Higgs ultraviolet softening

Abstract

Abstract

We analyze the leading effective operators which induce a quartic momentum dependence in the Higgs propagator, for a linear and for a non-linear realization of electroweak symmetry breaking. Their specific study is relevant for the understanding of the ultraviolet sensitivity to new physics. Two methods of analysis are applied, trading the Lagrangian coupling by: i) a "ghost" scalar, after the Lee-Wick procedure; ii) other effective operators via the equations of motion. The two paths are shown to lead to the same effective Lagrangian at first order in the operator coefficients. It follows a modification of the Higgs potential and of the fermionic couplings in the linear realization, while in the non-linear one anomalous quartic gauge couplings, Higgs-gauge couplings and gauge-fermion interactions are induced in addition. Finally, all LHC Higgs and other data presently available are used to constrain the operator coefficients; the future impact of pp [arrow right] 4 leptons data via off-shell Higgs exchange and of vector boson fusion data is considered as well. For completeness, a summary of pure-gauge and gauge-Higgs signals exclusive to non-linear dynamics at leading-order is included.

Published for SISSA by Springer Received: June 2, 2014

Revised: October 17, 2014 Accepted: November 10, 2014

Published: December 1, 2014

I. Brivioa O.J.P.bolib M.B. Gavelaa M.C. Gonzalez-Garcac,d,e L. Merloa and S. Rigolinf

aDepartamento de Fsica Terica and Instituto de Fsica Terica, IFT-UAM/CSIC, Universidad Autnoma de Madrid, Cantoblanco, 28049, Madrid, Spain

bInstituto de Fsica, Universidade de So Paulo,

So Paulo SP, Brazil

cC.N. Yang Institute for Theoretical Physics and Department of Physics and Astronomy, SUNY at Stony Brook, Stony Brook, NY 11794-3840, U.S.A.

dDepartament dEstructura i Constituents de la Matria and ICC-UB, Universitat de Barcelona, 647 Diagonal, E-08028 Barcelona, Spain

eInstituci Catalana de Recerca i Estudis Avanats (ICREA), Barcelona, Spain

f Dipartimento di Fisica e Astronomia G. Galilei, Universit di Padova and INFN Sezione di Padova,Via Marzolo 8, I-35131 Padua, Italy

E-mail: mailto:[email protected]

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

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

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

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

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

Web End [email protected]

Abstract: We analyze the leading e ective operators which induce a quartic momentum dependence in the Higgs propagator, for a linear and for a non-linear realization of electroweak symmetry breaking. Their specic study is relevant for the understanding of the ultraviolet sensitivity to new physics. Two methods of analysis are applied, trading the Lagrangian coupling by: i) a ghost scalar, after the Lee-Wick procedure; ii) other e ective operators via the equations of motion. The two paths are shown to lead to the same e ective Lagrangian at rst order in the operator coe cients. It follows a modication of the Higgs potential and of the fermionic couplings in the linear realization, while in the nonlinear one anomalous quartic gauge couplings, Higgs-gauge couplings and gauge-fermion interactions are induced in addition. Finally, all LHC Higgs and other data presently available are used to constrain the operator coe cients; the future impact of pp ! 4 leptons

data via o -shell Higgs exchange and of vector boson fusion data is considered as well. For completeness, a summary of pure-gauge and gauge-Higgs signals exclusive to non-linear dynamics at leading-order is included.

Keywords: Higgs Physics, Chiral Lagrangians, Technicolor and Composite Models

ArXiv ePrint: 1405.5412

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP12(2014)004

Web End =10.1007/JHEP12(2014)004

Higgs ultraviolet softening

JHEP12(2014)004

Contents

1 Introduction 1

2 Elementary Higgs: O 42.1 Analysis in terms of the LW ghost 42.2 Analysis via EOM 6

3 Light dynamical Higgs: P h 73.1 Analysis in terms of the LW ghost 83.2 Analysis via EOM 10

4 Chiral versus linear e ective operators 11

5 Signatures and constraints 125.1 E ects from O 125.2 E ects from P h and P610 13

6 Conclusions 18

A Analysis with a generic chiral potential V (h) 20

B Impact of O versus P h on ZZ ! ZZ scattering 22

C Chiral versus linear couplings 23

C.1 TGV couplings 23C.2 HVV couplings 24C.3 VVVV couplings 25

1 Introduction

A revival of interest in theories with higher derivative kinetic terms [1, 2] is taking place, as the increased momentum dependence of propagators softens the sensitivity to ultraviolet scales. Quadratic divergences are absent due to the faster fall-o of the momentum dependence of the propagators. For instance this avenue has been recently explored in view of an alternative solution to the electroweak hierarchy problem [3, 4].

Originally proposed by Lee and Wick [1, 2], a large literature followed to ascertain the eld theoretical consistency of this type of theories, in particular from the point of view of unitarity and causality. The issue is delicate as a second pole appears in the eld propagators, and this pole has a wrong-sign residue. Naively such theories are unstable and not unitary. The present understanding is that the S matrix for asymptotically free

1

JHEP12(2014)004

states may remain unitary, though, and acausality only occurs at the microscopic level while macroscopically and/or in any measurable quantity causality holds as it should.

For the computation of physical amplitudes, a modication of the usual rules to compute perturbative amplitudes was proposed [58] respecting the aforementioned desired properties. A more user-friendly eld-theory tool [3] to approach these theories consists in trading the higher derivative kinetic term by the presence of a new state with the same quantum numbers of the standard eld and quadratic kinetic energy, albeit with a wrong sign for both quadratic terms (kinetic energy and mass), i.e. a state of negative norm: a Lee-Wick (LW) partner or ghost. It corresponds to the second pole in the propagator, describing an unstable state that would thus not threaten the unitarity of the S matrix, as only the asymptotically free states participating in a scattering process are relevant for the latter.

In this paper, we focus on the study of a higher derivative kinetic term for the Higgs particle, in a model independent way. Although present Higgs data are fully consistent with the Higgs particle being part of a gauge SU(2) scalar doublet, the issue is widely open and all e orts should be done to settle it. Two main classes of e ective Lagrangians are pertinent, depending on how the Standard Model (SM) electroweak symmetry breaking (EWSB) is assumed to be realized in the presence of a light Higgs particle: linearly for an elementary Higgs particle [911] or non-linearly for a dynamical -composite- light one [1219]. The relevant couplings to be added to the SM Lagrangian will be denoted by

O = (D[notdef]D[notdef] ) (D D ) (1.1)

for linearly realized electroweak symmetry breaking (EWSB) scenarios, and

P h =

1

2 h h =

JHEP12(2014)004

1

2 (@[notdef]@[notdef]h) (@ @ h) (1.2)

if the light Higgs stems from non-linearly realized EWSB. In eq. (1.1) denotes the gauge SU(2) scalar doublet, which in the unitary gauge reads = 0, (v + h)/p2

[parenrightbig]

with v/p2

being the vacuum expectation value (vev) and h the Higgs excitation. D[notdef] stands for the

covariant derivative

D[notdef]

@[notdef] + igW[notdef] + i g[prime]2 B[notdef]

(1.3)

with W[notdef] W a[notdef](x)a/2 and B[notdef] denoting the SU(2)L and U(1)Y gauge bosons, respectively.

In equation (1.2), h denotes instead a generic scalar singlet, whose couplings are described by a non-linear Lagrangian (often dubbed chiral Lagrangian) and do not need to match those of a SU(2) doublet component.

Note that the operators O and P are but rarely [10] considered by practitioners

of e ective Lagrangian analyses, and almost never selected as one of the elements of the operator bases. They tend to be substituted instead by (a combination of) other operators which include fermionic ones because the bounds on exotic fermionic couplings are often more stringent in constraining BSM theories than those from bosonic interactions. Nevertheless, the new data and the special and profound theoretical impact of higher derivative kinetic terms deserve focalised studies, to which this paper intends to contribute.

2

In this context it is important to notice that, in order to have any impact on the hierarchy problem, the validity of the operators under study should be extrapolated into the regime E , which is beyond the usual regime where EFT description is valid. In

this sense, the SM Lagrangian with the addition of these operators can be treated as the complete Lagrangian in the ultraviolet.

Either in the linear or the non-linear realizations, the contribution to the Lagrangian of the e ective operators in eqs. (1.1) and (1.2) can be parametrised as

L = ciOi , (1.4)

with Oi [notdef]O , P h[notdef] respectively, with the parameters ci having mass dimension 2.1

The impact of O and P h appears as a correction in the propagator of the h scalar which

is quartic in four-momentum:

i

p2 m2h + ci p4

. (1.5)

This propagator has now two poles and describes thus two degrees of freedom. For instance for 1/ci m2h they are approximately located at [3]

p2 = m2h and p2 = 1/ci , (1.6)

which implies that the sign of the operator coe cient needs to obey ci < 0 in order to avoid tachyonic instabilities.

