# Bulk RS models, Electroweak Precision tests and the 125 GeV Higgs

Abhishek M Iyer<EMAIL_ADDRESS>Department of Theoretical
Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba,
Mumbai 400 005, India  K. Sridhar<EMAIL_ADDRESS>Department of
Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road,
Colaba, Mumbai 400 005, India  Sudhir K. Vempati<EMAIL_ADDRESS>Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012,
India

###### Abstract

We present upto date electroweak fits of various Randall Sundrum (RS) models.
We consider the bulk RS model, deformed RS and the custodial RS models. For
the bulk RS case we find the lightest Kaluza Klein (KK) mode of the gauge
boson to be $\sim 8$ TeV while for the custodial case it is $\sim 3$ TeV. The
deformed model is the least fine tuned of all which can give a good fit for KK
masses $<2$ TeV depending on the choice of the model parameters. We also
comment on the fine tuning in each case.

###### pacs:

73.21.Hb, 73.21.La, 73.50.Bk

††preprint: TIFR/TH/15-05

##  I Introduction

The discovery of the Higgs boson at $\sim 125$ GeV has firmly established the
status of the Higgs mechanism as the theory of electroweak symmetry breaking
physics. In addition it also fixes one of main unknown inputs of electroweak
precision fits. Electroweak precision measurements put very important and
sometimes very strong constraints on new physics models. In the present work
we focus on the Randall-Sundrum (RS) model RS  and its variations. We update
the constraints on the lightest Kaluza-Klein (KK) modes of RS scenarios in the
light of discovery of Higgs mass and improved measurements the W-boson mass
($m_{w}$) and top mass ($m_{t}$). Electroweak precision constraints played an
important role in the evolution of RS models and their phenomenology 111For a
recent review see Gher2 ; gherghetta. In the original standard proposal all
the standard model fields are localized on the IR brane

Later motivated by gauge coupling unification, gauge fields were moved to the
bulk while keeping the Higgs and the fermion fields on the brane Agashe:2002pr
. In both cases large contributions to the oblique $S$ and $T$ parameters were
noted Peskin:1991sw . resulting in bounds on the first KK mass in excess of 30
TeV Davou ; Hisano ; Huber:2000fh ; Csaki:2002gy ; Burdman:2002gr ;
Delgado:2007ne . Moving the fermions into the bulk served the following two
purposes:
1) It offered an elegant solution to the Yukawa hierarchy puzzle achieved by
localizing the fermions at different points in the bulk, resulting in
interesting flavour phenomenology in the hadronic sector AgasheSoni ; Huber1 ;
cedric ; neubert1 ; neubert2  and the leptonic sector gross ; Kitano ; Huber4
; Huber3 ; Huber2 ; Agashe ; Fitzpatrick ; Chen ; AgasheSundrum ; Huber1 ;
Archer:2012qa ; Iyer:2012db ; Iyer:2013hca ; Iyer:2013eka . For a detailed
description of RS phenomenology with bulk fields see Gherghetta ; Gher2 .
2) The constraints on the gauge KK states from the S parameter is
significantly weakened as all the light fermions except the top are localized
away from the IR brane and Higgs.
The constraints from T parameter, however remain strong as the Higgs doublet
is localized near the IR brane which is necessary for the solution to the
hierarchy problem. In view of this the following extensions were proposed: a
Models with bulk custodial symmetry Agashe:2003zs  The bulk gauge group in
question $SU(2)_{L}\times SU(2)_{R}\times U(1)_{X}$. In this case the
additional corrections to the T parameter due to new KK gauge bosons cancel
the volume enhanced contributions due to the KK states of the SM gauge bosons.
The T parameter vanishes at tree level and the limits on the KK mass of the
first gauge boson is mainly due to the S parameter. A straight forward
estimation of the S parameter results in a lower bound on the first KK mass to
be $\sim 4$ TeV for the point to lie inside the 3$\sigma$ region in Figure[1].
Taking into account the loop corrections to the T parameter, (in scenarios
with $Zbb$ protection ) it was found that one can lower the mass of the first
KK gauge boson to around 3 TeV at 3 $\sigma$. An additional alternative to
consider custodial models with gauge-Higgs unification which can address the
little hierarchy problem. Carena:2006bn . A global fit to the precision
observables in such models was performed in Carena:2007ua . Scenarios with
bulk Higgs was considered in Archer:2014jca. b Models with a deformed
metric:Falkowski:2008fz ; Cabrer1 ; Cabrer2  In this setup the bulk geometry
is RS like (AdS) near the UV brane while there is a deviation from AdS
geometry near the IR brane. Depending on the model parameters this often
results in a smaller volume factor as compared to the original RS setup. In
Cabrer3  the authors performed a fit to the data for different values of the
Higgs mass and evaluated the fine tuning required to fit that particular Higgs
mass. In Carmona:2011ib the authors studied the implications of the one loop
corrections to the T parameter on the fits. c Models with brane localized
kinetic terms for the gauge bosons Carena:2003fx . There were several previous
analyses where impact of bulk fields on oblique observables was studied. With
bulk fermions, the dominant constraint on the KK mass is due to the $T$
parameter which is enhanced due to the mixing of the zero mode gauge bosons
with the KK modes. This mixing is governed by the Higgs vev. As noted in
Archer:2012qa ,these constraints can be ameliorated with a Higgs vev not
strongly localized near the IR brane which in turn reduces the zero-KK mode
mixing. This set-up is particularly useful in a model with a deformed-RS
metric, where the KK constraints on the lowest KK masses can be reduced to
$\sim 2.5$ TeV. In Agashe:2013kxa the authors while updating the bounds on the
KK masses from precision electroweak data, also discuss the impact of the
future measurements of rare K and B decays on the parameter space of the
model. They also discuss the correlation between the these flavour
measurements and the limits from direct searches for current and future runs
of the LHC. Discussions of general composite Higgs models was done in
Fichet:2013ola of which RS was considered as an example. While discussing the
trilinear and quartic anomalous gauge couplings in these scenarios, they quote
limits on the mass of KK gauge resonance due to precise values of the $S$ and
$T$ parameters. In the absence of brane kinetic terms, the constraints on
custodial models with a brane Higgs was about 7 TeV while this could be
lowered to 6.6 TeV for pseudo Nambu goldstone Higgs at 95 % CL. Models with
bulk Higgs were also considered in Archer:2014jca where a detailed analysis
correlating signal strengths for different production mechanisms and decay
channels was performed as function of anarchic bulk Yukawa parameter, KK
masses and extent of the compositeness of the Higgs operator. In
Dillon:2014zea the authors showed that the inclusion of higher dimensional
operators in the bulk and on the branes can significantly reduce the
constraints on the T-parameter. These operators only require $\mathcal{O}(1)$
coefficients, and don’t contribute much to the other electroweak parameters.
We focus our attention on the scenarios a) and b) of the extensions to the RS
model. In both the scenarios the brane mass parameter of the Higgs doublet
plays an important role in determining constraint on the first KK mass of the
gauge boson. Additionally with the precise measurement of the Higgs mass, the
extent of fine tuning is related to the brane mass parameter $b$Cabrer1 ;
Cabrer2 ; Cabrer3 . We make explicit the interplay between the fine tuning
required to fit the Higgs boson mass and the $b$ parameter which gives the
best fit to the electroweak observables. Using the expansion formalism of
Wells  we determine the best fit points for the model parameters by using the
Standard $\chi^{2}$ analysis. We find that the KK mass of the first gauge
boson is lower than what was obtained in earlier analysis when naive bounds
from evaluation of S and T parameters were taken into account.

The paper is organized as follows: In Section[II] and Section[III] we outline
the formalism of Wells  thus providing the necessary background for the
analysis. in section[IV] we briefly review the bulk RS model and study the
bulk RS model with no additional gauge symmetries. In Section[V] and
Section[VI] we analyse RS model with deformed metric and custodial symmetry
respectively. In Section[VII] we conclude.

##  II Expansion formalism and the SM

In this section we briefly review the expansion formalism of Wells  which we
use for our analysis. There are numerous observables in the Standard Model
whose values have been very well measured. These observables are in general a
function of the following lagrangian parameters:

$p_{k^{\prime}}\equiv\\{g_{i},y_{t},v,\lambda\\}$ (1)

where $g_{i}$ are the gauge couplings, $y_{t}$ is top quark Yukawa coupling,
$v$ is vacuum expectation value (vev) and $\lambda$ is the quartic coupling.
These parameters are referred to as the ‘input parameters’. An $i^{th}$
observable, $\hat{O}_{i}^{SM}$ in the SM can be expressed as a function of
these parameters as

$\hat{O}_{i}^{SM}(\\{p_{k^{\prime}}\\})=\hat{O}_{i}^{ref}+\sum_{k^{\prime}}\frac{\partial\hat{O}_{i}^{SM}}{\partial
p_{k^{\prime}}}(p_{k^{\prime}}-p_{k^{\prime}}^{ref})+\ldots$ (2)

where the $\ldots$ denote higher orders. $\\{p_{k^{\prime}}^{ref}\\}$ is the
set of lagrangian parameters chosen at a reference value
$p^{ref}_{k^{\prime}}$ at which the evaluated expressions for the SM
observables match closely with experiment and
$\hat{O}_{i}^{ref}=\hat{O}_{i}^{SM}(\\{p_{k^{\prime}}^{ref}\\})$. Thus the
expansion in Eq.(2) is about the reference values ${p_{k^{\prime}}^{ref}}$ and
${p_{k^{\prime}}}$ is the allowed deviation about the reference values.

The Lagrangian parameters, however are not measured directly, but are
extracted from the measurements of certain observables. As a result it seems
logical to re-express the SM observables in terms of a few accurately measured
observables which will now serve as the input. One such list of input
observables is222A subset of these observables can be used to ‘determine’ the
input parameters.

$\hat{O}_{k^{\prime}}\equiv\\{m_{Z},m_{H},G_{F},\alpha(m_{Z}),\alpha_{s}(m_{Z}),m_{t}(m_{t})\\}$
(3)

In terms of the input observables, Eq.(2) can be re-expressed as follows:

$\hat{O}_{i}^{SM}(\\{\hat{O}_{k^{\prime}}\\})=\hat{O}_{i}^{ref}+\sum_{k^{\prime}}\frac{\partial\hat{O}_{i}^{SM}}{\partial\hat{O}_{k^{\prime}}}(\hat{O}_{k^{\prime}}-\hat{O}_{k^{\prime}}^{ref})+\ldots$
(4)

where $\hat{O}_{k^{\prime}}^{ref}$ is the experimentally measured central
value of the input observable and $\hat{O}_{k^{\prime}}$ quantifies the
deviation from the central value. Thus the deviation in $\hat{O}_{i}^{SM}$ can
be expressed in terms of experimental deviation of the input observables from
their central values. The relative deviation can be defined as

$\bar{\delta}^{SM}\hat{O}_{i}(\\{\hat{O}_{k^{\prime}}\\})=\frac{\hat{O}_{i}^{SM}(\\{\hat{O}_{k^{\prime}}\\})-\hat{O}_{i}^{ref}(\\{\hat{O}_{k^{\prime}}^{ref}\\})}{\hat{O}_{i}^{ref}}$
(5)

Defining

$c_{ik^{\prime}}=\frac{\hat{O}_{k^{\prime}}^{ref}}{\hat{O}_{i}^{ref}}\frac{\partial\hat{O}_{i}^{SM}}{\partial\hat{O}_{k^{\prime}}}$
(6)

we can express the relative deviation in Eq.(5) in the $i^{th}$ observable,
due to deviation from the central value of the input observables as

$\bar{\delta}O^{SM}_{i}=\sum_{i^{\prime}}c_{ii^{\prime}}\delta\hat{O}^{SM}_{i^{\prime}}$
(7)

Using this, Eq.(4) can be re-written as

$\hat{O}_{i}^{SM}(\\{\hat{O}_{k^{\prime}}\\})=\hat{O}_{i}^{ref}(1+\hat{\delta}^{SM}\hat{O}_{i})$
(8)

The deviation of all the SM observables can be quantified by constructing a
$\chi^{2}$ statistic defined as

$\chi^{2}(\hat{O}_{k^{\prime}})=\sum_{i}\left[\frac{\hat{O}_{i}^{SM}(\hat{O}_{k^{\prime}})-\hat{O}_{i}^{expt}}{\delta\hat{O}_{i}^{expt}}\right]^{2}$
(9)

It should be noted that while constructing the $\chi^{2}$ we assume that there
is no correlation Falkowski:2014tna  between the output observables. However
we note that taking into account correlation matrix for the output observables
will not significantly change the results of our analysis. The central values
and the allowed deviation for the input and output observables are given in
Table[1]. Using the Z pole observables and the W mass as the output
observables, we minimize the $\chi^{2}$ function in Eq.(9) by varying the
input observables within the experimentally allowed deviation given in
Table(1). The minimization is performed using minuit . We obtain
$\chi^{2}_{min}=24.54$ with the corresponding best fit values for the input
observables given in Table[2].

