Natural top quark condensation (a redux)

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Abstract

The Nambu–Jona-Lasinio (NJL) model involves a pointlike 4-fermion interaction. While it gives a useful description of chiral dynamics (mainly in QCD), it nonetheless omits the crucially important internal wave-function of a two-body bound state,

ϕ (r)

. This becomes significant near critical coupling where

ϕ (r)

extends to large distance, leading to dilution and suppression of induced couplings

\propto ϕ (0)

, such as the Yukawa and quartic couplings, as well as reduced fine-tuning of a hierarchy. In top quark condensation, where the Brout-Englert-Higgs (BEH) boson is a

\overline{t} t

bound state and we have a UV completion such as topcolor, we must go beyond the NJL model and include effects of

ϕ (r)

. We provide a formulation of this for the BEH boson, and find that it leads to an extended

ϕ (r)

, a significantly reduced and natural composite scale of

M_{0} \sim 6

TeV, a successful prediction for the quartic coupling, λ, and fine–tuning that is reduced to a few percent, providing a compelling candidate solution to the naturalness problem of the BEH boson. The theory is testable and the associated new physics may soon emerge at LHC energy scales.

1. Introduction

In the early 1990’s we proposed the idea of “top quark condensation,” i.e., that the Brout-Englert-Higgs (BEH) boson is composed of top + anti-top quarks [1]−[7]. The minimal model introduced a 4-fermion pointlike interaction, at large mass scale,

M_{0}

, amongst third generation quarks,(1)

\frac{g_{0}^{2}}{M_{0}^{2}} [{\overline{ψ}}_{i L} (x) ψ_{R} (x)] [{\overline{ψ}}_{R} (x) ψ_{L}^{i} (x)]; ψ_{L}^{i} = {(\begin{matrix} t \\ b \end{matrix})}_{L}, ψ_{R} = t_{R},

(where i is an electroweak

S U (2)

index, and

[. . .] =

denotes a sum over color). To treat this we deployed the Nambu-Jona-Lasinio (NJL) model [8], and introduced significant renormalization group (RG) improvement [3]. The NJL model led to a composite BEH electroweak isodoublet, described by a local field,

H^{i} (x) \sim [{\overline{ψ}}_{R} (x) ψ_{L}^{i} (x)]

. When tuned to the known electroweak scale,

v_{w e a k} = 175

GeV, the theory predicted the top quark and BEH boson masses.

The minimal top-condensation theory was one of the earliest composite BEH models, and was “philosophically successful” in that it non-trivially tied together unrelated parameters of the Standard Model (SM). However, the explicit predictions of the model were ultimately ruled out by subsequent experiment:

m_{t o p} \approx 220

GeV, (cf. 175 GeV, experiment) and

m_{B E H} \approx 260

GeV (cf. 125 GeV, experiment). The predicted

m_{B E H}

corresponded to a SM quartic coupling prediction of

λ \approx 1

, (cf.

λ \approx 0.25

, experiment), while the predicted

m_{t o p}

corresponded to a SM Yukawa coupling prediction of

g_{t o p} \equiv g_{Y} \approx 1.25

, (cf.

g_{Y} \approx 1.00

, experiment).

Moreover, to accommodate the electroweak scale, the theory required ultra-large

M_{0} \sim 10^{15}

GeV, which implied a drastic fine–tuning of the coupling

g_{0}^{2}

to its critical value

g_{c}^{2}

,(2)

(1 - \frac{g_{0}^{2}}{g_{c}^{2}}) \sim \frac{v_{w e a k}^{2}}{M_{0}^{2}} \sim 10^{- 26} (!)

So, the top condensation theory was directly testable by experiment and it evidently failed.

The NJL model was used in top condensation since it is concise, manifestly Lorentz invariant and provides a guide to the chiral symmetry breaking dynamics seen in QCD. There it leads to useful results, where the fundamental chiral current quarks dynamically become the heavy constituent quarks, yielding the chiral–constituent quark model [9]. The NJL model builds the pseudoscalar mesons and σ field by “integrating out” the light quarks which, in a sense, imitates quark confinement. It does well at explaining the Gasser-Leutwyler coefficients and other parameters [10][11] and it remarkably predicted a universal “chiral mass gap” in all heavy-light quark bound states, leading to long lived heavy-strange resonances, such as the

D_{s} (2317)

, [12].

