Right-handed currents and heavy neutrinos in high-energy ep and e+e− scattering

N P HUYCLEAR S I CS B

Nuclear Physics B 381 (1992) 109—128 North-Holland

_________________

Right-handed currents and heavy neutrinos in high-energy ep and e~e scattering W. Buchmüller Deutsches Elektronen Synchrotron DESY~Hamburg, Germany

C. Greub

*

Institutfür Theoretische Physik, Unirersität Zurich, Switzerland Received 4 March 1992 Accepted for publication 9 April 1992

Heavy Dirac or Majorana neutrinos can be produced via right-handed charged Currents which occur in extensions of the standard model with SU(2) 1 XSU(2)R XU(l)B_L gauge symmetry. Low-energy processes, Z precision experiments and direct search experiments in pp collisions are consistent with WR bosons heavier than 450 GeV, if the right-handed neutrinos are heavy. We study the production of heavy neutrinos via right-handed currents in e~e annihilation and ep scattering which appears particularly promising. At HERA heavy neutrinos and WR bosons can be discovered with masses up to 120 GeV and 700 GeV, respectively.

1. Introduction Within the standard model electroweak interactions are described by the gauge group SU(2)L X U(1)~,..However, despite the extraordinary phenomenological success of the standard model it is not excluded that new gauge interactions will become visible already at TeV energies. Such extended gauge theories have indeed been suggested on various theoretical grounds and particular attention has been given to models with right-handed currents based on the symmetry group SU(2)L x SU(2)~x U(l)BL [1] and to models with an additional U(1) symmetry contained in the unified group E6 [2]. Extended gauge theories also predict fermions in addition to the quarks and leptons of the standard model, since all gauged currents have to be anomaly free. In the minimal case, where the extended group is contained in the unified group SO(1O), the addition of one “right-handed” neutrino ~R for each quark-lepton family suffices to satisfy the requirement of anomaly freedom. The spontaneous breaking of the extended gauge group to the standard model group can induce *

Partially supported by Schweizerischer Nationalfonds.

0550-3213/92/$05.00 © 1992

—

Elsevier Science Publishers B.V. All rights reserved

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Right-handed currents

Majorana masses for the right-handed neutrinos, which satisfy an upper bound proportional to the mass of the additional neutral vector boson Z’ [3]. If a Z’ vector boson with mass of order 1 TeV is found, also heavy neutrinos with masses in the range from a few tens of GeV to a few hundred GeV are likely to exist. In this paper we shall study models with right-handed currents, and we shall determine the mass range of heavy neutrinos and “right-handed” charged vector bosons WR which can be explored at present and projected ep and e~e colliders. Of particular interest are electron—proton colliders where a single heavy neutrino can be produced without suppression by small mixing angles. This extends previous work [4] where models with an additional U(1) factor have been considered. Stringent bounds on the WR mass and the WR—WL mixing angle can be derived from various low-energy processes. In particular from the KL—Ks mass difference one obtains the bound mw> (1—3) TeV for the special case of left—right symmetry [5,6]. However, this bound does not hold for all models with SU(2)L>< SU(2)R X U(1)~~ gauge symmetry, but requires further assumptions on fermion mass matrices. In models with arbitrary Yukawa couplings the lower bound is reduced to 300 GeV [7]. As we shall see, constraints from LEP data on the mixing of the neutral vector bosons yield the lower bound of 450 GeV for mWR, which is close to the bound obtained for massless neutrinos [18]. Finally, the lower mass bound mW> 520 GeV has been obtained from direct production in proton—proton collisions for neutrinos with masses below 15 GeV [8]. The cross section for the production of heavy neutrinos via right-handed currents in ep scattering at HERA energies is known to be small [9], and because of the stringent lower bound on mWR in left—right symmetric models [5,6] the original interest [10] in this process at HERA waned. However, the current model-independent lower bound on mwR from low-energy processes and LEP data is only 450 GeV, and therefore it appears appropriate to investigate in some detail the production of right-handed neutrinos at HERA and also at future ep and e~e colliders. The paper is organized as follows: in sect. 2 we briefly describe the SU(2)L>< SU(2)R X U(1)B_L model, symmetry breaking and neutrino masses. Sect. 3 deals with mass bounds for neutrinos and vector bosons which follow from low-energy processes and from Z physics. In sect. 4 we compute heavy neutrino production cross sections in ep and e~e scattering and discuss discovery limits for HERA, LEP 0 LHC, LEP200 and the 500 GeV e~e LINAC. Sect. 5 contains a summary, and in appendix A we have listed the fully differential cross sections needed for Monte Carlo generators. 2. Models with right-handed currents In models with the gauge group SU(2)L x SU(2)R X U(1)BL left- and righthanded leptons and quarks transform as doublets under SU(2)L and SU(2)R,

