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Published for SISSA by Springer

Received: May 18, 2016 Revised: September 12, 2016 Accepted: November 2, 2016 Published: November 11, 2016

Vector and axial-vector resonances in composite models of the Higgs boson

Diogo Buarque Franzosi,a Giacomo Cacciapaglia,b Haiying Cai,b Aldo Deandreab and Mads Frandsenc

aII. Physikalisches Institut, Universitat Gottingen,

Friedrich-Hund-Platz 1, 37077 Gottingen, Germany

bUniv Lyon, Universit e Lyon 1, CNRS/IN2P3, IPNL,

F-69622, Villeurbanne, France

cCP 3-Origins & Danish Institute for Advanced Study DIAS, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark

E-mail: mailto:[email protected]

Web End [email protected] ,mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We provide a non-linear realisation of composite Higgs models in the context of the SU(4)/Sp(4) symmetry breaking pattern, where the e ective Lagrangian of the spin-0 and spin-1 resonances is constructed via the CCWZ prescription using the Hidden Symmetry formalism. We investigate the EWPT constraints by accounting the e ects from reduced Higgs couplings and integrating out heavy spin-1 resonances. This theory emerges from an underlying theory of gauge interactions with fermions, thus rst principle lattice results predict the massive spectrum in composite Higgs models. This model can be used as a template for the phenomenology of composite Higgs models at the LHC and at future 100 TeV colliders, as well as for other application. In this work, we focus on the formalism for spin-1 resonances and their bounds from di-lepton and di-boson searches at the LHC.

Keywords: Beyond Standard Model, Gauge Symmetry, Higgs Physics, Technicolor and Composite Models

ArXiv ePrint: 1605.01363

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2016)076

Web End =10.1007/JHEP11(2016)076

JHEP11(2016)076

Contents

1 Introduction 11.1 Vacuum alignment structure and fermion mass generation 3

2 E ective Lagrangian 52.1 Lagrangian 52.2 Other terms 8

3 Properties of vector states 93.1 Couplings 113.2 Electroweak precision tests 14

4 A minimal fundamental gauge theory 164.1 Scalar sector 174.2 Vector sector 174.3 Discrete symmetries 17

5 Phenomenology at the LHC and future 100 TeV colliders 195.1 Model implementation 195.2 LHC Run-II 205.3 Future 100 TeV proton colliders 24

6 Conclusions 27

A Explicit formulas 28A.1 Perturbative diagonalisation of the mass matrices 30A.2 EW precision parameters 33

B G-parity transformation 34

C Branching ratios 34

1 Introduction

E ective Lagrangian approaches have played a major role in various physics applications to model unknown sectors or situations that are simply di cult to treat. In more mature elds, they also provide a simple and more easily calculable tool describing a detailed and complex underlying theory. This is, for example, the case for the strong interactions of Quantum ChromoDynamics (QCD) that are described, at low energy, by a chiral Lagrangian incorporating the light composite degrees of freedom of the theory. The power of

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JHEP11(2016)076

the chiral Lagrangian stands on a well de ned expansion scheme that allows for very accurate calculations. A similar pattern can be followed in the electroweak sector, and attempts to describe the Standard Model (SM) in this way have been numerous since the the beginning [1, 2]. In particular, shortly after the theoretical establishment of the SM, the idea that QCD itself may play the role of a template for a composite origin of the electroweak symmetry breaking has been gaining popularity, leading to rescaled-QCD Technicolour models [3{5]. In this set up, the longitudinal degrees of freedom of the massive W and Z are accounted for as Goldstone bosons, i.e. pions, of the strong sector. While nowadays it is accepted that rescaled-QCD does not describe the physical reality, especially due to its Higgs-less nature, issues with generating quark masses [2, 6] and the correct avour structure [7] and precision tests [8], new versions of such theories are gaining momentum.

The main break-through can be traced back to the idea that the Higgs too can be described as a pseudo Nambu-Goldstone boson (pNGB) of an enlarged avour symmetry [9, 10], and the coset SU(5)/SO(5), containing a singlet and a 9-plet of the custodial SO(4)SU(2)L[notdef]SU(2)R together with the Higgs doublet, has been one of the rst candi

dates [11]. For recent reviews on the developments occurred in the last decade, we refer the reader to ref.s [12, 13]. The realisation that the minimal symmetry breaking, SO(5)/SO(4), embedding custodial symmetry, contains only a Higgs boson at low energy [14] has inspired the construction of e ective descriptions of the electroweak (EW) sector of the SM which do not contain more states [15{17]. Other composite states which are not pNGBs, like spin-1 resonances, or the so-called top partners [12] (needed in the partial compositeness scenario [18]), are typically heavier than a few TeV and thus their e ect at low energy can be embedded in higher order operators. Yet, at the energy reached by the LHC, their direct production is a crucial test of the theories. A very rich literature is already available, and we list here a forcibly incomplete list of papers addressing various issues on the experimental tests of composite top partners [19{27], aka vector-like fermions [28, 29], spin-1 resonances [30{32], or additional scalars [33, 34]. In particular, the search for top partners at both Run-I and Run-II has produced bounds on their masses which are now approaching 1 TeV.

Models beyond the minimal case are interesting as they contain additional light scalars [35{37], among which a Dark Matter candidate may arise [38, 39]. One possible way to discriminate among them is the requirement that they arise from an underlying theory consisting on a con ning fermionic gauge theory. In this sense, the coset SU(4)/Sp(4) (equivalent to SO(6)/SO(5)) can be considered the minimal one, also arising from a very simple underlying theory [40, 41] based on a con ning SU(2) Yang-Mills theory with 2 Dirac fermions in the fundamental representation. Besides being a template for a Composite Higgs model [42], this theory has also been used as a simple realisation of the SIMP Dark Matter candidate [43] (for a critical assessment, see [44]). In this paper we want to extend the e ective eld theory studies of this template by adding the lowest-lying spin-1 resonances, i.e. vector and axial-vector states [45, 46]. We follow the CCWZ [47, 48] prescription by employing the hidden symmetry technique [49]. While we focus on the Composite Higgs scenario with the scope of studying the phenomenology of such states at the LHC and at future higher energy colliders, our construction can be also applied to other phenomenological uses of this simple theory [43, 50{52].

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After reviewing the basic properties of the SU(4)/Sp(4) coset, in section 2 we construct an e ective Lagrangian for the spin-1 states. In section 3 we provide details of the properties of the physical states and connect them with the simplest underlying theory in section 4. Finally, we brie y study the collider phenomenology in section 5, focusing both on di-lepton constraints at the LHC and on prospects for the future 100 TeV proton collider.

1.1 Vacuum alignment structure and fermion mass generation

The vacuum structure of the SU(4)/Sp(4) model has already been extensively studied [35, 41, 53], so here we will brie y recap the main features. In this work, we will follow the prescription that the pNGB elds are de ned around a true vacuum which includes the source of electroweak symmetry breaking, as in ref.s [41, 42]. It can be thus shown that the vacuum alignment can be described in terms of a single parameter, , and in the SU(4) space it looks like

0 = cos (i2) sin (1)sin (1) cos (i2)

uTH( ) ; (1.1)

where uH is an SU(4) rotation along a direction in the space de ned by the generators transforming like a Brout-Englert-Higgs doublet. For = 0, i.e. uH(0) = 1, the electroweak symmetry is unbroken once the SU(2)L and U(1)Y generators are embedded in SU(4) as

Si = 12

i 0

0 0

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= uH( ) [notdef]

(i2) 0

0 (i2)

! [notdef]

; Y = S6 = 12

0 00 T3 !

: (1.2)

In the phase where is non-zero, the gauged generators of SU(4) are no more aligned with the 10 unbroken generators V a de ned as

V a [notdef] 0 + 0 [notdef] V aT = 0 ; Y a [notdef] 0 0 [notdef] Y aT = 0 ; (1.3)

where Y a are the 5 broken ones (explicit matrices can be found in [42]). As mentioned above, the pNGBs are de ned around the -dependent vacuum 0 as1

U = exp"ip2 f

XaaY a

; (1.4)

where the would-be Higgs boson is identi ed with h = 4, the singlet = 5, and the remaining 3 are exact Goldstones eaten by the massive W and Z. Also, f corresponds to the decay constant2 of the pNGBs, and it is related to the electroweak scale via :

f sin = vSM = 246 GeV: (1.5)

1Other parameterisation have been used in the literature where the pNGBs are de ned around the = 0 vacuum, and the \Higgs" one is then assigned a vacuum expectation value [53]. The main di erences lie in higher order interactions, see [39]. We prefer this approach because it sequesters the explicit breaking of the Goldstone shift symmetry to the potential terms that generate the pNGB masses.

2Here, we adopt the standard normalisation used in Technicolour literature, and other composite Higgs literature: the di erence with the f used in [42, 54] is f = 2p2f.

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The alignment along , together with the masses of the two physical pNGB, is then xed by a potential generated by explicit breaking terms of the SU(4) avour symmetry: the gauging of the electroweak symmetry, Yukawa couplings for the top (above all) and a mass for the underlying fermions (which is allowed by the symmetries as the underlying theory is vectorial).

