A calculation of Z0 → 2 gluinos

Nuclear Physics B217 (1983) 117-124 v: North-Holland Publishing C o m p a n y

A CALCULATION OF Z ° --, 2 G L U I N O S G.L. K A N E

Randall Laborato~' of Physics, Universi O, of Michigan, Ann Arbor. MI 48109, USA and

William B. R O L N I C K

Department of Physics, Wayne State Universi(v, Detroit, MI 48202, USA Received 7 December 1982

We report the results of a one-loop calculation of Z ° ~ ~ , where ~ is a gluino, the proposed supersymmetric partner of a gluon. Depending on the masses of the scalar quarks and of the top quark, the branching ratio for the decay is in the 10 5 to 10 - 4 range for gluino mass below about 40 GeV. The signature for gluinos should allow detection in this range.

1. Introduction

Many theorists feel that a supersymmetric theory is a very likely candidate for the next stage in particle physics, because it can accommodate, in a natural way, scalar bosons of mass small compared to the grand unified scale, and because it can connect particle physics to gravity. So far there is no hint of supersymmetry in n a t u r e - on the contrary, the supersymmetric partners of quarks, leptons, and gluons would have been observed if they had similar masses to those of the familiar particles. There are two very good ways to look for supersymmetric partners. (i) At any e+e collider the scalar partner of leptons can easily be detected up to the top machine energy. They give e+e - or #+# pairs with ½~/s missing energy. They have not been observed up to the top P E T R A / P E P energies. (ii) Because gluinos (~) are color octets they have large production cross sections at hadron machines. They would already have been observed [1]* if their mass was less than 3-5 GeV. Experiments at existing machines will allow detection of gluinos in the 10-20 GeV range (ISR, SPS collider). If they are not found in that range, the next stage (1987-1989) will be to look at the F N A L Tevatron collider, which may be able to * For the experimental limit of 3-5 GeV, see Ball et al. [la]. 117

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G.L. Kane, W.B. Rolnick / Z ~ ~ 2 gluinos

detect gluinos with masses up to 75 GeV, and at e+e - colliders (SLC, LEP) at the Z ° mass. There can be no tree graph (therefore no large decays) for Z ° ~ 2 gluinos since the gluinos cannot couple to the Z °, as they are SU(2)L ® U(1) singlets and the Z ° is a color singlet. At one loop there is a contribution from the quark-scalar quark intermediate state and that is what we have calculated in this work. The result is that the decay Z °-~ gg is expected to have a decay branching ratio above 10 -5 for an interesting range of parameters of the theory. While that is hardly a large rate, it is probably observable at an e+e - machine: a branching ratio of 10 5 at a machine with L = 6 × 103°/cruZ-sec corresponds to 30 events in 107 seconds. We will describe the details of the calculation below. The rate depends on the t-quark mass, the masses of the scalar partners of the quarks, and the gluino mass, so we can only give results for ranges of these. Further, there is as yet no satisfactory supersymmetric model, so we cannot be sure that our results would not change in a complete model. However, our results hold in a number of anomaly-free supersymmetric models [2] with spontaneously broken SU(3)c® SU(2)®U(1) symmetry, and are likely to give a reliable indication of the expected branching ratios. [We note that recently the rate for e ÷ e - ~ 2 gluinos at presently available energies at P E T R A / P E P has been calculated (ref. [3]) from one-loop diagrams, but not including the Z ° pole. They obtain ~< 10 -3 units of R, which will be negligible at the Z ° compared with the ¼ unit of R of our result.] We will also discuss the decay signatures. While they should be adequate for detection, they do require special detector characteristics, and we hope experimenters will plan to search for gluinos.