It is important to nd signals which discriminate among those two categories linear versus non-linear EWSB and this will be one of the main focuses of this paper for the higher derivative scalar kinetic terms considered. It will be shown that the e ects of the couplings in eqs. (1.1) and (1.2) di er on their implications for the gauge and gauge-Higgs sectors. The phenomenological analysis will be restricted to tree-level e ects and consistently to rst order in ci, and we will use two independent and alternative techniques, showing that they lead to the same results:

To trade the higher-derivative coupling by a LW ghost heavy particle, which is

subsequently integrated out.

To apply rst the Lagrangian equations of motion (EOM) to the operator, trading the

coupling by other standard higher-dimension e ective operators, which only require traditional elds and eld-theory methods.

Together with exploring the di erent physical e ects expected from the Higgs linear higher-derivative term O and the non-linear one P h, we will clarify their exact theoretical

relation, determining which specic combination of non-linear operators would result in the same physics impact than the linear operator O .

1From the point of view of the chiral expansion, P is a four-derivative coupling, and a slightly di erent

normalization (by a v2 factor) was adopted in ref. [20], using a dimensionless coe cient; the choice here allows to use the same notation for both expansions.

3

JHEP12(2014)004

The phenomenological analysis below includes as well a study of the impact of both operators in present and future LHC data. In the case of the LW version of the SM, it has been shown [21] that the measurements of the S and T parameters set very strong constraints on the gauge and fermionic LW partner masses, which need to exceed several TeV; this implies a sizeable tension with the issue of the electroweak hierarchy problem, as the LW partners induce a nite shift in the Higgs mass proportional to their own masses. On the contrary, the EW constraints are mild for the Higgs doublet LW partners, whose impact may be within LHC reach [22]. We explore the experimental prospects for O and

P h at rst order in the e ective operator coe cients, focusing only on the quark sector for simplicity as the extension to the lepton sector is straightforward.

The structure of the manuscript can be easily inferred from the table of Contents.

2 Elementary Higgs: O

The quark-Higgs sector of the SM Lagrangian supplemented by O will be considered in

this section:

L = (D[notdef] )D[notdef] [parenleftBig]

qL ~

YUuR + qL YDdR + h.c.

[parenrightBig]

+ c O V ( ) , (2.1)

where ~

i2 , and the Standard Model potential,

V ( ) =

v2 2

JHEP12(2014)004

2, (2.2)

can be rewritten for future convenience in the unitary gauge in terms of the Higgs particle mass, m2h = 2 v2 and the Higgs doublet vev [angbracketleft] [angbracketright] = v/p2 as

V (h) = m2h2 h2 +

m2h

2v h3 +

m2h8v2 h4 . (2.3)

2.1 Analysis in terms of the LW ghost

The Lee-Wick method for the case of a complex scalar doublet is applied next to the analysis of the operator O in eqs. (1.1) and (1.4), following ref. [3]. Dening an auxiliary

complex SU(2) doublet ', eq. (2.1) can be rewritten as a two-scalar-eld Lagrangian:

L = (D[notdef] )D[notdef] + (D[notdef]')D[notdef] + (D[notdef] )D[notdef]'

qL ~ YUuR + qL YDdR + h.c.

[parenrightBig] 1c

'' V ( ) .

(2.4)

The mass squared term for the auxiliary eld is given by 1/c , which requires c < 0

to avoid a tachyonic resonance. The kinetic energy terms can now be diagonalised via the simple eld redenitions ! [prime] '[prime], ' ! '[prime], and the mass terms can be diagonalised by

a subsequent symplectic rotation given by:

[prime]

'[prime]

[parenrightBigg]

= cosh sinh

sinh cosh

[parenrightBigg][parenleftBigg][parenleftBigg]

[prime][prime]

'[prime][prime]

[parenrightBigg]

, (2.5)

4

where

1 + 2x , with x c m2h/2 . (2.6)

Finally, dropping the primes on the eld notation, the scalar Lagrangian in eq. (2.4) can be rewritten as

L', =(D[notdef] )D[notdef] (D[notdef]')D[notdef]' + L'Y V ( , ') (2.7)

with

L'Y = (1 + x) [parenleftBig]

qL(~

~

')YUuR + qL( ')YDdR + h.c.

tanh 2 = 2x

JHEP12(2014)004

[parenrightBig]

, (2.8)

V ( , ') =

m2h 2

1 x +1 x

''

m2h2 (1 x)

( ')( ')[parenrightBig]2, (2.9)

expanded at order x, assuming small x values. The location of the minimum of the Higgs potential gets c corrections. For instance, for a BSM scale large compared with the

Higgs mass (i.e. x ! 0), the approximate location of the vacuum corresponds to:

! [angbracketleft] [angbracketright] +

+ m2h

2v2 (1 4x)

h

p2 , [angbracketleft] [angbracketright] =

v p2 [parenleftbigg]

1 + 15

2 x2[parenrightbigg]

+ O(x3) , (2.10)

v

p2(1 2x) + O(x3) , (2.11)

where h and [vector] are the eld excitation over the potential minima, and the exact potential has been retaken and terms up to x2 considered. In consequence, at leading order in c

the minimum of the Higgs potential remains unchanged. For the sake of comparison with the non-linear case in the next section, it is useful to write explicitly the potential restricted to the h and [vector] elds. After a further necessary diagonalization of the h and [vector] dependence, their scalar potential reads at rst order in x:

V (h, [vector]) = m2h2 (1 + 2x)h2 +

m2h 2

' ! [angbracketleft]'[angbracketright] +

[vector]

p2 , [angbracketleft]'[angbracketright] = x

1 + 2x 1 2x

[vector]2 + m2h2v (1 + 6x)(h [vector])3

(2.12)

+ m2h

8v2 (1 + 8x)(h [vector])4 .

Eqs. (2.7) and (2.12) illustrate that for small x the [vector] state exhibits a wrong sign in both the kinetic energy and the mass terms.

Integrating out the heavy scalar. At rst order in the operator coe cient c , the mixing in eq. (2.6) may be approximated by tanh 2x = c m2h, and the e ect of the

negative-norm heavy eld described by ' with absolute mass [notdef]c1 [notdef] can be integrated

out via its EOM:

'i = c

dRY DqL,i + qL,j"jiYUuR +m2h

v2 ( ) i[parenrightbigg]+ O c2

[parenrightbig]

, (2.13)

5

Throughout the paper we will work on the so-called Z-scheme of renormalization, in which the ve relevant electroweak parameters of the SM Lagrangian (neglecting fermion masses), gs, g, g[prime], v and the h self-coupling, are xed from the following ve observables: the world average value of s [23], the Fermi constant GF as extracted from muon decay [23], em extracted from Thomson scattering [23], mZ as determined from the Z lineshape at LEP I [23], and mh from the present LHC measurement [24, 25]. Eq. (2.13) above indicates that O will impact the renormalised fermion masses and the Higgs sector parameters.

Specically for the latter, while the electroweak vev v (p2GF )1/2 is not corrected, the

Higgs mass renormalization must absorb a correction

m2h m2h

= 2x . (2.14)

The resulting renormalized e ective Lagrangian reads (omitting again fermionic and gauge kinetic terms):

L = (D[notdef] ) D[notdef] + LY + L4F V , (2.15)

where

LY = [bracketleftBig]

qL ~

YUuR + qL YDdR + h.c.

[bracketrightBig] [parenleftbigg]

1 x

1 2

v2[parenrightbigg][parenrightbigg]unitary gauge

!

JHEP12(2014)004

v + h p2

LYUuR + dLYDdR + h.c.

[bracketrightbig] [parenleftbigg]1 + x

v2 (h2 + 2hv)[parenrightbigg]

, (2.16)

+ (RY UdL)( dLYUuR) + (RY UuL)(LYUuR)

+ (LYDdR)( dRY DuL) + ( dLYDdR)( dRY DdL)

+ n(LYUuR)( dLYDdR) (dLYUuR)(LYDdR) + h.c.

[bracerightBig][bracketrightBig]

, (2.17)

L4F = x

2 m2h

[bracketleftBig]

m2h

V =

2 (1 3x) +

m2h2v2 (1 6x)

2+ 2xm2h v4

3unitary gauge !

3h5 + 12v h6[parenrightbigg]

. (2.18)

It follow deviations from SM expectations in fermion-Higgs couplings, four-fermion interactions and scalar properties; in particular, the relation between the Higgs self-couplings and its mass is di erent from the SM one; this fact can be directly probed at the LHC and ILC [26]. Moreover, the Higgs potential exhibits now h5 and h6 terms not present in the SM, which require c < 0 for stability, consistently with the arguments given in the

Introduction. Note as well that, for the linear realization of EWSB under discussion, the couplings involving gauge particles are not modied with respect to their SM values.

