Comments on the Hierarchy Problem in Effective Theories
aa r X i v : . [ h e p - ph ] A p r Comments on the Hierarchy Problem inEffective Theories
Archil Kobakhidze and Kristian L. McDonald ARC Centre of Excellence for Particle Physics at the TerascaleSchool of Physics, The University of Sydney, NSW 2006, Australia
Abstract
We discuss aspects of the hierarchy problem in effective theories with light scalarsand a large, physical ultraviolet (UV) cutoff. We make two main points: (1) The (naive) fine-tuning observed in an effective theory does not automatically imply thatthe UV completion is fine tuned. Instead, it gives a type of upper bound on the severityof the actual tuning in the UV completion; the actual tuning can be less severe thanthe naive tuning or even non-existent. (2) Within an effective theory, there appear tobe two types of parameter relations that can alleviate the sensitivity of the scalar massto the cutoff — relationships among dimensionless couplings or relationships amongdimensionful parameters. Supersymmetric models provide symmetry-motivated exam-ples of the former, while scale-invariant models give symmetry-motivated examples ofthe latter. [email protected] [email protected] Introduction
The hierarchy problem [1] has driven much research into physics beyond the Standard Model(SM) in recent decades. When taken as an effective theory with a large cutoff, it is difficultto understand how the Higgs boson remains light relative to the large cutoff effects. Withthe experimental discovery of the Higgs boson at the LHC, the desire to understand howscalars can remain light, despite large cutoff-dependent corrections, has gained urgency inrecent years [2]-[5].In this work we discuss aspects of the hierarchy problem in effective theories with lightscalars and a large physical ultraviolet (UV) cutoff. We discuss two main points. Firstly,we note that the (naive) fine-tuning found in an effective theory does not automaticallyimply that the UV completion is fine tuned. Rather, it gives a sort of upper bound on theseverity of the actual tuning present in the underlying UV completion; the actual tuningdue to the new physics at the cutoff scale can be less severe than the naive tuning or evennon-existent. Secondly, we note that within an effective theory there appear to be two typesof parameter relations that can alleviate the sensitivity of the scalar mass to the cutoff; arelationship among dimensionless couplings ´a la the Veltman condition, or a relationshipamong dimensionful parameters. Supersymmetric (SUSY) models give symmetry-motivatedexamples in which a technically-natural Veltman-like condition arises, while scale-invariantmodels are symmetry-motivated examples where relations between dimensionful parametersare expected.In essence, these two points pertain to the two different aspects of the hierarchy problem,namely the naturalness of new particle thresholds and the tuning associated with “purecutoff” effects. If the UV physics takes the form of new thresholds, or behaves similar tosuch, the naive tuning can overestimate the actual tuning. This has implications for model-building approaches to beyond-SM physics [3, 5]. However, if the cutoff accurately representsthe behaviour of the UV physics, one expects a natural theory to shield the infrared (IR)sector. These points are elaborated within.The layout of this paper is as follows. In Section 2 we discuss the naive tuning in aneffective theory and consider a simple example where the actual tuning in the UV completioncan be less severe. In Section 3 we consider simple relationships among parameters in aneffective theory that can shield the IR sector from cutoff effects. Conclusions are drawn inSection 4.
