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Phenomenology of a Non-Standard Higgs

arXiv:hep-ph/9304293v2 3 May 1993BUHEP-93-7hep-ph/9304293March 19, 2021Phenomenology of a Non-Standard HiggsR. Sekhar Chivukula1, Vassilis Koulovassilopoulos2Physics Department, Boston University,590 Commonwealth Avenue,Boston, MA 02215 USAABSTRACTThe one-Higgs-doublet standard model is necessarily incomplete because of the trivial-ity of the scalar symmetry-breaking sector.

If the Higgs mass is approximately 600 GeVor higher, there must be additional dynamics at a scale Λ which is less than a few TeV.In this case the properties of the Higgs resonance can differ substantially from thosepredicted by the standard model. In this letter we construct an effective Lagrangiandescription of a theory with a non-standard Higgs boson and analyze the features ofa theory with such a resonance coupled to the Goldstone Bosons of the breaking ofSU(2) × U(1).

This Lagrangian describes the most general theory in which the Higgsand the Goldstone Bosons are the only particles with a mass small compared to Λ. Wecompute the leading corrections to the decay width of the Higgs boson and the contri-bution to the Peskin-Takeuchi parameter S and present results for the corrections toGoldstone Boson scattering. We find that the most prominent effects are due to onenew parameter, which is directly related to the Higgs boson width.1e-mail address: sekhar@weyl.bu.edu2e-mail address: vk@budoe.bu.edu

The standard one-doublet Higgs model predicts the existence of a neutral scalar par-ticle, the Higgs boson, whose couplings are related to its mass or to the masses of theelectroweak gauge bosons so as to preserve renormalizability. However, there are strongindications that, at least in the limit of vanishing gauge and Yukawa couplings, the scalarsymmetry-breaking sector is trivial [1, 2].

This means that in the continuum limit and foracceptable values of the bare scalar self-coupling λ0, the renormalized self-coupling λR iszero. This implies that the standard one-doublet Higgs model of electroweak symmetrybreaking is necessarily incomplete and can only be viewed as a low-energy effective theorybelow some finite cutoffΛ.If the Higgs mass is approximately 600 GeV or higher, there must be additional dy-namics at a scale which is less than a few TeV [3]: i.e.

the cutoffis low. In this casethe properties of the Higgs resonance can differ substantially from those predicted bythe standard model.

In this letter we construct an effective Lagrangian description ofa theory with a non-standard Higgs boson, and explore the properties of a theory withsuch a resonance coupled to the Goldstone Bosons of the breaking of SU(2) × U(1). ThisLagrangian describes the most general (custodially symmetric [4]) theory in which theHiggs and the Goldstone Bosons are the only particles with masses small compared toΛ.

We compute the leading corrections to the decay width of the Higgs boson and to thePeskin-Takeuchi parameter S [5]. We also present results for the corrections to GoldstoneBoson scattering.

Details of the calculations and an explicit model with a non-standardHiggs resonance will be presented elsewhere [6].The language of effective Lagrangians has been used extensively [7] to study a strongly-interacting symmetry-breaking sector. One usually assumes that the longitudinal gauge-bosons are the only particles in the symmetry-breaking sector with masses of less than aTeV and, therefore, (assuming a custodial symmetry) starts with an effective chiral La-grangian describing the Goldstone Bosons of the spontaneously broken SU(2)L × SU(2)Rchiral symmetry.

In the sense of the equivalence theorem [8], these Goldstone Bosonsrepresent the strongly-interacting longitudinal components of the weak gauge bosons.Instead, we wish to describe a theory in which an entire scalar-doublet φ is lighter thana TeV. We first consider theories in which the new dynamics is strongly interacting andin which, typically, the Higgs is a composite particle.

Examples of such theories includethe “Composite Higgs” models of Georgi and Kaplan [9], as well as “top-condensate” andrelated models [10].In these theories the usual “power-counting” rules of chiral perturbation theory [11, 12]1

apply: the coefficients of all chirally-invariant interactions in the effective Lagrangian areexpected to be of order one if (1) there is an overall factor of f 2Λ2, (2) each factor of theHiggs field appears with a factor of 1/f and (3) derivatives appear with a factor of 1/Λ.Here f is a measure of the amplitude for producing the Higgs [12] and Λ is the scale ofadditional, heavy, strongly interacting particles. For consistency, Λ must be less than oforder 4πf [11].

