11institutetext: INFN, Sezione di Firenze and Department of Physics and
Astronomy, University of Florence
Via G. Sansone 1, 50019 Sesto Fiorentino, Italy

# Spin-1 resonances

Stefania De Curtis 11<EMAIL_ADDRESS>Daniele Dominici 11
<EMAIL_ADDRESS>
###### Abstract

Spin-1 resonances are naturally present in composite Higgs frameworks. We
first review a model independent approach to parameterize a single additional
heavy triplet and then we consider more realistic models arising in composite
Higgs scenarios where a larger number of spin-1 resonances is expected. In
these cases, finite width and interference effects can heavily affect the
bounds extracted from the data.

##  1 Introduction

The presence of additional spin-1 particles, triplets or singlets of $SU(2)$,
is a common feature of several beyond the Standard Model (BSM) frameworks.
They appear as Kaluza-Klein excitations of $W$, $Z$ and $\gamma$ in flat or
warped extra-dimension extensions of the SM, as new vector and axial-vector
resonances in walking technicolor inspired effective Lagrangians, and are
naturally present in composite Higgs scenarios (for references see, for
example, the recent reviews Panico:2015jxa ; Cacciapaglia:2020kgq ). They are
also expected in weakly interacting BSM theories like the Little Higgs models
or Grand Unified theories. In this chapter we review some of the topics
related to the theoretical approaches to describe the interactions of the new
spin-1 resonances with the SM particles and with a composite Higgs. The new
vectors could give clear signals in the Drell-Yan channel at the LHC, in the
dilepton or in the diboson channels. From the LHC measurements one could then
derive bounds on their masses and couplings. A very convenient tool to get
these limits is a general model independent approach Pappadopulo:2014qza .
This allows, from a comparison with the data, to get bounds on the couplings
to fermions and bosons of a new spin-1 triplet at fixed mass. This simplified
approach however is not able to reproduce extended models containing more than
one triplet. We consider then possible proposals describing two new triplets
which can describe the low-lying resonances relevant at the LHC and outline
the importance of the interference and finite width effects.

In Section 2 we review a model independent approach, to parameterize the
presence of an additional heavy triplet of spin-1 resonances, that makes use
of an effective Lagrangian description. Experimental data from CMS and ATLAS
collaborations, using this parameterization, are used in Section 3 to get
bounds on two particular models. In Section 4 we review a general method to
build effective Lagrangians describing two new heavy triplets, replicas of $W$
and $Z$, analize two particular models and review the bounds on their
parameter space using the LHC measurements. Finally in Section 5 we outline,
by considering a composite Higgs model, the relevance of the interference and
finite width effects in the analysis of the data to extract bounds on the
model itself.

##  2 Heavy vector triplet simplified model and bounds from the LHC
measurements

Following Pappadopulo:2014qza  (see also deBlas:2012qp ) we can parameterize
the presence of a new heavy vector triplet, interacting with SM gauge bosons
and fermions, by the following simple effective phenomenological Lagrangian
containing operators up to dimension $d=4$

$\displaystyle\displaystyle{\mathcal{L}}_{V}$ $\displaystyle=$
$\displaystyle\displaystyle-\frac{1}{4}D_{[\mu}V_{\nu]}^{a}D^{[\mu}V^{\nu]\;a}+\frac{m_{V}^{2}}{2}V_{\mu}^{a}V^{\mu\;a}\displaystyle+\,i\,g_{V}c_{H}V_{\mu}^{a}H^{\dagger}\tau^{a}{\overset{{}_{\leftrightarrow}}{D}}^{\mu}H+\frac{g^{2}}{g_{V}}c_{F}V_{\mu}^{a}J^{\mu\;a}_{F}\vspace{2mm}$
$\displaystyle\displaystyle+\frac{g_{V}}{2}c_{VVV}\epsilon_{abc}V_{\mu}^{a}V_{\nu}^{b}D^{[\mu}V^{\nu]c}+g_{V}^{2}c_{VVHH}V_{\mu}^{a}V^{\mu
a}H^{\dagger}H-\frac{g}{2}c_{VVW}\epsilon_{abc}W^{\mu\nu
a}V_{\mu}^{b}V_{\nu}^{c}$

Here $V_{\mu}^{a}$, $a=1,2,3$, is the new heavy vector triplet, in the adjoint
representation of $SU(2)_{L}$ and with vanishing hypercharge, describing the
charged and the neutral heavy spin-1 particles: $V_{\mu}^{\pm}=(V_{\mu}^{1}\mp
iV_{\mu}^{2})/{\sqrt{2}}$ and $V_{\mu}^{0}=V_{\mu}^{3}$. The covariant
derivatives in Eq. (LABEL:sml) are defined by

$D_{[\mu}V^{a}_{\nu]}=D_{\mu}V^{a}_{\nu}-D_{\nu}V^{a}_{\mu}\,,\;\;\;\;\;D_{\mu}V_{\nu}^{a}=\partial_{\mu}V_{\nu}^{a}+g\,\epsilon^{abc}W_{\mu}^{b}V_{\nu}^{c}\,,$
(2)

where $g$ denotes the $SU(2)_{L}$ gauge coupling. The last two terms of the
first row of Eq. (LABEL:sml) contain the interactions of the new vector bosons
with the Higgs and the three Goldstones (encoded in $H$) and therefore, by the
Equivalence Theorem, with the longitudinal components of $W$ and $Z$ which are
their relevant degrees of freedom at high energies, and with the SM fermions.
Specifically:

$i\,H^{\dagger}\tau^{a}{\overset{{}_{\leftrightarrow}}{D}}^{\mu}H=i\,H^{\dagger}\tau^{a}D^{\mu}H\,-\,i\,D^{\mu}H^{\dagger}\tau^{a}H\,,$
(3)
$D^{\mu}H=(\partial^{\mu}+ig\frac{\sigma^{a}}{2}W^{a}-ig^{\prime}\frac{\sigma^{3}}{2}Y^{\mu})H\,,$
(4)

with $g^{\prime}$ the weak hypercharge coupling, while $J_{F}^{\mu\;a}$ are
the SM left-handed fermionic currents

$\displaystyle
J_{F}^{\mu\;a}=\sum_{f}\overline{f}_{L}\gamma^{\mu}\frac{\sigma^{a}}{2}f_{L}\,.$
(5)

The operators in the second line of Eq. (LABEL:sml) are not relevant for LHC
because they do not contain vertices with SM gauge bosons.

