Higgs Boson and Top-Quark Masses and Parity-Symmetry Restoration

The recent ATLAS and CMS experiments show the first observations of a new particle in the search for the Standard Model Higgs boson at the LHC. We revisit the scenario that high-dimensional operators of fermions must be present due to the theoretical inconsistency of the fundamental cutoff (quantum gravity) with the parity-violating gauge symmetry of the Standard Model. Studying the four-fermion interaction of the third quark family, we show that at an intermediate energy threshold E\approx 4.27 TeV for the four-fermion coupling being larger than a critical value, the spontaneous symmetry-breaking phase transits to the strong-coupling symmetric phase where composite Dirac fermions form fully preserving the chiral gauge symmetry of the Standard Model and the parity-symmetry is restored. Under this circumstance, we perform the standard analysis of renormalization-group equations of the Standard Model in the spontaneous symmetry-breaking phase. As a result, the Higgs boson mass m_H\approx 126.7 GeV and top-quark mass m_t\approx 172.7 GeV are obtained without drastically fine-tuning the four-fermion coupling.

in 1989 suggested that the symmetry breakdown of the Standard Model could be a dynamical mechanism of the Nambu-Jona-Lasinio or BCS type that intimately involves the top quark at a high-energy scale Λ. This dynamical mechanism leads to the formation of a low-energyttcondensate, which is responsible for the top quark, W ± and Z • gauge bosons masses, and a composite particle of the Higgs type. Since then, many models based on this idea have been proposed and studied [9]. For our following discussions, we will adopt the model for the minimal dynamical symmetry breaking via an effective four-fermion operator of the Nambu-Jona-Lasinio which was studied by Bardeen, Hill and Lindner (BHL) [8] in the context of a well-defined quantum field theory at the high-energy scale Λ. The fermion fields in L kinetic are massless, and the four- To achieve the low-energy electroweak scale for the top quark mass m t by the renormalization group equations [6,8,10], this model (1) requires Λ/m t ≫ 1 with a drastically unnatural fine tuning, which is known as the gauge hierarchy problem, and the top quark mass m t is determined by the infrared quasi-fixed point [10]. To have a natural scheme incorporating the effective fourfermion operator of the Nambu-Jona-Lasinio type (1), some strong technicolor dynamics at the TeV scale were invoked [11]. This scheme is preferentially coupled to the third quark family of top and bottom quarks. The possibility of the 126 GeV particle being a light pseudoscalar, such as the top-pion [10], seems unlikely because the loop-suppressed couplings of light pseudoscalars to the Standard Model gauge bosons are too small to generate the observed signal [12]. These discussions indicate that much effort is still required to study the issue of the minimal dynamical symmetry breaking that is preferentially associated with the top quark (the top-Higgs system) in the theoretical aspects of dynamics or/and symmetry (see for example [13]) to discover if the issue agrees with experiments.
Suppose that the effective high-dimensional operators of all fermion fields, for example Eq. (1), are generated by the new dynamics at the scale Λ, which will be discussed in the end of the Letter.
It is conceivable that the new dynamics at the scale Λ should be on an equal footing with all the fermions in the Standard Model because the scale Λ is much larger than the masses of all the fermions. This raises a neutral question: why should the new dynamics preferentially act on the top-quark alone? In our recent Letter [14], we understand, from the dynamical point of view, a compelling possible answer to this question by studying the following effective four-fermion operator: where a, b and i, j are, respectively, the color and flavor indexes of the top and bottom quarks, the left-handed doublet ψ L = (t L , b L ) and the right-handed singlet W ± µ and Z • . As more fermions acquire their masses by the spontaneous chiral symmetry breaking, more associated scalar and pseudoscalar modes are produced. As a result, the energetically favorable configuration is the one in which only one quark (the top quark) acquires its mass by the spontaneous chiral symmetry breaking, with three pseudoscalar modes as the longitudinal modes of the massive gauge bosons and a scalar particle of the Higgs type. In addition, we discussed the strong-coupling symmetric phase where composite Dirac fermions form and the vector-like feature of W ± -boson coupling, which leads to the explicit symmetry breaking for generating masses of other fermions.