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111

respectively:

(~)L~

(~0;

(~)L~ (2’

(~H

-1), 0;

(0; ~ -1),

~),

(~)R~0; ~

~).

(2.1) (2.2)

Here we have suppressed the index which labels the different generations of weak eigenstates. The symmetry group requires one right-handed neutrino L’R for each generation. In addition we introduce one gauge singlet ~L for each generation, ~

(0; 0; 0),

(2.3)

as suggested by the embedding into the unified group E6 where each quark-lepton generation is contained in a 27-plet. The minimal Higgs sector, which is required in order to generate masses for the fermions (2.1)—(2.3), contains the following doublet and triplet scalar fields (cf. refs. [1,11]):

~~G;

i*;0)

x~(O; ~ —1), (0; 1; 2)

(2.4) (2.5)

The vacuum expectation values of x and ~i break SU(2)L x SU(2)R X U(1)B_L to the standard model group which is then further broken to U(1)cm by the vacuum expectation value of 1~.The covariant derivatives of fermion and scalar fields are given by D~iLR

=

(a~+ ~

+

D~=a~+ ~g~raW~~_ DMX

=

=

(9,.~+

—

L)C~)~LR~

~

—

(a~+igC~)z.1+

i~(B

(2.7)

~

~ig~[~a,

(2.6)

(2.8) ~1]W~,

(2.9)

where 3~i~+ ~(T~+TTz~0).

L~

(2.10)

T

Here WL~,WR~and C~denote the SU(2)L, SU(2)R and U(I)RL vector fields, respectively, and g~,g~and ~ are the corresponding coupling constants. In eq.

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Right-handed currents

(2.10) the electric charges of the three components of ~1are indicated. Similarly, one has for the Higgs fields ~ and x:

x= 42

4~2

x

.

(2.11)

The vacuum expectation values =

K~°)~,~

(2.12)

=

break SU(2)L x SU(2)R x U(1)BL to the standard model group SU(2)L x which is further broken by V 1

—

\~Pi/O~

V2

—

\~P2/O

to the group U(1)em of electromagnetic interactions. The vacuum expectation values (2.12) and (2.13) must satisfy v’>> v, since the allowed contribution of right-handed currents to charged current processes is known to be very small. Given the Higgs fields ~ x and ct the masses of neutrinos and charged leptons are obtained from the following lagrangian 2J]I~g2(~

+~~thl(R

-~M =(~dI)g~~~ +e

+ ~i8~r

2~1h2 ~R +

2nLmnL +

h.c.,

(2.14)

where (2.15) Here g12, h12 and m are 3 x 3 complex matrices in generation space. The lagrangian (2.14) does not conserve lepton number. Hence, after spontaneous symmetry breaking, one will in general obtain nine Majorana neutrinos as mass eigenstates. In the following we shall restrict ourselves to the case where the additional neutrinos are heavy, since the smallness of the light neutrino masses is then naturally understood without requiring extremely tiny Yukawa couplings. Specifically, we shall assume mN> 15 GeV, so that WR masses below 520 GeV are still allowed by the present CDF bound [8]. For simplicity, let us now consider two special cases. If h2 0 and m 0, lepton number is conserved and the neutrino mass terms read =

~M

=“[j~D~R +n..mIvR +

b.c.,

=

(2.16)

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where m~=g

mi=hiv*.