2.2 Analysis via EOM

An avenue alternative to the LW method when working at rst order in the operator coe cient, and one which involves only standard elds and standard eld theory rules, is

6

m2h

m2h

2 h2 +

2v (1 + 4x) h3 +

m2h8v2 (1 + 24x) h4 + x

m2h 2v3

V[LParen1] [CapPhi] [CapPhi][RParen1]

v 2c [CapPhi] [Equal][Minus]10[Minus]3

[CapPhi] [CapPhi]

JHEP12(2014)004

[Minus] v

2

v

2

v 2c [CapPhi] [Equal]10[Minus]3

Figure 1. The scalar potential in the linear Lagrangian for di erent values of the coe cient c v2. The solid red line denotes the SM and the interline spacing is (v2c ) = 7.5 [notdef] 105.

to apply directly the EOM for the eld to the operator O in eq. (2.1):

i =

V ( ) i [parenleftBig]

dRY DqL,i + qL,j"jiYUuR

[parenrightBig]

, (2.19)

i = i

V

( ) [parenleftBig]

RY U"ijqL,j + qL,iYDdR

[parenrightBig]

. (2.20)

We have checked that this method leads to the same low-energy renormalized e ective Lagrangian than that in eqs. (2.15)(2.18), obtained via the Lee-Wick procedure involving a ghost eld.

Higgs potential. Figure 1 shows the dependence of the scalar potential on c : the points [notdef] [notdef] = [notdef]v/p2, corresponding to the SM vacuum, switch from stable minima to

maxima as c runs from negative to positive values. The location of Higgs vev for negative c is not modied at this order, see eq. (2.10).

3 Light dynamical Higgs: P h

This section deals with the alternative scenario of a light dynamical Higgs, whose CP-even bosonic e ective Lagrangian has been discussed in refs. [17, 20]. For simplicity and focus, the leading-order Lagrangian will be taken to be that of the SM modied only by the action of the operator P h in eq. (1.2). The scalar potential will thus be assumed as well

to take the SM form for h, to facilitate comparison with the linear case; nevertheless, in appendix A we discuss the extension to the case of a completely general potential for a singlet scalar eld h, showing that the conclusions obtained below are maintained.

7

The quark-Higgs sector of the Lagrangian reads then

L =

1

2@[notdef]h@[notdef]h

QLUYQR + h.c.) + c hP h V (h) , (3.1)

where V (h) takes the functional form in eq. (2.3). V[notdef] (D[notdef]U) U, where U(x) is a

dimensionless unitary matrix describing the longitudinal degrees of freedom of the EW gauge bosons:

U(x) = eiaa(x)/v , U(x) ! L U(x)R , (3.2) where L, R denotes SU(2)L,R global transformations, respectively. V[notdef] is thus a vector chiral eld belonging to the adjoint of the global SU(2)L symmetry, and the covariant derivative reads

D[notdef]U(x) @[notdef]U(x) + igW[notdef](x)U(x)

(v + h)2

4 Tr[V[notdef]V[notdef]]

v + h p2 (

JHEP12(2014)004

ig[prime]

2 B[notdef](x)U(x)3 . (3.3)

Note that eq. (3.1) is simply the SM Lagrangian written in chiral notation, but for the additional presence of the P h coupling.

3.1 Analysis in terms of the LW ghost

In parallel to the analysis in section (2.1), for c h < 0 the action of the operator P h in

the Lagrangian eq. (3.1) can be traded for that of an auxiliary SM singlet scalar eld [vector], and the Lagrangian in eq. (3.1) reads then

L =

1

2@[notdef]h@[notdef]h + @[notdef]h@[notdef][vector]

QLUYQR + h.c.) V (h, [vector]) , (3.4)

where the non-scalar kinetic terms were omitted and (see appendix A)

V (h, [vector]) = m2h2 h2 +

(v + h)2

4 Tr[V[notdef]V[notdef]]

v + h p2 (

m2h

2v h3 +

m2h8v2 h4 +

1

2c h

[vector]2 . (3.5)

The kinetic energy terms are diagonalised via the eld redenitions h ! h[prime] [vector][prime], [vector] ! [vector][prime], and the mass terms can be then diagonalised by a subsequent symplectic rotation analogous to that in eq. (2.5) (with and ' replaced by h and [vector], respectively), with a mixing angle given by

, with x c h m2h/2 . (3.6)

Finally, dropping the primes on the eld notation and omitting again fermionic and gauge kinetic terms, the Lagrangian reads:

Lh,[notdef] =

1

2@[notdef]h@[notdef]h

tanh 2 = 4x

1 4x

1

2@[notdef][vector]@[notdef][vector] + L[notdef]Y + L[notdef]gauge V (h, [vector]) , (3.7)

where, at rst order in x,

L[notdef]Y =

1 p2(

QLUYQR + h.c.) [v + (h [vector]) (1 + 2x)] , (3.8)

v2 + 2v(1 + 2x)(h [vector]) + (1 + 4x)(h [vector])2[bracketrightbig]

, (3.9)

while the scalar potential V (h, [vector]) coincides with that given in eq. (2.12).

8

L[notdef]gauge =

1 4Tr[V[notdef]V[notdef]]

Integrating out the heavy scalar. For small x (that is, [vector] mass large compared to the Higgs mass), the rst order EOM can be used to integrate out the LW partner,

[vector] = c h 2

p2( QLUYQR + h.c.) + Tr[V[notdef]V[notdef]](v + h) + m2h

v2 h2(h + 3v)[bracketrightbigg]+ O(c2 h) . (3.10)

While the masses of the gauge and fermion elds are una ected by the presence of P h, the Higgs mass renormalization absorbs the correction

m2h m2h

= 2x . (3.11)

The resulting e ective Lagrangian for the h eld is given by (omitting kinetic terms other than the Higgs one)

L h =

1

2@[notdef]h@[notdef]h + LY h + L4F h + Lgauge h V h(h) , (3.12)

with

LY h =

v + h p2

QLUYQR + h.c.

[parenrightbig] [parenleftBig]

1 + x

v2 (h2 + 2vh)

[parenrightBig]

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x m2h

(v + h)

p2 Tr[V[notdef]V[notdef]]

QLUYQR + h.c.

[parenrightbig]

, (3.13)

L4F h =

x 2m2h

QLUYQR + h.c.

2 , (3.14)

(v + h)2

4 Tr[V[notdef]V[notdef]]

Lgauge h =

1 + 2 x

v2 (h2 + 2vh)

[parenrightBig]

x4m2h

Tr[V[notdef]V[notdef]]2(v + h)2 , (3.15)

V h(h) =

3h5 + 12v h6[parenrightbigg]

. (3.16)

LY h above shows that anomalous gauge-fermion interactions weighted by Yukawas are ex

pected in the non-linear realization, in addition to the pure Yukawa-like corrections present in the linear expansion, see eq. (2.16). Furthermore, the potential V h(h) in eq. (3.16)

matches exactly the potential in eq. (2.18) for the linear case, as it should, exhibiting higher than quartic Higgs couplings that requires c h < 0 (i.e., x > 0) for the stability of the potential.

In summary, the resulting e ective Lagrangian for the non-linear case in eqs. (3.12) (3.16) shows deviations from SM expectations in fermion-Higgs couplings, four-fermion interactions and scalar properties, a pattern already found in the previous section for an elementary Higgs. Nevertheless, important distinctive features appear with respect to the case of a higher derivative kinetic term for a Higgs particle in linearly realised EWSB:

The number of e ective couplings modied is larger than in the linear case in

eqs. (2.15)(2.18), a characteristic feature already explored previously in other settings [20].

9

m2h

m2h

2 h2 +

2v (1 + 4x) h3 +

m2h8v2 (1 + 24x) h4 + x

m2h 2v3

V[LParen1] h[RParen1]

v 2c h [Equal][Minus]10[Minus]3

h

v 2c h [Equal]10[Minus]3

Figure 2. The scalar potential in the chiral Lagrangian for di erent values of the coe cient v2c h. The solid red line denotes the SM and the interline spacing is (v2c ) = 7.5 [notdef] 105.

Specically, couplings involving gauge particles are now modied with respect to their

SM values; in addition to anomalous gauge-fermion interactions, particularly interesting anomalous Higgs couplings to two (HVV) and three gauge bosons (HVVV), two Higgs-two gauge bosons (HHVV) and quartic gauge couplings (VVVV) are expected. The pure-gauge and gauge-Higgs anomalous couplings will be analyzed in detail in the next sections; they constitute a new tool to disentangle experimentally an elementary versus a dynamical nature of the Higgs particle, in the presence of higher-derivative kinetic terms.

3.2 Analysis via EOM

The alternative method of applying directly to the operator P h in the original non-linear

Lagrangian eq. (3.1) the standard eld theory EOM for the h eld,

h =

V (h)

JHEP12(2014)004

v + h

h

2 Tr[V[notdef]V[notdef]]

1 p2

QLUYQR + h.c.