In this Letter, we present the study of strong four-fermion interaction of the third quark family, and show that at an intermediate energy threshold E ≈ 4.27×10 3 GeV for the four-fermion coupling being larger than a critical value G crit = 2N c (π/Λ) 2 , the spontaneous symmetry-breaking phase transits to the strong-coupling symmetric phase where composite Dirac fermions form fully preserving the chiral gauge symmetry of the Standard Model and the parity-symmetry is restored. Taking duly into account this phase transition, we perform the standard analysis of renormalization-group equations of the Standard Model in the spontaneous symmetry-breaking phase. As a result, the Higgs boson mass m H ≈ 126.7GeV and top-quark mass m t ≈ 172.7GeV are obtained without drastically fine-tuning the four-fermion coupling. The natural unitsh = c = 1 are adopted, unless otherwise specified.
The weak-coupling phases. Employ the "large N c -expansion" for weak coupling G, i.e., keep GN c fixed and construct the theory systematically in powers of 1/N c . At the lowest order, one has the gap equation for the induced top-quark masses m t = −G t t : In addition to the trivial solution m t = 0, the gap equation (3) has a non-trivial solution m t = 0 when the coupling G ≥ G c ≡ 8π 2 /(N c Λ 2 ), where G c is the "critical" weak-coupling constant. The theory (1) is in the weak-coupling symmetric phase m t = 0 for G < G c , or in the symmetrybreaking phase m t = 0 for G > G c . The result (4) is the leading order of large N c -expansion, it becomes exact in the weak-coupling limit: The strong-coupling symmetric phase. In the strong-coupling limit Ga −2 ≫ 1, where we introduce the lattice spacing a ≡ (π/Λ), the theory (1) is in the strong-coupling symmetric phase (see Refs. [15,16]). Using the Lagrangian (2), we briefly review the strong-coupling symmetric phase based on Ref. [16]. In order to perform the strong coupling expansion in powers of 1/g, we rescaled all fermion fields to dimensionless fields, and rewritten the fermion action in terms of the dimensionless fields on the lattice where all weak gauge couplings are neglected. In the strong coupling limit ga 2 ≫ 1, treating the kinetic action S kinetic as a small perturbation, we calculated two-point function of fermion fields by the strong coupling (hopping) expansion in powers of 1/g. As a result, in the lowest non-trivial order we obtained the propagators (p µ a < 1) of the composite massive Dirac fermions: where the composite three-fermion states are: [Z S F ] and M are respectively the form-factor (wave-function renormalization) and mass of composite Dirac fermions. We need to stress that the composite Dirac fermion propagator (8), Z S F = 1 and M = 2ga/Z S F are obtained by considering S int of Eq. (7) and only one "hopping" step (1/g 1/2 ) of Eq. (6) at the cutoff scale a = π/Λ. It is difficult to do the calculations of many "hopping" steps to obtain the energy-momentum dependence of Z S F (p) and M (p) down to some scales µ smaller than the cutoff Λ.
In the strong-coupling symmetric phase, the three-fermion state Ψ ib . The discussions are the same for the bottom quark t Ra → b Ra . These three-fermion states (9) carry the appropriate quantum numbers of the chiral gauge group of the Standard Model that accommodates ψ ia L and t Ra . Therefore massive composite Dirac fermions are consistent with the chiral symmetry SU L (2) ⊗ SU R (2) [18], and their couplings to intermediate gauge bosons are vector-like [19,20], for example where g 2 is the SU L (2) coupling, U ij the CKM matrix, p, p ′ and q are respectively composite fermion and gauge boson momenta. The vector-like form factor f (p, p ′ ) of chiral-gauge coupling (10) is related to the chiral-symmetric mass M (p) in Eq. (8) by the Ward identity of chiral gauge symmetries. Consequently the parity-symmetry is conserved in this strong-coupling symmetric phase [16,19].
