Eur. Phys. J. C (2016) 76:416DOI 10.1140/epjc/s10052-016-4211-9
Regular Article - Theoretical Physics
http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-016-4211-9&domain=pdf
Web End = http://crossmark.crossref.org/dialog/?doi=10.1140/epjc/s10052-016-4211-9&domain=pdf
Web End = The complete HEFT Lagrangian after the LHC Run I
I. Brivio1,a , J. Gonzalez-Fraile2,b, M. C. Gonzalez-Garcia3,4,5,c, L. Merlo1,d
1 Departamento de Fsica Terica and Instituto de Fsica Terica, IFT-UAM/CSIC, Universidad Autnoma de Madrid, Cantoblanco,
28049 Madrid, Spain
2 Institut fr Theoretische Physik, Universitt Heidelberg, Heidelberg, Germany
3 C.N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, SUNY at Stony Brook, Stony Brook, NY 11794-3840, USA
4 Departament dEstructura i Constituents de la Matria and ICC-UB, Universitat de Barcelona, 647 Diagonal, 08028 Barcelona, Spain
5 Instituci Catalana de Recerca y Estudis Avanats (ICREA), Passeig de Llus Companys 23, 08010 Barcelona, Spain
Received: 4 May 2016 / Accepted: 14 June 2016 / Published online: 23 July 2016 The Author(s) 2016. This article is published with open access at Springerlink.com
Abstract The complete effective chiral Lagrangian for a dynamical Higgs is presented and constrained by means of a global analysis including electroweak precision data together with Higgs and triple gauge-boson coupling data from the LHC Run I. The operators basis up to next-to-leading order in the expansion consists of 148 (188 considering right-handed neutrinos) avour universal terms and it is presented here making explicit the custodial nature of the operators.This effective Lagrangian provides the most general description of the physical Higgs couplings once the electroweak symmetry is assumed, and it allows for deviations from the SU(2)L doublet nature of the Standard Model Higgs. The comparison with the effective linear Lagrangian constructed with an exact SU(2)L doublet Higgs and considering operators with at most canonical dimension six is presented. A promising strategy to disentangle the two descriptions consists in analysing (i) anomalous signals present only in the chiral Lagrangian and not expected in the linear one, that are potentially relevant for LHC searches, and (ii) decorrelation effects between observables that are predicted to be correlated in the linear case and not in the chiral one. The global analysis presented here, which includes several kinematic distributions, is crucial for reducing the allowed parameter space and for controlling the correlations between parameters. This improves previous studies aimed at investigating the Higgs Nature and the origin of the electroweak symmetry breaking.
a e-mail: mailto:[email protected]
Web End [email protected]
b e-mail: mailto:[email protected]
Web End [email protected]
c e-mail: mailto:[email protected]
Web End [email protected]
d e-mail: mailto:[email protected]
Web End [email protected]
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . 2
2 The complete HEFT Lagrangian . . . . . . . . . . . 32.1 The NLO Lagrangian . . . . . . . . . . . . . . 52.2 NLO basis: bosonic sector Lbos . . . . . . . 62.2.1 CP even bosonic basis L C Pbos . . . . . . 62.2.2 CP odd bosonic basis L
http://orcid.org/0000-0002-0396-5866
Web End = C Pbos . . . . . . 72.3 NLO basis: fermionic sector Lfer . . . . . . . 82.3.1 Single fermionic current L2F . . . . . 8Quark current operators . . . . . . . . . . . . . 8Leptonic current operators . . . . . . . . . . . 92.3.2 Four-fermion operators L4F . . . . . . 10Pure quark operators . . . . . . . . . . . . . . 10Pure leptonic operators . . . . . . . . . . . . . 10Mixed quarklepton operators . . . . . . . . . 102.4 Comparison with the SMEFT basis . . . . . . . 11
3 Phenomenology . . . . . . . . . . . . . . . . . . . 123.1 Physical parameters denitions . . . . . . . . . 123.2 Constraints from EWPD . . . . . . . . . . . . 133.3 Effects in Higgs physics . . . . . . . . . . . . 163.4 Triple gauge-boson couplings and Higgs interplay 184 Higher order operators and expansion validity . . . . 215 Conclusions . . . . . . . . . . . . . . . . . . . . . 23A Additional operators in the presence of RH neutrinos 24 B Removal of F(h) F(h) from the Higgs and fermions
kinetic terms . . . . . . . . . . . . . . . . . . . . . 25B.1 A practical example . . . . . . . . . . . . . . . 25
C Construction of the fermionic basis . . . . . . . . . 26C.1 Useful identities . . . . . . . . . . . . . . . . . 26C.2 Construction of L2F . . . . . . . . . . . . . 26C.3 Construction of L4F . . . . . . . . . . . . . 27Four-quark (lepton) operators . . . . . . . . . . 27Mixed quarklepton operators . . . . . . . . . 28
123
416 Page 2 of 45 Eur. Phys. J. C (2016) 76 :416
D Application of the EOMs . . . . . . . . . . . . . . . 28D.1 Operators that have been removed via EOM . . 29 Bosonic sector . . . . . . . . . . . . . . . . . . 29Fermionic sector . . . . . . . . . . . . . . . . . 29
E Feynman rules . . . . . . . . . . . . . . . . . . . . 30 FR: propagators . . . . . . . . . . . . . . . . . . . 31FR: Bosonic . . . . . . . . . . . . . . . . . . . . . 31FR: Fermionic . . . . . . . . . . . . . . . . . . . . 35
Single quark current . . . . . . . . . . . . . . . 35Single lepton current . . . . . . . . . . . . . . 39Four quarks . . . . . . . . . . . . . . . . . . . 41Four leptons . . . . . . . . . . . . . . . . . . . 42Two quark-two leptons . . . . . . . . . . . . . 43References . . . . . . . . . . . . . . . . . . . . . . . . 44
1 Introduction
The discovery of a resonance at LHC [1,2] compatible with the Standard Model (SM) scalar boson (Higgs for short) [35] opened a new era in particle physics. Now, the on going LHC measurements of the Higgs properties are a crucial step to understand the nature of the Higgs boson and of the Electroweak (EW) symmetry breaking (EWSB).
Without entering into details of specic scenarios, the formalism of Effective Field Theories (EFT) represents an optimal tool for studying the phenomenology of the Higgs sector. In particular, an appropriate description of scenarios in which the Higgs belongs to an elementary SU(2) doublet is provided by the Standard Model EFT (SMEFT). This consists of operators constructed with the SM spectrum, invariant under the Lorentz and SM gauge symmetries and respecting an expansion in canonical mass dimensions d. Assuming lepton and baryon number conservation, the rst corrections to the SM are provided by operators of dimension six [6,7], suppressed by two powers of the cut-off scale . Weakly coupled theories are the typical underlying scenarios that can be matched to the SMEFT (also referred to as linearLagrangian) at low energy.
Scenarios where the Higgs does not belong to an elementary exact SU(2)L doublet are still allowed within the current experimental accuracy. This is the case, for example, of composite Higgs models [812] or dilaton constructions [13,14].It is then fundamental and necessary to identify observables that allow one to disentangle these different possibilities.When the Higgs is not required to belong to an exact EW doublet, instead, a useful tool is the so-called Higgs EFT (HEFT) (also dubbed chiral Lagrangian). The main difference between SMEFT and HEFT resides in the fact that, in the latter formalism, the physical Higgs h and the ensemble of the three EW Goldstone bosons
are treated as independent objects, rather than being collectively described by the Higgs doublet. In particular, the physical Higgs h is assigned
to a singlet representation of the SM gauge groups. The Gold-stone bosons sector has been studied intensely in the past [1518] in the context of Higgs-less EWSB scenarios. These works were the rst to describe the GBs by means of a dimensionless unitary matrix transforming as a bi-doublet of the global symmetry SU(2)L SU(2)R,
U(x) eiaa(x)/f , U(x) LU(x)R, (1.1) being f the scale associated to the SM GBs, and L, R the SU(2)L,R transformations. After EWSB, the invariance under the group SU(2)L SU(2)R is broken down to the
diagonal SU(2)C, commonly called custodial symmetry, and explicitly broken by the gauging of the hypercharge U(1)Y and by the fermion mass splittings. It is customary to introduce two objects, the vector and scalar chiral elds, that transform in the adjoint of SU(2)L. They are dened, respectively, as
V (DU)U, T U3U, (1.2) where the covariant derivative is given by
DU(x) U(x) + igW(x)U(x)
ig
2 B(x)U(x)3.
(1.3)
Unlike V, T is not invariant under SU(2)C and can therefore be considered a custodial symmetry breaking spurion. The bosonic Higgs-less EW chiral Lagrangian can then be constructed with V, T and the gauge-boson eld strengths as building blocks, and the tower of invariant operators shall be organised according to a chiral (derivative) expansion [19].
In the last decade, the EW chiral Lagrangian has been extended with the introduction of a light physical Higgs h [2028], treated as an isosinglet of the SM gauge symmetries. The dependence on the h eld is customarily encoded in generic functions F(h), that are used as building blocks for
the construction of the effective operators. These functions are made adimensional by implicitly weighting the insertions of the Higgs eld with an opportune suppression scale fh, so that one may rewrite the dependence as F(h/fh). It is
worth underlining that the dependence on the structure (1 +
h/v), where v is the EW vacuum expectation value (vev), that characterises the SMEFT Lagrangian is lost in the HEFT and substituted by a generic h/fh expansion.
The typical underlying scenarios that can be described at low energy in terms of the matrix U(x), the Higgs functions
F(h) and the rest of the SM elds, are those of composite Higgs models [812,29]. These assume the existence of some strong (ultracolour) interaction at high energy, and initially invariant under some global symmetry group G. At
the scale s, the formation of ultracolour condensates breaks spontaneously this invariance, leaving a residual symmetry
H that can embed the EW group. This triggers the appear
ance of a certain number of Goldstone bosons, among which
123
Eur. Phys. J. C (2016) 76 :416 Page 3 of 45 416
three can be identied with the would-be GBs of the EW group and a fourth one with the Higgs. In such scenarios, all the SM scalars are naturally associated to the same scale f = fh f , with s 4 f . Spontaneous EWSB is
triggered by some explicit breaking of the H symmetry (pro
vided either by external symmetries [8] or by gauging the SM symmetry together with fermion interactions [11]) and takes place in a second stage. At this level, the Higgs eld acquires a vev h , which does not need to coincide with the EW scale
v, dened by the EW gauge-boson mass: the three quantities v, f and h are instead related by a model-dependent
function. The splitting between v and f constitutes the well-known ne-tuning of composite Higgs models. It is usually expressed in terms of the parameter by
v2f 2 , (1.4)
that substantially quanties the degree of non-linearity of the Higgs dynamics. The low-energy projection of composite Higgs models can be described by the HEFT Lagrangian [30,31] and the matching conditions allow one to write the low-energy effective operator coefcients in terms of the high-energy parameters, and the generic functions F(h)
as trigonometric functions of h/f . The HEFT Lagrangian can also be used to describe the SMEFT [2225,3032], after identifying the operator coefcients of the effective Lagrangians and writing all the F(h) functions in terms
of (1 + h/v). Dilaton constructions [13,14] or even more
exotic models, where the Higgs is an EW singlet, can also be described by the HEFT Lagrangian.
Without assuming any specic underlying scenario or comparing with SMEFT, the v/fh and v/f parameters are not physical and can be reabsorbed in the operators coefcients and in the coefcients of the F(h) functions. This is
tantamount to substituting f and fh by v, which ensures canonical kinetic terms for the GBs and xes the correct order of magnitude for the gauge bosons masses, without ne-tunings. This notation will be employed in the following, unless otherwise specied.
The disparities between the SMEFT and the HEFT originate from the different nature of the building blocks used in the construction of the effective operators. The independence between the GB eld U(x) and the physical h, together with the fact that h does not transform under the SM gauge symmetries, leads to a different ordering of the chiral effective operators compared to the linear ones. As a result, at any given order in the expansion the number of chiral independent operators is much larger than in the SMEFT case. The corresponding phenomenology, focussing on the bosonic part of the Lagrangian, has been studied in Refs. [24,25], where signatures that may allow one to discriminate between an elementary and a dynamical Higgs have also been identied. These signatures include sets of couplings that are pre-
dicted to be correlated in an elementary Higgs scenario but are generically decorrelated in the dynamical case, as well as effects that are expected to be suppressed in the linear realisation but may appear at the lowest order in the chiral expansion. These signatures are also typical in Dark Matter studies when the Higgs is not taken to be an exact SU(2)L doublet [33]. Complementary signatures that can distinguish between SMEFT and HEFT also include the scattering of the longitudinal components of the gauge bosons [3436].
The complete non-redundant HEFT Lagrangian including both bosonic and fermionic operators has been constructed in this work and is presented in Sect. 2, making explicit the custodial nature of the operators. The HEFT basis is formed by 148 independent avour universal operators altogether, whose extension to generic avour contractions is straightforward. The Lagrangian does not account for the presence of right-handed neutrinos, whose inclusion in the spectrum would imply the addition 40 extra operators to the basis, listed in Appendix A. Section 2 also contains a comparison between the HEFT Lagrangian and the SMEFT one, while a phenomenological analysis of the HEFT basis is presented in Sect. 3. The study considers all the available collider data, which includes electroweak precision measurements and Higgs and triple gauge-boson vertex (TGV) data from the LHC Run I. To the best of our knowledge, this is the rst time that such analysis has been done for the complete HEFT description. Finally, Sect. 4, contains a discussion of the impact of higher order operators: a set of invariants that may become relevant at the increased energies foreseen for the LHC and future colliders is also pointed out. The conclusions are presented in Sect. 5, while some more technical details are deferred to the appendices, together with the Feynman Rules for the CP even subset of HEFT operators.
2 The complete HEFT Lagrangian
In this section we review the construction of the HEFT Lagrangian, in a notation similar to that of Refs. [22 25,32,37]. The bosonic building blocks are the gauge eld strengths B, W, G, the vector and scalar chiral elds
V and T dened in Eq. (1.2) and the functions F(h) intro
duced in the previous section. The SM fermions are conveniently grouped into doublets of the global SU(2)L,R symmetries:
QL = U
L , QR = UR DR
, LL = L EL
,
L R =
0 ER
. (2.1)
This choice allows one to have a more compact notation for the fermionic operators. The SU(2)R doublet structure can easily be broken with the insertion of the custodial symmetry
123
416 Page 4 of 45 Eur. Phys. J. C (2016) 76 :416
breaking spurion T. Notice that the L R doublet only includes right-handed charged leptons. The inclusion of right-handed neutrinos requires an extension of the fermionic basis presented in Sect. 2.3 with the addition of the operators listed in Appendix A.
The HEFT Lagrangian can be written as a sum of two terms,
LHEFT L0 + L , (2.2)
where the rst term contains the leading order (LO) operators and the second one accounts for new interactions and for deviations from the LO.
The LO Lagrangian includes the kinetic terms for all the particles in the spectrum, the Yukawa couplings and the scalar potential1:
L0 =
1 4GG
plings have an analogous structure to FC(h):
YQ(h) diag nY (n)U
hnvn ,
n
Y (n)D
hn vn
,
YL (h) diag 0,
n
. (2.6)
The n = 0 terms yield fermion masses, while the higher
orders describe the interaction with n insertions of the Higgs eld h, accounting in general for non-aligned contributions.
The kinetic terms of the fermions and of the physical Higgs are not accompanied by any F(h) since, as shown in
Appendix B, it is always possible to reabsorb their contributions inside the generic functions FC(h) and YQ,L(h). This
can be done either via a eld redenition or, alternatively, applying the Equations of Motion (EOMs) (the two procedures are not equivalent in general, but lead to the same result at rst order in the deviations from the LO). Moreover, the kinetic terms of the gauge bosons in the rst line of Eq. (2.3) do not come associated with any F(h), assuming that the
transverse components of the gauge elds, described by the gauge eld strength, do not couple strongly to the Higgs sector. These couplings can be neglected at the LO and be considered, instead, at the next-to-leading order (NLO).
L contains higher order operators with respect to those appearing in L0. The precise ordering of these operators depends on the choice of a specic power counting rule. The HEFT can be seen as a fusion of two theories, the chiral perturbation approach associated to the SM GBsi.e. the longitudinal components of the gauge bosonsand the traditional linear description that applies to the transverse components of the gauge bosons and to fermions. The physical h should also undergo the chiral perturbation description as it enters in the Lagrangian via the adimensional functions
F(h): the latter can be interpreted as playing the same role as the adimensional GB matrix eld U(x). Indeed, in concrete composite Higgs models, the pseudo-GB nature of the Higgs forces the F(h) functions to take trigonometric struc
tures [30]. Being the HEFT a merging between linear and chiral descriptions, the counting rules which apply singularly to each of the expansions hold simultaneously for the HEFT [38]. As a result, the LO Lagrangian in Eq. (2.3) itself does not strictly respect the chiral expansion: L0 contains both operators with two derivatives and the gauge-boson kinetic terms, which has four derivatives; at the same time, some two-derivative operators have been excluded from the LO. On the other hand, L0 does not even follow an expansion in canonical dimensions, as for instance the Yukawa interactions and the gauge-boson mass term present an innite series of h legs, contrary to all the other terms in the LO Lagrangian.
Y (n)
hn vn
14 WaWa
14 B B
4 Tr(VV)FC(h) V (h)
+i QL /
DQL + i QR /
DQR + i LL /
DLL + i L R / DL R
v2 QL
+
1
2hh
v2
UYQ(h)QR + h.c.
v 2 L
L UYL(h)L R + h.c.
g2s162 s G G , (2.3)
where G 12 G . The rst line describes the kinetic
terms of the gauge bosons; the second line contains the Higgs and Goldstone bosons kinetic term, the scalar potential, and the mass terms for the EW gauge bosons; the third line presents the kinetic terms for all the fermions, while the fourth line accounts for the Yukawa interactions. Finally, the last line contains the theta term of QCD. The function FC(h)
appearing in the kinetic term for the GBs can be expanded as
FC (h) = 1 + 2aC
hv + bC
h2v2 + (2.4)
where the dots account for higher powers of (h/v). For the phenomenological analysis it is convenient to single out the BSM part of the coefcients aC, bC, using the notation
aC = 1 + aC, bC = 1 + bC, (2.5)
where aC, bC will be assumed to be of the same order as the coefcients accompanying the operators appearing in L . The functions YQ,L(h) appearing in the Yukawa cou-
1 Comments on the construction of the LO Lagrangian in Eq. (2.3) are given in Appendix B.
123
Eur. Phys. J. C (2016) 76 :416 Page 5 of 45 416
The renormalisability conditions are also different in the two descriptions. In the linear expansion an n-loop diagram containing one single d = 6 vertex generates divergent con
tributions that can be reabsorbed by other d = 6 operators
and do not require the introduction of any higher-dimensional operator. On the contrary, in the chiral case, 1-loop diagrams with n insertions of a two-derivative coupling, usually listed in the LO Lagrangian, produce divergences that require the introduction of operators with four-derivatives, which generically constitute the NLO Lagrangian.
Finally, the HEFT presents an additional aspect that makes it hard to identify a proper counting rule: the presence of multiple scales. Besides the cut-off of the theory , one should consider the presence of the GB scale f and of the h-scale fh. Although it may happen that the last two coincide with f = fh = f and that they are related to the rst one by
the constraint 4 f (which is the case in composite
Higgs models), the three scales are in principle independent and associated to different physical quantities. On top of this, one should not forget the ne-tuning associated to the EW scale v and parametrised by dened in Eq. (1.4). In practice, the counting rule associated to the HEFT depends on more than one expansion parameters and may vary depending on the typical energy scale of the observables considered in the phenomenological analysis.
In conclusion, rather than basing the choice of the NLO Lagrangian operators on a sophisticated counting rule whose applicability is not valid in full generality, here the selection is performed with the following strategy. An NLO operator should satisfy at least one of the criteria below:
It is necessary for reabsorbing 1-loop divergences arising from the renormalisation of L0.
It presents the same suppression as the operators in the rst class and receives nite 1-loop contributions: for instance, all the four-fermion operators are included in the NLO, in spite of the fact that only a subset of these is required to reabsorb 1-loop divergences.
It has been left out from the LO Lagrangian due to phenomenological reasons.