[parenrightbig]

, (3.17)

leads to the same e ective low-energy Lagrangian at rst order on c h than that in eqs. (3.12)(3.16), obtained above via the LW procedure, as it can be easily checked.

Again, the correction to the scalar potential requires to impose c h < 0 to ensure that the potential remains bounded from below.

Higgs potential. Figure 2 shows the dependence of the shape of the scalar potential on the perturbative parameter c h: for negative values the SM vacuum [angbracketleft]h[angbracketright] = 0 is still a

minimum, while for positive values the potential is not bounded from below and moreover the SM vacuum is turned into a maximum.

10

4 Chiral versus linear e ective operators

The linear operator O involves gauge elds in its structure - see eq. (1.1), contrary to

the chiral e ective operator P h dened in eq. (1.2). Nevertheless, the addition of the

former operator to the SM Lagrangian turned out to give no contribution to couplings involving gauge elds, while the chiral operator P h does. This seemingly paradoxical

state of a airs and the consistency of the results can be ascertained by establishing the exact correspondence between both operators, which we nd to be given by:

O =

1

2( h)2 +

(v + h)2

8 (Tr[V[notdef]V[notdef]])2 +

v + h

JHEP12(2014)004

2 Tr[V[notdef]V[notdef]]@ @ h Tr[V[notdef]V ]@[notdef]h@ h

(v + h)2

4 Tr[(D[notdef]V[notdef])2] (v + h)Tr[V D[notdef]V[notdef]]@ h (4.1)

= P h + v2 [parenleftbigg]

18P6 +

14P7 P8

14P9

1

2P10

linear F

.

The right hand-side of eq. (4.1) describes a combination of the non-linear operator P h and a particular set of independent e ective operators of the non-linear basis as determined in ref. [20], dened by

P6 = (Tr[V[notdef]V[notdef]])2 F6(h) , P7 = Tr[V[notdef]V[notdef]] F7(h) ,

P8 = Tr[V[notdef]V ] @[notdef]F8(h)@ F[prime]8(h) , P9 = Tr[(D[notdef]V[notdef])2] F9(h) ,

P10 = Tr[V D[notdef]V[notdef]] @ F10(h) ,

(4.2)

where the generic -model dependent- Fi(h) functions are often parametrised as [17, 20]

Fi(h) = 1 + 2ai h

v + bi

h2v2 . . . (4.3)

The subscript linear F in the right-hand side of eq. (4.1) indicates that the equality

holds when the arbitrary functions Fi(h) take the specic linear-like dependence see

ref. [20]2

F6(h) = F7(h) = F9(h) = F10(h) linear F= (1+h/v)2 , F8(h) = F[prime]8(h) linear F= (1+h/v) .

(4.4)

Strictly speaking, in a general chiral Lagrangian the denition of P h should also contain

a F h(h) factor on the right hand side of eq. (1.2) [19, 20]; it would be superuous to keep

track of F h(h) here, though, as we will restrain the analysis to couplings involving at

most two Higgs particles, which is tantamount to setting F h(h) = 1 in the phenomenological analysis.

Taken separately, P h as well as each of the ve operators in eq. (4.2) do induce

deviations on the SM expectations for couplings involving gauge bosons. Eq. (4.1) implies

2In that reference, powers of the parameter which refers to ratios of scales involved were extracted from the denition of the operator coe cients; we will refrain here from doing so, and adopt the simple notation in eq. (1.4).

11

nevertheless that the gauge contributions of these six operators will exactly cancel in any physical observable when their relative weights are given by

v2c h = 8c6 = 4c7 = c8 = 4c9 = 2c10 . (4.5)

We have explicitly checked such cancellations in several examples of physical transitions; appendix B describes the particular case of ZZ ! ZZ scattering, for illustration.

5 Signatures and constraints

Tables 1, 2, 3, and 4 list all couplings involving up to four particles that receive contributions from the e ective linear operator O or any of its chiral siblings P h and P610. We work

at rst order in the operator coe cients, which are left arbitrary in those tables; the Fi(h) functionals are also assumed generic as dened in eq. (4.3). For the sake of comparison, a SM-like potential is taken for both the linear and chiral operators; the extension to a general scalar potential for the chiral expansion can be found in appendix A and has no signicant impact.

It turns out that O gives no tree-level contribution to couplings involving gauge par

ticles as argued earlier, while instead P h and P610 are shown to have a strong impact on

a large number of gauge couplings. On the other side, anomalous four-fermion interactions are induced by both O and P h, even if with distinct patterns.

5.1 E ects from O

The only impact of O on present Higgs and gauge boson observables is to generate

the universal shift in the Higgs coupling to fermions shown in the rst line of table 1. Equivalently, in the notation in refs. [16, 24, 25, 27, 28], in which the deviations of the Yukawa couplings and the gauge kinetic terms from SM predictions were parametrised as

, (5.1)

, (5.2)

the shift induced by the operator O reads

c f 1 + f = 1 m2hc . (5.3)

while

a V 1 + V = 1 , b = 1 . (5.4)

In refs. [20, 29], a general analysis of the constraints on departures of the Higgs couplings strength from SM expectations used all available collider and EW precision data, and it was found that

0.55 f 0.25 , (5.5)

12

JHEP12(2014)004

LY ukawa

v p2

QLUYQR + h.c.

[parenrightbig] [parenleftbigg]

1 + ch

v + . . .

[parenrightbigg]

v2

Lgaugekinetic

4 Tr(V[notdef]V[notdef]) [parenleftbigg]

1 + 2ah

v + b

h2v2 + . . .

[parenrightbigg]

Fermionic couplings Coe . SM value Chiral Linear: O

hLYUuR + dLYDdR + h.c.

[parenrightbig]

1p2 1 m2hc h m2hc

h2LYUuR + dLYDdR + h.c.

[parenrightbig]

1 vp2

3m

2h

2 c h

3m

2h

2 c

Z[notdef]Z[notdef]LYUuR + dLYDdR + h.c.

[parenrightbig]

g2v

4p2c2

c h

W +[notdef]W [notdef]LYUuR + dLYDdR + h.c.

[parenrightbig]

g

2v

2p2

c h

(LYUuR)2 +

RY UuL

2 14 c h

(LYUuR)

RY UuL

[parenrightBig]

JHEP12(2014)004

12 c h 2c

dLYDdR

2 +

[parenleftBig]

dRY DdL

2 14 c h

dLYDdR

[parenrightbig] [parenleftBig]

dRY DdL

[parenrightBig]

1

2

c h 2c

(LYUuR)

[parenleftBig]

dRY DdL

[parenrightBig]

+ h.c. 12 c h

(LYUuR) dLYDdR

[parenrightbig]

+ h.c. 12 c h 2c

(LYDdR) dLYUuR

[parenrightbig]

+ h.c. 1 c

(LYDdR)

[parenleftBig]

dRY DuL

[parenrightBig]

+ dLYUuR

[parenrightbig] [parenleftBig]

RY UdL

[parenrightBig]

1

c

Table 1. E ective couplings involving fermions generated by the linear operator O and its

chiral siblings P h and P610. For illustration only the couplings involving quark pairs are listed,

although similar interactions involving lepton pairs are induced.

at 90% CL after marginalizing over all other e ective couplings. Eq. (5.5) constrains m2hc , in addition to any combination of coe cients of other dimension-six operators which may also modify universally the Higgs couplings to fermions, see for instance ref. [20].

When only O is added to the SM Lagrangian, eq. (5.5) translates into the bound

c [lessorsimilar] 1.6 [notdef] 105 GeV2. This constraint is quantitatively quite weak, a fact due to present

sensitivity. For illustration, it could be rephrased as a lower limit of 250 GeV on the Higgs doublet LW partner mass. It shows that the bound obtained is of the order of magnitude of the constraints established by previous analyses, which considered direct production in colliders and/or indirect contributions to EW precision data and avour data [21, 3036], setting a lower bound for the LW scalar partner mass of 445 GeV.

5.2 E ects from P h and P610Tables 2, 3 and 4 illustrate that P h generates tree-level corrections to the gauge boson self-

couplings, as well as to gauge-Higgs couplings. Note that some of these interactions would not be induced by any d = 6 operator of a linear expansion, an example being the ZZZZ interactions in table 2; other signals absent in both the SM and d = 6 linear expansions, and thus unique to the leading order chiral expansion, can be found in appendix C. They constitute a strong tool to disentangle a strong underlying EW dynamics from a linear one.

13

VV, TGV and VVVV

Coe .