However, in the case of top condensation one has a bound state of non-confined constituents. Hence the bound state couples to free unbound fermions that have the same quantum numbers as the constituents. Here we encounter fundamental physical limitations of the NJL model:

•
the NJL model is an effective pointlike 4-fermion interaction associated with a “large” mass scale $M_{0}$ , and the resulting bound states emerge as pointlike fields with mass $μ^{2} < M_{0}^{2}$ ;
•
in the NJL model the binding mechanism is entirely driven by quantum loop effects, while we see in nature that binding readily occurs semiclassically without quantum loops, such as the hydrogen atom;
•
Mainly, the NJL model lacks an internal wave-function $ϕ (r)$ . The inclusion of $ϕ (r)$ has significant impact upon the conclusions drawn from the model.

In the case of the hydrogen atom, before turning on the Coulomb interaction, there are open scattering states involving free protons and electrons.¹ As the interaction is turned on the lowest energy scattering states flow to become the bound states, while most scattering states remain unbound. The dynamics is governed by the non-relativistic semiclassical (tree level) Schrödinger equation [13], leading to normalizable yet spatially extended wave-functions on the scale

{(α m_{e})}^{- 1}

. The atom is described naturally in a configuration space picture. Quantum loop effects (such as the Lamb shift) are higher order corrections to this mostly semiclassical phenomenon.

In the NJL model the picture is substantially different. There is no semiclassical binding producing an extended bound state. Rather, the bound state is described by a local effective field, $Φ (x)$ , with its properties arising from quantum loops. The loops integrate out the constituent fermions from the large mass scale of the interaction, $M_{0}$ , down to an IR cut-off μ (e.g., $M_{0} \sim 1$ GeV and $μ \sim f_{π} \sim 100$ MeV in QCD). When the discussion is formulated in momentum space, treated in the large $N_{c o l o r}$ limit, bound states appear as poles in the S-matrix upon summing towers of fermion loop diagrams. With a large hierarchy,

, there are also large logarithms, and the sum of loop diagrams is best handled by using an effective action and the renormalization group (RG). $Φ (x)$ has only the minimal dynamical degrees of freedom of a pointlike field. Hence, the NJL model leads to a pointlike field theory description of a bound state, with boundary conditions on the RG running of its couplings at the scale $M_{0}$ .

Since the old top condensation theory used the pointlike NJL model it therefore omitted an internal bound state wave-function,

ϕ (r)

. The formulation of

ϕ (r)

, in a UV completion of the NJL model, has been developed in a recent work [14][15]. The omission of

ϕ (r)

is not expected to significantly affect QCD applications of the NJL model, due to confinement of quarks which would presumably cut off any wave-function spreading. However, if the NJL coupling constant is near its critical value for a non-pointlike and non-confining theory, then the low energy effective theory is approximately conformal. This implies that the internal wave-function

ϕ (r)

spreads significantly into empty space.

An extended wave function occurs for any localized potential with small eigenvalue for the Hamiltonian. The Dirac δ-function potential in

1 + 1

dimensions provides a typical example (Fig. 1). The Schrödinger equation is,(3)

- \frac{1}{2 m} \frac{d^{2}}{d x^{2}} ϕ (x) + V (x) ϕ (x) = E ϕ (x), where, V (x) = - α δ (x) .

The bound state solution (

- \infty < x < \infty

) is

ϕ (x) = - \sqrt{α m} \exp (- α m | x |)

, with eigenvalue

E = - α^{2} m / 2

. The bound state exists for any

α > 0

, with eigenvalue

E < 0

(analogous to the Coulomb potential in

1 + 3

dimensions). Hence the critical coupling, the value of α at which

E = 0

, is

α = 0

, and the external wave-function then coincides with the lowest energy

1 + 1

, non-normalizable, scattering state,

ϕ (x) \to (c o n s t a n t)

. Note that the transition from bound to unbound at

α = 0

is discontinuous (non-analytic) since the normalization, is finite (compact) for

α > 0

and divergent (non-normalizable) for

α < 0

[16]. This illustrates the general result that a near–critical bound state in a localized potential must always be an extended object, due to approximate scale invariance external to the potential.

