Resonance spectrum of a bulk fermion on branes

1.4 MB·

by  Yu-Peng Zhang

Resonance spectrum of a bulk fermion on branes
Yu-Peng Zhang∗, Yun-Zhi Du†, Wen-Di Guo‡, Yu-Xiao Liu §,
Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, China
It is known that there are two mechanisms for localizing a bulk fermion on a brane, one is the
well-known Yukawa coupling and the other is the new coupling proposed in [Phys. Rev. D 89,
086001 (2014)]. Inthispaper,weinvestigatelocalization andresonancespectrumofabulkfermion
onthesamebraneswiththetwolocalizationmechanisms. Itisfoundthatboththetwomechanisms
can result in a volcano-like effective potential of the fermion Kaluza-Klein modes. The left-chiral
fermion zero mode can be localized on the brane and there exist some discrete massive fermion
6
1 Kaluza-Kleinmodesthatquasilocalized onthebrane(alsocalled fermionresonances). Thenumber
0
of thefermion resonances increases linearly with thecoupling parameter.
2

r PACSnumbers: 04.50.-h,11.27.+d
a
M
I. INTRODUCTION of variousbulk matter fields on a brane have been inves-
0
tigated in five-dimensional models [1, 22, 26, 32–54] and
3
six-dimensional ones [55–60].

] It is well known that the Standard Model of parti- Sincetheelementarymattersconsistoffermions,local-
h clesandfieldsisnotsufficientforinterpretingsomeopen izationandresonancesofspin1/2fermionsareimportant
-t questions suchas the originofthe darkmatter,the huge inbranetheories. Inordertolocalizefermionsonathick
p hierarchy between the weak and Planck scales, and the brane of the RS-type, one usually needs to introduce
e cosmological constant problem. Recently, there were some other interactions with background fields besides
h
[ many models that interpret the hierarchy problem [1– gravity. One simple interaction in a five-dimensional
7], cosmological constant problem [8–14], and the dark brane model is the Yukawa coupling between fermions
2 matter [15–17] due to the existence of extra dimensions. and the background scalar fields [32, 43, 44, 50, 54, 61–
v
The extra dimension theory was proposed in the 1920s 72] when the scalar fields are odd functions of the extra
2
5 by Kaluza and Klein (KK), who tried to unify the elec- dimension. Withthiscoupling,theshapesoftheeffective
8 tromagnetism and Einstein’s gravity by constructing a potentialoftheleft-orright-chiralfermionKKmodecan
5 gravitationaltheoryinafive-dimensionalspacetimewith be classified as three types: volcano-like [43, 63, 65–69],
0 acompactextradimension[18–20]. Later,theKKtheory finite square well-like [39, 73], and harmonic potential-
1. wasdevelopedasbraneworldtheories[1,8,21,22],where like [44, 70]. Correspondingly, the spectra of the KK
0 our universe is considered as a hypersurface (brane) em- fermions are continuous, partially discrete and partially
6 bedded in the higher-dimensional spacetime. continuous, and discrete. For the last two types of effec-
1 tivepotentials,thefermionzeromodecanbelocalizedon
Inbraneworldscenarios,theparticlesandfieldsinthe
:
v StandardModel should be confined on the brane. While the brane without further condition. While for the first
i one, the localization of the fermion zero mode usually
X the massive resonant KK modes (new particles beyond
needs that the Yukawacoupling is largerthan some crit-
the Standard Model) can propagate into extra dimen-
r ical coupling (η >η ). In all cases, the localized fermion
a sions, which gives us the possibility of probing the extra c
zero mode is always chiral.
dimensions through their interaction with particles on
However,ifthebackgroundscalarfieldisanevenfunc-
thebrane[23–25]. Theappearanceoftheseresonancesis
tionoftheextradimension,theYukawacouplingmecha-
related to the structure of the brane and the interaction
nismwilldonotwork,sincetheZ reflectionsymmetryof
between the bulk fields and the background fields gen- 2
theeffectivepotentialsforthefermionKKmodescannot
erating the brane. Thus it is important and interesting
be ensured [74]. In this case, one should consider other
to study the localization mechanisms and resonant KK
mechanism. Recently,Liu,Xu, Chen,andWeipresented
modes on thick branes with different internal structures.
a new localization mechanism (shorted for the LXCW
Usually, a thick brane can be generated by scalar fields
localization mechanism) for localizing bulk fermions on
[26–30] or vector fields [31]. Localizationand resonances
the brane generated by an even scalar field φ [74]. The
coupling is givenby ηΨ¯γ ΓM∂ F(φ)Ψ, whichis usedto
5 M
describe the interaction between π-meson and nucleons
∗[email protected] inquantumfieldtheory. Therelatedstudycanbeseenin
†[email protected] Refs. [75,76]. Interestingly,thisnewlocalizationmecha-
‡[email protected] nismcanalsobeusedforthecaseofthebranegenerated
§[email protected],correspondingauthor by odd scalar fields.