SM value

Chiral

Linear: O

(@[notdef]Z[notdef])(@ Z )

g

2

2c2

c9

(@[notdef]W +[notdef])(@ W )

g2

c9

i(@[notdef]W [notdef])(Z W + ) + h.c.

e2g

c2

c9

i(@[notdef]W [notdef])(A W + ) + h.c.

eg2

c9

JHEP12(2014)004

(Z[notdef]Z[notdef])2

g4 32c4

v2c h + 8c6

2

W +[notdef]W [notdef]

g

221

m2W c h 2g2c6

W +[notdef]W [notdef]

[parenrightbig]

(Z Z )

g2c2 1

2Z

2 c h g

m

2

c4 c6

W +[notdef]Z[notdef]

[parenrightbig]

(W Z )

g2c2

1

e

2s2

c4 c9

W +[notdef]A[notdef]

[parenrightbig]

(W A )

e2g2

1

c9

W +[notdef]A[notdef]

[parenrightbig]

(W Z ) + h.c.

egc

1

e2c2 c9

Table 2. Anomalous pure-gauge couplings involving two, three and four gauge bosons, induced by the chiral operators P h and P610, in contrast with the non-impact of their linear sibling O .

The e ects stemming from the operators P610, which are also siblings of the linear

operator O , are displayed in these tables for gauge two-point functions (VV), triple gauge

vertices (TGV) and VVVV couplings. As previously discussed, the tree-level contributions to physical amplitudes induced by that set of chiral operators cancel if the conditions in eqs. (4.4) and (4.5) are satised. Notwithstanding, for generic values of the coe cients of P h and P610, some signatures characteristic of a non-linearly realised electroweak

symmetry breaking are expected, as those discussed next.

From tables 3 and 1 it follows that P h yields a universal correction to the SM Higgs

couplings to gauge bosons and fermions. Furthermore, in present Higgs data the Higgs state is on-shell and, in this case, P7 gives also a correction to the SM-like HVV couplings,

while the modications generated by P9 and P10 vanish for on-shell W and Z gauge bosons

or massless fermions. Thus these corrections can be cast as, in the notation of eqs. (5.1) and (5.2),

a V 1 + V = 1

m2h

v2 (v2c h + 4c7a7) , c f 1 + f = 1 m2hc h , (5.6)

with b7 = 0. The general constraints resulting from present Higgs and other data [20, 29]

14

HVV and HVVV

Coe .

SM value

Chiral

Linear: O

Z[notdef]Z[notdef]h

vg2

4c2

1

m2hc h

Z[notdef]Z[notdef] h

g

2

2c2

2c7a7 v

(@[notdef]Z[notdef])(@ Z )h

g

2

2c2

2c9a9 v

(@[notdef]Z[notdef])(Z @ h)

g

2

2c2

2c10a10 v

JHEP12(2014)004

W +[notdef]W [notdef]h

vg2 2

1

m2hc h

W +[notdef]W [notdef] h

g2

2c7a7 v

(@[notdef]W +[notdef])(@ W )h

g2

2c9a9 v

(@[notdef]W +[notdef])(W @ h) + h.c.

g

2

2

2c10a10 v

i(@[notdef]W [notdef])(Z W + )h + h.c.

e2g

c2

2c9a9 v

i(@[notdef]W [notdef])(A W + )h + h.c.

eg2

2c9a9 v

i(Z[notdef]W +[notdef])(W @ h) + h.c.

e

2g 2c

2c10a10 v

i(A[notdef]W +[notdef])(W @ h) + h.c

eg2 2

2c10a10 v

Table 3. Anomalous e ective couplings of the Higgs particle to two or three gauge bosons, induced by the chiral operators P h and P610, in contrast with the non-impact of their linear sibling O .

apply as well here. For instance, if the coe cients of operators contributing only to the SM-like HVV coupling such as c7a7 above cancel, the bound on V and f becomes, at 90% CL,

0.33 f = V 0.33 , (5.7)

which translates into a bound c h [lessorsimilar] 2.1 [notdef] 105 GeV2.

O -shell Higgs mediated gauge boson pair production. Potentially more interesting, P7 leads to a new contribution to the production of electroweak gauge-boson pairs ZZ

and W +W through

gg ! h[star] ! ZZ or W +W , (5.8) where the Higgs boson is o -shell [37, 38]. For the sake of illustration, we consider the ZZ pair production with one Z decaying into e+e while the other into [notdef]+[notdef]. The left panel

15

H2VV couplings

Coe .

SM value

Chiral

Linear: O

Z[notdef]Z[notdef]h2

g2 8c2

1

5m2hc h

Z[notdef]Z[notdef] (h2)

g

2

2c2

c7b7 v2

Z[notdef]Z @[notdef]h@ h

g

2

2c2

4c8a8a[prime]8 v2

(@[notdef]Z[notdef])(@ Z )h2

g

2

2c2

JHEP12(2014)004

c9b9 v2

(@[notdef]Z[notdef])(Z @ h)h

g

2

2c2

2c10b10 v2

W +[notdef]W [notdef]h2

g2 4

1

5m2hc h

W +[notdef]W [notdef] (h2)

g2

c7b7 v2

W +[notdef]W @[notdef]h@ h

g2

4c8a8a[prime]8 v2

(@[notdef]W +[notdef])(@ W )h2

g2

c9b9 v2

(@[notdef]W +[notdef])(W @ h)h + h.c.

g

2

2

2c10b10 v2

Table 4. Anomalous e ective couplings involving two Higgs particles and two gauge bosons, induced by the chiral operators P h and P610, in contrast with the non-impact of their linear

sibling O .

of gure 3 depicts the leading-order SM contribution to

pp ! e+e[notdef]+[notdef] ,

together with the SM higher-order and P7 contributions through the ZZ channel in eq. (5.8).

The results presented in this gure were obtained assuming a center-of-mass energy at the LHC of 13 TeV, and requiring that all leptons have transverse momenta in excess of 10 GeV, that they are central ([notdef] [notdef] < 2.5) and that the same-avour opposite-charge lepton pairs

reconstruct the Z mass ([notdef]M[lscript]

+[lscript] MZ[notdef] < 5 GeV). In presenting the P7 e ects a coupling

c7a7 = 0.5 was assumed, which is compatible with the presently available Higgs data. Also, since the goal here is to illustrate the e ects of P7, we did not take into account the SM

higher-order contribution to gg ! e+e[notdef]+[notdef] which interferes with the o -shell Higgs one;

for further details see ref. [39] and references therein.The results in the left panel of the gure 3 show that P7 leads to an enhancement of

the o -shell Higgs cross section with respect to the SM expectations at high four-lepton invariant masses. In fact, the scattering amplitude grows so fast that at some point unitarity is violated [37], and the introduction of some unitarization procedure will tend to

16

Figure 3. The left panel presents the four lepton invariant mass spectrum for the process pp !

e+e[notdef]+[notdef]. The right panel contains the W W transverse mass distribution of the process pp !

e+ e[notdef] . In both panels the black line stands for the SM leading-order contribution while the blue (red) one represents the SM (P7) higher-order contribution given by eq. (5.8). In this gure

we assumed a center-of-mass energy of 13 TeV and c7a7 = 0.5.

diminish the excess. Nevertheless, even without an unitarization procedure, the expected number of events above the leading order SM background induced by P7 is shown to be

very small, meaning that unraveling the P7 contribution will be challenging.

We have analyzed as well the process

pp ! e+ e[notdef] [notdef] ,

that can proceed via the W +W channel in eq. (5.8). In the right panel of gure 3 the corresponding cross section is depicted as a function of the W W transverse mass

MWWT =[bracketleftbigg][parenleftbigg][radicalBig]

(p[lscript]+[lscript][prime]T)2 + m2[lscript]+[lscript][prime] +

where [vector]

/

pT stands for the missing transverse momentum vector, [vector]p [lscript]+[lscript][prime]T is the transverse momentum of the pair [lscript]+[lscript][prime] and m[lscript]+[lscript][prime] is the [lscript]+[lscript][prime] invariant mass. Here [lscript] = e or [notdef].

The transverse momentum and rapidity cuts used were the same than those for the left panel. As expected, an enhancement of the gg ! e+ e[notdef] [notdef] cross section is induced by the

operator P7. Analogously to the case of ZZ production, the SM leading-order contribution

dominates but for large MWWT ; the expected signals from the excess due to P7 will be thus

very di cult to observe.

Corrections to four gauge boson scattering. As can be seen in tables 2 and 3 the combination v2c h + 8c6 generates the anomalous quartic vertex ZZZZ that is not present in the SM. Moreover, the same combination gives anomalous contributions to the

ZZW +W and W +W W +W . These are genuinely four gauge boson e ects which do not induce any modication to triple gauge boson couplings and, therefore, these coe cients are much less constrained at present.