1V1+g2V~’,

Since V’>> V, one naturally has m1 three Dirac neutrinos,

>>

rnD. One then obtains as mass eigenstates

(2.18)

—~~‘M=nLmNPR+h.c.,

rnN

=

m~+ 0(1/rn1),

nL=nL+PLmD—

and three massless Weyl neutrinos,

(2.17)

~L =

+

+

(2.19)

0(1/m~),

(2.20)

0(1/rn1), which are identified with

ye, I’~~’ ~‘r

If h2 ~ 0 lepton number is broken. The simplest case of this type corresponds to 0, where ~L represents three Majorana neutrinos which decouple from i~’L and ~ The remaining mass terms are (cf. eq. (2.17)) =

~‘M =P.~rnDvR +

2VRm2PR

+ b.c.,

(2.21)

where m2

=

%/~h2L~.

(2.22)

This leads, via the see—saw mechanism [12], to three heavy and three light Majorana neutrinos with mass matrices rnN

=

rn,=

rn2 + 0(1/rn2), 1 rnDmD-l-0(1/m2). rn 2

(2.23) (2.24)

The light Majorana neutrinos correspond to r~,t~ and v~.If V’, the scale of SU(2)R breaking, is of order 1 TeV, their masses are expected to be close to the present experimental upper bounds. The extended gauge symmetry leads to new charged and neutral current interactions which give the dominant contributions to the production of heavy neutrinos in ep and e~e scattering. Ignoring the 2, small onemixings obtains with standard model gauge bosons which are proportional to (V/V’) =J~W~+J~Wj~+J~LZ~, (2.25) ~

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Right-handed currents

where gR

1+y

l+~~

—

2

=

-

—

J~=g’~/i cot aY—

eVRy~

U +

2

(2.26)

1 2 (B—L) sin a

(2.27)

~t,

B-L 2

(2.28)

Here Y is the standard model hypercharge, g~and g’ are the SU(2)R and U(1)~ gauge couplings, and tan a is the ratio of the U(1)B_L and SU(2)R coupling constants. ~/t denotes the leptons and quarks listed in eqs. (2.1) and (2.2), TsR is the third component of the right-handed isospin, UR and yR are the hadronic and leptonic Kobayashi—Maskawa type matrices of the right-handed charged current. In the special case, where the gauge couplings g~and g~of the gauge groups SU(2)L and SU(2)R are equal, the neutral current reads ~/cos2O~ sin

_______

—

1 +

2

y~ -

~

—

—cli— 2 ~jcos2O~

B

—

2

L cli , (2.29)

where O~is the weak angle. In our discussion on heavy neutrino production in e~e scattering we will restrict ourselves to the case g~ ~

3. Bounds on the mass of WR The subject of this paper is the production of heavy neutrinos in ep and e~e collisions via right-handed charged and neutral currents. The production cross sections are strongly dependent on the WR and Z’ masses, which are constrained by low-energy processes and by precision measurements of Z boson properties. Bounds on the WR mass and the WL—WR mixing angle ~ have been studied in great detail by Langacker and Uma Sankar [7] and, in connection with CP violation in the B-system, by London and Wyler [13]. In the case of left—right symmetry, where the hadronic KM mixing matrices UL and UR for left- and right-handed currents are identical, the stringent bound rnw> 1—3 TeV has been obtained [5,6]. Of importance are also Bd—Bd mixing and the semileptonic branching ratio of b-decays. However, if one relaxes the strong assumption UL UR, all processes are compatible with rnw > 300 GeV [7]. =

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Right-handed Currents

115

For heavy Majorana neutrinos an important bound follows from the seach for neutrinoless double-beta decay (f3f3~,)of 76Ge [14]. From the analysis of Langacker and Uma Sankar [7] we obtain

(~ ~ (~ )4

<8.9

X

10_o( ~

_2(1024yr/Tl/

)4

2, (3.1) 2)~

where mLR are the WLR masses. We have included the sum of all intermediate heavy Majorana neutrinos N.. The J~’~ are the corresponding leptonic KM matrix elements and 1024 yr is the current experimental bound on the f3J3~,lifetime. Assuming J/~ ~IJ, rn~ rnR, .c~ c~and I U~ 1 one obtains the rather large lower bound rnR> 800 GeV. However, there can also be cancellations among the different contributions from heavy neutrinos and, hence, no model independent bound on rnR can be derived from eq. (3.1). For the simplest Higgs sector, described in the previous chapter, bounds on WL—WR and Z—Z’ mixings also imply bounds on the WR and Z’ masses. The mass matrices for charged and neutral vector bosons read (cf. eqs. (2.11), (2.12)): ‘~

