Naturalness redux
Abstract
The idea of naturalness, as originally conceived, refers only to the finite renormalization of the Higgs boson mass induced by the introduction of heavier states. In this respect, naturalness is still a useful heuristic principle in model building beyond the standard model whenever new massive states are coupled to the Higgs field. The most compelling case is provided by the generation of neutrino masses. In this paper we confront this problem from a new perspective. The righthanded sector responsible for the seesaw mechanism—which introduces a large energy threshold above the electroweak scale—is made supersymmetric to comply with naturalness while the standard model is left unchanged and nonsupersymmetric. Cancellations necessary to the naturalness requirement break down only at two loops, thus offering the possibility to increase the righthanded neutrino mass scale up to one order of magnitude above the usual values allowed by naturalness. If also the weak boson sector of the standard model is made supersymmetric, cancellations break down at three loops and the scale of new physics can be further raised. In the typeI seesaw, this implementation provides righthanded neutrino masses that are natural and at the same time large enough to give rise to baryogenesis (via leptogenesis). The model contains a dark matter candidate and distinctive new physics in the leptonic sector.
I Naturalness reconsidered
The idea of naturalness nat () is not so much about our distaste for finetuning dimensional quantities as about the decoupling of physics at different scales. What is felt to be unnatural is the cancellation among counterterms originating at different mass scales because it jeopardises the possibility of defining a lowenergy theory independently of the details of the higher scale physics. In the language of effective field theory, the naturalness requirement implies that no large threshold corrections be present. A problem of naturalness only arises if

There are (at least) two scales. The standard model (SM) taken by itself is natural. It is only when new physics is added at a higher scale that the problem arises. It consists in an unnaturally finetuned cancellation between the bare mass and the quantum corrections in the definition
(1) The correction is proportional to the mass of the new physics states and the coupling of these to the Higgs boson;

We can compute the effect of the new physics threshold. What cannot be calculated cannot give rise to a naturalness problem because it goes in the unknown (and divergent) part that is present in the quantum corrections. A case in point is gravitational physics: as long as we do not know the theory of quantum gravity, we cannot compute its contribution to the Higgs boson mass and therefore it belongs to the noncomputable part of the integral—the part that in dimensional regularization goes into the residue of the pole and about which we do not worry. The same holds for the strongly interacting regime at the Landau pole of the gauge interaction.
Even though naturalness as defined here is the mere restating of wellunderstood principles natgut (), the gist of it is often missed in many approaches (and even in some textbooks) because the renormalization of the Higgs boson mass is expressed in terms of a momentumdependent regularization where integrating over the degrees of freedom of the lowenergy theory (the usual loop integral of the quantum corrections) is entangled with the integrating out of the highscale modes producing the thresholds (in the running of the effective lowenergy parameters) Rothstein:2003mp (). The use of a cutoff regulator—when discussing naturalness, as a proxy for the masses at the higher physical scale—ultimately leads to incorrect results. Arguments based on generic quadratically (and cutoff dependent) divergent terms are misleading and suggest the mistaken conclusion that radiative corrections from loops of SM particles give rise to a naturalness problem. These corrections are only of the order of the SM particle masses and therefore included in the physical value of the Higgs boson mass without any fine tuning. In particular, there is nothing special in the large coupling of the top quark to the Higgs boson and loops of top quarks are not unnaturally large and need not be compensated by new states.
The requirement of naturalness has provided in the past and still provides today, once properly understood, a rather valuable heuristic principle in model building natural (). It tells us that whatever new physics we wish to include above the SM, to be natural, it must enter in a way that does no introduce large corrections to the Higgs boson mass that would make the electroweak (EW) vacuum unstable. On general grounds, whenever a threshold with particles of mass , coupled to the Higgs field with strength g is present, quantum corrections generate a contribution
(2) 
to the Higgs boson mass. It is possible to control the correction in eq. (2) in various way. In supersymmetry (SUSY), it vanishes because of the presence of bosonic and fermionic states with the same mass SUSYcanc (). In this paper we confront the problem with the same supersymmetric cancellation mechanism in mind but introducing a crucial difference with respect to the usual approach: it is not the SM that needs to be supersymmetrized but only the new physics sector. On the contrary, the SM breaks SUSY of the new physics explicitly. The scale of SUSY breaking is the EW scale.
