Page 1
arXiv:hep-ph/0305271v3 2 Jun 2003
Preprint typeset in JHEP style - HYPER VERSION
Domain wall generation by fermion self-interaction
and light particles
A. A. Andrianov♭♯, V.A.Andrianov♭, P. Giacconi♯, R. Soldati♯
♭V.A.Fock Department of Theoretical Physics, Sankt-Petersburg State University,
198504 Sankt-Petersburg, Russia
♯Dipartimento di Fisica, Universit´ a di Bologna and
Istituto Nazionale di Fisica Nucleare, Sezione di Bologna,
40126 Bologna, Italia
Abstract: A possible explanation for the appearance of light fermions and Higgs bosons
on the four-dimensional domain wall is proposed. The mechanism of light particle trapping
is accounted for by a strong self-interaction of five-dimensional pre-quarks. We obtain the
low-energy effective action which exhibits the invariance under the so called τ-symmetry.
Then we find a set of vacuum solutions which break that symmetry and the five-dimensional
translational invariance. One type of those vacuum solutions gives rise to the domain wall
formation with consequent trapping of light massive fermions and Higgs-like bosons as
well as massless sterile scalars, the so-called branons. The induced relations between low-
energy couplings for Yukawa and scalar field interactions allow to make certain predictions
for light particle masses and couplings themselves, which might provide a signature of
the higher dimensional origin of particle physics at future experiments.
translational symmetry breaking, eventually due to some gravitational and/or matter fields
in five dimensions, is effectively realized with the help of background scalar defects. As
a result the branons acquire masses, whereas the ratio of Higgs and fermion (presumably
top-quark) masses can be reduced towards the values compatible with the present-day
phenomenology. Since the branons do not couple to fermions and the Higgs bosons do not
decay into branons, the latter ones are essentially sterile and stable, what makes them the
natural candidates for the dark matter in the Universe.
The manifest
Keywords: eld.ssb.bsm.
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Contents
1.Introduction1
2.Fermion model with self-interaction in 5D4
3.
τ-symmetry breaking8
4. Domain walls: massless phase 10
5.Domain walls: Higgs phase 14
6. Manifest breaking of translational invariance17
7.Conclusions and discussion23
A. One-loop effective action 25
B. Spectral resolution for Schr¨ odinger operators27
C. Perturbation theory for the first excited state28
D. Perturbation theory in the presence of defects29
1. Introduction
The accommodation of our matter world on a four-dimensional surface – a domain wall
or a thick 3-brane – in a multi-dimensional space-time with dimension higher than four
has recently attracted much interest as a theoretical concept [1]-[6] promoting novel so-
lutions to the long standing problems of the Planck mass scale [3, 4], GUT scale [7, 8],
SUSY breaking scale [9, 10], electroweak breaking scale [11]-[14] and fermion mass hierar-
chy [15, 16]. Respectively, an experimental challenge has been posed for the forthcoming
collider and non-collider physics programs to discover new particles, such as Kaluza-Klein
gravitons [18, 19], radions and graviscalars [20, 21], branons [22]-[24], sterile neutrinos
[7, 25, 26, 27] etc., together with some other missing energy [28] or missing charge effects
[29]. The vast literature on those subjects and their applications is now covered in few
review articles [30]-[35]. Typically, the thick or fat domain wall formation and the trapping
of low-energy particles on its surface (layer) might be obtained [36]-[38] by a number of par-
ticular background scalar and/or gravitational fields living in the multi-dimensional bulk –
see however Ref. [39] – the configuration of which does generate zero-energy states localized
– 1 –
Page 3
on the brane. Obviously, the explanation of how such background fields can emerge and in-
duce the spontaneous symmetry breaking is to be found and the domain wall creation, due
to self-interaction of certain particles in the bulk with low-energy counterparts populating
our world, may be one conceivable possibility.
In this paper we consider and explore the non-compact 4 + 1-dimensional fermion
model with strong local four-fermion interaction that leads to the discrete symmetry break-
ing, owing to which the domain wall pattern of the vacuum state just arises and allows the
light massive Dirac particles to live in four dimensions1. In such a model the scalar fields
appear to be as composite ones. We are aware of the possible important role [31, 37] of a
non-trivial gravitational background, of propagating gravitons and gauge fields for issues
of stability of the domain wall induced by a fermion self-interaction. Nonetheless, in order
to keep track of the main dynamical mechanism, we simplify herein the fermion model just
neglecting all gravitational and gauge field interaction and, in this sense, our model might
be thought as a sector of the full Domain-Wall Standard Model. However, we believe that
the simplified model we treat in the present investigation will be able to give us the plenty
of quantitative hints on the relationships among physical characteristics of light particles
trapped on the brane.
Let us elucidate the domain wall phenomenology in more details and start from the
model of one five-dimensional fermion bi-spinor ψ(X) coupled to a scalar field Φ(X). The
extra-dimension coordinate is assumed to be space-like,
(Xα) = (xµ,z) ,(xµ) = (x0,x1,x2,x3) ,(gαα) = (+,−,−,−,−)
and the subspace of xµeventually corresponds to the four-dimensional Minkowski space.
The extra-dimension size is supposed to be infinite (or large enough). The fermion wave
function is then described by the Dirac equation
[iγα∂α− Φ(X)]ψ(X) = 0 ,
with γαbeing a set of four-dimensional Dirac matrices in the chiral representation.
The trapping of light fermions on a four-dimensional hyper-plane – the domain wall
– localized in the fifth dimension at z = z0 can be promoted by a certain topological,
z-dependent background configuration of the scalar field ?Φ(X)?0= ϕ(z), due to the ap-
pearance of zero-modes in the four-dimensional fermion spectrum. In this case, from the
viewpoint of four-dimensional space-time, Eq.(1.1) just characterizes the infinite set of
fermions with different masses that it is easier to see after the Fock-Schwinger transforma-
tion,
γα= (γµ,−iγ5) ,{γα,γβ} = 2gαβ,(1.1)
[iγα∂α+ ϕ(z)][iγα∂α− ϕ(z)]ψ(X) ≡ (−∂µ∂µ− ? m2
1One can find some relationship of this mechanism of domain-wall generation to that one of the Top-
Mode Standard Model [40] used to supply the top-quark with a large mass in four dimensions due to
quark condensation uniform in the space-time. However, in our case, the vacuum state will receive a scalar
background defect breaking translational invariance. As well, our model is taken five-dimensional and non-
compact as compared to six- or eight-dimensional extensions of the Top-Mode Standard Model [41, 42] with
compact extra dimensions and an essential role played by Kaluza-Klein gravitons and/or gauge fields.
z)ψ(X) ;
? m2
z= −∂2
z+ ϕ2(z) − γ5ϕ′(z) = ? m2
+PL+ ? m2
−PR,(1.2)
– 2 –
Page 4
where PL,R=1
operator consists of two chiral superpartners – in the sense of supersymmetric quantum
mechanics [43]-[45]
2(1±γ5) are projectors on the left- and right-handed states. Thus the mass
? m2
±= −∂2
+q+= q+? m2
z+ ϕ2(z) ∓ ϕ′(z) = [−∂z± ϕ(z)][∂z± ϕ(z)] ;
−,
? m2
(1.3)
? m2
−q−= q−? m2
+,q±≡ ∓∂z+ ϕ(z) .(1.4)
The factorization (1.3) guarantees the positivity of the mass operator – i.e. the absence
of tachyons – and the supersymmetry realized by the intertwining relations (1.4) entails
the equivalence of the spectra between different chiralities for non-vanishing masses. As a
consequence, the related left- and right-handed spinors can be assembled into the bi-spinor
describing a massive Dirac particle which, however, is not necessarily localized at any point
of the extra-dimension if the field configuration ϕ(z) is asymptotically constant. Indeed the
massive states will typically belong to the continuous spectrum – or to a Kaluza-Klein tower
for the compact fifth dimension – and spread out the whole extra-dimension. Meanwhile,
the spectral equivalence may be broken just by one single state, i.e. a proper normalizable
zero mode of one of the mass operators ? m2
q−ψ+
0(x,z) = 0 ,ψ+
±. Let us assume to get it in the spectrum of
? m2
+: then from Eq.s (1.3) and (1.4) it follows that
0(x,z) = ψL(x) exp
?
−
?z
z0
dwϕ(w)
?
,(1.5)
where ψL(x) = PLψ(x) is a free Weyl spinor in the four-dimensional Minkowski space.
Evidently, if a scalar field configuration has the following asymptotic behavior: namely,
ϕ(z)z→±∞
∼±C±|z|ν±, Reν±> −1 ,C±> 0 ,
then the wave function ψ+
handed fermion is a massless Weyl particle localized in the vicinity of a four-dimensional
domain wall. It is easy to check that in this case the superpartner mass operator ? m2
C±> 0 and ν±= 0, there is always a gap for the massive Dirac states. Further on we
restrict ourselves to this scenario.
The important example of such a topological configuration can be derived for the
system having the free mass spectrum – continuum or Kaluza-Klein tower – for one of
the chiralities, say, for right-handed fermions. This is realized by a kink-like scalar field
background
ϕ+= M tanh(Mz) .
0(x,z) is normalizable on the z axis and the corresponding left-
−does
not possess a massless normalizable solution and if ϕ(z) is asymptotically constant, with
(1.6)
The two mass operators have the following potentials
? m2
+= −∂2
z+ M2?1 − 2sech2(Mz)?;
? m2
−= −∂2
z+ M2, (1.7)
and the left-handed normalized zero-mode is properly localized around z = 0, in such a
way that we can set
ψ+
0(x,z) = ψL(x)ψ0(z) , ψ0(z) ≡
?
M/2 sech(Mz) .(1.8)
– 3 –
Page 5
As a consequence, the threshold for the continuum is at M2and the heavy Dirac particles
may have any masses m > M, the corresponding wave-functions being completely de-
localized in the extra-dimension.
On the one hand, if we investigate the scattering and decay of particles with energies
considerably smaller than M, we never reach the threshold of creation of heavy Dirac
particles with m > M and all physics interplays on the four-dimensional domain wall with
thickness ∼ 1/M. On the other hand, at extremely high energies, certain heavy fermions
will appear in and disappear from our world.
