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The photino mass and γγ˜γ˜ production in e+e− annihilation
Volume 139B, number 3
THE PHOTINO MASS AND ~ ~
PHYSICS LETTERS
10 May 1984
PRODUCTION IN e+e - ANNIHILATION
T. KOBAYASHI Laboratory o f International Collaboration on Elementary Particle Physics, University o f Tokyo, Tokyo, Japan and M. KURODA Institut ffftr Theoretische Physik, Universitiit Regensburg, Fed. Rep. Germany Received 29 December 1983 Revised manuscript received 10 February 1984
We present the exact calculation of the cross section for the process e+e- ~ ~,~"~with finite photino mass. The process is one of the best places to search for the stable photino at PETRA energy.
Supersymmetric theories (SUSY) [1] o f particle physics have been of interest recently, especially in connection with the understanding o f the gauge hierarchy problem [2]. Unfortunately none of the known elementary particles is the supersymmetric partner o f any other. Experimental searches of supersymmetric partners at PETRA and PEP have set a lower limit of about 20 GeV/c for the scalar electron (e-') mass, when the photino ( ~ mass is neglected [3]. However, it has been pointed out recently, that in evaluating such a production cross section o f supersymmetric particles, the photino mass m~, plays an important role [ 4 - 6 ] . When my = 0 and the right-handed and left-handed scalar electrons have the same mass, the cross section o(e+e - ~ R e R + ~Le'L + ~RfiL + ~L_e'R) shows an interesting peak near the threshold, mainly due to the s-wave nature of the process e+e - -+ e r e L + eLeR, which vanishes when = 0 [5]. On the other hand, when mTR ~ m~L, we expect presumably mgR ~ mgL and the process e+e - -+ e'ReR is energetically the most feasible among the scalar electron pair productions. In this case, for given m~'R, the cross section o(e+e - -+ ~Re'R) decreases as m~ increases from 0. There is, however, an absolute lower bound for the cross section at a certain value of m~ and as m~ increases further, the cross section starts to increase again (see fig. 2 of ref. [6]). This gives the possibility to set a model independent bound on the scalar electron mass (i.e. independent of the photino mass) when it is not experimentally detected. We address ourselves in this paper to another interesting process e+e - ~ ~ , ~ , and examine the effect of finite photino mass ,1. This process, although of higher order than the process e+e - -+ . ~ , is suitable for the detection o f stable (or almost stable) photinos, which usually escape detection ,2. There are six F e y n m a n diagrams for this process, which are shown in fig. 1. In the case m~ ~ E ~ mg, where E is the beam energy, the process e+e - -+ ~ is very analogous to the process e+e - ~ 7 ~ and in ref. [4], Ellis and Hagelin consider this process in this limiting case with a certain phenomenological correction for the finite photino mass. Since we are especially interested in the effect of finite photino mass, we perform in this paper the exact calculation of the cross section, keeping m~ finite throughout the calculation. 4:1 The process was f'trst considered by Fayet [7] for m~ = 0. 4:2 The unstable photino would be discovered, ff not too heavy to be produced, in the process e+e- ~ ~ by detecting the photon coming from the photino decay. See ref. [8] for experimental details. 208
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
==l
Volume 139B, number 3
PHYSICS LETTERS
"~(k) ~(kz)~(k 1)
"~(k) ~(k2)~(k 1) 1L-~_J1
i. .
.
.
.
.
.
.
.
.
e-(Pl)
e+(P2)
(a)
10 May 1984
(b)
(c)
e*(p~) e-(pl) (Q')
(b')
(c')
Fig. 1. Feynman diagrams for the process e+e- ~ 7"r"~.The primed diagrams a', b' and c' are obtained from the unprimed ones by interchanging the two photino momenta, k 1 and k 2.