If only the new physics sector is made supersymmetric, cancellations hold at oneloop level. They can be preserved to the next loop order by extending SUSY. In such a class of telescoping models, the large corrections are pushed to higher loop levels by extending SUSY to more and more sectors of the SM, and then to all loops by reintroducing the fully supersymmetric SM. In this very conservative and pragmatic approach, SUSY is not seen as a fundamental symmetry—its status resembling that of custodial symmetry in the SM. The extension of it—and the amount of superpartners—is determined by how severe the hierarchy problem is and how large a mass we want for the new physics states.
We implement this program in the case of the seesaw mechanism for neutrino masses seesaw () as a wellmotivated instance of new physics with an higher mass scale above the EW scale.^{1}^{1}1In Heikinheimo:2013xua () a similar approach was proposed to solve the dark matter (DM) problem. The problem in this example is the order of magnitude between the highest mass of the sterile neutrinos that is natural and the lowest possible value for this mass for a viable baryogenesis to take place by means of leptogenesis.
Ii The model
In this section we illustrate the idea outlined in the introduction with a specific example, namely the generation of neutrino masses via the typeI seesaw mechanism. First, in section II.1, we introduce a onegeneration toy model which serves as template for the more realistic implementation introduced in section II.2. Finally, in section II.3, we discuss the twoloops SUSY breaking effects generating corrections on the Higgs boson mass proportional to the large scale.
ii.1 From a toy model with one generation…
Let us consider first the (unrealistic) case of a single sterile neutrino with mass much larger than the SM scale—which we take to be GeV, the EW symmetry breaking scale. To protect the EW scale against quantum corrections, this state is introduced in a supersymmetric manner by means of the following chiral supermultiplet^{2}^{2}2Throughout this paper we adopt the notation of ref. Martin:1997ns ().
(3) 
which is a singlet under the SM gauge groups and is made of the fields encoding the new physics: we identify the bosonic component with an inert (complex) scalar singlet, the fermionic component with the Weyl component of a Majorana sterile neutrino.^{3}^{3}3Since we are working with lefthanded chiral superfields, should be interpreted as the charge conjugate of a righthanded sterile neutrino. Furthermore, we assign to a charge under a symmetry to prevent the presence of linear or trilinear terms in in the superpotential.
The supersymmetric sector is described by the usual WessZumino Lagrangian Wess:1973kz () density
(4) 
where the superpotential is given by
(5) 
The mass is the scale of new physics.
The equation of motion for the auxiliary field is
(6) 
At this stage the SM is described by the usual nonsupersymmetric Lagrangian. In order to couple it to the new physics sector we identify a supersymmetric chiral superfield inside the SM states, namely
(7) 
where is a doublet made of the SM Higgs boson and the lepton doublet field (the lefthanded electron and the corresponding neutrino). There is no contribution from to the superpotential because of gauge invariance. The multiplet components have their SM masses, the splitting of which is of the order of the EW scale and sets the SUSY breaking scale provided by the SM Lagrangian.
To protect the EW scale, the coupling must be such as that no corrections are introduced. It is not possible to couple the new physics chiral superfield and the SM chiral superfield in a supersymmetric manner because of the hypercharge assignments. There are two solutions to this problem and correspondingly two possible models. Either one works with just and introduces the interaction in the Kähler potential, or one introduces a new scalar multiplet with the opposite hypercharge—as it is done in the Minimal Supersymmetric Standard Model (MSSM). Both models produce the same Yukawa coupling between the sterile and the SM neutrino. They can be distinguished because they come with their own characteristic set of new states. To remain with the simplest case (that is, without the necessity of introducing higherdimensional operators in the Kähler potential), we explore in this paper the second possibility. Therefore, the chiral superfields are now three:
(8)  
(9)  
(10) 
where the index refers to the gauge group. We assign the following , , and charges:
(11)  
(12)  
(13) 
In eq. (9) represents a doublet of lefthanded Weyl fermions, while in eqs. (8)–(9) we adopt, as customary in models with two Higgs doublets (see section III.3), the following parametrization (before EW symmetry breaking) for the four neutral and two charged degrees of freedom:
(14)  
(15) 
where , are two mixing angles and where we identify the neutral component with the physical Higgs of the SM.
