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Quark and lepton flavor violating Z0 decays in E6
Volume 188, number 1
PHYSICS LETTERS B
2 April 1987
Q U A R K A N D L E P T O N FLAVOR V I O L A T I N G Z ° DECAYS IN E~ ~ Gad E I L A M 1 and Thomas G. R I Z Z O 2 Ames Laboratory and Department of Physics, Iowa State University, Ames, IA 50011, USA Received 29 December 1986 Appreciable lepton flavor violating decays of Z °, e.g. Z ° ~ #T, can occur in E 6 grand unified theories. We examine these decays and find, using the existing experimental limits on rare processes, that some m a y have branching ratios as large as a few times 10 -4. Flavor violating quark decays are also expected at the same level but only in the Q = - 1 / 3 sector (e.g. Z ° ~ bg, but Z ° ~ tc-) within the framework of E6; these quark decays are very difficult to observe experimentally.
Superstring theories [1], in a currently popular version, manifest themselves as a n E 6 grand unified theory ,1 below the Planck scale. At lower energies, one or more Z bosons as well as exotic fermions emerge, the latter transforming as (at least) three 27 representations of E613 ]. We concentrate here on implications of mixing between these new fermions and the usual leptons and quarks [3,4], which may cause detectable lepton and quark flavor changing neutral currents• Such currents are either nonexistent or unmeasurably small in the Standard Model, as they are absent from the lepton sector and occur only through loops for quarks [5]. In E 6 , the potentially large rates for decays such as Z ° ~ # ~ - or Z ° ~ b ~ , which proceed at the tree level, are suppressed only by unknown mixing angles, which we can limit using results of existing experiments on and B decays• The formalism leading to new exotic fermions, their mixing with the usual fermions and the fact that only exotic quarks with charge Q = - 1 / 3 * This work was supported by the US Department of Energy, Contract No. W-7405-Eng-82, Office of Energy Research (KA-01-01), Division of High Energy and Nuclear Physics. 1 On leave of absence from: Physics Department, Technion, Haifa 32000, Israel. 2 Address for period January 1, 1987-June 30, 1987: Center for Particle Theory, University of Texas - Austin, Austin, TX 78712, USA. ,1 For original references on E 6 grand unification see ref. [2].
(and not with Q = 2 / 3 ) emerge, has been already discussed [3,4]. The final results, relevant to the intergenerational mixing discussed here, are the following new neutral current interaction terms:
L,i(Q =
-ig - 1 ) = -~Cw 4y~(1 - Vs) 0
Lu( Q =
-1/3)
= -~g-clg~,T~(1 + Vs)
for charged leptons and Q = - 1 / 3 quarks, respectively, where U L, UR are 6 x 6 a priori unknown left and right mixing matrices, i and j (i 4=j) denote lepton or quark types, c w -=-cos 0w and x w -= sin20w = 0.22. Note the absence of a direct flavor changing electromagnetic term; therefore contributions to processes such as # --* ey will be highly suppressed (no tree, only loop diagrams are possible), while flavor changing processes mediated or initiated by a Z boson may proceed at the tree level• It is also clear that for such processes the contribution of a second Z ( Z ' ) can be safely neglected if Z ' is sufficiently massive. Let us first turn to lepton flavor violating transitions, with emphasis on Z ~ E , ~ + ~ - - ~ .