2
Since there aretwolocalizationmechanismsfor abulk EM EN ηM¯N¯ =gMN. Therelationbetweenthegamma
M¯ N¯
fermion, an interesting issue is whether the two mech- matrices (ΓM) in a five-dimensional curved spacetime
anisms give similar results of the fermion localization. andtheMinkowskianones ΓM¯ =(Γµ¯,Γ¯5)=(γµ¯,γ5) is
Therefore, the goal of this paper is to investigate local- given by ΓM =EM ΓM¯.
izationofabulkfermiononasamebranewiththeabove M¯ (cid:0) (cid:1)
The metric of the five-dimensional spacetime describ-
two localizationmechanisms that can yieldthe following
ing a static braneworld system is given by [1]
results: (1) The fermion zero mode can be localized on
thebrane. (2)Theeffectivepotentialoftheleft-orright- ds2 = g dxMdxN
MN
chiral fermion KK mode has the shape of volcano-like. = e2A(y)gˆ (xλ)dxµdxν +dy2, (4)
µν
(3) There exist fermion resonances that quasilocalized
on the brane. To this end, we will consider two kinds of whereystandsfortheextradimension,e2A(y)isthewarp
thick branes generated by one or at least one odd scalar factor,andgˆµν istheinducedfour-dimensionalspacetime
field. metric onthe brane. Here, both the warpfactor and the
Thispaperisorganizedasfollows. InSec. IIwereview scalar fields are supposed to be functions of y only, i. e.
the localization mechanisms and give the corresponding , A = A(y), φ = φ(y), χ = χ(y), , ρ = ρ(y). By
···
effective potentials ofthe fermionKK modes andthe so- performingthecoordinatetransformationdy =eAdz,the
lutions of the fermion zero mode. In Sec. III we investi- line-element (4) can also be rewritten as
gatelocalizationandresonancesofabulkfermionontwo
ds2 =e2A(z)(gˆ dxµdxν +dz2), (5)
kindsofthickbranes: asingle-scalar-field-generatedthick µν
braneandamulti-scalar-field-generatedone. We makea which is a conformal flat metric if gˆ = η . For the
µν µν
simple comparison of results with the two different cou- brane metric (5), non-vanishing components of the spin
pling mechanisms. Finally, we give a brief conclusion in connection (2) are ω = 1∂ Aγ γ + ωˆ , where ωˆ is
µ 2 z µ 5 µ µ
Sec. IV. derived from the four-dimensional metric gˆ (xλ).
µν
From the conformal metric (5) and expression of the
spin connection, the equation of motion of the five-
II. REVIEW OF LOCALIZATION MECHANISM dimensional Dirac fermion Ψ reads
γµ(∂ +ωˆ )+γ5(∂ +2∂ A)+η∂ F Ψ=0, (6)
In this section, we review the LXCW localization µ µ z z z 1
mechanism of a bulk fermion on a brane in five- whehre γµ(∂ +ωˆ ) is the dirac operator ion the brane.
µ µ
dimensional spacetime, which is realized by introducing While for the Yukawa coupling ηΨ¯F (φ,χ, )Ψ, the
2
the coupling between the fermion field and the back- − ···
corresponding Dirac equation is given by [44]
ground scalar fields generating the brane. The corre-
sponding five-dimensional action is given by [74] γµ(∂ +ωˆ )+γ5(∂ +2∂ A)+ηeAF Ψ=0. (7)
µ µ z z 2
h i
S = d5x√ g Ψ¯ΓM(∂ +ω )Ψ Next,wemakethefollowingchiraldecompositionforthe
1 M M
2 − five-dimensional Dirac field Ψ
Z h
+ηΨ¯ΓM∂MF1(φ,χ, ,ρ)γ5Ψ , (1) Ψ(x,z)=e−2A ψ (x)f (z)+ψ (x)f (z) ,
··· Ln Ln Rn Rn
where F1(φ,χ, ,ρ) is a function of muiltiple scalar Xn h i(8)
···
fields φ, χ, , ρ, and η is the coupling constant. A where ψ = γ5ψ and ψ = γ5ψ are the left-
five-dimensio·n··al Dirac fermion field is a four-component and righLt-nchira−l comLpnonents oRfnthe four-Rdnimensional ef-
spinorandthecorrespondinggammamatricesΓM satisfy fectiveDiracfermionfield,respectively. Substituting the
ΓM,ΓN = 2gMN, where the five-dimensional space- chiral decomposition (8) into Eq. (6), we get the four-
{ }
time indices are denoted by M, N, = 0, 1, 2, 3, 5. dimensional massive Dirac equations
···
The spin connection ω is defined as
M γµ(∂ +ωˆ )ψ (x)=µ ψ (x),
µ µ Ln n Rn (9)
ωM = 41ωMM¯N¯ΓM¯ΓN¯, (2) γµ(∂µ+ωˆµ)ψRn(x)=µnψLn(x),
and the following coupling equations of the KK modes
where f
Ln,Rn
1
ωMM¯N¯ = 2ENM¯(∂MENN¯ −∂NEMN¯) (∂z −η∂zF)fLn =+µnfRn, (10)
(∂ +η∂ F)f = µ f ,
z z Rn n Ln
1ENN¯(∂ E M¯ ∂ E M¯) −
−2 M N − N M where µn is the mass of the four-dimensional fermion