Without calculating the cross section in detail, we notice several general features o f the process. In the limit rn~e>> E, the diagrams b and b' are suppressed compared to the other diagrams because o f the extra scalar electron propagator. When rn~ = 0, there is no interference between the set o f diagrams (a, b, c) and the set (a', b', c'). The interference between these sets is therefore proportional to m 2, as we shall see later explicitly. In the propagator, eL as well as e'R can propagate. The matrix elements for e'L exchange are given from the matrix elements for e'R exchange by replacing I(1 + 75) with ~(1 x- 75) and m~"R with mgL. Due to the chiral coupling I(1 + 75) of the e~'~ vertex, there is no interference between the eR exchange and ~L exchange diagrams, when the electron mass is neglected, which we assume in this paper. F o r this reason, it is sufficient to consider only e'R exchange. The cross section for eL exchange is obtained from the cross section for eR exchange simply by replacing m~R by mgL. We shall show in figs. 2, 3 and 4, the cross section for eR exchange only. The experimentally observed cross section o(e+e - ~ 7~7) is then given by the sum of two cross sections, one coming from the e'R exchange and the other coming from the eL exchange, which are readily read out from figs. 2, 3 and 4. The matrix element of each F e y n m a n diagram (fig. 1) due to eR exchange is given as (the momenta of the particles are defined in fig. 1)
tl
(pb)
V~ = /+0GeV x = E~/E~ 0.25 IcosOl .< 0.8
1
~
=10
~
~ = ~OGeV
013020~~1~
O.Ol 0
10 rni(fieV/c2)
20
Fig. 2. The total cross section a(e+e - ~ "r~-"~)due to the eR exchange as a function of the photino mass m~. x/s = 2E = 40 GeV and the experimental cutsx = E~]E ~ 0.25 and I cos OI < 0.8 are applied to evaluate the cross section.
0
0.5
x = EIIE
Fig. 3. The differential cross section do(e+e - -* 77-'~)/dx for x/s = 40 GeV, Icos 01 ~ 0.8 and m~"R = 20 GeV/c 2. 209
V o l u m e 139B, n u m b e r 3
PHYSICS L E T T E R S
10 May 1984
do
dcosO \ (pb)
//
: a0 6eV
\
o .;20T,
10-1
10
10-2
15 I
10"3-1
0
Fig. 4. T h e d i f f e r e n t i a l cross s e c t i o n d o (e+e - --+ - r ~ ' ) / d cos 0 for x/~ = 4 0 GeV, x = Eq,/E >i 0.25 a n d m~"R = 20 G e V / c 2 .
m
cos E)
-2e 3 c~(a) = [°(P2) q~ (~b2 - ~) 3( 1 - 75) o(k2)] [ff(kl) ~(1 + ")'5) u(Pl)]
2(P2k)(A -- 2 p l k l ) '
2e3e" (kl - k2 - Pl + P2) c~(b) = [u(P2) ½(1 -- ")'5) o(k2)] [u(kl) 3( 1 + 3'5) U(Pl)] (A -- 2 P l k l ) ( A _ 2p2k2) ' 2e 3 c~(c) = [u(P2) ~-(1 -- 75) u(k2)] [ff(kl) -~(1 + 75)(/bl - ~) ~ u(Pl) ] 2 ( P l k ) ( A _ 2P2k2) , -
2
2
'
(1)
'
where A = rn~ -- m~.. The other three matrix elements QR (a), Off Co) and cTR(c') are obtained from the correspondr R ing unprimed matrix element by k 1 ~+ k 2 and by changing the overall sign due to the Fermi statistics. The general form of the differential cross section is given by do 1 ~ ' 1 Off 12, dx dz d cos 0 ckp - 64(2n) 4
(2)
where x = E T / E , z = E'~/E and 0 is the polar angle of the photon in the lab frame. The angle ~o is the angle between the plane (T77) and the plane (e+e-7), i.e. the azimuthal angle of the e -+beam in the frame where 777 is in the x - z plane with 7 in the z direction. Since two photinos produced in the process probably escape detection, the process looks like e+e - -+ 7 + missing energy and what are experimentally observable are the photon energy x and the photon scattering angle 0. Bearing in mind that a photino is a Majorana fermion, the differential cross section becomes do
_ °~3
z+
21r
0xdcos0 32.f ~ f z_
"[
,
~ Z~ ~12
0
(3)
where ~" means the spin average factor and the coupling constants are removed from [c-~ 12, and 2
2
z+ = 1 - (x/Z){1 -7- [1 - m.~[E (1 - x ) ]
1/2).
