The Nambu-Jona Lasinio mechanism and the electroweak symmetry breaking in the Standard Model

Abstract

This is a short report of the entire work developed during the study of the possible realizations of the "Top Mode" Standard Model. Here it is examined the breaking of internal symmetries from another point of view showing that is possible to reproduce the gauged electroweak panorama of the traditional Standard Model in a exhaustive and selfconsistent way. The result is reached applying the main futures of the Nambu-Jona Lasinio (NJL) mechanism to an electroweak invariant Lagrangian. In this context the use of functional formalism for composite operators naturally leads to a different dynamical approach. Meanwhile the Higgs mechanism acts on the Lagrangian form, a NJL like model looks directly at the physics of the system showing the real dynamical content hidden in the Green functions of the theory.

Full text

arXiv:hep-ph/9902367v2 7 May 1999
The Nambu-Jona Lasinio mechanism and
electroweak symmetry breaking in the
Standard Model
Edoardo Di Napoli
University of Rome ”La Sapienza” P.le Aldo Moro, 5 Rome Italy
Abstract
In this paper I examine the breaking of internal symmetries from another point of
view showing that is possible to reproduce the electroweak panorama of the tradi-
tional Standard Model in a exhaustive and self consistent way. The result is reached
applying the main futures of the old Nambu-Jona Lasinio (NJL) mechanism to an
electroweak invariant Lagrangian. In this context the use of functional formalism for
composite operators naturally leads to a different dynamical approach. While the
Higgs mechanism acts on the Lagrangian form, a NJL like model looks directly at
the physics of the system showing the real dynamical content hidden in the Green
functions of the theory.
1 Introduction
The aim of this model is to develop and analyze an alternative version of
the traditional electroweak sector of the Standard Model (SM). In the latter
the mechanism of mass ”generation”, for the gauge and matter field, is the
direct consequence of the insertion in the Lagrangian of a gauge invariant
scalar field with Yukawa coupling terms. Modifying suitably the potential
parameters, the appearance of a vacuum expectation value introduces mass
terms in the Lagrangian leading to a spontaneous symmetry breaking effect.
A different approach reaches the same results through a process that doesn’t
act on the Lagrangian form but directly on the Green functions. The starting
point of the whole mechanism draws inspiration from the basic work of Nambu
and Jona Lasinio (NJL) that gave origin to the homonymous model [1]. NJL,
aware of the tight analogy with the BCS theory of superconductivity, applied
Bogoliubov’s quasi particle method to the relativistic field theory. Through a
simple four fermion chirally invariant interaction, they obtained a gap equation
for an order parameter that is merely the dynamical fermion mass. Moreover
Preprint submitted to Elsevier Preprint 1 February 2008

the model points out the presence of induced relativistic bound states as much
as an unavoidable logarithmic dependence on the cut-off of the analytical
results which, paradoxically, enriches the theory. In the first part of this work
the fundamental ideas of this model are presented and then used with the
combined support of the composite operators functional formalism for effective
action and 1/N expansion techniques. I start by introducing a simplified four
fermion (4-F) interaction term in a Lagrangian invariant under the SUL(2)
⊗UY(1) symmetry group. Using stationary condition on the effective action
I find a Schwinger-Dyson (SD) equation for the top quark propagator in the
form of a self consistent condition for its dynamical mass. After having fine
tuned this condition, setting the system in the asymmetrical phase (breaking
chiral invariance), I proceed to extract the correct Bethe-Salpeter equations.
The latter show the presence of a series of bound states playing the role of
Goldstone and Higgs bosons of the traditional SM. Going on, calculating the
gauged SD equations, it is possible to ascertain the absorption of the pseudo-
Goldstone modes by the electroweak gauge propagators. In this way mass
terms are induced for W±and Z0, while the photonic propagator remains
mass free. Section 2 is a kind of toolbox in which a short explanation of the
mathematical instruments, used in the rest of the paper, is given. In section
3 I introduce a 4-F interaction and analyze the SD equations for the fermion
propagators. Section 4 is devolved to the relativistic bound states and to their
mass spectrum. In section 5 I take in consideration the SD equation for the
gauge electroweak propagators showing the mass inducing effect caused by
the pseudo-Goldstone amplitudes. Considerations and conclusions follow in
the last section.
2 Functional formalism toolbox
The first problem to be resolved is the possibility of using the same formalism
to analyze order parameters (as the mass operator functions) and bound states
amplitudes. Second, but not less important, is the quest for an equivalent
approximation order for all the results obtained in the course of the work.
The functional effective action for composite operators seemed to have all the
properties we were looking for. First of all its stationarity under variation with
respect to the second Legendre variable (on physical states) gives directly a SD
equation for a local propagator that includes all the corrections coming from
the interaction terms in the Lagrangian. Then, the second derivative respect
this propagator represents the inverse bilocal connected function of the theory,
that in simple words is the connected part of a four point particle scattering
amplitude. The second requirement on bound states is then accomplished. It
remains the approximation problem. It was evident that I should consider
some kind of cutting process so to make possible the practical handling of
2