There are many models of bound states with internal structure, such as [17][18][19] to name a few. We prefer, however, to focus on the well-defined NJL model [8] and generalize it to a non-pointlike theory. To understand the internal

ϕ (r)

in field theory we must first ask exactly how the local pointlike 4-fermion interaction is generated as the limit of a bilocal interaction,

V (x, y) \to V (x)

. With a bilocal interaction we must then replace the pointlike

H (x)

by a bilocal field

H (x, y)

. In the NJL model the natural candidate for this is “topcolor” [5] where the non–pointlike interaction arises from the exchange of a massive gluon-like object, of mass

M_{0}

and coupling

g_{0}

, called a “coloron” [7], [20]. In the large

M_{0}

limit the interaction recovers the pointlike NJL form, but the bilocal nature of

H (x, y)

is then established and remains so even in the pointlike

V (x, y) \to V (x)

limit!

This requires a general formulation of a bilocal field theory. The starting point for this begins with old ideas of Yukawa [21] of multilocal fields. We can modify and extend Yukawa’s bilocal fields to an action formalism. We then have a non-pointlike UV description of the physics as a generalization of the NJL model which we can rely upon for intuition. The semiclassical binding interaction is enhanced by a factor of

N_{c}

in analogy to BCS superconductivity [22]. The formalism then leads to a Schrödinger-Klein Gordon (SKG) equation that determines

ϕ (r)

with eigenvalue

μ^{2}

. Above critical coupling the eigenvalue,

μ^{2}

, becomes negative and spontaneous symmetry breaking (SSB) occurs.

In the symmetric phase of the model (before SSB) we find that bound states will form semiclassically,

{(ħ)}^{0}

, similar to the hydrogen atom, but relativistically in a configuration space picture. The resulting bound state near criticality, where the mass

| μ |

is small compared to the composite scale (the coloron mass

M_{0}

), will have an extended “tail” in its rest frame

ϕ (r) \sim e^{- | μ | r} / r

where

r = | (\vec{x} - \vec{y}) / 2 |

The major implication is that the wave-function spreading causes a significant “dilution” of

ϕ (0) \sim \sqrt{| μ | / M_{0}}

. The resulting top quark Yukawa coupling,

g_{Y} \propto ϕ (0)

, and quartic coupling,

λ \propto | ϕ (0) |^{4}

, are then determined by

ϕ (0)

, with its power law suppression, rather than the relatively slow RG evolution in the NJL model. By fitting the top quark Yukawa coupling to its known value

g_{Y} \approx 1

, and

μ^{2} = - (88)

GeV², the Lagrangian mass of the BEH boson in the symmetric phase of the SM, we can then readily determine

M_{0}

. The implied scale of compositeness of

H (x, y)

, i.e., the mass of the coloron, is then significantly reduced compared to the NJL model and we obtain a result:

M_{0} \sim 6

TeV(!). The SM parameters (Lagrangian BEH mass,

- μ^{2}

, electroweak VEV,

v_{w e a k}

, Yukawa coupling,

g_{Y}

, and (remarkably) the quartic coupling, λ) all become concordant with experiment. Moreover, the fine–tuning of the model is vastly reduced to a few %. The major prediction is the existence of a QCD color octet of colorons with mass

M_{0} \sim 6

TeV that may be accessible to the LHC.²

A core issue of a bilocal (or multilocal) description is the “relative time” problem [24]. In a two body bound state each particle carries its own clock, hence we have times

t_{1}

and

t_{2}

, therefore we have the “average time”

(t_{1} + t_{2}) / 2

and the “relative time”

(t_{1} - t_{2})

. This is endemic to non-relativistic, as well as relativistic systems. In the center of mass frame, which is the rest frame of the bound state (often called the “barycentric frame”) the relative time drops out of the kinetic terms. We can then integrate out the relative time and the interaction becomes a static potential in the rest frame. This requires a normalization of the bilocal field kinetic terms to establish the relevant normalized currents and charges when relative time is removed. The pointlike NJL theory avoids the relative time problem because it simplifies the interaction to a single point in spacetime, but one then misses the extended wave function

ϕ (r)

. For a relativistic system the reduction is done with Lorentz invariant constraints. While one loses manifest Lorentz invariance in the rest frame of the bound state, the overall Lorentz invariance of the theory is maintained, a procedure akin to gauge fixing in a gauge invariant field theory (see Appendices B and C for further discussion.)