As stated above, we are interested in the situation that Λ is of order oneor a few TeV, but we need not assume that f is that large.The effective Lagrangian describing the Higgs doublet in such a theory is convenientlywritten in terms of the matrix Φ = (iσ2φ∗, φ), which transforms3 as Φ →LΦR† underSU(2)L × SU(2)R. The most general custodially symmetric Lagrangian to order momen-tum squared isL = 12Tr"F Φ†Φf 2!∂µΦ†∂µΦ#−f 2Λ2Tr G Φ†Φf 2!,(1)where F and G are arbitrary (dimensionless) functions analytic4 at Φ†Φ = 0 and withF(0) = 1.In order for SU(2) × U(1) breaking to occur, the potential G must be minimizedfor Φ ̸= 0. We may analyze the Lagrangian (1) by expressing Φ in “polar” coordinates,Φ = ρΣ/√2.

Here ρ is real and positive, and Σ is a special unitary matrix. The Lagrangianto order momentum squared may be writtenL = 12∂µρ∂µρ + ρ24 A ρ2f 2!Tr (∂µΣ†∂µΣ) −Λ2f 2B ρ2f 2!,(2)where A and B are analytic functions related to the F and G above (in particular, F(0) = 1implies A(0) = 1).

We have not included a function multiplying the kinetic-energy termfor the ρ field because such a function can always be eliminated by a redefinition of theρ field. Writing ρ = ⟨ρ⟩+ H, expanding around the true vacuum, and keeping only thefirst few terms, we findL = 14(v2 + 2ξvH + ξ′H2 + ξ′′H36v ) Tr (∂µΣ†∂µΣ) + LH(3)3 As usual, SU(2)L will be identified with SU(2)weak and SU(2)R is the custodial SU(2) symmetrywhose τ3 component will be identified with hypercharge.4In Composite Higgs models the non-derivative interactions of the Higgs doublet in the potential arisefrom small chiral-symmetry violating interactions and therefore, we must also require that coefficients inthe expansion of G be small compared to 1.2

where ξ , ξ′ and ξ′′ are unknown coefficients, v2 = ⟨ρ⟩2A(⟨ρ⟩2/f 2) = (246 GeV)2, Σcontains the Goldstone Bosons waΣ = exp i⃗w · ⃗τv!,Tr (τ aτ b) = 2δaband LH is the isoscalar LagrangianLH = 12(∂µH)2 −V (H). (4)The leading terms in the scalar potential V (H) areV (H) = m22 H2 + λ3v3!

H3 + λ44! H4.

(5)The power-counting rules imply thatξ, ξ′ = 1 + O v2f 2!, ξ′′ = O v2f 2! (6)andλ3, λ4 = 3m2v2 + O v2f 2!.

(7)We recover the usual linear sigma model in the limit that f →∞. For models in whichthe Higgs resonance is heavy (600 GeV or higher), we expect Λ to be of order one or afew TeV.

In this case, v/f need not be small, and the ξ’s and λ’s can differ substantiallyfrom their standard model values.If, instead of being strongly coupled, the new dynamics is weakly coupled, non-derivative and derivative interactions are treated in the same way [12] and one writesan effective Lagrangian as an expansion in 1/Λ.The lowest order corrections to thestandard Higgs Lagrangian includes only dimension six operatorsXiciΛ2Oi(8)where a minimal set of custodial SU(2)R preserving {Oi} is [13]O1=φ†φ∂µφ†∂µφ −14∂µφ†φ∂µ φ†φO2=φ†φ3(9)3

There are only two additional parameters here, instead of the five above. The relationshipbetween the ci and the parameters of eq.

(3) isξ −1=−34 c1v2Λ2ξ′ −1=3 c1v2Λ2ξ′′=12 c1v2Λ2λ3=3m2v2 +"32 m22v2!c1 −6c2# v2Λ2λ4=3m2v2 +"3 m22v2!c1 −36c2# v2Λ2(10)We may now consider the phenomenology of a non-standard Higgs resonance. At treelevel, the Higgs boson decay width to Goldstone Bosons isΓ(0)H = 3m332πv2ξ2(11)Note that ξ is the only parameter which appears [16, 17].The other parameters in eq.

(3) appear at one-loop. As usual, loops induce infinitieswhich can be absorbed in the effective Lagrangian in the traditional way [11, 18]: namelythe infinities associated with non-derivative interactions are absorbed in the renormaliza-tion of the scalar self couplings in eq.