In order to determine the critical value g crit that separates the strong-coupling symmetric phase from the symmetry-breaking phase (m t = 0), we calculated the two-point functions of composite boson fields H i by the strong coupling (hopping) expansion in powers of 1/g [16]. As a result, in the lowest non-trivial order we obtained the propagator of massive composite bosons H i (q µ a < 1), where [Z S H ] 1/2 = 1 and µ H respectively are the form-factor and mass of composite bosons. Thus, µ 2 H HH † gives the mass term of the composite bosons H in the effective Lagrangian.
In the lowest non-trivial order of the strong-coupling expansion, the contribution to the 1PI vertex-coupling λ 0 of the self-interacting term (HH † ) 2 is suppressed by (1/g) 2 . The 1PI vertexcoupling λ 0 > ∼ 0 is small, but positive, the energy of ground states of the theory is bound from the bellow. The mass term µ 2 H HH † changes sign from µ 2 H > 0 to µ 2 H < 0, indicating a spontaneous symmetry breaking SU (2) → U (1) occurs, and non-zero vacuum expectational value (v = 0) is developed. Eq. (11) for µ 2 H = 0 gives rise to the critical strong-coupling G crit : where the second order phase transition from the strong-coupling symmetric phase to the symmetry-breaking phase takes place. Note that the inequality is valid, considering that G c should be calculated for N c ≫ 1.
Because we are not able to obtain the energy-momentum dependence of form-factors and masses of composite fermions and bosons, and other 1PI-functions of high-dimensional operators, as well as their renormalization group equations in the neighborhood of the second order phase transition, therefore we cannot give a detailed description of the dynamics occurring at the phase transition.
However, we conceive that fermion energy-momenta (p, p ′ ) and energy transfer (q) decrease down to the certain energy threshold E, the effective interacting vertex Γ (4) (p, p ′ , q) of Eq. (2) becomes small enough that the binding energies E bind [G(a), a] of the three-fermion bound states (9) vanish. As a result, these bound states (9) dissolve into their constituents [21], the mass term M (p) vanishes and the vector-like form factor f (p, p ′ ) → P L = (1 − γ 5 )/2. This restores the chiral-gauged fermion spectra and couplings, as described by the Standard Model [22]. We postulated [16,19] that the where v is the electroweak breaking scale. Numerical non-perturbative calculations are required to verify the postulation.
Renormalization-group boundary condition at high energies. First let us consider the four-fermion coupling G of the quantum field theory (1) defined at the high-energy scale Λ is smaller than the critical value G crit , i.e. G < G crit . Therefore the theory is in the symmetrybreaking phase, contains the spectra of fundamental fermions ψ and composite bosons H. Following the prescription of Ref. [8], at a renomalization scale µ below the scale Λ the effective Lagrangian of the theory is written as where g 2 t0 (Λ)/m 2 H (Λ) = G and m H (Λ) = Λ. The conventional renormalization Z ψ = 1 for fundamental fermions and the unconventional wave-renormalization Z H = 1 for composite Higgs bosons H are adopted. Thus the coupling constants, such asḡ t andλ are renormalized at the scale μ where Z HY and Z 4H are proper renormalization constants of the Yukawa-coupling and quartic vertex in Eq. (14). The Higgs field H is dynamical with a vanishing wave-function renormalization constant at the scale Λ, leading to the following boundary conditions In Eq. (14), the transformation H → H/ḡ t (µ) transforms the conventional normalization into the that required by Eq. (16), one thus has where the tilde will henceforth denote the normalization convention appropriate for compositeness.
