
Bottomup Naturalness as a Guide to New Physics
Anirban Kundu*
Department of Physics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700009, India
*akphy@caluniv.ac.in
ABSTRACT
Without any mechanism to protect its mass, the selfenergy of the Higgs boson diverges quadratically, leading to the hierarchy, finetuning, or naturalness problem. One bottomup solution is to postulate some yettobe discovered symmetry which forces the sum of the quadratic divergences to be zero, or almost negligible; this is known as the Veltman Condition. It is well known that in the Standard Model (SM), the Veltman Condition is badly violated. Thus, one needs additional degrees of freedom, either in an ultraviolet complete theory or in some effective theory, to alleviate the problem. In this article, both ways are discussed; first, with the addition of some new scalars, and second, with the addition of some dimension6 effective operators. The parameter spaces are consistent with all theoretical and experimental bounds, and should act as a guide to model builders.
INTRODUCTION
The 125 GeV resonance, first announced on July 4, 2012 [1, 2], is the only fundamental spin0 particle found till date. All its properties, including the decay widths into different channels and the quantum numbers like spin and parity, indicate that this is indeed the Higgs boson [3], or the celebrated "incomplete scalar multiplet" predicted by Peter Higgs fifty years ago. It is high time that we ask the question: is it the only fundamental scalar, or just the tip of the iceberg, the first member of a big family of fundamental scalars?
Fermion and gauge boson masses are protected by chiral and gauge symmetry respectively, and so the selfenergy is only logarithmically diverging. There is no symmetry that protects the scalar mass. The mass term, m^{2} Î¦^{†} Î¦ does not break any symmetry that is respected by the potential Î»(Î¦^{†} Î¦)^{2}. So the scalar selfenergy is quadratically divergent, Λ^{2}, where Λ is the cutoff scale for the theory. The only reason why the Higgs mass is not at the Planck scale M_{Pl} ~ 𝒪(10^{19}) GeV, but at the electroweak scale, is that there must be a tremendous cancellation between the huge quantum correction and the bare mass, leaving a tiny nonzero contribution. This is the finetuning or naturalness problem.
There is another way to look at the finetuning problem. Suppose the bare mass squared of the unrenormalized theory is _{}. We add a counterterm and get the renormalized mass squared, _{}. Now one changes _{} to _{}+Î´_{}, with Î´_{}/ _{}~ 𝒪(1). This changes _{} to _{}+Î´_{}. If Î´_{}/ _{} ≫ 1, the theory is findtuned. In this case, _{}/Î´_{} may be taken as an approximate measure of finetuning. For Λ = 2 TeV, just outside the reach of the Large Hadron Collider (LHC), this is about one or a few per cent, not at all uncomfortable, but higher values of Λ definitely brings back the finetuning problem, maybe in a softened way.
When we talk about the degree of divergence, it is always better to use the cutoff regularisation, which clearly brings out how badly divergent an amplitude is. In dimensional regularisation, all divergences are lumped into 1/ϵ, which is good if we want to subtract the divergent part and work with the finite part, but not so helpful if we want to know which divergence is worse, scalar selfenergy or electron selfenergy. Cutoff regularisation is not Lorentz invariant, but it is undoubtedly the best way to feel the badness of a divergence.
There are several ways to address the finetuning problem. One might bring in more degrees of freedom, invoking a perfect symmetry between bosons and fermions, so that the divergence coming from a bosonic loop is almost exactly cancelled by that coming from a fermionic loop (Supersymmetry). One might lower the Planck scale somehow so that the finetuning problem loses its severity (Extra dimensions). The Higgs might not be a fundamental scalar after all but just a composite object made up of fermions (Technicolour or topcondensate), or it might be the (pseudo)Goldstone boson of a higher symmetry group which keeps its mass small (Little Higgs models). There is always be the anthropic principle; bizarre as it may sound, it is perhaps no more bizarre than to ask for an almost complete cancellation between two huge and uncorrelated terms.
In this article, we will not talk about any possible ultraviolet complete (UVC) theory, like super symmetry, that may solve the hierarchy problem. We will, rather, demand that perhaps due to some yettobediscovered symmetry, the quadratically divergent contributions to the Higgs mass add up to zero, or a very small value. This is known as the Veltman Condition (VC) [4].
Thus, if we confine ourselves to oneloop diagrams only, the renormalised Higgs mass squared is given by
 (1)