The interactions among the three chiral superfields , and in this model can be written in a supersymmetric manner, and the superpotential is given by
(16) 
where is a generic coupling, while indicates the invariant product (for simplicity, we take and reals). All other terms are forbidden by gauge or symmetry because of our assignment in eqs. (11)–(13). There is a gauge anomaly that can be compensated by introducing an extra state with the appropriate hypercharge and mass —so as not to introduce new highscale masses. The presence of this additional Higgsinolike state has important phenomenological consequences that will be discussed in the next section.
By adding canonical kinetic terms for , the model is described by the following Lagrangian
(17)  
The equations of motion of the auxiliary fields are now
(18) 
and replacing them in eq. (17) gives rise to the following kinetic and interaction Lagrangians
The Yukawa coupling —together with the Majorana mass term for —generates the usual typeI seesaw mechanism. In eq. (II.1) the quartic term with both scalar fields and the mixed term proportional to guarantee, at the oneloop level, the typical cancellation characteristic of the unbroken WessZumino model: the sum of the corrections to the Higgs boson mass vanishes as long as the components of the new physics multiplet have the same mass.
For example the correction to the Higgs boson mass coming from the loop with the SM lepton and the sterile neutrino is cancelled by the sum of the diagram with the scalar singlet in the loop and the one with the scalar singlet and the fermionic neutral component of the chiral superfield . We summarize in fig. 1, from a diagrammatic point of view, the cancellation mechanism at the oneloop level. Notice that at this stage we are working in the unbroken EW phase since soft terms (like the masses of SM particles) only introduce natural corrections proportional to the EW scale.
Let us close this section with one final technical remark that will be useful in section III. Starting from the Lagriangian in eqs. (II.1)–(II.1), it is straightforward to integrate out the heavy fields and . We find
where the first term (last three terms) in the square bracket is (are) generated by integrating out (). The dim operator is the familiar (in spinor notation) Weinberg operator for neutrino masses. From eq. (II.1) is clear that all the effects due to the heavy chiral superfield decouple for large values of . Finally notice that, as customary in SUSY, all the effective operators in eq. (II.1) are controlled by the same coupling constant, namely the Yukawa coupling of the righthanded neutrino.
ii.2 …to a more realistic implementation
A realistic implementation requires more than one generation of neutrinos. Every supersymmetrized leptonic multiplet brings into the model an extra gauge doublet of scalar leptons which plays the role of the Higgs boson doublet in the first generation multiplet. The chiral superfields in eq. (8) and eq. (10) are now replaced by
(22) 
where plays the role of the Higgs boson doublet in section II.1, while are scalar leptons doublets. The multiplet is left unchanged. There are three kinds of sterile neutrinos and, correspondingly, three kinds of additional heavy scalar singlets.^{4}^{4}4A simpler setup would be with just two kinds of sterile neutrinos Frampton:2002qc ().
The superpotential becomes
(23) 
and the relevant interaction Lagrangian is therefore
(24) 
The first term in eq. (24) gives the seesaw Yukawa coupling between sterile neutrinos and leptons, while the other terms give new physics beyond the SM, namely the Yukawa terms for the two scalar leptons interacting with sterile neutrinos and Higgsinos. These interactions always involve one heavy state—either or —and are not relevant for lowenergy phenomenology. We discuss some aspects of this phenomenology in section III.
ii.3 SUSY breaking at two and three loops
The SM is not supersymmetric. It couples to the supersymmetric sector via the EW gauge interactions which therefore break explicitly SUSY at the twoloop level. This can be understood by looking at diagrams in which a loop of gauge particles is added to the basic oneloop diagrams of the previous section. Representative diagrams of this kind are shown in the upper panel of fig. 2.
Since the quantity we are interested in—the twopoint function of the Higgs boson with zero external momentum in the massless EW theory—is gauge invariant, the computation can be performed in the Landau gauge, where the only relevant diagrams are those in the upper panel of fig. 2. Neglecting all masses at the EW scale we find, considering for simplicity the model with one generation
(25)  
where is the gauge coupling and the Weinberg angle. In eq. (25) the first (second) integral comes from the first (second) diagram in the upper panel of fig. 2, where the sum over all the particles exchanged in the loop is understood.