It turns out that the real fermions – quarks and leptons living on the domain wall by
assumption – are mainly massive. Therefore, for each light fermion on a brane one needs
at least two five-dimensional proto-fermions ψ1(X),ψ2(X) in order to generate left- and
right-handed parts of a four-dimensional Dirac bi-spinor as zero modes. Those fermions
have clearly to couple with opposite charges to the scalar field Φ(X), in order to produce
the required zero modes with different chiralities when ?Φ(X)?0= ϕ+(z): namely,
[i ?∂ − τ3Φ(X)]Ψ(X) = 0 ,?∂ ≡ ? γα∂α, Ψ(X) =
ψ1(X)
ψ2(X)
, (1.9)
where ? γα≡ γα⊗12are Dirac matrices and τa≡ 14⊗σa, a = 1,2,3 are the generalizations
In this way one obtains a massless Dirac particle on the brane and the next task is to
supply it with a light mass. As the mass operator mixes left- and right-handed components
of the four-dimensional fermion it is embedded in the Dirac operator (1.9) with the mixing
matrix τ1mfof the fields ψ1(X) and ψ2(X). At last, following the general Standard Model
mechanism of fermion mass generation by means of the Higgs scalars, one can introduce
the second scalar field H(x) to make this job, replacing the bare mass τ1mf−→ τ1H(x).
Both scalar fields might be dynamical indeed and their self-interaction should justify the
spontaneous symmetry breaking by certain classical configurations allocating light massive
fermions on the domain wall. From the previous discussion it follows that the minimal set of
five-dimensional fermions has to include two Dirac fermions coupled to scalar backgrounds
of opposite signs. In addition to the trapping scalar field, another one is in order to supply
light domain wall fermions with a mass. Thus we introduce two types of four-fermion
self-interactions to reveal two composite scalar fields with a proper coupling to fermions.
As we shall see, these two scalar fields will acquire mass spectra similar to fermions with
light counterparts located on the domain wall. The dynamical mechanism of creation of
domain wall particles turns out to be quite economical and few predictions on masses and
decay constants of fermion and boson particles will be derived.
of the Pauli matrices σaacting on the bi-spinor components ψi(X).
2. Fermion model with self-interaction in 5D
Let us consider the model with the following Lagrange density
L(5)(Ψ,Ψ) = Ψ i?∂Ψ +
g1
4NΛ3
?Ψτ3Ψ?2+
g2
4NΛ3
?Ψτ1Ψ?2,(2.1)
– 4 –
Page 6
where Ψ(X) is an eight-component five-dimensional fermion field, see Eq.(1.9) – either
a bi-spinor in a four-dimensional theory or a spinor in a six-dimensional theory – which
may also realize a flavor and color multiplet with the total number N = NfNcof states.
The ultraviolet cut-off scale Λ bounds fermion momenta, as the four-fermion interaction
is supposed here to be an effective one, whereas g1and g2are suitable dimensionless and
eventually scale dependent effective couplings.
This Lagrange density can be more transparently treated with the help of a pair of
auxiliary scalar fields Φ(X) and H(X), which eventually will allow to trap a light fermion
on the domain wall and to supply it with a mass
L(5)(Ψ,Ψ,Φ,H) =
Ψ(i?∂ − τ3Φ − τ1H)Ψ −NΛ3
g1
Φ2−NΛ3
g2
H2. (2.2)
In this model the invariance (when it holds) under discrete τ-symmetry transformations
Ψ −→ τ1Ψ ;
Ψ −→ τ2Ψ ;
Ψ −→ τ3Ψ ;
Φ −→ −Φ ;
Φ,H −→ −Φ,−H ;
H −→ −H ,
(2.3)
(2.4)
(2.5)
does not allow the fermions to acquire a mass and prevents a breaking of translational
invariance in the perturbation theory. This τ-symmetry can be associated to the chiral
symmetry in four dimensions2.
However for sufficiently strong couplings, this system undergoes the phase transition
to the state in which the condensation of fermion-anti-fermion pairs does spontaneously
break – partially or completely – the τ-symmetry.
The physical meaning of the scale Λ is that of a compositeness scale for heavy scalar
bosons emerging after the breakdown of the τ-symmetry. In order to develop the infrared
phenomenon of τ-symmetry breaking, the effective Lagrange density containing the essen-
tial low-energy degrees of freedom has to be derived.
To this concern we proceed – only in this Section – to the transition to the Euclidean
space, where the invariant four-momentum cut-off can be unambiguously implemented.
Within this framework, the notion of low-energy is referred to momenta |p| < Λ0as com-
pared to the cut-off Λ ≫ Λ0. However, after the elaboration of the domain wall vacuum,
we will search for the fermion states with masses mfmuch lighter than the dynamic scale
Λ0, i.e. for the ultralow-energy physics. Thus, eventually, there are three scales in the
present model in order to implement the domain wall particle trapping.
Let us decompose the momentum space fermion fields into their high-energy part
Ψh(p) ≡ Ψ(p)ϑ(|p| − Λ0)ϑ(Λ − |p|), their low-energy part Ψl(p) ≡ Ψ(p)ϑ(Λ0− |p|) and
integrate out the high-energy part of the fermion spectrum, ϑ(t) being the usual Heaviside
step distribution. More rigorously, the above decomposition of the fermion spectrum should
2It can be also related to the chiral symmetry in a six-dimensional space-time where from our five-
dimensional model can be derived by dimensional reduction.
– 5 –
Page 7
be done covariantly, i.e. in terms of the full Euclidean Dirac operator,
D ≡ i(?∂ + τ3Φ + τ1H) .(2.6)
Nevertheless, as we want to concentrate ourselves on the triggering of τ-symmetry break-
ing by fermion condensation, we can safely assume to neglect further on the high-energy
components of the auxiliary boson fields, what is equivalent to the use of the mean-field or
large N approximations. Then the low-energy Lagrange density, which solely accounts for
low-energy fermion and boson fields, can be written as the sum of the expression in Eq.
(2.2) and the one-loop effective Lagrange density: namely,
L(5)
low(Ψl,Ψl,Φ,H) = L(5)
E(Ψl,Ψl,Φ,H) + ∆L(5)(Φ,H), (2.7)
where the tree-level low-energy Euclidean Lagrange density reads
L(5)
E(Ψl,Ψl,Φ,H) = iΨl(?∂ + τ3Φ + τ1H)Ψl+NΛ3
g1
Φ2+NΛ3
g2
H2. (2.8)
The one-loop contribution of high-energy fermions is given by
∆L(5)(Φ,H) = C − (N/2)tr?X|A|X? ,
A ≡ ϑ(Λ2− D†D)lnD†D
(2.9)
Λ2
− ϑ(Λ2
0− D†D)lnD†D
Λ2
0
,
where the normalization constant C is such that ∆L(5)(0,0) = 0 and the tr operation
stands for the trace over spinor and internal degrees of freedom. In Eq.(2.9) the choice of
normalizations in the logarithms ensures the continuity of spectral density at the positions
of cut-offs Λ and Λ0. Thereby the spectral flow through the spectral boundaryis suppressed.
In the latter operator A we have incorporated the cut-offs which select out the high-energy
region defined above [46].
From the conjugation property D†= τ2Dτ2, it follows that the Euclidean Dirac operator
is a normal operator, which has to be implemented in order to get a real effective action
and to define the spectral cut-offs with the help of the positive operator
D†D = −∂µ∂µ− ∂2
M2(X) ≡ Φ2(X) + H2(X) − τ3?∂Φ(X) − τ1?∂H(X) .
One can see that, in fact, the scale anomaly only contributes into ∆L(5), i.e. that part
which depends on the scales. Thus, equivalently,
?Λ
As we assume that the scalar fields carry momenta much smaller than both scales Λ ≫ Λ0,
then the diagonal matrix element in the RHS of Eq.(2.11) can be calculated either with
the help of the derivative expansion of the master representation [46]
?+∞
×
Q2
z+ M2(X) = −∂α∂α+ M2(X) ,
(2.10)
∆L(5)(Φ,H) = C + N
Λ0
dQ
Q
tr?X|ϑ(Q2− D†D)|X? .(2.11)
?X|ϑ(Q2− D†D)|X? = Qn
?
−∞
dt
2πi
exp{it}
t − iε
dnk
(2π)nexp
?
−it
?k2+ 2iQkα∂α− ∂α∂α+ M2(X)??
,(2.12)
– 6 –
Page 8
where n is a number of dimensions of the Euclidean space, or by means of the heat kernel
asymptotic expansion. For n ≤ 5, only three heat kernel coefficients at most are propor-
tional to non-negative powers of the large parameter Q (see Appendix A)
?X|ϑ(Q2− D†D)|X? = C0Qn+ C1Qn−2M2(X)
+Qn−4?C2[M2(X)]2+ C3∂α∂α[M2(X)]?+ O?Qn−6?,
where, for n < 6 and large scales Λ0< Q < Λ, the neglected terms rapidly vanish. For the
given operator ϑ(Q2− D†D), the coefficients Citake the following values
1
n 2n−1πn/2Γ(n/2);
1
2nπn/2Γ(n/2)= −n
n − 2
(2.13)
C0=
C1= −
2C0< 0 ;
C2=
2n+2πn/2Γ(n/2)=n(n − 2)
n − 2
8
C0> 0 ;
C3= −
3 · 2n+2πn/2Γ(n/2)= −n(n − 2)
24
C0< 0 ,(2.14)
where Γ(y) stands for the Euler’s gamma function. For different possible definitions of
the effective Lagrange density, involving operators and regularization functions other than
those ones of Eq.(2.11), one might obtain in general different sets of Ci, albeit their signs
are definitely firm. As we shall see further on, the negative sign of C1catalyzes the chiral
symmetry breaking at sufficiently strong coupling constants. Taking the little trace and
integrating the RHS of Eq.(2.13), from Eq.(2.11) one finds, up to a total n-divergence and
setting C = C0(Λn
0− Λn)N/n,
∆L(5)(Φ,H)Λ→∞
∼− A1
+ A2
?Λn−2− Λn−2
?Λn−4− Λn−4
0
??Φ2+ H2?
???Φ2+ H2?2?
?Λn−4− Λn−4
0
??(∂αΦ)2+ (∂αH)2?
+ A2
0
, (2.15)
where
A1=
N
(n − 2)2n−3πn/2Γ(n/2)
N(n − 2)
(n − 4)2n−1πn/2Γ(n/2)
n↑5
−→
N
9π3;
N
4π3.
