Nonlocal Quark Model Description of a Composite Higgs Particle1
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ISSN 1547-4771, Physics of Particles and Nuclei Letters, 2017, Vol. 14, No. 3, pp. 429–434. © Pleiades Publishing, Ltd., 2017. PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY Nonlocal Quark Model Description of a Composite Higgs Particle1 Aliaksei Kachanovicha, * and David Blaschkea, b, c, ** aInstitute of Theoretical Physics, University of Wroclaw, PL-50204 Wroclaw, Poland bBogoliubov Laboratory of Theoretical Physics, JINR Dubna, RU-141980 Dubna, Russia cNational Research Nuclear University (MEPhI), RU-115409 Moscow, Russia *e-mail: [email protected] **e-mail: [email protected] Received December 13, 2016 Abstract—We propose a description of the Higgs boson as top-antitop quark bound state within a nonlocal relativistic quark model of Nambu–Jona-Lasinio type. In contrast to models with local four-fermion inter- action, in the nonlocal generalization the mass of the scalar bound state can be lighter than the sum of its con- stituents. A simultaneous description of the experimentally determined values for both, the top quark mass and the scalar Higgs boson mass, is achieved by adjusting the interaction range and the value of the coupling constant. DOI: 10.1134/S1547477117030098 1. INTRODUCTION technicolor scalar within a walking technicolor approach see, e.g., [20]. The Higgs boson is a key ingredient of the Standard A different approach is the minimal supersymmet- Model (SM), because it provides a description of ric standard model (MSSM) [21]. Supersymmetry × spontaneous electro-weak SU(2)LY U (1) symmetry could be a solution of the naturalness problem, breaking [1–3], for a review see [4]. Almost half a cen- because radiative corrections of fermions are divergent tury of experimental quest for this last undiscovered only logarithmically. Through supersymmetry trans- piece of the SM ended in 2012 when ATLAS and CMS formations scalar loop corrections follow the same experiments at CERN announced the discovery of pattern. However, a general problem with supersym- new particle, which matches requirements for the metric models is that they predict superpartner parti- Higgs boson [5, 6]. The measured decay rates, and in cles for all known bosons and fermions with the same particular two photon branching ratio, suggest that it is mass spectrum, distinguished from each other only by a short lived state of spin 0 and mass 125 GeV. Despite their spin. This has not been experimentally observed this obvious success, the theoretical interpretation of yet. This problem can be avoided by a very high this scalar field as a fundamental object leads to a num- (@ 1 TeV) supersymmetry violation scale. There are ber of conceptual problems: naturalness problem [7], various supersymmetric models which predict mea- problem of triviality [8] and the hierarchy problem. surable phenomena (see, e.g., [22, 23]). An alternative to the elementary Higgs was pro- There are also many exotic approaches to Higgs posed independently by Weinberg [9] and Susskind boson ranging from interpretations as a topological [10]. They proposed new fermions which were cou- object to models which avoid the naturalness prob- pled by a new strong force into a scalar particle. The lem by introducing extra dimensions (for a review interaction and the fermions were named technicolor see, e.g., [24]). and techniquarks, respectively. The theory predicts The first attempts to relate the symmetry break- masses of the “ordinary” quarks and leptons, which down of the standard model with a dynamical mecha- match phenomenological data after complicated nism that generates the quark masses and describes the manipulations [11, 12]. Even after introducing these Higgs as a scalar composite particle have been made by extensions technicolor had problems with flavour Terazawa et al. [25] in the attempt to unify particles changing neutral currents (FCNC) which have been and forces on the subquark level [26] by employing a solved only for the first two families by introducing model of the Nambu–Jona-Lasinio type [27, 28]. the walking coupling constant [13–15]. With techni- Focussing on the dynamical chiral symmetry breaking colour there occurred also a problem with precision in the top quark sector Nambu [29], Miransky et al. electroweak measurements [16–19]. For a recent [30, 31] and Bardeen, Hill and Lindner [32] developed description of the 125 GeV Higgs boson as a light further the idea to use the local Nambu–Jona-Lasinio (NJL) model for a description of the mechanism for 1 The article is published in the original. generating the bare mass mt of the top quark and the 429 430 KACHANOVICH, BLASCHKE Higgs boson as a scalar tt bound state with a mass of The quark mass is described by a