The twoloop integrals can be computed using dimensional regularization Jurcisinova:2010zz (); we find
in dimensions. The integrals over the Feynman parameters can be evaluated by means of the ChengWu theorem Smirnov:2012gma () and, in the limit, we find the following exact result
(27) 
where is the EulerMascheroni constant and
(28) 
Disregarding the UV divergent part, and taking the renormalization scale , we find
(29) 
where we approximate the numerical combination in parenthesis with a factor. The generalization of this formula to the case of three generation would just introduce a more complicated factor with the general structure (assuming, without loss of generality, a diagonal ). Since we are interested more in an orderofmagnitude estimate rather than in a precise numerical analysis, we will employ eq. (29) for the purposes of our discussion.
Before illustrating the implication of eq. (29), let us notice that the cancellation observed at one loop (as discussed in section II.1 and fig. 1) can be further extended at two loops by supersymmetrizing the gauge sector of the SM. In this case corrections to proportional to arise only at three loops via exchange of SM quarks and leptons, as shown in the bottom panel of fig. 2.
Among these threeloop dcontributions, the doublebubble diagrams in fig. 3 deserve special attention. These diagrams are proportional to the fourth power of the Yukawa top coupling , and—if proportional to —they would generate a correction to comparable in magnitude with the twoloop result in eq. (29). We find
(30)  
The first two integrals in eq. (30)—which correspond, respectively, to the scalar part of the two loops over and in the internal bubbles in fig. 3—generate, if computed individually, a correction proportional to . However their sum, protected by SUSY, vanishes. The third integral in eq. (30)—which corresponds to the tensor part of the loop over in the left diagram in fig. 3—gives only a correction proportional to the external momentum (that is, a contribution to the wavefunction renormalization). Therefore, the class of doublebubble integrals in fig. 3 cannot generate large corrections if SUSY protects the Higgs twopoint function in the internal bubble.
ii.4 On the largest mass scale allowed by naturalness
As first discussed in Vissani:1997ys (), the typeI seesaw mechanism is natural for sterile neutrino masses up to GeV.^{5}^{5}5The case of lowscale (resonant) leptogenesis Pilaftsis:2003gt () is natural because it can be realized with heavy Majorana neutrinos as light as the EW scale (Pilaftsis:2005rv (); for a complete analysis including all flavor effects, see e.g. Dev:2014laa ()). Up to this value, the Higgs boson mass can have its physical value without requiring any finetuned cancellation. In fact, the oneloop correction to the Higgs boson mass is given
(31) 
where stands for the Yukawa couplings of the sterile neutrinos and leptons. These couplings are fixed by the neutrino mass value and given by
(32) 
By taking eV, and stretching the naturalness condition to its upper bound by taking , we find
(33) 
Notice that GeV corresponds to a Yukawa coupling . The result of Vissani:1997ys () can be reproduced by taking, more conservatively, .^{6}^{6}6 Notice how, by following the cutoff prescription discussed in the introduction, in this case we would have been misled to consider the loop of the top quark and write and erroneously find 1 TeV as the largest value for having a natural seesaw.
On the other hand, in order for baryogengesis to proceed via leptogenesis the mass of the sterile neutrinos must be of the order of GeV—the specific number depending on the assumptions on the initial abundance and the hierarchy in the righthanded neutrino states Davidson:2002qv (). This tension has been confirmed recently in Clarke:2015gwa () for the case with three generations of neutrinos.
The model introduced in the previous section helps in solving this problem. The coupling in eq. (II.1) can be identified with the Yukawa coupling previously introduced. The cancellation taking place at the oneloop level pushes the first contribution to the Higgs boson mass at the twoloop level. This suppression can be used to rise the scale of the heavy states. As we have seen in the previous section, the twoloop corrections gives a shift in the Higgs boson mass similar to that above but rescaled, namely
(34) 
which is natural as long as it is of the order of the EW scale and at most 1 TeV. We now find that the sterile neutrino mass can be as large as
(35) 
and therefore close to the lower bound on baryogenensis—the actual number depending on the details of the implementation.
