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Triviality Bounds in Two-Doublet Models

arXiv:hep-ph/9301222v1 8 Jan 1993BUHEP-93-2hep-ph/9301222January 8, 1993Triviality Bounds in Two-Doublet ModelsDimitris Kominis∗andR. Sekhar Chivukula†Dept.

of Physics, Boston University, 590 Commonwealth Avenue,Boston, MA 02215AbstractWe examine perturbatively the two-Higgs-doublet extension of the StandardModel in the context of the suspected triviality of theories with fundamental scalars.Requiring the model to define a consistent effective theory for scales below a cutoffof 2π times the largest mass of the problem, as motivated by lattice investigationsof the one-Higgs-doublet model, we obtain combined bounds for the parameters ofthe model. We find upper limits of 470 GeV for the mass of the light CP–evenneutral scalar and 650–700 GeV for the other scalar masses.∗e-mail address: kominis@budoe.bu.edu†e-mail address: sekhar@weyl.bu.edu

1IntroductionIn the Standard one-doublet Higgs Model of electroweak interactions the scalar potentialisV = 12m20 Φ†Φ + 14λ0 (Φ†Φ)2(1)where Φ is a complex doublet and m20, λ0 are bare parameters. There are strong indications[1, 2] that, in four dimensions and in the limit of vanishing gauge and Yukawa couplings,this defines a trivial field theory in the continuum limit.

This means that for any physicallyacceptable value of the bare coupling λ0, the renormalized self-coupling λR is forced tolie in a narrow range of values which shrinks to the point λR = 0 at the limit of infinitecutoff.Equivalently, a non-zero running coupling develops a Landau pole at a finitemomentum scale.Yukawa and gauge couplings are not expected to alter this picture[3, 4, 5].Consequently the Standard one-doublet Model can only be accepted as aneffective low energy theory valid up to some finite cutoffΛ. The value of the renormalizedcoupling is thus allowed to be non-zero, but is bounded from above.This can be illustrated perturbatively by integrating the one-loop β-function for thescalar self-coupling.

The result, ignoring gauge and Yukawa couplings, is1λ(µ) =1λ(Λ) +32π2 ln Λµ(2)Here λ(Λ) is the bare coupling and µ is some low energy renormalization scale. Sinceλ(Λ) ≥0, it follows thatλR ≡λ(µ) ≤2π231ln(Λ/µ)(3)For a given cutoffΛ, the mass MH of the Higgs boson is also found to be bounded fromabove [6, 2, 3, 7].

In lowest-order perturbation theory this is a consequence of the relationM2H = 2λR v2(4)where v is the vacuum expectation value of the Higgs field.Various physically motivated choices of Λ have been made leading to different boundson MH [8, 9, 10, 11, 12]. These bounds generally increase with decreasing Λ.

For theeffective theory to make sense, the cutoffΛ must be at least of order MH [6]. This placesan “absolute” upper bound on the mass of the Higgs boson, which has been estimated[2, 7, 12] to be about 600–700 GeV, for small Yukawa and gauge couplings.2

The purpose of this paper is to extend these considerations to models with two Higgsdoublets and derive bounds on the masses of the scalar particles of these models. Ourresults are obtained using perturbative arguments.

We believe they convey the right quali-tative picture and, in the light of their agreement with other, non-perturbative approachesin the case of the one-Higgs model, we expect they may also have some quantitative va-lidity.In Section 2 we briefly review the two-doublet extension of the Standard Model. InSection 3 we describe our calculation and in Section 4 we present and discuss our results.For completeness, we list the renormalization group equations for the couplings of themodel in the appendix.2The two-doublet modelThe scalar sector contains two electroweak doublets Φ1, Φ2, both with hypercharge Y =1.

A discrete symmetry must be imposed in order to eliminate flavor changing neutralcurrents at tree level. The two-doublet models fall in two broad categories according tothe way this discrete symmetry is implemented [13]:• Model I:Φ2 →−Φ2;dRi →−dRi• Model II:Φ2 →−Φ2(5)(dRi (i = 1, 2, 3) are the right-handed negatively charged quarks.) The Lagrangian isL = Lkin + LY −Vwhere Lkin contains all the covariant derivative terms, V is the scalar potential and LYcontains the fermion-scalar interactions.