Consider the effective theory for a self-interacting scalar field S , with Lagrangian L Λ S = ∂ µ S ∗ ∂ µ S − ¯ m S (Λ) | S | − λ S | S | + . . . (1)The parameters depend on the cutoff scale Λ, and the dots denote non-renormalizable irrel-evant IR operators with mass-dimension d >
4, suppressed by factors of Λ − d . Calculating1he one-loop corrected mass in the effective theory gives m S = ¯ m S (Λ) + δm ≡ ¯ m S (Λ) + λ S { Λ + ¯ m S (Λ) log( ¯ m S (Λ) / Λ ) } . (2)For large values of Λ, the theory appears to have a hierarchy problem, with small values of m S ≪ Λ requiring a fine-tuning between ¯ m S (Λ) and Λ. The effective theory is said to befine-tuned because the origin of this cancellation is understood within the effective theory.It is important to distinguish two qualitatively different contexts in which Eq. (2) isinterpreted. In the first case, Λ is merely as a tool to regularize the divergent loop-integral.Then, both Λ and the Λ-dependent bare parameters are regarded as unphysical quantities.In fact, one must take the limit Λ → ∞ and remove all divergences by renormalizingthe unphysical bare masses and couplings, leaving finite physical parameters. From thesymmetry perspective, a theory with an unphysical cut-off possesses a scale invariance thatis softly broken by the explicit mass terms and the logarithmic quantum corrections [6]. Thesoftness of the breaking is reflected in the infrared fixed-point structure of the mass RGEs [7].The second interpretation of Eq. (2), of interest here, applies when the cut-off Λ is physicaland therefore associated with a new scale in the UV-completion of the effective theory. Inthis case the effective theory has a hierarchy problem [1], and the cut-off Λ, as well as theΛ-dependent bare parameters and the set of irrelevant operators, fully encode informationabout the UV theory. Absent knowledge of the underlying theory, one cannot renormalizeaway the Λ-dependence. We refer to the fine-tuning associated with this hierarchy problemas the “naive fine-tuning of the effective theory,” or more simply as “the naive tuning.” Fora given fixed value of λ S = O (1) (for example), this naive tuning requires two parameters of O (Λ ) to cancel out at a precision of O ( m S / Λ ). With m S ≪ Λ this tuning is severe.One can phrase the hierarchy problem of Eq. (2) in the following way. Consider valuesof the parameters in the effective theory that generate some fixed value for the scalar mass m S . Now shift the bare mass ¯ m S (Λ) as follows:¯ m S (Λ) → ¯ m S (Λ) + δ ¯ m S (Λ) , with δ ¯ m S (Λ)¯ m S (Λ) . O (1) . (3)This will, in general, induce a shift in the scalar mass: m S → m S + δm S . (4)The effective-theory has a hierarchy problem if δm S /m S ≫ O (1) — i.e. if small changes in thebare mass create a large change in the scalar mass. For generic couplings, an effective theorywith m S ≪ Λ is expected to have a hierarchy problem. One must fine-tune ¯ m S (Λ) ≃ O (Λ )against the Λ -term to enable m S ≪ Λ . Thus, shifts of δ ¯ m S (Λ) / ¯ m S (Λ) = O (1) give δ ¯ m S (Λ) = O (Λ ), which in turn gives m S → m S + O (Λ ) ≫ m S . (5)The hierarchy problem manifests through this extreme sensitivity of the physical scalar massto small changes in the effective-theory mass-parameter. We suppress numerical loop-factors in this section. L Λ S , one can investigate the originof the naive tuning by calculating the “actual tuning” in the UV completion. One wouldlike to know if the naive tuning accurately encodes the actual tuning. Our main point inthis section is that the actual tuning can be less severe than the naive tuning and in somecases may even be absent. One should think of the naive tuning as representing a worst casescenario for the severity of the tuning associated with the new physics at the scale Λ. Wedemonstrate this with a simple example.The scale Λ is assumed physical, in the sense that new physics appears at this scale.There are, in principle, two classes of UV completions that one could consider. In one class,the scalar S persists as a physical degree of freedom beyond the scale Λ, and the new physicstakes the form of additional degrees of freedom with mass M H ∼ Λ. A typical exampleoccurs