Input observables | $m_{Z}$ | 91.1876(21) | ALEPH:2005ab  | $G_{F}$ | $1.1663787(6)\times 10^{-5}$ | Beringer:1900zz
---|---|---|---|---|---|---
$\alpha(m_{Z})$ | $7.81592(86)\times 10^{-3}$ | Beringer:1900zz  | $m_{t}(m_{t})$ | 173.34(75) | ATLAS:2014wva
$\alpha_{s}(m_{Z})$ | 0.1185(6) | Beringer:1900zz  | $m_{H}$ | 125.9(4) | Beringer:1900zz
Output observables | $m_{W}$ | 80.385(15) | Group:2012gb  | $\Gamma_{Z}$ | 2.4952(23) | ALEPH:2005ab
$\sigma_{had}$ | 41.541(37) | ALEPH:2005ab  | $R_{e}$ | 20.804(50) | ALEPH:2005ab
$R_{\mu}$ | 20.785(33) | ALEPH:2005ab  | $R_{tau}$ | 20.764(45) | ALEPH:2005ab
| $R_{b}$ | 0.21629(66) | ALEPH:2005ab  | $R_{c}$ | 0.1721(30) | ALEPH:2005ab
| $sin^{2}\theta_{e}$ | 0.23153(16) | ALEPH:2005ab  | $sin^{2}\theta_{b}$ | 0.281(16) | ALEPH:2005ab
| $sin^{2}\theta_{c}$ | 0.2355(59) | ALEPH:2005ab  | $A^{e}_{FB}$ | 0.0145(25) | ALEPH:2005ab
| $A^{b}_{FB}$ | 0.0992(16) | ALEPH:2005ab  | $A^{c}_{FB}$ | 0.0707(35) | ALEPH:2005ab
| $A_{b}$ | 0.923(20) | ALEPH:2005ab  | $A_{c}$ | 0.670(27) | ALEPH:2005ab
Table 1: Experimentally measured central values for the input and output observables along with the Standard Deviation Input observables | $m_{Z}$ | 91.188 | $G_{F}$ | $1.16638\times 10^{-5}$
---|---|---|---|---
$\alpha(m_{Z})$ | $7.81589\times 10^{-3}$ | $m_{t}(m_{t})$ | 173.59
$\alpha_{s}(m_{Z})$ | 0.118567 | $m_{H}$ | 125.89
Output observables | $m_{W}$ | 80.366 $\pm 0.005$ | $\Gamma_{Z}$ | 2.4957$\pm 0.0006$
$\sigma_{had}$ | 41.472 $\pm 0.04$ | $R_{e}$ | 20.7427$\pm 0.03$
$R_{\mu}$ | 20.7428$\pm 0.03$ | $R_{tau}$ | 20.7897 $\pm 0.03$
| $R_{b}$ | 0.215822 $\pm 0.00005$ | $R_{c}$ | 0.17209$\pm 0.000007$
| ${\rm sin}^{2}\theta_{e}$ | 0.23161 $\pm 0.000002$ | ${\rm sin}^{2}\theta_{b}$ | 0.2329$\pm 0.00001$
| ${\rm sin}^{2}\theta_{c}$ | 0.2315$\pm 0.000002$ | $A^{e}_{FB}$ | 0.0160$\pm 0.000004$
| $A^{b}_{FB}$ | 0.1025$\pm 0.00002$ | $A^{c}_{FB}$ | 0.0732$\pm 0.00001$
| $A_{b}$ | 0.9346$\pm 0.00005$ | $A_{c}$ | 0.6675$\pm 0.000008$
Table 2: Best fit values for the input and output observables for the SM fit
with $\chi^{2}_{min}=24.54$

##  III New Physics

The expansion formalism presented in Section[II] can be extended to include
new physics effects. Assuming the nature of new physics is such that it
modifies mostly the oblique parameters, the new physics effects can be
parametrized by the introduction of higher dimension operators in the
lagrangian. These operators can give corrections to any of the input and the
output observables. 333In the calculations used for our analysis only tree
level effective theory operators are considered. Things could get more
stringent if one loop effective theory operators are considered Alonso:2013hga
; Elias-Miro:2013eta ; Henning:2014wua . For a detailed analysis of precision
observables using Standard Model effective field theory see Trott. In the
presence of new physics Eq. (7) becomes

$\delta^{NP}\hat{O}_{i}^{th}(\\{\hat{O}_{k^{\prime}}\\},NP)=\bar{\delta}\hat{O}^{SM}_{i}+\zeta_{i}$
(10)

where
$\zeta_{i}(\\{\hat{O}_{k^{\prime}}\\},NP)\equiv\frac{\delta^{NP}\hat{O}_{i}^{th}(\\{\hat{O}_{k^{\prime}}\\},NP)}{\hat{O}_{i}^{ref}}$
parametrizes the relative contribution to the $i^{th}$ observable due to
higher dimension operators. Using Eq.(7), Eq.(10) can be written as

$\displaystyle\delta^{NP}\hat{O}_{i}^{th}(\\{\hat{O}_{k^{\prime}}\\},NP)$
$\displaystyle=$
$\displaystyle\sum_{i^{\prime}}c_{ii^{\prime}}\delta\hat{O}^{SM}_{i^{\prime}}+\zeta_{i}$
(11) $\displaystyle=$
$\displaystyle\sum_{i^{\prime}}c_{ii^{\prime}}\delta\hat{O}^{th}_{i^{\prime}}+\bar{\delta}^{NP}\hat{O}_{i}$

where
$\bar{\delta}^{NP}\hat{O}_{i}=\zeta_{i}-\sum_{i^{\prime}}c_{ii^{\prime}}\zeta_{i^{\prime}}$.
Here the superscript ’th’ denotes SM in addition to new physics. Note that
from Eq.(6), the matrix of co-efficients $c_{ik^{\prime}}$ is a unit matrix
for the input observables i.e.
$c_{i^{\prime}k^{\prime}}=\delta_{i^{\prime}k^{\prime}}$. Thus any new physics
effects to the input observables are adjusted such that the net shift is zero,
which is apparent in Eq.(11). This adjustment is however propagated in the
evaluation of the output observables through Eq.(11).

In many scenarios, new physics is such that their dominant contribution to the
various SM observables is only through the self energy corrections to the
various gauge boson propagators given below:

$\pi_{\alpha\beta}\equiv\\{\pi_{ZZ},\pi^{\prime}_{ZZ},\pi_{\gamma
Z},\pi^{\prime}_{\gamma\gamma},\pi_{WW},\pi^{0}_{WW}\\}$ (12)

The primed quantities denotes differentiation with respect to $q^{2}$, where
$q$ is the four momentum. Note that the corrections to the fermion coupling to
the gauge bosons are universal. In this case the new physics contribution to
the input observables in Eq.(11) can be re-expressed as

$\delta^{NP}\hat{O}_{i}^{th}=\sum
b_{i,\alpha\beta}\delta^{NP}\pi_{\alpha\beta}$ (13)

where it is understood that the sum extends over the list in Eq.(12) while the
coefficients $b_{\alpha\beta}$ are evaluated in Wells .

In such models the corrections to the gauge boson propagators can be encoded
in oblique parameters S and T 444The contribution to U is suppressed as only
dimension 8 operators contribute to it Peskin:1990zt ; Peskin:1991sw . These
oblique parameters are related to the new physics effects to the self energy
correction as follows Ellis ;

$\displaystyle\delta^{NP}\pi_{ZZ}$ $\displaystyle=$
$\displaystyle-\alpha(m_{Z})T+\frac{\alpha(m_{Z})}{2}S$
$\displaystyle\delta^{NP}\pi^{\prime}_{ZZ}$ $\displaystyle=$
$\displaystyle\frac{\alpha(m_{Z})}{2}S$ $\displaystyle\delta^{NP}\pi_{\gamma
Z}$ $\displaystyle=$ $\displaystyle-\frac{\alpha(m_{Z})}{4{\rm
sin}^{2}\theta_{W}}{\rm cos}2\theta_{W}{\rm tan}\theta_{W}S$
$\displaystyle\delta^{NP}\pi^{\prime}_{\gamma\gamma}$ $\displaystyle=$
$\displaystyle-\frac{\alpha(m_{Z})}{2}S$

We use Eq.(13) in Eq.(11) to construct the $\chi^{2}$ for the output
observables at the Z peak along with the $W$ mass. Using the results of the
analysis in Wells , the expression for the $\chi^{2}$ statistic defined in
Eq.(9) is given as

$\chi^{2}=25.0898+1102.39~{}S^{2}+28.746~{}S-72.0085~{}T-2256.69~{}ST+1377.07~{}T^{2}$
(14)

The input observables were fixed to their experimentally measured central
values while obtaining the above expression. Using this we obtain the S-T plot
in Fig.[1] in which the 68%,95% and 99% confidence level allowed regions are
depicted by red, blue and orange regions respectively. Our analysis is
performed by fixing $U=0$. Additionally the best fit point for
$m_{h}^{ref}=125$ GeV and $m_{t}^{ref}$ =173 GeV was obtained to be $S=0.08\pm
0.1$ and $T=0.09\pm 0.1$ with the correlation coefficient between the $S$ and
the $T$ parameter to be $+0.89$. In addition to $S$ and $T$, the other
floating point parameters were $m_{Z}=91.1876\pm 0.0021$,
$\alpha_{s}(M_{Z})=0.1185\pm 0.0006$ and $\alpha(M_{Z}^{2})=7.81596(86)\times
10^{-3}$ These results are to be compared with the Gfitter analysis
Baak:2012kk where the $\\{m_{Z},\alpha,\Delta\alpha\\}$ were considered as the
floating point parameters with $\Delta\alpha(M_{Z}^{2})=0.02757\pm 0.0001$.
They obtained $S=0.06\pm 0.09$ and $T=0.1\pm 0.07$ with a correlation co-
efficient $+0.91$.

Figure 1: The red,blue and orange regions denote the 68%,95% and 99%
confidence level allowed regions in the ST parameter space.

For a particular model of new physics, the oblique parameters S,T depend on
the model parameters. A given set of model parameters is valid only if the
corresponding S,T observables computed for that set lie at least within the
orange ellipse in Fig.[1]. Thus a very small contribution, for example to the
S parameter would necessitate T to also be very small so as to lie within the
bottom left portion of the ellipse. However an increasing S can admit larger
values of the T parameter corresponding to moving towards the top right
portion of the ellipse. Thus we can use Fig.[1] to constrain the model
parameters. We now use this analysis to obtain constraints on various Randall-
Sundrum models.