The suppression factor of each operator is determined using the NDA master formula, rst proposed in Ref. [39] and later modied in Refs. [38,40]. Following the notation of Ref. [38]:
4 162
Np
4
N
4 A
NA
Ny , (2.7)
where represents either the SM GBs or h, a generic fermion, A a generic gauge eld, g the gauge couplings
and y the Yukawa couplings. All the operators appearing in the LO Lagrangian in Eq. (2.3) are normalised according to this formula, apart from the operators providing gauge-boson masses, (v2/4)Tr(VV)FC(h), and
fermions masses (v2)
LUY(h)R, which are multi
plied by powers of the EW scale v and not by or f as expected. This is due to the well-known ne-tuning, typical of theories where the EWSB sector is non-linearly realised. Notice that with these conventions all the kinetic terms are canonically normalised, differently from the following using the original version of the NDA master formula from Ref. [39].
The master formula also ensures that the operators belonging to the NLO Lagrangian are typically suppressed with respect to those of L0 by powers of (4)(n2), reecting the renormalisation of the chiral sector, and/or by powers of (n2), associated to possible new physics contributions.
Different cases will be discussed when necessary.
2.1 The NLO Lagrangian
The second part of the HEFT Lagrangian, L , contains in general all the invariant operators appearing beyond the leading order. They include corrections to the interactions contained in L0 as well as completely new couplings. This
Lagrangian can be generically written as a sum of two parts
L = Lbos + Lfer, (2.8)
where Lbos contains all the purely bosonic operators, while Lfer accounts for the interactions that involve fermions.
In this work, L will be restricted to the NLO, dened according to the rules presented in the previous section. An alternative construction of a NLO Lagrangian was derived in Ref. [26]. We present a set of invariants that forms a complete, non-redundant basis at this order in the effective expansion, which has been constructed identifying rst a complete basis for each of the two sectors individually (bosonic and fermionic) and subsequently employing the EOMs to remove redundant terms.
Given the large number of invariants, the operators are classied as follows: the bosonic basis is split into CP conserving and CP violating subsets (the eld h is assumed to be a CP even scalar):
Lbos = L C Pbos + L
4
3/2
N
g 4
Ng
y 4
C Pbos , (2.9)
while in the fermionic sector the distinction is between fermionic single- and double-current structures:
Lfer = L2F + L4F. (2.10)
The operators are named differently according to the category to which they belong and each of them includes a function
Fi (h) conventionally parametrised as
123
416 Page 6 of 45 Eur. Phys. J. C (2016) 76 :416
Fi (h) = 1 + 2ai
hv + bi
h2v2 + (2.11) Moreover, each effective operator is multiplied with a real coefcient, indicated with a lowercase letter (c, c, n, r)
associated to each class. The following table denes the notation and summarises the number of independent invariants for each set, in the absence of right-handed neutrinos and after the application of the EOMs.
L Sub-category Notation # operators L C Pbos cj Pj 26 L
with
j = {T, B, W, G, DH, 1 6, 8, 11 14, 17, 18, 20 24,
26, W W W, GGG} (2.13) where all the operators contain four derivatives, with the exception of
PT (h) =
v2
4 Tr(TV)Tr(TV)FT (2.14) and
PW W W (h) =
4abc
2 WaWbWcFW W W ,
PGGG(h) =
4 f
C P bos
cj Sj 16
L2F Quark current nQj N Qj 36
Lepton current n j N j 14 L4F Four quarks rQj RQj 26
Four leptons r j R j 7
Two quarks and two leptons rQ j RQ j 23 Tot 148
Forty additional operators should be considered if right-handed neutrinos are added to the spectrum: 17 in L2F, eight four-lepton interactions and 15 mixed two-quarktwo-lepton terms.
The complete list of NLO operators is provided in the following: Sects. 2.2 and 2.3 are, respectively, dedicated to the bosonic and fermionic sectors. Further details of the construction of the invariants and of how the EOMs have been employed to remove redundant terms can be found in Appendices C and D. The Feynman rules of the complete CP conserving basis are reported in Appendix E, in unitary gauge and for vertices with up to four legs.
2.2 NLO basis: bosonic sector Lbos
At NLO in the chiral expansion, the Lagrangian Lbos contains purely bosonic operators. Complete bases for the CP even and CP odd sectors have been already constructed in Refs. [22,24,25] respectively. In this work only a subset of those ensembles are retained as, once the fermionic sector is introduced, some of the terms become redundant and can be removed using the EOMs (see Appendix D). Nonetheless, the original numeration of the operators has been kept, in order to simplify the comparison with the literature. Finally, the explicit formal dependence on h in the generic functions
Fi (h) is dropped in the following for brevity.
2.2.1 CP even bosonic basis L C Pbos
The CP even NLO Lagrangian reads
L C Pbos =
j
2 GGG FGGG, (2.15)
where f denotes the structure constants of SU(3).
The two-derivative operator PT (h) is very similar to
v2Tr(VV)FC and, therefore, it could have been included
in L0 a priori. However, it is customary to move it to L because the bounds existing on its coefcient are quite strong: cT [lessorsimilar] 102. In fact, this operator violates the custodial symmetry and contributes to the T parameter, which is constrained to a high accuracy by electroweak precision data (EWPD). In order to avoid irrelevant contributions to the EOMs, this operator has been moved to the NLO, which is justiable assuming an approximately preserved custodial symmetry.2 The two operators PW W W (h) and PGGG(h) are
not required to absorb divergences due to the 1-loop renormalisation. However, they can be listed among the NLO operators: containing only the transverse components of the gauge bosons, they follow the linear description; then they come suppressed by 2, on the same foot as the four-fermion operators. It will be shown in the following that they have a non-trivial impact at the phenomenological level.
The remaining 23 operators in L C Pbos, in the numeration of Ref. [24], are the following:
PB(h) =
14 B BFB,
PW (h) =
14 WaWaFW ,
PG(h) =
14 GaGaFG,
PDH (h) =
FDH (h)F DH(h) 2,
P1(h) = BTr(TW)F1,
P2(h) =
i4 BTr(T[V, V])F2,
2 Although the T parameter only constrains the h-independent coupling of PT (h), the whole operator has been moved to the NLO Lagrangian.
This follows the basic assumption that for a given operator the hn>0
coefcients are of the same order as the h0 coefcient. Indeed, if an operator is suppressed due to a symmetry principle, this applies to any of the hn0 couplings.
cj Pj (h), (2.12)
123
Eur. Phys. J. C (2016) 76 :416 Page 7 of 45 416
P3(h) =
i4 Tr(W[V, V])F3,
P4(h) =
i4 BTr(TV)F4,
P5(h) =
i4 Tr(WV)F5,
P6(h) =
1(4)2 (Tr(VV))2F6,
P8(h) =
1(4)2 Tr(VV)F8F 8,
P11(h) =
1(4)2 (Tr(VV))2F11,
P12(h) = (Tr(TW))2F12,
P13(h) =
i4 Tr(TW)Tr(T[V, V])F13,
P14(h) =
4 Tr(TV)Tr(VW)F14,
P17(h) =
i4 Tr(TW)Tr(TV)F17,
P18(h) =
1(4)2 Tr(T[V, V])Tr(TV)F18,
P20(h) =
1(4)2 Tr(VV)F20F 20,
P21(h) =
In the CP odd sector the bosonic Lagrangian contains 16 operators: according to Ref. [25],
L
C P bos
1(4)2 (Tr(TV))2F21F 21
P22(h) =
cj Sj ,
j = {2D, B, W, G, 1 9, 15,
=
j
W W W, GGG}, (2.16) where, as for L C Pbos, all the operators have four derivatives, with the exception of
S2D(h) i
v2
1(4)2 Tr(TV)Tr(TV)F22F 22,
P23(h) =
1(4)2 Tr(VV)(Tr(TV))2F23,
P24(h) =
1(4)2 Tr(VV)Tr(TV)Tr(TV)F24,
P26(h) =
4 Tr T V
F2D (2.17)
and
1(4)2 (Tr(TV)Tr(TV))2F26.
As anticipated in the previous section, while the kinetic terms for the gauge bosons are listed at the LO, the interactions obtained after introducing the dependence on h are reported in the list of NLO operators, under the assumption that the coupling of the transverse components of the gauge elds with the Higgs sector is a subleading effect.
It is also worth commenting on the operators P1(h) and
P12(h): these two structures, including the terms without h insertions, are customarily listed among the NLO terms despite their similarity with the gauge-boson kinetic terms. This is justied, a posteriori, by the fact that they contribute to the S and U parameters, respectively (see Sect. 3.2), which are strongly constrained. In this sense, their treatment is analogous to that of PT (h).
The operators PC(h) and PH(h) of Ref. [24] have not
been included in this list, as their effects can be reabsorbed in redenitions of the arbitrary functions FC(h) and YQ,L(h)
appearing in L0 in Eq. (2.3) (see Appendix B). Moreover,
compared to Ref. [24], a different normalisation for the operators has been chosen: the 4 suppression factors determined by the NDA master formula in Eq. (2.7) have been made explicit (see Ref. [38] for details of the advantages of the NDA normalisation), while the dependence on the coupling constants has been removed, in order to emphasise the generality of the EFT approach. It is customary, indeed, to include in the denition of the HEFT operators the numerical factors arising from the 1-loop renormalisation procedure (see Refs. [41,42] for a general discussion in the SMEFT case): for instance, the operator P1(h) is often dened proportion
ally to gg /(4)2 [17,18,22,24]. However, in principle the coefcients ci account not only for renormalisation effects, but also for possible external contributions, originating by sources that do not need to share the same dependence on the gauge couplings. This normalisation choice is common in many EFTs, such as Fermis theory, the EFT for mesons processes and the SMEFT.
2.2.2 CP odd bosonic basis L
C P bos
S
W W W (h) =
4abc
2
WaWbWcF
W W W ,
S GGG
GGG F GGG. (2.18)
The rest of operators entering L
C Pbos are
(h) =
4 f
2
S B
(h) B
S
W (h) Tr W
B F B ,
W
F
W ,
S G
(h) Ga
Ga F G ,
S1(h)
BTr
TW
F1
,
S2(h)
i 4
B Tr
T V
F2,
S3(h)
i4 Tr
W V
F3,
S4(h)
14 Tr W
V
Tr (T V) F4,
S5(h)
i(4)2 Tr V
V
Tr
T V F5,
123
416 Page 8 of 45 Eur. Phys. J. C (2016) 76 :416
S6(h)
i(4)2 Tr V
V
Tr
T V F6,
S7(h)
14 Tr T W
, V
F7,
S8(h) Tr T
W
Tr
T W F8 ,
S9(h)
i4 Tr T
W
Tr
T V F9,
F15 .
As for the CP even part of the bosonic basis, the explicit dependence on the gauge couplings is not part of the definition of the operators, while the 4 factors are reported according to Eq. (2.7).
The operator S2D(h) deserves a special remark. Being a
two-derivative operator, it would be naturally listed at the LO. However, restricting for simplicity the discussion to the unitary gauge, S2D(h) introduces a mixing between the gauge
boson Z and the physical h, that can be rotated away via a proper redenition of the Goldstone bosons matrix, as detailed in Refs. [25,43]:
U exp ia2D c2D
hv 3 . (2.19)
At leading order in the effective coefcients, the effects of this operator are eventually recast into CP odd contributions to the Yukawa couplings with arbitrary number of h legs and to the vertices Zhn, n 2. Furthermore, S2D(h) induces, at
1-loop, corrections to the Higgs gauge-boson couplings that are bounded by the strong experimental limits on fermionic EDMs, as discussed in Ref. [25]. For this reason, it is considered as a NLO operator, similarly to PT (h).
Finally, the two operators P W W W (h) and P GGG
[ , ]. Finally, the mark
CP on the
left of an operator indicates that it is intrinsically CP odd.
2.3.1 Single fermionic current L2F
The operators with a single fermionic current and up to two derivatives (including those in V) are contained in the
Lagrangian
L2F =
8
S15(h)
i(4)2 Tr T V Tr T V 2
j=1nQjN Qj +
28
j=91 (nQj + i nQj)N Qj
36
+
j=294 (nQj + i nQj)N Qj
2
+
j=1n jN j +
11
j=31 (n j + i n j)N j
j=124 (n j + i n j)N j + h.c., (2.20)
where we recall that the coefcients nQj, n j, nQj, n j are real
and smaller than unity.
The terms with two derivatives have overall canonical mass dimension 5 and are therefore suppressed by 1.
Moreover, they necessarily require chirality-ipping (scalar or tensor) Lorentz structures. These structures do not have denite CP character, as the scalar (
) and pseudo-scalar
14
+
(h) are the
CP odd counterparts of PW W W (h) and PGGG(h); comments
similar to those given for the latter apply here too.
2.3 NLO basis: fermionic sector Lfer
The fermionic Lagrangian at NLO is constituted by single-current operators with up to two derivatives and by four-fermion operators. Flavour indices are left implicit, unless necessary for the discussion. This section presents a set of independent terms that completes the NLO basis in the bosonic sector Lbos: some redundant structures have been removed using the EOMs, as detailed in Appendix D. Only baryon and lepton number conserving operators are considered (see Ref. [44] for the baryon and lepton number violating basis). Moreover, as already stated in the previous sections, right-handed neutrinos are not considered in the present description. Their inclusion in the spectrum would require an extension of the basis presented in this section, with the addition of the operators in Appendix A.
The numbering of the functions Fi(h) is dropped in the
following for brevity. The Pauli matrices that act on the SU(2)L components are denoted by i, while the Gell-Mann matrices that contract colour indices are indicated by A.
Whenever they are not specied, the colour (uppercase) and isospin (lowercase) contractions are understood to be diagonal. Flavour contractions are also assumed to be diagonal. The tensor structure entering the dipole operators is dened as =
i 2
i5) contractions have opposite parity. As a consequence, each SU(2) structure yields two contributions with opposite CP properties, which have been parameterised by two independent real coefcients: for the quark bilinears, the terms nQj(N Qj + h.c.) with the N Qjs dened below are CP
even, while the combinations nQj(iN Qj + h.c.) are CP odd.
A similar notation has been adopted for the lepton bilinears.
Quark current operators
All the non-redundant terms that can be constructed coupling one derivative or one chiral vector eld V to a fermionic bilinear necessarily have a vector-axial Lorentz structure, that preserves chirality. For the quarks case, they are:
N Q1(h) i QL V QL F,
(
123
Eur. Phys. J. C (2016) 76 :416 Page 9 of 45 416
N Q2(h) i QR UVU QR F,
CP N Q3(h) QL [V, T] QL F,
CP N Q4(h) QR U[V, T]U QR F, N Q5(h) i QL {V, T} QL F,N Q6(h) i QR U{V, T}U QR F, N Q7(h) i QL TVT QL F,N Q8(h) i QR UTVTU QR F.
Invariants with a derivative acting on a fermion eld or on a F(h) function are redundant upon application of the
EOMs and integration by parts, and they have therefore been removed from the nal basis.
Operators with two derivatives require a fermionic current with an even number (zero or two) of gamma matrices: therefore only chirality-ipping Lorentz structures are allowed. All the operators with a scalar structure are required as counter-terms in the 1-loop renormalisation of L0:
N Q9(h) QL U QR FF , N Q10(h) QL TU QR FF ,
N Q11(h) QL VU QR F,
N Q12(h) QL {V, T}U QR F,
N Q13(h) QL [V, T]U QR F,
N Q14(h) QL TVTU QR F,
N Q15(h) QL VVU QR F,
N Q16(h) QL VVTU QR F,
N Q17(h) QL TVTVU QR F,
N Q18(h) QL TVTVTU QR F,
N Q19(h) QL VTVU QR F,
N Q20(h) QL VTVTU QR F.
Operators with tensor structure are also included in the NLO basis, although they are not needed to reabsorb the 1-loop divergences of L0, as the loop diagrams that generate them in the EFT are nite. Nonetheless, these interactions may result from the (tree-level) exchange of a heavy BSM resonance and therefore they may be as relevant as those in the previous lists:
N Q21(h) QL VU QR F,
N Q22(h) QL [V, T]U QR F,
N Q23(h) QL {V, T}U QR F,
N Q24(h) QL TVTU QR F,
N Q25(h) QL VTVU QR F,
N Q26(h) QL VTVTU QR F,
N Q27(h) QL [V, V]U QR F,
N Q28(h) QL [V, V]TU QR F,
N Q29(h) ig QL U QR BF,
N Q30(h) ig QL TU QR BF,
N Q31(h) igs QL GU QR F,
N Q32(h) igs QL GTU QR F,
N Q33(h) ig QL WU QR F,
N Q34(h) ig QL {W, T}U QR F,
N Q35(h) ig QL [W, T]U QR F,
N Q36(h) ig QL TWTU QR F.
Leptonic current operators
Leptonic bilinears can be constructed along the same lines as the quark ones. The absence of right-handed neutrinos, however, reduces notably the number of independent invariants. Making use of Eq. (D.14), only two independent operators can be constructed with the insertion of a single derivative or V:
CP N 1(h) LL [V, T] LL F,
N 2(h) i L R U{V, T}U L R F.
Notice that, if avour effects are also taken into consideration, two other structures should be considered:
i LLi VLL j F, i LLi {T, V}LL j F. (2.21) only for the case with i = j. Indeed, as shown in Eq. (D.14),
the avour-diagonal contractions do not represent independent terms as they are related via EOMs to bosonic operators that have been retained in the basis.
With two derivatives, two V or a combination of them, the following structures can be constructed:
N 3(h) LL U L R FF ,
N 4(h) LL {V, T}U L R F,
N 5(h) LL [V, T]U L R F,
N 6(h) LL VVU L R F,
N 7(h) LL TVTVU L R F,
N 8(h) LL [V, T]U L R F,
N 9(h) LL {V, T}U L R F,
N 10(h) LL VTVU L R F,
N 11(h) LL [V, V]U L R F,
N 12(h) ig LL U L R BF,N 13(h) ig LL WU L R F,N 14(h) ig LL [W, T]U L R F.
where, as explained above, all these operators are required as counter-terms in the 1-loop renormalisation of L0 with the exception of those with tensor structure, that correspond
123
416 Page 10 of 45 Eur. Phys. J. C (2016) 76 :416
to nite contributions. It is also worth recalling that all the chirality-ipping structures listed here are CP even in the combination (N j +h.c.) but independent CP violating terms
of the form (iN j + h.c.) should also be considered.
2.3.2 Four-fermion operators L4F
Four-fermion operators can be classied into four-quark, four-lepton and two-quarktwo-lepton sets. The overall Lagrangian reads
L4F =
(4)2 2
8 rQj + i rQj
RQj +26
j=9 rQjRQj
7
+
r 1 + i r 1
R 1 +
j=2 r jR j
+ 6 rQ j + i rQ j
RQ j+23
j=7rQ jRQ j + h.c.
.
(2.22)
Details of the construction and reduction of this subset of operators can be found in Appendix C.3. As for the bilinears case, the chirality-ipping contractions (
LR)(
LR)
listed here are CP even in the combination (R fj + h.c.) but
independent CP violating terms of the form (i R fj + h.c.)
should also be considered.
Pure quark operators
The only four-quark operators required to remove divergences originating at one loop are the following:
RQ1(h) ( QL U QR )( QL U QR )F, RQ2(h) ( QL iU QR )( QL iU QR )F,
RQ3(h) ( QL U QR )( QL TU QR )F, RQ4(h) ( QL TU QR )( QL TU QR )F,
RQ5(h) ( QL AU QR )( QL AU QR )F, RQ6(h) ( QL AiU QR )( QL AiU QR )F
RQ7(h) ( QL AU QR )( QL ATU QR )F, RQ8(h) ( QL ATU QR )( QL ATU QR )F.
A large number of additional structures can be constructed, that are listed below and included in the basis. Although they do not correspond to counter-terms in the renormalisation of L0, they are potentially generated by the exchange of BSM resonances:
RQ9(h) ( QL QL )( QL QL )F, RQ10(h) ( QL QL )( QL T QL )F,
RQ11(h) ( QL T QL )( QL T QL )F, , RQ12(h) ( QL j QL )( QL j QL )F,
RQ13(h) ( QR QR )( QR QR )F, RQ14(h) ( QR QR )( QR UTU QR )F,
RQ15(h) ( QR UTU QR )( QR UTU QR )F, RQ16(h) ( QR j QR )( QR U jU QR )F,
RQ17(h) ( QL QL )( QR QR )F, RQ18(h) ( QL QL )( QR UTU QR )F,
RQ19(h) ( QL T QL )( QR QR )F, RQ20(h) ( QL T QL )( QR UTU QR )F,
RQ21(h) ( QL i QL )( QR UiU QR )F, RQ22(h) ( QL A QL )( QR A QR )F,
RQ23(h) ( QL A QL )( QR AUTU QR )F, RQ24(h) ( QL AT QL )( QR A QR )F,
RQ25(h) ( QL AT QL )( QR AUTU QR )F, RQ26(h) ( QL Ai QL )( QR AUiU QR )F.