2. Calculation Fig. la shows the contribution Z ° -~ qq -~ ~ , and fig. lb the contribution from the scalar quark (q) intermediate state. In all models of which we are aware [2] there are no other particles that are non-singlet under both SU(3)c and SU(2)L ® U(1), so no other diagrams occur at one-loop. They are summed over q, Cl to give the full amplitude. The part of the Z ° coupling which can produce two identical Majorana fermions (gluinos) is the axial vector current ),~,5T3. This axial vector decay is p-wave. Whenever weak isospin is unbroken, summing over the intermediate quark-doublets yields zero (Tr T3 = 0), as expected since this contribution is proportional to the triangle anomaly. This is a useful check on our results. For physical q and ci, the factor multiplying T3 will depend o n mq (and rhq) so that the sum over the doublet will no longer be proportional to Tr T3. This results in an incomplete cancellation of the two contributions. To get the full amplitude one sums fig. la over quarks and scalar quarks of all known flavors, and subtracts the crossed diagram, repeats this procedure for fig. lb,

G. L. Kane, W.B. Rolnick / Z ° ---, 2 gluinos

119

Z°

5°

/

q.,

I \

',q

j

k1

kz

(a)

k1

k2

(b)

Fig. 1. One-loop contributions to Z ° ~ ~.

and adds the results. Since only the axial Z ° coupling survives, all terms are proportional to ~q, and no sin20w dependence enters. The amplitude is also identically zero when mq = m~ for each supersymmetric pair, even if weak isospin is broken so m t =~ m b. This is less obvious to interpret; it appears to be a supersymmetric remnant of the fact that an axial vector is forbidden to decay to two massless gluons. Thus to get a non-zero answer both the gauge symmetry and the supersymmetry must be broken, as is the case in nature. Lacking a realistic supersymmetric theory, we cannot perform a consistent calculation with physical masses included for both quark and scalar quark. We imagine that the masses effectively enter softly in the lagrangian as mass terms so that we can calculate at the one-loop level by simply using "physical'' masses in the Feynman diagrams (even when we do not know the actual numerical values for m t or rh, the scalar quark mass). Since we will be satisfied with a sufficiently accurate calculation to determine whether an experimental search is feasible, this should be an adequate approximation. The vertex factors are for qqg:

~'v~g3PM,

(1)

where g3 is the Q C D coupling, g 2 / 4 ~ r = ~ , P is a left- or right-handed projection operator, and ½X" is an SU(3)c generator; for qqZ°:

g2 Tq3y. PL ,

(2)

where g2 is the electroweak coupling, g 2 / 8 M ~ v = G F / V / 2 c o s 2 O w , T 3 is the weak isospin eigenvalue of the quark, and PL is the left-handed projection operator; for q~tZ°"

g2T~3(2k + k 2 - k l ) u ,

(3)

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G.L. Kane, W.B. Rolnick / Z °---, 2 gluinos

where the momenta are labeled in fig. lb, Tfl = + ½ for left-handed scalar partners, and T3 = 0 for right-handed scalar partners. If we write the full matrix element as

M = Aff(k,)y~"[sv(k2)e,,

(4)

where G is the Z ° polarization vector, then (including the color factor and identical particle effects) summing over all spins, the partial width is

r(z o

= A M#12 .

(5)

To see the order of magnitude expected for the result, we note that (see the Feynman integral below) A will be of the form A ~

g2g

2~

2A / ( 2 )4 o ~ ,

(6)

where A 0 is the dimensionless sum of various integrals. For physical quark masses and for scalar quark masses of order rn~, ]A0l2 is of order unity. Using Frot -GFM3/Tr, this gives a branching ratio BR(Z ° ~ ~ ) ~ ¢2 IAo12a2/48~r 2 = 8 >< 10 51Aol2 Now we write some details of A for fig. la; the procedure for fig. lb is similar.

M~(k~ k2)= 2~g2g2i3f

d4k

[U(kl)PLVa~PRI)(k2)/DaL --~(kl)PRVa~PLV(k2)/D~R],

(8)

where V~= (y" k + y" k, + mq)(Tq3y~'PL - sin20w'~Oq)(T • k - 3/. k 2 + mq), Dac,R = (k 2 - rn2,R)((k + k , ) : - m2q)((k - k2) 2 - rn~),

(10)

and the scalar partners of the left- or right-handed quarks have mass mL. R. We assume for simplicity that m L = m R - r h and that rh is constant in each quark doublet; this is one way in which model dependence enters. (For two reasons, the current prejudice is that scalar quarks of all flavors and both chiralities will have very nearly the same mass. First [4], arguments based on the absence of flavor changing neutral currents or of large parity violating effects in nuclei require very nearly equal masses (Ar~2/rh 2 << 1) in the absence of protection from subtle symme-