(4)
-- m .2~ o l ( P i , pj; kl, k m ) ] ,
(5)
After lengthy calculations, we found
~"lOffl 2 = 4 ~
(i,j) (l,m)
210
[oO(Pi,Pj;kl, krn)
Volume 139B, number 3
PHYSICS LETTERS
10 May 1984
where (i,/) and (l, m) run over the combination (1,2) and (2,1) independently, o0 and o 1 are given as
(k2k)(klPl) - o0(Pl, P2; kl, k2) = (P2k)( A _ 2 p l k l ) 2 2(klPl){(p2k2)[4E2(1
+
2 ( k 2 P 2 ) ( k l P l ) [2E2(2 - x) - m 2 - (Pl k2) - (P2kl)] (A -- 2p2k2)2(A -- 2 P l k l ) 2
- x ) - 2 p l k 2 + p2 k] +
2 m~(p2~:)}
(P2k)(A -- 2plk1)2(A -- 2p2k2) +
( k l P l ) { 4 E 4 ( 1 - x ) + ( k z p l ) [ ( k p 2 ) - 2E2]) - E 2 x ( k l P l ) ( k 2 P 2 ) + (1 ~ 2) 2(p 1k)(P2 k) (A -- 2Pl k l ) ( A -- 2p2k2)
kPl
(6)
2E4(x + m 2 /E 2)
o l ( P l , P2; klk2) = 2(kp2)(A _ 2 P l k l ) ( A _ 2Plk2) + (A -- 2P2k2)(A -- 2 P l k l ) ( A -- 2p2kl)(A -- 2Plk2)
-
2E 2 [2E2(l - x) + (P2k)] + (P2kl)(,Plk2) - ( P l k l ) ( P 2 k 2 ) (P2k)(A -- 2 P l k l ) ( A -- 2 p 2 k l ) ( A - 2plk2)
2(1 - x)
+
x 2 sin20 (A - 2p2k2)( A - 2 p l k l ) "
(7)
The Lorentz scalar quantities (pik/), etc., are expressed in terms of x, O, ~o and z as follows
( k i k ) = 2E2(x + x i - 1), (t71 (2), k) = E 2 x ( 1 T-cos 0 ) ,
(Pl (2), ki) = E 2 x i ( 1 ¥ [3i sin 0 cos ~p sin 0 i ¥ fli cos 0 cos 0 i ) ,
(8)
with
. 2 tr:.2x2 fli = 1 --rrr~l r, i ,
c o s 0 i = [2(1 - - x - - x i ) + x x i ] / X X i f l i ;
x 1 =z,
x2 = 2 - - x - - z .
(9)
As before, A = m 2 _ m 2 and E is the beam energy. Fig. 2 shows the expected total cross section o(e+e - -~ 3~7) at PETRA (E = 20 GeV) with cut ,3 for x and cos O;x >1 0.25 and 0.8 ~> Icos 0 I. We observe a substantial effect of the finite photino mass. The cross section decreases with increasing m~; the cross section for rr~ = 10 GeV/c 2 is a factor of ~ 3 smaller than the cross section for m~ --- 0. Fig. 3 shows do/dx with cos 0 integrated between - 0 . 8 and +0.8 and fig. 4 shows do/d cos 0 with x integrated over x ~> 0.25. The cross section o(e+e - ~ 73'7) is of order 0 . 0 1 - 0 . 1 pb and with the integrated luminosity of ~ 1 0 0 pb - 1 at PETRA, we expect about 1 - 1 0 events. Possible backgrounds to the process are e+e - ~ 7 ~ , 777, e+e-7 and cosmic ray events, but any of them, except for the first one, could well be discriminated easily. The first process [9] e+e - ~ 7v~, which can be potentially a dominant background to the process e+e - -+ T~7 at high energy, is still not so serious at PETRA energy, x/s ~ 40 GeV (in which we are mainly interested in this paper) and for a number of neutrino species of order 3 and for rrrgeg 50 GeV/c 2. We compared our exact calculation with the soft photon approximation given in ref. [4]. Although there remains some doubt in using the soft photon approximation up to the hard photon, we found that the results of Ellis and Hagelin reproduce an almost correct m~ dependence. Their cross section is, however, underestimated, especially at large x: e.g., x 2 0.5, in which we look for the photon as a signal for the process and where the soft photon approximation starts to break down, their cross section is typically by 220% too small ,4. Finally we would like to stress the importance of the search for stable photinos in the present process, in which presumably we have the best chance to discover it at PETRA.
