# Bootstrap Dynamical Symmetry Breaking

Authors: Wei-Shu Hou
Journal: Advances in High Energy Physics
 (2013)

---

## Abstract

Despite the emergence of a 125 GeV Higgs-like particle at the LHC, we explore the possibility of dynamical electroweak symmetry breaking by strong Yukawa coupling of very heavy new chiral quarksQ. Taking the 125 GeV object to be a dilaton with suppressed couplings, we note that the Goldstone bosons G exist as longitudinal modes VL of the weak bosons and would couple to Q with Yukawa coupling λQ. With mQ≳700 GeV from LHC, the strong λQ≳4 could lead to deeply bound QQ¯ states. We postulate that the leading “collapsed state,” the color-singlet (heavy) isotriplet, pseudoscalar QQ¯ meson π1, is G itself, and a gap equation without Higgs is constructed. Dynamical symmetry breaking is affected via strong λQ, generating mQ while self-consistently justifying treating G as massless in the loop, hence, “bootstrap,” Solving such a gap equation, we find that mQ should be several TeV, or λQ≳4π, and would become much heavier if there is a light Higgs boson. For such heavy chiral quarks, we find analogy with the π−N system, by which we conjecture the possible annihilation phenomena of QQ¯→nVL with high multiplicity, the search of which might be aided by Yukawa-bound QQ¯ resonances.


---

## Body

## 1. Introduction and Motivation

The field of particle physics is in a state of both jubilation and anxiety. On one hand, the long-awaited Higgs boson seems to have finally emerged [1, 2] at ~125 GeV at the LHC, where a hint appeared already with 2011 data [3, 4]. The case has been further strengthened with more data added by end of 2012, and the final results for the 7-8 TeV run would be revealed by the Moriond meetings in 2013. On the other hand, there appears to be no New Physics below the TeV scale, and one is worried what really stabilizes the Higgs mass at 125 GeV.As much as the existence of a 125 GeV boson is beyond doubt, we note that it is not yet experimentally established that itis the Standard Model (SM) Higgs boson. In part because of the enhancement in γγ mode—for both ATLAS and CMS, and for both 7 and 8 TeV data—and also because the subdominant production channels are not yet experimentally firm, the 125 GeV object could still be a dilaton [5–7]. The dilaton couples like a Higgs boson, but with couplings suppressed by v/f, where v is the observed vacuum expectation value (v.e.v.) of electroweak symmetry breaking (EWSB), and f is the “dilaton decay constant” that is related to scale-invariance violation. The dilaton couplings to gg and γγ, however, are determined by the trace anomaly of the energy-momentum tensor and would depend on UV details. Keeping v/f and effective gg and γγ couplings free, it is found [6] that a “dilaton” interpretation is as consistent as an SM-Higgs. To reject the dilaton, one has to establish the subdominant vector boson fusion (VBF)  and Higgsstrahlung, or associated production (VH) processes at the expected SM level. Judging from the results available at the end of 2012, it seems [8] that the issue would have to await the restart of the LHC at 13 TeV.Regardless of whether the 125 GeV object is the SM Higgs boson or a dilaton, the Higgs mechanism is an experimental fact. That is, the electroweak (EW) gauge symmetry is experimentally established, while the gauge bosons, as well as the chirally charged fermions, are all found to be massive, in apparent violation of the gauge symmetry. Thus, the Goldstone particle of EWSB gets “eaten” by the EW gauge bosons, which become massive (the Meissner effect), as has been experimentally established since 30 years. The v.e.v.v is simply related to the venerable Fermi constant GF.With both quarks and gauge bosons massive, a heuristic argument was used to demonstrate [9] that, starting from the left-handed vector gauge coupling, the longitudinal component of the EW gauge boson, that is, equivalently the Goldstone boson G, couples to quarks by the SM Yukawa coupling, with both left- and right-handed components. Thus, Yukawa couplings are experimentally established. Furthermore [9], much of flavor physics and CP violation (CPV) studies probe the effects of Yukawa couplings, providing ample and highly nontrivial support for their “complex” existence.A natural question to ask is as follows: given three quark generations already, could there be a fourth copy? Does it carry its ownraison d’être? It is not our purpose to discuss in detail the issues, merits, and demerits regarding this possible fourth generation (4G), which we refer to [10]. At face value, we admit that the observed 125 GeV new boson would pose a difficulty. The main issue is not so much the existence of the Higgs boson, but one as light as 125 GeV. The gluon-gluon fusion production, through the top loop in SM, is now augmented by 4G quarks t′ and b′ in the loop, leading to an enhancement of order (1+1+1)2=9 in production cross-section, which does not seem reflected by data. In fact, searches [11] assuming this enhancement factor rule out a Higgs boson in the full mass range up to 600 GeV. We are reminded, however, that by simply extending from 3→4 generations, the effective CPV increases [12] by a thousand-trillion-fold or more and may provide enough CPV to satisfy the Sakharov conditions for baryogenesis. Given that the three-generation Kobayashi-Maskawa model [13] falls far short of the needed CPV, Nature might use such an enhancement factor. Furthermore, before one rules out the dilaton possibility, the premise for the order of magnitude enhancement in production may not stand.Fourth generationt′ and b′ quarks have been pursued vigorously at the LHC, as it should be done for a hadron collider, independent of the Higgs situation. The current bound [14–20] is
(1)mQ≳700GeV,
where Q stands for a chiral quark doublet, in which the limit is from both t′ and b′ search. We shall assume “heavy isospin” symmetry, IQ, and treat the doublet Q as degenerate, which can be viewed as part of the custodial SU(2) symmetry. What is important is that the current mQ bound is already above the nominal perturbative partial wave unitarity bound (UB) of 550 GeV [21]. The Yukawa coupling λQ≡2mQ/v≳4λt (where λt≃1) has already entered the strong coupling regime; αQ≡λQ2/4π≳1.With the ever increasing bound onmQ, the fourth generation may well not exist. But being beyond UB, a new question is as follows: could the strong Yukawa coupling of Q generate [22] EWSB itself? This is an intriguing conjecture and provides a second reason for having a fourth generation. Along this line, a gap equation, given symbolically in Figure 1, was constructed [9] without ever invoking the Higgs doublet, that is, the Higgs boson field of SM.Figure 1
Gap equation for generating heavy quark mass from Goldstone boson loop.The logic or philosophy went as follows. The Goldstone bosonG is viewed as a tightly bound QQ- state, bound by the Yukawa coupling itself. Guided by a Bethe-Salpeter equation study [23], strong Yukawa binding could lead to state collapse; that is, the bound state turns tachyonic, which is taken as suggestive of triggering EWSB itself. For further elucidation, see [24]. Reference [9] went one step further to postulate that the leading collapsed state, the color-singlet isosinglet pseudoscalar QQ- meson, π1, is the Goldstone bosonGitself. With no New Physics in sight at the LHC, not even the heavy chiral quark Q itself, the loop momentum integration runs up to roughly 2mQ (when the Goldstone boson ceases to exist), without the need to add any further effects (in the ladder approximation of truncating corrections to G propagation and GQQ vertex). This is therefore a “bootstrap” gap equation, in that the strong Yukawa coupling itself is the source of EWSB, or mass generation for quark Q, which simultaneously justifies keeping the Goldstone G in the loop (the 125 GeV object is assumed to be the dilaton). The existence of a large Yukawa coupling λQ is used as input, without a theory forλQ itself. There is no attempt at UV completion.It is important then to investigate whether one could find a solution to such a gap equation. If so, we would have demonstrated the case for dynamical symmetry breaking (DSB) and the potential riches that could follow. This paper offers a comprehensive elucidation of the line of thought from the aforementioned, through the demonstration of DSB, and linking the need of 2-3 TeV quarks to the possible new phenomenon ofQQ-→nVL production, the search of which could be aided by Yukawa resonances.In Section2, we trace the arguments that set up the bootstrap gap equation, including the postulate of the Goldstone boson G as a “collapsed state.” In Section 3, we formulate this gap equation more clearly, and demonstrate that numerical solution does exist [8]. A similar gap equation was formulated by Hung and Xiong [25], where G in Figure 1 is replaced by a massless Higgs doublet field. We will compare and offer a critique. Our numerical solution [8] suggests mQ to be in the 2-3 TeV range, corresponding to λQ≳4π. Even for mQ lower than this range, one may ask whether the usual assumption of pp→QQ-+X followed by free Q decay would hold, as assumed so far in direct searches [14–20]. The large GQQ Yukawa coupling now resembles the πNN coupling in strength; to our surprise, we find gπNN≃λπNN≡2mN/fπ~4π. We tap into the known pp-→nπ phenomena and conjecture that QQ-→nVL, with multiplicity ~6–12, and with a characteristic temperature of order v that is to be measured. This is discussed in Section 4. We end with further discussions and offer a conclusion in Section 5.


## 2. A Gap Equation without Higgs

My own interest in the 4th generation was revitalized by a hint of possible New Physics in electroweak penguin contributions to direct CPV inB→Kπ decay (e.g., see, the account given in [12]). In between 2007 and 2008, my interest turned to direct search for b′ and t′ at the LHC with the CMS experiment. Although initially I wished for mb′≲300 GeV for sake of the rich phenomenology [26, 27], I found the link [22] of strong Yukawa coupling with EWSB rather intriguing. As the direct search limits on t′ and b′ rose, I began to find the usual Higgs mechanism via an elementary Higgs doublet more and more problematic. Having just μ2<0 in the Lagrangian lacked dynamics (it is only a description), while Nature seems to be saying something through the absence of elementary scalars in QED and QCD, where the dynamics are better understood than EWSB. Furthermore, most problems such as the hierarchy problem arose by treating the Higgs doublet field as elementary.The following is how curiosity led the way from a flavor/CPV entry into dynamical EWSB, without invoking the existence of an elementary Higgs doublet field.


### 2.1. Yukawa Coupling from Gauge Coupling

Holding the SM-Higgsinterpretation of the 125 GeV particle observed [1, 2] at LHC as still suspect, let us recall the firm facts from experiment.First, we know [28] that all observed quarks q and charged leptons l are pointlike to smaller than 10-18 m, and they are governed by the SU(3)C⊗SU(2)L⊗U(1)Y gauge dynamics. Chromodynamics would not be our concern, but it is important to emphasize that, unlike the 1970s and early 1980s, the SU(2)L⊗U(1)Y  chiral gauge dynamics is now experimentally established. We know that quarks and leptons come in left-handed weak doublets and right-handed singlets, and for each given electric charge, they carry different hypercharge Y.Second, the weak bosons are found [28] to be massive; MW=(1/2)gv, where g is the measured SU(2)L weak coupling, and v2=1/2GF the vacuum expectation value. Hence, spontaneous breaking of SU(2)L⊗U(1)Y symmetry (SSB) is also experimentally established.Third, all fermions are observed [28] to be massive. These masses also manifest EWSB, since they link left- and right-handed fermions of the same electric charge, but different SU(2)L and U(1)Y charges. We shall not invoke the elementary Higgs boson for mass generation, as this is the question we explore, and because the dilaton possibility has to be ruled out by experiment.At this point, we need to acknowledge the important theoretical achievement of renormalizability [29] of non-Abelian gauge theories, which allowed theory-experiment correspondence down to per mille level precision, especially for the extensive work [28] done at the LEP, SLC, as well as TeVatron colliders. The proof of renormalizability is based on Ward identities, hence, [30] unaffected by SSB; that is, the underlying symmetry properties are not affected. From this, we now demonstrate [9, 31] the existence of Yukawa couplings as experimental fact.With proof of renormalizability, we choose the physical unitary gauge; hence, there are no would-be Goldstone bosons (or unphysical scalars), only massive gauge bosons (strictly speaking, only the on-shellS-matrix is finite in U-gauge, we thank T. Kugo for the comment and take it as a limiting case of the Rξ gauge for sake of illustration). Longitudinal W boson (the would-be Goldstone bosons that got “eaten”) propagation is via the ((kμkν)/MW2) part of the W boson propagator. If we take a kμ/MW factor and contract with a dj→ui charged current, as illustrated in Figure 2, simple manipulations give (dropping Vij for convenience)
(2)g2kMWL=g2pi-pjMWL=g2miL-mjRMW=2(mivL-mjRv)≡λiL-λjR,
where we have inferred
(3)λQ≡2mQv
as the effective WL, or Goldstone boson G coupling to quarks, which is nothing but the familiar Yukawa coupling. We have used the equation of motion, that is, the Dirac equation, on dj and ui quarks in the second step of (2), but this is justified since we exist in the broken phase for the real world, and we know experimentally that all quarks are massive.Figure 2
Deriving the Yukawa coupling, that is, Goldstone boson coupling to quarks, from purely left-handed gauge couplings, where E.o.M. stands for equation of motion.From Figure2 and equations (2) and (3), we see that from the experimentally well-established left-handed gauge coupling, the Goldstone boson couples via the usual Yukawa coupling. The Goldstone bosons of EWSB pair with the transverse gauge boson modes to constitute a massive gauge boson, the Meissner effect, but the important point is that we have not introduced a physical Higgs boson in any step. Unlike the Higgs boson, SSB of electroweak symmetry is an experimentally established fact. The Goldstone bosons couple with Yukawa couplings proportional to fermion mass, independent of whether it arises from an elementary Higgs doublet.We have kept a factorVij in Figure 2. Recall that the Kobayashi-Maskawa (KM) formalism [13] for quark mixing deals with massive quarks, or equivalently the existence of Yukawa matrices, and the argument remains exactly the same. Vast amount of flavor and CP violation (CPV) data overwhelmingly supports [28] the 3 generation KM picture. For example, the unique CPV phase with 3 quark generations can so far explain all observed CPV phenomena. These facts further attest to the existence of Yukawa couplings from their dynamical effects but again do not provide any evidence for the existence of the SM Higgs boson.


### 2.2. A Postulate: Collapse of Yukawa Bound State

Based on experimental facts and the renormalizability of electroweak theory, we have “derived” Yukawa couplings from purely left-handed gauge couplings in the previous section, without invoking an explicit Higgs sector, at least not at the empirical, heuristic level. We turn now to a more hypothetical situation: could there be more chiral generations? Since we already have three, the possibility that there exists a fourth generation of quarks should not be dropped in a cavalier way. We note again that increasing to 4 generations, one may have sufficient amount of CPV for baryogenesis [12] from the KM mechanism. There has also been some resurgent recent interest [10, 14–20]. As argued in the Introduction, although the spirit may have been dampened by the 125 GeV Higgs-like object, one should press on with the direct search.What we do know is that thet′ and b′ quarks should be suitably degenerate to satisfy electroweak constraints on the S and T variables [32]. A “heavy isospin” is in accordance with the custodial SU(2) symmetry.


#### 2.2.1. Relativistic Expansion versus Bethe-Salpeter Equation

Withmt′≅mb′≡mQ≳700 GeV [14–20], λQ is already 4 times stronger than the top quark, hence, stronger than all gauge couplings. There have been two complementary studies of strong Yukawa bound states. The first approach is along traditional lines of relativistic expansion [33]. Ignoring all gauge couplings except QCD, and taking the heavy isospin limit, with Q representing the 4th generation quark doublet, and G the triplet of Goldstone bosons, the t- and s-channel Goldstone exchange diagrams are depicted in Figure 3, with corresponding diagrams for g as well as H exchange ([33] did not put in s-channel gluon exchange).Figure 3
Scattering diagrams fort- and s-channel Goldstone boson G exchange between Q and Q- quarks. Analogous diagrams can be drawn for g (and H) exchange.The heavyQ-Q mesons form isosinglets and isotriplets and can be color singlet or octet. We borrow the notation from hadrons and call these states η1, ω1, π1, ρ1, and η8, ω8, π8, ρ8, respectively. Reference [33] used a variational approach, with radius a0 as parameter. It was found that, for color singlet ω1(ρ1), a0/aQCD~1 for mQ below 400 (540) GeV, but above which a0 suddenly precipitates towards tiny values. For η1(π1), the radius mildly decreases (increases) from 1, with trend reversed for binding energy; hence, it remains QCD-bound.To understand this, note that thet-channel Goldstone exchange for η1 is repulsive, while the s-channel Goldstone exchange, contributing only to π1, is also repulsive. However, the sudden drop in ω1 and ρ1 radii is due to the trial wave function sensing suddenly a lower energy at tiny radius due to t-channel Goldstone exchange; the strong Yukawa coupling has wrested control of binding from the Coulombic QCD potential. QCD binding energy is only a couple of GeV, but the sudden drop in radius leads to a sharp rise in binding energy, hence, a kink. One finds that the relativisticv/cexpansion fails just when it starts to get interesting. For color octet states, QCD is repulsive; so, η8 does not bind. In [33], the ω8 and π8 states are degenerate, with sudden shrinking of radius occurring around 530 GeV, but the s-channel QCD effect, left out in [33], should push the ω8 upwards; the ρ8 state does not shrink until later.Given that the relativistic expansion breaks down, a truly relativistic approach is needed. Such a study, based on a Bethe-Salpeter (BS) equation [23], was pursued around the time of demise of the SSC, for very heavy chiral quark doublets. The BS equation is a ladder sum of t- and s-channel diagrams of Figure 3, where the Q-Q pair forms a heavy meson bound state. While the ladder sum of t-channel diagrams is intuitive, a problem emerges for the s-channel, which contributes only to π1, ω8, and σ1 (same quantum numbers as G, g, and H, resp.). Rather than a triangle loop, the s-channel loop appears like a self-energy, hence, potentially divergent, while the momentum carried by the exchanged boson is the bound state mass itself. One could not formally turn the integral equation into an eigenvalue problem. This was resolved in [23] by a subtraction at fixed external momentum, which in effect eliminates all s-channel diagrams. Reference [23] then solved the BS equation numerically using several different approximations, which, in addition to the approximate nature of the BS equation itself, illustrates the uncertainties. Still, unlike [33], the bound state masses drop smoothly below 2mQ as mQ increases, showing no kink, which is an improvement from a relativistic expansion. However, a generic feature is collapse; bound state masses tend to drop sharply to zero at some high mQ and would naively turn tachyonic.


#### 2.2.2. Postulate for Leading Collapsed State

Here, we do not pursue the more conservative phenomenology for precollapse Yukawa couplings as in [24] but wish to address more fundamental issues. Although the de factos-channel subtraction made by [23] appeared reasonable on formal grounds, the contrast with the relativistic expansion is striking; the Goldstone G exchange in s-channel led to a specific repulsion [33] for π1 heavy mesons, disallowing it to shrink suddenly like the otherwise analogous ω1. But after subtracting the s-channel, the π1 becomes [23] the most attractive channel  (MAC), more so than ω1. Together with the tendency towards collapse for large enough mQ (equivalently λQ), this means that the π1 meson would be the first to drop to zero and turn tachyonic. That this occurs for the channel that experiences repulsion when 2mQ is far lower than collapse values (à la ω1 which has no s-channel effect) seems paradoxical. Does this falsify the whole approach, or else what light does this shed? And how is it related to s-channel subtraction?With experimental bounds [14–20] for 4th generation quarks entering the region of deep(er) binding, we offer a self-consistent view that may seem radical. Clearly, around and below 500 GeV mass, or 2mQ≲1 TeV, there could still be some repulsion due to G exchange in s-channel. But since we did not introduce any elementary Higgs doublet, the Goldstone boson G should perhaps be viewed as a Q-Q bound state. Hence, we Postulate(π1≡G). Collapse is a precursor to dynamical EWSB, and the first mode to collapse becomes the Goldstone mode.Although the full validity of the BS equation may be questioned, it is known [34] that “the appearance of a tachyonic bound state leads to instability of the vacuum,” which is “resolved by condensation into the tachyonic mode.” Our postulate removes the equation for π1 self-consistently (there is another aspect on the self-consistency of this subtraction/postulate, if one did not remove the π1 equation by invoking this subtraction, the leading collapse state of strong Yukawa coupling would be the ω1, condensation in this channel would break Lorentz invariance) and provides some understanding of the s-channel subtraction; a π1/G boson carrying p2~(2mQ)2 would no longer be a bound Goldstone boson in s-channel. Without an elementary Higgs boson, there is no σ1 channel subtraction, while for heavy enough mQ (so π1 has turned Goldstone), one can treat QCD effects as a correction, after solving the ω8 bound state problem, without need of subtracting s-channel g exchange. The self-consistent MAC behavior of the π1 channel seems like a reasonable outcome of the Goldstone dynamics, as implied by the gauge dynamics.It may now appear that EWSB is some kind of a “bootstrap” from “massive chiral quarks” with large Yukawa coupling as seen in broken phase.