The form of the latter is the following:• Model ILY = g(u)ij ψLiΦc1uRj + g(d)ij ψLiΦ2dRj + h.c. + leptons(6)• Model IILY = g(u)ij ψLiΦc1uRj + g(d)ij ψLiΦ1dRj + h.c. + leptons(7)i.e. in Model I Φ1 gives mass to up-type quarks and Φ2 to down-type quarks while inModel II only Φ1 couples to quarks.3

The results we present were derived using Model II. Since the dominant fermion effectsare due to the top quark whose couplings are the same in both models, no substantialchanges are expected in Model I.The scalar potential isV=µ21 Φ†1Φ1 + µ22 Φ†2Φ2 + λ1 (Φ†1Φ1)2 + λ2(Φ†2Φ2)2 + λ3 (Φ†1Φ1)(Φ†2Φ2)+λ4 (Φ†1Φ2)(Φ†2Φ1) + 12λ5 [(Φ†1Φ2)2 + (Φ†2Φ1)2](8)Note that by absorbing a phase in the definition of Φ2, we can make λ5 real and negative1:λ5 ≤0(9)The most interesting case arises when both doublets acquire non-zero vacuum expectationvalues (vevs).

To avoid spontaneous breakdown of the electromagnetic U(1), the vacuumexpectation values must have the following form:⟨Φ1⟩= 1√2 0v1!⟨Φ2⟩= 1√2 0v2! (10)where v21 + v22 ≡v2 = (246 GeV)2.

The choice (9) ensures that v1 and v2 are relativelyreal. (v1 can be chosen to be real by an SU(2) × U(1) rotation.) This configuration isindeed a minimum of the tree level potential ifλ1≥0λ2≥0λ4 + λ5≤04λ1λ2≥(λ3 + λ4 + λ5)2(11)The spectrum of the scalar sector contains three Goldstone bosons, to be eaten bythe W’s and the Z; two neutral CP–even scalars, denoted by h, H; one neutral CP–oddscalar ζ; and two charged scalars G±.

It is customary to introduce two angles α and β:β (0 < β < π/2) rotates the CP–odd and the charged scalars into their mass eigenstateswhile α (−π/2 ≤α < π/2) rotates the neutral scalars into their mass eigenstates. Thetree level expressions for the masses and angles are the following:1This pushes all potential CP violating effects into the Yukawa sector.4

tan β=v2v1(12)sin α=−(sgn C)12q(A −B)2 + 4C2 −(B −A)q(A −B)2 + 4C21/2(13)cos α=12q(A −B)2 + 4C2 + (B −A)q(A −B)2 + 4C21/2(14)M2G±=−12(λ4 + λ5) v2(15)M2ζ=−λ5 v2(16)M2H,h=12A + B ±q(A −B)2 + 4C2(17)whereA = 2λ1 v21;B = 2λ2 v22;C = (λ3 + λ4 + λ5) v1 v2We emphasize that, as is the case in the one-doublet model, all masses get their scalesfrom the vevs, with multiplicative factors that are functions of the quartic couplings. Ifconsiderations of triviality put bounds on the couplings (which they do), then these willautomatically translate into bounds for the masses.

The two-doublet models are describedby 7 independent parameters which can be taken to be α, β, MG±, Mζ, Mh, MH and thetop quark mass given byMt = gt v cos β(18)where gt is the top quark Yukawa coupling. The light quark and lepton couplings areinessential to our analysis and we ignore them.3Triviality and stability constraintsWe wish to determine when a given set of parameters {α, β, MG±, Mζ, MH, Mh, Mt} definesa valid, consistent low energy effective theory.

By ‘valid’ we mean the following: SupposeΛ is a finite cutoffscale beyond which new phenomena appear. Any physical quantity5

calculated using the two-doublet model as described in Section 2, will differ from its‘true’ value by terms of order p2i /Λ2, M2j /Λ2 where pi are typical external momenta of theprocesses under consideration and Mj are the masses of the particles in the problem. Weshall define our theory to be a valid effective theory if all masses satisfyMjΛ ≤12π(19)This convention corresponds to a Higgs correlation length M−1H = 2 (in lattice units), andis widely used in lattice investigations of the problem of triviality and Higgs mass bounds[2].

