The Minimal Scale Invariant Extension of the Standard Model
aa r X i v : . [ h e p - ph ] A ug MAN/HEP/2010/10June 2010
The Minimal Scale Invariant Extension ofthe Standard Model
Lisa Alexander-Nunneley and Apostolos Pilaftsis
School of Physics and Astronomy, The University of Manchester,Manchester M13 9PL, United Kingdom
ABSTRACT
We perform a systematic analysis of an extension of the Standard Model that includes a complexsinglet scalar field and is scale invariant at the tree level. We call such a model the MinimalScale Invariant extension of the Standard Model (MSISM). The tree-level scale invariance ofthe model is explicitly broken by quantum corrections, which can trigger electroweak symmetrybreaking and potentially provide a mechanism for solving the gauge hierarchy problem. Eventhough the scale invariant Standard Model is not a realistic scenario, the addition of a complexsinglet scalar field may result in a perturbative and phenomenologically viable theory. We presenta complete classification of the flat directions which may occur in the classical scalar potentialof the MSISM. After calculating the one-loop effective potential of the MSISM, we investigate anumber of representative scenarios and determine their scalar boson mass spectra, as well as theirperturbatively allowed parameter space compatible with electroweak precision data. We discussthe phenomenological implications of these scenarios, in particular, whether they realize explicitor spontaneous CP violation, neutrino masses or provide dark matter candidates. In particular,we find a new minimal scale-invariant model of maximal spontaneous CP violation which can stayperturbative up to Planck-mass energy scales, without introducing an unnaturally large hierarchyin the scalar-potential couplings.PACS numbers: 12.60.Fr, 11.15.Ex, 11.10.Hi, 14.60.St, 14.80.Ec Introduction
The Standard Model (SM) [1] is a renormalizable theory with a minimal particle contentwhich realizes the famous Higgs mechanism [2] to account for the origin of mass of thecharged fermions and the W ± and Z bosons. Despite intense scrutiny, the SM remainsresilient to new physics and appears to describe the data collected over the years at the LEPcollider, TEVATRON and in a number of low-energy experiments with remarkable success.Nevertheless, the SM predicts the existence of the Higgs boson which is associated with themechanism of electroweak spontaneous symmetry breaking (EWSSB), but which so far hasremained elusive. A natural realization of the EWSSB mechanism requires the presence ofa negative mass parameter, − m , in the Higgs potential. The negative mass parameter isthe source of the infamous gauge hierarchy problem, in which quantum corrections lead toquadratically divergent terms proportional to Λ , where Λ is an ultra-violet (UV) cut-offscale. This UV cut-off scale is usually associated with the scale of a possible higher-energy theory in which the SM might be embedded, such as Grand Unified Theory (GUT).In the SM, with no intermediate mass scale or theory between the electroweak (EW) andPlanck scale M Planck ≈ . × GeV, the cancellation of the divergent terms requiresexcessive fine-tuning. The avoidance of this fine-tuning problem has been the motivationfor many studies beyond the SM, including supersymmetry (SUSY). In SUSY this problemis naturally solved, provided the SUSY-breaking mass scale, M SUSY , stays close to the EWscale, e.g. M SUSY . m ,whose absence from the Higgs potential renders the complete tree-level Lagrangian of theSM scale invariant (SI). However, as first discussed by Coleman and E. Weinberg [3] andlater by Gildener and S. Weinberg [4], quantum corrections generate logarithmic termswhich explicitly break the scale invariance of the theory and can trigger EWSSB. Unfortu-nately, a perturbative SI version of the SM is not both theoretically and phenomenologicallyviable. Specifically, a perturbative SI version of the SM cannot accommodate the LEP2limit on the Higgs-boson mass, m H SM > . m term from the Higgs potential. However, its absence alone does not solve the gaugehierarchy problem as Λ terms can still be generated by quantum corrections in a UV cut-offscheme of regularization. This happens because the UV cut-off scheme introduces counter-2erms which explicitly violate the symmetry of classical scale invariance that governs thebare Lagrangian. Following the arguments of [9, 11], one has to therefore adopt a regula-rization scheme which does not break the classical symmetries of the local classical action,in this case scale invariance. Dimensional regularization (DR) [12] is such a SI schemewithin which the vanishing of the m term is maintained to all orders in perturbationtheory. Consequently, the scheme of DR will be used throughout this paper.An inherent field-theoretic difficulty of a SI model is the incorporation of gravitywhich requires the introduction of a dimensionful parameter, the Planck mass M Pl , intothe theory. The presence of the Planck mass explicitly breaks the classical symmetry of scaleinvariance, thereby reintroducing the issue of quadratic divergences in the theory. Eventhough addressing this problem lies beyond the scope of this paper, we note that attemptshave been made in the literature to provide SI descriptions of quantum gravity [9, 13, 14].In this paper we study in detail a minimal SI extension of the SM augmented by acomplex singlet scalar field, S . We call this model the Minimal Scale Invariant extensionof the Standard Model (MSISM). Unlike previous analyses [6–10], we impose no additionalconstraints on the theory, such as a U(1) symmetry or some specific discrete symmetryacting on S . Hence, the MSISM potential contains all possible interactions allowed bygauge invariance: V (Φ , S ) = λ † Φ) + λ S ∗ S ) + λ Φ † Φ S ∗ S + λ Φ † Φ S + λ ∗ Φ † Φ S ∗ + λ S S ∗ + λ ∗ SS ∗ + λ S + λ ∗ S ∗ , where the quartic couplings λ , ,..., are all dimensionless constants and Φ is the usual SMHiggs doublet. Note that the imposition of scale invariance forbids the appearance ofdimensionful mass parameters or trilinear couplings in the potential .The tree-level SI scalar potential can possess a large number of different phenomeno-logically viable flat directions, which may be classified into three major categories: Type I,Type II and Type III. Flat directions of Type I are characterized by a singlet field S withvanishing vacuum expectation value (VEV), whereas in flat directions of Type II both S and Φ possess non-zero VEVs. Finally, in flat directions of Type III the SM Φ has a zeroVEV, which makes it somehow difficult to naturally realize EWSSB and therefore we donot study them in detail in this paper.In our analysis of the MSISM effective potential, we follow the perturbative approachintroduced by Gildener and S. Weinberg (GW) [4]. With the aid of this approach we cananalytically calculate the scalar boson mass spectrum and determine the allowed range of For recent studies of non-SI models with dimensionful self-couplings and with real or complex scalarsinglet extensions see [15, 16]. S , T and U [18, 19]. Of the electroweak oblique parameters, S and T (thelatter associated with Veltman’s ρ parameter [20]) yield the strongest constraints on therange of the scalar-potential quartic couplings.An interesting feature of the MSISM is that it can be naturally extended by right-handed neutrinos in a SI way, such that a singlet Majorana mass scale, m M , can be gener-ated if the complex scalar S possesses a VEV [10, 21]. The expected size of m M is typicallyof the EW scale. This can give rise to a low-scale seesaw mechanism [22], which in turn canoffer a natural explanation for the smallness in mass of the light neutrinos as observed in thelow-energy neutrino data. Moreover, unlike the SM, the MSISM can realize both explicitand spontaneous CP violation. Of particular interest is a new minimal model of maximalspontaneous CP violation along a maximally CP-violating flat direction of Type II, whichcan stay perturbative up to energy scales of order M Planck , without the need to introducea large hierarchy among the scalar-potential quartic couplings or between the VEVs of theΦ and S fields [23]. The new CP-violating phase could act as a source for creating the ob-served Baryon Asymmetry in the Universe (BAU), e.g. via a strong first-order electroweakphase transition. Finally, the MSISM can predict stable scalar states that could qualify asDark Matter (DM) candidates.This paper is set out as follows. In Section 2 we review the basic properties of aSI classical action and derive the Ward identity which is obeyed by the tree-level scalarpotential. This Ward identity for scale invariance is then used to define the flat directionin the scalar potential. In Section 3, we review the EWSSB mechanism in multi-scalarSI models following the formalism outlined in [4]. In Section 4, we present the generalLagrangian describing the MSISM. Furthermore, we present a general classification of theflat directions that may occur in the tree-level scalar potential and then calculate the one-loop effective potential. We also discuss the possible phenomenology of the different flatdirections. Section 5 investigates models having Type I flat directions in both the U(1)invariant limit and the general non-invariant scenario. Likewise, Section 6 investigatesmodels that realize flat directions of Type II, in the U(1) invariant limit and a simplifiednon-invariant scenario. In Section 7, we discuss extensions of the MSISM that include theinteractions of the complex singlet field S and its complex conjugate S ∗ to right-handedneutrinos. Technical details of all our calculations have been relegated to a number ofappendices. Finally, Section 8 summarizes our conclusions.4 The Ward Identity for Scale Invariance
In this section we derive the Ward identity (WI) that results from imposing the property ofscale invariance on a theory. The WI for scale invariance will then be used to consistentlydefine the flat directions as local minima of the scalar potential.To start with, let us consider a simple model with one real scalar field, Φ( x ), describedby the Lagrangian: L = 12 ∂ xµ Φ( x ) ∂ µx Φ( x ) + 12 m Φ ( x ) − λ Φ ( x ) , (2.1)with the notation ∂ µx ≡ ∂∂x µ . Under a scale transformation, the scalar field Φ( x ) transformsas Φ( x ) → Φ ′ ( x ) = σ Φ( σx ) , (2.2)where σ = e ǫ >
0. We note that a general scale transformation is defined as Φ( x ) → Φ ′ ( x ) = e ǫa Φ( e ǫ x ), where a is the scaling dimension of the field Φ( x ). At the classicallevel the scaling dimension takes the value a = 1, if Φ( x ) is a boson, and the value a = ,if Φ( x ) is a fermion. The effect of the scale transformation (2.2) of the scalar field Φ( x ) onthe classical action S [Φ( x )] = Z d x L [ ∂ µ Φ( x ) , Φ( x )] (2.3)is to give rise to a transformed action given by S [ σ Φ( σx )] = Z ∞−∞ d x (cid:20) σ ∂ xµ Φ( σx ) ∂ µx Φ( σx ) + 12 m σ Φ ( σx ) − λσ Φ ( σx ) (cid:21) = Z σ ( ∞ ) σ ( −∞ ) d ( σx ) (cid:20) ∂ ( σx ) µ Φ( σx ) ∂ µ ( σx ) Φ( σx ) + 12 σ − m Φ ( σx ) − λ Φ ( σx ) (cid:21) . (2.4)Obviously, the transformed action S [ σ Φ( σx )] is equal to the original one S [Φ( x )], providedthe dimensionful parameter m vanishes, i.e. the absence of the m term results in a SItheory.Having gained some insight from the above simple model, we now consider a generaltheory, where Φ( x ) represents the generic field of the theory, which could be a scalar,fermion or vector boson. The variation δS [Φ( x )] of the classical action (2.3) under a scaletransformation is calculated as δS [Φ( x )] = Z d y (cid:20) δ Φ i ( y ) δδ Φ i ( y ) + δ Φ † i ( y ) δδ Φ † i ( y ) + δ (cid:0) ∂ µ Φ i ( y ) (cid:1) δδ (cid:0) ∂ µ Φ i ( y ) (cid:1) + δ (cid:0) ∂ µ Φ † i ( y ) (cid:1) δδ (cid:0) ∂ µ Φ † i ( y ) (cid:1) (cid:21) Z d x L [Φ( x )] , (2.5)5here summation over repeated indices is implied for all the fields in the theory. Given δ Φ( x ) = ǫ (cid:0) a Φ( x ) + x µ ∂ µ Φ( x ) (cid:1) for an infinitesimal scale transformation, the variation δS [Φ( x )] is found to be δS [Φ( x )] = ǫ Z d x (cid:20) a ∂ L [Φ( x )] ∂ Φ i ( x ) Φ i ( x ) + a Φ † i ( x ) ∂ L [Φ( x )] ∂ Φ † i ( x ) + (1 + a ) ∂ L [Φ( x )] ∂ (cid:0) ∂ µ Φ i ( x ) (cid:1) (cid:0) ∂ µ Φ i ( x ) (cid:1) +(1 + a ) (cid:0) ∂ µ Φ † i ( x ) (cid:1) ∂ L [Φ( x )] ∂ (cid:0) ∂ µ Φ † i ( x ) (cid:1) − L [Φ( x )] (cid:21) + ǫx µ L [Φ( x )] | x µ →±∞ . (2.6)In the above, the last term is a surface term which we assume that vanishes at infinity.Requiring that δS [Φ( x )] = 0, as it should for a SI theory, we derive the WI for scaleinvariance:4 L [Φ( x )] = a " ∂ L [Φ( x )] ∂ Φ i ( x ) Φ i ( x ) + Φ † i ( x ) ∂ L [Φ( x )] ∂ Φ † i ( x ) +( a + 1) " ∂ L [Φ( x )] ∂ (cid:0) ∂ µ Φ i ( x ) (cid:1) (cid:0) ∂ µ Φ i ( x ) (cid:1) + (cid:0) ∂ µ Φ † i ( x ) (cid:1) ∂ L [Φ( x )] ∂ (cid:0) ∂ µ Φ † i ( x ) (cid:1) . (2.7)If the scalar potential V (Φ) of a theory is SI at tree-level then the WI (2.7) impliesthat ∂V tree (Φ) ∂ Φ i Φ i + Φ † i ∂V tree (Φ) ∂ Φ † i = 4 V tree (Φ) . (2.8)For notational simplicity we hereafter suppress the x -dependence of the scalar field Φ,i.e. Φ = Φ( x ). From the context it should be clear whether we refer to the x -dependentquantum field excitation or to its stationary and x -independent background field value.If Φ = ( φ , φ , . . . , φ n ) is a vector whose components represent all the scalar fields of thetheory as real degrees of freedom, the WI (2.8) straightforwardly generalizes to Φ · ∇ V tree ( Φ ) = 4 V tree ( Φ ) , (2.9)where ∇ ≡ (cid:0) ∂∂φ , ∂∂φ , · · · , ∂∂φ n (cid:1) . Moreover, the dot indicates the usual scalar product ofvectors in an n -dimensional vector space spanned by all n real scalar fields of the theory.The WI (2.9) can be applied to a specific direction in the n -dimensional field space.To this end, we may parametrize the field vector Φ as Φ = ϕ N , where N is a fixed given n -dimensional unit vector in the field space and ϕ is the radial distance from the origin ofthe field space. In this case, we may rewrite (2.9) as ϕ N · ∇ V tree ( ϕ N ) = ϕ d Φ dϕ · ∇ V tree ( ϕ N ) = ϕ dV tree ( ϕ N ) dϕ = 4 V tree ( ϕ N ) . (2.10)The condition for V tree ( ϕ N ) to have a flat direction along a given unit vector N = n is dV tree ( ϕ n ) dϕ = 0 . (2.11)6n account of the WI (2.10), the latter condition is equivalent to V tree ( ϕ n ) = 0. Inaddition, the condition for this flat direction to be an extremal or stationary line is ∇ V tree ( Φ ) (cid:12)(cid:12)(cid:12) Φ = ϕ n = . (2.12)In order for this extremal line to be a local minimum of the potential, one has to requirethat ( v · ∇ ) V tree ( Φ ) (cid:12)(cid:12)(cid:12) Φ = ϕ n ≥ , (2.13)for any arbitrary vector v belonging to the n -dimensional field space. Finally, one has toensure that the scalar potential is BFB, i.e. V tree ( N ) ≥