### 2.3. A Bootstrap Gap Equation without Higgs

Motivated by the previous heuristic discussion, we construct a gap equation for the dynamical generation of heavy quark mass without invoking the Higgs boson.For a long time, asmQ limit rose, I focused on breaching the UB, which the experimental pursuit would not be concerned about. Although UB violation (UBV) occurs at far higher energy, once the experimental limits breach the bound, some form of strong interactions would take over. I asked myself how UBV might be amended by Nature. The study of Yukawa bound states [24] arose from this pursuit, but did not shed sufficient light on the issue, except approaching the abyss of state “collapse”, as described earlier.A mindset change occurred when one connected two of theQ or Q- lines in Figure 3. One gets the self-energy for Q by G exchange and readily arrives at the gap equation [9] as depicted in Figure 1. If quark mass mQ, represented by the cross X, could be nonzero, then one has dynamical chiral symmetry breaking, which is equivalent to EWSB! Both Q and G are treated as massless at the diagrammatic level, since the gap equation is summing over all possible momenta q carried by G as it mediates—dominates—QQ-→QQ- scattering. But if |q| reaches 2mQ, there would no longer be a Goldstone boson G (assuming G is a QQ- bound state), or it would be resolved in the t-channel. Thus, the summation over q2 should not exceed (2mQ)2 at the heuristic level. The whole picture is heuristic, but realistic in the experimental sense, since the gap equation integrates over a large momentum range of q2, where we have no other New Physics that enter, as indicated by LHC data.Such a gap equation was constructed recently from a different, and in our view moread hoc, theoretical argument. In [25], an elementary Higgs doublet is assumed together with a 4th generation. Motivated by their earlier study [35, 36], where some UV fixed point (UVFP) behavior was conjectured, these authors pursued dynamical EWSB via a Schwinger-Dyson equation that is rather similar to our Figure 1. However, perhaps in anticipation of the UVFP that might develop at high energy [35, 36], they put in by hand a massless Higgs doublet, hence, a scale invariant theory to boot. It is the massless Higgs doublet that runs in the loop, replacing our Goldstone boson G. The massless nature of the Higgs doublet appears ad hoc, and the paper defers the discussion of the physical Higgs spectrum for a future work.In contrast, our Goldstone bosonG, identified as the collapsed π1 state as it turns tachyonic, is strictly massless in the broken phase. In the gap equation of Figure 1, we speculate that the loop momentum should be cut off around 2mQ, rather than some “cut-off” scale Λ>2mQ. In so doing, we bypass all issues of triviality that arise from having Λ approaching 2mQ. What happens at scales above 2mQ is to be studied by experiment.Here, we remark that the first, elementary Higgs of [25], π and σ are our bound state Goldstone bosons, and indeed we should have a σ-like massive broad bound state [24] that could mimic the heavy Higgs boson. Their second Higgs doublet, in the form of t′ and b′ bound states, would be excitations above the π1 and σ1 for us, likely rather broad. We think that their claimed third doublet, that of bound τ′ and ντ′, may not be bound at all, as their Yukawa couplings may not be large enough.The gap equation illustrated in Figure1 actually links to a vast literature on strongly coupled, scale-invariant QED. It is known that such a theory could have spontaneous chiral symmetry breaking when couplings are strong enough. We turn to our numerical study [8] in the next section, where we also offer our critique of [25].Although our line of thought may seem constructed, we have developed a self-consistent picture where EWSB from large Yukawa coupling may be realized with some confidence—all without assuming an elementary Higgs boson. We have not yet really touched on theσ1 meson, which would be the heavy Higgs boson. However, our scenario is to have Goldstone bosons as strongly (and tightly) bound “Cooper pairs” of very heavy quarks, which may please Nambu. For the 125 GeV Higgs-like object, the inherent scale-invariant nature of our gap equation allows the possibility of a dilaton interpretation, as we will discuss later.


## 2.1. Yukawa Coupling from Gauge Coupling

Holding the SM-Higgsinterpretation of the 125 GeV particle observed [1, 2] at LHC as still suspect, let us recall the firm facts from experiment.First, we know [28] that all observed quarks q and charged leptons l are pointlike to smaller than 10-18 m, and they are governed by the SU(3)C⊗SU(2)L⊗U(1)Y gauge dynamics. Chromodynamics would not be our concern, but it is important to emphasize that, unlike the 1970s and early 1980s, the SU(2)L⊗U(1)Y  chiral gauge dynamics is now experimentally established. We know that quarks and leptons come in left-handed weak doublets and right-handed singlets, and for each given electric charge, they carry different hypercharge Y.Second, the weak bosons are found [28] to be massive; MW=(1/2)gv, where g is the measured SU(2)L weak coupling, and v2=1/2GF the vacuum expectation value. Hence, spontaneous breaking of SU(2)L⊗U(1)Y symmetry (SSB) is also experimentally established.Third, all fermions are observed [28] to be massive. These masses also manifest EWSB, since they link left- and right-handed fermions of the same electric charge, but different SU(2)L and U(1)Y charges. We shall not invoke the elementary Higgs boson for mass generation, as this is the question we explore, and because the dilaton possibility has to be ruled out by experiment.At this point, we need to acknowledge the important theoretical achievement of renormalizability [29] of non-Abelian gauge theories, which allowed theory-experiment correspondence down to per mille level precision, especially for the extensive work [28] done at the LEP, SLC, as well as TeVatron colliders. The proof of renormalizability is based on Ward identities, hence, [30] unaffected by SSB; that is, the underlying symmetry properties are not affected. From this, we now demonstrate [9, 31] the existence of Yukawa couplings as experimental fact.With proof of renormalizability, we choose the physical unitary gauge; hence, there are no would-be Goldstone bosons (or unphysical scalars), only massive gauge bosons (strictly speaking, only the on-shellS-matrix is finite in U-gauge, we thank T. Kugo for the comment and take it as a limiting case of the Rξ gauge for sake of illustration). Longitudinal W boson (the would-be Goldstone bosons that got “eaten”) propagation is via the ((kμkν)/MW2) part of the W boson propagator. If we take a kμ/MW factor and contract with a dj→ui charged current, as illustrated in Figure 2, simple manipulations give (dropping Vij for convenience)
(2)g2kMWL=g2pi-pjMWL=g2miL-mjRMW=2(mivL-mjRv)≡λiL-λjR,
where we have inferred
(3)λQ≡2mQv
as the effective WL, or Goldstone boson G coupling to quarks, which is nothing but the familiar Yukawa coupling. We have used the equation of motion, that is, the Dirac equation, on dj and ui quarks in the second step of (2), but this is justified since we exist in the broken phase for the real world, and we know experimentally that all quarks are massive.Figure 2
Deriving the Yukawa coupling, that is, Goldstone boson coupling to quarks, from purely left-handed gauge couplings, where E.o.M. stands for equation of motion.From Figure2 and equations (2) and (3), we see that from the experimentally well-established left-handed gauge coupling, the Goldstone boson couples via the usual Yukawa coupling. The Goldstone bosons of EWSB pair with the transverse gauge boson modes to constitute a massive gauge boson, the Meissner effect, but the important point is that we have not introduced a physical Higgs boson in any step. Unlike the Higgs boson, SSB of electroweak symmetry is an experimentally established fact. The Goldstone bosons couple with Yukawa couplings proportional to fermion mass, independent of whether it arises from an elementary Higgs doublet.We have kept a factorVij in Figure 2. Recall that the Kobayashi-Maskawa (KM) formalism [13] for quark mixing deals with massive quarks, or equivalently the existence of Yukawa matrices, and the argument remains exactly the same. Vast amount of flavor and CP violation (CPV) data overwhelmingly supports [28] the 3 generation KM picture. For example, the unique CPV phase with 3 quark generations can so far explain all observed CPV phenomena. These facts further attest to the existence of Yukawa couplings from their dynamical effects but again do not provide any evidence for the existence of the SM Higgs boson.


## 2.2. A Postulate: Collapse of Yukawa Bound State

Based on experimental facts and the renormalizability of electroweak theory, we have “derived” Yukawa couplings from purely left-handed gauge couplings in the previous section, without invoking an explicit Higgs sector, at least not at the empirical, heuristic level. We turn now to a more hypothetical situation: could there be more chiral generations? Since we already have three, the possibility that there exists a fourth generation of quarks should not be dropped in a cavalier way. We note again that increasing to 4 generations, one may have sufficient amount of CPV for baryogenesis [12] from the KM mechanism. There has also been some resurgent recent interest [10, 14–20]. As argued in the Introduction, although the spirit may have been dampened by the 125 GeV Higgs-like object, one should press on with the direct search.What we do know is that thet′ and b′ quarks should be suitably degenerate to satisfy electroweak constraints on the S and T variables [32]. A “heavy isospin” is in accordance with the custodial SU(2) symmetry.


### 2.2.1. Relativistic Expansion versus Bethe-Salpeter Equation

Withmt′≅mb′≡mQ≳700 GeV [14–20], λQ is already 4 times stronger than the top quark, hence, stronger than all gauge couplings. There have been two complementary studies of strong Yukawa bound states. The first approach is along traditional lines of relativistic expansion [33]. Ignoring all gauge couplings except QCD, and taking the heavy isospin limit, with Q representing the 4th generation quark doublet, and G the triplet of Goldstone bosons, the t- and s-channel Goldstone exchange diagrams are depicted in Figure 3, with corresponding diagrams for g as well as H exchange ([33] did not put in s-channel gluon exchange).Figure 3
Scattering diagrams fort- and s-channel Goldstone boson G exchange between Q and Q- quarks. Analogous diagrams can be drawn for g (and H) exchange.The heavyQ-Q mesons form isosinglets and isotriplets and can be color singlet or octet. We borrow the notation from hadrons and call these states η1, ω1, π1, ρ1, and η8, ω8, π8, ρ8, respectively. Reference [33] used a variational approach, with radius a0 as parameter. It was found that, for color singlet ω1(ρ1), a0/aQCD~1 for mQ below 400 (540) GeV, but above which a0 suddenly precipitates towards tiny values. For η1(π1), the radius mildly decreases (increases) from 1, with trend reversed for binding energy; hence, it remains QCD-bound.To understand this, note that thet-channel Goldstone exchange for η1 is repulsive, while the s-channel Goldstone exchange, contributing only to π1, is also repulsive. However, the sudden drop in ω1 and ρ1 radii is due to the trial wave function sensing suddenly a lower energy at tiny radius due to t-channel Goldstone exchange; the strong Yukawa coupling has wrested control of binding from the Coulombic QCD potential. QCD binding energy is only a couple of GeV, but the sudden drop in radius leads to a sharp rise in binding energy, hence, a kink. One finds that the relativisticv/cexpansion fails just when it starts to get interesting. For color octet states, QCD is repulsive; so, η8 does not bind. In [33], the ω8 and π8 states are degenerate, with sudden shrinking of radius occurring around 530 GeV, but the s-channel QCD effect, left out in [33], should push the ω8 upwards; the ρ8 state does not shrink until later.Given that the relativistic expansion breaks down, a truly relativistic approach is needed. Such a study, based on a Bethe-Salpeter (BS) equation [23], was pursued around the time of demise of the SSC, for very heavy chiral quark doublets. The BS equation is a ladder sum of t- and s-channel diagrams of Figure 3, where the Q-Q pair forms a heavy meson bound state. While the ladder sum of t-channel diagrams is intuitive, a problem emerges for the s-channel, which contributes only to π1, ω8, and σ1 (same quantum numbers as G, g, and H, resp.). Rather than a triangle loop, the s-channel loop appears like a self-energy, hence, potentially divergent, while the momentum carried by the exchanged boson is the bound state mass itself. One could not formally turn the integral equation into an eigenvalue problem. This was resolved in [23] by a subtraction at fixed external momentum, which in effect eliminates all s-channel diagrams. Reference [23] then solved the BS equation numerically using several different approximations, which, in addition to the approximate nature of the BS equation itself, illustrates the uncertainties. Still, unlike [33], the bound state masses drop smoothly below 2mQ as mQ increases, showing no kink, which is an improvement from a relativistic expansion. However, a generic feature is collapse; bound state masses tend to drop sharply to zero at some high mQ and would naively turn tachyonic.


### 2.2.2. Postulate for Leading Collapsed State

Here, we do not pursue the more conservative phenomenology for precollapse Yukawa couplings as in [24] but wish to address more fundamental issues. Although the de factos-channel subtraction made by [23] appeared reasonable on formal grounds, the contrast with the relativistic expansion is striking; the Goldstone G exchange in s-channel led to a specific repulsion [33] for π1 heavy mesons, disallowing it to shrink suddenly like the otherwise analogous ω1. But after subtracting the s-channel, the π1 becomes [23] the most attractive channel  (MAC), more so than ω1. Together with the tendency towards collapse for large enough mQ (equivalently λQ), this means that the π1 meson would be the first to drop to zero and turn tachyonic. That this occurs for the channel that experiences repulsion when 2mQ is far lower than collapse values (à la ω1 which has no s-channel effect) seems paradoxical. Does this falsify the whole approach, or else what light does this shed? And how is it related to s-channel subtraction?With experimental bounds [14–20] for 4th generation quarks entering the region of deep(er) binding, we offer a self-consistent view that may seem radical. Clearly, around and below 500 GeV mass, or 2mQ≲1 TeV, there could still be some repulsion due to G exchange in s-channel. But since we did not introduce any elementary Higgs doublet, the Goldstone boson G should perhaps be viewed as a Q-Q bound state. Hence, we Postulate(π1≡G). Collapse is a precursor to dynamical EWSB, and the first mode to collapse becomes the Goldstone mode.Although the full validity of the BS equation may be questioned, it is known [34] that “the appearance of a tachyonic bound state leads to instability of the vacuum,” which is “resolved by condensation into the tachyonic mode.” Our postulate removes the equation for π1 self-consistently (there is another aspect on the self-consistency of this subtraction/postulate, if one did not remove the π1 equation by invoking this subtraction, the leading collapse state of strong Yukawa coupling would be the ω1, condensation in this channel would break Lorentz invariance) and provides some understanding of the s-channel subtraction; a π1/G boson carrying p2~(2mQ)2 would no longer be a bound Goldstone boson in s-channel. Without an elementary Higgs boson, there is no σ1 channel subtraction, while for heavy enough mQ (so π1 has turned Goldstone), one can treat QCD effects as a correction, after solving the ω8 bound state problem, without need of subtracting s-channel g exchange. The self-consistent MAC behavior of the π1 channel seems like a reasonable outcome of the Goldstone dynamics, as implied by the gauge dynamics.It may now appear that EWSB is some kind of a “bootstrap” from “massive chiral quarks” with large Yukawa coupling as seen in broken phase.


## 2.2.1. Relativistic Expansion versus Bethe-Salpeter Equation

Withmt′≅mb′≡mQ≳700 GeV [14–20], λQ is already 4 times stronger than the top quark, hence, stronger than all gauge couplings. There have been two complementary studies of strong Yukawa bound states. The first approach is along traditional lines of relativistic expansion [33]. Ignoring all gauge couplings except QCD, and taking the heavy isospin limit, with Q representing the 4th generation quark doublet, and G the triplet of Goldstone bosons, the t- and s-channel Goldstone exchange diagrams are depicted in Figure 3, with corresponding diagrams for g as well as H exchange ([33] did not put in s-channel gluon exchange).Figure 3
Scattering diagrams fort- and s-channel Goldstone boson G exchange between Q and Q- quarks. Analogous diagrams can be drawn for g (and H) exchange.The heavyQ-Q mesons form isosinglets and isotriplets and can be color singlet or octet. We borrow the notation from hadrons and call these states η1, ω1, π1, ρ1, and η8, ω8, π8, ρ8, respectively. Reference [33] used a variational approach, with radius a0 as parameter. It was found that, for color singlet ω1(ρ1), a0/aQCD~1 for mQ below 400 (540) GeV, but above which a0 suddenly precipitates towards tiny values. For η1(π1), the radius mildly decreases (increases) from 1, with trend reversed for binding energy; hence, it remains QCD-bound.To understand this, note that thet-channel Goldstone exchange for η1 is repulsive, while the s-channel Goldstone exchange, contributing only to π1, is also repulsive. However, the sudden drop in ω1 and ρ1 radii is due to the trial wave function sensing suddenly a lower energy at tiny radius due to t-channel Goldstone exchange; the strong Yukawa coupling has wrested control of binding from the Coulombic QCD potential. QCD binding energy is only a couple of GeV, but the sudden drop in radius leads to a sharp rise in binding energy, hence, a kink. One finds that the relativisticv/cexpansion fails just when it starts to get interesting. For color octet states, QCD is repulsive; so, η8 does not bind. In [33], the ω8 and π8 states are degenerate, with sudden shrinking of radius occurring around 530 GeV, but the s-channel QCD effect, left out in [33], should push the ω8 upwards; the ρ8 state does not shrink until later.Given that the relativistic expansion breaks down, a truly relativistic approach is needed. Such a study, based on a Bethe-Salpeter (BS) equation [23], was pursued around the time of demise of the SSC, for very heavy chiral quark doublets. The BS equation is a ladder sum of t- and s-channel diagrams of Figure 3, where the Q-Q pair forms a heavy meson bound state. While the ladder sum of t-channel diagrams is intuitive, a problem emerges for the s-channel, which contributes only to π1, ω8, and σ1 (same quantum numbers as G, g, and H, resp.). Rather than a triangle loop, the s-channel loop appears like a self-energy, hence, potentially divergent, while the momentum carried by the exchanged boson is the bound state mass itself. One could not formally turn the integral equation into an eigenvalue problem. This was resolved in [23] by a subtraction at fixed external momentum, which in effect eliminates all s-channel diagrams. Reference [23] then solved the BS equation numerically using several different approximations, which, in addition to the approximate nature of the BS equation itself, illustrates the uncertainties. Still, unlike [33], the bound state masses drop smoothly below 2mQ as mQ increases, showing no kink, which is an improvement from a relativistic expansion. However, a generic feature is collapse; bound state masses tend to drop sharply to zero at some high mQ and would naively turn tachyonic.


## 2.2.2. Postulate for Leading Collapsed State

Here, we do not pursue the more conservative phenomenology for precollapse Yukawa couplings as in [24] but wish to address more fundamental issues. Although the de factos-channel subtraction made by [23] appeared reasonable on formal grounds, the contrast with the relativistic expansion is striking; the Goldstone G exchange in s-channel led to a specific repulsion [33] for π1 heavy mesons, disallowing it to shrink suddenly like the otherwise analogous ω1. But after subtracting the s-channel, the π1 becomes [23] the most attractive channel  (MAC), more so than ω1. Together with the tendency towards collapse for large enough mQ (equivalently λQ), this means that the π1 meson would be the first to drop to zero and turn tachyonic. That this occurs for the channel that experiences repulsion when 2mQ is far lower than collapse values (à la ω1 which has no s-channel effect) seems paradoxical. Does this falsify the whole approach, or else what light does this shed? And how is it related to s-channel subtraction?With experimental bounds [14–20] for 4th generation quarks entering the region of deep(er) binding, we offer a self-consistent view that may seem radical. Clearly, around and below 500 GeV mass, or 2mQ≲1 TeV, there could still be some repulsion due to G exchange in s-channel. But since we did not introduce any elementary Higgs doublet, the Goldstone boson G should perhaps be viewed as a Q-Q bound state. Hence, we Postulate(π1≡G). Collapse is a precursor to dynamical EWSB, and the first mode to collapse becomes the Goldstone mode.Although the full validity of the BS equation may be questioned, it is known [34] that “the appearance of a tachyonic bound state leads to instability of the vacuum,” which is “resolved by condensation into the tachyonic mode.” Our postulate removes the equation for π1 self-consistently (there is another aspect on the self-consistency of this subtraction/postulate, if one did not remove the π1 equation by invoking this subtraction, the leading collapse state of strong Yukawa coupling would be the ω1, condensation in this channel would break Lorentz invariance) and provides some understanding of the s-channel subtraction; a π1/G boson carrying p2~(2mQ)2 would no longer be a bound Goldstone boson in s-channel. Without an elementary Higgs boson, there is no σ1 channel subtraction, while for heavy enough mQ (so π1 has turned Goldstone), one can treat QCD effects as a correction, after solving the ω8 bound state problem, without need of subtracting s-channel g exchange. The self-consistent MAC behavior of the π1 channel seems like a reasonable outcome of the Goldstone dynamics, as implied by the gauge dynamics.It may now appear that EWSB is some kind of a “bootstrap” from “massive chiral quarks” with large Yukawa coupling as seen in broken phase.