G. L. Kane, W.B. Rolnick / Z ° ~ 2 gluinos

12I

tries (assuming the scalar quark masses are of order mw). Second [5]* recent models suggest that scalar quarks that get mass radiatively will get a large, flavor independent, contribution with only small percentage corrections from flavor dependent loops. The scalar quarks get much of their mass from supersymmetry breaking, while quarks get mass by gauge symmetry breaking.) As discussed above the full matrix element is

M = M,( kl, k 2 ) - - M a ( k 2 , k,) + Mb( k., k z ) - M b ( k 2 , k , ) .

(11)

After manipulation, we have

Ma( k, , k2 ) = - i3 e~g~g2Tq3 ( M2zJ' + 2 J " + m2qI ) gys y,v,

(12)

(2¢r) 4

so its contribution to A in eq. (4) can be read. The integrals J', J " and I are obtained from

I=fd4k/D.,

(13)

= fd"t:k.kJD. =J(k,ukl~+k2~,k2~)+J'(k,,k2~+k,~k2,)+J"g~,~.

(14)

Similar integrals occur for Mb,

2i3e

Mb--

zT-_3

( 2 ~ ) 4 q gttu-~t,'~/x~ ,

(15)

where

L"'= f d4kk~kJDb= L(kl.k" + k2.k2~) + L (kl>k2v+ klvk21z) + "ggv. (16) this gives

A = ½dab2i3g~g2Tq3Z(2L '' (2~r) 4

* See also the recent modelsof ref. [2].

q

--

2J"

-m2zJ,_mZi)

(17)

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G.L. Kane, W . B . Rolnick /

Z u -* 2 gluinos

where a, b are the color indices of the gluinos in the color-octet. The integrals are finite except for J " and L" whose singular parts cancel in A. (The cancellation is easily handled using dimensional regularization, the pole terms at n = 4 cancelling since they are independent of mq and r~.) The remaining integrals can be done exactly for m~ = 0, with results expressed in terms of Spence functions. (We employ the appendix to ref. [6] to calculate Spence functions of complex arguments in terms of Bernoulli numbers.) We examine numerically the change in our results for m~ ~ 0.

3. R e s u l t s

In practice all contributions except those from the (t,b) doublet either are numerically small or cancel. The relevant parameter is ( m q / m z ) 2, which is very small for the first two generations. Since we do not know the value of m t, the scalar quark mass r~, or the gluino mass m~ we give results for a range of their values. We calculate our final branching ratios by summing over the gluino octet. Figs. 2a, b show the branching ratio as a function of m t for m~ = 0 and two values of rh, figs. 3a, b,c show the branching ratio as a function of r~ for m~ = 0 and three values of m t, and fig. 4 shows the decrease of the branching ratio with increasing m~; the decrease is slower than might be expected from phase space because the terms in the numerator which vanish when mg = 0 enter effectively with a relative plus sign and increase the amplitude. (Note that the denominators also change.) Over a wide range of physically interesting parameters the branching ratio may be large enough to observe. It is interesting to turn things around. Suppose gluinos are observed in other experiments and mg < ½m z. Then the size of the branching ratio here gives a sum rule or inequality (if this mode is not seen at a certain level) which provides

1.0 .8

(a)

1.0

(b)

.8 !

~t .6 >< o~ a~4

.2

OZO '

/ ,4

.2

Y

4'0 ' 6'0 ' 8'0~ 020 ' 4'0 ' 6'0 ' 8'0 rn t (Gev) m t (BEY)

'

Fig. 2. (a) Branching ratio for Z °--, ~,~ versus m t for mg = 0 and r~ = 50 GeV. (b) Branching ratio for Z ° ~ ~g versus m t for m~ = 0 and rh = 100 GeV.

123

G.L. Kane, W.B. Rohfick / Z ° --* 2 glumos

lo!r

(a)

tos[

(b)

lo

81

.8

%6 g

(c)

.6i~ '

4 2

.