17

JHEP12(2014)004

q/

p2T + m2[lscript]+[lscript][prime]

2 ([vector]p [lscript]+[lscript][prime]T + [vector]

/

pT )2

1/2, (5.9)

Nowadays the most stringent bounds on the coe cients of these operators are indirect, from their one-loop contribution to the electroweak precision data [40], in particular to T which at 90% CL imply

0.23

18v2c h + c6 0.26 . (5.10)

At the LHC with 13-14 TeV center-of-mass energy, they can be detected or constrained by combining their impact on the VBF channels

pp ! jjW +W and pp ! jj(W +W + + W W ) , (5.11) where j stands for a tagging jet and the nal state W s decay into electron or muon plus neutrino [41]; the attainable 99% CL limits on these couplings are

1.2 [notdef] 102

18v2c h + c6 < 102 . (5.12)

Disregarding the contribution from c6, this would translate into c h [lessorsimilar] 1.3 [notdef] 106 GeV2,

which would suggest a sensitivity to the mass of the LW partner for the singlet Higgs in the chiral EWSB realization up to 887 GeV.

Strictly speaking, the relevant four gauge boson cross-section also receives modications induced by those operators which correct the HVV and TGV vertices when the Higgs boson or a gauge boson is exchanged in the s, t or u channels. In principle, these triple vertex e ects can be discriminated from the purely VVVV e ects by their di erent dependence on the scattering angle of the nal state gauge bosons. In practice, a detailed simulation will be required to establish the nal sensitivity to all relevant coe cients.

6 Conclusions

An e ective coupling for bosons which is tantamount to a quartic kinetic energy is a full-rights member of the tower of leading e ective operators accounting for BSM physics in a model-independent way. This is so in both the linear and non-linear realizations of electroweak symmetry breaking, or in other words irrespective of whether the light Higgs particle corresponds to an elementary or a composite (dynamical) Higgs. The corresponding higher derivative kinetic couplings, denoted here O and P h, respectively, eqs. (1.1)

and (1.2), are customarily not considered but traded by others (e.g. fermionic ones) instead of being kept as independent elements of a given basis.

It is most pertinent to analyze those couplings directly, though, as they are related to intriguing and potentially very important solutions to ultraviolet issues, such as the electroweak gauge hierarchy problem. The eld theory challenges they rise constitute as well a fascinating theoretical conundrum. Their theoretical impact is diluted and hard to track, though, when they are traded by combinations of other operators. On top of which, the present LHC data o er increasingly rich and precise constraints on gauge and gauge-Higgs couplings, up to the point of becoming competitive with fermionic bounds in constraining BSM theories; this trend may be further strengthened with the post-LHC facilities presently under discussion.

18

JHEP12(2014)004

We have analyzed and compared in this paper O and P h, unravelling theoretical

and experimental distinctive features.

On the theoretical side, two analyses have been carried in parallel and compared:i) the Lee-Wick procedure of trading the second pole in the propagator by a ghost scalar partner; ii) the application of the EOM to the operator, trading it by other e ective operators and resulting in an analysis which only requires standard eld-theory tools. Both paths have been shown to be consistent, producing the same e ective Lagrangian at leading order in the operator coe cient dependence.

A most interesting property is that the physical impact di ers for linearly versus nonlinear EWSB realizations: departures from SM values for quartic-gauge boson, Higgs-gauge boson and fermion-gauge boson couplings are expected only for the case of a dynamical Higgs, i.e. only from P h while not from O ; in addition, they induce a di erent pattern

of deviations on Yukawa-like fermionic couplings and on the Higgs potential.

Note that these distinctive signals of a dynamical origin of the Higgs particle would be altogether missed if a d = 6 linear e ective Lagrangian was used to evaluate the possible impact of an underlying strong dynamics, showing that in general a linear approach is not an appropriate tool to the task. Indeed, for completeness we identied all TGV, HVV and VVVV experimental signals which are unique in resulting from the leading chiral expansion, while they cannot be induced neither by SM couplings at tree-level nor by d = 6 operators of the linear expansion: the TGV couplings g 6, gZ5 and gZ6, the HVV couplings g(4)HV V , g(5)HV V and g(6)HV V and the VVVV couplings g(1)ZZ and g(5) Z, with the quartic kinetic energy coupling for non-linear EWSB scenarios P h contributing only

to g(1)ZZ among the above. The experimental search of that ensemble of couplings and their correlations (see tables 5, 6 and 7 in appendix C), constitute a superb window into chiral dynamics associated to the Higgs particle.

To tackle the origin of the di erent physical impact of quartic derivative Higgs kinetic terms depending on the type of EWSB, we have explored and established the precise relation between the two couplings: it was shown that O corresponds to a specic

combination of P h with ve other non-linear operators.On the phenomenological analysis, the impact of O , P h and P610 has been scru

tinised. All LHC Higgs and other data presently available were used to constrain the

O and P h coupling strengths. Moreover, the impact of future 14 TeV LHC data on

pp ! 4 leptons has been explored; the operators under scrutiny intervene in the process

via o -shell Higgs mediation in gluon-gluon fusion, gg ! h[star] ! ZZ or W +W , inducing

excesses at high four-lepton invariant masses via the ZZ channel, and at high values of the W W invariant mass in the W W channel. The corrections expected at LHC through their impact on four gauge boson scattering, extracted combining information from vector boson fusion channels, pp ! jjW +W and pp ! jj(W +W + + W W ), has been also

discussed. The possibility that LHC may shed light on Lee-Wick theories through the type of analysis and signals discussed here is a fascinating perspective.

19

JHEP12(2014)004

Acknowledgments

We thank especially J. Gonzalez-Fraile for early discussions about the presence of new o -shell Higgs e ects. We acknowledge partial support of the European Union network FP7 ITN INVISIBLES (Marie Curie Actions, PITN-GA-2011-289442), of CiCYT through the project FPA2009-09017, of CAM through the project HEPHACOS P-ESP-00346, of the European Union FP7 ITN UNILHC (Marie Curie Actions, PITN-GA-2009-237920), of MICINN through the grant BES-2010-037869, of the Spanish MINECO Centro de Excelencia Severo Ochoa Programme under grant SEV-2012-0249, and of the Italian Ministero dellUniversit e della Ricerca Scientica through the COFIN program (PRIN 2008) and the contract MRTN-CT-2006-035505. The work of I.B. is supported by an ESR contract of the European Union network FP7 ITN INVISIBLES mentioned above. The work of L.M. is supported by the Juan de la Cierva programme (JCI-2011-09244). The work of O.J.P.E. is supported in part by Conselho Nacional de Desenvolvimento Cientco e Tecnolgico (CNPq) and by Fundao de Amparo Pesquisa do Estado de So Paulo (FAPESP). The work of M.C.G-G is supported by USA-NSF grant PHY-09-6739, by CUR Generalitat de Catalunya grant 2009SGR502, by MICINN FPA2010-20807 and by consolider-ingenio 2010 program CSD-2008-0037.

A Analysis with a generic chiral potential V (h)

In the analysis performed in this paper the e ective operators P h and O are assumed

to be the only departures from the Standard Model present in the chiral and linear Lagrangians, respectively. However, the choice of a SM-like scalar potential might not appear satisfactory for the chiral case: a priori V (h) is a completely generic polynomial in the singlet eld h, and the current lack of direct measurements of the triple and quartic self-couplings of the Higgs boson leaves room for a less constrained parametrization.

Therefore, it can be interesting to test the stability of our results against deviations of the scalar potential from the SM pattern. To do this, we apply the Lee-Wick method to the Lagrangian in eq. (3.1) although with the SM-like potential in eq. (2.3) replaced by a generic one,

V (h) = a1h + m2h2 a2h2 +

JHEP12(2014)004

m2h

m2h8v2 a4h4 , (A.1)

where we choose to omit higher h-dependent terms, as the analysis remains at tree level and limited to interactions involving at most two Higgs particles. The correction factor a2 can always be reabsorbed in the denition of mh, and will thus be xed from the start to

a2 = 1 .

The comparison with the case described in section 3 is straightforward choosing, in addition, a3 = a4 = 1 and a1 = 0. The resulting mass-diagonal Lagrangian containing the LW eld [vector] is:

L[notdef] = (kin. terms) + L[notdef]Y + L[notdef]gauge V (h, [vector]) , (A.2)

20

2v a3h3 +

with

L[notdef]Y =

1 p2(

QLUYQR + h.c.) [1 + (1 + 2x)(h [vector])] , (A.3)

v2 + 2v(1 + 2x)(h [vector]) + (1 + 4x)(h [vector])2[bracketrightbig]

, (A.4)

V (h, [vector]) = a1(1 + 2x)(h [vector]) +

L[notdef]gauge =

1 4Tr[V[notdef]V[notdef]]

m2h

2 (1 + 2x)h2 +

m2h 2

1 12x + 2x

[vector]2

+ m2h

2v a3(1 + 6x)(h [vector])3 +

m2h8v2 a4(1 + 8x)(h [vector])4 , (A.5)

JHEP12(2014)004

where x = c hm2h/2 > 0.