0370-2693/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
91
Volume 188, number 1
PHYSICS LETTERS B
These decays have the following advantages as probes for exciting new physics: (1) The Standard Model does not accommodate flavor changing lepton transitions either on a tree or a loop level. Therefore, any nonzero signal will indicate important deviations from the Standard Model [6] .2 If heavy neutrinos exist, they can, through loops, induce a Z ~ 4 4 transition. However, based on extensive experience from investigations of Z--+ qiglj processes [5], we do not expect branching ratios in excess - optimistically - of 10 - 7 , with the major suppression factors (even for neutrino masses in the hundreds of GeV range) coming from loop integrals and mixing angles. A calculation is certainly worthwile, and should not be very difficult as it will follow a well-beaten track [5]. (2) The experimental signal is reasonably clear, and the background is easily estimated, as demonstrated below. (3) Based on experimental limits from lepton decays, we find that measurable rates are permissible, if one of the outgoing leptons is a
2 April 1987
Turning to Z --+ e/*, the branching ratio is given by
41Gol 2
B R ( Z ~ e/*) = B R ( Z ~ e + e - )
1 + (1
= 0.121Cj,~ 12,
-
4Xw) 2
(7)
where in the last step three generations were assumed to be accessible in Z decays. The tiny branching ratio in eq. (3) then follows. Turning to lepton flavor violating decays involving r, it is not unnatural to expect that universality will be violated with C~l>> Cu~ ( l ~ r ) . Without a deeper understanding of the mixing, we can only use the following experimental limits [8] on rare ~ decays: B R ( , --+ 3/*) ~< 4.9 × 10 -4, B R ( r --+ 3e) ~< 4.0 x 10 -4, BR($ -+ 2/*0 ~< 3.3 x 10 -4, B R ( r ~ 2e/*) ~< 4.4 x 10 -4, to extract the bounds on the values of the Cu; we obtain [ C,, [ z ~< 3.5 x 10-3, {C,e 12 ~ 2.9 x 10-3, from which we derive, similarly to the derivation of eq. (3),
'r.
B R ( Z -~ re) ~< 3.4 X 10 -4,
(8)
It is easy to see, based on [7] B R ( / * - + e - e + e - ) ~< 2.4 × 10 -12 that, for example,
B R ( Z ~ r/*) ~< 4.2 X 10 -4.
(9)
B R ( Z --+ e/*) ~< 7.2 X 10 -13,
(3)
which is unmeasurable. To this end, we write the contribution of Z to/* ~ eee [see eq. (1)] as dF dx -
2 5 Gf mu
24~r 3
IC"e{2[X(3xZ-5x3)+ Yx3]'
(4)
where x =- E~+/rn~,, X = ½11 + (1 - 4Xw)], 2 Y = ½(1 - 4Xw) and =
A possible experimental signature is Z ~ r f ~ ~rv, d or Ov, g. An obvious background to Z ~ rd (d4= r) is Z -~ r r ~ rv, ved. To estimate its importance, consider the differential rate for the background process in the Z rest frame,
dr
dE e dz
_ G~ [3m2(l_flz)E2 24rr 3 -4(1 -
3 flz) 2 E, Ee],
(10)
where m t << M z is assumed, fl is the velocity of the r, z - cos 0e with 0e being the angle between the ~ m o m e n t u m and the initial ~ direction. The energy of ~ is limited by the constraint
o
(here 4 = e, ~. =/*). We then find BR(/*- --, e - e + e - ) --- 0.4 1C~ 12 ~ 2.4 x 10 -~2.
(6)
0 ~< Eel< 2E,(1 -
flz)"
(11)
Assuming that E e can be measured to an accuracy :~3of 1 GeV, we find that the branching ratio :~2
92
The authors of ref. [6] discuss lepton flavor changing Z ° decays in a model for which the rates are extremely small. Even with the most optimistic number of Z ° bosons expected, it will be practically impossible to detect such small rates, due to background problems.
