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ISSN 1547-4771, Physics of and Nuclei Letters, 2017, Vol. 14, No. 3, pp. 429–434. © Pleiades Publishing, Ltd., 2017.

PHYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY

Nonlocal 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 as top-antitop quark within a nonlocal relativistic of Nambu–Jona-Lasinio type. In contrast to models with local four- 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 mass and the scalar 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 (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 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 , which matches requirements for the metric models is that they predict parti- Higgs boson [5, 6]. The measured decay rates, and in cles for all known and fermions with the same particular two 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” and , 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 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 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. 1. Dimensionless mass of the top quark vs. dimensionless coupling constant GΛ2 . The square boxes indicate the values of the dimensionless coupling for which the ratio of Higgs mass to top quark mass assumes the physical value, see Fig. 2.

The scalar meson mass is obtained by the mass pole Higgs boson as a composite of two top quarks is a condition defined with the polarization function for a strongly bound state! In the following section we per- meson at rest (q = 0 ) as form the corresponding numerical solutions for this model for three examples of formfactor functions −Π, =. (12) 12GmHH (0) 0 defining the nonlocal model and demonstrate that After evaluating the traces implied in the defini- indeed the wanted solutions can be found. tion (11) and introducing the notation for averages of a momentum-dependent quantity Ap() (for details, see [34]), 3. RESULTS ⎡⎤2 In the chiral limit, quarks remain massless until a 2 gp() Ap () Adpp= ⎢⎥∫ critical value for the dimensionless coupling GΛ 2 is ⎣⎦Ep()Ep22()− m 4 H Λ 2 −1 (13) reached. This critical value Gc depends on the ⎡⎤2 choice of the form factor. In our model calculations we ×,2 gp() 1 ⎢⎥∫ dpp 22 will employ two types of form factors, namely the Ep() ()− 4 ⎣⎦Ep mH Gaussian and generalized Lorentzian type, we obtain the Higgs boson mass formula 2 ⎛⎞k 22=− 2 −. 2 gk()=− exp⎜⎟ , (15) mmH 4 (0) 4 m (0) mp ( ) (14) G Λ ⎝⎠G This Higgs boson mass formula (14) is the key result of this paper. It explains how for a nonlocal =,1 gkL() 2α (16) quark model the mass of the scalar bound state can be ⎛⎞k 1 + ⎜⎟ lighter than the sum of the masses of its quark constit- ⎝⎠Λ uents due to their momentum dependence. L When the quark mass function drops with increas- where Λ is the effective range of the interaction in ing momentum as in the cases of the example form momentum space and α is a parameter regulating the factors of our nonlocal quark model, the second term shape of the generalized Lorentzian formfactor. The in Eq. (14) leads to a reduction of the scalar meson latter facilitates the regularization of the otherwise mass and makes it a true bound state with a finite divergent loop integrals for the original Lorentzian binding energy. This fact allows the simultaneous formfactor (α=1 ). description of the Higgs boson as a scalar bound state The solutions of the quark mass gap equation (8) of a top-antitop quark pair with their physical masses, are shown in Fig. 1 for the top quark mass =. ==== for which holds mmH 0718 t . This means that the mmkt (0)(0) gvv in units of the interaction

PHYSICS OF PARTICLES AND NUCLEI LETTERS Vol. 14 No. 3 2017 432 KACHANOVICH, BLASCHKE

2.0 Gaussian form factor 1.8 Lorentzian form factor with α = 2 Lorentzian form factor with α = 1.01 1.6 Higgs/top quark Physical value

t 1.4 m /

H 1.2 m

1.0 Ratio Ratio 0.8 GΛ2 = 3.35 GΛ2 = 7.74

0.6 GΛ2 = 6.01

0.4

0.2 1 2 3 4 5 6 7 8 9 10 Coupling constant GΛ2

Fig. 2. Ratio of the Higgs boson and top quark masses in dependence of the dimensionless coupling GΛ2 for three form factor models. The square boxes denote the values of the dimensionless coupling for which the mass ratio assumes the physical value, shown by the thin, red dash-double-dotted line. range Λ. The values of the critical coupling are indi- by a thin red dash-double dotted line. We show in this cated by the fancy cross in Fig. 1 and their values are figure the possible values of the ratio mmH t for differ- given in the first column of table. ent formfactors. For all formfactors one finds a value Only massive quarks can form a bound state. The for GΛ 2 matching the experimental constraint. Using maximal mass of a true bound state is given by the sum in addition the dependence of the dimensionless top of the masses of its constituents which for a quark- Λ Λ 2 antiquark bound state is twice the constituent mass. quark mass mt on G as shown in Fig. 1, one can Λ The ratio of the Higgs boson mass to the top quark fix the cut-off parameter and the coupling constant =. G . The resulting values are presented in table. mass is mmH t 0 718 . In Fig. 2 this value is marked The values of coupling constant G are about two orders of magnitude larger than the Fermi constant of Λ2 =. × −−52 Dimensionless critical coupling constant Gc for the onset the weak interaction GF 1 664 10 GeV . This of spontaneous chiral symmetry breaking (first column), fact may be attributed to the nonlocality of the model the value of the dimensionless coupling for which the physical as opposed to the local Fermi model. It would be ratio of Higgs to top quark mass is obtained (second column, interesting in an extension of this study to investigate Λ see Fig. 2), the cut-off parameter which follows for this value other formfactors of the nonlocal NJL model and the physical top quark mass (third column) and corre- approach. sponding coupling constant G (last column). All values are given for three different form factors gx() 4. CONCLUSIONS Λ, G , Λ2 Λ2 Gc G GeV –3 –2 We have revisited the dynamical top quark mass 10 GeV generation mechanism that was studied by Bardeen, Hill and Lindner within the local NJL model now Lorentzian, α=101 . 1.70 3.35 42.21 1.88 within a nonlocal generalization and obtained a com- posite Higgs boson as a scalar top-antitop quark bound state with a mass well below that of the local Lorentzian, α=200 . 2.10 7.74 42.85 4.16 = NJL mass formula for which mmHt2 before renor- malization-group correction. We isolated the effect of Gaussian 3.28 6.01 58.10 1.79 nonlocality in a generalized Higgs boson mass formula that allowed us to describe simultaneously the now

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PHYSICS OF PARTICLES AND NUCLEI LETTERS Vol. 14 No. 3 2017