(2.16)
A2=
n↑5
−→
(2.17)
As Λ0≪ Λ one can neglect the Λ0-dependence in Eq.(2.15) for n > 4, whereas near four
dimensions the pole in A2together with the cut-off dependent factor generate the coefficient
∼ ln(Λ/Λ0). For n = 5 we eventually find
∆L(5)(Φ,H)Λ→∞
NΛ
4π3
∼
?(∂αΦ)2+ (∂αH)2?−NΛ3
9π3
?Φ2+ H2?+NΛ
4π3
?Φ2+ H2?2.(2.18)
– 7 –
Page 9
Although the actual values of the coefficients A1and A2might be regulator-dependent, as
already noticed, the coefficients of the kinetic and quartic terms of the effective Lagrange
density are definitely equal, no matter how the latter is obtained from the basic Dirac
operator of Eq.(2.6). This very fact is at the origin of the famous Nambu relation between
the dynamical mass of a fermion and the mass of a scalar bound state in the regime of
τ-symmetry breaking, as we will see in the next Section.
3. τ-symmetry breaking
The interplay between different operators in the low-energy Lagrange density (2.7) may
lead to two different dynamical regimes, depending on whether the τ-symmetry is broken
or not. Indeed, going back to the Minkowski five-dimensional space-time, the low-energy
Lagrange density can be suitably cast in the form
L(5)
low(Ψl,Ψl,Φ,H) = Ψl(X)[i?∂ − τ3Φ(X) − τ1H(X)]Ψl(X)
+NΛ
?N
−NΛ
4π3∂αΦ(X)∂αΦ(X) +NΛ
9π3−N
g1
4π3∂αH(X)∂αH(X)
?N
+
?
Λ3Φ2(X) +
9π3−N
g2
?
Λ3H2(X)
4π3[Φ2(X) + H2(X)]2. (3.1)
We notice that the quartic operator in the potential is symmetric and positive, whilst the
quadratic operators have different signs for weak and strong couplings, the critical values
for both couplings being the same within the finite-mode cut-off regularization of Eq.(2.18):
namely,
gcr
i= 9π3,i = 1,2 . (3.2)
Let us introduce two mass scales ∆iin order to parameterize the deviations from the critical
point
∆i(gi) =2Λ2
9gi
(gi− gcr
i) .(3.3)
Taking into account that gi≥ 0 we have
∆i(gi) ≤ (2/9)Λ2
(3.4)
and the effective Lagrange density for the scalar fields takes the simplified form
L(5)
scalar(Φ,H) =NΛ
4π3
?(∂αΦ)2+ (∂αH)2+ 2∆1Φ2+ 2∆2H2− (Φ2+ H2)2?
So far two constants gi, and thereby ∆i, play an equivalent role and the related vertices
are invariant under the replacements Ψl?−→ τ2Ψland subsequent reflection of the scalar
fields – see Eq.(2.5). Therefore, without loss of generality, one can always choose
.(3.5)
∆1(g1) > ∆2(g2) . (3.6)
– 8 –
Page 10
Whenever both couplings gi are within the range 0 < gi < gcr
∆i(gi) < 0 and consequently the potential has a unique symmetric minimum. If instead
one at least of the couplings gidoes exceed its critical value, then the symmetric extremum
at Φ = H = 0 is no longer a minimum, though either a saddle point for ∆2(g2) < 0 < ∆1(g1)
or even a maximum for 0 < ∆2(g2) < ∆1(g1). If ∆1(g1) > 0, then the true minima appear
at a non-vanishing vacuum expectation value of the scalar field Φ(X): namely,
i
= 9π3, then we have
(I)ΦI≡ ?Φ(X)?0= ±
?
∆1(g1) ,HI≡ ?H(X)?0= 0 .(3.7)
This follows from the stationary point conditions for constant fields
?∆1(g1) − Φ2− H2?Φ = 0 ,
?∆2(g2) − H2− Φ2?H = 0(3.8)
and from the positive definiteness of the second variation of the boson effective action for
constant boson fields. As a matter of fact, if we set
S ≡ (S1,S2) = (Φ,H) ,
V[S] ≡NΛ
4π3
SI≡
i(X) + [Si(X)Si(X)]2?
?
±
?
∆1,0
?
,
?
d5X
?−2∆iS2
,(3.9)
we readily find
V[S] − V [SI] =1
2
?
d5X
?
d5Y si(X)sj(Y )
δ2V[S]
δSi(X)δSj(Y )
????
S=SI
+ ... ,
=NΛ
4π3
?
d5X si(X)sj(X)
MI
ij+ ... , (3.10)
where
si(X) ≡ Si(X) − SI,i = 1,2, (3.11)
whereas the Hessian mass matrix MI:
MI=
4∆1
0
0 2∆1− 2∆2
≡
M2
1
0
0 M2
2
,(3.12)
is manifestly positive definite and determines the mass spectrum of the five-dimensional
scalar excitations.
A further constant solution of Eq.(3.8) does exist for ∆2 > 0, i.e. ?Φ?0 = 0 and
?H?0= ±√∆2. However, it corresponds to a saddle point of the potential, as it can be
seen from Eq.(3.10) for ∆1 > ∆2. Likewise, if ∆1 > ∆2 > 0, then the matrix M is
negative definite at the symmetric point ?Φ?0= ?H?0= 0 which corresponds thereby to a
maximum. The degenerate situation – i.e. the valley – actually occurs for ∆1= ∆2> 0,
when the rotational τ2-symmetry is achieved by the Lagrange density but is spontaneously
broken. The massless scalar state in Eq.(3.12) arises in full accordance with the Goldstone’s
theorem.
– 9 –
Page 11
The corresponding dynamical effect for the fermion model of Eq.(2.1) consists in the
formation of a fermion condensate and the generation of a dynamical fermion mass M –
see Eq.s (2.2) and (2.7) – that breaks the τ2-symmetry. Its ratio to the heaviest scalar
mass just obeys the Nambu relation
M1≡ 2M > M2,M ≡ ?Φ?0=
?
∆1, (3.13)
the second, lighter composite scalar being a pseudo-Goldstone state. We notice that the
above relationship holds true independently of the specific values of the coefficients Aiin
Eq.(2.15) and, consequently, it takes place in four and five dimensions. However, if we
properly re-scale the scalar fields according to
Φ(X)√Λ ≡ ±M√Λ + ν u(X) ,
in such a way that dim[u] = dim[υ] = 3/2, then the low-energy Lagrange density (3.1) can
be suitably recast in the form
H(X)√Λ ≡ ν υ(X) ,ν ≡
?
2π3/N ,(3.14)
L(5)
low(Ψl,Ψl,u,υ) = M4Λ/2ν2+ iΨl(X)?∂Ψl(X) − MΨl(X)τ3Ψl(X)
+ (1/2)∂αu(X)∂αu(X) − 2M2u2(X)
+ (1/2)∂αυ(X)∂αυ(X) − 2(M2− ∆2)υ2(X)
− νΛ−1/2?Ψl(X)τ3Ψl(X)u(X) − Ψl(X)τ1Ψl(X)υ(X)?
∓ 2νMΛ−1/2[u2(X) + υ2(X)]u(X)
− (ν2/2)Λ−1[u2(X) + υ2(X)]2.
As a consequence, one can see that all the low-energy effective couplings for fermion and
boson fields do rapidly vanish in the large cut-off limit. In particular, if the cut-off Λ is
much larger than the energy range of our physics, then we are dealing with a theory of
practically free, non-interacting particles. Moreover, this pattern of τ-symmetry breaking
does not provide the desired trapping on a domain wall: the heavy fermions and bosons live
essentially in the whole five-dimensional space. From now on we shall proceed to consider
another type of vacuum solutions, which break the five-dimensional translational invariance
and give rise to the formation of domain walls.
(3.15)
4. Domain walls: massless phase
The existence of two minima in the potential of Eq.(3.9) gives rise to another set of vacuum
solutions [47] - [51], which connect smoothly the minima owing to the kink-like shape of
Eq.(1.6) with M =√∆1. On variational and geometrical grounds one could expect that
certain minimal solutions are collinear, just breaking the translational invariance in one
direction. We specify this direction along the fifth coordinate z. Then one can discover
two types of competitive solutions [49]-[51],
(J)?Φ(X)?0≡ ΦJ(z) = ±Mtanh(Mz) ,
?H(X)?0≡ HJ(z) = 0 ;
?Φ(X)?0≡ ΦK(z) = ±Mtanh(βz) ,
?H(X)?0≡ HK(z) = ±µsech(βz) .
(4.1)
(K)
(4.2)
– 10 –
Page 12
Further on we select out only positive signs in vacuum configurations to analyze the scalar
fluctuations around them, having in mind that our analysis is absolutely identical around
other configurations. When we insert the second solution (K) into the stationary point
conditions
2?M2− Φ2− H2?Φ = ∂α∂αΦ ,
2?∆2− H2− Φ2?H = ∂α∂αH ,
?
(4.3)
we find
µ =
2∆2− M2,β =
?
M2− µ2. (4.4)
The solution (K) exists only for ∆2< M2< 2∆2and it coincides with the extremum (J)
in the limit ∆2→ M2/2, µ → 0, β → M. The question arises about which one of the two
solutions is a true minimum and whether they could coexist if µ > 0. The answer can be
obtained from the analysis of the second variation of the bosonic low-energy effective action.
The corresponding relevant second order differential operator can be suitably written in
terms of the notations introduced in Eq.(3.9): namely,
The stationary point solutions (J) and (K) are true minima iff the matrix-valued mass
operator M[S(X)] becomes positive semi-definite at the extrema
D2
X
ij≡ − δij?x−
M[S(X)]
ij
,
?x≡ ∂µ∂µ,(4.5)
M[S(X)]
ij≡ δij
?−∂2
z− 2∆i+ 2Sk(X)Sk(X)?+ 4Si(X)Sj(X) .(4.6)
SJ= (MtanhMz,0) ,SK= (Mtanhβz,µsechβz) .(4.7)
Now, at the stationary point (J) the matrix-valued mass operator
MJ(z) ≡ M[SJ(z)](4.8)
turns out to be diagonal with entries
M11
M22
M12
J ≡ −∂2
J ≡ −∂2
J = M21
z+ 4M2− 6M2sech2(Mz) ,
z+ 2M2− 2∆2− 2M2sech2(Mz) ,
J= 0.
(4.9)
(4.10)
(4.11)
Both components do represent one-dimensional Schr¨ odinger-like operators, the eigenvalue
problem of which can be exactly solved analytically. The Schr¨ odinger-like operators (4.9)
and (4.10) can be presented in the factorized form similar to that one of Eq.(1.3): namely,
M11
M22
J= [−∂z+ 2Mtanh(Mz)][∂z+ 2Mtanh(Mz)] ;
J= M2− 2∆2+ [−∂z+ Mtanh(Mz)][∂z+ Mtanh(Mz)]
Therefrom it is straightforward to check that the ground states of the operator MJ(z) are
described by the real, node-less in z and normalized wave functions
(4.12)
(4.13)
M11
φJ(z) ≡ sech2(Mz)
JφJ(z) = 0 ,M22
?