−21EPM¯EQN¯EMR¯(∂PEQR¯ −∂QEPR¯).(3) rfieewldrsittψeLnna,Rsnt(hxe).ScThhr¨oedainbgoevre-ltikweoeeqquuaatitoionnss can also be
Hloecrael LthoerenlettzteirnsdiMc¯es, aN¯n,d··t·hearveietlhbeeinfivEe-Mdimesnastiiosfineasl [[−∂∂2z2++VVL((zz))]]ffLn == µµ22nffLn, ((1121))
M¯ − z R Rn n Rn
3
with the effective potentials where z = 10z and the parameter z could be cho-
max b b
senasthecoordinateofthemaximumofthecorrespond-
VL,R(z)=(η∂zF1)2 ∂z(η∂zF1). (13) ingeffective potential. Since the potentialsconsideredin
±
this paper are symmetric, the wave functions are either
Note that the Schr¨odinger-like equations (11) and (12)
even or odd. Hence, we can use the following boundary
can be transformed as
conditions to solve the differential equation (11) numer-
U†Uf =µ2f ically [66]:
Ln n Ln (14)
UU†f =µ2f
Rn n Rn f (0) = 0, f′ (0)=1, odd KK modes,(21)
Ln,Rn Ln,Rn
wmiatshstshqeuaorpeerisatnoornU-n=ega∂tziv−e,ηi∂.ez.F, ,µw2 hic0h.insure that the fLn,Rn(0) = 1, fL′n,Rn(0)=0, even KK modes.(22)
For the Yukawa coupling ηF (φn,≥χ, )Ψ¯Ψ, the cor-
2 Nextwe will investigatelocalizationandresonancesof
− ···
responding effective potentials in the Schr¨odinger-like
the fermions in two thick braneworld models.
equations are [44]
V (z)=(ηeAF )2 ∂ (ηeAF ). (15)
L,R 2 z 2
± III. FERMION LOCALIZATION AND
Inordertoobtainthefour-dimensionaleffectiveaction RESONANCES
for the massless and massive Dirac fermions
S = d4x gˆ ψ¯ γµ(∂ +ω ) µ ψ , (16) A. Single-scalar-field thick brane
eff n µ µ n n
− −
n Z
X p (cid:2) (cid:3) In this subsection, we investigate localization and res-
the KK modes f should satisfy the orthonormality
Ln,Rn onances of a bulk fermion on a single-scalar-field thick
conditions
brane in a five-dimensional spacetime [28, 39]. The ac-
+∞ +∞ tion of this system reads
f f dz = f f dz =δ ,
Lm Ln Rm Rn mn
Z−+∞∞ Z−∞ S = d5x√ g M53R 1∂Mφ∂Mφ V(φ) , (23)
− 4 − 2 −
fLnfRndz = 0, (17) Z (cid:20) (cid:21)
Z−∞ where R is the five-dimensionalscalar curvature and the
which are important to check whether the KK modes fundamental mass scale M will be set to 1 for conve-
5
can be localizedon the brane. From Eq. (10), the corre- nience.
sponding chiral zero-modes read The line element (4) can be rewritten as
fL0,R0 ∝e±ηRdz ∂zF1 =e±ηF1. (18) ds25 =e2Ads24+dy2, (24)
While for the Yukawa coupling mechanism, the zero- where ds2 stands for the line element on the brane. For
4
modes are the warped thick branes, ds2 has the following forms
4
f e±ηRdz eAF2(z) =e±ηRdy F2(y). (19)
L0,R0 ∝ ds2 = −dt2+e2βt(dx21+dx22+dx23) dS4 brane, (25)
In the following we will use the subscripts “new” and 4 (cid:26)e−2βx3(−dt2+dx21+dx22)+dx23 AdS4 brane.
“Yuk”forthe LXCWandYukawacouplingmechanisms,
Here β is a parameter related to the four-dimensional
respectively.
cosmological constant of the dS or AdS brane: Λ =
The massive KK modes can be obtained by solving 4 4 4
3β2 or Λ = 3β2 [70, 77].
Eqs. (11) numerically. Inspired by the results of Ref. 4 −
[44], Almeida ea al investigated the issue of localization A brane solution was found in Ref. [28]:
of a bulk fermion on a brane and firstly suggested that
3
largepeaksinthe distributionofthenormalizedsquared V(φ) = a2(1+Λ ) 1+(1+3s)Λ cosh2(bφ)
4 4
4
wavefunction f (0)2 as a function of m would reveal
the existence |ofLf,Rermi|on resonant states [65]. However, 3a2(1+Λ4)2(cid:2)sinh2(bφ), (cid:3) (26)
−
this method is effective only for evenfermion resonances 1
φ(y) = arcsinh(tany¯), (27)
because f (0)= 0 for any odd wavefunction. In order b
L,R
to find all fermion resonances,Liu et al presented a new 1
A(y) = ln[sa2(1+Λ )sec2y¯], (28)
method called relative probability, which is defined as −2 4
[66]:
where y¯ a(1+Λ )y, the parameter s (0,1], a is a
4
P = zR−m−zzambzxbax|f|LfnL,nR,nR(nz()z|)2|d2zdz, (20) rtoeaslepeatrha≡amtettehre, tahnidckbb=ranqe i3s(12e+(x1(1t++eΛns4)d)Λe4d).∈inItthies oinbtveirovuasl
R
4
y π , π . The relation between the
∈ −|2a(1+Λ4)| |2a(1+Λ4)|
conf(cid:16)ormal and physical coord(cid:17)inates reads
y 1 cos 1y¯ +sin 1y¯
z = e−A(yˆ)dyˆ= ln 2 2 , (29)
Z0 h cos(cid:0)21y¯(cid:1)−sin(cid:0)12y¯(cid:1)!