expressions in terms of the most relevant contributions. Moreover the order of
approximation had to be the same for every explicit formula. In this case the
use of functional formalism points out, as natural solution, the approximation
in series of 1/Nc(where Ncis number of colors) showing the high level of self
consistency of this approach.
For simplicity let me introduce the functional formalism for composite opera-
tors in case of a scalar field φ(x). As we already know, the application of the
Legendre transform to the usual generating functional Z(J), that depends on
the local source J(x), gives the generator of the 1-particle irreducible graphs
of the theory. If we formally define a generating functional that also depends
on a bilocal source K(x, y)
Z(J, K) = exp i
~W(J, K)=Zdφ exp i
~Zd4x(L(φ) + φ(x)J(x)) +
+ZZ d4xd4y φ(x)K(x, y)φ(y).(1)
we can imagine to recover an analogous generator with a similar transform. I
will call the latter second Legendre transform,
Γ(φc, G) = W(J, K)−Zd4x φc(x)J(x)−1
2ZZ d4xd4y φc(x)K(x, y)×
×φc(y)−1
2~ZZ d4xd4y G(x, y)K(x, y) (2)
with
δW (J, K)
δJ (x)=φc(x) = h0|φ(x)|0i
δW (J, K)
δK(x, y)=1
2[φc(x)φc(y) + ~G(x, y)] = h0|φ(x)φ(y)|0i.(3)
The functional so obtained is consequently the generator of the 2-particle irre-
ducible Green functions 1with all the internal lines expressed in terms of the
complete propagator G(x, y). The word complete here is used because G(x, y)
automatically takes into account the corrections coming from the interaction
terms in the Lagrangian. This functional together with its two stationary con-
ditions
δΓ(φc, G)
δφc(x)=−J(x)−Zd4y K(x, y)φc(y) (4)
1We call 2 point irreducible a fuction whose graph remains connected cutting any
couple of internal lines.
3

δΓ(φc, G)
δG(x, y)=−1
2~K(x, y).
give the Jackiw, Cornwall and Tomboulis[2] variational method for compos-
ite operators. On physical states, when all the sources are equal to zero, (4)
become a system of equations from which is formally possible to extract the
expressions for φcand G. In particular the latter equation, depending on self-
energy, represents essentially the SD equation for G. In order to extract an
explicit expression for Gwe have to obtain a practical method that gives
formulas to be handled with approximation procedures. One of the most
known is the loop expansion[2][4]. Referring to the usual scalar field action
S(φ) = Rd4xL(φ(x)) and defining
iD−1(φ;x, y) = δ2S(φ)
δφ(x)δφ(y)=iD−1(x−y) + δ2Sint(φ)
δφ(x)δφ(y)
with D−1the inverse free propagator, it is possible to write the effective action
as
Γ(φc, G) = S(φc) + 1
2i~Tr hln DG−1+D−1(φc)G−1i+ Γ2(φc,)G. (5)
An infinite number of terms are stored in Γ2and its definition is possible
through a slightly different version of the classic action where the field φ
is translated by the quantity φc. This defines a new interaction Sint (φ;φc)
whose vertices explicitly depend on φc. With this premise Γ2is made by the
collection of all 2-particle irreducible vacuum graphs determined by a theory
with the interaction Sint (φ;φc) and all the internal lines equal to G. These
graphs are easily classifiable in a loop series giving a coherent expression for
(5). This argument is equally reproducible for gauge boson and fermion fields
with the exceptions that for the latter ψc=h0|ψ|0iis identically zero (with
the consequence that D−1is equal to D−1) and all the - 1
2factors must be
replaced with 1. Let us imagine, now, to set the first stationary condition such
that φc= 0, the functional W(0, K) = 1
iln Z(0, K) becomes the generator of
the bilocal connected Green functions
W(0, K) =
∞
X
n=0
(i)n−1
n!G(n)
c(x1y1;...;xnyn)K(x1;y1). . . K(xn;yn) (6)
with
G(n)
c(x1y1;...;xnyn) = δnW(K)
δK(x1;y1). . . δK (xn;yn)
while Γ(0, G) is the sum of all 2-particle irreducible (2PI) vacuum graphs of
4