JhJ(z) = m2
3M/4 ,
hhJ(z) ,
hJ(z) ≡ sech(Mz)
m2
h≡ M2− 2∆2;
?
(4.14)
M/2 ;(4.15)
– 11 –
Page 13
in such a way that we can suitably parameterize the shifts of the scalar field with respect
to the background vacuum solution (J) – see Eq.s(3.11) and (4.1) – by the following two
eigenstates of the mass matrix MJ(z): namely,
0
Ωφ
J(X) = φ(x)
φJ(z)
ν
√Λ
,Ωh
J(X) = h(x)
0
hJ(z)
ν
√Λ
, (4.16)
where φ(x),h(x) do eventually represent the ultralow-energy scalar fields on the Minkowski
space-time, as we shall better see below on. As a consequence, the spectrum of the second
variation is positive if M2> 2∆2and, in this case, the solution (K) does not exist whilst
the scalar lightest states are localized on the domain wall. More precisely – see Appendix
B – the first boson Φ has two states on the brane: a massless state and a heavy massive
state of mass√3M. The existence of a massless scalar state around the kink configuration
(J) is a consequence of the spontaneous breaking of translational invariance – see next
Section. Other heavy states belong to the continuous part of the spectrum with threshold
at 2M. The second boson H has only one state on the brane of mass√M2− 2∆2and its
continuous spectrum starts at√2M2− 2∆2≥ M.
Since the vacuum expectation value of the scalar field ?Φ(X)?0= Mtanh(Mz) has a
kink shape, its coupling to fermions induces the trapping of the lightest, massless fermion
state on the domain wall: namely,
ψ2R(x)
Ψ0(X) =
ψ1L(x)
ψ0(z) ,ψ0(z) = sech(Mz)
?
M/2 ,(4.17)
see Eq.s (1.8) and (1.9). The continuum of the heavy fermion states begins at M and
involves pairs of heavy Dirac fermions.
In conclusion, at ultralow energies much smaller than M, the physics in the neighbor-
hood of the vacuum (J) is essentially four-dimensional in the fermion and boson sectors.
It is described by the massless Dirac fermion
ψR(x)
ψ(x) =
ψL(x)
,(4.18)
with ψL(x) and ψR(x) being the two-component non-trivial parts of ψ1L(x) and ψ2R(x)
respectively, in such a way that we can set
ψR(x)
Ψl(X) =
ψL(x)
ψ0(z) .(4.19)
In result one has two four-dimensional scalar bosons, a massless one and a massive one,
provided M2− 2∆2≪ M2otherwise there is decoupling.
The matrix τ3does not mix the two types of fermions ψ1Land ψ2R, but the related
Yukawa vertex in the Lagrange density (2.2) mixes left- and right-handed components of
each of them. As a consequence the massless scalar field φ(x) does not couple directly to a
light fermion-anti-fermion pair. Its coupling to fermions involves inevitably heavy fermion
degrees of freedom. Therefore, the ultralow-energy effective action does not contain a
Yukawa-type vertex for the field φ(x), which appears thereby to be sterile.
– 12 –
Page 14
On the other hand, the interaction between light fermions and the second scalar field
h(x) on the domain wall does achieve the conventional Yukawa form. Indeed, once they
are projected on the zero-mode space of ψ(x)ψ0(z), the matrices τ1, τ2and τ3act as the
Dirac matrices in the chiral representation: namely,
?
1 0
τ1−→ γ0=
0 1
?
;τ3−→ γ5=
?
1 0
0 −1
?
;τ2−→ iγ0γ5;(4.20)
where the 2 ×2 unit matrix 1 acts on Weyl components ψL(x) and ψR(x). Meanwhile the
matrices in the kinetic part of the Dirac operator are projected onto the zero-mode space
as
?
0 −σk
where the Pauli matrices σkact on the two-component spinors ψL(x) and ψR(x). As a
consequence, the matrix τ1−→ γ0just induces the Yukawa mass-like vertex in the effective
action on the domain wall.
As a final result, in the vicinity of the vacuum solution (J) of Eq.(4.1), the ultralow-
energy effective Lagrange density for the light states on the four-dimensional Minkowski
space-time comes out from Eq.s(3.1),(4.15),(4.16),(4.19) and reads3
? γ0? γk−→
σk
0
?
≡ γ0γk, (4.21)
L(4)
J
?ψ,ψ,φ,h?=
?+∞
= iψ(x)γµ∂µψ(x) +1
−∞
dz L(5)
low
?
Ψl(X),Ψl(X),Ωφ
J(X),Ωh
J(X)
?
(4.22)
2∂µφ(x)∂µφ(x)
2m2
+1
2∂µh(x)∂µh(x) −1
− gfψ(x)ψ(x)h(x) − λ1φ4(x) − λ2φ2(x)h2(x) − λ3h4(x) ,
h≡?M2− 2∆2
?
NΛ
−∞
π3
NΛ
−∞
35ΛN
π3
NΛ
−∞
π3
NΛ
−∞
hh2(x)
with the scalar mass m2
by
?and the ultra-low energy effective couplings given
?+∞
?+∞
?+∞
?+∞
gf=
2π3
dz hJ(z)ψ2
0(z) =π
4
?
Mπ3
ΛN
,
λ1=
dz φ4
J(z) =18Mπ3
,
λ2=dz h2
J(z)φ2
J(z) =2Mπ3
5ΛN
,
λ3=
dz h4
J(z) =Mπ3
3ΛN.
(4.23)
Herein the ultralow-energy fields φ(x) and h(x) only have been retained, whilst the heavy
scalars and fermions with masses ∼ M have been decoupled.
3To be precise, the ultralow-energy effective Lagrange density for the light states on the four-dimensional
Minkowski space-time is well defined up to subtraction of an infrared divergent constant.
– 13 –
Page 15
Quite remarkably, the domain wall Lagrange density (4.23) has a non-trivial large cut-
off limit provided that the ratio M/Λ < 1 is fixed. The four-dimensional ultralow-energy
theory happens to be interacting with the ratios,
g2
f: λ1: λ2: λ3∼ 6 : 5 : 4 : 3 ,(4.24)
being independent of high-energy scales and regularization profiles – here we leave aside
the issues concerning the renormalization group improvement.
However the solution (J) is not of our main interest because the vacuum expectation
value of the field H vanishes and does not supply the domain wall fermion with a light
mass.
5. Domain walls: Higgs phase
Evidently, the domain wall solution of Eq.(4.1) as well as the constant background solution
of Eq.(3.7) just break the τ1- and τ2-symmetries of the Lagrange density (2.2), whereas
they keep the τ3-invariance untouched – see Eq.(2.5). Meanwhile, the second domain wall
background of Eq.(4.2) does break all the τ-symmetries, i.e. it realizes a different phase in
which the masses for light particles are naturally created. We notice however that for the
kink solution (K) realizing the space defect in the z-direction the combined parity under
the transformations,
z −→ −z,
Ψl(x,−z) −→ ˆ γ5τ3Ψl(x,z),
Φ(x,−z) −→ −Φ(x,z),H(x,−z) −→ H(x,z),
¯Ψl(x,−z) −→ −¯Ψl(x,z)ˆ γ5τ3, (5.1)
remains unbroken.
As we will see below the mass scale for light particles is controlled by the parameter
µ =√2∆2− M2, which describes the deviation from the critical scaling point where the
two regimes of Eq.s (4.1) and (4.2) melt together.
As we want to supply fermions with masses much lower than the threshold of penetra-
tion into the fifth dimension, to protect the four-dimensional physics, we assume further on
that µ ≪ M. At the solution (K): S(X) = SK(z), the 2 × 2 matrix-valued mass operator
M[S(X)] of Eq.(4.6) for scalar excitations gets the following entries,
M11
M22
M12
K= −∂2
K= −∂2
K= M21
z+ 4M2+ 2(µ2− 3M2)sech2(βz) ,
z+ M2− µ2− 2(M2− 3µ2)sech2(βz) ,
K= 4Mµsinh(βz)sech2(βz) ,
(5.2)
(5.3)
(5.4)
the positive quantities µ and β being defined in Eq.(4.4). As this mass operator is non-
diagonal it mixes the scalar fields Φ and H. However, this mixing does fulfill the combined
symmetry,
s(x,z) = ±σ3s(x,−z) ,
because the diagonal elements of M[SK(z)] are even and the off-diagonal ones are odd with
respect to the reflection z → −z. This symmetry allows to classify the non-degenerate
M[SK(z)] = σ3M[SK(−z)]σ3,(5.5)
– 14 –
Page 16
eigenstates of the operator MK(z) ≡ M[SK(z)] according to their parity. Actually, they
may consist of even-odd or odd-even pairs of components
S(X) − SK=
s1(x,z)
s2(x,z)
.
One more symmetry can be revealed in respect to the simultaneous change in the sign of
µ and the reflection z → −z, that means
M[SK(z,µ)] = M[SK(−z,−µ)] .(5.6)
This formal symmetry turns out to be crucial in perturbation theory to build up eigenstates
of MK(z) for small µ ≪ M. Indeed, after a proper normalization of the eigenstates, the
expansion in powers of µ of the latter ones is constrained as follows: the parity-even
components of s(x,z) have to contain even powers of µ, whilst the parity-odd ones have to
contain odd powers of µ.
As a matter of fact, for µ ≪ M the operator MK(z) can be suitably decomposed
into its diagonal part M(0)
Kwith the very same diagonal matrix elements of Eq.s (4.9) and
(4.10), in which M is replaced by β and µ = 0, plus the small perturbation ∆MK,
MK= M(0)
K+ ∆MK,(5.7)
where
M(0)
K≡
−∂2
z+ 4β2− 6β2sech2(βz)
0
0
−∂2
z+ β2− 2β2sech2(βz)
,
whereas
∆MK≡ 4β2ǫ
ǫtanh2(βz)
√1 + ǫ2tanh(βz)sech(βz)
ǫsech2(βz)
√1 + ǫ2tanh(βz)sech(βz)
.