(cid:0) (cid:1) (cid:0) (cid:1)
from which we have
FIG.1: Plotsoftheeffectivepotentials(32)withtheYukawa
1 π coupling (F2 = arcsinh2q−1(bφ)) for different values of Λ4.
y = 2arctan ehz . (30)
a(1+Λ4) − 2 The parameters are set to a = s = q = −η = 1 and Λ4 =
h (cid:0) (cid:1) i 0.2 (thickred line), 0 (black line), −0.2 (thin blue line).
where h 1+Λ4. Note that the conformal coordinate
≡ s
z will trendqto infinite as y π .
→|2a(1+Λ4)|
We can substitute the relation (30) into Eqs. (28) and
(26) to obtain the corresponding warp factor and scalar
field in the coordinate z [39]:
1 h
φ(z) = arcsinh[sinh(hz)]= z,
b b
FIG.2: Plotsoftheeffectivepotentials(32)withtheYukawa
1
A(z) = −2ln a2s(1+Λ4)cosh2(hz) . (31) coupling(F2 =arcsinh2q−1(bφ))fordifferentvaluesofq. The
(cid:2) (cid:3) parameters are set to a = s = q = −η = 1, Λ4 = 0, and
Note that Λ =0 corresponds to the flat brane case.
4 q=1 (thick black line), 2 (red line), 3 (thin blue line).
1. Yukawa coupling mechanism potentials for different values of Λ4 and q are shown in
Figs. 1 and 2, respectively.
In this subsection we mainly consider localization and
Firstly, we consider the Yukawa coupling mechanism
resonances of a bulk fermion on the flat brane (Λ = 0)
forthefermionlocalizationonthesingle-scalar-fieldthick 4
withdifferentvaluesofthepositiveintegerq(corresponds
brane. The authors of Ref. [39] investigated localization
todifferentcouplingfunctions). Forsimplicity,wechoose
of fermion on the brane with the forms of F = φ and
2
a=s=1, for which y¯ ay(1+Λ )=y. For q =1, the
F = sinh(bφ), for which the effective potentials are re- 4
2 ≡
effective potentials have the form
spectively volcano-type and PT-type, but they did not
consider fermion resonances. Here, we would like to in- 1
vestigatefermionresonanceson the brane. The coupling VLY,uRk(z) = ± η sech2z √1+z2 ±η arcsinh2z
function can be chosen as F = λ φ + λ sinh(bφ) + (cid:18)
2 1 2
λ3arcsinh2q−1(bφ) with a structure parameter q (posi- arcsinhzsinhz . (34)
tive integer). For simplicity, we only consider the case −
(cid:19)
of λ = λ = 0, λ = 1. From Eqs. (15) and (19), the
1 2 3 The behaviors of the effective potentials (34) for the
effective potentials and zero modes are given by
Yukawa coupling mechanism are
VYuk(z) = η sech(hz)arcsinh2q−2(hz) (2q−1)h VLY,uRk(0) = ±η, (35)
L,R ± a2s(1+Λ4) b√1+h2z2 VLY,uRk(z →∞) → ∓4 η e−zln(2z)→0±, (36)
η arcsinph2q(hz)sech(hz) where we have considered η < 0 in the formula (36) in
order to localize the left-chiral fermion zero mode. The
± a2s(1+Λ )
4 form of F =arcsinh2q−1(bφ) is
2
p
h arcsinh(hz)tanh(hz) , (32) F (y) = arcsinhz. (37)
2
− !
The zero modes read
and arcsinh(z)
fYuk (z) exp η dz . (38)
L0,R0 ∝ ± cosh(z)
arcsinh2q−1(hz) (cid:20) Z (cid:21)
fYuk (z) exp η dz .
L0,R0 ∝ "∓ Z a2s(1+Λ4)cosh(hz) # Itnheofrodlelorwtiongobntoarimnatlhizeafteiromnicoonndzeitrioonmsosdheosulodnbtehseabtirsafineed,
(33)
p ∞
Notethattheparametersη, q, aandΛ affecttheheight
4 f 2dz < . (39)
andwidthoftheeffectivepotentials. Plotsoftheeffective | L0,R0| ∞
Z−∞
5
Because the integral arcsinh(z)dz 2ln(2z)e−zdz = with τ = 1 [36, 65]. Here, we only list the lifetimes and
cosh(z) → Γ
2Ei( z) 2e−zln(2z) 0, where the function Ei(z) = mass spectrum of the resonances for the case of q = 2
∞−e−−t/tdtandEiR(→ )=0,thezeRromodestrendto in Table I. The corresponding resonances for the case of
a−co−nzstant at infinite.−S∞o the above normalization con- q =1 and η = 7 is shown in Fig. 6.