=

M2

~g~i2

—

w

—

2

~g~(V

0

g~j2

M~=~ —g~g~t’2

g~(L2+L~2)

0 V2

—g~tan

L’

(3 . 2 )

2 —g~g~1~V + V 2 + 2V~2)

—gLg~L’1L2

2+

V

11

2,

—g~ tan at”2

al)’2 V’2

,

(3.3)

g~ tan2aV~2 V~I2+4L2.

(3.4)

21

Here tan a is the ratio of the U(1)BL and SU(2)R gauge coupling constants. For 1) 0, diagonalization of the W~—Csubmatrix of (3.3) yields one massless vector boson which couples to the standard model hypercharge current and the heavy vector boson Z’ which couples to the neutral current J~ (cf. eq. (2.27)). The mixing ~ between Z’ and the standard model neutral vector boson Z, and the mixing ~w between WL and WR are of order (P2/LI2), and therefore small. One easily finds =

2

2

(gL+gR

I~I~

2

sin a)

1/2

P

2

—~

gR gL

I~wI~

3

cosa

(3.5)

V

1V 11)21

Iv~I2+2L,~2~

(3.6)

116

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W. Buchmüller, C. Greub

Right-handed currents

The corresponding vector boson masses are m~ ~g~v2,

rn~ ~g~( I

m~—(g~+g~ sin2a)V2,

~

2

+ 2V~2),

(lV;I2+4L~2),

where we have neglected terms of relative order In the special case = one has cos2a = 1 eqs. (3.7) and (3.8) yields the inequalities

(3.7)

(3.8)

p2/pI2~

—

tan2Ow) ~

m~ —~--

rn

©

1

—

—

tan2O~which, together with

tan2Ow.

(3.9)

The Z—Z’ mixing angle reads in this case mz

I~~I =~cos2O~ rn

2

.

(3.10)

7 From an analysis of recent LEP data the lower bound rn~>800 Gev has been derived [15,16] which, together with the inequalities (3.9) implies mR>45OGeV.

(3.11)

Hence, LEP data yield a more stringent lower bound on the WR mass than low-energy processes.

4.

Production of heavy neutrinos

Production and decay of heavy Dirac or Majorana neutrinos in ep scattering proceeds through the charged current processes shown in fig. 1. The production cross section is essentially identical to the one for heavy Majorana neutrino

Fig. 1. Production and decay of heavy neutrinos in ep scattering.

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W. BuchmOller, C. Greub

Right-handed currents

117

production via WL exchange which has been studied in detail in ref. [4]. For gL = gR, one easily obtains dci dxdy

____

G~ =

m~

23T (ys+rn~) 2

+

[(~—m~)(u(x,

(1 -y)(~(1 -y)

+c(x,

~2)

rn~)(J(x, ~2)

-

+ ~(x,

p2))

(4.1)

~2))J

where x, y and ~ = xs are the usual kinematical variables (cf. fig. 1) s

= (P

+

Q2

k)2,

—q2,

=

X

2P~q’

=

(4.2)

~ = ~.

which are restricted to the intervals rn~ ——~x~1,

rn~ 0
(4.3)

xs

S

u, c, d and s are the densities of up, charm, down and strange quarks in the proton, which depend on the renormalization scale ~i. Note, that the cross section is not suppressed by small mixing angles but rather by the WR mass which enters the propagator and which has to be larger than 450 GeV (cf. eq. (3.11)). The total cross section is easily obtained after performing numerical integration over x and y. The result is shown in figs. 2—4 as function of rn~for different WR masses and for three different center-of-mass energies which correspond to HERA,

I

80

e

p

I NR

—-->

x

I

I

[VS

I

=

I

I

314 GeV]

60

—

=

‘N N .~

40—

20

0

I

460 GeV

———mw~ = 600 GeV ‘~,

—

—

‘--.

I

-

100

I

I

120

Fig. 2. Total cross section for ep

—

I

140 m,~[GeV] —~

160

180

200

NX at HERA for different values of the WR mass.