the same theory. If we substitute now the scalar field with a generic spinorial
one, the last formula do not involve any ansatz on ψc, but are a natural
consequences of the antisymmetric character of the field. In this scenario the
combined effect of the two formula
δW (K)
δK(x;y)=−G(x;y)δΓ(G)
δG(x;y)=K(x;y)
leads to the integro differential equation
ZZ d4x d4yδ2Γ
δGαβ (x1;y1)δGγδ (x;y)
δ2W
δKδγ(x;y)δKηθ (x2;y2)=
=−[δαηδβ θ δ(x1−x2)δ(y1−y2)]
where the Greek indices are spinorial. This demonstrates that Γ(2)(x1y1;x2y2)
is the inverse of the four point connected Green function G(2)
c(x1y1;x2y2).
Using now the loop expansion to calculate Γ(2), we simply recover a compact
expression for G(2)
cin momentum space
G(2)
c;αβ,γδ (p;q;P) = G(2)
0;αβ,γδ (p;q;P) + 1
(2π)8ZZ d4q′d4p′G(2)
0;αβ,ρτ (p;q′;P)×
×Kρτ ,ση (p′;q′;P)G(2)
c;ση,γ δ (q′;q;P) (7)
where
−iδ2Γ2
δGδG (p;q;P) = K(p;q;P)
G(2)
0;αβ,γδ (p;q;P) = (2π)4δ(p−q)Gαδ (p+ηP )Gγ β (p−(1 −η)P).
This is exactly the Bethe-Salpeter (BS) equation cited at the beginning of the
section. For cases in which its expression is relatively simple, this equation is
the starting point from which to detect the presence of bound states. In the
present one the expansion in series of 1
Ncof the kernel K(p;q;P), permits to
solve it in an elegant and self consistent way.
3 The breaking of chiral invariance
After the technical premises of the previous section we can enter the core of
the model. Let us consider a local SUL(2)⊗UY(1) gauge invariant Lagrangian
divided in the following groups of terms: a pure gauge electroweak sector LB, a
5

pure fermionic sector LFincluding all three families of quarks and leptons and
a 4-F interaction L4−F(note that we are not including any Higgs or Yukawa
term). A sufficiently general 4-F interaction expression could be
L4−F=Ka,b
α,β hΨα
Lψa
Rψb
RΨβ
Li (8)
α,βand a, b being respectively family and isospin indexes. In order to be gauge
invariant, the possible indexes of this expression are constrained by the values
of the weak hypercharge quantum numbers. This implies that both the pairs
α, β and a, b must be at the same time quark like or lepton like. Moreover
aand bmust always refer to the same isospin orientation. We need now to
make a reasonable hypothesis, on a phenomenological basis, so as to restrict
the number of possible interaction terms. A strong indication comes from the
abnormal heaviness of the quark top. It could be a significative signal of its
major role in comparison with the other lighter fermions suggesting to ignore
all other interaction terms apart from those involving it directly. This doesn’t
mean that all the other terms are canceled, but that, for the moment, we
consider a simplified model with
L4−F=Kt,t
3q,3qhΨ3q
LtRtRΨ3q
Li (9)
=G
Nc(1
2tt2
+1
2tiγ5t2
+"b(1 + γ5)
2tt(1 −γ5)
2b#)
where bindicates quark bottom and tquark top. Expression (9) is crucial for
the determination of Γ2in the loop series expansion of the effective action
Γ (Gt, Gb) = iNcX
a=b,t ZZ d4x d4yTr hln S−1
a(x−y)Ga(y;x) +
−S−1
a(x−y)Ga(y;x) + 1i+ Γ2(Gt, Gb) (10)
where every Nccomes from the sum over color for each quark loop considered
and S−1
ais an inverse free propagator of the theory. The loop expansion of Γ2
gives schematically
Γ2(Gt, Gb) = NcF−1(TrGt) + F0(TrGt+ Tr (Gt;Gb) + ...) +
+1
Nc
F1(TrGt+ Tr (Gt;Gb) + ...) + ...
with
F−1=−G Zd4x(Tr 1
2Gt(x;x)2
+Tr 1
2iγ5Gt(x;x)2)
6