##  IV Randall-Sundrum Models

Randall-Sundrum model is a model of a single extra-dimension compactified on
an $S_{1}/Z_{2}$ orbifold RS . The five dimensional gravity theory is defined
by the following line element:

$\displaystyle ds^{2}=e^{-2A(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}-dy^{2}$ (15)

Two opposite tension branes are located at the two fixed points of the
orbifold. The space between the branes is endowed with a large negative bulk
cosmological constant making it a slice of AdS. The presence of brane
localized sources of energy results in zero cosmological constant being
induced on the branes. In the original setup $A(y)=ky$ where k is reduced
Planck scale. Identifying the scale of physics on the $y=0$ brane as $M_{IR}$,
the effective UV scale induced at the $y=\pi R$ brane owing to geometry is
given as

$M_{IR}=e^{-kR\pi}k$ (16)

where $R$ is the compactification radius. Choosing $kR\sim 12$, will result in
$M_{IR}\sim 200$ GeV owing to large exponential warping. Any radiative
instability to the masses of fundamental scalars in the theory can be warped
down to the electroweak scale thus solving the gauge hierarchy problem. In the
original setup, with the exception of gravity all the SM fields were localized
on the brane at $y=y_{1}=\pi R$ also referred to as the IR brane. Here we
consider a generalization of the original setup where particles of all types
of spin are allowed to propagate in the bulk.

A bulk field $\Psi^{s}(x^{\mu},y)$ with spin $s$ can be expanded in the KK
basis as follows:

$\Psi^{s}(x^{\mu},y)=\frac{1}{\sqrt{\pi
R}}\sum^{\infty}_{n=0}\psi^{(n)}_{s}(x^{\mu})f_{s}^{(n)}(y)$ (17)

The zero modes for the fields are identified as the SM fields. While the zero
mode for the gauge bosons are flat at leading order, the ones for the scalars
and the fermions are controlled by the brane and bulk mass terms respectively.
They are parametrized as $m^{s}_{brane}=bk$ and $m^{f}_{bulk}=ck$ where $b,c$
are dimensionless $\mathcal{O}$(1) quantities. The normalized profiles for the
fields are given as

$\displaystyle f^{(0)}_{0}(b,y)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{2(b-1)kR\pi}{e^{2(b-1)kR\pi}-1}}e^{(b-1)ky}$
$\displaystyle f^{(0)}_{1/2}(c,y)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{(1-2c)kR\pi}{e^{(1-2c)kR\pi}-1}}e^{(0.5-c)ky}$
$\displaystyle f^{(0)}_{1}(y)$ $\displaystyle=$ $\displaystyle 1$ (18)

where the normalization conditions are given as

$\frac{1}{\pi R}\int_{0}^{\pi R}(f^{(0)}_{s}(y))^{2}dy=1$ (19)

$c<0.5$ and $b>1$ ($c>0.5$ and $b<1$ ) correspond to the fields being
localized towards the IR(UV) brane respectively. The KK modes of all fields
are however localized near the IR brane. We note here that while the profiles
of the gauge boson fields are flat at leading order, it receives corrections
due to the mixing of the KK mode with the zero mode. The mixing is
proportional to the vacuum expectation value (vev) and is given as

$a_{01}=\frac{v^{2}}{(kR\pi)M^{2}_{KK}}\int_{1}^{z_{IR}}z^{2}f^{(0)}_{0}(z)^{2}f_{1}(z)^{(0)}f_{1}(z)^{(1)}$
(20)

where $z=e^{kRy}$ is the conformal co-ordinate. $f^{(0)}_{0,1}$ are given in
Eq.(18), while $f^{(1)}_{1}$ is the profile of the first KK mode of the gauge
boson. Detailed review about bulk RS models can be found in Gher2 ;
gherghetta. Thus diagonalizing the mass matrix of KK modes and zero mode, will
result in the lightest state (identified as the SM boson) having a small KK
component proportional to Eq.(20). As a result the coupling of the fermions to
the SM boson will have a non-universal component which is a function of its
localization parameter $c$. The $c$ parameters will be in general different
for different fermionic generations to generate the required hierarchy in the
Yukawa parameters. For the light fields with the exception of the top it is
fair to assume $c>0.5$ to reduce the overlap with the Higgs. For $c>0.5$, non-
universal component of the coupling is very small and can be neglected Gher2 ;
Hewett:2002fe . This is enough to evade bounds from FCNC processes which can
occur at tree level.

Solution of the hierarchy problem requires the Higgs zero mode to be localized
very close to the IR brane. It corresponds to a choice $b\geq 2$ for the brane
mass parameterLuty:2004ye ; Cabrer2 555Realization of EWSB also requires
$b>2$Davoudiasl:2005uu .. For a bulk scalar field with a massless zero mode
the brane mass parameter $b$ is related to the bulk mass parameter $a$ as
666Bulk scalar mass is parametrized as $m^{s}_{bulk}=ak$

$b=2\pm\sqrt{4+a}$ (21)

Henceforth, we will drop the $b=2-\sqrt{4+a}$ solution as it will never lead
to $b\geq 2$ necessary for the solution to the hierarchy problem. The zero
mode increasingly becomes sharply localized near the IR brane as $b$
increases. However an increase in $b$ is only facilitated by the corresponding
increase in $|a|$. Depending on the value of $\frac{k}{M_{pl}}$, the bulk mass
parameter cannot be increased indefinitely as the product $ak$ will become
greater than the 5D Planck scale. Fig.[2] shows a plot of $b$ as function of
$a$. Depending on the value of $\frac{k}{M_{pl}}$, the plot is terminated on
the right at which $ak=M_{Pl}$. For instance for $\frac{k}{M_{pl}}=0.1$, the
plot (blue curve) is terminated at $b=5.74$ while for $\frac{k}{M_{pl}}=0.25$
the plot (red curve) is terminated $b=4.82$.

Figure 2: Maximum allowed value of $b$ for $\frac{k}{M_{pl}}=0.25$ (red) and
$\frac{k}{M_{pl}}=0.1$(blue)

Thus the case with brane localized Higgs will be treated separately and not as
a limiting case where the bulk Higgs field tends to a brane localized one.

We now proceed to study the impact of EWPT in various RS models.

###  IV.1 Bulk Higgs with no additional symmetries

This model is the same same set-up discussed above. All the fermionic fields
except the top are localized near the UV brane. This is sufficient to fit the
masses of all fermions except the top. Due to localization of all the fermions
near the UV brane the vertex corrections are very small and universal, thus
the new physics effects can be parametrized in terms of the oblique operators
S and T which are given as Cabrer1 ; Cabrer2 ; Cabrer3

$\displaystyle\alpha T$ $\displaystyle=$ $\displaystyle\frac{{\rm
sin}^{2}\theta_{W}m_{Z}^{2}y_{1}k^{2}e^{-2kR\pi}}{\Lambda^{2}_{IR}}(\alpha_{hh}-2\alpha_{hf}+\alpha_{ff})$
$\displaystyle\alpha S$ $\displaystyle=$ $\displaystyle\frac{8{\rm
sin}^{2}\theta_{W}{\rm
cos}^{2}\theta_{W}m_{Z}^{2}y_{1}k^{2}e^{-2kR\pi}}{\Lambda^{2}_{IR}}(\alpha_{ff}-\alpha_{hf})$

where $y_{1}$ denotes the position of the IR brane and $\alpha_{ij}$ are
parameters involving the bulk propagators of the bulk gauge fields with $(++)$
boundary conditions, where $+$ denotes Neumann boundary condition. They are
given as Cabrer; Cabrer2

$\displaystyle\alpha_{hh}$ $\displaystyle=$ $\displaystyle\int
e^{2A(y)}\left(\Omega_{h}-\frac{y}{y_{1}}\right)^{2}$
$\displaystyle\alpha_{hf}$ $\displaystyle=$ $\displaystyle\int
e^{2A(y)}\left(\Omega_{h}-\frac{y}{y_{1}}\right)\left(\Omega_{f}-\frac{y}{y_{1}}\right)$
$\displaystyle\alpha_{ff}$ $\displaystyle=$ $\displaystyle\int
e^{2A(y)}\left(\Omega_{f}-\frac{y}{y_{1}}\right)^{2}$ (23)

where $\Omega_{f,h}(y)=\frac{1}{y_{1}}\int_{0}^{y}dyf^{2}_{f,h}(y)$ and the
profiles $f^{\prime}s$ are given by Eq.(18) For the case where the fermions
are localized on the UV brane $\Omega_{f}=1$. These co-efficients are a
function of the localization of the zero mode of the fermionic and the Higgs
field. For a fixed KK scale, the co-efficients increase as the fields move
closer to the IR brane due to larger overlap of the zero mode with the KK
modes.

For the oblique $T$ parameters, the co-efficient $\alpha_{hh}$, also
contributes in addition to $\alpha_{hf,ff}$. Owing to the localization of the
Higgs very close to the IR brane, $\alpha_{hh}$ will be enhanced as compared
to $\alpha_{hf,ff}$, which is smaller as the fermions are closer to the UV
brane. As a result in this scenario the contributions to the $T$ parameter is
large. In this case the oblique observables primarily depend on two
parameters:
a) The localization parameter $b$ for the bulk Higgs field.
b) First KK scale of the gauge boson.
To extract the parameter space of these two parameters which are consistent
with the constraints on the S and T parameters, a scan is performed over the
following ranges

$b\equiv[2,5]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Lambda_{IR}\equiv[1250,10000]$ (24)

Fig.[3] shows the $3\sigma$ region in the $b-\Lambda_{IR}$ plane.

Figure 3: $3\sigma$ allowed parameter space in the $b-\Lambda_{IR}$ plane for
regular bulk RS. $\Lambda_{IR}$ is in GeV.

The first KK mass of the gauge boson is related to the IR scale as
$m^{(1)}_{KK}\sim 2.44\Lambda_{IR}$. We see $\Lambda_{IR}$ is lowered as $b$
approaches 2 corresponding to shifting of the Higgs away from the IR brane.
However $b\geq 2$ must be maintained for the model to serve as solution to the
hierarchy problem Luty:2004ye ; Cabrer2 . Table[3] gives the fit values when
all input observables along with $b$ and $\Lambda_{IR}$ are varied
simultaneously to minimize the $\chi^{2}$ in Eq.(9). From the plot in Fig.[3]
we find that the lowest value of $\Lambda_{IR}$ possible is around 3.4 TeV
corresponding a first KK mass for the gauge boson to be around 8 TeV. The plot
is highly concentrated around $b=2$ since the coupling of the SM fields to the
KK states is small as compared to higher values of $b$. This point corresponds
to the case where the mass of the first KK gauge boson is minimum.

Input observables | $m_{Z}$ | 91.1813 | $G_{F}$ | $1.1663784\times 10^{-5}$
---|---|---|---|---
$\alpha(m_{Z})$ | $7.81611\times 10^{-3}$ | $m_{t}(m_{t})$ | 173.05
$\alpha_{s}(m_{Z})$ | 0.119101 | $m_{H}$ | 126.3
Output observables | $m_{W}$ | 80.411 | $\Gamma_{Z}$ | 2.4983
$\sigma_{had}$ | 41.479 | $R_{e}$ | 20.7472
$R_{\mu}$ | 20.7473 | $R_{tau}$ | 20.7941
| $R_{b}$ | 0.2158 | $R_{c}$ | 0.1712
| ${\rm sin}^{2}\theta_{e}$ | 0.2313 | ${\rm sin}^{2}\theta_{b}$ | 0.2327
| ${\rm sin}^{2}\theta_{c}$ | 0.2312 | $A^{e}_{FB}$ | 0.0164
| $A^{b}_{FB}$ | 0.1038 | $A^{c}_{FB}$ | 0.0742
| $A_{b}$ | 0.9347 | $A_{c}$ | 0.6683
Model Parameters | b | 2.00 | $m^{1}_{kk}$ | 8.3 TeV
Table 3: Fit values for the input and output observables with a bulk Higgs.
Input observables are free in the fit and are varied within their 1 $\sigma$
allowed experimental deviation. b=2.00 and $m^{1}_{kk}=8.3$ TeV is obtained
for the fit.