118

W. BuchmOller, C. Greub

I

80

I~E~F

e 60

p

I I

I

I

I~T~J

x [Vs

NR

—-->

/ Right-handed currents

mw,

—

I

I

I

450 GeV]

-

-

600 GeV

=

900 GeV

=

———mw

I

-

1200 GeV

=

40— b

2:

~

100

150

200

[0eV]

1~N

250

Fig. 3. Total cross section for ep —* NX at a HERA upgrade (see text) for different values of the WR mass.

(V.~

314 GeV), an upgraded version of HERA, (%/i~= 450 GeV) and LEP 0 LHC 1300 GeV). We have used set 1 of Duke—Owens densities [17] with scales corresponding to the average transverse momentum of the produced heavy neu= =

100 I

e

I p

I

NR

——>

80-\ •

X [Vs

—mw~

=

I =

I

I

1300 GeV]

1000 0eV

mw~ = 1500 GeV —--—mw,~= 2000 GeV •

60

b

I

40—

20

0

-~

~

200

Fig. 4. Total cross section for ep

400 ma [GeV] -.

600

800

NX at LEP®LHC for different values of the W~mass.

W Buchmüller, C. Greub

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119

TABLE 1 Discovery limits for WR masses (mR) and heavy neutrino masses (mN) for three ep colliders with different center-of-mass energies and integrated luminosities per year. 5 events are required

HERA 314 200

Vi~[GeV] 1/yj L [pb mR

1GeV]

450 600 900 1200 1500 2000

--

trino, i.e. =

5

><

HERA upgrade 450 4000

~2

m~1GeV] 120 80

--

—

—

270 220 170

—

—

—

—

—

=

LEP®LHC 1300 2000

iO~GeV2 (i/v

i0~GeV2

=

314 GeV),

~2

=

— —

750 640 540 350

2>< i0~GeV2

(V~

=

450 GeV),

(~/~ = 1300 GeV). At the three machines one year of running is

expected to yield the integrated luminosities 200 pb’, 4000 pb’ and 2000 pb’, respectively. A rough estimate of the discovery limits for neutrino and WR masses is obtained by requiring five events. The corresponding values of rnN and m~ are listed in table 1 for the three different machines. A thorough study of the discovery limits has to take the decays of the heavy neutrinos into account. For mN
:~:.

IoI

e

N W~ Nc Ib)

N

e’~ cl

Fig. 5. Pair production of heavy Dirac (a, b) or Majorana neutrinos (a, b, c) in e~e

annihilation.

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Right-handed currents

fully differential cross section for the production of a heavy neutrino followed by its three-body decay. In e~e scattering heavy Majorana neutrinos can be pair produced via the charged and neutral current processes depicted in fig. 5. We consider the case g~ g~for which the relevant couplings are given by eqs. (2.25)—(2.28). From the different s-, t- and u-channel contributions one obtains the differential cross section: =

24 GFrnL

dodt

1 t—rn~

4~s2

=

+

(

2

U

—

s

—

1 2

umR

t

2

1

+

where s,

~

)2

2 2

—

A2

s—rn~~

—

—

rnR

2

37

(s

rn~s

22

+

—

2

((U—mN)

2

—SmN)

A2 2 ((t_m~)2_Srn~) rn~~)

(4.4)

,

tmR

and u are the kinematical variables (cf. fig. 5) s

2

(p

=

2

2

= (p1 —p4) , ti = (p2 —p4) 2 O~, A=2cos2Ow~ sin 37=~—A.

1 +p2)

t

,

(4.5) (4.6)

For heavy Dirac neutrinos, and assuming LN Le~, only the processes shown in fig. 5a, Sb contribute to the production cross section, and one obtains (cf. eq. (4.6)) =

dci dt

__

G~m~ 2

u

1 —m~

___

-

s—m~

2

22

(f-mN)

2irS

A2

Compared to interference corresponding the Majorana + (sterms case, mz’) 2are eq. 2 (U missing. (4.4), rn~)2 the . t-channel contribution and the -