Analysis at ATLAS and CMS on new heavy vector triplet models are performed by
using the first line operators, in terms of $m_{V}$ and two parameters $g_{H}$
and $g_{F}$ defined as

$g_{H}=g_{V}c_{H},\,\,\,\,g_{F}=\frac{g^{2}c_{F}}{g_{V}}\,.$ (6)

In Fig. 1 the CMS bounds on $(g_{H},g_{F})$, obtained by using an integrated
luminosity of 35.6 fb-1 at the LHC with a center of mass energy of 13 TeV
Sirunyan:2019vgt  are shown. A combination of searches for a heavy triplet of
resonances decaying into pairs of vector bosons, a vector boson and a Higgs
boson, two Higgs bosons, or pairs of leptons, has been considered. The bounds
are obtained for narrow resonances, the areas bounded by the thin gray contour
lines correspond to regions where the resonance widths are predicted to be
larger than the average experimental resolution (5%). The red dot and the
violet cross correspond to model A and B that we review in the next
subsection. A similar analysis has been performed by ATLAS Aaboud:2018bun .

Figure 1: Exclusion limits on the couplings of heavy vector resonances to
fermions $g_{F}$ and SM vector bosons and the Higgs boson $g_{H}$ for narrow
resonances obtained from the statistical combination of all the channels
(solid lines). The dilepton (dotted lines) and the diboson searches (dashed
lines) constrain different regions. Three resonance masses hypotheses (3.0,
4.0, and 4.5 TeV) are considered. The hatched bands indicate the regions
excluded. The analysis uses an integrated luminosity of 35.9 fb-1. From
Sirunyan:2019vgt .

Figure 2: Expected (dashed line) and observed (solid line) 95%CL limits on the
coupling strength $g_{W^{\prime}}/g_{W}$ as functions of the $W^{\prime}$
mass, for the muon channel. The area above the limit line is excluded. The SSM
$W^{\prime}$ couplings are shown as a dotted line. The analysis uses an
integrated luminosity of 35.9 fb-1. Similar bounds are extracted from the
electron channel. From Sirunyan:2018mpc .

In Fig. 2 we present a further result from a CMS analysis corresponding to the
same integrated luminosity of 35.9 fb-1, where only the charged channel in
Drell-Yan process of new resonances has been considered to obtain 95%CL limits
on the coupling strength $g_{W^{\prime}}/g_{W}$ as a function of the
$W^{\prime}$ mass Sirunyan:2018mpc . Notice that the for the Sequential
Standard Model (SSM), where the coupling of the $W^{\prime}$ is set at the SM
value, the bound on $M_{W^{\prime}}$ is around 5 TeV.

##  3 Two reference models

In this section we review two reference models that have been proposed in the
past to describe additional triplets and have been used in the ATLAS and CMS
analyses. They are a model, (A), with an extended electroweak symmetry
Barger:1980ix  and a strongly coupled composite model (B)Contino:2011np .

###  3.1 Model A

This model corresponds to an extension of the electroweak symmetry to
$SU(2)_{1}\otimes SU(2)_{2}\otimes U(1)_{Y}$ broken to $SU(2)\otimes
U(1)_{Y}$. Fermions are supposed to transform only under $SU(2)_{1}\otimes
U(1)_{Y}$. Two scalar fields are introduced: the SM Higgs $H$, a doublet of
$SU(2)_{1}$, and an additional one, $\Phi$, belonging to the representation
$\bf{(2,2)}$ of $SU(2)_{1}\otimes SU(2)_{2}\otimes U(1)_{Y}$, the latter
acquiring a vacuum expectation value:

$\langle\Phi\rangle=\begin{pmatrix}f&0\\\ 0&f\end{pmatrix}.$ (7)

This bidoublet breaks the $SU(2)_{1}\otimes SU(2)_{2}$ symmetry to the
diagonal $SU(2)$. The gauge coupling constants are identified as
$g_{V}=g_{2}$, $1/g^{2}=1/g_{1}^{2}+1/g_{2}^{2}$, with $g_{1,2}$ the gauge
couplings of $SU(2)_{1,2}$. Other details can be found in Barger:1980ix  and
Pappadopulo:2014qza . In Fig. 1 the violet cross represents this model. Only
masses of the new resonances higher than 4.5-5 TeV are allowed.

###  3.2 Model B

This is the simplest model where the Higgs boson is realized as a pseudo
Nambu-Goldstone from the breaking, at the scale $f$, of an extended symmetry,
$SO(5)\to SO(4)$, the so-called Minimal Composite Higgs Model Contino:2011np .
The construction of the Lagrangian is based on the Callan-Coleman-Wess-Zumino
formalism. The Higgs is identified as a (2,2) under the unbroken
$SU(2)_{L}\otimes SU(2)_{R}\sim SO(4)$ and new vectors are introduced as
representations of the unbroken $SO(4)$. In particular a new triplet field
$\rho_{\mu}$, $\bf{(3,1)}$ of $SU(2)_{L}\otimes SU(2)_{R}\sim SO(4)$ is
present in the spectrum. The Lagrangian describing the Higgs gauge boson
sector is given by

$\mathcal{L}_{\rho}=-\frac{1}{4\tilde{g}^{\prime
2}}(B_{\mu\nu})^{2}-\frac{1}{4\tilde{g}^{2}}(W^{a}_{\mu\nu})^{2}+\frac{f^{2}}{4}d_{\mu}^{i}d^{\mu
i}-\frac{1}{4g_{\rho}^{2}}({\rho}_{\mu\nu}^{a})^{2}+\frac{m_{\rho}^{2}}{2g_{\rho}^{2}}\left({\rho}_{\mu}^{a}-e_{\mu}^{a}\right)^{2}.$
(8)

The $\rho$ field strength is given by
$\displaystyle{\rho}_{\mu\nu}^{a}=\partial_{\mu}{\rho}_{\nu}^{a}-\partial_{\nu}{\rho}_{\mu}^{a}-\epsilon^{abc}{\rho}_{\mu}^{b}{\rho}_{\nu}^{c}$,
while the $d$ and ($\rho-e$) terms for the $SO(5)/SO(4)$ coset, in the large
$f$ limit, are given by (see also Ref. DeSimone:2012fs ):