,3 The process has a mass singularity and an infrared singularity in the approximation m e = 0. ,4 See also ref. [10]. 211
Volume 139B, number 3
PHYSICS LETTERS
l0 May 1984
The authors would like to thank Professor S. Yamada for the useful discussions and suggestions. TK thanks Professor M. Koshiba for support and he is indebted to the DESY directorate for their hospitality.
References [1] J. Wess and B. Zumino, Nucl. Phys. B70 (1974) 39; Phys. Lett. 49B (1974) 52; A. Salam and B. Strathdee, Phys. Rev. D l l (1975) 1521; P. Fayet and S. Ferrara, Phys. Rep. 32C (1977) 249. [2] S. Weinberg, Phys. Lett. 82B (1979) 387; A. Buras et al., Nucl. Phys. B135 (1978) 66; L. Susskind, Phys. Rev. D20 (1978) 2619. [3] JADE Collab., D. Cords, Proc. XXth Intern. Conf. on High energy physics, eds. L. Durand and L.G. Pondrom (Wisconsin, 1980) p. 590; CELLO Collab., H.J. Behrend et al., Phys. Lett. l14B (1982) 287; MARK J Co[lab., B. Adeva et al., Phys. Lett. 115B (1982) 345; TASSO Collab., R. Bandelik et al., Phys. Lett. l17B (1982) 365. [4] J. Ellis and J.S. Hagelin, Phys. Lett. 122B (1983) 303. [5] M. Gliick and E. Reya, Phys. Lett. 130B (1983) 423. [6] T. Kobayashi and M. Kuroda, Phys. Lett. 134B (1984) 271. [7] P. Fayet, Phy~ Lett. 117B (1982) 460. [8] CELLO CoUab., H.J. Behrend et al., Phys. Lett. 123B (1983) 127. [9] E. Ma and J. Okada, Phys. Rev. Lett. 41 (1978) 287; Phys. Rev. D18 (1978) 4219; K.J.F. Gaemers, R. Gastman and F.M. Renard, Phys. Rev. D19 (1979) 1605. [10] Yu.A. Gnedov, K.G. Klimenko and F.F. Tikhonin, Serpukhov preprint 83-89.
212
THE PHOTINO MASS AND ~ ~
PHYSICS LETTERS
10 May 1984
PRODUCTION IN e+e - ANNIHILATION
T. KOBAYASHI Laboratory o f International Collaboration on Elementary Particle Physics, University o f Tokyo, Tokyo, Japan and M. KURODA Institut ffftr Theoretische Physik, Universitiit Regensburg, Fed. Rep. Germany Received 29 December 1983 Revised manuscript received 10 February 1984
We present the exact calculation of the cross section for the process e+e- ~ ~,~"~with finite photino mass. The process is one of the best places to search for the stable photino at PETRA energy.