Mainly, we propose in the present paper a new version of the top condensation idea, a “redux,” which relies on a “topcolor” interaction that generates the UV completion of the NJL scheme and provides the binding mechanism through a bilocal interaction

V (x, y)

. Though topcolor was previously introduced in the 1990’s, much of its structure carries over for us presently [5], [6]. Here we are invoking it as the primary binding mechanism of the BEH boson (replacing, e.g., “technicolor,” rather than “assisting technicolor”).

We begin with a quick summary of key features in the old top condensation NJL model. We then give a simple example of a bilocal formulation of the non-relativistic hydrogen atom, which illustrates the formal issues and the problem of relative time. To provide orientation, we then follow with a lightning summary of the composite BEH theory. The full technical details, some of which we think are rather stunning, are then given in the bulk of the paper.

1.1. Nambu–Jona-Lasinio model application to top condensation

We will rely heavily on intuition from the NJL model, so we provide this quick summary (more details appear in [14], [15]). The “old” NJL model of top condensation assumes chiral fermions, with

N_{c} = 3

“colors” and a pointlike 4-fermion interaction, hence we have:(4)

where i is an isospin index,

[. .]

implies color singlet combination, and

ψ_{R, L} = (1 \pm γ^{5}) ψ / 2

The NJL interaction is invariant under

S U {(3)}_{Q C D} \times S U (2) \times U (1)

gauge symmetry. The fields and covariant derivatives are defined in the standard model (for simplicity we don’t display the color indices on the quark fields):(5)

ψ_{L}^{i} = \frac{(1 - γ^{5})}{2} (\begin{matrix} t \\ b \end{matrix}), ψ_{R} = \frac{(1 + γ^{5})}{2} t, D_{L μ} = \partial_{μ} - i g_{2} W_{μ}^{A} \frac{τ^{A}}{2} - i g_{1} B_{μ} \frac{Y_{L}}{2} - i g_{3} G^{A} \frac{χ^{A}}{2}, D_{R μ} = \partial_{μ} - i g_{1} B_{μ} \frac{Y_{R}}{2} - i g_{3} G^{A} \frac{χ^{A}}{2},

where the QCD gluons are

G^{A}

, the weak hypercharges

Y_{L} = 1 / 3

Y_{t R} = 4 / 3

, and

Y_{b R} = - 2 / 3

, and the electric charges are as usual:

Q = I_{3} + \frac{Y}{2}

(e.g., for the

b_{L}

quark,

Q = - \frac{1}{2} + \frac{1}{2} \frac{1}{3} = - \frac{1}{3}

, while for

b_{R}

Q = 0 - \frac{1}{2} \frac{2}{3} = - \frac{1}{3}

, etc.).

An equivalent form of the interaction can be written by introducing an auxiliary isodoublet field

H^{i} (x)

:(6)

The “equation of motion” for

H^{i} (x)

is then:(7)

M_{0}^{2} H^{i} (x) = g_{0} [{\overline{ψ}}_{R} (x) ψ_{L}^{i} (x)] .

H (x)

will become the bound state field. Note that

H (x)

is a pointlike field since the 4-fermion interaction is pointlike.

Following Wilson [25] we view eqs. (6), (7) as the effective action at the high scale $m = M_{0}$ . We integrate out the fermions to obtain the effective action for the bound state field $H (x)$ at a lower scale

:(8)where,(9) $μ^{2} = M_{0}^{2} - \frac{g_{0}^{2} N_{c}}{8 π^{2}} M_{0}^{2}, Z = \frac{g_{0}^{2} N_{c}}{8 π^{2}} \ln (M_{0} / m), λ = \frac{g_{0}^{4} N_{c}}{4 π^{2}} \ln (M_{0} / m) .$ We see, from Feynman loops, that $H (x)$ acquires a kinetic term with the covariant derivative,(10) $D_{H μ} = \partial_{μ} - i g_{2} W_{μ}^{A} \frac{τ^{A}}{2} - i g_{1} B_{μ} \frac{Y_{H}}{2},$ where the gluons cancel, and the weak hypercharge becomes $Y_{H} = - 1$ , apropos the BEH boson of the SM.