$d_{\mu}^{i}d^{\mu\,i}=\frac{4}{f^{2}}|D_{\mu}H|^{2}+\frac{2}{3f^{4}}\left[(\partial_{\mu}|H|^{2})^{2}-4|H|^{2}|D_{\mu}H|^{2}\right]+O(\frac{1}{f^{6}})\,,$
(9)

and

$\rho^{a}_{\mu}-e_{\mu}^{a}=\rho_{\mu}^{a}+W_{\mu}^{a}-\frac{i}{f^{2}}H^{\dagger}\tau^{a}\overset{\leftrightarrow}{D_{\mu}}H+\frac{i}{f^{4}}|H|^{2}H^{\dagger}\tau^{a}\overset{\leftrightarrow}{D_{\mu}}H+O(\frac{1}{f^{6}}).$
(10)

The heavy vector triplet $V_{\mu}^{a}$ of Eq. (LABEL:sml), is identified as

$V_{\mu}^{a}\equiv\rho_{\mu}^{a}+W_{\mu}^{a}.$ (11)

After the breaking of $SU(2)\otimes U(1)_{Y}$, the couplings of the physical
Higgs to pairs of SM gauge bosons are rescaled by

$\sqrt{1-\xi},\,\,\,{\rm
with}\,\,\,\,\,\xi=\frac{v^{2}}{f^{2}},\,\,\,\,\,v=245~{}{\rm GeV}.$ (12)

The current limit on $\xi$, using $k_{V}=1.05\pm 0.04$ from ATLAS Aad:2019mbh
, is $\xi\lessapprox 0.06$ at 95%CL.

The coupling $g_{V}$ is here identified with $g_{\rho}$, and it turns out
Pappadopulo:2014qza  that

$c_{H}\sim c_{F}\sim 1\,.$ (13)

The red dot of Fig. 1 corresponds to $c_{H}=-1$ and $g_{V}=3$, for which
resonances ligher than 4-4.5 TeV are excluded. Other vector resonances like
$\rho_{R}$ belonging to the representation $\bf{(1,3)}$ of $SU(2)_{L}\otimes
SU(2)_{R}$ could be included in the model, see for instance Marzocca:2012zn .

##  4 Effective Lagrangians for new spin-1 triplets

A general procedure for building effective Lagrangians describing new vector
resonances is the so-called hidden local symmetry approach Bando:1984ej ;
Casalbuoni:1985kq ; Casalbuoni:1988xm . We review here the main ingredients
for two new triplets of vector resonances. For similar proposals in the
framework of walking technicolor models see Appelquist:1999dq ; Foadi:2007ue .
In these approaches the Higgs is described as an isosinglet scalar state.

We start considering the following group structure:
$G^{\prime}=SU(2)_{L}\otimes SU(2)_{R}\otimes SU(2)_{L}^{\prime}\otimes
SU(2)_{R}^{\prime}$ broken to the diagonal $SU(2)_{V}$. The nine Goldstone
bosons resulting from the spontaneous breaking can be described by three
independent $SU(2)$ fields $L(x)$, $R(x)$ and $M(x)$, transforming under the
extended symmetry group $G^{\prime}$ as follows:

$L^{\prime}=g_{L}Lh_{L},~{}~{}~{}~{}~{}~{}R^{\prime}=g_{R}Rh_{R},~{}~{}~{}~{}~{}~{}M^{\prime}=h_{R}^{\dagger}Mh_{L}\,,$
(14)

with $g_{L,R}\in SU(2)_{L,R}$ and $h_{L,R}\in SU(2)_{L,R}^{\prime}$.

In order to derive the most general Lagrangian up to second order in the
derivatives, it is convenient to consider the following covariant quantities

${\cal V}_{0}^{\mu}=L^{\dagger}D^{\mu}L,\,\,\,{\cal
V}_{1}^{\mu}=M^{\dagger}D^{\mu}M,\,\,\,{\cal
V}_{2}^{\mu}=M^{\dagger}(R^{\dagger}D^{\mu}R)M\,,$ (15)

with

$D_{\mu}L=\partial_{\mu}L-L{\bf
L}_{\mu},\,\,\,\,D_{\mu}R=\partial_{\mu}R-R{\bf
R}_{\mu},\,\,\,D_{\mu}M=\partial_{\mu}M-M{\bf L}_{\mu}+{\bf R}_{\mu}M$ (16)

where ${\bf L}_{\mu}=ig_{L}L_{\mu}^{a}{\tau^{a}}/{2}$, ${\bf
R}_{\mu}=ig_{R}R_{\mu}^{a}{\tau^{a}}/{2}$ are gauge fields of
$SU(2)_{L}^{\prime}\otimes SU(2)_{R}^{\prime}$. Using Eq. (15) we can build
six independent invariants under the transformations of Eq. (14):

$\textrm{Tr}{\cal V}_{0}^{2},\,\,\,\,\textrm{Tr}{\cal
V}_{1}^{2},\,\,\,\textrm{Tr}{\cal V}_{2}^{2},\,\,\,\textrm{Tr}({\cal
V}_{0}{\cal V}_{1}),\,\,\,\textrm{Tr}({\cal V}_{1}{\cal
V}_{2}),\,\,\,\textrm{Tr}({\cal V}_{0}{\cal V}_{2})\,.$ (17)

In conclusion, the most general Lagrangian for the vector boson interactions
is a combination of these six invariants. The gauging of the Lagrangian with
respect to the electroweak symmetry is obtained by substituting the covariant
derivatives of Eq. (16) with

$D_{\mu}L\to D_{\mu}L+{\tilde{\bf{W}}}_{\mu}L,\,\,\,D_{\mu}R\to
D_{\mu}R+{\tilde{\bf{Y}}}_{\mu}R,\,\,\,\,D_{\mu}M\to D_{\mu}M\,,$ (18)

where $\tilde{\bf{W}}_{\mu}=i\tilde{g}{\tilde{W}}_{\mu}^{a}{\tau^{a}}/{2}$,
$\tilde{\bf{Y}}_{\mu}=i{{\tilde{g}}^{\prime}}{\tilde{Y}}_{\mu}{\tau^{3}}/{2}$,
and the ”tildes” indicate quantities which are not the SM ones due to mixing.
The non-linear formulation of the SM is obtained in terms of the matrix
$U\equiv LM^{\dagger}R^{\dagger}$ and the scalar singlet $h$. The effective
Higgs Lagrangian is then written in terms of the quantities given in Eq. (15)
as

