K+→π++γ̃γ̃ in spontaneously broken supersymmetric models

Volume 123B, number 3,4

K + -* n + + ~

PHYSICS LETTERS

31 March 1983

IN S P O N T A N E O U S L Y B R O K E N S U P E R S Y M M E T R I C M O D E L S

Mary K. GAILLARD, Yeong-Chuan KAO, I-Hsiu LEE and Mahlko SUZUKI Lawrence Berkeley Laboratory and Department of Phystcs, Universtty of Cahfornta, Berkeley, CA 94 720, USA Received 20 December 1982

We study the decay K + ~ r r + ~ ' m models with spontaneous supersymmetry breaking and find that It 1s generally suppressed relative to the decay K* A rr+v'v of the conventional model, except possibly for a class of models where the squark masses are generated by radiative correctmns from a much larger supersymmetry breaking scale. For a small range of squark and p h o t m o parameters, the cascade process K +---,rr+rr° ~ rr+~'~"will become dominant over the 7v mode We also comment on the posslblhty of probing the neutrino mass through the K + ~ rr+rc° --' rr"ffv cascade decay

Supersymmetrlc variants [ 1 ] of the standard electroweak gauge theory [2] have recently become very fashionable, motivated primarily by the long-standing gauge hierarchy problem of grand unified theories. In these theories there is a superpartner for each known or predicted particle species of the conventional electroweak theory, namely spin zero squarks ~ and sleptons ~" as superpartners of quarks q and leptons ~, and spin 1/2 gauginos and htggsinos as superpartners of the gauge bosons and scalar Hlggs particles, respectively At present there is no evidence for their existence, and little evidence against them. This gives model budders a free rein for speculation, and, short of discovering such a particle, it is important to exclude as wide a range of mass parameters as one can. Direct pmr production in e+e - annihilation [3] gives a lower bound on smuon and selectron masses, m ~ , m ; . ~ > 16 GeV ,

(1)

under the assumption that the slepton decays into a lepton and a neutral m o which does not decay in the apparatus. Bounds similar to (1) have also been extracted [4] from the upper limit on the difference between the experimental and theoretical values o f g ~ 2. Analyses [5] lndacate that the gluxno must be heavier than -~ 2 GeV. At present, the only bound on the photlno mass is based on cosmological arguments [6] which exclude a small window, allowing m - < 30 eV 3'

and

m~ > 0.3 MeV . 3"

(2)

0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

If supersymmetry is spontaneously broken, photlnos and g l u l n o s c a n acquire masses only through radiative corrections, and, in many models, they are expected to be much lighter than squarks and sleptons Under this assumption the selectron mass bound has been recently increased [7] to rn;- ~> 19.2 G e V ,

(3)

by looking for single selectron production In e-3' colhslons. In this paper we shall consider only the class of models in which the photlno is very light 1

m~ < ~ r n K ,

(4)

and study the sensitivity of the decay rate * 1 K + -+ rr+ + ~,~,

(5)

to the masses of the exchanged squarks. In the past, analyses of rare K decays have provided a powerful tool for constraining gauge theories of weak interactions. They provided a successful estimate [9] of the charmed quark mass, constraints on the top quark mass [ 10] and Kobayashi-Maskawa mixing matrix parameters [11 ], and have all but excluded extended technicolor theories [ 12]. Previous applications [ 13] of these analyses to supersymmetrlc models have proven less powerful. The reason is that If one replaces internal quark and W lines by squark and ,1 An order-of-magnitude estimate is given m ref. [8].

241

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lines in a typical diagram, the usual GIM [14] suppression factor

(Am2q/m2) sin 0 c cos 0 c + top

quark correction ,

(6)

becomes replaced simply by

(Am2/m}) sin 0"c cos 0"c + ....