## 2.3. A Bootstrap Gap Equation without Higgs

Motivated by the previous heuristic discussion, we construct a gap equation for the dynamical generation of heavy quark mass without invoking the Higgs boson.For a long time, asmQ limit rose, I focused on breaching the UB, which the experimental pursuit would not be concerned about. Although UB violation (UBV) occurs at far higher energy, once the experimental limits breach the bound, some form of strong interactions would take over. I asked myself how UBV might be amended by Nature. The study of Yukawa bound states [24] arose from this pursuit, but did not shed sufficient light on the issue, except approaching the abyss of state “collapse”, as described earlier.A mindset change occurred when one connected two of theQ or Q- lines in Figure 3. One gets the self-energy for Q by G exchange and readily arrives at the gap equation [9] as depicted in Figure 1. If quark mass mQ, represented by the cross X, could be nonzero, then one has dynamical chiral symmetry breaking, which is equivalent to EWSB! Both Q and G are treated as massless at the diagrammatic level, since the gap equation is summing over all possible momenta q carried by G as it mediates—dominates—QQ-→QQ- scattering. But if |q| reaches 2mQ, there would no longer be a Goldstone boson G (assuming G is a QQ- bound state), or it would be resolved in the t-channel. Thus, the summation over q2 should not exceed (2mQ)2 at the heuristic level. The whole picture is heuristic, but realistic in the experimental sense, since the gap equation integrates over a large momentum range of q2, where we have no other New Physics that enter, as indicated by LHC data.Such a gap equation was constructed recently from a different, and in our view moread hoc, theoretical argument. In [25], an elementary Higgs doublet is assumed together with a 4th generation. Motivated by their earlier study [35, 36], where some UV fixed point (UVFP) behavior was conjectured, these authors pursued dynamical EWSB via a Schwinger-Dyson equation that is rather similar to our Figure 1. However, perhaps in anticipation of the UVFP that might develop at high energy [35, 36], they put in by hand a massless Higgs doublet, hence, a scale invariant theory to boot. It is the massless Higgs doublet that runs in the loop, replacing our Goldstone boson G. The massless nature of the Higgs doublet appears ad hoc, and the paper defers the discussion of the physical Higgs spectrum for a future work.In contrast, our Goldstone bosonG, identified as the collapsed π1 state as it turns tachyonic, is strictly massless in the broken phase. In the gap equation of Figure 1, we speculate that the loop momentum should be cut off around 2mQ, rather than some “cut-off” scale Λ>2mQ. In so doing, we bypass all issues of triviality that arise from having Λ approaching 2mQ. What happens at scales above 2mQ is to be studied by experiment.Here, we remark that the first, elementary Higgs of [25], π and σ are our bound state Goldstone bosons, and indeed we should have a σ-like massive broad bound state [24] that could mimic the heavy Higgs boson. Their second Higgs doublet, in the form of t′ and b′ bound states, would be excitations above the π1 and σ1 for us, likely rather broad. We think that their claimed third doublet, that of bound τ′ and ντ′, may not be bound at all, as their Yukawa couplings may not be large enough.The gap equation illustrated in Figure1 actually links to a vast literature on strongly coupled, scale-invariant QED. It is known that such a theory could have spontaneous chiral symmetry breaking when couplings are strong enough. We turn to our numerical study [8] in the next section, where we also offer our critique of [25].Although our line of thought may seem constructed, we have developed a self-consistent picture where EWSB from large Yukawa coupling may be realized with some confidence—all without assuming an elementary Higgs boson. We have not yet really touched on theσ1 meson, which would be the heavy Higgs boson. However, our scenario is to have Goldstone bosons as strongly (and tightly) bound “Cooper pairs” of very heavy quarks, which may please Nambu. For the 125 GeV Higgs-like object, the inherent scale-invariant nature of our gap equation allows the possibility of a dilaton interpretation, as we will discuss later.


## 3. Solving Bootstrap Gap Equation

The question now is whether the symbolic gap equation of Figure1 affords actual solutions. Towards finding a solution, in this section, we briefly review the Nambu-Jona-Lasinio model, where one sets up a gap equation with its well-known solution. We then turn to the so-called strongly coupled scale-invariant QED, which is closer to our gap equation. By recounting some major steps, we also set up our notation for later usage. We find a coupled set of integral equations, which is more complicated than strong QED in Landau gauge. But a numerical solution is found [8], with λQ≳4π, where we compare and contrast with [25].


### 3.1. NJL Model:p-Independent Self-Energy

The Nambu-Jona-Lasinio model [37] is the earliest, explicit model of DSB, where the breaking of global chiral symmetry leads to generation of nucleon mass, and the pion as a (pseudo-) Goldstone boson [38–40]. (Strictly speaking, the Goldstone boson should really be called the Nambu-Goldstone (NG) boson, since [38] predated [39]. But for sake of notation, and because of more common usage, we shall use G and Goldstone boson throughout our paper).The model is depicted in Figure4, where a four-fermion interaction is introduced (the blob on the right-hand side). The nucleon mass, represented by a cross, is self-consistently generated. One easily finds the gap equation
(4)mN=NC8π2G∫0Λ2dq2q2mNq2+mN2=NC8π2GΛ2(1-mN2Λ2log(1+Λ2mN2))mN,
where G here is the four-fermi coupling and Λ is the cutoff. Since mN on both sides factor out, one has
(5)1-GcritG=mN2Λ2log(1+Λ2mN2),
which admits a solution for G>Gcrit, with
(6)Gcrit=8π2NCΛ2.Figure 4
The gap equation for the Nambu-Jona-Lasinio model for generating nucleon mass.To understand what is happening, one can iterate the cross of the left-hand side of Figure4 on the right-hand side and see that it constitutes an infinite number of diagrams. This effectively puts the original self-energy diagram into the denominator. In the end, one trades the parameters G and Λ for the physical mN and the pion-nucleon coupling. At the more refined level and using the quark language, one can show further that the emergent Goldstone boson, the pion, is in fact a ladder sum of the quark-level four-fermi interaction.We will return at the end to discuss the similarity and differences of the NJL model with our gap equation.


### 3.2. Strong QED:p-Dependent Self-Energy

We note that the self-energy bubble of Figure4 does not depend on external momentum p, and so at the superficial level, Figure 4 is quite different from Figure 1. We now turn to QED, where there is much closer similarity.The general gap equation for QED [41] can be written in the form of the Schwinger-Dyson (SD) equation
(7)Σ(p)=ie2∫‍d4q(2π)4γμDμν(q)S(p-q)Γν(p,q),
where Σ=p-m-S(p)-1 is the electron self-energy with S the (full) electron propagator, Dμν is the (full) photon propagator, and Γ is the full vertex.


#### 3.2.1. Ladder Approximation

Truncating the exact, full vertex and photon propagator by the approximation(8)Γν(p,q)=γν,(9)Dμν(q)=-gμν+qμqν/q2q2-ξqμqνq4,
called the ladder (or rainbow) approximation, but retaining the electron self-energy, the gap equation becomes
(10)S(p)-1=p-ie2∫‍d4q(2π)4γμDμν(p-q)S(q)γν,
with Dμν now given in (9), and we have set m=0, that is, massless QED at Lagrangian level. Pictorially, this is represented as in Figure 5, where we note that, compared with the four-fermion coupling of the NJL model depicted in Figure 4, the external momentum p now flows into the self-energy loop. The question now is whether chiral symmetry can be dynamically broken, that a nontrivial solution to the self-energy could be generated at some strong coupling α=e2/4π?Figure 5
Gap equation for QED in the ladder approximation.We define the electron propagator as [41]
(11)S(p)=1A(p2)p-B(p2),
where the A term corresponds to wave function (w.f.) renormalization related to the usual Z-1 factor. A finite pole of the propagator, p2=B2(p2)/A2(p2), would give the dynamical effective mass. Our aim is therefore solving A and B from the gap equation of (10). Inserting (9) and after some algebra, one finds
(12)B(p2)=e2∫‍id4q(2π)4(-4+1-ξk2)B(q2)A2(q2)q2-B2(q2),A(p2)=1-e24p2tr∫‍id4q(2π)4γμDμν(ξ)A(q2)qA2(q2)q2-B2(q2)γνp.Simplification can be achieved in the Landau gauge,ξ=0, where one finds A(p2)=1. Since A relates to w.f. renormalization, it satisfies the Ward-Takahashi identity even under the ladder approximation. The gap equation is simplified to a single equation
(13)B(p2)=-3e2∫‍id4q(2π)4B(q2)(p-q)2(q2-B2(q2)).
After Wick rotation, and using
(14)∫‍d4qf(q2)(p-q)2=π2∫‍dq2q2[θ(p-q)p2+θ(q-p)q2]f(q2),
one obtains
(15)B(x)=3α4π(1x∫ΛIR2xdyyB(y)y+B2(y)+∫xΛ2dyB(y)y+B2(y)),
where x=pE2 is the Euclidean momentum squared, and Λ2, ΛIR2 are the ultraviolet (UV) and infrared (IR) cutoffs, respectively.The integral equation can be changed to a differential equation by noting(16)dB(x)dx=3α4π(-1x2)∫ΛIR2x‍dyyB(y)y+B2(y),d(xB(x))dx=3α4π∫xΛ2dyB(y)y+B2(y).
One obtains the differential equation
(17)xd2B(x)dx2+2dB(x)dx+3α4πB(x)x+B2(x)=0,
together with the IR and UV boundary conditions (B.C.)
(18)dB(x)dx|x=ΛIR2=0,d(xB(x))dx|x=Λ2=0.
If IR cutoff is 0, the B.C. for IR should be replaced by 2BdB/dx=-3α/4π at x=0.


#### 3.2.2. Solution in Landau Gauge

The coupled integral equations are simplified and put into a differential equation with B.C. In order to study qualitative features, let us first find an approximate solution. For a special range for the IR cutoff, let us takeB(x)=ΛIR<x1/2<Λ; then, (17) is simplified to
(19)x2d2B(x)dx2+2xdB(x)dx+3α4πB(x)=0.
Inserting B(x)=Cxλ, the characteristic equation is
(20)λ(λ-1)+2λ+3α4π=0,
with discriminant 1-3α/π; hence, the behavior is different for α>αc and α<αc, where αc=π/3. The analytical solution under the approximation is given as
(21)B=b1x(-1+1-α/αc)/2+b2x(-1-1-α/αc)/2≡b1xc1+b2xc2,
and the boundary conditions can be written as
(22)(c2c1ΛIR2(c1-c2)-1Λ2(c1-c2))(b1b2)=(00).
The determinant must vanish for nontrivial b1,b2; hence,
(23)1-α/αc-11-α/αc+1=(Λ2ΛIR2)1-α/αc.
To satisfy this condition, α>αc=π/3 is needed, and for given Λ/ΛIR, α takes on discontinuous values
(24)nπ+tan-11α/αc-1=α/αc-1logΛΛIR.
One sees that α→αc for Λ/ΛIR→∞, the nominal “continuum limit.” For α<αc, the only solution that satisfies the B.C. is the trivial B(x)=0.The approximate solution for small IR cutoff can now be obtained by replacingB(x)=m (constant) for small x. The differential equation becomes
(25)xd2B(x)dx2+2dB(x)dx+3α4πB(x)x+m2=0,
where B(x) is a hypergeometric function. Namely,
(26)B(x)=mF(12-iγ,12+iγ,2;-xm2),
where
(27)iγ=121-ααc,αc=π3,
and F is a hypergeometric function. Checking the asymptotic behavior for x≫m2, the power behavior for α<αc cannot satisfy the UV boundary condition.The gap equation has a nontrivial or oscillatory solution forα>αc, where the critical coupling is αc=π/3 for QED. The dynamical effective mass is obtained by solving x=B2(-x). Note that the pole of the propagator should be given in the time-like region; so, one has to make analytical continuation in order to obtain a physical mass, which can be done smoothly for ΛIR→0.An issue arises in that the dynamical mass(28)m=4Λe-π/α/αc-1
is proportional to the UV cutoff Λ. For m to be physical, however, it should not depend on Λ. Miransky suggested [42] that α→αc as Λ→∞; that is, αc is a nontrivial UV fixed point. A related issue, which we would not go into, is whether there would be a dilaton associated with breaking of scale invariance [43, 44]. (These two are also the two best references for gauged NJL model).


### 3.3. Bootstrap Gap Equation for EWSB

Our recapitulation of strong QED is for the purpose of setting up our approach for solving the bootstrap gap equation [9] with large empirical Yukawa coupling, where we will continue to follow the Fukuda-Kugo approach.In Landau gauge (ξ=0), the propagations of (would-be) Goldstone modes and gauge bosons are properly separated, and the gap equation becomes equivalent to our discussion if one takes the g→0 limit for the gauge coupling. The Goldstone bosons G± and G0 couple to fermions with the familiar Yukawa couplings, which we have argued [9] as experimentally established. They are also unaltered by the g→0 limit, as seen in (2). The main assumption is the addition of a new (heavy) chiral quark doublet Q, where the heavy isospin IQ symmetry implies that λU=λD≡λQ, that is, equality of the IQ3=±1/2 Yukawa couplings. We ask whether large λQ could be the source of EWSB, through the conceptual gap equation of Figure 1.


#### 3.3.1. Ladder Approximation

In any case, we do not know the full propagator and vertex functions. We approximate the Goldstone-fermion vertex as undressed, and the Goldstone propagator remains asi/k2, analogous to the QED treatment in previous section. This is in part aided by the insight that G is a very tight QQ- bound state, which would be massless as long as the symmetry remains spontaneously broken. Similar to Figure 5, the gap equation for large Yukawa (vanishing g) coupling is depicted in Figure 6. Again, the external momentum p flows through the loop.Figure 6
Gap equation for large Yukawa coupling in the ladder approximation, where vanishingm0 is indicated.The change from the symbolic or conceptual equation of Figure1 is that, even with bare mass forbidden by gauge invariance (m=0), there is a p part for Q propagation, and we aim at solving for the quark propagator S(p), given as
(29)S(p)-1=A(p2)p-B(p2),
which is of course of the same form as (11) for QED. The Goldstone boson G is colorless, and, unlike the massless photon from electromagnetic gauge invariance for QED case, its masslessness is “bootstrapped” into the gap equation itself. Following similar steps as in Section 2, and assuming λU=λD=λQ degeneracy, we obtain
(30)Bp2=3λQ22∫‍d4qi(2π)41(p-q)2Bq2q2Aq22-Bq22-λQ22∫‍d4qi(2π)41(p-q)2-mh2Bq2q2Aq22-Bq22,(31)Ap2=1+3λQ22p2∫‍d4qi(2π)4p·q(p-q)2Aq2q2Aq22-Bq22+λQ22p2∫‍d4qi(2π)4p·q(p-q)2-mh2Aq2q2Aq22-Bq22,
where placement of i anticipates the Wick rotation, and Bq2 stands as shorthand for B(q2), and likewise for A. We have already used ξ=0; so, compared to massless QED, we now have to consider A, or wave function renormalization effects; hence, we have to face a coupled set of equations for A(p2) and B(p2). Note that we have kept a “Higgs” term for calculational purposes, applying Standard Model Higgs boson, h0, couplings. This is for purpose of later comparison with the work of Hung and Xiong [25], as well as for discussion of the dilaton case.For our aim of bootstrap DSB, we simply drop the second term (no physical Higgs, or takingmh~∞), and so only Goldstone modes propagate in both equations. After angular integration and Wick rotation, one gets
(32)B(x)=κb(B(y)yA2(y)+B2(y)1x∫ΛIR2x‍dyyB(y)yA2(y)+B2(y)+∫xΛ2‍dyB(y)yA2(y)+B2(y)),A(x)=1+κa(A(y)yA2(y)+B2(y)1x2∫ΛIR2x‍dyy2A(y)yA2(y)+B2(y)+∫xΛ2‍dyA(y)yA2(y)+B2(y)),
where
(33)κb=2κa=(3/2)λQ216π2=(3/2)αQ4π(noHiggs).
Taking the limit mh→0 in (30) and (31) to mimic the Hung-Xiong approach of massless scalar doublet,
(34)κb=κa=λQ216π2=αQ4π(masslessHiggs).
We keep the notation of κb and κa in (33) and (34) to cover these two cases.Analogous to Section2, (32) can be put into differential form
(35)xB′′+2B′+κbBxA2+B2=0,(36)xA′′+3A′+2κaAxA2+B2=0,
with the boundary conditions
(37)B′(x)|x=ΛIR2=0,[xB′(x)+B(x)]|x=Λ2=0,(38)A′(x)|x=ΛIR2=0,[x2A′(x)+A(x)]|x=Λ2=1,
where prime stands for x-derivative.At this point, we note that, if one ignores wave function renormalization, that is, (36), while forcefully setting A(x)=1 in (35), then one has the same solution as in QED, with the change in critical coupling
(39)αQc=2π3,
which is twice as high as for QED, and
(40)λQc=2π23≃5.13;
hence, superficially a “critical mass” mQc≃890 GeV, which is above current LHC limits [14–20]. But it should be clear that the wave function renormalization A(p2) term cannot be neglected.


