decay properties of massless bound gluinos

Volume 124B, number 3,4 PHYSICS LETTERS 28 April 1983

DECAY PROPERTIES OF MASSLESS BOUND GLUINOS

Enrico FRANCO Istituto di Fisiea Guglielmo Marconi, Rome, Italy

Received 4 January 1983

Massless gluinos are confined into R-hadrons, together with gluons and quarks. We estimate the life-time of such R-hadrons on the basis of some model previously developed to describe ordinary hadrons. The energy spectrum of the emitted nuino is also derived. Using the known limitations on the parameters of broken supersymmetric models, the lower bound to the life-time of R-hadrons is derived: r > 2.3 × 10 - 1 4 (1 GeV/M) s s.

Until now there is no experimental evidence of supersymmetry in nature. This means that supersymmetry is broken in such a way that we can see only one side of the particle spectrum easily. Previous anal- yses based on different schemes of broken supersymmetry [1,2], seem to indicate a low mass for the gluino (the supersymnaetric partner of the gluon), suggesting that gluinos could be observed in present experiments. The gluino could be a promising particle in order to see for the first time a piece of the other side of the theory.

The gluino is a color octet, flavour singlet particle. It interacts strongly with gluons and quarks, combin- ing with them into R-hadrons: ~g and gqgt states.

A previous analysis [3] has estimated the gluino life-time considering the case of a massive gluino bound to a gluon or to light quarks. The decay of the R-ha@on is then essentially due to the decay of the massive gluino.

In this paper we consider the case of a massless gluino confined into an R-hadron.

We know that beam dump experiments [4] seem to indicate rather heavy R-hadrons [5], suggesting a massive gluino, but we think that the possibility of the R-hadron mass arising purely from confinement is still not excluded.

In the latter case the rate of~g can be computed in a simple model, similar to that used in ref. [6], with the following steps: (i) One estimates the probabil i ty for the R-hadron to dissociate into a gluino with in-

variant mass W and a massless gluon. (ii) One assumes the decay rate for the gluino to be the same as that for a free gluino of mass W. (iii) Finally one integrates over W.

The probabil i ty in step one can be related to deep inelastic structure functions. As a guide to the evaluation of the g~ structure functions, we shall use a phe- nomenological model applied previously to the pion (i.e. q~) structure functions, which agrees with the available data.

The rate for the ~qC: 1 can be computed in a similar manner. In this case there is also a possibility for the R-hadron to dissociate into a massless q (q) and a pair ~q ( ~ ) with invariant mass W, followed by annihilation of the pair.

Similarly to the g~ case we shall use a model applied to proton structure functions in deep inelastic scatter- ing [7] as a guide to the evaluation of the structure functions of the R-hadron. We assume the usual semi- realistic broken supersymmetric model [8] for the interaction of the virtual gluino. We obtain two results: one is the energy spectrum of the emitted nuino (photino or goldstino) which can be used to analyse the calorimetric searches for gluinos; the other is a lower bound to the life-time of the R-hadrons:

r > 2.3 X 10 -14 (1 G e g / m ) 5 s , (1)

where M is the mass of the R-hadron, which can be used as a guide in the planning of emulsion searches for R-hadrons. The lower bound (1) results from the

0 031-9163 /83 /0000-0000 /$ 03.00 © 1983 North-Holland 271


Fig. 1. Relevant diagram in the interaction of the gluino. A nuino is a photino or a goldstino.

known limitations on the masses of scalar quarks and on the value of the supersymmetry-breaking order parameter, d [9].

As a preliminary step, we recall the evaluation of the massive gluino decay rate. The relevant diagrams are shown in fig. 1. The rate for the decay with photino emission is:

p = (asM~e2/19 2 112) s . ~(l[M4 + 1/M4t + 2/M4t), (2)

where we have summed over the charges of up and down quarks. Ms, Mt, Mst are the masses of s and t scalar quarks, including a possible mixing term.

The rate corresponding to goldstino emission can be computed from the effective ~ - q - ~ - n u i n o interaction derived from general current algebra considerations [9].

The total rate for goldstino emission is

F = asM~/192 7r2d 2 . (3)

The same result can of course be derived from the ex- plicit evaluation of the diagrams of fig. 1.

An upper bound for the previous rates arises as fol- lows. From PETRA experiments [ 10] there is a lower bound to the eigenvalues of the mass matrix of the scalar quarks: m 2 > (16 GeV) 2, i = 1, 2. This means that

1/M 4 + 1/M 4 + 2/M4st > 4 (1/16 GeV) 4 .

From astrophysical considerations [ 11 ] one finds a lower bound on d: d > 2.6 X 10 3 GeV 2.