where the ubiquitous coefficient of 1/16𝜋^{2} comes from the evaluation of the loop, and f(g_{i}) is a function of relevant scalar, Yukawa, and gauge couplings. The VC demands that f(g_{i}) should be zero, or extremely tiny, so that m_{h}_{,0} is not too much away from the electroweak scale. The logarithmically divergent as well as the finite terms coming from the loop diagrams have been neglected, and denoted by the trailing ellipses.
One may argue that f(g_{i}) need not be exactly zero; in fact, f(g_{i}) ~ 16𝜋^{2} _{}/ Λ^{2} should be perfectly acceptable. However, with all the masses known, the VC fails badly with respect to the SM [5, 6]. There are numerous attempts in the literature to make f(g_{i}) ≈ 0 by introducing more particles, like extra scalars or fermions [713]. While these attempts were more or less successful and provided some important constraints on the parameter space, the VC could hardly be stabilised over the entire energy scale from v to Λ if one considers the renormalisation group (RG) evolution of the couplings. This remains one of the major shortcomings of the bottomup approach.
There are, therefore, two ways to cancel the quadratic divergences. One may introduce some new degrees of freedom below the cutoff scale Λ, resulting in more dimension4 operators in the Lagrangian (d < 4 terms are not relevant). Alternatively, one may assume that whatever New Physics (NP) exists there at the high energy scale can be effectively integrated out at the scale Λ to give us the SM, plus some effective operators involving only the SM fields, which is known as the SM Effective Field Theory (SMEFT). In the second approach, the exact nature of the UVC is irrelevant. All the UVC information is incorporated in the Wilson coefficients (WC) of the effective operators.
In the first part of this article, we will discuss two extensions of the SM, namely, by a singlet scalar, followed by another Higgs doublet; we then explore the parameter spaces of these models for a possible solution of the VC. In the second part, we will focus upon the SMEFT approach.
In SMEFT, the first interesting higher dimensional operators come at d = 6. There are many equivalent bases to express the complete set of d = 6 operators. We will use the basis given in Ref. [14]. Only a handful among the 59 dimension6 operators contribute to the quadratically divergent part of the scalar selfenergy.
An ndimensional operator can at most result in a divergence in Higgs selfenergy that goes as Λ^{n2}. As these operators are suppressed by Λ^{n4}, one expects contributions to f(g_{i}) from all orders. Ref. [15] explains why it is enough to consider d = 6 operators.
Thus, we will focus only on an effective theory with a schematic Lagrangian
 (2)

where c_{4}_{i} and c_{6}_{i} are dimensionless constants. The VC now takes the form
 (3)

Our aim will be to find out the parameter space for the c_{4}_{i} and c_{6}_{i} coefficients.
The article is arranged as follows. In Section 2, we discuss the VC in the SM (with only d = 4 operators). In the next two Sections, Sec. 3 and Sec. 4, we discuss the d = 4 extensions, first with one (or more) gauge singlet scalar(s), and then in a twoHiggs doublet model. In Section 5, we discuss the VC in SMEFT with dimension6 operators. Section 6 concludes the paper.
VELTMAN CONDITION IN THE SM
The scalar potential of the Standard Model is given by
 (4)

with Î¼^{2}, Î» > 0. This ensures the stability of the potential (a welldefined vacuum state) as well as spontaneous symmetry breaking. We set the vacuum expectation value (VEV) of Î¦ as <Î¦> = v /_{}, so that v = 246 GeV.
At oneloop, the Higgs selfenergy receives a quadratically divergent correction
 (5)

where g_{1} and g_{2} are the U(1)_{Y} and SU(2)_{L} gauge couplings, and g_{t} = _{}m_{t} /v is the top quark Yukawa coupling. One can safely treat all other fermions as massless. As advocated earlier, we use cutoff regularisation.
Thus, the Veltman Condition, in its original form, is written as
 (6)

In terms of the masses, this reads
 (7)

