A.A. Iikhoded Z° Decays Due to the Three-Boson Interaction

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INSTITUTE FOR fflGH ENERGY PHYSICS

ШЕР 90-92 TD

A.A. Iikhoded

Z° decays due to the three-boson interaction

Submitted to Pkys. Lett. В

Protvino 1990 ГОК 538.1.01

Abstract

А.А. Iikhoded ft decays due to the three-boson interaction: ШЕР Preprint 90-92. — Protvino, 1990. — p. 16, refe. 17.

The e*e~-annihilation reactions with the three-vector-boson interaction and a final state meson have been discussed The possibility of making use of these processes for the exploration of the three-vector-boson reactions was estimated for the mesons corresponding to the diagonal Cabibbo-Kobayashi-Maskawa matrix elements (1Г, д А, Д, Tt) at the present and planned luminosity of the LEPl collider (y/S~ 100 GeV).

Аннотация

A.A. Лиходед. Распады Z"-6oooHa, обусловленные трехбоооиныи взаимодействием : Препринт ИФВЭ 90-92. — Протвино, 1990. —16 с, бнблиогр.: 17.

Рассмотрены реакции б+е'-анивгидяцни с трехбозонным взаимодействием и ме- ооноы в конечном состоянии. Дла случая обраоовашы мезонов, соответствующих диагональным элементам матрацы Кабнббо-Кобаяши-М&скава (iT,ft Д ,Д,/7), оце- нена вооможность использования данного класса реакций для изучения трехбооон- ното взаимодействия при существующей и возможной светнмостях коллайдера LEP1 (/5~ loo )

i — Institute for High Energy Physics, 1990. Introduction

The main two aspects of the W-pair production in the reaction of e+e~- annihilation exist in the physics of today. The first one is a precise de- termination of the W mass and width, W-boson coupling constants with different quark flavours, i.e., the Cabibbo-Kobayashi-Maskawa matrix elements. The precise measurement of the W mass is particularly important for testing the Standard Model at the loop level. Second, the reactions of the W-pair production provide the best opportunity to measure directly the three-vector-boson couplings 7WW, ZWW via s - channel in e+e~- annihilation and , as it is shown in [1] , [2], even a small deviation of these couplings from their guage theory values can lead to observable effects. However, because of the low LEP1 energy the detailed investigation of the three-vector-boson vertex is the LEP2 collider problem. Small cross- sections of e+e~ —* W~{l~v)W+(l+v) [3] processes, in fact, exclude such attempts at LEP1 with yfs ~100 GeV. The coupling constants measurement is also possible in e+e~ —* W(qq)W reactions, whose cross-sections are higher than those of processes with the leptonic modes of W-boson decay. Although the cross-sections of e+e~ —+ W{qq)W are too small [2] 6 (Г(г°-Уг+д-1/зЬ/з)/Г{г°->ц+»-) ~ 10" [3]), nevertheless, according to the recent proposals [4] on increasing the LEP luminosity, it seems possible to suggest a simple way for making the process of e+e~-annihilation with W-boson and quark-pair in the final state experimentally observable. The aim of this work is to investigate the quark-pair production processes with the subsequent hadronization of this pair into a meson (see Fig.l).

Fig.l The cross-sections of the processes with the production of тг±,р±,Аг and (cs) - mesons, corresponding to the diagonal Cabibbo-Kobayashi- Maskawa matrix elements will be evaluated. As the estimates presented bellow are mainly of a demonstrative character, we will omit the small background (7- exchange, see Fig.2) and interference (jZ) terms at en- ergies near the Z°-resonance pole,« q

Fig.2

1 Vertex of the WWV—interaction.

The general form of the three-vector-boson interaction of two charged vector bosons with a neutral vector boson has been much spoken about in literature [1,5 and refs. therein]. In the momentum space the relevant WWV-vertices can be written as follows:

Fig.3 аР = ГЛя-яУя - iyAw

!~Я)Р (1)

where V = "f,Z°. The form-factors ff introduced here are dimensionless functions of p2. In the first order in p2. these form-factors can be represented in the following form:

Л = -V f! = 9^ + kv + Xy (2)

/6 -

fv 1 \ /7 = ~2Av > and the coupling constants gv/wv for 7-quant and Z°-boson are

gww-r = —e , gwwz = —e • cot^w , (3) where e is the positron charge, в\у is the Weinberg angle. For coupling a charge vector boson with 7-quantum there exists a simple interpretation of parameters contained in (1): g7 defines the W charge, ky is generally termed the "anomalous" magnetic moment of W [6], A7 is connected [7] with the magnetic moment nw and the electric quadrupole moment Qw of W-boson.

