R-baryon masses

Volume 153B, number 4,5

PHYSICS LETTERS

4 April 1985

R-BARYON MASSES Franco BUCCELLA lnstituto di Fisica Teorica, Universita di Napoli, Naples, Italy and INFN, Sezione di Napoli, Naples, Italy

Glennys R. F A R R A R Institute for Advanced Study, Princeton, NJ 08540, USA and Department of Physics, Rutgers University, New Brunswick, NJ 08903, USA

and Alessandra P U G L I E S E Dipartimento di Fisica, Universita di Roma I, Rome, Italy and INFN, Sezione di Roma, Rome, Italy

Received 9 November 1984

The MIT bag model is used to calculate masses of (R-)baryons, composed of three quarks and a gluino. If the gluino mass is small, the lightest of these, a flavor singlet, could be long-lived or even absolutely stable. The next lightest, the R-nucleons, probably have only weak decays, while all others are likely to decay strongly. This physical picture is not ruled out experimentally.

It has recently been observed [1 ] that previous lower limits on the masses of R-hadrons (hadrons con. 1 raining a gluino, the spin i supersymmetry partner of a gluon) must be discarded if the R-hadrons do not decay rapidly into photinos plus ordinary hadrons. This revives as a phenomenologically acceptable option the possibility that gluinos are light, say as light or lighter than the strange quark. Knowledge of the mass spectrum of the R-hadrons is of crucial importance in devising suitable strategies for detecting them, especially since their splittings play an important role in determining their lifetimes. Masses of R-mesons ( q - - ~ gluino bound states) have been calculated by Chanowitz and Sharpe [2] : if the gluino is light, as of interest here, their calculation indicates that the Rmesons are heavy enough to decay strongly to ordinary mesons and a gluino-gluon bound state, making them difficult to detect. This note presents a bag model calculation of the masses of the iightest R-baryons,

those which are most likely to be stable against strong decay and hence most readily detected. We extend the results of ref. [1] for the J = 0 and J = 2 R-baryons, where the gluino mass was approximated to be zero, to include the J = 1 states and massive gluinos. In the MIT bag model the mass o f an R-hadrron is given by [3] M = ( 2 . 0 4 / r ) N 0 +iVm w ( m , r ) + ~B1rr 3 + , \i>/

(1) Here o i (h i) and s (A) are the spin (color) matrices for the three quarks (i = 1,2, 3) and gluino. The terms of eq. (1) are, in following order, the quark and gluino kinetic energies (N O is the number o f massless fermions, N m , the number of fermions of mass m), the bag energy, the quark-quark and quark-gluino 311

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contributions to the hyperfine splitting, and the "zero point energy". The radius, r, is determined for each hadron by aM/~r = 0. DeGrand et al. [3] fit the ordinary baryon masses to ftx B I/4 = 0.145 GeV -1 , Z 0 = 1.84, and m s = 0.279 GeV, assuming m u = m d = 0. K is determined below by fitting the N--A splitting. The three quarks of an R-baryon are in a color octet state, so that Fermi statistics requires them to form a flavor octet if their spin is 3/2 (always taking L = 0). If their total spin is 1[2, they can be in a flavor singlet, octet, or decuplet. Evaluation of the expectation values of the hyperfine splitting operator in the necessary configurations of color, spin, and flavor is in general a non-trivial problem, except for the J = 0 and J = 2 [1 ] states, for which the calculation can be recast into a much simpler form providing a useful check on the calculational method necessary for the J = 1 case. Define

4 April 1985

Expectation values of the Casimirs relevant for our calculation are, for SU(2): C{1) = 0, C ( 2 ) = ¼, C ( 4 ) = i s4 , and for SU(3): C ( 1 ) = 0, C{3) -"g* , c ( 8 } = 3 , C{ 10) = 6. Evaluating the N--A mass splitting in terms of K/r yields t 1C mzx - m N = (K/r) l ( C f A -- CfN + ~CsA -- ~ sN ) = g[r.