,3 We thank J. Hauptman for pointing out to us that this is a conservative estimate, based on proposals for the L 3 detector in LEE
Volume 188, number 1
PHYSICS LETTERS B
for the background process is 3 × 10 -6 (2 × 10 -7) for an accuracy of 10 mrad (1 mrad), respectively, in Ov Obviously, a problem arises if the signal is of order 10 -7 or smaller. Turning to quark flavor changing decays Z qg~j (i 4:j), we emphasize again that in E6, transitions to charge 2 / 3 quarks are forbidden at the tree level, which leaves us with decays like Z ~ bg, bd, etc. Background problems are more severe here than in the leptonic case, since hadrons are produced, and jets are difficult to identify without a great loss in branching ratios and efficiency. Furthermore, the Standard Model gives a nonzero (though small) contribution through loops [5]. Luckily, an equivocal signature for E 6 will be the observation of Z ~ b~ + b q (q 4: b) at the level of 10-4 and the absence (down to a very low level, as given by the Standard Model at the one-loop level [5]) of Z ~ t~t + tq with q 4: t. Such an encouraging level is possible as shown by the following considerations. Assume that b---, q#+/~- is mediated by Z with coupling as described above. Then, the limit on the branching ratio [9] BR(b --~ q/~+/~-) = 9 . 0 ~ ] {Cbq I 2 ~ 7 X 10 -3 q
(12) leads to BR(Z ~ bF:1 + b q ) ~< 2.8 × 10 - 4 , where we take [Vcb [ = 0.07, and three generations are assumed to be available in Z decays. Unfortunately, Z ~ qYtj decays are very difficult to disentangle from the background, and even with rates of 10 - 4 the outlook for their detection is pessimistic. Finally, let us comment that other exotic decays such as Z ~ Ld or Nv, where L (N) are fourth-generation charged leptons (neutrinos), are also possible within E6, with branching ratios which are a priori unrestricted, as long as a reliable mixing scheme is unavailable. Furthermore, new processes such as e+e - --* f/~, qiF:lj (for Q = - 1 / 3 quarks only) are also possible. We can estimate that the contribution to R is given by the Z pole only (note again: no photon contributions!):
2 April 1987
R ( e + e - ~ T+/L+) = 0.54 [ C,, I 2 s2
×(s_M~)2+M~F~ ,
(13)
which results in R(e+e - ~ ~'/~) ~< 9.7 × 10 -4, 1.0 × 10 -1, 3.1 × 10 -3 at v~-=60, 100, 200 GeV, respectively, and R ( e + e - --, bq + bq) = 1 . 6 2 E I Cbn I 2 q s2 X
(s - M z ) 2 +
2 ~ 2 MzF
(14) leading to limits for R ( e + e - ~ bq) which are = 30% lower than the limits for e+e - ~ ~-/z. To summarize, it was demonstrated that superstring motivated E 6 theories may induce measurable lepton and quark neutral current flavor changing transitions with unique signatures, which are distinctively different from predictions of the Standard Model or other extensions of it. Though it may seem that without introducing a specific model, such as E6, limits on ~" decays may be directly translated to limits for lepton flavor nonconserving Z ° decays, E 6 dictates very specific couplings which are employed here. Experimentalists are urged to place limits on flavor changing Z decays and e+e - transitions (where for the latter even current experiments may provide useful information) to both leptons and quarks. Obviously, improvement of the current limits on rare ~- and B decays are more than welcome. One of us (G.E.) would like to thank the memers of the High Energy Physics Group at Ames Laboratory for their warm hospitality.
References [1] E. Witten, Nucl. Phys. B 258 (1985) 75; P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46; M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett. B 150 (1985) 55; M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. [2] E.g.Y. Achiman and B. Stech, Phys. Lett. B 77 (1978) 389;
93
Volume 188, number 1
PHYSICS LETTERS B