In particular, note the behavior of the composite BEH boson mass, $μ^{2}$ , of eq. (9) due to the loop contribution, $- g_{0}^{2} N_{c} M_{0}^{2} / 8 π^{2}$ , (we use a UV cut-off $M_{0}^{2}$ on the fermion loops to imitate a softening of the interaction on scale

). The NJL model therefore has a critical value of its coupling, $g_{c}$ , defined by the vanishing of $μ^{2}$ :(11) $\frac{g_{c}^{2} N_{c}}{8 π^{2}} = 1 .$ We can renormalize, $H \to {\sqrt{Z}}^{- 1} H$ , to obtain the full renormalized effective Lagrangian. The notable feature here is that the renormalized couplings evolve logarithmically in the RG “running mass” m:(12) $g_{Y}^{2} = \frac{g_{0}^{2}}{Z} = \frac{4 π^{2}}{N_{c} \ln (M_{0} / m)}, λ_{r} = \frac{λ}{Z^{2}} = \frac{16 π^{2}}{N_{c} \ln (M_{0} / m)} .$ These are the solutions to the RG equations in the large $N_{c}$ limit, keeping only fermion loops, [4]. Eq (12) implies the renormalized couplings have Landau poles, i.e., $(g_{Y}^{2} (m), λ_{r} (m))$ blow up logarithmically as $m \to M_{0}$ . This defines “compositeness boundary conditions” on H for the RG running. We can then use the full RG equations, including QCD and electroweak interactions, to obtain precise low energy predictions [4]. In particular, the Yukawa coupling $g_{Y}$ approaches the IR fixed point value [26]. Results are shown in Fig. 2 and Table 1.

Table 1. Results for the top quark mass, m_top, determined by RG running from Landau pole in g_Y at M₀, to v_weak.

M₀ GeV	10¹⁹	10¹⁵	10¹¹	10⁷	10⁵
m_t (GeV) Fermion Loops	144	165	200	277	380
m_t (GeV) Planar QCD	245	262	288	349	432
m_t (GeV) Full RG	218	229	248	293	360
m_BEH (GeV) Full RG	239	256	285	354	455

For super-critical NJL coupling,

g^{2} > g_{c}^{2}

, we see that the (renormalized) Lagrangian mass,

μ_{r}^{2} = μ^{2} / Z < 0

, implying there will be a vacuum instability. The effective action, with the induced quartic coupling

\sim λ_{r} {(Φ^{†} Φ)}^{2}

term, yields the usual sombrero potential, and the

S U (2) \times U (1)

symmetry is spontaneously broken, and the neutral component of the BEH field H acquires a VEV,(13)

〈 H^{0} 〉 = v_{w e a k} = \frac{| μ_{r} |}{\sqrt{λ_{r}}} .

The top quark then acquires mass

m_{t} = g_{Y} v_{w e a k}

(for light quark and lepton masses and mixings we would rely upon higher dimension “Extended Technicolor” (ETC) operators [27]).

The solutions for the NJL based top quark mass are shown in Table 1. At the time the model was proposed there were upper bounds on the top quark and BEH boson masses of order several hundred GeV. We see that, to obtain a top quark mass

GeV, we require very large $M_{0}$ due to the slow running of the RG and its fixed point. With a choice of e.g., of $M_{0} \sim 10^{15}$ GeV, we obtain from eq. (9):(14) $μ_{r}^{2} \sim M_{0}^{2} (1 - \frac{g_{0}^{2} N_{c}}{8 π^{2}}) = M_{0}^{2} (1 - \frac{g_{0}^{2}}{g_{c}^{2}}) .$ We see that small BEH mass, $μ_{r}^{2}$ , mandates the fine–tuning of $g_{0}^{2} / g_{c}^{2}$ at the level of $\sim 10^{- 26}$ :(15) $\frac{δ g_{0}^{2}}{g_{c}^{2}} \sim \frac{| μ_{r}^{2} |}{M_{0}^{2}} \sim 10^{- 26}!$

While the top condensation theory was directly testable by experiment, it evidently failed. However, the difficulties with the old top condensation theory stem from the limitations of the Nambu–Jona-Lasinio model.

1.2. The hydrogen atom as a bilocal field theory

As a warm-up example we presently give a derivation of the well-known Schrödinger equation for the hydrogen atom, using bilocal field techniques that we develop for a composite BEH boson. This illustrates issues that will arise subsequently for the problem of bound states of pairs of chiral fermions and our generalization of the NJL model.

We can represent the hydrogen atom in terms of proton and electron fields (scalars for present purposes),

ψ_{p} (y)

and

ψ_{e} (x)

, where

(x^{μ}, y^{μ})

are 4-vectors. We then introduce a bilocal field describing a proton-electron pair,(16)

Φ (x, y) = ψ_{p} (y) ψ_{e} (x) .