gap equation and the mass of mesonic bound states is obtained from the 2mt before renormalization-group corrections (see, e.g., Ref. [33] for a review on this model for dynamical Bethe-Salpeter equation in the corresponding interac- chiral symmetry breaking and mesonic bound state tion channel. For the derivation of these equations we generation in the light quark sector of low-energy use standard procedures [33] (for the nonlocal gener- QCD). They also described the quark loop corrections alization see, e.g., [34]). ± to the W and Z 0 boson masses within the renormal- Using Hubbard–Stratonovich bosonisation ization-group setting of the standard model and gave [39, 40], we obtain the action in the form predictions for both, the Higgs boson and the top 2 (4) =−⎡⎤ γμ + σ +σ . quark masses. At the time when Ref. [32] was written, the SVtr{} ln⎣⎦ ( kμ ) gk ( ) (3) mass of Higgs boson and the mass of the top quark were 4G unknown. Nowadays it is plain that the application of the where tr{… } = N dk4 {}… strands for the trace in c ∫ (2π )4 trD local NJL model is not consistent with the data, basically color, Dirac and 4-momentum spaces, respectively. because of the scalar meson bare mass formula which Next we expand the effective action in powers of the yields 2mt for the composite Higgs mass. fluctuation δσ of the scalar meson field σ around its In this paper we propose to revisit the idea of a vacuum expectation value v, composite Higgs boson within a nonlocal generaliza- σ=v +δσ, (4) tion of the Nambu model, which describes a scalar quark-antiquark bound state with a mass that is below and obtain up to quadratic order the sum of the constituent quark masses. This has been 2 δσ 2 (4)=+vvδσ +() −⎡⎤− 1 , SVtr⎣⎦ ln GMF ( kk0 ) demonstrated for the light quark sector in ref. [34]. In 42GG 4 G this paper we shall apply the nonlocal NJL model for −,δσtr[] ( ) ( ) (5) the first time to the problem of spontaneous top quark GkkgkMF 0 mass generation and composite Higgs boson as a sca- −,δσ,δσ,1 tr()[]gkGMF ( kk00 ) gkG () MF ( kk ) lar tt bound state. We shall demonstrate within a sin- 2 gle-flavor model that the physical top and Higgs where the inverse mean field propagator is masses as they are known now from experiment can be −μ1 ,=γ− , described simultaneously. The generalization of such a GkkMF ()0 kmkμ () (6) model to the flavor doublet structure of the standard with the dynamical quark mass model is the straightforward along the lines of [32]. mk()=. gk ()v (7) This next step is deferred to future work. In order to assure stationarity of the action (5) with respect to small fluctuations about v, the contribution 2. NONLOCAL NAMBU QUARK MODEL linear in δσ has to vanish. This condition defines the FOR SCALAR BOUND STATE meanfield gap equation which after performing Dirac trace, color summation and k0 integration [33] takes In order to introduce the nonlocal effective theory the form we consider the ansatz of a local current-current ver- 3 tex, but with nonlocal particle currents. Early versions =,dk 2 v v 2()GNc ∫ g k (8) of the nonlocal generalization of the NJL model have (2π )3 Ek() been given, e.g., in [34–36], see also [37]. We will fol- low here the introduction of the covariant separable where N c is the number of colors, gk() is the form fac- model in [38], but specialize it to the instantaneous tor, which depends of a regularization method, and case as in [34]. The effective action for non-local Ek() is the relativistic dispersion relation Nambu model has the form Ek()=+ k22 m () k . (9) 4 ⎡⎤ S=−∂−,∫ dxqx()( i )() qxG JxJx ()() (1) The scalar quark-antiquark bound state which we ⎣⎦⎢⎥2 shall denote as Higgs boson is described by the Bethe- where as the main difference between to the local NJL Salpeter equation for the nonlocal theory. The inverse models for the current Jx() a nonlocal generalization −1 propagator GqqH0(), of the Higgs boson is defined by is introduced in the form [38] the terms in the action (5) which are quadratic in the sigma field fluctuations Jx()=+−,∫ dzgzqx4 ()( zz) qx( ) (2) 22 − Gqq1(),=−Π,,1 () qq (10) with gz() being a formfactor which is responsible for H02G H0 the spatial nonlocality of the current. Local models are where the polarisation function is defined as a limiting cases of this form for gz()=δ(4) () z. We will Π,()tr()()()qq =[ gkG kk , gk + q choose specific ansätze for this form factor when per- H0MF 0 (11) ×+,+ forming the numerical solutions in Sect. 3. GkqkqMF ().00] PHYSICS OF PARTICLES AND NUCLEI LETTERS Vol. 14 No. 3 2017 NONLOCAL QUARK MODEL DESCRIPTION OF A COMPOSITE HIGGS PARTICLE 431 25 Gaussian form factor Lorentzian form factor with α = 2 Lorentzian form factor with α = 1.01 20 Higgs value Critical couplings Λ / t m 15 10 Top quark mass Top 5 0 1 2 3 4 5 6 7 8 9 10 Coupling constant GΛ2 Fig.