when the theory is UV completed by a new heavy scalar, H . If the two scalars couplevia a quartic interaction, λ mix S H , the UV completion generates a mass correction for thelight scalar of the form δm S,H = λ mix M H log( M H /µ ) , (6)where µ denotes a renormalization scale in the UV completion. With regard to fine-tuningdue to H , one can differentiate three cases: • For λ mix ∼ O (1), there is a hierarchy problem and a light scalar with m S ≪ M H requires a fine-tuned UV completion. The required tuning is at the level of O ( m S / Λ ),occurring between quantities of O (Λ ). In this case the actual tuning agrees with thenaive tuning of the IR observer. • For m S /M H ≪ λ mix ≪
1, there is still a hierarchy problem and the UV completionremains fine-tuned. However, now the actual tuning is at the level of O ( m S / [ λ mix M H ]),between quantities of O ( λ mix M H ) ≪ Λ . The actual tuning due to the physics at Λ isless severe than the naive tuning. • For λ mix . m S /M H , there is no hierarchy problem. The loop-correction does not exceedthe physical scalar mass and the theory is technically natural. This case is contrary tothe naive expectation of the IR observer; the naively-tuned effective-theory possessesa technically-natural UV completion.Note that the second and third cases are only possible if values of λ mix ≪ λ mix → L Λ S , there is a transition to a new theory atthe scale Λ, such that the degrees of freedom are different and the scalar S ceases to exist. For our purpose in this section it suffices to take the cutoff for the UV theory merely as a regulator. Wefurther discuss cutoff effects in the next section. S is a low-energy composite object and the transition to more-fundamental “quarks” occurs at Λ, or similarly if Λ is a minimal length-scale in the UVtheory, or a type of non-locality scale such as in string theory. In these examples the actualtuning required in the UV completion is expected to be as severe as the naive tuning.Note that an IR observer cannot differentiate between these two classes of UV comple-tions, absent information about the UV physics. When faced with a naively-tuned effectivetheory an IR observer can, at best, conclude that the underlying UV completion may be finetuned, due to new physics at the scale Λ. The naive tuning provides a type of upper boundfor the severity of the actual tuning in the UV completion as a result of the new physics atthe scale Λ; the actual tuning in the completion can be less severe or in some cases evennon-existent.The physical origin of the fine-tuning in the two classes of UV completions is distinct.The Λ -term in (2) encodes the mass correction to the IR scalar S , due to the UV (Euclidean)momentum modes with | p E | ∼ Λ for the IR degree of freedom. In the first class of models,where heavy new physics with mass M H ∼ Λ appears, the origin of the actual tuning differsfrom the source of the naive tuning. The actual tuning, if present, is due to mass correctionsfrom the heavy UV physics, while the naive tuning results from UV momentum modes forthe IR fields. These two effects have distinct physical meanings and this is why the naivetuning and the actual tuning can differ. One should think of the naive tuning as being aproxy for the actual tuning — the existence of a naive tuning in the IR theory indicates thatthe UV completion may contain an actual tuning. Note that, from the perspective of theUV theory, there is nothing special about modes with | p E | ∼ Λ for the IR scalar; these justhappen to have the same momentum as the new physics scale.In the case where S does not persist in the UV, the momentum modes with | p E | ∼ Λ forthe IR scalar S are “special” in the sense that modes with | p E | > Λ simply do not exist. Nowthe Λ -term has a clear physical meaning and one understands why the modes | p E | ∼ Λ wouldgive a mass correction of greater physical significance than modes with, e.g., | p E | ∼ Λ ′ ≪ Λ.In this case there is no obvious reason why the bare mass and the cutoff should cancel-out soprecisely; the natural value for the scalar mass is expected to be m S ∼ Λ. Said differently,the naive tuning is expected to well-approximate the actual-tuning.