$\displaystyle\frac{v^{2}}{4}(1+2a\frac{h}{v}+b\frac{h^{2}}{v^{2}})$
$\displaystyle\textrm{Tr}(D^{\mu}U)^{\dagger}D^{\mu}U+...$ $\displaystyle=$
$\displaystyle-\frac{v^{2}}{4}(1+2a\frac{h}{v}+b\frac{h^{2}}{v^{2}})\textrm{Tr}[({\cal
V}_{0}-{\cal V}_{1}-{\cal V}_{2})^{2}]+...$ (19)

The SM couplings correspond to $a=b=1$ and dots stand for interaction terms
between the scalar singlet $h$ and the other invariants of Eq. (17). For
example, by adding the two invariants $\textrm{Tr}{\cal V}_{0}^{2}$ and
$\textrm{Tr}{\cal V}_{2}^{2}$, one obtains a model with two spin-1 vector
triplets, with ${\bf L}_{\mu}$ and ${\bf R}_{\mu}$ unmixed, when neglecting
weak interactions.

Following the hidden local symmetry approach, here briefly reviewed, the SM
fermions are coupled to the extra spin-1 resonances via mixing with the SM
gauge bosons. In addition, direct couplings, for example, of the $\psi_{L}$
fermions to ${\bf L}_{\mu}$ are allowed Casalbuoni:1985kq . In fact, for each,
$\psi_{L}$ we can construct a $SU(2)$ doublet

$\chi_{L}=L^{\dagger}\psi_{L}\,,\,\,$ (20)

and consider the following invariant term

$b_{L}\bar{\chi}_{L}i\gamma^{\mu}(\partial_{\mu}+i{\bf
L}_{\mu}+\frac{i}{2}{{\tilde{g}}^{\prime}}(B-L)Y_{\mu})\chi_{L}$ (21)

where $b_{L}$ is a dimensionless parameter. In the unitary gauge $L=I$ and,
shifting $\psi_{L}\rightarrow\psi_{L}/{\sqrt{1+b_{L}}}$, one gets a fermion-
heavy triplet interaction term

$-\frac{b_{L}}{{1+b_{L}}}\bar{\psi}_{L}\gamma^{\mu}{\bf L}_{\mu}\psi_{L}$ (22)

contributing, together with the mixing term, to $g_{F}=g^{2}c_{F}/g_{V}$ in
Eq. (LABEL:sml). The procedure can be generalized to several triplets
Casalbuoni:2005rs .

###  4.1 Explicit models with two new spin-1 triplets

In the following we will concentrate on two different models describing two
extra triplets of spin-1 resonances. We have seen that their interactions are
described, in general, by six independent invariants. They are reduced to four
by requiring the invariance under the discrete left-right transformation,
denoted by $P:L\leftrightarrow R,~{}M\leftrightarrow M^{\dagger}$. Namely, the
most general $G^{\prime}\otimes P$ invariant Lagrangian is given by
Casalbuoni:1988xm

${\cal
L}_{G}=-\frac{v^{2}}{4}[a_{1}I_{1}+a_{2}I_{2}+a_{3}I_{3}+a_{4}I_{4}]+{\cal
L}_{kin}$ (23)

with

$\displaystyle I_{1}$ $\displaystyle=$ $\displaystyle\textrm{Tr}[({\cal
V}_{0}-{\cal V}_{1}-{\cal V}_{2})^{2}],\,\,\,I_{2}=\textrm{Tr}[({\cal
V}_{0}+{\cal V}_{2})^{2}],$ $\displaystyle I_{3}$ $\displaystyle=$
$\displaystyle\textrm{Tr}[({\cal V}_{0}-{\cal
V}_{2})^{2}],\,\,\,\,I_{4}=\textrm{Tr}[{\cal V}_{1}^{2}]$ (24)

and the kinetic terms (we take $g_{L}=g_{R}={{g^{\prime\prime}}}/{\sqrt{2}}$):

${\cal L}_{kin}=\frac{1}{{g^{\prime\prime}}^{2}}\textrm{Tr}[F_{\mu\nu}({\bf
L})]^{2}+\frac{1}{{g^{\prime\prime}}^{2}}\textrm{Tr}[F_{\mu\nu}({\bf
R})]^{2}\,.$ (25)

####  4.1.1 4-Site Model with a Composite Higgs

As said, new vector resonances appear in five-dimensional extensions of the SM
as Kaluza-Klein (KK) excitations of the SM gauge bosons Agashe:2003zs ;
Csaki:2003dt ; Csaki:2003zu . When deconstructed ArkaniHamed:2001ca ;
ArkaniHamed:2001nc ; Hill:2000mu ; Cheng:2001vd , these theories appear as
gauge theories with extended $SU(2)$ symmetries.

We review here a simple four-dimensional model Accomando:2008jh ;
Accomando:2010ir ; Accomando:2011eu ; Accomando:2011xi ; Accomando:2012yg ;
Fedeli:2012cn , corresponding to a deconstruction including two additional
copies of the $SU(2)$ symmetry, namely $G^{\prime}=SU(2)_{L}\otimes
SU(2)_{R}\otimes SU(2)_{1}\otimes SU(2)_{2}$ with the particular choice:

$a_{1}=0,\,\,\,a_{2}=a_{3}=\frac{2f_{1}^{2}}{v^{2}},\,\,\,\,a_{4}=\frac{4f_{2}^{2}}{v^{2}},\,\,\,\,\frac{4}{v^{2}}=\frac{2}{f_{1}^{2}}+\frac{1}{f_{2}^{2}}\,.$
(26)

Following the general construction illustrated above, we add to the Lagrangian
the scalar-scalar and scalar-vector interactions as in Eq. (4) (with scalar we
here refer both the Higgs particle and the Goldstones):

$\displaystyle{\cal L}_{hG}$ $\displaystyle=$
$\displaystyle(2a_{h}\frac{h}{v}+b_{h}\frac{h^{2}}{v^{2}})f_{1}^{2}[\textrm{Tr}(D_{\mu}\Sigma_{1})^{\dagger}D^{\mu}\Sigma_{1}+\textrm{Tr}(D_{\mu}\Sigma_{3})^{\dagger}D^{\mu}\Sigma_{3}]$
(27) $\displaystyle+$
$\displaystyle(2c_{h}\frac{h}{v}+d_{h}\frac{h^{2}}{v^{2}})f_{2}^{2}\textrm{Tr}(D_{\mu}\Sigma_{2})^{\dagger}D^{\mu}\Sigma_{2}\,,$

with the identification of the chiral fields:
$\Sigma_{1}=L,\,\,\Sigma_{2}=M^{\dagger},\,\,\Sigma_{3}=R$. The covariant
derivatives are defined as in Eq. (18) with the additional gauge fields
identified as: ${\bf L}_{\mu}\to-{\bf\tilde{A}}_{\mu}^{1}$, ${\bf
R}_{\mu}\to-{\bf\tilde{A}}_{\mu}^{2}$,
${\bf\tilde{A}}_{\mu}^{i}=ig_{i}\tilde{A}_{\mu}^{ia}\tau^{a}/2$ and we assume
$g_{1}=g_{2}$ for the couplings.

