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.

Page 2

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 –

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