CP violating effects in the decay Z -> mu^+mu^-gamma induced by ZZgamma and Zgammagamma couplings

0.14 MB·English

by  M. A. Perez

CP violating effects in the decay Z µ+µ−γ induced by ZZγ and Zγγ couplings
→
M. A. P´erez∗ and F. Ram´ırez-Zavaleta†
Departamento de F´ısica, CINVESTAV, Apartado Postal 14-740, 07000, M´exico, D. F., M´exico
(Dated: February 2, 2008)
We analyze possible CP-violating effects induced in the Z decay with hard photon radiation by
γZZ and γγZ anomalous vertices. Weestimate thesensibility of futurelinear collider experiments
onthesecouplingscomingfromCP-oddasymmetriesassociated toangularcorrelationsofthethree
particle final state in e+e− Z µ+µ−γ. We find that a linear collider with an integrated
luminosityof500fb−1 and√→s=0.0→5TeVcanplacethebound hγ,Z <0.92atthe90%confidence
| 1 |
level for these couplings.
PACSnumbers: 14.70.Pw,13.38.Dg
5
0 Trilineargaugebosonself-couplingshavebeenstudiedasanewsourceofCP-violatinginteractionsathighenergies
0 [1,2,3,4]. Inparticular,anomalousneutraltriplegaugecouplings(NTGC),whicharenotpresentattreelevelinthe
2 StandardModel(SM),mayinduceCP-violatingeffectsinprocessessuchasγe Ze[5],e+e− γZ [6],e+e− ZZ
n [7], γγ tt¯[8] and pp¯ Zγ [4]. → → →
→ →
a InthepresentpaperweareinterestedinstudyingtheCP-violatingeffectsinducedbyNTGCinthe radiativedecay
J Z µ+µ−γ. The contributing diagrams to this decay mode are depicted in Fig. 1. Fig. 1(a) refers to the SM
→
1 contribution, the other diagrams include CP-violating effective vertices denoted by heavy dots. To lowest order, the
1 CP-violating effects in this reaction are induced by the interference between the SM contribution and any one of