#### 3.3.2. Numerical Solution

Redefiningp2=x=e2t, our coupled differential equations with B.C. become
(41)B¨+2B˙+4κbBA2+B2e-2t=0,A¨+4A˙+8κaAA2+B2e-2t=0,B˙(tIR)=0,B˙(tUV)+B(tUV)=0,A˙(tIR)=0,14A˙(tUV)+A(tUV)=1,
where dot represents t-derivative, and etUV=ΛUV=Λ, etIR=ΛIR.Asymptotic Properties and Critical Coupling. Due to scale invariance, the differential equations are invariant under
(42)x→a2x(t→t+loga),ΛUV,IR→aΛUV,IR,B→aB,A→A.
As a result, the solutions of the differential equations depend only on ΛUV/ΛIR (=etUV-tIR) and mdyn≡B(tIR)/A(tIR) for given κa and κb. Thus, mdyn is a kind of integration constant. We will see that the most important feature of the solutions is that only special values (discontinuous values) of κa and κb are allowed for given boundary conditions.If we takeκa=0, the equations can be solved analytically, and the solution can be described as in the case of strong coupling QED with Landau gauge, (21)–(23). The property illustrated for κa=0 should hold also for κa≠0, where, depending on ΛUV/ΛIR, κa should take on special discontinuous values to satisfy the gap equation for the cases of κa/κb=1 or 1/2.To see this, we note that theB2e-2t term in the denominators is irrelevant. This is because for t≪logm, the t dependence of A and B is negligible due to the boundary conditions A˙(tIR)=B˙(tIR)=0. Therefore, to understand the behavior of the solution, one can take tIR~logm and consider the differential equation with B2e-2t dropped as follows:
(43)B¨+2B˙+4κbBA2=0,A¨+4A˙+8κaA=0,
where the equation for A becomes independent from B, but its solution affects B.The solution ofA is obtained analytically as follows:
(44)A(x)=A0ΛUV4x2e-ζ2κaπ,ζ=Erf-1[1A0(1-x2ΛUV4)+Erf(1κa)],
where A0 is an integration constant which can be fixed by the B.C. for IR. Using the analytical solution, one can show that only discontinuous values of κa,b can satisfy the B.Cs. The “critical” value of κb can be easily obtained numerically, even without neglecting B2e-2t term in the denominators. Of course, since the solutions are found numerically, it is not a proof that one really has a “critical” value. The upshot is that only special values of the coupling can satisfy the coupled equations for given values of ΛUV/ΛIR, for mdyn<ΛIR (or ΛUV/mdyn for mdyn>ΛIR) and mdyn=B(ΛIR)/A(ΛIR).Our numerical solution gives(45)κbc≃1.4,(κb=2κa=3αQ8π),κbc≃13.7,(κb=κa=αQ4π),
corresponding to
(46)λQc≃12,(κb=2κa=3αQ8π),(47)λQc≃46,(κb=κa=αQ4π),
where the latter case is much higher. Here c stands for “critical,” and our numerical values are extracted in the large ΛUV/ΛIR and ΛUV/mdyn limit. Note that for the artificial case of κa=0 (i.e., A=1), the critical value was λQc≃5.1, (40); hence, the effect of considering A or wave function renormalization is quite nontrivial.The values in (46) and (47) correspond to effectively taking ΛUV→∞, which is certainly not the range of validity for (46) as a descendent of Figure 1. That is, the conceptual foundation for Figure 1 is that, for momentum roughly up to somewhere below 2mQ, corrections to the Goldstone boson propagator and vertex have been ignored. Nevertheless, at face value, if we naively apply the physical v≃246 GeV, then (46) and (47) imply the mass values
(48)mQc>2.1TeV(noHiggs)
and 8.1 TeV, respectively, which are rather high. We drop the latter, “massless (or light) Higgs” case, not only because it is clearly out of reach for the LHC, but for theoretical reasons as we explain later. The lower bound nature of (48) would be explained subsequently.


### 3.4. Discussion

The large critical coupling ofλQ≳4π of (46) is somewhat surprising, resulting in the rather high critical mass of (48). We note that this is without the contribution of a light SM Higgs boson, which would put mQc much higher. Before we touch phenomenology of such high chiral quark masses, and in so doing find an independent justification for such high critical Yukawa coupling, we offer some further discussion.


#### 3.4.1. Comparison with Hung and Xiong

We can now make some comparison with, and offer a critique of, the approach of Hung and Xiong [25] (HX). We have mimicked the concept of HX by keeping a “Higgs” scalar contribution in (30) and (31), which resulted in the second case of (34) in the limit of mh→0, as compared with our case of interest, (33), where we drop the mh-dependent term.HXassumed the existence of a massless Higgs doublet, where our (35) and (36) with (34) should be a faithful representation. However, not only is the massless doublet assumed, HX also ignored wave function renormalization, that is, the A term, in the treatment of their gap equation. Furthermore, the sign of our second integral in (30) disagrees with HX, suppressing the coefficient of λQ2 in comparison. In the earlier work [9] that set up the gap equation of Figure 1, taking the numerics of HX, the estimate of λQc=2π (compare (40)) gave mQc≃770 GeV, which is not too far above the current LHC bound. But with our sign for the second integral in (30), we would arrive at λQc=2π, or mQc~1.1 TeV, that is, a factor of 2 higher. With our sign, the Higgs boson effect would cancel out part of the Goldstone effect, hence, requiring stronger Yukawa coupling.But we have argued that it is not justified to ignore the wave function renormalization effect ofA(p2). After all, the boson loop has momentum dependence, and so it would necessarily affect the Z-1 factor. Thus, the previous simple numerics are incorrect. Keeping A in our numerical study, hence, the coupled A-B equations, a considerably higher critical mQc is found. For the case of taking κb=κa=λQ2/16π2, (34), where we mimic HX’s massless Higgs doublet effect, the critical λQc of (47) is almost 4 times as high as that for (46). In fact, we obtain the A(ΛUV2)=2.7, which is quite different from 1.Our criticism goes far deeper. Taking a scalar doublet as massless, such that superficially one has “scale-invariance” as in strong QED, is totallyad hoc. Effectively one has to hold the parameters of the Higgs potential such that the Higgs field always remains massless. However, there is no principle by which this scale-invariance or masslessness of the Higgs field can be maintained. After all, one is invoking large Yukawa couplings, which feed the notorious divergent quadratic corrections to the Higgs boson mass. The one-loop two-point and four-point functions with quark Q in the loop would generate effective μ2 and λ self-coupling terms for the Higgs field. With no explicit dynamical principle (such as gauge invariance for the case of QED), the assumption of a massless Higgs doublet as the agent of DSB is not only ad hoc, but clearly unsustainable.In contrast, we see the merit, as well as the meaning, of our “bootstrap” gap equation. As long as we are in the broken phase, there is a massless Goldstone boson that couples with the Yukawa couplingλQ. Treating λQ as large, if a nontrivial solution to the gap equation is found (as we have illustrated earlier in the previous subsection), it in turn justifies the use of a massless Goldstone boson in the gap equation, which is why we affix the name “bootstrap.” In fact, the physical argument [9] was to view the Goldstone boson G as an extremely tight ultrarelativistic bound state of heavy Q and Q- from the broken phase, while G enters the “bootstrap” gap equation to dynamically generate mQ, hence, break the symmetry, and in the same stroke justify its own existence.We also see that this argument for our bootstrap gap equation puts the existence of a light Higgs boson in doubt. That is, it seems rather difficult to keep it light, and one cannot ignore corrections to the Higgs propagator. At the foundation level [9], unlike the Goldstone boson, the experimental basis for the Higgs boson is rather recent. Even with the newly discovered [1, 2] 125 GeV boson, one has to distinguish between an SM Higgs boson versus a dilaton. Our numerical study also shows that keeping the Higgs term tends to raise the critical Yukawa coupling considerably, implying 2mQ higher than the LHC collision energy. Thus, the light Higgs case of κb=κa is rather problematic in bootstrap DSB. We will return to discuss the dilaton possibility later.


#### 3.4.2. Physical Cutoff

The previous comparison with [25], and in particular the bootstrap nature of the Goldstone boson G in the gap equation, illustrates that the limit ΛUV→∞ would take us outside the range of validity of the gap equation itself. It is clear that, for timelike q2>(2mQ)2, there would no longer be a Goldstone boson. Thus, for the cutoff Λ, one should not use the traditional language of ΛUV. One probably should not contemplate “UV-completion" at the current stage. The bootstrap gap equation is rather heuristically or physically argued. But it does not provide a theory of the heavy quark Yukawa coupling λQ, employing it for DSB instead. Thus, we suggest a cutoff Λ<2mQ<ΛNP, where ΛNP is some true New Physics scale where the origin of Yukawa couplings may be contemplated, but it is out of reach from the bootstrap DSB approach.Viewed from a slightly different angle, this cutoff is related to the restoration of symmetry. The Goldstone boson couples to broken currents, but theZ factor of the Goldstone boson may vanish at some scale related to the scale where the Goldstone boson becomes unbounded, and symmetry is restored. Rather than a true ΛUV that in principle extends to infinity, there exists a cutoff Λ of the gap equation. We are in effect summing over the Goldstone boson correction to the self-energy of the quark Q, when the Goldstone boson is still defined. As noted in [9], this picture does receive experimental support, in that no New Physics seem to be there below ~TeV scale. So, one sums up only the effect of the Goldstone boson, and nothing else.The gap equation sums up the effect, from low to high momentum, of the correction by the Goldstone bosonG to the quark self-energy. Equation (46) reflects taking this sum to ΛUV→∞. Since the summation is accumulative, if now one sums only to some Λ≲2mQ, as we have argued, then the cumulative effect is less than summing to very large momentum above 2mQ. We note that 2mQ is a physical scale parameter that is external to the scale-invariant gap equation. Nevertheless, we can plot the dependence of λQc on the cutoff Λ. From Figure 7, one can see that, for a lower cutoff, λQc has to be higher than in (46). This is because, as the integration range is smaller, a larger λQ value is needed to compensate. Thus, (46) gives a lower bound on mQ. If we take Λ=2mQ, then λQc~17.7, and
(49)mQ~3TeV(noHiggs;Λ=2mQ),
which is higher than (48). We will return to discuss whether this could be an overestimate later.Figure 7
Cutoff dependence of Yukawa coupling, where the cutoffΛ is in units of the dynamical quark mass mdyn.


#### 3.4.3. Future Work

Up to now, we have been cavalier in the relation betweenλQc and mQ(c), treating it as 2mQ(c)=λQcv. But v is a physically measured quantity, and mQ is yet to be experimentally measured. If electroweak symmetry is indeed dynamically broken by the large Yukawa coupling of a new heavy chiral quark Q, when Q is discovered in the future, likely mQ>mQc. We then see that the actual v.e.v. value, v, may not correspond to the critical value λQc. This brings about the question of how scale invariance is actually broken in our gap equations, and whether there might be a dilaton [5–7, 45–49]. (When the 125 GeV hint emerged from 2011 LHC data, [45–47] already suggested that it might be a dilaton rather than the SM Higgs boson, while [48, 49] are from the (walking) technicolor perspective).Rather than to approach this deeper problem, we try to obtain the decay constantfG of the Goldstone boson, which should be the same as the vacuum expectation value, v. Following the Pagels-Stokar formula [50] naively, we obtain
(50)fG2=NC4π2∫0Λ2dxxB(x)2-(1/4)x(B(x)2)′(xA(x)2+B(x)2)2.
More generally, we write
(51)fG2=∫0Λ2‍dxℱ[A(x),B(x)]=mQ2∫0Λ2/mQ2‍dx^ℱ^[A(x^),B(x^)/A(x^)/mQ],
where mQ≡B(0)/A(0), and we have scaled by mQ (and redefined the function ℱ), treating it as physical.We can get back the “Yukawa” couplingλ (it should really be denoted as Y, and the question is whether Y=λ) for the input to the gap equation, or
(52)12λ2=∫0Λ2/mQ2‍dx^ℱ^[A(x^),B(x^)/A(x^)/mQ].
If the system is really scale-invariant, the r.h.s. of (52) is a function of Λ/mQ and λ. In order to satisfy the gap equation, λ is obtained as a function of Λ/mQ. Namely, it should not depend on mQ explicitly. Taking the cutoff Λ=2mQ, the equation becomes iterative for λ. Therefore, solving the gap equation, we obtain a prediction for the heavy quark mass. But technical issues remain. Is mQ a physical mass? What about the infrared cutoff? Can our assumption of λ(ΛUV/mQ)=Y(ΛUV/mQ) be maintained self-consistently? Our work is far from complete, and we leave these theoretical questions to the future.


## 3.1. NJL Model:p-Independent Self-Energy

The Nambu-Jona-Lasinio model [37] is the earliest, explicit model of DSB, where the breaking of global chiral symmetry leads to generation of nucleon mass, and the pion as a (pseudo-) Goldstone boson [38–40]. (Strictly speaking, the Goldstone boson should really be called the Nambu-Goldstone (NG) boson, since [38] predated [39]. But for sake of notation, and because of more common usage, we shall use G and Goldstone boson throughout our paper).The model is depicted in Figure4, where a four-fermion interaction is introduced (the blob on the right-hand side). The nucleon mass, represented by a cross, is self-consistently generated. One easily finds the gap equation
(4)mN=NC8π2G∫0Λ2dq2q2mNq2+mN2=NC8π2GΛ2(1-mN2Λ2log(1+Λ2mN2))mN,
where G here is the four-fermi coupling and Λ is the cutoff. Since mN on both sides factor out, one has
(5)1-GcritG=mN2Λ2log(1+Λ2mN2),
which admits a solution for G>Gcrit, with
(6)Gcrit=8π2NCΛ2.Figure 4
The gap equation for the Nambu-Jona-Lasinio model for generating nucleon mass.To understand what is happening, one can iterate the cross of the left-hand side of Figure4 on the right-hand side and see that it constitutes an infinite number of diagrams. This effectively puts the original self-energy diagram into the denominator. In the end, one trades the parameters G and Λ for the physical mN and the pion-nucleon coupling. At the more refined level and using the quark language, one can show further that the emergent Goldstone boson, the pion, is in fact a ladder sum of the quark-level four-fermi interaction.We will return at the end to discuss the similarity and differences of the NJL model with our gap equation.


## 3.2. Strong QED:p-Dependent Self-Energy

We note that the self-energy bubble of Figure4 does not depend on external momentum p, and so at the superficial level, Figure 4 is quite different from Figure 1. We now turn to QED, where there is much closer similarity.The general gap equation for QED [41] can be written in the form of the Schwinger-Dyson (SD) equation
(7)Σ(p)=ie2∫‍d4q(2π)4γμDμν(q)S(p-q)Γν(p,q),
where Σ=p-m-S(p)-1 is the electron self-energy with S the (full) electron propagator, Dμν is the (full) photon propagator, and Γ is the full vertex.


### 3.2.1. Ladder Approximation

Truncating the exact, full vertex and photon propagator by the approximation(8)Γν(p,q)=γν,(9)Dμν(q)=-gμν+qμqν/q2q2-ξqμqνq4,
called the ladder (or rainbow) approximation, but retaining the electron self-energy, the gap equation becomes
(10)S(p)-1=p-ie2∫‍d4q(2π)4γμDμν(p-q)S(q)γν,
with Dμν now given in (9), and we have set m=0, that is, massless QED at Lagrangian level. Pictorially, this is represented as in Figure 5, where we note that, compared with the four-fermion coupling of the NJL model depicted in Figure 4, the external momentum p now flows into the self-energy loop. The question now is whether chiral symmetry can be dynamically broken, that a nontrivial solution to the self-energy could be generated at some strong coupling α=e2/4π?Figure 5
Gap equation for QED in the ladder approximation.We define the electron propagator as [41]
(11)S(p)=1A(p2)p-B(p2),
where the A term corresponds to wave function (w.f.) renormalization related to the usual Z-1 factor. A finite pole of the propagator, p2=B2(p2)/A2(p2), would give the dynamical effective mass. Our aim is therefore solving A and B from the gap equation of (10). Inserting (9) and after some algebra, one finds
(12)B(p2)=e2∫‍id4q(2π)4(-4+1-ξk2)B(q2)A2(q2)q2-B2(q2),A(p2)=1-e24p2tr∫‍id4q(2π)4γμDμν(ξ)A(q2)qA2(q2)q2-B2(q2)γνp.Simplification can be achieved in the Landau gauge,ξ=0, where one finds A(p2)=1. Since A relates to w.f. renormalization, it satisfies the Ward-Takahashi identity even under the ladder approximation. The gap equation is simplified to a single equation
(13)B(p2)=-3e2∫‍id4q(2π)4B(q2)(p-q)2(q2-B2(q2)).
After Wick rotation, and using
(14)∫‍d4qf(q2)(p-q)2=π2∫‍dq2q2[θ(p-q)p2+θ(q-p)q2]f(q2),
one obtains
(15)B(x)=3α4π(1x∫ΛIR2xdyyB(y)y+B2(y)+∫xΛ2dyB(y)y+B2(y)),
where x=pE2 is the Euclidean momentum squared, and Λ2, ΛIR2 are the ultraviolet (UV) and infrared (IR) cutoffs, respectively.The integral equation can be changed to a differential equation by noting(16)dB(x)dx=3α4π(-1x2)∫ΛIR2x‍dyyB(y)y+B2(y),d(xB(x))dx=3α4π∫xΛ2dyB(y)y+B2(y).
One obtains the differential equation
(17)xd2B(x)dx2+2dB(x)dx+3α4πB(x)x+B2(x)=0,
together with the IR and UV boundary conditions (B.C.)
(18)dB(x)dx|x=ΛIR2=0,d(xB(x))dx|x=Λ2=0.
If IR cutoff is 0, the B.C. for IR should be replaced by 2BdB/dx=-3α/4π at x=0.


### 3.2.2. Solution in Landau Gauge

The coupled integral equations are simplified and put into a differential equation with B.C. In order to study qualitative features, let us first find an approximate solution. For a special range for the IR cutoff, let us takeB(x)=ΛIR<x1/2<Λ; then, (17) is simplified to
(19)x2d2B(x)dx2+2xdB(x)dx+3α4πB(x)=0.
Inserting B(x)=Cxλ, the characteristic equation is
(20)λ(λ-1)+2λ+3α4π=0,
with discriminant 1-3α/π; hence, the behavior is different for α>αc and α<αc, where αc=π/3. The analytical solution under the approximation is given as
(21)B=b1x(-1+1-α/αc)/2+b2x(-1-1-α/αc)/2≡b1xc1+b2xc2,
and the boundary conditions can be written as
(22)(c2c1ΛIR2(c1-c2)-1Λ2(c1-c2))(b1b2)=(00).
The determinant must vanish for nontrivial b1,b2; hence,
(23)1-α/αc-11-α/αc+1=(Λ2ΛIR2)1-α/αc.
To satisfy this condition, α>αc=π/3 is needed, and for given Λ/ΛIR, α takes on discontinuous values
(24)nπ+tan-11α/αc-1=α/αc-1logΛΛIR.
One sees that α→αc for Λ/ΛIR→∞, the nominal “continuum limit.” For α<αc, the only solution that satisfies the B.C. is the trivial B(x)=0.The approximate solution for small IR cutoff can now be obtained by replacingB(x)=m (constant) for small x. The differential equation becomes
(25)xd2B(x)dx2+2dB(x)dx+3α4πB(x)x+m2=0,
where B(x) is a hypergeometric function. Namely,
(26)B(x)=mF(12-iγ,12+iγ,2;-xm2),
where
(27)iγ=121-ααc,αc=π3,
and F is a hypergeometric function. Checking the asymptotic behavior for x≫m2, the power behavior for α<αc cannot satisfy the UV boundary condition.The gap equation has a nontrivial or oscillatory solution forα>αc, where the critical coupling is αc=π/3 for QED. The dynamical effective mass is obtained by solving x=B2(-x). Note that the pole of the propagator should be given in the time-like region; so, one has to make analytical continuation in order to obtain a physical mass, which can be done smoothly for ΛIR→0.An issue arises in that the dynamical mass(28)m=4Λe-π/α/αc-1
is proportional to the UV cutoff Λ. For m to be physical, however, it should not depend on Λ. Miransky suggested [42] that α→αc as Λ→∞; that is, αc is a nontrivial UV fixed point. A related issue, which we would not go into, is whether there would be a dilaton associated with breaking of scale invariance [43, 44]. (These two are also the two best references for gauged NJL model).