Upon integrating out the heavy LW ghost, the following renormalized Lagrangian results:

L h =

1

2@[notdef]h@[notdef]h

1 4Z[notdef] Z[notdef]

1

2W +[notdef] W [notdef] + i

Q /

DQ +Lfer. h + Lgauge h V h(h) , (A.6)

where

Lfer. h =

1 p2

QLUYQR + h.c.

[parenrightbig] [bracketleftbigg]

v + (1 + 2x) h + 3a3xh2v + a4x

h3 v2

[bracketrightbigg]

(A.7)

x 2m2

QLUYQR + h.c.

2

x

m2h

v + h

p2 Tr[V[notdef]V[notdef]]

QLUYQR + h.c.

[parenrightbig]

,

Lgauge h =

1 4Tr[V[notdef]V[notdef]][bracketleftbigg]

(v + h)2

1 + 4xh

v + 2xh2[parenrightbigg]

(A.8)

+ 2x(v + h)h2

v2 (3v(a3 1) + h(a4 1))

[bracketrightbigg]

x 4m2h

(v + h)2 Tr[V[notdef]V[notdef]]2 ,

V h(h) =

m2h

2 h2 + a1(1 + 2x)h +

m2h 2v

a3(1 + 4x) + 2a1x

m2hv(a4 + a3 3a23)[bracketrightbigg]

h3 (A.9)

9a23 + a1a4m2v (2 3a3)

[parenrightbigg][bracketrightbigg]

h4 + 3a3a4m2hx2v3 h5 +a24m2hx 4v4 h6 .

Phenomenological impact. Assuming that the departures from unity of the ai parameters are small (of order c h at most), we can replace

a1 ! a1 , ai ! 1 + ai , i = 3, 4 (A.10) and expand the renormalized Lagrangian (A.6) up to rst order in x and in the is. Restricting for practical reasons to vertices with up to four legs, the list of couplings that are modied is very reduced and only includes terms in the scalar potential:

+ m2h 8v2

a4(1 + 6x) + 2x

m2h

2v (1 + 4x + a3)h3 ,

(A.11)

In consequence, upon the assumption that possible departures of the scalar potential from a SM-like form are quantitatively at most of the same order as c h, those contributions would not a ect the numerical analysis presented in the text.

21

m2h8v2 (1 + 24x + a4)h4 ,

a1h .

B Impact of O versus P h on ZZ ! ZZ scatteringThis appendix provides an illustrative example of how the contributions of the chiral operators P h , P610 to physical amplitudes combine to reproduce those of the linear operator

O , once the conditions (4.5) and (4.4) are imposed.

Let us consider the elastic scattering of two Z gauge bosons. This process is not a ected by O , therefore the corrections induced by the six chiral operators are expected

to cancel exactly, upon assuming (4.5) and (4.4).

Assuming the external Z bosons are on-shell, the only Feynman diagrams containing deviations from the Standard Model are the following

As + At + Au = hZ2

JHEP12(2014)004

Z1

Z3

Z1

Z3

Z1

Z4

+ h

Z2

+ h

Z2

(B.1)

Z4

Z4

Z3

Z2

Z4

A4Z =

(B.2)

Z1

Z3

For the amplitudes depicted in (B.1), the relevant couplings are ZZh and ZZ h (see table 3), and the contributions from each channel turn out to be

As = ("1 [notdef] "2)(" 3 [notdef] " 4)

i

s m2h

4m4Z v2

1 2m2hc h +8s

v2 c7a7[parenrightbigg]

, (B.3)

At = ("1 [notdef] " 3)("2 [notdef] " 4)

i

t m2h

4m4Z v2

1 2m2hc h +8s

v2 c7a7[parenrightbigg]

, (B.4)

Au = ("1 [notdef] " 4)(" 3 [notdef] "2)

1 2m2hc h +8s

v2 c7a7[parenrightbigg]

, (B.5)

where "1, "2 denote the polarizations of the incoming Z bosons, and " 3, " 4 those of the outgoing ones.

Imposing the constraints c7 = v2c h/4, from eq. (4.5) and a7 = 1 from eq. (4.4), the dependence on the exchanged momentum drops from the non-standard part of the amplitudes:

Ah = As + At + Au =

i

u m2h

4m4Z v2

4im4Z v2

("1 [notdef] "2)(" 3 [notdef] " 4)

s m2h+ ("1 [notdef] " 3)("2 [notdef] " 4)

t m2h+ ("1 [notdef] " 4)("2 [notdef] " 3) u m2h [bracketrightbigg]

h("1 [notdef] "2)(" 3 [notdef] " 4) + ("1 [notdef] " 3)("2 [notdef] " 4) + ("1 [notdef] "4)("2 [notdef] " 3)[bracketrightBig]

.(B.6)

The diagram (B.2) contains only the four-point vertex ZZZZ (see table 2), and gives

A4Z = 32im4Z v4

8im4Z

v2 c h

c6 + v28 c h

[parenrightbigg] [bracketleftBig]("1 [notdef] "2)(" 3 [notdef] " 4) + ("1 [notdef] " 3)("2 [notdef] " 4) + ("1 [notdef] "4)("2 [notdef] " 3)[bracketrightBig]

= 8im4Z

v2 c h

h("1 [notdef] "2)(" 3 [notdef] " 4) + ("1 [notdef] " 3)("2 [notdef] " 4) + ("1 [notdef] "4)("2 [notdef] " 3)[bracketrightBig]

.

(B.7)

In the second line the condition (4.5) has been assumed, which imposes v2c h = 8c6.

22

The neat correction to the Standard Model amplitude for ZZ scattering induced by the chiral operators P h , P610 is nally proved to vanish, as

A = Ah + A4Z = 0 . (B.8)

C Chiral versus linear couplings

In this appendix, we gather the departures from SM couplings in TGV, HVV and VVVV vertices, which are expected from the leading order tower of chiral scalar and/or gauge operators (which includes P h and P610 discussed in this manuscript), as well as from

any possible chiral or d = 6 linear coupling which may a ect those same vertices at leading order of the respective e ective expansions. Their comparison allows a straightforward identication of which signals may point to a strong dynamics underlying EWSB, being free from SM or d = 6 linear operators contamination. In tables 5, 6 and 7 below:

The O , P h and P610 operators are dened as in eqs. (1.1), (1.2)and (4.2), while

for all other couplings mentioned linear or chiral the naming follows that in ref. [20], to which we refer the reader.

All operator coe cients appearing in the tables below are dened as in eq. (1.4).

In comparison with the denitions in ref. [20] this means that: i) the coe cient of the chiral operator P h has been rescaled, see footnotes 1 and 2; ii) the d = 6

linear operator coe cients fi in refs. [20, 29] are related to those in the tables below as follows:

As discussed in the text, new anomalous vertices related to a quartic kinetic energy for the Higgs particle include as well HHVV couplings and new corrections to fermionic vertices. We leave for a future publication the corresponding comparison between the complete linear and chiral bases. When referring below to the SM, only tree-level contributions are considered.

C.1 TGV couplings

The CP-even sector of the Lagrangian that describes TGV couplings can be parametrized as

LWW V = igWW V

where V [notdef] , Z[notdef] and gWW e = g sin W , gWW Z = g cos W . The SM values for

the phenomenological parameters dened in this expression are gZ, 1 = Z, = 1 and gZ, 5 = gZ, 6 = 0. The resulting TGV corrections are gathered in table 5. For instance, while g 6 and gZ6 cannot be induced by any linear d = 6 operators, they receive contributions from the operators P610 discussed in this manuscript. Barring ne-tunings and one-loop

e ects, a detection of such couplings with sizeable strength would point to a non-linear realization of EWSB.

23

JHEP12(2014)004

ci = fi/ 2 . (C.1)

W +[notdef] W [notdef]V W +[notdef]V W [notdef] [parenrightBig]+ V W +[notdef]W V [notdef] (C.2)

igV5 [epsilon1][notdef] W +[notdef]@W W @W +[notdef] [parenrightbig]

gV1

V + gV6 @[notdef]W +[notdef]W @[notdef]W [notdef]W +

[parenrightbig]

V

,

Coe . Chiral Linear

e2/s2 [notdef]v2 1 2c1 + 2c2 + c3 4c12 + 2c13

18 (cW + cB 2cBW )

g 6 1 c9

gZ1

1 c2

s22 4e2c2 cT + 2s

2

c2 c1 + c3

18 cW +

s2 4c2 cBW

s22 16e2c2 c ,1

Z 1

s2

e2c2 cT + 4s

c2 c1 2s

2

2

ct2 c2 + c3 4c12 + 2c13

18 cW

s2 8ct2 cB +

s2 2c2 cBW

s2 4e2c2 c ,1

JHEP12(2014)004

gZ5

1c2 c14

gZ6

1c2 s2 c9 c16

Table 5. E ective couplings parametrizing the V W +W vertices dened in eq. (C.2). The coe cients in the second column are common to both the chiral and linear expansions. The third column lists the specic contributions from the operators in the chiral basis. For comparison, the last column exhibits the corresponding contributions from linear d = 6 operators.