Pure leptonic operators
The set of independent four-lepton operators is considerably smaller than that with four quarks, due to the absence of right-handed neutrinos and of colour charges. Only one operator is required as a 1-loop counter-term:
R 1(h) ( LL U L R )( LL U L R )F.
Six additional structures, that are not required as counter-terms, complete the list of possible invariants:
R 2(h) ( LL LL )( LL LL )F, R 3(h) ( L R L R )( L R L R )F,
R 4(h) ( LL LL )( LL T LL )F, R 5(h) ( LL T LL )( LL T LL )F,
R 6(h) ( LL LL )( L R L R )F, R 7(h) ( LL T LL )( L R L R )F.
Mixed quarklepton operators
Finally, barring any B or L violation effects, mixed four-fermion operators can only contain two quarks and two lep-tons in either of the current structures LL QQ and L Q QL.
Among the constructed invariants, the following are required to reabsorb 1-loop divergences:
123
Eur. Phys. J. C (2016) 76 :416 Page 11 of 45 416
RQ 1(h) ( LL U L R )( QL U QR )F, RQ 2(h) ( LL U QR )( QL U L R )F,
RQ 3(h) ( LL U L R )( QL TU QR )F, RQ 4(h) ( LL TU QR )( QL U L R )F,
RQ 5(h) ( LL iU L R )( QL iU QR )F,RQ 6(h) ( LL iU QR )( QL iU L R )F,while the remaining correspond to nite diagrams and are included for completeness:
RQ 7(h) ( LL LL )( QL QL )F, RQ 8(h) ( L R L R )( QR QR )F,
RQ 9(h) ( LL LL )( QL T QL )F, RQ 10(h) ( L R L R )( QR UTU QR )F,
RQ 11(h) ( LL T LL )( QL QL )F, RQ 12(h) ( LL T LL )( QL T QL )F,
RQ 13(h) ( LL i LL )( QL i QL )F, RQ 14(h) ( LL LL )( QR QR )F,
RQ 15(h) ( QL QL )( L R L R )F, RQ 16(h) ( LL T LL )( QR QR )F,
RQ 17(h) ( QL T QL )( L R L R )F, RQ 18(h) ( LL LL )( QR UTU QR )F,
RQ 19(h) ( LL T LL )( QR UTU QR )F, RQ 20(h) ( LL j LL )( QR U jU QR )F,
RQ 21(h) ( QL LL )( L R QR )F, RQ 22(h) ( QL T LL )( L R QR )F,
RQ 23(h) ( QL j LL )( L R U jU QR )F.
2.4 Comparison with the SMEFT basis
The comparison with the SMEFT is crucial for the identication of signals able to shed some light on the Higgs nature.
For the bosonic sector, the relation between the HEFT and its linear counterpart has already been identied in Ref. [24], adopting the so-called HISZ basis [45,46], which is also used in Refs. [4749]. Those results still hold here, up to the fact that some operators have been traded for fermionic ones: the correspondence is summarised in Table 1, where the relation to the basis of Ref. [7] is also reported. The fermionic sector of the HEFT has also been matched with the linear bases of Refs. [7,4749], as indicated in Table 2.
It is worth pointing out a few points that should be kept into account when performing this comparison:
In the HEFT, right-handed fermions are grouped in the SU(2)R doublets, L R and QR, and the different com-
ponents of each bilinear fermionic structure are disentangled inserting UTU = 3 or U jU. Each lin
ear operator, written in the traditional notation, is then easily matched with a linear combination of HEFT invariants. The adimensional scalar eld T corresponds, in the linear context, to a quadratic combination of Higgs doublets. As a consequence, the counterparts of fermionic invariants containing T are mostly linear operators of dimension d > 6, which are therefore not present in the list of Refs. [7,48,49].
The insertions of T into right-handed currents, mentioned in the previous point, represent an exception. In fact, in these cases T appears in the combination UTU = 3, that does not contain any eld and in fact
is not associated to dimensional objects in the linear language. The adimensionality of T also leads to the presence of CP-odd operators in L , whose corresponding structures in the SMEFT would appear only at d > 6. An example is the operator N Q3(h) that has been already
studied in Ref. [23,37] for its impact on avour physics. The two-derivative object VV is typically described, in the SMEFT, by a quantity proportional to D D , which has canonical dimension 4. Thus, fermionic bilinears containing this structure correspond to SMEFT operators with d 7.
Tables 1 and 2 summarise the relations between operators of the HEFT, dened in the previous section, and those of the SMEFT from Refs. [7,48,49]. The only differ-
Table 1 Correspondence between the SMEFT operators from Refs. [7,48,49], and the HEFT terms presented here for the bosonic sector. The refers to the absence of an equivalent operator. The use of the Qi notation for the second column means that a particular operator
does not explicitly appear in Refs. [48,49], but it anyway enters the SMEFT basis and is dened as in Ref. [7]. Numerical coefcients and signs in the combinations of the HEFT operators are not indicated
Ref. [7] Refs. [48,49]
HEFT Ref. [7] Refs.
[48,49]
HEFT
Q O ,3 scalar pot. Q O 2 FC + FY (PH ) QD O ,1 PT QG OGG PG
QW OW W PW QB OB B PB
QW B OBW P1 OB P2 + P4 OW P3 + P5
QG QG PGGG QW OW W W PW W W
Q
G Q G
S G Q B
Q B
S B
Q
W Q
W S W Q
W B Q
W B S1
Q G
Q G
P GGG
Q
W Q
W P
W W W
123
416 Page 12 of 45 Eur. Phys. J. C (2016) 76 :416
Table 2 Correspondence between the SMEFT operators from Refs. [7,48,49], and the HEFT terms presented here for the fermionic sector.The refers to the absence of an equivalent operator. The use of the Qi notation for the second column means that a particular operator
does not explicitly appear in Refs. [48,49], but it anyway enters the SMEFT basis and is dened as in Ref. [7]. Flavour indices are omitted, unless explicitly indicated. Numerical coefcients and signs in the combinations of the HEFT operators are not indicated
Ref. [7] Refs. [48,49] HEFT Ref. [7] Refs. [48,49] HEFT
Qu Ou YU (h) Qe Oe YE (h) Qd Od YD(h) Q(1)l,ii
Q(1)q O(1) Q N Q5 Q(1)l,i j O(1) L,i j i LLi {T, V}LL
j
F
Q(3)q O(3) Q N Q1 Q(3)l,ii
Qu O(1) u N Q2 + N Q6 + N Q8 Q(3)l,i j O(3) L,i j i LLi VLLjF
Qd O(1) d N Q2 + N Q6 + N Q8 Qe O(1) e N 2 Qud O(1) ud N Q2 + N Q8
QuG QuG N Q31 + N Q32
QdG QdG N Q31 + N Q32
QuW QuW N Q33 + N Q34 + N Q35
QdW QdW N Q33 + N Q34 + N Q35 QeW QeW N 13
QuB QuB N Q29 + N Q30
Qd B Qd B N Q29 + N Q30 QeB QeB N 12
Q(1)qq Q(1)qq RQ9 Qll Qll R 2 Q(3)qq Q(3)qq RQ12 Q(1)lq Q(1)lq RQ 7
Quu Quu RQ13 + RQ14 + RQ15 Q(3)lq Q(3)lq RQ 13
Qdd Qdd RQ13 + RQ14 + RQ15 Qee Qee R 3
Q(1)ud Q(1)ud RQ13 + RQ15 Qeu Qeu RQ 8 + RQ 10 Q(8)ud Q(8)ud RQ13 + RQ16 + RQ15 Qed Qed RQ 8 + RQ 10 Q(1)qu Q(1)qu RQ17 + RQ18 Qle Qle R 6Q(8)qu Q(8)qu RQ22 + RQ23 Qlu Qlu RQ 14 + RQ 18 Q(1)qd Q(1)qd RQ17 + RQ18 Qld Qld RQ 14 + RQ 18
Q(8)qd Q(8)qd RQ22 + RQ23 Qqe Qqe RQ 15
Q(1)quqd Q(1)quqd RQ1 + RQ2 Qledq Qlelq RQ 21 + RQ 22
Q(8)quqd Q(8)quqd RQ5 + RQ6 Q(1)lequ Q(1)lequ RQ 2 + RQ 6
Q(3)lequ Q(3)lequ RQ 1 + RQ 2 + RQ 3 + RQ 5 + RQ 6
ence between these two linear bases (the rst two columns in both tables) lies in the choice of two invariants: in Refs. [48,49] the EOMs have been used for removing the fermionic terms corresponding to Q(1)l,ii and Q(3)l,ii
in Ref. [7], replacing them with the bosonic operators
OB and OW . In the HEFT construction, the EOMs have
been applied analogously to Refs. [48,49], namely retaining PB and PW , rather than two leptonic invariants (see
Eq. (D.14)).
All the HEFT operators that do not appear in this list have SMEFT counterparts (dubbed also linear siblings) of dimension larger than six and therefore are not contained in the bases of Refs. [7,48,49].
3 Phenomenology
3.1 Physical parameters denitions
The phenomenological analysis is carried out in the Z-scheme, dened by the following set of observables, that are taken as input parameters:
s world average [50],GF extracted from the muon decay rate [50], em extracted from Thomson scattering [50],
MZ extracted from the Z lineshape at LEP I [50], Mh measured at LHC [51].
(3.1)
123
Eur. Phys. J. C (2016) 76 :416 Page 13 of 45 416
All the other quantities appearing in the Lagrangian will be implicitly interpreted as corresponding to the combinations of experimental inputs as follows:
e2 = 4em, sin2 W =
1
2
Fermionic couplings:
It is convenient to adopt the following compact notation:
g1 = cT + 3222
v2 2 (r 2 r 5),
1 1 4em
2GF M2Z
c2c2 cT + t2c1 2c12 +
3222c2
c2
,
gW =
v2 2 (r 2 r 5),
v2 =
1 2GF
, g =esin W , g =e cos W
g2 = s2
W , e as above
.
s2
s2
t
(3.2)
The trigonometric functions sin W , cos W will be conveniently shortened to s, c.
The kinetic terms are made canonical and diagonal with the following eld redenitions:
A A 1 + s2c1 + 2s2c12
1
2(c2cB + s2cW )
s22 2c2
cT +2c1s2 + 3222 v2 2 (r 2 r 5) , (3.6)
where g1 accounts for the renormalisation of Z, g and c in the combination gZ/c; gW for the renormalisation of W and g in the combination gW; g2 for the renormalisation of s2 and for the contribution to the Z couplings that comes from the redenition of the photon eld:
A A + Z (see Eq. (3.3)). With this notation, the renor
malisation of Z couplings to left-handed and right-handed fermions, g fL = (T f3 s2Q f ) and g fR = s2Q f , and of the
W to left-handed fermions can be written as
g fL,R = g fL,R g1 + Q f g2 g f f
=
+ Z 2 c2c1 + s2 c12 +
cB cW 4
+ O(c2i)
Z Z 1 s2c1 + 2c2c12
1
2
c2cW + s2cB
= gW , (3.7)
where Q f and T3 f are, respectively, the electric and isospin charges of the fermion f , and where the W couplings to left-handed fermions is normalised to 1 in the SM.
The next sections are dedicated to the discussion of the constraints imposed on the operator coefcients considering respectively electroweak precision data, Higgs results from the LHC and the Tevatron, and measurements of the triple gauge-bosons couplings. For the sake of simplicity we will assume fermion universality as well as the absence of new sources of avour violation.
3.2 Constraints from EWPD
After accounting for nite renormalisation effects in the gauge bosons wavefunctions and couplings as well as for direct contributions to the vertices, 12 operators modify the Z and W gauge-boson couplings to fermions with the same Lorentz structure as the SM and the W mass, which correspondingly lead to linear modications of the EWPD.
Five operators, PT (h), P1(h), P12(h), R 2(h), R 5(h) give
tree level contributions to universal modications of the couplings and of the W mass, which can be recast in terms of the oblique S, T, U parameters [52,53] and of the shift in the Fermi constant GF . In particular
S = 8scc1, T = 2cT , U = 16s2c12,
GF
GF = 642
2 v2
2 (r 2 r 5), (3.8)
W
+ O(c2i)
W+ W+ 1
1
2cW + O(c2i). (3.3)
The contributions to the input parameters at rst order in the effective coefcients read
em
em 2s2c1 + 4s2c12 c2cB s2cW ,
GF
GF 64
22 v2
2 (r 2 r 5),
MZ
MZ cT s2c1 + 2c2c12
1
2(c2cW + s2cB),
MhMh 0. (3.4)
The resulting shifts for the W mass and fermion couplings to gauge bosons with respect to their corresponding SM expectations due to these nite renormalisation effects are summarised below:W mass:
MW
MW =
c2c2 cT +
s2
c2 c1 2c12
+
3222s2 c2
v2 2 (r 2 r 5). (3.5)
123
416 Page 14 of 45 Eur. Phys. J. C (2016) 76 :416
so, for example, the correction to the W mass in Eq. (3.5) reads
MW
MW =
c22c2 T
14c2 S +
1 8s2
U
s2 2c2
GF
GF .
(3.9)
The other seven operators, N Q1(h), N Q2(h), N Q5(h), N Q6(h), N Q7(h), N Q8(h), N 2(h), give fermion dependent contribu
tions to the W and Z couplings. Altogether the shifts to the SM Z couplings can be written as
g fL,R = g fL,R g1 + Q f g2 + g fL,R, (3.10)
where the nite renormalisation shifts of the fermion couplings in Eq. (3.6) can be rewritten as:
g1 =
1
2
T GF GF
,
g2 =
s2 c2
c2
T GFGF 1 4s2
, (3.11)
while the fermion dependent modication of the couplings read3
guL =nQ1 + 2nQ5+nQ7, guR =nQ2 + 2nQ6 + nQ8,
gdL =nQ1 + 2nQ5 nQ7, gdR =nQ2 + 2nQ6 nQ8,
gL =0, gR =0,
geL =0, geR =2n 2. (3.12)
The corresponding shifts to the W couplings to left-handed fermions (normalised to 1 in the SM) are
g f f
W
= gW + g f f
S
W , (3.13)
with the universal shift due to the nite renormalisation dened in Eq. (3.6) given by
gW =
MW
MW
1
2
GF
GF , (3.14)
and the fermion dependent shifts induced by the fermionic operators by
gudW = 2nQ1 2nQ7, geW = 0. (3.15)
There are two main differences with respect to the corresponding contributions to EWPD obtained assuming a linear realisation of the SU(2)L U(1)Y gauge symmetry break
ing with operators up to dimension six (see for example Refs. [54,55]).
3 One could expect g,eL to have a similar contributions as gu,dL.
This is not the case as the corresponding leptonic operators have been removed from the basis by using the EOMs, as discussed in Eq. (D.14). This choice simplies the renormalisation procedure as g,eL are van
ishing.
First, in the SMEFT no contribution to the U parameter is generated at dimension six, while a contribution is generated in the HEFT at NLO, O(p4).
Second, in the linear description and assuming universality, the fermion dependent shifts of the W couplings to fermions are directly determined by those of the Z as there are only ve independent dimension-6 operators entering those vertices with SM Lorentz structure (which can be chosen for example to be O(3)q, O(1)q, Ou, Od,
Oe in the notation of Ref. [7]). In the chiral description at order p4 the fermion dependent contributions come in contrast from the seven operators given above, of which six combinations contribute independently to EWPD.
So altogether 10 combinations of the 12 operator coefcients can be determined by the analysis of EWPD which have been chosen here to be cT , c1, c12, (r 2 r 5), nQ1, (nQ2 + nQ8), nQ5, nQ6, nQ7 and n 2. In order to obtain the corre
sponding constraints on these 10 parameters a t including 16 experimental data points is performed. These are 13 Z observables: Z, 0h, Ppol, sin2 eff, R0l, Al(SLD), A0,lFB,
R0c, R0b, Ac, Ab, A0,cFB, and A0,bFB from SLD/LEP-I [56], plus
three W observables: the average of the W-boson mass, from [57], the W width, W , from LEP-II/Tevatron [58], and the leptonic W branching ratio, BreW, for which the average in
Ref. [50] is taken. The correlations among the inputs can be found in Ref. [56] and have been taken into consideration in the analysis. As mentioned above, unlike in the ts to dimension-6 SMEFT operators, the independent experimental information on the W couplings to fermions have been included in the present study: this is done by considering in the t the leptonic W branching ratio, as it is measured independently of the total W width, which is determined from kinematic distributions. The corresponding predictions for the observables in the analysis in terms of the shifts of the SM couplings dened above are given by
Z = 2 Z,SM
f (g fL g fL + g fR g fR)N fC
f (|g fL |2 + |g fR|2)N fC
, (3.16)
0h = 20h,SM
(geL geL + geR geR)
|geL|2 + |geR|2
+
q (gqL gqL + gqR gqR)
q (|gqL|2 + |gqR|2)
Z Z,SM
, (3.17)
R0l
hadZ
lZ
= 2R0l,SM
q (gqL gqL + gqR gqR)
q (|gqL|2 + |gqR|2)
(glL glL + glR glR)
|glL|2 + |glR|2
, (3.18)
123
Eur. Phys. J. C (2016) 76 :416 Page 15 of 45 416
R0q
qZ
hadZ
= 2R0q,SM
(gqL gqL + gqR gqR)
|gqL|2 + |gqR|2
q (gq
L gq
+ gq
R gq
R )
L
1 4s2
U (3.28)
the contributions from T , U, and GF cancel both in the Z observables and in MW .Therefore, along this direction in the parameter space, the bounds on these three quantities come from the contribution of GF to W and BreW in Eq. (3.15), but these observables are less precisely determined.
It is important to notice that this weakening arises even if the n fi coefcients, that is all the fermion dependent contributions, but the four-fermionic ones, are set to zero and only the four contributions c1, cT , c12 and r 2 r 5 are retained. In
this particular case, the result of the t is
S = 0.1 0.1, T = 0.43 2.86, U = 0.3 2.4,
GF
GF = (0.26 2.0) 102, (3.29)
to be compared with Eq. (3.26). On the contrary, in the framework of linear dimension-6 operators, the condition U = 0
makes this cancellation not possible, so bounds on the corresponding operator coefcients are generically stronger. In other words, when making the EWPD analysis in the context of HEFT at O(p4) the bounds on the operators contributing
to T and U are generically weaker by more than one order of magnitude.
The fermionic operators can also lead to modications of the semileptonic decay amplitudes used to determine the elements of the CKM matrix and to test its unitarity. In particular, N Q1(h), N Q7(h), R 2(h), R 5(h), RQ 13(h) induce linear
shifts to the corresponding amplitudes (normalised to GF as determined from decay) which can be parameterised as a shift in the effective CKM matrix,
VCKMi j
= VCKM,SMi j 6422
v2 2 rQ 13 + gudW
, (3.19)
GF =
q (|gq
L|2 + |gq
R |2)
g fR
sin2 leff = sin2 leff,SM
glL
glL glR
g fR
g fL
g fL
,
(3.20)
A f = 4A f,SM
g fL g fR
|g fL |4 |g fR|4
g fR g fL g fL g fR
,
(3.21)
Ppol = Al, (3.22)
A0, fFB = A0, fFB,SM
Al Al
+
A f Af
, (3.23)
W = W,SM
43 gudW +
2
3 geW + MW , (3.24)
43 geW . (3.25)
When performing the t within the context of the SM the result is 2EWPD,SM = 18.3, while when including the
10 new parameters it gets reduced to 2EWPD,min = 6.
The results of the analysis are shown in Fig. 1 which displays the 2EWPD dependence of the 10 independent operator coefcients. In each panel 2EWPD is shown after marginalising over the other nine coefcients. The gure shows the corresponding 95 % allowed ranges given in Table 3: the only operator coefcient not compatible with zero at 2 is nQ2 + nQ8, a result driven by the 2.7 dis
crepancy between the observed A0,bFB and the SM expectation.