In the unitary gauge, this model predicts two new triplets of gauge bosons,
which acquire mass through the same non-linear symmetry breaking mechanism
giving mass to the SM gauge bosons. For large $g_{1}$, $W^{\pm},Z$ mass
eigenvalues are given by

$M^{2}_{W^{\pm}}\simeq\frac{\tilde{g}^{2}v^{2}}{4}\left[1-\frac{\tilde{g}^{2}}{2g_{1}^{2}}(1+z^{4})\right],\,\,\,\,M^{2}_{Z}\simeq\frac{(\tilde{g}^{2}+{\widetilde{g}}^{\prime
2})v^{2}}{4}\left[1-\frac{\tilde{g}^{2}}{2g_{1}^{2}}z_{\zeta}\right]\,,$ (28)

where

$z=\frac{f_{1}}{\sqrt{f_{1}^{2}+2f_{2}^{2}}},\,\,\,z_{\zeta}=\frac{1}{2}\frac{z^{4}+\cos^{2}2\widetilde{\theta}}{\cos^{2}\widetilde{\theta}},\,\,\,\tan\widetilde{\theta}=\frac{{{\tilde{g}}^{\prime}}}{\tilde{g}}\,.$
(29)

The heavy vector triplets, ${\bf A}^{1}=(W_{1}^{\pm},Z_{1})$ and ${\bf
A}^{2}=(W_{2}^{\pm},Z_{2})$, have masses approximately given by $M_{1},M_{2}$
with

$M_{1}\simeq\frac{g_{1}v}{\sqrt{2(1-z^{2})}},\,\,\,\,M_{2}\simeq\frac{1}{z}M_{1}\,,$
(30)

As said, direct couplings of the extra spin-1 resonances to the SM fermions
are allowed by the symmetries. They are present also in the small mixing limit
and can be introduced by following the construction described above.

From Eq. (27) we derive the couplings of the Higgs $h$ to the charged gauge
bosons:

$\frac{2h}{v}\left[aM^{2}_{W}W^{+}W^{-}+a_{h}M_{1}^{2}W^{+}_{1}W^{-}_{1}+(a_{h}z^{2}+c_{h}(1-z^{2})M_{2}^{2}W^{+}_{2}W^{-}_{2}\right]\,,$
(31) $a=a_{h}(1+z^{2})+c_{h}z^{2}\,,$ (32)

and similarly for the neutral sector. The SM limit is obtained for $z\to 1$
(namely $g_{1},f_{1}\to\infty$), $c_{h}\to 1$ and $a_{h}\to 0$. In this limit
$a\to 1$.

Bounds on $hW^{+}W^{-}$, $hZZ$, $h\gamma\gamma$ couplings from recent analysis
at LHC, can be translated in limits on $a_{h},c_{h}$ for different values of
$z$. Assuming no deviation in the top Higgs couplings ($c_{t}=1$), we can
compute Bellazzini:2012tv

$\Gamma/\Gamma_{SM}(h\to\gamma\gamma)\sim\left[1+\frac{9}{8}[a_{h}(1-z^{2})+c_{h}(1-z^{2})]+\frac{9}{7}(a-1)\right]$
(33)

and use ATLAS measurements Aad:2019mbh  on $k_{V}=1.05\pm 0.04$,
$k_{\gamma}=1\pm 0.06$ based on 79.8 fb-1 of integrated luminosity to get the
95%CL bounds shown in Fig. 3 (Left).

Figure 3: Left: 95%CL bounds on the plane $(a_{h},c_{h})$ for $z=0.95$
(continuous blue) and $z=0.9$ (dashed green) from ATLAS measurements
Aad:2019mbh  on $k_{V}=1.05\pm 0.04$, $k_{\gamma}=1\pm 0.06$ based on 79.8
fb-1. Right: 5$\sigma$-discovery plot from D-Y production at LHC with L=100
fb-1 for $z=0.8$ in the plane $(a^{c}_{2},M_{2})$. Inside the grey (dashed)
(central white) regions both $W_{1,2}$ (only $W_{2}$) (no resonance) can be
detected. From Accomando:2008jh .

Unitarity bounds on the 4-Site model extra spin-1 resonance masses are
approximately 2-4 TeV depending on the $z$ value. For $z=0.95$ unitarity
bounds increase to approximately 4 TeV Accomando:2012yg .

Bounds from $S,T,U$ parameters on the charged couplings of the heavy triplets
to SM fermions, $a_{1}^{c}$ and $a_{2}^{c}$, and on the corresponding neutral
couplings, can be found in Accomando:2008jh ; Accomando:2010ir ;
Accomando:2011eu ; Accomando:2011xi ; Accomando:2012yg ; Fedeli:2012cn . For
example, for $z=0.95$, $M_{W_{1}}=2$ TeV, one obtains $-0.35\leq a_{1}^{c}\leq
0.4$. An extended analysis of Drell-Yan production of these new vectors in the
dilepton and diboson channel at LHC was also performed in Accomando:2008jh ;
Accomando:2010ir ; Accomando:2011eu ; Accomando:2011xi ; Accomando:2012yg ;
Fedeli:2012cn . The 5$\sigma$-discovery contours for a foreseen LHC integrated
luminosity of 100 fb-1 in the plane $(a^{c}_{2},M_{2})$, for $z=0.8$, where
$a_{2}^{c}$ is the $W_{2}$ charged coupling and $M_{2}$ is approximately
$W_{2}$ mass, are plotted in Fig. 3 (Right). The upper and lower parts of the
plot are excluded by the EWPTs. Inside the grey regions both $W_{1,2}$ are
visible; inside the dashed ones only $W_{2}$ can be detected. Inside the
central uncolored region no resonance is visible in the Drell-Yan channel.