This needs m_{h} = 316 GeV, so in the Standard Model, the VC is far from being satisfied.
There are a number of points worth discussing.
• We have not talked about any fundamental theory or mechanism that keeps the Higgs light, so is the idea at all appealing? Not if one knows very definitely that one of the few beyond Standard Model ideas that have been proposed must be the truth. Unfortunately, we do not know this, we are groping in the dark, and this is perhaps the best that a cartographer in a newly discovered continent can do. That is why this is a venture off the beaten track: the goal is to first find the minimal extension of the Standard Model that solves the finetuning problem, and then try to embed the solution in some concrete, wellmotivated model. In a sense, supersymmetry implements the Veltman Condition, but some underlying dynamics of the theory ensure that the condition is satisfied, at least in the limit of exact supersymmetry.
• The Veltman Condition is not satisfied by cutoff regularisation; does dimensional regularisation help? As we have noted, dimensional regularisation does not separate quadratic and logarithmic divergences, and dumps everything into the coefficient of the ubiquitous 1/ϵ. We get a slightly different correction with dimensional regularisation [6]:
 (8)

So even this does not help much.
• Is the VC gauge invariant? The answer is yes, as can be explicitly checked by working out the quadratic divergences in Landau and 't HooftFeynman gauges. However, one may ask what happens in the unitary gauge, as the gauge propagator has a leading momentum dependence of k^{0}. While one may question the justification of using the unitary gauge as the VC is relevant only for Λ ≫ v where the electroweak symmetry is still unbroken, all particles are massless, and the condition is formulated in terms of the couplings only, it would nevertheless be satisfactory to see that nothing catastrophic happens in the unitary gauge. For example, one may think of having a quartic divergence, ~ Λ^{4}, coming from the gauge loop, as the gauge propagator is not momentum suppressed. At the same time, one has to remember that electroweak symmetry is broken, and there are generic Higgsgaugegauge vertices in the theory. One can have a selfenergy contribution with two such vertices, which again is quartically divergent. It has been explicitly checked by us [15] that these quartic divergences cancel; for the Wloop, the amplitude with the fourpoint vertex gives g^{2} Λ^{4}/(128𝜋^{2} m_{W}^{2}), which is exactly cancelled by the amplitude with two threepoint vertices. The latter also gives a quadratic divergence, which is needed to restore gauge invariance.
• Is a strict implementation of Eq. (6) necessary? There are at least two reasons why we do not expect this. First, the higher loop effects are there; what has been shown is only the oneloop result. Compared to the leading order, the higher loops are suppressed by further powers of log (Λ^{2} / m^{2}) / 16𝜋^{2}, so their contributions have to be subleading, but definitely at the level of a few per cent. Second, we can always accommodate some cancellation between the bare Higgs mass and the radiative corrections. If Î´_{} ~ _{}, there is no finetuning, so this is perfectly acceptable. If Î´_{} ~ 10_{}, there is a cancellation of one in 10, or a tuning of 10%, which is nothing to worry about. Without any finetuning, i.e. Î´_{} ≤ _{}, one finds
 (9)

This inequality is clearly not satisfied in the SM for v^{2} / Λ^{2} ≤ 0.1, or Λ ≥ 760 GeV, and onset of NP at such a low scale is almost ruled out by the LHC. If we allow a finetuning of 1 in N , the scale goes up by a factor of _{}. But that is what we expect: if there is some NP at a few TeV, no one bothers about the finetuning problem.
• One might also expect the Veltman Condition to be satisfied for every energy scale for which the theory remains valid. This upper range might be the Planck scale, the GUT scale, or some intermediate scale where the theory ceases to be valid. In other words, the stability condition in the Standard Model is
 (10)

where t = log(Q^{2}/Î¼^{2}). If we use only the oneloop Î²functions, this becomes [7]
 (11)