(5), while the ones associated with vertices involvingderivatives are absorbed in the counterterms of order p4. In general, these introduce fur-ther unknown parameters in our amplitudes.

We compute the leading corrections in theMS scheme, setting the O(p4) counterterms to zero when the renormalization scale µ isequal to Λ. These results include the so-called “chiral logarithms”, which are the leadingcontributions if p2/Λ2 is sufficiently small [19], and in any case are expected to be compa-rable to the full O(p4) corrections [11].

In addition, when the parameters take the valuesof the linear sigma model (f →∞in eqs. (6) and (7)), the µ-dependence disappears(as it must for a renormalizable theory) and we show that our results reduce to thosepreviously computed in the standard Higgs model [20, 21].The one-loop corrections to the Higgs boson decay width in eq.

(11), written as ΓH =Γ(0)H + Γ(1)H , areΓ(1)HΓ(0)H=18π2(m2v2 (1 + L) + ξ′λ32ξ" π√3 −1#+ λ32v24m2"1 −2π√39#+ ξ′′m2ξv2 L4

+ ξ′3m22v213 + L+ ξ2m22v2"π26 −4 −L#−ξλ32"π√3 −3 −2π29#)(12)where L = 1 −ln(m2/µ2). While this result is µ-dependent, we can estimate the effect ofhigher-order interactions by setting µ = Λ.

In the linear sigma model limit our calculationreproduces the one-loop result of ref. [20] whereΓ(1)HΓ(0)H=m22π2v2 1916 −3√3π8+ 5π248!

(13)To get a feeling for the magnitude of the deviations, we set the parameters of eq. (12) tothe values5m = 750 GeV , ξ = 0.6 , ξ′ = −0.26 , ξ′′ = 0.66 , λ4 = 3.34 , λ3 = 20 , Λ = 2 TeV(14)The one-loop contribution increases the width by 24% while the corresponding increasefor the linear sigma model is only 8%.We now investigate the behavior of such a non-standard Higgs resonance in longitu-dinal Goldstone Boson scattering.

By expanding the Lagrangian in eq. (3), the tree-levelamplitude for w+w−→zz isAtr = sv2 − ξ2v2!s2s −m2 −Σ(s)(15)In order to consistently count powers of λ (≡m2/2v2) in the resonance region, we useΣ(s) = iImΠone−loopH(s), where ΠH is the Higgs boson self-energy function.

The first termcomes from the gauge boson contact interaction and corresponds precisely to the lowenergy theorems [14], while at somewhat higher energies deviations from the standardHiggs model emerge.Next, we compute the one-loop contributions to the w+w−→zz scattering amplitude.The full analytical expression will be presented in ref. [6].

Here we provide the result forthe I = J = 0 partial wave defined asa00 =116πsZ 0−s dt [3A(s, t, u) + A(t, s, u) + A(u, t, s)](16)where A(s, t, u) is the w+w−→zz amplitude, which includes the tree-level expressiongiven in eq. (15) (here with Σ(s) = Πone−loopH(s) + iImΠtwo−loopH(s)) and the one-loopcorrections calculated in the MS scheme.5These values correspond to a representative choice of parameters in the model described in [6].5

At energies small compared to the mass of the Higgs, the one-loop amplitude is :A(s, t, u) = sv2 +1(4πv2)2 M(17)M=s22 ln µ2−s + 16t(s + 2t) ln µ2−t + 16u(s + 2u) ln µ2−u+ s2 P + Q (t2 + u2) + R ln m2µ2(18)whereP=59 −4918ξ4 + ξ3λ34λ + ξ24 λ4λ −λ232λ2!+ ξ252ξ′ −3718+ ξ ξ′λ32λ + 2ξ′′ −λ32λ!+ ξ′ −λ44λ + λ238λ2 π√3 −1! (19)Q=1318 −119 ξ2 + 518ξ4(20)R=s2"λ44λ −ξ′ 1 + ξ′2!+ ξ λ32λ −2ξ′′!+ ξ2 76 −ξ′ −λ44λ!−ξ3λ32λ + 53ξ4#+ ξ232 −ξ2 t2 + u2(21)This reduces to the appropriate expression in the linear-sigma model limit [21].In Fig.

1 we show the modulus of the I = J = 0 partial wave as a function of √s forour benchmark parameter values eq. (14).