It is interesting to notice that the resulting constraints on the coefcients contributing to T , U and GF are considerably weaker than what one would obtain in the standard three-parameter ts to S, T , U. Quantitatively, the results of the 10-parameter analysis performed here give the following 1 ranges for S, T, U and GF :
S = 0.45 0.37, T = 0.3 2.8, U = 0.1 2.5,
GF
GF = (0.08 2.2) 102, (3.26) to be compared with the results of the standard three-parameter t for S, T, U [55],
S = 0.08 0.1, T = 0.1 0.12, U = 0.0 0.09.
(3.27)
While the range for S is only about 4 times broader when including the effects of all the additional operators, the bounds on T and U are weakened by more than a factor20. The main reason is that when GF is also included in the
analysis cancellations can occur. In particular as can be seen in Eq. (3.9)(3.11) for
T =
GF
BreW = BreW,SM
43 gudW +
GF GF
,
(3.30)
and which can lead to violations of unitarity of the CKM matrix which are strongly constrained. In the case of SMEFT with operators up to dimension six, three operators enter this observable after equivalent application of the EOMs [54,55] (which can be chosen for example to be O(3)q, Oll, and, O(3)lq
Ref. [7]). From the global analysis in Ref. [50]
i
|Vui|2 1 = 2 6422
v2 2 rQ 13 + gudW
GF GF
= (1 6) 104. (3.31)
123
416 Page 16 of 45 Eur. Phys. J. C (2016) 76 :416
Fig. 1 Dependence of 2EWPD+CKM (= 2EWPD for
all but last panel) on the 11 independent operator coefcients as labelled in the gure. In each panel 2EWPD+CKM is shown after
marginalising over the other undisplayed parameters
In combination with the analysis of the EWPD, this allows for constraining the coefcient of an 11th operator RQ 13(h).
Adding this data point to the 16 of the EWPD allows one to construct 2EWPD+CKM, which is now a function of 11 param
eters (with 2EWPD+CKM,SM = 18.4 and 2EWPD+CKM,min = 6). The marginalised distributions verify 2EWPD+CKM(x)
= 2EWPD(x) for the rst 10 parameters, i.e. the inclusion
of the CKM unitarity constraint has no impact in the previous analysis as long as rQ 13 is allowed to vary free in the t. The new 2EWPD+CKM(rQ 13) is shown in the curve in the last
panel in Fig. 1 and its 95 % CL range is listed in the last row in Table 3.
3.3 Effects in Higgs physics
This section is dedicated to the study of the current bounds stemming from the Higgs searches at the LHC. Restricting the analysis to the subset of C and P even operators,4 the focus is on those terms that contribute to the trilinear Higgs interactions with fermions and gauge bosons (deviations in the Higgs triple vertex will only become observable in the future). The deviations on Higgs quartic vertices (H V f f )
generated by some of the single fermionic current operators
4 The extension of the analysis to C P odd non-linear operators could be performed after the inclusion of C P sensitive observables; see Ref. [25].
123
Eur. Phys. J. C (2016) 76 :416 Page 17 of 45 416
Table 3 95 % allowed ranges for the combinations of operator coefcients entering the EWPD analysis and the CKM unitarity test
Coupling 95 % allowed range
c1 (0.66, 2.7) 103
cT (0.023, 0.021)
c12 (0.011, 0.011)
v2 2 (r 2 r 5) (4.9, 4.7) 105
nQ1 (4.9, 2.0) 103 nQ2 + nQ8 (22, 1.5) 103 nQ5 (1.6, 1.2) 103 nQ6 (0.025, 8.8) 103 nQ7 (4.2, 2.7) 103 n 2 (0.2, 1.1) 103
v2 2 rQ 13 (7.1, 6.6) 105
The 13 parameters in this Lagrangian can be re-written in terms of the following 10 coefcients5:
aC, aB, aG, aW , a4, a5, a17, Y (1)t, Y (1)b, Y (1), (3.33)
and explicitly they read
gHgg =
1
2v aG,
g(1)H Z =
gs 4vc
a5 + 2cs a4 + 2a17 ,
g(2)H Z =
sc
v (aB aW ) ,
g(1)H Z Z =
g 4v
2 sc a4 a5 2a17 ,
g(2)H Z Z =
1
2v (s2aB + c2aW ),
g(3)H Z Z = M2Z(2GF )1/2 (1 + aC) ,
gH =
1
2v (s2aW + c2aB),
have been omitted from this analysis. Those contributions to Higgs physics could also be studied at the LHC [5961] and, if analysed in combination with gauge-fermion data, they would potentially improve the comparison between linear and non-linear scenarios [60,61]. Nevertheless the generalisation of the analysis with the inclusion of these effects is out of the scope of the present study. The list of operators analysed includes then PT (h), PB,G,W (h) and P1,4,5,12,17(h),
in addition to the contributions from Y (1)U, Y (1)D, Y (1) and to the deviations in the GBs kinetic term parameterised by aC. This set can be further reduced considering the strong constraints imposed on PT,1,12(h) by the global analysis
of EWPD at the Z pole: the impact of these operators on Higgs physics can be safely neglected, given the accuracy at which these observables are currently measured. Moreover, the current Higgs searches are only sensitive to H f f vertices with f = t, b, (the addition of to the analysis
will be straightforward once the sensitivity to this coupling increases). Therefore, only a subset of 10 operators is relevant for the analysis of the available Higgs data. Their contributions to the several Higgs trilinear interactions can be illustrated with the usual HVV phenomenological Lagrangian in the unitary gauge:
L = gHgg HGaGa + gH H A A
+g(1)H Z A Z H + g(2)H Z H A Z
+g(1)H Z Z Z Z H + g(2)H Z Z H Z Z
+g(3)H Z Z H ZZ
+g(1)H W W W+W H + h.c.
+g(2)H W W H W+W + g(3)H W W H W+W
+
f =,b,t
g(1)H W W =
g4v a5, g(2)H W W =
1v aW ,
g(3)H W W = 2M2W (2GF )1/2 (1 + aC) ,
g f =
Y (1)f
2 . (3.34)
The anomalous Higgs interactions described by these 10 operators can be studied and constrained in a model independent way by means of a global analysis of all the Higgs experimental measurements that were performed at the LHC during the Run I. This includes not only event rate data in several Higgs production and decay categories, but also some kinematic distributions, that have an interesting phenomenological impact, as shown in the context of SMEFT in Ref. [6267]. Indeed, they are important for allowing one to obtain nite constraints in the large-dimensional parameter space spanned in the global analysis [62]. Moreover, they make it possible to disentangle the non-SM Lorentz structures from the SM-like shifts.
The global analysis of all Run I Higgs, data using the SFitter framework [6872] for the SMEFT [48,49], has been presented in Ref. [62]: in that case, the 13 parameters of the phenomenological Lagrangian in Eq. (3.32) received contributions from nine linear operators. Here, that analysis is extended to account for the 10th coefcient a17. All the details regarding the data set and the kinematic distributions analysed, as well as the statistical treatment performed in this log-likelihood analysis follow exactly the description presented in Ref. [62] and will not be repeated here.
The results of the global analysis on the parameters in Eq. (3.33) using the available Higgs data, including all the
5 Notice the implicit redenitions ai ciai for the bosonic operators.
g f H fL fR + h.c. . (3.32)
123
416 Page 18 of 45 Eur. Phys. J. C (2016) 76 :416
Table 4 Best t and 95 % CL allowed ranges of the coefcients of the operators contributing to Higgs data (aG,
aW , aB, a4, a5, a17, aC, Y (1)t,
Y (1)b and Y (1)) and to TGV analyses (c2, c3 and cW W W ).
Y (1)t, Y (1)b and Y (1) are normalised to the SM expectation
Best t 95% CL region aG
-0.0125 -0.0030
0.00290.0123
(0.018, 0.0080) (0.0054, 0.0058)
(0.0091, 0.017) aW -0.017 (0.11, 0.088)
aB 0.0052 (0.025, 0.041)
a4 0.041 (0.85, 1.1)
a5 0.13 (0.81, 0.60)
aC -0.13 (0.30, 0.23)
a17 0.055 (0.52, 0.65)
Y (1)t/Y (0)t -1.11 (1.7, 0.53)
1.31 (0.56, 1.7)
Y (1)b/Y (0)b -0.70 (1.7, 0.39)
0.66 (0.35, 1, 7)
Y (1)/Y (0) -0.94 (1.37, 0.63)
0.82 (0.66, 1.47) c2 0.041 (0.24, 0.27)
c3 0.15 (0.093, 0.39)
cW W W 0.006 (0.013, 0.018)
kinematic distributions described in Ref. [62], are reported in Table 4. On the right gure we graphically display the corresponding values where error bars refer to the 95 % CL allowed ranges, obtained proling for each coefcient on the other nine parameters that are included in the global analysis. The off-shell m4 distributions, which have been implemented in Ref. [62], are not included here, as their impact in the present analysis is subdominant with respect to the rest of kinematic distributions considered.
The addition of the extra parameter, a17, has enlarged the allowed range for all the rest of coefcients contributing to the bosonic Higgs trilinear interactions (a4, a5, aW , aB and aC) in comparison with the results in Refs. [62,73] (after taking into account the different normalisations used between the two analyses). This was expected given the larger dimensionality of the parameter space analysed here. The new contributions from P17(h) are consequently strongly correlated
to some of the other operators, as illustrated in Fig. 2, where the two-dimensional planes aB vs. a17 and a4 vs. a17 are shown, after proling on the rest of undisplayed coefcients for each of the panels.
In the present analysis the addition of kinematic distributions is crucial both for closing the allowed regions on
all the considered parameters, and for controlling the correlations among the anomalous couplings [62]. To the best of our knowledge, the results derived here present the most complete set of Higgs based constraints on the set of operators of the HEFT Lagrangian. They highlight, in addition, the potential of the EFT expansion to describe and study the Higgs interactions at the LHC.
3.4 Triple gauge-boson couplings and Higgs interplay
The study of triple gauge-boson vertices is complementary to the analysis of Higgs physics, and it is fundamental for obtaining a more complete description of the EWSB sector. Focusing again on the C and P even operators and after including the strong constraints from EWPD, only four operators, P2(h), P3(h), P13(h) and PW W W (h), enter this anal
ysis.6 They can give observable deviations from the SM predictions for the triple gauge-boson vertices W W Z and W W . These anomalous contributions can be parameterised
6 An additional operator, P14(h), generates a C P conserving but C and
P violating coupling, whose effects and numerical analysis have been discussed in Refs. [24,74] and also hold here.
123
Eur. Phys. J. C (2016) 76 :416 Page 19 of 45 416
Fig. 2 Results of the global analysis of LHC Higgs run I data, including kinematic distributions, for {aB, a4, a17}, proling on the undisplayed
parameters. The colours refers to the different CL regions: from the inner to outer, 68, 90, 95, 99 % CL
in terms of the usual phenomenological TGV Lagrangian presented in Ref. [75]:
W+WV W+VW
+ V W+WV +
V 2m2W
LW W V
= igW W V gV1
W+WV
,
(3.35)
with deviations from the SM predictions, gZ1 = Z = =
1, = Z = 1,
gZ1 = gZ1 1
g 4c2
ig
2 (D )W(D ),
OB=
ig
2 (D )B(D ),
OW W W=
ig3
OW
=
c3,
Z = Z 1
g4 (c3 + 2c13 2tc2) , (3.36)
8 Tr W
WW
, (3.37)
where the notation of the original papers has been kept.
As pointed out in Ref. [24], comparing the interactions generated by these three operators with those induced by the relevant operators in the HEFT basis, one nds two differences: (i) for the TGV phenomenology OW and OB give
corrections to the vertices equivalent to those induced by
P2(h) and P3(h), while for the HVV couplings their effects
are equivalent to those of P4(h) and P5(h); (ii) the O(p4)
chiral operator P13(h) has no equivalent in the linear expan
sion at dimension 6.
In other words, (i) implies that, as is well known from the pre-LHC times [77], and recently emphasised in some of the postHiggs discovery analyses [49,66,78], the operators
OW and OB lead at the same time to anomalous contributions
to both Higgs physics and TGV anomalous measurements. Thus, any deviation generated by them should be correlated in data from both sectors, and consequently the combined analysis of Higgs data and TGV measurements becomes
= 1
g 4
c3 + 2c13 + 2 c2 t
,
6 g v2
2 cW W W .
Electromagnetic gauge invariance enforces g1 = 1, both in
the SM and in the presence of the new operators. In Eq. (3.35), V {, Z}, gW W = e, gW W Z = g cos W , and W and
V refer exclusively to the kinetic part of the gauge eld strengths.
The combination of all the most sensitive searches for anomalous TGV deviations in W V diboson production has been performed in Ref. [76], presenting the results obtained in the SMEFT framework. These results show that at present the most stringent constraints on the anomalous TGV are set by the LHC Run I searches, whose combined sensitivity has clearly surpassed that of LEP. Even more relevant is the fact that, while the LHC Higgs data and gauge-boson pair
production searches are able to separately set stringent constraints on the HEFT operators, the combined study of the two sets of data could be used to improve the understanding of the nature of the Higgs boson state, as already emphasised in Ref. [24].
In brief, three CP even SMEFT operators with d = 6
can lead to sizeable corrections to the TGV vertices after considering all bounds from EWPD [4749,62,76]:
= Z
123
416 Page 20 of 45 Eur. Phys. J. C (2016) 76 :416
Fig. 3 Present bounds on B, W , B and W (see the text for the details of their denition) as obtained from the most recent combined global analysis of Higgs and TGV data. The rest of the undisplayed parameters spanned in the global analysis
( aC, aB, aG, aW , , a17, Y (1)t, Y (1)b, Y (1) and cW W W ) have been proled. The black dots signal the (0, 0) point, while the stars signal the current best t point obtained in the analysis
mandatory in order to obtain constraints as strong as possible on their coefcients [76]. Conversely, in the HEFT case, the anomalous TGV deviations induced by OW and OB are
generated by P2(h) and P3(h), while their effects on Higgs
physics originate from P4(h) and P5(h). Therefore, devia
tions in TGV and in Higgs physics could remain completely uncorrelated in the HEFT context [24]. This means that the nature of the Higgs boson can be directly probed by testing the presence of this (de)-correlated pattern of interactions in the event of an anomalous observation in any of the two sectors.
To illustrate the present status of such comparison, a global analysis of the data available both on the Higgs interactions and on the searches for anomalous TGV has been performed. The analysis spans the 10 coefcients relevant for Higgs physics in the HEFT scenario; see Eq. (3.33), together with the three parameters relevant for the TGV sector, which have an equivalent in the SMEFT Lagrangian, c2, c3 and cW W W
(i.e. setting c13 to zero).7In what respects the TGV analysis, the simulation of the
relevant distributions and the statistical t follow those of Ref. [76]. The best t values and 95 % CL intervals obtained for c2, c3 and cW W W are quoted for completeness in Table 4.
As can be seen comparing the results in Table 4 with Table 4 of Ref. [24], derived considering only the LEP based TGV bounds on c2 and c3, the new combination of LHC Run I
7 Notice that the operator belonging to the SMEFT expansion which contains the same interactions described by P13(h), also called linear
sibling, arises only at d = 8.
searches is able to improve substantially the constraints on
P2(h) and P3(h).
It was already shown in Ref. [24] that four specic combinations of the coefcients P2(h), P3(h), P4(h) and P5(h)
are meaningful for illustrating the Higgs+TGV results:8
B
1gt (2c2 + a4), W
1
2g (2c3 a5), (3.38)
1
2g (2c3 + a5).
These four parameters were dened in such a way that, at d = 6 order in the SMEFT expansion, the two s are zero
because of gauge invariance and of the doublet nature of the Higgs, B = W = 0. On the other hand, the operators OW
and OB contribute to the s leading to B = v2 fB 2 and
W = v2 fW 2 , being fi the associated Wilson coefcients.
In contrast, the HEFT operators could generate independent modications to each of these four variables. Figure 3 shows the current status of the bounds on the two relevant planes of coefcients after taking into consideration all the Higgs measurements included in the presented Higgs global analysis (based on Ref. [62]), together with the most recent combination of TGV searches presented in the previous subsection (based on Ref. [76]).
As described in Ref. [24], in the left panel of Fig. 3 the (0, 0) point corresponds to no deviation from the SM, while
8 For the sake of comparison with Ref. [24], the four combinations have been dened to be quantitatively equivalent to those in Ref. [24], in spite of the different normalisation for the ci and ai coefcients used here.
B
1gt (2c2 a4), W
123
Eur. Phys. J. C (2016) 76 :416 Page 21 of 45 416
in the right one it represents the limit in which TGV and HVV couplings show a SMEFT-like correlation. Therefore, any deviation from (0, 0) in the left panel would indicate BSM physics irrespective of the nature of the EWSB real-isation, while a similar departure in the right panel would disfavour a linear EWSB. As the s and the s are orthogonal combinations of parameters, the two panels of Fig. 3 are in principle independent of each other. In particular, deviations from (0, 0) may occur arbitrarily in only one plane or in both at the same time.
The constraints of B, W , B and W shown in Fig. 3 present a signicant improvement with respect to the bounds previously shown in Fig. 2 of Ref. [24]. The reason for such a sizeable improvement relies on two key points. First, the strength of the derived results is increased by the inclusion of the more complete set of run I LHC Higgs event rate measurement and by the addition of relevant kinematic distributions, that are sensitive to the anomalous SM Lorentz structures generated by a3 and a5 [62]. Second, the combination of the signicant LHC Run I diboson production analysis as described in Ref. [76] also has a huge impact in the analysis. The combination of these two ameliorations enhances signicantly the accuracy of the combined results shown in Fig. 3, in spite of the larger dimensionality of the parameter space considered in the present study with respect to the global analysis in Ref. [24].
4 Higher order operators and expansion validity
An important issue for numerical analyses performed in an EFT approach is that of establishing whether the EFT description is valid at the typical energies of the processes considered. The task is particularly relevant when collider data is included in the analysis, as the corresponding measurements are typically taken at energies signicantly higher than the EW scale.
In general, the validity of the expansion can be discussed studying the impact of operators which belong to different expansion orders. In the context of the SMEFT, this is tantamount to analysing operators with dimension d > 6. As discussed in Refs. [41,7982], this analysis sets different constraints on the cut-off of the theory, depending on the observables and of the operators considered: the strongest bounds are associated to observables that receive contributions from d = 8 operators with a larger number of derivatives, as they
induce a strong energy-dependence.Similar general considerations also apply to the HEFT.
However, in this case the discussion is complicated by the simultaneous presence of several characteristic scales and, consequently, of multiple expansion parameters. Although the only physical scales of the HEFT are and v, as explained in Sect. 1, it is useful to keep momentarily the scale f (
4 f ) as an independent quantity. The limit f v will be
discussed later on.
In realistic composite Higgs models, that can be considered as a benchmark for understanding the role played by each scale, v, f and enter the low-energy Lagrangian in three different combinations: v/f = , 1/4 f/ 1,
and E/ , where E is the characteristic energy scale of a given observable. As shown in Ref. [38], cross sections of physical processes only depend on scale suppressions: the generic expression, adopting the NDA normalisation of Eq. (2.7), is given by
(4)2 E2
E2 2
N , (4.1)
where (N ) is the number of powers of that suppress an
interaction term. The NDA master formula takes automatically care of all the 4 factors appearing in the cross-section (see Ref. [38] for further details and for generalisations), so that (N ) actually counts both powers of and of f indif
ferently. As a result, the only quantities that can be considered as proper suppression factors are and E/ . The physical relevance of a given cross-sections is basically determined by its dependence on these two parameters.
While the dependence on 1/ is explicit in HEFT operators, it is less trivial to trace that on = v/f . To this aim,
it is useful to recall (see Sect. 1) that f is the scale associated to both the SM GBs and the Higgs and, as such, it is always hidden inside the GB matrix U(x) and the generic Higgs functions F(h). The dependence on f can be made
explicit expanding these structures:
U = 1 + 2i
aa
f + , F(h) = 1 + 2a
hf + . (4.2)
Within V and upon going to unitary gauge, the powers on 1/f are converted into factors of . This is due to the fact that, in the kind of scenarios considered here, represents a ne-tuning that necessarily weights insertions of longitudinal components of the gauge bosons [38]. This indeed occurs in composite Higgs models (see Refs. [30,31]), where analogous conclusions are found to hold also for F(h).