We finally note that for the brane localized case, a minimum KK mass of 13.6
TeV is required for the model to be consistent with the data. The fit values
for the input and the output observables are given in Table[4]

Input observables | $m_{Z}$ | 91.1856 | $G_{F}$ | $1.16637854\times 10^{-5}$
---|---|---|---|---
$\alpha(m_{Z})$ | $7.816649\times 10^{-3}$ | $m_{t}(m_{t})$ | 172.33
$\alpha_{s}(m_{Z})$ | 0.118657 | $m_{H}$ | 126.295
Output observables | $m_{W}$ | 80.402 | $\Gamma_{Z}$ | 2.4976
$\sigma_{had}$ | 41.477 | $R_{e}$ | 20.744
$R_{\mu}$ | 20.744 | $R_{tau}$ | 20.7916
| $R_{b}$ | 0.2158 | $R_{c}$ | 0.17229
| ${\rm sin}^{2}\theta_{e}$ | 0.2314 | ${\rm sin}^{2}\theta_{b}$ | 0.2327
| ${\rm sin}^{2}\theta_{c}$ | 0.2313 | $A^{e}_{FB}$ | 0.0164
| $A^{b}_{FB}$ | 0.1036 | $A^{c}_{FB}$ | 0.07412
| $A_{b}$ | 0.9347 | $A_{c}$ | 0.6682
Table 4: Fit values for the input and output observables. Input observables
are free in the fit and are varied within their 1 $\sigma$ allowed
experimental deviation. $m^{1}_{kk}=13.6$ TeV is obtained for the fit.

It is to be noted that KK scales in excess on 20 TeV is required when
constraints from FCNC like $\mu\rightarrow e\gamma$ are taken into account
Iyer:2012db . Implementation of bulk flavour symmetries with the imposition of
the Minimal Flavour Violation (MFV) ansatz helps in substantially reducing the
KK mass to around $\sim 3$ TeV AgasheSoni ; perez ; Fitzpatrick ; Iyer

Fine Tuning: It is well known that the requirement of massless zero mode for
bulk scalar field requires the bulk mass $m_{bulk}=ak$ and the brane mass
$m_{brane}=bk$ be related by the following relation777The relation
$m_{brane}=2k-\sqrt{4k^{2}+m^{2}_{bulk}}$ is relevant for a Higgs field
localized away from the IR brane and is not relevant to the discussion here.:

$m_{brane}=2k+\sqrt{4k^{2}+m^{2}_{bulk}}$ (25)

Any misalignment between the brane and the bulk masses will result in a non-
zero mass for the zero mode. In a realistic model with electroweak symmetry
breaking, the Higgs boson is massive and is related to the misalignment
asCabrer1 ; Cabrer2 ; Quiros:2013yaa :

$m_{H}^{2}=4(b-1)(m_{brane}-m^{\prime}_{brane})\frac{\Lambda^{2}_{IR}}{k}$
(26)

where $m^{\prime}_{brane}=bk$ is value of the brane mass when the zero mode is
massless. As given in Table[3], $\Lambda_{IR}=2.41$ TeV for the model to
consistent with the electroweak precision data. As a result for $b=2$ a
cancellation upto the fourth decimal between $m^{\prime}_{brane}$ and
$m_{brane}$ is required to fit a Higgs mass of 126 GeV. The level of tuning
increases as the Higgs field is pushed further near the IR brane corresponding
to an increase in $b$. Due to a direct dependence on the brane mass parameter
$b$ Cabrer1 ; Cabrer2 ; Fitzpatrick:2013twa , it is fair to expect that the
Higgs boson mass is best fit by $b\sim 2$. In the dual theory this corresponds
to the Higgs field is a partial composite state with relevant coupling between
the source and the CFT (conformal field theory) sectors. As $b$ increases,
this coupling increases and the state is fully composite of the CFT, thus
recovering the original RS setup.

##  V Deformed RS model

The expression for the S and T parameters in Eq.(LABEL:oblique) can be re-
expressed as Cabrer1 ; Cabrer2

$\displaystyle\alpha S$ $\displaystyle=$ $\displaystyle 8{\rm
cos}^{2}\theta_{W}{\rm
sin}^{2}\theta_{W}\frac{m_{Z}^{2}}{\Lambda^{2}_{IR}}\frac{1}{Z}I$
$\displaystyle\alpha T$ $\displaystyle=$ $\displaystyle{\rm
sin}^{2}\theta_{W}\frac{m_{Z}^{2}}{\Lambda^{2}_{IR}}\frac{ky_{1}}{Z^{2}}I$
(27)

where $y_{1}$ is the position of the IR brane and the dimensionless integral
$I$ and $Z$ are defined as

$\displaystyle
I=k\int_{0}^{y_{1}}\left[(k\left(y_{1}-y\right))^{2}\right]e^{2A(y)-2A(y_{1})}dy$
$\displaystyle
Z=k\int_{0}^{y_{1}}dy\frac{h^{2}(y)}{h^{2}(y_{1})}e^{-2A(y)+2A(y_{1})}$ (28)

$h(y)$ is the profile of the vacuum expectation value and is given as

$h(y)=h(y_{1})e^{bk(y-y_{1})}$ (29)

We find the $T$ parameter is enhanced by the volume factor in addition to
being suppressed by two powers of $Z$. In RS models where $A(y)=ky$, $Z=0.5$
for $b=2$ and becomes smaller as the Higgs field approaches the IR brane
($b\rightarrow\infty$). This results in the enhancement of the $T$ parameter
leading to stringent constraints in the KK scale. As a result, the authors in
Cabrer1 ; Cabrer2 ; Cabrer3  considered an alternative solution by considering
modification of the line element in Eq.(15) where $A(y)$ is now given as

$A(y)=ky-\frac{1}{\nu^{2}}\log(1-\frac{y}{y_{s}})$ (30)

Note that $\nu\rightarrow\infty$ results in RS limit. A consequence of this
metric is that the singularity at the IR brane is shifted outside the patch
between IR and UV brane at $y_{s}=y_{1}+\Delta$. $\Delta$ is the distance of
the singularity from the IR brane. For the case where the hierarchy problem is
solved i.e. $A(y_{1})\sim 36$, the position of the IR brane in the bulk
$y_{1}$ is a function of $\nu,\Delta$. Smaller $\nu$ will in general result in
a smaller volume factor $y_{1}$ and helps in ameliorating the constraints on
the KK mass from the T parameter. Additionally as noted in Cabrer1 ; Cabrer2
this setup results in large values of $Z$ for certain choices of parameters
$\nu,\Delta,b$ which help in reducing the KK scales so as to be within the
reach of LHC.

As before we perform an analysis to determine the parameter space of the
$b-\Lambda_{IR}$ plane. We choose two sets of $(\nu,\Delta)$ as follows:
a) $\nu=0.8$ and $\Delta=1$.This corresponds to $y_{1}=30.60/k$ so that
$A(y_{1})\equiv 36$
b) $\nu=1$ and $\Delta=0.1$.This corresponds to $y_{1}=30.28/k$ so that
$A(y_{1})\equiv 36$

|
---|---
Figure 4: Allowed parameter space in the $b-\Lambda_{IR}$ plane for deformed
metric. $\Lambda_{IR}$ is in GeV. The left panel corresponds to $\nu=0.8$ and
$\Delta=1$ while the right panel corresponds to $\nu=1$ and $\Delta=0.1$

For the deformed metric, the $\Lambda_{IR}$ is related to the first $m_{kk}$
scale from the following relationCabrer1 ; Cabrer2

$m^{1}_{kk}\sim j_{0,1}\frac{A^{\prime}(y_{1})}{k}\Lambda_{IR}$ (31)

where $j_{0,1}$ is the first zero of Bessel function $J_{0}(x)$. We scan the
$b$ parameter from 2 to 5 and $\Lambda_{IR}$ is scanned from 50 to 587 GeV for
case a) while it is scanned from 50 to 120 GeV for case b). The upper limit on
$\Lambda_{IR}$ corresponds to a KK mass of $\sim 3$ TeV. From Fig.(4), for the
left panel, a lowest value of $\Lambda_{IR}=472.6$ GeV is obtained for b=2
which corresponds to a first KK mass of about 2.3 TeV. While for case b)
depicted in the right panel of Fig.(4), a lowest value of $\Lambda_{IR}=71.10$
GeV is obtained again for $b=2$. This corresponds to a first KK mass of about
1.7 TeV for the gauge boson. Thus we see that for certain choices of the
metric depending on the values $(\nu,\Delta)$, the first KK mass of the gauge
boson can be below 2 TeV. The fit values for the input and output observables
are given in Table[5]. Case b) offers an advantage over Case a) in terms of
being a less fine tuned model since the $\Lambda_{IR}$ for the fit is small.
The analysis can be repeated for different values of $\nu$ and $\Delta$. For
our analysis we fit the top quark mass by $c_{Q_{3}}\sim 0.475$ and
$c_{t}\sim-1$ with a choice of $\mathcal{O}(1)$ Yukawa $\sim 4$.

The localization of the top doublet relatively near the UV brane is to
minimize the correction to the $Zbb$ vertex.

Input observables | $m_{Z}$ | 91.1813 | $G_{F}$ | $1.1663785\times 10^{-5}$
---|---|---|---|---
$\alpha(m_{Z})$ | $7.81663\times 10^{-3}$ | $m_{t}(m_{t})$ | 173.12
$\alpha_{s}(m_{Z})$ | 0.119118 | $m_{H}$ | 126.29
Output observables | $m_{W}$ | 80.419 | $\Gamma_{Z}$ | 2.498
$\sigma_{had}$ | 41.486 | $R_{e}$ | 20.7381
$R_{\mu}$ | 20.7382 | $R_{tau}$ | 20.785
| $R_{b}$ | 0.215 | $R_{c}$ | 0.171
| ${\rm sin}^{2}\theta_{e}$ | 0.2314 | ${\rm sin}^{2}\theta_{b}$ | 0.2328
| ${\rm sin}^{2}\theta_{c}$ | 0.2313 | $A^{e}_{FB}$ | 0.0162
| $A^{b}_{FB}$ | 0.1032 | $A^{c}_{FB}$ | 0.0737
| $A_{b}$ | 0.9346 | $A_{c}$ | 0.6679
Model Parameters | b | 2.00 | $m^{1}_{kk}$ | 2.3 TeV
Table 5: Fit values for the input and output observables for the deformed RS
case. $b=2.0006$ and $m^{1}_{kk}=2.3$ TeV is obtained for the fit. $\nu=0.8$
and $\Delta=\frac{1}{k}$ are chosen for the fit

Fine Tuning: Due to the deformation in the metric, the Higgs mass in Eq.(26)
can be generalized to Cabrer1 ; Cabrer2 ; Quiros:2013yaa :

$m_{H}^{2}=\frac{2}{Z}(m_{brane}-m^{\prime}_{brane})\frac{\Lambda^{2}_{IR}}{k}\;\;\;\text{where}~{}Z=\int_{0}^{y_{1}}\frac{h^{2}(y)}{h^{2}(y_{1})}e^{-2A(y)+2A(y_{1})}$
(32)

For the normal RS case $A(y)=ky$ and $Z=\frac{1}{2(a-1)}$, thus reducing to
Eq.(26). In comparison to RS where $Z<1$, certain choices of $\nu$ and
$\delta$ result in $Z>1$ which not only lowers the contribution to the T
paramter in Eq.(27) but also helps in reducing the fine tuning to obtain the
Higgs mass. For instance for the parameters in Table.[5], $Z=2.6$, the tuning
reduces to 0.018.

##  VI Custodial RS

The custodial Randall Sundrum set up [Agashe:2003zs ] contains an enlarged
bulk gauged symmetry given by

$SU(2)_{L}\times SU(2)_{R}\times U(1)_{X}$ (33)

which restores the custodial symmetry in the RS setup for the Higgs potential.
The corresponding gauge bosons are denoted by
$W^{1,2,3}_{L\mu},W^{1,2,3}_{R\mu},X_{\mu}$ with $g_{5L,5R,5X}$ denoting the
corresponding five dimensional gauge couplings. In updating the electroweak
constraints in this setup we follow the notation of Carena:2006bn .