If the supersymmetric sector of the SM is extended, by telescoping this model, to include also the EW gauge bosons, this value can be increased by one order of magnitude to GeV and thus comfortable within the viable range for leptogenesis.^{7}^{7}7In order to produce thermally the heavy neutrino states a reheating temperature of the Universe after inflation of is required; excessively high values of typically lead to an overproduction of gravitinos in the early Universe, in tension with Big Bang Nucleosynthesis (BBN) constraints Kawasaki:1994af (). However, reheating temperatures up to GeV—compatible with a mass scale — are not excluded because the gravitino density is still too small to cause disruption of BBN Davidson:2002qv (). Since our SUSY scenario does not descend from a supergravity model, we do not expect in any case to encounter the gravitino problem. Larger scales can be achieved by extending SUSY to the SM fermions and then to the colored gauge states. In the latter case, we recover the full MSSM and cancellations to all loop orders. From this perspective, our approach completely overturns the usual conclusion (based on the cutoff regularization) according to which, in order to cancel the contribution of the top loop, one is forced to introduce, first and foremost, colored top partners in order to naturally justify a new physics scale above the TeV.
Iii New physics
In this section we discuss some lowenergy implication of the setup outlined in section II. In particular, in section III.1 we highlight the new interactions characterizing the leptonic sector of the SM Lagrangian, and in section III.2 we discuss the role of extra scalar states; in section III.3, we briefly mention the properties related to the existence of two Higgs doublets and conclude, in section III.4, reviewing possible DM candidates.
The aim of this discussion is not to be exhaustive, since a careful phenomenological analysis is beyond the scope of this paper.
iii.1 Leptonic sector
At low energy the SM Lagrangian is enlarged by new degrees of freedom originating from the supersymmetric sector, .
Focusing on the leptonic sector, in addition to the usual SM Yukawa coupling, the model features the Yukawa interaction involving the righthanded neutrinos in eq. (24) together with their mass and kinetic term
(36) 
The role of the SM Higgs doublet is played, as already mentioned, by . Furthermore, without loss of generality, we work in the basis in which is diagonal. At this stage, lepton number is an accidental symmetry of the SM Lagrangian with the usual assignments. As a consequence in eq. (36) lepton number is violated by the Majorana mass term for two units as customary in the typeI seesaw. Notice that in our model we are not trying to relate the lepton number to the charge of the supersymmetric sector. On the contrary, lepton number is badly broken by the interactions in eq. (24). The important point to keep in mind is that all the interactions in eq. (24) always involve one component of the heavy chiral superfield and, apart form neutrino masses, they decouple from lowenergy phenomenology. Since we are dealing with sterile neutrino masses as large as GeV, in this realization all the effects related to the mixing between light and heavy neutrinos are severely suppressed.
iii.2 Extra scalars
With reference to eq. (22), the model—considering the realization discussed in this paper—provides for extra scalar doublets with hypercharge , namely . The Lagrangian , therefore, will contain their kinetic and bare mass terms
(37) 
where, in a diagonal basis, . This mass term is protected by the same mechanism protecting that of the Higgs boson. Additional couplings and mixings, allowed by Lorentz and gauge symmetries, might be present; for the sake of simplicity, we limit our analysis to a minimal set of operator. Since have gauge interactions, they can be i) constrained using LEP data via direct searches and EW precision observables and ii) produced directly at the LHC via EW DrellYan processes mediated by channel or exchange. In both cases it is crucial to discuss the mass splitting between the neutral and the charged component of the doublet.

LEP searches for both neutral and charged scalar particles place a lower limit on the value of their mass; as a rough estimate, we can use a representative lower limit from direct slepton searches LEPIISUSY (). As far as the EW precision observables are concerned, the presence of the scalar doublet (we omit the index hereafter) affects the oblique parameters , and Peskin:1990zt (). These parameters enter in the vacuum polarization amplitudes of the EW gauge bosons, and are severely constrained by LEPI and LEPII results Barbieri:2003pr (). For the scalar doublet we find the contributions Blum:2011fa ()
(38) (39) (40) where is the finestructure constant. Of particular concern are the and corrections since they measure the amount of weak isospin breaking, here represented by the mass splitting .
In fig. 4 we show the 1 and 3 exclusion limits on the mass spectrum of the scalar doublet obtained from LEPI and LEPII data; mass splittings are in tension with EW precision data. Considering the Lagrangian in eq. (37), the two components of the scalar doublet are degenerate in mass at the tree level. However, EW loop corrections induce a mass splitting such that the charged component turns out to be slightly heavier than the neutral one. Explicitly, we have Cirelli:2005uq () where, omitting the UV divergent part, with . This would correspond to a mass splitting MeV for GeV. Thanks to this mass splitting, the charged component of is not stable; the dominant decay channels are . The first one turns out to be dominant Cirelli:2005uq (); FileviezPerez:2008bj (), with a typical decay length if is of the order of a few hundreds MeV’s.