## 3.2.1. Ladder Approximation

Truncating the exact, full vertex and photon propagator by the approximation(8)Γν(p,q)=γν,(9)Dμν(q)=-gμν+qμqν/q2q2-ξqμqνq4,
called the ladder (or rainbow) approximation, but retaining the electron self-energy, the gap equation becomes
(10)S(p)-1=p-ie2∫‍d4q(2π)4γμDμν(p-q)S(q)γν,
with Dμν now given in (9), and we have set m=0, that is, massless QED at Lagrangian level. Pictorially, this is represented as in Figure 5, where we note that, compared with the four-fermion coupling of the NJL model depicted in Figure 4, the external momentum p now flows into the self-energy loop. The question now is whether chiral symmetry can be dynamically broken, that a nontrivial solution to the self-energy could be generated at some strong coupling α=e2/4π?Figure 5
Gap equation for QED in the ladder approximation.We define the electron propagator as [41]
(11)S(p)=1A(p2)p-B(p2),
where the A term corresponds to wave function (w.f.) renormalization related to the usual Z-1 factor. A finite pole of the propagator, p2=B2(p2)/A2(p2), would give the dynamical effective mass. Our aim is therefore solving A and B from the gap equation of (10). Inserting (9) and after some algebra, one finds
(12)B(p2)=e2∫‍id4q(2π)4(-4+1-ξk2)B(q2)A2(q2)q2-B2(q2),A(p2)=1-e24p2tr∫‍id4q(2π)4γμDμν(ξ)A(q2)qA2(q2)q2-B2(q2)γνp.Simplification can be achieved in the Landau gauge,ξ=0, where one finds A(p2)=1. Since A relates to w.f. renormalization, it satisfies the Ward-Takahashi identity even under the ladder approximation. The gap equation is simplified to a single equation
(13)B(p2)=-3e2∫‍id4q(2π)4B(q2)(p-q)2(q2-B2(q2)).
After Wick rotation, and using
(14)∫‍d4qf(q2)(p-q)2=π2∫‍dq2q2[θ(p-q)p2+θ(q-p)q2]f(q2),
one obtains
(15)B(x)=3α4π(1x∫ΛIR2xdyyB(y)y+B2(y)+∫xΛ2dyB(y)y+B2(y)),
where x=pE2 is the Euclidean momentum squared, and Λ2, ΛIR2 are the ultraviolet (UV) and infrared (IR) cutoffs, respectively.The integral equation can be changed to a differential equation by noting(16)dB(x)dx=3α4π(-1x2)∫ΛIR2x‍dyyB(y)y+B2(y),d(xB(x))dx=3α4π∫xΛ2dyB(y)y+B2(y).
One obtains the differential equation
(17)xd2B(x)dx2+2dB(x)dx+3α4πB(x)x+B2(x)=0,
together with the IR and UV boundary conditions (B.C.)
(18)dB(x)dx|x=ΛIR2=0,d(xB(x))dx|x=Λ2=0.
If IR cutoff is 0, the B.C. for IR should be replaced by 2BdB/dx=-3α/4π at x=0.


## 3.2.2. Solution in Landau Gauge

The coupled integral equations are simplified and put into a differential equation with B.C. In order to study qualitative features, let us first find an approximate solution. For a special range for the IR cutoff, let us takeB(x)=ΛIR<x1/2<Λ; then, (17) is simplified to
(19)x2d2B(x)dx2+2xdB(x)dx+3α4πB(x)=0.
Inserting B(x)=Cxλ, the characteristic equation is
(20)λ(λ-1)+2λ+3α4π=0,
with discriminant 1-3α/π; hence, the behavior is different for α>αc and α<αc, where αc=π/3. The analytical solution under the approximation is given as
(21)B=b1x(-1+1-α/αc)/2+b2x(-1-1-α/αc)/2≡b1xc1+b2xc2,
and the boundary conditions can be written as
(22)(c2c1ΛIR2(c1-c2)-1Λ2(c1-c2))(b1b2)=(00).
The determinant must vanish for nontrivial b1,b2; hence,
(23)1-α/αc-11-α/αc+1=(Λ2ΛIR2)1-α/αc.
To satisfy this condition, α>αc=π/3 is needed, and for given Λ/ΛIR, α takes on discontinuous values
(24)nπ+tan-11α/αc-1=α/αc-1logΛΛIR.
One sees that α→αc for Λ/ΛIR→∞, the nominal “continuum limit.” For α<αc, the only solution that satisfies the B.C. is the trivial B(x)=0.The approximate solution for small IR cutoff can now be obtained by replacingB(x)=m (constant) for small x. The differential equation becomes
(25)xd2B(x)dx2+2dB(x)dx+3α4πB(x)x+m2=0,
where B(x) is a hypergeometric function. Namely,
(26)B(x)=mF(12-iγ,12+iγ,2;-xm2),
where
(27)iγ=121-ααc,αc=π3,
and F is a hypergeometric function. Checking the asymptotic behavior for x≫m2, the power behavior for α<αc cannot satisfy the UV boundary condition.The gap equation has a nontrivial or oscillatory solution forα>αc, where the critical coupling is αc=π/3 for QED. The dynamical effective mass is obtained by solving x=B2(-x). Note that the pole of the propagator should be given in the time-like region; so, one has to make analytical continuation in order to obtain a physical mass, which can be done smoothly for ΛIR→0.An issue arises in that the dynamical mass(28)m=4Λe-π/α/αc-1
is proportional to the UV cutoff Λ. For m to be physical, however, it should not depend on Λ. Miransky suggested [42] that α→αc as Λ→∞; that is, αc is a nontrivial UV fixed point. A related issue, which we would not go into, is whether there would be a dilaton associated with breaking of scale invariance [43, 44]. (These two are also the two best references for gauged NJL model).


## 3.3. Bootstrap Gap Equation for EWSB

Our recapitulation of strong QED is for the purpose of setting up our approach for solving the bootstrap gap equation [9] with large empirical Yukawa coupling, where we will continue to follow the Fukuda-Kugo approach.In Landau gauge (ξ=0), the propagations of (would-be) Goldstone modes and gauge bosons are properly separated, and the gap equation becomes equivalent to our discussion if one takes the g→0 limit for the gauge coupling. The Goldstone bosons G± and G0 couple to fermions with the familiar Yukawa couplings, which we have argued [9] as experimentally established. They are also unaltered by the g→0 limit, as seen in (2). The main assumption is the addition of a new (heavy) chiral quark doublet Q, where the heavy isospin IQ symmetry implies that λU=λD≡λQ, that is, equality of the IQ3=±1/2 Yukawa couplings. We ask whether large λQ could be the source of EWSB, through the conceptual gap equation of Figure 1.


### 3.3.1. Ladder Approximation

In any case, we do not know the full propagator and vertex functions. We approximate the Goldstone-fermion vertex as undressed, and the Goldstone propagator remains asi/k2, analogous to the QED treatment in previous section. This is in part aided by the insight that G is a very tight QQ- bound state, which would be massless as long as the symmetry remains spontaneously broken. Similar to Figure 5, the gap equation for large Yukawa (vanishing g) coupling is depicted in Figure 6. Again, the external momentum p flows through the loop.Figure 6
Gap equation for large Yukawa coupling in the ladder approximation, where vanishingm0 is indicated.The change from the symbolic or conceptual equation of Figure1 is that, even with bare mass forbidden by gauge invariance (m=0), there is a p part for Q propagation, and we aim at solving for the quark propagator S(p), given as
(29)S(p)-1=A(p2)p-B(p2),
which is of course of the same form as (11) for QED. The Goldstone boson G is colorless, and, unlike the massless photon from electromagnetic gauge invariance for QED case, its masslessness is “bootstrapped” into the gap equation itself. Following similar steps as in Section 2, and assuming λU=λD=λQ degeneracy, we obtain
(30)Bp2=3λQ22∫‍d4qi(2π)41(p-q)2Bq2q2Aq22-Bq22-λQ22∫‍d4qi(2π)41(p-q)2-mh2Bq2q2Aq22-Bq22,(31)Ap2=1+3λQ22p2∫‍d4qi(2π)4p·q(p-q)2Aq2q2Aq22-Bq22+λQ22p2∫‍d4qi(2π)4p·q(p-q)2-mh2Aq2q2Aq22-Bq22,
where placement of i anticipates the Wick rotation, and Bq2 stands as shorthand for B(q2), and likewise for A. We have already used ξ=0; so, compared to massless QED, we now have to consider A, or wave function renormalization effects; hence, we have to face a coupled set of equations for A(p2) and B(p2). Note that we have kept a “Higgs” term for calculational purposes, applying Standard Model Higgs boson, h0, couplings. This is for purpose of later comparison with the work of Hung and Xiong [25], as well as for discussion of the dilaton case.For our aim of bootstrap DSB, we simply drop the second term (no physical Higgs, or takingmh~∞), and so only Goldstone modes propagate in both equations. After angular integration and Wick rotation, one gets
(32)B(x)=κb(B(y)yA2(y)+B2(y)1x∫ΛIR2x‍dyyB(y)yA2(y)+B2(y)+∫xΛ2‍dyB(y)yA2(y)+B2(y)),A(x)=1+κa(A(y)yA2(y)+B2(y)1x2∫ΛIR2x‍dyy2A(y)yA2(y)+B2(y)+∫xΛ2‍dyA(y)yA2(y)+B2(y)),
where
(33)κb=2κa=(3/2)λQ216π2=(3/2)αQ4π(noHiggs).
Taking the limit mh→0 in (30) and (31) to mimic the Hung-Xiong approach of massless scalar doublet,
(34)κb=κa=λQ216π2=αQ4π(masslessHiggs).
We keep the notation of κb and κa in (33) and (34) to cover these two cases.Analogous to Section2, (32) can be put into differential form
(35)xB′′+2B′+κbBxA2+B2=0,(36)xA′′+3A′+2κaAxA2+B2=0,
with the boundary conditions
(37)B′(x)|x=ΛIR2=0,[xB′(x)+B(x)]|x=Λ2=0,(38)A′(x)|x=ΛIR2=0,[x2A′(x)+A(x)]|x=Λ2=1,
where prime stands for x-derivative.At this point, we note that, if one ignores wave function renormalization, that is, (36), while forcefully setting A(x)=1 in (35), then one has the same solution as in QED, with the change in critical coupling
(39)αQc=2π3,
which is twice as high as for QED, and
(40)λQc=2π23≃5.13;
hence, superficially a “critical mass” mQc≃890 GeV, which is above current LHC limits [14–20]. But it should be clear that the wave function renormalization A(p2) term cannot be neglected.


### 3.3.2. Numerical Solution

Redefiningp2=x=e2t, our coupled differential equations with B.C. become
(41)B¨+2B˙+4κbBA2+B2e-2t=0,A¨+4A˙+8κaAA2+B2e-2t=0,B˙(tIR)=0,B˙(tUV)+B(tUV)=0,A˙(tIR)=0,14A˙(tUV)+A(tUV)=1,
where dot represents t-derivative, and etUV=ΛUV=Λ, etIR=ΛIR.Asymptotic Properties and Critical Coupling. Due to scale invariance, the differential equations are invariant under
(42)x→a2x(t→t+loga),ΛUV,IR→aΛUV,IR,B→aB,A→A.
As a result, the solutions of the differential equations depend only on ΛUV/ΛIR (=etUV-tIR) and mdyn≡B(tIR)/A(tIR) for given κa and κb. Thus, mdyn is a kind of integration constant. We will see that the most important feature of the solutions is that only special values (discontinuous values) of κa and κb are allowed for given boundary conditions.If we takeκa=0, the equations can be solved analytically, and the solution can be described as in the case of strong coupling QED with Landau gauge, (21)–(23). The property illustrated for κa=0 should hold also for κa≠0, where, depending on ΛUV/ΛIR, κa should take on special discontinuous values to satisfy the gap equation for the cases of κa/κb=1 or 1/2.To see this, we note that theB2e-2t term in the denominators is irrelevant. This is because for t≪logm, the t dependence of A and B is negligible due to the boundary conditions A˙(tIR)=B˙(tIR)=0. Therefore, to understand the behavior of the solution, one can take tIR~logm and consider the differential equation with B2e-2t dropped as follows:
(43)B¨+2B˙+4κbBA2=0,A¨+4A˙+8κaA=0,
where the equation for A becomes independent from B, but its solution affects B.The solution ofA is obtained analytically as follows:
(44)A(x)=A0ΛUV4x2e-ζ2κaπ,ζ=Erf-1[1A0(1-x2ΛUV4)+Erf(1κa)],
where A0 is an integration constant which can be fixed by the B.C. for IR. Using the analytical solution, one can show that only discontinuous values of κa,b can satisfy the B.Cs. The “critical” value of κb can be easily obtained numerically, even without neglecting B2e-2t term in the denominators. Of course, since the solutions are found numerically, it is not a proof that one really has a “critical” value. The upshot is that only special values of the coupling can satisfy the coupled equations for given values of ΛUV/ΛIR, for mdyn<ΛIR (or ΛUV/mdyn for mdyn>ΛIR) and mdyn=B(ΛIR)/A(ΛIR).Our numerical solution gives(45)κbc≃1.4,(κb=2κa=3αQ8π),κbc≃13.7,(κb=κa=αQ4π),
corresponding to
(46)λQc≃12,(κb=2κa=3αQ8π),(47)λQc≃46,(κb=κa=αQ4π),
where the latter case is much higher. Here c stands for “critical,” and our numerical values are extracted in the large ΛUV/ΛIR and ΛUV/mdyn limit. Note that for the artificial case of κa=0 (i.e., A=1), the critical value was λQc≃5.1, (40); hence, the effect of considering A or wave function renormalization is quite nontrivial.The values in (46) and (47) correspond to effectively taking ΛUV→∞, which is certainly not the range of validity for (46) as a descendent of Figure 1. That is, the conceptual foundation for Figure 1 is that, for momentum roughly up to somewhere below 2mQ, corrections to the Goldstone boson propagator and vertex have been ignored. Nevertheless, at face value, if we naively apply the physical v≃246 GeV, then (46) and (47) imply the mass values
(48)mQc>2.1TeV(noHiggs)
and 8.1 TeV, respectively, which are rather high. We drop the latter, “massless (or light) Higgs” case, not only because it is clearly out of reach for the LHC, but for theoretical reasons as we explain later. The lower bound nature of (48) would be explained subsequently.


## 3.3.1. Ladder Approximation

In any case, we do not know the full propagator and vertex functions. We approximate the Goldstone-fermion vertex as undressed, and the Goldstone propagator remains asi/k2, analogous to the QED treatment in previous section. This is in part aided by the insight that G is a very tight QQ- bound state, which would be massless as long as the symmetry remains spontaneously broken. Similar to Figure 5, the gap equation for large Yukawa (vanishing g) coupling is depicted in Figure 6. Again, the external momentum p flows through the loop.Figure 6
Gap equation for large Yukawa coupling in the ladder approximation, where vanishingm0 is indicated.The change from the symbolic or conceptual equation of Figure1 is that, even with bare mass forbidden by gauge invariance (m=0), there is a p part for Q propagation, and we aim at solving for the quark propagator S(p), given as
(29)S(p)-1=A(p2)p-B(p2),
which is of course of the same form as (11) for QED. The Goldstone boson G is colorless, and, unlike the massless photon from electromagnetic gauge invariance for QED case, its masslessness is “bootstrapped” into the gap equation itself. Following similar steps as in Section 2, and assuming λU=λD=λQ degeneracy, we obtain
(30)Bp2=3λQ22∫‍d4qi(2π)41(p-q)2Bq2q2Aq22-Bq22-λQ22∫‍d4qi(2π)41(p-q)2-mh2Bq2q2Aq22-Bq22,(31)Ap2=1+3λQ22p2∫‍d4qi(2π)4p·q(p-q)2Aq2q2Aq22-Bq22+λQ22p2∫‍d4qi(2π)4p·q(p-q)2-mh2Aq2q2Aq22-Bq22,
where placement of i anticipates the Wick rotation, and Bq2 stands as shorthand for B(q2), and likewise for A. We have already used ξ=0; so, compared to massless QED, we now have to consider A, or wave function renormalization effects; hence, we have to face a coupled set of equations for A(p2) and B(p2). Note that we have kept a “Higgs” term for calculational purposes, applying Standard Model Higgs boson, h0, couplings. This is for purpose of later comparison with the work of Hung and Xiong [25], as well as for discussion of the dilaton case.For our aim of bootstrap DSB, we simply drop the second term (no physical Higgs, or takingmh~∞), and so only Goldstone modes propagate in both equations. After angular integration and Wick rotation, one gets
(32)B(x)=κb(B(y)yA2(y)+B2(y)1x∫ΛIR2x‍dyyB(y)yA2(y)+B2(y)+∫xΛ2‍dyB(y)yA2(y)+B2(y)),A(x)=1+κa(A(y)yA2(y)+B2(y)1x2∫ΛIR2x‍dyy2A(y)yA2(y)+B2(y)+∫xΛ2‍dyA(y)yA2(y)+B2(y)),
where
(33)κb=2κa=(3/2)λQ216π2=(3/2)αQ4π(noHiggs).
Taking the limit mh→0 in (30) and (31) to mimic the Hung-Xiong approach of massless scalar doublet,
(34)κb=κa=λQ216π2=αQ4π(masslessHiggs).
We keep the notation of κb and κa in (33) and (34) to cover these two cases.Analogous to Section2, (32) can be put into differential form
(35)xB′′+2B′+κbBxA2+B2=0,(36)xA′′+3A′+2κaAxA2+B2=0,
with the boundary conditions
(37)B′(x)|x=ΛIR2=0,[xB′(x)+B(x)]|x=Λ2=0,(38)A′(x)|x=ΛIR2=0,[x2A′(x)+A(x)]|x=Λ2=1,
where prime stands for x-derivative.At this point, we note that, if one ignores wave function renormalization, that is, (36), while forcefully setting A(x)=1 in (35), then one has the same solution as in QED, with the change in critical coupling
(39)αQc=2π3,
which is twice as high as for QED, and
(40)λQc=2π23≃5.13;
hence, superficially a “critical mass” mQc≃890 GeV, which is above current LHC limits [14–20]. But it should be clear that the wave function renormalization A(p2) term cannot be neglected.


## 3.3.2. Numerical Solution

Redefiningp2=x=e2t, our coupled differential equations with B.C. become
(41)B¨+2B˙+4κbBA2+B2e-2t=0,A¨+4A˙+8κaAA2+B2e-2t=0,B˙(tIR)=0,B˙(tUV)+B(tUV)=0,A˙(tIR)=0,14A˙(tUV)+A(tUV)=1,
where dot represents t-derivative, and etUV=ΛUV=Λ, etIR=ΛIR.Asymptotic Properties and Critical Coupling. Due to scale invariance, the differential equations are invariant under
(42)x→a2x(t→t+loga),ΛUV,IR→aΛUV,IR,B→aB,A→A.
As a result, the solutions of the differential equations depend only on ΛUV/ΛIR (=etUV-tIR) and mdyn≡B(tIR)/A(tIR) for given κa and κb. Thus, mdyn is a kind of integration constant. We will see that the most important feature of the solutions is that only special values (discontinuous values) of κa and κb are allowed for given boundary conditions.If we takeκa=0, the equations can be solved analytically, and the solution can be described as in the case of strong coupling QED with Landau gauge, (21)–(23). The property illustrated for κa=0 should hold also for κa≠0, where, depending on ΛUV/ΛIR, κa should take on special discontinuous values to satisfy the gap equation for the cases of κa/κb=1 or 1/2.To see this, we note that theB2e-2t term in the denominators is irrelevant. This is because for t≪logm, the t dependence of A and B is negligible due to the boundary conditions A˙(tIR)=B˙(tIR)=0. Therefore, to understand the behavior of the solution, one can take tIR~logm and consider the differential equation with B2e-2t dropped as follows:
(43)B¨+2B˙+4κbBA2=0,A¨+4A˙+8κaA=0,
where the equation for A becomes independent from B, but its solution affects B.The solution ofA is obtained analytically as follows:
(44)A(x)=A0ΛUV4x2e-ζ2κaπ,ζ=Erf-1[1A0(1-x2ΛUV4)+Erf(1κa)],
where A0 is an integration constant which can be fixed by the B.C. for IR. Using the analytical solution, one can show that only discontinuous values of κa,b can satisfy the B.Cs. The “critical” value of κb can be easily obtained numerically, even without neglecting B2e-2t term in the denominators. Of course, since the solutions are found numerically, it is not a proof that one really has a “critical” value. The upshot is that only special values of the coupling can satisfy the coupled equations for given values of ΛUV/ΛIR, for mdyn<ΛIR (or ΛUV/mdyn for mdyn>ΛIR) and mdyn=B(ΛIR)/A(ΛIR).Our numerical solution gives(45)κbc≃1.4,(κb=2κa=3αQ8π),κbc≃13.7,(κb=κa=αQ4π),
corresponding to
(46)λQc≃12,(κb=2κa=3αQ8π),(47)λQc≃46,(κb=κa=αQ4π),
where the latter case is much higher. Here c stands for “critical,” and our numerical values are extracted in the large ΛUV/ΛIR and ΛUV/mdyn limit. Note that for the artificial case of κa=0 (i.e., A=1), the critical value was λQc≃5.1, (40); hence, the effect of considering A or wave function renormalization is quite nontrivial.The values in (46) and (47) correspond to effectively taking ΛUV→∞, which is certainly not the range of validity for (46) as a descendent of Figure 1. That is, the conceptual foundation for Figure 1 is that, for momentum roughly up to somewhere below 2mQ, corrections to the Goldstone boson propagator and vertex have been ignored. Nevertheless, at face value, if we naively apply the physical v≃246 GeV, then (46) and (47) imply the mass values
(48)mQc>2.1TeV(noHiggs)
and 8.1 TeV, respectively, which are rather high. We drop the latter, “massless (or light) Higgs” case, not only because it is clearly out of reach for the LHC, but for theoretical reasons as we explain later. The lower bound nature of (48) would be explained subsequently.