(7)

Under the plausible assumption that rnv~ ~> row, this simply means that squared mass differences of squarks Am 2 and mixing angles 0" are constrained to be slmdar to tliose for quarks, which is an automatic feature of realistic models. In fact, they are Identical to the corresponding quantities of quarks at the tree level in realistic models. The potential advantage of K + -+ rr+~7 is that in comparison to the analogous weak process

(8)

K +-+ rr+ + ~ v ,

the squark line in, for example, the box dmgram of fig. l a replaces a W line in the analogous diagram for (8). Then the effective GIM cancellatmn is again provided by the known quark mass difference and we expect the suppression factor (6), applicable to K + -+ rr+-~v,to become

(Am2/m2) s m

0 c cos 0 c ,

(9)

for K + ~ n+q~. This would allow a probe of the squark mass up to a value comparable to the W mass in an experimental search for K + ~ n + + "nothing", which is sensitive [ 15] to the branching ratm of 10 -10 preS

P U,C W

~=

u,c

[

U,C i i i

31 March 1983

dinted [8] for K + -+ n+Fv m the standard model. Unfortunately, the rate o f K + -+ rr+~7 is further reduced relative to that of K + -+ 7r+~u because of a few numerreal factors; an electromagnetic couphng factor e 2 = g2 sin20w replacing one weak coupling factor g 2, a logarithmic enhancement factor In (rn2/m2) which is slightly smaller than In (m2/m 2) o f K + -+ n+~u, and a smaller overall numerical coefficient. We thus obtain a branching ratio B R ( K + -+ rr+ + ~ )

= 0.70 X 10 -10

(10) X {1 + 0.43 In (m~.(GeV)/20]} 2 [20/rn~.(GeV)] 4, which does not allow a probe ofsquark masses beyond the existing limit (3) for a branching ratio measurement of 10 -10. However, there is some chance to enhance the K + ~ r r + ~ decay rate through a large logarithmic factor if supersymmetry is broken at a mass scale much hagher than m w and xf the supersymmetric cancellation occurs by an interplay of partmles with largely different masses. In the case that 1 (ll) m~3' ~ ~mlro , there is an additional contribution to the K + ~ 7r+~'~ decay, namely the cascade decay

K+ -+ rr+ + 7r0

~.

Whale providing a smaller window over which to probe the p h o t m o mass, the process (12) is not GIM suppressed, and in fact has the empirical nonleptomc enhancement factor; it could be observable in an experiment of the sensitivity discussed above for a limited range of squark and photmo masses. We find

I

BR(K+-+ n+ + "~7)cascade -~ 1.6 X 10 -11 (a)

S

- - -..7 -.7, . . . . U,C M,C U,C

(b)

Fig. 1. Box dmgrams for s--, d + ~ + 7. The pactures are drawn m the unitary gauge. 242

(12)

X (1

(13)

2 2 1/2 [20/m~.(GeV)] 4 [m~.(MeV)] 2. - 4mg/m,O

Thus the cascade decay would dominate over the dlrect decay for 2 MeV < rn~. < :1 rn,ro and even over the K + -+ n+-~v decay for some range of values of m~- and m~.. While the box diagrams of fig. 1 give the most important contribution, there are many vertex correction diagrams whach potentially give a sizeable contr> butlon to the amphtude for the direct weak decay of K + -+ lr+~,, and a full renormalization must be per-

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formed for them. Some approximations must be made to make the calculation feasible. We shall first describe the qualitative features of this calculation, and then the results of exphclt evaluation. Finally we shall present the calculation o f the cascade decay (12). Within the class of models which satisfy (4), there are certain approximations which follow simply as a matter o f consistency. We of course neglect the photino mass throughout in the direct decay. In this class of models the W - W mass difference is expected to be of the same order of magnitude as the photlno mass, so we set rn~ = mw. We further neglect mLxlng between qL and qR, 1.e. those squarks which form charal supermultlplets with left- and right-handed quarks, respectively. A significant mass mixing tnLR ve 0 is a measure o f R non-xnvarlance which could also be expected to govern the photino mass. Moreover, terms which are first order in t~2LR are always accompanied by mu, c and, because o f chirahty conservation, by masses of external lines. Therefore, two photinos in the final state always have opposite hehcltles and all diagrams are to be added coherently. The effective four-quark operator so obtained takes the form, after Fierz transformation, [37,(1 - ~'5)s] ( ' ~ 7 " 7 5 ~ ) + h.c.