The origin of the potential and the top mass is only relevant for the current study in so far as it modi es the spin-1 phenomenology. The top mass for this model may arise as in extended technicolor (ETC) descriptions via 4-fermion operators bilinear in the top-quark. This is discussed for the SU(4)=Sp(4) coset in e.g. [41, 42]. These bilinear 4-fermion interactions may arise from the exchange of heavy spin-1 bosons [2, 6] or heavy scalars [55] external to the strongly interacting sector considered here. A recent explicit example employing a chiral gauge theory is provided in [56]. These ETC interactions induce direct couplings of the spin-1 resonances with SM fermions and these couplings can provide a welcome negative contribution to the electroweak S-parameter [57]. However these couplings are typically negligible relative to those induced by the mixing between the heavy spin-1 resonances, especially for the light SM fermion generations, and the e ects on S consequently small. We therefore ignore these small e ects in the current study. Alternatively the top mass may also arise via fermion partial compositeness [18], a mechanism that requires the presence of fermionic bound states that mix linearly to the elementary elds. The realisation of this mechanism in terms of explicit 4d gauge theories with fermions, relevant for our coset, has been studied in [34, 58, 59]. Top partners may play a dual role of generating the top mass and stabilising the Higgs potential, in which case one would expect them to be parametrically lighter than the typical resonance scale [60]. Else, other spurions like a mass for the underlying fermions [42] can be used as a stabiliser, and the top partner can be heavy and irrelevant for the phenomenology of the Higgs3 and vector resonances. In both mechanisms sketched above, the model needs to face severe constraints from avour observables, especially in the form of Flavour Changing Neutral Currents induced by four-fermion operators at the avour scale. The usual way out relies on the presence of a Conformal Theory in the UV, that generates large anomalous dimensions responsible for the separation of the scale of avour violation from the scale of compositeness and the generation of hierarchies in the fermion masses. Obtaining large anomalous dimensions, however, has been proven to be very challenging for scalar operators [61{63] (as needed in the case of bilinear 4-fermion operators), as well as for fermionic ones (as in partial compositeness, see for instance [64, 65]). A complete theory of avour is thus still beyond the horizon.

In the following, we will assume the case of 4-fermion interaction, or of heavy top partners, and only focus on the dynamics responsible for the composite Higgs. As realising partial compositeness requires extending the strongly interacting sector (need additional coloured techni-fermions), we leave this possibility and the study of the interplay between top partners and vectors for a future study.

3For the potential, details can be found in [42] for the case where the fermion mass is used as a stabiliser, and in [53] for the case where top partners are present and used to ne tune the top loops.

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2 E ective Lagrangian

To describe the new strong sector and remain as general as possible, a chiral-type theory can be constructed on the basis of custodial symmetry and gauge invariance. The simplest construction one can imagine uses a local copy of the global SU(2) \chiral" symmetry and builds the relevant invariants [66]. The same results follow from the hidden gauge symmetry approach [49]. Furthermore the global avour symmetry can be enlarged in di erent ways, depending on the required model-building features (see for example the early attempts in [67, 68]). To this basic idea one can add the Higgs boson as a pseudo-Nambu-Goldstone boson [9, 10] or as a massive composite state, or as a superposition of both [42]. In the following we shall consider a model with vector and axial-vector particles: for a template description of these resonances based on the SU(2) group see [69]. In order to describe this kind of spectrum, we introduce a local copy of the global symmetry. When the new vector and axial-vector particles decouple, one obtains the non-linear sigma-model Lagrangian, describing the Goldstone bosons associated to the breaking of the starting symmetry to a smaller one. The approach we use is the standard one of the hidden gauge symmetry [49] (for an alternative, equivalent, way, see [70]).4

In our speci c case, i.e. the minimal model with a fermionic gauge theory as underlying description, the global symmetry SU(4)/Sp(4) is extended in order to contain, initially, two SU(4)i, i = 0; 1. The SU(4)0 corresponds to the usual global symmetry leading to the Higgs as a composite pNGB, and the electroweak gauge bosons are introduced via its partial gauging. The new symmetry SU(4)1 allows us to introduce a new set of massive \gauge" bosons, transforming as a complete adjoint of SU(4), which correspond to the spin-1 resonances in this model.

2.1 Lagrangian

Following the prescription of the hidden gauge symmetry formalism, we enhance the symmetry group SU(4) to SU(4)0 [notdef] SU(4)1, and embed the SM gauge bosons in SU(4)0 and

the heavy resonances in SU(4)1. The low energy Lagrangian is then characterised in terms of the breaking of the extended symmetry down to a single Sp(4): the SU(4)i are spontaneously broken to Sp(4)i via the introduction of 2 matrices Ui containing 5 pNGBs each.

The remaining Sp(4)0 [notdef] Sp(4)1 is then spontaneously broken to Sp(4) by a sigma eld K,

containing 10 pNGBs corresponding to the generators of Sp(4).

The 5+5 pNGBs associated to the generators in SU(4)/Sp(4) are parameterised by the following matrices:

U0 = exp

that transform nonlinearly as

Ui ! U[prime]i = giUih(gi; i) ; (2.2)

4In that approach only one set of pNGBs is used for the CCWZ prescription, and the mass term for the axial-vectors

a2 2 (gaa d)2 will give rise to a bilinear mixing of a4@h.

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JHEP11(2016)076

"ip2f05

"ip2f15

Xa=1(a0Y a)

Xa=1(a1Y a)

; U1 = exp

; (2.1)

where gi is an element of SU(4)i and h the corresponding transformation in the subgroup Sp(4)i. It is convenient to de ne the gauged Maurer-Cartan one-forms as

!R i; = UiD Ui ; (2.3)

where D are the appropriate covariant derivatives

D U0 = (@ ig

fW ig[prime]B )U0 ; (2.4) D U1 = (@ i

egV i

egA )U1 : (2.5) The spin-1 elds are embedded in SU(4) matrices as

B = B S6;

fW =

where

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3 fW a Sa; V =

Xa=1

Va Va; A =

Xa=1

Aa Ya ; (2.6)

fW k (k = 1; 2; 3) and B are the elementary electroweak gauge bosons associated with the SU(2)L and U(1) hypercharge groups. The vector Vj (j=1 to 10) and axial-

vector Al (l=1 to 5) are the composite resonances generated by the strong dynamics and

associated to the unbroken Va and broken Ya generators as de ned in eq. (1.3). The projections to the broken and unbroken generators are de ned respectively by

p i = 2

Tr (Ya!R i; ) Ya ; (2.7)

v i = 2

Tr (Va!R i; ) Va ; (2.8)

so that v i transforms inhomogeneously under SU(4)i

v i ! v[prime] i = h(gi; i) (v i + i@ ) h(gi; i) ; (2.9)

while p i transforms homogeneously

p i ! p[prime] i = h(gi; i) p i h(gi; i) (2.10)

and can be used to construct invariants for the e ective Lagrangian.

The K eld is introduced to break the two remaining copies of Sp(4), Sp(4)0 [notdef] Sp(4)1

to the diagonal nal Sp(4):

K = exp [ikaV a=fK] ; (2.11)

and it transforms like

K ! K[prime] = h(g0; 0) K h(g1; 1) ; (2.12)

thus its covariant derivative takes the form

D K = @ K iv0 K + iKv1 : (2.13) The 10 pions contained in K are needed to provide the longitudinal degrees of freedom for the 10 vectors Vj , while a combination of the other pions i act as longitudinal degrees of

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freedom for the Al . It should be reminded that out of the 5 remaining scalars, 3 are exact

Goldstones eaten by the massive W and Z bosons, while 2 remain as physical scalars in the spectrum: one Higgs-like state plus a singlet .

To lowest order in momentum expansion, and including the scalar singlet , the e ective Lagrangian is given by

L =

2g2 Tr

2g[prime]2 Tr B B

F ()

eg2 Tr F F

+12 G0()f20 Tr p0 p 0 +12 G1()f21 Tr p1 p 1 + r()f21 Tr p0 Kp 1K

+12 K()f2K Tr D K D K +

2@ @ V()

+Lfermions : (2.14)

We have introduced the singlet eld for generality, as it may be light in some theories, via generic functions in front of the operators in the strong sector: in the following, however, we will be interested to the case where its heavy and thus we will replace the functions by the rst term in the expansion, i.e. X() = 1 and r() = r. The eld strength tensors are de ned by

V = @ V @ V i[V ; V ] (2.15)

for V = B ;

fW ; and F = V +A . The canonically normalised elds are g[prime] B , g

fW k ,

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eg Vk and eg Ak .

Due to the presence of the r term in the Lagrangian, the pions 0;1 do not have proper kinetic terms. Calling the normalised elds A and B, they are given by

a0 = aA

p1 + r f1=f0

aB p2

p1 r f1=f0; (2.16)

a1 = aA

p1 + r f1=f0+ aB p2

p1 r f1=f0: (2.17)

As already mentioned, a linear combination of the two sets of 5 pions is eaten by the vector states A once they pick up their mass. The eaten Goldstones aU, and the 5 physical ones

aP before the EW gauging, are given by

aA = cos aP sin aU; (2.18)

aB = sin aP + cos aU; (2.19)

where the mixing angle is

tan =

s1 + rf1=f01 rf1=f0: (2.20)

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Combining the above rede nitions, we get5

a0 = aP 1

p1 r2f21=f20; (2.21)

a1 = aU aP

rf1=f0

p1 r2f21=f20: (2.22)

Note that only the pions associated with Y 4 and Y 5 are physical, as the remaining 3 are exact Goldstones eaten by the W and Z. In the following, we will associate one with the Higgs boson, 4P = h, and the other with the additional singlet 5P = of the

SU(4)/Sp(4) coset.

2.2 Other terms

The previous Lagrangian contains the low energy composite sector in terms of e ective elds using the CCWZ formalism and the hidden symmetry one, allowing for a description of composite spin-0 pNGB and spin-1 vector and axial-vector resonances. The interactions among these states are, to a large extent, described by this formalism, however some extra terms can potentially be added. While we leave a detailed study to a future work, it is worth mentioning how they can a ect the phenomenology when added.

A rst set of contributions are those induced by the Wess-Zumino-Witten (WZW) anomaly [71, 72]. In our case, these can be added in a similar way to what is done for chiral Lagrangians to describe, for example, the decay of a neutral pion into two photons. These terms are relevant for di-boson nal states, allowing a scalar pNGB to couple to the SM gauge bosons. Furthermore, anomalous couplings of the vectors will also be generated, thus potentially providing new decay channels.

Another set of possible terms are the ones allowing the spin-1 vector and axial-vector resonances to couple directly to fermions instead of getting their coupling to the SM fermions only by mixing e ects with the SM gauge bosons. Such couplings are allowed by the symmetries of the Lagrangian, as in a similar way to the SM, the fermionic current couples to weak SU(2) gauge triplets. However the phenomenological constraints indicate that the new direct coupling should be small. Nevertheless this new source of direct decay to fermions for the new composite vectors can have a non-negligible impact on phenomenology. Finally, in analogy with QCD, the new and of the underlying strong dynamics require a detailed study. This part of the scalar sector is not the main focus here and a detailed description can be found in [54]. Other possible items in this list are symmetry breaking terms and kinetic mixing terms. All these points will not be discussed further here, and may deserve a separate study.