0, for all possible directions N . Here we review the GW perturbative approach [4] to EWSSB that occurs in generic multi-scalar SI models. We also discuss the scalar mass spectrum of these models. The analyticresults presented here will be used in the next section to study the EWSSB in the MSISMand to calculate its scalar mass spectrum.According to the GW approach, the minimization of the full potential, V = V tree + V − loopeff + . . . , is performed perturbatively along an extremal (minimal) flat direction asdefined in the previous section. This approach is only valid if the theory is weakly coupled,which constitutes the regime of validity for our investigations.Let us consider a renormalizable gauge field theory with an arbitrary set of n realscalars φ i (with i = 1 , , . . . , n ) which represent the components of an n -dimensional fieldmultiplet Φ (see also Section 2). We assume that the theory is SI at tree-level so that itsscalar potential is generically given by V tree ( Φ ) = 14! f ijkl φ i φ j φ k φ l , (3.1)where summation over repeated indices is implied and f ijkl stands for the quartic couplingsof the potential; f ijkl is fully symmetric in all its indices. Notice that (3.1) is a generalsolution to the WI for SI given in (2.9).As we discussed in the previous section, the potential (3.1) may have a non-trivialcontinuous local minimum along the ray Φ = ϕ N , in a given direction N = n of theunit vector and at a specific renormalization group (RG) scale µ = Λ. To find this localminimum one first needs to identify all the flat directions present in the potential by solvingthe equation: V tree ( N ) = 14! f ijkl ( µ ) N i N j N k N l = 0 , (3.2)7here we have explicitly displayed the dependence of the quartic couplings f ijkl on the RGscale µ . Suppose that this condition is met for a particular unit vector N = n and forthe specific value of the RG scale, µ = Λ. According to (2.11), one then has V tree ( Φ ) = 0everywhere along the ray Φ flat = ϕ n , which represents the flat direction.The next step is to ensure that the flat direction Φ flat , as determined above, representsa stationary line. This leads to the condition ∂V tree ( N ) /∂N i | N = n = 0, and hence to theconstraint f ijkl (Λ) n j n k n l = 0 . (3.3)Observe that this constraint is equivalent to the condition (2.12). It should also be notedthat (3.3) imposes a single constraint on the parameters f ijkl , independent of how manyparameters f ijkl contains and specifically only at the RG scale Λ. Finally, one needs toimplement the condition (2.13), i.e. the stationary line is a local minimum line. Therefore,one has to require that the Hessian matrix, defined as( P ) ij ≡ ∂ V tree ( N ) ∂N i ∂N j N = n = 12 f ijkl n k n l , (3.4)is non-negative definite, i.e. the n × n -dimensional matrix P has either vanishing or positiveeigenvalues.Since V tree ( N ) vanishes along the flat direction Φ flat , the full potential of the theorywill be dominated by higher-loop contributions along Φ flat and specifically by the one-loopeffective potential, V − loopeff ( Φ ). Adding higher order quantum corrections gives a smallcurvature in the radial direction Φ flat = ϕ n , which picks out a specific value, v ϕ , alongthe ray as the minimum. In addition, a small shift may also be produced in a direction δ Φ = v ϕ δ n perpendicular to the flat direction n , i.e. n · δ n = 0. We may now extend thestationary condition (2.12) to the one-loop corrected scalar potential, i.e. ∇ (cid:16) V tree ( Φ ) + V − loopeff ( Φ ) (cid:17) (cid:12)(cid:12)(cid:12) Φ = v ϕ ( n + δ n ) = . (3.5)According to the GW perturbative approach, one has to consistently expand this lastexpression to the first loop order, by treating the perpendicular shift δ Φ as an one-looporder parameter. In this way, we find v ϕ P · δ Φ + ∇ V − loopeff ( Φ ) (cid:12)(cid:12)(cid:12) Φ = v ϕ n = , (3.6)where the dot indicates the usual matrix multiplication of the Hessian P with the vector δ Φ .The perturbative minimization condition (3.6) uniquely determines δ Φ , except fordirections along eigenvectors of P with zero eigenvalues. These zero eigenvectors includethe flat direction n itself, since n · P = by virtue of (3.3) and (3.4). They also include8he Goldstone directions that may result from the spontaneous symmetry breaking of anycontinuous symmetries. Therefore, we may eliminate the first term in (3.6) by contractingthe relation (3.6) from the left with n . Thus, we get the minimization condition along theradial direction: n · ∇ V − loopeff ( Φ ) (cid:12)(cid:12)(cid:12) Φ = v ϕ n = dV − loopeff ( ϕ n ) dϕ (cid:12)(cid:12)(cid:12)(cid:12) ϕ = v ϕ = 0 . (3.7)Here it is useful to remark that this condition will be used to fully specify the VEV of φ to one-loop order in perturbation theory.Along the flat direction Φ flat = ϕ n , the one-loop effective potential, V − loopeff ( ϕ n ),takes the general form: V − loopeff ( ϕ n ) = A ( n ) ϕ + B ( n ) ϕ ln ϕ Λ , (3.8)where the n -dependent dimensionless constants A and B are given in the MS scheme by A = 164 π v ϕ (cid:26) Tr (cid:20) m S (cid:18) −
32 + ln m S v ϕ (cid:19) (cid:21) + 3Tr (cid:20) m V (cid:18) −
56 + ln m V v ϕ (cid:19) (cid:21) − (cid:20) m F (cid:18) − m F v ϕ (cid:19) (cid:21) (cid:27) ,B = 164 π v ϕ (cid:18) Tr m S + 3Tr m V − m F (cid:19) , (3.9)where m S,V,F are the tree-level scalar, vector and fermion mass matrices, respectively, whichare evaluated at v ϕ n and the trace is taken over the mass matrix and over all internaldegrees of freedom . Analytic results for the tree-level mass matrices m S,V,F will be givenin the next section, where we will calculate the one-loop effective potential of the MSISMfollowing the GW approach.Minimizing (3.8) according to (3.7) shows that the potential has a non-trivial station-ary point at a value of the RG scale Λ, given byΛ = v ϕ exp (cid:18) A B + 14 (cid:19) . (3.10)Note that since the effective-potential coefficients A and B are of the same loop order, theRG scale Λ and the absolute minimum v ϕ are expected to be of comparable order as well.Thus, a natural implementation of the breaking of the scale symmetry can be obtained in Note that the internal degrees of freedom for Majorana fermions are half of those of the Dirac fermions.Consequently, if the fermion F is of the Majorana type, the pre-factor − − /v ϕ ) can be keptunder control.The relation (3.10) can now be used to find the form of the one-loop effective potentialalong the flat direction in terms of the one-loop VEV v ϕ , V − loopeff ( ϕ n ) = B ( n ) ϕ (cid:18) ln ϕ v ϕ − (cid:19) . (3.11)Even though the above substitution has made the explicit dependence of V − loopeff ( ϕ n )on Λ to disappear, there still exists an implicit dependence of the kinematic parametersin B ( n ) and the flat direction ϕ on the RG scale Λ. On the other hand, in order for v ϕ n to be a minimum, V − loopeff ( v ϕ n ) must be less than the value of the potential at theorigin ϕ = 0, hence it must be negative. From (3.11), it is easy to see that this can onlyhappen if B >
0. Moreover, this constraint ensures that the potential is BFB, i.e. theone-loop effective potential remains non-negative for infinitely large values of ϕ in any fielddirection N .At the tree-level, the squared masses of the scalar bosons are given by the eigenvaluesof the matrix, ( m S ) ij = ∂ V tree ( Φ ) ∂φ i ∂φ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Φ = v ϕ n = v ϕ ( P ) ij . (3.12)From our discussion above, it is clear that the Hessian matrix P has positive definiteeigenvalues, except for a set of zero eigenvalues due to the Goldstone bosons associatedwith the spontaneous symmetry breaking of compact symmetries of the theory and onezero eigenvalue due to flat direction. Hence the model contains a set of massive scalars,a set of massless Goldstone bosons and a single massless scalar, which we denote as h ,associated with the spontaneous symmetry breaking of scale invariance.The single massless scalar does not remain massless beyond the tree approximation.In detail, the one-loop correction V − loopeff to the scalar potential shifts the mass matrix to( m S + δm S ) ij = ∂ (cid:0) V tree ( Φ ) + V − loopeff ( Φ ) (cid:1) ∂φ i ∂φ j Φ = v ϕ ( n + δ n ) . (3.13)To first order in a perturbative expansion, this becomes( δm S ) ij = ∂ V − loopeff ( Φ ) ∂φ i ∂φ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Φ = v ϕ n + v ϕ f ijkl n k δφ l . (3.14)In order to remove the second term in (3.14), we contract ( δm S ) ij with n i and n j . Thus,the mass of the field h is calculated to be m h = n i n j ( δm S ) ij = n i n j ∂ V − loopeff ( Φ ) ∂φ i ∂φ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Φ = v ϕ n = d V − loopeff ( ϕ n ) dϕ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ = v ϕ = 8 Bv ϕ , (3.15)10here we have used (3.8) and (3.10) to arrive at the last equality in (3.15). The field h iscommonly called the pseudo-Goldstone boson of the anomalously broken scale invariance,since it is massless at tree-level when scale invariance holds, but acquires a non-zero massat the one-loop level once scale invariance is broken by quantum corrections.The remaining massive scalar states of the theory can be easily determined provided( δm S ) ij remains a small effect compared to the tree-level mass matrix ( m S ) ij . In this case,their masses are determined from the relation: m H = ˜ n i ˜ n j ∂ V tree ( Φ ) ∂φ i ∂φ j Φ = v ϕ n = ˜n · P · ˜n , (3.16)where the massive scalar directions are defined similarly to Φ flat as Φ H = ϕ ˜n , where ˜n isa generic unit vector perpendicular to n . The Goldstone bosons remain massless provided V − loopeff ( Φ ) respects the same global symmetries as V tree ( Φ ). In this section we use the analytic results presented in the previous two sections to study themechanism of EWSSB in the Minimal Scale Invariant extension of the Standard Model.First, we briefly review the general Lagrangian describing the MSISM. We then discussthe parameterization of the flat directions and present a general classification of the flatdirections that may occur in the tree-level scalar potential. We also present the one-loopeffective potential for the MSISM, from which we derive its scalar mass spectrum. Finally,we briefly discuss the generic phenomenological features of the different realizations of flatdirections in the MSISM. A detailed investigation of the physically viable flat directions inthe MSISM is deferred to Sections 5 and 6.
The Lagrangian defining the MSISM can be written as a sum of five terms: L MSISM = L inv + L GF + L FP + L ν − V tree (Φ , S ) , (4.1)where L inv , L GF and L FP are the gauge-invariant, gauge-fixing and Faddeev–Popov La-grangians, respectively, and a detailed description of these Lagrangians is given in Ap-pendix A. The term L ν is the right-handed neutrino Lagrangian which is discussed sep-arately in Section 7. The last term, V tree (Φ , S ), is the tree-level potential of the MSISM,11hich is given by V tree (Φ , S ) = λ † Φ) + λ S ∗ S ) + λ Φ † Φ S ∗ S + λ Φ † Φ S + λ ∗ Φ † Φ S ∗ + λ S S ∗ + λ ∗ SS ∗ + λ S + λ ∗ S ∗ . (4.2)where for simplicity the x -dependence of the fields has been suppressed and will continueto be suppressed unless distinction is required between the field φ ( x ) and the flat directioncomponent φ . As usual, we may linearly decompose the SU(2) L scalar doublet Φ and thecomplex singlet field S as follows:Φ = G +1 √ ( φ + iG ) ! , S = 1 √ σ + iJ ) , (4.3)where φ and σ ( G and J ) are CP-even (odd) real scalar fields and G + is the chargedwould-be Goldstone boson.In order to provide a stable minimum for the scalar potential, we must ensure that V tree is BFB. This can be achieved by placing a set of constraining conditions on the quarticcouplings λ , ,..., . These conditions can be determined by analyzing the potential in termsof the two real and independent gauge-invariant field bilinears, Φ † Φ and S ∗ S . To convert(4.2) into this representation, we re-express the field S as S = | S | e iθ S , where θ S is the phaseof the complex field and S ∗ S = | S | . The tree-level scalar potential can then be rewrittenin the form V tree = 12 (cid:16) Φ † Φ , S ∗ S (cid:17) Λ Φ † Φ S ∗ S ! , (4.4)where Λ is a real symmetric matrix with the elements:Λ = λ , Λ = Λ = λ + λ e iθ S + λ ∗ e − iθ S , Λ = λ + 2 λ e iθ S + 2 λ ∗ e − iθ S + λ e iθ S + λ ∗ e − iθ S . (4.5)Since the two bilinears Φ † Φ and S ∗ S are both positive-definite by definition, the requirementfor V tree to be BFB depends exclusively on the matrix elements of Λ . In detail, the followingtwo conditions are required to keep V tree BFB:(i) Tr Λ ≥ , (ii) ( Λ ≥ , if Λ = 0 or Λ = 0Det Λ ≥ , if Λ = 0 and Λ = 0 . (4.6)The above conditions must hold for all directions in the bilinear vector space, includingthe flat directions. Obviously, these conditions explicitly depend on the phase θ S through12he matrix elements of Λ given in (4.5). This phase determines the direction of a rayin the σ - J plane within the entire real scalar field space. It is therefore essential thatthe conditions (4.6) hold true for all values of θ S , ensuring that V tree remains BFB in allpossible field directions.It is now instructive to show that the angle θ S is SI. We can prove this by using theWI (2.8) for scale invariance. We first note that the derivatives of the tree-level potential V tree with respect to the different representations, real fields, complex fields and bilinears,are related through: ℜ G + ∂V tree ∂ ℜ G + + ℑ G + ∂V tree ∂ ℑ G + + G ∂V tree ∂G + φ ∂V tree ∂φ = ∂V tree ∂ Φ Φ + Φ † ∂V tree ∂ Φ † = 2Φ † Φ ∂V tree ∂ (Φ † Φ) ,σ ∂V tree ∂σ + J ∂V tree ∂J = S ∂V tree ∂S + S ∗ ∂V tree ∂S ∗ = 2 S ∗ S ∂V tree ∂ ( S ∗ S ) , (4.7)with ℜ G + = √ ( G + + G − ) and ℑ G + = i √ ( G − − G + ). The second equation in (4.7)involving the complex singlet field S was derived by employing the relations: S ∗ S ∂V tree ∂ ( S ∗ S ) + ∂V tree ∂ (2 iθ S ) = S ∂V tree ∂S , S ∗ S ∂V tree ∂ ( S ∗ S ) − ∂V tree ∂ (2 iθ S ) = S ∗ ∂V tree ∂S ∗ . (4.8)Hence, the WI (2.8) can be re-expressed in terms of derivatives with respect to bilinearsonly, i.e. S ∗ S ∂V tree ∂ ( S ∗ S ) + Φ † Φ ∂V tree ∂ (Φ † Φ) = 2 V tree . (4.9)Evidently, the absence of a derivative term with respect to the phase θ S implies that θ S isa truly SI quantity in the MSISM.A comment regarding the predictive power of the Higgs sector of the MSISM isin order. The MSISM potential contains several quartic couplings that would seem toimply that the MSISM will be less predictive than the SM. However, imposing the flatdirection condition (3.3) and possible additional symmetries, such as a U(1) or a Z discretesymmetry acting on S , reduces the number of the independent parameters significantly. Infact, most of the generic cases that we will be studying have only two or three independentquartic couplings, thereby making the MSISM a rather predictive theory. Following the approach presented in Sections 2 and 3, we parametrize the flat directionas an n -dimensional vector, whose components represent all real degrees of freedom of the13calars fields in the theory. For the MSISM, the flat direction lies in the vector spacespanned by the real scalar fields, {ℜ G + , ℑ G + , G, φ, σ, J } . Without loss of generality, we may exploit the SM gauge symmetry to set ℜ G + = ℑ G + = G = 0 and restrict the field space to the neutral fields φ , σ and J , which may developan electrically neutral VEV. Thus, the general flat direction Φ flat can be dimensionallyreduced to Φ flat = ϕ n φ n σ n J = φσJ , (4.10)where the components n φ,σ,J satisfy the unit-vector constraint: n φ + n σ + n J = 1. Observethat v ϕ n φ ≡ v φ , v ϕ n σ ≡ v σ and v ϕ n J ≡ v J , at the minimum of the one-loop effectivepotential.In order that the flat directions represent minimal lines of the tree-level potential,we need to require that all the derivatives of V tree with respect to the fields φ , σ and J ,or equivalently with respect to the fields Φ and S , vanish when evaluated along the flatdirection [cf. (2.12)]. In this way, the following two complex tadpole conditions need to besatisfied: ∂V tree ∂ Φ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Φ flat = Φ † h λ (Λ)Φ † Φ + λ (Λ) S ∗ S + λ (Λ) S + λ ∗ (Λ) S ∗ i = 0 , (4.11) ∂V tree ∂S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Φ flat = S ∗ h λ (Λ) S ∗ S + λ (Λ)Φ † Φ + 3 λ (Λ) S + λ ∗ (Λ) S ∗ i + S h λ (Λ)Φ † Φ + 2 λ (Λ) S i = 0 , (4.12)where Φ flat is defined in (4.10). As we will discuss in more detail below, there are threedistinct ways to satisfy the above minimization conditions, which generically lead to threedifferent types of flat directions: Type I, Type II and Type III. Along the Type I flat direction, the scalar doublet Φ develops a VEV, but not the complexfield S , i.e. the flat direction components σ and J in (4.10) are both zero. If S = 0, theminimization condition (4.12) is automatically satisfied, whilst the condition (4.11) forcesus to set λ (Λ) = 0. The values of the other quartic couplings are constrained by the BFBconditions (4.6), such that Λ > > S has a vanishing VEV, the flat direction (4.10) gets dimen-sionally reduced to Φ flat = ϕ n φ = φ , (4.13)with n φ = 1. This implies that the flat direction lies directly along the φ axis and that thequantum field φ corresponds exactly to the massless scalar field h , which is the pseudo-Goldstone boson associated with broken scale invariance (see our discussion in Section3). Along the Type II flat direction, both the doublet Φ and the singlet S fields developnon-zero VEVs. This implies (4.11) and (4.12) can only be satisfied if specific relationsamong the quartic couplings are met at some RG scale Λ. For instance, consider a U(1)-invariant MSISM scalar potential which is invariant under U(1) rephasings of the field S → e iα S , where α is an arbitrary phase. As a consequence of the U(1) invariance thequartic couplings λ , , vanish. Moreover, the minimization conditions (4.11) and (4.12)lead to the constraint: Φ † Φ S ∗ S = n φ n σ + n J = − λ (Λ) λ (Λ) = − λ (Λ) λ (Λ) . (4.14)In addition, in order to satisfy the above relation and the BFB condition (4.6), we mustdemand that λ > λ > λ < σ and J will develop VEVs, since S is acomplex field. However, if a U(1) symmetry is acting on the scalar potential, any possiblephase of S can be eliminated through a U(1) rephasing, such that S is real and J =0. Consequently, for the U(1) invariant scenario, the flat direction is reduced to a twocomponent vector and applying the constraints (4.14) and n φ + n σ = 1 yields Φ flat = ϕ q − λ (Λ) λ (Λ) − λ (Λ) q λ (Λ) λ (Λ) − λ (Λ) = φ q λ (Λ) − λ (Λ) ! . (4.15)Since the U(1)-invariant Type II flat direction is composed of both the φ and σ fields,there will be mixing between the two CP even states in the mass basis, where the mass basisis defined by the field along the flat direction and those fields along directions perpendicularto it. Thus, for the U(1) invariant scenario, the mass eigenstates are the massless Goldstoneboson J associated with the spontaneous breaking of the U(1) symmetry and the massivescalar states h and H , given by h = cos θ φ + sin θ σ , H = − sin θ φ + cos θ σ , (4.16)15here cos θ = − λ (Λ) / [ λ (Λ) − λ (Λ)].The general U(1) non-invariant scenario is much more involved and will be discussedin detail in Section 6.2. In the U(1) non-invariant scenario, the flat direction is in general athree component vector. Hence, unless S is either real or imaginary and so preserves the CPsymmetry, all three quantum fields φ , σ and J will mix together to form the scalar-bosonmass eigenstates. The third type of flat direction is characterized by Φ = 0. However, a zero VEV for the Φdoublet is not phenomenologically viable, since it is difficult to realize successful EWSSB.In particular, the electroweak gauge bosons remain massless at the tree-level. Beyond thetree approximation, there will be a small shift in the direction of the flat direction, but thisturns out to be generically too small to account for the W ± - and Z -boson masses, unlessa large hierarchy between the VEVs of Φ and S fields is introduced [23]. Therefore, we donot study the Type III flat direction in this paper.It is important to note here that the three types of flat directions described abovegive a complete classification of the flat directions in the MSISM. However, each type maycontain several different variations. For example, consider the U(1) non-invariant Type IIflat direction. It requires (4.11) and (4.12), but places no explicit constraints on how thequartic couplings of the scalar potential satisfy them. Each choice provides a unique validflat direction which gives rise to a vast number of possible variants. We do not intendto go through each such variant, but rather concentrate on a few representative scenarioswhich appear to be physically interesting, in terms of new sources of CP violation, neutrinomasses and DM candidates. We now present the general one-loop effective potential of the MSISM. This has beencomputed in terms of Φ and S in Appendix B, where the full one-loop renormalized effectivepotential V − loopeff is given in (C.14). Along the minimum flat direction, the RG scale takesthe specific value µ = Λ and V − loopeff can be put in a form similar to the one in (3.8), i.e. V − loopeff ( φ ) = α φ + β φ ln φ Λ . (4.17)16he coefficients α and β are dimensionless parameters and are given in the MS scheme by α = 164 π v φ (cid:20) X i =1 m H i (cid:18) −