F0=GZd4x(Tr "1
2iγ5Gt(x;x)2#+ Tr "1
2Gt(x;x)2#+
+ Tr "(1 −γ5)
2Gt(x;x)(1 + γ5)
2Gb(x;x)#)+...
Taking in consideration the highest order in the 1
Ncexpansion we finally arrive
to
Γ (Gt, Gb) = iNCX
a=b,t ZZ d4x d4yTr hln S−1
a(x−y)Ga(y;x) +
−S−1
a(x−y)Ga(y;x) + 1i− G Zd4x NC(Tr 1
2Gt(x;x)2
+
+Tr 1
2iγ5Gt(x;x)2).(11)
As we can see this expression contains terms of the same order and so its
solution will be completely self-consistent. But before starting to search for it,
is necessary to make some remarks. Let’s go back to the stationary conditions
(4) and analyze their meanings on the mass shell
δΓ(φc, G)
δφc(x)= 0 (12)
δΓ(φc, G)
δG(x, y)= 0.
In the traditional SM the Lagrangian depends on an explicit complex scalar
doublet, invariant under global SUL(2) transform. The effective action is then
the first Legendre transform of the functional action Z(J), meanwhile the first
stationary condition (12) (the only one in this case) determines univocally φc.
It is just a positive value of φc, on the mass shell, that breaks the global SUL(2)
leading the electroweak breaking. In the present case the attention is, instead,
focused on the second relation of (12). The main target is to find a solution
for Gtthat breaks chiral invariance spontaneously giving a dynamical value to
the mass operator. As the global symmetry breaking in the Higgs mechanism,
the chiral symmetry breaking is just a means trough which it is possible to
reach the electroweak breaking. We can now impose on (11) the second of the
(12), calculated for the quark top propagator, remembering that the breaking
of a loop, in the derivation process, carries a multiplicative 1
Ncterm
δΓ(Gt, Gb)
δGt(t;z)=iG−1
t(p)−iS−1
t(p)−1
2
G
(2π)4Zd4kTr [Gt(k)] = 0.(13)
7

A possible solution for Gtis obtained with the help of the additional hypothesis
of linearity `a la Hartree-Fock. Writing G−1
t(p) = S−1
t(p)−Σt(p), the (13)
becomes an integro-differential equation for the mass operator Σt(p)
Σt(p) = iG
2(2π)4Zd4kTr "kαγα+ Σt(k)
k2−Σ2
t(k)#.(14)
Because of momentum independence, Σt(p) can now be written as mt. This
fact transforms the last equation in a self-consistent condition that will be
frequently used in the following analysis of the bound states
1 = 2iG
(2π)4Zd4k
k2−m2
t
.(15)
After regularization, (15) becomes an important expression linking together
cut off, dynamical mass and 4-F coupling constant G
G−1=Λ2
8π2"1−m2
t
Λ2ln Λ2
m2
t!#.(16)
This is the first important result and needs some observations. First it contains
two cut off dependent terms, one logarithmically and the other quadratically
divergent. In some way it resembles the quadratic divergence of the Higgs
mechanism where a fine tuning of the Yukawa and scalar quartic coupling
constants were required. Second, for positive dangerous values of m2
tit implies
that K=G
Gc≥1 giving to Gc=8π2
Λ2the significance of a critical coupling
constant, dividing the symmetrical from the asymmetrical phase 2. This means
that even the (16) needs a fine tuning process in order to respect the hierarchy
of scales mt≪Λ, together with the continuum limit lim
Λ2
m2
t
→∞
K= 1. From this
point of view the spontaneous and dynamical symmetry breaking mechanisms
resolve their divergence problems in a similar way with the only difference
that from the point of view of renormalization group the continuum limit for
Kcan be interpreted as an ultraviolet stable point
lim
Λ2
m2
t
→∞
βK=dK
dln Λ = 0.
2For what concerns Gb, its solution here is not considered for two main reasons:
the dependence on powers of 1
Ncis of higher order respect that of Gtand the
experimental values of mtand mbsuggest to work in the condition of maximal
isospin violation.
8