I

0 L 40 80 120 r~ (6eV)~-

0 40

6

~

.4 .2

80 120 160 20 40 60 80 ~ (6ev)~ • (Gev)--~.-

Fig. 3. (a) B r a n c h i n g r a t i o for Z ° ~ g g versus dl for m~ = 0 a n d m t = 20 GeV. (b) B r a n c h i n g r a t i o for Z ° --+ ~ versus rk for m g = 0 a n d m t = 50 G c V . (c) B r a n c h i n g r a t i o for Z ° ---, ~ versus tk for m g = 0 a n d m t = 80 G e V .

I.D~., 0

10

20

\x 30

m)"(GeV)

40

Fig. 4. S u p p r e s s i o n f a c t o r for rn~ * 0, i n c l u d i n g p h a s e space. T h e d a s h e d c u r v e is f o r m t = 20 G c V a n d r;7 = 50 G e V a n d the solid c u r v e is for m t = 50 G e V a n d rk = 100 GeV.

information about m t and about scalar quark masses. Alternatively, every supersymmetric model with mg < ½Mz predicts a definite branching ratio for Z ° --+ gg as one of its tests; as people produce workable models their predictions can be written down by comparison with our formulas.

4. Signature In all models gluinos are expected to have a decay mode ~ ~ qq~, where ? is the supersymmetric partner of the photon (the photino). In most models this mode will dominate. The photinos will escape from collider detectors without interacting. Then the decay Z ° --+ ~ will have the unique signature of qqqq with _{ of the total energy and momentum missing, and no prompt hard charged leptons. If m~ is near ±M 2 z, the qqqq are highly spherical, and there may be a significant PT imbalance. If there is considerable energy available in the decay the gluino decays give jets and a clear PT imbalance. In the latter case, for larger m~ the final quark jets should be

124

G. L. Kane, W.B. Rolnick / Z ° ~ 2 gluinos

resolvable while if rn~ is below about 15-20 GeV each qq will be an unresolved wider jet. While such events will not be obviously and dramatically "something new", they should be detectable. Indeed, if gluino masses are in the 20-40 GeV range this may be the best way to see them. We are grateful to J. Leveille, who was a collaborator in an early stage of this work, to S. Raby for discussions and suggestions, and to D.R.T. Jones and M. Einhorn for helpful comments. This research was supported in part by the U.S. DoE contracts number DE-AC02-76ER02302 and DE-AC02-76ER01112. References [1] G.R. Farrar and P. Fayet, Phys. Lett. 76B (1978) 442, 575; G.L. Kane and J.P. Leveille, Phys. Lett. l12B (1982) 227 [la] R.C. Ball, et al., preprint UM HE 82-21, submitted to the 21st Annual Int. Conf. on High-Energy Physics, Paris, 1982 [2] P. Fayet and J. Iliopoulos, Phys. Lett. 31B (1974) 461; P. Fayet, Phys. Lett. 69B (1977) 489; 70B (1977) 461; C.R. Nappi and B.A. Ovrut, Phys. Lett. 113B (1982) 175; M. Dine and W. Fischler, Institute for Advanced Study preprint Dec. 1981; Nucl. Phys. B204 (1982) 346 L. Alvarez-Gaume, M. Claudson and M.B. Wise, preprint HUTP-81/A063, 12/81; J. Ellis, L.E. Ibanez, and G.G. Ross, Phys. Lett. I I3B (1982) 283; S. Dimopoulos and H. Georgi, Nucl. Phys. B193 (1981) 150; N. Sakai, Tohoku University report no. YU/81/225, unpublished; L. Hall and I. Hinchliffe, preprint LBL-14020, Feb. 1982; S. Weinberg, Phys. Rev. D26 (1982) 287; B.A. Ovrut, in Third Workshop on Grand Unification, University of North Carolina, Chapel Hill (April 1982) [3] P. Nelson and P. Usland, Harvard preprint HUTP-82/A017, Gluino pair production in electronpositron annihilation [4] M. Suzuki, Berkeley preprints UCB-PTH-82/8, 82/7; J. Ellis and D.V. Nanopoulos, Phys. Lett. 110B (1982) 44 [5] S. Raby, private communication [6] G. 't Hooft and M. Veltman, Nucl. Phys. B153 (1979) 365