashion. This minimum quickly becomes theglobal minimum and the corresponding instability scaledrops below the usual standard-model metastabilityscale – hence invalidating the original motivation. Weconclude that the dimension-5 portal coupling cannotsignificantly stabilize the Higgs-potential. This is due toits power-counting dominated suppression towards theIR in the vicinity of the Gaußian fixed point. The samebehavior was already found in the previous analysisof higher-order operators in the scalar and Yukawapotential [50, 58].

Further, we supplement the simple Higgs-Yukawamodel with an approximation of the gauge structure ofthe standard model, for which it was confirmed in [18]that the running of couplings is quantitatively very closeto that of the full standard model. We explicitly verifythat all the above statements carry over to this gauge-supplemented model. This allows us to quantitativelyanalyze the influence of fermionic dark matter on theHiggs metastability-bound. We find that the regime inwhich the Higgs potential can at least be stabilized byone order of magnitude relies on a very small portal cou-pling λhψ < 10−8 GeV−1. This is in conflict with theperturbative relic-density constraint λhψ > 10−3 GeV−1

for dark fermion masses > 100 GeV obtained in, e.g. [84].In turn, our study suggests that if fermionic dark mat-

ter obeying the relic-density constraint should be de-tected through its portal interaction with the Higgs,it would quickly lead to an increase of the scalar self-interactions encoded in the Higgs potential. This woulddrive the quartic coupling out of the perturbative regime.In this case, we find the formation of Landau-pole typestructures in the couplings within our nonperturbativecalculations. The presence of fermionic dark matter thusrequires additional new physics at scales no higher thanabout Λ(fdmSM)

new-phys ≈ 1/λhψ. This is particularly intrigu-ing because it suggests that fermionic dark matter eitherleads to a novel non-perturbative scaling regime, such asthe asymptotically safe fixed-point proposal in [39], orrequires an extended dark sector with additional degreesof freedom.

Acknowledgements: We are grateful to A. Eichhornand H. Gies for fruitful discussions and comments.A. Held acknowledges support by the Studienstiftung desdeutschen Volkes as well as by the Emmy-Noether grantof the DFG awarded to Astrid Eichhorn under grant no.EI/1037-1. A. Held also acknowledges the hospitality ofPerimeter Institute for Theoretical Physics during the fi-

nal stages of this work. Research at Perimeter Instituteis supported by the Government of Canada through theDepartment of Innovation, Science, and Economic De-velopment, and by the Province of Ontario through theMinistry of Research and Innovation. RS acknowledgessupport by a postdoc fellowship of the Carl-Zeiss founda-tion as well as by the DFG under Grants No. Gi328/9-1.

Appendix A: Comparison with the optimizedtruncation order

In this appendix, we address the reliability of the low-est non-trivial truncation of the polynomial approxima-tion, (Nu, Nh, Nψ) = (2, 0, 1), regarding the parameterspace of fermionic dark matter of Sec. III. As discussedin detail for the simple toy model of Sec. II, the scheme(Nu, Nh, Nψ) = (4, 2, 1) captures all relevant informa-tions of the polynomial flows regarding the IR masses ofthe particles. We observe the same behavior in the modelwith gauge boson contributions as well. Due to slight cor-rections of the higher-dimensional operators on the run-ning of the quartic coupling, we obtain the same quali-tative results but minor quantitative corrections on log-arithmic scales. For instance, the instability scale Λ(SM)

metadecreases as the quartic coupling is slightly driven fastertowards zero mainly by the higher-dimensional operatorsof the Yukawa sector. As the quartic coupling runs loga-rithmically this causes a diminishing of the metastabilityscale by less than an order of magnitude. More precisely,we have log10 Λ(SM),(4,2,1)

meta / log10 Λ(SM),(2,0,1)meta = 0.91.

As the basic stabilizing mechanisms take place in theUV near the cutoff scale of the model, we get the sameorder of magnitude of the relative increase (Λ(fdmSM)

meta −Λ(SM)

meta)/Λ(SM)meta , if we choose the same UV initial condi-

tions in the dark fermion sector for the different trunca-tion schemes. However, due to the difference in the ref-erence scale Λ(SM)

meta , we obtain quantitative corrections inthe IR physics. This manifests as a shift of the contourstowards smaller masses and larger portal couplings in thelog-log graph of Fig. 3. This shift is almost universal forall test points in the plot. As a rule of thumb, the con-tours are shifted by ≈ −0.87 in the horizontal direction(dark fermion mass) and ≈ 0.89 in the vertical direction(portal coupling) on the presented log-log scale, i.e., byless than one order of magnitude. These results are sta-ble with respect to a further increase of the truncationorders (Nu, Nh, Nψ).Thus, we are still confronted with the fact that

the relic-density constraint excludes the possibility offermionic dark matter to stabilize the Higgs potentialwithout introducing a Landau-pole-like instability belowΛ(SM)

meta for the advanced analysis. Therefore, we refrainfrom performing the analysis of Sec. III for the com-putationally expensive advanced truncation as the phe-nomenological interesting region in parameter space isstill out of reach and the same conclusions remain.

11

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