the other diagrams shown in Fig. 1. Figs. 1(b) and 1(c) correspond to the contributions arising from the electric
4
and weak dipole moments of the muon. Since these moments are tightly bounded [9], their contribution exclude
v
any observable CP-violating effect in the decay Z µ+µ−γ [4, 10]. The possible CP-violating effects induced by
2
→
1 the contact interaction µµγZ shown in fig. 1(d) have been studied in Ref. [10]. However, it has been pointed out
2 [4, 11] that a fully gauge invariant effective Lagrangian that induces this local interaction also generates the CP-
0 violating vertices µµZ and µµγ shownin Fig. 1(b) and 1(c). When all these new vertices are takeninto account, the
1 interference between the lowest order SM diagram and the new CP-violating contributions vanishes for the process
4 e+e− µ+µ−γ and one is left with no CP-odd effects. Nonetheless, the L3 collaboration searched for CP-violating
0 →
effectsinthedecayoftheZ bosonwithhardphotonradiationandtheycouldsetanupperboundontheCP-violating
/
h coupling µ+µ−γZ [12].
p We will assume that possible CP-violating effects in the decay Z µ+µ−γ are induced by the effective vertices
- →
p ZZγ and Zγγ shown in Fig. 1(e). In Ref. [10], it was argued that is not necessary to study the CP-violating effects
e induced in this decay mode by the NTGC since the Lorentz structure of the respective amplitude reduces to that
h obtainedforthe contactinteractionµ+µ−γZ,Fig. 1(d). Ourpointofviewinthe presentpaperisthatthisargument
v: is not enough to disregard the contribution of the TNGC to the CP-violating effects in the decay Z µ+µ−γ. It
→
i has been pointed out [3, 13] that these effective vertices have a rather rich structure in the framework of the effective
X
Lagrangian formalism. They may be generated in both the linear and non-linear realization of the SU(2) U(1)
L
r gaugesymmetry. We haveshown[3]thatinbothrealizations,the NTGC induce the sameLorentzstructureb×ut that
a
this structure may be induced by operators of dimension 6 in the non-linear scenario or operators of dimension 8 in
the case of the linear realization of the SU(2) U(1) symmetry. As a consequence, any one of these constructions
L
may generate CP-odd observables which could b×e independent of those induced by the µ+µ−γZ contact interaction
or the weak and electric dipole moments of the muon.
The most generalform of the Zα(p)Vβ(q)γµ(k) vertex function, with Vβ =γ,Z,when Zα and γµ are on-shelland
which respects Lorentz and electromagnetic gauge invariance is given by [2, 3]
ΓZV∗γ(p,q,k) = ie [hV(kαgµβ kβgµα)+ hV2 qα(q kgβµ kβqµ)
αβµ m2 1 − m2 · −
Z Z
hV
+hVǫαβµνk + 4 qαǫβµνρp q ](q2 m2), (1)
3 ν m2 ν ρ V − V
Z
wherem istheZ-bosonmassandthefirsttwotermsin(1)areCP-violatingandtheothertwoareCP-conserving. The
Z
∗E-mail:mperez@fis.cinvestav.mx
†E-mail:rzf@fis.cinvestav.mx
2
µ+ µ+ µ+
Z γ Z γ Z γ
µ− µ− µ−
(a) (b) (c)
γ
µ+
Z γ Z V∗ µ+
µ−
µ−
(d) (e)
FIG.1: .FeynmandiagramscontributingtoCP-violatingeffectsinthedecayZ µ+µ−γ. TheheavydotsdenoteCP-violating
effective vertices. Crossed diagrams are not shown, V∗ stands for Z or γ. →
FIG.2: Energydistributionsofthephotonemittedine+e− µ+µ−ZforSM(solidline)andTNGB(dottedline)contributions
with hV =1. →
1
respective couplings hγ,Z have been bounded in e+e− and pp¯collisions [9, 14]: hZ [ 015,014], hZ [ 009,008],
hZ [ 022,011], hZi [ 007,015],hγ [ 006,006], hγ [ 005,002], hγ 1[∈0−06,·0004·],hγ 2 [∈ 0−00·4,00·42].
3 · · 4 · · 1 · · 2 · · 3 · · 4 · ·
F∈ollo−wing Ref. [10, 1∈2],−we will use the ∈ang−ular correlation∈so−f the three partic∈les−in the final state∈in−e+e− Z
µ+µ−γ in order to test CP-violating effects induced by the anomalous ZZγ/Zγγ couplings. The following C→P-od→d
3
FIG. 3: Distributionsfor theCP-odd observables V (solid line) and T (dotted line) corresponding to theinterference between
theSM and TNGB contributions(hV =1).
1
0.0025
0.002
0.0015
V)
A(
T) /
A(
0.001
0.0005
0
0 0.2 0.4 0.6 0.8 1
|T| / |V|
FIG. 4: Asymmetries for theCP-odd observables V (dotted line) and T (solid line) corresponding to theinterference between
theSM and TNGB contributions(hV =1).
1
observables have been proposed [10] to search for CP-violation in this radiative decay
T = (kbµ+ −kbµ−)·pbe+(kbµ+ ×kbµ−)·pbe+
V = (kbµ+ ×kbµ−)·pbe+. (2)
where kbµ+ andkbµ− areboth the unitary vectorsof muonandantimuon,respectively,pbe+ is the directionof colliding
positron.
4
3
2.8
2.6
2.4
2.2
2
1.8
m 1.6
lih1 1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1
|T| / |V|
FIG. 5: The 90% C. L. contours on hV1 from the asymmetries AT (solid line) and AV (dotted line) as function of the CP-odd
variables T and V.
We use the narrow-width approximation and in order to suppress the SM contributions from radiative Z decay,
we impose the cuts: PTγ 10 GeV, PTµ− 20 GeV, PTµ+ 20 GeV [4, 12]. In Fig. 2 we present the energy
≥ ≥ ≥
distributions for the emitted photon when we consider separately the SM and TNGB contributions. While the SM
contributionispeakedinthe forwarddirection,theCP-odddistributionhasamaximumatlargevaluesofthephoton
energy. Fig. 3 shows the distributions expected in this Z radiative decay for the observables T and V obtained with
a Monte Carlo program for the numerical integration using hV. It is observed a rather slight asymmetry between
1
negativeandpositivevaluesofT andV. InordertoestimatethesensitivityoftheTESLAcollidertothehV couplings,
1
we will make use of the relative asymmetries between negative and positive values of the observables T and V [12]:
NT>0 NT<0
A = − (3)
T
NT>0+NT<0
NV>0 NV<0
A = − . (4)
V
NV>0+NV<0
In Fig. 4 these asymmetries are shown for intervals in T and V , respectively. Using these results, we can obtain
90 % CL limits on the hV vertices as a function of T a|nd| V (|Fi|g. 5). We find that the best limits are obtained
1
for V 1, which correspond to hV 0.92. These|lim|its co|rre|spond to a future linear collider with √s=500 GeV
1
and|an|∼integrated luminosity of 5|00|f≤b−1. We found no difference for the asymmetries induced by the hZ and hγ
1 1
couplingsduetotheLorentzstructureshowninEq. (1),whiletherespectiveCP-oddeffectsinducedbythecouplings
hV are even more suppressed due to the extra suppression factor m2 in Eq. (1).
2 Z
In conclusion, our numerical study has shown that anomalousCP-violatingvertices ZZγ and Zγγ also give rise to
CP-odd asymmetries associated to angular correlations of the three particle final state. While rare Z decays may be
sensitivetoboththe CP-conservingandCP-violatinghV couplings[16],the detectionoftheA andA asymmetries
i T V
in the radiative decay Z µ+µ γ may be used to get specific constraints on the CP-violating hV couplings. We
→ − i
have derived sensitivity limits expected in a future linear collider for the CP-violating vertices ZZγ and Zγγ. These
sensitivity limits onhV are aboutone orderofmagnitude below thanthose obtainedfromCP-conservingobservables
1
in e+e− experiments [14, 15] or from future experiments in e+e− γZ with a transverse beam polarization [7].
→
However,the analysis presented in this paper has shown that an improvementin the luminosity expected in a future
TeV linear collider [17] may impose better and independent bounds on the CP-odd couplings hV.
1
The calculations involved in the present work needed a considerable symbolic and numerical computation. We
achieved this objective by using FeynCalc into the Mathematica for symbolic manipulations as reduction of traces
of Dirac matrices, simplification of large expressions, etc. Fortran program code was used as translator from results
obtained by VEGAS to graphical environment by program PAW++. The numerical phase space integrations were
done with VEGAS program into the COMPHEP, like also Monte Carlo events generation.
5
Acknowledgments
We acknowledge support from Conacyt and SNI (M´exico). We would like to thank G. Tavares-Velascofor collabo-
ration in the early stages of the present work and H. Castilla and A. Sa´nchez for useful discussions.
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