Again, this is far from being satisfied. Of course, twoloop calculations exist. It was shown in [6, 16] that in any generic Yukawa theory, if the Higgs mass correction is to remain zero at all scales (f_{h} = 0, f_{h}' ≡ df_{h }/dt = 0) using oneloop Î²functions, it is precisely the same condition as for quadratic divergences at twoloop to vanish^{1}. Recently, it was pointed out in [17] that the correction to the Higgs mass changes sign at a very high scale, the exact value of which again depends on the top quark mass, which led to the conjecture [18] that the sign reversal may trigger electroweak symmetry breaking. However, Jones [19] has disputed the claims of [17].
It is clear that one needs more bosonic degrees of freedom to satisfy the Veltman Condition. One can try to have more gauge bosons; however, more scalars are needed anyway to give them gauge invariant masses. So let us just try to enhance the scalar sector, adding scalars that couple to the Standard Model doublet Î¦. In fact, all multiplets do that, because 𝒮^{†} 𝒮Î¦^{†} Î¦ is always a gauge singlet, no matter how 𝒮 transforms. There are other nontrivial constraints though. The first is the stability of the scalar potential: there cannot be any direction in the field space where the potential becomes unbounded from below, or breaks U(1)_{em}^{2}. The second one is that the new scalars must also have their own Veltman Conditions satisfied if they should remain at the electroweak scale. In fact, if they are very heavy, one can always integrate them out at the electroweak scale and the Higgs finetuning problem will be back. Thus, we have three necessary conditions:
•There must be new scalars.
•The scalar potential must be stable.
•The new scalars must also couple to fermions^{ 3}.
In the next two Sections, we will concentrate on two different models, each formulated with a completely different motivation:
• Models with one or more singlet scalar: This is by far the minimal extension of the Standard Model. With enough singlets, the electroweak phase transition can be made firstorder. However, singlets do not couple to chiral fermions, so we need vectorial fermions to address their own finetuning.
• TwoHiggs doublet models (2HDM): There are four different types of 2HDM that do not allow any FCNC, and there are more if controllable FCNC are allowed. Except for the TypeI 2HDM, both doublets couple to ordinary fermions, so we do not need any extra fermions.
^{1 }We do not expect this to hold exactly. First, f(g_{i}) may not be exactly zero. Second, the result was derived for a nongauge Yukawa theory. Third, this gives only the leading divergence but not the subleading ones, e.g. for twoloop, terms proportional to Λ^{2} log(Λ^{2 }/^{ }m^{2}) but not terms proportional to Λ^{2} only. However, the subleading terms can be neglected compared to the oneloop contributions.
^{2 }If there are coloured scalars in the theory, one must ensure SU(3)c invariance too.
^{3 }Some of the new scalar couplings might be negative and still consistent with the second condition, but they are usually not enough to avoid the necessity of fermionic couplings. We will see some examples later on.
THE MINIMAL EXTENSION: ONE REAL SCALAR SINGLET
The singlet extension of the scalar sector is the most economical option. Moreover, it does not run afoul of the ρparameter, and can also be a very good cold dark matter candidate if it does not mix with the doublet Higgs. The role of one (or more) extra singlet(s) has been extensively considered in the literature [710, 20]. Here, we will mostly follow Ref. [10]. We will also talk only about the oneloop analysis; that is easiest to follow.
Let us first consider the SM augmented with one real singlet scalar S. The potential is
 (12)

We will take both Î¼^{2}, M^{2} > 0 to start with. The remnant of Î¦ after spontaneous symmetry breaking is the Higgs boson h. There might be a cubic term cS^{3} in the potential, but that would not affect the subsequent analysis. The linear terms in S giving rise to tadpole diagrams are assumed to cancel out and they will remain so even after the quantum corrections. This happens if we take the tadpole potential to be
 (13)

with Î±_{1} + _{} Î±_{2} v^{2} = 0. There might also be a discrete symmetry, like S → S, preventing odd terms. For N singlets with an O(N) symmetry, the potential looks like
 (14)

With one extra singlet S, as in Eq. (12), the VC is modified to
 (15)

If there are N number of identical singlets (i.e., an O(N) symmetric singlet sector), the last term is replaced by N_{a}.
For N = 1, we find that a = 4.17, which is quite large even if not nonperturbative (nonperturbativity sets in when a coupling is at about 4𝜋 ≈ 12.56). More singlets bring down the value to 4.17/N. This is to be taken as an indicative value only, as there is no reason why this value would be absolutely stable if one takes higherorder corrections (for an estimate, see [9], where it can be seen that such corrections bring a marginal change.) To be consistent, we will use only oneloop renormalization group (RG) equations to calculate the evolution of the couplings.
The singlet VC, independent of whether the singlet develops a VEV or not, reads
 (16)

