Heavy neutral scalar decays into electroweak gauge bosons in the Littlest Higgs Model

0.22 MB·

by  J. I. Aranda

Heavy neutral scalar decays into electroweak gauge bosons in the Littlest Higgs Model
(a)J. I. Aranda, (a,b)I. Cort´es-Maldonado,(a)S. Montejo-Montejo, (a)F. Ram´ırez-Zavaleta, (a)E. S. Tututi
(a)Facultad de Ciencias F´ısico Matema´ticas, Universidad Michoacana de San Nicol´as de Hidalgo,
Avenida Francisco J. Mu´jica S/N, 58060, Morelia, Michoaca´n, M´exico.
(b)C´atedras Consejo Nacional de Ciencia y Tecnolog´ıa, M´exico.
(Dated: January 26, 2017)
We study the heavy neutral scalar decays into standard model electroweak gauge bosons in the
contextoftheLittlestHiggsmodel. WefocusourattentionontheΦ0 WW,γV processesinduced
at theone-loop level, with V =γ,Z. Sincethebranchingratios of the→Φ0 γV decaysresult very
suppressed, only the Φ0 WW process is analyzed in the framework →of possible experimental
→
scenarios by using heavy scalar masses between 1.6 TeV until 3.3 TeV. The branching ratio for
7 the Φ0 WW decay is of the order of 10−3 throughout the interval 2 TeV < f < 4 TeV, which
→
1 represent the global symmetry breaking scale of the theory. Thus, it is estimated the associated
0 production cross section for the pp→Φ0X →WW, finding around ten events for mΦ0 ≈1.6 TeV
2 at best.

n
a
I. INTRODUCTION
J

8
1 At the LHC, the work of many experimental collaborations that have been dedicated their efforts to the search
for the Higgs boson was succeeded [1, 2]. This particle represents the first evidence of detection of a spin-0 particle,

] which in principle seems to be the Higgs boson of the SM. In this direction, one could think that in the fundamental
h
theory (at high energies), there may exist couplings between new scalar particles with all the particles detected. In
p
- addition,sincetheLHCisthefirstmachinethatoperatesatthescaleofTeVs,itishopedtohavegoodpossibilitiesfor
p detection of new particlesat this energy scale. Therefore,it is expected that in the following yearsthe LHC can offer
e
different possibilities ofdetection of this type ofparticles or discardtheir existence atthe scale ofTeVs. At least, the
h
scenario is optimistic since the experimental collaborations ATLAS and CMS have been able to construct a reliable
[
detectionmachineryreferringtospin-0resonances. Indeed,ATLASandCMScollaborationshavecontinuedsearching

1 for new exotic particles, such as the Randall-Sundrum spin-2 boson [3] or new heavy scalar particles [4]. Thus, the
v
possibility to search for new physics phenomena at this energy scale is higher than past years with the upgrade of
2
the CMS and ATLAS detectors [5–8]. Abreast, in the literature there are theoretical proposals for the search of new
5
9 heavy scalar particles [9]. In more recentworks,with the aim of the explaining a possible resonance around750 GeV
4 in the diphoton channel, which was misunderstood due to statistical fluctuation, a lot of phenomenological studies
0 were done [10–17].
. Several extensions of the standard model (SM) predict the existence of new particles with masses of the order
1
0 of TeVs [18–26]. The exhaustive search for new scalar particles at the LHC give us the possibility to explore new
7 physicsprocessesrelatedtoextendedHiggssectors. Inthissense,therearemanyalternativeformulationsthatpredict
1 more content of scalar particles such as the two-Higgs doublet models (2HDMs) [18, 27], three-Higgs doublet model
: (3HDM) [28], Higgs-singlet extension model [29], little Higgs models (LHMs) [30], etc. Among the wide variety of
v
i LHMs, the case of the littlest Higgs model (LTHM) is of peculiar interest since there is no new degrees of freedom
X
beyond the SM under TeV scale together with its reduced spectrum of new scalar particles [31]. Furthermore, this
r typeofmodelsofferapossiblesolutiontohierarchyproblem,whichtakesplacewhentheHiggsbosonmassisaffected
a
by corrections at one-loop level.
The LHMs also arise as an alternative for the study of the electroweaksymmetry breaking [30, 31], on the basis of
dimensionaldeconstruction[32,33],withfeatureofcancelingquadraticdivergences. Infact,the quadraticdivergence
generated at the one-loop level through the SM gauge bosons is canceled via the quadratic divergence introduced by
the new heavy gauge bosons at the same perturbative level. The one-loop quadratic divergence induced in the SM
Yukawa sector is removed by introducing new heavy fermions such that it cancels the quadratic divergence coming
from the top quark. In LHMs, the new Higgs field acquire mass becoming pseudo-Goldstone bosons in accordance
with the breaking of a globalsymmetry at the energy scale of TeVs, where a massless Higgs appears. The quadratic-
divergent corrections to the Higgs mass arise at loop level, therefore, this warrants a light Higgs. With regard to
LTHM[34],thenewstatesarisingattheTeVenergyscaleformanewsetoffourgaugebosonswiththesamequantum
numbers as the SM gauge bosons, namely, AH, ZH, and WH±, an exotic quark with the same charge as the SM top
quark,and a new heavyscalartriplet, which containssix physicalstates: double-chargedscalarsΦ , single-charged
±±
scalars Φ , a neutral scalar Φ0, and a neutral pseudo scalar Φp. A detailed presentation of LTHM model can be
±
found in Ref. [34].