5Note that for r = 0, we would have P = 0 and U = 1, as expected seen that SU(4)1 is associated with the massive vectors. Thus, r parameterises the mixing between the two sectors. The rotation de ned for P and U is divergent for r = f0/f1, this point is not physical since it will lead to v = 0 thus no EWSB can be generated.

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3 Properties of vector states

The intrinsic properties of the 15 spin-1 states introduced in eq. (2.6) determine the structure of masses, mixing, couplings and their contributions to electroweak precision tests. The vector elds can be organised as a matrix in SU(4) space, de ned by

F = V + A =

Xa=1Aa Ya: (3.1)

where the generators in the general vacuum, V a and Y a, are de ned in eq. (1.3). Under the unbroken Sp(4), the two multiplets transform as a 10 and a 5 respectively. It is however more convenient to classify the states in terms of their transformation properties under a subgroup SO(4)Sp(4), which corresponds to the custodial symmetry SU(2)L[notdef]SU(2)R of

the SM Higgs sector in the limit ! 0:

V ! 10Sp(4) = (3; 1) (1; 3) (2; 2) ; A ! 5Sp(4) = (2; 2) (1; 1) : (3.2)

Physically, however, the SM custodial symmetry is broken to the diagonal SU(2)V in a generic vacuum alignment, under which symmetry the physical spectrum contains 4 triplets, plus additional singlets. A complete list of the states, and their classi cation, can be found in table 1. We would like to remind the reader, here, of the generic properties of the vector resonances in a minimal case of composite EWSB: in fact, any model of compositeness necessarily contains the spontaneous breaking of SU(2)L[notdef]SU(2)R ! SU(2)V , under which

one expects to have a vector triplet

! and an axial-vector triplet

!a . In our case, more states are present (shown explicitly in their SU(4) embedding in eq. (A.1)), however one can always identify states corresponding to the minimal case. In fact, the triplet v0;[notdef] can always be associated to the

! of the \vector" SU(2); on the other hand the interpretation of the axial-vector depends on the speci c realisation of the model. On one hand, in the Technicolor limit, = =2, we nd that a0;[notdef] , which has a component of axial-vector SU(2)A proportional to sin , can be associated to

Xa=1Va Va +

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!a ; on the other hand, in the pNGB Higgs limit, ! 0, it is s0;[notdef] , having a SU(2)A component proportional to cos ,

that transforms like the axial-vector states. All the above states mix with the elementary gauge bosons of the SM. Simpli ed models describing vector triplets have been used in the composite Higgs literature: for instance, two triplets corresponding to our v0;[notdef] and s0;[notdef] are usually considered in the minimal SO(5)=SO(4) model [30], while in more simpli ed cases a single triplet is accounted for [31, 73{75]. Although our complete Lagrangian contains such states, it is not possible to nd limits where the other states decouple.6 Thus, simpli ed models can only partially describe the phenomenology of the model under study.

The additional states

es[notdef];0 , ev0 , x0 and ex0 do not mix with the elementary gauge bosons:

their masses are given by

0 = MA : (3.3)

6For instance, one may decouple the A states by sending f1 ! 1 (and r ! 0), however the0,[notdef] and

~v0 will remain light.

M[tildewide]s

= M[tildewide]v

0 = MV and Mx0 = M[tildewide]x

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SU(2)V SU(2)L[notdef]SU(2)R TC CH

v0;[notdef] 3 (3,1) (1,3)

s0;[notdef] 3

V 0;[notdef] 3 (2,2)~v0 1

a0;[notdef] 3 (2,2)

x0 1

~x0 1 (1,1)

Table 1. Classi cation of the spin-1 resonances in the model.

where the mass parameters MA and MV are de ned in terms of Lagrangian parameters as

egfKp2 : (3.4)

The other states mix among themselves and with the SM weak bosons (

fW i and B ).

egf1p2 and MV

The mass mixing Lagrangian is

Lmass =

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W a v s

M2C

fW + a+

+ 12

fW 3 a0 v0 s0

M2N

fW 3 a0

: (3.5)

The matrices M2C and M2N are given in eqs. (A.4){(A.5).

Upon diagonalisation, the interaction eigenstates are rotated to the physical vector bosons

fW + a+

: (3.6)

Approximate expressions for C and N are given in appendix A.1 in an expansion for large

eg. The eigenstate in the neutral sector which is exactly massless is identi ed to be the photon, and it is related to the interactions eigenstates (exactly in

eg) as

= C

W +

A+

V +

S+

;

fW 3 a0

= N

Z A0

V 0

A = e g

fW 3 + e

g[prime] B +

p2e

eg v0 (3.7)

with

1=e2 = 1=g[prime]2 + 1=g2 + 2=

eg2 : (3.8)

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Besides the photon, all the massive states mix with each other with mixing angles typically of order 1=

eg, with the exception of v and s whose mixing is controlled by the angle . For instance, in the charged sector, see eq. (A.4), it is clear that the combination cos v[notdef] s[notdef]

decouples from the other states and has a mass equal to MV . A similar situation is realised in the neutral sector where, however, residual mixings suppressed by 1=

eg are present.

Approximate expressions for the masses of the charged states are given below, including leading corrections in 1=

eg2:

M2W = 14g2v2 "

s2 + 2

+ O 1=

eg4

; (3.9)

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M2A+ = M2A

"1 + 12 g

2r2s2 + O 1=

eg4

; (3.10)

M2S+ = M2V ; (3.11)

M2V+ = M2V"1 + 12 g

2(2 s2 ) + O 1=

eg4

: (3.12)

Similarly, in the neutral sector, we nd:

M2Z = 14(g2 + g[prime]2)v2

1 + (g2 + g[prime]2)2(1 r2)s2 2(g4 + g[prime]4)

2(g2 + g[prime]2)

eg2 + O 1=

eg4

; (3.13)

M2A0 = M2A

1 + r2(g2 + g[prime]2)s2 2

eg2 + O 1=

eg4

; (3.14)

M2V0=S0 = M2V

1+ g2 + g[prime]2 4

eg2

1+ c2 [notdef] q1 + 2(g[prime]46g[prime]2g2+g4)(g2+g[prime]2)2c2 + c4

+ O (1=

eg)4

: (3.15)

In all above expressions, s = sin and c = cos . Furthermore, v = 246 GeV is the value of the e ective EW scale, obtained from the de nition of the Fermi decay constant as:

1p2GF =

4g2 W

+W (0) =

4 g2

hM2C1i11

M2Ar2)s2 =2 : (3.16)

Replacing the masses with the Lagrangian parameters, we also obtain the relation

v2 = f20 r2f21

sin2 = f2 sin2 (3.17)

where f =

pf20 r2f21 is the decay constant of the SU(4)/Sp(4) pions, as in eq. (1.5). As a consistency check, note that f = f0 for r = 0 and, as mentioned in the previous section, f = 0 for r = f0=f1.

3.1 Couplings

We assume here that the SM fermions only couple to the SM weak bosons,

fW and B : this is a reasonable assumption, as direct couplings to the composite resonances can only be induced by interactions external to the dynamics. The interaction with the heavy vectors, therefore, are generated via mixing terms. For the charged currents, we have

LCC = g

Xi;f

C1i f R+i; f[prime] + h:c: ; (3.18)

{ 11 {

= 2(M2V f20

f2K

where R[notdef]i; = (W [notdef] ; A[notdef] ; V [notdef] ; S[notdef] ) and f labels all the SM fermions. For the neutral currents

LNC = 12Xi;fR0i; f [(gfLiPL + gfRiPR)] f ; (3.19)

where R0i; = (A ; Z ; A0 ; V 0 ; S0 ), f is a SM fermion, PL;R = (1 5)=p2, andgLj = gT 3N2j + g[prime]YLN1j; gRj = g[prime]YRN1j; (3.20)

with T 3 being the weak isospin and YL;R the hypercharge of the left-handed doublet and the right handed singlet respectively. All the neutral vector couplings can be expressed like this, but for the photon gauge invariance requires that

g[prime]N11 = gN21 = e : (3.21)

Note that eq. (3.18) and eq. (3.19) encode corrections to the couplings of SM fermions with respect to the SM predictions, that are strongly constrained by EW precision observables and can be encoded in the oblique parameters, as discussed in the following section.

The Higgs couplings to weak bosons are phenomenologically important because they can constrain the model parameters, both from direct measurements and from its contribution to the electroweak parameters. Schematically, the couplings can be written as

Lh = chR+i Rj h R+ ;iR; j +

2chR

JHEP11(2016)076

0i R0j h R0 ;iR0; j ; (3.22)

where R[notdef]i and R0i encode all the charged and neutral vectors. In the gauge interaction basis, the couplings are provided in appendix A, while in the mass eigenbasis we calculated expressions at leading order in 1=

eg. We nd that the Higgs couplings to at least one photon are automatically zero at the tree level, with the other couplings given by

2M2Zv c = cSMhZZc ; (3.23)

in agreement with previous studies [42], while the couplings to the resonances are (we list only the diagonal ones and the ones with one SM gauge boson)

chW+W [similarequal]

2M2Wv c = cSMhW

+W c ; chZZ [similarequal]

g2M2Ar2s2

g2M2V s2

chA+A [similarequal]

eg2v ; chV

+V [similarequal]

eg2v ;

chW+S [similarequal]gM2V (r2 1)s2

egv ; chW

+V [similarequal]

gM2V (r2 1)s2 2

egv ; (3.24)

chA0A0 [similarequal](g[prime]2 + g2)M2Ar2s2

g2M2V s2

eg2v ; chV

0V 0

[similarequal]

eg2v ; chS

0S0

[similarequal]

g[prime]2M2V s2

eg2v

0 pg[prime]2 + g2M2V (r2 1)s2 2

egv : (3.25)

The charged heavy vector states contribute to the decay of the Higgs boson into two photons via loops. In general, computing loops of heavy resonances is not reliable, nevertheless we can approximate the contribution of the strong dynamics to h ! by computing