It is worth noticing that, while U(x) and F(h), consid
ered globally, are adimensional quantities, their expansions contain terms with different canonical dimensions that come suppressed by powers of f . As a result, the leading terms of V and F(h), obtained applying one derivative to the
series of Eq. (4.2), have canonical dimension two: one dimension being associated to the derivative and the other to the rst non-vanishing term in the expansion of either U or F(h).
This observation can be generalised introducing the primary dimension dp, dened in Ref. [38] as the canonical dimension of the leading term in the expansion of a given object. For fundamental elements, such as derivatives, gauge elds and fermions, the primary dimension coincides with the tra-
123
416 Page 22 of 45 Eur. Phys. J. C (2016) 76 :416
Table 5 Different HEFT building blocks and their primary dimensions.
The two last columns report the suppression factors associated to each object
Building block dp Factors of Factors of p/
U(x) 0 1 1
F(h) 0 1 1 1 1 (p/ )
3/2 1 (p/ )3/2
X 2 1 (p/ )2
V 2 (p/ )
F(h) 2 (p/ )
ditional canonical dimension. Table 5 contains a summary of the primary dimensions for the building blocks used in the construction of the HEFT Lagrangian, together with the associated suppression factors. It follows from the discussion above that a term suppressed by /2(p/ ) must have
dp = + .
With the information provided by Table 5, it is easy to infer the dependences for all the HEFT operators, that can be thus organised in a two-parameter expansion as indicated, schematically, in Table 6. The colours discriminate between two sets of operators: the structures reported in the cyan boxes correspond to the NLO Lagrangian considered in this work; the structures in the white cells, instead, are customarily considered as higher order terms, but their impact may be compa-
rable to that of the NLO terms for sufciently high energies. Depending on the observables considered, it may be necessary to include (part of) the second set of operators into the phenomenological analysis (see also Ref. [83]), even if this would mean working with a ill-dened basis from a renormalisation point of view. This should not be seen as a concern, as, even considering a complete, non-redundant basis at NNLO, only the subcategories listed in Table 6 would be physically relevant. Effects due to operator mixing under the renormalisation group running are also expected to be completely negligible at the experimental sensitivities foreseen for the near future.
In the limit f v, the dependence on does not rep
resent a suppression anymore and the physical impact of an operator is determined only by the factors of p/ . In this case, one recovers a pure chiral expansion, which is organised horizontally in the representation of Table 6.
On the contrary, in the limit p/ , all the opera
tors with the same dp are equally suppressed and therefore one recovers, altogether, the linear expansion organised in canonical (or primary) dimensions. In this case, all the operators in the white boxes of Table 6 should be considered. This condition is for instance fullled for = 10 TeV and
E 1 TeV, which is within the range of energies that are
relevant for processes to be observed at LHC13.
The introduction of the primary dimension, i.e. of a counting on explicit and implicit scale suppressions, allows one to link the particular structure of an operator to the strength
(F)2(V)2 (V)4
(F)4
3/2
Table 6 HEFT operators distributed according to their and p/ suppressing factors. A schematic notation has been adopted for categorising the operators based on the building blocks they contain. The terms appearing in the cyan boxes correspond to the NLO operators listed in the previous sections. The other terms refer to operators that usually belong to higher Lagrangian orders, but that can have an impact similar to that of the NLO ones for sufciently high energies. EOMs have been employed to remove redundant structures
2
(F)(V)(
)
(V2)(X)2
(F)(V)(X)
(V)2(
)
(F)2(
(X)2(V)2
(F)(V)(X)2 (F)2(X)2
)
(V)(X)(
)
(V)(
)2
(F)(
(
)(V)
(F)(X)(
)
)2
(
)2
(X)3
(X)(
)2
(X)4
1
(X)2
(X)(
)
(X)2(
)
1
p
p
2
p
3
p
4
123
Eur. Phys. J. C (2016) 76 :416 Page 23 of 45 416
of a physical signal in terms of cross sections. Indeed, if an observable receives contributions from a single operator, then the corresponding cross section is uniquely determined by the primary dimension of that operator, according to Eq. (4.1).As a consequence, the dp is a useful phenomenological tool to indicate whether the strength of an observable, that receive contributions only from operators belonging to higher expansion orders, is expected to be of the same order or more suppressed with respect to the other processes already considered in the phenomenological analysis.
An interesting application of the primary dimension is that if the dp of an HEFT operator is smaller than the canonical dimension of the corresponding linear sibling, then the processes described by these operators represent smoking guns to test the linearity of the EWSB realisation. This is the case of the operator P14(h) discussed in Ref. [38]: it induces an
anomalous TGV, commonly called gZ5, that is expected to be strongly suppressed in the SMEFT description, but not in the
HEFT one.
5 Conclusions
The complete effective Lagrangian for a non-linear realisation of the EWSB (shortened into HEFT) has been presented. It provides the most general description of the Higgs couplings and it can be used for investigating a large spectrum of distinct theories, ranging from the SM to technicolour constructions, including composite Higgs realisations and dilaton-like frameworks. In contrast with the effective Lagrangian for a linearly realised EWSB (also SMEFT), in which the Higgs belongs to an exact SU(2)L doublet, in the
HEFT the physical Higgs is assigned to a singlet representation of the EW group and it is treated as an object independent of the Goldstone bosons matrix.
Assuming invariance under the Lorentz and SM gauge symmetries, as well as the conservation of baryon and lepton numbers, the complete chiral basis at the next to leading order contains a total number of 148 independent, avour universal terms. When extending the SM spectrum to include three right-handed neutrinos, 40 more operators enrich the basis. The generalisation to arbitrary avour contractions is straightforward.
Conversely, the SMEFT basis up to d = 6 consists of only
59 avour universal terms, in absence of right-handed neutrinos. The different number of operators and of building blocks used for the construction of the two bases lead to fundamental differences between the SMEFT and the HEFT. The possibility of distinguishing between them has been discussed performing a global t including all the available data from colliders, including EWPD, Higgs and TGV measurements taken at the LHC Run I. The main outcomes are summarised in the following points:
The Electroweak precision data analysis together with the study of the CKM matrix unitarity allows one to constrain 11 parameters of the HEFT Lagrangian. The corresponding value of the 2 at the minimum is 6. This can be compared with the corresponding analysis within the SM, whose 2 is 18.4.
The results for the S, T andU parameters are signicantly different from the standard analysis in the SMEFT with operators up to dimension 6, due to the presence of extra free parameters: the allowed range for S is about 4 times broader, while the bounds on T and U are about 20 times weaker.
The analysis of Higgs data depends on a total of 10 parameters, with one bosonic operator more compared to the same analysis in the SMEFT case at dimension six. Although the nal results are quite similar to those obtained for the SMEFT, the addition of the extra parameter broadens the allowed range for the remaining nine coefcients, as expected.
The interplay between triple gauge-boson vertices and Higgs couplings provides an interesting way of investigating the nature of EWSB. Although this analysis is not conclusive yet due to the limited sensitivity on the observables considered, the introduction of kinematic distributions is seen to improve considerably the results. Would the accuracy of Higgs measurements improve signicantly in the future, this kind of analysis may reveal signatures of non-linearity in the Higgs sector.
It has been underlined that with the increase in energy at colliders, it may be necessary to consider several operators that, in spite of being usually considered as higher order effects, may have a non-negligible phenomenological impact. The list of the relevant structures has been given in Table 6.
In summary, this work extends the chiral basis of Refs. [24,25] with the introduction of fermionic operators. Moreover, the analysis presented here updates and extends that contained in Ref. [24] with the inclusion of more recent collider data and of fermionic observables. A strategy for disentangling the nature of the EWSB has been discussed, based on the presence of new anomalous signals and of decorrelations among observables. It has also been discussed how the phenomenological analysis should be modied when higher energy data is kept into account, specifying the relevant operator structures that should be added to the basis in this case. The analysis presented here represents the rst phenomenological study performed with the complete HEFT Lagrangian and it could be taken as a reference for dedicated experimental analyses aimed at shedding light on the Electroweak symmetry breaking sector and the Higgs nature.
123
416 Page 24 of 45 Eur. Phys. J. C (2016) 76 :416
Acknowledgments We thank M. B. Gavela for useful discussions during the development of the project, as well as Anja Butter, Tilman Plehn and Michael Rauch for their crucial assistance with both the Higgs and the TGV analyses. We also thank E.E. Jenkins, A.V. Manohar andM. Trott for interesting comments on the manuscript. I.B. research was supported by an ESR contract of the EU network FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442). M.C.GG is supported by USA-NSF grant PHY-13-16617, by grants 2014-SGR-104 and by FPA2013-46570 and consolider-ingenio 2010 program CSD-2008-0037. L.M. acknowledge partial support of CiCYT through the project FPA2012-31880 and of the Spanish MINECOs Centro de Excelencia Severo Ochoa Programme under grant SEV-2012-0249. M.C.G-G and L.M. acknowledge partial support by FP7 ITN INVISIBLES (PITN-GA-2011-289442), FP10 ITN ELUSIVES (H2020-MSCA-ITN-2015-674896) and INVISIBLES-PLUS (H2020-MSCA-RISE-2015-690575)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/
Web End =http://creativecomm http://creativecommons.org/licenses/by/4.0/
Web End =ons.org/licenses/by/4.0/ ), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP3.
A Additional operators in the presence of RH neutrinos
Adding right-handed neutrinos to the spectrum amounts to declaring a non-zero upper component for the L R doublet, which shall be dened as L R = (NR, ER)T . Consequently,
the lepton Yukawa matrix in the LO Lagrangian Eq. (2.3) has to be generalised to account for the masses and interactions of the neutrinos with the Higgs
YL (h) diag n
Y (n)
hnvn ,
n
. (A.1)
In addition, the fermionic basis presented in Sect. 2.3 must be enlarged in order to account for the increased number of possible invariants, as follows:
L2F =
Y (n)
hn vn
17 n j N j +
28
j=181
n j + i n j N j
+
31
j=294
n j + i n j N j, (A.2)
L4F =
(4)2 2
10 r j + i r j
R j +15
j=1r j R j
+ 29 rQ j + i rQ j
RQ j +38
j=30rQ j RQ j
.
(A.3)
The complete list of additional operators is provided in this appendix.
Single leptonic current operators
With one derivative
N 15(h) i L R UVU L R F,
CP N 16(h) L R U[V, T]U L R F,
N 17(h) i L R UTVTU L R F.
With two derivatives
N 18(h) LL TU L R FF , N 19(h) LL VU L R F,
N 20(h) LL TVTU L R F,
N 21(h) LL VVTU L R F,
N 22(h) LL TVTVTU L R F,
N 23(h) LL VTVU L R F,
N 24(h) LL VTVTU L R F,
N 25(h) LL VU L R F,
N 26(h) LL TVTU L R F,
N 27(h) LL VTVTU L R F,
N 28(h) LL [V, V]TU L R F,
N 29(h) ig LL TU L R BF,N 30(h) ig LL {W, T}U L R F, N 31(h) ig LL TWTU L R F.
Four-fermion operators
Additional operators with four leptons:
R 8(h) ( LL iU L R )( LL iU L R )F, R 9(h) ( LL U L R )( LL TU L R )F,
R 10(h) ( LL TU L R )( LL TU L R )F,
R 11(h) ( L R L R )( L R UTU L R )F, R 12(h) ( L R UTU L R )( L R UTU L R )F,
R 13(h) ( LL LL )( L R UTU L R )F, R 14(h) ( LL T LL )( L R UTU L R )F,
R 15(h) ( LL i LL )( L R UiU L R )F.
Additional mixed operators with two quarks and two lep-tons
RQ 24(h) ( LL U QR )( QL TU L R )F, RQ 25(h) ( LL TU L R )( QL U QR )F,
RQ 26(h) ( LL TU L R )( QL TU QR )F, RQ 27(h) ( LL TU QR )( QL TU L R )F,
RQ 28(h) ( LL iTU L R )( QL iU QR )F, RQ 29(h) ( LL iTU QR )( QL iU L R )F,
RQ 30(h) ( L R UTU L R )( QR QR )F, RQ 31(h) ( L R UTU L R )( QR UTU QR )F,
123
Eur. Phys. J. C (2016) 76 :416 Page 25 of 45 416
RQ 32(h) ( L R U jU L R )( QR U jU QR )F, RQ 33(h) ( QL QL )( L R UTU L R )F,
RQ 34(h) ( QL T QL )( L R UTU L R )F, RQ 35(h) ( QL j QL )( L R U jU L R )F,
RQ 36(h) ( QL LL )( L R UTU QR )F, RQ 37(h) ( QL T LL )( L R UTU QR )F,
RQ 38(h) ( QL jT LL )( L R U jU QR )F
B Removal of F(h) F(h) from the Higgs and fermions
kinetic terms
All the kinetic terms in the LO Lagrangian, Eq. (2.3), are canonically normalised, despite the fact that the singlet nature of the h eld in principle allows one to couple them to a function F(h). In the case of the gauge-boson kinetic term, the
absence of a Higgs-dependence is justied in the assumption that the transverse components of the gauge elds do not interact with the Higgs sector at LO. On the other hand, in the cases of the Higgs and of the fermions kinetic terms, the dependence F(h) is completely redundant, as it can be
removed via a eld redenition (analogously to what was done in Ref. [84]. See also Ref. [26]). This appendix provides more details about this redenition.
The coupling of the fermionic kinetic term to a generic Higgs function would be of the form
i 2
/
In this case, the FH(h) function can be removed by the eld
redenition
h
h
1 + FH(s) ds; (B.6)
in fact
1
2h h =
0
1 + FH(h) 2 = PH(h). (B.7)
Although this redenition looks quite involved, it clearly induces modications of all the Higgs couplings in the Lagrangian. As these are always described by arbitrary coefcients, the redenition (B.6) can be entirely reabsorbed into redenitions of the functions Fi(H), which appear in the
Lagrangian. As seen for the case of F(h) above, the pres
ence of FH(h) in the Higgs kinetic term is redundant within
the LO Lagrangian chosen in Eq. (2.3).
B.1 A practical example
In order to give a practical illustration, one can consider a specic function
FH (h) = 2aH
hv . (B.8)
Then the equation
h =
h
1
2
h
2aH h
D
/ 1 + F
1 + 2aHs/v ds =v 3aH
(h)
, (B.1)
where = {Q, L} and
F (h) = c + 2a
hv + b
v + 1 3/2 1
(B.9)
0
h2v2 + (B.2) The dependence of F(h) can therefore be removed via the
redenition
1 + F
(h)
can be solved analytically, obtaining
h =
v 2aH
3aHv h + 1 2/3 1 . (B.10)
Plugging this result into FC(h) and re-expanding in h /v, it
gives9:
FC (h) = 1 + 2aC
h v + (bC aCaH)
1/2 . (B.3)
As this substitution is applied to the whole Lagrangian, it induces a modication of all the couplings between fermionic and Higgs elds, which can be reabsorbed in redenitions of the functions Fi(h), which accompany fermionic operators.
In particular, this is also true for the LO Yukawa couplings, as they are accompanied by arbitrary polynomials YQ,L(h). In
conclusion, the insertion of a function F(h) in the fermionic
kinetic term is redundant in the LO Lagrangian of Eq. (2.3).The Higgs kinetic term may also be written as
1
2hh (1 + FH(h)) , (B.4)
with
FH (h) = cH + 2aH
hv + bH
h v
2, (B.11)
so that the impact of the redenition can be entirely reabsorbed dening primed coefcients,
a C = aC, b C = bC aCaH. (B.12)
An analogous redenition allows one to reabsorb inside the function Y(n) the effects on the Yukawa interactions.
9 In this computation aH > 0 is assumed. For negative values the third roots give some complications.
h2v2 + (B.5)
123
416 Page 26 of 45 Eur. Phys. J. C (2016) 76 :416
C Construction of the fermionic basis
This appendix provides additional information about the construction of the fermionic basis specifying, in particular, the relation between the structures present in the operators presented in Sect. 2.3 and those that have been removed. In the following, generic fermion elds are denoted by = {Q, L}
while stands for an arbitrary SU(2) structure, combination of the blocks {T, V, D, j}. The Lorentz contractions are
always explicited and, whenever they are not specied, chiralities are arbitrary. The correspondence between classes of operators is indicated schematically by an arrow (); signs
and numerical coefcients are not specied in these relations.
C.1 Useful identities
A list of useful identities is provided below. Since the building blocks A = {T, V, DV} are traceless, they can be
generically rewritten as A =
1
2 Tr[Aa]a. This yields the
relations:
[T, V] =
i2i jkTr(Ti)Tr(V j)k, (C.1)
{T, V} = Tr(TV)1, (C.2) TVT =
1
2
Whenever they are applied to SU(2) doublets (and SU(3) triplets), these identities must be applied together with the completeness relations for the generators of SU(2) (and of SU(3)),
ai jamn = 2inmj i jmn, (C.11)
Ai jAmn = 2inmj
2
3i jmn, (C.12)
in order to recover the correct gauge contractions. For example, combining Eq. (C.8) with Eqs. (C.11) and (C.12), the scalar identity for quark doublets reads
( Q1L Q2R)( Q3R Q4L)
=
112( Q1LQ4L)( Q3R Q2R)
112( Q1Lk Q4L)( Q3R k Q2R)
18( Q1LA Q4L)( Q3R A Q2R)
18( Q1LAk Q4L)( Q3R Ak Q2R). (C.13)
C.2 Construction of L2F
Since for traceless matrices Tr(AB)1 = {A, B}, the
operators of the type
Tr( 1 2)F with i = {T, V, DV} are always equivalent to the bilinears { 1, 2}F.
Bilinears with a derivative on the fermion eld and a vector current (e.g. (D
Tr(TV)Tr(Ti) Tr(Vi) i
= TTr(TV) V. (C.3)
The properties of the SU(2) generators additionally lead to the following identities:
TVV = VVT, TV[TV] = V[TV]T,
TV[TV]T = V[TV], (C.4)
T[V, V] = [V, V]T 2V[TV].
The transformation properties of T and V ensure
DT = [V, T], (C.5)
V = DV DV = igW ig B/2 + [V, V].(C.6)
The Fierz identities for chiral (anticommuting) elds have been employed for the reduction of the four-fermion basis
(L BR)( CL DR) =
1
2(L DR)( CL BR)
) X) can been removed via integration by parts and application of the EOMs (see Appendix D). Operators with a derivative on the fermion eld but with no gamma matrices (of the type
D) are removed using the relation g = { , }/2, integration by parts
and the EOMs:
D = g D = / ( /D)
+
/
(D)
=
/
( /
D) (
/
( /
D /
)
/
+
D + i
D .
18(L DR)( CL BR), (C.7)
(C.14)
Bilinears with the structure DV can be reduced to a combination of dipole operators (containing eld strengths), terms with the structure VV and bilinears with the direct contraction DV. In fact:
DV = g
i
1
2(LDL) ( CR BR), (C.8)
(LBL)( CL DL) = (LDL) ( CL BL), (C.9)
(RBR)( CR DR) = (RDR) ( CR BR). (C.10)
(L BR)( CR DL) =
DV
= DV
i2V (C.15)
123
Eur. Phys. J. C (2016) 76 :416 Page 27 of 45 416
where Eq. (C.6) shall be applied on the latter term. The former can also be removed using the EOMs. the commutator [D, D] is always vanishing when
applied to SU(2) invariants (right-handed fermions, B and G elds, F(h) functions), while it is traded for a
eld strength when it acts on a quantity X with non-trivial isospin transformations: [D, D]X = ig[W, X].
further combinations of T and V that do not appear in the basis reported in Sect. 2.3 have been traded for others using the identities (C.1) and (C.4).
C.3 Construction of L4F
Further details of the construction and of the reduction of the four-fermion operators basis are provided in this section. None of the terms of L4F have been removed via the EOMs, while the Fierz identities (C.7)(C.10) have been extensively employed for removing redundant structures. In particular, operators with tensor currents ((
v2 s2 ,
(C.18)
where we have used the compact notation s sin
(| |/v), c cos(| |/v). From Eq. (C.18) it follows
immediately that
RQ1 RQ2
2 = RQ1 c2 + |
|2
)2) were
not included in the nal basis, as they are always equivalent to combinations of scalar contractions via the Fierz identity (C.7). Similarly, operators with the scalar contraction (
|2
v2 s2
RL) have been traded for terms with the vector structure (
LR)(
LL)(
R R) employing the iden-
tity (C.8).