Supersymmetric theories (SUSY) [1] o f particle physics have been of interest recently, especially in connection with the understanding o f the gauge hierarchy problem [2]. Unfortunately none of the known elementary particles is the supersymmetric partner o f any other. Experimental searches of supersymmetric partners at PETRA and PEP have set a lower limit of about 20 GeV/c for the scalar electron (e-') mass, when the photino ( ~ mass is neglected [3]. However, it has been pointed out recently, that in evaluating such a production cross section o f supersymmetric particles, the photino mass m~, plays an important role [ 4 - 6 ] . When my = 0 and the right-handed and left-handed scalar electrons have the same mass, the cross section o(e+e - ~ R e R + ~Le'L + ~RfiL + ~L_e'R) shows an interesting peak near the threshold, mainly due to the s-wave nature of the process e+e - -+ e r e L + eLeR, which vanishes when = 0 [5]. On the other hand, when mTR ~ m~L, we expect presumably mgR ~ mgL and the process e+e - -+ e'ReR is energetically the most feasible among the scalar electron pair productions. In this case, for given m~'R, the cross section o(e+e - -+ ~Re'R) decreases as m~ increases from 0. There is, however, an absolute lower bound for the cross section at a certain value of m~ and as m~ increases further, the cross section starts to increase again (see fig. 2 of ref. [6]). This gives the possibility to set a model independent bound on the scalar electron mass (i.e. independent of the photino mass) when it is not experimentally detected. We address ourselves in this paper to another interesting process e+e - ~ ~ , ~ , and examine the effect of finite photino mass ,1. This process, although of higher order than the process e+e - -+ . ~ , is suitable for the detection o f stable (or almost stable) photinos, which usually escape detection ,2. There are six F e y n m a n diagrams for this process, which are shown in fig. 1. In the case m~ ~ E ~ mg, where E is the beam energy, the process e+e - -+ ~ is very analogous to the process e+e - ~ 7 ~ and in ref. [4], Ellis and Hagelin consider this process in this limiting case with a certain phenomenological correction for the finite photino mass. Since we are especially interested in the effect of finite photino mass, we perform in this paper the exact calculation of the cross section, keeping m~ finite throughout the calculation. 4:1 The process was f'trst considered by Fayet [7] for m~ = 0. 4:2 The unstable photino would be discovered, ff not too heavy to be produced, in the process e+e- ~ ~ by detecting the photon coming from the photino decay. See ref. [8] for experimental details. 208
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
==l
Volume 139B, number 3
PHYSICS LETTERS
"~(k) ~(kz)~(k 1)
"~(k) ~(k2)~(k 1) 1L-~_J1
i. .
.
.
.
.
.
.
.
.
e-(Pl)
e+(P2)
(a)
10 May 1984
(b)
(c)
e*(p~) e-(pl) (Q')
(b')
(c')
Fig. 1. Feynman diagrams for the process e+e- ~ 7"r"~.The primed diagrams a', b' and c' are obtained from the unprimed ones by interchanging the two photino momenta, k 1 and k 2.
Without calculating the cross section in detail, we notice several general features o f the process. In the limit rn~e>> E, the diagrams b and b' are suppressed compared to the other diagrams because o f the extra scalar electron propagator. When rn~ = 0, there is no interference between the set o f diagrams (a, b, c) and the set (a', b', c'). The interference between these sets is therefore proportional to m 2, as we shall see later explicitly. In the propagator, eL as well as e'R can propagate. The matrix elements for e'L exchange are given from the matrix elements for e'R exchange by replacing I(1 + 75) with ~(1 x- 75) and m~"R with mgL. Due to the chiral coupling I(1 + 75) of the e~'~ vertex, there is no interference between the eR exchange and ~L exchange diagrams, when the electron mass is neglected, which we assume in this paper. F o r this reason, it is sufficient to consider only e'R exchange. The cross section for eL exchange is obtained from the cross section for eR exchange simply by replacing m~R by mgL. We shall show in figs. 2, 3 and 4, the cross section for eR exchange only. The experimentally observed cross section o(e+e - ~ 7~7) is then given by the sum of two cross sections, one coming from the e'R exchange and the other coming from the eL exchange, which are readily read out from figs. 2, 3 and 4. The matrix element of each F e y n m a n diagram (fig. 1) due to eR exchange is given as (the momenta of the particles are defined in fig. 1)
tl
(pb)
V~ = /+0GeV x = E~/E~ 0.25 IcosOl .< 0.8
1
~
=10
~
~ = ~OGeV
013020~~1~
O.Ol 0
10 rni(fieV/c2)
20
Fig. 2. The total cross section a(e+e - ~ "r~-"~)due to the eR exchange as a function of the photino mass m~. x/s = 2E = 40 GeV and the experimental cutsx = E~]E ~ 0.25 and I cos OI < 0.8 are applied to evaluate the cross section.