The bulk symmetry is broken down to the Standard Model by considering the
following boundary conditions for the gauge fields

$W^{1,2,3}_{L\mu}(++)\;\;\;B_{\mu}(++)\;\;\;W^{1,2}_{R\mu}(-+)\;\;\;Z^{\prime}_{\mu}(-+)$
(34)

with $+(-)$ denoting Neumann(Dirichlet) boundary conditions as before. The
gauge fields $B_{\mu}$ and $Z^{\prime}_{\mu}$ are defined as

$B_{\mu}=\frac{g_{5X}W^{3}_{R\mu}+g_{5R}X_{\mu}}{\sqrt{g^{2}_{5R}+g^{2}_{5X}}}\;\;\;\;\;\;;Z^{\prime}_{\mu}=\frac{g_{5R}W^{3}_{R\mu}-g_{5X}X_{\mu}}{\sqrt{g^{2}_{5R}+g^{2}_{5X}}}$
(35)

The the $W^{1,2,3}_{L\mu}$ and $B_{\mu}$ possess zero modes corresponding to
the SM $SU(2)_{L}$ and the $U(1)_{Y}$ gauge boson respectively. The
hypercharge coupling is given by $Y=2(T^{3}_{R}+Q_{X})$. After electroweak
symmetry breaking, the electromagnetic charge is given by
$Q_{em}=T^{3}_{R}+T^{3}_{L}+Q_{X}$. On the other hand, owing to the mixed
boundary conditions of $W^{1,2}_{R\mu}$ and $Z^{\prime}_{\mu}$, they do not
possess a zero mode.

The presence of new gauge bosons induce additional corrections to the T
parameter but the S parameter remains unchanged. It is given as Cabrer2 ;
shrihari ; Delgado:2007ne ; neubert1 ; Casagrande:2010si

$\alpha T=\frac{{\rm
sin}^{2}\theta_{W}m_{Z}^{2}y_{1}k^{2}e^{-2kR\pi}}{\Lambda^{2}_{IR}}\left(\alpha_{hh}-\alpha^{\prime}_{hh}-2\alpha_{hf}+\alpha_{ff}\right)$
(36)

where the $\alpha^{\prime}_{hh}$ is the bulk propagator for the bosons with
$(-+)$ boundary conditions and is given as

$\alpha^{\prime}_{hh}=\int_{0}^{y_{1}}e^{2A(y)}(1-\Omega_{h})^{2}$ (37)

The contribution to the $T$ parameter due to the KK states of the SM as well
as the new gauge bosons are very similar in magnitude. Recalling that the
dominant contribution to the $T$ parameter is due to $\alpha_{hh}$, the
presence of $\alpha^{\prime}_{hh}$ nearly cancels this contribution, thus
significantly lowering the constraint on the first KK scale of the gauge
boson. As a result, the dominant constraint to the KK scale is due to the $S$
parameter. To see the effects of a negligible T parameter on the fits we
assume UV localized fermions to begin with. To obtain the plot for
$b-\Lambda_{IR}$ parameter space in the presence of custodial symmetry, we
first evaluate the constraints at tree level. From the $S-T$ plot in Fig.[1],
the region around $T\sim 0$ would also necessitate the $S$ parameter to be
small thereby pushing the KK scale up. Indeed, as is noted in the left panel
of Fig.[5], a lowest value of $\Lambda_{IR}=1659$ GeV is obtained which
translates into a lowest KK mass of around $4$ TeV. While this case does
better than the normal bulk RS scenario the first KK mass is still out of
reach of LHC.

Input observables | $m_{Z}$ | 91.1938 | $G_{F}$ | $1.1663787\times 10^{-5}$
---|---|---|---|---
$\alpha(m_{Z})$ | $7.81509\times 10^{-3}$ | $m_{t}(m_{t})$ | 173.3499
$\alpha_{s}(m_{Z})$ | 0.119003 | $m_{H}$ | 125.40
Output observables | $m_{W}$ | 80.347 | $\Gamma_{Z}$ | 2.495
$\sigma_{had}$ | 41.471 | $R_{e}$ | 20.736
$R_{\mu}$ | 20.736 | $R_{tau}$ | 20.783
| $R_{b}$ | 0.215 | $R_{c}$ | 0.171
| ${\rm sin}^{2}\theta_{e}$ | 0.231 | ${\rm sin}^{2}\theta_{b}$ | 0.233
| ${\rm sin}^{2}\theta_{c}$ | 0.2317 | $A^{e}_{FB}$ | 0.0155
| $A^{b}_{FB}$ | 0.100 | $A^{c}_{FB}$ | 0.072
| $A_{b}$ | 0.934 | $A_{c}$ | 0.666
Model Parameters | b | 2.00 | $m^{1}_{kk}$ | 2.88 TeV
Table 6: Fit values for the input and output observables. The corresponding RS parameters are $b=2.0004$ and $m^{1}_{kk}=2.9$. TeV. $\Lambda_{IR}$ is in GeV. The loop contribution to the $T$ parameter $\sim 0.06$ |
---|---
Figure 5: Left panel shows the $b-\Lambda_{IR}$ parameter space when just the
tree level computations of $S-T$ are taken into account. In the right panel,
the loop contributions to the $T$ parameter are also included. $\Lambda_{IR}$
is in GeV.

The assumption of UV localized fermions is not sufficient to fit the top quark
mass as it would result in large $\mathcal{O}(1)$ Yukawa coupling. As a result
the zero mode top doublet and the singlet must be moved closer to the IR brane
($c<0.5$) to increase overlap with the Higgs. This results in the shift of the
coupling of $b_{L}$ to the Z boson. The relative shift to the $b_{L}$ is given
as ponton ; Carena:2006bn ; Delgado:2007ne

$\frac{\delta
g_{b_{L}}}{g_{b_{L}}}=-v^{2}\frac{\left(g_{L}^{2}T^{3}_{L}-g_{R}^{2}T^{3}_{R}\right)\alpha^{\prime}_{hf}+\left(g_{L}^{2}T^{3}_{L}-(g^{\prime})^{2}Y\right)\left(\alpha_{hf}-\alpha^{UV}_{hf}-\alpha^{\prime}_{hf}\right)}{1-\frac{2}{3}\sin^{2}\theta_{W}}$
(38)

The first term involving to $\alpha^{\prime}_{hf}$ will be significant in this
case as the third generation doublet is localized closer to the IR brane to
fit the top quark mass. In the second term however the presence of
$\alpha^{\prime}_{hf}$ and $\alpha_{hf}$ with a relative minus sign softens
the impact of localization of the third generation on $\frac{\delta
g_{b_{L}}}{g_{b_{L}}}$. The current constraints on the corrections to the
$Zb_{L}b_{L}$ coupling pushes the limit obtained in the left panel of Fig.[5]
to beyond 5 TeV. However it was observed in zbb , that the dominant
contribution due to the first term in Eq.(38), can be removed by assuming
$T^{3}_{L}=T^{3}_{R}$ and $g_{L}=g_{R}$ thus significantly softening the
constraints on the KK mass from corrections to the $Zb\bar{b}$ vertex. This
implies that the left handed bottom must belong to bi-doublets of
$SU(2)_{L}\times SU(2)_{R}$. The bi-doublets induce large negative
contributions to the T parameter in most regions of the parameter space. The
contribution is a function of $c_{Q_{3},t}$ which are the localization
parameters for the bi-doublet and the singlet $t_{R}$. It was noted in
Carena:2006bn  that the negative contribution decreases as the doublet and/or
singlet are localized away from the IR brane. However, the top quarks mass as
a function of $c_{Q_{3},t}$ is given as

$m_{top}=\tilde{Y}^{u}_{3,3}\frac{v}{\sqrt{2}(\pi R)^{3/2}}\int_{0}^{\pi
R}f^{(0)}_{0}(b,y)f^{(0)}_{1/2}(c_{Q_{3}},y)f^{(0)}_{1/2}(c_{t},y)$ (39)

where $Y^{u}_{3,3}$ is dimensionless $\mathcal{O}$(1) parameter. Choosing the
bi-doublet and the singlet to be localized away from the IR brane, will result
in the choice of large $Y^{u}_{3,3}>10$ to fit the top quark mass and is not
feasible. As s result the combination of $c_{Q_{3},t}$ which induces a
positive contribution to the T parameter is when $c_{Q}\leq 0$ and
$c_{t}\sim[0.4,0.5]$. As shown by Carena:2006bn , this choice of $c$
parameters not only fits the top quark mass but also gives non-negative
contribution to the T parameter.

A non-zero positive contribution to the $T$ parameter would correspond to
moving vertically up in the $S-T$ plane in Fig.[1]. Owing the tilted
orientation of the ellipse, it makes it possible to accommodate larger values
of $S$ thereby helping in reducing the lower bound on the first KK mass. The
right panel of Fig.[5] corresponds to the case where the loop level
contributions to the $T$ parameter Carena:2006bn  have been turned on. In the
figure we have assumed $\Delta T_{loop}\leq 0.1$. We find that a minimum of
$\Delta T_{loop}\sim 0.06$ for $b=2$ is required to lower the scale of the
first KK gauge boson below 3 TeV. As $b$ increases corresponding to Higgs
moving further towards the IR brane, one can expect minimum $\Delta T_{loop}$
required to keep the KK scale below 3 TeV to increase. The fit values for the
input and output observables is given in Table.[6].

Fine tuning: In this case too we find that the best fit to the precision data
is when the brane mass parameter $b=2$. The KK mass is lowered to 3 TeV thus
reducing the fine tuning by an order of magnitude. As a result the
cancellation between $m_{brane}$ and $m^{\prime}_{brane}$ is of the order of
$0.002$

##  VII Conclusions

There is a perception that in order that the Randall-Sundrum model
successfully address the gauge-hierarchy problem the Higgs ought to be
localised on the IR brane. It has been noted earlier Luty:2004ye ;
Quiros:2013yaa  that this is not the case and our results bear this out. In
fact, we find that even if we move the Higgs field off the IR brane, a
solution to the gauge-hierarchy problem is obtained as long as we have $b\geq
2$. Further, electroweak fits and fine tuning argument seem to be preferring a
$b$ value very close to 2. In the dual CFT terminology, the Higgs field is a
partially composite state batellgherghetta1 ; batellgherghetta2 . This has to
do with the exponential form of the scalar profiles which get pushed close to
the IR brane for values of $b$ greater than 2, so that from the point of view
of the gauge-hierarchy the Higgs is essentially IR-localized. However, such a
bulk Higgs differs from the brane-localised Higgs in the freedom that it
offers in exploring the parameter space of the model when confronted with
electroweak precision constraints.

Model | $m_{KK}(TeV)$ | b
---|---|---
Normal RS | 5.9 | 2.00
Deformed RS
($\nu=0.8,\Delta=1$) | 2.3 | 2.00
Deformed RS
($\nu=1,\Delta=.1$) | 1.7 | 2.00
Custodial RS | 2.88 | 2.00
Table 7: Summary of results for the various models at 3$\sigma$

A few remarks about the collider implications of bulk RS models are in order.
Generically, in these models the gauge boson KK modes provide the most
interesting signals and the KK gluon is, of these, the most important
Agashe:2006hk ; Agashe:2007ki ; Agashe:2008jb . The production cross-section
of the KK gauge boson modes is very small partly because of the couplings of
these modes to the SM particles but also because of the strong constraints on
the masses of the KK modes coming from electroweak and flavour constraints.
The cross-sections for other KK modes, like those of the fermions, are even
smaller than that of the gauge boson KK modes (except in some versions of the
RS model where the Higgs is treated as a pseudo-Nambu Goldstone boson). The
collider tests of the bulk RS models are therefore difficult and several
studies which propose probing alternative production channels have been
presented Guchait:2007jd ; Allanach:2009vz  but the range of masses probed by
these processes is just marginally larger than that allowed by precision
constraints. In view of this, our results of the global fit for the deformed
metric case are very encouraging. Unlike the custodial symmetry case for which
the global fits yield a bound on the mass of the first KK mode of about 2.9
TeV, one gets a lower bound of around 2.3 TeV at 3 $\sigma$ for the case of
the deformed metric for $\nu=0.8~{}\text{and}~{}\Delta=1$. This bound reduces
to about 1.7 TeV $\nu=1~{}\text{and}~{}\Delta=0.1$ Table[7] gives a summary of
the results obtained. A collider analysis for such class of models was done in
deBlas:2012qf. The deformed metric model then is testable at the LHC at a
statistically significant level and a more detailed study of the collider
implications of this model is called for.