## 3.4. Discussion

The large critical coupling ofλQ≳4π of (46) is somewhat surprising, resulting in the rather high critical mass of (48). We note that this is without the contribution of a light SM Higgs boson, which would put mQc much higher. Before we touch phenomenology of such high chiral quark masses, and in so doing find an independent justification for such high critical Yukawa coupling, we offer some further discussion.


### 3.4.1. Comparison with Hung and Xiong

We can now make some comparison with, and offer a critique of, the approach of Hung and Xiong [25] (HX). We have mimicked the concept of HX by keeping a “Higgs” scalar contribution in (30) and (31), which resulted in the second case of (34) in the limit of mh→0, as compared with our case of interest, (33), where we drop the mh-dependent term.HXassumed the existence of a massless Higgs doublet, where our (35) and (36) with (34) should be a faithful representation. However, not only is the massless doublet assumed, HX also ignored wave function renormalization, that is, the A term, in the treatment of their gap equation. Furthermore, the sign of our second integral in (30) disagrees with HX, suppressing the coefficient of λQ2 in comparison. In the earlier work [9] that set up the gap equation of Figure 1, taking the numerics of HX, the estimate of λQc=2π (compare (40)) gave mQc≃770 GeV, which is not too far above the current LHC bound. But with our sign for the second integral in (30), we would arrive at λQc=2π, or mQc~1.1 TeV, that is, a factor of 2 higher. With our sign, the Higgs boson effect would cancel out part of the Goldstone effect, hence, requiring stronger Yukawa coupling.But we have argued that it is not justified to ignore the wave function renormalization effect ofA(p2). After all, the boson loop has momentum dependence, and so it would necessarily affect the Z-1 factor. Thus, the previous simple numerics are incorrect. Keeping A in our numerical study, hence, the coupled A-B equations, a considerably higher critical mQc is found. For the case of taking κb=κa=λQ2/16π2, (34), where we mimic HX’s massless Higgs doublet effect, the critical λQc of (47) is almost 4 times as high as that for (46). In fact, we obtain the A(ΛUV2)=2.7, which is quite different from 1.Our criticism goes far deeper. Taking a scalar doublet as massless, such that superficially one has “scale-invariance” as in strong QED, is totallyad hoc. Effectively one has to hold the parameters of the Higgs potential such that the Higgs field always remains massless. However, there is no principle by which this scale-invariance or masslessness of the Higgs field can be maintained. After all, one is invoking large Yukawa couplings, which feed the notorious divergent quadratic corrections to the Higgs boson mass. The one-loop two-point and four-point functions with quark Q in the loop would generate effective μ2 and λ self-coupling terms for the Higgs field. With no explicit dynamical principle (such as gauge invariance for the case of QED), the assumption of a massless Higgs doublet as the agent of DSB is not only ad hoc, but clearly unsustainable.In contrast, we see the merit, as well as the meaning, of our “bootstrap” gap equation. As long as we are in the broken phase, there is a massless Goldstone boson that couples with the Yukawa couplingλQ. Treating λQ as large, if a nontrivial solution to the gap equation is found (as we have illustrated earlier in the previous subsection), it in turn justifies the use of a massless Goldstone boson in the gap equation, which is why we affix the name “bootstrap.” In fact, the physical argument [9] was to view the Goldstone boson G as an extremely tight ultrarelativistic bound state of heavy Q and Q- from the broken phase, while G enters the “bootstrap” gap equation to dynamically generate mQ, hence, break the symmetry, and in the same stroke justify its own existence.We also see that this argument for our bootstrap gap equation puts the existence of a light Higgs boson in doubt. That is, it seems rather difficult to keep it light, and one cannot ignore corrections to the Higgs propagator. At the foundation level [9], unlike the Goldstone boson, the experimental basis for the Higgs boson is rather recent. Even with the newly discovered [1, 2] 125 GeV boson, one has to distinguish between an SM Higgs boson versus a dilaton. Our numerical study also shows that keeping the Higgs term tends to raise the critical Yukawa coupling considerably, implying 2mQ higher than the LHC collision energy. Thus, the light Higgs case of κb=κa is rather problematic in bootstrap DSB. We will return to discuss the dilaton possibility later.


### 3.4.2. Physical Cutoff

The previous comparison with [25], and in particular the bootstrap nature of the Goldstone boson G in the gap equation, illustrates that the limit ΛUV→∞ would take us outside the range of validity of the gap equation itself. It is clear that, for timelike q2>(2mQ)2, there would no longer be a Goldstone boson. Thus, for the cutoff Λ, one should not use the traditional language of ΛUV. One probably should not contemplate “UV-completion" at the current stage. The bootstrap gap equation is rather heuristically or physically argued. But it does not provide a theory of the heavy quark Yukawa coupling λQ, employing it for DSB instead. Thus, we suggest a cutoff Λ<2mQ<ΛNP, where ΛNP is some true New Physics scale where the origin of Yukawa couplings may be contemplated, but it is out of reach from the bootstrap DSB approach.Viewed from a slightly different angle, this cutoff is related to the restoration of symmetry. The Goldstone boson couples to broken currents, but theZ factor of the Goldstone boson may vanish at some scale related to the scale where the Goldstone boson becomes unbounded, and symmetry is restored. Rather than a true ΛUV that in principle extends to infinity, there exists a cutoff Λ of the gap equation. We are in effect summing over the Goldstone boson correction to the self-energy of the quark Q, when the Goldstone boson is still defined. As noted in [9], this picture does receive experimental support, in that no New Physics seem to be there below ~TeV scale. So, one sums up only the effect of the Goldstone boson, and nothing else.The gap equation sums up the effect, from low to high momentum, of the correction by the Goldstone bosonG to the quark self-energy. Equation (46) reflects taking this sum to ΛUV→∞. Since the summation is accumulative, if now one sums only to some Λ≲2mQ, as we have argued, then the cumulative effect is less than summing to very large momentum above 2mQ. We note that 2mQ is a physical scale parameter that is external to the scale-invariant gap equation. Nevertheless, we can plot the dependence of λQc on the cutoff Λ. From Figure 7, one can see that, for a lower cutoff, λQc has to be higher than in (46). This is because, as the integration range is smaller, a larger λQ value is needed to compensate. Thus, (46) gives a lower bound on mQ. If we take Λ=2mQ, then λQc~17.7, and
(49)mQ~3TeV(noHiggs;Λ=2mQ),
which is higher than (48). We will return to discuss whether this could be an overestimate later.Figure 7
Cutoff dependence of Yukawa coupling, where the cutoffΛ is in units of the dynamical quark mass mdyn.


### 3.4.3. Future Work

Up to now, we have been cavalier in the relation betweenλQc and mQ(c), treating it as 2mQ(c)=λQcv. But v is a physically measured quantity, and mQ is yet to be experimentally measured. If electroweak symmetry is indeed dynamically broken by the large Yukawa coupling of a new heavy chiral quark Q, when Q is discovered in the future, likely mQ>mQc. We then see that the actual v.e.v. value, v, may not correspond to the critical value λQc. This brings about the question of how scale invariance is actually broken in our gap equations, and whether there might be a dilaton [5–7, 45–49]. (When the 125 GeV hint emerged from 2011 LHC data, [45–47] already suggested that it might be a dilaton rather than the SM Higgs boson, while [48, 49] are from the (walking) technicolor perspective).Rather than to approach this deeper problem, we try to obtain the decay constantfG of the Goldstone boson, which should be the same as the vacuum expectation value, v. Following the Pagels-Stokar formula [50] naively, we obtain
(50)fG2=NC4π2∫0Λ2dxxB(x)2-(1/4)x(B(x)2)′(xA(x)2+B(x)2)2.
More generally, we write
(51)fG2=∫0Λ2‍dxℱ[A(x),B(x)]=mQ2∫0Λ2/mQ2‍dx^ℱ^[A(x^),B(x^)/A(x^)/mQ],
where mQ≡B(0)/A(0), and we have scaled by mQ (and redefined the function ℱ), treating it as physical.We can get back the “Yukawa” couplingλ (it should really be denoted as Y, and the question is whether Y=λ) for the input to the gap equation, or
(52)12λ2=∫0Λ2/mQ2‍dx^ℱ^[A(x^),B(x^)/A(x^)/mQ].
If the system is really scale-invariant, the r.h.s. of (52) is a function of Λ/mQ and λ. In order to satisfy the gap equation, λ is obtained as a function of Λ/mQ. Namely, it should not depend on mQ explicitly. Taking the cutoff Λ=2mQ, the equation becomes iterative for λ. Therefore, solving the gap equation, we obtain a prediction for the heavy quark mass. But technical issues remain. Is mQ a physical mass? What about the infrared cutoff? Can our assumption of λ(ΛUV/mQ)=Y(ΛUV/mQ) be maintained self-consistently? Our work is far from complete, and we leave these theoretical questions to the future.


## 3.4.1. Comparison with Hung and Xiong

We can now make some comparison with, and offer a critique of, the approach of Hung and Xiong [25] (HX). We have mimicked the concept of HX by keeping a “Higgs” scalar contribution in (30) and (31), which resulted in the second case of (34) in the limit of mh→0, as compared with our case of interest, (33), where we drop the mh-dependent term.HXassumed the existence of a massless Higgs doublet, where our (35) and (36) with (34) should be a faithful representation. However, not only is the massless doublet assumed, HX also ignored wave function renormalization, that is, the A term, in the treatment of their gap equation. Furthermore, the sign of our second integral in (30) disagrees with HX, suppressing the coefficient of λQ2 in comparison. In the earlier work [9] that set up the gap equation of Figure 1, taking the numerics of HX, the estimate of λQc=2π (compare (40)) gave mQc≃770 GeV, which is not too far above the current LHC bound. But with our sign for the second integral in (30), we would arrive at λQc=2π, or mQc~1.1 TeV, that is, a factor of 2 higher. With our sign, the Higgs boson effect would cancel out part of the Goldstone effect, hence, requiring stronger Yukawa coupling.But we have argued that it is not justified to ignore the wave function renormalization effect ofA(p2). After all, the boson loop has momentum dependence, and so it would necessarily affect the Z-1 factor. Thus, the previous simple numerics are incorrect. Keeping A in our numerical study, hence, the coupled A-B equations, a considerably higher critical mQc is found. For the case of taking κb=κa=λQ2/16π2, (34), where we mimic HX’s massless Higgs doublet effect, the critical λQc of (47) is almost 4 times as high as that for (46). In fact, we obtain the A(ΛUV2)=2.7, which is quite different from 1.Our criticism goes far deeper. Taking a scalar doublet as massless, such that superficially one has “scale-invariance” as in strong QED, is totallyad hoc. Effectively one has to hold the parameters of the Higgs potential such that the Higgs field always remains massless. However, there is no principle by which this scale-invariance or masslessness of the Higgs field can be maintained. After all, one is invoking large Yukawa couplings, which feed the notorious divergent quadratic corrections to the Higgs boson mass. The one-loop two-point and four-point functions with quark Q in the loop would generate effective μ2 and λ self-coupling terms for the Higgs field. With no explicit dynamical principle (such as gauge invariance for the case of QED), the assumption of a massless Higgs doublet as the agent of DSB is not only ad hoc, but clearly unsustainable.In contrast, we see the merit, as well as the meaning, of our “bootstrap” gap equation. As long as we are in the broken phase, there is a massless Goldstone boson that couples with the Yukawa couplingλQ. Treating λQ as large, if a nontrivial solution to the gap equation is found (as we have illustrated earlier in the previous subsection), it in turn justifies the use of a massless Goldstone boson in the gap equation, which is why we affix the name “bootstrap.” In fact, the physical argument [9] was to view the Goldstone boson G as an extremely tight ultrarelativistic bound state of heavy Q and Q- from the broken phase, while G enters the “bootstrap” gap equation to dynamically generate mQ, hence, break the symmetry, and in the same stroke justify its own existence.We also see that this argument for our bootstrap gap equation puts the existence of a light Higgs boson in doubt. That is, it seems rather difficult to keep it light, and one cannot ignore corrections to the Higgs propagator. At the foundation level [9], unlike the Goldstone boson, the experimental basis for the Higgs boson is rather recent. Even with the newly discovered [1, 2] 125 GeV boson, one has to distinguish between an SM Higgs boson versus a dilaton. Our numerical study also shows that keeping the Higgs term tends to raise the critical Yukawa coupling considerably, implying 2mQ higher than the LHC collision energy. Thus, the light Higgs case of κb=κa is rather problematic in bootstrap DSB. We will return to discuss the dilaton possibility later.


## 3.4.2. Physical Cutoff

The previous comparison with [25], and in particular the bootstrap nature of the Goldstone boson G in the gap equation, illustrates that the limit ΛUV→∞ would take us outside the range of validity of the gap equation itself. It is clear that, for timelike q2>(2mQ)2, there would no longer be a Goldstone boson. Thus, for the cutoff Λ, one should not use the traditional language of ΛUV. One probably should not contemplate “UV-completion" at the current stage. The bootstrap gap equation is rather heuristically or physically argued. But it does not provide a theory of the heavy quark Yukawa coupling λQ, employing it for DSB instead. Thus, we suggest a cutoff Λ<2mQ<ΛNP, where ΛNP is some true New Physics scale where the origin of Yukawa couplings may be contemplated, but it is out of reach from the bootstrap DSB approach.Viewed from a slightly different angle, this cutoff is related to the restoration of symmetry. The Goldstone boson couples to broken currents, but theZ factor of the Goldstone boson may vanish at some scale related to the scale where the Goldstone boson becomes unbounded, and symmetry is restored. Rather than a true ΛUV that in principle extends to infinity, there exists a cutoff Λ of the gap equation. We are in effect summing over the Goldstone boson correction to the self-energy of the quark Q, when the Goldstone boson is still defined. As noted in [9], this picture does receive experimental support, in that no New Physics seem to be there below ~TeV scale. So, one sums up only the effect of the Goldstone boson, and nothing else.The gap equation sums up the effect, from low to high momentum, of the correction by the Goldstone bosonG to the quark self-energy. Equation (46) reflects taking this sum to ΛUV→∞. Since the summation is accumulative, if now one sums only to some Λ≲2mQ, as we have argued, then the cumulative effect is less than summing to very large momentum above 2mQ. We note that 2mQ is a physical scale parameter that is external to the scale-invariant gap equation. Nevertheless, we can plot the dependence of λQc on the cutoff Λ. From Figure 7, one can see that, for a lower cutoff, λQc has to be higher than in (46). This is because, as the integration range is smaller, a larger λQ value is needed to compensate. Thus, (46) gives a lower bound on mQ. If we take Λ=2mQ, then λQc~17.7, and
(49)mQ~3TeV(noHiggs;Λ=2mQ),
which is higher than (48). We will return to discuss whether this could be an overestimate later.Figure 7
Cutoff dependence of Yukawa coupling, where the cutoffΛ is in units of the dynamical quark mass mdyn.


## 3.4.3. Future Work

Up to now, we have been cavalier in the relation betweenλQc and mQ(c), treating it as 2mQ(c)=λQcv. But v is a physically measured quantity, and mQ is yet to be experimentally measured. If electroweak symmetry is indeed dynamically broken by the large Yukawa coupling of a new heavy chiral quark Q, when Q is discovered in the future, likely mQ>mQc. We then see that the actual v.e.v. value, v, may not correspond to the critical value λQc. This brings about the question of how scale invariance is actually broken in our gap equations, and whether there might be a dilaton [5–7, 45–49]. (When the 125 GeV hint emerged from 2011 LHC data, [45–47] already suggested that it might be a dilaton rather than the SM Higgs boson, while [48, 49] are from the (walking) technicolor perspective).Rather than to approach this deeper problem, we try to obtain the decay constantfG of the Goldstone boson, which should be the same as the vacuum expectation value, v. Following the Pagels-Stokar formula [50] naively, we obtain
(50)fG2=NC4π2∫0Λ2dxxB(x)2-(1/4)x(B(x)2)′(xA(x)2+B(x)2)2.
More generally, we write
(51)fG2=∫0Λ2‍dxℱ[A(x),B(x)]=mQ2∫0Λ2/mQ2‍dx^ℱ^[A(x^),B(x^)/A(x^)/mQ],
where mQ≡B(0)/A(0), and we have scaled by mQ (and redefined the function ℱ), treating it as physical.We can get back the “Yukawa” couplingλ (it should really be denoted as Y, and the question is whether Y=λ) for the input to the gap equation, or
(52)12λ2=∫0Λ2/mQ2‍dx^ℱ^[A(x^),B(x^)/A(x^)/mQ].
If the system is really scale-invariant, the r.h.s. of (52) is a function of Λ/mQ and λ. In order to satisfy the gap equation, λ is obtained as a function of Λ/mQ. Namely, it should not depend on mQ explicitly. Taking the cutoff Λ=2mQ, the equation becomes iterative for λ. Therefore, solving the gap equation, we obtain a prediction for the heavy quark mass. But technical issues remain. Is mQ a physical mass? What about the infrared cutoff? Can our assumption of λ(ΛUV/mQ)=Y(ΛUV/mQ) be maintained self-consistently? Our work is far from complete, and we leave these theoretical questions to the future.


## 4. New Phenomena:QQ-→nVL

The stringent limit ofmQ≳700 GeV/c2 [14–20] is already above the perturbative, tree-level partial wave unitarity bound (UB) that is nominally around 550 GeV/c2 [21]. With TeV scale heavy quark masses, the actual UB violation (UBV) in the high energy limit for QQ- scattering may be out of reach. Instead, the question to ask is should the current search strategy for ultraheavy quarkQat the LHC be modified? Of course, with the advent of the 125 GeV boson at the LHC, the search for the four generation quarks has been blunted, and the search effort is being reshuffled to search for vector-like quarks, where flavor changing neutral current (FCNC) decays to Z and Higgs bosons need to be studied. We maintain, however, that SM has built-in absence of FCNC, both for the Z (only chiral fermions) and Higgs (single Higgs doublet). There is some merit to stick to this, while our bootstrap equation has even disposed of the Higgs doublet altogether.So, what if there are heavy chiral quarks in Nature at the 2-3 TeV level? Can they be discovered at the LHC? We draw on the analogy with the proton to argue [52] that QQ-→nG (G≡VL is the longitudinal component of vector boson) may be the new signature at the LHC.TheπNN coupling gπNN2/4π≃14 [53, 54] gives gπNN≃13, which is very large and quite close to the πNN “Yukawa coupling,” λπNN≡2mN/fπ≃14, which is slightly larger than 4π. Although λQ≡2mQ/v≳3.5 (v≅246 GeV is the electroweak symmetry breaking scale) from the current mQ bound is not yet as large, drawing analogy with pp- annihilation, we expect that QQ-→nG may be the dominant process for mQ∈(1,2) GeV. It is also realizing the fact that λπNN≃gπNN is of similar strength to 12≲λQc≲17, that such strong couplings already exist in Nature, which gives some confidence to our very large critical Yukawa coupling strength.