We turn to the situation that the strong-coupling symmetric phase appears for G > G crit . In this phase, according to the massive spectra (8,11) of composite Dirac fermions and bosons we obtained, the effective Lagrangian can be written as (µ 2 H > 0) at the scale µ (E < µ < Λ) being smaller than the scale Λ but larger than an intermediate energy we cannot calculate their evolutions with the scale µ from Λ to E. The boundary condition (16) at the scale Λ should be modified into the following boundary condition at the scale E, The top-quark and Higgs boson masses. In the Standard Model of particle physics, using the full one-loop β-functions (neglect light-quark masses and mixings), the renomalization-group equations for running couplingsḡ t (µ 2 ) andλ(µ 2 ) are where A = 1 4ḡ and, for running gauge couplings of SU c (3), SU L (2) and U Y (1) are with where N g = 3 is the number of fermion families and t = ln µ. Adopting M z ≈ 91.2GeV, M w ≈ 80.4GeV, v ≈ 239.5GeV, gauge couplingsḡ 2 1 (M z ) ≈ 0.13,ḡ 2 2 (M z ) ≈ 0.45 andḡ 2 3 (M z ) ≈ 1.5, we use the mass-shell condition to determine the top-quark mass and the Higgs boson mass The determined energy threshold E value, which is about 18 times larger than the electroweak breaking scale v, has some physical consequences. The quadratic divergence Λ 2 in the gap-equation where G c ≈ 8π 2 /N c E 2 . The unnatural fine-tuning problem is greatly soften by setting the fourfermion coupling G/G c = 1 + O(m 2 t /E 2 ) and m 2 t /E 2 ≈ 1.64 × 10 −3 , instead of the drastically fine-tuning the four-fermion coupling, G/G c = 1 + O(m 2 t /Λ 2 ) for Λ ≫ m t . In this case, one can have the physically sensible formula that connects the pseudoscalar (coupling to the longitudinal W and Z) decay constant f π to the top-quark mass (see [8]): without a drastic fine-tuning, where G F = 1/ √ 2v 2 is the Fermi constant.
Some remarks. We are not able to non-perturbatively calculate the energy threshold E, where σ L (σ R ) is the cross-section of high-energy (> E) particle colliding with left-handed (righthanded) polarized particles. The signal A LR → 0 indicates the restoration of the parity-symmetry.
To end our Letter, we present a brief discussion what is the possible dynamics at high-energy scale for the origin of effective high-dimensional operators of all fermions fields. Usually composite models for top-quark and Higgs scalar are based on an extended gauge group (strong technicolor) at a higher scale (see for example Ref. [11]). What is a possible completion of the theory in this Letter at an even higher scale? We present, on the basis of our previous works on this issue, a brief discussion on the origin of high-dimensional operators of all fermion fields due to the quantum gravity at the Planck length (a pl ∼ 10 −33 cm, Λ pl = π/a pl ∼ 10 19 GeV). Studying the quantum Einstein-Cartan theory in the framework of Regge calculus [24,25], we recently calculated this minimal length a ≈ 1.2 a pl [26]. This discrete space-time provides a natural regulator for local quantum field theories of particles and gauge interactions. Based on low-energy observations of parity violation, the Lagrangian of Standard Model was built in such a way as to preserve the exact chiral gauge symmetries SU L (2) ⊗ U Y (1) that are accommodated by left-handed fermion doublets and right-handed fermion singles. However, a profound result, in the form of a generic no-go theorem [27,28], tells us that there is no consistent way to straightforwardly transpose on a discrete space-time the bilinear fermion Lagrangian of the continuum theory in such a way as to preserve the chiral gauge symmetries exactly, one is led to consider at least quadrilinear fermion interactions to preserve the chiral gauge symmetries. For example, the four-fermion operator in the Einstein-Cartan theory can be obtained by integrating over static torsion fields at the Planck scale [29]. The very-small-scale structure of space-time and high-dimensional operators of fermion interactions must be very complex as functions of the space-time spacingã and the gravitational gauge-coupling g grav between fermions and quantum gravity at the Planck scale. We are bound to find an ultra-violet fix point of the gravitational gauge-coupling [30]. As the running gravitational gauge-coupling g grav (ã) is approaching to its ultra-violet critical point g crit grav forã → a pl , physical scale Λ = ξ −1 [g grav (ã),ã] ≪ã −1 should satisfy the renormalization group invariant equation in the neighborhood of the ultra-violet fix point, where the irrelevant high-dimensional operators of fermion interactions are suppressed at least by O(Λ/Λ pl ); only the relevant operators receives anomalous dimensions and become renormalizable dimension-4 operators at the scale Λ and their effective couplings is larger than the critical value (12). This is a complicate and difficult issue and needs non-perturbative calculations to show such scaling phenomenon.