4 Vertex operators
The breaking of chiral invariance, just described, is an indispensable ingredi-
ent for the development of the entire model. In particular, as already stated,
the use of the (15) in the (7) eliminates quadratic divergences from vertex
amplitudes and makes possible the individuation of poles in their spectrum.
Depending on the interaction form there are three kinds of vertices: a scalar,
a pseudoscalar one and two mixed. In all the typologies the basic instrument
of research is the BS equation. In fact the vertices operators can be written as
e
ΓS(x, y) = G
2NC
h0|T[t(0)t(0)t(x)t(y)] |0ic=−G(2)
c;αβ,γδ(xy; 00) G
2NC
δγδ
e
ΓP S (x, y) = G
2NC
h0|T[t(0)γ5t(0)t(x)t(y)] |0ic=−G(2)
c;αβ,γδ(xy; 00) G
2NC
(γ5)γδ
e
ΓM+=G
4NC
h0|Thb(1 + γ5)tbti|0ic=−G(2)
c;αβ,γδ(xy; 00) G
4NC
(1 + γ5)γδ
e
ΓM−=G
4NC
h0|Tht(1 −γ5)btbi|0ic=−G(2)
c;αβ,γδ(xy; 00) G
4NC
(1 −γ5)γδ
so that their Fourier transforms can be inserted in the (7). Again the principal
dynamical informations that enter in the equations are collected in Γ2in the
form of a second derivative kernel. The problem, here, resides in the fact
that the higher order in powers of 1
Ncis not easily given by the first graphs
of the loop expansion of Γ2. In fact its double derivative carries extra 1
Nc
factors that forced us to take into account an infinite series of graphs with
the consequence of complicate combinatorial calculus. Moreover we should
take great care of the spinorial indices that often are the keys of the right
approximation. For reasons of convenience is preferable to treat the scalar
and pseudoscalar vertices independently from the mixed ones. Thus we have
for the kernel
Kαβ,γ δ =−iδ2Γ2(Gt, Gb)
δGt;αβ δGt;γδ
=iG
2NChδαβδγδ + (iγ5)αβ (iγ5)γδ i+
iG
2NC(δδαδβγ "1 +
∞
X
n=1
[FS]n#+ (iγ5)δα (iγ5)β γ "1+
∞
X
n=1
[FP S ]n#)+
+0 1
N2
C!(17)
with
9

FS(P) = iG
2 (2π)4Zd4qTr Gtq+P
2Gtq−P
2 (18)
FP S (P) = −iG
2 (2π)4Zd4qTr γ5Gtq+P
2γ5Gtq−P
2.
The next step, after the insertion of Kαβ ,γδ , consists in an infinite reiteration
process of the BS equations. The spinorial indices play, in this case, a major
role in selecting which reiterated terms must be kept or just ignored. The
point of the question resides in the rising of spinorial traces of propagators,
each carrying an Ncfactor with it. Thus only the first term of Kαβ,γ δ is kept
producing a great simplification of the final expressions
ΓS;αβ(P) = −G
2NC
δαβ (1+
∞
X
n=1
[FS(P)]n)(19)
ΓP S;αβ (P) = −G
2NC
(γ5)αβ (1+
∞
X
n=1
[FP S (P)]n).
After regularization, it is always possible to locate a set of values of P2for
which F(P)<1, the last expressions becoming a geometrical series and so
can be formally rewritten as
ΓS;αβ(P) = −G
2NC
δαβ [1 −FS(P)]−1(20)
ΓP S;αβ (P) = −G
2NC
(γ5)αβ [1 −FP S (P)]−1.
Calculating (18) explicitly it is easy to express (20) suitably, showing they
contain a term equal to the self-consistent equation (15), and arrive to the
final expressions for the vertex functions
ΓS(P) = −i
2NC
(4m2
t−P2)
(2π)4Zd4k1
h(k+P)2−m2
ti[k2−m2
t]

−1
ΓP S (P) = −iγ5
2NC
P2
(2π)4Zd4k1
h(k+P)2−m2
ti[k2−m2
t]

−1
.(21)
A similar procedure takes to the evaluation of the mixed vertex functions.
The differences, in this case, are a slight change of spinor indices in the BS
equation and a more complicate kernel
Kαβ,γδ [q; (p−k)] = −iG
4NC
(1 + γ5)δα (1 −γ5)βγ (1+
∞
X
n=1
[FM(P)]n)
10