{ 12 {

chZS0 [similarequal]

pg[prime]2 + g2M2V (r2 1)s2 2

egv ; chZV

[similarequal]

loops of the lightest spin-1 resonances. While this is not a complete calculation, it can provide an estimate of the additional contributions and allow us to test their impact. The partial width, including new physics e ects, can be written as,

h! =

2m3h2563v2 [notdef]NcQ2top tAf(t) + W AV (W ) + resAV (1)[notdef]2; (3.26)

where we approximate the amplitude of the heavy states to the asymptotic value AV (1) =

7 (Af(f) and AV (W ) being the standard amplitudes [76]), and t;W are the modi cation of the fermion and vector couplings of the Higgs normalised by the SM expectation (in our case, W cos , while t depends on the mechanism providing a mass for the top and is

equal to t = cos in the simplest case):

W = chW

v ; res =

Xi=2;3;4

chR+

2M2W

JHEP11(2016)076

i Ri

2M2Ri

v ; (3.27)

where both couplings and masses are de ned in the mass eigenstate basis. By analysing the mass and coupling matrices, we found that the following sum rule holds, at all orders in 1=

eg:

W + res = v2Tr

hchV +V [notdef] M2C

= cos ; (3.28)

where chV +V are the couplings of the Higgs in the interaction basis (see eq. (A.6)) and we used exact matrices. At leading order in 1=

eg, the sum rule is saturated by the W coupling W = cos , as shown in eq. (3.23). However, corrections arise at order 1=

eg2:

using eq. (3.24), we nd

2(r2 1) sin2 : (3.29)

We expect, therefore, the contribution of the mixing with the heavy resonances to be very small, as it is suppressed by sin2 , 1=

eg2, and it also vanishes for r = 1: the latter is a reminder of the fact that the two SU(4)s decouple in this limit. This analysis shows that the e ect of the heavy resonances on the Higgs properties can be neglected, thus the bounds from the measured Higgs couplings are the same as in [54], and they are typically less constraining than electroweak precision tests.

For completeness, a similar analysis can be done in the neutral sector, where we de ne

Z = chZZ 2M2Z

v ; 0;res =

res = cos W [similarequal]

eg2

Xi=3;4;5chR0i R0i2M2R0iv ; (3.30)

and the exact mass and coupling matrices entail the following sum rule:

Z + 0;res = v2Tr

h chV 0V 0 [notdef]

= cos ; (3.31)

where the reduced coupling,

chV 0V 0 , and mass matrices,

M2N, are 4[notdef]4 matrices obtained

from the complete ones in eq. (A.7) and eq. (A.5) by removing the photon, i.e. the zero-mass eigenstate. Like for the charged case, Z is the deviation of the Higgs couplings to

{ 13 {

the Z boson normalised by the SM value, while 0;res encodes the contribution of the heavy resonances:

0;res = cos Z =Xi=3;4;5chR0i R0iv2M2R0i[similarequal]g2 + g[prime]2

2(r2 1) sin2 : (3.32)

We can see that the custodial violation due to the gauging of the hypercharge emerges here.

The o -diagonal Higgs and gauge couplings will contribute to the H-Z- vertex [77], but no bound is available at current LHC precision [78]. Other relevant interactions involve the state, which couples to the \tilded" vectors. The production of heavy vector states can go through a cascade of decays with rich phenomenology. The interaction Lagrangian involving and vector elds is given, at leading order in g=

eg and sin , by

L ;C = g r2 1

eg2

JHEP11(2016)076

s2 M2V p2egv

eS+ W ; +

M2A M2V

rs v

eS+ A;

g2M2V s2 p2eg2v

eS+ V ; + h:c (3.33)

L ;N =

pg[prime]2 + g2M2V r2 1

eS0 Z +

M2A M2V

rs v

p2egv

eS0 A0;

g2M2V s2 p2eg2v

eS0 V 0;

g[prime]2M2V s2 p2eg2v

eS0 S0; +

M2A M2V

rs p2v

eV 0 X0; (3.34)

An interesting collider signature would be the production of A with subsequent decay into

eS + , then eS ! + Z and the two resonances decay for instance into top pairs.3.2 Electroweak precision tests

The precise measurements near the Z-pole performed at several high energy experiments, especially at LEP [79], are crucial tests for any kind of model of New Physics. These e ects can be parameterised via the so-called oblique parameters [80, 81], expressed explicitly in terms of the weak boson self energies in eq. (A.55).

At tree level, the vector contribution to the oblique parameters are given by

S =

g2 r2 1

eg2 + g2

2 + (r2 1)s2 ; (3.35)

W = g2M2W

r2M2V M2A + 2M2A

; (3.36)

M2 AM2V

(r2 1) s2 + 2 + 2

eg2

Y = g[prime]2M2W

r2M2V M2A + 2M2A

(r2 1) s2 + 2 ; (3.37)

X = gg[prime]s2 M2W M2A r2M2V

AM2V

M2 AM2V

2 eg2 + g[prime]2

(r2 1) s2 + 2 + 2

2eg2

eg2

+ g[prime]2

(r2 1) s2 + 2

; (3.38)

where the other EW observables vanish, ^

T = 0, ^

U = 0. For = =2 these expressions agree with [82] once one identi es 1 ~ = r and sets the hyper-charge y = 0. In our analysis,

{ 14 {

we are going to use the notation adopted by the Particle Data Group (PDG) and rescale S = 4s2W ^

U= EW . Note that the above contributions can replace the contribution of the strong dynamics, estimated in [8, 54] as a loop of the underlying fermions. For r 1 the S parameter vanishes and higher order parameters,

W ,Y and X will play the dominant role. This situation is similar to the Custodial Vector Model described in [67, 83].

Additionally, deviations in the Higgs coupling w.r.t. the SM ones also bring additional contributions to the S and T parameters. The modi cation in the Higgs coupling, eq. (3.24), produce approximately the following deviations in the S and T parameters.

S = 1

(1 2V ) log mh + log mhmh;ref ; (3.39)

T =

38 cos2 W

eg4. Pure loops of the heavy resonances may be unsuppressed, however their e ect should be small as the dynamics is custodial invariant. Furthermore, due to the intrinsic strong interactions among resonances, such loop calculations are not reliable in general because perturbative expansions cannot be trusted. In this paper, therefore, we follow the philosophy of Vector Meson Dominance (which is experimentally tested in QCD) and assume that the tree level exchange is the dominant contribution, while e ects due to loops or higher order operators can be neglected. Alternative proposals have been put forward in the literature in order to render the theory more calculable, see for instance [30]. The main idea is to assume that the lowest lying resonances are weakly coupled to themselves and the rest of the dynamics, so that loops can be reliably calculated, together with the assumption of negligible higher order operators. As a result, potential cancellations have been observed that can relax the constraints from EW precision tests [84, 85], including loop contributions from light top partners: these results, however, are not generic. Loop corrections can be considered a modelling of the contribution of the strong dynamics (see also [86, 87]). In principle, improved calculability in ref. [30] can be imposed on our model, however we prefer to stay with a more conservative bound, as an order of magnitude estimate of the resonance e ects.

The experimental values from PDG for S and T (leaving U to be free), with a strong correlation coe cient 0:90, at 1 deviation are [88]:

S = 0:03 [notdef] 0:10 ; T = 0:01 [notdef] 0:12 : (3.41) The corresponding limits on the model parameters are shown in gure 1. For r [greaterorsimilar] 1 the vector partially cancels the Higgs contribution allowing a larger parameter space: in some areas of the parameter space, therefore, the most constraining bound comes form the measurements of the Higgs couplings, which give constraints on of the order of

{ 15 {

S= EW , T = ^

T = EW and U = 4s2W

(1 2V ) ln mh + log mhmh;ref : (3.40)

In the above formulas, the couplings of the SM gauge bosons to the Higgs, V , include corrections up to order 1=

eg2 from eq. (3.29) in order to be consistent with the tree-level e ects (for simplicity, we neglect the term in g[prime]2 so that V W Z). In principle, the

heavy resonances also contribute at one-loop level: naively, the loops with a Higgs boson have an additional suppression 1=

eg2 [notdef] m2W =M2V 1=

JHEP11(2016)076

1.0

S-T exclusion

r=0.1

r=0.6

r=0.9

r=1.1

0.8

0.6

0.4

JHEP11(2016)076

0.2

0.0

2 4 6 8 10

Figure 1. The S-T bound on the parameter space at 99% con dence level in the plane of (

eg) for r = 0:1; 0:6; 0:9; 1:1, with the cut o scale = p2v= sin . The region above the curves is excluded by EWPT.

[lessorsimilar] 0:6 [54]. Another e ect that may signi cantly modify the EWPT is the presence of the state, that will in general mix with h, with an un-calculable mixing angle , and can potentially alleviate the constraints from EWPT [54].

4 A minimal fundamental gauge theory

The e ective model characterised in the previous sections, can originate from a very simple scalar-less underlying theory [40, 41]: it consists of a gauged and con ning GHC = SU(2)

with two light Dirac avours transforming as the fundamental representation. Following the notation of [40, 42], the 2 Dirac fermions, U and D, can be arranged in a avour SU(4) multiplet as

Qi;a =

UL DL

fUL

fDL

; (4.1)

where is the spin Lorentz index, i is a avour index and a is a hyper-colour index. The tilded elds are left-handed spinors containing the right-handed components of the Dirac elds, i.e.L = i2U R and

~DL = i2D R.

Following the embedding of the EW symmetry we chose in this work, the pair (UL; DL)

transforms as a doublet of the weak isospin SU(2)L, while the other two (L; ~DL) as an anti-doublet of the custodial SU(2)R.

{ 16 {

4.1 Scalar sector

The scalar sector of the SU(4)=Sp(4) models was studied in [42]. In general we can write a scalar matrix

Mij = Q AiQ Aj = Q BiQ AjIAB (4.2)

where greek letters are Lorentz indices, capital letter are hyper-colour gauge indices and lower case latin are avour indices. The gauge group invariant IAB depends on the gauge group and fermion representation. If the gauge representation is pseudo-real, like in our case, IAB is antisymmetric (for fundamentals of SU(2)HC, IAB = AB). Accordingly, M is avour anti-symmetric, and it transforms as a 6SU(4). In general, this matrix contains both the light pNGBs and heavier scalar resonances.