= RQ1 1 |
|2
v2 +
43 |
|4
v4 +
RQ5 RQ6
2 = RQ2 c2 + |
|2
v2 s2
= RQ2 1 |
Four-quark (lepton) operators
There are four independent SU(2) contractions of four quarks that can be constructed with the scalar structure (
|2
v2 +
43 |
|4
v4 + (C.19)
R 1 R 8
2 = R N c2 + |
|2
v2 s2
LR)(
LR). They are easily identied in unitary gauge by the U(1)em invariants
(uu)(uu), (dd)(dd), (uu)(dd), (ud)(du).
Keeping colour contractions into account, the total number of independent operators in this category is eight. With four leptons there is only one invariant with this Lorentz structure, due to the absence of right-handed neutrinos: (ee)(ee).
We do not provide the expressions of all the possible SU(2) structures in terms of the invariants selected for the basis of Sect. 2.3. However, it is worth commenting on two contractions that can be constructed without the explicit insertion of Goldstone bosons: in the four-quark case they are
RQ1 = i jab( QL i QR a)( QL j QR b)F,RQ2 = i jab( QL iA QR a)( QL jA QR b)F. (C.16)
In the four-lepton case, it is possible to introduce a structure analogous to the rst one, but only in the presence of right-handed neutrinos. This would read
R N = i jab( LL i L R a)( LL j L R b)F. (C.17)
The operators of Eqs. (C.16) and (C.17) are redundant in the basis of Sect. 2.3: in fact, exploiting the properties of the Pauli matrices and the completeness relation (C.11) one has
UiaUjb (Uk)ia(Uk)jb = 2i jab c2 + |
v4 + .
Therefore, the interactions contained in RQ1, RQ2 and R N are already described by linear combinations of operators in the basis. The class of four-fermion operators with two left-handed currents contains four independent operators in both the four-quark and the four-lepton cases:
( QL QL )2: (uu)(uu), (dd)(dd),
(uu)(dd), (ud)(du),( LL LL )2: ()(), (ee)(ee), ()(ee),
(e)(e).
Notice that in this case the octet colour contraction ( QL A QL )2 is not independent. In fact it is equiv
alent to a combination of invariants with singlet colour contractions. Using Eqs. (C.9) and (C.12):
= R N 1 |
|2
v2 +
43 |
|4
123
416 Page 28 of 45 Eur. Phys. J. C (2016) 76 :416
( QL a QL )( QL a QL )
13( QL QL )( QL QL )
= +( QL j QL )( QL j QL ). (C.20)
An analogous relation holds for the structures with right-handed currents. The class of four-fermion operators with two right-handed currents contains four independent operators in the four-quark case but only one in the four-lepton sector:
( QR QR )2: (uu)(uu), (dd)(dd), (uu)(dd),
(ud)(du),( L R L R )2: (ee)(ee).
Finally, there are ve independent SU(2) contractions for quark vector currents of opposite chirality (
LL)
R R), to be doubled when including octet colour contractions:
(uu)(uu), (dd)(dd), (uu)(dd), (dd)(uu),(ud)(du) + (du)(ud).
The four-lepton counterpart, instead, contains two invariants corresponding to the interactions
(ee)(ee), ()(ee).
Mixed quarklepton operators
Operators with the scalar contraction (
(
v2YQ(h)UL, (D.1)
(DW)a =
=Q,L
g2
LaL
DR =
LR)(
LR)
can have either the structure ( QQ)( L L) or ( QL)( L Q).
Each of these yield three independent invariants, that are most easily identied in unitary gauge by the interactions:
( QQ)( LL): (uu)(ee), (dd)(ee), (du)(e),
( QL)( L Q): (ue)(eu), (de)(ed), (de)(u).(C.21)
The two combinations ( QLQL)( LL LL ),
( QLLL )( LL QL) are related by the Fierz iden
tity (C.9), and therefore only the former structure has been retained. The same holds for the analogous terms constructed with right-handed currents, that are connected by Eq. (C.10).
This class includes ve independent left-handed invariants, identied by the hermitian combinations
(uu)(ee), (dd)(ee), (uu)(), (dd)(),(du)(e) + (ud)(e),
and two right-handed ones:
(uu)(ee), (dd)(ee).
Operators with one left-handed and one right-handed current can be constructed in either of the combinations ( QL QL )( L R L R ), ( LL LL )( QR QR ) and
( QL LL )( L R QR ). These provide, respectively, 2
+ 5 + 3 independent interactions:
( QQ)( LL): (uu)(ee), (dd)(ee),
( L L)( QQ): (ee)(uu), (ee)(dd), ()(uu),
()(dd), (e)(du) + (e)(ud),
( L Q)( QL): (eu)(ue), (ed)(de), (u)(de).(C.22)
D Application of the EOMs
Given the LO Lagrangian in Eq. (2.3), the elds satisfy the following EOMs:
i /
DL =
v 2UY(h)R,
i /
igv2
4 Tr[Va]FC(h), (D.2)
B = gc
i=L,R
=Q,L
+
ihi i
igcv2
4 Tr[TV]FC(h), (D.3)
v2
4 Tr[VV]F C(h)
=Q,L
h = V (h)
v 2
LUY (h)R + h.c.
, (D.4)
where hi are the hypercharges in the 2 2 matrix notation:
hQL = diag (1/6, 1/6) , hQR = diag (2/3, 1/3),
hLL = diag (1/2, 1/2) , hL
R
= diag (0, 1), (D.5)
and the prime denotes the rst derivative with respect to h. A consequence of Eqs. (D.2) and (D.1) is
123
Eur. Phys. J. C (2016) 76 :416 Page 29 of 45 416
D VFC =iv2 D
=Q,L
L j L
belong to the basis. The following CP even terms have been eliminated, in the notation of Ref. [24]:
P9(h) = Tr((DV)2)F, P10(h) = Tr(VDV)F, P15(h) = Tr(TDV)Tr(TDV)F, (D.11) P16(h) = Tr([T, V]DV)Tr(TV)F, P19(h) = Tr(TDV)Tr(TV)F.
Analogously, ve CP odd operators have been traded for others: in the notation of Ref. [25] they are
S10
j
=
12v
=Q,L
L jUY(h)R
RY(h)U jL
j,
(D.6)
which can be recast in the form
Tr( jDV)F(h) =
2 v
RY(h)U jL
Tr( jV)F(h), (D.7) which is valid order by order in the h expansion.
D.1 Operators that have been removed via EOM
The EOMs relate the purely bosonic and the fermionic sectors, and they have been used to eliminate operators that are redundant when both sectors are considered at the same time. In this section we list the categories of operators that have been removed.
Bosonic sector
Operators containing F(h).
Applying the EOM for the Higgs, Eq. (D.4), these terms can be traded for a combination of other bosonic operators plus fermionic bilinears and four-fermion operators. The following CP even terms have been removed, compared to the basis of Ref. [24]:
P H (h) =
=Q,L
L jUY(h)R
= iTr(VDV)Tr(TV)F,
S11
= iTr(TDV)Tr(VV)F, S12= iTr([V, T]DV)F, (D.12)
S14= iTr(TDV)F(h)F ,
S16
= iTr(TDV)Tr(TV)Tr(TV)F.
Fermionic sector
Bilinears of the type
F.
Applying the EOMs for fermions (Eq. (D.1)), these operators can be schematically rewritten as
F = /
D F (D )F
/
DF
(D )F
F. (D.13)
Bilinears containing F.
Operators in this category are removed applying the EOM for the Higgs eld, Eq. (D.4) and traded for other bilinears plus four-fermion operators. invariants containing DV
As in the bosonic sector, these operators are removed applying the identity (D.7). and traded for other bilinears plus four-fermion operators. Finally, the EOMs for the gauge (Eqs. (D.2), (D.3)) and Higgs (Eq. (D.4)) elds imply the following additional relations (signs and numeric coefcients not specied):
PB
h hv2 F,
P7(h) = Tr(VV) F, (D.8) P25(h) = Tr(TV)Tr(TV) F,
and the CP odd operator
S13= Tr(TV) F. (D.9)
Operators containing DV.
Rewriting the traceless matrix DV as
DV =
a
2 Tr(aDV) (D.10)
and applying the identity (D.7), these bosonic operators can be traded by combinations of fermion bilinears, four-fermion operators and other bosonic terms that already
+ P1 + P2 + P4 + PT
i LLi {V, T}LLi F + N Q5 + N Q6,
PW
+ P1 + P3 + P5 + Tr(VV)F
i LLi VLLi F + N Q1, (D.14)
PT
+ P1 + P3 + P12 + P13 + P17
i LLi TVTLLi F + i LLi VLLi F + N Q7
+ N Q1.
123
416 Page 30 of 45 Eur. Phys. J. C (2016) 76 :416
These have been employed to remove the three (avour-diagonal contractions of the) leptonic operators specied on the right-hand side. This choice simplies the renormalisation procedure.
E Feynman rules
This appendix provides a complete list of all the Feynman rules resulting from both fermionic and bosonic operators considered in the present work and listed in Sects. 2.2 and 2.3.For compactness we omit CP violating terms, that are not relevant for the phenomenological study presented. The rules are derived in unitary gauge and only vertices with up to four legs are shown. The SM contribution and the renormalisation effects are also included, up to rst order in the effective coefcients. The latter are sometimes encoded in the quantities g1, g2, gW and MW dened in Eqs. (3.5), (3.6)
in the text.A few comments about the notation and conventions used:
All momenta are owing inwards and the convention ip has been used in the derivation.
We use a shorthand notation for the products ciai: for the bosonic operators, we replace aici ai and bici ai.
For the fermionic operators, we write a fi n fi (na) fi .
The structure
FF gives couplings hh f f with the
coefcients n fi a fi a fi . This notation has been shortened in (naa ) fi . For the coefcients of the function FC(h),
dened in Eq. (2.4), the notation aC = 1 + aC, bC =
1 + bC is adopted.
We have xed VCKM = 1 for compactness. At the same
level, all the effective coefcients are implicitly taken to be avour diagonal. In the vertices with a single fermion current the spin contractions are obvious. For those with four fermions we use a notation with square brackets and lowercase indices: for example [PR]ab[PL]cd means that the right chirality pro
jector contracts the spins of the a and b particle, and the left chirality one shall be inserted between the c and d elds. Uppercase indices indicate colour and are assumed to be summed over when repeated. Whenever they are not specied, the colour (and avour) contractions go with those of the spin.
123
Eur. Phys. J. C (2016) 76 :416 Page 31 of 45 416
t +c 3+2c 13
( g p Z g p Z ) ( 1 + 2 ( t c 1 2 c 12 )
(p Zp )p + (p Zp +)p
5)
r
2
v2
2(r
64 2 2 s 2
c 2
2c 2
(p Ap )p + (p Ap +)p
c 2 4c 12+
g ((p +p )p Z (p Zp )p + )+g
g
4
2c Tc2
t +2c 12 +
gc 3
4c2
(p 3p 2)p 1 (p 3p 1)p 2
1 + 2 t 2 c 1 +
c 1
12
(1+c G)
p A p + p p A p + p
g 2 ( g ( p + p ) g p + g p + ) 1 +
p Z p + p p Z p + p
p 3 p 1 p 2 p 3 p 1 p 2
(isthegaugexingparameter)
g (p +p ) g p +g p + (g p A g p A )
g ((p +p )p A (p Ap )p + )+g
2c WWW
g (p 2p 1) +g (p 1p 3) +g (p 3p 2)
g ((p 1p 2)p 3 (p 3p 2)p 1 )+g
2
Zc
2
= m
M2
W
M2
W
1 +
Zc
; M
(0)
f
(p Ap +)p (p +p )p A ++
(p Zp +)p (p +p )p Z +
(p 3p 1)p 2 (p 1p 2)p 3 +
2
24ic
2
=m
g
4(2t c 2+c 3+2c 13)
2
W
g2 c 14
4c
p p
p2
p p
p2
v
2Y
g (1)
g (1)
(p + p )
W
1 c2
2f;mf=
p p
m2 Z
p p
M2
ABC
1 + g 1 +
2c GGGf
+g
g
g
2c WWW
24is
(p 1) g s f ABC
+g
p2m
i(/p+m f)
Z
2
W
2
h
24
p2M
+g
2
p2
i p2m
p2m
igc
+
ie
i
i
i
ig
p2
+
(p 2)
W+
W+
GA
(FR.4)G
Z
W
h
f
(p 3)
GB
FR:propagators
(FR.1)
(FR.7)A
W
(FR.8)Z
W
(FR.2)
(FR.3)
(FR.5)
(FR.6)
FR:Bosonic
(FR.9)GC
123
416 Page 32 of 45 Eur. Phys. J. C (2016) 76 :416
(c 11+c 24)
g2
162(2c 6+c 11)
g2
162c4
p + (p A 1+p A 2) +p (p A 1+p A 2)
g g (p A 1+p A 2)(p ++p )+g g (p A 1p +p A 2p +)+g g (p A 1p ++p A 2p )
g g (p Z 1+p Z 2)(p ++p )+g g (p Z 1p +p Z 2p +)+g g (p Z 1p ++p Z 2p )
p A 2 (p +p ) p A 1 p + p A 2 p +
p A 2 (p p+) p A 2 p p A 1 p
g g +g g
p Z 1 (p ++p ) +p Z 2 (p ++p )
p Z 2 (p +p ) p Z 1 p + p Z 2 p +
p Z 2 (p p+) p Z 2 p p Z 1 p
p Z (p ++p ) +p A (p ++p )
1+
g g (p Z+p A)(p ++p )+g g (p Zp +p Ap +)+g g (p Zp ++p Ap )
g g (p 1+p 2)(p 3+p 4)+g g (p 1p 4+p 2p 3)+g g (p 1p 3+p 2p 4)
eg
p A (p +p ) p Z p + p A p +
p A (p p+) p A p p Z p
g g +g g
p 2 (p 3p 4) p 1 p 3 p 2 p 3
2 c 14
2c
p 1 (p 3+p 4) +p 2 (p 3+p 4)
1
(c 6+c 23)
g g +g g
p 2 (p 4p+) p 2 p 4 p 1 p 4
g
p + (p Z 1+p Z 2) +p (p Z 1+p Z 2) +
g
g
p + (p Z+p A) +p (p Z+p A) +
162c4
g
g2
162c 11
g2
g
g
+g (p A 1 (p p +) p A 1 p p A 2 p )+g
+g (p Z 1 (p p +) p Z 1 p p Z 2 p )+g
p 3 (p 1+p 2) +p 4 (p 1+p 2) +
1
p A 1 (p +p ) p A 2 p + p A 1 p + +
g
2c WWW
p A 1 (p ++p ) +p A 2 (p ++p )
p Z 1 (p +p ) p Z 2 p + p Z 1 p + +
+g (p Z (p p +) p Z p p A p )+g
1+
p Z (p +p ) p A p + p Z p + +
g
+g (p 1 (p 4p 3) p 1 p 4 p 2 p 4 )+g
24ies
2g g
2g g
2g g
g g +g g ]
p 1 (p 3p 4) p 2 p 3 p 1 p 3 +
gc 3
2c2
4c2
g3 c 13
gc 3
g 2+
g 2+
4c 12+
2 c2
1 c2
1 + 2 g 1 +
g
g
gc 3
2
iegc 1 + g 1 +
g
[2g g
2c WWW
2c WWW
1+2 g W+
24igc2
2c WWW
2 +g
+g
2 c2
+g
24iec
+g
24ig
+g
ie
ig
+
ig2
(p 4)
W+
(p 3)
W+
W
W+
W
W+
W
W
Z
A (p A 1)
Z (p Z 1)
(p 1)
W
(p 2)
(FR.10)
A (p A 2)
(FR.11)
Z (p Z 2)
(FR.12)
A
(FR.13)
W+
123
Eur. Phys. J. C (2016) 76 :416 Page 33 of 45 416
g (p 2 p 1 +p 3 p 4 )+g (p 1 p 2 +p 4 p 3 )+g (p 1 p 2 +p 3 p 4 )+(g g g g )(p 1p 2+p 3p 4)
+f
g (p 1 p 3 +p 2 p 4 )g (p 1 p 3 +p 2 p 4 )+g (p 1 p 3 p 1 p 3 )+g (p 2 p 4 p 2 p 4 )
+g (p 4 p 2 +p 1 p 3 )
g (p 2 p 3 +p 1 p 4 )g (p 1 p 4 +p 2 p 3 )+g (p 2 p 3 +p 1 p 4 )+g (p 2 p 3 +p 1 p 4 )
+g (p 1 p 4 p 1 p 4 )
(1+c G)
a 5
4gv2
]
ADX fBCX (g g g g )