The fey and A7 constants at parity violating terms are connected with the electric dipole dw and magnetic quadrupole Qw moments of W-boson as Let us discuss the three-vector-boson interaction in the approximation of the Standard Model. As shown in [1], in the Standard Model there exist sufficiently rigorous restrictions on form-factors (2) /ft.) = 1 + O{a) /ft.) = O(a) , (6) /ft.) = 2 + O(a) both for 7-quantum and for Z°-boson. It is worth mentioning that the «-dependence of form-factors arises only at the terms of the order of a. These restrictions can be written as

kv = l + O(a) , Xv = O(a) . (7)

Thus, in the Standard Model the vertex of the three-vector-boson interaction can be rewritten in the following form:

• (8)

In the calculations presented below expression (8) will be used to repre- sent the three-boson vertex in the momentum space.

2 W-pair production with final state meson.

The W-pair production processes in the e+e -annihilation in resonance Z° mode with a final state meson are shown on the diagram

Fig.4

4 .where C{ is a W-meson(Afj) coupling constant. Let us choose among the mesons produced after quark-pair hadronization the mesons corresponding to the large Cabibbo-Kobayashi-Maskawa matrix elements

\Vril = 0.9747±0.0011 [8], \Va\ > 0.66 [9] or \VC.\ = 0.88±0.10 [10].

(ud)-mesons

Let us discuss the case with (ud)-mesons and perform evaluations for the тг-meson production process. The coupling constant of W* with 7ri

/x (see Fig.5) is well known from the experiment [11]

/, = 131.69 ± 0.15 MeV Fig.5

The matrix element of the process shown in Fig.6

Fig.6 can be written as

- ft W"

where g and 3 are the guage constants of the weak neutral and charged

currents, фт is a pseudoscalar 7r-meson wave function normalized by unity, 2 \Vud\ = cos9c, eu is a polarization vector of on-shell W-boson, £ = sin dw Calculating the matrix element squared , summing over the initial and averaging over final particle polarizations one obtains

dt ~ 64тг ' v 8\/2

(Aijs)? + A2(s)t

where g = ~^~, Ai(s) are the functions of s obtained as a result of tracing ( the analytic forms of Ai are represented in Appendix A). The 2 numeric values of Ai(s,тпж, Mz, Mw) at s =(91.2 GeV) , Mz = 91.2

GeV, Mw = 81 GeV and mT = 0.14 GeV are shown bellow: GeV2 6 4 A2 = -1.667-10 GeV . (11) 11 6 A3 = 1.036 • 10 GeV The total cross-section

* = Т%^ (12)

Integration yields

At the highest LEP1 luminosity ~ 5 • lO^cm'V"1 such a small cross- section makes this process unobservable. However, when detecting the final meson states other channels, for example, processes with the p, A\ and (cs)-mesons production can contribute. The coupling constant of p-meson with W can be obtained from the r-lepton decay.

9рФа , (14)

Fig.7

where фа is the wave function of the vector p-meson and

2 др is the scalar parameter of m dimensionality. To determine gp one can note that in Fig.7 gp = /, where / is the value of the loop integral, and we can write шГ^Г ' (15) where 7 is a dimensionless parameter, which from a detailed e+e~-annihilation treatment is — = 2.36 ±0.18 . (16) 4тг Thus,

2 д р 2 dt 64x ' V 8^/2 f m p

2 2 2 (17) - M zf + M zT z)((ml - where Bt are analogous to A{ in (10) and B{ ~ A{ (see Appendix A):

2 2 Bx = -9.58 • 10 GeV 6 4 J2 = -1.68 10 GeV . (18) 11 6 B3 = 1.03 • 10 GeV Then the cross-section of />-meson production can easily be evaluated

111 p

Making use of (16) and taking into account the p-meson mass, one obtains

— 2-145 •CT(e+e-_jj»--.w(,)w) • (20)

