Phenomenology of Vector Like Quarks in Composite Higgs Models

Michael Eggleston Department of Physics, Duke University (Dated: October 17, 2017)

Abstract: Composite Higgs Models ﬁt inside one of many frameworks proposed to explain physics beyond the Standard Model. Such models predict a new strong force under which heavy, vector-like quarks (VLQs) interact with the Higgs boson, the vector Gauge bosons, and the third generation of quarks. A consequence of this interaction also shows a degree of composite structure of the Top quark. This will be shown to address the Naturalness Problem. The focus of this paper will introduce phenomenology of the generic framework, and examine several particles common to speciﬁc models. We will discuss the most probable Standard Model decay channels in which a search could be performed at the Large Hadron Collider, looking for mass resonances in the 1 – 7 TeV range.

I. INTRODUCTION

The Standard Model (SM) has provided the ability to perturbatively calculate precise predictions of fundamental measureable quantities such as decay rates and cross sections. One element that has evaded precise prediction however, is the Higgs boson mass. A scalar spin 0 particle, the Higgs boson has a problem when the mass is calculated in a perturbative expansion. As explained in Ref. [1], a perturbative expansion of the Higgs mass introduces corrective terms that are quadratic in the energy scale of the SM, denoted ΛSM . When the 19 scale is taken to a large value, for example on the order of the Planck mass MP ∼ 10 GeV, the mass must be adjusted to a precision on the order of 10−38. Otherwise the mass will only be calculable to the scale of ΛSM . Because the mass is known to be approximately 125

GeV, the next suggestion is that ΛSM must take a smaller value. In such a case, the mass hierarchy problem can be posed as the question: what new physics describes the region between ΛSM and MP ? One such collection of models that aim to explain the relatively small Higgs mass, posit that the Higgs boson is a pseudo Nambu-Goldstone boson (pNGB), arising from explicit and spontaneous symmetry breaking of a “Composite group” G. These are called the Composite Higgs Models (CHMs) [2]. As a consequence of the necessarily 2 large dimension of G, there also arise heavy vector-like quarks that interact via a new strong force. While the VLQs may have several various fractional electric charges, they are all generically referred to as top partners, as will be explained in Sec. II.

The evidence of a pNGB Higgs is most readily supplied by observing decays of the VLQs. This will be addressed in Section II after summarizing the theory, and providing a qualitative argument for how the heavy top partners can produce a natural Higgs mass. Section III provides details on how a search for the VLQs will be impacted by the mass range being considered, in addition to the exclusions calculated from Run One of the LHC.

II. PHENOMEONOLOGY

From the Goldstone theorem, massless bosons exist for every generator that sponta- neously breaks a continuous symmetry [3]. The quintessential example of this theorem is illustrated by SU(2) transformations of the Higgs scalar doublet, where one of four vacuum states is not invariant under weak interactions, forming the nonzero Higgs vacuum expecta- tion value (vev), while the remaining three represent the Goldstone bosons. However, when the symmetry is also explicitly broken, the bosons need not be massless.

Most often considered in the literature are CHMs with a spontaneous SO(5) → SO(4) breaking, also called Minimal CHMs (MCHM) [4]. This spontaneous breaking occurs at a scale called the Goldstone decay constant, denoted f. In the language of Sec. I, SO(5) is the composite group G, under which heavy fermionic operators transform. Moreover, we call SO(4) the “Elementary group”. While models are differentiated in large part by the multiplet representations tranforming under the composite and elementary groups, there are also some cases where higher dimensional breaking is considered, such as SO(6) → SO(4) among others. Regardless of the model specifics, the goal becomes finding the intersection between the composite and elementary groups, which also contain as a subgroup the SM

SUc(3)×SUL(2)×UY (1) group. This is to say using the MCHM example, we are concerned with ﬁnding the group elements common to SO(5), and SO(4) that also have a subset of representations identical to those of the Standard Model group. This places a requirement that the VLQs interact with and decay into SM particles. 3

The Lagrangian we wish to study must contain couplings between the left and right handed heavy fermionic fields, and the elementary quarks. When the fields themselves are vector-like, the couplings may be assumed to enter as linear parameters. Furthermore, because the heavy fermionic operators live in the same composite space as the pNGB Higgs, there is also a direct coupling between the Higgs and fermion fields [5]. Following the same procedure as with the SM by diagonalizing the Lagrangian, the physical mass eigenstates are obtained. However, with the previous couplings in mind, the final outcome is a physical top partially composed of the elementary tops and partially mixed with its composite partners. The degree to which one is favored over any other is parameterised by a mixing angle, in a measure called “Partial Compositeness”.