However, in presence of several new triplets, as in this model, we cannot
extract bounds on the couplings of the new resonances from Fig. 1 and Fig. 2
because as shown in Accomando:2011eu ; Accomando:2013sfa ; Accomando:2016mvz
there are significant interference effects that cannot be neglected. In
Section 5 we review these effects in a composite Higgs model.

####  4.1.2 Linear degenerate BESS

A second interesting choice in the general parameterization given in Eq. (23)
is $a_{1}=1$, $a_{4}=0$, $a_{2}=a_{3}$. The Lagrangian then reads

${\cal
L}_{G}=\frac{v^{2}}{4}\\{\textrm{Tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U)+2~{}a_{2}~{}[\textrm{Tr}(D_{\mu}L^{\dagger}D^{\mu}L)+\textrm{Tr}(D_{\mu}R^{\dagger}D^{\mu}R)]\\}\,.$
(34)

In this model, before turning on electro-weak interactions, the new triplets
${\bf L}_{\mu}$ and ${\bf R}_{\mu}$ are degenerate in mass (from here the name
degenerate Breaking the Electro-Weak Symmetry Strongly (BESS) Model). Notice
that each of the three terms in the above expression is invariant under an
independent $SU(2)\otimes SU(2)$ group

$U^{\prime}=\omega_{L}U\omega_{R}^{\dagger},~{}~{}~{}~{}~{}~{}L^{\prime}=g_{L}Lh_{L},~{}~{}~{}~{}~{}~{}R^{\prime}=g_{R}Rh_{R}\,,$
(35)

so that the symmetry of the Lagrangian is enlarged to $G_{max}=[SU(2)\otimes
SU(2)]^{3}$. As a consequence of this enlarged symmetry or equivalently of the
degeneracy of ${\bf L}_{\mu}$ and ${\bf R}_{\mu}$ fields (or the combination
corresponding to the vector and axial-vector fields) the correction to the $S$
parameter at leading order is zero Casalbuoni:1995qt . The possibility of an
enhanced symmetry in near conformal theories and the emergence of parity
doublet was suggested also in Appelquist:1999dq  (see also Foadi:2007ue ) in
the framework of walking technicolor.

The degenerate BESS model has also a linear (renormalizable) formulation
Casalbuoni:1996wa ; Casalbuoni:1997rs . It is based on a gauge group
$G=SU(2)_{L}\otimes U(1)\otimes SU(2)_{L}^{\prime}\otimes SU(2)_{R}^{\prime}$
and has a scalar sector consisting of scalar fields
$\tilde{L}\in(2,0,2,0),\tilde{U}\in(2,2,0,0),\tilde{R}\in(0,2,0,2)$ of the
group $G$ (it is a generalized version of Model A in Sect. 3.1). The scalar
fields break the gauge symmetries through the following chain

$\begin{matrix}SU(2)_{L}\otimes U(1)\otimes SU(2)_{L}^{\prime}\otimes
SU(2)_{R}^{\prime}\cr\downarrow u\cr SU(2)_{\rm weak}\otimes
U(1)_{Y}\cr\downarrow v\cr U(1)_{\rm em}\end{matrix}$ (36)

The two breakings are induced by the expectation values
$\langle\tilde{L}\rangle=\langle\tilde{R}\rangle=u$, and
$\langle\tilde{U}\rangle=v$ respectively and we assume $u\gg v$. The first two
expectation values make the breaking $SU(2)_{L}\otimes SU(2)^{\prime}_{L}\to
SU(2)_{\rm weak}$ and $U(1)\otimes SU(2)^{\prime}_{R}\to U(1)_{Y}$, whereas
the second breaks in the standard way $SU(2)_{\rm weak}\otimes U(1)_{Y}\to
U(1)_{\rm em}$. The parameter $a_{2}$ of Eq. (34) is related to $u$ via
$a_{2}=u^{2}/(2v^{2})$.

Standard fermions are supposed to couple only to $\tilde{U}$. The model, in
the limit of large $u$, satisfies decoupling giving back the SM with a light
Higgs and suppressed contributions to the parameters $S,T,U$.

We parameterize the scalar fields as $\tilde{L}=\tilde{\rho}_{L}L$,
$\tilde{R}=\tilde{\rho}_{R}R$, $\tilde{U}=\tilde{\rho}_{U}U$, with
${L}^{\dagger}{L}=I$, ${R}^{\dagger}{R}=I$ and ${U}^{\dagger}{U}=I$. The
Lagrangian for their kinetic terms and interactions with the gauge bosons is
given by (${\bf L}_{\mu}\to{\bf\tilde{A}}_{\mu}^{1}$, ${\bf
R}_{\mu}\to-{\bf\tilde{A}}_{\mu}^{2}$)

${\cal L}_{\rho
G}=\frac{1}{4}\left[\textrm{Tr}(D_{\mu}\tilde{U})^{\dagger}(D^{\mu}\tilde{U})+\textrm{Tr}(D_{\mu}\tilde{L})^{\dagger}(D^{\mu}\tilde{L})+\textrm{Tr}(D_{\mu}\tilde{R})^{\dagger}(D^{\mu}\tilde{R})\right]\,,$
(37) $\displaystyle
D\tilde{L}=\partial\tilde{L}+\tilde{\bf{W}}_{\mu}\tilde{L}-\tilde{L}{\bf\tilde{A}}_{\mu}^{1}$
$\displaystyle
D\tilde{R}=\partial\tilde{R}+\tilde{\bf{Y}}_{\mu}\tilde{R}-\tilde{R}{\bf\tilde{A}}_{\mu}^{2}$
$\displaystyle
D\tilde{U}=\partial\tilde{U}+\tilde{\bf{W}}_{\mu}\tilde{U}-\tilde{U}\tilde{\bf{Y}}_{\mu}$
(38)

The scalar potential is expressed in terms of three Higgs fields:

$V=2\mu^{2}(\tilde{\rho}_{L}^{2}+\tilde{\rho}_{R}^{2})+\lambda(\tilde{\rho}_{L}^{4}+\tilde{\rho}_{R}^{4})+2m^{2}\tilde{\rho}_{U}^{2}+h\tilde{\rho}_{U}^{4}+2f_{3}\tilde{\rho}_{L}^{2}\tilde{\rho}_{R}^{2}+2f\tilde{\rho}_{U}^{2}(\tilde{\rho}_{L}^{2}+\tilde{\rho}_{R}^{2})$
(39)

After shifting the fields by their VEV’s $u,v\neq 0$, the squared mass
eigenvalues of the Higgses are:

$\displaystyle m_{\rho_{U}}^{2}\simeq
8\left(h-2\frac{f^{2}}{\lambda+f_{3}}\right)v^{2},~{}~{}~{}m_{\rho_{L}}^{2}\simeq
8{(\lambda-f_{3})}{u^{2}},~{}~{}~{}m_{\rho_{R}}^{2}\simeq
8({\lambda+f_{3}}){u^{2}}.$ (40)

Concerning the gauge sector, let’s call the mass eigenstates of the new spin-1
resonances ${\bf A}^{i}=(W_{i}^{\pm},Z_{i})$, $i=1,2$.

The fields ${W}_{2}^{\pm}$ are unmixed and their squared mass is given by

$M^{2}_{W_{2}}=\frac{1}{4}g_{1}^{2}u^{2}\equiv M^{2}$ (41)

where $g_{1}$ is the common coupling constant of the extra gauge symmetry. The
other two charged eigenvalues, in the limit of small $r=g^{2}v^{2}/(4M^{2})$,
are

$M^{2}_{{W}}\simeq\frac{\displaystyle{v^{2}}}{4}g^{2}(1-rs_{\varphi}^{2}),\,\,\,\,M^{2}_{W_{1}}\simeq\frac{M^{2}}{c_{\varphi}^{2}}\,,$
(42)

with $\varphi$ defined by the relation $g=g_{1}s_{\varphi}$, with
$1/g^{2}=1/\tilde{g}^{2}+1/g_{1}^{2}$. Notice that the SM gauge boson masses
receive corrections $O(r)$ due to mixing.

In the charged sector, at the first order in $r$, the couplings are given by

${\cal L}_{\rm fermions}^{\rm
charged}=-(a_{W}W_{\mu}^{-}+a_{1}^{c}W^{-}_{1\mu})J_{L}^{\mu-}+h.c.,$ (43)

with

$a_{W}=\frac{g}{\sqrt{2}}(1-s_{\varphi}^{2}r),\,\,\,\,a_{1}^{c}=-\frac{g}{\sqrt{2}}(1+c_{\varphi}^{2}r)\tan\varphi,$
(44)

and $J^{\pm}_{L}=\bar{\psi}_{L}\gamma^{\mu}\tau^{\pm}\psi_{L}$. There is no
coupling of $W_{2}^{\pm}$ to fermions, because they do not mix with $W^{\pm}$.
The fermionic couplings of the neutral gauge boson sector are given in
Casalbuoni:1997rs . The heavy gauge bosons are coupled to fermions only
through mixing with the SM ones, namely these couplings vanish for
$s_{\varphi}\to 0$. Nevertheless one could consider also direct couplings to
fermions as shown in the derivation of the general effective Lagrangian.

Figure 4: 95%CL bounds on the plane $(M,\tan\varphi)$ from electro-weak
precision measurements (continuous line) compared with bounds (dotted) from
the muon channel at LHC extracted from the analysis of Fig. 2.

Since in this set-up, only the charged gauge bosons $W_{1}^{\pm}$ are coupled
to fermions, there is no the interference effect between $W_{1}^{\pm}$ and
$W_{2}^{\pm}$ and we can use the analysis presented in Fig. 2 for a single
heavy triplet resonances to extract bounds on the linear degenerate BESS
model. Results on the plane $(M,\tan\varphi)$ are shown in Fig. 4.

In the neutral sector both $Z_{1}$ and $Z_{2}$ are coupled to fermions and a
dedicated analysis is necessary to get bounds from LHC results. Couplings of
the SM-like Higgs $\rho_{U}$ to SM fermions are rescaled by $c_{\alpha}$,
where $\alpha$ is the mixing angle in the scalar sector, while the couplings
of $\rho_{U}$ to SM gauge bosons are given by

$\rho_{U}\Big{[}\frac{v}{2}g^{2}c_{\alpha}(1-s_{\varphi}^{4}x^{2})W^{+}W^{-}+\frac{v}{4c_{\theta}^{2}}g^{2}c_{\alpha}[1-2x^{2}\frac{s_{\varphi}^{4}}{c_{\theta}^{4}}(1-2c_{\theta}^{2}s_{\theta}^{2})]Z^{2}\Big{]}\,.$
(45)

Due to the stringent limits on the plane $(M,\tan\varphi)$, current bounds
from LHC on the parameter $k_{V}$ Aad:2019mbh  can be directly translated on a
bound on $\alpha$, namely: $0.97\leq c_{\alpha}\leq 1$ at 2$\sigma$ level.

##  5 Drell-Yan production and interference effects in models with multi-
triplet structure

Drell-Yan production in the dilepton channels is a very good tool for
searching new heavy triplets as we have already discussed in Section 2. A
class of BSM scenarios, like composite Higgs models and the SM model in five
dimensions, predict several new spin-1 resonances. In these cases the
interpretation of the experimental results and related bounds on the parameter
space of the models is complicated by the presence of finite width and
interference effects. We briefly review in this Section some of such effects
for the D-Y process of the neutral channels (new $Z^{\prime}$s) in composite
Higgs models Accomando:2016mvz . The four-dimensional composite Higgs model
(4DCHM), that we are going to review, is the one proposed in DeCurtis:2011yx ,
based on two non-linear $\sigma$ models, one for $SO(5)/SO(4)$ and the second
for $SO(5)_{L}\otimes SO(5)_{R}/SO(5)_{L+R}$. After the breaking one is left
with 4 neutral and 6 charged spin one heavy particles, or two triplets
(3,1)+(1,3), almost degenerate in mass, and, in addition, two neutral and one
charged coset resonances.