leading to
ΓM±;αβ(P) = −G
4NC
(1 ±γ5)αβ −iG
4NC
(1 ±γ5)αβ
(2π)4(1+
∞
X
n=1
[FM(P)]n)×
×Zd4q NCTr h(1 ∓γ5)Gt(q+P) ΓM±(P)Gb(q)i(22)
with
FM(P) = iG
4 (2π)4Zd4lTr [(1 −γ5)Gt(l+P) (1 + γ5)Gb(l)] .
Unfortunately the reiteration of (22) doesn’t eliminate any term and produces
a final expression that, only under the right decomposition, becomes
ΓM±;αβ(P) = −G
4NC
(1 ±γ5)αβ "FM(P)2
1−FM(P)#(23)
with
1−FM(P) = 1
2−iG
(2π)4Zd4k(k2−P2)
(P+k)2(k2−m2
t)−→
P2→00 (24)
FM(P)2=(−1
2−iG
(2π)4Zd4k(k2−P2)
(P+k)2(k2−m2
t))2
−→
P2→01.
Collecting results from (21), (23) and (24) it is straight now to observe that the
scalar vertex presents a pole for a transferred momentum P= 2mt, while the
pseudoscalar and mixed ones have poles at P2= 0 corresponding to bosonic
bound states on their mass shell. In particular the scalar channel represents
a top-antitop state that could be compared to a sort of pseudo-Higgs boson
because of its extremely massive characteristic. Moreover the absence of a
negative mass squared bound state is a first indication of the stability of
the asymmetrical phase. The natural conclusion is that the pseudoscalar and
mixed resonances are a sort of pseudo-Goldstone bosons coming from the
breaking of the chiral invariance. At this point the logic hope is that these
latter amplitudes shall be absorbed in some ways in the gauge electroweak
propagators in the form of mass terms. The analogy with the Higgs mechanism
would be, then, complete.
11

5 Electroweak symmetry breaking
Having all the basic prerequisites in our hand it is now possible to approach the
main target of this work. Because the SD equation coming from the stationary
condition of the effective action is the main tool used in here, let us recall the
loop expansion in terms of gauge propagators
Γ(Dµν , Gt, Gb) = i
2X
i=W,A,Z0ZZ d4x d4yTr nln hDi
µν (x;y)Di−1
µν (y;x)i+
+Di−1
µν (x;y)Di
µν (y;x)−1o+ Γ2(Di
µν , Gt, Gb)i.(25)
We obviously didn’t mention terms dependent exclusively on fermion prop-
agators because they would be eliminated by the stationary condition after
the derivation respect Di
µν . Again the richness of the theory is enclosed in
Γ2and so depends on the form of interaction terms. In this case such terms
are not only given by the 4-F interaction, but depends directly on the form
of the neutral and charged currents Jµ
3=1
4htγµ(1 −γ5)t−bγµ(1 −γ5)bi,
Jµ
Y=h2
3(tγµt)−1
3bγµb−Jµ
3iand Jµ=hb(1 + γ5)γµtiwith its complex
conjugate. Consequently the combinatorics of 2PI vacuum graphs becomes
more complicate and necessitates an accurate evaluation that keeps an infi-
nite number of terms of the same order in power of Nc. Taking into account
only the higher order terms, the final expression, in the case of charged weak
propagators is
Γ2DW
µν , Gt, Gb=−g2
2
16NCZZ d4x d4y{Tr [Gt(y;x)γν(1 −γ5)×
×Gb(x;y) (1 + γ5)γµ]DW
µν (x;y)o+
−ig2
2G
64 NC"∞
X
n=1
F(n)
WDW
µν , Gt, Gb#(26)
with
F(n)
W=iG
4n−1Z···Zd4xd4z1. . . d4znd4y{Tr [Gb(x;z1) (1 + γ5)Gt(z1;x)×
×γν(1 −γ5)] ×Tr [Gb(z2;z1) (1 −γ5)Gt(z1;z2) (1 + γ5)] ×...
...×Tr [Gb(zn;zn−1) (1 −γ5)Gt(zn−1;zn) (1 + γ5)] ×
Tr [Gt(y;zn) (1 + γ5)Gb(zn;y)γµ(1 −γ5)] DW
µν (x;y)o.
Applying the stationary condition δΓ(DW
µν ,Gt,Gb)
δDW
µν = 0 to the (25) one directly
obtains the SD equation for the complete propagator DW
µν as a complicate
12

function of the free one DW
µν and the fermionic Gt,Gb, that demonstrate the
complete absorption of the mixed vertex functions
DW−1
µν =DW−1
µν +iNCg2
2
8{Tr [Gtγν(1 −γ5)Gb(1 + γ5)γµ] + iR (FM(P)) ×
×Tr hGtγν(1 −γ5)GbΓM+iTr hΓM−GtGbγµ(1 −γ5)io.(27)
After a long and accurate regularization [3] that doesn’t break the gauge in-
variance one recover an expression that makes evident how the absorption
process is a concrete ”mass generation” mechanism giving body to the elec-
troweak symmetry breaking
1
g2
2
DW−1
µν =iPµPν
P2−gµν  1
e
g2
2(P2)P2−e
h2P2!(28)
with
1
e
g2
2(P2)=1
g2
2
+NC
48π2

ln Λ2
m2
t
+P2
∞
Z
m2
t
d4k
k2(k2−P2) 1−m2
t
k2!2 1 + 2m2
t
k2!


e
h2P2=NCm2
t
32π2

ln Λ2
m2
t
+P2
∞
Z
m2
t
d4k
k2(k2−P2) 1−m2
t
k2!