−
R
ditions can not be satisfied for both the two zero modes,
which means that the zero modes can not be localized
on the brane for the case of λ = λ = 0. So, in order
1 2
to localize the left- or right-chiral fermion zero mode on
the brane, we need to keep λ or λ , or both or them.
1 2
However, since our interesting is fermion resonances, we
can neglect them for simplicity. Surely, if consider non-
vanishing λ or λ , the following numerical results will
1 2 (a)η=−3 (b)η=−3
be different, but our conclusion will not change.
We canobtainthe solutionofthe massivefermionKK
modes by solving numerically the Schr¨odinger-likeequa-
tions (11) and (12) with the potentials (32). First, we
choose q = 1, 2 and η = 3, 5, 7, and the corre-
− − −
spondingplotsofthepotentialsareshowninFigs. 3and
4. From these figures, we can see that the depth of the
quasi-well of the effective potential V (z) increases with
R
the parameters η and q. This appearance of the quasi- (c)η=−5 (d)η=−5
| |
wellwouldresultinquasi-localizedmassiveKKfermions
(also called fermions resonances).
(e)η=−7 (f)η=−7
FIG. 5: Plots of the probabilities PL,R (as a function of m2)
for the Yukawa coupling mechanism, which include the even
FIG.3: PlotsoftheeffectivepotentialsVL,R(z)(32)withthe
parity(bluedashedline)andoddparity(redline)massiveKK
Yukawa coupling for q = 1 and different values of η. The
modes of the left-chiral (up channel) and right-chiral (down
parameters are set to a = s= 1, Λ4 = 0, and η = −3 (thick
channel) fermions. The parameters are set to a = s = 1,
black line), −5 (red line), −7 (thin blueline).
q=1, and Λ4 =0.
FIG.4: Plotsoftheeffectivepotentials(32)withtheYukawa
(a)fL(z) (b)fR(z)
coupling for q =2 and different values of η. The parameters
FIG. 6: Plots of the resonances of the left- and right-chiral
are set to a = s = 1, Λ4 = 0, and η = −3 (thick black KKfermionsfortheYukawacouplingmechanismwithη=−7
line), −5 (red line), −7(thin blue line).
and q=1.
We find that, if we fix a = s = 1, Λ = 0, q = 1
4
or q = 2, then there are no resonances for the case of
η = 1. So we only consider the cases of q = 1, 2 and
η = −3, 5, 7. Plots of the relative probability (20) 2. LXCW coupling mechanism
− − −
are shownin Figs. 5 and 7, which show that the number
oftheresonantKKfermionsincreaseswithqand η . We In this subsection, we consider the LXCW coupling
| |
can obtain the corresponding lifetimes τ of the fermion mechanism reviewed in Sec. II. Since the coupling func-
resonancesbythewidth(Γ)athalfmaximumofthepeak tion F should be an even function of the extra dimen-
1
6
η chirality parity m2n mn Γ τ P
odd 2.2336 1.4945 1.244×10−2 80.36 0.600
L even 5.5950 2.3654 0.156 6.409 0.210
-3
even 2.2336 1.4945 1.261×10−2 79.29 0.927
R odd 5.5825 2.3622 0.176 5.675 0.396
ood 3.2302 1.7973 2.058×10−4 4857 0.811
L even 9.0201 3.0033 7.491×10−3 133.5 0.552
odd 13.865 3.7236 0.0762 13.13 0.276
-5
even 3.2302 1.7973 1.975×10−4 5063 0.981
R odd 9.0200 3.0033 7.491×10−3 133.5 0.835
even 13.8700 3.7242 0.0751 13.31 0.427
odd 4.0289 2.0072 1.171×10−6 8.541×105 0.906
L even 11.6927 3.4195 1.535×10−4 6513 0.740
odd 19.0650 4.3664 3.264×10−3 306.4 0.497
even 26.0660 5.1059 0.0352 28.37 0.340
-7
even 4.0289 2.0072 1.196×10−6 8.363×105 0.994
R odd 11.6927 3.4195 1.550×10−4 6452 0.938
even 19.0650 4.3664 3.206×10−3 311.9 0.750
odd 26.0670 5.1056 0.0351 28.53 0.429
TABLEI:Themassspectrumm2n andmn,width(Γ),lifetime(τ),andrelativeprobability(P)oftheleft-andright-chiralKK
fermion resonances for the Yukawacoupling. The parameters are set to q =2, a=s=1, Λ4 =0. The symbols L and R are
short for left-chiral and right-chiral, respectively.
sion,wechooseF asF =arcsinh2p(bφ)withastructure in the single-scalar-field thick brane model do not affect
1 1
parameterp(positive integer). Thenthe effective poten- the structures ofthe effective potentials (40), and only η
tials are given by and p have affect. Plots of the effective potentials with
differentvaluesofcouplingparameterspandηareshown
in Figs. 8 and9, respectively. It can be seenfrom Fig. 8
2pηh2arcsinh2p−2(hz)
VLn,eRw(z)=± (1+h2z2)3/2 tthhaetptohteernetiwailllbaaprrpieearsrafoqruVasi-wwelilllfoinrcVrReawseheqnuipck≥ly2wainthd
L,R
the parameter p. The large coupling ( η) also results in
(2p 1) 1+h2z2 hz arcsinh(hz) −
× − − a quasi-well for VR and large potential barriers for VL,R
(cid:18)
p (see Fig. 9).