Since we typically define normalizations as

\int d^{3} x | ψ |^{2} = 1

, therefore the ψ fields have mass-dimension

1 / \sqrt{V} \sim M^{3 / 2}

. We will presently assume Φ has mass dimension

\sim M^{3}

A dimensionless bilocal action can be written as:(17)

\int d^{4} x d^{4} y (i Z Φ^{†} \frac{\partial}{\partial x^{0}} Φ + i Z Φ^{†} \frac{\partial}{\partial y^{0}} Φ - \frac{1}{2 m_{p}} Z | \partial_{\vec{y}} Φ |^{2} - \frac{1}{2 m_{e}} Z | \partial_{\vec{x}} Φ |^{2} - e^{2} D (x - y) | Φ |^{2}) .

While this is non-relativistic we have kept the 4-volumes associated with each coordinate

(x, y)

. We have included a normalization factor, Z, of mass dimension ∼M on the bilocal kinetic terms which will become clear momentarily. The interaction is generated by a single photon exchange between the non-relativistic charge densities of the proton and electron, and using eq. (16):(18)

e^{2} | ψ_{e} (x) |^{2} | ψ_{p} (y) |^{2} D (x - y) = - e^{2} | Φ (x, y) |^{2} \int \frac{d^{4} q}{{(2 π)}^{4}} \frac{e^{i q_{μ} {(x - y)}^{μ}}}{q_{0}^{2} - {\vec{q}}^{2} + i ϵ} .

It is useful to go to coordinates where the heavy proton is located at

X^{μ}

and the electron at

X^{μ} + ρ^{μ}

as,(19)

X = y; ρ = x - y; then, \partial_{x} = \partial_{ρ}; \partial_{y} = \partial_{X} - \partial_{ρ}; with Jacobian: | \frac{\partial (X, ρ)}{\partial (x, y)} | = 1,

and the action becomes, with

Φ^{'} (X, ρ) = Φ (X + ρ, X)

,(20)

\int d X^{0} d^{3} X d ρ^{0} d^{3} ρ (i Z Φ^{'}^{†} \frac{\partial}{\partial ρ^{0}} Φ^{'} + i Z Φ^{'}^{†} (\frac{\partial}{\partial X^{0}} - \frac{\partial}{\partial ρ^{0}}) Φ^{'} - \frac{1}{2 m_{p}} Z | (\partial_{\vec{X}} - \partial_{\vec{ρ}}) Φ^{'} |^{2} - \frac{1}{2 m_{e}} Z | \partial_{\vec{ρ}} Φ^{'} |^{2} - e^{2} D (ρ) | Φ^{'} |^{2}) .

We see in eq. (20) that the derivative

\partial / \partial ρ^{0}

cancels. This is also central to a relativistic formalism:

ρ^{0}

is the “relative time” and is seen to drop out of the action in the rest frame. The only time degree of freedom then carried by the system is

X^{0}

and we can therefore integrate out

ρ^{0}

. This reveals the purpose of the normalization factor Z, and we define:(21)

\int d ρ^{0} Z = 1 .

Z acts only on the kinetic terms and is needed to insure these are canonical, e.g., that they generate properly normalized Noether currents, etc. Note that we could alternatively define

Z = δ (ρ^{0})

We therefore assume that Φ has no dependence upon

ρ^{0}

(this can be done in the relativistic case with Lorentz invariant constraints). We can then integrate over

ρ^{0}

in the interaction which then becomes the standard Coulomb form,(22)

e^{2} \int d ρ^{0} D_{F} (ρ) | Φ^{'} (\vec{ρ}) |^{2} = - e^{2} \int d ρ^{0} \frac{d^{4} q}{{(2 π)}^{4}} \frac{1}{q^{2}} e^{i q_{μ} ρ^{μ}} | Φ^{'} (\vec{ρ}) |^{2} = e^{2} \int \frac{d^{3} q}{{(2 π)}^{3}} \frac{1}{\vec{q}^{2}} e^{- i \vec{q} \cdot \vec{ρ}} | Φ^{'} (\vec{ρ}) |^{2} = \frac{e^{2}}{4 π | \vec{ρ} |} | Φ^{'} (\vec{ρ}) |^{2} .