4.2 Vector sector

The composite spin-1 states can be de ned in terms of the underlying fermions via the avour adjoint left-current:

(F )ji

(4.4)

Eab (4.5)

where Eab is the antisymmetric tensor making a hyper-colour singlet. Note the rst line

is non-standard notation for the left bilinears, but the one that directly implements the the avour transformation structure F ! g [notdef] F [notdef] g. The last line is the standard bilinear

notation. After some current algebra, the components in the vector matrix from table 1 can be associated to currents in terms of the underlying quarks, as detailed in table 2, where the notation is used: [Rfractur](J ) = 12(J + J ) and [Ifractur](J ) = i2(J J ).

We rst notice that this decomposition matches with the interpretation we provided at the beginning of the previous section: the triplet

!v corresponds to the \vector" current

!a contain an \axial" current component, associated with a meson in QCD, proportional to the cos and sin respectively. More precisely, due to the symmetry relating the Technicolor limit to the composite Higgs limit, the cos component of s[notdef];0 and ~x0 can be exactly mapped from the sin component of a[notdef];0 and ~v0 .

4.3 Discrete symmetries

The action of space-time discrete symmetries on the composite states can be derived from the transformation properties of the underlying quarks. However, the gauging of the EW interactions break P and C individually, but preserves CP in the strong con ning sector. Under CP , the bound state elds transform as

M CP

! M ; F

{ 17 {

JHEP11(2016)076

Q i

14 ji Q k

(4.3)

= Q ai Qb j 14 ij Qa k Qb k Eab

Qj Q i 14 ij Qk Qk

Q Q, typically associated to the meson in QCD, while

!s and

! (1) (F )T ; (4.6)

Field Fermion currents P C G GP

Massive spin-1 V (unbroken generators) v+ D U

v0 1

p2 U U D D

v U D

~v0 p2 cos [Ifractur] UT C D

+ 1

p2 sin U U + D D

+ i2 sin

UT C 5U D C 5DT s0 1p2 cos U 5U D 5D

p2 sin [Ifractur] UT C 5D

s+ cos D 5U

JHEP11(2016)076

+ + + +

s cos U 5D

+ i2 sin

U C 5UT DT C 5D

UT C 5U + D C 5DT

0 p2 [Rfractur] UT C 5D

+ i

U C 5UT + DT C 5D

Massive spin-1 A (broken generators) a+ i

2 cos UT C 5U D C 5DT sin D 5U

a0 p2 cos [Ifractur] UT C 5D

+ 1

p2 sin U 5U D 5D

+ + + +

a i

2 cos

U C 5UT DT C 5D sin U 5D

x0 p2 [Rfractur] UT C D

+ +

Table 2. Classi cation of composite vectors in terms of the underlying fermionic currents, using a notation C = 0 2. We quote transformation properties in terms of spacial parity P, charge conjugation C, and pion parity G de ned from listed currents. The combination GP is a good symmetry from the strong dynamics. In our notation, P-parity is de ned in spacial direction, i.e. (1) is \" parity.

where (1) = 1 for = 0, and 1 on spacial directions. The parities associated with the

spin-1 resonances are summarised in table 2: in our notation, a vector has CP = +, while CP = for a pseudo-vector. In the scalar sector, as expected, the Higgs h is de ned as a

scalar, while transforms as a pseudo-scalar, see table 3.

However, CP is not a convenient symmetry to label states as it maps charged states in their complex conjugate (particles into anti-particles). In terms of composite states, it is thus convenient to de ne a new parity G, de ned as C plus an internal rotation in the avour symmetry, which corresponds to an SU(2) rotation in our case:

U G

! 2D ; D

~x0 1

p2 cos U U + D D

p2 sin [Ifractur] UT C D

! 2U : (4.7)

with its action on fermion current illustrated in appendix B. Once combined with P , the

{ 18 {

Field Fermion currents P C G GP

Scalar pNGBsh 12 cos UU + DD

+ sin [Ifractur] UT CD

+ + + +

[Rfractur] UT CD

Table 3. Classi cation of the pNGB states in terms of the underlying fermionic currents, and their parities.

new symmetry de nes a parity acting as:

M GP! GP [notdef] M [notdef] TGP ; F

! (1) GP [notdef] (F )T [notdef] GP ; GP =

2 0

0 2

!: (4.8)

From table 2 and table 3 we see that all the tilded elds are odd under GP , as well as , thus this is the symmetry preventing decays of such eld directly into SM ones. Note, however, that GP is violated by the anomalous WZW term which generates decays for . Additional decay channels will also be generated for the vectors.

5 Phenomenology at the LHC and future 100 TeV colliders

5.1 Model implementation

To study the phenomenology of this model, we implemented the Lagrangian in Mad-Graph [89], using the Mathematica package FeynRules [90]. The implementation of the FeynRules model le is sketched in this section, while the model les are publicly available on the HEPMDB website.7

The neutral resonances A, Z, A0, V 0, S0,0 and ~V 0 are introduced as one particle class of VN, with two additional neutral states X0, ~X0 into another particle class of VX. The charged resonances W +, A+, V +, S+ and+ are put into one particle class of VC. The e ective Lagrangian for the strong sector is written in terms of physical pions, thus in the Unitary gauge, and vector bosons in gauge basis, with the latter rotated to their mass eigenstates via mixing matrices. The quarks and leptons only couple to

fW and B, thus the Yukawa structure is exactly the same as in the Standard Model. In the model implementation, the rotation matrices CM4[notdef]4 and NM5[notdef]5 are provided in two independent

Les Houches blocks of VCMix and VNMix as external parameters, whose numerical values are calculated by a speci c Fortran routine. Note that for the rotation, the eigenstates are ordered such that the diagonal element in CM4[notdef]4 or NM5[notdef]5 are maximal in each corresponding

column, to make sure each one carries the largest component of the original gauge state as

described in eq. (3.6).Five model parameters MV , MA, r, and

JHEP11(2016)076

eg are introduced into the Les Houches block DEWSB, with all associated decay constants f1 = p2MA=

eg, fK = p2MV =

eg and

qv2= sin2 + f21r2 de ned as internal parameters in the model le. Note that the

7http://hepmdb.soton.ac.uk/hepmdb:0416.0200

Web End =http://hepmdb.soton.ac.uk/hepmdb:0416.0200 .

{ 19 {

f0 =

1.0

V V S

0.2 0.4

0.8

0.6

(ppR)(fb)

0.4

0.2

0.0

10 2000 2500 3000 3500 4000 4500 5000

M (GeV)

0.5 1.0 1.5 2.0 2.5 3.0

JHEP11(2016)076

a) b)

Figure 2. Left: Drell-Yan cross section of composite states for MA = 3 TeV,

eg = 3:0, r = 0:6 and = 0:2 at LHC Run-II with ps = 13 TeV. Right: the largest =M of composite resonances as a function of r, with MV = 2:5 TeV, MA = 3 TeV,

eg = 3:0 (solid line) and

eg = 6:0 (dashed line).

latter relation derives directly from xing the value of GF (i.e. the EW scale v), as shown in eq. (3.16). Furthermore, we have imposed the SM values of e and MZ into the following analytic expressions to calculate g[prime] and g in terms of the independent model parameters.

1e2 =

1g[prime]2 +

1g2 +

eg2 ; det M2N M2ZI5

= 0 (5.1)

The model le is loaded using FeynRules package which exports the Lagrangian into UFO format [91]. We implement one python code as the parameter card calculator, to conduct the numerical rotation and write all block information into a param card.dat.

5.2 LHC Run-II

At the LHC Run-II, several resonances may be produced via the Drell-Yan production mechanism, with qq[prime] as initial states, therefore unfolding a delighting and rich phenomenology just like hadron spectroscopy in QCD but with completely new challenges and opportunities. Here we brie y discuss what would be the rst probable observations in the vector sector of our model, by investigating cross sections and experimental bounds from a ps =13 TeV LHC. The calculation is conducted in MadGraph 5 [89], using the PDF set NN23LO [92].

We present the cross section for each resonance at the LHC Run II in gure 2a by varying the parameter of MV , with xed MA = 3 TeV, r = 0:6 and = 0:2. The leading production channel is for the resonance V [notdef], followed by the neutral resonances V 0 and S0. The vector resonance S[notdef] is de ned as the one with largest portion of s+ state, with exact mass of MV . This state is rotated out from the matrix Ca in eq. (A.10), as a linear

combination of v+ and s+, thus it can not be directly produced due to current model set up. Increasing

eg will result in a smaller cross section as the couplings to quarks, generated by the mixing, are suppressed. Furthermore, only A[notdef],0 shows clear dependence on the other parameters, and r, and in the case of MV < MA, A[notdef],0 will always be subleading to

{ 20 {

V [notdef],0;S0 by several orders of magnitude. Note that for the \axial" resonances A[notdef] and A0, the cross sections turn out to be zero at the point of MV = MA, since the mixing does not contain any component of

fW and B and they decouple from SM quarks. We also check the parameter space where the narrow width approximation (NWA) can be used, as shown in gure 2b where we nd that the relevant parameter is r. We set the benchmark point to be MV = 2:5 TeV and MA = 3 TeV, and vary the other parameters (

eg; ; r) to inspect the region where the largest =M among all resonances is less than 50%. Generally, in order to use NWA as an approximate analysis for the event line shapes (e.g. di-lepton invariant mass distribution) we require =M < 10% so that interference e ects with Z; can be safely neglected. Due to the small mass split between many resonances, o -diagonal width e ects may also be important [93, 94]. Furthermore, the small width region will be favoured in order to resolve the compressed multi-peaking structure in the spectrum. According to this criterion, for a small = 0:2, the NWA applies very well for 0 < r < 2:0, but with a larger value = 0:4, the resonance will become broad and we need at least to tune

eg > 6:0

for the NWA to be e ective.