32 + ln m H i v φ (cid:19) + 6 m W (cid:18) −
56 + ln m W v φ (cid:19) + 3 m Z (cid:18) −
56 + ln m Z v φ (cid:19) − m t (cid:18) − m t v φ (cid:19) − X i =1 m Ni (cid:18) − m Ni v φ (cid:19)(cid:21) ,β = 164 π v φ (cid:18) X i =1 m H i + 6 m W + 3 m Z − m t − X i =1 m Ni (cid:19) . (4.18)In the above, we have neglected all light fermions, except of the top quark and the pos-sible presence of heavy Majorana neutrinos N , , [cf. (C.14)]. The parameters m X , with X = { H , , W, Z, t, N } , are the tree-level particle masses. These are given by the massparameters M X , defined in Appendix B, evaluated at the minimum φ = v φ ≡ v SM , where v SM ≈
246 GeV is the VEV of the SM Higgs doublet Φ.Notice that the one-loop effective potential V − loopeff (Λ) in (4.17) can be written downentirely in terms of φ and v φ , without the need to involve the other flat direction components σ and J . This is possible, since either σ = J = 0 along the Type I flat direction, or σ and J are related to φ along the Type II flat direction. In this context, it can be shown that theMSISM effective potential (4.17) can be written in the general form of (3.8). To make thisexplicit, we employ the fact that φ = ϕ n φ in (4.17), which allows us to make the followingobvious identifications for the parameters A and B : A = α n φ + β n φ ln n φ , B = β n φ . (4.19)Substituting the above expressions for A and B in (3.15) and (3.10), we may readily obtainthe analytic dependence of the Higgs-boson mass m h and the minimization RG scale Λ onthe effective potential coefficients α and β : m h = 8 β n φ v φ , (4.20)Λ = v φ exp (cid:18) α β + 14 (cid:19) . (4.21)We may now employ the relation (4.21) to eliminate the explicit dependence of the effectivepotential V − loopeff in (4.17) on the RG scale Λ, V − loopeff ( φ ) = β φ (cid:18) ln φ v φ − (cid:19) , (4.22)where all kinematic quantities on the RHS of (4.22), such as β , φ and v φ , are evaluated atthe RG scale Λ [cf. (3.11)]. Hence, the size of the radiative corrections along the minimum17at direction is determined by the effective potential coefficient β and is therefore highlymodel-dependent. In our analysis of the specific flat directions of Type I and Type II, wewill use the two formulae for m h and Λ given in (4.20) and (4.21), respectively.Along the minimum flat direction, the scalar mass spectrum of the MSISM generallyconsists of two massive states H , with masses m H , , and one massless state h corre-sponding to the pseudo-Goldstone of the anomalously broken scale invariance at the treelevel. The would-be Goldstone bosons associated with the EWSSB of the SM gauge groupreceive gauge-dependent masses along the minimum flat direction, e.g. see (A.6). However,these gauge-dependent mass terms do not contribute to the one-loop effective potential V − loopeff (Λ), since they cancel against the gauge-dependent part of the gauge-boson andghost contributions. More technical details are given in Appendix B.Given the analytic form of the effective potential coefficient β in (4.18), it is nowinteresting to see why a SI version of the SM cannot be phenomenologically viable. In a SIextension of the SM, we expect that the Higgs boson H SM is massless at the tree level, butacquires an one-loop radiatively generated mass given by (4.20). This implies that the SMHiggs-boson mass m H SM ≡ m h is explicitly dependent on β , i.e. β = 164 π v φ (cid:18) m W + 3 m Z − m t (cid:19) . (4.23)Considering the presently well-known experimental values of the top-quark, W ± - and Z -boson masses, the coefficient β turns out to be negative, giving rise to an unphysicallytachyonic mass, in gross violation to the LEP2 limit [5]: m H SM > . β and B are negative, the SI limit of the SM also fails to realize a scalar potential which is BFB,according to our discussion in Section 3. As was already mentioned in the introduction, the MSISM provides a conceptually veryminimal solution to the gauge-hierarchy problem, with a minimal set of new fields and newcouplings. Following a bottom-up approach, it is interesting to analyze the phenomeno-logical features of the different variants of the MSISM. In particular, we are interested inscenarios which include new sources of CP violation, provide massive DM candidates andcan incorporate a natural mechanism for generating the small light-neutrino masses, suchas the seesaw mechanism [22].In the MSISM, naturally small Majorana masses for the light neutrinos can be gen-erated via the seesaw mechanism, only if there exist SI interactions of S with right-handedneutrinos and the singlet field S possesses a non-zero VEV, S = 0. Hence, as we have listed18(1) Invariant CP Violation Massive DM SeesawCandidate NeutrinosFlat Direction of Type I S = 0 Yes None Yes No S = 0 No Explicit Yes NoFlat Direction of Type II S = real Yes None No Yes S = real No Explicit Model YesDependent S = imaginary No Explicit Model YesDependent S = complex No Explicit or Model YesSpontaneous DependentTable 1: Taxonomy of all possible U(1)-invariant and U(1) non-invariant realizations thatmay occur within the MSISM, in terms of their potential to realize explicit or spontaneousCP violation, massive DM candidates and possible implementation of the seesaw mechanismfor naturally explaining the small light-neutrino masses.
19n Table 1, only Type-II flat directions have the ability to realize the seesaw mechanism.In addition, we have presented in Table 1 the scenarios of the MSISM, which can containboth explicit or spontaneous CP violation through complex quartic couplings λ , , or acomplex VEV for the field S , respectively. Notice that the Type-II flat direction along animaginary S does not violate CP spontaneously, since one may redefine S as S ′ ≡ iS torender this flat direction real, without introducing any new phase in the quartic couplingsof the scalar potential. Finally, Table 1 shows the different variants of the MSISM, whichhave the potential to predict a massive stable scalar particle that could qualify as a DMcandidate. As was pointed out in [24], a natural way to have a massive stable scalar bosonis to impose a parity symmetry on the scalar potential. Such parity symmetries could be: σ → − σ , J → − J , or σ ↔ ± J . Therefore, as we comment in Table 1, the existence of aDM candidate is model-dependent and requires further constraints on the theory.In the next two sections, Sections 5 and 6, we discuss in more detail the phenomeno-logy of a few representative scenarios of the MSISM, without the inclusion of right-handedneutrinos. A detailed analysis of the MSISM augmented by right-handed neutrinos is givenin Section 7. In this section, we investigate the MSISM which realizes a Type I flat direction, i.e. theVEV of the complex singlet field S is zero at the tree level. In detail, we determine theperturbative values of the quartic couplings of the potential and consider their effect onthe scalar mass spectrum. We then further constrain the theoretically allowed parameterspace by applying the experimental limits on the electroweak oblique parameters S , T and U [18] and the LEP2 limit [5]: m H SM > . S = 0, we must have λ (Λ) = 0 to satisfy the tree-levelminimization condition (4.11). Moreover, in the Type-I MSISM, the flat direction lies alongthe φ axis, as given in (4.13), with n φ = 1, so the quantum field φ can be identified withthe pseudo-Goldstone boson h of the anomalously broken scale invariance.20 .1 The U(1) Invariant Limit Assuming that the theory is U(1) symmetric and imposing the constraint λ (Λ) = 0 at agiven RG scale Λ, the tree-level potential (4.2) for the Type-I MSISM reduces to V tree (Λ) = λ (Λ)2 ( S ∗ S ) + λ (Λ) Φ † Φ S ∗ S , (5.1)where λ (Λ) and λ (Λ) should both be positive owing to the BFB conditions (4.6). Eventhough the scalar potential (5.1) depends on the two independent parameters λ (Λ) and λ (Λ), it is not difficult to show that the tree-level scalar masses and the renormalizationscale Λ are fully determined by one single parameter, the quartic coupling λ (Λ). Moreexplicitly, by setting S = 0 and λ (Λ) = 0 in the general squared scalar mass matrix M S given in (B.9), we obtain that the only non-zero elements of M S at φ = v φ are the followingentries: m σ = m J = λ (Λ)2 v φ . (5.2)Hence, the scalar spectrum consists of the mass eigenstates φ ≡ h , σ ≡ H and J ≡ H , where the latter two states are degenerate, with equal masses m H , = m σ = m J ,proportional to p λ (Λ). The first state h corresponds to the pseudo-Goldstone bosonof the anomalously broken scale invariance, which receives its mass m h at the one-looplevel, by means of (4.20). The h -boson mass squared is directly proportional to β , since n φ = 1. Consequently, m h is fully specified by the coupling λ (Λ) through the scalarmasses m H , = m σ = m J . Likewise, the renormalization scale Λ, as was evaluated in(4.21), depends on m H , through the coefficients α and β , and hence its exact value is alsofixed by λ (Λ).From the above discussion, it is now obvious that possible theoretical constraints on λ (Λ) will directly translate into limits on the scalar mass spectrum and the RG scale Λ.An upper theoretical constraint on the value of λ (Λ) originates from the requirementthat the theory remains perturbative at the scale Λ. We may enforce this constraint byrequiring that β λ ≤ , (5.3)where λ denotes a generic coupling of the MSISM, i.e. λ = { λ , ,..., , g ′ , g, g s , h e,u,d } , and β λ is the one-loop RG beta-function for the generic coupling λ . A complete list of all theone-loop beta functions β λ of the MSISM is presented in Appendix C. Assuming λ (Λ) issmall and setting λ (Λ) = 0, we find that the most stringent upper limit on λ (Λ) comesfrom demanding that β λ ≤ µ = Λ. This implies that2 λ (Λ) + 1 . λ (Λ) ≤ π , (5.4)and an upper limit of λ (Λ) ≤ .
84 is deduced, for m W = 80 . m Z = 91 .
19 GeV and m t = 171 . λ (Λ) is non-negligible, the upper limit on λ (Λ) decreases. The21ower theoretical constraint is determined by requiring that the potential remains BFB.This is assured if the coefficient β of the effective potential is positive, thus giving rise toa lower theoretical bound of λ (Λ) > . λ (Λ) can be derived from experimentaldata of direct Higgs searches and the electroweak oblique parameters S , T and U . Analyticresults of the S , T and U parameters in the MSISM are presented in Appendix D. Usingthese results, we may place additional limits on λ (Λ) from experiment. In the U(1)-invariant Type-I MSISM, only the h boson interacts with the photon and the W ± and Z bosons. As a consequence, the shifts, δS , δT and δU , to the electroweak oblique parametersevaluated in the MSISM with respect to the SM will result from the h interactions. Sincethese interactions are identical to those of the SM Higgs boson H SM , the shift parameters δS , δT and δU only depend on the difference between the two masses, m h and m H SM . Assumingthat δS , δT and δU fall within their 95% CL interval for a fixed given SM Higgs-boson masse.g. m H SM = 117 GeV [17], we find that the limits from δS and δT require the respectiveconstraints: λ (Λ) < .
12 and λ (Λ) < .
28, however the prediction for δU lies entirelyinside the range δU exp , even for large values λ (Λ) < m H SM = m h > . h boson, we obtain the constraint: λ (Λ) > .
29, which lies slightly outsidethe perturbative limit of λ (Λ) ≤ .
84. In this context, we note that the highest RG scalefor a Landau pole to appear for λ (Λ) ≈ . µ Landau ∼ GeV, which is obtained for λ (Λ) = 0.In Fig. 1, we display the dependence of the scalar-boson masses m h and m σ,J on thequartic coupling λ (Λ), for which the Type-I flat-direction condition λ (Λ) = 0 is realized.The solid (black) β λ < . < λ (Λ) ≤ .
84. The continuation ofthese lines into dashed (grey) β λ > λ (Λ) > .
84. The area between the horizontal blue LEP line and the horizontalred δS line indicates the combined experimental limit on λ (Λ), i.e. 6 . ≤ λ (Λ) < . δT line is excluded by the δT limit. It isinteresting to remark here that unlike the well-known “chimney plot” [25] which constrainsthe SM Higgs-boson mass to an allowed band by considerations of triviality and vacuumstability [26], Fig. 1 shows an exact value for the physical scalar masses m h,σ,J against thequartic coupling λ (Λ) which is related to the RG scale Λ, see Fig. 2.Fig. 2 shows the dependence of the RG scale Λ on the quartic coupling λ (Λ). Thesame line colour convention as in Fig. 1 is used, only now the horizontal LEP, δS and δT lines are vertical. We observe that as λ (Λ) approaches its minimum value, the coefficient β gets close to zero, and so the RG scale Λ tends to infinity. However, this area is not22
10 10 Λ H L L m h H G e V L LEP ∆ T ∆ S Β Λ < Β Λ >
11 10 10 Λ H L L m Σ , J H G e V L LEP ∆ T ∆ S Β Λ < Β Λ > Figure 1:
Numerical estimates of m h (upper plot) and m σ,J (lower plot) as functions of λ (Λ) in the U(1)-symmetric Type-I MSISM. The solid/black β λ < line shows the per-turbative values of λ (Λ) ≤ . , whilst the dashed/gray β λ > line shows the non-perturbative values of λ (Λ) > . . The area between the horizontal blue LEP line and thehorizontal red δS line is allowed by experimental considerations of the LEP2 mass limit onthe SM-like h boson and the δS parameter respectively. The area above the horizontal red δT line is excluded by the δT parameter constraint.
10 10 Λ H L L L H G e V L LEP ∆ T ∆ S Β Λ < Β Λ > Figure 2:
The RG scale Λ as a function of λ (Λ) in the U(1)-symmetric Type-I MSISM.The solid/black β λ < line shows the perturbative values of λ (Λ) ≤ . , whilst thedashed/gray β λ > line shows the non-perturbative values. The areas lying to the right ofthe red δS and δT lines are excluded, and similarly to the left of the blue LEP line is alsoexcluded by the LEP2 Higgs mass limit. physically viable, as has already been excluded by the LEP limits.If we interpret λ (Λ) ≈ . ≈
294 GeV as the most experimentallyfavourable value of this quartic coupling within the U(1)-invariant Type-I MSISM, we arethen able to offer a sharp prediction for the masses of the heavier degenerate scalar bosons σ and J . Specifically, by virtue of (5.2), we find that m σ,J ≈
437 GeV. The fields σ and J areboth stable and can qualify as DM candidates in the so-called “Higgs-portal” scenario [24].A detailed study of the DM relic abundances of σ and J is beyond the scope of this paperand will be given elsewhere.Since the h -boson couplings to fermions and electroweak gauge bosons have exactlythe SM form, its phenomenological distinction from the SM Higgs boson itself will bedifficult. One possibility would be to look for the presence of large hσ - and hJ -couplingsat the International e + e − Linear Collider (ILC), along the lines studied in [27]. Moreover,even though the trilinear and quadrilinear h self-couplings are absent at the tree level, thelarge hσ - and hJ -couplings can give sizable contributions at the one-loop quantum level.Therefore, precision Higgs experiments at the ILC might be able to distinguish the MSISMfrom the SM.From the analysis given above, it is clear that in spite of being very predictive, theU(1)-invariant Type-I MSISM has a number of weaknesses. This scenario satisfies all24xperimental limits for a large quartic coupling λ ≈ .
3, which is close to the boundary ofnon-perturbative dynamics. Another problematic feature is that it exhibits a Landau poleat energy scales of order 10 GeV, which is many orders of magnitude below the standardGUT ( M GUT ≈ × GeV) and Planck ( M Planck ≈ . × GeV) mass scales. Therefore,in the next section, we relax the constraint of U(1) invariance, and investigate whether ageneral Type-I MSISM can be perturbative up to the GUT and Planck scales.