Acknowledgements A.I. and K.S. would like to thank the Centre for High Energy
Physics, IISc for its hospitality during his visit where part of the
discussions were conducted.

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# Generalizations of the Bassian and co-Bassian
Properties for Abelian Groups

Andrey R. Chekhlov  Department of Mathematics and Mechanics, Tomsk State
University, 634050 Tomsk, Russia<EMAIL_ADDRESS><EMAIL_ADDRESS>,
Peter V. Danchev  Institute of Mathematics and Informatics, Bulgarian Academy
of Sciences, 1113 Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Patrick W. Keef  Department of Mathematics, Whitman College, Walla Walla,
WA 99362, USA<EMAIL_ADDRESS>
###### Abstract.

Trying to finalize in some way the present subject, this paper targets to
generalize substantially the notions of Bassian and co-Bassian groups by
introducing the so-called finitely (co-)Bassian groups, semi (co-)Bassian
groups, fully generalized (co-)Bassian groups, absolutely generalized
(co-)Bassian groups and establishing their crucial properties and
characterizations. In fact, some of the concepts give nothing new by
coinciding in the reduced case with the well-known (co-)Bassian property.
However, in some of the definitions, the situation is slightly more
complicated and we obtain a few new and interesting things by showing that the
extensions of the Bassian and co-Bassian properties are totally distinct each
other.

###### Key words and phrases:

Abelian groups; (co-)Bassian groups; Generalized (co-)Bassian groups, Finitely
(co-)Bassian group, Semi (co-)Bassian groups, Fully generalized (co-)Bassian
groups, Absolutely generalized (co-)Bassian groups

###### 2010 Mathematics Subject Classification:

20K10

(Dedicated to the anniversary of the 100th birthday of László Fuchs)

##  1\. Definitions and Motivations

All groups considered in the current paper will be Abelian and additively
written. Our notations and terminology, as usual, will mainly agree with the
classical books [7] and [8], respectively. For instance, if $G$ is a group,
the letter $T=T(G)$ will denote its maximal torsion subgroup with $p$-primary
component $T_{p}=T_{p}(G)$. Likewise, if $x\in G$, then the symbol $|x|_{p}$
will denote its $p$-height as computed in $G$. Also, for any integer $n\geq
1$, the notation $\mathbb{Z}(p^{n})$ will stand for the traditional cyclic
group of order $p^{n}$.

The following concepts are central to our investigations (see [1], [2], [6],
[3] as well): a group $G$ is said to be Bassian if it cannot be embedded in a
proper homomorphic image of itself or, equivalently, if the existence of an
injective homomorphism $G\to G/N$, for some subgroup $N$ of $G$, forces that
$N=\\{0\\}$.

Generally, if this injection implies that $N$ is a direct summand of $G$, then
$G$ is named generalized Bassian and, in a more general form, if $N$ is
essential in a direct summand of $G$, then $G$ is termed semi-generalized
Bassian.

Restated slightly, the Main Theorem of [1] asserts that a group $G$ is Bassian
if, and only if, it has finite torsion-free rank and, for all primes $p$,
$T_{p}$ is also finite. Thus, it is pretty clear that the standard quasi-
cyclic $p$-group $\mathbb{Z}(p^{\infty})$ is definitely not Bassian.

Unfortunately, the generalized Bassian groups were not completely
characterized in [2], but it was shown there that if $G$ is generalized
Bassian, then $pT_{p}$ must be finite for all primes $p$ ([2, Lemma 3.1]) as
well as it was established in [6] that such a group $G$ has finite torsion-
free rank.

We now can state the following four new specifications of these Bassian
properties.

######  Definition 1.1.

A group $G$ is called finitely Bassian provided that, for any subgroup $N$ of
$G$, whenever there is an injection $G\to G/N$, then $N$ is finite.

It is pretty clear that Bassian groups are finitely Bassian, and also it is
obvious that torsion-free finitely Bassian groups are Bassian. Generally, we
shall curiously illustrate in what follows even that the classes of finitely
Bassian and Bassian groups do coincide in the reduced case.

We just now recall that an arbitrary subgroup $E$ of a group $G$ is essential
in $G$ if, for any non-zero subgroup $S$ of $G$, the intersection between $E$
and $S$ is also non-zero. It is an elementary exercise to see that every
subgroup will be essential as a subgroup of itself and thus, in particular,
the zero subgroup $\\{0\\}$ is essential in itself too.

So, we introduce the following.

######  Definition 1.2.

A group $G$ is called semi Bassian provided that, for any subgroup $N$ of $G$,
whenever there is an injection $G\to G/N$, then either $N=\\{0\\}$ or $N$ is
essential in $G$.

Thus, under these circumstances, each Bassian group is semi Bassian, but we
shall demonstrate in the sequel that the converse manifestly fails.

We recall that a subgroup $F$ of a group $G$ is fully invariant if
$\phi(F)\subseteq F$ for all endomorphism $\phi$ of $G$. So, we arrive at the
following.

######  Definition 1.3.

A group $G$ is called fully generalized Bassian provided that, for any
subgroup $N$ of $G$, whenever there is an injection $G\to G/N$, then $N$ is a
fully invariant direct summand of $G$.

It is quite apparent that these groups are situated between Bassian groups and
generalized Bassian groups. We, however, will show in the sequel that the
converse manifestly fails, i.e., these three classes are independent each
other, thus motivating the study in-depth of this interesting class of groups.

We recall that a subgroup $A$ of a group $G$ is its absolute direct summand
if, for any $A$-high subgroup $V$ of $G$, it must be that $G=A\oplus V$. By
definition, such a group $V$ is maximal with respect to $V\cap A=\\{0\\}$.

######  Definition 1.4.

A group $G$ is called absolutely generalized Bassian provided that, for any
subgroup $H$ of $G$, whenever there is an injection $G\to G/H$, then $H$ is an
absolute direct summand of $G$.

It is rather evident that these so-defined groups are situated between Bassian
groups and generalized Bassian groups. We, however, will show in the sequel
that in two of situations the converse manifestly fails, namely that the
classes of semi-Bassian and absolutely generalized Bassian groups are
independent each other, thus motivating their study in-depth. Unfortunately,
the fully generalized Bassian groups are precisely the Bassian ones.

On the other vein, in [9] was introduced two important classes of groups like
this: a group $G$ is said to be co-Bassian or, respectively, generalized co-
Bassian if, for all subgroups $H\leq G$, whenever there is an injection
$\phi:G\to G/H$ either $\phi(G)=G/H$ or, respectively, the group $\phi(G)$ is
a direct summand of the factor-group $G/H$. Fortunately, these groups were
completely described in terms of invariants.

More generally, in [4] was considered and examined the case of groups $G$
where $\phi(G)$ is essential in a direct summand of the quotient-group $G/H$,
calling them semi-generalized co-Bassian and succeeded to characterize all
such $p$-groups and groups of finite torsion-free rank.

Nevertheless, it is rather interesting to know what happens when the image
$\phi(G)$ of the group $G$ is either of finite index in $G/H$, or is an
essential subgroup of $G/H$, or is a fully invariant direct summand of $G/H$,
or is an absolute direct summand of $G/H$, respectively.

Precisely, we state the following four new notions.

######  Definition 1.5.

A group $G$ is called finitely co-Bassian provided that, for any subgroup $H$
of $G$, whenever there is an injection $\phi:G\to G/H$, then $\phi(G)$ has
finite index in $G/H$.

######  Definition 1.6.

A group $G$ is called semi co-Bassian provided that, for any subgroup $H$ of
$G$, whenever there is an injection $\phi:G\to G/H$, then $\phi(G)$ is
essential in $G/H$.

We, however, will show in the sequel that these two notions curiously do
coincide and will classify them accordingly.

######  Definition 1.7.

A group $G$ is called fully generalized co-Bassian provided that, for any
subgroup $H$ of $G$, whenever there is an injection $\phi:G\to G/H$, then
$\phi(G)$ is a fully invariant direct summand of $G/H$.

######  Definition 1.8.

A group $G$ is called absolutely generalized co-Bassian provided that, for any
subgroup $H$ of $G$, whenever there is an injection $\phi:G\to G/H$, then
$\phi(G)$ is an absolute direct summand of $G/H$.

And so, our basic motivation to explore significantly these groups is that the
class of generalized Bassian groups is far from being completely characterized
(cf. [1], [2] and [6]) as well as that the class of generalized co-Bassian
groups is only numerically characterized (see [9]), and thus the given above
specifications of the property ”direct summand” in the corresponding
definitions are worthy of consideration and investigation.

That is why, our further work is subsequently organized in the following
manner: in the next two sections, we state and prove our major results, which
motivate us to write this research. The first one contains four subsections
pertaining to generalizations of the Bassian property in which the main
achievements are Theorems 2.3, 2.4, 2.6 and 2.8, respectively, as well as the
second one also contains three subsections pertaining to generalizations of
the co-Bassian property in which the chief achievements are Theorems 3.4, 3.5
and 3.6, respectively.

##  2\. Generalizing the Bassian Property

For the convenience of the reader, we include the following.

######  Lemma 2.1.

[9, Lemma 2.4] Suppose $G$ has finite (torsion-free) rank, $N$ is a subgroup
of $G$, $\pi:G\to\overline{G}:=G/N$ is the usual epimorphism and
$\phi:G\to\overline{G}$ is an injection. Then, $N\subseteq T$, so that
$\overline{T}=T/N$ is the torsion subgroup of $\overline{G}$. In addition,
$\pi$ induces an isomorphism $\pi^{\prime}:G/T\cong\overline{G}/\overline{T}$
and $\phi$ induces an embedding
$\phi^{\prime}:G/T\to\overline{G}/\overline{T}$. Finally, if $G/T$ is
divisible, then $\phi^{\prime}$ is an isomorphism.

###### Proof.

Let $S$ be the torsion subgroup of $\overline{G}$; clearly,
$\phi(T)=\phi(G)\cap S$ and $\pi(T)\subseteq S$. So, $\phi$ and $\pi$ induce
injective and surjective homomorphisms

$\phi^{\prime},\pi^{\prime}:G/T\to\overline{G}/S,$

respectively.

Since both $G/T$ and $\overline{G}/S$ are torsion-free of finite rank, it
easily follows that

$\pi^{\prime}:G/T\to\overline{G}/S$

is an isomorphism. And since

$\pi^{\prime}([N+T]/T)=[\pi(N)+S]/S=\\{0\\},$

we have $[N+T]/T=\\{0\\}$, i.e., $N\subseteq T$, so that $S=\overline{T}$.
Finally, if $G/T$ is divisible, then an easy verification shows that
$\phi^{\prime}$ is an isomorphism, as required. ∎

We will make frequent use of the following construction.

######  Lemma 2.2.

Suppose $G$ is a group that has subgroups $N\leq M$ such that $M/N$ is
divisible and there is an injection $M\to M/N$. Then, there is a splitting
$G/N\cong(M/N)\oplus(G/M)$ and an injective homomorphism $\phi:G\to G/N$.

###### Proof.

Since $M/N$ is divisible, the splitting

$G/N\cong(M/N)\oplus(G/M)$

is immediate. Note that the injection $M\to M/N$ will extend to a homomorphism
$\gamma:G\to M/N$. For each $x\in G$, defining $\phi(x)=(\gamma(x),x+M)$ can
readily be seen to be injective. ∎

Here we completely describe the afore-defined three kinds of generalized
Bassian groups as follows.