In order to generate a Higgs mass, the Lagrangian symmetries must be broken. This is achieved in the same manner as in the SM, with yukawa couplings of fermions to the Higgs. Because the elementary particles are not invariant under the composite group, they break the symmetry, and produce a Higgs potential. By considering a top quark loop, the potential takes on a quartic form to generate ElectroWeak Symmetry Breaking (EWSB). Moreover, the general form of the potential has two competing terms, one being quadratic and the other quartic. Without deriving the expressions here (see Ref. [4] for details), by comparing the newly formed Higgs potential with its mass mH derived from the SM scalar

ﬁeld Lagrangian, it becomes possible to relate mH to the composite top Yukawa coupling.

The ﬁnal result of MCHM shows a linear relation between mH and the mass of the lightest top partner, modulo factors of π, and the Goldstone decay constant f.

Of importance in the treatment of the Higgs potential is to constrain the quadratic and quartic contributions so as to achieve the EWSB potential, with a properly displaced global minimum. This is made possible by scanning in the region of EWSB by ﬁne tuning a parameter ξ ≡ sin2(v/f) ≈ v2/f 2 1, where v is the Higgs vev [2]. Typical values used in plotting the model dependent parameter space, constrained by ElectroWeak precision tests, are in the ξ = [0, 0.2] region. As an example calculation, by taking the Higgs vev v = 246 GeV, and for a small value ξ = 0.01, we can estimate f ≈ 2.5 TeV. This value represents a crude upper bound to the lightest resonance masses.

Because inividual models are largely diﬀerentiated by considering the various multiplet 4 representations that transform under the SO(5) generators, there exist many model dependent fermionic states. Common to most models however, are the fourplet and singlet states. While the fourplet contains two partners with quantum numbers the same as the mass eigenstate top and bottom, there are also two exotic top partners in the multiplet.

It is perhaps more intuitive to further break the fourplet down into two SUL(2) doublets, containing the resonances labeled X5/3, X2/3, T , B. Here the subscripts are electric charge. The 5/3 and 2/3 particles form one doublet, while the T and B form the second. The latter two have identical quantum numbers to the t,b quark doublet, as their labels suggest, while having larger masses than the X doublet.

III. VLQ PRODUCTION

In general, it is agreed that the X5/3 is the lightest top partner, with the X2/3 the next most massive state. As such, these represent the most readily measureable resonances at the LHC. Notice that the exotic charge state can only experience charged ﬂavor changing + weak current, or decay into a W t when interacting with SM particles. The X2/3 however may experience neutral interaction, in association with either a Z or Higgs boson, into an elementary top [2]. The allowable interactions are fairly straight forward to list for any given top partner, keeping in mind charge and parity conservation.

Central to examining the production mechanisms is to notice that the fermions have color charge, just like their elementary counterparts. As such, this shows that production by QCD interactions will have leading order contributions in the pair-wise and single channels. Figures 1 and 2 show the relevant Feynman diagrams for each of these processes, with VLQs denoted by ψ.

The most recent searches performed at the ATLAS detector from LHC Run One have placed bounds on pair production of the fourplet and singlet states at a largest value of approximately 900 GeV. Single production has also been excluded up to this range, however it is expected to become the dominant process for masses above 1 TeV [4]. 5

ψ q ψ

ψ¯ q¯ ψ¯

(a) Gluon fusion as a leading order contribution. (b) qq¯ annihilation with gluon mediator.

FIG. 1: Two of three leading order contributions to pair produced VLQ ﬁnal states.

q q0 q q0

V V

ψ t¯(¯b)

t¯(¯b) ψ

(a) Charged or neutral weak current flavor (b) Charged or neutral current flavor changing change. pair into single VLQ final state.

FIG. 2: Leading order contributions for single production in the ﬁnal state.

CONCLUSION

We have shown that a composite sector giving rise to a pNGB Higgs and heavy VLQs, can produce a light and natural Higgs. This is due to the composite structure of the top quark, and relatively small scale at which the new strong dynamics could explain physics beyond the Standard Model. This point is most essential, because the EWSB is predominantly controlled by the lightest top partners. We have further illustrated the most likely channels to search for signatures of VLQs at the LHC, which will beneﬁt from focus on the single production for masses above the current exclusion limits.

[1] L. Susskind, Phys. Rev. D 20, 2619 (1979). 6

[2] G. Panico and A. Wulzer, “The Composite Nambu–Goldstone Higgs,” arXiv:1506.01961v2 [hep- ph] (2015). [3] R. Tenchini and C. Verzegnassi, The Physics of the Z and W Bosons (World Scientiﬁc, 2008). [4] O. Matsedonskyi, G. Panico, and A. Wulzer, JHEP 01, 164 (2013). [5] C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40, 100001 (2016).