g (p 1 p 2 p 1 p 2 )+g (p 3 p 4 p 4 p 3 )g (p 1 p 2 +p 3 p 4 )
](2a 4+t (a 5+2a 17))
+a W+4t a 14a 12
h)(p ++p )p hg
p Ap hg
p
+p
h
+p
ACX fBDX (g g g g )+f
](2s a 4c (a 5+2a 17))
a Bt2
a W
g2v2+[(p
h
]
Ap
+g (p 2 p 3 p 2 p 3 )+(g g g g )(p 1p 4+p 2p 3)
Z2
[p
[g g +g g +g g ](c 6+c 11+2c 23+2c 24+4c 26)
+g (p 1 p 3 +p 2 p 4 )+(g g g g )(p 1p 3+p 2p 4)
+f
Z1p
ig
4v
]
][4a 1c 2 +(4a 12+a Ba W)s 2 ]+
2
(a W4a 12)
p Z 1 p Z 2 p
+p
(p + p )2p
a B4t a 1+t
(p Z 1+p Z 2)p hg
g2v2[g
2c4
+[ 2 g
fABX fCDX (g g g g )+f
(1+ a C 2(a T c T ))+
]
M2
W
M2
W
]a G
fABX fCDX
A2
G2
A1p
h
g 1 + a C +
Z2p
Z
Ap
G1p
p A 1 p A 2 p
+p
Z1p
2c GGG
ACX fBDX
ADX fBCX
(p A p Z )p
p G 1 p G 2 p
h
24ig s
v2[p
g
[g
4gs4
2
v
c
162c4
Z
2 v
[g
Zc
2
+
2
v
[g
ig2
s
(p 3)
(p 4)
A (p A 1)
i2c
Z (p Z 1) 2 i m
G (p G 1)
i2
v
i2m
Z
ig4
Z
GC
GD
A
i
v
W+
(p 1)
GB
(p 2)
Z
(FR.16)h
A (p A 2)
(FR.17)h
Z (p Z 2)
(FR.18)h
Z
(FR.19)h
W
(FR.20)h
G (p G 2)
(FR.14)
Z
(FR.15)
GA
123
416 Page 34 of 45 Eur. Phys. J. C (2016) 76 :416
t +a 3+2a 13
(p Ap )p + (p Ap +)p
2a 2
g
4
+b W+4t b 1b 12
t +
2a 1
b 5+2b 17
g ((p +p )p A (p Ap )p + )+g
s
a W4a 12
[g p Z +g p Z ]
[g p A +g p A ]
(p Zp )p + (p Zp +)p
(p 3p 2)p 1 (p 3p 1)p 2
c
2b 4
[p + p ]a 14
A(p h 1+p h 2)
]
p A p + p p A p + p
b Bt2
](a 8+2a 22)
p Z p + p p Z p + p
(p Z 1+p Z 2)(p h 1+p h 2)g
2igc
v
ig
4v2[p
]
Z2
Z1p
g ((p +p )p Z (p Zp )p + )+g
](4b 1c 2 +(4b 12+b Bb W)s 2 )+
p 3 p 1 p 2 p 3 p 1 p 2
h2
g2
2vc
2
(b W4b 12)
p Z 1 p Z 2 p
ga 3
2va WWW
g (p 2p 1) +g (p 1p 3) +g (p 3p 2)
h1p
4c
+p
[g (p +p ) +g p +g p + ]
g ((p 1p 2)p 3 (p 3p 2)p 1 )+g
2ie
v
a W+
h2
[g (p +p ) +g p +g p + ]a W
ga 18
2
h1p
g
4(a 32a 2t +2a 13)+a W4a 12+2a 1t
[p
2a 17+
b B4t b 1+t
v2[g
Z2(p h 1+p h 2)
ig2
[g p h g p h ]a 5+
(p Ap +)p (p +p )p A +
](2b 4+t (b 5+2b 17))
2c2
42v2c2
v ie2
4vc [g p h g p h ] a 5 +
(p Zp +)p (p +p )p Z +
48is
(p 3p 1)p 2 (p 1p 2)p 3 +
(1+ b C 2(b T c T ))+i
g p h 1p h 2(a 20+2a 21)+
1 s2
]
A2
+p
A1p
Z
Ap
ABC
p A 1 p A 2 p
Z1(p h 1+p h 2)
2va WWW
(p A p Z )p
p A(p h 1+p h 2)g
(p 1) 2 g s a G f ABC
+g
48
2va GGGf
ieg
4v
2ie
2igc
48ic
+g
+g
2 v2[g
Z
v2g
ie
4v2[p
ig2
22v2c2
2
v
v2[g
Z (p Z 1) i 2 m
+
+
+
+
(p 2)
A (p A 1)
i2c
A (p A 2)
Z (p Z 2)
W+
W
W+
W
GA
GB
A
i
Z
(p 3)
h
h
h
h
h
h
(FR.21)
A
(FR.22)
Z
(FR.23)
GC
(FR.24)
h
(FR.25)
h
(FR.26)
h
123
Eur. Phys. J. C (2016) 76 :416 Page 35 of 45 416
22v2g p h 1p h 2a 20
ig2
]b 5+
P L
(p ++p )(p h 1+p h 2)g
P L
+nQ
7)
7)
]b W
+nQ
+p
2nQ
+2nQ
5
5
(p + p )2p
P R +
5
P R
5
1
1
2
p Z
2 8)
p Z
32s
8)
+nQ
34s
6
+nQ
6
2
2
2
g 2+
30)t
6 30)t
2nQ
6
nQ
2
g 2+
+2nQ
+nQ
i
v2[2g
29
29
32s
+
(nQ
34s
(nQ
36)2(nQ
2
4
2
]a 8
]b G
36)+2(nQ
2
1 + g 1 +
3 2s2
3 s2
M2
W
M2
W
+p
h1p
G1p
1 + g 1 +
h2
G2
g 2
+nQ
g 2
+nQ
1+ b C+
+(p h 1+p h 2)
+p
+2nQ
1 s2
1 s2
34
h2
h1p
p G 1 p G 2 p
34
2nQ
1 + g 1
s2
(nQ
33
1 + g 1
2s2
+
33
42v2[p
1
2
(nQ
2
v2g
ig
4v2[p
(1)
D
(1)
U
1
2
ig2
2
2
Zc
v2[g
i
2Y
i
2Y
ig
c
s2
4c
ig
c
2s2
+4c
2
+
i2m
+
+
G (p G 1)
i2
G (p G 2)
d
u
W+
W
d
u
Singlequarkcurrent
h
h
(FR.29)h
d
(FR.30)h
u
(FR.31)Z
d
(FR.32)Z
u
(FR.27)
h
(FR.28)
h
FR:Fermionic
123
416 Page 36 of 45 Eur. Phys. J. C (2016) 76 :416
5
h
5
p
p h
14)
24)
5
5
+(na)Q
+(na)Q
p W
p W
12
23
36)
36)
2(na)Q
2(na)Q
nQ
nQ
+8(nQ
+8(nQ
ig
vc ((na)Q
v((na)Q
33
33
11
21
igc
p W
P R
5
p W
8)
P R 16nQ
35
35
P R +16nQ
+(na)Q
p Z
2
30)t
5
2(na)Q
(na)Q
6
8)
8)
5
5
5
nQ
nQ
p A
29
p A
36)+2((na)Q
36)
2
2(nQ
2
2(nQ
2
36)
p G
+nQ
p G
nQ
10)
10)
+
+
32)(A ) BC
(naa )Q
+(naa )Q
P L +((na)Q
34
+2nQ
34
32)(A ) BC
7)
nQ
7)
nQ
+2nQ
+(na)Q
33
nQ
+nQ
7)
8i
v2 p h 1p h 2((naa )Q
9
9
33
1+ g W+2(nQ
1
nQ
+nQ
1+ g W+2(nQ
1
A ) BC +16g s (nQ
8i
v2 p h 1p h 2((naa )Q
31
31
+(na)Q
34
+2nQ
30
2nQ
30
2(na)Q
5
29
12(2nQ
+6(2nQ
29
A ) BC +16g s (nQ
2(na)Q
((na)Q
33
P L
P L
(2)
D
(2)
D
((na)Q
1
8igc
v
vY
vY
ig
2
ig
2
2ie
3
2 i
2 i
d 2 ig
vc
+
dC
ig s(
uC
ig s(
d
u
d
ie
3
u
d
d
u
u
d
h
h
h
(FR.33)W+
(FR.34)W
(FR.35)A
d
(FR.36)A
u
(FR.37)GA
(FR.38)GA
u
d
dB
uB
(FR.39)
h
(FR.40)
h
(FR.41)
Z
123
Eur. Phys. J. C (2016) 76 :416 Page 37 of 45 416
5
h
5
p
p h
14)
24)
h
5
h
5
+(na)Q
+(na)Q
p
p
14)
14)
(na)Q
(na)Q
p h
p h
+2(na)Q
+2(na)Q
12 23
22
22
11
v((na)Q
(na)Q
v((na)Q
(na)Q
11
ig
vc ((na)Q
11 21
v((na)Q
2 ig
p W
2 ig
p W
35
+
35 +
igc
+
P R
8(na)Q
P R
8(na)Q
P R
5 +
8)
8)
8)
+(na)Q
(na)Q
2 2 ig
v
(na)Q
2 2 ig
v
p Z
+2i(na)Q
4
5 +
4
5
5
5
2
+2(na)Q
6 30)t
+(na)Q
2i(na)Q
p A
p A
p h
p h
36)
36)
24)
36)2((na)Q
P L +((na)Q
(na)Q
(na)Q
+(na)Q
P L +((na)Q
P L +((na)Q
2
24)
29
(na)Q
2
2
34
34
((na)Q
+2(na)Q
+2(na)Q
21
21
5
5
7)
7)
+(na)Q
((na)Q
7)
(na)Q
33
+(na)Q
(na)Q
33
p G
p G
p W
p W
(na)Q
+(na)Q
+2(na)Q
+2i(na)Q
2i(na)Q
32)(A ) BC
32)(A ) BC
34
36)
36)
+2(na)Q
5
(na)Q
(na)Q
3
3
30
2(na)Q
30
+2(na)Q
33
33
33
((na)Q
(na)Q
+(na)Q
p
h
((na)Q
1
13
13
v(na)Q
8((na)Q
8((na)Q
p
h
29
((na)Q
1
((na)Q
1
v(na)Q
v(2(na)Q
v(2(na)Q
29
v((na)Q
31
31
8igc
v
2 2 ig
v
2 2 ig
2 2 ig
v
2 2 ig
v((na)Q
2 ig
v
2 ig
v
2ig
vc
8ie
8ie
+
u
dC
16g S
dB
uC
16g S
uB
u
u
d
u
d
d
d
u
u
h
h
h
h
h
h
h
(FR.42)
Z
(FR.43)
W+
(FR.44)
W
(FR.45)
A
(FR.46)
A
(FR.47)
GA
(FR.48)
GA
123
416 Page 38 of 45 Eur. Phys. J. C (2016) 76 :416
5
5
27
27
nQ
+nQ
28
5
5
28
nQ
nQ
28)
28)
2 ig 2
c2
nQ
+nQ
2 ig 2
c2
27
27
+2(nQ
+2(nQ
35
35
nQ
26
26
8 2 ig 2 c
nQ
nQ
nQ
8 2 ig 2 c
25
25
36)+nQ
36)nQ
5
5
+nQ
+nQ
34
34
36)
36)
nQ
2nQ
+2nQ
nQ
33
33
33
33
(nQ
8(nQ
8(nQ
(nQ
4 2 ig 2 c
4 2 ig 2 c
ig2
2
ig2
2
20)g
20)g
20)g +
20)g
35
g +
g
35
nQ
+nQ
+nQ
nQ
nQ
nQ
17) 5
8 2 ige
8 2 ige
19
19
17) 5
nQ
+nQ
5
+nQ
nQ
19
19
nQ
nQ
nQ
18
18
5
5
+nQ
18
18
+nQ
nQ
+(nQ
(nQ
20
20
17
17
32)f AB X(X ) CD
36)
36)
+nQ
+nQ
nQ
nQ
17
17
nQ
nQ
nQ
nQ
19
19
nQ
16
16
+nQ
16
16
33
nQ
+nQ
nQ
nQ
18
nQ
18
(nQ
(nQ
33
15
15
(nQ
(nQ
15
15
31
4 2 ige
4 2 ige
2 (nQ
2 (nQ
s
(nQ
ig2
2c2
ig2
2c2
ig2
ig2
dD
8ig2
dC
d
ig2
2 c
u
u
ig2
2 c
d
d
u
u
d
d
d
u
u
d
d
u
u
Z
Z
A
A
Z
Z
W
W
GA
(FR.49)
W+
(FR.50)
W
(FR.51)
W+
(FR.52)
W
(FR.53)
Z
(FR.54)
Z
(FR.55)
W+
(FR.56)
W+
(FR.57)
GB
123
Eur. Phys. J. C (2016) 76 :416 Page 39 of 45 416
5
9p h
P R
v (na)
2igc
2
n
2 s2
g 2+
p Z 5 +
1 s2
1 + g 1
2
12t
P R
P L
(na) 13 + 2 ( na )
2
p W
p W
14)
14)
8(n 13 2 n
8(n 13 2 n
8igc
v
5
5
h 5+
2
12s
2
5
+
3
4p A
4p
32)f AB X(X ) CD
1+ g 1+
8i
v2 p h 1p h 2(naa )
1
1
1+ g W4in
1+ g W+4in
p Z
13)
2ig
vc (na)
n
2
12t
12
+(2n
P R +
+nQ
+s
n 13 + 2 n
2 P L(1+ g 1)
2
P L
2
1
2
P L
s
(nQ
4ig
vc (na)
31
(1)
E
ie
(2)
E
2
i
2Y
ig
c
4c
vY
ig
ig
2
ig
2
2 i
uD
8ig2
uC
l+
l+
l+
l+
l+
l
l+
l
Singleleptoncurrent
GA
h
h
(FR.59)h
l
(FR.60)Z
l
(FR.61)Z
(FR.62)W+
(FR.63)W
(FR.64)A
l
l
(FR.58)
GB
(FR.65)
h
(FR.66)
Z
123
416 Page 40 of 45 Eur. Phys. J. C (2016) 76 :416
P R
P L
8p h
8p h
14)p W +(na)
14)p W +(na)
P R
P L
5
4((na) 13 2 ( na )
4((na) 13 2 ( na )
11)
11)
11)
14)+n
14)+n
+2n
10
5
2n
2n
+n
13)p A
13
13
13
2 2 ig
v
2 2 ig
v
2
(n
2
(n
P R
P L
hP R
hP L+
(na)
n 7 g 2 ( 4 c
n 7 g 2 ( 4 c
14)
14)
7)g (8n
5p
5p
2n
2n
7)g
12
v (2(na)
+n
v (na)
v (na)
13
13
(n 6 n
6
2 2 ig
(n
(n
2 2 ig
ig2
2 c
8ie
ig2
2 c
4ige 2
(n
4ige 2
ig2
2
ig2
2c2
+
l+
l
l+
l
l+
l
l+
l
l+
l
l+
l
h
h
h
Z
Z
A
A
W
Z
(FR.67)
W+
(FR.68)
W
(FR.69)
A
(FR.70)
W+
(FR.71)
W
(FR.72)
W+
(FR.73)
W
(FR.74)
W+
(FR.75)
Z
123
Eur. Phys. J. C (2016) 76 :416 Page 41 of 45 416
P L ] ab [ P L ] de
P R ] ae [ P L ] db )
DB([P L] ae[P L] db+[P R] ae[P R] db)
P L ] ab [ P L ] de
P R ] ae [ P L ] db )
P L ] ae [ P R ] db +[
Q
9)([
12)([
DB([P L] ae[P L] db+[P R] ae[P R] db)
+r
+rQ
4)([P L] ab[P L] de+[P R] ab[P R] de+[P L] ae[P L] db+[P R] ae[P R] db)
4)([P L] ab[P L] de+[P R] ab[P R] de+[P L] ae[P L] db+[P R] ae[P R] db)
11
11
+rQ
+rQ
P L ] ae [ P R ] db +[
+rQ
P L ] de )
10
10
rQ
P L ] de )
a
P R ] ae [ P R ] db )+4i(rQ
P R ] ab [ P L ] de +[
P R ] ab [ P L ] de +[
9
9
AE
a
AE
P R ] ab [
DE([P L] ab[P L] de+[P R] ab[P R] de)+
a
P R ] ab [
a
P L ] ab [ P R ] de +[
DE([P L] ab[P L] de+[P R] ab[P R] de)+
P R ] de +[
P R ] ae [ P R ] db )+4i(rQ
P R ] de +[
P L ] ab [
P L ] db )
P L ] ab [
P L ] db )
P R ] ab [ P R ] de +[
DE([
P R ] ab [ P R ] de +[
P L ] ab [ P R ] de +[
P R ] ae [
DE([
P R ] ae [
a
a
a AB
a AB
21)([
21)([
a AB
P R ] db +[
a AB
P R ] db +[
a
a
26)
26)
+rQ
+rQ
16)([
+rQ
+rQ
+rQ
+rQ
16)([
8)
25
8)
25
+rQ
3 +rQ
+rQ
20 +rQ
+rQ
3 +rQ
+rQ
20 +rQ
+rQ
P L ] ae [ P L ] db )
rQ
24
P L ] ae [
15
24
P L ] ae [
rQ
+rQ
+rQ
1 +rQ
P L ] ae [ P L ] db )
15
2 +rQ
19 +rQ
7 +rQ
2 +rQ
19 rQ
7 rQ
+rQ
23
14
23
14
+rQ
18 +rQ
6 +rQ
DB([
1 rQ
rQ
18 +rQ
6 rQ
DB([
2 i ( r Q
2 i ( r Q
a
17
22
+4i(rQ
+8i(rQ
+2i(rQ
+8i(rQ
+8i(rQ
13
17
AE
a
5
+2i(rQ
22
13
162
2
+[
+8i(rQ
AE
+4i(rQ
5
a
a
+
162
2
+[
+
uD
d
uE
e
dD
d
dE
e
uA
a
dA
a
Fourquarks
(FR.76)
uB
b
(FR.77)
dB
b
123
416 Page 42 of 45 Eur. Phys. J. C (2016) 76 :416
P L ] de
P L ] ab [ P L ] de )
P R ] ab [
P L ] ae [ P L ] db )
[
1 2
12)([
P R ] ab [ P L ] de
DB([P L] ae[P L] db+[P R] ae[P R] db)
DE
a
AB
2([P L] ae[P L] db+[P R] ae[P R] db)
P L ] ab [ P L ] de +[
rQ
P R ] ae [ P L ] db )
a
26)
rQ
25
11
rQ
21)[
rQ
rQ
rQ
24
P R ] ae [ P R ] db )+4i(rQ
P L ] ae [ P R ] db +[
9
a
20
rQ
AE
23
a
+rQ
rQ
19
6
5)([
22
+rQ
18
+r
4)([P L] ab[P L] de+[P R] ab[P R] de)+4irQ
P L ] ab [ P R ] de +2i(rQ
DE([P L] ab[P L] de+[P R] ab[P R] de)+16irQ
1([P L] ab[P L] de+[P L] ae[P L] db+[P R] ab[P R] de+[P R] ae[P R] db)
P R ] de +8i(rQ
P L ] ae [ P L ] db )
P L] de
4
17
P L ] db )
r
2
16([
P R ] ab [ P R ] de )+8irQ
P R ] ae [ P L ] db )
P L ] ab [
P R ] ae [
[
P R ] ae [ P R ] db )+4i(r
P R ] ab [ P L ] de +[
P R ] ab [
P R ] db +[
DE
7)[
AB
21)[
P L ] ab [ P L ] de +[
a
+r
P L ] ab [ P R ] de +[
a
26)
6
rQ
rQ
rQ
P L ] ae [ P R ] db +[
20
AB
P L ] ae [
P L ] ab +(r
a
rQ
25
rQ
16)([
P L ] ae [ P L ] db )
a
8)
5)([
19
rQ
24
+rQ
rQ
+rQ
DB([
P R ] ab [ P R ] de +[
+r
5)[
rQ
a
2
rQ
18
AE
4
r
rQ
rQ
6 rQ
7)([
15
23
a
+r
2 i ( r Q
1
12([
21([
26
2(r
2
2
13
17
+4i(rQ
+8irQ
+2i(rQ
+8i(rQ
+8i(rQ
+16irQ
22
r
+4irQ
2 ir
3([
6
4 i ( r
2 i
5
162
2
162
2
+4ir
+2i(r
dD
d
dE
e
l
d
l+
e
d
162
2
e
d
162
2
e
uA
a
l+
a
a
l+
a
(FR.78)
uB
b
Fourleptons
(FR.79)
l
b
(FR.80)
b
(FR.81)
l
b
123
Eur. Phys. J. C (2016) 76 :416 Page 43 of 45 416
P L] de
P L] de
P R ] ab [
P R ] ab [
P R ] ab [ P R ] de
P R ] ab [ P L ] de
P R ] ab [ P R ] de
P R ] ab [ P L ] de
P R ] ab [ P L ] de
P R ] ae [ P L ] db )
20)[
20)[
+rQ
rQ
[
20
P L ] ab [ P L ] de +4irQ
19
19
20)[
20)[
+rQ
rQ
+rQ
rQ
rQ
+rQ
P R ] ae [ P L ] db
P L ] ae [ P R ] db +[
18
18
+rQ
rQ
19
19
P L ] ab [ P L ] de +2i(rQ
P L ] ab [ P L ] de +2i(rQ
8 rQ
8 +rQ
16
16
+rQ
+rQ
18
23)[
rQ
P R ] ae [ P L ] db )
18
+rQ
14
14
P L ] ab +(rQ
P L ] ab +(rQ
rQ
5)([P L] ab[P L] de+[P R] ab[P R] de)
5)([P L] ab[P L] de+[P R] ab[P R] de)
[
P L ] ab [ P R ] de +2i(rQ
6)([P L] ae[P L] db+[P R] ae[P R] db)
16
16
22
rQ
23([
rQ
13 +rQ
14
P L ] ae [ P R ] db +[
5([P L] ab[P L] de+[P R] ab[P R] de)+4irQ
14
21
6)[P R] ae[P R] db+i(rQ
13)[
13)[
+rQ
rQ
P L ] ab [ P R ] de +2i(rQ
13)[
13)[
rQ
+rQ
12
12
12
12
+rQ
rQ
rQ
+rQ
11
11
rQ
11
11
rQ
+rQ
23)([
+rQ
+rQ
rQ
9
17)[
+rQ
9
+rQ
rQ
rQ
+rQ
rQ
+rQ
3 +rQ
9 +rQ
3 rQ
9 rQ
4
rQ
(rQ
7
+rQ
4
22
1 rQ
(rQ
7
i ( r Q
+2i(rQ
+2i(rQ
+2irQ
1 7 15 i ( r Q
+2i(rQ
+2i(rQ
2
+i(rQ
7 15 2 21 2 i
2 i
2 ir Q
d 16 2
2
162
2
162
2
+i(rQ
+i(rQ
l
d
l+
e
l
d
l+
e
d
162
2
e
d
162
2
e
l+
e
Twoquark-twoleptons
u a
d a
u a
d a
d a
(FR.82)
u b
(FR.83)
d b
(FR.84)
u b
(FR.85)
d b
(FR.86)
u b
123
416 Page 44 of 45 Eur. Phys. J. C (2016) 76 :416
References
1. ATLAS Collaboration, G. Aad et al., Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B 716, 129 (2012). http://arxiv.org/abs/1207.7214
Web End =arXiv:1207.7214
2. CMS Collaboration, S. Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC.Phys. Lett. B 716, 3061 (2012). http://arxiv.org/abs/1207.7235
Web End =arXiv:1207.7235
3. F. Englert, R. Brout, Broken symmetry and the mass of gauge vector mesons. Phys. Rev. Lett. 13, 321323 (1964)
4. P.W. Higgs, Broken symmetries, massless particles and gauge elds. Phys. Lett. 12, 132133 (1964)
5. P.W. Higgs, Broken symmetries and the masses of gauge bosons.
Phys. Rev. Lett. 13, 508509 (1964)
6. W. Buchmuller, D. Wyler, Effective Lagrangian analysis of new interactions and avor conservation. Nucl. Phys. B 268, 621653 (1986)
7. B. Grzadkowski, M. Iskrzynski, M. Misiak, J. Rosiek, Dimension-six terms in the standard model Lagrangian. JHEP 10, 085 (2010). http://arxiv.org/abs/1008.4884
Web End =arXiv:1008.4884
8. D.B. Kaplan, H. Georgi, SU(2) U(1) breaking by vacuum mis-
alignment. Phys. Lett. B 136, 183 (1984)9. D.B. Kaplan, H. Georgi, S. Dimopoulos, Composite Higgs scalars.