.
The (28) has, evidently, a pole in correspondence of M2
W=P2=e
g2
2(P2)e
h2(P2)
that is just the gauge mass term searched.
A similar procedure, even if a bit more lengthy, applies for the neutral gauge
propagators. After the additional transform that rotate fields from W3µ,Bµ
to Z0
µ,Aµchanging the interaction terms, the relevant difference is contained
in the explicit expressions of the Γ2expansion
Γ2= ΓA
2+ ΓZ0
2(29)
where the photonic part is
ΓA
2=−2
9e2NCZZ d4x d4ynTr [Gt(y;x)γµGt(x;y)γν]DA
µν (x;y)o+
−ie2G
9NC("∞
X
n=1
F(n)
A,S#+"∞
X
n=1
F(n)
A,P S #) (30)
with
13

F(n)
A,S =iG
2n−1Z···Zd4xd4z1. . . d4znd4y{Tr [Gt(x;z1)γµGt(z1;x)] ×
×Tr [Gt(z2;z1)Gt(z1;z2)] ...Tr [Gt(zn;zn−1)Gt(zn−1;zn)] ×
×Tr [Gt(y;zn)Gt(zn;y)γν]DA
µν (x;y)o
F(n)
A,P S =iG
2n−1Z···Zd4xd4z1. . . d4znd4y{Tr [Gt(x;z1)γµGt(z1;x)γ5]×
×Tr [γ5Gt(z2;z1)γ5Gt(z1;z2)] ...Tr [γ5Gt(zn;zn−1)γ5Gt(zn−1;zn)] ×
×Tr [Gt(y;zn)γ5Gt(zn;y)γν]DA
µν (x;y)o
meanwhile the Z0part has the following form
ΓZ0
2=−g2
2NC
32 cos2θWZZ d4xd4yTr Gt(y;x)γµ1−8
3sin2θW−γ5×
×Gt(x;y)1−8
3sin2θW−γ5DZ0
µν −ig2
2G
64 cos2θW
NC×
×("∞
X
n=1
F(n)
Z0,S #+"∞
X
n=1
F(n)
Z0,P S #) (31)
with
F(n)
Z0,S =iG
2n−1Z···Zd4xd4z1. . . d4znd4yDZ0
µν (x;y)×
×Tr Gt(x;z1)γµ1−8
3sin2θW−γ5Gt(z1;x)
×Tr [Gt(z2;z1)Gt(z1;z2)] ...Tr [Gt(zn;zn−1)Gt(zn−1;zn)] ×
×Tr Gt(y;zn)Gt(zn;y)γν1−8
3sin2θW−γ5
F(n)
Z0,P S =iG
2n−1Z···Zd4xd4z1. . . d4znd4yDZ0
µν (x;y)×
×Tr Gt(x;z1)γµ1−8
3sin2θW−γ5Gt(z1;x)γ5
×Tr [γ5Gt(z2;z1)γ5Gt(z1;z2)] ...Tr [γ5Gt(zn;zn−1)γ5Gt(zn−1;zn)] ×
×Tr Gt(y;zn)γ5Gt(zn;y)γν1−8
3sin2θW−γ5.
Again considering the stationary conditions for the neutral gauge boson prop-
agators lead us to
14

DA−1
µν (P) = DA−1
µν (P) + 4iNCe2
9{Tr [GtγµGtγν] +
−iΓSTr [GtγµGt] Tr [GtγvGt] +
−iΓP S Tr [Gtγ5Gtγµ] Tr [Gtγ5Gtγν]}(32)
and
DZ0−1
µν (P) = DZ0−1
µν (P) + iNCg2
2
16 cos2θWTr Gtγµ1−8
3sin2θW−γ5×
×Gtγν1−8
3sin2θW−γ5+
−iΓSTr GtGtγµ1−8
3sin2θW−γ5×
×Tr GtGtγv1−8
3sin2θW−γ5+
−iΓP S Tr Gtγ5Gtγµ1−8
3sin2θW−γ5×
×Tr Gtγ5GtγGv 1−8
3sin2θW−γ5 (33)
giving expressions in which vertex functions appear naturally. Explicit calcu-
lations bring us to
1
g2
2
DZ0−1
µν (P) = iPµPν
P2−gµν  1
g2
2(P2)P2−e
f2P2!(34)
with
1
g2
2(P2)=1
g2
2
+NC
96π2cos2θW 1−8
3sin2θW2
+ 1!"ln Λ2
m2
t
+
+P2
∞
Z
4m2
t
d4k
k2(k2−P2) 1−4m2
t
k2!1
2 1 + 2m2
t
k2!


