Constraints on supersymmetry from rare kaon decays

Volume 174, number 3 PHYSICS LETTERS B 10 July 1986

C O N S T R A I N T S ON S U P E R S Y M M E T R Y F R O M RARE KAON DECAYS

Stefano B E R T O L I N I and An ton io M A S I E R O 1

Physics Department, New York University, New York, NY 10003, USA

Received 27 March 1986

The supersymmetric contributions to K + ---, ~r + + '" nothing" decays are analyzed in the context of low energy spontaneously broken N = 1 supergravity models. It is found that the leading supersymmetric contributions to K + ~ ~r + up can be at most of the same order as the standard model contributions with three generations. On the other hand a large enhancement for K + ---, ~r + ~ (photino) is present. The constraints that the planned BNL experiment can impose on this class of models are commented on.

Low energy physics of rare processes has proven so far to be useful in establishing bounds on the masses, or mass relations, of the supersymmetric partners of ordinary particles. We consider here the possibility that the BNL experiment of this year [ 1 ], with the expected sensitivity of 10-10 for the branching ratio K + ~ ~r + + "nothing", may constitute a new test of low energy supersymmetric models.

The standard model with three generations of fermions predicts the branching ratio for the process K + ~ 7r+u~ to be in the range 4 X 1 0 - 1 1 - 1 0 - 1 0 ,

where the bounds on the product sin 02 sin 03 (with 0 2 and 03 angles of the Kobayashi-Maskawa (KM) matrix), coming from the long B lifetime, are taken into account [2 -4 ] . Therefore, if the BNL experiment runs with the planned sensitivity of one K + rr+uF event per 10-10 BR no detection of the K + 7r + + "nothing" decay is expected.

In the case of a positive BR(K L ~ ~+/~-) must be bility one is led to consider is certainly the presence of a fourth generation [5]. The new freedom coming from the enlargement of the KM matrix and the extra neutrino family allows, indeed, for an enhancement of the K + ~ ~r+~,~ - decay. However, one must be care- ful not to exceed the experimental bound in the tightly correlated process K L ~/a+/a - : the short-

I On leave of absence from INFN, Sezione di Padova, 1-35131 Padua, Italy.

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

distance contributions to BR (K L -+ ~ + p - ) must be less than 6 X 10 - 9 [6]. It seems possible to achieve this result in the enlarged 4 X 4 KM matrix [5].

The next candidate to be put under scrutiny is low energy supersymmetry. There have already been some analyses in the past of the process K + --, lr+~'~ taking here the photino to be the lightest superpartner [4,7]. Their conclusion was that this process cannot greatly exceed the rate for K + -+ rr+uV as predicted in the standard model. However in this last couple of years we have achieved a much better understanding of the structure of the low energy supersymmetric models coming from spontaneously broken N = 1 supergravity theories and, in particular, much important work has been done in the sector of the flavour changing processes mediated by neutral fermionic superpartners [8], namely photino (~) , zino ('z), gluinos (g) or higgsino (~'0).

Taking into account all this more recent progress, we reanalyze the K + -~ lr + + "nothing" process in the context of low energy supersymmetric theories coming from spontaneously broken N = 1 supergravity. Our two major conclusions are the following:

(i) when "nothing" denotes ug, the new supersymmetric contributions can be at most of the same order as the SM contribution for superpartner masses below the W mass; the K L -~ t~+~ - bound does not put new constraints on the SUSY masses, at least if we take the limits on squark and gluino masses as derived

343


from the p-~- collider physics; (ii) if "nothing" repre- sents a pair of the lightest superpartner and for defini- tiveness we take it to be the photino, then, due to the presence of large effects in flavour changing quark- squark-phot ino couplings, we claim that the present experimental upper bound (BR(K + -+ ~+ + "nothing")

10 - 7 , constrains the down-squark masses to be larger than 30 GeV and K + -+ 7 r + ~ will be indeed de- tectable at BNL, if the squark masses are smaller than 70 GeV. Obviously the conclusion (ii) rests on the assumption that the lightest superpartner can be suf- ficiently light to render this K + decay kinematically possible. On the other hand, there are well known cosmological arguments [9] that require the photino to be havier than 3 0 0 - 5 0 0 MeV (or very light ~< 100 eV). We think that no matter how reliable they are, it is always better to have some direct confirmation from experiment (so far no experimental bound on the photino mass has been given) and, moreover, these cosmological bounds suppose the usual discrete R-symmetry to be exact and so the ~ to be absolutely stable. Alternative schemes with decaying ~ ' s have been propose d [ 10]. Experimentally, photino masses <~100 eV are not ruled out, although such values are disfavoured in most models so far proposed.