So, only with the singlet, we need a large (and definitely nonperturbative) and negative quartic coupling, and the potential develops a minimum unbounded from below in the direction Î¦ = constant and S → ∞. Thus, this solution is clearly unacceptable.
While one needs some negative contribution to Eq. (16), one notes that this cannot come from chiral fermions of the SM as they do not couple to S. Thus, one is led to introduce vector fermions — either singlets or doublets under SU (2). This introduces further terms in the potential:
 (17)

and the mass of F is m_{F} + Î¶_{F}_{ }<S>. Note that a symmetry like S → S implies F → iÎ³5F and hence forbids the bare mass term, unless the symmetry is explicitly broken. We will not pursue this possibility further as we show explicitly that a nonzero VEV to the singlet is disfavored, so the vector fermions must get their masses from the bare mass term. For this case, i.e., M^{2} < 0 and <S> = 0, there is no mass term from the Yukawa couplings. Direct searches at the LHC put a lower limit of the order of 500 GeV on the mass of vector quarks.
For simplicity, we assume a complete generation of vector fermions (N, E), (U, D) with all Yukawa couplings Î¶i to be the same (the heavy neutrino N is not to be confused with N, the number of singlets). One can, in principle, consider only one such fermion in the spectrum, or only the lepton or quark doublet. The VC for S now reads
 (18)

where
 (19)

N_{c} being the color of the corresponding fermions. While this does not guarantee a degenerate generation, that remains a distinct possibility, and thus one can avoid the strong constraints coming from oblique S and T parameters—because of the vectorial nature and degeneracy.
If there is only one singlet and M^{2} > 0, the minimization conditions are
 (20)

where <S> = v'. The mass term can be written as
 (21)

The condition for both masses to be real is
 (22)

For Î» ≈ 0.13 and a ≈ 4, this makes _{} very large (≥ 33) and clearly nonperturbative. While this by itself may still be acceptable, the fact that all the scalar couplings hit their respective Landau poles almost right at the electroweak scale rules this option out.
One might wonder whether the situation improves in the largeN limit. However, if the original scalar sector has an O(N) symmetry, a spontaneous symmetry breaking will result in N  1 Goldstone bosons, which couple to the doublet Higgs h, and therefore will give a very large invisible decay width of h. This is again unacceptable from the measurement of the branching fractions of the Higgs at the LHC.
Thus, we are forced to take M^{2} < 0, so that the singlet does not develop a VEV and there is no singletdoublet mixing. This is true even if there are more than one such singlets. The masssquared of the singlet is given by
 (23)

so for N = 1, the lowest possible mass for the singlet is about 500 GeV, and goes down as 1/_{}. This means no change in the decay pattern of the 125 GeV scalar from the SM Higgs, and no h → SS invisible decay unless N is so large that m_{S} < m_{h}_{ }/2 (this happens for N ≥ 66).
Unfortunately, the singlet scalar model ceases to be valid much before the Planck scale. This can be understood from the relevant oneloop Î²functions:
 (24)