We now turn to a different aspect of the hierarchy problem. Having focused mainly on newthreshold effects in the preceding, we now focus on the Λ -term, assuming that it encodes areal physical effect that must be dealt with, for the theory to be natural. We consider thesimplest possibilities for alleviating the cutoff sensitivity within the effective theory, namelyby parameter relations that shield the light scalar mass from cutoff effects. To discuss thismatter it is helpful to consider a more detailed IR sector. In general, one would expect more IR composite states than a single scalar for this case. Analogous to the role played by the inter-atomic spacing when describing spin-correlation functions ofa magnetic system by an effective scalar field theory. S = ( s , s , ... ) T , gauge fields V , and Weyl fermions F , assumed valid up to a large UV cut-off Λ, which is understoodin the Wilsonian sense. If present, a hierarchy problem would manifest in the relevantoperators S a S b that appear in the quantum-corrected effective action. The coefficients ofthese operators are given by the non-derivative part of the 2-point functions Γ (2) ab . In the1-loop approximation they are: (cid:0) m S (Λ , µ ) (cid:1) ab = (cid:0) ¯ m S (Λ) (cid:1) ab + X A = S,V,F ( − J A (2 J A + 1) ( g A ) abcd π (cid:20) Λ δ cd − (cid:0) ¯ m A (Λ) (cid:1) cd ln Λ µ (cid:21) , (7)where ¯ m A (Λ) is the effective bare-mass for the field A of spin J A , and µ is an arbitraryrenormalization scale, | m A | < µ < Λ . Here g A denotes the matrix of dimensionless couplingsbetween the field A and the scalars S , defined through the interaction terms as14! ( g S ) abcd S a S b S c S d ,
12 ( g V ) abcd V a V b S c S d and ( y F ) abc ¯ F a F b S c , (8)where ( y F ) abk ( y F ) cdk = ( g F ) abcd . As before, for light values of m S ≪ Λ , the effective theoryhas a hierarchy problem.How can one remove this sensitivity to the cutoff? From the perspective of the effectivetheory, there appear to be two types of relationships that could remove the cutoff sensitivityin Eq. (7). One could consider a relationship among dimensionless couplings that cancelsthe Λ -term, or a relationship among dimensionful quantities that relates the bare mass tothe UV scale. For such a relationship to provide a viable explanation (i.e. not transfer thetuning to a different sector), it should be motivated by a symmetry. We discuss these twocases in turn. First consider the case where the Λ -dependence is removed by a relationship among dimen-sionless couplings. We discuss two examples, differentiated by the absence/presence of anunderlying symmetry. The Veltman Condition.
The Λ -term in Eq. (7) disappears if the dimensionless cou-plings in the theory are related: X A = S,V,F ( − J A (2 J A + 1) ( g A ) abcc = 0 . (9)This possibility was introduced by Veltman [9], and studied prior-to (after) the Higgsdiscovery in Ref. [10] ([11]). In the SM, this can be converted into a mass relation: m h + 2 m W + m Z − m t = 0 , (10)5iving m h ∼
300 GeV, in conflict with the data. However, this relation might be satisfiedin the UV, perhaps with additional beyond-SM fields also participating [11]. If the Veltmancondition is realized, the Λ -term in Eq. (7) cancels out and the bare Higgs mass is similar tothe observed mass, m h / ¯ m h (Λ) = O (1). Consequently, small changes to the bare mass givesmall changes to the Higgs mass, and the theory appears natural. However, the (generalized)Veltman condition is not motivated by any symmetry. From the perspective of the effectivetheory, one cannot understand why such a relationship exists, nor how it remains radiativelystable. If a Veltman-like relation holds, the hierarchy problem is revealed by looking atsmall shifts in individual dimensionless couplings. Under a small shift g A → g A + δg A for aparticular coupling g A , the cancellation of the Λ -term ceases to function and the need forfine-tuning is manifest. Thus, the hierarchy problem of the effective theory is not resolved;one has simply exchanged a tuning among mass-parameters for a tuning amongst couplings. Supersymmetry.