We can then approximate the proton as infinitely heavy, since the proton kinetic term is suppressed as

1 / m_{p} \sim 0

. In the rest frame the proton is a zero-momentum plane wave with “volume normalization,”

ψ_{p} (x) = e^{i E X^{0}} / \sqrt{V}

and we can integrate it out with

\int d^{3} X = V

. We then have

Φ^{'} (X, ρ) \to e^{- i E X^{0}} ψ_{e} (\vec{ρ})

, where the “clock time” is

X^{0} = t

, and we have a static electron wave-function

ψ_{e} (\vec{ρ})

. The bilocal action then becomes the usual single particle form for the electron:(23)

\int d t d^{3} ρ (E | ψ_{e} (\vec{ρ}) |^{2} - \frac{1}{2 m_{e}} | \partial_{\vec{ρ}} ψ_{e} (\vec{ρ}) |^{2} + \frac{α}{| \vec{ρ} |} | ψ_{e} (\vec{ρ}) |^{2}) .

In spherical coordinates, integrating by parts and extremalizing the action gives the Schrödinger equation,(24)

- \frac{1}{2 m_{e}} \nabla_{\vec{ρ}}^{2} ψ_{e} (\vec{ρ}) - \frac{α}{| \vec{ρ} |} ψ_{e} (\vec{ρ}) = E ψ_{e} (\vec{ρ}),

where E is the eigenvalue, and

α = e^{2} / 4 π

This illustrates that the natural starting point for a two-body bound state is a bilocal field,

Φ (x, y)

, and schematically anticipates the treatment we will use below for pairs of chiral fermions [14]. To give another less trivial example, we give a relativisitic bilocal field theory composed of scalar fields in ref. ([15]). We now turn to the UV completion and bilocalization of the NJL model.

1.3. Outline of a bilocal BEH boson theory

We presently give a brief summary of our theory of a composite BEH boson arising in “natural” top quark condensation. This will illustrate the principles of the construction and one will see similarities with the simple hydrogen atom example given above. This preliminary summary omits the technical details that follow in the bulk of the paper.

In the NJL top condensation model the pointlike 4-fermion effective interaction of eq. (1) can be viewed as a Fierz rearrangement of a color-current interaction (the Fierz rearrangement is derived in Appendix C of [14]):(25)

- \frac{g^{2}}{M_{0}^{2}} [{\overline{ψ}}_{i L} ψ_{R}] [{\overline{ψ}}_{R} ψ_{L}^{i}] = \frac{g^{2}}{M_{0}^{2}} ({\overline{ψ}}_{i L} γ_{μ} \frac{χ^{A}}{2} ψ_{L}^{i}) ({\overline{ψ}}_{R} γ^{μ} \frac{χ^{A}}{2} ψ_{R}) + O (1 / N_{c}),

where

N_{c} = 3

is the number of colors. This is exactly the form (including the sign) induced by a massive color octet vector boson exchange, and leads to topcolor models [5].

In topcolor, QCD is embedded into an

S U {(3)}_{1} \times S U {(3)}_{2}

gauge group at higher energies. The second (weaker)

S U {(3)}_{2}

gauge interactions acts upon the first and second generation quarks while the (stronger)

S U {(3)}_{1}

interaction acts upon the third generation and drives the formation of the BEH bound state, H. Additional dynamics is also incorporated to disallow the formation of a second BEH boson,

H^{'}

, containing

b_{R}

, (usually achieved by introducing a heavy

Z^{'}

so the

{\overline{ψ}}_{L} b_{R}

channel is subcritical; this can be accommodated in extension of the present minimal model). In the following minimal model we simply omit the

b_{R}

quark from the binding dynamics.

The gauge structure at high energies (ignoring any

Z^{'}

interactions) is therefore [5]:(26)

S U {(3)}_{1} \times S U {(3)}_{2} \times S U {(2)}_{L} \times U {(1)}_{Y} \to S U {(3)}_{Q C D} \times S U {(2)}_{L} \times U {(1)}_{Y} .