### 4.1. Phenomenology ofpp-→nπ

The phenomenon forpp- annihilation is unfamiliar to the average particle physicist. One tends to think of annihilation into photons (much suppressed by α of QED) or gluons. But even for the latter, it is not practical with total energy of ~1.88 GeV and in consideration of detection of final states. Certainly, pp- would annihilate into mesonic resonances. However, all mesonic resonances basically end up as pions. Thus, one can safely conclude that pp-→nπ in general.Indeed, it is observed thatpp- annihilation [51, 55] goes mainly into n pions. However, the features are rather surprising, even for the nuclear and hadron physicists that worked on the subject from the 1950s to the 1990. It is found that pp-→nπ goes via a “fireball.” The salient features of the annihilation “fireball” are (see Figure 8)(i)
size of order1/mπ;(ii)
temperatureT≃120MeV;(iii)
average number of emitted pions〈nπ〉≃5;(iv)
a soft-pionpπ2/Eπ2 factor modulates the Maxwell-Boltzman distribution for the pions.Figure 8
Illustration forpp-→nπ for n=〈nπ〉≅5 and the analog of QQ-→nG, where G=VL is the Goldstone boson of EWSB. Depending on mQ=1-2 GeV, 〈nG〉 could go from 6 to 12 (see (58) and Table 1). The gluon line indicates the relatively soft shedding of color.It is worthwhile to elucidate these features a little further. The size1/mπ means that, when the p and p- meet to annihilate, destined to shed the p and p- content, the region of annihilation extends over a region ~1/mπ. The system seems to thermalize to a temperature Tpp- of order 120 MeV, hence, “loses memory” of its origins, and the emitted pions carry momenta that satisfy a thermal distribution. This rapid thermalization probably takes place due to the rather large πNN (as well as ππ) coupling, while the pπ2/Eπ2 suppression [56] (satisfied rather well by data; see Figure 12 of [51]) for low pion momentum from the thermal distribution reflects nothing but the Goldstone nature of pion couplings. That is, the π as Goldstone boson couples derivatively, hence, cannot get emitted at zero momentum. This seems to explain the enhancement factor of 1.3 for the mean kinetic energy 〈Kπ〉≡〈Eπ〉-mπ beyond equipartition expectation of (3/2)Tpp-. Thus, the relatively high 〈Eπ〉~370MeV gives rise to 〈nπ〉≃5.1, as compared to the maximal allowed number of pions, 2mN/mπ~13.At a more refined level, it is found that〈nπ±〉≃3.1 and 〈nπ0〉≃2.1, with 2〈nπ0〉/〈nπ±〉>1; that is, more neutral pions are emitted than charged ones. Remarkably, the pion multiplicity distribution appears Gaussian,
(53)P(nπ)=12πσe-(nπ-〈nπ〉)2/2σ2,
with σ~1. More specifically [57],
(54)σ≃12〈nπ〉
is argued from statistical models [58]. Thus, σ≃1.13 gives a good fit to data [51], which is given in the first row of Table 1. Note the minute pp-→ππ 2-body fraction, while the pion multiplicity cuts off above 8.Table 1
Sample multiplicity distributions:Ppp-(n) is the observed distribution for pp-→nπ [51], while PQQ-1(2)(n) is the QQ-→nG distribution for mQ=1 (2) TeV according to (58), where G≡VL is the electroweak Goldstone boson.
P
(
n
)
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
P
p
p
-
0.4%
8%
18%
46%
22%
6%
0.3%
—
P
Q
Q
-
1
0.1%
1%
6%
19%
31%
27%
12%
3%
0.4%
—
P
Q
Q
-
2
—
0.2%
0.9%
3%
8%
16%
22%
22%
16%
8%
3%
0.9%
0.2%This successful “statistical model” which accounts for gross features ofpp-→nπ annihilation goes back to Fermi [59], who considered a system of noninteracting pions. It has been refined through the years, and the strong interactions of the pions do play a major role. One final aspect is a focusing of incoming waves by attractive potential that leads to strong absorption in a smaller region than originally suggested.


### 4.2.QQ-→nVL Analog

We mean byQ a left-handed chiral doublet (with corresponding right-handed weak singlets) that is degenerate in mass, thereby possessing a heavy isospin symmetry IQ, much like the nucleon N. To draw true analogy with the πNN case, the GQQ Yukawa coupling λQ should be of order 13-14; that is, mQ≳2 TeV, in accordance with findings of the bootstrap gap equation. However, for sake of phenomenology, we will assume that analogous phenomenon already appears for 1 TeV; hence, we consider mQ∈(1,2) TeV.With Higgs mechanism already established, the Goldstone bosonG≡VL carries with it an effective length scale 1/MW    (assuming IQ symmetry, we take MZ=MW and ignore weak gauge couplings g, g′ compared with λQ), which should define the size of the QQ- annihilation fireball. Compared with mπ∝mu+md, here MW∝g where g is the weak gauge coupling. Thus, the size of the fireball is controlled by random parameters such as mq or g that are unrelated to the underlying annihilation dynamics.The fireball temperatureTQQ- is harder to assess. Noting that Tpp-~120MeV<TcQCD~170MeV, likely T<TcEW, where TcEW is the electroweak transition temperature. By this analogy, however, one notes that TcQCD arises from the detailed underlying theory for hadron phenomena (which includes pp-→nπ), QCD. Even if EWSB arises from strong Yukawa coupling [8, 9], λQ, we do not yet have an underlying theory for λQ itself. Thus, we do not have a good handle on TQQ-, except that it is in the 100 GeV scale, of order v.The GoldstonepG2/EG2 factor should still modulate the thermal pG distribution. But because of the smallness of MW2 compared with 4mQ2, the modulation is considerably milder than the pp-→nπ case, and so 〈KG〉 should be closer to (3/2)TQQ-. We shall therefore take as nominal
(55)TQQ-~23v~160GeV,
which can be interpreted as either 1.3×(1/2)v (here 1.3 corresponds to Tpp-/fπ) or v×Tpp-/TcQCD. The latter would give 170 GeV, which is not so different from (55). We stress, however, that the fireball temperature could be 1.5, even twice as high, and should be determined eventually by experiment.Assuming (55) but without applying the 1.3 enhancement factor over equipartition (as was done for the case of pp-→nπ), we take 〈KG〉~(3/2)TQQ-~240 GeV; hence, 〈EG〉~320 GeV, or
(56)〈|pG|〉~310GeV,
with γG~4. For mQ=1(2) TeV, or 2mQ=2(4) TeV, and this corresponds to
(57)〈nG〉~6.25(12.5),
where we artificially keep three digits of significance for generating a “realistic” multiplicity distribution. Assuming (53) and (54), we have σ≃〈nG〉/2~1.25(1.77), and the multiplicity distribution is
(58)P(nG)≃0.319e-(nG-6.25)2/3.13(0.226e-(nG-12.5)2/6.25),
for mQ=1(2) TeV. We note that a higher fireball temperature TQQ- would result in lower average multiplicity 〈nG〉 and a narrower distribution (controlled by σ).We illustrate theQQ-→nG process in Figure 8 (gluon emission discussed later) and tabulate the multiplicity distributions in Table 1. For mQ=1 TeV (second row of Table 1 marked by PQQ-1), about 90% of QQ- annihilations go into 5–8 prongs of VL≡G. Several VLs should be considerably above 300 GeV momentum, while 4-prong events (at 6%) are in general composed of VLs with momentum ~500 GeV. Therefore, W-tagged “fat” jets, jW, should become a useful tool for identifying these multi-VL events. For mQ=2 TeV (third row of Table 1 marked by PQQ-2), again over 90% of QQ- annihilations go into 10–15 prong VLs, which is a rather large number. For 9–12 prong events (at ~50%), a significant number of VLs would have momentum above 400 GeV, while for higher multiplicity, many should still carry momentum higher than the mean, (56). These high multiplicity nVL events would be possible hallmark for heavy QQ- production.


### 4.3. LHC Prospects

At the moment, conventional wisdom has turned against searching for fourth generation quarks per se, because of the SM-like Higgs signal, even though a dilaton interpretation is still possible. Our discussion, however, point towards a possible new type of signature.If our analogy withpp- annihilation is already realized for mQ=1 TeV, then even at 8 TeV running of LHC, where of order 20fb-1 data is expected in 2012 for both ATLAS and CMS, one could already get a hint. The cross-section is of order a couple fb; so, one might observe some number of 4 or more W-tagged jet (jW) events, with additional jet multiplicity that are less well W-tagged. The competing modes would be regular QQ- production, followed by “free quark decay,” for example, (assuming mb′>mt′) b′b′-→tt-W+W-→bb-WWWW, or t′t′-→bb-W+W- [60]; we shall assume CKM hierarchy for simplicity; hence, b′→c, u and t′→s, d transitions are ignored. We see that the W-jet multiplicity is lower, associated with isolated high pTb-jets, and practically no Z-jets or Z→ℓ+ℓ-. In contrast, QQ-→nVL does not have isolated b-jets (b-jets would come in pairs at lower fraction, to form a Z from Z→bb-), and W-jet multiplicity is higher and tends to have Z-jets (the analogy with pp-→nπ suggests a slight excess of π0 over (half the) π±; however, the Z mass is heavier than the W, and so this point is to be determined by experiment). We expect that the QQ-→nVL fireball process would dominate over the QQ-→bb-WW(WW) free quark decay process, as we would argue shortly.We have demonstrated in Section3 that if the heavy chiral quarks Q themselves are responsible [9] for dynamical EWSB, then mQ>1 TeV is required [8]. Our earlier analogy with the πNN “Yukawa” coupling suggests mQ~2 TeV. If so, the prospects for the 2012 LHC run at 8 TeV are not good, and one would have to wait for the 13-14 TeV run, expected by late 2014. Running the HATHOR code [61] for QQ- production, we estimate the 14 TeV cross sections to be of order 50–60 fb for mQ=1 TeV, dropping to ~3 fb for mQ=1.5 TeV, and 0.2-0.3 fb for mQ=2 TeV. From 2mQ=2 to 4 TeV, one quickly runs out of parton luminosity. Note that qq-→QQ- production dominates over gg→QQ- production, as the valence quark supplies the needed large parton momentum fraction.From the cross section and expected LHC luminosities, formQ≲1.5 TeV, again we do not foresee a problem for discovery. Note that assuming IQ symmetry, that is, near degeneracy of t′ and b′, then (i)
t′(b′)→b′(t′)+W* decay:suppressed by both phase space and small Goldstone momentum;(ii)
t′(b′)→b(t)+W:suppressed by CKM element |Vt′b| (|Vtb′|), and with no sign of New Physics in Bs→J/ψϕ, Bs→μ+μ-, and Bd→K*0μ+μ-, one expects [62, 63] such CKM elements to be less than 0.1.In contrast, onceqq-→QQ- pulls the heavy quark pair out of the vacuum, the QQ- pair “sees" a cross section of order 1/MW2, which is at the μb level. With qq-→QQ- production, there is Yukawa attraction [24] between QQ- that mimics the focusing attraction for pp-→nπ. Thus, there is good likelihood that QQ-→nG, that is, nVL, would dominate over free quark decay.We comment that the producedQQ- is likely in a color-octet state; hence, it would need to shed color. However, gluons have no way to sense the T~160 GeV (or higher) of the fireball, which is of electroweak nature. Instead, the heaviness of Q means gluon radiation is 1/mQ suppressed (heavy quark symmetry). We illustrate gluon radiation in Figure 8 but expect the associated gluon-jet to be soft and does not provide a discriminant.In casemQ≳1.5 TeV, one quickly runs out of parton luminosities (higher energy would be preferred!); hence, one would need high luminosity running of LHC at 14 TeV. However, the situation needs not be so pessimistic; the very large Yukawa coupling suggests the existence of bound states below 2mQ. For example, as discussed in [24], there is likely an isosinglet, color-octet ω8 resonance that can be produced via qq-→ω8. How ω8 decays would depend on more details of the QQ- bound state spectrum and properties. The beauty of our analogy with pp-→nπ annihilation is precisely the thermal nature of this fireball process [51, 55], with little “remembrance,” either of the initial pp- state or detailed resonances in the hadron spectrum. Thus, we make no assertion on ω8 decay properties here, except that it offers hope for an enhanced production cross section.If the decay of theω8 is analogous to the fireball picture, then by mω8<2mQ and the resonance production nature, there is good hope for earlier discovery. If ω8 decays through similar chains as discussed in [24], then it might lead to the discovery of several resonances. The study of [24] was done with 500 GeV < mQ < 700 GeV in mind, to avoid issues of boundstate collapse [9, 23]. But since this region is now close to being ruled out, a numerical update, in particular also on obtaining the spectrum, is certainly called for. This, however, requires nonperturbative solutions for strong Yukawa coupling.An offshoot study of [24] provides an interesting contrast. If free quark decay is suppressed by very small Vt′b, and some kinematic selections are operative, it is argued that ω8→π8+W (π8 is some isotriplet, color-octet Yukawa-bound “meson”), followed by π8→WT+g, where WT is transverse. The upshot is the signature of ω8→WWg. This is an exception to our fireball discussion, in that (1) it is effectively 2-body, WW, in vector bosons, and one of which is transverse; (2) the gluon is very energetic. If reconstructed [64], one could find two resonances simultaneously. These signatures arise from special conditions that are unlikely to hold in general. Furthermore, [24] did not consider fireball-like decays of QQ- annihilation by multi-Goldstone radiation.Multiple weak boson production has been considered belowQQ- threshold [65]. There is a rise of high multiplicities as one starts to approach threshold of high mQ. But that would be out of the range of validity for the gg→nG amplitude via virtual Q loop considered. We remark that our multi-VL signature is in principle quite distinct from micro-blackhole production [66]. Micro-blackholes in essence emit all types of particles democratically. In contrast, our fireball is heated in the electroweak sense and prefers emitting by far the strongly coupled weak Goldstone bosons VL≡G. However, since searches so far are based only on the simplified signature of high jet multiplicities, a refined search is needed to separate micro-blackholes from QQ- fireballs.We give a final remark onVLVL scattering. If a very heavy chiral quark doublet Q exists above the TeV scale, it would not be easily compatible with a light Higgs because of the very large quadratic corrections to the light Higgs mass, as we have already remarked. One would have to check whether the 125 GeV boson is more consistent with a SM Higgs boson, or something like a dilaton. If the latter emerges, a corollary of our argument would then suggest that the traditional VLVL→VLVL scattering study for heavy or strongly coupled Higgs case may be the wrong place to search for New Physics enhancement. Instead, one should again watch out for VLVL scattering to high(er) multiplicity of VLs. In general, one should treat the UBV in VLVL and QQ- scattering as one single problem.


## 4.1. Phenomenology ofpp-→nπ

The phenomenon forpp- annihilation is unfamiliar to the average particle physicist. One tends to think of annihilation into photons (much suppressed by α of QED) or gluons. But even for the latter, it is not practical with total energy of ~1.88 GeV and in consideration of detection of final states. Certainly, pp- would annihilate into mesonic resonances. However, all mesonic resonances basically end up as pions. Thus, one can safely conclude that pp-→nπ in general.Indeed, it is observed thatpp- annihilation [51, 55] goes mainly into n pions. However, the features are rather surprising, even for the nuclear and hadron physicists that worked on the subject from the 1950s to the 1990. It is found that pp-→nπ goes via a “fireball.” The salient features of the annihilation “fireball” are (see Figure 8)(i)
size of order1/mπ;(ii)
temperatureT≃120MeV;(iii)
average number of emitted pions〈nπ〉≃5;(iv)
a soft-pionpπ2/Eπ2 factor modulates the Maxwell-Boltzman distribution for the pions.Figure 8
Illustration forpp-→nπ for n=〈nπ〉≅5 and the analog of QQ-→nG, where G=VL is the Goldstone boson of EWSB. Depending on mQ=1-2 GeV, 〈nG〉 could go from 6 to 12 (see (58) and Table 1). The gluon line indicates the relatively soft shedding of color.It is worthwhile to elucidate these features a little further. The size1/mπ means that, when the p and p- meet to annihilate, destined to shed the p and p- content, the region of annihilation extends over a region ~1/mπ. The system seems to thermalize to a temperature Tpp- of order 120 MeV, hence, “loses memory” of its origins, and the emitted pions carry momenta that satisfy a thermal distribution. This rapid thermalization probably takes place due to the rather large πNN (as well as ππ) coupling, while the pπ2/Eπ2 suppression [56] (satisfied rather well by data; see Figure 12 of [51]) for low pion momentum from the thermal distribution reflects nothing but the Goldstone nature of pion couplings. That is, the π as Goldstone boson couples derivatively, hence, cannot get emitted at zero momentum. This seems to explain the enhancement factor of 1.3 for the mean kinetic energy 〈Kπ〉≡〈Eπ〉-mπ beyond equipartition expectation of (3/2)Tpp-. Thus, the relatively high 〈Eπ〉~370MeV gives rise to 〈nπ〉≃5.1, as compared to the maximal allowed number of pions, 2mN/mπ~13.At a more refined level, it is found that〈nπ±〉≃3.1 and 〈nπ0〉≃2.1, with 2〈nπ0〉/〈nπ±〉>1; that is, more neutral pions are emitted than charged ones. Remarkably, the pion multiplicity distribution appears Gaussian,
(53)P(nπ)=12πσe-(nπ-〈nπ〉)2/2σ2,
with σ~1. More specifically [57],
(54)σ≃12〈nπ〉
is argued from statistical models [58]. Thus, σ≃1.13 gives a good fit to data [51], which is given in the first row of Table 1. Note the minute pp-→ππ 2-body fraction, while the pion multiplicity cuts off above 8.Table 1
Sample multiplicity distributions:Ppp-(n) is the observed distribution for pp-→nπ [51], while PQQ-1(2)(n) is the QQ-→nG distribution for mQ=1 (2) TeV according to (58), where G≡VL is the electroweak Goldstone boson.
P
(
n
)
n
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
P
p
p
-
0.4%
8%
18%
46%
22%
6%
0.3%
—
P
Q
Q
-
1
0.1%
1%
6%
19%
31%
27%
12%
3%
0.4%
—
P
Q
Q
-
2
—
0.2%
0.9%
3%
8%
16%
22%
22%
16%
8%
3%
0.9%
0.2%This successful “statistical model” which accounts for gross features ofpp-→nπ annihilation goes back to Fermi [59], who considered a system of noninteracting pions. It has been refined through the years, and the strong interactions of the pions do play a major role. One final aspect is a focusing of incoming waves by attractive potential that leads to strong absorption in a smaller region than originally suggested.