The main objective of this manuscript is to study the relevance of the Φ0 WW,γV (V = γ,Z) decays in the
→
context of the linearized theory of the LTHM, which refers to make a first-order expansion of the Σ field around its
2
vacuumexpectationinpowersofv/f [34]. Currently,theparameterspaceoftheLTHMhasbeenseverelyconstrained
by the Higgs discovery channels and electroweak precision observables [35], therefore, the processes in question serve
as a way to test this model. Our analysis considers pooled results from experimental and phenomenological studies
where an experimental scenarioas realistic as possible is set [35]. The viable interval of experimental analysis for the
energy scale f comes from a phenomenologicalstudy based on the experimental searchesof a scalar boson consistent
withSM-Higgsbosonbymakinguseofthesignalstrengthmodifieralongwithelectroweakprecisiondata. Inorderto
be consistent with electroweakprecision data, a lower limit for the energy scale f around 2-4 TeV is established [35].
As anaddedvalue,we presenta phenomenologicalscenarioontheWW associatedproductionatLHC resultingfrom
the hypothetical Φ0 resonance with pp collisions at center of mass energy of 14 TeV. Thereby, the phenomenological
implications of a heavy neutral scalar boson with mass around 1 TeV decaying into two W bosons are discussed in
the LHC setting.
The present paper is organized as follows. In section II, we present an overview of the littlest Higgs model with
special attention on the scalar sector. In section III, we analyze the Φ0 WW,γV decays at one-loop level. In
→
section IV, the numerical results are discussed. Finally, the conclusions are presented in section V.
II. MODEL FRAMEWORK
The LTHM is constructed on the basis of a nonlinear sigma model with SU(5) global symmetry together with the
gaugedgroup[SU(2) U(1) ] [SU(2) U(1) ][31,34]. TheSU(5)groupisspontaneouslybrokentoSO(5)atthe
1 1 2 2
⊗ ⊗ ⊗
energy scale f, where f is constrained to be of the order of 2–4 TeV [35]. At the same time, the [SU(2) U(1) ]
1 1
⊗ ⊗
[SU(2) U(1) ]groupisalsobrokentoitssubgroupSU (2) U (1),whichisidentifiedastheSMelectroweakgauge
2 2 L Y
⊗ ⊗
group. Theglobalsymmetrybreakingpatternleaves14GoldstonebosonswhichtransformundertheSU (2) U (1)
L Y
groupasarealsinglet1 ,arealtriplet3 ,acomplexdoublet2 ,andacomplextriplet3 [31,34]. Atthe⊗scalef,
0 0 1 1
±2 ±
the spontaneous globalsymmetry breaking of the SU(5) groupis generatedby the vacuum expectation value (VEV)
of the Σ field, denoted as Σ [34]. The Σ field is explicitly given by
0
Σ=eiΠ/fΣ eiΠT/f, (1)
0
with
0 0 1
2 2 2 1 2 2
Σ = 0 × 1× 0 × (2)
0 1 2 1 2
1 × 0 0 × 
2 2 2 1 2 2
× × ×
 
and Π being the Goldstone boson matrix having the following form
0 h /√2 φ
2 2 † †
Π= h/√×2 0 h /√2 . (3)
∗
 
φ hT/√2 0
2 2
×
 
Here, h is a doublet and φ represents a triplet under the SU (2) U (1) SM gauge group [34]
L Y
⊗
φ++ φ+
h=(h+,h0), φ= √2 . (4)
φ+ φ0 !
√2
By the spontaneous symmetry breaking (SSB), both the real singlet and the real triplet are absorbed by the longi-
tudinal components of the gauge bosons at the energy scale f. At this scale, the complex doublet and the complex
triplet remain massless. The complex triplet acquires a mass of the order of f by means of the Coleman-Weinberg
type potential when the global symmetry of the group SO(5) breaks down. The complex doublet is identified as the
SM Higgs field.
The effective Lagrangianinvariant under the [SU(2) U(1) ] [SU(2) U(1) ] group is [34]
1 1 2 2
⊗ ⊗ ⊗
= + + + V , (5)
LTHM G F Σ Y CW
L L L L L −
where represents the gauge bosons kinetic contributions, the fermion kinetic contributions, the nonlinear
G F Σ
L L L
sigma model contributions of the LTHM, the Yukawa couplings of fermions and pseudo-Goldstone bosons, and
Y
L
the last term symbolizes the Coleman-Weinberg potential.