R.W. Robinett and J.L. Rosner, Phys. Rev. D 26 (1982) 2396; R.W. Robinett, Phys. Rev. D 26 (1982) 2388; P.H. Frampton and T.W. Kephart, Phys. Rev. D 25 (1982) 1459. [3] J.L. Rosner, Enrico Fermi Institute report EFI-85-34 (1985); R.W. Robinett, Phys. Rev. D 33 (1986) 1908; S.M. Barr, Phys. Rev. Lett. 55 (1985) 2278; M. Dine, V. Koplunovsky, M. Magano, C. Nappi and N. Seiberg, Nucl. Phys. B 259 (1985) 549; E. Cohen, J. Ellis, K. Enquist and D.V. Nanopoulos, Phys. Lett. B 161 (1985) 85; A. Sen, Phys. Rev. Lett. 55 (1985) 33; L.S. Durkin and P. Langacker, University of Pennsylvania report DPR-0287-T (1985); M. Drees, N.K. Falck and M. Ghick, University of Dortmund report DO-TH-85/25 (1985); J.P. Derendinger, L.E. Ibh~aez and H.P. Nilles, CERN report CERN-TH-4228/85 (1985); J.L. Hewett, T.G. Rizzo and J.A. Robinson, Phys. Rev. D 33 (1986) 1476; Ames Lab report IS-J-2042 (1986). [4] V. Barger, N.G. Deshpande and K. Whisnant, Phys. Rev. Lett. 56 (1986) 30;
94
[5]
[6] [7] [8] [9]
2 April 1987
V. Barger, N.G. Deshpande, R.J.N. Phillips and K. Whisnant, Phys. Rev. D 33 (1986) 1912; T.G. Rizzo, Ames Lab reports IS-J-1975, 2036, 2040, (to be published in Phys. Rev. D); Ames Lab reports IS-J-2037, 2116, 2167, 2168 (1986). G. Eilam, in: Proc. XVIlth Recontre de Moriond, ed. Tran Thanh Van (1982) p. 141; Phys. Rev. D 28 (1983) 1202; M. Clements, C. Footman, A. Kronfeld, S. Narasimhan and D. Photaidis, Phys. Rev. D 27 (1983) 570; V. Ganapathi, T. Weiler, E. Laermann, I. Schmitt and P.M. Zerwas, Phys. Rev. D 27 (1983) 579; A. Axelrod, Nucl. Phys. B 209 (1982) 349; K.-I. Hikasa, Phys. Lett. B 149 (1984) 221; W.-S. Hou, N.G. Deshpande, G. Eilam and A. Soni, preprint U C L A / 8 6 / T E P / 1 2 , PT8608-86, to be published. T.K. Kuo and N. Nakagawa, Phys. Rev. D 32 (1985) 306. W. Bertl et al., Nucl. Phys. B 260 (1985) 1. K.G. Hayes et al., Phys. Rev. D 25 (1982) 2869. B. Adeva et al., Phys. Rev. Lett. 50 (1983) 799; W. Bartel et al., Phys. Lett. B 132 (1983) 241.
PHYSICS LETTERS B
2 April 1987
Q U A R K A N D L E P T O N FLAVOR V I O L A T I N G Z ° DECAYS IN E~ ~ Gad E I L A M 1 and Thomas G. R I Z Z O 2 Ames Laboratory and Department of Physics, Iowa State University, Ames, IA 50011, USA Received 29 December 1986 Appreciable lepton flavor violating decays of Z °, e.g. Z ° ~ #T, can occur in E 6 grand unified theories. We examine these decays and find, using the existing experimental limits on rare processes, that some m a y have branching ratios as large as a few times 10 -4. Flavor violating quark decays are also expected at the same level but only in the Q = - 1 / 3 sector (e.g. Z ° ~ bg, but Z ° ~ tc-) within the framework of E6; these quark decays are very difficult to observe experimentally.
Superstring theories [1], in a currently popular version, manifest themselves as a n E 6 grand unified theory ,1 below the Planck scale. At lower energies, one or more Z bosons as well as exotic fermions emerge, the latter transforming as (at least) three 27 representations of E613 ]. We concentrate here on implications of mixing between these new fermions and the usual leptons and quarks [3,4], which may cause detectable lepton and quark flavor changing neutral currents• Such currents are either nonexistent or unmeasurably small in the Standard Model, as they are absent from the lepton sector and occur only through loops for quarks [5]. In E 6 , the potentially large rates for decays such as Z ° ~ # ~ - or Z ° ~ b ~ , which proceed at the tree level, are suppressed only by unknown mixing angles, which we can limit using results of existing experiments on and B decays• The formalism leading to new exotic fermions, their mixing with the usual fermions and the fact that only exotic quarks with charge Q = - 1 / 3 * This work was supported by the US Department of Energy, Contract No. W-7405-Eng-82, Office of Energy Research (KA-01-01), Division of High Energy and Nuclear Physics. 1 On leave of absence from: Physics Department, Technion, Haifa 32000, Israel. 2 Address for period January 1, 1987-June 30, 1987: Center for Particle Theory, University of Texas - Austin, Austin, TX 78712, USA. ,1 For original references on E 6 grand unification see ref. [2].