Herein the dimensionless parameter ǫ = µ/β ≪ 1 controls the deviation from the scaling
point where the two regimes (J) and (K) do coincide and both scalars are massless. One
can choose the parameters β and ǫ as independent ones, characterizing the mass unit
and a scaling-point deviation. Then the initial mass scales M and µ can be conveniently
expressed in terms of the latter ones as
?
Now, it turns out that the operator M(0)
Khas two zero-modes, as it immediately follows
from the comparison with the related operators (4.9) and (4.10) in which M → β and
M2− 2∆2is omitted. The corresponding massless eigenfunctions are obviously given by
Eq.s (4.15) and (4.16) up to the replacement M → β. The deviation from the scaling point
is therefore fully generated by the perturbation term ∆MK, as it does.
The masses of the lightest localized scalar states can be obtained again – see Eq.(4.16)
– from the Schr¨ odinger equation with a matrix-valued potential: namely,
?
M = β
1 + ǫ2,µ = βǫ . (5.8)
j=1,2
Mij
KΩj
K(X) = m2Ωi
K(X) , i = 1,2.(5.9)
– 15 –
Page 17
It is easy to find the massless state as a zero-mode of the operator in Eq.(5.7). Indeed,
differentiating with respect to z the equations (4.3) of the stationary configuration, one
obtains the normalized zero-mass solution from the derivative of the kink-like solution
SK(z), with the upper signs, in the form
?
Ωφ
K(X) = φ(x)
K(z)
φ−
K(z)
φ+
K(z)
φ−
K(z)
ν
√Λ
,
φK(z) ≡
φ+
4 + 6ǫ2
≡ ∂zSK(z)
?
3
β3(4 + 6ǫ2)
=
3β
√1 + ǫ2sech2(βz)
−ǫ sinh(βz)sech2(βz)
,(5.10)
according to the translation symmetry breaking in the kink background [47]. Evidently,
this eigenfunction is related to the unperturbed zero-mode of Eq.(4.15) at µ = 0 which is
parity-even. Respectively, it consists in turn of the parity-even upper component φ+
and the parity-odd lower component φ−
K(z), in accordance to the combined symmetry
of Eq.(5.5), because the perturbation term does not change the parity properties of an
eigenstate. Finally, one can convince oneself that, for a given β, the expansion in ǫ ↔ µ
contains only even powers of this parameter for the upper, even component and only its
odd powers for the lower, odd component in accordance with the µ-symmetry of Eq.(5.6).
The second light scalar state Ωh
K(X) arises from the perturbation of the zero-mass
state of Eq.(4.16) for µ = 0. Therefore, it contains a parity-odd upper component h−
that can be expanded into odd powers of ǫ, together with a parity-even lower component
h+
K(z) which can be expanded in even powers of ǫ, in such a way that the exact normalized
second light scalar state can be written as
K(z)
K(z)
Ωh
K(X) ≡ h(x)
h−
K(z)
h+
K(z)
ν
√Λ
. (5.11)
To the first order in the ǫ expansion – see Appendix C – the localization functions read
2
hK(z) ≡
h−
K(z)
h+
K(z)
=
?
β
−ǫβz sech2(βz)
sech(βz)
+ O(ǫ2) .(5.12)
The mass of this scalar state, to the first order in ǫ expansion, is given by
mh= β ǫ√2 + O(ǫ2) ≃ µ√2 .(5.13)
One can see that now the mass eigenstates enter both in the upper component s1(X) =
Φ(X)−?Φ(X)?0and in the lower component of s2(X) = H(X)−?H(X)?0of the low-energy
scalar field: namely,
?
s2(X) =φ(x)φ−
s1(X) =φ(x)φ+
K(z) + h(x)h−
K(z)
?ν
?ν
√Λ
,
?
K(z) + h(x)h+
K(z)
√Λ
.(5.14)
– 16 –
Page 18
However, the admixture of the opposite-parity states is strongly suppressed in the ultralow-
energy effective Lagrange density in the Minkowski space-time, owing to the integration
over the extra-dimension and to the high order in ǫ.
As well as for the solution (J), in the vicinity of the vacuum solution (K) of Eq.(4.2),
the ultralow-energy effective Lagrange density for the light states can be obtained from
Eq.(3.1) in a similar form. In the leading and next-to-leading order of ǫ-expansion, one
can show that the only difference concerns the appearance of the mass for light fermions and
of a cubic scalar interaction that we calculate for positive values of ?Φ(X)?0and ?H(X)?0
in Eq. (4.2): namely,
?ψ,ψ,φ,h?= L(4)
the fermion mass being determined by the expression
?+∞
where Eq.s (4.2) and (4.17) have been employed, whereas
?
ΛN
−∞
?
We stress that a generally a priori possible 3-point vertex φ2(x)h(x) does actually receive
mutually compensating contributions from different mixing terms.
decay of the massive Higgs-like boson h into a pair of massless branons [24] is suppressed
and the low-energy Standard Model matter turns out to be stable.
To sum up, in the present model the ratio of the dynamical fermion mass (presumably
the top quark one [40]) to the Higgs-like scalar mass is firmly predicted to be
mh: mf= 4√2 : π ≃ 1.8 ,
which is somewhat less than such a ratio as predicted in the Top-mode Standard Model –
the Nambu relation gives mh: mf= 2.
Finally, concerning the coupling parameters g2
scribed by Eq.s (4.23), up to the two orders in ǫ-expansion, the corrections being of the
order O(ǫ2). We would like to stress that the coupling of fermions to the massless scalar
φ(x) does no longer appear even in the next order ∼ ǫ2, because its mixing form factor
φ−
K(z) in Eq.(5.14) is an odd function, making thereby the relevant integral of Eq. (4.23)
identically vanishing.
L(4)
K
J
?ψ,ψ,φ,h?− mfψ(x)ψ(x) − λ4h3(x) + O(ǫ2) , (5.15)
mf≡
−∞
dz ψ0(z)HK(z)ψ0(z) =π
4β ǫ =π
4µ ,(5.16)
λ4=
8π3
?+∞
Mπ
ΛN.
dz
?
h+
K(z)
?2 ?ΦK(z)h−
K(z) + HK(z)h+
K(z)?
≈ µπ
(5.17)
Thereby the direct
(5.18)
f, λ1, λ2, λ3, they are essentially de-
6. Manifest breaking of translational invariance
One can conceive that in reality the translational invariance in five dimensions is broken not
only spontaneously but also manifestly due to the presence of a gravitational background,
– 17 –
Page 19
of other branes etc. In a full analogy with the Goldstone boson physics one can expect [24]
that the manifest breaking of translational symmetry is such to supply the branons with a
mass.
In the model presented in our paper the natural realization of the translational symme-
try breaking can be implemented by the inhomogeneous scalar background fields coupling to
the lowest-dimensional fermion currents. Let us restrict ourselves to the scenario of the type
(K) and introduce two scalar field defects with the help of the functions FΦ(z) and FH(z),
which are supposed to be quite small, i.e., |FΦ(z)| ≪ |?Φ(X)?0| and |FH(z)| ≪ |?H(X)?0|.
These scalar defect fields catalyze the translational symmetry breaking and the domain
wall formation by means of their interactions with the fermion currents,
L(5)
F= − FΦ(z)Ψ(X)τ3Ψ(X) − FH(z)Ψ(X)τ1Ψ(X) .
When supplementing the Lagrangian (2.2) under the Hubbard-Stratonovich transforma-
tion, one can see that those background defect fields do actually couple to the auxiliary
scalar fields in Eq. (2.2) after the replacements
(6.1)
Φ ?−→ Φ − FΦ(z) ,H ?−→ H − FH(z) .(6.2)
The latter redefinition of the auxiliary fields reveals the explicit coupling of defect fields to
the auxiliary scalars, which dictates in turn the eventual change of the scalar field Lagrange
density (3.1): namely,
L(5)?Ψ,Ψ,Φ,FΦ(z),H,FH(z)?= L(5)(Ψ,Ψ,Φ,H) +
g1
g2
2NΛ3
ΦFΦ(z) +2NΛ3
H FH(z) −NΛ3
g1
F2
Φ(z) −NΛ3
g2
F2
H(z) .(6.3)
In order to prepare the scalar part of the low-energy effective action in comparable units
let us re-scale,
FΦ(z) ≡
g1µ3
4π3Λ2fΦ(z) ,FH(z) ≡
g2µ3
4π3Λ2fH(z) , (6.4)
where we have kept in mind that g1≃ g2≃ gcr= 9π3. If we assume the dimensionless
functions fΦand fHto be O(1), then the last two terms in Eq. (6.3) are of the order µ/Λ
and thereby negligible within the approximation adopted in this paper.
Taking the notations of Eq. (4.4) into account, it is not difficult to show that the
effective Lagrange density for low-energy scalar fields in the presence of defects can be cast
in the form
L(5)
+ (∂αΦ)2+ (∂αH)2+ 2M2Φ2+ (M2+ µ2)H2− (Φ2+ H2)2?
The stationary vacuum configurations of scalar fields obey now the modified set of Eq. (4.3):
namely,
∂α∂αΦ = µ3fΦ(z) + 2Φ?M2− Φ2− H2?
∂α∂αH = µ3fH(z) + H
scalar[Φ,H,fΦ(z),fH(z)] =NΛ
4π3
?
2µ3fΦ(z)Φ + 2µ3fH(z)H
.(6.5)
,
?M2+ µ2− 2H2− 2Φ2?
.(6.6)
– 18 –
Page 20
Let us search for the solutions of these equations generating a domain wall of type (K)
which preserves the symmetry (5.1). Evidently, the latter ones can be actually achieved if
both the scalar defect fields and the very vacuum solutions have the same definite parity
properties, i.e., fΦ(z) and ΦK(z) being odd functions of z, while fH(z) and HK(z) being
even ones. Thus we tune ourselves to what we could call the collinear mechanism of a brane
creation. As we shall see here below, it allows to formulate within the perturbation theory
the self-consistent catalyzation of a domain wall in the presence of some weak defects, i.e.,
for sufficiently small defect functions fΦ(z) and fH(z).