We now lift the constraint of U(1) invariance from the scalar sector of the Type-I MSISM.The tree-level scalar potential of the general Type-I MSISM then reads: V tree (Λ) = λ (Λ)2 ( S ∗ S ) + λ (Λ) Φ † Φ S ∗ S + λ (Λ) Φ † Φ S + λ ∗ (Λ) Φ † Φ S ∗ + λ (Λ) S S ∗ + λ ∗ (Λ) SS ∗ + λ (Λ)2 S + λ ∗ (Λ)2 S ∗ . (5.5)Exactly as we did for the U(1)-invariant scenario, we can show that the tree-level scalar-boson masses and the RG scale Λ do not depend on all the couplings but only on λ (Λ)and the modulus | λ (Λ) | of the generally complex quartic coupling λ (Λ). In order to showthis, we first notice that by substituting S = 0 and λ (Λ) = 0 into the squared scalar-bosonmass matrix M S given in (B.9), we obtain only three non-zero matrix elements, i.e. m σ = 12 (cid:16) λ (Λ) + λ (Λ) + λ ∗ (Λ) (cid:17) v φ ,m J = 12 (cid:16) λ (Λ) − λ (Λ) − λ ∗ (Λ) (cid:17) v φ ,m σJ = i (cid:16) λ (Λ) − λ ∗ (Λ) (cid:17) v φ . (5.6)If λ (Λ) is complex, the scalar-pseudoscalar mass term, m σJ , gives rise to explicit CPviolation. In this case, the scalar mass spectrum consists of the fields: h ≡ φ , H = cos θ σ + sin θ J , H = − sin θ σ + cos θ J . (5.7)If the theory preserves CP, we have that H = σ and H = J are CP-even and CP-oddscalar fields, respectively. In the general case, however, the mass eigenstates H , haveindefinite CP parities, with their tree-level masses given by m H = 12 (cid:16) λ (Λ) + 2 | λ (Λ) | (cid:17) v φ , m H = 12 (cid:16) λ (Λ) − | λ (Λ) | (cid:17) v φ , (5.8)where cos θ = ( m σ − m H ) / ( m H − m H ). Hence, the scalar-boson masses m H , dependon only two coupling parameters, λ (Λ) and | λ (Λ) | . For the same reason, the two effective25otential coefficients α and β also depend on λ (Λ) and | λ (Λ) | through the scalar-bosonmasses m H , . It is therefore not difficult to see that the one-loop induced h -boson massand the RG scale Λ also depend only on λ (Λ) and | λ (Λ) | , by means of (4.20) and (4.21).The fact that the scalar-boson masses m H , have to be positive leads to the constraint: λ (Λ) ≥ | λ (Λ) | > . (5.9)This constraint automatically enforces the second condition in (4.6) for any value of θ S ,such that the potential remains BFB, i.e. Λ ≥
0, for Λ = 0. The first BFB conditionin (4.6) is only fulfilled, if Λ ≥
0. This restricts the allowed parameter space of the othercouplings, λ , λ and λ . In order for the first BFB condition to hold for any possible valueof the phase θ S , we must require that λ (Λ) ≥ | λ (Λ) | + 2 | λ (Λ) | > . (5.10)As in the U(1)-invariant scenario, we may derive additional theoretical limits on λ (Λ) and | λ (Λ) | , by demanding that the couplings remain perturbative at Λ and that the one-loopeffective potential V − loopeff is BFB. The best theoretical upper limit on λ (Λ) and | λ (Λ) | is obtained by requiring that β λ ≤ λ (Λ), λ (Λ) and λ (Λ) arenegligible. This implies that2 λ (Λ) + 8 | λ (Λ) | + 1 . λ (Λ) ≤ π . (5.11)Correspondingly, a lower theoretical limit may be obtained by requiring that β >
0, whichtranslates into the constraint: λ (Λ) + 4 | λ (Λ) | ≥ . . (5.12)Experimental data encoded as constraints on the electroweak oblique parameters S , T and U provide complementary limits on the quartic couplings λ (Λ) and | λ (Λ) | . Exactlyas in the U(1)-invariant scenario, only the h boson interacts with the SM particles, withcouplings of the SM form. Therefore, as before, useful perturbative constraints on λ and | λ (Λ) | can only be derived from the 95% CL interval of the electroweak oblique parameters δS and δT for m H SM = 117 GeV. Since the h boson has standard interactions, the LEP2lower limit on the SM Higgs boson applies in full, giving rise to the constraint: λ (Λ) + 4 | λ (Λ) | > . . (5.13)In Fig. 3 we present numerical estimates for the scalar-boson masses m h (upper panel), m H (middle panel) and m H (lower panel), as functions of the quartic coupling λ (Λ), afterincorporating all the aforementioned theoretical and experimental limits. The perturbative26
10 10 Λ H L L m h H G e V L LEP ∆ T ∆ S Β Λ < Β Λ > Λ = Λ = Λ Λ H L L m H H G e V L LEP ∆ T ∆ S Β Λ < Β Λ > Β= Λ = Λ = Λ Λ H L L m H H G e V L LEP ∆ T ∆ S Β Λ < Β Λ > Λ = Β= Figure 3:
Numerical estimates of m h (upper panel), m H (middle panel) and m H (lowerpanel) versus λ (Λ) in the general Type-I MSISM. The white area between the black linesshow the regions which correspond to perturbative values of λ (Λ) and | λ (Λ) | and positivescalar masses (5.9), whilst the gray-shaded areas show their non-perturbative regions. Theareas lying to the right of the red lines for δS and δT are excluded. Likewise, the area leftof the blue LEP line is ruled out by the LEP2 Higgs-mass limit.
10 10 Λ H L L L H G e V L LEP ∆ T ∆ S Β Λ < Β Λ > Λ = Λ Λ = Figure 4:
The RG scale Λ as a function of λ (Λ) in the general Type-I MSISM. The whitearea between the black lines shows the region that corresponds to perturbative values of λ (Λ) and | λ (Λ) | , whilst the gray-shaded area shows the non-perturbative region. The areabetween the red δS and blue LEP lines is permitted by the oblique parameters and the LEP2Higgs-mass limit. The area to the right of the red δT line is excluded by the δT limit. areas which also contain positive scalar masses (5.9) are given by the white regions betweenthe black lines, whereas their non-perturbative extrapolations are shaded grey with blackdashed border lines. The LEP2, δT and δS limits are shown as the blue and two red lines,respectively. The areas to the right of the δS and δT lines are excluded by the respective95% CL limits on δS exp and δT exp . Just like the U(1)-invariant scenario, the experimentallypermitted regions lie between the LEP and δS lines, for quartic couplings which are slightlyoutside the boundary of perturbative dynamics. From the middle and lower panels of Fig. 3,we see that the preferred values of the H and H masses which correspond to m h ∼ . < ∼ m H < ∼
519 GeV , ≤ m H < ∼
436 GeV . (5.14)As in the U(1)-invariant scenario, the distinction of the Higgs sector of the general Type-IMSISM from that of the SM might require precision Higgs experiments at the ILC.In Fig. 4 we display the dependence of the RG scale Λ on the quartic coupling λ (Λ)and include both the theoretical and experimental limits, using the same line colour con-vention as in Fig. 3. From Fig. 4, we see that the RG scale Λ is of the electroweak order,lying in the range: 293 GeV < Λ <
359 GeV, for perturbative λ (Λ) couplings, once theLEP2 Higgs-boson mass limit is taken into account.28n the general Type-I MSISM, the H boson is a stable particle in all the allowed rangeof the quartic couplings. Therefore, it can represent a viable cold DM candidate, providedthe H boson is sufficiently massive, e.g. for m H > ∼
30 GeV. If m H is small, then it opensa new decay channel for the SM-like Higgs boson h via h → H and for certain regionsof parameter space can be the dominant mode of decay over h → bb when m h < m W .However, for m h > m W , the h -boson decay into W + W − or ZZ still dominates. In additionto the H boson, the heaviest H boson might also become a stable particle and so a validDM candidate, if its decay via the quartic interaction H H is kinematically forbidden,i.e. as long as m H < m H .The general CP-violating Type-I MSISM shares the same weakness as the U(1)-invariant Type-I MSISM. It turns out that it also generates a Landau pole at a maximumof 10 GeV, far below the GUT and Planck scales. Unlike the U(1)-invariant scenario,the general model contains new sources of CP violation, which might be of particularimportance for realizing electroweak baryogenesis. However, one serious drawback of theType-I MSISM is that it cannot provide a natural implementation of the seesaw mechanism.Since the VEV of the complex singlet scalar vanishes, i.e. S = 0, no Majorana mass termscan be generated in this scenario. We therefore turn our attention in the next section tothe Type-II MSISM, where S = 0. In this section we study the MSISM that realizes a Type II flat direction along whichboth the Higgs doublet Φ and the complex singlet scalar S develop non-zero VEVs. Weinvestigate the Type-II MSISM in two distinct cases: (i) the U(1)-invariant limit and(ii) a U(1) non-invariant scenario where CP is maximally broken spontaneously along theflat direction σ = J . For these two scenarios, we determine the perturbative values of thequartic couplings of the potential and the limits that these set on the scalar mass spectrum.Once these limits are considered, we find that the electroweak oblique parameters S , T and U give no further constraints on the model parameters. On the other hand, as we willsee, the LEP2 Higgs-boson mass limit does produce useful limits on the quartic couplingsand the scalar-boson mass spectrum. Unlike in the Type-I MSISM, the pseudo-Goldstone h boson in the Type-II case is in general a linear composition of all the neutral fields φ , σ and J . As a consequence, it is possible for all the Higgs mass eigenstates h , H or H to couple to the Z boson, but with reduced strength compared to the SM Higgs-bosoncoupling. 29 .1 The U(1) Invariant Limit In the U(1) invariant limit, the Type-II MSISM tree-level potential takes on the simpleform: V tree (Λ) = λ (Λ)2 (Φ † Φ) + λ (Λ)2 ( S ∗ S ) + λ (Λ) Φ † Φ S ∗ S . (6.1)Imposing the minimization conditions (4.11) and (4.12) on the tree-level potential (6.1),one gets a minimal flat direction at a given RG scale Λ, provided the following relationsamong the VEVs of the scalar fields and quartic couplings are simultaneously met: φ σ = n φ n σ = − λ (Λ) λ (Λ) = − λ (Λ) λ (Λ) , (6.2)where we have made use of the U(1) symmetry to set the VEV of the S field real. Hence,the flat direction Φ flat becomes a two-dimensional vector with components φ and σ , givenby (4.15). Moreover, as stated after (4.14), the quartic couplings should lie in the ranges: λ (Λ) > λ (Λ) > λ (Λ) < λ (Λ) and λ (Λ). Instead, the quartic coupling λ (Λ)may eliminated in favour of the relation: λ (Λ) = [ λ (Λ)] /λ (Λ). Consequently, the scalarmasses and the RG scale Λ can be expressed entirely in terms of λ (Λ) and λ (Λ). Takingthe relations (6.2) into account, the scalar mass matrix given in (B.9) has the followingnon-zero entries: m φ = λ (Λ) v φ , m σ = − λ (Λ) v φ , m φσ = − p − λ (Λ) λ (Λ) v φ . (6.3)We note that the U(1)-invariant Type-II MSISM cannot realize CP violation in the Higgssector. Explicitly, the scalar mass spectrum consists of the mass eigenstates h = cos θ φ + sin θ σ , H ≡ H = − sin θ φ + cos θ σ , H ≡ J , (6.4)where cos θ = − λ (Λ) / [ λ (Λ) − λ (Λ)]. The h and H ≡ H bosons are CP even and the J ≡ H boson CP odd. The CP-odd scalar J is the massless Goldstone boson, associatedwith the spontaneous symmetry breaking of the U(1) symmetry. At the tree-level, the onlymassive scalar is the H boson, whose mass squared is given by m H = (cid:2) λ (Λ) − λ (Λ) (cid:3) v φ . (6.5)Since m H depends solely on the combination λ (Λ) − λ (Λ), so do the two effective potentialcoefficients α and β . Likewise, the RG scale Λ also depends on the combination λ (Λ) − λ (Λ), through (4.21). However, the one-loop contribution to m h , given in (4.20), dependson λ (Λ) as well, through the flat direction component n φ = cos θ , given in (4.15). Thus,30 - - - -
40 1 2 3 4 Λ H L L Λ H L L LEP Β = Β Λ = Β Λ = - - - - Λ H L L Λ H L L LEP Β= Β Λ = Figure 5:
Theoretical and experimental exclusion contours in the λ (Λ) - λ (Λ) parameterspace in the U(1)-invariant Type-II MSISM. The upper panel shows the full perturbativeparameter space, whilst the lower panel focuses on the region with small λ (Λ) . The theo-retically allowed areas are enclosed by the black lines which correspond to keeping β λ , ≤ , β > and λ (Λ) ≤ . The LEP2 limit is given by the blue (grey) LEP line and above(below) is excluded for the upper (lower) panel. The blue and grey shaded areas are allowedby the theoretical constraints, the LEP2 Higgs-mass limit and the oblique parameters. Theregion of parameter space which remains perturbative to GUT (Planck) scale is enclosed bythe solid (dashed) green Pert lines. λ (Λ) − λ (Λ) and λ (Λ),or equivalently on λ (Λ) and λ (Λ).The full theoretical and experimental limits on the two quartic couplings λ (Λ) and λ (Λ) are displayed in Fig. 5. The top panel displays the full range, whilst the lowerpanel focuses on a very narrow region, which is viable for very small values of λ (Λ). Astheoretical constraints, we require that the model remains perturbative at the RG scale Λ,i.e. β λ , (Λ) ≤
1, which is represented by the black β λ , = 1 lines in Fig. 5. From theseconsiderations, we find the upper limits λ (Λ) < .
45 and λ (Λ) > − .
29. Another usefultheoretical constraint is obtained by requiring that the one-loop effective potential remainsBFB ( β > λ (Λ) − λ (Λ) > . , (6.6)which is indicated by the black β = 0 lines in Fig. 5. Thus, the theoretically admissibleregion is the one enclosed by the β λ , = 1, β = 0 and λ (Λ) = 0 lines in the upper paneland by the β = 0, β λ = 1 and λ (Λ) = 0 lines in the lower panel.The λ (Λ)- λ (Λ) parameter space may be further constrained by experimental LEP2limits on the Higgs-boson mass and by the electroweak oblique parameters S , T and U . Wefind that the 95% CL limits on S , T and U parameters provide no additional constraints onthe theoretically admissible region. Instead, the LEP2 Higgs-boson mass limits significantlyrestrict the λ (Λ)- λ (Λ) parameter space. To properly derive these limits, we first observethat the pseudo-Goldstone boson h and the heavy H -boson interact with reduced couplings g hV V and g HV V with respect to the SM coupling of H SM to a pair of vector bosons V = W ± , Z . The squared reduced couplings g hV V and g HV V are given by g hV V = cos θ = − λ (Λ) λ (Λ) − λ (Λ) , g HV V = sin θ = λ (Λ) λ (Λ) − λ (Λ) , (6.7)satisfying the identity: g hV V + g HV V = 1. Since the reduced hZZ -coupling can be muchsmaller than the SM one, the SM Higgs-boson mass limit m h > . ξ h ≡ g hV V and the scalar mass m h ,which are presented in Fig. 10(a) of Ref. [5]. We perform a polynomial fit up to order 10on the LEP2 data to obtain a reliable constraint on ξ h ( m h ), which in turn restricts the λ (Λ)- λ (Λ) parameter space. This constraint is represented by the blue (grey) LEP linein the upper (lower) panel of Fig. 5, where the blue (grey) shaded region respect both thetheoretical constraints and the LEP2 Higgs-mass limit. As there are two distinct shadedregions blue and grey, which correspond respectively to higher and lower values of m h , weshall consider each scenario separately. 32 .1.1 The Electroweak Mass h -Boson Scenario We first consider the higher mass h -boson scenario represented by the shaded blue areain Fig. 5, which is dominated by large values of λ (Λ) and λ (Λ). In Fig 6 we show thedependence of the scalar boson masses m h and m H on the quartic coupling λ (Λ). Theareas enclosed by the black lines are the regions which respect the theoretical constraintsi.e. β λ , ≤ β > λ (Λ) ≤
0. Including the LEP2 Higgs-mass limit, we obtain theshaded blue areas, corresponding to an electroweak mass h boson, with mass in the range:111 . < m h ≤ . H boson, with mass in the interval:593 GeV < m H ≤
627 GeV. In addition, the gray shaded areas correspond to a very light h boson, which is not clearly visible on the lower frame of Fig. 6, as it follows the λ = 0 line.This ultra-light h -boson mass scenario will be discussed in more detail in Subsection 6.1.2.The electroweak mass h boson could be detected at the CERN Large Hadron Collider(LHC), through the decay channel h → γγ . The observation of the H boson may proceedvia the so-called “golden channel,” H → ZZ → l . However, in the region λ (Λ) ≈ − λ (Λ),we have g hV V ≈ g HV V ≈ .