The SM fermions are assigned to

(S U {(3)}_{1}, S U {(3)}_{2}, S U (2), Y)

, where

Q = I_{3} + \frac{Y}{2}

, as follows:(27)

{(t, b)}_{L} \sim (3, 1, 2, 1 / 3) {(t)}_{R} \sim (3, 1, 1, 4 / 3) [{(b)}_{R} \sim (3, 1, 1, - 2 / 3)] {(ν_{τ}, τ)}_{L} \sim (1, 1, 2, - 1) {(τ)}_{R} \sim (1, 1, 1, - 2) {(u, d)}_{L}, {(c, s)}_{L} \sim (1, 3, 2, 1 / 3) {(u)}_{R}, {(c)}_{R} \sim (1, 3, 0, 4 / 3) {(ν_{e}, e)}_{L}, {(ν_{μ}, μ)}_{L} \sim (1, 1, 2, - 1) {(ℓ)}_{R}, {(μ)}_{R} \sim (1, 1, 1, - 2) .

The

S U {(3)}_{1} \times S U {(3)}_{2}

extended color interaction is broken to the diagonal

S U {(3)}_{Q C D}

(this is described elsewhere [5][7]) leading to the massive octet of “colorons,”

G_{μ}^{' A}

, and the massless octet of the gluons of QCD,

G_{μ}^{A}

Integrating out the heavy colorons and Fierz rearranging gives a bilocal interaction:(28)

S^{'} = g_{0}^{2} \int d^{4} x d^{4} y [{\overline{ψ}}_{i L} (x) ψ_{R} (y)] D_{F} (x - y) [{\overline{ψ}}_{R} (y) ψ_{L}^{i} (x)] .

We therefore introduce a color singlet bilocal BEH field of mass dimension 1, (analogous to eqs. (7), (16):(29)

\sqrt{N_{c}} M_{0}^{2} H^{i} (x, y) = [{\overline{ψ}}_{R} (x) ψ_{L}^{i} (y)],

(here

(i, j)

are electroweak indices,

[. .]

denotes color indices summed, and

H^{i}

is conventionally color normalized, as in Section 2.1 below). The fields appearing on the rhs of eq. (29) are those that will form the bound state when the interaction is turned on, generally the low momentum scattering states. The interaction then becomes bilocal,(30)

g_{0}^{2} M_{0}^{4} N_{c} \int d^{4} x d^{4} y D_{F} (x - y) | H (x, y) |^{2}, where, D_{F} (x - y) = - \int \frac{1}{q^{2} - M_{0}^{2}} e^{i q (x - y)} \frac{d^{4} q}{{(2 π)}^{4}},

where, due to a color singlet normalization of H, an

N_{c}

enhancement occurs in analogy to BCS theory [22] (as detailed in Section 2). The interaction also generates Yukawa couplings of H to fields that remain free fermions,(31)

g_{0}^{2} M_{0}^{2} \sqrt{N_{c}} \int d^{4} x d^{4} y D_{F} (x - y) {[{\overline{ψ}}_{L}^{i} (y) ψ_{R} (x)]}_{f} H_{i} (x, y) + h . c .

We can then construct the Lorentz invariant action that yields the equations of motion by variation:(32)

S_{K} = M_{0}^{4} \int d^{4} x d^{4} y (Z | D_{R}^{†} H (x, y) |^{2} + Z | D_{L} H (x, y) |^{2} + g_{0}^{2} N_{c} D_{F} (x - y) | H (x, y) |^{2}),

where the covariant derivatives are as defined in the NJL model:(33)

D_{L μ} = \frac{\partial}{\partial y_{μ}} - i g_{2} W_{μ}^{A} (y) \frac{τ^{A}}{2} - i g_{1} B_{μ} (y) \frac{Y_{L}}{2}; D_{R μ}^{†} = \frac{\partial}{\partial x_{μ}} + i g_{1} B_{μ} (x) \frac{Y_{R}}{2} .

Note that

D_{L}

(

D_{R}^{†}

) acts at coordinate y (x), and

D_{R}^{†}

acts on

{\overline{t}}_{R}

, hence the sign flip in the gauge field terms (note the derivative

D^{†}

acts in the forward direction as we have written the kinetic term eq. (32)).

We now pass to barycentric coordinates,(34)

X^{μ} = \frac{x^{μ} + y^{μ}}{2}, r^{μ} = \frac{x^{μ} - y^{μ}}{2}, \partial_{x} = \frac{1}{2} (\partial_{X} + \partial_{r}), \partial_{y} = \frac{1}{2} (\partial_{X} - \partial_{r}) with Jacobian: {| \frac{\partial (X, ρ)}{\partial (x, y)} |}^{- 1} \equiv J = 2^{4} .

We then use Wilson lines to “pull-back” the gauge couplings from <span id=%2

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