A lower bound of the same order of magnitude is obtained also from muon anomalous magnetic moment analysis [ 12].

In conclusion for a free massive gluino we find

r > 4 . 5 X 10 -15 (1 GeV/M) 5 s.

We turn now to the case of the massless gluino, considering first the g~ case. In the frame P ~ ~' (P is the momentum of the R-hadron), we call L(x) the

probability density to find an on-shell gluon with momentum xP and a gluino with momentum (1 - x) P. We assume for simplicity that transverse momenta are negligible.

The invariant mass of the virtual gluino is then W 2

= M2(1 - x ) and the probability distribution for the R-hadron to emit a gluino with invariant mass W is simply given by

dP(W2/M 2) = L(1 - WZ/M 2) dW2/M 2 = L(x) dx , (4)

where M is the mass of the R-hadron. Assuming that the decay of the off-shell gluino is

the same as that of a massive gluino with mass W, the energy spectrum of the nuino in the frame where the gluino is at rest is

d F 0 ~dE 0 = ( 16 F0/W 5 )(3 W2E 2 - 4 WE 3) 0 (~ W - EO) (5)

The spectrum in the frame where the R-hadron is at rest, is obtained with the following transformation:

dr__ w__L 2 [+ dr0 dE 2pEw E_ J E° dE° '

E, =EW/(E w + p ) , Ew = (W 2 + p 2 ) 1 / 2 , (6)

where p is the momentum of the gluino. From eqs. ( 2 ) - (6 ) , we obtain

I" = (asM5B/192 7r 2)

X [Se2(1/M4 + 1/M4t + 2/M4st)/(2/d2)l ,

= f L(x) dx, 0

1 d F _ 2 ]- L(x] ( 1 - x) 1/2 r d e 8 0 J " " ( - 1 Z - x - ~

X { 3 e 2 [ ( K 2 - 1)Ix + 2 - x l ( 1 - x ) 1/2

- ~e3[(K 3 - 1)Ix + 3 - 3x +xZl ( l - x ) -1/2} d x ,

K = I i f x < l - e ,

K = ( 1 - x ) / e i f x > l - e , e = 2 E / M . (7)

The total rate coincides with the decay rate of a free massive gluino, with an effective mass: Mef f = MB 1/5.

To obtain a numerical estimate we use for L(x) the expression given in ref. [13] for the structure func

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1 tlI' F de

.2 .4 .6 .8 1

Fig. 2. Relative energy spectrum of the nuino in the decay of ~g (full line). Also is given the spectrum of the nuino emitted in the decay of a free massive gluino (dashed line).

tions of ordinary mesons ,1 . The nuino spectrum thus obtained is plot ted in fig. 2. The corresponding value o r b is 0.26. In the same figure we report (dashed line) the spectrum of the nuino emitted from a free massive gluino with mass M *~. The spectrum of the massless bound gluino is considerably softer.

As an indication we also give the value o r B for two extreme shapes of L(x):

L(x) = 6(x - 1-) B = 0.167 = 6 -1

L(x) = 1 , B = 0.280 ~ (3.57) -1 .

The effective gluino mass is rather insensitive to the exact shape of L(x).

We consider now the case ~qC: 1. There are two mechanisms for the decay: in the first the R-hadron emits a q (or a ?:1) on-shell and then the gluino annihilates with the remaining ?:1 (q); alternatively the R-hadron emits a qC: 1 pair on-shell and the gluino decays alone.

In the first case the rate of decay is an average over W of the annihilation rate of a g?:l (gq) pair with mass W: F(W) = o(s = W 2) P0 where o is the annihilation cross section, and P0 is the square of the wave func-

~:1 This should be a not too bad approximation since both the mesons and R-hadrons are bound states of two massless, colored objects.

*2 The real spectrum of massive gluino bounded to a gluon, will be less sharp at the end-point because of the Fermi- motion inside the R-hadron. Anyway we think that the difference between the two cases remains well appreciable.

tion in the origin of the system g?t (or ~q). The expression for o is

o(W 2) = lasW2 [O2e2(1/M4 + 1/M 4 + 2/M4t)/(2/d2)] ,

(8)

where Q is the quark charge. The energy spectrum of the nuino in the frame

with the R-hadron at rest is obtained via the previous transformation:

1 d P l _ 1 ~ ( l - x ) 3 / 2

F 1 de m 1 -e x ( 1 - x / 2 ) L(x) d x '

1 f (1 -- X) 3/2

m = - - d x - - ~ , 0 ( 1 - x / 2 ) L(x) ~ 2

e = 2E/M. (9)

The total rate is

1 2 P l = aasM Pom

× [Q2e2(1/M4 s + 1/M 4 + 2/M4st)(2/d2)] .