0
0.5
x = EIIE
Fig. 3. The differential cross section do(e+e - -* 77-'~)/dx for x/s = 40 GeV, Icos 01 ~ 0.8 and m~"R = 20 GeV/c 2. 209
V o l u m e 139B, n u m b e r 3
PHYSICS L E T T E R S
10 May 1984
do
dcosO \ (pb)
//
: a0 6eV
\
o .;20T,
10-1
10
10-2
15 I
10"3-1
0
Fig. 4. T h e d i f f e r e n t i a l cross s e c t i o n d o (e+e - --+ - r ~ ' ) / d cos 0 for x/~ = 4 0 GeV, x = Eq,/E >i 0.25 a n d m~"R = 20 G e V / c 2 .
m
cos E)
-2e 3 c~(a) = [°(P2) q~ (~b2 - ~) 3( 1 - 75) o(k2)] [ff(kl) ~(1 + ")'5) u(Pl)]
2(P2k)(A -- 2 p l k l ) '
2e3e" (kl - k2 - Pl + P2) c~(b) = [u(P2) ½(1 -- ")'5) o(k2)] [u(kl) 3( 1 + 3'5) U(Pl)] (A -- 2 P l k l ) ( A _ 2p2k2) ' 2e 3 c~(c) = [u(P2) ~-(1 -- 75) u(k2)] [ff(kl) -~(1 + 75)(/bl - ~) ~ u(Pl) ] 2 ( P l k ) ( A _ 2P2k2) , -
2
2
'
(1)
'
where A = rn~ -- m~.. The other three matrix elements QR (a), Off Co) and cTR(c') are obtained from the correspondr R ing unprimed matrix element by k 1 ~+ k 2 and by changing the overall sign due to the Fermi statistics. The general form of the differential cross section is given by do 1 ~ ' 1 Off 12, dx dz d cos 0 ckp - 64(2n) 4
(2)
where x = E T / E , z = E'~/E and 0 is the polar angle of the photon in the lab frame. The angle ~o is the angle between the plane (T77) and the plane (e+e-7), i.e. the azimuthal angle of the e -+beam in the frame where 777 is in the x - z plane with 7 in the z direction. Since two photinos produced in the process probably escape detection, the process looks like e+e - -+ 7 + missing energy and what are experimentally observable are the photon energy x and the photon scattering angle 0. Bearing in mind that a photino is a Majorana fermion, the differential cross section becomes do
_ °~3
z+
21r
0xdcos0 32.f ~ f z_
"[
,
~ Z~ ~12
0
(3)
where ~" means the spin average factor and the coupling constants are removed from [c-~ 12, and 2
2
z+ = 1 - (x/Z){1 -7- [1 - m.~[E (1 - x ) ]
1/2).