C.2 HVV couplings

The Higgs to two gauge bosons couplings can be phenomenologically parametrized as

LHVV gHgg Ga[notdef] Ga[notdef] h + gH A[notdef] A[notdef] h + g(1)HZ A[notdef] Z[notdef]@ h + g(2)HZ A[notdef] Z[notdef] h

+ g(1)HZZ Z[notdef] Z[notdef]@ h + g(2)HZZ Z[notdef] Z[notdef] h + g(3)HZZ Z[notdef]Z[notdef]h + g(4)HZZ Z[notdef]Z[notdef] h

+ g(5)HZZ @[notdef]Z[notdef]Z @ h + g(6)HZZ @[notdef]Z[notdef]@ Z h (C.3)

+ g(1)HWW W +[notdef] W [notdef]@ h + h.c.

[parenrightbig]

+ g(2)HWW W +[notdef] W [notdef] h + g(3)HWW W +[notdef]W [notdef]h

+ g(4)HWW W +[notdef]W [notdef] h + g(5)HWW @[notdef]W +[notdef]W @ h + h.c.

+ g(6)HWW @[notdef]W +[notdef]@ W h ,

where V[notdef] = @[notdef]V @ V[notdef] with V = [notdef]A, Z, W, G[notdef]. Separating the contributions into SM

ones plus corrections, g(j)i [similarequal] g(j)SMi + g(j)i , it turns out that

g(3)SMHZZ =

2m2Zc2

v , (C.4)

while the tree-level SM value for all other couplings in eq. (C.3) vanishes.

While P h may induce a departure from SM expectations in two HVV couplings,

g(3)HZZ and g(3)HWW , table 6 shows that those signals could be mimicked by some d = 6 linear operators. On the contrary, a putative detection of g(4)HV V couplings may arise from the P7 operator discussed in this manuscript while neither from the SM not any linear d = 6

couplings, and would thus be a smoking gun for a non-linear nature of EWSB realization; the same applies to g(5)HV V from P10, and to g(6)HV V from P9.

24

m2Z

v , g(3)SMHWW =

Coe . Chiral Linear

e2/4v [notdef]v2

gHgg g

2

s

e2

2cGaG 4cGG

gH 1 2(cBaB + cW aW ) + 8c1a1 + 8c12a12 (cBB + cWW ) + cBW g(1)HZ

1s2 8(c5a5 + 2c4a4) 16c17a17 2(cW cB)

g(2)HZ

c

s 4s

2

c2 cBaB 4cW aW + 8c2 c2 c1a1 + 16c12a12 2s

2

c2 cBB 2cWW + c2 c2 cBW

g(1)HZZ

1c2 4c

2

s2 c5a5 + 8c4a4 8c

2

s2 c17a17

c2

s2 cW + cB

4

c4 cBaB + 2cW aW + 8s

g(2)HZZ c

2

s2 2s

2

c2 c1a1 8c12a12

s4

c4 cBB + cWW + s

2

c2 cBW

JHEP12(2014)004

g(3)HZZ

m2Z e2

2cH + 2cC(2aC 1) 8cT (aT 1) 4m2hc h c ,1 + 2c ,4 2c ,2 g(4)HZZ 1s22 16c7a7 + 32c25a25

g(5)HZZ 1s22 16c10a10 + 32c19a19

g(6)HZZ 1s22 16c9a9 + 32c15a15

g(1)HWW

1s2 4c5a5 cW

g(2)HWW

1s2 4cW aW 2cWW

g(3)HWW

c2 c ,1 + 4c ,4 4c ,2 + 4e2c2 cBW

g(4)HWW 1s2 8c7a7

g(5)HWW 1s2 4c10a10

g(6)HWW 1s2 8c9a9

m2Z c

2

2

e2

4cH + 4cC(2aC 1) + 32e2c2 c1 + 16c

c2 cT 8m2hc h 32e2s2 c12

2(3c2 s

2 )

Table 6. Higgs-gauge bosons couplings as dened in eq. (C.3). The coe cients in the second column are common to both the chiral and linear expansions.The third column lists the specic contributions from the operators in the chiral basis. For comparison, the last column exhibits the corresponding contributions from linear d = 6 operators.

C.3 VVVV couplings

The e ective Lagrangian for VVVV couplings reads

L4X g2

g(1)ZZ(Z[notdef]Z[notdef])2 + g(1)WW W +[notdef]W +[notdef]W W g(2)WW (W +[notdef]W [notdef])2

+ g(3)VV [prime] W +[notdef]W V[notdef]V [prime] + V [prime][notdef]V

[parenrightbig]

g(4)VV [prime] W + W V [notdef]V [prime][notdef]

(C.5)

+ ig(5)VV [prime] e[notdef] W +[notdef]W VV [prime]

,

where V V [prime] = [notdef] , Z, ZZ[notdef]. At tree-level in the SM, the following couplings are non-

vanishing:

g(1)SMWW =

1

2 , g(2)SMWW =

1

2 , g(3)SMZZ =

c2

2 , g(3)SM =

s2

2 ,

(C.6)

table 7 shows the impact on the couplings in eq. (C.5) of the leading non-linear versus linear operators. While P h and P6 may induce g(2)WW and g(4)ZZ couplings, the table shows

25

g(3)SMZ =

2 , g(4)SMZZ = c2 , g(4)SM = s2 , g(4)SMZ = s2 ,

s2

Coe . Chiral Linear e2/4s2 [notdef]v2 g(1)WW 1

c2 c1 + 4c3 + 2c11 16c12 + 8c13

cW

s22

e2c2 cT + 8s

2

2 + s

2

c2 cBW

s22

4c2 e2 c 1

g(2)WW 1

s22

e2c2 cT + 8s

2

c2 c1 + 4c3 4c6 v22 c h 2c11 16c12 + 8c13

cW

2 + s

2

c2 cBW

s22

4c2 e2 c 1

g(1)ZZ

1c4 c6 + v28 c h + c11 + 2c23 + 2c24 + 4c26

g(3)ZZ

s22 c

2

e2c2 cT + 2s

22

c2 c1 + 4c2 c3 2s4 c9 + 2c11 + 4s2 c16 + 2c24

cW c2

2 + s

22

4c2 cBW s

22 c

2

4e2c2 c 1

1 c2

JHEP12(2014)004

g(4)ZZ

1 c2

2s22 c

2

e2c2 cT + 4s

22

c2 c1 + 8c2 c3 4c6 v22 c h 4c23 cW c2 + 2 s

22

4c2 cBW s

22 c

2

2e2c2 c 1

g(3) s2 2c9 g(3) Z

s c

c2 c1 + 4c3 + 4s2 c9 4c16

cW

s22

e2c2 cT + 8s

2

2 + s

2

c2 cBW

s22

4c2 e2 c 1

g(4) Z

s c

2s22

e2c2 cT + 16s

c2 c1 + 8c3 cW + 2 s

2

c2 cBW

2

s22

2c2 e2 c 1

g(5) Z

s

c 8c14

Table 7. E ective couplings parametrizing the vertices of four gauge bosons dened in eq. (C.5). The third column lists the specic contributions from the operators in the chiral basis. For comparison, the last column exhibits the corresponding contributions from linear d = 6 operators.

that those signals could be mimicked by some d = 6 linear operators. On the contrary, the 4Z coupling g(1)ZZ is induced by P h, while it vanishes in the SM and in any linear d = 6

expansion. A detection of g(1)ZZ would thus be a beautiful smoking gun of a non-linear nature of EWSB realization, which may simultaneously indicate a quartic kinetic energy for the Higgs scalar of LW theories (although g(1)ZZ may also be induced by other chiral operators, including P6 as discussed towards the end of section 5).

Summarising this appendix, some experimental signals are unique in resulting from the leading chiral expansion, while they cannot be induced neither by the SM at tree-level nor by d = 6 operators of the linear expansion; among those analyzed here they are

the TGV couplings g 6, gZ5, and gZ6,

the HVV couplings g(4)HV V , g(5)HV V , and g(6)HV V ,

the VVVV couplings g(1)ZZ, and g(5) Z,

with the quartic kinetic energy coupling for non-linear EWSB scenarios P h contributing

only to g(1)ZZ among the above. g(3) does not receive contributions from d = 6 linear operators, but it is induced by three-level SM e ects. The experimental search of that ensemble of couplings, with the correlations among them following from tables 5, 6 and 7, constitute a fascinating window into chiral dynamics associated to the Higgs particle.

26

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

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

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