Indeed, let us accept the latter assumption and write down the trial solutions as the
sum of the unperturbed functions ΦK(z) ≡ Mtanh(βz), HK(z) ≡ µsech(βz) given in
Eq.s (4.2) and the corresponding small deviations δfΦ(z),δfH(z) due to the presence of a
weak defect, so that
H(fz)HK(z) + δfH(z)
Sf(z) =
Φf(z)
=
ΦK(z) + δfΦ(z)
≡ SK(z) + δfS(z) .
fH(z)
(6.7)
To the first order in deviation functions, Eq.s (6.6) entail
M[SK(z)] δfS ≡ MK(z)δfS = µ3f , f(z) ≡
fΦ(z)
,(6.8)
where the second variation operator MK(z), already introduced in Eq. (5.7), does appear
in this inhomogeneous equation. Its solution can be suitably presented by means of the
spectral decomposition for the operator MK(z) itself. In fact, as it was elucidated in the
preceding Section, its spectral resolution contains two sets of eigenfunctions with opposite
parity properties. In particular, once we search for solutions of odd upper component
and even lower component type, the relevant part of the spectral decomposition of the
operator MK(z) consists of the projector onto the bound state, Eq. (5.12), belonging to
the eigenvalue m2
starting from the eigenvalue M2≫ 2µ2: namely,
?∞
where?Ph,?Pσ are the projectors onto the proper and improper states with the relevant
of the spectrum and ̺(σ) the corresponding density of the states. After inversion of the
operator (6.9) the approximate solution of Eq. (6.8) reads
?1
Thus, taking into account that µ2≪ M2< σ, a finite solution ∼ µ exists iff the projection
of the defect function f(z) onto the bound state is not vanishingly small. Then the first
order approximate solution is given by
?2
h≃ 2µ2, together with the projectors onto the continuum improper states,
MK(z) = m2
h?Ph+
M2dσ ̺(σ)σ?Pσ+ irrelevant ,(6.9)
parity properties respectively, whereas σ denotes the invariant mass of the continuous part
δfS ≃ µ
2
?Phf +
?∞
M2dσ ̺(σ)µ2
σ
?Pσf
?
.(6.10)
δfS(z) = ξ µ
βhK(z) + O?µǫ2,µξ2?
,(6.11)
– 19 –
Page 21
where the dimensionless parameter ξ is introduced by means of the relation,
?
8
−∞
ξ ≡
β
?+∞
dz
?
h−
K(z)fΦ(z) + h+
K(z)fH(z)
?
. (6.12)
Let us first assume that ǫ ≪ |ξ| ≪ 1. Then, for a wide variety of scalar defects f(z), the
dynamical mechanism of the domain wall formation is essentially triggered by the light
bound state of the mass operator MK, up to the first order approximation in perturbation
theory. Hence, for this variety, the manifest breaking of translational invariance is labeled
by a one-parameter family of weak defect amplitudes
in the calculation of the branon mass the next-to-leading terms – quadratic in ξ – are indeed
required. The latter ones are more sensitive to the high-frequency modes in the spectrum
of the operator MK, owing to the non-linear interaction in Eq.s (6.6). Consequently, they
appear to be dependent upon the details of the defect function f(z). The analysis concerning
the general solution will not be addressed here, instead of, let us make the ansatz for f(z)
which allows to solve Eq.s (6.6) exactly. Firstly, one is evidently free to restrict the allowed
set of defect functions in such a way that the solution (6.11) keeps itself reliable all over
the range ǫ ≪ |ξ| < 1. In particular, it is provided by the ansatz (A) :
?8
6ΦK(δfΦ)2+ 4HKδfH δfΦ + 2(δfΦ)3+ 2(δfH)2δfΦ ;
?8
6HK(δfH)2+ 4ΦKδfΦδfH + 2(δfH)3+ 2(δfΦ)2δfH ,
|ξ| ≤ 1. However, it turns out that
µ3f(A)
Φ
≃ ξ µ3
βh−
K+
µ3f(A)
H
≃ ξ µ3
βh+
K+
(6.13)
where ΦK,HKare given by Eq.s (4.2) and the deviations δfΦ,δfH are taken from Eq. (6.11).
For such a defect, to all orders in the weak defect amplitude, the modified kink configuration
of the scalar fields reads
ǫ(1 + ξ)sech(βz)
S(A)
K(z) ≈ β
?1 +1
2ǫ2?tanh(βz) − ξǫ2βz sech2(βz)
.(6.14)
As the lightest domain wall fermion gets its mass only from the lower component of S(A)
then the mass correction is just reproduced by the following multiplicative factor of the
unperturbed mass (5.16),
mf≈π
In terms of β and ǫ, the mass operator for scalar particles receives an additional perturba-
tion as compared to Eq.s (5.7): namely,
K(z),
4µ (1 + ξ) . (6.15)
M(A)
K
∆MA≡ 4β2ǫξ sech2(βz)
×
= M(0)
K+ ∆MK+ ∆MA,(6.16)
×
?1 + O?ǫ2??
ǫ?1 +1
2ξ − 3βz tanh(βz)?
,
sinh(βz)
2ξ − βz tanh(βz)?
sinh(βz)ǫ?3 +3
– 20 –
Page 22
where the basic definition of the second variation operator Eq. (4.6) has been used. There-
from one obtains the masses of scalar states – see Appendix D. In particular, the lightest
branon is no longer massless
?
and its localization function turns out to be
?
4
m(A)
φ
?2
≈4
5µ2ξ2
(6.17)
φA(z) ≈
3β
sech2(βz)
−ǫ (1 + ξ) sinh(βz)sech2(βz)
.(6.18)
Respectively, the correction to the Higgs boson mass reads,
?
m(A)
h
?2
≈ 2µ2
?
1 + 3ξ +5
3ξ2
?
, (6.19)
whilst the corresponding localization function is given by
hA(z) ≈
?
β
2
−ǫ(1 + ξ) βz sech2(βz)
sech(βz)
.(6.20)
We see that the regular defect does not rigidly polarize the domain wall vacuum (K) in
a fixed direction – the sign of ξ can be not only positive but also negative while delivering
a local minimum of the induced scalar field Hamiltonian, i.e. physical masses for scalar
bosons. For the latter case, ξ < 0, one can reduce the mass of Higgs particle as compared
to the fermion (“ top-quark”) mass,
?
m(A)
h
?2
: m2
f=32(1 + 3ξ + 5ξ2/3)
π2(1 + ξ)2
. (6.21)
For instance, if ξ ≃ −0.4 and the fermion mass is assumed to be of order of the top-quark
mass ∼ 175 GeV, then the Higgs mass is estimated to be ∼ 135 GeV, which is acceptable
from the phenomenological viewpoint [52]. For the same choice the branon mass is found
to be ∼ 100 GeV. However, we stress that the predictions for scalar masses are essentially
based on contributions which are quadratic in ξ, especially for the branon mass. But the
latter terms depend on a model for the space defect.
Both the localization functions (6.18), (6.20) contain corrections linear in ξ which are
the only ones relevant for calculation of the induced 3-point and 4-point vertices for low-
energy scalar fields. The coupling constants g2
domain wall are built of the components φ+
at the leading order in ǫ2and thereby do not depend on a small background defect. The
coupling constant λ4in the 3-point vertex h3(x) is described by the integral in Eq. (5.17)
which is evidently homogeneous in the multiplier (1+ξ). Hence, in the presence of a defect,
this coupling is simply renormalized as follows,
f,λ1,λ2,λ3 in the 4-point vertices on the
A, which are universally given by Eq.s (4.23)
A,h+
λ(f)
4≈ (1 + ξ)λ4≈ (1 + ξ)µπ
?
Mπ
ΛN.
(6.22)
– 21 –
Page 23
Another possible induced coupling constant in the vertex φ2(x)h(x) is also homogeneous
in the multiplier (1 + ξ). Thus it remains vanishing, the Higgs boson does not decay into
branons and the Standard Model matter is stable.
The previous ansatz (A) has been a representative of non-topological or regular defects
with background functions decreasing at the infinity. Now we shall introduce and discuss
another kind of defect of a topological type, the influence of which on the branon masses
is drastically different. Let us take the following ansatz (B) for the defect f(z) providing
the analytical solution for vacuum configurations,
µf(B)(z) =
M γ tanh(¯βz)
µκsech(¯βz)
, (6.23)
where γ,κ are dimensionless parameters. Notice that the components are just borrowed
from the unperturbed solution (4.2). We accept that the upper component can be enhanced
including the normalization on M rather than on µ – otherwise it becomes irrelevant as it
will be clear later on. We also remark that, on the one hand, the related function is not
square integrable and lies outside the domain spanned by the normalizable eigenfunctions
of the operator MK. On the other hand, it represents a genuine topological defect whereas
the lower, square integrable component does not influence the field topology, i.e., it is a
regular defect.
The exact solution of Eq.s (6.6) keeps the form of Eq. (4.2), although its coefficients
are now controlled by γ,κ: namely,
S(B)
K(z) =
M atanh(¯βz)
µ(1 + ξ)sech(¯βz)
. (6.24)
The constants a,ξ,¯β obey the relations,
2(a2− 1)a =
?
¯β2= M2a2− µ2(1 + ξ)2≈ β2
γǫ2
1 + ǫ2,
?
a ≈ 1 +1
4γǫ2, (6.25)
2ξ + ξ2+γ
2a
(1 + ξ) = κ , (6.26)
?
1 + ǫ2
?1
2γ − 2ξ − ξ2
??
. (6.27)
From the last equation one can reveal a certain interplay between the enhanced topological
defect and the regular defect. Instead of the inversion of Eq. (6.26) we prefer to use ξ as
an independent parameter, thereby applying this relation as a definition of κ.
Let us expand now the solution (6.24) in powers of ǫ,
S(B)
K(z) ≈
(6.28)
β
?
1 +1
4ǫ2(2 + γ)
?
tanh(βz) +1
4ǫ2(γ − 4ξ − 2ξ2)βz sech2(βz)
ǫ(1 + ξ)sech(βz)
.
Here one can see that the linear analysis in Eq. (6.11) is again valid and entails ξ =
(κ/2)−(γ/4) in accordance with the linear part of the exact solution (6.26). As the lower,
– 22 –
Page 24
even component of Eq. (6.28) coincides with that one in Eq. (6.14), the shift in the fermion
mass remains the same as in Eq. (6.15).
The mass operator for this ansatz (B) gets a similar functional structure albeit different
coefficients: namely,
M(B)
K
∆M11
+3(γ − 4ξ − 2ξ2)βz tanh(βz) + 3γsinh2(βz)?
∆M12
∆M22
+(γ − 4ξ − 2ξ2)βz tanh(βz) + γsinh2(βz)?
The corresponding mass spectrum of lightest scalar states is calculated in Appendix D. In
particular, the branon mass is no longer governed by terms quadratic in ξ,
= M(0)
B≡ β2ǫ2sech2(βz)?4ξ + 2ξ2
B≡ ∆M21
B≡ β2ǫ2sech2(βz)?12ξ + 6ξ2
K+ ∆MK+ ∆MB,
+ O?ǫ4?