(4)
-- m .2~ o l ( P i , pj; kl, k m ) ] ,
(5)
After lengthy calculations, we found
~"lOffl 2 = 4 ~
(i,j) (l,m)
210
[oO(Pi,Pj;kl, krn)
Volume 139B, number 3
PHYSICS LETTERS
10 May 1984
where (i,/) and (l, m) run over the combination (1,2) and (2,1) independently, o0 and o 1 are given as
(k2k)(klPl) - o0(Pl, P2; kl, k2) = (P2k)( A _ 2 p l k l ) 2 2(klPl){(p2k2)[4E2(1
+
2 ( k 2 P 2 ) ( k l P l ) [2E2(2 - x) - m 2 - (Pl k2) - (P2kl)] (A -- 2p2k2)2(A -- 2 P l k l ) 2
- x ) - 2 p l k 2 + p2 k] +
2 m~(p2~:)}
(P2k)(A -- 2plk1)2(A -- 2p2k2) +
( k l P l ) { 4 E 4 ( 1 - x ) + ( k z p l ) [ ( k p 2 ) - 2E2]) - E 2 x ( k l P l ) ( k 2 P 2 ) + (1 ~ 2) 2(p 1k)(P2 k) (A -- 2Pl k l ) ( A -- 2p2k2)
kPl
(6)
2E4(x + m 2 /E 2)
o l ( P l , P2; klk2) = 2(kp2)(A _ 2 P l k l ) ( A _ 2Plk2) + (A -- 2P2k2)(A -- 2 P l k l ) ( A -- 2p2kl)(A -- 2Plk2)
-
2E 2 [2E2(l - x) + (P2k)] + (P2kl)(,Plk2) - ( P l k l ) ( P 2 k 2 ) (P2k)(A -- 2 P l k l ) ( A -- 2 p 2 k l ) ( A - 2plk2)
2(1 - x)
+
x 2 sin20 (A - 2p2k2)( A - 2 p l k l ) "
(7)
The Lorentz scalar quantities (pik/), etc., are expressed in terms of x, O, ~o and z as follows
( k i k ) = 2E2(x + x i - 1), (t71 (2), k) = E 2 x ( 1 T-cos 0 ) ,
(Pl (2), ki) = E 2 x i ( 1 ¥ [3i sin 0 cos ~p sin 0 i ¥ fli cos 0 cos 0 i ) ,
(8)
with
. 2 tr:.2x2 fli = 1 --rrr~l r, i ,
c o s 0 i = [2(1 - - x - - x i ) + x x i ] / X X i f l i ;
x 1 =z,
x2 = 2 - - x - - z .
(9)
As before, A = m 2 _ m 2 and E is the beam energy. Fig. 2 shows the expected total cross section o(e+e - -~ 3~7) at PETRA (E = 20 GeV) with cut ,3 for x and cos O;x >1 0.25 and 0.8 ~> Icos 0 I. We observe a substantial effect of the finite photino mass. The cross section decreases with increasing m~; the cross section for rr~ = 10 GeV/c 2 is a factor of ~ 3 smaller than the cross section for m~ --- 0. Fig. 3 shows do/dx with cos 0 integrated between - 0 . 8 and +0.8 and fig. 4 shows do/d cos 0 with x integrated over x ~> 0.25. The cross section o(e+e - ~ 73'7) is of order 0 . 0 1 - 0 . 1 pb and with the integrated luminosity of ~ 1 0 0 pb - 1 at PETRA, we expect about 1 - 1 0 events. Possible backgrounds to the process are e+e - ~ 7 ~ , 777, e+e-7 and cosmic ray events, but any of them, except for the first one, could well be discriminated easily. The first process [9] e+e - ~ 7v~, which can be potentially a dominant background to the process e+e - -+ T~7 at high energy, is still not so serious at PETRA energy, x/s ~ 40 GeV (in which we are mainly interested in this paper) and for a number of neutrino species of order 3 and for rrrgeg 50 GeV/c 2. We compared our exact calculation with the soft photon approximation given in ref. [4]. Although there remains some doubt in using the soft photon approximation up to the hard photon, we found that the results of Ellis and Hagelin reproduce an almost correct m~ dependence. Their cross section is, however, underestimated, especially at large x: e.g., x 2 0.5, in which we look for the photon as a signal for the process and where the soft photon approximation starts to break down, their cross section is typically by 220% too small ,4. Finally we would like to stress the importance of the search for stable photinos in the present process, in which presumably we have the best chance to discover it at PETRA.
,3 The process has a mass singularity and an infrared singularity in the approximation m e = 0. ,4 See also ref. [10]. 211
Volume 139B, number 3
PHYSICS LETTERS
l0 May 1984
The authors would like to thank Professor S. Yamada for the useful discussions and suggestions. TK thanks Professor M. Koshiba for support and he is indebted to the DESY directorate for their hospitality.
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