(and not with Q = 2 / 3 ) emerge, has been already discussed [3,4]. The final results, relevant to the intergenerational mixing discussed here, are the following new neutral current interaction terms:
L,i(Q =
-ig - 1 ) = -~Cw 4y~(1 - Vs) 0
Lu( Q =
-1/3)
= -~g-clg~,T~(1 + Vs)
for charged leptons and Q = - 1 / 3 quarks, respectively, where U L, UR are 6 x 6 a priori unknown left and right mixing matrices, i and j (i 4=j) denote lepton or quark types, c w -=-cos 0w and x w -= sin20w = 0.22. Note the absence of a direct flavor changing electromagnetic term; therefore contributions to processes such as # --* ey will be highly suppressed (no tree, only loop diagrams are possible), while flavor changing processes mediated or initiated by a Z boson may proceed at the tree level• It is also clear that for such processes the contribution of a second Z ( Z ' ) can be safely neglected if Z ' is sufficiently massive. Let us first turn to lepton flavor violating transitions, with emphasis on Z ~ E , ~ + ~ - - ~ .
0370-2693/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
91
Volume 188, number 1
PHYSICS LETTERS B
These decays have the following advantages as probes for exciting new physics: (1) The Standard Model does not accommodate flavor changing lepton transitions either on a tree or a loop level. Therefore, any nonzero signal will indicate important deviations from the Standard Model [6] .2 If heavy neutrinos exist, they can, through loops, induce a Z ~ 4 4 transition. However, based on extensive experience from investigations of Z--+ qiglj processes [5], we do not expect branching ratios in excess - optimistically - of 10 - 7 , with the major suppression factors (even for neutrino masses in the hundreds of GeV range) coming from loop integrals and mixing angles. A calculation is certainly worthwile, and should not be very difficult as it will follow a well-beaten track [5]. (2) The experimental signal is reasonably clear, and the background is easily estimated, as demonstrated below. (3) Based on experimental limits from lepton decays, we find that measurable rates are permissible, if one of the outgoing leptons is a
2 April 1987
Turning to Z --+ e/*, the branching ratio is given by
41Gol 2
B R ( Z ~ e/*) = B R ( Z ~ e + e - )
1 + (1
= 0.121Cj,~ 12,
-
4Xw) 2
(7)
where in the last step three generations were assumed to be accessible in Z decays. The tiny branching ratio in eq. (3) then follows. Turning to lepton flavor violating decays involving r, it is not unnatural to expect that universality will be violated with C~l>> Cu~ ( l ~ r ) . Without a deeper understanding of the mixing, we can only use the following experimental limits [8] on rare ~ decays: B R ( , --+ 3/*) ~< 4.9 × 10 -4, B R ( r --+ 3e) ~< 4.0 x 10 -4, BR($ -+ 2/*0 ~< 3.3 x 10 -4, B R ( r ~ 2e/*) ~< 4.4 x 10 -4, to extract the bounds on the values of the Cu; we obtain [ C,, [ z ~< 3.5 x 10-3, {C,e 12 ~ 2.9 x 10-3, from which we derive, similarly to the derivation of eq. (3),
'r.
B R ( Z -~ re) ~< 3.4 X 10 -4,
(8)
It is easy to see, based on [7] B R ( / * - + e - e + e - ) ~< 2.4 × 10 -12 that, for example,
B R ( Z ~ r/*) ~< 4.2 X 10 -4.