2pη 1+h2z2 arcsinh2p(hz) . (40)
± The solution of the zero modes are
(cid:19)
p
By defining z¯ = hz, Vnew = h−2Vnew, and µ¯ = µ/h,
L,R L,R
the Schr¨odinger-like equations (11) and (12) become fnew (z) = N exp( ηF )
L0,R0 L,R ± 1
−∂z¯2+VL,R(z¯) fLn,Rn(z¯)=µ¯2nfLn,Rn(z¯). (41) = NL,Rexp ±η arcsinh2p(hz) , (44)
There(cid:2)fore, the role of(cid:3)the parameter h is just a rescaling (cid:0) (cid:1)
of the mass spectrum. Thus, we take h = 1 without where N are normalization constants. Since
L,R
lossofgenerality. Theasymptoticbehaviorsoftheabove arcsinh2p(z) at z , in order to localize the
effective potentials are described as follows massless left-→chir∞al fermi→on±on∞the brane, the parameter
η should be negative, for which the massless right-chiral
2η, p=1
Vnew(0) = ± (42) fermion can not be localized. The normalization condi-
L,R ( 0, p 2 tion (17) for the massless left-chiral mode is
≥
Vnew( ) 0. (43)
L,R ∞ →
Now it is clear that for the LXCW coupling mechanism ∞
f (z)2dz < (45)
with F1 = arctan2p(bφ), the parameters a, b, s, and Λ4 Z0 | L0 | ∞
7
PL PR
0.5 0.8
0.4 0.6
0.3
0.4
0.2
0.1 0.2
m2 m2
2 4 6 8 10 2 4 6 8 10
(a)η=−3 (b)η=−3
(a)p=1 (b)p=1
PL PR
0.8 1.0
0.8
0.6
0.6
0.4
0.4
0.2 0.2
m2 m2
5 10 15 20 5 10 15 20
(c)η=−5 (d)η=−5
(c)p=2 (d)p=2
PL PR
1.0
0.8
0.8
0.6
0.6
0.4 0.4
0.2 0.2
m2 m2
10 20 30 40 10 20 30 40
(e)η=−7 (f)η=−7
FIG. 7: Plots of the probabilities PL,R for the Yukawa cou- (e)p=3 (f)p=3
plingmechanism,whichincludetheevenparity(bluedashed FIG.8: Plotsoftheeffectivepotentials(40)withtheLXCW
line) andoddparity(redline) massiveKKmodesoftheleft- coupling for different valuesof p and η=−1.
chiral(upchannel)andright-chiral(downchannel)fermions.
The parameters are set toa=s=1, q=2, and Λ4 =0.
is satisfied because
f (z )2 N exp 2η ln2p(hz)
L0 L
| →∞ | →
1(cid:2) −2ηln2p−1(h(cid:3)z)
= N
L
hz
(cid:18) (cid:19)
1 +∞ FIG.9: Plotsoftheeffectivepotentials(40)withtheLXCW
→ NL hz . (46) couplingforp=1anddifferentvaluesofη(−1forthickblack
(cid:18) (cid:19)
line, −3 for red line, and −5for thin blueline).
Which means that we can get the four-dimensional
massless left-chiral fermion on the thick brane. From
the asymptotic behaviors (42) and (43), we require
Vnew(0) < 0 in order to localize the massless left-chiral
L
fermion. Therefore, the parameter η should be negative, TABLEII.Becausetheprecisionoftheprogramwolfram
which is consistent with the above result (45). mathematicaisnotenoughwedonotgivethelifetime of
Because the depth of the potentials increases quickly KK mode with m2 =27.996951884884and η = 5.
−
with the parameter p for fixed coupling constant η, the
numberofmassiveKKmodesarealsoincreasingquickly, Before closing this subsection, we give a brief sum-
inthe followingwe mainly considerthe effect ofη onthe mary. We have considered localization and resonances
massiveKKmodeswithp=1. Plotsoftherelativeprob- of left- and right-chiral fermions on a single-scalar-field
ability with different values of η and the corresponding thick brane with two kinds of scalar-fermion couplings.
wave functions with η = 5 are shown in Figs. 10 and Themassspectraandlifetimesofthefermionresonances
−
11. Furthermore,weobtainthecorrespondingmassspec- for both the left- and right-chiralfermions are the same.
trum of KK modes, and the lifetimes τ of fermion reso- The number of the fermion resonances increases with
nances can be calculated by the width (Γ) at half maxi- the scalar-fermion coupling parameter η and the brane
mum of the peak with τ = 1 [36, 65], which are listed in structure parameter. | |
Γ
8
η chirality parity m2n mn Γ τ P
odd 9.9697 3.1575 1.172×10−3 853.37 0.948
L even 15.6585 3.9573 0.150 6.684 0.373
-3
even 9.9697 3.1575 1.172×10−3 853.37 0.987
R odd 15.6327 3.9535 0.142 7.029 0.499
odd 17.9915 4.2416 9.666×10−9 1.035×108 0.999
L even 31.9057 5.6485 2.656×10−4 3765 0.961
odd 41.3497 6.4304 0.050 20.10 0.605
-5
even 17.9915 4.2416 9.548×10−9 1.047×108 0.999
R odd 31.9057 5.6485 4.957×10−4 2017 0.987
even 41.3471 6.4302 0.042 23.83 0.721
TABLE II: The mass spectrum m2n and mn, width (Γ), lifetime (τ), and relative probability (P) of the left- and right-chiral
KK fermion resonances for theLXCW coupling. The parameters are set to p=1, a=s=1, and Λ4 =0.