,
B= 4ξβ2ǫsech(βz)tanh(βz) + O?ǫ3?,
+ O?ǫ4?
. (6.29)
?
m(B)
φ
?2
≈ γµ2. (6.30)
Now, the striking evidence occurs for a strong polarization effect induced by a topologi-
cal defect: the local minimum is guaranteed only for asymptotics at infinities which are
coherent in their signs, i.e., for positive γ.
The Higgs mass encodes both topological and non-topological vacuum perturbations,
?
As γ > 0 the topological defect makes the Higgs particle heavier. However, once again, the
sign of ξ is not fixed by the requirement to provide a local minimum. Therefore one can
find a window for a relatively light Higgs scalar. If branons are very light, viz. ξ ≫ γ, then
one can neglect the last term in Eq. (6.31) and reproduce the Higgs mass (6.19) of ansatz
(A) with a reasonable precision.
As to the localization functions for scalar states, they coincide with those ones of ansatz
(A). Thus they are universally parameterized by terms linear in ξ, to the leading orders
in ǫ-expansion. Respectively, the induced coupling constants g2
universal as well.
?
m(B)
h
?2
≈ µ2
2 + 6ξ + 3ξ2+1
2γ
?
.(6.31)
f,λ1,λ2,λ3,λ4appear to be
7. Conclusions and discussion
The main task of the present work has been the explanation of how light fermions and
Higgs bosons may be located on the four-dimensional domain wall representing our Uni-
verse. The five-dimensional fermions with their strong self-interactions have been used to
discover the vacuum that breaks spontaneously the translation invariance at low energies.
Its remarkable feature is the appearance of a topological defect which induces the light
particle trapping. Depending on the relation of the four-fermion coupling constants, the
light fermion states may remain massless or achieve the masses ∼ µ, supposedly, much less
– 23 –
Page 25
than the characteristic scale M of the dynamical symmetry breaking. These two phases
differ in the level of symmetry breaking. Namely, the internal discrete τ-symmetry of the
initial fermion Lagrange density is generated by the algebra of Pauli matrices and this sym-
metry can be broken partially or completely. In the latter case, the light fermions acquire
a dynamical mass which is assumed to be ∼ µ ≪ M and the ratio of the Higgs boson mass
to this fermion mass is ∼ 1.8 close to the ratio in the Top-mode Standard Model [40].
As a consequence of spontaneous breaking of the τ-symmetry and of translational
invariance two composite scalar bosons arise, one of which is certainly a Higgs-like massive
state whilst the other one is a Goldstone-like state – a massless domain-wall excitation
called branon [23, 24]. On the one hand, the latter particles turn out to be sterile because
in the ultra-low energy world they do not couple directly to the light fermions. On the other
hand, their original coupling to fermions in Eq. (3.1) mixes light fermions and the ultra-
heavy fermion states with masses ∼ M. Evidently, the amplitude of branon production in
the light fermion annihilation due to exchange by an ultra-heavy fermion state ∼ g2
(see Fig. 1) is strongly suppressed if the domain wall thickness ∼ 1/M is much less than
the Compton length for light particles, 1/M ≪ 1/mf.
However, it is quite plausible that the
translation and τ-symmetries are manifestly
broken just because the five-dimensional world
contains a classical background of gravitational
and various matter fields with a net result
similar to a crystal defect. In the present
paper we have assumed the scalar nature of
a five-dimensional space-time defect, which
triggers the domain-wall formation through
its coupling to the low-dimensional fermion
operators. The two types of such defects have
been investigated.
The first type is a non-topological one, the shape of which belongs to the Hilbert space
of square integrable functions. In this case it happens that the only relevant dimensionless
parameter ξ governing the manifest symmetry breaking is given by a projection of the
defect profile function on the Higgs boson wave function. The occurrence of such a defect
supplies the branons with a small mass ∼ ξµ though its numerical value is rather model
dependent. Meantime, the polarization of the domain wall background, i.e. its sign, is not
correlated with the sign of a non-topological defect – for any sign, the local minimum of
the low-energy effective action is attained.
The second type of defects is a topological one, with different non-vanishing asymp-
totics at the infinity. It makes much stronger the catalyzation of a domain wall, namely,
the minimum and respectively the physical branon masses are reached only for coherent
signs of the profiles of defect and domain-wall. In the latter case, again, the branons are
supplied with a small mass.
Concerning the Higgs boson masses, their ratio to the fermion (∼ top-quark) mass
can be substantially reduced with an appropriate choice of a non-topological part of the
fm2
f/M2
f
_
f
f
f
_
Figure 1:
fermions f,¯f due to super-heavy fermion Ψ ex-
change.
Branon φ creation from light
– 24 –
Page 26
defect ∼ ξ. In particular, the Higgs masses may be well adjusted to a phenomenologically
acceptable [52] value ∼ 135 GeV for a reasonably small value of a defect ξ ∼ 0.4.
In all scenarios the ratios of the induced coupling constants in the four-dimensional
effective action for light particles are predicted unambiguously and in a model-independent
way. Their absolute values are universally parameterized by the ratio of the symmetry
breaking scale M to the high-energy (compositeness) scale Λ – a cutoff in the four-fermion
interaction. This ratio M/Λ < 1 can be thought of as an empirical input which must be
established from the relevant experiments by measuring, for instance, the Yukawa coupling
constant gfin the direct Higgs production via the fusion mechanism.
However, the branon matter is much more elu-
f
sive for experiments because it is not produced by
fermions at the leading order in the Yukawa cou-
pling and does not appear as a decay product of
the Higgs bosons.
(see, Fig. 2) may result in missing energy events
due to radiative creation of branon pair mediated
by Higgs bosons. The corresponding Feynman in-
tegral is well convergent and therefore, at very low
momenta, the amplitude of this process can be esti-
mated to be of order g2
if they are produced by very light fermions, e.g. e+e−, the very last ratio is negligibly small
putting such signals into the experimental background. Nonetheless, for top-quarks (again
in the fusion production) there might be a possibility to discover branon pair signals for
sufficiently light branons. More qualitative estimates will be presented elsewhere. Anyhow,
the sterile branons seem to be good candidates for saturation of the dark matter of our
Universe – see Ref. [24] for a more detailed motivation.
f
_
h
h
Figure 2: Missing energy in fermion an-
nihilation due to branon emission.
Still the fermion annihilation
f·λ2·m2
f/µ2. We notice that
Acknowledgments
One of us (A.A.) is grateful to V.A.Rubakov for multiple discussions during his visit to
the University of Barcelona which were indeed stimulating to start the present work. We
also acknowledge the long-term correspondence with R. Rodenberg whose erudition helped
us a lot to get orientated in the abundant literature about the particle physics from extra
dimensions. This work is supported by Grant INFN/IS-PI13. A. A. and V. A. are also
supported by Grant RFBR 01-02-17152, INTAS Call 2000 Grant (Project 587) and The
Program ”Universities of Russia: Fundamental Investigations” (Grant UR.02.01.001).
A. One-loop effective action
Let us consider the second order matrix-valued positive elliptic differential operator
D†D = −∂µ∂µ− ∂2
M2(X) ≡ Φ2(X) + H2(X) − τ3?∂Φ(X) − τ1?∂H(X) ,
z+ M2(X) = −∂2+ M2(X) ,
(A.1)
– 25 –
Page 27
acting on a n-dimensional flat Euclidean space (eventually n = 5). Our aim is to evaluate
the matrix element of the distribution of the states operator: namely,
?X|ϑ(Q2− D†D)|Y ? =
?c+i∞
c−i∞
dt
2πi
exp{tQ2}
t
?X|exp{−tD†D}|Y ? , c > 0 .(A.2)
As a matter of fact, if the positive operator D†D is also of the trace class, then the trace of
ϑ(Q2− D†D) does represent the number of the eigenstates of D†D up to the momentum
square Q2. It is convenient to write the heat kernel in the form
?X|exp{−tD†D}|Y ? ≡ (4πt)−n/2exp
?
−(X − Y )2
4t
?
Ω(t|X,Y ) ,(A.3)
where Ω(t|X,Y ) is the so called transport function which fulfills
?
lim
t↓0Ω(t|X,Y ) = 1 ,
∂t+X · ∂
t
+ D†
XDX
?
Ω(t|X,Y ) = 0 , (A.4)
(A.5)
in such a way that
lim
t↓0?X|exp{−tD†D}|Y ? = 1δ(n)(X − Y ) .(A.6)
If we insert Eq.(A.3) in Eq.(A.2) and change the integration variable we obtain
?X|ϑ(Q2− D†D)|Y ? = 2Qn(4π)−1−n/2
?c+i∞
c−i∞
dt t−1−n/2
× exp
?
t −Q2(X − Y )2
4t
−iπ
2
?
Ω
?
t
Q2|X,Y
?
.(A.7)
Now, if we write
Ω
?
t
Q2|X,Y
?
=
[n/2]
?
k=0
tkak(X,Y )Q−2k+ R[n/2]+1
?
t
Q2|X,Y
?
,(A.8)
it turns out that, by insertion of Eq.(A.8) into the integral (A.7), the last term in the RHS
of the above expression becomes sub-leading and negligible in the limit of very large Q. As
a consequence, the leading asymptotic behavior of the matrix element (A.2) in the large Q
limit reads [53]
?X|ϑ(Q2− D†D)|Y ?
Q→∞
∼ (4π)−n/2
[n/2]
?
dt
2πitk−1−n/2exp
k=0
ak(X,Y )Qn−2k
×
?c+i∞
c−i∞
?
t −Q2(X − Y )2
4t
?
X→Y
∼ (4π)−n/2
[n/2]
?
k=0
ak(X,Y )
Qn−2k
Γ(1 − k + n/2).(A.9)
– 26 –
Page 28
In the diagonal limit X = Y , for n = 4,5 the relevant coefficients of the heat kernel
asymptotic expansion take the form [54]
a0(X,X) ≡ 1 ;
a2(X,X) =1
a1(X,X) = −M2(X) ;
2[M2(X)]2−1
6∂2M2(X) ,(A.10)
in such a way that we can finally write the dominant diagonal matrix element – leading
eventually to the five-dimensional Euclidean effective Lagrange density – as
?X|ϑ(Q2− D†D)|X?
Q5
60π3
Q→∞
∼
?
1 −5
2Q−2M2(X) +15
8Q−4[M2(X)]2−5
8Q−4∂2M2(X)
?