Similarly, one can evaluate the contribution from an axial Лх(1260)- meson. The ^iW^-constant can be obtained from the Weinberg sum rule applying it to the r —• vT + (2re + 1)тг decay [12]

lAi Ip and taking into account the difference of A\- and p-meson masses, we get

9\ mAj

= ^-.a{eU-^Zo^w(p)w) (21) ~ 0.79-

(cs)- mesons

The large (~1) matrix element |V^S| corresponds to (cs)-mesons and D*(1969) among them. The order of magnitude of the Df -mesons production cross-oection Fig.8 can be estimated in the same way as in the case with тг, А\ and />-inesons. Estimates of the pseudoscalar X^-meson constant were obtained from the QCD sum rules [13] fo, — 220 MeV and from the lattice calculations of

UCLA [14] fD, = 234 ± 46 ± 55 MeV and ELC [15] fD, = 215 ± 17 MeV. Then the cross-section of the process shown in Fig.8 is

2 \Vc,\ fh. (22)

Since 1, then

2.1 (23)

In additon to the pseudoscalar Df -mesons a system of с and s quarks also produces vector and axial mesons. The vector D*±(2112)-meson was observed in experiments. However, in contrast to the reactions discussed, the investigation of the D*± production processes has some difficulties due to the lack of direct indications on the value of the gjy constant. Nevertheless, it seems possible to make use of the following ideas for the e+e~ —• W(D*)W process: the matrix element of a neutral vector meson V° decay into e+e~-pair

Fig.9

9 will be written as

П 1 Mtf = - К~ие7ЧГе -. ^JJ.-fV. , (24) 7v° q2 iv° where q is the 4-momentum of the virtual boson and q2 = my. Hence,

ТПу , V^"/

and for the decay width ГУ°-»е+е- one has

Туо_е<-е- = - • -у- . (26)

The vector meson constant gy± is connected with jyo by the simple rela- tion (27)

where <5e// is the effective charge of a quark-antiquark pair. For the heavy vector mesons (0(s?)(lG2O), J/*(cc)(3097),T(&5)(9460)) one can make use of the fact, that within the experimental accuracy [17]

Ц^- = const , (28) while const can easily be determined from the experimental data on ф, J/Ф, T - mesons. Then, using (26) and (27), one obtains

const , const ~ 0.01215 . (29)

It leads to go- ~ 0.717 GeV2, which, as it was expected, lies between the proper values corresponding to ф and J/Ф (дф ~ 0.240 GeV2, a 1.273 GeV2). Thus, for the e+e~ -* W(D*)W process we get

— П mD-J*2 > 4.8 •

10 3 Summary.

Thus, taking into account the contribution from тг*, р±, Af, Df, D*'h -mesons into the cross-section of processes with the three-vector- boson interaction and a final state meson one gets from (13). (20), (21), (23), (30) :

~ 2 l {r+)w-)0 + 2.145 + 0.79 + 2.1 + 4.8) ~ (31) ~ 22 • &(e+e-

It means that for e+e~ -» W(meson)W a > 0.43 • 10~39cm2 while 40 2 2 Z° —> W(meson)W. Nevertheless, presently there exist some proposals to raise the LEP luminosity by an order of magnitude ("Pretzel"-scheme) [4], which allows us to observe tens of events from the three-vector-boson interaction processes. In the calculations presented cross-sections of the processes with mesons corresponding to the nondiagonal Cabibbo-Kobayashi-Maskawa matrix elements were neglected because

« 1 \Vu\ Besides, the contribution from the following diagrams was ommitted

Fig.10

11 The cross-sections of these processes are

< Ю~ ~ 10~42СШ2 . (32)

For example, in the тг-ineson cases

- 0.27 • 10"46cm2 ( ~ 0.86 • 10-42cm2 , (33)

2 2 2 which is due to the factor ~ g cos 6cf for the diagram (a) in Fig.lO, and arising form-factor squared F2(s) ~ 1/(1 + -jjf )2 for the diagram (b). The detailed calculations of the processes shown in (a) and (b) (Fig.lO) for the case of pion-pair production are given in Appendix B.

Acknowledgements

In conclusion the author «would like to express his gratitude to A.K. Likhoded for his having inspired this work and also to S.S. Gershtein and G.V. Djikia for their fruitful discussions arid valuable remarks.