Models with exact SUSY possess an equal number of bosonic andfermionic degrees of freedom (i.e., P A = S,V,F ( − J A (2 J A + 1) = 0). Furthermore, SUSYforces relationships among dimensionless coupling constants, and requires multiplets to bemass-degenerate, namely X A = S,V,F ( − J A (2 J A + 1) ( g A ) abcc = 0 (11)and X A = S,V,F ( − J A (2 J A + 1) ( g A ) abcd ( m A ) cd = 0 . (12)These equations reflect the perturbative non-renormalization theorem [8], according to whichonly wavefunction renormalization is required in N=1 SUSY theories. Eq. (11) solves thehierarchy problem because it forces the coefficient of the Λ -term to vanish in Eq. (7). Notethe similarity with the Veltman condition; it is clear that SUSY models can be thought ofas symmetry-motivated examples where a Veltman-like condition is automatically achieved.In realistic applications, effective theories describing supersymmetric extensions of theSM cannot possess exact SUSY. However, Eq. (11) also holds in softly -broken SUSY theories,since the soft SUSY-breaking terms are dimensionful parameters that do not appear in (11).On the other hand, Eq. (12) is modified because the soft-breaking terms lift the fermion-boson mass degeneracy, and set the SUSY-breaking scale, X A = S,V,F ( − J A (2 J A + 1) ( g A ) abcd ( m A ) cd ∼ M SUSY . (13)Then, provided M SUSY ≪ Λ, light scalars with mass m S = O ( M SUSY ) are technicallynatural in softly-broken SUSY models. In terms of the bare scalar mass, this corresponds to¯ m S (Λ) . O ( M SUSY ), so a small shift induces the change δm S . O ( M SUSY ) in the physicalmass, manifesting naturalness.If SUSY is broken in the UV completion, then presumably the fermion-boson massdegeneracy is broken by large O (Λ) effects in some heavy (hidden) sector of the theory.6he light (visible) sector must couple to the heavy sector with sufficiently weak couplings toenable m S ∼ M SUSY ≪ Λ . This is what happens in most of the realistic particle physicsmodels, where SUSY is spontaneously broken at a high energy scale in a heavy hidden sector,and feebly communicated to the visible sector.In the case of hard SUSY-breaking, Eq. (11) is also violated and the SUSY-breaking scaleis set by X A = S,V,F ( − J A (2 J A + 1) ( g A ) abcc Λ ∼ M SUSY . (14)Light scalars can only be accommodated if the hard SUSY-breaking dimensionless couplingsare sufficiently small, i.e. the relation (11) is satisfied with sufficient accuracy. We stressthat hard
SUSY-breaking also allows technically-natural light scalars, as the limit when Eq.(11) is strictly satisfied corresponds to an increased symmetry in the theory. In fact, small, hard
SUSY-breaking terms frequently appear in SUSY models without causing problems.Recall that, although the Λ -term canceled out in models with a Veltman condition,the hierarchy problem manifested under small shifts to the dimensionless couplings, g A → g A + δg A . SUSY cures this problem by demanding that the shift g A → g A + δg A is accompaniedby complimentary shifts g ′ A → g ′ A + δg ′ A in any couplings related to g A through SUSY. Onemust enforce a relationship among dimensionless couplings to ensure that Eq. (11) is satisfied.To vary a single coupling without varying the SUSY-related couplings would amount to adeparture from the physical content of the theory. This symmetry-motivated origin for thecoupling-relation ensures its stability under radiative corrections and tells us the couplingsare likely born in some related way.To summarize, in the presence of two (or more) sectors with hierarchically differentmasses, SUSY can ensure stability of the hierarchy if it is broken softly with all the soft-breaking mass parameters being of order M SUSY ≪ Λ, as in softly-broken SUSY GUTs. Inother cases, the heavy hidden sector should couple to the light visible sector very weakly.The latter case may or may not be technically natural, depending on details of the model [5].
SUSY models and the Veltman-condition both cancel out the quadratic divergences via arelationship among dimensionless couplings. The other possibility is that the sensitivity tothe cutoff in Eq. (7) is alleviated by a relationship among dimensionful parameters in theeffective theory. Indeed, the hierarchy problem of the SM manifests as a tuning betweenthe bare scalar-mass and the UV scale — this sensitivity between the only two dimensionfulparameters of the effective theory may suggest a deeper connection.
Relationship without symmetry.
In analogy with the Veltman condition, one canimagine a UV completion that triggers a relationship between the dimensionful parametersin the effective theory. However, for arbitrary relations among dimensionful parameters one7annot understand the origin of the relation within the effective theory, and the relation isequivalent to a tuning. For example, if the UV completion triggers the relation( ¯ m S (Λ)) ab + X A = S,V,F ( − J A (2 J A + 1) 116 π ( g A ) abcc Λ = M ab , (15)for some fixed M ab ≪ Λ , this would “remove” the quadratic divergence. However, theIR observer could not distinguish this from a fine-tuning — this relation is the standardexpression for the fine-tuning! As with the Veltman condition, one cannot understandhow such a relationship remains stable under radiative corrections in the effective theoryframework, so the tuning persists. This is evidenced by the fact that a small shift in thebare-mass induces a large shift in the physical mass. Scale invariance.