where Î²_{h}_{ }≡ dh / dt, and t ≡ ln(Q^{2 }/^{ }Î¼^{2}). The scalar quartics are coupled and hence all of them hit the Landau pole almost simultaneously; this we may take as the range of validity of the model. For the details, we refer the reader to Ref. [10].
Fig. 1: The energy scales where (i) the scalar quartic couplings hit the Landau poles (assuming oneloop RG equations to be still valid) [upper curve] and (ii) at least one of the scalar couplings ceases to be perturbative (≤ 4𝜋) [lower curve].
TWOHIGGS DOUBLET MODELS
The twoHiggs doublet models (2HDM) [21] are one of the most widely investigated scenarios that go beyond the Standard Model. Any 2HDM consists of five physical scalars: two CPeven neutral h and H, one CPodd neutral A, and two charged bosons H^{Â±}. The CP quantum numbers are, of course, assigned with the assumption that the scalar potential is CP conserving and hence the mass eigenstates are also CP eigenstates. However, a generic 2HDM suffers from large flavourchanging neutral currents (FCNC); to prevent this, one invokes the GlashowWeinbergPaschos theorem [22, 23]. The theorem states that there will be no treelevel FCNC if all righthanded fermions of a given electric charge couple to only one of the doublets. This can be achieved in 2HDMs by introducing discrete symmetries for fermions or scalars.
Let us denote the two doublets by Î¦_{1} and Î¦_{2}, and invoke a Z_{2} symmetry Î¦_{1 }→ Î¦_{1}, Î¦_{2 }→ Î¦_{2}. There are four types of 2HDM, depending on the transformation of the fermions under this Z_{2}, for which there will be no treelevel FCNC. They are: (i) Type I, for which all fermions couple with Î¦_{2} and none with Î¦_{1}; (ii) Type II, for which uptype quarks couple to Î¦_{2} and downtype quarks and charged leptons couple to Î¦_{1} (this is the type that is embedded in the minimal supersymmetric SM (MSSM) and hence has received the most attention); (iii) Type Y (sometimes called Type III or Flipped), for which uptype quarks and charged leptons couple to Î¦_{2} and downtype quarks couple to Î¦_{1}, and (iv) Type X (sometimes called Type IV or Leptonspecific), for which all charged leptons couple to Î¦_{1}, and all quarks couple to Î¦_{2}.
We do not need any extra fermions for 2HDM; apart from TypeI, both the doublets couple to fermions. However, the extra constraint regarding the Veltman Conditions implies that tan Î² = v_{2 }/v_{1} is no longer a free parameter; it will be determined by the scalar quartic couplings. Consequences of applying the VC to 2HDMs were discussed in Ref. [24] and then later discussed in more detail in Ref. [25]. We will base our discussion on a paper by us [11].
2HDM in brief
We will follow the notations and conventions of Ref. [21]. There are two scalar doublets, Î¦_{1} and Î¦_{2}, with hypercharge +1. The lower components, which are electrically neutral, have nonzero VEV:
 (25)

with tan Î² = v_{2 }/v_{1} and _{}. The CPconserving scalar potential can be written as
 (26)

where m_{12}^{2} softly breaks the Z_{2} symmetry. The two CPeven neutral states ρ_{1} and ρ_{2}, which are components of Î¦_{1} and Î¦_{2} respectively, are not mass eigenstates. The corresponding mass matrix can be diagonalized through a rotation by an angle Î±, and the mass eigenstates are
 (27)

where h(H) is the lighter (heavier) eigenstate. Note that if Î±  Î² = 𝜋/2(0), h(H) will be the SM Higgs boson, with a VEV of _{}. For example, the hhVV^{*} (HHVV^{*}) coupling is just the SM coupling times sin^{2} (Î±  Î²) (cos^{2} (Î±  Î²)), where V is any weak gauge boson. The CPodd scalar A does not couple to gauge bosons.
Before we proceed any further, let us note that one should formulate the VCs for h and H. However, if we demand the quadratic divergences for both h and H to vanish, we might as well formulate them for ρ_{1} and ρ_{2}. This is what we will do in our subsequent discussion, and perform the entire analysis in terms of the couplings and not the masses. While the propagators are illdefined in the Î¦_{1 } Î¦_{2} basis, this does not affect our analysis as long as we focus on purely the divergent terms.
The most generic Yukawa interactions for these four models can be written as [21]
 (28)

where _{}j = iτ_{2} Î¦j^{*}, Q_{L}, L_{L}, d_{R}, u_{R} and l_{R} are generic doublet quarks, doublet leptons, singlet downtype and singlet uptype quarks, and singlet charged leptons respectively. Y_{j}^{d}, Y_{j}^{u}, Y_{j}^{e} are 3Ã—3 complex matrices, containing Yukawa couplings for the down, up, and leptonic sectors respectively. In our analysis we will consider only top, bottom, and τ Yukawa couplings to be nonzero.
Stability conditions
The requirement that the scalar potential always remains bounded from below leads to the following stability conditions:
 (29)

Thus, Î»_{3}, Î»_{4}, and Î»_{5} can potentially be negative. There can be chargebreaking or CPbreaking stable points of the potential; however, if the normal minimum is deeper, such stable points can at best be saddle points. The last condition shows that Î»_{5 }= 0 leads to the most stable configuration for a given set of the other quartic couplings.
Veltman conditions
If the Yukawa couplings are neglected, the VCs for ρ_{1} and ρ_{2} are the same for all 2HDMs. The selfenergy corrections are^{ 4}
 (30)

 (31)