5, on account of (6.7), which means that both decays will givereduced signals compared to the SM Higgs signals. Moreover, the heavier H boson maypredominantly decay invisibly into a pair of U(1) Goldstone bosons J [28], thanks to therelatively large quartic couplings. This last characteristic makes the U(1) Type-II MSISMdistinguishable from the corresponding Type-I one.In Fig. 7 we present the dependence of the RG scale Λ as a function of the quartic cou-pling λ (Λ). The area within the black lines respects the theoretical constraints, β λ , ≤ β > λ (Λ) ≤
0. The areas which also respectthe LEP2 limit are shaded blue, which correspond to the electroweak h -boson scenario, andgrey, which correspond to a scenario with a very light h boson. The latter region is verynarrow and not clearly visible in the figure, since it very closely follows the λ = 0 line.If λ (Λ) and λ (Λ) are in the blue shaded region and remain perturbative, then the RGscale Λ is of the order of the EW scale and lies in the range 390 . ≤ Λ < . × GeV. h -Boson Scenario Another experimentally and theoretically viable region of the λ (Λ)- λ (Λ) parameter spacecorresponds to a very small quartic coupling λ (Λ), giving rise to an ultra-light h boson.The relevant region is shaded grey in the lower panel of Fig. 5. We will not present adetailed phenomenological analysis of this scenario, but rather highlight its key features.33 Λ H L L m h H G e V L LEPLEP Β Λ = Β Λ =
10 1 2 3 4200300400500600700 Λ H L L m H H G e V L LEPLEP Β Λ = Β Λ = Λ = Β= Figure 6:
Predicted numerical values of m h (upper panel) and m H (lower panel) as afunction of λ (Λ) in the U(1)-symmetric Type-II MSISM. The areas within the black linesshow the regions which respect the theoretical constraints i.e. keeping β λ , ≤ , the potentialBFB and λ (Λ) ≤ . The blue (electroweak m h ) and grey (ultra-light m h ) shaded regions(denoted LEP) are permitted by the LEP2 Higgs-mass limit and the theoretical constraints. Λ H L L L H G e V L LEPLEP Β Λ = Β Λ = Λ = Figure 7:
Predicted numerical values of Λ as a function of λ (Λ) in the U(1)-invariant Type-II MSISM. The areas within the black lines show the regions which respect the theoreticalconstraints i.e. keeping β λ , ≤ , the potential BFB and λ (Λ) ≤ . The blue (electroweak m h ) and grey (ultra-light m h ) shaded regions (denoted LEP) are permitted by the LEP2Higgs-mass limit and the theoretical constraints. As can be seen from Fig. 5, the LEP2 Higgs-mass limit puts an upper bound on − λ (Λ) < ∼ . h -boson mass is m h < ∼ . h -boson scenario, thereduced hZZ -coupling is rather suppressed, with g hV V ≤ . h boson difficult to detect at the LHC.The other CP-even H boson has almost a SM-like coupling to the vector bosons, with g hV V ≈
1. Its mass may range for perturbative values of λ (Λ), between 315 GeV < m H <
458 GeV. This range is given by the λ (Λ) = 0 line in Fig. 6. Since the HJ J -coupling isproportional to the small λ (Λ) coupling it is suppressed and the SM-like H boson wouldmost likely be detected via the “golden channel,” H → ZZ → l .An interesting feature of the ultra-light h -boson scenario is the existence of a regionthat remains perturbative to higher scales than the previously considered models. This isindicated by the area enclosed by the solid and dashed green lines in Fig. 5. Specifically,within the allowed region, the model becomes non-perturbative at energies of order 10 GeVand develops a Landau pole at energies 10 GeV, which is higher than the electroweakmass h -boson scenario. In conclusion, it worth reiterating that the U(1)-invariant Type-IIMSISM has no new source of CP violation beyond the standard Kobayashi–Maskawa (KM)35hase [29] and predicts no massive DM candidate (cf. Table 1). In spite of these drawbacks,the model does have the ability to generate Majorana neutrino masses through the seesawmechanism, as we will discuss in more detail in Section 7. In the following section, weconsider a minimal U(1)-violating Type-II MSISM which realizes maximal spontaneousCP violation (SCPV). Without the restriction of U(1) invariance, the tree-level scalar potential (4.2) of the generalType-II MSISM contains a total of 9 real quartic couplings which results in a multitude ofvalid solutions that all satisfy the minimization requirements (4.11) and (4.12). However,not all of these possible cases are phenomenologically interesting. Therefore, we havefocused our investigation on a single U(1) non-invariant scenario that minimally realizesmaximal spontaneous CP violation, i.e. it has a flat direction along the σ = J field line.The tree-level scalar potential of such a scenario is given by V tree = λ † Φ) + λ S ∗ S ) + λ Φ † Φ S ∗ S + λ S + S ∗ ) , (6.8)where λ = λ = 0 and λ is real as a consequence of CP invariance. In addition to CPsymmetry, the tree-level scalar potential (6.8) is invariant under the Z discrete symmetry: S → S ′ = ωS and Φ → Φ ′ = Φ, with ω = 1. The CP symmetry and the Z discretesymmetry are sufficient to uniquely fix the form of the tree-level scalar potential V tree given in (6.8).Minimizing the tree-level potential at the RG scale Λ, by means of (4.11) and (4.12),we find the following relations for the flat direction: φ σ = n φ n σ = − λ (Λ) λ (Λ) = − (cid:2) λ (Λ) − λ (Λ) (cid:3) λ (Λ) , σ = J , n σ = n J . (6.9)Note that the second (or third) condition implies a flat direction that triggers maximalspontaneous CP violation with θ S = π/
2. Combining (6.9) with the BFB condition (4.6)requires that λ (Λ) > λ (Λ) < λ (Λ) − λ (Λ) >
0, where the signs of λ (Λ) and λ (Λ) individually remain undetermined. Another choice of a maximally CP-violating flatdirection would be to have θ S = 3 π/
4, i.e. σ = − J . However, such a choice does not affectthe scalar masses or the phenomenology of the model in an essential manner.Other solutions to the minimization conditions, (4.11) and (4.12), are possible butthey either reduce the potential to the U(1) invariant scenario ( λ (Λ) = 0) or modify it toa Type I flat direction ( σ = J = 0), both of which have been previously investigated inSections 6.1 and 5, respectively. 36he flat direction for this model can be expressed as a 3-dimensional vector, withnon-zero φ , σ and J components, i.e. Φ flat = ϕ q (cid:2) λ (Λ) − λ (Λ) (cid:3) p − λ (Λ) p λ (Λ) p λ (Λ) = φ p − λ (Λ) p − λ (Λ) p λ (Λ) p λ (Λ) . (6.10)Considering the relations (6.9), the scalar mass matrix elements in (B.10) become m φ = λ (Λ) v φ , m σ = m J = (cid:2) λ (Λ) + 2 λ (Λ) (cid:3) v σ ,m σJ = (cid:2) λ (Λ) − λ (Λ) (cid:3) v σ , m φσ = m φJ = λ (Λ) v φ v σ . (6.11)Note that the elements m φJ and m σJ are CP-violating. In terms of the quantum fields φ , σ and J , the mass eigenstates h , H and H are given by h = r − λ λ − λ φ + s λ λ − λ ) ( σ + J ) ,H = r λ λ − λ φ − s − λ λ − λ ) ( σ + J ) ,H = 1 √ − σ + J ) . (6.12)Correspondingly, their tree-level masses squared are given by m h = 0 , m H = (cid:2) λ (Λ) − λ (Λ) (cid:3) v φ , m H = 4 λ (Λ) λ (Λ) − λ (Λ) v φ , (6.13)where we employed the relation λ (Λ) = [ λ (Λ) /λ (Λ)] + 2 λ (Λ), as can easily be derivedfrom (6.9). In order for the H -boson mass squared m H to be positive, we require that λ (Λ) >
0, implying λ (Λ) > λ (Λ), λ (Λ) and λ (Λ). The quartic coupling λ (Λ) can be eliminatedin favour of λ (Λ), by means of (6.9). Explicitly, the scalar masses m H and m H depend onthe three quartic couplings λ , , (Λ), as can be seen from (6.13). Likewise, the RG scale Λdetermined in (4.21) depends on the effective potential coefficients α and β that are bothfunctions of m H and m H . Finally, according to (4.20), the pseudo-Goldstone h boson de-pends on β and n φ . Nevertheless, from (6.10), we see that n φ = p − λ (Λ) / [ λ (Λ) − λ (Λ)].Consequently, the entire scalar-boson mass spectrum of the model only depends on the threequartic couplings λ , , (Λ).We may now exploit the extra freedom of the three independent quartic couplingsto identify theoretically and experimentally viable regions of the parameter space which37 (Λ) m h m H m H Λmin max min max min max min max0.2 54 78 155 181 783 1110 490 6750.1 34 56 155 180 703 1110 444 6740.05 21 39 154 179 607 1110 395 6740.02 11 25 154 178 515 1110 350 675Table 2:
Minimum and maximum values of m h , m H , m H and Λ as determined by theLEP2 Higgs-mass limit and the theoretical constraint α ≤ for a range of λ (Λ) . remain perturbatively renormalizable up to Planck-mass energy scales. To be precise, werequire that β λ , , , ( M Planck ) ≤ λ ( M Planck ) > λ ( M Planck ) − λ ( M Planck ) > λ ( M Planck ) <
0. Moreover, ifwe assume λ (Λ) ≪ λ (Λ), such that λ (Λ) ≈ λ (Λ) we find that the quartic couplings λ , (Λ) are restricted to the intervals,0 . < ∼ λ (Λ) < ∼ . , < λ (Λ) < ∼ . , (6.14)for − . < ∼ λ (Λ) <
0. From (6.13), we observe that in the limit λ (Λ) →
0, the H -bosonmass m H becomes infinite. Therefore, to obtain an upper limit on λ (Λ), we require thatthe coefficients α and β of the one-loop effective V − loopeff in (4.17) are small, e.g. α, β ≤ α ≤
1, which is comparable to the constraint β ≤
1. Fordefiniteness, we choose two representative values of λ (Λ): λ (Λ) = 0 .
02 and λ (Λ) = 0 . m h , m H , andthe RG scale Λ, as functions of the the quartic coupling λ (Λ). The solid and dashedblack line enclose the regions permitted by considering the theoretical bounds which mosttightly constrain the values of λ (Λ) namely, β λ ( M Planck ) ≤ λ ( M Planck ) ≥ β >
0, for λ (Λ) = 0 . h -boson mass m h for λ (Λ) = 0 . λ (Λ) considered. As a result, the λ (Λ) coupling has to take small absolutevalues, with λ (Λ) > ∼ − .
02. The solid (dashed) red lines represent the theoretical limit α ≤ λ (Λ) = 0 . . α = 1 lines is excluded. The greyshaded regions are the areas which respect all the theoretical constraints and the LEP2limit. Finally, the electroweak oblique parameters offer no useful constraints, within thetheoretically allowed parameter space. In Table 2, we present the upper and lower limits onthe masses of the h and H bosons and on the RG scale Λ for different values of λ (Λ). The38 - - - - - Λ H L L m h H G e V L Λ = Λ = Α= Α= - - - - - Λ H L L m H H G e V L Λ = Λ = Α= Α= - - - - - Λ H L L L H G e V L Λ = Λ = Α= Α= Figure 8:
Numerical estimates of m h (top panel), m H (middle panel) and the RG scale Λ (lower panel) as functions of λ (Λ) in a Type-II MSISM of maximal SCPV. The areasbetween the solid and dashed black lines correspond to the masses, for which β λ ( M Planck ) ≤ , λ ( M Planck ) ≥ and β > with λ (Λ) = 0 . and . respectively. The solid and dashedblue lines represent the LEP2 Higgs-mass limit below which are excluded. The solid anddashed red lines represent the constraint α ≤ and above each of the lines is excluded. Thegrey regions correspond to areas that respect the theoretical and LEP2 limits. The solidlines correspond to λ (Λ) = 0 . whilst the dashed lines correspond to λ (Λ) = 0 . . α ≤ H -boson mass m H as a function ofthe λ (Λ) coupling. The black lines correspond to values of the quartic couplings whichrespect the limits λ ( M Planck ) > β λ ( M Planck ) < β >
0. Even though the H -boson mass m H evaluated in (6.13) does not explicitly depend on λ (Λ), the LEP2 limitapplied to m h and the theoretical constraint α ≤ α = 1 lines are allowedby the respective constraints for λ (Λ) = 0 . m H , whilst the α = 1 constraint gives a lower limit. These upper andlower limits on the H -boson mass are exhibited in Table 2, for various values of the λ (Λ)coupling.In spite of the additional quartic coupling λ (Λ), the interactions of the h and H scalars to a pair of V = W ± , Z bosons are very similar to the U(1)-invariant scenario. Thereduced hV V - and H V V -couplings are given by g hV V = − λ (Λ) λ (Λ) − λ (Λ) , g H V V = λ (Λ) λ (Λ) − λ (Λ) . (6.15)In the Type-II MSISM of maximal SCPV under study, the h boson has a large componentfrom the heavy H scalar and so it can generically be heavier than the respective h in theU(1)-invariant model, this allows it to comfortably evade detection at the LEP2. On theother hand, the H boson has a SM-like coupling to the electroweak vector bosons and wouldagain most likely be detected through the standard discovery channel H → ZZ → l . Inaddition to the standard discovery channel, H → ZZ → l , the H boson may now decayfavourably to a pair of h bosons, i.e. H → hh , if kinematically allowed. Then, each ofthe h bosons may decay into a pair of τ leptons or b quarks. A detailed phenomenologicalstudy of this detection channel for the LHC is beyond the scope of this paper.The minimal Type-II MSISM of maximal SCPV gives rise to rich phenomenology. Asmentioned previously, the model spontaneously and maximally violates the CP symmetry.Since the complex singlet S has a non-zero VEV, the model can also generate naturally smallneutrino masses through the seesaw mechanism. Moreover, the presence of a permutationparity symmetry, σ ↔ J , which remains intact after EWSSB, renders the massive H boson stable, with vanishing VEV. Hence, the H boson could act as a cold DM, accordingto the Higgs-portal scenario [24]. In general, there are two parity symmetries that couldbe imposed on a general Type-II MSISM with SCPV, they are: σ ↔ J and σ ↔ − J .Both symmetries lead to similar mass spectra, so we do not discuss them separately. Also,both symmetries trigger maximal SCPV, which might open up the possibility for successfulelectroweak baryogenesis in this scenario. 40 - - - - - Λ H L L m H H G e V L Λ = Β Λ = Λ = Α= - - - - - Λ H L L m H H G e V L Λ = Β= Β Λ = Λ = Α= Figure 9:
Predicted numerical values of m H as a function of λ (Λ) for λ (Λ) = 0 . (upperpanel) and λ (Λ) = 0 . (lower panel), in a Type-II MSISM of maximal SCPV. The blacklines corresponds to masses restricted by the conditions: β λ ( M Planck ) ≤ , λ ( M Planck ) ≥ and β > (lower panel only). The grey shaded regions correspond to the areas permittedby the LEP2 limit (solid and dashed blue lines) and the α ≤ limit (solid and dashed redlines). In summary, the Type-II MSISM of maximal SCPV is a theoretically and experimen-tally viable scenario. The quartic couplings of the model can remain perturbative up toPlanck energy scales and its scalar-boson spectrum is compatible with limits from LEP2Higgs searches and the S , T and U oblique parameters. Most importantly, the model doesnot require additional theory to stay perturbatively renormalizable up to the standard41uantum gravity scale, i.e. M Planck . Since the addition of right-handed neutrinos can havea significant impact on the one-loop effective potential V − loopeff and on the phenomenologyof the model in general, we analyze in detail such a scenario in the next section. In order to account for the observed non-zero neutrino masses, we extend the MSISM withthree right-handed neutrinos, ν , , R . As was already mentioned in Section 4.4, the Type-IMSISM cannot realize the seesaw mechanism since the VEV of the S field is zero alongthe minimal flat direction. The only way of introducing neutrino masses in a SI fashioninto the Lagrangian is through the hugely suppressed neutrino Yukawa couplings of order10 − , which are about 6 orders of magnitude smaller than the electron Yukawa coupling.Obviously, such a scenario has the difficulty of naturally explaining the smallness of thelight neutrino masses. Moreover, the Type-I MSISM with right-handed neutrinos is a highlyuninteresting scenario as the actual effect of the very small neutrino Yukawa couplings onthe scalar potential is negligible.We therefore turn our attention to the Type-II MSISM. The Lagrangian term L ν in (4.1), which describes the dynamics of the right-handed neutrinos, is given by L ν = ¯ ν iR iγ µ ∂ µ ν iR − h νij ¯ L iL ˜Φ ν jR − h ν † ij ¯ ν iR ˜Φ † L jL − h Nij ¯ ν CiR Sν jR − h N † ij ¯ ν iR S ∗ ν CjR − ˜h Nij ¯ ν iR Sν CjR − ˜h N † ij ¯ ν CiR S ∗ ν jR . (7.1)where the usual summation convention over repeated indices is implied, with i, j = 1 , , e , µ and τ , respectively. In (7.1), h νij are the Dirac-neutrinoYukawa couplings of the SM Higgs doublet Φ to the lepton doublets L iL , as defined inAppendix A. In addition, h Nij and ˜h Nij are the two possible Majorana-neutrino Yukawacouplings of the singlet field S to the right-handed neutrinos ν , , R . Note that h N and ˜h N are symmetric 3 × h N = h N T , ˜h N = ˜h N T . Since the Majorana-neutrinoYukawa couplings h Nij and ˜h Nij can be sizeable, we need to calculate their effect on the flat-directions and the one-loop β functions. Technical details of such calculations are given inAppendices B and C.Since S = 0 along the Type-II flat direction, the following neutrino mass terms aregenerated: L Mass ν = −
12 (¯ ν iL , ¯ ν CiR ) Dij m TDij m Mij ! ν CjL ν jR ! + H . c . (7.2)42ith m D = φ √ h ν , m M = 1 √ h σ ( h N + ˜h N † ) + iJ ( h N − ˜h N † ) i . (7.3)Without loss of generality, we can assume a weak basis, in which m M is diagonal, real andpositive, whilst h N , ˜h N and m D are in general 3 × × L Mass ν can be block-diagonalized via a unitary matrix U asfollows: U T D m TD m M ! U = m ν
00 m N ! . (7.4)To leading order in an expansion in powers of m D m − M , we obtain the standard seesawformulae: m ν = − m D m − M m TD , m N = m M . (7.5)where m ν is a 3 × ν , , and m N is the heavy neutrino mass matrix, predicting new heavy Majorananeutrinos, which we denote hereafter as N , , .As we will see in this section, the heavy Majorana neutrinos N , , in the Type-II MSISM are typically not much heavier than the EW scale. In the standard seesawframework [22], all Dirac-neutrino Yukawa couplings h νij have to be less than about 10 − ,e.g. of order the electron Yukawa coupling. However, the possible presence of approximateflavour symmetries in m D and/or m M [31–33] are sufficient to relax this constraint forsome of the Dirac-neutrino Yukawa couplings h νij and render them sizeable of order 10 − –1 [34, 35]. Even though we keep the analytic dependence of our results on h ν , we assumethat all h νij < ∼ .