e
f2P2=NCm2
t
32π2cos2θW

ln Λ2
m2
t
+P2
2
∞
Z
4m2
t
d4k
k2(k2−P2) 1−4m2
t
k2!−1
2


giving to the Z0field a mass term M2
Z0=P2=e
g2
2(P2)e
f2(P2). Meanwhile
for the photon propagator one obtains naturally no mass correction but only
a coupling constant renormalization.
1
e2DA−1
µν (P) = iPµPν−gµν P2 1
e
e2(P2)!(35)
15

with
1
e
e2(P2)=1
e2+2NC
27π2

ln Λ2
m2
t
+P2
∞
Z
4m2
t
d4k
k2(k2−P2) 1−4m2
t
k2!1
2 1 + 2m2
t
k2!

.
As we have already noted, in (27), (32) and (33) the inverse gauge propagators
are corrected by means of terms that involve vertex functions. In a Feynman
graph representation these terms contain internal lines corresponding to the
resonance channels found in the section 4. In other words corrections to the
gauge propagators imply inevitably the exchange of bound state particles.
Moreover calculation details show [3] that the only vertices that contribute to
the mass generation are those relative to the pseudo-Goldstone modes, while
the scalar amplitude remains outside the process: a strong hint towards the
identification of latter’s resonance with a composite Higgs boson.
6 General considerations and conclusions
This work certainly doesn’t constitute the only attempt to build a model that
try to justify the largeness of the quark top mass and use it as a breakthrough
in the electroweak symmetry breaking. An entire class of so called ”Top Mode
Standard Model” (TMSM) were written in the years that divide the last two
decades[5]. A second phase for TMSM arrived after a period of five years when
a series of works tried to improve the mechanism giving to it the label ”renor-
malizable”[6][7][8]. Despite their results and conclusions on the gauged NJL
model and its renormalization, I would like to emphasize some aspects of the
approach exposed here. Forgetting for the moment the divergencies included in
the self-consistent equation (16), I show the importance of the chiral symmetry
breaking as a means through which obtain a gauge symmetry breaking. The
appearance of corrections depending on bound states amplitudes, in the SD
equations for the inverse gauge propagators, makes the dynamical ”absorption
process” a real alternative model to the static Higgs mechanism. This result is
not conclusive obviously: the expressions obtained in section 5 gives in terms
of 1
Ncpower series the following behavior
Dµν = 0 (1) + 1
Nc+. . . .
This implies that in the SD equation for Gtthe neglect of boson contributions
to Γ2wasn’t the correct procedure. At least we should take in consideration
the gauge propagator contribution to graphs in planar approximation which,
in fact, carries the same order (in 1
Nc) of the 4-F interaction. This was not done
16

for two main reasons: the preponderance of 4-F induced terms due to the over-
critical value of Grespect to g1and g2. Second, because the 4-F hypothesis was
analyzed in a low-energy regime (the SD without gauge contributions could be
considered the zero approximation for mtin a P2power series expansion) the
”generation” of gauge masses would be then justified as a tree-level approx-
imation. In this sense the next logical step would be the inclusion of gauge
corrections to all the equations of the model. The program is compatible with
fermionic SD and BS equations, but becomes really complex for gauge SD
equations because of the combinatorics of the graphs which clearly complicate
the localization of the ”generated” mass terms.
Second important remark concerns the supposed predictability this model can
give of the SM mass spectrum. In section 3, in fact, I ignored some interaction
terms in order to simplify the analysis of the successive sections. Even if not
mentioned here I have studied [3] the consequences, on the fermion mass sector,
of the insertion of additional 4-F interactions in the Lagrangian. In analogy
with the work of Hasenfratz et Al. [9], I found that on this side gauged NJL
models don’t give any additional previsional information compared with the
traditional Higgs mechanism. This should not divert the attention from the
fundamental dynamical content that could help to enlighten several aspects of
existing problems. The NJL model is still a fruitful argument of analysis and
will be again a starring of future researches.
References
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[3] E. Di Napoli, Laurea thesis, 20th of March 1997
[4] R.W.Haymarker, Riv. Nuovo Cim.N8, 14 (1991)
[5] V.A. Miransky, M.Tanabashi and K.Yamawaki, Phys. Lett. B221, 177 (1989).
V.A.Miransky, M.Tanabashi and K.Yamawaki, Mod. Phys. Lett., 1043 (1989).
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17