3
The standard form of the Lagrangianof the nonlinear sigma model is
f2
= tr Σ2, (6)
Σ µ
L 8 |D |
where the covariant derivative is written as
2 3
DµΣ=∂µΣ−i "gj Wµaj QajΣ+ΣQajT +gj′Bµj YjΣ+ΣYjT #. (7)
j=1 a=1
X X (cid:0) (cid:1) (cid:0) (cid:1)
Here,Wa aretheSU(2)gaugefields,B aretheU(1)gaugefields,Qa aretheSU(2)gaugegroupgenerators,Y are
µj µj j j
the U(1)gaugegroupgenerators,g arethe couplingconstantsofthe SU(2)group,andg arethe couplingconstants
j j′
ofthe U(1)group[34]. After SSB aroundΣ ,itis generatedthe masseigenstatesoforderf forthe gaugebosons[34]
0
W = cW +sW , (8)
µ′ − µ1 µ2
B = c +s , (9)
µ′ − ′Bµ1 ′Bµ2
W = sW +cW , (10)
µ µ1 µ2
B = s +c , (11)
µ ′ µ1 ′ µ2
B B
3
where W Wa Qa and B Y for j =1,2; c= g / g2+g2, c = g / g2+g2, s= g / g2+g2, and
µj ≡ µj j Bµj ≡ µj j 1 1 2 ′ 1′ 1′ 2′ 2 1 2
a=1
s = g / g2+Pg2. Notice that Σ field has been expanded aropund Σ holding dopminant terms in p[34]. At this
′ 2′ 1′ 2′ 0 LΣ
stage of SSB the B and W fields remain massless.
µ µ
p
The SSB at the Fermi scale provides mass to the SM gauge bosons (B and W) and induces mixing between heavy
and light gauge bosons. The arising masses at the leading order (neglecting terms of order v2 , with v being the
O f2
vacuum expectation value at the Fermi scale) are (cid:16) (cid:17)
gf
m = , (12)
ZH 2sc
g f
′
m = , (13)
AH 2√5sc
′ ′
gf
m = . (14)
WH 2sc
As it is known, c=m /m andtakes the value equals to one atthe leadingorder[34, 36], inorderto have values
WH ZH
for the masses of the weak gauge bosons not very different, as it occurs in the electroweak sector of the SM [36].
In LTHM the Higgs potential is generated by one-loop radiative corrections at the leading order. This potential
contains the contributions coming from gauge boson loops and fermion loops. When the Σ field is expanded into the
nonlinear sigma model it is obtained the associated Coleman-Weinberg potential [37]
V =λ f2Tr(φ φ)+iλ f(hφ hT h φ h ) µ2hh +λ4(hh )2, (15)
CW φ2 † hφh † − ∗ † † − † h †
wherein the λ’s coefficient are given by
a g2 g2
λ = + ′ +8aλ2,
φ2 2 s2c2 s2c2 ′ 1
(cid:20) ′ ′ (cid:21)
a (c2 s2) (c2 s2)
λhφh = −4 g2 s2−c2 +g′2 ′s2−c2′ +4a′λ21,
(cid:20) ′ ′ (cid:21)
a g2 g2 1
λh4 = 8 s2c2 + s2′c2 +2a′λ21 = 4λφ2, (16)
(cid:20) ′ ′ (cid:21)
where c, s (c, s) are the gauge coupling constants of the SU(2) (U(1)) symmetry group. The a and a parameters
′ ′ ′
represent the unknown ultraviolet (UV) physics at the cutoff scale Λ . Their values depend on the details of the UV
S
completion atthe cutoff scale Λ [34]. As far as the µ2 parameterrefers, it is a free parameter which receivesequally
S
significant contributions from one-loop logarithmic and two-loop quadratically divergent parts [34]. The VEV v (v )
′
of the doublet (of the triplet) is obtained after minimizing the V potential, which fulfill the following relations
CW
µ2 λ v2
v2 = , v = hφh . (17)
λ2 ′ 2λ f
λ hφh φ2
h4 − λφ2
4
ThemassesoftheheavyscalarsareobtainedbydiagonalizingtheHiggsmassmatrix[34]. Thus,thegaugeeigenstates
of the Higgs sector can be written in terms of the mass eigenstates in the following way
1 i
h0 = c H s Φ0+v + c G0 s ΦP ,
0 0 P P
√2 − √2 −
1 (cid:0) (cid:1) i (cid:0) (cid:1)
φ0 = s G0+c ΦP s H +c Φ0+√2v ,
P P 0 0 ′
√2 − √2
h+ = c G(cid:0)+ s Φ+, (cid:1) (cid:16) (cid:17)
+ +
−
1
φ+ = s G++c Φ+ ,
+ +
i
Φ+(cid:0)+ (cid:1)
φ++ = , (18)
i