Within an effective theory, scale-invariance appears to be badly broken.In addition to the bare masses ¯ m A (Λ), and the logarithmic quantum-anomalous terms, whichbreak scale-invariance softly, one encounters hard -breaking relevant operators ∼ Λ , whichintroduce the quadratic sensitivity of the light masses to the cut-off scale. Nevertheless, theeffective theory can describe an underlying UV theory that maintains scale invariance, atleast at the classical level. The renormalized mass terms computed (within the perturbativeframework) in a scale-invariant theory are necessarily zero because all bare mass terms areabsent. On the other hand, the mass terms computed in a scale-invariant UV theory shouldmatch the corresponding mass terms m S (Λ , µ ), computed in the low-energy effective theory,at the matching scale defined by the effective theory cut-off, µ = Λ [12]. The IR effectivetheory for such UV theories automatically satisfies the relation,( ¯ m S (Λ)) ab + X A = S,V,F ( − J A (2 J A + 1) 116 π ( g A ) abcc Λ = 0 , (16)ensuring the effective theory accurately matches the UV theory at the matching scale Λ.Thus, only a logarithmic sensitivity to the UV scale remains:( m S (Λ , µ )) ab = X A = S,V,F ( − J A (2 J A + 1) 116 π ( g A ) abcd ( m A ) cd ln Λ µ . (17)Eq. (16) is a concrete example of a symmetry-motivated relationship among dimensionfulparameters that removes the quadratic cutoff dependence of the scalar mass. In the previoussection, the hierarchy problem in the effective theory was revealed by varying the bare masswhile keeping the UV scale and the dimensionless couplings fixed; small changes in the baremass induced large changes in the physical mass. However, in a scale-invariant theory, a shiftmade to the bare mass while keeping the other parameters fixed amounts to a departure fromthe physical content of the UV theory. That is, if the UV and IR scales are born in somecommon way, due to an underlying scale invariance, the shift¯ m S (Λ) → ¯ m S (Λ) + δ ¯ m S (Λ) , with δ ¯ m S (Λ)¯ m S (Λ) . O (1) , (18) A number of works studied scale-invariance in relation to the hierarchy problem [14]-[20], and discussionof quadratic divergences appears in Ref. [13]. → Λ + δ Λ , with δ Λ Λ . O (1) , (19)to ensure an accurate low-energy effective-theory description. If the couplings are held fixed,a small change in the bare mass of the form (18) must come partnered with a small changein the UV scale of the form (19) to ensure that Eq. (16) holds. More generally, the shift inthe bare mass should be partnered with a compensating shift in the couplings and/or theUV-scale to ensure they remain related by Eq. (16). Only then does the effective theoryaccurately encode the relationship between the two scales that is inherent in the parenttheory.Note that the soft-breaking masses ¯ m A in scale invariant theories emerge from the sponta-neous breaking of scale invariance, the mechanism known as dimensional transmutation [21].Thus, for a single source of symmetry breaking, all such masses are proportional to the scaleof this breaking, M SI . As with SUSY models, we again encounter two distinct possibilities:(i) M SI ≪ Λ and all the dimensionless couplings, and hence masses, are of the same orderof magnitude [15], or (ii) M SI ∼ Λ and a hierarchy of masses exists due to a hierarchyamong dimensionless coupling-constants; that is, hierarchically separated sectors of thetheory interact sufficiently weakly to preserve the hierarchical scales [5, 16]. Finally, we note the similarity between the SUSY and scale-invariant narratives. If theSM is an effective theory that is UV completed by the MSSM, new symmetry-motivateddegrees of freedom appear at the cutoff scale to enable a Veltman-like condition amongdimensionless couplings. This protects the weak scale from quadratic divergences. In thescale-invariant case, a new scale appears in the UV which enables a symmetry-motivatedrelationship among dimensionful parameters, ensuring the IR scale of the effective theoryis related to the UV scale. Quantization of course breaks scale-invariance softly via ln Λoperators, but this only affects marginal, d = 4 operators in the low-energy effective theory. In this work we discussed aspects of the hierarchy problem in effective theories with alight scalar. We sought to make two main points, namely: (1) The naive tuning in aneffective theory can be less severe than the actual tuning in the UV completion; the naivetuning gives a type of upper bound for the severity of the actual tuning associated with thenew physics at the cutoff scale. (2) There appear to be two classes of parameter relationsthat can alleviate the quadratic cutoff-dependence in an effective theory; relations amongdimensionless couplings (of the Veltman type), which ensure that the quadratic divergencescancel out, or relations among dimensionful parameters, which indicate the IR and UV Recent works have focused on threshold effects, assuming the quadratic divergences are dealt with by anas-yet unknown mechanism [3, 5] (the so-called “miraculous cancellation” [4]). Scale-invariance provides asymmetry-based rationale for the neglect of quadratic divergences, offering a motivation for this perspective.