To start with, we consider the supersymmetric contributions to K + -+ rr+uV. The supersymmetriza- tion of the SM box diagram, fig. l a, yields the amplitude

S L '~ d L

vj ~ -- vj IlL

(o)

S L "~ - - J < d L

~ _ UiL t ~ -

/ ' / ' L ~ . . . . . ~ ' /U" L

(b) Fig. 1. Supersymmettization of the SM box contributions to K + ~ ~+v~ and K L ~ ~+/~-.

344

y { (box) dig+ ~ .a-+V~-) SUSYU'-

~-- (_ 1/r~ 2 ) (g4/32rr2) U~s Utd

/ (1)

where Uts and Utd are KM matrix elements,

g(.y,x) = (y - x) -1 { tv/(y - 1)] 2 l ny

- [x/(x - 1)] 21nx - ( y - 1) - 1 +(x - 1) -1} ,

(2) andx ' t ~2 ~2 _ ~2 ~2 ~2 ~2 = m t / m w , ~ " u ~1 .= - r n u / m w , m~y/m w.

Henceforth the tilde will denote the superpartners of the standard particles.

The SM contribution to K + --* 7r+ub - from the analogous box diagram is [3]

(box)Iv+ ._~ SM U" zr+uv-) = (-1/m2)(g4/16~2)U*tsUtd

X ~ [go'j, x t) - gO'/, x u)] (x L 7u dE) (FL/3 'u VL/), / (3)

2 2 2 2 w h e r e x t = m t / m w , X u = m u / m w andyj = m~j/m 2 In eqs. (1) and (3) the leading " top" contributions

are reported. It is worth recalling that FCNC processes strongly constrain the mass splittings among squarks of the same charge. For instance in (1)r~ 2 - r~ 2 is of O(m2). This degeneracy among squark masses becomes obviously more exact the higher their masses are. As a consequence, the super-GIM mechanism becomes more efficient than the usual GIM mechanism. Even considering quite low values o f the SUSY par- ticle masses, for instance r~ u ~-- 25 GeV, r~ t ~ 50 GeV r ~ ~ 20 GeV, mW ~-- 3 0 - 4 0 GeV, the SUSY contribution turns out to be at most of the same order as the SM one. On the other hand, we must be always concerned with what happens for the correlated K L /a+# - decay: in the SM, indeed, there is a factor 4 of suppression in the amplitude for this latter process [2] with respect to K + -+ 7r+vF, whereas such a factor, due to the Dirac algebra, is not present in the SUSY case. Moreover, the presence of~"s instead of charged sleptons in the box of fig. lb can further increase the SUSY contribution. Therefore if we want to respect the experimental bound on K L ~ ¢t+/a- we can ex- clude a major contribution from these box diagrams to K + --* 7r+vV.


7, Z

l - / ~ . d i L

. I /

(a) dE

7,Z

i / \ / x

$L g dc SL g

(b) (c)

×,Z

dL

Fig. 2. Penguin diagrams with gluino exchange.

We come now to the more promising situation of the penguin contributions. First we consider the exchange of gluinos. As is by now well known, gluinos can mediate flavour changing processes [8]. The mixings which appear at the vertices correspond to the usual KM matrix elements. If we consider for instance the diagrams in fig. 2, and if we take a i to be the b squark, then we get the UtdUts mixing.

If the gauge boson in fig. 2 is a photon, then clearly no contribution to K L ~ #+t~- or K + ~ n+uv results, but we can still study the interesting K L n+e+e - process for which the experimental branching ratio is (2.6 -+ 0.5) X 10- 7 Conservation of the elec- tric charge tells us that the contribution must be proportional to (qUqV _ q2gUV) where q is the momentum of the photon. We find

s~ SUsy(K + ~ zr+e+e - )

_ 4 ~ 2 2 2 2 * -~(1 /mg)( ,gse /16rr )UtsUtd

X [f(X'b) -- f(x'd )] (SLT, dL)(~-Tue)

where ," b = mb/rng,X d - 2 --2 -- = r~2/r~2 and

f ( x ) = (x - 1) - 3 [½(x 2 +x + 1 ) - ~(x + 1)