—

In figs. 6 and 7 the total cross sections are shown as function of the neutrino ~ 200 GeV and different WR masses with rnz~ 2 mR, for Dirac neutrinos and Majorana neutrinos, respectively. The cross section for Dirac neutrinos is slightly larger than the one for Majorana neutrinos. The difference is mass rnN for

=

=

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Right-handed currents

121

100 e

-

e~

NR NR

-

80

200

=

GeV]

Dirac NR

—

mw = 450 0eV ~--~--—-—mw=600GeV

60

b

[VS

-

40—

20

—

~

—

I

40

——

I I

I

I

I I

50

lillil

60

“

ill I

70 mN [0eV]

I

I

I

80

I

I

I

—

Iii~

90

100

Fig. 6. Total pair production cross section for Dirac neutrinos at LEP200.

particularly significant for neutrino masses close to the kinematic limit mN = due to the well known f33 factor in the Majorana case. The corresponding cross sections for %I~= 500 GeV are shown in figs. 8 and 9 for different values of mN.

100

Tl~I

I

I I

80

I

I

I I

I

[VS

I =

I I

I

~]FI~I

200 GeV]

‘-.

I -

—

“~.

~==_.

o

I

mw~= 450 GeV ———mw~ = 600 GeV

‘~-

40—

20

I

Majorana NR

60

b

I

e~ -s NR NR

e

-

I

40

I

50

I

I

IllIll I

60 mN

I

70 [GeV]

—

____________

80

90

100

Fig. 7. Total pair production cross section for Majorana neutrinos at LEP200.

/

W. Buchmüller, C. Greub

122 40

II~~I~I I

e

e~ -s

I

NR NR

Right-handed currents

I

I~F~j

[VS

I

I

I

I

500 GeV]

=

Dirac NR 30

—

—m~~

-

mw~ = 1500 GeV —---—m~~ = 2000 0eV

0

100

1000 GeV

=

150

—

-

—

200

250

ms~[0eV] Fig. 8. Total pair production cross section for Dirac neutrinos at NLC.

Estimates of discovery limits for neutrino masses and WR masses for LEP200 200 GeV, L 500 pb~/y) and NLC (~/~500 GeV, L 10 fb’/y) can be obtained from figs. 6—9 by requiring 10 events. Some representative values are listed in table 2. =

=

40 I

I I

e

-

30

=

e~

-~

=

I

N~ N~

I

[Vs

=

I

~ ———m~

-

I

I

I

-

500 GeV]

Majorana NR mw~ = 1000 GeV

—

I

-

—

=

1500 GeV

-

=

2000 GeV

-

a

100

150

200 ma, [GeV]

Fig. 9. Total pair production cross section for Majorana neutrinos at NLC.

250

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TABLE

Right-handed currents

123

2

Discovery limits for WR masses (ms) and heavy Dirac and Majorana neutrino masses (mN) for two e~e colliders with different center-of-mass energies and integrated luminosities per year. 10 events are required

v’i~[GeV] L[pb~/y]

LEP200

NLC

200 500

500 10000

m~[GeV] 450 600 1000 1500 2000

mN [GeV] Dirac

Majorana

90 70

80 50

—

—

—

—

—

—

Dirac

-

Majorana —

—

250 240 170

—

240 200 130

5. Conclusions New gauge interactions beyong the strong and electroweak forces with gauge are predicted by all unified theories, and it is group SU(3)~x SU(2)L x U(1)~~ conceivable that an extension of the standard model gauge group becomes visible already at TeV energies. In the past particular attention has been given to models with right-handed currents based on the electroweak symmetry group SU(2)R X SU(2)R X U(l)BL which also predict right-handed neutrinos. Low-energy processes require the charged WR boson to be heavier than 300 GeV. From Z precision experiments one finds the lower bound of 450 GeV for equal gauge couplings gL and g~of SU(2)L and SU(2)R, respectively; in this case the Z—Z’ mixing angle is fixed in terms of the neutral vector boson masses. The direct search for WR bosons in pp collisions yields the current lower bound of 520 GeV for right-handed neutrinos lighter than 15 GeV. Our present knowledge about WR bosons and heavy neutrinos will be significantly improved by current and projected ep and e~e colliders. HERA can search for WR bosons with masses up to 700 GeV and for heavy neutrinos with masses up to 120 GeV. At LEP ® LHC ~ and t/R masses of order 1 TeV appear accessible. LEP200 can test for WR masses up to 600 GeV and the projected 500 GeV e~e linear collider (NLC) should be able to reach I TeV for the WR boson. In the search for right-handed currents and heavy neutrinos ep colliders have two main advantages compared to e~e colliders: First, in ep scattering only a single heavy neutrino is produced and, consequently, much larger neutrino masses are accessible. Second, in ep collisions a violation of lepton number in the decays of Majorana neutrinos can be discovered by just measuring a positive charge of the final state lepton, whereas in e~e annihilation the angular distribution of the final state leptons has to be studied.