12 Appendix A

The functions Ai used in (10) are the dimensional functions of S, and Mw> where Mx is a pseudoscalar meson mass :

А (

2 2 - 2M\(S + 3M XS + 2MX)) ,

w 3 2 S + MXS - MXS - Mx) -

2 - Mb(2Mx + 3S + 8MXS) - 3 2 - MX(2S + 5S MX - ЗМ£) +

+ 6SMX - 2MX)) ,

w 3 2 {2S + 5S MX + MXS - 2MX) 2 - 6M| + 6MXS + UMXS ) -

Correspondingly, for the case with a vector meson, functions B{ in (17) are:

r» 4Ao . ,-, w2\2 i -кг

- USMX + 11MX) - M$r(UMx - 3S)) ,

w 2 - 2Mb(2S + ЪМХ) + M^(25 - Mx)) , where My is a vector meson mass and

AQ = 1 — 4$in2ew + 8sin*6w

13 In these calculations the electron mass was neglected.

Appendix В

The matrix elements of processes with я -meson pair production (the relevant diagrams are shown in Fig.10) can be written as

2 gcos ew((Pl - 9

~ ~щп , g

(^г)Та(~1 + 4£ — 7s)£/(fci)) — ; (? ~ *

where F(«) is the function of the total energy squared (analogous to the pion formfactor):

Mp is p - meson mass, pi and p2 are pion momenta, ф^+ and фг- are their wave functions. Calculating the matrix element squared, summing over the initial and averaging over final particle polarizations and integrating over phase space at y/s ~ 91.2 GeV one can easily obtain the cross-section estimates of the processes discussed above (see (33)).

14 References

[1] K.Hagiwara, R.D.Peccei, D.Zeppenfeld, DESY Preprint, DESY 86- 058 (1986); F.Jegerlehner in Rad. Corr. e+e~ Coll., Proc. Int. Workshop, Te- gernsel, April 3-7, 1989, Berlin. [2] V.Barger and T.Han, Preprint of University of Wisconsin, Madison, USA MAD/PH/548, 1989. [3] E.firanco in Physics at LEP, CERN Report 86-02, eds. J.Ellis and R.Peccei, v.l,p.l87.

[4] D.Treille in Rad. Corr. e+e~ Coll., Proc. Int. Workshop, Tegernsel, April 3-7, 1989, Berlin, p.323-347; J.MJowett, CERN LEP-TH/89-17. [5] G.Barbellini et al., in Physics at LEP, CERN Report 86-02, eds. J.Ellis and R.Peccei, v.2, p.l. [6] T.D.Lee and CN.Yang, Phys.Rev. 128(1962)885.

[7] N.Cabibo and R.Gatto, Phys.Rev. 124(1961)1577; H.UberaU, Nucl.Phys. 58(1964)625; H.Aronson, Phys.Rev. 186(1969)1434; K.J.Kim and Y.-S.Tsai, Phys.Rev. D7(1973)3710.

[8] W.J.Marciano and A.Sirlin, Phys.Rev.Lett. 56(1986)22; A.Sirlin and R.Zucchini, Phys.Rev.Lett. 57(1986)1994.

[9] T.M.AUev et al., Yad.Fiz. 40 (1984)823 (Sov.J.Nucl.Phys. 40(1984)527); M.Bauer, B.Stech and M.Wirbel, Z.Phys. C29(1985)637.

[10] K.Kleinknecht in Flavour Mixing in Weak Interactions, ed. by L.L.Chau (Plenum Press, N.Y. 1984). [11] Review of Particle Properties, PDG Report, ed. by R.Gatto et sl.,Phys.Lett. B204(1988)165.

15 [12] "Leptons and quarks", L.B.Okun'., Moscow.: "Nauka" (1981) p.102.

[13] S.Narison, Phys.Lett. B197(1987)423; C.A.Dominguez and N.Paver, Phys.Lett В 197(1987)423.

[14] G.Bernard et al., Phys.Rev. D38(1988)354.

[15] M.B.Gavela et al., Phys.Leti. В20в(1988)113. [16] Review of Particle Properties, PDG Report, ed. by R.Gatto et al.,Phys.Lett. B204(1988)107.

[17] Review of Particle Properties, PDG Report, ed. by R.Gatto et al,Phys.Lett. B204(1988)28.

Received July, 02, 1990.

16 А.А.Лиходед Распады Z°-6o3OHa, обусловленные трехбозонным взаимо- действием. Редактор А.А.Антипова. Технический редактор Л.П.Тимкина. Подписано к печати 18.07.90. Т-09833 Формат 60x90Д6. Офсетная печать. Печ.л. 1.0. Уч.-изд.л. 1,2. Тираж 270. Заказ Ш9. Индекс 3649. Цена 18 коп. Институт физики высоких энергий, 142284, Протвино Мос- ковской обл. 18 кос. Индекс 3649

ПРЕПРИНТ 90-92, Л Ф В Э, 1990