where H is the Higgs boson, Φ0 is a new neutral scalar, ΦP is a neutral pseudoscalar, Φ+ and Φ++ are the charged
and doubly charged scalars, and G+ and G0 are the Goldstone bosons that are eaten by the massless W and Z
bosons [34]. At the leading order in the theory, the masses of the new heavy scalar particles are degenerate [34]
√2m f
H
m = , (19)
Φ
1 y2 v
− v
where y =4v f/v2. This mass expression is positive depfinite if
v ′
v2 v2
′
< . (20)
v2 16f2
Additionally, the generic expression for the Higgs mass is given by
m2 2 λ λ2 /λ v2 =2µ2. (21)
H ≃ h4 − hφh φ2
Continuing with the description of the model,(cid:0)the LTHM incor(cid:1)porates new heavy fermions which couple to Higgs
field in a such way that the quadratic divergence of the top quark is canceled [31, 34]. In particular, this model
introduces a new set of heavy fermions arranged as a vector-like pair (t˜,t˜c) with quantum numbers (3,1) and
′ Yi
(3¯,1) , respectively. The new Yukawa interactions are proposed to be
−Yi
1
= λ fǫ ǫ χ Σ Σ uc+λ ft˜t˜c+H.c., (22)
LY 2 1 ijk xy i jx ky ′3 2 ′
where χ = (b ,t ,t˜); ǫ and ǫ are antisymmetric tensors for i,j,k = 1,2,3 and x,y = 4,5 [31]. Here, λ and λ
i 3 3 ijk xy 1 2
are free parameters, where the λ parameter can be fixed such that, for given (f,λ ), the top quark mass adjust to
2 1
its experimental value [35].
Expanding the Σ field and retaining terms up to (v2/f2) after diagonalizing the mass matrix, it can be obtained
O
the massstatest , tc, T ,andTc,whichcorrespondto SMtopquarkandthe heavytopquark,respectively[34,35].
L R L R
The explicit remaining terms of the Lagrangian as well as the complete set of new Feynman rules can be
LTHM
L
found in Ref. [34].
III. THE STUDY OF THE Φ0 WW,γV DECAYS
→
To study the one-loop level Φ0 WW,γV decays we begin with the calculation of the total decay width of the
Φ0 (Γ ) which includes only SM fi→nal states. The dominantcontributions to Γ arethe tree-leveldecays ofΦ0 into
Φ0 Φ0
HH,ZZ,ZZH. However,wewillalsocomputethesubdominantcontributionsoffinalstatessuchast¯t,WWZ,WWH.
1. Tree-level decays
TheFeynmandiagramsfortwo-andthree-bodydecayscontributingtoΓ areshowninFigs.1and2,respectively.
Φ0
According to Fig. 1, for two-body decays we employ the vertices Φ0tt¯,Φ0ZZ,Φ0HH given in Ref. [34]. Moreover, in
the contextof the linearizedtheory of LTHM the Φ0Wµ±Wν∓ and Φ0WH±µ Wν∓ vertexesvanish [34]. Keeping in mind
5
Φ0(p) Z(k1) Φ0(p) t(k1)
Z(k2) t¯(k2)
Φ0(p) H(k1)
H(k2)
FIG. 1: Feynman diagrams contributingto theΦ0 tt¯,ZZ,HH decays at treelevel.
→
this facts, we now proceed to present the Φ0 tt¯decay width, which can be written as
→
N m m2 m2 3/2
Γ(Φ0 t¯t)= c Φ0 t 1 4 t , (23)
→ 64πf2 − m2
(cid:18) Φ0(cid:19)
wherem istheheavyscalarmass,N isthecolorfactorequalto3forquarksandf istheglobalsymmetrybreaking
Φ0 c
scale.
The Φ0 ZZ width decay is given as follows
→
g4 m3 v4 m2 1/2 m2 m4
Γ(Φ0 ZZ)= Φ0 1 4 Z 1 4 Z +12 Z . (24)
→ 84πc4 f2 m4 − m2 − m2 m4
W Z (cid:18) Φ0(cid:19) (cid:18) Φ0 Φ0(cid:19)
The Φ0 HH decay width is expressed as
→
m3 m2 1/2
Γ(Φ0 HH)= Φ0 1 4 H . (25)
→ 256πf2 − m2
(cid:18) Φ0(cid:19)
Regarding to the three-body decays, these processes originate from a four-point vertex, as well as from Feynman
diagramsmediatedbyneutralgaugebosonsandheavyscalarbosons. Explicitly,thedominantthree-bodydecaysthat
contribute to Γ are Φ0 ZZH, Φ0 WWZ and Φ0 WWH. The respective Feynman diagrams are depicted
Φ0
→ → →
in Fig. 2.
W+(k1),W−(k2) Z(k1),Z(k2)
W+(k1),Z(k1)
W−(k2),W+(k1) Z(k2),Z(k1)
φ0(p) W−(k2),Z(k2) φ0(p) φ0(p)
H(k3) φ−,φ+ H(k3) Z,ZH H(k3)
Z(k1),Z(k2) W+(k1),W−(k2) Z(k3)
Z(k2),Z(k1) W−(k2),W+(k1) W−(k2)
φ0(p) φ0(p) φ0(p)
φp H(k3) φ−,φ+ Z(k3) Z,ZH W+(k1)
FIG. 2: Feynman diagrams representing tree-level contributions toΓΦ0.
The decay width of the Φ0 scalar boson decaying to three bodies can be obtained by using the following equation
dΓ(Φ0 ABC) m
→ = Φ0 (Φ0 ABC)2, (26)
dx dx 256π3|M → |
a b
6
where 2 is the squared amplitude, ABC represents the SM final states for each process presented in Fig. 2. The
|M|
amplitudes were calculated employing the Feynman rules given in Ref. [34]. The explicit form of these amplitudes is
presented below
ig2v (kµ+2kµ)(2kν +kν +2kν) (kµ+2(kµ+kµ))(kν +2kν)
M(Φ0 →WWH)= 8√2f"2gµν + 1 (k13+k3)12−m22Φ0 3 + 1 (k2+2 k3)23−m22Φ0 3 #ǫ∗µ(k1)ǫ∗ν(k2). (27)
It must be recalled that mΦ0 =mΦ+ =mΦ−.
ig2v (kµ+2kµ)(2kν +kν +2kν) (kµ+2(kµ+kµ))(kν +2kν)
(Φ0 ZZH) = c2 24gµν +8 1 3 1 2 3 +8 1 2 3 2 3
M → 32√2c4Wf" W (k1+k3)2−m2Φ0 (k2+k3)2−m2Φ0 !
gµν (k1µ+k3µ)(k1ν+k3ν) gµν (k2µ+k3µ)(k2ν+k3ν)
+ 4g2v2 − m2Z + − m2Z
(k +k )2 m2 (k +k )2 m2
1 3 − Z 2 3 − Z !
gµν (k1µ+k3µ)(k1ν+k3ν) gµν (k2µ+k3µ)(k2ν+k3ν)
+ g2(c2c−2ss22)2v2 (k−+k )2m2ZHm2 + (k−+k )2m2ZHm2 ǫ∗µ(k1)ǫ∗ν(k2), (28)
1 3 − ZH 2 3 − ZH !#
g3v2 gαν(kµ+2(kµ+kµ)) gαµ(2kν +kν +2kν) 2f2gαβ
(Φ0 WWZ) = if2 1 2 3 + 1 2 3 +
M → 8√2cWf3"− (k2+k3)2−m2Φ0 (k1+k3)2−m2Φ0 ! (k1+k2)2−m2Z
(k +k )(k +k )
1β 2β 1γ 2γ g (kγ kγ)gµν (kµ+2kµ)gγν +(2kν +kν)gγµ
× m2Z − βγ!(cid:18) 2 − 1 − 1 2 1 2 (cid:19)
(c2 s2)2v2gαβ (k +k )(k +k )
− 1β 2β 1γ 2γ g (kγ kγ)gµν (kµ+2kµ)gγν
− (k1+k2)2−m2ZH m2ZH − βγ!(cid:18) 2 − 1 − 1 2
+ (2kν +kν)gγµ ǫ (k )ǫ (k )ǫ (k ). (29)
1 2 ∗µ 1 ∗ν 2 ∗α 3
(cid:19)#
To obtain the decay widths, we square the decay amplitudes with the aid of FeynCalc package [38]. The integrations
over three-body phase space were numerically performed.
2. One-loop level Φ0 WW decay
→
Since the Φ0WW coupling is absent at the tree-level in the LTHM, it is interesting to study the one-loop level
Φ0 WW process. IntheLTHM,thisdecayreceivesdominantcontributionsfromSMtopquarkandtheHiggsboson;
→
when they circulate in the loops. The dominant Feynman diagrams can be appreciated in Fig. 3. After performing
dimensional regularizationfor the one-loopamplitudes coming from Feynman diagramsin Fig. 3(a), we arrive to UV
divergent contributions which cannot be analytically canceled, so it is necessary to implement the renormalizationof
the one-loop amplitudes in order to obtain finite contributions at the leading order. This is achieved by adding the
corresponding counterterms to the ultraviolet infinite amplitude in d=4 [39]. In Fig. 3(b), we present the Feynman
diagramwhichrepresentscountertermsthatexactlycancelultravioletdivergentcontributionsoftheone-loopΦ0WW
vertex.
The amplitude for the Φ0 WW decay can be written as
→
g2N V2m2m4
(Φ0 WW) = c tb t W AWW(kµkν +kµkν)+BWW(kµkν +kµkν)
M → 16√2π2fm2 m2 4m2 2 1 1 2 2 1 2 2 1
Φ0 Φ0 − W h
+ CWWm2Φ0gµν (cid:0)ǫ∗µ(k1)ǫ∗ν(k2),(cid:1) (30)
i
7
where
AWW = 4 m2 C + B (2) B (3) 2(1+(m2+m2)C )
" W 0 0 − 0 − b t 0
(cid:16) (cid:17)
(m2 m2)
+ b − t B (2)+B (3) 2B (1)+(m2 m2)C
m2W 0 0 − 0 b − t 0 #
(cid:16) (cid:17)
(m2 m2)
+ 2 1 m2 C 2 B (2) B (3) m2C + b − t
" − W 0− 0 − 0 − b 0 m2W
(cid:0) (cid:1)
m2
3B (1)+B (3) 4B (2)+(m2 m2)C Φ0
× 0 0 − 0 b − t 0 #m2W
(cid:16) (cid:17)
(m2 m2) m4
b − t (B (1) B (2)) m2 C (m2 m2)C Φ0, (31)
− " m2W 0 − 0 − W 0− b − t 0#m4W
W(k1) W(k2)
t
Φ0(p) Φ0(p)
b H
t W(k2) W(k1)
(a)
W(k1)
Φ0(p)
W(k2)
(b)
FIG. 3: Feynmandiagrams contributingtotheΦ0 WW decay at one-loop level. (a) Dominant contributions. (b) Feynman
→
diagram corresponding to counterterm contributions.
BWW = 4 2+2(m2+m2)C B (2)+B (3) 4m2 C
−( b t 0− 0 0 − W 0
(cid:16) (cid:17)
(m2 m2)
4 b − t B (2)+B (3) 2B (1)+(m2 m2)C
− m2 0 0 − 0 b − t 0
W
(cid:16) (cid:17)
m4
B (2) B (3)+(m2 3m2 m2 )C 1 Φ0
− 0 − 0 t − b − W 0− m4
W
(cid:16) (cid:17)
(m2 m2)
2 b − t B (1)+B (2) 2(B (3)+(m2 m2)C )
− " m2W 0 0 − 0 b − t 0
(cid:16) (cid:17)
m2
B (2) B (3) 2(2m2+m2)C 3 Φ0 , (32)
− 0 − 0 − b t 0− #m2W )
(cid:16) (cid:17)
(4m2 m2 ) m2 (m2 m2)
CWW = W − Φ0 4 Φ0 +2(B (2) B (3)) b − t
2m2W " − m2W 0 − 0 m2W
m2 m2
b − t +1 (m2 2(m2+m2 m2))C . (33)
− (cid:18) m2W (cid:19) Φ0 − t W − b 0#
Notice that the form factors AWW,BWW, and CWW are finite, being, B (1) = B (0,m2,m2),B (2) =
0 0 b t 0
B (m2 ,m2,m2),B (3) = B (m2 ,m2,m2) and C = C (m2 ,m2 ,m2 ,m2,m2,m2), the Passarino-Veltman scalar
0 W b t 0 0 Φ0 t t 0 0 W W Φ0 t b t
8
functions. Thus, the decay width of the Φ0 WW process is
→
m2 4m2
Γ(Φ0 WW)= Φ0 − W (Φ0 WW)2. (34)
→ q 16πm2 |M → |
Φ0
Letusmentionthatthe contributiontothe one-loopamplitude arisingfromthe Higgsloopis exactlycanceledbythe
respective counterterm.
3. One-loop level Φ0 γV decays
→
We will now describe the analytical expressions for the decay amplitudes of the Φ0 γV process, which were
→
computed in the unitary gauge. In the LTHM these decays are only mediated by SM quarks and a new exotic top
quark T. In specific, the heavy top quark circulating inside the fermionic-triangle loop for the Φ0 γZ process
→
induces anon-zerocontribution. However,intheamplitude, the contributionscomingfromthisnew exotictopquark
are suppressed at least by two orders of magnitude with respect to the SM top quark contributions, which are the
dominant ones between SM quarks. Therefore, we do not include this new exotic contribution in our calculations. In
Fig. 4weshowthe FeynmandiagramscorrespondingtoΦ0 γγ andΦ0 γZ decays;insidethe loopsiscirculating
SM top quark. The contribution to Φ0 γγ with only fluc→tuating T qua→rk is null since there is no present the Ttγ
→
coupling.
γ(k1) γ(k2)
t t
Φ0(p) Φ0(p)
t t
t γ(k2) t γ(k1)
(a)
γ(k1) Z(k2)
t t
Φ0(p) Φ0(p)
t t
t Z(k2) t γ(k1)
(b)
FIG. 4: Dominant Feynman diagrams contributingto theΦ0 γγ and Φ0 γZ decay at one-loop level.
→ →
The amplitude for the Φ0 γγ process is given by
→
(Φ0 γγ)=Aγγ(k k gµν kνkµ)ǫ (k )ǫ (k ). (35)
M → 1· 2 − 1 2 ∗µ 1 ∗ν 2
The coefficient Aγγ is expressed in terms of Passarino-Veltman(PV) scalar functions as it is observed below
g2N s2 m2
Aγγ = c W t ((m2 4m2)C (1)+2), (36)
9√2π2fm2 Φ0 − t 0
Φ0
where C (1) = C (m2,0,0,m2,m2,m2) is the three-point PV scalar function. After performing some algebraic
0 0 Φ t t t
operations, the decay width of the Φ0 γγ process can be expressed as
→
Aγγ 2m3
Γ(Φ0 γγ)= | | Φ0. (37)
→ 64π
Let us study the Φ0 γZ decay. The respective amplitude after some algebraic manipulations can be written as
→
M(Φ0 →γZ) = AγZ(k1·k2gµν −k1νk2µ)ǫ∗µ(k1)ǫ∗ν(k2). (38)
The coefficient AγZ is explicitly given by
g2N s (3 8s2 )m2m2 m2 m2
AγZ = c W − W t Z (B (3) B (4))+ Φ0 − Z(C (2)(4m2+m2 m2 )+2) , (39)
36√2π2c f(m2 m2)2 0 − 0 2m2 0 t Z − Φ0
W Φ0 − Z (cid:16) Z (cid:17)
9
where B (4) = B (m2,m2,m2) and C (2) = C (m2 ,m2,0,m2,m2,m2). Finally, the decay width for this process
0 0 Z t t 0 0 Φ0 Z t t t
can be read as follows
AγZ 2(m2 m2)3
Γ(Φ0 γZ)= | | Φ0 − Z . (40)
→ 32πm3
Φ0
It should be stand out that the Aγγ and AγZ coefficients are free of UV divergences and the Lorentz structures
appearing in Eqs. (35) and (38) satisfy gauge invariance. The numerical evaluation of the branching ratios for the
Φ0 γV processes was carried out by means of LoopTools package [40].
→
IV. NUMERICAL RESULTS
In this work we propose a scenario where m is around 1 TeV so that the one-loop level decays of the scalar into
Φ0
WW,γγ and γZ could help us to discern on the validity of the model or whether this model should be ruled out.
In the plots that Fig. 5 shows, it is clearly observed that the decay widths of the Φ0 WWZ and Φ0 ZZH
→ →
processes are not sensible to c parameter variations. In Fig. 5, a variation of the c parameter between 0.1 and 0.9 is
performed, for three distinct energy scales, i.e. f = 2 TeV, f = 3 TeV and f = 4 TeV. Since only the Φ0 WWZ
and Φ0 ZZH processes depend of c and these are not the dominant decay modes of the Φ0 scalar p→article, it
→
is concluded that essentially the total decay width of the Φ0 boson will not depend of the c parameter. Thus,
we will assume a specific value of c in such a way that the Z contribution to Φ0 WWZ and Φ0 ZZH is
H
→ →
magnified. This election seeks to avoidthe decoupling in the Φ0Z Z, HZ Z andZ WW interactions. Accordingly
H H H
with Fig. 5, it is assumed c = 0.85, which corresponds to a small mixing angle and is consistent with a enhanced
contribution of the Z boson to Γ(Φ0 WWZ) and Γ(Φ0 ZZH). In Fig. 6(a), the decay widths for the
H
Φ0 HH,ZZ,tt,WWZ,WWH,ZZH,W→W,γγ,γZ process ar→e shown. In Fig. 6(a), it can be observed that the
dom→inantcontributions come from two-body decaysof Φ0 into HH and ZZ,which areof the orderof unities of GeV
on the interval 2 TeV < f < 4 TeV. The two-body decay widths of the Φ0 boson were calculated for the first time
in Ref. [41], however, these were not studied as a function of the energy scale f. We can observe that Γ(Φ0 WW)
is of the order of 10 2 GeV over the interval 2 TeV < f < 4 TeV. Conversely, the decay widths of the Φ→0 γγ
−
and Φ0 γZ processes are of the order of 10 7 GeV along the interval 2 TeV < f < 2.3 TeV. Abreast,→their
−
→
respective branching ratios as a function of the energy scale f are depicted in Fig. 6(b). Before continuing with the
analysis of results, we recall that m is a function of the energy scale f at which the global symmetry is broken.
Φ0
Notice that the energy scale f is the only free parameter with which we can play. The symmetry breaking scale is
constrained by the experimental data being restricted to lower limits around 2-4 TeV [35]. Since the mass of the Φ0
scalarbosonis a monotonousincreasingfunction in their dependence onthe f parameter,inour analysiswe consider
the range of study to be 2 TeV< f <4 TeV. Using this information, we obtain that the mass of the heavy scalar
is ranging from 1.66 TeV to 3.32 TeV. The total decay width of the Φ0 boson includes the following decay modes:
HH,ZZ,¯tt,WWZ,WWH,ZZH.