(9)
B R ( Z --+ e/*) ~< 7.2 X 10 -13,
(3)
which is unmeasurable. To this end, we write the contribution of Z to/* ~ eee [see eq. (1)] as dF dx -
2 5 Gf mu
24~r 3
IC"e{2[X(3xZ-5x3)+ Yx3]'
(4)
where x =- E~+/rn~,, X = ½11 + (1 - 4Xw)], 2 Y = ½(1 - 4Xw) and =
A possible experimental signature is Z ~ r f ~ ~rv, d or Ov, g. An obvious background to Z ~ rd (d4= r) is Z -~ r r ~ rv, ved. To estimate its importance, consider the differential rate for the background process in the Z rest frame,
dr
dE e dz
_ G~ [3m2(l_flz)E2 24rr 3 -4(1 -
3 flz) 2 E, Ee],
(10)
where m t << M z is assumed, fl is the velocity of the r, z - cos 0e with 0e being the angle between the ~ m o m e n t u m and the initial ~ direction. The energy of ~ is limited by the constraint
o
(here 4 = e, ~. =/*). We then find BR(/*- --, e - e + e - ) --- 0.4 1C~ 12 ~ 2.4 x 10 -~2.
(6)
0 ~< Eel< 2E,(1 -
flz)"
(11)
Assuming that E e can be measured to an accuracy :~3of 1 GeV, we find that the branching ratio :~2
92
The authors of ref. [6] discuss lepton flavor changing Z ° decays in a model for which the rates are extremely small. Even with the most optimistic number of Z ° bosons expected, it will be practically impossible to detect such small rates, due to background problems.
,3 We thank J. Hauptman for pointing out to us that this is a conservative estimate, based on proposals for the L 3 detector in LEE
Volume 188, number 1
PHYSICS LETTERS B
for the background process is 3 × 10 -6 (2 × 10 -7) for an accuracy of 10 mrad (1 mrad), respectively, in Ov Obviously, a problem arises if the signal is of order 10 -7 or smaller. Turning to quark flavor changing decays Z qg~j (i 4:j), we emphasize again that in E6, transitions to charge 2 / 3 quarks are forbidden at the tree level, which leaves us with decays like Z ~ bg, bd, etc. Background problems are more severe here than in the leptonic case, since hadrons are produced, and jets are difficult to identify without a great loss in branching ratios and efficiency. Furthermore, the Standard Model gives a nonzero (though small) contribution through loops [5]. Luckily, an equivocal signature for E 6 will be the observation of Z ~ b~ + b q (q 4: b) at the level of 10-4 and the absence (down to a very low level, as given by the Standard Model at the one-loop level [5]) of Z ~ t~t + tq with q 4: t. Such an encouraging level is possible as shown by the following considerations. Assume that b---, q#+/~- is mediated by Z with coupling as described above. Then, the limit on the branching ratio [9] BR(b --~ q/~+/~-) = 9 . 0 ~ ] {Cbq I 2 ~ 7 X 10 -3 q
(12) leads to BR(Z ~ bF:1 + b q ) ~< 2.8 × 10 - 4 , where we take [Vcb [ = 0.07, and three generations are assumed to be available in Z decays. Unfortunately, Z ~ qYtj decays are very difficult to disentangle from the background, and even with rates of 10 - 4 the outlook for their detection is pessimistic. Finally, let us comment that other exotic decays such as Z ~ Ld or Nv, where L (N) are fourth-generation charged leptons (neutrinos), are also possible within E6, with branching ratios which are a priori unrestricted, as long as a reliable mixing scheme is unavailable. Furthermore, new processes such as e+e - --* f/~, qiF:lj (for Q = - 1 / 3 quarks only) are also possible. We can estimate that the contribution to R is given by the Z pole only (note again: no photon contributions!):
2 April 1987
R ( e + e - ~ T+/L+) = 0.54 [ C,, I 2 s2
×(s_M~)2+M~F~ ,
(13)
which results in R(e+e - ~ ~'/~) ~< 9.7 × 10 -4, 1.0 × 10 -1, 3.1 × 10 -3 at v~-=60, 100, 200 GeV, respectively, and R ( e + e - --, bq + bq) = 1 . 6 2 E I Cbn I 2 q s2 X
(s - M z ) 2 +
2 ~ 2 MzF
(14) leading to limits for R ( e + e - ~ bq) which are = 30% lower than the limits for e+e - ~ ~-/z. To summarize, it was demonstrated that superstring motivated E 6 theories may induce measurable lepton and quark neutral current flavor changing transitions with unique signatures, which are distinctively different from predictions of the Standard Model or other extensions of it. Though it may seem that without introducing a specific model, such as E6, limits on ~" decays may be directly translated to limits for lepton flavor nonconserving Z ° decays, E 6 dictates very specific couplings which are employed here. Experimentalists are urged to place limits on flavor changing Z decays and e+e - transitions (where for the latter even current experiments may provide useful information) to both leptons and quarks. Obviously, improvement of the current limits on rare ~- and B decays are more than welcome. One of us (G.E.) would like to thank the memers of the High Energy Physics Group at Ames Laboratory for their warm hospitality.