Even if we neglect the gauge couplings, there are no solutions consistent with Eq. (29), except the trivial solution Î»_{i} = 0. Note that there is no term proportional to Î»_{5}; the quadratically divergent contributions cancel out.
With the introduction of the Yukawa couplings (only for t, b, and τ), the corrections turn out to be as follows.
• Type I:
 (32)

• Type II:
 (33)

• Leptonspecific:
 (34)

• Flipped:
 (35)

Thus, the complete oneloop quadratically divergent corrections are
 (36)

and the strict enforcement of the VCs require f_{ρ}_{1 }= 0, f_{ρ}_{2} = 0. In addition, if they are to hold at all energy scales, we also need df_{ρ}_{1 }/d(ln q^{2}) = 0, df_{ρ}_{2 }/d(ln q^{2}) = 0. We will not show the RG equations here; they can be found in [21] or in [11].
^{4 }We are not in the mass basis, so these are, strictly speaking, the corrections to the 2point Green's functions.
tan Î² as a function of quartics
We are not going into any detailed discussion here, but let us just focus on the major outcomes. Note that 2HDMs are unlike the singlet extension; the 125 GeV Higgs is potentially a mixture of ρ_{1} and ρ_{2}. However, this scalar behaves exactly like the SM Higgs, within the margin of error, so cos (Î±  Î²) ≈ 0. There are more constraints: the parameters should be so adjusted as to produce a light CPeven state at about 125 GeV, and the charged Higgs should not be lighter than 300 GeV for TypeII and flipped models. This bound is independent of the precise value of tan Î² as long as tan Î² > 1 and comes from the rate of the radiative decay b → sÎ³. There is no such bound on the charged Higgs in the leptonspecific model.
Let us note here that the Yukawas for f_{ρ}_{1} and f_{ρ}_{2} should be of the same order, given all other quartic and gauge couplings. This forces large values of tan Î². For TypeII and Flipped, the lowest possible value of tan Î² is about 31.5 amd 42.5 respectively and increases almost linearly with Î»_{1}. There is of course a band in the allowed values of tan Î², but that is very narrow, and depends on how strictly we impose the Veltman Conditions. In the strictest limit, the width goes to zero. The lowest value of tan Î² comes from the fact that stability prevents Î»_{1 }< 0. For leptonspecific models, the lowest value of tan Î² is about 140. Thus, the Yukawa couplings are so large that the potential quickly becomes unstable, at about a scale of 1 TeV (remember that the Yukawa couplings drive the scalar quartic couplings towards negative).
This, in some sense, is the most interesting result: if we demand a solution to the naturalness problem, we are forced to have large values of tan Î². Needless to say, this will affect the search strategies.
There is still a large parameter space consistent with the stability of potential as well as the direct and indirect constraints where both the Veltman Conditions are satisfied. Unlike the singlet case, the theory does not blow up before the Planck scale for most of this parameter space. Again, this is true only for TypeII and Flipped models.
What about the stability of f_{ρ}_{1} and f_{ρ}_{2}? Everything would have been perfect if the Veltman Conditions were absolutely stable with the scale variation. Unfortunately, this is not so. But this may be due to the simplistic approach of keeping only the oneloop terms; at higherorders, we might expect a scale independence.
Veltman Condition with dimension6 operators
We will use the SMEFT basis as in Ref. [14]. Keeping in mind that only operators with two or more Higgs fields are relevant and the divergence should be quartic, the relevant operators are as follows:
 (37)

where
 (38)

g, g' being the SU(2)_{L} and U(1)_{Y} gauge couplings respectively, and Î»^{a}, σ^{a} are the GellMann and Pauli matrices. Note that the mixed gauge operator O_{BW} = Î¦^{†}_{} Î¦ cannot generate a selfenergy amplitude, either at one or at twoloop.
With these set of nine operators, we can write the dimension6 part of Eq. (2) as
 (39)

and Eq. (5) takes the form
 (40)

where f_{i} and g_{i} terms come respectively from the oneloop and twoloop quartic divergences with the insertion of the operator O_{i}, any of the nine dimension6 operators listed above. The contributions are given by
 (41)