###  2.1. Finitely Bassian groups

We begin this section with the following quite surprising assertion by showing
that, actually, there is nothing new in Definition 1.2 concerning the reduced
case. In other words, the classes of reduced semi-Bassian groups and reduced
Bassian groups curiously do coincide.

However, the case of divisible groups is also of some interest being non-
trivial. For example, consider
$G={\mathbb{Z}}(2^{\infty})\oplus{\mathbb{Z}}(3^{\infty})$. If $N\leq G$ and
$\phi:G\to G/N$ is injective, it plainly follows that $N$ is finite and even,
in fact, cyclic. So, this $G$ is finitely Bassian, as expected.

Concretely, we offer the following.

######  Theorem 2.3.

The group $G$ is finitely Bassian if, and only if, $G=A\oplus C$, where $A$ is
a divisible hull of a finite group and $C$ is a Bassian group.

###### Proof.

Suppose first that $G=A\oplus C$ is as above, $N$ is a subgroup of $G$ and

$\phi:G\to G/N=:\overline{G}$

is an injection. So, by Lemma 2.1, it must be that $N\leq T$. If
$\pi:G\to\overline{G}$ is the canonical epimorphism, then for every prime $p$,
the maps $\pi,\phi$ induce, respectively, surjective and injective
homomorphisms $T_{p}\to\overline{T}_{p}$. This automatically implies that
$T_{p}$ and $\overline{T}_{p}$ have the same $p$-rank. Now, if $N_{p}$ was
infinite, we could conclude that it is not reduced, which would imply that
$\overline{T}_{p}\cong T_{p}/N_{p}$ would have $p$-rank strictly less than the
$p$-rank of $T_{p}$. Since this cannot happen, each $N_{p}$ is finite.

For any prime $p$ such that $A_{p}=\\{0\\}$, $T_{p}$ is finite, so that it is
Bassian. Therefore, for such $p$, $N_{p}=\\{0\\}$. And since this is true for
all but finitely many primes, we can conclude that $N$ is finite.

Conversely, suppose that $G$ is finitely Bassian. Let $A$ be the maximal
divisible subgroup of $T$, so that $G=A\oplus C$ for some subgroup $C$. If $A$
decomposed into an infinite number of non-zero terms, it would follow that
there was an infinite subgroup $N\leq A$ such that $A\cong A/N$ (for example,
$N$ could be the socle of $A$). This would immediately imply an isomorphism

$\phi:G=A\oplus C\to(A/N)\oplus C,$

so that $G$ is not finitely Bassian. Consequently, $A$ must be a divisible
hull of a finite subgroup, for instance, its socle.

Let $S$ be the torsion subgroup of $C$. If $p$ is a prime, we want to show
that $S_{p}$ is finite. We first claim that it is bounded: if this failed,
then $S_{p}$ would have a countable unbounded reduced subgroup $M$. This $M$,
in turn, will have a basic subgroup $N$ such that $M/N$ has countably infinite
rank. Since there is clearly an injection $M\to M/N$, thanks to Lemma 2.2,
there is an injection

$\phi:G\to G/N\cong(M/N)\oplus(G/M).$

And since $N$ is infinite, this contradicts that $G$ is finitely Bassian.

Therefore, $S_{p}$ is bounded. If it were infinite, then for some integer $k$
there would be a decomposition

$G=(\oplus_{i<\omega}\mathbb{Z}(p^{k})_{i})\oplus G^{\prime},$

where each $\mathbb{Z}(p^{k})_{i}$ is a copy of $\mathbb{Z}(p^{k})$. Clearly,
if $N=\oplus_{i<\omega}\mathbb{Z}(p^{k})_{2i}$, then $N$ is infinite and
$G/N\cong G$, contradicting that $G$ is finitely Bassian.

Thus, each $S_{p}$ is finite, so we must show that $G$ has finite rank. If
this failed, we could find a free subgroup $F\leq G$ of countably infinite
rank, and a subgroup $N\leq F$ such that $F/N$ is divisible of countably
infinite rank. Since there is clearly an injection $F\to F/N$, using Lemma
2.2, there is an injection

$\phi:G\to(F/N)\oplus(G/F).$

And since $N$ is infinite, this contradicts that $G$ is finitely Bassian, as
needed. ∎

###  2.2. Semi Bassian groups

We characterize the semi Bassian groups in the next result.

######  Theorem 2.4.

The group $G$ is semi Bassian if, and only if, either
$G\cong\mathbb{Z}(p^{\infty})$ or $G$ is a Bassian group.

###### Proof.

Suppose first that $G$ is semi Bassian. If $T$ is not reduced, then
$G=G^{\prime}\oplus Z$, where $Z\cong\mathbb{Z}(p^{\infty})$. Since $G\cong
G/Z[p]$, we could conclude that $Z[p]$ is essential in $G$. This clearly
implies that $G^{\prime}=\\{0\\}$, so that we find
$G=Z\cong\mathbb{Z}(p^{\infty})$. So, we may assume $T$ is reduced.

If $p$ is a prime, we want to show that $T_{p}$ is finite. We first claim that
it is bounded: if this failed, then $T_{p}$ would have a countable unbounded
reduced subgroup $M$. This $M$, in turn, will have a basic subgroup $N$ such
that $M/N$ has countable rank. Since there is an obvious injection $M\to M/N$,
appealing to Lemma 2.2, there is an injection

$\phi:G\to G/N\cong(M/N)\oplus(G/M).$

And since $N$ is not essential in $M$, it is apparently not essential in $G$
too; this surely contradicts that $G$ is semi Bassian.

Therefore, $T_{p}$ is bounded. If it were infinite, then, for some integer
$k$, there would be a decomposition

$G=(\oplus_{i<\omega}\mathbb{Z}(p^{k})_{i})\oplus G^{\prime},$

where each $\mathbb{Z}(p^{k})_{i}$ is a copy of $\mathbb{Z}(p^{k})$.
Evidently, if $N=\oplus_{i<\omega}\mathbb{Z}(p^{k})_{2i}$, then $N$ is clearly
not essential in $G$ and $G/N\cong G$, contradicting that $G$ is finitely
Bassian.

Thus, each $T_{p}$ is finite, so to show $G$ is Bassian we must prove that $G$
has finite (torsion-free) rank. However, if this failed, we could find a free
subgroup $F\leq G$ of countably infinite rank, and a subgroup $N\leq F$ such
that $F/N$ is divisible of countably infinite rank. Since there is plainly an
injection $F\to F/N$, using Lemma 2.2, there is an injection

$\phi:G\to(F/N)\oplus(G/F).$

And since, again, $N$ is not essential in $M$, it is directly not essential in
$G$ as well. This again contradicts that $G$ is semi Bassian, as needed.

Since a quasi-cyclic or a Bassian group is immediately semi-Bassian, the
converse is automatic. ∎

It is worthy of noticing that if we consider the following modified version of
Definition 1.2, there will be nothing new and interesting. In fact, let us
call a group $G$ almost Bassian provided that, for any non-zero subgroup $N$
of $G$, whenever there is an injection $G\to G/N$, then $N$ is essential in
$G$.

In fact, $G$ will be “almost Bassian” exactly when it is “semi Bassian”:
indeed, if $P(\phi)$ is the statement “$\phi$ is an injection”, $Q(N)$ is the
statement “$N=\\{0\\}$” and $R(N)$ is the statement “$N$ is essential in $G$”,
then the statement “$G$ is semi Bassian” can be written symbolically as

$\forall(N\leq G)\forall(\phi:G\to G/N)(P(\phi)\Rightarrow(Q(N)\lor R(N))$

and “$G$ is almost Bassian” can be written symbolically as

$\forall(N\leq G)\forall(\phi:G\to G/N)((P(\phi)\land\lnot Q(N))\Rightarrow
R(N)).$

But, it is readily to see that the stuff inside the quantifiers is
tautologically equivalent.

###  2.3. Fully generalized Bassian groups

We start in this section with the following quite surprising assertion by
demonstrating that, actually, there is nothing new in Definition 1.3. In other
words, the classes of fully generalized Bassian groups and Bassian groups
curiously do coincide.

To show that, we first recall that a group $G$ is B+E if it is isomorphic to
the direct sum $B\oplus E$, where $B$ is a Bassian group and $E$ is an
elementary group.

We now have the following helpful assertion.

######  Corollary 2.5.

([6, Corollary 3.1]) Every generalized Bassian group is B+E.

We are now in a position to establish the promised above equivalence.

######  Theorem 2.6.

The group $G$ is fully generalized Bassian if, and only if, $G$ is a Bassian
group.

###### Proof.

Certainly, by definition, any Bassian group is fully generalized Bassian, so
assume $G$ is fully generalized Bassian. It follows that $G$ is generalized
Bassian, and hence B+E as stated in the previous statement. Hence, write
$G=B\oplus E$ with $B$ Bassian and $E$ elementary.

If we can show, for each $p$, that $E_{p}$ has finite $p$-rank, then it will
follow that $G$ is also Bassian, as required. So, by way of contradiction,
suppose $p$ is some prime such that $E_{p}$ has infinite $p$-rank. It readily
follows that $G=G^{\prime}\oplus E_{p}$ for some subgroup $G^{\prime}\leq G$.
Let $N$ be a non-zero cyclic direct summand of $E_{p}$. It must be that
$E_{p}\cong E_{p}/N$, so that $G\cong G/N$. However, since $N$ is obviously
not fully invariant in $G$, we deduce that $G$ is not fully generalized
Bassian, giving our contradiction, as pursued. ∎

###  2.4. Absolutely generalized Bassian groups

First of all, we are prepared to use the following well-known statement.

######  Lemma 2.7.

[7, §10, Exersize 9] The direct summand $B$ of an arbitrary group $A$ is its
absolute direct summand if, and only if, $B$ is either a divisible group or
the quotient $A/B$ is a torsion group such that the $p$-component of which is
equal to zero at multiplying by $p^{k}$ if $B\setminus pB$ has an element of
order $p^{k}$.

Now, we are ready to prove here the following assertion.

######  Theorem 2.8.

The group $G$ is absolute generalized Bassian if, and only if, exactly one of
the following two conditions holds:

(1) $G$ is a torsion group with elementary or finite $T_{p}$ for each prime
$p$.

(2) $G$ is a Bassian non-torsion group.

###### Proof.

The sufficiency being an almost straightforward observation, we shall attack
the necessity. So, assume $G$ is absolute generalized Bassian, but not
Bassian; thus, we need to show that $G$ is a torsion group satisfying the
required property (1).

To that goal, since $G$ is absolute generalized Bassian, it is generalized
Bassian, and hence B+E as already noticed above. Therefore, for any prime $p$,
$T_{p}$ is the direct sum of an elementary and a finite group. Suppose $p$ is
any prime such that $T_{p}$ (and hence $G$) has a maximal elementary direct
summand $E_{p}$ of infinite $p$-rank; since $G$ is not Bassian, such a $p$
exists. If $G=E_{p}\oplus G^{\prime}$ and $N$ is a cyclic direct summand of
$E_{p}$, then it is also a direct summand of $G$. Again, there is an
isomorphism $E_{p}\cong E_{p}/N$ which extends to an isomorphism $G\to G/N$.
Therefore, since $G$ is absolutely generalized Bassian, $N$ must be an
absolute direct summand of $G$. By usage of Lemma 2.7, this implies that
$G^{\prime}$ is a torsion group with no $p$-torsion. However, since the above
argument works for all such primes $p$, it clearly implies that $G$ is of the
form presented in (1), as needed. ∎

So, we arrived at the interesting fact that the class of absolute generalized
Bassian groups does not coincide with the class of Bassian groups and properly
lies between the two classes of Bassian groups and generalized Bassian groups.

##  3\. Generalizing the co-Bassian Property

Here we completely describe the afore-defined two kinds of generalized co-
Bassian groups as follows.