(The external momenta pi should also satisfy a similar relation, but this is irrelevanthere.) Thus, given a set of parameters, we define a cutoffΛ = 2π max {MG±, Mζ, MH, Mh, Mt, MZ}(20)MZ being the Z-boson mass, and require, for consistency of the theory, the followingconditions to be true:(i) No coupling should develop a Landau pole at a scale less than Λ;(ii) The effective potential should be stable for all field values less than Λ.2The last requirement is satisfied ifλ1(µ)≥0λ2(µ)≥0(21)˜λ(µ)≥−2qλ1(µ) λ2(µ)for all µ ≤Λ, where˜λ(µ) =(λ3(µ) + λ4(µ) + λ5(µ)ifλ4(µ) + λ5(µ) < 0λ3(µ)ifλ4(µ) + λ5(µ) ≥0(22)Our numerical procedure was the following: a set of parameters {α, β, MG±, Mζ, MH,Mh, Mt} was chosen at random.By inverting the relations (12)–(18) the scalar andYukawa couplings were calculated.

It was assumed that the tree-level expressions (12)–(18) approximate best the physical values when the renormalization scale at which thecouplings are evaluated is taken to beµ = max {MG±, Mζ, MH, Mh, Mt, MZ}(23)2For field values greater than Λ the cutoffeffects are large and the renormalized effective potentialis meaningless. If a one-component Higgs-Yukawa system is well defined as a bare theory, then it doesnot develop a vacuum instability [4].

If this is the case in this model too, then the inequalities (21) areequivalent to the condition that the theory exists as a bare theory.6

Note that (11) are automatically satisfied if all masses are real.The coupled renormalization group equations [14] for the scalar, gauge and top Yukawacouplings were evolved up to the scale defined by eq. (20).

(In practice, Λ was taken to beat least 1 TeV which is the lowest scale at which one would expect new phenomena.) If anyof the couplings became unbounded during this evolution or if the stability constraints (21)were violated, this set of parameters was rejected; otherwise it was accepted.

Subsequentlya new set was chosen and the procedure repeated. In the end, a large set of randomlygenerated ‘points’ in parameter space was accumulated.An envelope to these pointsrepresents the combined bounds we are seeking.4Results and discussionIn Figures 2–8 we display projections of the allowed volume of parameters on selectedtwo-dimensional planes.

For comparison, in Fig. 1 we show the bounds for the Standardone-doublet Model particles obtained using the same method3.The absolute boundson the masses of the scalar particles in the two-doublet model are about 650–700 GeV(roughly the same as the one-doublet model Higgs mass bound), with the exception ofthe light neutral scalar which is constrained to be lighter than about 470 GeV.

Upperbounds on the top quark are somewhat looser than in the Standard one-doublet Model.We estimate the numerical errors in the calculation of the bounds to be not more than afew GeV, which is insignificant given the largely qualitative nature of our computation.Experimental and other theoretical bounds are not shown in these figures. The upperlimits on some splittings among the scalar masses that arise from the precise measurementof the electroweak ρ-parameter [15, 16] are hardly more stringent than our trivialitybounds.

Most other reported bounds are lower bounds and do not interfere with ourconclusions.It is not possible to give a description of the exact shape of the bounding surface in theparameter space. We will simply mention some broad qualitative features: The boundsdepend strongly on the angle β; because of (18) the stability (lower) bounds becomestricter as β becomes large at fixed Mt.

It is also found that for both small and large βthe triviality bounds are stricter than they are for moderate β; the precise way in whichthis happens depends on the values of the other parameters. The dependence on α isnot as strong.

Stability bounds on the scalars are strictest when α takes values close tozero (for a fixed top quark mass.) The bounds on (MG±, Mζ) are largely insensitive tothe values of (MH, Mh) for a large range of these values, but shrink sharply outside that3Note the close agreement with the results of ref.

[2] where a relation equivalent to (19) was used.7

range —and vice-versa— much like fig. 6 shows.The angle α−β is of phenomenological significance since it governs the couplings ofthe neutral scalars to the W’s and the Z.

We examined the bounds on the neutral scalarmasses as a function of cos2(α−β), projecting out all the other parameters, and found nosignificant variation.There is a way in which most of these bounds can be avoided, still within the contextof two-doublet models. A quadratic termµ23 Φ†1Φ2 + h.c.can be added to the scalar potential (8).

This violates the discrete symmetry (5) butonly softly, so that flavor changing neutral currents still do not appear at tree level. Inthis case all scalar particle masses but Mh are increasing functions of |µ23|; since µ23 is notconstrained from triviality considerations, we can only impose bounds on Mh.

As |µ23|grows from zero, we expect the bounds on MG±, Mζ and MH to become gradually weaker.For large |µ23| there is a hierarchy between the scales M2h and |µ23|; the latter determinesthe other scalar masses. Below |µ3| the theory looks like the one-Higgs model; insistingthat the theory makes sense as a two-doublet model requires an effectively Standard Modelquartic coupling to remain finite up to a scale of order 2π|µ3| rather than 2πMh; hencewe expect much stricter bounds than those exhibited in Fig.