(14)

We assume that Am 2 = Arn 2 ~ m 2, while as usual we Cl ~t neglect mu,d, s. We also neglect the top flavor contribution wtuch can easily be incorporated by writing the amplitude as A = UscUdcF(me) • 2 - UstU~tF(m2),

31 March 1983

The evaluation of the box diagrams of fig. 1 is now straightforward. Since a posteriori we find that we can probe squark masses only to a small fraction of the W mass, we calculate only to the lowest order in m 2 / 2 . This allows us to eliminate all box diagrams with q mw more than one internal W or W line. We are still left with a large number o f diagrams arising from vertex and propagator renormalization. Their evaluation is equivalent to the evaluation of the effective d~'~ and gd'~ vertices, which we argue must vanish in the supersymmetric limit at least for zero external momenta, consistent with our approximation that neglects external fermlon masses. The argument is as follows' The .q3'q ... coupling without a derivative is obtained as the D term of the super-space coupling cb~,egV~,,, but this same couphng also contains the off-~dlagon~ charge coupling forbidden by gauge invarlance. The nonrenormallzation theorem of supersymmetry allows only D type terms for effective interactions obtained from loop diagrams, when external momenta are set equal to zero m the supersymmetrlc limit. One might hope that this result would suffice to eliminate the diagrams of fig. 2 on the ~round that they contain an additional factors (m 2 - m2)/m. 2, when supersymmetry is broken q q w Untortunately, this is not the case. Those diagrams

s _- @ - - ~ I

I~ Id I

(15)

where Uqq, are appropriate elements o f the K.M. matrix and the first term in (15) is the result with the to~ quark neglected (UscUde ~ cos 0 e sin 0c) and m 2 >> mc>> m2,d,s is used. In order to make an exphclt evaluation o f diagrams, we need to specify a model in a little more detail. A pair of Higgs doublets that couple to quarks and leptons are asusmed to develop vacuum expectation values of the same magnitude, so that their contribution to the D terms of SU(2)L × U(1) are zero, keeping supersymmetry unbroken. Including the surviving Higgs in the computation is necessary in order to be consistent with supersymmetry and to check the cancellatlon among various diagrams in the s y m m e m c limit. Right- and left-chlral squarks are assumed to be degenerate in mass even after supersymmetry is broken.

Ca) $

a~

@

Q a-

,

tb)

7

Fig. 2 (a) Vertex type diagrams which include wave-function renormahzatlon. (b) Flavor-changing mass diagrams Corresponding diagrams for fermlons do not contribute to the leading order m our approximation. 243

Volume 123B, number 3,4

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which become singular at mq and/or ~q~m~ -+ 0 are suppressed only by a factor of (m q2 - m~)[m~ in the broken symmetry. It is, however, still a useful check to make sure that the cancellations are realized in our calculation if we go to the supersymmetric limit. The effective four-quark operator of the direct decay is written m the form •/~eff

=

2-l/2G(c~fi r) cos 0 c sin Oc(m2/m 2)

× C[dTu(1 - 75)s] ('~7u75~), 4

2

(16a) 2

C = [~ ln(m~c/m2) - ~ + O(m-~/mw)] + ~ 0 ( m 2 /m 2

,,

(16b)

to lowest order m the four-quark model. Here the first square bracket in (16b) comes from the box diagrams and the rest comes from the vertex and selfenergy type corrections. No diagram produces an enhancement factor of In (m2/m2)for the coefficient C of (16b). The leading enhancement factor of In (m 2 / m 2) comes from the box diagrams. The next leading 2 tm2., enhancement factor of In ~~'mw/ ~-) is produced by numerous diagrams of vertex and self-energy types, but they cancel among themselves after all the diagrams are summed over. The decay rate due to (16) can be readily compared with the decay rate for K + ~ n o + e + + ve. Taking account of ~ being a majorana fermion, we find for the direct decay F ( K + ~ ~r+ + ~ ) F(-~-~+e+--~e)

O(mw) or less. In models where the supersymmetry breaking scale is much larger than m w and the cancellatlon is realized by particles with vastly different masses, it is possible that a contribution like In (M2/m2),

(18)