###  3.1. Finitely and semi co-Bassian groups

In [9], a $p$-group $T_{p}$ was said to have generalized finite $p$-rank if
there are elements of $\omega\cup\\{\infty\\}$,
$\alpha_{1}<\alpha_{2}<\cdots<\alpha_{n}$ and cardinals $\rho_{i}$
($i=1,2,\dots,n$) such that

$T_{p}\cong\mathbb{Z}(p^{\alpha_{1}})^{(\rho_{1})}\oplus\cdots\oplus\mathbb{Z}(p^{\alpha_{n}})^{(\rho_{n})},$

where $\rho_{j}$ is finite whenever $j>1$.

We now recollect the following two statements.

######  Theorem 3.1 ([9], Theorem 2.5).

The group $G$ is generalized co-Bassian if, and only if, one of the following
two conditions holds:

(1) $G$ is divisible;

(2) $G/T$ is divisible of finite rank and, for each prime $p$, $T_{p}$ has
generalized finite $p$-rank.

######  Theorem 3.2 ([9], Theorem 2.6).

The group $G$ is co-Bassian if, and only if, $G/T$ is divisible of finite rank
and, for each prime $p$, $T_{p}$ has finite $p$-rank.

We pause for a simple but useful observation.

######  Lemma 3.3.

Suppose $N\leq G$ and $\overline{G}=G/N$ decomposes as
$(\oplus_{i<\omega}A_{i})\oplus G^{\prime}$, where each $A_{i}$ is a copy of a
single non-zero group, and $\phi:G\to\overline{G}$ is an injective
homomorphism. Then, $G$ will be neither finitely co-Bassian nor semi co-
Bassian.

###### Proof.

Let $\gamma:\overline{G}\to\overline{G}$ be the identity on $G^{\prime}$, and
map each $A_{i}$ isomorphically onto $A_{2i}$. It, thus, follows at once that
$\phi^{\prime}:=\gamma\circ\phi$ is injective and hence a simple verification
guarantees that

$\phi^{\prime}(G)\subseteq(\oplus_{i<\omega}A_{2i})\oplus G^{\prime}$

is neither essential in $\overline{G}$, nor is the quotient
$\overline{G}/\phi^{\prime}(G)$ finite, as wanted. ∎

This brings us to our characterization of finitely co-Bassian and semi co-
Bassian groups, which classes curiously coincide.

######  Theorem 3.4.

For a group $G$, the following are equivalent:

(a) $G$ is finitely co-Bassian;

(b) $G$ is semi co-Bassian;

(c) $G$ has finite torsion-free rank and each $T_{p}$ has finite $p$-rank.

###### Proof.

We will show that if $G$ is of the form specified in (c), then it satisfies
both (a) and (b); and conversely, if it is not of that form, then both (a) and
(b) fail.

Suppose first that $G$ satisfies (c), and let $N\leq G$,
$\pi:G\to\overline{G}:=G/N$ be the canonical epimorphism and
$\phi:G\to\overline{G}$ be an injective homomorphism. It follows from Lemma
2.1 that $N\leq T$ and $T/N=:\overline{T}$ is the torsion subgroup of
$\overline{G}$. In addition, $\phi$ and $\pi$ restrict to injective and
surjective homomorphisms $T\to\overline{T}$, respectively. However, by Theorem
3.2, $T$ is co-Bassian. Therefore, $\overline{T}=\phi(T)$.

Consider now the commutative diagram with short-exact rows, whose columns are
induced by the injection $\phi$:

$\begin{matrix}\\{0\\}&\to&T&\to&G&\to&G/T&\to&\\{0\\}\cr&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\phi_{T}$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\phi$}}$\hss}&&\downarrow\hbox
to0.0pt{$\vbox{\hbox{$\scriptstyle\phi^{\prime}$}}$\hss}&&\cr\\{0\\}&\to&\overline{T}&\to&\overline{G}&\to&\overline{G}/\overline{T}&\to&\\{0\\}\cr\end{matrix}$

Again referring to Lemma 2.1, we detect that $\phi^{\prime}$ will be
injective. So, since $G/T$ and $\overline{G}/\overline{T}$ are torsion-free
groups of the same finite rank and $\phi_{T}$ is onto, it elementarily follows
that $\phi(G)$ is essential in $\overline{G}$, proving (b).

In addition, the above commutative diagram implies an isomorphism

$G/\phi(G)\cong(\overline{G}/\overline{T})/\phi^{\prime}(G/T).$

And on the other hand, since $\phi^{\prime}$ is injective, and with the aid
Lemma 2.1 we represent $G/T\cong\overline{G}/\overline{T}$, one can conclude
that

$\phi^{\prime}(G/T)\cong G/T\cong\overline{G}/\overline{T}.$

Recall that, whenever $X$ is a torsion-free group of finite rank and $Y$ is a
subgroup of $X$ that is isomorphic to $X$, then we can infer that $X/Y$ is
finite. Therefore, $(\overline{G}/\overline{T})/\phi^{\prime}(G/T)$ is finite,
so that $G/\phi(G)$ is finite, which proves (a).

Conversely, suppose $G$ does not satisfy (c); we want to show that (a) and (b)
fail, as well. First, if $p$ is a prime, $n<\omega$ and the $n$th Ulm
invariant $(p^{n}T)[p]/(p^{n+1}T)[p]$ is infinite, then Lemma 3.3 (with
$N=\\{0\\}$, $\pi=1_{G}=\phi$ and $A_{i}\cong\mathbb{Z}(p^{n+1})$) implies
that $G$ does not satisfy either (a) or (b). So, we may assume this does not
hold for any prime $p$.

It follows that if $T_{p}$ does not have finite rank, then it has an countable
unbounded reduced subgroup $M$. If $N$ is a basic subgroup of $M$ such that
$M/N$ has countably infinite rank, then it is clear that $M$ embeds in $M/N$.
So, with Lemma 2.2 at hand, $G$ embeds in

$G/N\cong(M/B)\oplus(G/M),$

so that Lemma 3.3 guarantees that $G$ does not satisfy (a) or (b), as pursued.

Thus, we may assume that $T_{p}$ has finite rank for all primes $p$. It will
suffice, therefore, to show that it $G$ has finite rank. If this failed, then
$G$ would have a free subgroup $F$ of countable rank, and $F$ would have a
subgroup $N$ such that $F/N$ is a torsion-free divisible group of countably
infinite rank. As it is again clear that $F$ embeds in $F/N$, by Lemma 2.2,
$G$ embeds in

$G/N\cong(F/N)\oplus(G/F),$

so that by Lemma 3.3, $G$ does not satisfy (a) or (b), as desired. ∎

###  3.2. Fully generalized co-Bassian groups

We are now planning to establish the following result.

######  Theorem 3.5.

The group $G$ is fully generalized co-Bassian if, and only if, it is co-
Bassian, i.e., $G/T$ is divisible of finite rank and, for each prime $p$,
$T_{p}$ has finite $p$-rank.

###### Proof.

It is elementary that if $G$ is co-Bassian, then it is fully generalized co-
Bassian.

Conversely, suppose $G$ is fully generalized co-Bassian. In particular, it
immediately follows that $G$ is generalized co-Bassian, so that Theorem 3.1 is
applicable.

Note that if $G=G^{\prime}\oplus(\oplus_{i<\omega}A_{i})$, where each $A_{i}$
is a copy of a single non-zero group $A$, then $G$ cannot be fully generalized
co-Bassian: indeed, for $N=\\{0\\}$, there is an obvious injection $\phi:G\to
G\cong G/N$ such that

$\phi(G)=G^{\prime}\oplus(\oplus_{i<\omega}A_{2i})\subseteq G.$

But $\phi(G)$ is certainly not fully invariant in $G\cong G/N$, as asked for.

So, if Theorem 3.1(a) applies, i.e., $G$ is divisible, then it routinely
follows that each divisible $T_{p}$ will have finite $p$-rank and $G/T$ will
be divisible of finite rank, completing the proof.

On the other hand, suppose Theorem 3.1(b) applies. We already know that $G/T$
will be divisible of finite rank. And if $p$ is a prime, then we know that
$T_{p}$ must have generalized finite $p$-rank. For such a prime, if $T_{p}$
did not actually have finite rank, then we could conclude that $G$ has a
direct summand of the form ${\mathbb{Z}}(p^{\alpha})^{(\omega)}$, where
$\alpha$ is in $\omega\cup\\{\infty\\}$. But, again by the above observation,
this implies that $G$ is not fully generalized co-Bassian, as expected. ∎

###  3.3. Absolutely generalized co-Bassian groups

We now have all the ingredients necessary to attack the truthfulness of the
following affirmation.

######  Theorem 3.6.

The group $G$ is absolutely generalized co-Bassian if, and only if, it is
generalized co-Bassian, i.e., $G$ is either divisible, or $G/T$ is divisible
of finite rank and each $T_{p}$ has generalized finite rank.

###### Proof.

Certainly, if $G$ is absolutely generalized co-Bassian, then it must be
generalized co-Bassian. So, assume $G$ is generalized co-Bassian. So, if

$\pi,\phi:G\to G/N=:\overline{G}$

are surjective and injective homomorphisms, we need to verify that $\phi(G)$
is an absolute direct summand of $\overline{G}$.

Clearly, if $G$ is divisible, then so is $\phi(G)$, so that it is an absolute
direct summand, as required. So, we may assume that $G/T$ is divisible of
finite rank and each $T_{p}$ has generalized finite rank. By Lemma 2.1, $\phi$
induces an isomorphism $\phi^{\prime}:G/T\cong\overline{G}/\overline{T}$ on
the corresponding quotients. In addition, $N\subseteq T$ and, consequently,

$\overline{T}=T/N\cong\oplus_{p}T_{p}/N_{p}\cong\oplus_{p}\overline{T}_{p}$

will be the torsion subgroup of $\overline{G}$.

Since $G$ is generalized co-Bassian, it must be that
$\overline{G}=\phi(G)\oplus C$. Furthermore, as $\phi$ induces an isomorphism
$\phi^{\prime}:G/T\cong\overline{G}/\overline{T}$ on the corresponding
quotients, it follows that $C\subseteq\overline{T}$, so that $C$ must be a
torsion group.

To prove $\phi(G)$ is an absolute direct summand of $\overline{G}$, let $p$ be
some prime. Since $T_{p}$ has generalized finite rank, there is a
decomposition $T_{p}=H_{p}\oplus F_{p}$, where either $H_{p}$ is divisible and
$F_{p}=\\{0\\}$, or there is an $k<\omega$ such that
$H_{p}\cong\mathbb{Z}(p^{k})^{(\gamma_{p})}$ and $F_{p}$ has finite $p$-rank
and no direct summand isomorphic to $\mathbb{Z}(p^{j})$ for any $j\leq k$.

If, however, $T_{p}=H_{p}$ is divisible, then there are no elements in
$\phi(G)\setminus p\phi(G)$ of order $p^{j}$ for any $j<\omega$. So, suppose
$T_{p}=H_{p}\oplus F_{p}$ and $k<\omega$ is as above. It follows that the maps
$\pi,\phi$ restrict to surjective and injective homomorphism $p^{k}T_{p}\to
p^{k}\overline{T}_{p}$, respectively. Consequently, since $p^{k}T_{p}$ is a
$p$-group of finite rank, so that it is co-Bassian, we have
$\phi:p^{k}\phi(T_{p})=p^{k}\overline{T}_{p}$. This implies that
$p^{k}C_{p}=\\{0\\}$, so that by virtue of Lemma 2.7, the image $\phi(G)$ is
an absolute direct summand of $\overline{G}$, as stated. ∎

Funding: The work of the first-named author, A.R. Chekhlov, is supported by
the Ministry of Science and Higher Education of Russia under agreement No.
075-02-2023-943. The work of the second-named author, P.V. Danchev, is
partially supported by the Junta de Andalucía, Grant FQM 264.

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