The peak in the one-loop curve is ∼9% lower(as well as being wider) than the tree-level peak. The corresponding curves for the SMHiggs are shown in Fig.

2. Note that both the tree-level and one-loop amplitudes in thelinear and nonlinear models violate (or, at least, do not saturate) unitarity on the peak.This is an indication that perturbation theory is not very accurate on the peak: eventhough the amplitudes obey unitarity to the appropriate order in perturbation theory 6,the higher order corrections are large.Qualitatively, however, for gauge-boson scattering below a TeV, the width appearsto be the most important feature differentiating a standard from a non-standard Higgsresonance7.6This has been studied in detail in the linear sigma model in [22].7For SSC phenomenology, the potentially non-standard coupling of the Higgs boson to the top-quarkwill also be important since it will affect the Higgs production rate.6

The sharp fall in the amplitude in the region above the peak in Fig. 1 can be understoodby noticing that for ξ < 1 the tree amplitude in eq.

(15) vanishes at some energy greaterthan m2 (if one does not include a finite width). This only signals that higher order effectsare expected to be significant there.

Also, far above the peak the amplitude presented isnot trustworthy due to the breakdown of the expansion in powers of 1/Λ.So far, we have ignored all gauge corrections. These are small in longitudinal gaugeboson scattering, as shown by the Equivalence Theorem.However, our non-standardHiggs boson does contribute to electroweak radiative corrections, including the Peskin-Takeuchi parameter S [5]S = −16π ddq2 Π3Y (q2)q2=0(22)Again, we will only compute the leading corrections [23] to S.Since Σ transforms under SU(2)L × SU(2)R as Σ →LΣR† the gauge bosons areincluded in the theory by replacing the ordinary derivative by the covariant oneDµΣ = ∂µΣ + ig2⃗τ · ⃗WµΣ −ig′2 BµΣτ3(23)The diagram shown in Fig.

3 contributes to Π3Y . We calculate this diagram using dimen-sional regularization (MS) and absorb the infinities in the p4 counterterms.

Then, aftersubtracting the contribution from “known” physics (the Standard Model with a Higgsboson of mass m0, usually taken to be 1 TeV [5]) we obtainS =112π(ln m2m20+1 −ξ2 "16 −ln m2µ2#)(24)where µ is the renormalization scale. For our benchmark parameters (14), m0 = 1 TeV,and µ = Λ we find S = +0.021.

If ξ were greater than 1, the contribution to S could evenbe negative, and slightly larger in absolute value than the corresponding contribution inthe standard model with the same Higgs mass.In this letter we have constructed an effective Lagrangian description of a non-standardHiggs boson. At lowest order in momentum and up to one-loop this description includes5 new parameters.

We have calculated the leading corrections to the Higgs boson width,w+w−→zz scattering, and the Peskin-Takeuchi S parameter.We have found that the most prominent effects are due to one new parameter, namelyξ, which is directly related to the Higgs boson width. The one-loop corrections do notqualitatively change the features of longitudinal gauge boson scattering, although a de-tailed analysis can only be done in a specific model.

If a “Higgs” is discovered, it will be7

important to see whether or not its properties are those predicted by the standard model.It remains to be seen how well the parameter ξ can be measured.AcknowledgmentsWe would like to thank E. H. Simmons, H. Georgi, M. Golden, D. Kominis, and S. Selip-sky for useful discussions and suggestions. R.S.C.

acknowledges the support of an Al-fred P. Sloan Foundation Fellowship, an NSF Presidential Young Investigator Award, aDOE Outstanding Junior Investigator Award, and a Superconducting Super Collider Na-tional Fellowship from the Texas National Research Laboratory Commission. This workwas supported in part under NSF contract PHY-9057173 and DOE contract DE-FG02-91ER40676, and by funds from the Texas National Research Laboratory Commissionunder grant RGFY92B6.8

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Figure captions• Figure 1 : The absolute value of the I = J = 0 partial wave for gauge bosonscattering with a non-standard Higgs boson (using the values shown in eqn. (14))at tree-level (solid) and one-loop (dotted).• Figure 2 : The absolute value of the I = J = 0 partial wave for gauge bosonscattering with a standard Higgs boson at tree-level (solid) and one-loop (dotted).• Figure 3 : The one-loop diagram contribution to the Peskin-Takeuchi S parameter.12

W 3B 3 Figure 3µνHw

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