Is added to Cin (16b). Here,M ('~ m~.) denotes a mass of a very heavy particle needed to realize the cancellation. Such models resemble models of soft and explicit breaking of supersymmetry, and some of the models that generate squark masses through loop diagrams from a very large mass scale may have such features. When the logarithmic term (18)is really large, this can be viewed as the renormallzatlon of the diagonalization matrix for the d, ~" squarks, which generates effectively flavor-changing neutral interactions of photino and gluinos. If this happens, the K + 7r+~'~ decay occurs effectively at the tree level and its decay rate may increase by more than one order of magnitude. In models of soft and explicit symmetry breaking, the logarithmic divergences generally remain uncancelled m the off-diagonal squark mass and the K + -> 7r+'~ decay rate may come close to the experimental lower bound O(10 -7 ) [16]. The decay of rr0 -->~ m the cascade process occurs through the diagram drawn m fig. 3. The decay rate is easily computed by keeping the photmo mass this time. The result is given by F(n 0 -->"~'~) = -~7ra2(1 - 4m 2/m2]l/2(f2rn 2 Im 4 ]m

/~z\2 / m e \4 = 2C 2 ~-~) coS20ct ~ ) , (17)

which leads to the branching ratio given in (10) by use o f m c = 1.5 GeV and the experimental value, BR(K + -+ 7r0 + e + + %) = 0.048. In the supersymmetric models, the decay rate of K + -~ 7r+ + Fv is also modified, but the leading correction is, relative to the standard result, of order of In ( m 2 / m c ) rather than In (mw/mc) 2 2 since new diagrams added have (~, ~') squark lines instead of (u, c) quark lines. Therefore, the K + ~ n + + ~v decay rate remams roughly the same as in the standard model wathout supersvmmetry for winch Zt_ e uBR(K + ~ 7r+ +~ivi) ~- 10 -10 with the four quarl~. A remark is m order on this estimate. The evaluation of the branching ratio eq. (17) is vahd in a class of models where the cancellation of divergences takes place among internal particle lines with masses of 244

31 March 1983

-

(19)

whe re f~r = 93 MeV, m~. is the mass of ~- and d-squarks and m~- is the photino mass. By substituting the experimental value BR(K + ~ lr+Tr0) = 0.21, we obtam the estimate of(14). The cascade decay is clearly distmgnishable from the direct decay through a monochromatic signature of the 7r+ in K + ~zr + + "nothing". However, we should be aware of the competing cascade decay

Fig. 3. Decay of no into two photmos.

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PHYSICS LETTERS

References

K + ~ rr+ + zr0 L.~ ~ v ,

(20)

whach may be enhanced for massive neutrinos. In the standard theory [2], this decay branctung ratio is calculable through the neutral weak current wathout ambiguity o f lepton flavor mixing parameters. The result is BR(K + ~ rr+ + ~b')cascade ~-- 3.5 × 10 -13 [mv(MeV)] 2(1 -

31 March 1983

am2/m2) 1/2 .(21) --,,

~V/-.

-~.

",

With the sensitivity o f 10 -10, one can probe the neutrino mass m the range of 17 MeV < m v < 67 MeV. This process deserves a serious consideration as a search for massive neutrinos, m particular, because of its independence o f lepton mixing parameters. In conclusion, we have found that the direct decay o f K + -+ rr+ + ~ can only be a small fraction o f the K + -+ 7r+ + ~v decay in the standard model. It wdl not serve the purpose o f constraining the squark mass, although it might serve to constrain the supersymmetry breaking scale M in some class o f models w l t h M >> m~-. On the positive side, however, future experimental tests for K ÷ -+ 7r+ + ~v through K + D 7r+ + " n o t h i n g " will not be disturbed by K + ~ rr+ + ~ . If masses of squarks and the photino are just in the right range of value, the cascade decay mode o f K + ~ 7r+ + ~ " may become appreciable. We thank our colleagures for friendly chscussions. This work was supported in party by the National Science Foundation under Research Grant Nos. PHY81-18547 and PHY-82-03424; and in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Dwlslon of High Energy Physics and the US Department of Energy, under Contract DE-AC03-76SF-00098.

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