References [1] E. Witten, Nucl. Phys. B 258 (1985) 75; P. Candelas, G.T. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B 258 (1985) 46; M. Dine, R. Rohm, N. Seiberg and E. Witten, Phys. Lett. B 150 (1985) 55; M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. [2] E.g.Y. Achiman and B. Stech, Phys. Lett. B 77 (1978) 389;
93
Volume 188, number 1
PHYSICS LETTERS B
R.W. Robinett and J.L. Rosner, Phys. Rev. D 26 (1982) 2396; R.W. Robinett, Phys. Rev. D 26 (1982) 2388; P.H. Frampton and T.W. Kephart, Phys. Rev. D 25 (1982) 1459. [3] J.L. Rosner, Enrico Fermi Institute report EFI-85-34 (1985); R.W. Robinett, Phys. Rev. D 33 (1986) 1908; S.M. Barr, Phys. Rev. Lett. 55 (1985) 2278; M. Dine, V. Koplunovsky, M. Magano, C. Nappi and N. Seiberg, Nucl. Phys. B 259 (1985) 549; E. Cohen, J. Ellis, K. Enquist and D.V. Nanopoulos, Phys. Lett. B 161 (1985) 85; A. Sen, Phys. Rev. Lett. 55 (1985) 33; L.S. Durkin and P. Langacker, University of Pennsylvania report DPR-0287-T (1985); M. Drees, N.K. Falck and M. Ghick, University of Dortmund report DO-TH-85/25 (1985); J.P. Derendinger, L.E. Ibh~aez and H.P. Nilles, CERN report CERN-TH-4228/85 (1985); J.L. Hewett, T.G. Rizzo and J.A. Robinson, Phys. Rev. D 33 (1986) 1476; Ames Lab report IS-J-2042 (1986). [4] V. Barger, N.G. Deshpande and K. Whisnant, Phys. Rev. Lett. 56 (1986) 30;
94
[5]
[6] [7] [8] [9]
2 April 1987
V. Barger, N.G. Deshpande, R.J.N. Phillips and K. Whisnant, Phys. Rev. D 33 (1986) 1912; T.G. Rizzo, Ames Lab reports IS-J-1975, 2036, 2040, (to be published in Phys. Rev. D); Ames Lab reports IS-J-2037, 2116, 2167, 2168 (1986). G. Eilam, in: Proc. XVIlth Recontre de Moriond, ed. Tran Thanh Van (1982) p. 141; Phys. Rev. D 28 (1983) 1202; M. Clements, C. Footman, A. Kronfeld, S. Narasimhan and D. Photaidis, Phys. Rev. D 27 (1983) 570; V. Ganapathi, T. Weiler, E. Laermann, I. Schmitt and P.M. Zerwas, Phys. Rev. D 27 (1983) 579; A. Axelrod, Nucl. Phys. B 209 (1982) 349; K.-I. Hikasa, Phys. Lett. B 149 (1984) 221; W.-S. Hou, N.G. Deshpande, G. Eilam and A. Soni, preprint U C L A / 8 6 / T E P / 1 2 , PT8608-86, to be published. T.K. Kuo and N. Nakagawa, Phys. Rev. D 32 (1985) 306. W. Bertl et al., Nucl. Phys. B 260 (1985) 1. K.G. Hayes et al., Phys. Rev. D 25 (1982) 2869. B. Adeva et al., Phys. Rev. Lett. 50 (1983) 799; W. Bartel et al., Phys. Lett. B 132 (1983) 241.