Alternatively the composite resonances can be produced via vector boson fusion (VBF) [73, 95, 96], with the production cross sections shown in gure 3 for the same benchmark scenario. In the calculation for pp ! R+2 jets (R = A0;[notdef]; V 0;[notdef]; S0), we consider all pure EW diagrams which form a gauge invariant set with the VBF topology, including diagrams with one t-channel weak boson exchange following a composite resonance emitted from a quark line. Although the signal de nition is ambiguous we expect that the VBF topology dominates. It was required pT (j) > 20 GeV in order to avoid the t-channel singularity of a photon exchange. The longitudinal weak bosons, WL, coupled to composite vectors through partial compositeness, play a less important role due to small mixing angles. This is noticed that in the right hand panel of gure 3, the cross section decreases with

eg. Therefore, in general VBF is a subdominant production mechanism. As previously remarked in [73], the exception to this trend occurs in the special parameter space region MA [similarequal] MV , where the Drell-Yan production of A0;[notdef] is highly suppressed.

In gure 4, we show the typical branching ratios for V0 and A0, with all the decays into SM fermions drawn in dotted line (V +; A+ show similar decay pattern). The entry in the legend is well patterned, each mode arranged in the same colour and line-style in order to easily compare the di erences in each scenario, with top standing for t

t, light quarks for

u1;2

u1;2+d1;2

d1;2 (Cabibbo CKM mixing used), leptons for l+l and neutrino for

. For

the decay of A0 in the case of MV < MA, we draw the mode with V [notdef];0 in the solid line while the mode with S[notdef];0 is in the dash-dotted line, since there is certain overlap between the decay modes of V 0h and S0h, similar for V [notdef]W and S[notdef]W , in the low region, but start to split from [greaterorsimilar] 0:3. An analogous situation happens to the decay of V 0 in the case of MA < MV , where branching ratios into A0Z and X0h, mostly overlap with each other in the range of 0 < < 0:8 due to the global symmetry. We also explore the branching ratios as a function of r: the fermions spectrum goes to a maximum at r = 1, while the W W or hZ spectrum, instead, goes to a minimum since the coupling is / (r2 1) at 1=

JHEP11(2016)076

eg order.

In either vector or axial resonance dominant case, the lower mass state displays a larger branching ratio into l+l and W +W rather than into nal states containing a composite

{ 21 {

V V S A A

VBF =0.2

V V S A

(ppR)(fb)

JHEP11(2016)076

10 2000 2500 3000 3500 4000 4500 5000

M (GeV)

4 6 8 10

Figure 3. Vector Boson Fusion (VBF) cross sections of composite states for MA = 3 TeV, r = 0:6 and = 0:2 at LHC Run-II with ps = 13 TeV. On the left:

eg = 3:0 is xed and MV varies; on the

right: MV = 2:5 TeV and

eg varies.

Leptons Neutrinos Light quarks Top-

W hZ

Leptons Neutrinos Light quarks Top

S h

V h

W hZ

X X') BR(V

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

X X') BR(A

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(a) (b)

Leptons Neutrinos Light quarks Top-

W hZ

Leptons Neutrinos Light quarks Top

A h

A Z

X X') BR(A

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

X X') BR(V

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 4. Branching ratios (BR) of composite states V 0 and A0, with the dependence on for

eg = 3:0, r = 0:6, MV = 2:5 TeV (top row) and MV = 3:5 TeV (bottom row), with xed MA = 3 TeV.

{ 22 {

0.6

exclusion

0.5

1600 1800 2000 2200 2400 2600 2800 3000

( )

Figure 5. Excluded region with MV = MA, recast from the 95% observed limit for the di-lepton l+l and di-boson W Z channel measured by ATLAS at LHC Run-II. The l+l exclusion is in solid lines, and the W Z exclusion in dashed ones. The parameter r is varied in the range of [0; 1], with contours in the (

eg MV ) plane on the left and in the ( MV ) plane on the right. The region with =M > 0:1, where NWA is not applicable, is identi ed by dotted contour lines.

vector, therefore we can exploit the most recent LHC Run-II results to constrain the model parameters.

Since our model provides several candidates as a heavy Z[prime] or W [prime], the LHC measurement for the Drell-Yan process and di-boson process would impose a stringent constraint on the parameter space. We calculate the theoretical cross section for pp ! R0 ! l+l and pp ! R[notdef] ! W Z in this SU(4)=Sp(4) model and compare them with the 95% upper bound

observed from the latest ATLAS measurement [97, 98]. Similar results can be obtained by using the corresponding CMS searches [99, 100]. The single lepton plus MET process is expected to require similar constraints to the di-lepton ones, thus we do not consider the l channel in detail for simplicity. We derive the exclusion limits in the parameter space speci ed by (

eg, , r ) after assuming MV = MA. Since we have not included the acceptance factor into this analysis, our result would be stronger than the exact 95% exclusion from the LHC Run-II searches. We show the exclusion contours from l+l and W Z in gure 5, with the di-lepton bound drawn in solid line, and the di-boson bound in dashed line. The plot shows that the two channels are complementary to each other. Notice that for an increasing r (in range of [0; 1]), the di-lepton channel imposes a stronger exclusion limit than the di-boson. The left panel of gure 5 shows that the lower limit for the mass of the resonances approaches MV [greaterorsimilar] 2:5 TeV for a coupling constant

eg [similarequal] 3:0 and small angle = 0:2. The right panel of gure 5 also shows that the di-lepton limit is more sensitive to the small area, while the di-boson channel mainly probes the large area. To summarise, for small < 0:2, as expected in composite pNGB Higgs limit of the model [54], the dilepton searches impose a lower bound on MV between 2 and 2:5 TeV, depending on the value of r.

{ 23 {

0.4

0.3

0.2

JHEP11(2016)076

0.1

1600 1800 2000 2200 2400 2600 2800 3000

( )

5.3 Future 100 TeV proton colliders

As shown in the previous section, current LHC bounds on the resonances range in the2 TeV ballpark. However, the naive expectation is that the resonances populate this mass range only in the Technicolour limit, where f v, in the composite pNGB limit, all

the resonances masses would be enhanced by a factor 1= sin due to the increase in the compositeness scale. Thus, the most natural mass range seem to lie above tens of TeV, thus more relevant for a future 100 TeV collider than for the LHC. For the simplest underlying gauge theory realising SU(4)=Sp(4) global symmetry, namely SU(2) with 2 Dirac fermions, lattice results have recently been published [101], providing a rst numerical prediction for the masses of the spin-1 resonances, found to be MA = 3:5 TeV= sin( ) and MV =

3:2 TeV= sin( ), far from LHC reach in the small limit. Thus, a machine colliding protons at ps = 100 TeV would be a perfect stage to probe its vast spectrum. It should be noted, however, that the masses can be lighter in di erent underlying gauge theories. In such case, even though the mass scales as 1= sin , the resonances might be at the reach of LHC.

The Drell-Yan production of the states V 0;[notdef], S0 and A0;[notdef] are shown in the top row of gure 6 as functions of and

eg. When we use Madgraph for simulation, only the PDF of the rst two generations of quarks are taken into account. However, at the high energy collider, the top and bottom quark PDFs can be important and need to be included to conduct a reliable prediction at 100 TeV [102]. Nonetheless, the cross sections present here can serve as a guideline. Similarly to the scenario described in the last section, the production rate for these states with r 1 is not large, around O(1) fb for 0:2, since

the resonance coupling to SM quarks are generated via mixing.

At 100 TeV, Vector Boson Fusion plays a more important role due to the enhancement of collinear radiated weak bosons from the spectator quarks, which translates into a large e ective luminosity of weak bosons inside the proton in the language of the E ective W approximation [103]. Indeed, the importance of VBF can be appreciated in the bottom row of gure 6, which shows that for V 0;[notdef] and A0;[notdef] resonances the VBF cross section is dominant over the Drell-Yan production, with very mild dependence on r. However the S0 production is much more sensitive to r. Since the W +W S0 coupling almost vanishes at the point of r = 1:0, with the main contribution to VBF from the V [notdef]W S0 fusion, this makes the S0 production particularly small. But once departing from r = 1:0, the W +W S0 fusion turns back to be important, therefore the VBF cross section for S0 is actually two orders of magnitude larger in the case of r = 1:1.

It is also important to note that SM physics, jets, top production and other important background for the process will present quite peculiar aspects at a 100 TeV collider (see e.g. [104]) and must be taken into consideration for a more precise phenomenological analysis.

The value of r is constrained by perturbativity of the e ective description. The consistent region is illustrated in gure 7, with the largest ratio of width over mass extracted in the plane of (

eg r). We nd that the region of r close to one is where all the resonances are narrow, thus it is valid to apply the NWA for event analysis. For r [negationslash]= 1 the coupling of

heavy vector to longitudinal bosons rapidly grows as the width of the resonance approaches

its mass, jeopardising perturbativity and the validity of the description [49].

{ 24 {

JHEP11(2016)076

104

1000

= 0.2MV 16.1 TeV

100

[ ]

0.01

~ = 3.0

V S A V A

JHEP11(2016)076

10 0.1 0.2 0.3 0.4 0.5

-5

0.1

4 6 8 10

(a) (b)

VS , r= 1.1

A V A

106

S , r =1.1 A

VBF g

~ = 3.0

VBF

=0.2 MV

16.1 TeV MA

17.6 TeV

104

1000

[ ]

100

[ ]

0.001

0.01

0.1 0.2 0.3 0.4 0.5

4 6 8 10

Figure 6. Drell-Yan (plots (a) and (b)) and Vector Boson Fusion (plots (c) and (d)) cross sections of composite states with MV = 3:2= sin( ) TeV, MA = 3:5= sin( ) TeV at ps = 100 TeV. The left panel shows the dependence on for

eg = 3:0, r = 1:0. The right panel shows the dependence on

eg for r = 1:0 and = 0:2. For the S0 VBF production, the dot-dashed cyan line is using r = 1:1, compared with the solid cyan line of r = 1:0.