Note that P 1 depends very weakly on the shape of L(x) , which, in this case, is related to the probabil i ty for finding the "specta tor" quark in the R-hadron with a fraction x of the total momentum (in the [P[ -+ oo frame).

The spectrum of the nuino is obtained using for L(x) an expression derived for the proton structure function [7]. We plot in fig. 3 the relative spectrum. Note that the spectrum is peaked at the end-point. The height of the peak depends essentially on how

1 dE de

I It

.2 .4 .6 .8 1

Fig. 3. Re la t ive energy spec t rum o f the n u i n o in the decay o f ~qq in the ann ih i l a t i on c h a n n e l

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dif ferent L(x) is f rom zero near the origin. In the ex-

t reme case o f L ( x ) = 6(x - 1/3) the spec t rum is flat

be tween x = 2/3 and x = 1.

Consider now the virtual gluino decay. We must use

L(Xl, x 2 ) which is the probabi l i ty for the R-hadron to

dissociate into a q wi th m o m e n t u m x 1P and a 7:t wi th

m o m e n t u m x2P (in the frame IP[ + oo), bo th with van-

ishing mass. The rate 1" 2 is similar to that o f ~g, but

n o w

1 (1 _ X l _X2)3 . . . .

B= f (i ---"~-1/i~'~-~2/2) L(xl 'x2) axl dx2 ' 0

where L is normal ized to uni ty and 021 = 022 = 1/3.

For two " e x t r e m e " shapes o f L we have:

L = 6(x 1 - 1/3) 5(x 2 - I / 3 ) , g ~- (18) -1 ,

L = 2 0 ( 1 - X l - X 2 ) , B ~ ( 8 . 2 ) - 1 .

The decay rate for this mechanism is thus about

one tenth o f the free decay rate. To see which o f the two mechanisms is dominan t ,

we consider the ratio

1"1/P2 = 32 7r2B-1Po/M 3' ~ 3.1 X 103 Po/M 3 . (10)

As a rough est imate o f P0 we can use a formula derived 1 r 2 M in the non-relativist ic model : P0 = i i~ R q, where Mq

is the mass o f the massive quark. I f we use f R "~ f n

= 140 MeV, Mq = 300 MeV, the ratio is F 1 / P 2 -~ 1.5

(1 G e V / M ) 3 .

The use o f the non-relativistic mode l is rather crude

but wi th this est imate and wi th the fact that very prob-

a b l y M > 1 GeV, if we put P 1 ~ [ '2 we are likely to

overes t imate the total rate. This is compat ib le wi th

the fact that we are looking for an upper bound to

the rate. The presence o f two channels compensates

the bigger factor B - I , wi th respect to the previous

case. In conclusion for bo th g~ and ~qq we have

r . > 2.3 W-14(1 G e V / M ) 5 s .

I would like to thank L. Maiani and N. Cabibbo

for useful discussions.

References

[1] P. Fayet and J. Iliopoulos, Phys. Lett. 51B (1974) 461. [2] E. Witten, Nucl. Phys. B188 (1981) 513;

S. Dimopoulos and P. Raby, Nucl. Phys. B192 (1981) 353.

[3] G. Farrar and P. Fayet, Phys. Lett. 76B (1978) 575. [4] P. Bosetti et al., Phys. Lett. 74B (1978) 143;

T. Hansl et al., Phys. Lett. 74B (1978) 139; P. Alibran et al., Phys. Lett. 74B (1978) 134.

[5] G. Farrar and P. Fayet, Phys. Lett. 79B (1978) 442. [6] G. AltareUi et al., Nucl. Phys. B208 (1982) 365. [7] G. Altarelli et al., Nucl. Phys. B69 (1974) 531. [8] P. Fayet, Phys. Lett. 69B (1974) 489. [9] P. Fayet, in: Unification of the fundamental particle

interactions, eds. S. Ferrara, J. Ellis and P. van Nieuwen- huizen (Plenum, New York, 1980) p. 587.

[10] D. Barber et al., Phys. Rev. Lett. 45 (1980) 1904. [11] M. Fukugita and N. Sakai, KEK preprint KEK-th 42. [12] R. Barbieri and L. Maiani, Phys. Lett. l17B (1982) 203;

J. Ellis, J. Hagelin and D.V. Nanopoulos, Phys. Lett. l16B (1982) 283.

[13] G. Altarelli et al., Nucl. Phys. B92 (1975) 413.

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decay properties of massless bound gluinos

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