(a)η=−1 (b)η=−1
(a)fL1(z) (b)fR1(z)
(c)η=−3 (d)η=−3
(c)fL2(z) (d)fR2(z)
(e)η=−5 (f)η=−5
FIG. 10: Plots of the probabilities PL,R for the LXCW cou- (e)fL3(z) (f)fR3(z)
plingmechanism,whichincludeevenparity(bluedashedline) FIG. 11: Plots of the resonances of the left- and right-chiral
andoddparity(redline)massiveKKmodesoftheleft-chiral KKfermionsfortheLXCWcouplingmechanismwithη=−5
(up channel) and right-chiral (down channel) fermions. The and p=1.
parameters are set to a=s=1, p=1, and Λ4 =0.
action for the brane system is given by [27, 67]
B. Multi-scalar-field flat thick brane
M3 1
In this subsection, we mainly investigate localization S = d4x dy√ g 5R ∂ φ∂Mφ
M
− 4 − 2
and resonances of fermion based on the new (LXCW) Z (cid:18)
coupling mechanism on a multi-scalar-field flat thick 1 1
∂ χ∂Mχ ∂ ρ∂Mρ V(φ,χ,ρ) . (47)
brane embedded in a five-dimensional spacetime. The − 2 M − 2 M −
(cid:19)
9
We also set M = 1. The scalar potential V is taken as wegetasingle-well(see Fig. 12). The appearanceofthe
5
the following form [27] well in V (y) is decided by the two parameters a and η.
R
When p 2, the effective potentials will vanish at y =0
V = 1 4a2φ2(ρ2+χ2)+ 1 φ2+a(ρ2+χ2) 2 and ther≥e are a double-well and a single-well for VL(y)
2 − and V (y), respectively (see Fig. 13). The appearance
(cid:20) (cid:21) R
4 1 (cid:2) 2 (cid:3) of the well in VR(y) may result in fermion resonances.
φ2 1 φ2 a(χ2+ρ2) . (48) The solution of the zero modes based on (18) are
− 3 − 3 −
(cid:20) (cid:21)
±2ηtanh2p(ky)
A brane solution was found in Refs. [27]: 1
f 2 sech(ky) . (53)
L0,R0
∝ ra − !
φ(y) = tanh(ky),
1 Since sech(ky) 2e−k|y| and tanh2p(ky) 1 as y
χ(y) = 2cos(θ)sech(ky), , the asympto→tic behaviors of the zero m→odes are| | →
ra − ∞
1 fL0,R0(y)y→∞ e∓2ηk|y|. (54)
ρ(y) = 2sin(θ)sech(ky), | ∝
a −
r Therefore, the normalization conditions are
1
A(y) = (1 3a)tanh2(ky) 2lncosh(ky) ,(49) ∞ ∞
9a − − f (z)2dz = f (y)2e−Ady
L0,R0 L0,R0
(cid:2) (cid:3) | | | |
where k = 2a, the domains of parameters a and θ are Z0 Z0
∞
0of<thaes<ca1la/r2φa(nyd)aθn∈dt[h0,e2lπum).pFcoorntfihgeurkaintikoncsoonffiχgu(yr)atainodn → Z0 e(94∓4ηk)|y|dy <∞,(55)
ρ(y), we choose the function F =φ2pln[χ2+ρ2], where
1 whichareturnedouttobeη > 1 andη < 1 forthe
p (positive integer) is the structure parameter. The cor- 18a −18a
left-andright-chiralfermionzeromodes,respectively. In
responding potentials (13) read
what follows, we take η > 1 in order to obtain local-
18a
VL,R = 8aη tanh2p−2(ky)sech4/9a(ky)e2(19−a3a)tanh2(ky) iozfedthleefzte-crhoirmalodfeersmwioitnhzderiffoemreondtevaolnuetsheofbaraanne.d pPlaortes
± 9
shown in Figs. 12 and 13, respectively. From Fig. 12, it
2tanh4(ky) 1 9aηtanh2p(ky) can be seen that the parameter a affects the localization
× ( ± position of the left-chiral zero mode: when a is small,
(cid:0) (cid:1) there is a large potential barrier aroundthe originof ex-
+ sech2(ky)tanh2(ky) 9(a+4ap)
tra dimension and the left-chiral zero mode is localized
−
2(1+9a) (y)+(6ah 2)tanh2(ky) at two sides; when a becomes large, there is a potential
− F − well around the origin and the left-chiral zero mode is
36aη (y)tanh2p(ky) localizedthere. From Fig. 13, one can see that there are
∓ F
two wells and one well for p 2 and p=1, respectively.