The extra neutral resonances, that couple to fermions, are denoted by $Z_{2}$,
$Z_{3}$, $Z_{5}$ with squared masses, at order $O(\xi)$, given by

$\begin{split}\begin{split}&M^{2}_{Z_{2}}\simeq\frac{m_{\rho}^{2}}{c_{\psi}^{2}}\left(1-\frac{s_{\psi}^{2}c_{\psi}^{4}}{4c_{2\psi}}\xi\right),\\\
&M^{2}_{Z_{3}}\simeq\frac{m_{\rho}^{2}}{c_{\theta}^{2}}\left(1-\frac{s_{\theta}^{2}c_{\theta}^{4}}{4c_{2\theta}}\xi\right)\\\
&M^{2}_{Z_{5}}\simeq
2m_{\rho}^{2}\left[1+\frac{1}{16}(\frac{1}{c_{2\theta}}+\frac{1}{2c_{2\psi}})\xi\right],\end{split}\end{split}$
(46)

with $\xi=v^{2}/f^{2}$, $\tan\theta=\tilde{g}/g_{\rho}$ and
$\tan\psi=\sqrt{2}\tilde{g}^{\prime}/g_{\rho}$ where $g_{\rho}$ is the gauge
coupling of the $SO(5)$ group, $f$ is the scale of the spontaneous strong
symmetry breaking and $m_{\rho}=fg_{\rho}$. The $Z_{5}$ resonance, having a
mass $\sim\sqrt{2}$ higher than $Z_{2}$ and $Z_{3}$, gives a negligible
contribution to the Drell-Yan process at LHC RunII and it has been neglected
in the analysis. The change in the differential D-Y cross section in the
dilepton invariant mass produced by the interference is shown in Fig. 5 for
the benchmark F, corresponding to $M_{Z_{2}}=2192$ GeV and $M_{Z_{3}}=2258$
GeV. The interference effects produce the dip before the resonant peak, and
spoil the analysis performed within the narrow width approximation (NWA) or in
the finite width approach (FWA). The same conclusion can be derived by
plotting the ratio of the cross section over the NWA as a function of $\delta
m=M_{ll}-M_{Z^{\prime}}$. In the NWA the signal rate is typically estimated
modelling the signal as a Breit-Wigner function convoluted with a Gaussian
function, which is used to describe the dilepton mass resolution. This example
shows that one should avoid to use NWA or FWA within the 4DCHM or similar
models like the 4-Site model of Section 4.1 to extract bounds on $Z^{\prime}$
couplings and masses.

The two resonances $Z_{2}$ and $Z_{3}$ are, in some regions of the parameter
space, almost degenerate. Therefore they could appear as a single bump in the
dilepton invariant mass after convoluting the signal with a gaussian
distribution to simulate detector resolution. This is shown in Fig. 6 (Left)
for the benchmark F of Table 1. In Fig. 6 (Right) the two resonances
corresponding to he benchmark G of Table 1 are separated enough so that the
two peak structure is not washed out after the inclusion of the detector
smearing.

Benchmark |  $f$ [GeV] | $g_{\rho}$ |  $M_{Z_{2}}$ [GeV] |  $M_{Z_{3}}$ [GeV] |  $M_{Z_{5}}$ [GeV]
---|---|---|---|---|---
F | 1200 | 1.75 | 2192 | 2258 | 2972
G | 2900 | 1.00 | 3356 | 3806 | 4107
Table 1: 4DCHM parameter space points associated to the benchmarks F, G
mentioned in the text. From Accomando:2016mvz .

Figure 5: (Left) Differential cross section in the dilepton invariant mass for
the benchmark point F within the 4DCHM i.e. double-resonant $Z_{2,3}$
scenario. (Right) Ratio of the full signal cross section for the $Z_{2,3}$
bosons corresponding to benchmark F within the 4DCHM scenario (dashed line)
and the two resonances FWA (solid line) over the NWA result as a function of
the symmetric integration interval around the peak. The vertical red line
represents the CMS adopted optimal cut, $\delta m=|M_{ll}-M_{Z^{\prime}}|\leq
0.05E_{LHC}$, which keeps the interference and FW effects below 10% in the
case of narrow single $Z^{\prime}$ models Accomando:2013sfa . From
Accomando:2016mvz .

Figure 6:  Left(Right): Differential cross section in the dilepton invariant
mass for the BP F(G) after the smearing due to finite detector resolution. The
width of the gaussian is assumed 26 GeV (38 GeV). The statistical error,
visualized as the blue bands, is evaluated for 30 fb-1. From Accomando:2016mvz
.

Figure 7: 95%CL exclusion limits from D-Y production in dilepton channel at
the LHC RUNII with 13 TeV for different values of integrated luminosity. The
blue (red) contour refers to the electron (muon) channel. From
Accomando:2016mvz .

In Fig. 7 we show the exclusion limits at LHC RUN II for different values of
integrated luminosity (30-3000 fb-1) in the parameter space $(f,g_{\rho})$.

The 4DCHM can be considered as a concrete realization of the Model B,
described in subsection (3.2) with the feature that the two almost degenerate
triplets of spin-1 resonances corresponding to the $SO(4)$ symmetry (broken by
the EW interactions) are both phenomenologically relevant. Their interference
cannot be neglected as well as their widths which is naturally non-narrow. A
direct comparison of the results shown in Fig. 7 with the results shown in
Fig. 1 is not possible, because the analysis of CMS refers to a single heavy
vector triplet and takes into account, in addition to the D-Y processes,
diboson and H-boson productions.

##  6 Conclusions

The presence of new spin-1 particles, heavier than $W$ and $Z$, is a common
feature of several proposals of BSM physics, like for example extensions of
the SM in flat or warped extra dimensions, composite Higgs or technicolor
models. In this note we have first reviewed a general model independent
approach based on a simplified effective Lagrangian describing the
interactions of a new heavy triplet with the SM particles. This
parameterization has been used, using CMS and ATLAS results, to get bounds on
the parameters of the effective Lagrangian. We have then reviewed a general
approach to built effective Lagrangians describing the interactions in
presence of two new triplets of heavy resonances and considered two particular
models and their present limits. Finally we have discussed, in the framework
of a composite Higgs model the relevance of the interference and finite width
effects, in the Drell-Yan differential cross section in the dilepton neutral
channel. The interference effect, coming from the contributions of the neutral
spin-1 resonances exchanged in the s-channel, generate a dip before the
resonant peak, that spoils the simple analysis performed within the narrow
width approximation or in the finite width approach. In these cases a model
dependent approach is therefore necessary to get bounds on the parameters of
the model.

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