01, such that their numerical impact on the one-loop effective potential andthe electroweak oblique parameters can be safely ignored.In the following, we study several representative scenarios within the framework of theType-II MSISM with right-handed neutrinos. First, we consider a U(1)-symmetric theorythat preserves the lepton number. We then consider a benchmark scenario of Type-IIMSISM with maximal SCPV and analyze two variants of such a scenario. The first variantassumes a CP-symmetric neutrino Yukawa sector, where the CP invariance is only violatedspontaneously by the ground state of the theory. The second variant promotes a paritysymmetry present in the scalar potential of the model to the neutrino Yukawa sector, thusgiving rise to a massive stable scalar particle. This stable scalar particle could act as apotential candidate to solve the cold DM problem.
We now consider the effect of including right-handed neutrinos in the the U(1)-invariantType-II MSISM. The imposition of U(1) symmetry on the neutrino Yukawa sector is equiv-43lent to lepton-number conservation, where the right-handed neutrinos ν , , R carry thelepton number +1 and the singlet field S the lepton number −
2. As a consequence oflepton-number conservation, the Majorana Yukawa coupling ˜h N vanishes and the heavy-neutrino mass matrix along the Type-II flat direction is given by m N = σ √ h N , (7.6)where we have set J = 0 by virtue of a U(1) rotation.With the aid of (6.2), we may now express the light- and heavy neutrino mass matrices, m ν and m N , in terms of the SM VEV v φ : m ν = − s − λ (Λ)2 λ (Λ) v φ h ν ( h N ) − h νT , m N = s λ (Λ) − λ (Λ) v φ h N . (7.7)where h N is a real and diagonal matrix. For simplicity, we assume that three heavyMajorana neutrinos N , , are nearly degenerate, specifically by assuming that h N = h N is SO(3) symmetric. The perturbativity constraint on the Yukawa couplings h N may betranslated into the inequality, Tr (cid:0) β † h N β h N (cid:1) ≤
3, at the RG scale Λ. This constraint leadsto the upper bound, h N (Λ) < .
0, for a perturbative theory up to the EW scale. If weinsist that the Majorana Yukawa couplings h Nij stay perturbative up to the Planck scale,we find the tighter upper limit: h N (Λ) ≤ .
89. Finally, the condition that the one-loopscalar potential be BFB, i.e. β >
0, along with the perturbativity conditions, β λ , , ≤ h N , h N (Λ) < .
5, at the EW scale.Fig. 10 shows the allowed parameter space of the h -boson mass and the Majorana-neutrino Yukawa coupling h N , compatible with the LEP2 Higgs-mass limit. The maximumperturbative value is represented by the black m max h line, such that the area between theblack line and the m h = 0 line corresponds to perturbative masses. The maximum perturba-tive value for m h depends on the perturbatively allowed values for λ (Λ), λ (Λ) and h N , i.e β λ , (Λ) ≤ β h N (Λ) ≤
1. Since right-handed neutrinos induce a negative contributionto the coefficient β defined in (4.18) and so to m h in (4.20), m max h decreases as the right-handed neutrino Yukawa coupling h N increases. In Fig. 10, the areas which are permittedby the LEP2 Higgs-mass limit are shaded blue and grey, for the electroweak mass andthe ultra-light h -boson scenarios, discussed in Subsections 6.1.1 and 6.1.2, respectively. Inthe electroweak mass h -boson scenario, where λ (Λ) ≈ −
3, the Majorana-neutrino Yukawacoupling h N is restricted to be: h N < .
40. Instead, for the ultra-light h boson scenario(with λ (Λ) ≈ − . h N < . h N on the RG scale Λ is not significant, as wefind Λ ≈
464 GeV for h N max = 1 .
40. We also verified that all values of λ (Λ) and λ (Λ)which respect β λ , (Λ) ≤ δS exp , δT exp and δU exp , usingthe limits for m H SM = 117 GeV. 44 h N H L L m h H G e V L LEPLEP m h max Figure 10:
Predicted numerical values of the LEP2 Higgs-mass limit allowed range of m h asa function of h N in the Type-II U(1)-invariant MSISM with right-handed neutrinos. Theblue and grey shaded areas correspond to those regions allowed by the LEP2 limit, for theelectroweak and ultra-light h -boson scenarios, respectively. The black m max h line representsthe maximum perturbatively attainable values of m h . In Fig. 11 we display the allowed parameter space of m N and h N , for all perturbativevalues of λ (Λ) and λ (Λ), under the constraint: | β λ , | <
1. The allowed space is given bythe area enclosed by the two black β λ , = 1 lines. The blue and grey shaded areas indicatethe parameter space which is allowed by the LEP2 Higgs-boson mass limit. As can be seenfrom Fig. 11, the resulting allowed areas set upper limits on the heavy Majorana neutrinomasses, m N <
244 GeV and m N <
274 GeV, for the electroweak and the ultra-light h -boson scenarios, respectively. Depending on the strength of the neutrino Yukawa couplings h νij , such heavy Majorana neutrinos can be produced at the LHC [36], leading to like-signdilepton signatures without missing energy.As was discussed in Section 6.1, the U(1)-invariant Type-II MSISM predicts no mas-sive stable scalar particle that could play the role of the cold DM. In fact, the presenceof the Majorana neutrinos, ν , , and N , , , leads to new decay channels for the scalarparticles h and H , such as h → ( ν i N j , N i N j ) and H → ( ν i N j , N i N j ) [32]. Moreover, theinclusion of right-handed neutrinos does not change the UV behaviour of the model whichbecomes non-perturbative and develop a Landau pole far below M GUT and M Planck . Forthis reason, we turn our attention to the Type-II MSISM of maximal SCPV, which doesnot exhibit this weakness. 45 h N m N H G e V L LEPLEP Β Λ = Β Λ = Figure 11:
Perturbatively allowed values of m N against h N in the Type-II U(1) symmet-ric MSISM with right-handed neutrinos. The perturbatively allowed parameter space of ( h N , m N ) is given by the area between the black β λ , = 1 lines. The internal blue andgrey shaded areas represents the regions allowed by the LEP2 Higgs-mass limit, for theelectroweak and ultra-light h -boson scenarios, respectively. We now consider an extension of the Type-II MSISM presented in Section 6.2, by addingright-handed neutrinos. The Type-II flat direction of this scenario is given by σ = J ,which leads to maximal SCPV in the one-loop scalar potential. Along this flat direction,the heavy Majorana neutrino mass matrix m M takes on the form: m M = σ √ h (1 + i ) h N + (1 − i ) ˜h N † i . (7.8)Since the Majorana Yukawa couplings, h N and ˜h N , may contain large number of indepen-dent parameters, we will investigate two simple variants of the model. In the first variant,we assume that both h N and ˜h N are real, i.e. there is no sources of explicit CP violation inneutrino Yukawa sector. The second variant makes use of a parity symmetry, which givesrise to a massive stable scalar particle that could qualify as DM. In the CP symmetric limit of the theory, the Yukawa couplings h Nij and ˜h Nij are all real.In the weak basis, where m M is real and diagonal, one then gets the constraint: h N = ˜h N .46mplementing this last constraint along the Type-II flat direction σ = J , the neutrino massmatrices read: m ν = − s − λ (Λ) λ (Λ) v φ h ν ( h N ) − h νT , m N = s λ (Λ) − λ (Λ) v φ h N . (7.9)Assuming a universal scenario with three degenerate heavy neutrinos, with h N = h N ,the coupling parameter h N has to be less than 2 . h N ≤ .
52 and h N ≤ .
47, respectively.This model depends on four independent theoretical parameters, namely λ (Λ), λ (Λ)(or λ (Λ)), λ (Λ) and h N . As particular viable benchmark models, we consider the followingthree cases: Case A : λ (Λ) = 0 . , λ (Λ) = − . , Case B : λ (Λ) = 0 . , λ (Λ) = − . , Case C : λ (Λ) = 0 . , λ (Λ) = − . . (7.10)In Fig. 12 we present the allowed parameter space in the h N - m h plane, for the Cases A,B and C given in (7.10). The area between the black lines is allowed by the considerations: β λ ( M Planck ) < λ ( M Planck ) > λ ( M Planck ) − λ ( M Planck ) > β > α = 1 line isexcluded, because it violates perturbative unitarity in the MSISM Higgs sector [30]. ForCase A and C, the α = 1 line is above the allowed region and has not been displayed. Wefind that, within the theoretically allowed areas, the predictions for the electroweak obliqueparameters S , T and U fall within the 95% CL intervals for the three scenarios considered.The region below the grey dashed line is excluded by the LEP-2 Higgs-mass limit applied tothe h -boson mass m h . As a consequence, the grey shaded areas correspond to the regionswhich are allowed by our theoretical considerations and the LEP2 and oblique paameters.The presence of the right-handed neutrinos does not greatly affect m h , except when h N approaches its maximum allowed value which reduces the prediction for m h , as shown inFig. 12. The other scalar masses, m H , , are not affected by the inclusion of neutrinos, sincethey are independent of h N at the tree level.Fig. 13 displays the allowed parameter space spanned by the Majorana-neutrinoYukawa coupling h N and the universal right-handed neutrino mass m N for the three bench-mark scenarios listed in (7.10). As before, the area between the black lines is permitted bythe considerations: β λ ( M Planck ) < λ ( M Planck ) > λ ( M Planck ) − λ ( M Planck ) > β > α = 1 line violates47 h N H L L m h H G e V L LEP Β Λ = Λ = h N H L L m h H G e V L LEP Β Λ = Λ = Λ - Λ = Α=
10 0.1 0.2 0.3020406080 h N H L L m h H G e V L LEP Β Λ = Λ = Figure 12:
Numerical estimates of m h as a function of h N (Λ) in the minimal Type-IIMSISM with maximal SCPV and massive Majorana neutrinos for Cases A, B and C definedin (7.10). The area between the black lines show the regions which correspond to imposing β λ ( M Planck ) < , λ ( M Planck ) > and λ ( M Planck ) − λ ( M Planck ) > in Case B or β > in Cases A and C. The area above the red α = 1 line is excluded. The area below the greydashed LEP line is excluded by LEP2 Higgs-mass limit. The grey shaded areas correspondto the regions allowed by theory and experiment. h N H L L m N H G e V L LEP Β Λ = Λ = Β = Case A0 0.1 0.2 0.30200400600800 h N H L L m N H G e V L LEP Β Λ = Λ = Λ - Λ = Α= h N H L L m N H G e V L LEP Β Λ = Λ = Β= Figure 13:
Numerical estimates of m N as a function of h N (Λ) in the minimal Type-IIMSISM with maximal SCPV and massive Majorana neutrinos for Cases A, B and C definedin (7.10). The area between the black lines show the regions corresponding to the constraints: β λ ( M Planck ) < , λ ( M Planck ) > and λ ( M Planck ) − λ ( M Planck ) > in Case B or β > in Cases A and C. The region above the red α = 1 line is excluded. The area below the greydashed LEP line is excluded by LEP2 Higgs-mass limit. The grey shaded areas correspondto the regions allowed by both theory and experiment. m h . The grey shadedregion is permitted by theory and the LEP2 limit. Comparing the three cases, we observethat if λ (Λ) decreases or λ (Λ) increases, both the upper limits on m N and h N increase.From Fig. 8, we see that if λ (Λ) increases λ (Λ) also needs to increase to remain withinthe theoretical and LEP2 limits and so the two effects cancel and we assume the maximalvalues of m N and h N do not vary significantly from the values given in Case B. Within thisbenchmark scenario, we can then derive approximate upper limits on the values of m N and h N . Thus, from the middle panel of Fig. 13, we observe that the heavy Majorana neutri-nos can generically have masses up to TeV scale, i.e. m N < ∼ h N must remainrelatively small in order for the one-loop effective potential to be BFB, i.e. h N < ∼ . H boson, is no longer stable, since it can decay to ν i N ∗ j , where N ∗ j is an off-shellheavy Majorana neutrino, which can subsequently decay into off-shell W ± and Z bosonsand charged leptons and light neutrinos. The decay of the H boson is a consequence of theviolation of the parity symmetry, σ ↔ J , in the Majorana-neutrino Yukawa sector. In thefollowing, we consider a minimal Type-II MSISM, where the parity symmetry is elevatedto an exact global symmetry acting on the complete Lagrangian of the theory. H Boson as a Cold DM Candidate
As mentioned above, in the absence of right-handed neutrinos, the scalar potential of theType-II MSISM with maximal SCPV possesses the permutation symmetry: σ ↔ J . Underthe action of this symmetry, the scalar field H = ( J − σ ) / √ H → − H . Thisparity symmetry remains unbroken after the EWSSB, leading to a massive stable scalarparticle, which could play the role of the cold DM in the Universe.We may now extend the above permutation or parity symmetry to neutrino Yukawasector of the model, which implies that h N = − i ˜h N † . As a consequence, the H bosonwill not interact with the neutrinos, so it will remain a massive stable particle which canpotentially act as DM particle. Given the relation h N = − i ˜h N † , the light- and heavy-neutrino mass matrices become m ν = − s − λ (Λ) λ (Λ) v φ h ν (Re h N ) − h νT , m N = 2 s λ (Λ) − λ (Λ) v φ Re h N , (7.11)where Re h N = − Im h N in the weak basis, in which m M is real. Assuming a universalMajorana flavour structure with h N = h N , we find that Re h N must be less than 2.1in order to be perturbative at the RG scale Λ and less than 0.37 and 0.33 to remainperturbative at the GUT and Planck scales, respectively.50 h N H L L m h H G e V L LEP Β Λ = Λ = h N H L L m h H G e V L LEP Β Λ = Λ = Α= h N H L L m h H G e V L LEP Β Λ = Λ = Figure 14:
Numerical estimates of m h as a function of Re h N (Λ) in the minimal Type-IIMSISM with maximal SCPV, massive Majorana neutrinos and a scalar DM, for Cases A,B and C defined in (7.10). The area between the black lines correspond to regions allowedby β λ ( M Planck ) < , λ ( M Planck ) > and the potential BFB ( β > ). The region above thered α = 1 line is excluded. The area below the grey dashed LEP line is excluded by LEP2Higgs-mass limit. The grey shaded areas correspond to the regions allowed by both theoryand the LEP2 limit. h N H L L m N H G e V L LEP Β Λ = Λ = Β= h N H L L m N H G e V L LEP Β Λ = Λ = Β= Α= h N H L L m N H G e V L LEP Β Λ = Λ = Β= Figure 15:
Numerical estimates of m N as a function of Re h N (Λ) in the minimal Type-IIMSISM with maximal SCPV, massive Majorana neutrinos and a scalar DM, for Cases A,B and C defined in (7.10). The area between the black lines show the regions which satisfy: β λ ( M Planck ) < , λ ( M Planck ) > and β > . The red α = 1 line excludes the region abovethis line. The area below the grey dashed LEP line is excluded by LEP2 Higgs-mass limit.The grey shaded areas correspond to the regions allowed by both theory and the LEP2 limit. h -boson masses and the real part ofthe Majorana Yukawa coupling Re h N (Λ), for the three Cases A, B and C defined in (7.10).The area enclosed by the black lines is theoretically favoured by the perturbative andBFB conditions: β λ , ( M Planck ) < λ ( M Planck ) > β > α = 1 line is disfavoured, becauseit violates perturbative unitarity in the Higgs sector. Above the grey dashed LEP linescorrespond to the regions which are also permitted by the LEP2 Higgs-mass limit applied to m h , whereas constraints from the S , T and U parameters play no role in the theoreticallyallowed parameter space. The grey shaded regions are theoretically and experimentallypermitted. From Fig. 14, we observe that the h -boson mass has a similar range of valuesas the CP-symmetric MSISM discussed in the previous subsection.In Fig. 15 we display the allowed parameter space of the universal right-handed neu-trino Majorana mass m N and Re h N (Λ), for the three different Cases A, B and C. As before,we consider the following theoretical conditions: β λ ( M Planck ) < λ ( M Planck ) > β > α ≤