+ (y)sech4(ky) 9a(2pi 1)+(2 6a)tanh2(ky) But the left-chiral zero mode≥is always localized around
F − −
the origin of extra dimension.
h
18aη (y)tanh2p(ky) , (50) Next,wecometomassiveKKmodes. Sincethepoten-
± F ) tialsvanishattheboundariesofextradimension,thereis
i
noboundmassiveKKmodes. Butwecanalsousetherel-
where (y) pln (1−2a)sech2(ky) . Theasymptoticbe- ativeprobabilityto findthe fermionresonances. Plotsof
F ≡ a the relativeprobabilitiesP (20) areshowninFigs.14
L,R
haviors of the abo(cid:16)ve effective pote(cid:17)ntials are described as and 15, from which we can see that the number of the
follows
fermion resonances increasing with the coupling param-
eters η and p since they increase the quasi-potentialwell
8aηln(1−2a), p=1
VL,R(0) = ± a (51) ofVR. Massspectrum,width,lifetime,andrelativeprob-
( 0, p 2 abilityoftheleft-andright-chiralfermionresonancesare
≥
V ( ) 0. (52) listed in Table III for a=0.45, p=1, and η =3, 5. For
L,R
∞ → simplicityweonlygivethe fermionresonantstateinFig.
In the following discussion, we take positive coupling η 16 with the case of p=1 and η =5.
inordertolocalizetheleft-chiralfermionzeromode. We In this subsection, we considered localization and res-
can see that the parameter a will affect the asymptotic onances of a bulk fermion on a multi-scalar-field thick
behavior (51) if p = 1. In order to investigate the prop- brane with the LXCW coupling. The conclusion is simi-
ertyofthepotentialsaty =0,wecalculatethefirst-order lar to the case of the single-scalar-fieldthick brane. The
andsecond-orderderivativesofV (0)withrespecttoa. resonances of fermion based on the Yukawa coupling
L,R
Wefindthatwhentheparametera (0,0.412141),there mechanism had been investigated in Ref. [67]. The au-
is a double-well for V (y), while fo∈r a [0.412141,0.5) thorsconsideredthecouplingfunctionasF =φqχρwith
L 2
∈
10
η chirality parity m2n mn Γ τ P
L odd 13.2466 3.6396 2.20×10−3 454.9 0.996
even 22.8513 4.7803 9.62×10−2 10.38 0.661
3
R even 13.2465 3.6396 2.10×10−3 475.8 0.997
odd 22.8437 4.7795 9.95×10−2 10.05 0.720
odd 23.1095 4.8072 6.86×10−7 1.46×106 0.999
L even 43.3217 6.5819 6.08×10−4 1.65×103 0.998
odd 59.6198 7.7214 2.65×10−2 37.66 0.930
5
even 23.1085 4.8071 7.90×10−7 1.27×106 0.999
R odd 43.3215 6.5819 6.08×10−4 1.65×103 0.998
even 59.6165 7.7212 2.66×10−2 37.66 0.949
TABLE III: Mass spectrum m2n and mn, width (Γ), lifetime (τ), and relative probability (P) of the left- and right-chiral KK
fermion resonances. The parameters are set to a=0.45, p=1, and η=3, 5.
VLHyL,fL0HyL VRHyL,fR0HyL
3 3.0
2 2.5
2.0
1
1.5
y
-10 -5 5 10 1.0
-1 0.5
-2 y
-10 -5 5 10
(a)a=0.02 (b)a=0.02
(a)p=1 (b)p=1
VLHyL,fL0HyL VRHyL,fR0HyL
3 3.0
2.5
2
2.0
1 1.5
y 1.0
-10 -5 5 10
0.5
-1 y
-10 -5 5 10
(c)a=0.35 (d)a=0.35
(c)p=2 (d)p=2
VLHyL,fL0HyL VRHyL,fR0HyL
3 3.0
2.5
2
2.0
1 1.5
y 1.0
-10 -5 5 10
0.5
-1 y
-10 -5 5 10
(e)a=0.45 (f)a=0.45
(e)p=3 (f)p=3
FIG.12: Plotsoftheeffectivepotentials(blackline)andzero
FIG.13: Plotsoftheeffectivepotentials(blackline)andzero
modes (dashed red line) for themulti-scalar-field model with
modes (dashed red line) for the second model with different
different values of the parameter a. The parameters are set
valuesof p. The parameters are set toη=1 and a=0.45.
to p=1 and η=1.
figurationsofχandρandthe correspondinglocalization
odd q 1 and found that the left-chiral zero mode can
≥ condition is η > 1 . For the Yukawa coupling mecha-
not be localized on the brane because of the lump con- 18a
nismtheparameterainthebranesolutiondoesnotaffect
figurations of χ and ρ. In order to localize the left-chiral
thelocalizationandlocalizationpositionoftheleft-chiral
zeromodeonthebranetheauthorsadoptedthecoupling
F = φq and found the localization condition is η > 2. zero mode [67], but for the LXCW coupling mechanism
2 9 it does (see Fig. 12).
In the LXCW coupling mechanism the left-chiral zero
mode can be localized on the brane with the lump con- The relationships between the number n of the reso-