Table Of ContentDiboson excess and Z(cid:48)–predictions via left-right non–linear Higgs
Jing Shu1,∗ and Juan Yepes1,†
1State Key Laboratory of Theoretical Physics and Kavli Institute for Theoretical Physics China (KITPC)
Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, P. R. China
The excess events reported by the ATLAS Collaboration in the WZ–final state, and by the
CMS Collaboration in the e+e−jj, Wh and jj–final states, may be induced by the decays of a
heavy boson W(cid:48) in the 1.8–2 TeV mass range, here modelled via the larger local group SU(2) ×
L
SU(2) ×U(1) in a non–linear dynamical Higgs scenario. The W(cid:48)–production cross section at
R B−L
the13TeVLHCisaround700–1200fb. ThisframeworkalsopredictsaheavyZ(cid:48) bosonwithamass
of2.5–4TeV,andsomedecaychannelstestableintheLHCRunII.Wedeterminethecrosssection
times branching fractions for the dijet, dilepton and top–pair Z(cid:48)–decay channels at the 13 TeV
LHC around 2.3, 7.1, 70.2 fb respectively, for MZ(cid:48) = 2.5 TeV, while one/two orders of magnitude
smallerforthedijet/dileptonandtop–pairmodesatMZ(cid:48) =4TeV.Non-zerocontributionsfromthe
6 effectiveoperators,andtheunderlyingHiggssectorofthemodel,willinducesizeableenhancement
1 in the W+W− and Zh–final states that could be probed in the future LHC Run II.
0
2
n I. INTRODUCTION b) can be understood [8, 12–14] through the process
a pp → W(cid:48) → Ne → eejj [15], and for a charged gauge
J TantalizingdeviationsfromtheSMpredictionshavebeen boson mass MW(cid:48) ∼ 2 TeV, with gR < gL at the TeV-
6 recently reported by the ATLAS and CMS Collabora- scale [8]. Finally, the dijet excess (item d) may simply
2 tionsaroundinvariantmassof1.8–2TeV,andareclaim- be yielded by W(cid:48) →jj.
ing for: Theobservedexcesseventsareinterpretedinthiswork
h] asbeinginducedbythedecaysofaheavybosonW(cid:48) with
p a) 3.4σ local (2.5σ global) excess in the ATLAS a mass range 1.8–2 TeV, where the underlying frame-
- search[1](CMSreportsaslightexcessatthesame work relies in a non–linearly realized left–right model
p
mass [2]) for a heavy resonance W(cid:48) decaying as coupled to a light Higgs particle. Calling for the larger
e
h W(cid:48) →WZ →JJ, where J stands for two colinear localgroupG =SU(2)L×SU(2)R×U(1)B−L inanelec-
[ jets from a W or Z–boosted decay; troweak non–linear σ–model, the Goldstone bosons are
parametrized as customarily via the dimensionless uni-
1
b) 2.8σ excess in the CMS search [3] for a heavy right tarymatricesU (x)andU (x)forthesymmetrygroup
v L R
handed boson W(cid:48) decaying into an electron and a SU(2) ×SU(2) , and defined as
1 L R
9 right handed neutrino N, as W(cid:48) →Ne→eejj;
8 UL(R)(x)=eiτaπLa(R)(x)/fL(R), (1)
6 c) 2.2σ excess in the CMS search [4] for W(cid:48) → Wh,
0 with a highly boosted SM Higgs boson h decaying with πLa(R)(x) the corresponding GB fields suppressed
. as h→b¯b and W →(cid:96)ν (with (cid:96)=e,µ); by their associated non–linear sigma model scale f .
1 L(R)
In addition, this non–linear effective set–up is coupled a
0
6 d) 2.1σ excess in the CMS dijet search [5]. posteriori to a Higgs scalar singlet h through powers of
1 h/f [16], via the generic light Higgs polynomial func-
L
: InspiteofrequiringmorestatisticsattheLHCRunIIto tions F(h) [17]
v
shed light on their real origin, and being not significant
i
X enough to point out BSM new phenomenon, it is worth- h h2 (cid:16)h3(cid:17)
F (h)≡1+2a +b +O . (2)
r whiletoexplorewhichfeaturesaremotivatedbysuchde- i i fL i fL2 fL3
a viationsinagiventheoreticalframework. Inthisregard,
many models and scenarios have been proposed. Among This work is split into: Sect. II describes the EW ef-
them, the left–right EW symmetric model, based on the fective Lagrangian following the light dynamical Higgs
gauge group G = SU(2) ×SU(2) ×U(1) [6, 7], picture in [17–21] (see also Ref. [22–24] and [25] for a
L R B−L
seems to address properly the observed excesses in all Higgs portal to scalar dark matter in non-linear EW ap-
the mentioned decay channels. Indeed, the WZ excess proaches),focusedonlyintheCP–conservingbosonicop-
(item a) and Wh excess (item c) can be tackled [8–10] erators1. The mixing effects for the gauge masses trig-
viaW(cid:48) →WZ, Wh,astheimpliedcouplingsarisenatu- geredbytheLRHoperatorsandthecorrespondinggauge
rallyinthesemodels(see[11]forsomealternativeexpla- physical masses are also analysed there. Sect. III anal-
nations of the diboson excess). The eejj excess (item yses the W(cid:48)–production and the constraints on the pa-
rameter space of our scenario entailed by the reported
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] 1 See [20,23,26,27]fornon–linearanalysisincludingfermions.
2
excesses in the WZ and Wh–final states. Sect. IV ex- introduced in the scenario once the SM local symmetry
plores the prediction of a heavy boson Z(cid:48) in the model, group G is extended to G. The associated fermion ki-
SM
its possible mass range and the implied dijet, dilepton netictermsaredescribedbythe3rdand2ndlinesin(4)-
and top–pair decay channels. The less dominant decays (5) respectively, with the quark and lepton doublets qi
Z(cid:48) →{W+W−, Zh},andthesizeableenhancementthey and li (i stands for fermion generations) defined as
cansufferbythephysicalimpactofnon-zerocontribution
from the effective non–linear operators is also analysed. (cid:18)ui (cid:19) (cid:18)νi (cid:19)
qi = L ∼(2,1,1/6), li = L ∼(2,1,−1/2),
Finally, Sect. V summarizes the main results. L di L ei
L L
II. EFFECTIVE LAGRANGIAN qi =(cid:18)uiR (cid:19)∼(1,2,1/6), li =(cid:18)NRi (cid:19)∼(1,2,−1/2),
R di R ei
R R
(8)
The NP departures with respect to the SM Lagrangian
L andwillbeencodedinthisworkthroughtheeffective where it have been specified the transformation prop-
0
Lagrangian erties under the group G corresponding to the usual
fermion representation for the left-right models. The
Lchiral =L0 + L0,R + ∆LCP + ∆LCP,LR. (3) right-handed neutrinos NRi acquire masses at the TeV
scalethroughthemechanismofRef.[28]. Thescalarsec-
The first three pieces in Lchiral read as torincludesingeneralanSU(2)RdoubletχRwhoseVEV
around several TeV triggers the breaking of SU(2) ×
R
L = U(1) downtotheSMhyperchargegroupU(1) ,plus
0 B−L Y
− 41BµνBµν − 14Wµaν,LWLµν,a− 14GaµνGµν,a+ abrbeaidkoinugbleattΣthwehwoeseakVsEcValetr(igsegeer[s2t9h]efoSrUm(2o)rLe×deUta(i1l)sY).
The corresponding covariant derivatives are given by
1 f2 (cid:16) (cid:17)(cid:18) h (cid:19)2
+ 2(∂µh)(∂µh)−V(h)− 4LTr VLµVµ,L 1+ fL + Dµψχ ≡ ∂µψχ + 2i gχWχµ,aτχaψχ+ig(cid:48)BµYB−Lψχ,
+iq¯ D/q +i¯l D/l , (9)
L L L L where τa and Y correspond to the SU(2) and
(4) χ B−L χ
U(1) generators, with χ ≡ L, R, and the fermion
B−L
field ψ standing for ψ ≡ q, l. Other fermion ar-
L = rangements, dictated either by leptophobic, hadropho-
0,R
1 f2 (cid:16) (cid:17)(cid:18) h (cid:19)2 bic, fermionphobic [30–32], ununified [33] or non-
− Wa Wµν,a − R Tr Vµ V 1+ + universal [34] are also possible and are beyond the scope
4 µν,R R 4 R µ,R f
L of this work.
+iq¯ D/q +i¯l D/l , Operators mixing the LH and RH-covariant are also
R R R R constructable in this approach via the proper insertions
(5)
oftheGoldstonematricesU andU ,morespecifically,
where the adjoints SU(2) –covariant vectorial Vµ L R
L(R) L(R) through the following definitions [21]
and the covariant scalar T are defined as
L(R)
Vµ ≡(DµU ) U† , T ≡U τ U† , (6) V(cid:101)χµ ≡U†χVχµUχ, T(cid:101)χ ≡U†χTχUχ, (10)
χ χ χ χ χ 3 χ
withχ=L,Randthecorrespondingcovariantderivative
for both of the Goldstone matrices UL(R)(x) introduced W(cid:102)χµν ≡U†χWχµνUχ, (11)
as
whereWµν ≡Wµν,aτa/2. Non–zeroNPdepartureswith
χ χ
DµU ≡∂µU + i g Wµ,aτaU −i g(cid:48)BµU τ3 (7) respect to those described in L0 + L0,R + L0,LR will
χ χ 2 χ χ χ χ 2 χ be parametrized through the remaining last two pieces
in (3), i.e. ∆L and ∆L . The former contains
CP CP,LR
where the SU(2) , SU(2) and U(1) gauge fields
L R B−L LH and RH covariant objects up to the p4–order as
are denoted by Waµ, Waµ and Bµ correspondingly, and
L R
the associated gauge couplings gL, gR and g(cid:48) respec- ∆L =∆L +∆L . (12)
tively. The scale factor of Tr(VµV ) entails GB– CP CP,L CP,R
L µ,L
kinetic terms canonically normalized, in agreement with
The latter can be further written down as
the U –definition in (1). The corresponding SU(2) –
L R
counterparts for the strength gauge kinetic term and the 26
(cid:88) (cid:88)
custodial conserving operator at the Lagrangian L are ∆L =α P (h)+ α P (h)+ α P (h)
0 CP,L B B i i,L i i,L
parametrizedbyL0,R in(5),entailingthusanadditional i={W,C,T} i=1
scale f that encodes the new high energy scale effects (13)
R
3
∆LCP,R = (cid:88) βiPi,R(h) + (cid:88)26 βiPi,R(h). (14) Ttehrme coofm∆pLlete set ohfavoepebreaetnorfsulPlyi(ja)n,LdRl(isht)edininth[2e1s]e.cTonhde
CP,LR
i={W,C,T} i=1 corresponding CP–violating counterparts of ∆L and
CP
∆L havebeencompletelylistedandstudiedin[37].
Themodel–dependentconstantcoefficientsα andβ are CP,LR
i i Notice that in the unitary gauge, non-zero mass mix-
denoting correspondingly the weighting coefficients for
ing terms among the LH and RH gauge fields are trig-
the LH and RH operators, whilst the first two terms of
gered by the operator P (h), leading to diagonalize
∆L in (13) and the first term in (14) can be jointly C,LR
CP,L the gauge sector in order to obtain the required physical
written as
gauge masses.
g(cid:48)2
P (h) =− B BµνF (h),
B 4 µν B
g2
P (h) =− χ Wa Wµν,aF (h),
W,χ 4 µν,χ χ W,χ
(15)
f2 (cid:16) (cid:17) A. Charged and neutral gauge masses
P (h) =− χTr VµV F (h),
C,χ 4 χ µ,χ C,χ
P (h) = fχ2 (cid:16)Tr(cid:16)T Vµ(cid:17)(cid:17)2F (h), The gauge basis is defined by
T,χ 4 χ χ T,χ
W3
withsuffixχlabellingagainasχ=L,R,andthegeneric W± µ,L
Falil(thh)e–foupnecrtaiotonrsoffotlhloewsincagladrefisninitgiloent h(2)i.s Ninotrgolduuocneidc ofopr- W(cid:99)µ± ≡Wµ±,L , N(cid:98)µ ≡Wµ3,R (18)
erator has been included in ∆LCP,L. The contribution µ,R Bµ
∆L hasalreadybeenprovidedin[17,18]inthecon-
CP,L
text of purely EW chiral effective theories coupled to a
light Higgs, whereas part of ∆LCP,L and ∆LCP,R were where the charged fields Wµ±,χ are introduced as usual
partially analysed for the left–right symmetric frame-
works in [35, 36], and finally completed in [21]. W1 ∓ iW2
Finally, ∆L parametrizes any possible mixing W± ≡ µ,χ √ µ,χ , χ=L, R. (19)
CP,LR µ,χ
2
interacting term between the SU(2) and SU(2) –
L R
covariant objects up to the p4–order in the Lagrangian
expansion, permitted by the underlying left–right sym- The mass eigenstate basis is defined as
metry, and encoded through
A
(cid:88) (cid:88)26 (cid:32) W± (cid:33) µ
∆LCP,LR =i={W,C,T}γiPi,LR(h)+i=2,i(cid:54)=4γi(j)Pi(j),LR(h) Wµ± ≡ Wµ(cid:48)± , Nµ ≡ Zµ , (20)
µ
(16) Z(cid:48)
µ
where the index j spans over all the possible operators
that can be built up from the set of 26 operators P (h)
i,χ and it can be linked to the gauge basis through the fol-
in (13)–(14), and here labelled as P (h) together
i(j),LR lowing field transformations
withtheircorrespondingcoefficientsγ . Thefirstterm
i(j)
in ∆L encodes the non-linear mixing operators
CP,LR
W(cid:99)µ± ≡RWWµ±, N(cid:98)µ ≡RN Nµ. (21)
1 (cid:16) (cid:17)
PW,LR(h) =−2gLgRTr W(cid:102)LµνW(cid:102)µν,R FW,LR(h),
The mass matrices for the charged and neutral sector in
the gauge basis are
1 (cid:16) (cid:17)
PC,LR(h) = 2fLfRTr V(cid:101)LµV(cid:101)µ,R FC,LR(h), g2 f2 1+αC −√γCλ g2 f
M = L L , λ≡ L (cid:15)2, (cid:15)≡ L .
PT,LR(h) = 12fLfRTr(cid:16)T(cid:101)LV(cid:101)Lµ(cid:17)Tr(cid:16)T(cid:101)RV(cid:101)µ,R(cid:17)FT,LR(h) W 4 −√γCλ (1+λβC) gR2 fR
(17) (22)
4
(cid:16) (cid:17)
1+α −√γ −g(cid:48) 1+α− fRγ
λ gL fL
(cid:16) (cid:17)
MN = gL24fL2 −√γλ 1+λβ gLg√(cid:48)λ γ− ffRL (1+β) (23)
−g(cid:48) (cid:16)1+α− fRγ(cid:17) g√(cid:48) (cid:16)γ− fR (1+β)(cid:17) g(cid:48)2 (cid:16)1+α− 2fRγ+ fR2 (1+β)(cid:17)
gL fL gL λ fL gL2 fL fL2
with the definitions
α≡α −2α , β ≡β −2β , γ ≡γ +2γ . (24)
C T C T C T
Therotationmatrixforthechargedsectorcanbewritten out to be depending on the masses ratio M2 /M2
W W(cid:48)
down as through the parameter λ and the mixing coefficient γ
C
in (22) as
c −s
ζ ζ
√
RW = , cζ ≡cosζ, sζ ≡sinζ. λ g2 f2 M2
s c tanζ =− γ , λ≡ L L (cid:39) W . (31)
ζ ζ 1−λ C g2 f2 M2
(25) R R W(cid:48)
For the neutral sector the rotation is dictated by the The neutral gauge masses are
Euler-angles parametrization in terms of three angles:
the Weinberg mixing angle θW, and the analogous mix- M2 (cid:39) MW2 , M2 (cid:39) MW2 (cid:48) (cid:0)1−2s2 (cid:15)γ(cid:1) (32)
ing angle θR for the SU(2)R×U(1)B−L subgroup, both Z c2W Z(cid:48) c2R R
defined as
g g with the coefficient γ introduced in (24). The well mea-
cosθW ≡cW = (cid:112)g2L+g2 , sinθW ≡sW = (cid:112)g2Y+g2 sured MZ–mass strongly constrains additional contribu-
L Y L Y tions from the operators P (h) and P (h) in (32).
(26) C,L T,L
Similarly, the M –mass bounds tightly constrains the
W
cosθR ≡cR = (cid:112) gR , sinθR ≡sR = (cid:112) g(cid:48) . conAtsriibtuctaionnboefnPoCt,icRe(dh)frionm(3(03)2.), the Z(cid:48)-mass turns out
g2 +g(cid:48)2 g2 +g(cid:48)2
R R tobelargerwithrespecttotheW(cid:48)-mass,i.eM >M .
(27) Z(cid:48) W(cid:48)
In addition, a mass range for the neutral gauge field Z(cid:48)
The third angle φ can be linked to the latter two up to
can be predicted in terms of the W(cid:48)–mass and the gauge
O((cid:15)2γ2)–contributions through
couplings g and g , via the mixing angle θ in (27)
R Y R
tanφ (cid:39) (cid:15) gL cR (cid:0)(cid:15)s2 −γ(cid:1) . (28) and the link among the SU(2)L, U(1)B−L and the SM
g c R hypercharge gauge couplings as
R W
The rotation matrix for the neutral sector becomes 1 1 1
+ = . (33)
parametrized then as g2 g(cid:48)2 g2
R Y
s c (cid:15)c gL (cid:0)γ−(cid:15)s2(cid:1)
W W R gR R The observed excess at the ATLAS and CMS Collabora-
RN (cid:39)cW sR −sRsW − ggRL ccW2R (cid:15)γ cR(cid:16)1− ggRL sRcWsW (cid:15)γ(cid:17) . tpiroentsedatrooubnedinidnvuacreidanbtymaWass(cid:48)–ocfon1t.8ri–b2utTioenV. cTahnebcoeuipnltienrg-
g will determine the strength of the couplings among
c c c (cid:16)−s + gL sR (cid:15)γ(cid:17) −s − gL sWc2R (cid:15)γ thRe W(cid:48) and fermions fields, and therefore it will control
R W R W gRcW R gR cW aswelltheproductionrateofW(cid:48)–resonancesviathepro-
(29)
cess pp→W(cid:48) analysed in the following section.
with the coefficient γ encoding the contributions in-
duced by the left–right custodial conserving and custo-
dial breaking operators P (h) and P (h) respec-
C,LR T,LR III. W(cid:48)–PRODUCTION
tively (defined in (24)). Such contributions are sup-
pressed by the scale ratio (cid:15). In the limit f (cid:28) f , the
L R
charged gauge masses are By considering the charged currents from the La-
grangians L and L in (4) and (5) respectively, we
0 0,R
1 (cid:16) (cid:17) 1 (cid:16) (cid:17)
M2 (cid:39) g2 f2 1−λγ2 , M2 (cid:39) g2 f2 1+λγ2 . have
W 4 L L C W(cid:48) 4 R R C
(30) 1 (cid:16) √ (cid:17)
L =−√ u¯γµ g λγ P −g P dW(cid:48) + h.c.,
where the masses have been expanded up to M2 /M2 - udW(cid:48) 2 L C L R R µ
W W(cid:48)
terms. The mixing angle ζ for the charged sector turns (34)
5
section productions are
• AtM ∼1.8TeV,around∼0.25pb, 1.2pb, 1.5pb
√W(cid:48)
at s =8-13-14 TeV respectively;
• At M ∼2TeV, around ∼0.13pb, 0.7pb, 0.9pb
W(cid:48)
and at the same c.o.m energies correspondingly.
Thecouplingg canbedeterminedfromthecrosssection
R
required to produce the dijet resonance near M . The
W(cid:48)
CMSdijetexcess[39]atamassinthe1.8–1.9TeVrange
indicates that the W(cid:48) production cross section times the
dijet branching fraction is in the 100–200 fb range (this
FIG. 1: W(cid:48)-production cross section via the process pp→W(cid:48)
is consistent with the ATLAS dijet result [40], which
as a function of MW(cid:48), for gR = 0.5 and at the 8-13-14 TeV shows a smaller excess at 1.9 TeV). This was assumed
LHC (black, blue and red curves respectively). Departures with
in Refs. [8, 10] to be the range for σ(pp → W(cid:48) → jj),
respect to the vanishing γC-case are suppressed by MW/MW(cid:48) where j is a hadronic jet associated with quarks or anti-
andcanbeentirelyneglectedfromtheproductioncrosssection.
quarksotherthanthetop. BycomparingtheW(cid:48)produc-
tion cross section to the CMS dijet excess, the coupling
g was determined in the range g ≈ 0.45−0.6 [8]. A
R R
where a flavour diagonal couplings have been assumed similar range is obtained by computing the dijet decay
and the family indices are implicit, with P ≡ channel of a W(cid:48) in our scenario, and it will be assumed
L(R)
(cid:0)1∓γ5(cid:1)/2. The W(cid:48)-production cross section through henceforth. Such range, together with a W(cid:48)–boson mass
the process pp → W(cid:48) can be computed from the La- nearby 1.8–2 TeV, can be translated via the W(cid:48) mass
grangianin(34)byusingMadGraph5andimplementing formula in (30) into the relation
the scale-dependent K-factors calculated in [38]. They
√
are in the ranges K ∈ [1.32, 1.37] at s = 8 TeV 3.6–4TeV
f ≈ ≈ 6−8TeV. (35)
and K ∈ [1.23, 1.25] at 13-14 TeV. Fig. 1 shows the R g
R
W(cid:48)-production cross section for g = 0.5 at the center-
R
of-mass (c.o.m) energies 8-13-14 TeV LHC (black, blue The W(cid:48)-production via the decay modes pp → W(cid:48) →
and red curves respectively). The coefficient γ is run- WZ and pp → W(cid:48) → Wh, together with the observed
C
ning as γ = −1.0, 0, 1. In general, departures with excesses in the WZ and Wh–final states at ATLAS and
C
respec√t to the vanishing γC-case are suppressed by the CMS, allow us to infer ranges for the strength of the
ratio λ(cid:39)M /M ,andtheycanbeneglectedforthe associates operators contributing to those channels. The
W W(cid:48)
W(cid:48)-production. As it can be seen from Fig. 1, the cross latter can be described by the effective Lagrangians
(cid:16) (cid:17)
L =i g(1) W† W(cid:48)νZµ + g(2) W(cid:48)† WνZµ + g(3) Z Wµ†W(cid:48)ν + h.c. , (36)
WW(cid:48)Z WW(cid:48)Z µν WW(cid:48)Z µν WW(cid:48)Z µν
1 (cid:16) (cid:17) (cid:16) (cid:17)
L =− g(1) W† W(cid:48)µνh + h.c. + g(2) M W†W(cid:48)µh + h.c. , (37)
hWW(cid:48) M hWW(cid:48) µν hWW(cid:48) W µ
W
with V ≡ ∂ V −∂ V , for V ≡ W, W(cid:48), Z. The cor- the RH gauge filed content is integrated out from the
µν µ ν ν µ
responding couplings are collected in Table I. Only the physical spectrum [41]. We will keep henceforth the La-
LO Lagrangian L + L in (4)-(5) and the opera- grangiansin(4)-(5)andtheoperatorssetin(15)and(17)
0 0,R
tors set in (15) and (17) have been kept for simplicity. for the analysis below.
AdditionalcontributionsfromtheoperatorsP (h)and
i,L
P (h) (3rd and 2nd terms in Eq. (13)-(14)), and the
i,R
operators P (h) (2nd term in Eq.(16)) would lead
i(j),LR A. WZ and Wh excesses
toalargerparameterspaceanditisbeyondthescopeof
this work. Many of those operators are also irrelevant at
For a charge resonance around the TeV scale, the ratios
lowenergiesastheircontributionbecomenegligibleonce
M2/M2 and M2/M2 turns out to be negligible and
Z W(cid:48) H W(cid:48)
6
W(cid:48) →WZ Coeff. 100fb 200fb
<0 [−0.11, −0.06] [−0.07, −0.04]
gW(1)W(cid:48)Z 4ce2W (cid:16)seR2 γW + 2scWW MMWW(cid:48) γC(cid:17) γC >0 [0.06, 0.11] [0.04, 0.07]
(cid:16) (cid:17) <0 [−0.026, −0.018] [−0.018, −0.013]
gW(2)W(cid:48)Z −4se2W seR2 γW − 2csWW MMWW(cid:48) γC γW >0 [0.018, 0.026] [0.013, 0.018]
g(3) − e MW γ
WW(cid:48)Z cWsW MW(cid:48) C TABLE II: Allowed negative and positive ranges for the coef-
W(cid:48) →Wh ficients γC and γW (upper and lower rows) and for the values
σ (W(cid:48)) ∼ 100−200fb [40] (3rd & 4th columns). The values
jj
gh(1W)W(cid:48) 4cWes3Rs2W γ(cid:101)W σσWWZZ((WW(cid:48)(cid:48)))∼and3−th1e0fcboe[4ffi2c]i,etnhteseaqCu,LivRal=encaeWr,eLlRat=ion1/σ2W,hw(eWre(cid:48))im≈-
(cid:20) (cid:21) plemented for the W(cid:48) → WZ and W(cid:48) → Wh–decay widths
gh(2W)W(cid:48) −sWe MMWW(cid:48) γC+ MMW2W2(cid:48) (γ(cid:101)C−γC) in (38)-(39) with the relations in (40)-(41).
TABLE I: Effective couplings encoded by the Lagrangians
the coefficients (γ , γ ) to vary simultaneously, we ob-
LWW(cid:48)Z and LhWW(cid:48) in (36) and (37) respectively. The re- tain the allowed pCaramWeter space in Fig 2. The ranges
lations g = e , g = e , g(cid:48) = e have been imple-
L sW R cWsR cRcW arebasicallyofthesameorderofmagnitudesuggestedby
mented through all the couplings, with e the electromagnetic
theranges−0.02<γ <0.02and−0.016<γ <0.018
coupling constant. The coefficient γ stands for γ ≡ a γ , C W
(cid:101)i (cid:101)i i i obtained from the stringent EW constrains on the Z-
with i=C, W and a coming from the F(h)–definition in (2).
i
gauge masses and the S and T parameter bounds in [41]
respectively.
therefore the decay width for the processes W(cid:48) → WZ It is worth to point out the dependence of the ranges
and W(cid:48) →Wh become written as inTableIIandtheparameterspaceinFig2ontheHiggs
coefficientsa =a =1/2enteringinthehWW(cid:48)–
C,LR W,LR
c2 M5 (cid:16) (cid:17)2 couplingsthroughthelightHiggsfunctionin(2). Larger
Γ(W(cid:48) →WZ)= W W(cid:48) g(2) , (38)
192π M4 WW(cid:48)Z values aC,LR = aW,LR ∼ 1 will reduce (enhance) the al-
W
lowed positive (negative) ranges of γ by one order of
W
magnitude with respect to those in Table II in the range
g(1) (cid:18) M2 (cid:19)M5 σjj(W(cid:48))∼150−200fb, whereas part of the ranges of γC
Γ(W(cid:48) →Wh)= hWW(cid:48) g(1) + g(2) W W(cid:48) . willbeslightlymodifiedandsomeothercanreachsmaller
48π hWW(cid:48) hWW(cid:48) M2 M4
W(cid:48) W values close to zero for small values of γ . The limiting
(39) W
case a = a ∼ 0 enhances the γ –ranges in-
The cross sections for the processes pp → W(cid:48) → WZ C,LR W,LR W
stead, but keeping the same order of magnitude of the
and pp → W(cid:48) → Wh can be computed in terms of the
ranges in Table II though.
corresponding one for the decay pp→W(cid:48) →jj as
σ (W(cid:48)) Γ(W(cid:48) →WZ) σ (W(cid:48)) Γ(W(cid:48) →Wh)
WZ = , Wh = IV. Z(cid:48)–PREDICTIONS
σ (W(cid:48)) Γ(W(cid:48) →jj) σ (W(cid:48)) Γ(W(cid:48) →jj)
jj jj
(40)
A mass prediction for the neutral gauge field Z(cid:48) can be
with σ (W(cid:48)) ≡ σ(pp → W(cid:48) → XX). Neglect-
XX inferred from the relation (32) in terms of the W(cid:48)–mass
ing the M /M –corrections induced by the operators
W W(cid:48) andthegaugecouplingsg andg ,viathemixingangle
P (h) and P (h) (see Eq. (34)), the width for the R Y
C,LR T,LR θ in (27) and the relation in (33). Assuming the cou-
decay W(cid:48) → jj can be related to the process W(cid:48) → t¯b R
plingg intherangeg ≈0.45−0.6asdeterminedin[8]
through the Lagrangian in (34) as R R
and g ∼0.36, it is possible to predict the mass range
Y
Γ(W(cid:48) →jj)(cid:39)2Γ(W(cid:48) →t¯b)∼ gR2 M . (41) 2.5TeV < MZ(cid:48) < 4TeV. (42)
8π W(cid:48)
The prospectives in detecting a Z(cid:48)-signal in the futures
The Goldstone equivalence theorem requires Γ(W(cid:48) → colliderexperimentscanbetackledthroughthefermionic
Wh) (cid:39) Γ(W(cid:48) → WZ) up to kinematic factors. In decay channels Z(cid:48) → {νLν¯L, NRN¯R, (cid:96)+(cid:96)−, tt¯, jj}, and
this case the pp → W(cid:48) → Wh cross section satisfies via the gauge-scalar modes Z(cid:48) →{W+W−, Zh} as well,
σ (W(cid:48)) ≈ σ (W(cid:48)). Implementing in addition the and will be analysed in the following section.
Wh WZ
results in (38)-(40), and requiring the cross section val-
ues σ (W(cid:48)) ∼ 3−10fb implied by the ATLAS search
WZ
for pp → W(cid:48) → WZ → JJ [42] and σ (W(cid:48)) ∼ 100− A. Z(cid:48)-production decay modes
jj
200fb [40], we obtain the ranges for the coefficients γ
C
(γ =0)andγ (γ =0)inTableIIandassumingthe ByconsideringtheneutralcurrentsfromLagrangiansL
W W C 0
Higgs coefficient values a = a = 1/2. Letting and L in (4) and (5) respectively, it is possible to
C,LR W,LR 0,R
7
f gfLfLZ(cid:48) gfRfRZ(cid:48)
(cid:16) (cid:17) (cid:16) (cid:17)
u e sR −γMW 2c2W+1 e sR − 3cR +4γMW sW
6cW cR MZ(cid:48) cWsW 6cW cR sR MZ(cid:48) cW
(cid:16) (cid:17) (cid:16) (cid:17)
d e sR +γMW c2W+2 e c2R+2 −2γMW sW
6cW cR MZ(cid:48) cWsW 6cW cRsR MZ(cid:48) cW
N 0 − e
2cRcWsR
(cid:16) (cid:17)
ν − e sR +γMW 1 0
2cW cR MZ(cid:48) cWsW
(cid:16) (cid:17) (cid:16) (cid:17)
e − e sR −γ MW c2W e cR − sR −2γMW sW
2cW cR MZ(cid:48) cWsW 2cW sR cR MZ(cid:48) cW
TABLE III: Z(cid:48)-fermion-couplings from the Lagrangian in (43).
The relation Q = 1T3 + 1T3 +Y , with T3 ≡ 1τ3 ,
2 L 2 R Q L(R) 2 L(R)
emergesnaturallyfromthe fermion–photoncoupling inoursce-
nario and it has been employed in all the listed couplings. In
addition, the relations g = e , g = e , g(cid:48) = e have
L sW R cWsR cRcW
also been used, with e the electromagnetic coupling constant.
FIG. 2: Allowed parameter space (γC, γW) by combining the Notation c2W ≡cos(2θW) and c2R ≡cos(2θR) is implicit.
W(cid:48) → WZ and W(cid:48) → Wh–decay widths in (38)-(39) to-
gether with the relations in (40)-(41). The cross section val-
ues σ (W(cid:48)) ≈ σ (W(cid:48)) ∼ 3−10fb and σ (W(cid:48)) ∼ 100− Z(cid:48) →W+W−
Wh WZ jj
150−200fb (brown, orange and red charts respectively) have
(cid:16) (cid:17)
been implemented and assuming the Higgs coefficient values g(1) e MW MW sR −γ cW
a =a =1/2. WWZ(cid:48) 2cW MZ(cid:48) MZ(cid:48) cR sW
C,LR W,LR
g(2) e (cid:104)γ e2cR − MW (cid:16)MW sR −γ cW(cid:17)(cid:105)
WWZ(cid:48) cW W 2sRs2W MZ(cid:48) MZ(cid:48) cR sW
describe fermionic decay modes through Z(cid:48) →Zh
LffZ(cid:48) =f=u(cid:88),d,N,ν,ef¯γµ (gfLfLZ(cid:48)PL+gfRfRZ(cid:48)PR) fZµ(cid:48) . gh(1Z)Z(cid:48) 2cWe3scRRs2W γ(cid:101)W
(43) g(2) 2e (cid:104)sRsW +γ MW (cid:16)c2Rs2W +1(cid:17)− MZ(cid:48) (γ +γ)(cid:105)
ThecouplingsgfLfLZ(cid:48) andgfRfRZ(cid:48) arelistedinTableIII. hZZ(cid:48) sW cRcW MZ(cid:48) c2R c2W MW (cid:101)C
The self gauge and gauge-Higgs Lagrangians accounting
for the gauge–scalar modes will be described by TABLE IV: Effective couplings encoded at the Lagrangians
L = LWWZ(cid:48) and LhZZ(cid:48) in (44) and (45) respectively. The coef-
WWZ(cid:48) ficient γ(cid:101)i stands for γ(cid:101)i ≡ aiγi, with i = C, W and ai the
coefficient introduced in the F(h)–definition of (2).
(cid:16) (cid:17)
i g(1) W† WνZ(cid:48)µ + h.c. + ig(2) Z(cid:48) Wµ†Wν,
WWZ(cid:48) µν WWZ(cid:48) µν
(44) (couplings g(2) and g(1) ). These particular features
WWZ(cid:48) hZZ(cid:48)
enhance the corresponding leading order branching ra-
tios of Z(cid:48) → W+W− and Z(cid:48) → Zh for a non–vanishing
1 g(2)
LhZZ(cid:48) =−2MZ gh(1Z)Z(cid:48)ZµνZ(cid:48)µνh + hZ2Z(cid:48) MZZµZ(cid:48)(µ4h5). lefTt–hreighbtraonpcehriantgorfsra{cPtiCo,nLsR(ohf)t,hPeTZ,L(cid:48)R(bho)s,oPnWfo,LrRM(hW)}(cid:48).=
1.8−2 TeV, with g ≈ 0.45−0.6 and assuming a right
The corresponding couplings are collected in Table IV. R
handed neutrino mass2 m = m = m = 1.5
Contributions induced by the left–right custodial con- NRe NRτ ND
TeV,hasbeencomputedforthefermionicdecaychannels
serving operator PC,LR(h) and the custodial breaking Z(cid:48) → {ν ν¯ , N N¯ , (cid:96)+(cid:96)−, tt¯, jj}, and for the gauge-
P (h) (encoded by the coefficient γ) are suppressed L L D D
T,LR scalarmodesZ(cid:48) →{W+W−, Zh}inFig.3(upperplot).
bythemassesratioM /M foralltheZ(cid:48)–fermioncou-
W Z(cid:48) The Z(cid:48)–production cross section times branching frac-
plingsinTableIII.Suchcontributionsturnouttobesup-
tionsarecomputedatthe13TeVLHCandaredisplayed
pressedbyonefactorofM /M lesswithrespecttothe
W Z(cid:48) in Fig. 3 (lower plot). The coefficients γ and γ have
leading order terms for the pure gauge and gauge–Higgs C W
couplings in Table IV, but for the coupling g(2) , whose
hZZ(cid:48)
last term is enhanced by M /M due to the longitudi-
Z(cid:48) W
nal helicity components in the decay Z(cid:48) → Zh. On the 2 The Majorana masses mNe and mNτ turns out to be equal as
otherhand,thecontributionsinducedbythekineticleft– the Ne and Nτ–fields formR a DiracRfermion (see [29] for more
R R
right operator PW,LR(h) are not MW/MZ(cid:48)–suppressed details).
8
• At M = 4TeV, the cross sections of
Z(cid:48)
{0.2, 0.04, 0.73}fb for fermionic decay modes cor-
respondingly, and 0.01fb for gauge–scalar modes.
The total Z(cid:48)–production cross sections of ∼1.0fb
at M =4TeV, is dominated mainly by the dijet
Z(cid:48)
channel (71.7%) with complementary small contri-
butionsfromthetop–pairmode(4.6%)andlepton–
pair channel (20.7%), plus the W–pair and Zh
modes (1.4% both).
As it was pointed out before, and according to the cou-
plings in Table III, the fermionic decay channels are
slightly modified by the modifications induced by the
operators {P (h), P (h)} as the involved effec-
C,LR T,LR
tive couplings are suppressed by M /M . Nonethe-
W Z(cid:48)
less,sizeablecontributionsaretriggeredonthegaugeand
gauge-Higgsdecaymodesoncetheeffectiveoperatorsare
switched on (Table IV). Fig. 4 shows the induced effects
on the Z(cid:48)-production cross sections for a vanishing op-
erators {P (h), P (h)} but P (h), at the 13
T,LR W,LR C,LR
TeV LHC for M = 1.9 TeV. In particular, the corre-
W(cid:48)
sponding coefficient γ runs over the allowed parameter
C
space in Fig. 2 for γ =0 and at σ (W(cid:48))∼200fb (left
W jj
and right red charts), i.e, γ running over the ranges
C
[−0.07, −0.04] (upper plot) and [0.04, 0.07] (lower plot)
from Table II. We predict then
• In the negative range γ = [−0.07, −0.04], a
C
total Z(cid:48)–production cross sections of 68.1–66.2fb
at M = 2.5, TeV and 1.86–1.58fb at M =
FIG. 3: Branching fractions (upper plot) and Z(cid:48)–production Z(cid:48) Z(cid:48)
4TeV. There is an enhancement of (18.8–10.9)%
cross section times branching fractions (lower plot) at the 13
and (38.7–21.5)% in the W–pair and Zh modes
TeV LHC for MW(cid:48) = 1.8−2 TeV, with gR ≈ 0.45−0.6 and respectively at M = 2.5, TeV, while a raise of
assuming a right handed neutrino mass m = 1.5 TeV. The Z(cid:48)
ND (38.9–24.4)% and (82.6–49.7)% correspondingly at
coefficients γ and γ have been set to zero. All the bands
C W
correspond to the mass range MW(cid:48) =1.8−2 TeV (central line MZ(cid:48) = 4, TeV. This leads to an associated en-
ineachofthemcorrespondstoMW(cid:48) =1.9TeV).Thejj–bandis hancement in the total Z(cid:48)–production cross sec-
thesumofthepartialwidthsfor{uu¯, dd¯, ss¯, cc¯, b¯b},whileνν¯– tions of (6.9–3.9)% at M =2.5, TeV and (60.4–
Z(cid:48)
bandisthesumofpartialwidthsintoSMneutrinos. Thebands 36.8)%atM =4, TeVwithrespecttothevanish-
Z(cid:48)
labelled with several decay modes stand for individual channels. ing operator case (thick lines in Fig. 4 upper plot).
• In the positive range γ = [0.04, 0.07], a to-
C
tal Z(cid:48)–production cross sections of 64.6–66.7fb at
been set to zero. All the bands in both plots correspond M =2.5, TeV and 1.86–1.58fb at M =4TeV.
Z(cid:48) Z(cid:48)
to the mass range MW(cid:48) = 1.8−2 TeV (central line in Anenhancementof(2.2–10.3)%and(9.6–29.5)%in
each of them corresponds to MW(cid:48) = 1.9 TeV). Fig 3 the W–pair and Zh modes respectively at MZ(cid:48) =
shows a preferred dijet decay channel rather than the 2.5, TeV, while a raise of (11.4–28.9)% and (33.4–
top and lepton pair final states respectively. We predict 73.7)% correspondingly at M = 4, TeV. Con-
Z(cid:48)
for MW(cid:48) =1.9TeV sequently, an enhancement is observed in the to-
tal Z(cid:48)–production cross sections of (1.4–4.7)% at
• Z(cid:48)-production cross sections of {2.3, 7.1, 70.2}fb
M =2.5, TeV and (22.2–51)% at M =4TeV
Z(cid:48) Z(cid:48)
at M =2.5TeV, through the lepton–pair, top–
Z(cid:48) with respect to the vanishing operator case (thick
pair and dijet channels Z(cid:48) → {(cid:96)+(cid:96)−, tt¯, jj} re-
lines in Fig. 4 lower plot).
spectively,while0.98fbforthegauge–scalarmodes
Z(cid:48) → {W+W−, Zh}. The total Z(cid:48)–production SmalldeviationsfromtheGoldstoneequivalencetheorem
cross section of 81.7fb at M = 2.5TeV re- in the decay widths Γ(Z(cid:48) →W+W−) and Γ(Z(cid:48) →Zh)
Z(cid:48)
spectively, mainly dominated by the dijet chan- areinducedbythenon-zerocontributionsoftheeffective
nel (86%) with complementary small contributions operators {P (h), P (h)}. In addition, sizeable
C,LR W,LR
from the top–pair mode (8.7%) and lepton–pair enhancementistriggeredinthosechannelsduetotheef-
channel (2.8%), plus the W–pair and Zh modes fective operators contribution. Such departures become
(1.2% both). negligibleforsmallcoefficientsγ andγ , whoseranges
C W
9
V. CONCLUSIONS
The small mass peaks observed at ATLAS and CMS
near the 1.8-2 TeV is described here via a W(cid:48)–model in-
spiredbythelargerlocalgroupG =SU(2) ×SU(2) ×
L R
U(1) in a non–linear EW dynamical Higgs scenario.
B−L
The W(cid:48)–production cross section at the 13 TeV LHC is
around 700–1200 fb. We analysed the W(cid:48)–production
and the constraints on the parameter space of our sce-
nario entailed by the reported excesses in the WZ and
Wh–final states (Table II and Fig. 2). We predict the
existence of a heavy gauge boson Z(cid:48) in the 2.5–4 TeV
mass range as well as some of its decay channels testable
intheLHCRunII.Wedeterminethecrosssectiontimes
branching fractions, shown in Fig. 3, for the dijet, dilep-
ton and top–pair Z(cid:48)–decay channels at the 13 TeV LHC
around 2.3, 7.1, 70.2 fb respectively, for M =2.5 TeV,
Z(cid:48)
while one/two orders of magnitude smaller for the di-
jet/dilepton and top–pair modes at M = 4 TeV. Non-
Z(cid:48)
zero contributions from the effective operators, and the
underlying Higgs sector of the model, will induce size-
able enhancement in the W+W− and Zh–final states
that could be probed in the future LHC Run II.
Acknowledgements
Theauthorsofthisworkacknowledgevaluablecomments
fromJ.Gonzalez-Fraile. J.Y.alsoacknowledgesKITPC
financial support during the completion of this work.
FIG. 4: Z(cid:48)–production cross section times branching fractions
at the 13 TeV LHC for MW(cid:48) = 1.9 TeV, and γC running over
the ranges [−0.07, −0.04] (upper plot) and [0.04, 0.07] (lower
VI. W(cid:48) HEAVY BOSON DECAY WIDTHS
plot) following the values in Table II and the allowed parameter
space for γ = 0 and σ (W(cid:48)) ∼ 200fb (left and right red
W jj
chartsinFig.2). ThickcurvescorrespondtoγC =0,whilstthe From the Lagrangian LudW(cid:48) in (34), one has
line,dashedanddottedcurvesstandforthelower,intermediate
g2 + g2
and upper γC–values according to the allowed ranges. Γ(cid:0)W(cid:48) →ud¯(cid:1)= uLdLW(cid:48) uRdRW(cid:48) M . (46)
16π W(cid:48)
Thisdecaywidthalsoappliesforthefinalstatecs¯,while
for t¯b one has
Γ(cid:0)W(cid:48) →t¯b(cid:1)= gt2LbLW(cid:48) + gt2RbRW(cid:48) (cid:18)1− 3 m2t (cid:19) M
16π 2 M2 W(cid:48)
are determined by the WZ and Wh excesses in the W(cid:48)– W(cid:48)
(47)
decays studied in Sect. IIIA (Table II and Fig. 2). The
Theinvolvecouplingsabovearegivenbythecorrespond-
effective coefficients a from the Higgs sector introduced
i ing ones in (34) as
in the F(h)–definition of (2), in particular a and
C,LR
a ,will fix the allowed parameter space (γ , γ ). √ M
W,LR C W g =g λγ (cid:39)g W γ , g =−g
Larger values aC,LR, aW,LR ∼ 1 will reduce (enhance) uLdLW(cid:48) L C L MW(cid:48) C uRdRW(cid:48) R
the allowed positive (negative) γ –ranges by one or- (48)
W
der of magnitude, whereas part of the γ –ranges can Extending the Lagrangian L to the lepton–W(cid:48) in-
C udW(cid:48)
reach smaller values close to zero for small values of γ . teractions, one has
W
This feature would favour coefficients a and a
C,LR W,LR g2
of order 1 in case of observing tiny departures with re- Γ(cid:0)W(cid:48) →ν ¯l(cid:1)= νllLW(cid:48) M , l=e, µ, τ (49)
spect to the cross sections for the gauge–scalar modes l 48π W(cid:48)
Z(cid:48) → {W+W−, Zh} in Fig. 3. Sizeable deviations, spe-
cially for a larger MZ(cid:48)–values, would point towards in- Γ(cid:0)W(cid:48) →N ¯l(cid:1)= gN2llW(cid:48) (cid:18)1− 3MN2D(cid:19)M , l=e, τ
termediate values a ∼ a ∼ 1/2 (as shown in D 48π 2 M2 W(cid:48)
C,LR W,LR W(cid:48)
Fig. 4) or smaller ones. (50)
10
The decay width for the Nµµ¯–final state is not reported Alltheinvolvecouplingsin(51)-(54)g andg
R fLfLZ(cid:48) fRfRZ(cid:48)
asnoµµjj–signalhasbeenobervedsofar. Thecouplings with f = u,d,N,ν,e, are listed in Table III. From the
g andg correspondtothecouplingsin(48) effectiveLagrangianL in(44)onehas,fortheW–
νLlLW(cid:48) NllRW(cid:48) WWZ(cid:48)
respectively. The decay widths for the final states WZ pair final state
and Wh have been given in (38)-(39).
VII. Z(cid:48) HEAVY BOSON DECAY WIDTHS (cid:16) (cid:17)2
Γ(cid:0)Z(cid:48) →W+W−(cid:1)= gW(2)WZ(cid:48) MZ4(cid:48) M . (55)
The Z(cid:48)-heavy boson decays are reported here for the 192π M4 Z(cid:48)
W
fermionicchannelsaswellasthegaugeandgauge–scalar
modes. FromtheeffectiveLagrangianL in(43)itis
ffZ(cid:48)
possible to compute for the leptonic pair final states
g2 + g2
Γ(cid:0)Z(cid:48) →l+l−(cid:1)= lLlLZ(cid:48)24π lRlRZ(cid:48) MZ(cid:48), l=e, µ, τ TnahleheexlitcriatyfaccotmorpoMnZe4n(cid:48)/tMinW4 tchoemdeescafryomZ(cid:48)th→e loWng+itWud−i-,
(51)
being compensated by the quadratic inverse term from
Γ(Z(cid:48) →ν ν¯)= gν2lνlZ(cid:48) M (52) bgW(l2e)WIVZ)(cid:48).foAranovna-nziesrhoinogpoerpaetroartocroncotrnitbruibtiuotniolnea(dlosotkoaatdTdai--
l l 24π Z(cid:48)
tionaltermsenhancedbytheextrafactorasitisreflected
in Fig 4. Finally, for the Zh–final state, one has
(cid:115)
g2 M2 (cid:18) M2 (cid:19)
Γ(cid:0)Z(cid:48) →N N¯ (cid:1)= NlNlZ(cid:48) 1−4 ND 1− ND M
D D 24π M2 M2 Z(cid:48)
Z(cid:48) Z(cid:48)
(53)
g(1) (cid:18) M2 (cid:19) M4
For the quark–antiquark final states one has Γ(Z(cid:48) →Zh)(cid:39) hZZ(cid:48) g(1) + Z g(2) Z(cid:48) M .
192πc2 hZZ(cid:48) M2 hZZ(cid:48) M4 Z(cid:48)
W Z(cid:48) Z
g2 + g2 (56)
Γ(Z(cid:48) →qq¯)= qLqLZ(cid:48) qRqRZ(cid:48) M , q =u, d.
8π Z(cid:48)
(54) The involve couplings are listed in Table IV.
[1] G. Aad et al. [ATLAS Collaboration], arXiv:1506.00962 and J. H. Yu, Phys. Rev. D 92, 055030 (2015)
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