(4)

- ln(x)/(x - 1) + 3]. (5)

The ratio c/~ (K + ~ n+e+e - ) of the SUSY penguin contribution versus the SM amplitude [2] becomes then

sin2a r~2 / .~2a c'~ (K+ ~ n+&-e- )= (4~S/C 0 uWt,,,W/,,,g)

X [f(~b) -- f (~d) ] / ln ( m2/m 2 ) , (6)

where ol S is the strong coupling constant. In our analysis a s is evaluated at the typical scale of the supersymmetric particles in the loop, namely a S ~- 0.1

Before plugging numbers, two remarks concerning fig. 2 are in order. We consider only d L and s L in the external legs and not d R and s R because the domi- nant contribution to flavour changing g lu ino -qua rk - squark vertices comes from the left-handed sector. Indeed, at tree level, the q and q mass matrices are simultaneously diagonalized by the same rotation on the q and q current eigenstates and, analogously to what happens for the T and Z vertices in the SM, no off-diagonal couplings are possible. However the renormalization of the couplings in the q'L sector spoils this simultaneous diagonalization: the 3 X 3 mass matrix in the ~L sector becomes m2/21 + MD3~ D + cMuM~j , where M D and M U are the d- and u-quark mass matrices respectively and the term proportional to c signals the presence of the renormalization effects. An exact determination o f c requires a quite complicate solution of the coupled system of renormalization group equations. Taking rn3/2 in the 40 GeV range leads to values o f c between 0.5 and 0.7 [11]. The second remark concerns the rt~ b, r~ d mass difference. Due to the above mentioned structure of the ~L mass matrix we get r~b 2 - r~d2 ~-- cm 2. Taking the ~ andg" masses around m t --~ 40 GeV leads to c~ (K+ ~ n+e+e-) ~_ 1. Therefore, no new constraints on SUSY come from K + + rr+e+e - .

Replacing 7 by Z in fig. 2 can lead in principle to the K L ~/a+/J - and K + ~ n+vF processes. The self- energy diagrams, figs. 2b and 2c, yield

1 . 2 * i ~ = (ig2/4n2)(g/cos 0W) ( - ½ + g san Ow)UtsUtd

X [A 1 (~b) -- A 1 (~d)] (SLTUdL) , (7)

where

1 Al(X)= [ 1 - ( x - 1) - 2 ] l n x + ( x - 1) - 1 - 5 . (8)

The vertex function in fig. 2a, however, gives a contribution which, for zero external momenta, exactly cancels (7). The reason is that we are using only dE, s L external legs so that Z is forces in fig. 2 to couple only to dL, s L or ~L particles. Therefore, again Ward identities warn us that the contributions in fig. 2 must vanish even when Z replaces 7. Clearly the way of communicating the information of the SU(2)L × U(1) breaking, and so the realization of different

345


Z

S, t] d L (a)

Z Z

SL ¢J dE S L ~ dL

(b) (c)

? 7 " ÷ ; . _ : * 7 * ' : ; $L g d L S, 'g dl.

(d) (e)

Fig. 3. Penguin contributions with b L-b R mass insertions.

couplings of Z to ~'L and ~'R particles, requires the insertion of ~L--~'R mixings in the squark propagator line. To be sure, we insist in getting dL--S L external lines, we must make two L - R insertions as depicted in fig. 3. In the minimally constrained ver- sion of models derived from spontaneously broken N = 1 supergravity, the mixing ~L--~R is given by A m 3 / 2 M D where M D is the down-quark mass matrix and A is a numerical coefficient O(1).

In order to obtain a quantitative result we consider the b L - b R 2 X 2 mass matrix:

( m 2 / 2 + m 2 + cm 2 A m 3 / 2 m b ) (9) A m 3 / 2 m b m2/2 + m 2 / "

In this way we are clearly neglecting flavour mixing and, thus, we are led to some upper bound of the supersymmetric contribution. The diagonalization of (9) yields the eigenvalues

1 2 ~2 ,2 = m2/2 + m 2 + ~cm t

_+ ~' [(cmt)2 2 + 4A2m~/2m2] l /2 , (10a)

and the eigenstates are given in terms of the mixing angle 13 defined as

tg 213 = 2Am3/2mb/cm 2, 0 <. 13 <~ n/2 . (10b)

The computation of the graphs depicted in fig. 2 leads to

(ig2/8rr2)(g/cos 0w) sin 2 213 U~sUtd

X [¼A 1(~1) + ¼A l('X2) - {A 2(~1, x"2)] (SLTu dL), (11)

where~i=~n2/~n2, i = 1,2 and

A 2 ( x , y ) = (v - x) - 1 {[y2/(y _ 1)] l ny

l (12) - [ x 2 / ( x - 1)1 lnx} a •

In eq. (1 I) we have included the angles UAd,Uts • . ~ ~ +

which appear in the vertices g dLb L and gs t b E In this way we expect our estimate to be correct up to a factor O[(cm2/m2/2)2 ] ~ 1. It is easy to check that for ~'1 = 2"2 eq. (11) vanishes. The eigenvalues X'l andx" 2 can be written indeed as~'l , 2 =~'b + e where

Xb (m2/2 + m 2 , 2 ~2 = + ~ cmt )/mg ,

2e = [(era2) 2 + (2Am3/zmb) 2] 1/2/r~2 •

For the mass range considered in our analysis e < 1. By expanding now, in powers of e, the function in square brackets in eq. (11) we find that the linear term also vanishes, due to the x 1 +~x 2 symmetry, signaling, as expected, the necessity of a double L - R insertion. The e 2 term yields

(ig2/24rr2) (g/cos . ~2 2 Ow) UtsUtd(Am 3 / 2mb/rn g)

X C ('Xb) (~L'yu dL) , (13)

where

C(x) = (x - 1)-2 (½ -- 2/(x - 1) + [3/(x - 1) 2] lnx

- 1 / x ( x - 1 ) } , (14)

-~2~2 The and we used e 2 sin 2 213 = (Am3/2mb/mg j . simple L - R insertion calculation of the diagrams of fig. 3 leads indeed to the same result.

The SUSY amplitude for the process K L -+ O+/J- with gluino exchange is, therefore, given by

-~ SUsy(KL -+/~+/~-)

"" -- (GF/V~) (2aS/370 U~sUtd(Am 3] 2mb/r~2) 2

× [C(~ b) - (md/mb)2C(~d)](gLTudL)( f fLTvVL), (15)

346

Volume 174, number 3 PHYSICS LETTERS B

where we used the relation GF/X/~ = g 2 / 8 m 2 and we took into account that only the axial part of the Z- current contributes to decay. Comparing with the SM [3]

~ s M ( K L -+/J+U')

= (GF/V~)(a/rr sinZ0w) U~sUtd

X C(xt)CffLT, dL)(fiL3,lat.tL) , (16) ~',Z

where_ 3 ~ l

C ( x ) = 3[x/(x - 1)] 2 lnx + 4 x - - ~ x / ( x - 1) (17) s,

leads, for the SUSY to SM amplitudes, to the bound

~(K L ---> U+/~-) ~ 2(aS/a).sin20 w

X (Am 3 /2mb/m 2 )2 iC('~b)/~ (xt) I , (I 8)

For ~ng ~< ~/b ~ ~t3/2 ~ mt ~ 40 GeV, eq. (18) gives c)~ (K L _+ ~+~l- ) ~ 1. Analogous results for the process K ÷ ~ 7r+vV follow by changing ffL3' ~IL to

_ _ ,12

--UL"/uP L in eq. (15) and C (xt)(~LTu/JL) to --D(y/, x t)(gLTuUL) in eq. (16), with D(y/ , xt) the function defined in eq. (2.15) of ref. [3]. We can therefore write

c~ (K + ~ 7r+W) = I C (x t ) /D@ j, xt)lc'~ (K L -+/J+/a-) . (19)

Neglecting the masses of the charged leptons and as- suming x t = 0.25, we find IC/1)[ -- 5, giving therefore a further suppression factor for the supersymmetric contributions to K + -+ 7r+ub -. It is worth mentioning that eq. (19) does not depend on the Dirac or Majorana nature of the neutrinos (in the latter case,

~-L'),u p L -+ --~-T#,75p) ' We consider now the possibility of replacing the

gluino by a higgsino, i.e. a mass eigenstate which is mainly composed of the fermionic components of the Higgs superfields. In this case no GIM suppression is present. The neutral higgsino, however, can be readily dismissed because of the unavoidable presence of the small Yukawa couplings of the d and s quarks.

The most attractive situation occurs when instead of the neutral higgsino we consider a charged higgsino: to the advantage of absence of super-GIM effects we can now add the fact that the Yukawa coupling of the top quark is present. The sum of the diagrams in fig. 4, with a photon in the external leg, vanishes for

10 July 1986

~',Z zZ

q

SL h+ du SL t'. OL

(a) (b)

~Z

- t .--4. r.

t I . ~ I L .

:h+ OL S L '~+ d L

(c) (d)

Fig. 4. Penguin diagrams with the exchange of a charged higgsino.

zero external momenta. We checked that this cancellation entails the cancellation of the same graphs when the Z is present. Again we must invoke L - R insertions. However, in this case, differently from what occurred in the gluino case, we do not have to go to radiative effects to induce a flavour change; therefore, no matter what the value of the c coefficient is, we do not have any suppression effect from the require- ment of off-diagonal couplings. The direct evaluation of the diagrams in fig. 4 yields

-i(ht/x/2)2(1/32rr2)(g/cos Ow ) U~sUtd

X( ~ [¼AI('Xi) -BI('~i) +B2(Yi) l i=1,2

+ ~A2(Xl,X" 2) (SLTudL) , (20)

where h t = gmt/m w is the Yukawa coupling of the top quark,

1 BI(X) = [ x 2 / ( x - 1) 2 ] lnx +x/(1 - x ) + i , (21a)

B2(x ) = [ 2 x / ( x - 1) 2] lnx + 2/(1 - x ) , (21b)

- ~2 ~2 • and x i = m i / m h, t = 1,2 refer to the eigenvalues of the 2 × 2 matr ix '~L- '~ R which is analogous to (9) with b ~ t. In this case cm 2 ~ 2Arn3/2m t and the diagonalization proceeds through a 45 ° rotation. The SUSY amplitude for K L ~/l+/a - , from penguin diagrams with the exchange of a charged higgsino, is

347


therefore given by

MSUsy(K L ~ ~+/a-)

= (GF/X/~)(~/47r s in2Ow)(m2t/m2)U;sUtd

X F('x I , "x2)(XLT~dL)(~-L~'U/aL) , (22)

where F ( ~ I , "x2) is defined as the function inside the large brackets in eq. (20). It is worth mentioning that, since we want to investigate SUSY masses in the region below row, the naive mass insertion calculation does not represent, in this case, a good approximation (e ~ A m 3 / 2 m t / r ~ 2 ~ O(1)).

By comparing with eq. (16)we find, indeed, that even for light SUSY masses, m h ~ m3/2 ~ m t ~ 40 GeV, the contribution of eq. (22) does not exceed the SM prediction. Afort iori , due to eq. (19), this result holds for the SUSY contributions to K + ~ 7r+vV.

Finally, we come to the analysis of the process K + --~ 7r + + LS + LS where LS denotes the lightest superpartner. In particular we consider here the possibility that LS is the photino. It is easy to construct box diagrams with gluino exchange leading to K + --* 7r+~'~ ". It is however strikingly evident that the hunt for the leading contribution does not require to go so far; we have indeed tree level diagrams, fig. 5a, contributing to this process. The amplitude for the photino production is readily computed:

S L '~ l "~ 1

I

Cl k , I ,

(a)

s -_ ~ n I I

t J

d - I it ~o

( b )

F i g . 5 . T r e e l e v e l c o n t r i b u t i o n s t o K + --* ~r + ~ ' ~ " a n d K + --* ~ o ~ ' o .

s~ (K + -~ n + ~ )

1 1 2 * = ~ ( g v ~ e ) [cm2/(m4/2 + cm2m2/2)] UtsUtd

X (X L ")'~ d L) (~-7u3'5 ~ ) , (23)

where the Majorana nature of the photino has been taken into account. By comparing eq. (23) with the SM contribution to K + ~ 7r+uu, properly multiplied by a factor x/'J in order to include the contributions from the three neutrino families (we consider here the neutrinos to be Majorana particles as well), we find that the present experimental bound of I0 - 7 on the BR(K + ~ ~+ + "nothing") implies squark masses ~>30 GeV. A negative result of the BNL experiment, probing a branching ratio BR < 10-9 , would require instead the squarks to be heavier than 70 GeV.

The analogous tree level decay into neutral higgsinos (fig. 5b) presents the same kind of suppres- sions that we mentioned when describing penguin diagrams with neutralino exchange. It turns out indeed to be negligible even with respect to the K + n+uF decay.

This concludes our analysis of the major supersymmetric contributions to the process K + ~ 7r + + "nothing". If "nothing" denotes uV, we find indeed that, even for light SUSY masses, the leading contributions do not exceed the SM prediction with three generations. On the other hand, if "nothing" denotes a light superparticle, there is a large enhancement in the photino case and the BNL results can provide stringent bounds on squark masses. For neutral higgsino (as well as for sneutrinos) the leading contributions arise at the one-loop level and they are not expected to exceed significantly the K + ~ 7r+uF decay channel.

So far the K 0 - K 0 system has provided the most interesting constraints on SUSY coming from low energy physics. We think that, given the sensitivity which is now obtainable at BNL, the rare kaon decays can play a major role in probing the SUSY models; this suggests that improving the experimental bounds on rare decays of heavy mesonic systems (for instance B), as well as of the Z, can turn out to be relevant for further search of SUSY signals. A more detailed analysis of this latter aspect is in progress.

348


This work was s t imula ted by a conversa t ion wi th

W.J. Marc iano and A. Sanda w h o also c o n t r i b u t e d

w i th in te res t ing fu r the r discussions. We are also thank-

ful to F. Borzumat i , G. Giudice and A. Sirlin. This re-

search was suppo r t ed in par t by the Na t iona l Science

F o u n d a t i o n u n d e r g ran t no. PHY 81 16102. One of

the au tho r s (S.B.) acknowledges rece ip t fo A. Della

Riccia Fel lowship .

References

[1] Y. Asano et al., Phys. Lett. B 107 (1981) 159; L.S. Littenberg, Brookhaven National Laboratory Report No. BNL - 35086, unpublished.

[2] M.K. Gaillaxd and B.W. Lee, Phys. Rev. D10 (1974) 897. [3] T. Inami and C.S. Lira, Progr. Theor. Phys. 65 (1981)

297. [4] J. Ellis and J.S. Hagelin, Nucl. Phys. B 217 (1983) 189;

F.J. Gilman and J.S. Hagelin, Phys. Lett. B 133 (1983) 443.

[5] U. TiJrke, Phys. Lett. B 168 (1986) 296; W.J. Marciano, talk given at the DESY Workshop (October 1985), DESY internal report; W.J. Marciano and Z. Parsa, Ann. Rev. Nucl. Phys., to be published.

[6] R.E. Shrock and M.B. Voloshin, contrib, paper Photon lepton Symp. (Fermilab, August 1973); M.B. Voloshin, Soy. J. Nucl. Phys. 24 (1976) 422;

A.D. Dolgov, V.I. Zakharov and L.B. Okun, Sov. Phys. Usp. 15 (1972) 404; M.B. Voloshin and E.P. Shabalin, JETP Lett. 23 (1976) 107.

[7] M.K. Galliard, Y.-C. Kao, I.-H. Lee and M. Suzuki, Phys. Lett. B 123 (1983) 241.

[8] J.F. Donoghue, H.P. Nilles and D. Wyler, Phys. Lett. B 128 (1983) 55; M.J. Duncan, Nucl. Phys. B221 (1983) 285; M.J. Duncan and J. Trampetic, Phys. Left. B 134 (1984) 439; J.M. Gerard, W. Grimus, A. Masiero, D.V. Nanopoulos and A. Raychaudhuri, Phys. Lett. B141 (1984) 79; Nucl. Phys. B235 (1985) 93; R. Langacher and R. Sathiapalan, Phys. Lett. B 144 (1984) 401; A. Bouquet, J. Kaplan and C.A. Savoy, Phys. Lett. B 148 (1984) 69; M. Dugan, B. Grinstein and L. Hall, Nucl. Phys. B255 (1985) 413.

[9] H. Goldberg, Phys. Rev. Lett. 50 (1983) 1419; L.M. Krauss, Nucl. Phys. B227 (1983) 556; J. Ellis, J. Hagelin, D.V. Nanopoulos and M. Srednicki, Phys. Lett. B 127 (1983) 233.

[10] L. Hall and H. Suzuki, Nucl. Phys. B 231 (1984) 419; I. Lee, Nucl. Phys. B246 (1984) 120; J. Ellis, G. Gelmini, C. Jaxlskog, G.G. Ross and J. Valle, Phys. Lett. B 150 (1985) 142; G. Ross and J. Valle, Phys. Lett. B 151 (1985) 375.

[ 11 ] A. Sanda, private communication.

349

Documents

Constraints on supersymmetry from rare kaon decays