We show the BR of V 0 state as a function of in the left panel of gure 8. S0 has similar decay structure as V 0, while charged states present similar pattern, thus we do not show them here. At r = 1 they mainly decay into SM fermions, in particular into di-jets. There will be small di erences in the BR spectrum between V 0 and S0. We nd that, in the channel of di-leptons, V0 decays at 10% and S0 at 40%. Moreover, for V0, the

decay into W +W is larger than hZ, while for S0 the decay of hZ turns to dominate over W +W . Varying r to be slightly larger than 1, notable changes happen as the di-boson and hZ channels rapidly overcome the fermion ones. For r = 1:1 the branching ratios are close to 45%, equally split between W +W and hZ at small . Only small variation can be observed in [greaterorsimilar] 0, but the two channels will start to split exactly till [lessorsimilar] 0:8.

{ 25 {

1.4

1.2

1.0

0.8

0.6

JHEP11(2016)076

4 6 8 10

Figure 7. The largest ratio of width over mass max( R=MR) among all the resonances. For large r, some resonances become too broad. The value of is set to 0.2, and the dependence on is very small in the perturbative region.

Leptons Neutrinos Light quarks Top

S h

W hZ

Leptons Neutrinos Light quarks Top

S h

V h

W hZ

X X') BR(V

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

X X') BR(A

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

(a) (b)

Leptons Neutrinos Light quarks Top-

W hZ

Leptons Neutrinos Light quarks Top

S h

V h

W hZ

X X') BR(V

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

X X') BR(A

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 8. Branching ratios (BR) of composite states A0 and V0, with the dependence on for MV = 3:2= sin( ) TeV, MA = 3:5= sin( ) TeV,

eg = 3:0 and r = 1:0 (top row) and r = 1:1

(bottom row).

{ 26 {

The decay pattern of the A0 resonance is shown in the right panel of gure 8, with more channels opened. At r = 1, the fermion channel is subdominant, while the W W , hZ channels almost disappear. The decay into V [notdef]=S[notdef]W and V 0=S0h become competitive, and we can observe dominant decays into , with further decaying into a pair of tops, or gauge bosons via the WZW anomaly term (as discussed in [54]). Since the decay is nearly 100% to Z for the lattice benchmark point, this will give rise to novel collider signature of 4t + Z nal states. For r = 1:1 the di-boson and Higgs-strahlung channels enter into play, but this does not alter the picture dramatically.

It has been argued that the luminosity of this future machine should be at least a factor 50 larger than the LHC luminosity in order to pro t from its full potential to nd new physics [105, 106]. An integrated luminosity of 3 30 ab1 per year is therefore expected,

leading to several heavy vector bosons produced and a promising phenomenology. We also stress that probing masses up to 50 TeV indirectly tests the models at small values of

0:05, where the high level of ne tuning renders the models unnatural and unappealing.

While an ultimate exclusion is not possible due to a decouplings limit ! 0 (like in

supersymmetry), in our opinion a 100 TeV collider can ultimately probe the \motivated" region of the parameter space in this class of composite Higgs models.

6 Conclusions

In the present work we construct an e ective Lagrangian that allows to describe vector spin-1 resonances in composite models of the Higgs boson. The framework adopted is the one of the hidden gauge symmetry approach, and we focus on a case with global symmetry structure based on the minimal case of an SU(4) symmetry broken to Sp(4). The chosen coset both satis es the requirement of a custodial Higgs sector and allows for a fundamental composite description of the new resonances in terms of fermionic bound states. The SU(4) structure is promoted to SU(4)0[notdef]SU(4)1 in order to apply the hidden gauge symmetry idea

and to obtain the vector and axial-vector states in the adjoint of the second SU(4). The paper discusses in detail the e ective Lagrangian for these states and their properties including mass matrices, mixing and couplings. The underlying fundamental realisation of the theory in terms of fermionic bound states is also discussed, together with the associated discrete symmetries, such as parity.

Schematically, the model contains 3 triplets that mix with the standard model gauge bosons, plus additional states that do not mix. Therefore, the phenomenology is much richer than in the minimal case containing just a single isospin triplet. We outline the main properties of the spin-1 states and their role in the phenomenology of the basic model. At the LHC, the most sensitive channel for searching for the new resonances is dilepton, which already imposes a bound on their mass around 2 TeV. The unmixed states, on the other hand, tend to decay into the singlet pion, , thus providing new signatures compared to the minimal cases studied in the literature. Furthermore, in the case of a pseudo-Goldstone Higgs, where the compositeness scale is raised, the masses are expected to be higher, in the 10 TeV range. We show that a future 100 TeV collider may be able to probe the most interesting parameter space for naturalness. We focus on a minimal

{ 27 {

JHEP11(2016)076

underlying description, where the masses have been computed on the Lattice, and detail the cross sections and branching ratios. This scenario can thus be one of the benchmark models for the 100 TeV collider.

This overview of the model, and its phenomenology, that we present is a template for the study of fundamental strong dynamics in the electroweak sector. Besides the speci c case under study, which corresponds to the minimal fundamental model, it can be applied to other scenarios like, for instance, the case of composite strongly interacting Dark Matter candidates.

Acknowledgments

We thank Marc Gillioz for collaboration at an early stage of this work. GC, HC and AD acknowledge partial support from the Labex-LIO (Lyon Institute of Origins) under grant ANR-10-LABX-66 and FRAMA (FR3127, F ed eration de Recherche \Andr e Marie Amp ere"). MTF is partially funded by the Danish National Research Foundation, grant number DNRF90.

A Explicit formulas

The explicit embedding of the vectors in SU(4) matrix form, in terms of charge eigenstates (see table 1), is

JHEP11(2016)076

F +

eF ; (A.1)

with

s a0 c s0 + v0 2p2

c s+ s a+ + v+ 2

c a+ + s s+

c a0 + s s0 ix0 2p2

s a + c s + v

c s0 s a0 v0 2p2

c a0 + s s0 + ix0

2p2

c a s s2

F =

c a + s s

c a0 + s s0 ix02p2

c s0 s a0 + v02p2

s a + c s v2

, (A.2)

c a0 + s s0 + ix0

2p2

c a+ s s+2

s a+ + c s+ v+2

c s0 s a0 + v0 2p2

ev0 + c

ex02p2 0

+2c

ev0 s

ex0 + i es0 2p2

0 c

ex0 + s

ev02p2 c

ev0 + s

ex0 + i es0 2p2

es 2

eF =

. (A.3)

es2 c ~v0 + s

ex0 i

es02p2 c

ex0 + s

ev02p2 0

es0 2p2

es+2 0 c

ev0 s

ex0 i

ex0 s

ev0 2p2

{ 28 {

In the gauge eigenbasis, the vector mass matrices in the charged MC and neutral MN

sectors are

M2C =

grM2As p2eg

gM2V p2eg

gM2V c p2eg

g2M2V (1 + !s2 )

eg2

grM2As p2eg

M2A 0 0

gM2V p2eg

; (A.4)

0 M2V 0

gM2V c p2eg

0 0 M2V

JHEP11(2016)076

g[prime]2M2V (1 + !s2 )2

g[prime]gM2V !s2

g[prime]M2Ars p2~g

g[prime]M2V p2~g

g[prime]c M2V p2~g

g[prime]gM2V !s2 2

g2(1 + !s2 )M2V 2

grM2As p2~g

gM2V p2~g

gc M2V p2~g

M2N =

g[prime]M2Ars p2~g

gM2Ars

p2~g M2A 0 0

; (A.5)

g[prime]M2Vp2~g

gM2Vp2~g 0 M2V 0

g[prime]c M2V p2~g

gc M2Vp2~g 0 0 M2V

where 2! = f20=f2K 1.

In the same basis, we provide the couplings of one Higgs with charged vector bosons

chV +V =

p2g2M2

V ! cos sin

pM2V (2!+1)M2Ar 2

g(M2V M2A)r cos

2 pM2V (2!+1)M2Ar2 0 0

(M2AM2V )r p2pM2

V (2!+1)M2Ar

2 pM2V (2!+1)M2Ar2 0

g(M2V M2Ar2) sin 2

pM2V (2!+1)M2Ar 2

(A.6)

0 0 0 0

g(M2V M2Ar2) sin 2

pM2V (2!+1)M2Ar 2

(M2AM2V )r p2pM2

V (2!+1)M2Ar

2 0 0

and for the neutral ones

chV 0V 0 =

p2g[prime]2M2V ! cos sin

pM2

V (2!+1)M

g[prime](M2

V M

2 2) sin

Ar2

p2g[prime]gM2V ! cos sin

pM2

V (2!+1)M

Ar2

A)r cos 2

pM2

V (2!+1)M

2 Ar2 0

g[prime](M2

V M

pM2

2 V (2!+1)M

Ar2

g(M2

p2g[prime]gMV ! cos sin

pM2

V (2!+1)M

Ar2

p2g2MV ! cos sin

pM2

V (2!+1)M

Ar2

2 Ar2 0

g(M2

V M

2 2) sin 2

pM2

V (2!+1)M

Ar2

A)r cos 2

pM2

V (2!+1)M

Ar2

g[prime](M2

V M

g(M2

2 Ar2 0 0

(M2

V )r p2pM2V (2!+1)M

Ar2

. (A.7)

0 0 0 0 0

g[prime](M2

V M

2 2) sin

(M2

pM2

2 V (2!+1)M

Ar2

g(M2

V M

2 2) sin 2

pM2

V (2!+1)M

Ar2

V )r p2pM2V (2!+1)M

2 Ar2 0 0

{ 29 {

Similarly, the -V -V interaction in gauge eigenstate are provided below:

LG ;C =

es+ ; +

g sin2 M2V M2Ar2 p2egv

sin M2A M2V

es+ a; + h:c

LG ;N =

g[prime] sin2 M2V M2Ar2 p2egv

es0 B

g sin2 M2V M2Ar2 p2egv

es0

fW 3;

+sin r M2A M2V

es0 a0; +

sin r M2A M2V

ev0 x0; : (A.8)

The above couplings are provided in the gauge eigenbasis, so one need to include the mixing matrices in order to extract couplings in the mass eigenstate basis. Approximate expressions for the mixing matrices are provided in the following section.