Phys. Lett. B 136, 187 (1984)
10. T. Banks, Constraints on SU(2) U(1) breaking by vacuum mis-
alignment. Nucl. Phys. B 243, 125 (1984)11. K. Agashe, R. Contino, A. Pomarol, The minimal composite Higgs model. Nucl. Phys. B 719, 165187 (2005). http://arxiv.org/abs/hep-ph/0412089
Web End =arXiv:hep-ph/0412089
12. B. Gripaios, A. Pomarol, F. Riva, J. Serra, Beyond the minimal composite Higgs model. JHEP 04, 070 (2009). http://arxiv.org/abs/0902.1483
Web End =arXiv:0902.1483
13. E. Halyo, Technidilaton or Higgs? Mod. Phys. Lett. A 8, 275284 (1993)
14. W.D. Goldberger, B. Grinstein, W. Skiba, Distinguishing the Higgs boson from the dilaton at the large hadron collider. Phys. Rev. Lett. 100, 111802 (2008). http://arxiv.org/abs/0708.1463
Web End =arXiv:0708.1463
15. T. Appelquist, C.W. Bernard, Strongly interacting Higgs bosons.
Phys. Rev. D 22, 200 (1980)
16. A.C. Longhitano, Heavy Higgs bosons in the WeinbergSalam model. Phys. Rev. D 22, 1166 (1980)
17. A.C. Longhitano, Low-energy impact of a heavy Higgs boson sector. Nucl. Phys. B 188, 118 (1981)
18. F. Feruglio, The chiral approach to the electroweak interactions.
Int. J. Mod. Phys. A 8, 49374972 (1993). http://arxiv.org/abs/hep-ph/9301281
Web End =arXiv:hep-ph/9301281
19. S. Weinberg, Phenomenological Lagrangians. Physica A 96, 327 (1979)
20. B. Grinstein, M. Trott, A HiggsHiggs bound state due to new physics at a TeV. Phys. Rev. D 76, 073002 (2007). http://arxiv.org/abs/0704.1505
Web End =arXiv:0704.1505
21. R. Contino, C. Grojean, M. Moretti, F. Piccinini, R. Rattazzi, Strong double Higgs production at the LHC. JHEP 05, 089 (2010). http://arxiv.org/abs/1002.1011
Web End =arXiv:1002.1011
22. R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin, J. Yepes, The effective chiral Lagrangian for a light dynamical Higgs particle. Phys.Lett. B 722 (2013) 330335. http://arxiv.org/abs/1212.3305
Web End =arXiv:1212.3305 [Erratum: Phys. Lett.B 726, 926 (2013)]
23. R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin, J. Yepes, Flavor with a light dynamical Higgs particle. Phys. Rev. D 87(5), 055019 (2013). http://arxiv.org/abs/1212.3307
Web End =arXiv:1212.3307
24. I. Brivio, T. Corbett, O. boli, M. Gavela, J. Gonzlez-Fraile et al., Disentangling a dynamical Higgs. JHEP 1403, 024 (2014). http://arxiv.org/abs/1311.1823
Web End =arXiv:1311.1823
25. M.B. Gavela, J. Gonzlez-Fraile, M.C. Gonzlez-Garca, L. Merlo,S. Rigolin, J. Yepes, CP violation with a dynamical Higgs. JHEP 10, 44 (2014). http://arxiv.org/abs/1406.6367
Web End =arXiv:1406.6367
26. G. Buchalla, O. Cat, C. Krause, Complete electroweak chiral Lagrangian with a light Higgs at NLO. Nucl. Phys. B 880, 552573 (2014). http://arxiv.org/abs/1307.5017
Web End =arXiv:1307.5017
27. J. Yepes, Spin-1 resonances in a non-linear left-right dynamical Higgs context. http://arxiv.org/abs/1507.03974
Web End =arXiv:1507.03974
28. J. Yepes, R. Kunming, J. Shu, CP violation from spin-1 resonances in a left-right dynamical Higgs context. Commun. Theor. Phys. 66, 93103 (2016). http://arxiv.org/abs/1507.04745
Web End =arXiv:1507.04745
29. F. Feruglio, B. Gavela, K. Kanshin, P.A.N. Machado, S. Rigolin,S. Saa, The minimal linear sigma model for the Goldstone Higgs. JHEP 06, 038 (2016). http://arxiv.org/abs/1603.05668
Web End =arXiv:1603.05668 30. R. Alonso, I. Brivio, B. Gavela, L. Merlo, S. Rigolin, Sigma decomposition. JHEP 12, 034 (2014). http://arxiv.org/abs/1409.1589
Web End =arXiv:1409.1589
31. I.M. Hierro, L. Merlo, S. Rigolin, Sigma decomposition: the CP-odd Lagrangian. JHEP 04, 016 (2016). http://arxiv.org/abs/1510.07899
Web End =arXiv:1510.07899
32. I. Brivio, O.J.P. boli, M.B. Gavela, M.C. Gonzlez-Garcia, L. Merlo, S. Rigolin, Higgs ultraviolet softening. JHEP 12, 004 (2014). http://arxiv.org/abs/1405.5412
Web End =arXiv:1405.5412
33. I. Brivio, M.B. Gavela, L. Merlo, K. Mimasu, J.M. No, R. del Rey,V. Sanz, Non-linear Higgs portal to dark matter. JHEP 04, 141 (2016). http://arxiv.org/abs/1511.01099
Web End =arXiv:1511.01099 34. H. Murayama, V. Rentala, J. Shu, Probing strong electroweak symmetry breaking dynamics through quantum interferometry at the LHC. Phys. Rev. D 92(11), 116002 (2015). http://arxiv.org/abs/1401.3761
Web End =arXiv:1401.3761
35. R.L. Delgado, A. Dobado, F.J. Llanes-Estrada, One-loop WL WL and ZL ZL scattering from the electroweak Chiral Lagrangian with a light Higgs-like scalar. JHEP 02, 121 (2014). http://arxiv.org/abs/1311.5993
Web End =arXiv:1311.5993
36. R.L. Delgado, A. Dobado, M.J. Herrero, J.J. Sanz-Cillero, One-loop W+L WL and ZL ZL from the electroweak
Chiral Lagrangian with a light Higgs-like scalar. JHEP 07, 149 (2014). http://arxiv.org/abs/1404.2866
Web End =arXiv:1404.2866 37. R. Alonso, M.B. Gavela, L. Merlo, S. Rigolin, J. Yepes, Minimal avour violation with strong Higgs dynamics. JHEP 06, 076 (2012). http://arxiv.org/abs/1201.1511
Web End =arXiv:1201.1511
38. B.M. Gavela, E.E. Jenkins, A.V. Manohar, L. Merlo, Analysis of general power counting rules in effective eld theory. http://arxiv.org/abs/1601.07551
Web End =arXiv:1601.07551
39. A. Manohar, H. Georgi, Chiral quarks and the nonrelativistic quark model. Nucl. Phys. B 234, 189 (1984)
40. A.G. Cohen, D.B. Kaplan, A.E. Nelson, Counting 4 Pis in strongly coupled supersymmetry. Phys. Lett. B 412, 301308 (1997). http://arxiv.org/abs/hep-ph/9706275
Web End =arXiv:hep-ph/9706275
41. A. Biektter, A. Knochel, M. Krmer, D. Liu, F. Riva, Vices and virtues of Higgs effective eld theories at large energy. Phys. Rev. D 91, 055029 (2015). http://arxiv.org/abs/1406.7320
Web End =arXiv:1406.7320
42. R. Contino, A. Falkowski, F. Goertz, C. Grojean, F. Riva, On the validity of the effective eld theory approach to SM precision tests. http://arxiv.org/abs/1604.06444
Web End =arXiv:1604.06444
43. H. Georgi, D.B. Kaplan, L. Randall, Manifesting the invisible axion at low-energies. Phys. Lett. B 169, 73 (1986)
44. L. Merlo, S. Saa, M. Sacristan, B and L non-conserving effective Lagrangian for a dynamical Higgs (to appear)
45. K. Hagiwara, S. Ishihara, R. Szalapski, D. Zeppenfeld, Low-energy effects of new interactions in the electroweak boson sector. Phys. Rev. D 48, 21822203 (1993)
46. K. Hagiwara, T. Hatsukano, S. Ishihara, R. Szalapski, Probing nonstandard bosonic interactions via W boson pair production at lepton colliders. Nucl. Phys. B 496, 66102 (1997). http://arxiv.org/abs/hep-ph/9612268
Web End =arXiv:hep-ph/9612268
47. T. Corbett, O.J.P. boli, J. Gonzlez-Fraile, M.C. Gonzlez-Garcia, Constraining anomalous Higgs interactions. Phys. Rev. D 86, 075013 (2012). http://arxiv.org/abs/1207.1344
Web End =arXiv:1207.1344
48. T. Corbett, O.J.P. boli, J. Gonzlez-Fraile, M.C. Gonzlez-Garcia, Robust determination of the Higgs couplings: power to the data. Phys. Rev. D 87, 015022 (2013). http://arxiv.org/abs/1211.4580
Web End =arXiv:1211.4580
123
Eur. Phys. J. C (2016) 76 :416 Page 45 of 45 416
49. T. Corbett, O.J.P. boli, J. Gonzlez-Fraile, M.C. Gonzlez-Garcia, Determining triple gauge boson couplings from Higgs data. Phys.Rev. Lett. 111, 011801 (2013). http://arxiv.org/abs/1304.1151
Web End =arXiv:1304.1151
50. Particle Data Group Collaboration, K.A. Olive et al., Review of particle physics. Chin. Phys. C 38, 090001 (2014)
51. ATLAS, CMS Collaboration, G. Aad et al., Combined measurement of the Higgs boson mass in pp collisions at s = 7 and
8 TeV with the ATLAS and CMS experiments. Phys. Rev. Lett. 114, 191803 (2015). http://arxiv.org/abs/1503.07589
Web End =arXiv:1503.07589 52. M.E. Peskin, T. Takeuchi, A new constraint on a strongly interacting Higgs sector. Phys. Rev. Lett. 65, 964967 (1990)
53. M.E. Peskin, T. Takeuchi, Estimation of oblique electroweak corrections. Phys. Rev. D 46, 381409 (1992)
54. A. Pomarol, F. Riva, Towards the ultimate SM t to close in on Higgs physics. JHEP 01, 151 (2014). http://arxiv.org/abs/1308.2803
Web End =arXiv:1308.2803
55. M. Ciuchini, E. Franco, S. Mishima, M. Pierini, L. Reina, L. Silvestrini, Update of the electroweak precision t, interplay with Higgs-boson signal strengths and model-independent constraints on new physics, in International Conference on High Energy Physics 2014 (ICHEP 2014) Valencia, Spain, July 29, 2014 (2014). http://arxiv.org/abs/1410.6940
Web End =arXiv:1410.6940
56. SLD Electroweak Group, DELPHI, ALEPH, SLD, SLD Heavy Flavour Group, OPAL, LEP Electroweak Working Group, L3 Collaboration, S. Schael et al., Precision electroweak measurements on the Z resonance. Phys. Rep. 427 (2006) 257454. http://arxiv.org/abs/hep-ex/0509008
Web End =arXiv:hep-ex/0509008
57. CDF, D0 Collaboration, T.E.W. Group, 2012 Update of the combination of CDF and D0 results for the mass of the W boson. http://arxiv.org/abs/1204.0042
Web End =arXiv:1204.0042
58. Tevatron Electroweak Working Group, CDF, DELPHI, SLD Electroweak and Heavy Flavour Groups, ALEPH, LEP Electroweak Working Group, SLD, OPAL, D0, L3 Collaboration, L.E.W. Group, Precision electroweak measurements and constraints on the standard model. http://arxiv.org/abs/1012.2367
Web End =arXiv:1012.2367
59. G. Isidori, A.V. Manohar, M. Trott, Probing the nature of the Higgs-like boson via h V F decays. Phys. Lett. B 728, 131135 (2014).
http://arxiv.org/abs/1305.0663
Web End =arXiv:1305.0663 60. G. Isidori, M. Trott, Higgs form factors in associated production.
JHEP 02, 082 (2014). http://arxiv.org/abs/1307.4051
Web End =arXiv:1307.4051
61. M. Gonzalez-Alonso, A. Greljo, G. Isidori, D. Marzocca, Electroweak bounds on Higgs pseudo-observables and h 4 decays.
Eur. Phys. J. C 75, 341 (2015). http://arxiv.org/abs/1504.04018
Web End =arXiv:1504.04018 62. T. Corbett, O.J.P. boli, D. Goncalves, J. Gonzlez-Fraile, T. Plehn,M. Rauch, The Higgs legacy of the LHC Run I. JHEP 08, 156 (2015). http://arxiv.org/abs/1505.05516
Web End =arXiv:1505.05516 63. E. Mass, V. Sanz, Limits on anomalous couplings of the Higgs boson to electroweak gauge bosons from LEP and the LHC. Phys.Rev. D 87(3), 033001 (2013). http://arxiv.org/abs/1211.1320
Web End =arXiv:1211.1320
64. S. Banerjee, S. Mukhopadhyay, B. Mukhopadhyaya, Higher dimensional operators and the LHC Higgs data: the role of modied kinematics. Phys. Rev. D 89(5), 053010 (2014). http://arxiv.org/abs/1308.4860
Web End =arXiv:1308.4860
65. J. Ellis, V. Sanz, T. You, Complete Higgs sector constraints on dimension-6 operators. JHEP 07, 036 (2014). http://arxiv.org/abs/1404.3667
Web End =arXiv:1404.3667
66. J. Ellis, V. Sanz, T. You, The effective standard model after LHC Run I. JHEP 03, 157 (2015). http://arxiv.org/abs/1410.7703
Web End =arXiv:1410.7703
67. R. Edezhath, Dimension-6 operator constraints from boosted VBF Higgs. http://arxiv.org/abs/1501.00992
Web End =arXiv:1501.00992
68. R. Lafaye, T. Plehn, M. Rauch, D. Zerwas, M. Duhrssen, Measuring the Higgs sector. JHEP 08, 009 (2009). http://arxiv.org/abs/0904.3866
Web End =arXiv:0904.3866
69. M. Klute, R. Lafaye, T. Plehn, M. Rauch, D. Zerwas, Measuring Higgs couplings from LHC data. Phys. Rev. Lett. 109, 101801 (2012). http://arxiv.org/abs/1205.2699
Web End =arXiv:1205.2699
70. T. Plehn, M. Rauch, Higgs couplings after the discovery. Europhys. Lett. 100, 11002 (2012). http://arxiv.org/abs/1207.6108
Web End =arXiv:1207.6108
71. M. Klute, R. Lafaye, T. Plehn, M. Rauch, D. Zerwas, Measuring Higgs Couplings at a Linear Collider. Europhys. Lett. 101, 51001 (2013). http://arxiv.org/abs/1301.1322
Web End =arXiv:1301.1322
72. D. Lopez-Val, T. Plehn, M. Rauch, Measuring extended Higgs sectors as a consistent free couplings model. JHEP 10, 134 (2013). http://arxiv.org/abs/1308.1979
Web End =arXiv:1308.1979
73. T. Corbett, O.J.P. Eboli, D. Goncalves, J. Gonzalez-Fraile, T. Plehn, M. Rauch, The non-linear Higgs legacy of the LHC RunI. http://arxiv.org/abs/1511.08188
Web End =arXiv:1511.08188 74. O.J.P. Eboli, J. Gonzalez-Fraile, M.C. Gonzalez-Garcia, Scrutinizing the ZW+W- vertex at the large hadron collider at 7 TeV. Phys. Lett. B 692, 2025 (2010). http://arxiv.org/abs/1006.3562
Web End =arXiv:1006.3562
75. K. Hagiwara, R.D. Peccei, D. Zeppenfeld, K. Hikasa, Probing the weak boson sector in e+e W+W. Nucl. Phys. B 282, 253
(1987)76. A. Butter, O.J.P. boli, J. Gonzlez-Fraile, M.C. Gonzlez-Garcia, T. Plehn, The gauge-Higgs legacy of the LHC Run I. http://arxiv.org/abs/1604.03105
Web End =arXiv:1604.03105
77. K. Hagiwara, R. Szalapski, D. Zeppenfeld, Anomalous Higgs boson production and decay. Phys. Lett. B 318, 155162 (1993). http://arxiv.org/abs/hep-ph/9308347
Web End =arXiv:hep-ph/9308347
78. A. Falkowski, M. Gonzalez-Alonso, A. Greljo, D. Marzocca, Global constraints on anomalous triple gauge couplings in effective eld theory approach. Phys. Rev. Lett. 116(1), 011801 (2016). http://arxiv.org/abs/1508.00581
Web End =arXiv:1508.00581
79. A. Drozd, J. Ellis, J. Quevillon, T. You, Comparing EFT and exact one-loop analyses of non-degenerate stops. JHEP 06, 028 (2015). http://arxiv.org/abs/1504.02409
Web End =arXiv:1504.02409
80. M. Gorbahn, J.M. No, V. Sanz, Benchmarks for Higgs effective theory: extended Higgs sectors. JHEP 10, 036 (2015). http://arxiv.org/abs/1502.07352
Web End =arXiv:1502.07352
81. J. Brehmer, A. Freitas, D. Lopez-Val, T. Plehn, Pushing Higgs effective theory to its limits. Phys. Rev. D 93, 075014 (2016). http://arxiv.org/abs/1510.03443
Web End =arXiv:1510.03443
82. A. Biektter, J. Brehmer, T. Plehn, Pushing Higgs effective theory over the edge. http://arxiv.org/abs/1602.05202
Web End =arXiv:1602.05202
83. O.J.P. boli, M.C. Gonzalez-Garcia, Mapping the genuine bosonic quartic couplings. Phys. Rev. D 93, 093013 (2016). http://arxiv.org/abs/1604.03555
Web End =arXiv:1604.03555
84. G.F. Giudice, C. Grojean, A. Pomarol, R. Rattazzi, The strongly-interacting light Higgs. JHEP 06, 045 (2007). http://arxiv.org/abs/hep-ph/0703164
Web End =arXiv:hep-ph/0703164
123
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer
The European Physical Journal C is a copyright of Springer, 2016.
Abstract
(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)
The complete effective chiral Lagrangian for a dynamical Higgs is presented and constrained by means of a global analysis including electroweak precision data together with Higgs and triple gauge-boson coupling data from the LHC Run I. The operators' basis up to next-to-leading order in the expansion consists of 148 (188 considering right-handed neutrinos) flavour universal terms and it is presented here making explicit the custodial nature of the operators. This effective Lagrangian provides the most general description of the physical Higgs couplings once the electroweak symmetry is assumed, and it allows for deviations from the ...... doublet nature of the Standard Model Higgs. The comparison with the effective linear Lagrangian constructed with an exact ...... doublet Higgs and considering operators with at most canonical dimension six is presented. A promising strategy to disentangle the two descriptions consists in analysing (i) anomalous signals present only in the chiral Lagrangian and not expected in the linear one, that are potentially relevant for LHC searches, and (ii) decorrelation effects between observables that are predicted to be correlated in the linear case and not in the chiral one. The global analysis presented here, which includes several kinematic distributions, is crucial for reducing the allowed parameter space and for controlling the correlations between parameters. This improves previous studies aimed at investigating the Higgs Nature and the origin of the electroweak symmetry breaking.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer