Lepton Polarization Asymmetries of $H\to\gamma\tau^+\tau^-$ Decays in Standard Model

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by  Rabia Akbar

Prepared for submission to JHEP
Lepton Polarization Asymmetries of H → γτ+τ−
Decays in Standard Model
4 Rabia Akbar,a,b Ishtiaq Ahmed,a,b M. Jamil Aslamb
1
0 aNational Centre for Physics,
2
Quaid-i-Azam University Campus, Islamabad 45320, Pakistan

n bDepartment of Physics,
a
Quaid-i-Azam University, Islamabad 45320, Pakistan
J

4 E-mail: [email protected], [email protected], [email protected]

]
h Abstract: Recently, CMS and ATLAS collaborations at LHC announced a Higgs like particle
p
with mass near 125GeV. Regarding this, to explore its intrinsic properties, different observables
-
p are needed to be measured precisely at the LHC for various decay channels of the Higgs. In
e
this context, we calculate the final state lepton polarization asymmetries, namely, single lepton
h
[ polarization asymmetries (Pi) and double lepton polarization asymmetries (Pij) in the SM for

radiative semileptonic Higgs decay H →γτ+τ−. In the phenomenological analysis of these lepton
1
v polarization asymmetries both tree and loop level diagrams are considered and it is found that
3 these diagrams give important contributions in the evaluation of said asymmetries. Interestingly,
1
it is found that in P the tree level diagrams contribute separately, which however, are missing
8 ij
0 in the calculations of Pi and the lepton forward-backward asymmetries (AFB). Similar to the
. other observables such as the decay rate and the lepton forward-backward asymmetries, the τ-
1
0 leptonpolarizationasymmetrieswouldbeinterestingobservables. Theexperimentalstudyofthese
4 observables will provide a fertile ground to explore the intrinsic properties of the SM Higgs boson
1
and its dynamics as well as help us to extract the signatures of the possible new physics beyond
:
v the SM.
i
X
r Keywords: Higgs particle: radiative decay, polarization asymmetries, lepton, CMS, ATLAS
a
Contents
1 Introduction 1
2 Formulation of the Amplitude 3
3 Observables 4
4 Numerical Analysis 7
4.1 Single Lepton Polarization Asymmetries 7
4.2 Double Lepton Polarization Asymmetries 9
5 Conclusion 12
A Appendix 12
1 Introduction
InJuly2012,thediscoveryofanewheavyHiggslikeparticlewithamassaround125GeVannounced
atCERNbytheATLASandCMSexperiments[1,2],ledustoanexcitingeraoftheparticlephysics.
This is a great leap towards the success of a theory proposed by Glashow, Salam and Weinberg in
1970,whereitwasrealizedthatthereisaclosetiesbetweentheelectromagneticandtheweakforce
andthesearethemanifestationofasingleunderlyingforce,theelectroweakforce. Theelectroweak
unificationoftheforcesispresentlyknownastheStandardModel(SM).Sincelastfourdecadesthe
SMhasbeentestedbymanyexperimentsandithasbeenshowntosuccessfullydescribehigh-energy
particle interactions. The simplest and most elegant way to construct the SM would be to assert
that all fundamental particles are massless, and by the virtue of a Higgs mechanism the photon
remains massless while its close cousins, the W and Z bosons, acquire a mass some 100 times that
of a proton mass. Experimentally these bosons were discovered with the mass predicted in the SM
and the discovery of the Higgs boson is a main missing chunk of this model.
With the advent of a Higgs like boson, the next goal of the particles physics is to understand
its nature to be sure if it is the Higgs boson of the SM or a new scalar particle. This can be done
by studying the different possible decays of the Higgs boson and it is an important task for the
theoretical physics. A numerous amount of literature is already present on the theoretical study of
the SM Higgs boson. On the other hand in experimental analysis of different decay channels of a
newly observed particle at the ATLAS [1] and CMS [2] experiments suggest that it is most likely a
signal of the SM Higgs boson. However, it is still needed to be confirmed whether it is a SM Higgs
boson or not? because of the excess events of H → γγ decay. In this context, the diphoton decay
channel of Higgs is studied in different new physics scenarios [3–28], but due to the limitation of
data it will take sometime to distinguish these beyond SM signatures.
Furthermore, to have more insight about the Higgs boson properties, besides the H → γγ
decay channel, a complementary radiative decay channel H → γ(cid:96)+(cid:96)− with (cid:96) = e, µ or τ got some
attentions[29–34]. Inthesestudies,theemphasisisontheanalysisofthedecayrates,invariantmass
distributions and leptons forward-backward asymmetry (A ). This channel is actually induced
FB
by H →γγ through the internal conversion i.e. the decay of a virtual photon to a pair of leptons.
For the numerical calculations of aforementioned observables performed in refs. [32–34] the value
– 1 –
of the mass of the Higgs particle is set to be 125GeV. Furthermore, Sun et. al. [34] focused on
the calculation of the lepton forward-backward asymmetries (A ) in H → γ(cid:96)+(cid:96)− decays. They
FB
have shown that due to the parity odd decay of the Higgs H →γZ∗, the lepton forward-backward
asymmetryisexpectedtobenonzerointhesemileptonicHiggsbosondecays. ThevaluesoftheA
FB
in case of electrons and muons as final state leptons come out to be of the order of 10−2 which will
be a challenging task to measure at the LHC. This motivated us to look for the other asymmetries
which may have larger magnitude than the A . Going along this direction we performed the
FB
detailed study of a single and the double lepton polarization asymmetries for H → γτ+τ− decays
by closely following the scheme of the study of A performed in ref. [34]. In principle one can
FB
include the case where electrons and µ’s appear as the final state leptons but it has some technical
issues.
ThefirstoneisthatattheLHCdetectors(CMSandATLAS)thereisapossibilitytoreconstruct
the polarization of a particle if it is unstable because the spin direction is inferred from the decay
distribution [35, 36]. In this way the electrons are stable so their polarization can not be detected
via energy measurement of final state. Also muons produced in the Higgs decay are penetrating
and they can travel an average distance of 400km, so its reconstruction inside the LHC detectors
is impossible. The second one is due to the low event rates of the H → γe+e− and H → γµ+µ−
decay channels. To elaborate this point, if one assume that the Higgs production is mainly driven
by the gluon-gluon fusion, then even for LHC operating at its full energy, i.e. 14 TeV, the roughly
estimates cross-sections are the
σ(pp→H →γe+e−) ≈ 3×10−2fb
σ(pp→H →γµ+µ−) ≈ 5×10−2fb
σ(pp→H →γτ+τ−) ≈ 4.2fb. (1.1)
The integrated luminosity of 300fb−1 is expected from the upcoming run period, therefore, we
can expect O(10) events of H → γe+e− and H → γµ+µ− decays, even without considering the
background effects. These number of events are too few for any appreciable measurements of an
observable such as the polarization asymmetry. Due to these reasons, the polarizations of electron
and muon are not measured at the ATLAS and CMS detectors, therefore, we will not add their
numerical analysis in the forthcoming study.
Thesituationfortheτ leptonisdifferentbecauseatfirstitisasequentialleptonsoitsdecayis
maximallyparityviolating. Intheτ restframe,parityviolationdeterminestheangulardistribution
oftheτ productwithrespecttoτ helicity. Whenboostedintothelabframethisangulardistribution
is manifest in the form of the energy and angular distribution of the decay products which can be
measured. Thus the energy and angular distributions of τ decay products is a τ polarimeter.
√
Also, the τ decay length at the centre of mass energy of the Higgs mass s = M is around
H
3.6mm and this ensure that τ decays are easily contained within the detector. Just as an example,
τ polarization has been measured by the ATLAS in τ hadronic decays with a single final state
charged particle [38]. In addition, compared to the electrons and muons as the final state leptons,
the Higgs decaying to taus will give a thousand of events (c.f. Eq. (1.1)) and hence it is likely that
an analysis can be performed. Therefore, the precise measurements of these asymmetries in future
will not only increase our understanding of the properties of Higgs boson but will also give us a
valuable information about its various couplings.
The structure of the paper is as follows. In section 2, we present the theoretical framework
necessary for H →γτ+τ− decays. After defining the formulae of asymmetries under consideration
in section 3, we derive the expressions of these asymmetries. In section 4, the above mentioned
observableswillbeanalyzednumericallyandwillbediscussedinlength. Finally,inthelastsection
– 2 –
Figure 1: The tree level diagrams for H →γ(cid:96)+(cid:96)− decays, where (cid:96) corresponds to τ.
Figure 2: The Z∗ and γ∗ loop diagrams for H →γ(cid:96)+(cid:96)− decays, with (cid:96) corresponding to τ lepton.
we will summarize and conclude the main results. Appendix includes some definitions and the
fermion and W-boson loops functions.
2 Formulation of the Amplitude
The tree and the loop level Feynman diagrams for H →γτ+τ− decays are shown in Figs. 1 and 2,
respectively. The amplitudes for these diagrams can be expressed as [34]
(cid:18)2pν +γνk/ k/γν +2pν(cid:19)
M = C u¯(p ) 2 − 1 v(p )(cid:15)∗,
tree 0 2 2p k˙ 2p k˙ 1 ν
2 1
M = (cid:15)∗ν(k q −g (k.q))u¯(p )(C γµ+C γµγ5)v(p )
Loop µ ν µν 2 1 2 1
+(cid:15) (cid:15)∗νkαqβu¯(p )(C γµ+C γµγ5)v(p ), (2.1)
µναβ 2 3 4 1
with
−2παm
C = τ ,
0 m sinθ
W W
1 1 1
C = −( −sin2θ )P Π − Π , C = P Π ,
1 4 W Z sγZ s γγ 2 4 Z sγZ
1 1
C = −( −sin2θ )P Π C = P Π . (2.2)
3 4 W Z aγZ 4 4 Z aγZ
Here, m denotes the mass of τ lepton, p , p , k and q are the momenta of τ−, τ+, γ and the
τ 1 2
virtualparticlesγ∗ orZ∗,respectively. Thefinestructureconstantisαwhileθ istheelectroweak
W
– 3 –
mixing angle. The definitions of the P , Π , Π and Π are given in ref. [34], however, for
Z sγZ γγ aγZ
the sake of completeness we have recollected them in the Appendix A.
Using these amplitudes, we have derived the expression of the the decay rate for the process
H → γτ+τ−. Except a negative sign in the third term of the numerator of B defined below, our
expression of the differential decay rate of H →γτ+τ− decay is similar to the one given in ref. [34]
and it can be expressed as follow:
d2Γ (m2 −s)
= H v∆, (2.3)
dsdx 512π3m3
H
with
∆ = |C |2A+2Re(C C∗)B+2Re(C C∗)C
0 0 1 0 4
+(|C |2+|C |2)D+(|C |2+|C |2)E +2Re(C C∗+C C∗)F, (2.4)
1 3 2 4 1 4 2 3
and
1
A = [m4 +s2+32m4 −8m2s−8m2m2 −(m4 +s2−8m2s)v2x2],
(m2 −s)2(1−v2x2)2 H τ τ τ H H τ
H
m2 −s−sv2(1−x2)
B = 8m H ,
τ 1−v2x2
(m2 −s)
C = 8m H vx,
τ 1−v2x2
(m2 −s)2
D = H (s+4m2 +s2v2x2),
2 τ
(m2 −s)2
E = H sv2(1+x2),
2
F = (m2 −s)2svx. (2.5)
H
(cid:113)
Here, s is the square of momentum transfer i.e., s=q2, v = 1− 4m2τ and x=cosθ with θ is the
s
angle between Higgs boson and a lepton in the rest frame of dileptons. The limits on the phase
space parameters s and x are
4m2 ≤s≤m2 , −1≤x≤+1. (2.6)
τ H
In Eq. (2.5), the term A represents the contribution from the tree diagrams where as B and C
terms correspond to the interference between the tree and loop diagrams. The last three terms D,
E and F are originated purely from the loop diagrams.
3 Observables
It has already mentioned that the differential decay rate and the lepton forward-backward asym-
metry were under debate in literature [29–34] and the purpose of this study is to investigate the
polarization asymmetries of the final state τ leptons in H → γτ+τ− decay. To achieve this goal,
letusintroducetheorthogonalfourvectorsbelongingtothepolarizationofτ− andτ+,namelyS−
i
and S+, respectively [37, 39]. These polarization vectors can be defined as follow:
i
(cid:18) (cid:19)
p
S−α ≡(0,e ) = 0, 1 ,
L L |p |
1
(cid:18) (cid:19)
k×p
S−α ≡(0,e ) = 0, 1 ,
N N |k×p |
1
S−α ≡(0,e ) = (0,e ×e ), (3.1)
T T N L
– 4 –
where the subscripts L, N and T correspond to the longitudinal, normal and transverse polariza-
tions, respectively. Also, p , p andkaredenotingthethreemomentavectorsofthefinalparticles
1 2
τ−, τ+ and γ, respectively, in the centre of mass frame of τ+τ− system. It can be noticed that by
replacing p → p one can obtain the unit vectors for the polarizations of τ+. The longitudinal
1 2
unit vector S are boosted by Lorentz transformations in the CM frame of τ+τ−:
L
(cid:18) (cid:19)
|p | E p
S−α = 1 , l 1 . (3.2)
LCM m m |p |
τ τ 1
Withtheseunitvectors,letusdefinethesingleleptonpolarizationasymmetryinthefollowingway:
dΓ(n± =e±)− dΓ(n± =−e±)
P(±)(s) = ds i ds i , (3.3)
i dΓ(n± =e±)+ dΓ(n± =−e±)
ds i ds i
where e denotes the unit vector with subscript i corresponds to longitudinal (L), normal (N) and
i
transverse(T)leptonpolarizationsindex,andn± isthespindirectionofτ±. Thedifferentialdecay
rate for the polarized lepton τ± in H →γτ+τ− decay along the spin direction n± is related to the
unpolarized decay rate by the following relation:
dΓ(n±) = 1(cid:18)dΓ(cid:19)(cid:2)1+(P±e±+P±e± +P±e±)·n±(cid:3). (3.4)
ds 2 ds L L N N T T
Finally, by using the definitions (3.3), the expressions for the longitudinal (P ), normal (P ), and
L N
transverse (P ) lepton polarizations can be written as
T
P (s) = 1(cid:20)4sv(cid:0)m2 −s(cid:1)2[2Re(C C∗)+2Re(C C∗)]− 4mτ(cid:0)m2H −s(cid:1){tanh−1(v)(cid:0)4m2 −sv2+s(cid:1)
L ∆ 3 H 1 2 3 4 sv2 τ
(cid:21)
+s2v(cid:0)v2−1(cid:1)2Re(C C∗)−{2(cid:0)sv2+s(cid:1)tanh−1(v)+sv(v2−1)}2Re(C C∗)} ,
0 3 0 2
(3.5)
√ √ √ √
1(cid:20) 4πm2m2 (cid:0)2v2+ 1−v2+1(cid:1) π s(cid:0) 1−v2+1(cid:1)(cid:0)4m2 +m2 −s(cid:1) 2 s(cid:0)m2 −2s(cid:1)
P (s) = −2{ τ H√ √ − τ H + H
N ∆ sv2 1−v2 v2 v
√ √
4m2 s(cid:0)2 1−v2+πv(cid:1) √
+ τ √ − s(cid:0)8m2 +(2+π)m2 −(π−4)s(cid:1)}Re(C C∗)
v 1−v2 τ H 0 2
+{ 2√π (cid:0)4m2(cid:0)m2 +s(cid:1)+s(cid:0)m2 −3s(cid:1)(cid:1)(cid:16)4m2 −s(cid:16)v2+2(cid:112)1−v2−1(cid:17)(cid:17)}Re(C C∗)
s3/2v2 1−v2 τ H H τ 0 3
√ (cid:21)
+πm s(cid:0)m2 −s(cid:1)2[2Re(C C∗)] , (3.6)
τ H 1 3
P (s) = 1(cid:20)πv√s(cid:0)m2 −s(cid:1)(cid:8)2(cid:2)2Im(C∗C )+2Im(C∗C )(cid:3)−m (cid:0)m2 −s(cid:1)(cid:2)2Im(C∗C )+2Im(C∗C )(cid:3)(cid:9)(cid:21),
T ∆ H 1 0 4 0 τ H 4 1 2 3
(3.7)
where C ,...,C are given in Eq. (2.2) and the ∆ used in the above equations is defined in Eq.
0 4
(2.4).
To calculate the double-lepton-polarization asymmetries, we consider the polarizations of both
τ− andτ+ simultaneouslyandintroducethefollowingspinprojectionoperatorsfortheτ− andτ+,
1(cid:16) (cid:17)
Λ = 1+γ S/− ,
1 2 5 i
1(cid:16) (cid:17)
Λ = 1+γ S/+ , (3.8)
2 2 5 i
– 5 –
wherei=L,T andN againdesignatethelongitudinal, transverseandnormalleptonpolarizations
index, respectively. In the rest frame of the τ−τ+ one can define the following set of orthogonal
vectors Sα:
(cid:18) (cid:19)
p
S−α = (0,e−)= 0, 1 ,
L L |p |
1
(cid:18) (cid:19)
k×p
S−α = (0,e−)= 0, 1 ,
N N |k×p |
1
S−α = (0,e−)=(cid:0)0,e− ×e−(cid:1), (3.9)
T T N L
(cid:18) (cid:19)
p
S+α = (0,e+)= 0, 2 ,
L L |p |
2
(cid:18) (cid:19)
k×p
S+α = (0,e+)= 0, 2 ,
N N |k×p |
2
S+α = (0,e+)=(cid:0)0,e+ ×e+(cid:1).
T T N L
Just like the single lepton polarizations, through Lorentz transformations we can boost the longi-
tudinal component in the CM frame of τ−τ+ as
(cid:32) (cid:33)
(cid:0)SL−α(cid:1)CM = |mp1l|,mEl(cid:12)(cid:12)pp1−(cid:12)(cid:12) ,
(cid:32) (cid:33)
(cid:0)SL+α(cid:1)CM = |mp2l|,−mEl(cid:12)(cid:12)pp2+(cid:12)(cid:12) . (3.10)
The normal and transverse components remain the same under Lorentz boost. We now define the
double lepton polarization asymmetries as
(cid:0)dΓ(cid:0)S−,S+(cid:1)− dΓ(cid:0)−S−,S+(cid:1)(cid:1)−(cid:0)dΓ(cid:0)S−,−S+(cid:1)− dΓ(cid:0)−S−,−S+(cid:1)(cid:1)
P (s)= ds i j ds i j ds i j ds i j , (3.11)
ij (cid:0)dΓ(cid:0)S−,S+(cid:1)− dΓ(cid:0)−S−,S+(cid:1)(cid:1)+(cid:0)dΓ(cid:0)S−,−S+(cid:1)− dΓ(cid:0)−S−,−S+(cid:1)(cid:1)
ds i j ds i j ds i j ds i j
wherethesubscriptsiandj correspondtotheτ− andτ+ polarizationsindices, respectively. Using
thesedefinitionsthevariousdoubleleptonpolarizationasymmetriesasafunctionofscanbewritten
as
(cid:20)
1 32
P (s) = {tanh−1(v)(−64m6s+8m4(m4 +7s2)−4m2s(m4 −m2 s+5s2)
LL ∆ s2v3(m2 −s)2 τ τ H τ H H
H
+s2(m4 +2m2 s(v2−1)+s2(3−2v2)))−sv(16m4s−2m2(m4 +3s2)
H H τ τ H
(m2 −s)2
+s(m4 −m2 s+s2))}|C |2+ H {(8m4 +s2(v2−1))(|C |2+|C |2)
H H 0 3m2 τ 1 3
τ
8
+s2v2(v2−1)(|C |2+|C |2)}− {m2sv(2sv2−2(m2 −s)−v3(m2 −s))
4 2 m sv3 τ H H
τ
+tanh−1(v)(64m6 −16m4s−8m2s2−s2(v2−1)(2s+v3(m2 −s)))}(2R (C C∗))
τ τ τ H e 0 1
(cid:21)
2
+ {(m2 −s)(s(4m2 +sv5/2−sv3)−tanh−1(v)((4m2 +s)2−s2v3))(2R (C C∗))} ,
m s H τ τ e 0 4
τ
(3.12)
(cid:20)
1 32
P (s) = {(2(m2 −s)(2m2 −s)+s2v2(v2+1))tanh−1(v)+sv(m2 +s(v2−1)}|C |2
NN ∆ v(m2 −s)2 H τ H 0
H
(cid:21)
2 8m
+ sv2(m2 −s)2{(|C |2+|C |2)−(|C |2+|C |2)}− τ{8m2tanh−1(v)+s(π−2v)}(2R (C C∗)) ,
3 H 1 3 4 2 v τ e 0 1
(3.13)
– 6 –
(cid:20)
1 32
P (s) = − (16m4s+2m2(m4 −4m2 s−s2)+m2 s2)(sv−s(v2+1)tanh−1(v))|C |2
TT ∆ s2v3(m2 −s)2 τ τ H H H 0
H
16m
+ (2m2tanh−1(v)(−m2 +2sv2+s)−s2v3)(2Re(C C∗))
sv3 τ H 0 1
(cid:21)
+2s(m2 −s)2(cid:0)(2−v2)(|C |2+|C |2)−v2(|C |2+|C |2)(cid:1) , (3.14)
3 H 1 3 4 2
1(cid:20) π √ (cid:112) 2m2 8m2
P (s) = s(m2 −s){−4 1−v2( τ +v2)− τ +2v2+2}((2Im(C C∗))
LN ∆ 2v2 H s s 0 2
√ (cid:112) (cid:21)
+ 2π s( 1−v2−1)(m2 −s)(2Im(C C∗)) =−P (s), (3.15)
H 0 3 NL
(cid:20)
1 16πm
P (s) = − τ |C |2
LT ∆ √s(cid:113) 1 −1 0
v2
√ √
+1πm √sv(cid:0)m2 −s(cid:1)2(2Re(C C∗)+2Re(C C∗))− 2π s(cid:0) 1−v2−1(cid:1)(cid:0)m2H −s(cid:1)(2Re(C C∗))
2 τ H 1 4 2 3 v 0 1
+(cid:0)m2H −s(cid:1){π(cid:8)−16m4 +8m2s+s2(cid:0)v4+3(cid:1)(cid:9)−8πm s(cid:112)2m2 +sv2}(2Re(C C∗))(cid:21)P , (3.16)
2s3/2v2 τ τ τ τ 0 4 TL
(cid:20) (cid:21)
P (s) = 8mτ (cid:0)m2 −s(cid:1)(cid:0)(cid:0)v2−1(cid:1)tanh−1(v)+v(cid:1)(2Im(C C∗))+(cid:0)4m2tanh−1(v)−sv(cid:1)(2Im(C C∗))
NT ∆v2 H 0 3 τ 0 2
= −P , (3.17)
TN
withC ,...,C and∆usedinaboveequationsaredefinedinEq. (2.2)andEq. (2.4), respectively.
0 4
Table 1: Default values of input parameters used in the calculations
m =125 GeV, m =172 GeV, m =0.106 GeV,
H t µ
m =0.51×10−3 GeV, m =1.77 GeV, α(M )−1 =128,
e τ Z
m =91.18 GeV, Γ =2.48 GeV.
Z Z
4 Numerical Analysis
Intheprevioussectionwehavepresentedtheexpressionsofthesingleanddoubleleptonpolarization
asymmetries in the SM for H → γτ+τ− decay by considering both tree and loop diagrams. To
proceed with the numerical analysis of these physical observables, the numerical values of different
input parameters in the SM are listed in Table 1.
4.1 Single Lepton Polarization Asymmetries
Similar to the case of leptons forward-backward asymmetries A calculated in ref. [34], one
FB
can see from Eqs. (3.5, 3.6, 3.7) that the single lepton polarization asymmetries are not separately
dependentonthetreeleveldiagrams. However,theydependontheinterferenceofthecontributions
fromdifferentloopdiagrams(c.f. Figure2)aswellasontheinterferenceoftreeandloopdiagrams.
Moreover, for the longitudinal polarization asymmetry P the contributions generated from the
L
interference between tree and loop diagrams C C∗ and C C∗ are m suppressed.
0 3 0 2 √ τ
The longitudinal lepton polarization P as a function of s in H → γτ+τ− is drawn in Fig.
L
3. It is expected from Eq. (3.5) that the major contribution is coming through the interference of
– 7 –
PLΤ(cid:43)Τ(cid:45)
0.0(cid:72)2 (cid:76)
sGeV
40 60 80 100 120
(cid:45)0.02
(cid:45)0.04
Figure 3: The longitudinal lepton polarization asymmetry P (s) in H → γτ+τ− as a function
√ L
of the momentum transfer s. The solid line correspond to the total contribution, the dashed
line correspond to the contribution through the interference between the tree diagram and Z∗ pole
diagrams and the dotted line corresponds to the contribution from the Z∗−Z∗ interference while
the red line corresponds to the contribution from the γ∗−Z∗ interference.
PN&PT(cid:72)Τ(cid:43)Τ(cid:45)(cid:76)
0.10
0.05
sGeV
40 60 80 100 120
(cid:45)0.05
(cid:45)0.10
Figure 4: The Normal lepton polarization asymmetry P (s) in H →γτ+τ− as a function of the
√ N
momentum transfer s. The solid line corresponds to the P and the dashed line is for P .
N T
s −s (GeV2) 102−302 302−502 502−702 702−902 902−1102 full Phase space
min max
H →γτ+τ− 3.3% 4.8% 5.4% 7.9% -1.3% 0.02%
Table 2: The average Normal polarization asymmetries (cid:104)P (cid:105) for some cuts on s in H → γτ+τ−
N
decays .
differentloopdiagrams. IfwerecallEq. (2.2)forthedefinitionsofC −C thentherearetwotypes
1 4
ofcontributions,namely,γ∗−Z∗ interference(referredastheoverlapbetweentheγ∗ poleamplitude
andtheZ∗ poleamplitude)andthesecondoneisZ∗−Z∗ interference(i.e. theinterferenceamong
thedifferentamplitudesofZ∗ polediagrams). SimilartotheA [34]thelongitudinalpolarization
FB
asymmetry also has zero crossing because of the change of sign of γ∗−Z∗ interference at s=m2
Z
which is due to the sign change of the real part of P and it is evident from Fig. 3. It can also be
Z
seen from Figure 3 that throughout the allowed kinematical region the value of P in H →γτ+τ−
L
is of the order of 10−2 and to measure such a small number at the current colliders is not an easy
task.
In contrast to the longitudinal lepton polarization asymmetry (P ) and the leptons forward-
L
– 8 –
s −s (GeV2) 102−302 302−502 502−702 702−902 902−1102 full Phase space
min max
H →γτ+τ− 86.7% 80.8% 84.7% 84.6% 93.8% 99.6%
Table 3: The average polarization asymmetries (cid:104)P (cid:105) for some cuts on s in H →γτ+τ− decays .
LL
s −s (GeV2) 102−302 302−502 502−702 702−902 902−1102 full Phase space
min max
H →γτ+τ− -7% 18% 41.7% 64.6% 88.7% 99%
Table 4: The average polarization asymmetries (cid:104)P (cid:105) for some cuts on s in H →γτ+τ− decay.
TT
backwardasymmetry(A ), thecontributionsfromtheloopdiagrams(c.f. contributionsfromthe
FB
Z∗−Z∗ andγ∗−Z∗ interference)arem suppressedinthenormalleptonpolarizationasymmetry
τ
(P )andhencecanbesafelyignored. Thenormalleptonpolarization(P ),wherethecontribution
N N
is coming from the interference between the tree and Z∗ pole diagrams is displayed with solid line
in Figure 4.
Justliketheleptonsforward-backwardasymmetry(A )calculatedinref. [34],themagnitude
FB
of transverse lepton polarization asymmetry (P ) is very small in almost all the kinematical region
T
and the dahsed line in Figure 4 demonstrate this fact. By looking at the Eq. (3.7) it is even more
evidentthatitisproportionaltotheimaginarypartofthecontributionsfromdifferentinterference
diagrams which are too small, hence we ignored its detailed analysis over here.
Furthermore,wehavealsocalculatedtheaveragevalueofnormalpolarizationasymmetry(P )
N
in different bins of s and listed it in Tables 2 for H →γτ+τ−. It is clear from the Table 2 that by
scanning full phase space altogether this asymmetry might not be measurable quantities at LHC,
but by making an analysis in different bins of s the average value of normal lepton polarization
asymmetry may be measurable at the LHC.
4.2 Double Lepton Polarization Asymmetries
Now we discuss the double lepton polarization asymmetries P , where the indices i,j can be L,
ij
T and N. In Section 3, we have derived the expressions of different double lepton polarization
asymmetries (c.f. Eqs. (3.12-3.14)), where one can immediately notice that in contrast to the
above discussed single lepton polarization asymmetries and the forward-backward asymmetries,
discussed in length in ref. [34], some of the P ’s are explicit functions of the tree diagrams in
ij
addition to their interference with loop diagrams.
The double longitudinal lepton polarization asymmetry (P ) in H → γτ+τ− as a function
√ LL
of s is drawn in Fig. 5. Fig. 5 shows that in the H → γτ+τ− decay the major contribution is
coming from the tree diagram which is denoted by the red line in this figure. The average value of
P invariousbinsofsisgiveninTable3wherewecanseethatthisasymmetryhasasignificantly
LL
large value in the whole phase space range. From experimental point of view, it has already been
mentionedthatthereconstructionofthepolarizationispossibleonlyiftheparticleisunstableand
due to the decay of τ lepton to a final state charged hadron its spin direction is inferred from the
decay products distribution [35, 36] at the LHC. Therefore, the scanning of double longitudinal
lepton polarization asymmetry at the LHC will help us to dig out some intrinsic properties of the
Higgs particle and its dynamics.
√
Fig. 6 display the behavior of P as a function of s for H →γτ+τ− decay where the major
TT
contribution is coming from the tree diagram and it is denoted by the red line in Fig. 6. In Fig. 6
√ √
one can see that in the region 90GeV≤ s ≤ s the major contribution is coming only from
max
the tree diagrams and it can be seen that the average value of (cid:104)P (cid:105) enhanced upto 88% in this
TT
region (c.f. Table 5) which is in a measurable ball park of the LHC.
– 9 –