. (A.11)
B. Spectral resolution for Schr¨ odinger operators
In this Appendix we calculate the spectra and eigenfunctions of the Schr¨ odinger operators
M11
Jarising in Eq.s (4.9) and (4.10) and presented in the factorized form in Eq.s
(4.12) and (4.13).Let us choose the inverse mass units, z = y/M and introduce the
dimensionless operators
Jand M22
M11
J ≡ M2d+
Φ≡ M2q+
H≡ M2q+
Φq−
Hq−
Φ,(B.1)
M22
J− M2+ 2∆2≡ M2d+
H, (B.2)
q±
Φ= ∓∂y+ 2tanh(y) ,
q±
H= ∓∂y+ tanh(y) .
(B.3)
(B.4)
The two operators d+
adds their supersymmetric partners
Φand d+
Hcan be embedded into the supersymmetric ladder when one
d−
Φ≡ q−
Φq+
Φ= d+
H+ 3 ,d−
H≡ q−
Hq+
H= −∂2
y+ 1 ,(B.5)
the latter one describing a free particle propagation. The intertwining Darboux relations
read
q±
q±
q−
d+
q−
d+
Φd∓
Hd∓
Hq−
Φq+
Hd+
Hq+
Φ= d±
H= d±
Φd+
Φq+
H= (−∂2
H= q+
Φq±
Hq±
Φ,
H,
H(d+
Φq+
y+ 1)q−
H(−∂2
Φ= q−
H= q+
H+ 3)q−
H(−∂2
H,
Φ= (−∂2
y+ 4) ,
y+ 4)q−
Hq−
Φ,
y+ 1) .(B.6)
From these relations one concludes that the spectra of the operators d+
equivalent one to each other and, in turn, both equivalent to the free particle continuous
spectrum – up to a shift of the origin. In other words, they consist of the continuous part
and of the zero-modes of the intertwining operators q−
Φand d+
Hare almost
Hq−
Φand q−
Hrespectively. Owing to
– 27 –
Page 29
Eq.s (B.6) their eigenvalues and normalized improper eigenfunctions are related as follows:
namely,
ψΦ(ky).=q+
√k2+ 4
ψH(ky).=q+
√k2+ 1
1
√2πexp(iky) ,
The imprecise equality.= signifies a possible discrepancy due to the existence of zero-modes.
In particular, for the operator d+
mode h0(y) =?1/√2?sech(y) just coincides with the zero-mode of q−
zero-mode φ0(y) =?√3/2?sech2(y) coincides with the zero-mode of q−
φ1(y) =
q+
Φh0(y) =
ΦψH(ky)
.=
q+
Φq+
Hψ0(ky)
?(k2+ 1)(k2+ 4)
Ec
,Ec
Φ(k) = k2+ 4 ,
Hψ0(ky)
,
H(k) = k2+ 1 ,
ψ0(ky) =
k ∈ R .(B.7)
Hthe continuum starts from Ec
H(0) = 1 and the only zero-
H. Concerning the
Φ(0) = 4, whereas the
Φ. Furthermore, the
operator d+
Φ, one can easily verify that the continuum starts from Ec
second parity-odd proper eigenstate
?
1/√3
?
?
3/2tanh(y)sech(y) , (B.8)
does coincide with the zero-mode of q−
discrete eigenvalue E+
Hq−
Φand corresponds to the first non-vanishing,
Φ= 3.
C. Perturbation theory for the first excited state
In this Appendix we develop the perturbation theory for the first excited eigenstate and the
corresponding eigenvalue of the operator MKdefined in Eq. (5.7). As in the Appendix B
the inverse mass units z = y/β are chosen, so that the dimensionless operator Ξ ≡ MK/β2
has the following components in terms of the operators in Eq.s (B.1)-(B.4): namely,
H
The second, perturbation term contains the elements of different order in ǫ. This is the
ultimate reason why, in order to compute the corrections to the eigenvalues, we have to take
into account not only the eigenfunction of zeroth-order, but also the first-order correction
turns out to be necessary. Thus we use the eigenfunction hK(y) = [h−
Eq.(5.12), taking into account that, to the lowest order, the parity-odd function h−
of order ǫ and the parity-even one is h+
in ǫ, the function h+
K(y) is derived from the following equation
Ξ =
d+
Φ0
0 d+
+ 4ǫsech2(y)
ǫsinh2(y)
√1 + ǫ2sinh(y)
√1 + ǫ2sinh(y)
ǫ
.(C.1)
K(y),h+
K(y)] from
K(y) is
K(y) ≃ h0(y) =?1/√2?sech(y). To the first order
K(y) = −4ǫtanh(y)sech(y)h0(y) = −4ǫ
ǫ
√2q+
which follows from the basic definition ΞhK= EhhKand the estimation of Eh∼ ǫ2. Since
there are no normalizable zero-modes for the operator q+
d+
Φh−
√2tanh(y)sech2(y)
= −
Φsech2(y) ,(C.2)
Φone concludes that
q−
Φh−
K(y) = −ǫ
√2sech2(y) ,(C.3)
– 28 –
Page 30
which represents a first-order differential equation with the normalizable solution
h−
K(y) = −ǫ
√2y sech2(y) .(C.4)
Now, the eigenvalue Ehcan be obtained from its integral representation Eh= ?hK|Ξ|hK?
in terms of the normalizable function ?hK|hK? = 1. To the first order in ǫ it reads eventually
?+∞
Eh= 4
−∞
dy h0(y)?ǫ2sech2(y)h0(y) + ǫtanh(y)sech(y)h−
D. Perturbation theory in the presence of defects
K(y)?= 2ǫ2. (C.5)
The inclusion of a small regular defect is described by the scalar functions fΦ(z),fH(z) and
leads to the change of the mass matrix MKpresented in Eq. (6.16) for the ansatz (A) and in
Eq. (6.29) for the ansatz (B). If we introduce the dimensionless operators Ξj,
then according to the notations of Eq. (C.1) we obtain
j = A,B ,
Ξj= ΞK+ ∆j,
∆11
∆12
∆22
∆11
∆12
∆22
(D.1)
A≈ ǫ2sech2(y)[4ξ + 2ξ2− 12y tanh(y)] ,
A= ∆21
A≈ ǫ2sech2(y)[12ξ + 6ξ2− 4y tanh(y)] ,
B≈ ǫ2sech2(y)?4ξ + 2ξ2+ 3(γ − 4ξ − 2ξ2)y tanh(y) + 3γsinh2(y)?
B≈ ǫ2sech2(y)?12ξ + 6ξ2+ (γ − 4ξ − 2ξ2)y tanh(y) + γsinh2(y)?
One can see that again the perturbation matrices contain elements of different order in ǫ and
therefore the calculation of energy levels needs first to derive the perturbed eigenfunctions
to a relevant order in ǫ as it has been done in the Appendix C.
To the relevant order in ǫ, the eigenvalue equations for the scalar wave function com-
ponents s+
A(y) follow from the matrix elements of the operator in Eq. (D.1),
?
+4ǫ (1 + ξ) sech2(y) sinh(y) s−
?
+4ǫ (1 + ξ) sech2(y) sinh(y) s+
A≈ 4ǫξ sech2(y)sinh(y) ,(D.2)
,
B≈ ∆21
B= 4ξǫsech2(y)sinh(y) ,(D.3)
,
A(y),s−
(d+
Φ− Es) s+
j(y) +4ǫ2tanh2(y) + ∆11
j
?
?
s+
j(y)
j(y) = 0 ,
(d+
H− Es) s−
j(y) +4ǫ2sech2(y) + ∆22
j
s−
j(y)
j(y) = 0 .(D.4)
These equations have terms of different order in ǫ in two sectors of solutions.
Actually, for even-odd solutions with s±
φ+(0)+ ǫ2φ+(1)
j
+ O(ǫ4) contains the zero mode (5.10) of the operator d+
term. Meanwhile, the lower odd component φ−
unperturbed one – see Eq. (5.10) in its functional form. Indeed, as one expects that the
lightest eigenvalues are of the order Es ∼ ǫ2, then the dominating part of the second
Eq. (D.4) reads,
d+
j
+ 4ǫ (1 + ξ) tanh(y)sech(y)φ+(0)= 0.
j(y) = φ±
j(y), the upper even component φ+
j=
Φas a leading
jhas an order of ǫ and coincides with the
Hφ−(0)
(D.5)
– 29 –
Page 31
Comparing this equation with the unperturbed one, corresponding to ξ = 0, from Eq. (5.10)
one immediately finds that
?
4
On the other hand, as d+
contribution φ+(1)
j
into the even part of the wave function. Now this equation allows us to
obtain both the approximate eigenvalue and the correction term φ+(1)
onto the zero-mode φ+(0)
j
of the operator d+
and calculates eventually the branon mass: namely,
?+∞
+4ǫ(1 + ξ) sech2(y) sinh(y)φ−(0)(y)φ+(0)(y)
φ−(0)
j
(y) = −
3β
ǫ (1 + ξ) sinh(y)sech2(y) ≡ φ−(0)(y) . (D.6)
Φφ+(0)= 0 the first Eq. (D.4) determines the next-to-leading
j
. After projection
Φ, one gets rid of the unknown function φ+(1)
j
?
mj
φ
?2
= β2Eφ= β2
−∞
dy
??
4ǫ2tanh2(y) + ∆11
j
??
φ+(0)(y)
?
?2
. (D.7)
In the sector of odd-even solutions s±
opposite order in ǫ, i.e.,
j(y) = h∓
j(y) the lightest-state wave function has the
h−(0)
j
(y) ∼ ǫ, h+
j(y) = h+(0)+ ǫ2h+(1)
j
+ O(ǫ4) .
Thus the role of two Eq.s (D.4) is merely interchanged. In particular, it can be easily
derived that
(y) = − ǫ(1 + ξ) y sech2(y)/√2 ≡ h−(0)(y) ,
h+(0)(y) = sech(y)/√2 ,
h−(0)
j
(D.8)
to be compared with Eq. (5.12). Respectively, when we project onto the zero-mode h+(0)
of the operator d+
H, the second equation helps us to obtain the Higgs boson mass,
?+∞
+4ǫ(1 + ξ) sech2(y)sinh(y)h−(0)(y)h+(0)(y)
j
(y)
?
mj
h
?2
= β2Eh= β2
−∞
dy
??
4ǫ2sech2(y) + ∆22
j
??
h+(0)(y)
?
?2
. (D.9)
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