4.5
4.0
WWZ (f=2 TeV)
3.5 ZZH (f=2 TeV)
WWZ (f=3 TeV)
ZZH (f=3 TeV)
V] 3.0 WWZ (f=4 TeV)
e ZZH (f=4 TeV)
G
X) [ 2.5

0fi 2.0
F(
G 1.5
1.0
0.5
0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
c
FIG. 5: Decay widthsfor theΦ0 WWZ and Φ0 ZZH processes as a function of thec parameter for f =2,3,4 TeV.
→ →
FromFig.6(b), it is clearlyseenthatthe γγ andγZ branchingratiosaresuppressedaround6 ordersofmagnitude
compared to the dominant decay modes. In contrast, the associated branching ratio for the Φ0 WW process is
→
10
mF 0 [TeV] mF 0 [TeV]
1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2
101 100
100 10−1
10−1 10−2
0fi X) [GeV] 111000−−−432 WWWWWHZWHHZZtt 0FfiBr( X) 111000−−−543 WWWWWHZWHHZZtt
GF( 10−5 ZZgggHZ 10−6 ZZgggHZ
10−6 10−7
10−7 10−8
10−8 10−9
2.0 2.5 3.0 3.5 4.0 2.0 2.5 3.0 3.5 4.0
f [TeV] f [TeV]
(a) (b)
FIG. 6: (a) Decay widths for the Φ0 X processes as a function of the f parameter, where X =
HH,ZZ,WWZ,t¯t,WW,WWH,ZZH,γγ,γZ. (→b) Branching ratios for the Φ0 X processes depending on the f param-
→
eter.
of the order of 10 3 throughout the interval 2 TeV < f < 4 TeV. Therefore, for these one-loop level processes, the
−
decay channel corresponding to two SM W bosons would be of great interest for experimental studies in production
of hypothetical heavy neutral scalar bosons.
It is important to add that in the calculation of the total decay width of the Φ0 scalar boson it is found that the
subdominantdecaysare: Φ0 ZZH,Φ0 WWZ,Φ0 WWZ andΦ0 tt¯, withdecay widths being ofthe order
→ → → →
of 10 1 GeV for f between 2 TeV and 3 TeV.
−
IsrelevanthighlightthatintheLTHMthelowerlimitforthef parameterallowsustoexploreheavyneutralscalar
masses greater than 1 TeV.
A. Production of the heavy neutral scalar boson
ThissectionprovidestheresultsfortheproductioncrosssectionoftheheavyscalarΦ0 inthe contextoftheLTHM
at LHC. To carry out this task, we used the version2.1.1 of the WHIZARD package, which is a generic Monte-Carlo
eventgeneratorformulti-particleprocessesathigh-energycollisions[42]. Thiseventgeneratorwasusedinaprevious
workofusforthecalculationoftheassociatedproductioncrosssectionoftheZ neutralgaugebosonasaproofofthe
H
proper functioning of the WHIZARD package [36], where employed the CTEQ5 parton distribution function [43]. In
thiscalculationwesimulateproton-protoncollisionsatacenter-of-massenergyof14TeVwiththepartondistribution
function MSTW2008NLO [44].
In our simulations we calculate the cross secction as a function of the f parameter in the range values between 2
TeV and 2.5 TeV, as it is shown in Fig. 7. The top axis represents the heavy scalar mass throughout the interval
1.66 TeV< m < 2.1 TeV. In order to do this simulation we employed values from f = 2 TeV until f = 2.5 TeV.
Φ0
The behavior of the cross section is depicted in the Fig. 7, where we can see that the cross section is equal to 3.37
fb for f = 2 TeV and the corresponding value of the cross section for f = 2.5 TeV is equal to 1.66 TeV. As it was
shown before, the branching ratios for the Φ0 γγ,γZ decays are very suppressed with respect to dominant decays
→
of the Φ0 boson, therefore, these would not be of experimental interest. On the other hand, the branching ratio for
the Φ0 WW process is of the order of 10 3 between 2 TeV < f < 2.5 TeV, which results interesting because the
−
LHC int→egratedluminosityatthe finalstageofoperationisexpectedtobe 3000fb 1 ormaybemore[45]whichcould
−
reverse this plight. Continuing with the feasibility analysis of the LTHM, let us consider the smallest value studied
for the mass of the scalar boson Φ0 which corresponds to the maximum value analyzed for the Φ0 production cross
section σ(pp Φ0X). For these values, in the Gaussian approximation, it is found around unities of events for the
Φ0 WW d→ecay.
→
In this work is not intended to study in detail the mechanismof eventproduction since we are doing anestimation
exercise at best. Nevertheless, this analysis allows us to say that it would be difficult to observe a heavy neutral
scalar resonance of the LTHM decaying into two W bosons at the LHC. Thus, it would be of great interest to study
the production of the Φ0 scalar boson in the context of the Compact Linear Collider since it would offers cleanest
collisions with a considerably large integrated luminosity [46].