## 4.2.QQ-→nVL Analog

We mean byQ a left-handed chiral doublet (with corresponding right-handed weak singlets) that is degenerate in mass, thereby possessing a heavy isospin symmetry IQ, much like the nucleon N. To draw true analogy with the πNN case, the GQQ Yukawa coupling λQ should be of order 13-14; that is, mQ≳2 TeV, in accordance with findings of the bootstrap gap equation. However, for sake of phenomenology, we will assume that analogous phenomenon already appears for 1 TeV; hence, we consider mQ∈(1,2) TeV.With Higgs mechanism already established, the Goldstone bosonG≡VL carries with it an effective length scale 1/MW    (assuming IQ symmetry, we take MZ=MW and ignore weak gauge couplings g, g′ compared with λQ), which should define the size of the QQ- annihilation fireball. Compared with mπ∝mu+md, here MW∝g where g is the weak gauge coupling. Thus, the size of the fireball is controlled by random parameters such as mq or g that are unrelated to the underlying annihilation dynamics.The fireball temperatureTQQ- is harder to assess. Noting that Tpp-~120MeV<TcQCD~170MeV, likely T<TcEW, where TcEW is the electroweak transition temperature. By this analogy, however, one notes that TcQCD arises from the detailed underlying theory for hadron phenomena (which includes pp-→nπ), QCD. Even if EWSB arises from strong Yukawa coupling [8, 9], λQ, we do not yet have an underlying theory for λQ itself. Thus, we do not have a good handle on TQQ-, except that it is in the 100 GeV scale, of order v.The GoldstonepG2/EG2 factor should still modulate the thermal pG distribution. But because of the smallness of MW2 compared with 4mQ2, the modulation is considerably milder than the pp-→nπ case, and so 〈KG〉 should be closer to (3/2)TQQ-. We shall therefore take as nominal
(55)TQQ-~23v~160GeV,
which can be interpreted as either 1.3×(1/2)v (here 1.3 corresponds to Tpp-/fπ) or v×Tpp-/TcQCD. The latter would give 170 GeV, which is not so different from (55). We stress, however, that the fireball temperature could be 1.5, even twice as high, and should be determined eventually by experiment.Assuming (55) but without applying the 1.3 enhancement factor over equipartition (as was done for the case of pp-→nπ), we take 〈KG〉~(3/2)TQQ-~240 GeV; hence, 〈EG〉~320 GeV, or
(56)〈|pG|〉~310GeV,
with γG~4. For mQ=1(2) TeV, or 2mQ=2(4) TeV, and this corresponds to
(57)〈nG〉~6.25(12.5),
where we artificially keep three digits of significance for generating a “realistic” multiplicity distribution. Assuming (53) and (54), we have σ≃〈nG〉/2~1.25(1.77), and the multiplicity distribution is
(58)P(nG)≃0.319e-(nG-6.25)2/3.13(0.226e-(nG-12.5)2/6.25),
for mQ=1(2) TeV. We note that a higher fireball temperature TQQ- would result in lower average multiplicity 〈nG〉 and a narrower distribution (controlled by σ).We illustrate theQQ-→nG process in Figure 8 (gluon emission discussed later) and tabulate the multiplicity distributions in Table 1. For mQ=1 TeV (second row of Table 1 marked by PQQ-1), about 90% of QQ- annihilations go into 5–8 prongs of VL≡G. Several VLs should be considerably above 300 GeV momentum, while 4-prong events (at 6%) are in general composed of VLs with momentum ~500 GeV. Therefore, W-tagged “fat” jets, jW, should become a useful tool for identifying these multi-VL events. For mQ=2 TeV (third row of Table 1 marked by PQQ-2), again over 90% of QQ- annihilations go into 10–15 prong VLs, which is a rather large number. For 9–12 prong events (at ~50%), a significant number of VLs would have momentum above 400 GeV, while for higher multiplicity, many should still carry momentum higher than the mean, (56). These high multiplicity nVL events would be possible hallmark for heavy QQ- production.


## 4.3. LHC Prospects

At the moment, conventional wisdom has turned against searching for fourth generation quarks per se, because of the SM-like Higgs signal, even though a dilaton interpretation is still possible. Our discussion, however, point towards a possible new type of signature.If our analogy withpp- annihilation is already realized for mQ=1 TeV, then even at 8 TeV running of LHC, where of order 20fb-1 data is expected in 2012 for both ATLAS and CMS, one could already get a hint. The cross-section is of order a couple fb; so, one might observe some number of 4 or more W-tagged jet (jW) events, with additional jet multiplicity that are less well W-tagged. The competing modes would be regular QQ- production, followed by “free quark decay,” for example, (assuming mb′>mt′) b′b′-→tt-W+W-→bb-WWWW, or t′t′-→bb-W+W- [60]; we shall assume CKM hierarchy for simplicity; hence, b′→c, u and t′→s, d transitions are ignored. We see that the W-jet multiplicity is lower, associated with isolated high pTb-jets, and practically no Z-jets or Z→ℓ+ℓ-. In contrast, QQ-→nVL does not have isolated b-jets (b-jets would come in pairs at lower fraction, to form a Z from Z→bb-), and W-jet multiplicity is higher and tends to have Z-jets (the analogy with pp-→nπ suggests a slight excess of π0 over (half the) π±; however, the Z mass is heavier than the W, and so this point is to be determined by experiment). We expect that the QQ-→nVL fireball process would dominate over the QQ-→bb-WW(WW) free quark decay process, as we would argue shortly.We have demonstrated in Section3 that if the heavy chiral quarks Q themselves are responsible [9] for dynamical EWSB, then mQ>1 TeV is required [8]. Our earlier analogy with the πNN “Yukawa” coupling suggests mQ~2 TeV. If so, the prospects for the 2012 LHC run at 8 TeV are not good, and one would have to wait for the 13-14 TeV run, expected by late 2014. Running the HATHOR code [61] for QQ- production, we estimate the 14 TeV cross sections to be of order 50–60 fb for mQ=1 TeV, dropping to ~3 fb for mQ=1.5 TeV, and 0.2-0.3 fb for mQ=2 TeV. From 2mQ=2 to 4 TeV, one quickly runs out of parton luminosity. Note that qq-→QQ- production dominates over gg→QQ- production, as the valence quark supplies the needed large parton momentum fraction.From the cross section and expected LHC luminosities, formQ≲1.5 TeV, again we do not foresee a problem for discovery. Note that assuming IQ symmetry, that is, near degeneracy of t′ and b′, then (i)
t′(b′)→b′(t′)+W* decay:suppressed by both phase space and small Goldstone momentum;(ii)
t′(b′)→b(t)+W:suppressed by CKM element |Vt′b| (|Vtb′|), and with no sign of New Physics in Bs→J/ψϕ, Bs→μ+μ-, and Bd→K*0μ+μ-, one expects [62, 63] such CKM elements to be less than 0.1.In contrast, onceqq-→QQ- pulls the heavy quark pair out of the vacuum, the QQ- pair “sees" a cross section of order 1/MW2, which is at the μb level. With qq-→QQ- production, there is Yukawa attraction [24] between QQ- that mimics the focusing attraction for pp-→nπ. Thus, there is good likelihood that QQ-→nG, that is, nVL, would dominate over free quark decay.We comment that the producedQQ- is likely in a color-octet state; hence, it would need to shed color. However, gluons have no way to sense the T~160 GeV (or higher) of the fireball, which is of electroweak nature. Instead, the heaviness of Q means gluon radiation is 1/mQ suppressed (heavy quark symmetry). We illustrate gluon radiation in Figure 8 but expect the associated gluon-jet to be soft and does not provide a discriminant.In casemQ≳1.5 TeV, one quickly runs out of parton luminosities (higher energy would be preferred!); hence, one would need high luminosity running of LHC at 14 TeV. However, the situation needs not be so pessimistic; the very large Yukawa coupling suggests the existence of bound states below 2mQ. For example, as discussed in [24], there is likely an isosinglet, color-octet ω8 resonance that can be produced via qq-→ω8. How ω8 decays would depend on more details of the QQ- bound state spectrum and properties. The beauty of our analogy with pp-→nπ annihilation is precisely the thermal nature of this fireball process [51, 55], with little “remembrance,” either of the initial pp- state or detailed resonances in the hadron spectrum. Thus, we make no assertion on ω8 decay properties here, except that it offers hope for an enhanced production cross section.If the decay of theω8 is analogous to the fireball picture, then by mω8<2mQ and the resonance production nature, there is good hope for earlier discovery. If ω8 decays through similar chains as discussed in [24], then it might lead to the discovery of several resonances. The study of [24] was done with 500 GeV < mQ < 700 GeV in mind, to avoid issues of boundstate collapse [9, 23]. But since this region is now close to being ruled out, a numerical update, in particular also on obtaining the spectrum, is certainly called for. This, however, requires nonperturbative solutions for strong Yukawa coupling.An offshoot study of [24] provides an interesting contrast. If free quark decay is suppressed by very small Vt′b, and some kinematic selections are operative, it is argued that ω8→π8+W (π8 is some isotriplet, color-octet Yukawa-bound “meson”), followed by π8→WT+g, where WT is transverse. The upshot is the signature of ω8→WWg. This is an exception to our fireball discussion, in that (1) it is effectively 2-body, WW, in vector bosons, and one of which is transverse; (2) the gluon is very energetic. If reconstructed [64], one could find two resonances simultaneously. These signatures arise from special conditions that are unlikely to hold in general. Furthermore, [24] did not consider fireball-like decays of QQ- annihilation by multi-Goldstone radiation.Multiple weak boson production has been considered belowQQ- threshold [65]. There is a rise of high multiplicities as one starts to approach threshold of high mQ. But that would be out of the range of validity for the gg→nG amplitude via virtual Q loop considered. We remark that our multi-VL signature is in principle quite distinct from micro-blackhole production [66]. Micro-blackholes in essence emit all types of particles democratically. In contrast, our fireball is heated in the electroweak sense and prefers emitting by far the strongly coupled weak Goldstone bosons VL≡G. However, since searches so far are based only on the simplified signature of high jet multiplicities, a refined search is needed to separate micro-blackholes from QQ- fireballs.We give a final remark onVLVL scattering. If a very heavy chiral quark doublet Q exists above the TeV scale, it would not be easily compatible with a light Higgs because of the very large quadratic corrections to the light Higgs mass, as we have already remarked. One would have to check whether the 125 GeV boson is more consistent with a SM Higgs boson, or something like a dilaton. If the latter emerges, a corollary of our argument would then suggest that the traditional VLVL→VLVL scattering study for heavy or strongly coupled Higgs case may be the wrong place to search for New Physics enhancement. Instead, one should again watch out for VLVL scattering to high(er) multiplicity of VLs. In general, one should treat the UBV in VLVL and QQ- scattering as one single problem.


## 5. Discussion and Conclusion

Although the numbers arose from dynamical considerations—for sake of EWSB—rather than from constraints, from a phenomenological standpoint, our numerical value ofmQ~3 TeV in (49), even the 2 TeV value in (48), seems depressingly high. At the same time, these masses are for the “no Higgs” case, or that the Higgs boson should be very heavy hence the contribution in the loop is subdominant, and we assert that the role of Higgs mechanism can be taken over dynamically by effective QQ- condensation. But a 125 GeV boson has appeared at the LHC. Can a heavy chiral quark doublet still be viable? We offer  few remarks, first on how mQ might be lowered. We then address the 125 GeV Higgs boson issue.In the spirit of [9] and [24], the Goldstone boson G is the lowest or most tightly bound QQ- state through the Yukawa coupling itself. There should exist Yukawa-bound resonances above this isotriplet, color-singlet, pseudoscalar state. The leading ones should be the isotriplet, pseudoscalar π8 and the isosinglet, vector ω8, both of which are color-octet, and the isosinglet, color-singlet, vector ω1 “mesons.” We have not yet solved the strongly coupled bound state problem; so, we do not yet know the actual spectrum (i.e., how tightly they are bound below 2mQ), nor the “decay constants,” that is, how they couple to the heavy quark Q. But the couplings should be rather strong. The point is, as one integrates the Goldstone loop up to 2mQ, at some point these heavy mesons should also enter and contribute in the same spirit to the self-energy of Q.There is thus some hope that extra, attractive contributions could lowerλQc, coming as additional effects of the large λQ. But it also illustrates the limits of our bootstrap approach. The momentum integration for these extra contributions starts from the meson mass, up to 2mQ, but clearly the meson propagators and the meson-QQ vertices would be much more sensitive to q, compared to the Goldstone boson G. Even for G, which is the π1, as one approaches 2mQ, its bound state nature would lead to modifications of its propagator (even if symmetry remains broken hence it remains massless) and vertex, which we have ignored in the ladder (or rainbow) approximation.We have already offered our critique of the work of Hung and Xiong [25] and also showed numerically that the needed λQc becomes exorbitantly high if one retains the light Higgs scalar in the loop. A different question is whether our gap equation is actually equivalent to the NJL model. We have already commented that for the NJL model, the self-energy does not depend on momentum, and one simply cuts the loop momentum off at some Λ. The wave function part, A(p2), for our gap equation carries momentum dependence; that is, the Yukawa loop always modifies the Z factor. We already saw this in scale-invariant QED. For NJL, the cutoff Λ is traded, together with the associated dimension -2 coupling constant G, for the physical fπ and mN, although, depending on the cutoff, there is a critical coupling Gcrit (see (6)). For our case, one cannot take arbitrary values for the cutoff. Instead, we argued that, because the Goldstone boson would become unbounded at some scale, at least by 2mQ, the loop momentum has to be cut-off below this heuristic, finite value. Further similarities and differences are noted in [9].A fundamental difference from NJL model may be the implicit postulation that the dimension zero Yukawa coupling of the Goldstone boson to be the experimentally verified one related to the left-handed vectorgauge coupling g of massive quarks. If the Goldstone boson G is an ultratight QQ- bound state, it has turned the effective QQ- dimensionality to 1. In this sense, our gap equation may resemble the gauged NJL model [43, 44], in which the dimensionality of the bound state tends to 1 near the critical gauge coupling.Our approach is also conceptually different from those descended from the top condensation model [67, 68], in which the gauged NJL is applied to EWSB. The self-energy in the top condensation model incorporates both our Figure 4 (loop with four-quark operator) and Figure 5 (but with vector boson in the loop). The gap equation with the four-quark operator is equivalent to the minimization of the linear sigma model with a compositeness condition, such that the Yukawa and Higgs quartic couplings blow up at the composite scale. Naively speaking, our gap equation of Figure 6 corresponds to the linear sigma model with large Yukawa coupling. One can therefore read off the schematic correspondence between top condensation and our approach by replacing gauge coupling with Yukawa coupling, and four-quark operator loop by possible heavy bound state loop mentioned earlier. In the top condensation model, the four-quark interaction generates the large Yukawa coupling; hence, it is clearly rather different from our approach.We were intrigued to find that the physically measuredgπNN coupling (extracted via one-pion exchange in Born approximation) is consistent with λπNN≡2mN/fπand is rather similar in value to our finding in (46). This offers a totally separate argument that mQ is in fact above 2 TeV. Having made this analogy of π-N with G-Q, it is then interesting to ask whether the pion could really have been an NN- bound state, as originally conjectured by Fermi and Yang [69], through our gap equation. However, developments in hadron physics subsequent to the Fermi-Yang conjecture relatively quickly gave rise to meson states in the 500 to 800 MeV range (and corresponding baryon resonances), eventually exploding in the 1 to 2 GeV range, that is, below 2mN. Thus, our gap equation does not apply to the π-N and hadron system case. Put simply, in such a π-N gap equation, one cannot integrate the pion loop up to 2mN; new phenomena emerged when the loop momentum q reached ~ΛQCD, which is quite below 2mN. We know that the π and N are QCD bound states, with mesons formed by string breaking, and so the pion is not an ultratight bound state of N and N-. If the QQ- ultratight bound state picture for the Goldstone G could be realized according to our bootstrap gap equation, it would strengthen our reasoning [9] that the underlying theory for Yukawa couplings cannot be a simple mock-up of QCD.We do not have new insight on bound state phenomena, other than what is already discussed in [24]. Unfortunately, this reference was very conservative and did not discuss above mQ>700 GeV. The reason to keep this bound on mQ is that, above this value, the Bethe-Salpeter (BS) equation approach tends to have collapsed states. But this was in turn the foundation for the postulate in [9], that the leading collapsed state, π1, is precisely the Goldstone boson G, which in turn motivated the bootstrap gap equation study. We do not know at present whether our gap equation, with its numerical solution [8], could shed light on the QQ- bound state spectra, but the SD equation itself is “higher level” than the BS equation. If the BS equation Yukawa-boundstate approach can be a guide for mQ as high as 2 TeV, that is, with αQ>1, the noteworthy point is that the leading bound states ω1, ω8, and π8 are rather distinct from the ρ’s and η’s of technicolor (TC), which after all is an extension of QCD. Note that these bound states emerge from strong Yukawa coupling, even though we have not offered a theory of Yukawa couplings (nor did we touch on the flavor aspect of Yukawa couplings). We mention in passing that our gap equation can be easily extended to finite temperature, allowing one to potentially explore issues related to electroweak phase transition, which is a direction that we would take up in a subsequent work.We turn now to the difficult question of how to deal with the 125 GeV boson that has emerged [1, 2] at the LHC. It is rather likely that this is the long-awaited SM Higgs boson. There are two issues against the existence of a 4th generation chiral doublet if one has a 125 GeV Higgs boson, as compared with having the Higgs boson above 600 GeV or so (the original premise or working assumption in the formulation of the bootstrap gap equation). The first issue is the expected enhancement of gg→h production by roughly an order of magnitude due to 4th generation quarks seen by the gluons in the loop [70]. But data indicate [1, 2] that gg→h→VV seem not inconsistent with SM expectations. A second issue is if mQ≳700 GeV [14–20], which is 4 times the top mass, what keeps the Higgs mass light? The top quark contribution to the Higgs mass quadratic correction is already a major concern. An effect from the fourth generation quarks that is an order of magnitude stronger seems rather difficult to tame. From a theoretical standpoint, the second problem is more serious.It is possible to extend the Higgs sector, for example, to two Higgs doublets, where one could have one neutral Higgs boson light [71–73], and accommodate the Higgs data. Such phenomenological accommodation (where the Higgs fields could be composite), however, does not help our cause. Having two Higgs doublets introduces several more parameters, which we did not consider in the formulation of our gap equation. Furthermore, if these Higgs scalars couple with some modulated Yukawa coupling, our gap equations suggest that they raise the 4th generation quark mass (but the pseudoscalars may help). Conversely, these models would still have to face how to keep a neutral Higgs scalar light in the presence of strong Yukawa couplings. Quintessentially, however, our bootstrap gap equation is against the concept of an elementary Higgs field. In any case, having more than one Higgs doublet invalidates our gap equation formulation and would bring us out of scope.There has been discussion that a light Higgs boson could be a pseudo-Goldstone boson (PGH) from some underlying strong dynamics [74]. In these models, the PGH still has SM-like Higgs boson couplings. This would not suffice for our gap equation, since we have shown that the presence of a light Higgs, whether a PGH or not, would drive mQ too high for comfort. Such large mQ as implied by (47) would create its own “hierarchy” problem. We are therefore left with the dilaton option.We find it remarkable, within the strong dynamical EWSB setting, the dilaton is a recurring issue. And, if not fortuitous, as mentioned in the Introduction, that the enhancedγγ mode for the Higgs-like signal at the LHC permits a dilaton interpretation [5–7]. In our case, our gap equation has scale-invariance, which we have in fact used in (42) towards finding our numerical solution. Although there was no attempt at the UV theory, it seems that our bootstrap gap equation permits a connection with an underlying scale-invariant or conformal theory (the AdS-CFT or holographic link; see [6] and references therein). It should be noted that our goal was dynamical EWSB, or generating v.e.v., v. But since we tacitly used λQ=2mQ/v throughout; v generation is equivalent with mQ generation, hence, breaking of the scale-invariance of the gap equation. But since we did not attempt any theory of Yukawa couplings, λQ, the actual source of scale-invariance violation is left for the theory of Yukawa couplings. It should be noted that if the 125 GeV object is a dilaton, its couplings are modified by v/f, where f is the dilaton decay constant (hence, VBF and VH production processes would be suppressed). Taking the bound that v2/f2≲1/20 (probably considerably smaller), the dilaton contribution in our gap equation would be self-consistently subdominant.In conclusion, the ever-increasing mass bound on heavy sequential chiral quarkQ stimulates the question whether strong Yukawa coupling itself can be the source of electroweak symmetry breaking. A dynamical gap equation is argued, treating the Goldstone as massless inside the loop, coupling with chiral quarks Q with the usual Yukawa couplings. By some analogy with scale-invariant QED, numerical solutions are found, such that dynamical EWSB is demonstrated. The resulting quark mass mQ seems to be in the 2-3 TeV range. We favor the heavy or no Higgs scenario since mQ would become exorbitantly high if a light Higgs boson is kept in the loop. Despite our rather high mQ value, LHC might still shed light on it. By analogy with the observed pp-→nπ fireball annihilation, we conjecture that such very heavy chiral quarks that induce EWSB by their very large (≳4 π) Yukawa coupling may annihilate via QQ-→nVL with high multiplicity. Even with 2-3 TeV quark mass, discovery may be aided by Yukawa-bound QQ- resonances such as the color-octet (heavy) isosinglet vector meson ω8, if it undergoes similar annihilation. With a 125 GeV boson that has emerged at LHC, to reconcile with a very heavy chiral quark doublet, that is, a 4th generation, it must be a dilaton [5–7] of scale-invariance breaking.



---

*Source: 650617-2013-03-28.xml*