1. The theoretically favoured regions are those, which are enclosed by the black β λ < λ > β >
0) lines. The grey shaded areas correspond to the regionswhich are also permitted by the LEP2 Higgs-mass limit applied to m h . In all the threebenchmark scenarios considered, the heavy Majorana neutrino mass scale m N stays belowthe TeV scale and the value of Re h N (Λ) is constrained to be: Re h N < ∼ . H boson becomes astable particle and so could play the role of the cold DM in the Universe. Second, thepresent model can implement an electroweak seesaw mechanism to provide naturally smallneutrino masses. It contains a new source of spontaneous CP violation, thereby enablingus to address the problem of the baryon asymmetry in the Universe. The model success-fully passes all obvious experimental constraints from LEP2 Higgs and other electroweakprecision data. Finally, of particular interest is the existence of a significant region of thetheoretical parameter space, within which the model can stay perturbative up to Planck-mass energy scales. We have performed a systematic analysis of an extension of the Standard Model thatincludes a complex singlet scalar field S and is scale invariant at the tree level. We havecalled such a model the Minimal Scale Invariant extension of the Standard Model (MSISM).Quantum corrections explicitly break the scale invariance of the classical Lagrangian of the53odel and may trigger EWSSB. Even though the scale invariant SM is not a realisticscenario, the MSISM may result in a perturbative and phenomenologically viable theorythat may potentially solve the gauge hierarchy problem.We have presented a complete classification of the flat directions which may occurin the classical scalar potential of the MSISM. Employing the perturbative GW approachto EWSSB, we have calculated the one-loop effective potential along the different flatdirections and derived the necessary and sufficient conditions for the scalar potential tobe BFB [cf. (4.6)]. In addition, we have computed the scalar-boson masses, includingtheoretical constraints from the validity of perturbation theory, as well as phenomenologicallimits from electroweak precision data and direct Higgs-boson searches at LEP2.The different flat directions in the MSISM can be classified in three major categories:Type I, Type II and Type III. In the Type-I MSISM, the singlet scalar S has a zeroVEV at the tree level, whereas in the Type-II MSISM both the VEVs of S and the SMHiggs doublet Φ are non-zero. In Type-III MSISM, the Higgs doublet Φ has a vanishingVEV at the tree-level, which makes it somewhat difficult to naturally realize EWSSB.Therefore, our analysis has focused only on scenarios realizing Type-I and Type-II flatdirections. We have found that the general Type-I MSISM is perturbative only up to theEW scale and exhibits a Landau pole at energy scales ∼ GeV. Likewise, we have foundthat the U(1)-invariant Type-II MSISM is perturbative up to energies ∼ GeV anddevelops a Landau pole at energy scales ∼ GeV. In this respect, our results are inqualitative agreement with [8]. As we have shown, however, this is not an indispensableproperty of a general Type-II MSISM. Moving away from the model-building constraintof U(1) invariance, we have explicitly demonstrated that a minimal Type-II MSISM ofmaximal SCPV can stay perturbative up to the Planck scale, without the need to introduceunnaturally large hierarchies between the scalar-potential quartic couplings, or between theVEVs of the Φ and S fields which may reintroduce an additional hierarchy problem.In the present study, we have taken the view that the generation of the electroweakscale M EW is the result of the breaking of the scale invariance of the Higgs sector of theMSISM. Instead, we have tacitly assumed that quantum gravity effects are small and donot destabilize the gauge hierarchy. As was argued in [9, 11], for example, the latter maybe the consequence of a conformally UV complete theory of quantum gravity, which we arecurrently lacking. However, a necessary ingredient for such a theory to succeed appears tobe the absence of any additional scale between M EW and M Planck . It is therefore importantthat the quartic couplings remain perturbative up to the Planck scale, without the presenceof a Landau pole which could introduce an additional unwanted higher scale in the theory,through non-perturbative effects that could dynamically break the scale invariance and sodestabilize the gauge hierarchy. 54e have investigated the phenomenological implications of the Type-I and Type-IIMSISM, in particular, whether they realize explicit or spontaneous CP violation, neutrinomasses or predict dark matter candidates. The key features of the different scenarios havebeen summarized in Table 1. To naturally account for the very small light-neutrino massesthrough the seesaw mechanism, we have extended the Type-II MSISM with right-handedneutrinos. Our analysis shows that the right-handed neutrino mass scale m N cannot bemuch higher than the TeV scale and so heavy Majorana neutrinos might lead to observablelike-sign dilepton effects at the LHC. On the other hand, the addition of right-handedneutrinos generically renders all scalar fields unstable and so prevents them from acting asDM particles. However, we have shown that this problem could be solved by promotinga parity symmetry present in the scalar potential of the model to the neutrino Yukawasector and to the complete Lagrangian. One of the scenarios satisfying this criterion is theType-II MSISM of maximal SCPV.There are several issues which are beyond the scope of the present paper, but needto be studied in greater detail. Specifically, it would be interesting to determine the pre-cise constraints on the parameter space derived from the predicted DM relic abundances.Similarly, additional constraints may be derived from considerations of the baryon asym-metry in the Universe. Finally, it would be interesting to investigate, whether the presenceof some of the quasi-flat directions in the MSISM could also serve to drive cosmologicalinflation. These are some of the issues that remain open within the MSISM, which we aimto address in the near future. Acknowledgements
L. A-N. thanks the participants of the “Workshop on Multi-Higgs Models” in Lisbon,Portugal (16 th − th September 2009) for useful discussions.55
The Yukawa and Gauge Sectors of the MSISM
Here we briefly discuss the Yukawa and electroweak gauge sectors of the MSISM, whichclosely resemble the SM. This brief exposition will enable us to set up the notation anddetermine the gauge-dependent masses and couplings that enter our calculations for theeffective potential, the anomalous dimensions and the electroweak oblique parameters.The gauge-invariant part of the Lagrangian describing the Yukawa and electroweakgauge sectors is given by L inv = − G aµν G a,µν − F iµν F i,µν − B µν B µν + ¯ ψiγ µ D µ ψ + ( D µ Φ) † ( D µ Φ) + ( ∂ µ S ∗ )( ∂ µ S ) − (cid:16) h uij ¯ Q iL ˜Φ u jR + h dij ¯ Q iL Φ d jR + h eij ¯ L iL Φ e jR + H . c . (cid:17) , (A.1)where G aµν = ∂ µ G aν − ∂ ν G aµ + g s f abc G bµ G cν , F iµν = ∂ µ A iν − ∂ ν A iµ + gε ijk A jµ A kν and B µν = ∂ µ B ν − ∂ ν B µ are the field strength tensors of the SU(3) c , SU(2) L and U(1) Y gauge fields, G aµ (with a = 1 , . . . , A iµ (with i = 1 , ,
3) and B µ , respectively. Correspondingly, g s , g and g ′ are the SU(3) c , SU(2) L and U(1) Y gauge couplings and D µ is the covariant derivativedefined as D µ = ∂ µ − ig s λ a G aµ − ig τ i A iµ − i Y g ′ B µ , where λ a ( τ i ) are the usual Gell-Mann(Pauli) matrices and Y is the U (1) Y weak hypercharge of the various fields, Y (Φ) = 1 , Y ( S ) = 0 , Y ( L L ) = − , Y ( e R ) = − ,Y ( Q L ) = 13 , Y ( u R ) = 43 , Y ( d R ) = − . (A.2)In (A.1), we have used ψ to collectively represent all the fermions of the model, Q iL = u i d i ! L , u iR , d iR , L iL = ν i e i ! L , e iR , (A.3)where the subscripts L and R denote the left- and right-handed chiralities of the fermion.Each type of fermion has three generations represented by i = 1 , ,
3, i.e. e i = ( e, µ, τ ). Thematrices h u,d,eij contain the Yukawa couplings for the SM up- and down-type quarks andcharged leptons. Finally, we denote the hypercharge conjugate field of the Higgs doubletΦ as ˜Φ = iτ Φ ∗ .A convenient gauge-fixing scheme to remove the tree-level mixing terms betweenthe Goldstone and gauge bosons is the R ξ class of gauges. Adopting this scheme anddecomposing linearly the neutral component of Φ about its one-loop induced VEV, as v φ + φ ,56e may write the gauge-fixing and the induced Faddeev–Popov Lagrangians as follows: L GF = − ξ h ( ∂ µ G aµ ) + ( ∂ µ A iµ ) + ( ∂ µ B µ ) i − i √ gv φ ( G − − G + ) ∂ µ A µ − √ gv φ ( G − + G + ) ∂ µ A µ + 12 gv φ G∂ µ A µ − g ′ v φ G∂ µ B µ − ˜ m G ± G + G − −
12 ˜ m G G , L FP = − ¯ η a ∂ µ ( ∂ µ δ ac − g s f abc G bµ ) η c + ω † i m fij ω j + ω † i m fi χ + χ † m fi ω i + χ † m f χ , (A.4)where η a ( a = 1 , ..., ω i ( i = 1 , ,
3) and χ are the SU(3) c , SU(2) L and U(1) Y ghost fields,respectively, and m fii = − ∂ µ ∂ µ − g ξv φ φ − g ξv φ , m f = − m f = g∂ µ A ,µ − g ξv φ G ,m f = − m f = − g∂ µ A ,µ − √ g ξv φ ( G − + G + ) ,m f = − m f = g∂ µ A ,µ + i √ g ξv φ ( G − − G + ) ,m f = − √ gg ′ ξv φ ( G − + G + ) , m f = i √ gg ′ ξv φ ( G − − G + ) ,m f = 14 gg ′ ξv φ φ + 14 gg ′ ξv φ , m f = − ∂ µ ∂ µ − g ′ ξv φ φ − g ′ ξv φ . (A.5)The would-be Goldstone bosons obtain gauge-dependent mass contributions due to thegauge fixing term L GF , given by m G ± = 14 g ξv φ , m G = 14 ( g + g ′ ) ξv φ . (A.6)Similarly, the ghosts also gain gauge-dependent mass eigenvalues from L GF , i.e. m ω ± = 14 g ξv φ , m ω Z = 14 ( g + g ′ ) ξv φ , m ω A = 0 , m η a = 0 , (A.7)where ω ± = √ ( ω ∓ iω ), ω Z = √ g + g ′ ( gω − g ′ χ ) and ω A = √ g + g ′ ( g ′ ω + gχ ).We should note that after EWSSB, all v φ -dependent masses and couplings affect theone-loop effective potential V − loopeff along the flat direction, but they do not influence theone-loop anomalous dimensions and β functions, which may be computed in the symmet-ric phase of the theory. In the same context, we also note that the v φ -dependent termscontribute to the electroweak oblique parameters, S , T and U , which are conventionallycalculated in the Feynman-’t Hooft gauge ξ = 1.57 The One-Loop Effective Potential of the MSISM
Here we calculate the one-loop effective potential of the MSISM. To this end, we use thefunctional expression [37, 38]: V − loopeff = − C s i ~ (cid:16) Tr ln H ϕ ϕ ( ϕ c ) − Tr ln H ϕ ϕ (0) (cid:17) , (B.1)where H ϕ ϕ is the second derivative of the classical action S = R d x L , i.e. H ϕ ϕ ( ϕ c ) = δ Sδϕ ( x ) δϕ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ϕ = ϕ c . (B.2)In the above, ϕ collectively denotes each of the fields, { Φ , S, A iµ , B µ , ω ± , ω Z , ω A , η a , u i , d i , e i , ν i , N i } where ϕ c is the classical field defined as the VEV of the operator ϕ in the presence of thesource J ( x ) and C s = +1 ( −
1) for fields obeying the Bose–Einstein (Fermi–Dirac) statistics.Moreover, the trace Tr in (B.1) acts over all space and internal degrees of freedom. For ourpurposes, a more convenient representation of (B.1) is V − loopeff = − C s i Z dx Tr (cid:20) H ϕ ϕ ( ϕ c ) − H ϕ ϕ (0) x ( H ϕ ϕ ( ϕ c ) − H ϕ ϕ (0)) + H ϕ ϕ (0) (cid:21) . (B.3)In momentum space of n = 4 − ε dimension, this last expression becomes V − loopeff = − C s i Z dx Z d n k (2 π ) n tr (cid:20) H ϕ ϕ ( ϕ c ) − H ϕ ϕ (0) x ( H ϕ ϕ ( ϕ c ) − H ϕ ϕ (0)) + H ϕ ϕ (0) (cid:21) (B.4)and tr now symbolizes the trace only over the internal degrees of freedom, e.g. over thepolarizations of the gauge fields, the spinor components of the fermions or the Yukawacoupling matrices.The one-loop effective potential of the MSISM can now be calculated by applying(B.4) to the scalars, gauge bosons (GB), ghosts, charged fermions (CF) and neutrinos (N)individually, i.e. V − loopeff = V − loopeff (Scalar) + V − loopeff (GB) + V − loopeff (Ghost)+ V − loopeff (CF) + V − loopeff (N) . (B.5)For the scalar contribution, this is a non-trivial derivation, since H ϕ ϕ ( ϕ c ) as definedin (B.2) is the 6 × H Φ † Φ H Φ † Φ † H Φ † S H Φ † S ∗ H ΦΦ H ΦΦ † H Φ S H Φ S ∗ H S Φ H S Φ † H SS H SS ∗ H S ∗ Φ H S ∗ Φ † H S ∗ S H S ∗ S ∗ . (B.6)58bserve that H Φ † Φ , H Φ † Φ † , H ΦΦ and H ΦΦ † are 2 × H SS , H SS ∗ , H S ∗ S and H S ∗ S ∗ are complex numbers, and the remaining entries, e.g. H Φ S , H Φ S ∗ etc, are two-dimensionalcomplex vectors. This internal matrix structure needs be treated with care and must bepreserved when determining the matrix, [ x ( H ϕ ϕ ( ϕ c ) − H ϕ ϕ (0)) + H ϕ ϕ (0)] − . Takingthis fact into account, the scalar contribution is found to be V − loopeff (Scalar) = 164 π (cid:20) M G ± (cid:18) − ε −
32 + ln M G ± ¯ µ (cid:19) + M G (cid:18) − ε −
32 + ln M G ¯ µ (cid:19) + X i =1 M H i (cid:18) − ε −
32 + ln M H i ¯ µ (cid:19) (cid:21) , (B.7)where ln ¯ µ = − γ +ln 4 πµ , γ ≈ . µ is ’t-Hooft’srenormalization scale. The Goldstone mass terms in the above equation are given by M G = M G ± = λ Φ † Φ + λ S ∗ S + λ S + λ ∗ S ∗ . (B.8)These mass terms vanish along the flat direction because of (4.11). However, after EWSSBthey obtain additional ξ -dependent contributions through the gauge fixing terms [cf. (A.6)].The masses M H , , appearing in (B.7) correspond to the eigenvalues of the matrix M S = M φ M φσ M φJ M φσ M σ M σJ M φJ M σJ M J , (B.9)where M φ = 32 λ φ + 12 ( λ + λ + λ ∗ ) σ + i ( λ − λ ∗ ) σJ + 12 ( λ − λ − λ ∗ ) J ,M σ = 12 ( λ + λ + λ ∗ ) φ + 32 ( λ + 2 λ + 2 λ ∗ + λ + λ ∗ ) σ +3 i ( λ − λ ∗ + λ − λ ∗ ) σJ + 12 ( λ − λ − λ ∗ ) J ,M J = 12 ( λ − λ − λ ∗ ) φ + 12 ( λ − λ − λ ∗ ) σ + 3 i ( λ − λ ∗ − λ + λ ∗ ) σJ + 32 ( λ − λ − λ ∗ + λ + λ ∗ ) J ,M φσ = φ h ( λ + λ + λ ∗ ) σ + i ( λ − λ ∗ ) J i ,M σJ = i "
12 ( λ − λ ∗ ) φ + 32 ( λ − λ ∗ + λ − λ ∗ ) σ − i ( λ − λ − λ ∗ ) σJ + 32 ( λ − λ ∗ − λ + λ ∗ ) J ,M φJ = φ h i ( λ − λ ∗ ) σ + ( λ − λ − λ ∗ ) J i . (B.10)59ote that M φ,σ,J reduce to the squared mass terms for the φ , σ and J fields, respectively,if all mixing terms M φσ,φJ,σJ between the scalar fields vanish along a given flat direction.In addition, we should remark here that one of the eigenvalues of the matrix (B.9) willalways be zero along a minimal flat direction, since it corresponds to the pseudo-Goldstoneboson h of scale invariance.We now turn our attention to the gauge-boson contribution in (B.5), which has beencalculated in the R ξ gauge. The gauge-boson contribution reads: V − loopeff (GB) = 164 π (cid:20) M W (cid:18) − ε −