A.1 Perturbative diagonalisation of the mass matrices

The label of the physical states, W + , A+ , V + and S+ in the charged sector, and A , Z , A0 , V 0 and S0 in the neutral sector (left hand side of eq. (3.5)), are de ned as the ones with predominant component of the corresponding interaction eigenstates,

fW + ,

a+ , v+ and s+ in the charged sector, and B ,

fW 3 , a0 , v0 and s0 in the neutral sector respectively. Therefore, in theory, the columns in C and N do not assume xed

expressions which can swap depending on the largest entry, i.e, the matrix is reorganised in such a way that the diagonal entry is the largest in each column. In practice, however, for the parameter values we consider, the columns 1 and 2 in C and 1,2 and 3 in N have

xed expressions, even though there are signi cant mixing between the photon, A and Z . On the other hand, the states V 0;[notdef] and S0;[notdef] are highly mixed, and columns 3 and 4 in

C and 4 and 5 in N can be swapped, depending on the parameters, to ful l our de nition

of these states.In the following we provide expressions for these mixing matrices, C and N , de ned

in eq. (3.5), keeping in mind that the last two columns may be swapped depending on the values of their entries.

The charged rotation matrix can be split like

C = CaCb (A.9)

where

Ca =

JHEP11(2016)076

1 0 0 0 0 1 0 0 0 0 cos

pcos2( )+1

(A.10)

0 0

pcos2( )+1

cos

pcos2( )+1

{ 30 {

rotates away a state with mass exactly MV . The other part Cb at leading order in g=

eg is

given by:

Cb11 = 1

1 4

2 cos2( ) + r2 sin2( ) + 1

(A.11)

Cb12 =

gr sin( )

egp2 (A.12)

Cb14 =

p1 + cos2( )

egp2 (A.13) Cb21 =gr sin( )

egp2 (A.14)

Cb22 = 1 +

1 4

JHEP11(2016)076

2 r2 sin2( )

(A.15)

Cb24 =

2 M2Ar sin( )

p1 + cos2( )2 M2A M2V

(A.16)

Cb41 =

p1 + cos2( )

egp2 (A.17)

Cb42 =

2 M2V r sin( )

p1 + cos2( )2 M2A M2V

(A.18)

Cb44 = 1 +

2 (1 + cos2( ))4 (A.19)

and Cb3i = Cbi3 = 0; i [negationslash]= 3, Cb33 = 1.For the neutral gauge bosons, we de ne:

N = N a [notdef] N b [notdef] N c (A.20)

At leading order in 1=, each matrix has the following explicit expression:

N a11 = 1

1 4

g[prime]

; (A.21)

N a21 = 0; (A.22)

N a31 =

g[prime]r sin( )

egp2 ; (A.23)

N a41 =

+ cos2( ) + r2 sin2( )

g[prime]

egp2; (A.24)

N a51 =

g[prime] cos( )

egp2 (A.25)

N a12 =

g[prime]g

eg2

sin2( ); (A.26)

N a22 = 1

1 4

2 1 + cos2( ) + r2 sin2( )

; (A.27)

gr sin( )

egp2 ; (A.28)

{ 31 {

N a32 =

egp2; (A.29) N a52 = g cos( )

egp2 (A.30)

N a13 =

g[prime]r sin( )

egp2 ; (A.31) N a23 =gr sin( )

egp2 ; (A.32)

N a33 = 1

1 4

N a42 =

g[prime]2 + g2

eg2 r2 sin2( ); (A.33)

N a43 =

JHEP11(2016)076

g2 g[prime]2

eg2 sin( )

M2V M2V M2A

; (A.34)

g[prime]2 + g2

N a53 =

eg2 sin( ) cos( )

M2V M2V M2A

(A.35)

g[prime]

egp2; (A.36) N a24 = g

egp2; (A.37)

N a34 =

N a14 =

g2 g[prime]2

eg2 sin( )

M2A M2V M2A

; (A.38)

g[prime]2 + g2

N a44 = 1

1 4

eg2 ; (A.39) N a54 = 0 (A.40)

N a15 =

g[prime] cos( )

egp2 ; (A.41) N a25 =g cos( )

egp2 ; (A.42)

N a35 =

g[prime]2 + g2

eg2 sin( ) cos( )

M2A M2V M2A

; (A.43)

N a45 =

g[prime]2 g2

eg2 cos( ); (A.44)

g[prime]2 + g2

N a55 = 1

1 4

eg2 cos2( ) (A.45)

pg[prime]2+g2

g[prime]

pg[prime]2+g2 0 0 0

g[prime]

pg[prime]2+g2

N b =

pg[prime]2+g2 0 0 0

0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

(A.46)

{ 32 {

: (A.47)

For the vector bosons V 0 and S0, we take a further approximation sin2 0, and we

de ne:

N c =

N a [notdef] N b will take the mass matrix into the following form:

0 0 0 0 0 0 M2Z 0 0 0 0 0 M2A0 0 0

0 0 0 1

eg2 M2V

g[prime]2

+ g2

+ 2

eg2

g[prime]2 g2

M2V cos( )

0 0 0 1

2 eg2 g[prime]2 g2

M2V cos( )

eg2 M2V

g[prime]2

+ g2

cos2( ) + 2

JHEP11(2016)076

1 0 0 0 0 0 1 0 0 0 0 0 1 0 0

0 0 0 (

: (A.48)

g[prime]2+g2)sin2

4p2(g[prime]2g2)

+ 1

p2 1p2

(g[prime]2+g2)sin2 4p2(g[prime]2g2)

0 0 0 1

(g[prime]2+g2)sin2 4p2(g[prime]2g2)

+ 1

The rotation N = N1 [notdef] N2 [notdef] N3 will fully diagonalize the mass matrix to be:

0 0 0 0 0 0 M2Z 0 0 0 0 0 M2A0 0 0

0 0 0 12 M2V

2 + g[prime]22 2 sin2

: (A.49)

0 0 0 0 12 M2V

2 + g22 2 sin2

A.2 EW precision parameters

The oblique parameters are related to the polarisation functions of the EW gauge bosons:

[prime]W3B(0) [prime]W+W (0)

; (A.50)

1 M2W

W3W 3 (0) W

+W (0) [prime]W+W (0)

; (A.51)

[prime]W3W 3 (0) [prime]W

+W (0) [prime]W+W (0)

; (A.52)

M2W 2

[prime][prime]W3W 3 (0) [prime]W+W (0)

; (A.53)

M2W 2

[prime][prime]BB(0) [prime]BB(0)

; (A.54)

M2W

[prime][prime]W3B(0)

2 q [prime]W+W (0) [prime]BB(0)

(A.55)

{ 33 {

B G-parity transformation

The convention we are using here are:

!; C = 0 2 =

2 2 !; (B.1)

= 1; i

; = 1; i

: (B.2)

with the conjugate of fermion currents derived to be:

U 5D

= D 5U (B.3)

UT C D

= D CUT (B.4)

UT CD

= DCUT ;

UT C 5D

= D C 5UT (B.5)

Using the de nition for the G-parity in eq. (4.7), we can derive its action on fermionic currents to be:

UU G

! DD; DD

! UU (B.6)

D U G

JHEP11(2016)076

! D 5U (B.7)

U U G

! D U; D 5U

! U U (B.8)

D 5D G

! D D; D D

! D 5D (B.9)

UT C 5U G

! U 5U; U 5U

! D C 5DT ; DT C 5D

! U C 5UT (B.10)

UT CD G

! (UT CD) (B.11) UT C D G

! (UT C D) (B.12) UT C 5D G

! (UT C 5D) (B.13)

<(UT CD) G! [Rfractur](UT CD); [Ifractur](UT CD)

! [Ifractur](UT CD) (B.14)

<(UT C D) G! [Rfractur](UT C D); [Ifractur](UT C D)

! [Ifractur](UT C D) (B.15)

<(UT C 5D) G! [Rfractur](UT C 5D); [Ifractur](UT C 5D)

! [Ifractur](UT C 5D) (B.16)

C Branching ratios

Since in section 5 we discussed the production and decay of V , S and A triplets, here in gure 9 and gure 10 we show some representative branching ratio distributions of the more exotic vector states.

The branching ratio of ~X0 is independent on, r and , and this state will decay into h ~V 0, Z0, W [notdef] in the ratio of (1 : 1 : 2).

{ 34 {

h 0

h Z

X X') S ~

BR(

X X') S ~

BR(

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

10 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

JHEP11(2016)076

Figure 9. Branching ratios (BR) of0 for MA = 3 TeV, MV = 3:5 TeV and

eg = 3. On the left as a function of with r = 0:6 and on the right as a function of r with = 0:2. The behaviour of the charged[notdef] is analogous, replacing Z by W [notdef] and A0 by A[notdef].

S Z

V Z

h 0

X X') BR(X

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

X X') V ~

BR(

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Figure 10. Left: ratios (BR) of ~V 0 for MA = 3 TeV, MV = 3:5 TeV. It is independent of

eg and r.

This state decays only to h ~X0 and X0. Right: branching ratios (BR) of X0 for MA = 3 TeV and MV = 2:5 TeV. It is independent of

eg and r.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Abstract

We provide a non-linear realisation of composite Higgs models in the context of the SU(4)/Sp(4) symmetry breaking pattern, where the effective Lagrangian of the spin-0 and spin-1 resonances is constructed via the CCWZ prescription using the Hidden Symmetry formalism. We investigate the EWPT constraints by accounting the effects from reduced Higgs couplings and integrating out heavy spin-1 resonances. This theory emerges from an underlying theory of gauge interactions with fermions, thus first principle lattice results predict the massive spectrum in composite Higgs models. This model can be used as a template for the phenomenology of composite Higgs models at the LHC and at future 100 TeV colliders, as well as for other application. In this work, we focus on the formalism for spin-1 resonances and their bounds from di-lepton and di-boson searches at the LHC.

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Title

Vector and axial-vector resonances in composite models of the Higgs boson

Author

Franzosi, Diogo Buarque; Cacciapaglia, Giacomo; Cai, Haiying; Deandrea, Aldo; Frandsen, Mads

Pages

1-42

Publication year

2016

Publication date

Nov 2016

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP11(2016)076

ProQuest document ID

1838563477

SISSA, Trieste, Italy 2016

Vector and axial-vector resonances in composite models of the Higgs boson

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