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W. Buchmüller, C. Greub

/ Right-handed currents

We would like to thank G. Ingelman for helpful discussions on Monte Carlo generators. We are also indebted to J. Polak and M. Zralek for pointing out a numerical error in the original version of the paper.

Appendix A THE FULLY DIFFERENTIAL CROSS SECTIONS

In this appendix we give formulae for the fully differential cross sections for the reactions ep—*XhN—’XhrW~ —sX~rq1~2,

(A.1)

ep—sXhN—sXhf~W~~~sX~(fq1~2.

(A.2)

In eqs. (Al) and (A.2) Xh denotes hadronic matter stemming from the proton remnants and the scattered quark (or antiquark). It turns out that the differential cross sections, when expressed in terms of variables defined in the centre-of-mass frame of the heavy neutrino, and also the kinematical ranges of these variables have a relatively simple form. For both processes we use the followingvariables: x, y deep-inelastic variables defined in eq. (4.2); E, energy of t’ in rest-frame of heavy neutrino; Eq energy of q2 in rest-frame of heavy neutrino; O~,O~ polar angles of ( and ~ 2 in rest-frame of heavy neutrino with respect to proton direction; ~ azimuthal angle of0t~ q2 O~, in rest-frame of heavyangle neutrino. 4~the azimutal 4, of 1~ in the For given values of E~,E0, rest-frame of the heavy neutrino N is then fixed to be either —

—

—

—

—

=

çli,~+

arccos C

or

=

+

2~ arccos C, —

(A.3)

with

~=

rn~—2mN(E(+EO)+2E(EO(1—cos O~ cos Oq) 2E,E 0 sin O~sin

(A4) .

0q’ 45q and by choosing çb~according to (A.3), the Hence, by specifying E1, E0, four-momenta of ~ and q ~ 2 in the rest-frame of N are uniquely fixed, and by energy-momentum conservation also the four-momentum of q1 is determined. In order to write down the cross section in a compact form, we also use (as auxiliary variables) the energy (EN), the momentum (PN) and the polar angle (0) of the heavy neutrino in the laboratory frame. x, y and the laboratory beam

W BuchmOller, C. Greub

/

Right-handed currents

125

energies Ee of the electron and E~of the proton fix longitudinal (p~) and transverse (p~)parts of the momentum of the heavy neutrino N: (p~)2=y[(1 -y)~-rn~], §y

m~ 4(1 —y)E2

+

—

4Ee

Hence, also 0,

and

PN

2

PN=[(P~)

e

§=xS=4xE~E~.

,

(A.5)

are fixed:

EN

L PN cos0=~,

2 1/2

+(p~)j

,

_________

EN=V’p~+rn~. (A.6)

We now give the fully differential cross section dci. for reaction (Al) with a negatively charged lepton in the final state. Note, that one obtains the same result for Dirac and Majorana neutrinos:

dx dy dE~ dE~d0~do 0 d~0 = VX{A[u(x~

~2)

+c(x,

~2)}

§(1—y)—rn~ B4d(xjc2)+~(x,/i2)} -

+

(A.7) with t~mNEO(mN 2E G~rn 0) sin 0~sin rn~) 04 2’ 6FNJ(xyS+ rn~)2(rn~ 2mNEC 16~—

—

(A.8)

—

A..=(~—rn~)—2cos0O[EC(pN+ENcos0)+xEP(pN—ENcos9)I —

B

-

=

§(1

—

2mN(xEP y)

—

—

4~,

Ee) sin 0 sin 04 cos

2 xE~cos 0~(p N

—

EN

cos 0)

—

(A.9)

2 xrn N E~sin 0 sin 0~cos

(A.10) J =

2[z

[cos(0 1

—

0~)

2E,E-—

—

z} ~

—

cos(0, +

o~)]~2,

2mN(E(+E-)+m~ 2E~E 0

(All) (A.12)

126

W~BuchmOller, C. Greub

/

Right-handed currents

P

//

pr~rcco S

Fig. Al. The shaded area denotes the allowed range for the variables

and 0,.