Eqs. (40) and (41) are the central results of this Section. The extra 1/16𝜋^{2} suppression tells us that we may neglect the g_{i} terms (and thus will be justified to neglect the dimension8 and other higherdimensional operators), unless we deal with pathological cases like Î£f_{i} ≈ 0, or all the WCs being zero except c_{∅,3}.
With only the f_{i} terms, the modified VC reads
 (42)

which immediately tells us that at least one, or perhaps more, WCs should be negative. As the operators do not contain strongly interacting fields (except O_{GG}), the running between Λ, which is the matching scale, and the electroweak scale, are controlled by electroweak radiative corrections only (at the leading order).
We will study the VC for two values of Λ, namely, 100 TeV and 10^{6} TeV. To start with, let us assume that only one of the eight SMEFT operators (neglecting O_{∅,3} which does not contribute to f_{i}) is present at the matching scale. We also need to evolve the SM couplings to that scale, for which we use the package SARAH v4.14.1 [26], with twoloop RG equations.
Taking Λ = 100 TeV, one gets, for exact cancellation of the quadratic divergence [15],
 (43)

and for Λ = 10^{6} TeV
 (44)

Of course, one may relax these numbers a bit if exact cancellation is not warranted. Note the large values for the weak gauge WCs; they stem from the definition of the corresponding f_{i}s in Eq. (41) which contain g^{2} or g'^{2}; the UVC need not be nonperturbative. On the other hand, if we take Λ = 2 TeV only, the corresponding exactcancellation values are
 (45)

This change is entirely due to the running of the SM couplings.
However, there is hardly any UVC theory that generates only one of these eight operators at the matching scale. As the sign of the WCs can be either positive or negative, the eight free parameters do not even give a closed hypersurface in the 8dimensional plot, and therefore marginalisation is of very limited use. We thus show in Fig. 2, two distinct cases where only a pair of WCs are nonzero at Λ. For the left panel of Fig. 2, we take c_{∅,2}, c_{∅,4} ≠ 0, while for the right panel, c_{WW} and c_{BB} are taken to be nonzero (an identical plot is obtained for c_{W} versus c_{B}).
The narrow lines, as shown in the plot, are obtained with the demand of an exact cancellation. They broaden out to bands if we allow a finite amount of finetuning, the bands getting narrower for higher values of Λ.
The SMEFT operators contribute to anomalous trilinear and quartic gaugegauge and gaugeHiggs couplings, as well as modified wavefunction renormalisation for the bosonic fields. It is indeed heartening to note that the parameter space that we obtain is consistent with all other theoretical and experimental constraints [27]. For other collider signatures of these d = 6 operators, like vector boson scattering and Higgs pair production at the LHC, we refer the reader to, e.g., Refs. [28] and [29].
Fig. 2: The parameter space for c_{∅,2} and c_{∅,4} (left), and c_{WW} and c_{BB} (right) that is needed for an exact cancellation of the quadratic divergence at the scale Λ. The red solid line is for Λ = 100 TeV, while the blue dashed line is for Λ = 10^{6} TeV. For the left plot, the two lines almost coincide.
CONCLUSION
In this article, we have discussed the Veltman Condition leading to the cancellation of the quadratic divergence of the Higgs selfenergy in the context of (i) extensions of the scalar sector with more d = 4 operators, and (ii) an SMEFT framework with d = 6 operators.
In the first category, we discuss extensions with one or more gauge singlet scalars, as well as the twoHiggs doublet models. The VC imposes strong constraints on the allowed parameter space. For example, it prefers large tan Î² regions for twoHiggs doublet models. In the second category, we find the set of d = 6 operators that one should consider, and the constraints imposed by the VC on the Wilson coefficients of those operators. It turns out that at least one of the WCs has to be negative, but they are all consistent with a highscale perturbative theory. Thus, this study should set a benchmark for the model builders.
Acknowledgements: This article is based on Refs. [10, 11, 15], and AK would like to thank and acknowledge his collaborators, Ambalika Biswas, Indrani Chakraborty, and Poulami Mondal. He also acknowledges the support from the Science and Engineering Research Board, Govt. of India, through the grants CRG/2019/000362, MTR/2019/000066, and DIA/2018/000003.
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Anirban Kundu is a Professor in the Physics Department of the University of Calcutta, India. He did his Ph.D from Saha Institute of Nuclear Physics, India, in 1996. He is a particle phenomenologist with research interests in B Physics, Higgs Physics, and Collider Studies. 