Acknowledgements
The authors thank R. Foot and R. Volkas. This work was supported in part by the AustralianResearch Council.
A Comments on the Standard Model
We can relate the observations of Section 2 to the hierarchy problem in the SM (treatedas an effective-theory, valid up to a large UV-scale Λ). The three cases correspond to thefollowing classes of UV completions for the SM: • λ mix ∼ O (1) = ⇒ The UV completion contains new physics at the scale Λ, with O (1)couplings to the SM. This case has a hierarchy problem and requires fine-tuning. Theactual tuning is as severe as the naive tuning in the effective theory. The prototypicalexample is the UV completion of the SM by a renormalizable Grand Unified Theory,with unification scale M GUT ∼ Λ ≫ GeV. • m h / Λ ≪ λ mix ≪ ⇒ The UV completion contains heavy physics that is weaklycoupled to the SM, yet gives a radiative correction to the Higgs mass that exceedsthe observed value. This case has a hierarchy problem and requires an unnaturalfine-tuning. However, the actual tuning of the parent theory is less severe than thenaive tuning, being between parameters of O ( λ mix Λ ) ≪ Λ , rather than parametersof O (Λ ). The prototypical example is the UV completion of the SM by heavy right-handed neutrinos, to generate neutrino masses via the seesaw mechanism, with heavyMajorana masses in the range ≫ ( M R / GeV) ≫ . Radiative corrections to theHiggs mass then exceed the observed value, necessitating a tuning between parametersof O ( y ν M R ) ≪ Λ , where y ν ⇔ λ mix is the Dirac Yukawa-coupling. • λ mix . m h / Λ = ⇒ The UV completion contains weakly-coupled UV physics that doesnot generate a hierarchy problem. This case is technically-natural, provided the limit λ mix → M R / GeV) . [5, 22].Radiative corrections to the Higgs mass are less than the observed value, and the The upper bound results from the standard seesaw expression, assuming O (1) Dirac Yukawa-couplingsand that the SM neutrinos have masses m ν ∼ . References [1] K. G. Wilson, Phys. Rev. D , 1818 (1971); L. Susskind, Phys. Rev. D , 2619 (1979);S. Weinberg, Physica A , 327 (1979).[2] S. Dubovsky, V. Gorbenko and M. Mirbabayi, JHEP , 045 (2013) [arXiv:1305.6939[hep-th]]; R. Barbieri, Phys. Scripta T , 014006 (2013) [arXiv:1309.3473 [hep-ph]];A. de Gouvea, D. Hernandez and T. M. P. Tait, arXiv:1402.2658 [hep-ph].[3] M. Farina, D. Pappadopulo and A. Strumia, JHEP , 022 (2013) [arXiv:1303.7244[hep-ph]].[4] G. F. Giudice, arXiv:1307.7879 [hep-ph].[5] R. Foot, A. Kobakhidze, K. L. McDonald and R. R. Volkas, arXiv:1310.0223 [hep-ph].[6] W. A. Bardeen, FERMILAB-CONF-95-391-T.[7] C. Wetterich, Phys. Lett. B , 215 (1984); DESY-87-154.[8] M. T. Grisaru, W. Siegel and M. Rocek, Nucl. Phys. B
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