For simplicity, in eq. (A.7) we have chosen the Kobayashi—Maskawa type matrices UR and yP~ equal to the unit matrix. The case of nonvanishing mixing angles can be easily implemented. The kinematically allowed ranges of x, y, E,, E4 and read: rn~ O
m~ —
_~—J,

E,~ [o~ rn~

E 4E

E

mN—

2

‘

(A.14)

‘

[0, 2ir}.

(A.15)

The boundaries of the polar angles 0, and 9~are cos(0,+04)
(A.13)

XS 2E,. mN

given

by the inequalities

cos(0,—O4)~Z,

(A.16)

with Z given in eq. (A.12). The allowed range in the (0,, Oq) plane corresponds to the shaded area in fig. Al. We now discuss the second process given in eq. (A.2), where the final-state lepton is positively charged. This case only occurs for heavy Majorana neutrinos. The corresponding fully differential cross section du~ is obtained from eq. (A.7) by replacing V, A and B by V~, A~ and B~, respectively. For these quantities we get: 0q

,

16IT6FNJ(xyS G~m~rnN(2E,+ + m~)2(rn~ 2E4 mN)2rnNE, sin 0, sin rn~)2 —

—

—

(A.17)

W. Buchmüller, C. Greub

/

Right-handed currents

127

A~=(s~—m~)(mN—E,—EO) cos 0,+E4 cos

cos 0) +xEP(pN—EN cos 0)]

6q)[Ee(PN+EN

2mN sin 0(xE~—E~)(E,sin 0, cos çb,+E 4 sin 0~cos

—

B~=.~(1

(A.18)

~i~’),

—y)(mN—E,—E4) 2xEP(pN—EN cos 0)(E, cos 0,+E

— 4 cos o~) 2xmNEP sin 0(E, sin 0, cos çb,+E 4 sin 04 cos —

i~). (A.19)

We now discuss how one has to generate the four-momenta of the final-state particles of an event in the laboratory frame. For a given “point”

4~)

(x, y, E,, E4, 0,, 04,

(A.20)

we first choose ~, according to eq. (A.3) with equal probability. Then the four-momenta of the final state particles originating from the heavy neutrino are uniquely fixed in the neutrino rest-frame. In order to get the corresponding four-momenta in the laboratory frame of the ep reaction one has to perform the following Lorentz transformation: L’LAB

= R~(

1)

R~(0)

(A.21)

L~VcM,

where v generically stands for the four-momentum of ~ q2 and q1. The rotations R~(cP),R~(0)and the boost L~read R ~ ~‘

= ‘

1 0 0

0 cos ~ sin ~

0 —sin ~ cos ~

0 0 0

0

0

0

1

R ~

=

~“ ‘

‘

1 0 0

0 cos 0 0

0 0 1

0 sin 0 0

0

—sin0

0

cosO (A.22)

EN

0 rn~4

0 0

PN 0

1 L~=—

0

mN

‘-I

ij

rnN

PN

0

0

~

‘_.~

(A.23)

.

EN

Here 0 and tl are polar and azimuthal angle of the heavy neutrino in the lab-frame, and EN and PN are its energy and momentum: p~=(EN,

PN

sin 0 cos

~,

PN sin

0 sin

~,

PN

cos 0),

PN

=

‘PNI’

(A.24)

128

W. Buchmüller, C. Greub

/

Right-handed currents

0, EN and PN are fixed by eqs. (A.5) and (A.6), whereas cP has to be chosen in the interval [0, 2ir] with equal probability. The absolute weight of the event is then given by dx dy dE, dEdO, do4 d~4 where the factors ~ and 1/(27r) are due to the choices of

(A.25)

4, and P, respectively.

References Ill

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