1. We have not examinedintermediate values of |µ23| in more detail.Bounds on the scalar particle masses from triviality considerations have previouslybeen reported in the literature.

The authors of ref. [17, 18] concentrate on very large cut-offs while in ref.

[19] a different definition of triviality, closely associated with perturbativeunitarity, is used. Our bounds are generally stricter than those imposed by perturbativeunitarity [19, 20].

The authors of ref. [21] adopt a similar, but stricter, approach thanours and obtain a bound of 475 GeV for the charged scalar mass MG±.According to triviality constraints, the scalar sector of the one-Higgs model is not al-lowed to become strongly interacting; even the heaviest possible Higgs will be light enoughto be detected as a relatively narrow resonance at the SSC.

We are currently investigat-ing the implications of the triviality and stability constraints on the phenomenology oftwo-doublet models.AcknowledgementsWe thank A. Cohen, K. Lane and Y. Shen for reading the manuscript. D. K. would liketo thank H. Larralde and C. Rebbi for suggestions in the computational part of the workand V. Koulovassilopoulos and Y. Shen for discussions.

R.S.C. acknowledges the support8

of an Alfred P. Sloan Foundation Fellowship, an NSF Presidential Young InvestigatorAward, a DOE Outstanding Junior Investigator Award, and a Superconducting SuperCollider National Fellowship from the Texas National Research Laboratory Commission.This work was supported in part under NSF contract PHY-9057173 and DOE contractDE-FG02-91ER40676, and by funds from the Texas National Research Laboratory Com-mission under grant RGFY92B6.9

AppendixIn this appendix we include the coupled renormalization group equations for the cou-plings of the two-doublet model [14]. The gauge couplings for the SU(3), SU(2) and U(1)groups are gc, g and g′ respectively.

For the other couplings we use the notation of thetext. We use the notationD ≡16π2 µ ddµDgc=−7g3cDg=−3g3Dg′=7g′3Dgt=gt−1712g′2 −94g2 −8g2c + 92g2tDλ1=24λ21 + 2λ23 + 2λ3λ4 + λ24 + λ25 −3λ1(3g2 + g′2) + 12λ1g2t+98g4 + 34g2g′2 + 38g′4 −6g4tDλ2=24λ22 + 2λ23 + 2λ3λ4 + λ24 + λ25 −3λ2(3g2 + g′2) + 98g4 + 34g2g′2 + 38g′4Dλ3=4(λ1 + λ2)(3λ3 + λ4) + 4λ23 + 2λ24 + 2λ25 −3λ3(3g2 + g′2) + 6λ3g2t+94g4 −32g2g′2 + 34g′4Dλ4=4λ4(λ1 + λ2 + 2λ3 + λ4) + 8λ25 −3λ4(3g2 + g′2) + 6λ4g2t + 3g2g′2Dλ5=λ5(4λ1 + 4λ2 + 8λ3 + 12λ4 −3(3g2 + g′2) + 6g2t )10

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Figure captions1. Triviality and stability bounds for the Standard Model Higgs and top quark massesMH, Mt.

The allowed region is inside the curve.2. Triviality and stability bounds in the two-doublet model, for the heavy neutral scalarH and the top quark t. All other parameters are projected on the (MH, Mt) plane:the region outside the curve is excluded whatever the values of the parameters notshown on the graph.

Constraints from the weak interaction ρ-parameter suggestthat Mt <∼250 GeV [22].3. Same as fig.

2, but projecting on the (Mh, Mt) plane.4. Same as fig.

2, but projecting on the (MG±, Mt) plane. A similar graph is obtainedin the (Mζ, Mt) plane, the bound on Mζ being slightly higher than the one on MG±.5.

Same as fig. 2, but projecting on the (MH, Mh) plane.6.

Same as fig. 2, but projecting on the (MH, Mζ) plane.

A similar plot is obtained forthe (MH, MG±) plane, with the bounds on MG± slightly lower than those on Mζ.7. Same as fig.

2, but projecting on the (MG±, Mζ) plane.8. Same as fig.

2, but projecting on the (Mh, Mζ) plane; as in figures 4 and 6, thebounds on Mζ are slightly higher than those on MG±.13

Figure 1Figure 2Figure 3Figure 4

Figure 5Figure 6Figure 7Figure 8

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