EW DrellYan processes have three potential signatures: monoX (where X can be a jet, a photon or an EW gauge boson), disappearing tracks and emission with subsequent lepton decay.
As far as monojet analyses are concerned, monojet searches have been carried out both at the Tevatron Abazov:2003gp () and the LHC ATLAS:2012ky (); Chatrchyan:2012me () in the context of large extra dimensions and DM via effective contact operators. Similar studies have been proposed in the context of simplified supersymmetric model Low:2014cba () and minimal DM Cirelli:2014dsa (). Even if these analyses cannot be directly applied to our setup, they show that values of the order of a few hundreds GeV’s are beneath the reach of highluminosity LHC with TeV.
As previously mentioned, the pions, electrons and muons produced via two and threebody decays are extremely soft (since MeV) and, as a consequence, invisible to the tracking system. Therefore, analysis based on emission with leptons in the final state (unless GeV, unrealistic in our minimal scenario) are disfavored. On the contrary, the footprint of the soft decays is represented by the presence of a disappearing charged track. At the LHC, ATLAS searches of disappearing tracks signatures in the context of simplified supersymmetric models with a chargino nearly massdegenerate with the lightest neutralino place a 95% C.L. exclusion limit for GeV with luminosity fb Aad:2013yna (). Similar studies with simplified supersymmetric spectra Low:2014cba () and minimal DM Cirelli:2014dsa () show that this bound can be further raised considering a center of mass energy of TeV and ab.
iii.3 Two Higgs doublets
The model features two Higgs doublets, () and (). The most general potential is Branco:2011iw ()
(41)  
Referring to eqs. (14)–(15), the two Higgs doublets are characterized by the usual Goldstone bosons , that gives the longitudinal component of the and gauge bosons, two CPeven scalars and , one CPeven scalar and a charged Higgs . In full generality, the breaking vacuum expectation values of the two doublets are , . The two Higgs doublet model described by the potential in eq. (41) has been studied in ref. Altmannshofer:2012ar (). Here, we just notice that our model is compatible with a simplified version of eq. (41) in which a discrete symmetry is enforced. Imposing the transformation rules , , it follows that . Moreover, since now is forced to be zero, the only breaking vacuum expectation value is . In this setup the Higgs sector recovers the socalled Type I twoHiggs doublet model Branco:2011iw () where —coupled to leptons, down and (via ) uptype quarks—plays the role of the SM Higgs while the second doublet is inert. The model predicts deviations of the tree level Higgs couplings to gauge bosons and fermions (, ) with respect to their SM values (, ). In particular we have
(42) 
The Higgs sector of the model offers a rich phenomenology that can be tested at the LHC considering both the existence of new additional degrees of freedom and deviation from the SM Higgs couplings.
iii.4 Dark matter
Does the model contain a DM candidate? The simplest possibility is to look for a viable DM candidate in the inert Higgs doublet as discussed in the context of the socalled Inert Doublet Model LopezHonorez:2006gr (). In this case DM can be identified either with or , with a mass around GeV Honorez:2010re ().
There also exists another possibility, related to the presence in the model of the fermionic Higgs partner (see eq. (9)). These Higgsinos are stable because the potentially dangerous decay channel arising—after EW symmetry breaking, and assuming the inert condition —form the effective operator in eq. (II.1) can be kinematically closed by appropriately choosing the mass spectrum of the inert doublet.
As already mentioned in section II.1, it is necessary to introduce an extra doublet with hypercharge , hereafter, in order to cancel the gauge anomaly. The state , together with , can provide a good DM candidate. To show it, we explicitly introduce the following gauge invariant soft mass term
(43) 
where, for reason of clarity, we used the following notation
(44) 
The mass term in eq. (43) defines a degenerate spectrum of one charged (Dirac) fermion plus two neutral fields
(45) 
obtained diagonalizing the mass matrix in the neutral sector by means of the unitary transformation . The gauge coupling to the boson