56 + ln M W ¯ µ (cid:19) + 2 ξ M W (cid:18) − ε −
32 + ln ξM W ¯ µ (cid:19) + 3 M Z (cid:18) − ε −
56 + ln M Z ¯ µ (cid:19) + ξ M Z (cid:18) − ε −
32 + ln ξM Z ¯ µ (cid:19) (cid:21) , (B.11)where M W = g † Φ , M Z = g + g ′ † Φ . (B.12)In the same class of R ξ gauges, the ghost contribution is given after EWSSB by V − loopeff (Ghost) = − π (cid:20) M ω ± (cid:18) − ε −
32 + ln M ω ± ¯ µ (cid:19) + M ω Z (cid:18) − ε −
32 + ln M ω Z ¯ µ (cid:19) (cid:21) , (B.13)where M ω ± = ξM W and M ω Z = ξM Z are the field-dependent ghost masses.Next, we calculate the charged fermion contribution to the effective potential (B.5).This is given by V − loopeff (CF) = − π (cid:20) X i =1 M ui (cid:18) − ε − M ui ¯ µ (cid:19) + 3 X i =1 M di (cid:18) − ε − M di ¯ µ (cid:19) + X i =1 M ei (cid:18) − ε − M ei ¯ µ (cid:19) (cid:21) , (B.14)where M fi ( f = u, d, e ) are the eigenvalues of the background Φ-dependent squared massmatrix for the f -type fermion: ( h f † h f ) Φ † Φ. Note the factor 3 in front of the up- anddown-type quark contributions which counts the SU(3) c colour degrees of freedom.If the MSISM is extended with right-handed neutrinos, these will give rise to addi-tional quantum effects on the one-loop effective potential (B.5). The contribution of thelight and heavy Majorana neutrinos to the effective potential is given by V − loopeff (N) = − π (cid:26) Tr (cid:20) ( M ν M † ν ) (cid:18) − ε − M ν M † ν ¯ µ (cid:19)(cid:21) + Tr (cid:20) ( M N M † N ) (cid:18) − ε − M N M † N ¯ µ (cid:19)(cid:21) (cid:27) , (B.15)60here M ν is the background Φ- and S -dependent light-neutrino mass matrix, M ν = (ΦΦ T ) h ν M − N h νT , (B.16)and M N is the respective S -dependent heavy-neutrino mass matrix: M N = h N S + ˜h N † S ∗ . (B.17)Finally, an important remark is in order. The one-loop effective potential V − loopeff is in general gauge dependent through (B.11) and after EWSSB through (B.13) and theGoldstone ξ -dependent mass terms in (B.7) as well. However, it is known that the effectivepotential becomes gauge-independent when evaluated at local extrema [39, 40]. Within thecontext of perturbation theory, the one-loop effective potential should be ξ -independent,if it is evaluated along a stationary flat direction [41]. This is exactly the case of the GWapproach to the effective potential (3.8). Therefore, as a consistency check, we have verifiedthat the ξ -dependent terms due to gauge, Goldstone and ghost contributions cancel againsteach other in the effective potential (B.5) when evaluated along a stationary flat direction. C One-Loop Anomalous Dimensions and β -Functions In this section, we calculate the one-loop anomalous dimensions of the fields and the β functions of couplings in the MSISM, within the MS scheme of renormalization in the R ξ class gauges. Our calculation is based on the so-called displacement operator formalism, or D -formalism in short, which was developed in [42] as an alternative approach to systemat-ically performing renormalization to all orders in perturbation theory. Since this is not acommon approach, we briefly review its basic features.According to the D -formalism, the renormalized one-particle irreducible n -point cor-relation functions, denoted hereafter with a script R , are related to the unrenormalizedones through: ϕ nR IΓ Rϕ n ( λ R , ξ R ; µ ) = e D (cid:16) ϕ nR IΓ ϕ n ( λ R , ξ R ; µ, ǫ ) (cid:17) , (C.1)where D is the displacement operator that takes the form, D = δϕ ∂∂ϕ R + δλ ∂∂λ R + δξ ∂∂ξ R , (C.2)where ϕ again represents all the fields in the model, λ all the coupling constants, i.e. λ i , g, g ′ , g s , h fij , and ξ is the gauge fixing parameter. In addition, the countertermrenormalizations, δϕ , δλ etc, are defined as, δϕ = ϕ − ϕ R = ( Z / ϕ − ϕ R , δλ = λ − λ R =( Z λ − λ R etc. 61e may now perform a loopwise expansion of the operator e D in (C.1), e D = 1 + D (1) + (cid:16) D (2) + 12 D (1)2 (cid:17) + . . . , (C.3)where the superscript ( n ) on D denotes the loop order, i.e. D ( n ) = δϕ ( n ) ∂∂ϕ R + δλ ( n ) ∂∂λ R + δξ ( n ) ∂∂ξ R . (C.4)Correspondingly, the parameter or counterterm shifts δϕ ( n ) , δλ ( n ) and δξ ( n ) are loopwisedefined as δϕ ( n ) = Z ( n ) ϕ ϕ R , δλ ( n ) = Z ( n ) λ λ R , δξ ( n ) = Z ( n ) ξ ξ R . (C.5)Applying the D -formalism to one-loop, we have ϕ nR Γ R (1) ϕ n ( λ R , ξ R ; µ ) = D (1) (cid:16) ϕ nR Γ (0) ϕ n ( λ R , ξ R ; µ ) (cid:17) + ϕ nR Γ (1) ϕ n ( λ R , ξ R ; µ, ǫ ) . (C.6)This last equation can be used to calculate the wavefunction and coupling constant renor-malizations, Z (1) ϕ and Z (1) λ . Having thus obtained Z (1) ϕ and Z (1) λ , we may compute theone-loop anomalous dimensions γ ϕ of the fields and the β λ functions of the couplings asfollows: γ ϕ ≡ − µ d ln ϕ R dµ = −
12 lim ε → X λ i ε d λ i λ iR ∂∂λ iR Z (1) ϕ ,β λ i ≡ µ dλ iR dµ = λ iR lim ε → X λ j ε d λ j λ jR ∂∂λ jR Z (1) λ i , (C.7)where ε d λ is the tree-level scaling dimension of the generic coupling λ in n = 4 − ε dimensions, with d λ i = 2 for the scalar quartic couplings, d g = d h = 1 for the gaugeand Yukawa couplings and d ξ = 0 for the gauge-fixing parameter. It is useful to remarkhere that the one-loop anomalous dimensions γ ϕ of the fields and the β λ functions can becalculated in the symmetric phase of the theory.Employing (C.6) and (C.7) in the MS scheme, we may calculate the one-loop anoma-lous dimensions and β functions in the R ξ gauge. More explicitly, we obtain for the anoma-62ous dimensions of the fields: γ Φ = 1(4 π ) (cid:20)
14 ( ξ − g + g ′ ) + T (cid:21) ,γ S = 1(4 π ) T , γ u L = 1(4 π ) (cid:20) (cid:16) h u h u † + h d h d † (cid:17) + ξ (cid:16) g s + 34 g + 136 g ′ (cid:17) (cid:21) , γ u R = 1(4 π ) (cid:20) h u † h u + 49 ξ (cid:16) g s + g ′ (cid:17) (cid:21) , γ d L = 1(4 π ) (cid:20) (cid:16) h u h u † + h d h d † (cid:17) + ξ (cid:16) g s + 34 g + 136 g ′ (cid:17) (cid:21) , γ d R = 1(4 π ) (cid:20) h d † h d + 19 ξ (cid:16) g s + g ′ (cid:17) (cid:21) , γ ν L = 1(4 π ) (cid:20) (cid:16) h e h e † + h ν h ν † (cid:17) + ξ (cid:16) g + g ′ (cid:17) (cid:21) , γ ν CL = 1(4 π ) (cid:20) (cid:16) h e ∗ h eT + h ν ∗ h νT (cid:17) + ξ (cid:16) g + g ′ (cid:17) (cid:21) , γ ν R = 1(4 π ) (cid:18) h ν † h ν + 12 h N † h N + 12 ˜h N ˜h N † (cid:19) , γ ν CR = 1(4 π ) (cid:18) h νT h ν ∗ + 12 h N h N † + 12 ˜h N † ˜h N (cid:19) , (C.8)where T = Tr (cid:0) h u h u † + 3 h d h d † + h e h e † + h ν h ν † (cid:1) and T = Tr (cid:0) h N † h N + ˜h N † ˜h N (cid:1) . Noticethat ( γ ν L ) ∗ = γ ν CL and ( γ ν R ) ∗ = γ ν CR , where we have used h N = h NT and ˜ h N = ˜ h NT ,which is a consequence of the Majorana constraint on the left-handed and right-handedneutrinos, ν iL and ν iR .Correspondingly, we start by listing the one-loop β functions of the scalar-potential63uartic couplings: β λ = 18 π (cid:20) λ + λ + 4 λ λ ∗ + 38 (cid:18) g + 2 g g ′ + g ′ (cid:19) − T − λ (cid:18) (cid:0) g + g ′ (cid:1) − T (cid:19) (cid:21) ,β λ = 18 π (cid:20) λ + 2 λ + 4 λ λ ∗ + 54 λ λ ∗ + 36 λ λ ∗ − Tr (cid:16) h N h N † h N h N † (cid:17) − (cid:16) ˜h N ˜h N † h N † h N (cid:17) − (cid:16) ˜h N † ˜h N h N h N † (cid:17) − Tr (cid:16) ˜h N † ˜h N ˜h N † ˜h N (cid:17) + λ T (cid:21) ,β λ = 18 π (cid:20) λ λ + 2 λ λ + 2 λ + 8 λ λ ∗ + 6 λ λ ∗ + 6 λ λ ∗ − (cid:16) h N † h N h ν † h ν (cid:17) − (cid:16) ˜h N ˜h N † h ν † h ν (cid:17) − λ (cid:18) (cid:0) g + g ′ (cid:1) − T − T (cid:19) (cid:21) , (C.9) β λ = 18 π (cid:20) λ λ + λ λ + 4 λ λ + 3 λ λ + 6 λ ∗ λ − (cid:16) ˜h N h N h ν † h ν (cid:17) − λ (cid:18) (cid:0) g + g ′ (cid:1) − T − T (cid:19) (cid:21) ,β λ = 18 π (cid:20) λ λ + 2 λ λ + 18 λ ∗ λ − Tr (cid:16) ˜h N † ˜h N h N ˜h N (cid:17) − Tr (cid:16) h N h N † h N ˜h N (cid:17) + λ T (cid:21) ,β λ = 18 π (cid:20) λ λ + 2 λ + 9 λ − Tr (cid:16) ˜h N h N ˜h N h N (cid:17) + λ T (cid:21) , where T = Tr (cid:0) h u h u † h u h u † + 6 h d h d † h d h d † + 2 h e h e † h e h e † + 2 h ν h ν † h ν h ν † (cid:1) . Note that theone-loop β functions of the complex conjugate quartic couplings, i.e. β λ ∗ , , , are given by β λ ∗ , , = ( β λ , , ) ∗ .For the one-loop β functions of the SU(3) c , SU(2) L and U(1) Y gauge couplings, weuse the well-established results: β g s = − π g s , β g = − π g , β g ′ = 18 π g ′ . (C.10)Next, we present the known one-loop β functions of the up-type and down-type quarkYukawa couplings β h u = 18 π (cid:20) − g ′ − g − g s + 12 T + 34 (cid:0) h u h u † − h d h d † (cid:1) (cid:21) h u , β h d = 18 π (cid:20) − g ′ − g − g s + 12 T + 34 (cid:0) h d h d † − h u h u † (cid:1) (cid:21) h d . (C.11)Finally, the one-loop β functions of the light- and heavy-neutrino Yukawa couplings are64alculated to be β ˜h N = 18 π (cid:20) ˜h N (cid:18) h N h N † + 14 h ˜N † h ˜N + 12 h νT h ν ∗ (cid:19) + (cid:18) h N † h N + 14 ˜h N ˜h N † + 12 h ν † h ν (cid:19) ˜h N + 14 ˜h N T (cid:21) , β h N = 18 π (cid:20) h N (cid:18) ˜h N ˜h N † + 14 h N † h N + 12 h ν † h ν (cid:19) + (cid:18) ˜h N † ˜h N + 14 h N h N † + 12 h νT h ν ∗ (cid:19) h N + 14 h N T (cid:21) , β h ν = 18 π (cid:20) h ν (cid:18) − g ′ − g + 12 T (cid:19) + 34 (cid:18) h ν h ν † − h e h e † (cid:19) h ν + 14 h ν (cid:18) h N † h N + ˜h N ˜h N † (cid:19) (cid:21) . (C.12)The one-loop anomalous dimensions and β functions can be used to verify the renor-malizability of V − loopeff . To be specific, the potential V = V tree + V − loopeff should be UVfinite after renormalization. In the so-called MS renormalization scheme [43], the one-loopUV counter-terms for the fields and coupling constants are explicitly given by δϕ (1) = Z (1) 1 / ϕ ϕ R = − (cid:18) ε − γ + ln 4 π (cid:19) γ ϕ ϕ R , δλ (1) = Z (1) λ λ R = 12 (cid:18) ε − γ + ln 4 π (cid:19) β λ . (C.13)Taking these relations into account, the one-loop MSISM effective potential can berenormalized in the MS scheme and its complete analytic form is given by V − loopeff = 164 π (cid:26) M G ± (cid:18) −
32 + ln M G ± µ (cid:19) + M G (cid:18) −
32 + ln M G µ (cid:19) + X i =1 m H i (cid:18) −
32 + ln m H i µ (cid:19) +6 M W (cid:18) −
56 + ln M W µ (cid:19) + 3 M Z (cid:18) −
56 + ln M Z µ (cid:19) − ξ M W (cid:18) −
32 + ln ξM W µ (cid:19) − ξ M Z (cid:18) −
32 + ln ξM Z µ (cid:19) − X i =1 M ui (cid:18) − M ui µ (cid:19) (C.14) − X i =1 M di (cid:18) − M di µ (cid:19) − X i =1 M ei (cid:18) − M ei µ (cid:19) − (cid:20) ( M ν M † ν ) (cid:18) − M ν M † ν µ (cid:19) (cid:21) − (cid:20) ( M N M † N ) (cid:18) − M N M † N µ (cid:19)(cid:21) (cid:27) , where the mass terms are defined in Appendices A and B. Notice that along a stationaryflat direction, µ → Λ and the ξ -dependent Goldstone-boson masses M G ± and M G , given65n (A.6), cancel against the ξ -dependent contributions from the W ± and Z bosons andtheir respective ghost fields. Hence, the complete one-loop renormalized effective potentialbecomes gauge independent in this case. D The Electroweak Oblique Parameters
In order to calculate the electroweak oblique parameters S , T and U , we adopt the notationand formalism developed in [18]. To this end, we first review the definitions of the S , T and U parameters and present their basic relations with the gauge-boson self-energies, whichwe will then use to determine the electroweak oblique parameters in the MSISM.In detail, the vacuum polarization amplitudes are defined as i Π µνXY ( q ) = ig µν Π XY ( q ) + ( q µ q ν terms) , (D.1)where XY = { , , , Q, QQ } andΠ XY ( q ) = Π XY (0) + q Π ′ XY ( q ) . (D.2)These vacuum polarizations are related to the one-particle irreducible self-energies of the A , W ± and Z gauge bosons through:Π AA = e Π QQ , Π W W = e sin θ w Π , Π ZA = e cos θ w sin θ w (cid:0) Π Q − sin θ w Π QQ (cid:1) , Π ZZ = e cos θ w sin θ w (cid:0) Π − θ w Π Q + sin θ w Π QQ (cid:1) , (D.3)where e is the electric charge and θ w is the electroweak mixing angle. One can now solve theabove system of linear equations for the vacuum polarization amplitudes Π XY and definethe so-called electroweak oblique parameters [18] in terms of them as follows: α em S = 4 e h Π ′ (0) − Π ′ Q (0) i ,α em T = e sin θ w cos θ w m Z h Π (0) − Π (0) i ,α em U = 4 e h Π ′ (0) − Π ′ (0) i , (D.4)where α em = e / (4 π ) is the electromagnetic fine structure constant. Noting the sin θ w de-pendence of Π , Π Q and Π QQ in Π ZZ (D.3), the S , T and U parameters can be determinedby calculating the ZZ and W W vacuum polarization amplitudes only.66 νφG/G ± p µ νφZ/W ± p µ νφp Figure 16:
Feynman diagrams pertinent to the scalar-boson contributions to the eletroweakgauge-boson vacuum polarization amplitudes.
Our interest is to find the difference in the predictions for the electroweak obliqueparameters in the MSISM from the corresponding ones in the SM, i.e. δP = P MSISM − P SM ,where P = { S, T, U } . As shown in Figure 16, the main loop effect beyond the SM arisesfrom the MSISM Higgs scalars h and H , that occur in the W W and ZZ self-energies.The sum of these three diagrams for each one of the three scalar bosons, h , H and H , isdenoted as e P . Specifically, the shifts δP are due to the Higgs scalar masses m h and m H , ,as well as their modified gauge couplings g hV V and g H , V V with respect to the SM coupling g H SM V V = 1, where
V V = { ZZ, W W } . Hence, the deviations of the electroweak obliqueparameters may be obtained by δP = g hV V e P ( m h ) + g H V V e P ( m H ) + g H V V e P ( m H ) − e P ( m H SM ) . (D.5)Here, the generic function e P ( m ) stands for the functions e S ( m ), e T ( m ), and e U ( m ), whichare defined as e S ( m ) = 112 π (cid:20) − ǫ −
12 + m ( m − m Z )( m − m Z ) ln (cid:18) m ¯ µ (cid:19) + m Z (3 m − m Z )( m − m Z ) ln (cid:18) m Z ¯ µ (cid:19) − m − m m Z + 5 m Z m − m Z ) (cid:21) , (D.6) e T ( m ) = 316 π sin θ w cos θ w m Z (cid:20) (cid:18) ǫ + 1 (cid:19) ( m Z − m W ) + m m W m − m W ln (cid:18) m ¯ µ (cid:19) − m m Z m − m Z ln (cid:18) m ¯ µ (cid:19) − m W m − m W ln (cid:18) m W ¯ µ (cid:19) + m Z m − m Z ln (cid:18) m Z ¯ µ (cid:19) (cid:21) , (D.7) e U ( m ) = 112 π (cid:20) m ( m − m W )( m − m W ) ln (cid:18) m ¯ µ (cid:19) − m ( m − m Z )( m − m Z ) ln (cid:18) m ¯ µ (cid:19) + m W (3 m − m W )( m − m W ) ln (cid:18) m W ¯ µ (cid:19) + m Z ( m Z − m )( m − m Z ) ln (cid:18) m Z ¯ µ (cid:19) (D.8) − m − m m W + 5 m W m − m W ) + 5 m − m m Z + 5 m Z m − m Z ) (cid:21) .
67n the above, we have followed the standard convention and calculated the electroweakoblique parameters in the Feynman-’t Hooft ξ = 1 gauge, in which m G = m Z and m G ± = m W ± . Moreover, it is important to note that δS , δT and δU are UV finite and independentof ¯ µ , as it can be easily checked by means of the coupling sum rule: g hV V + g H V V + g H V V = g H SM V V = 1.The theoretical predictions for δS , δT and δU in the MSISM are confronted withtheir experimental values [17]: δS exp = − . ± .
10 ( − . ,δT exp = − . ± .
11 (+0 . ,δU exp = 0 . ± .
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