Using (r) for the N - A system [3] = 5.25 GeV -1 gives K = 1.5. For R-hadrons we need to evaluate 02 as well. First note that because the R-hadron is a color singlet, A = - Z X i, and the color Casimir of the quarks alone is C c = 3. Two spin representations are easy to evaluate: J = 0, for which s = -Z/3= 1 o i and the quark spin Casimir C s -" ~, 3 and J = 2, which is totally symmetric. Judicious manipulations then yield 5

(02)j= 0

'8 +
(02)./= 2 = _ 34 "

where the color, flavor, and spin matrices in the fundamental representations are normalized to tr )'a "Xb _a - ~Sab, etc. (For color and flavor, a and b = 1 ..... 8, while for spin,a and b = 1,2, 3.) Fermi statistics for the quarks implies that for each pair, i], of quarks, peps_ tvf ., . i] i] - - i]' so m a t

More work is required to obtain the matrix elements of 0 2 for the J = 1 states. We begin by decomposing the quark states of the totally antisymmetric 816 of SU(18) into representations of SU(6)C_S X SU(3)F, SU(6)F_S X SU(3) C, and SU(3)C X SU(2) s X SU(3)F (table 1). We are interested in the color octets which are in the second row, which can be seen to be in the 56, 70, and 20 of SU(6)C_S, respectively for flavor singlet, octet, and decuplet. Next , i is proportional we note that the operator --3 .,.i=l o izAa to the generator of SU(6)C_S. We need only the F part of the matrix elements of this operator between the quark color-octet states, since the matrix element of the gluino operator is of pure F-type. These matrix elements are given in ref. [4] :

(9 1 =-41 ~ ( : 1

( 5 6 , 8 , 2 , S z = ' ~ [1

O 1 = i> ~] (°i'°/)(ki'kJ),

O2 = ~

(oi's)(~ki'A).

Evaluation of matrix elements of O i is facilitated by use of the quark permutation operators for color, flavor, and spin: pC - 1 1 i] - "~ -h 2~i.~J, p.f. tl :_ ~a + 2pi'p]' ps.il =- "~ ÷ 2°i'°J'

_._ 2pi.p] : ~O(.O] -- ~ki'X]).

(2)

The RHS of eq. (2) can be expressed in terms of the quadratic Casimirs of the quark representations under color, flavor, and spin for the baryonic states of interest, C c (ZXi) 2, etc.:

0

zXa [ 5 6 , 8 , 2 , S z = I ) =1

D + 52F ,

(70, 8, 4 , S z = ~lOzXa170, 8 , 4 , S z = ~) = F, (70, 8 , 4 , S z =1 i I°z Xa[70, 8, 2 , S z = ~ ) = D + ~ F 1 '

=

1

_

( 2 0 , 8 , 2 , S z = ~[OzXa[20,8,2,S z = ~ ) - O. Ce=4+2

cs

~cAi'x]), i>]

Cf=4+2

E(o%/),

~(lai'lj), i>]

From the above matrix elements and the WignerEckart theorem one then has:

(J= 2;8[O21J= 2;8) = - 6 ,

i>l

1

; J q -_$3; 8 l O 2 [ J = l ; J

so that (2) becomes simply:

(J=

-'21 ~ xcfl 1 (91 - ]"6 -~Cc

(J=l;Jq

312

_"1 C 12 s"

=

q

= $3; 8 ) = + ~ , _1

4

~;8102[J=l;J q ~;8)=+~,

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PHYSICS LETTERS

4 April 1985

Table 1 Decomposition o f the 816 of SU(18) with respect to S U ( 6 ) C _ S × SU(3) F, S U ( 6 ) F _ S × SU(3)C, and SU(3) C × SU(2) S × SU(3)F.

SU(6)C__S X SU(3)F

SU(3) C × S U ( 6 ) F _ S

(56, 1) (1, 56) (8, 70) (10, 20)

(8, 2, 1) (10, 4, 1)

( J = 1 ; J q -_~1; 8 l O 2 [ J = l ; J q (J

= 0;

(70, 8)

=1

~;8)=-~, 2

8lO2lJ = 0; 8) = +_6S'

(J= 1 a n d 0 ; 10lO21J= 1 a n d 0 ; 1 0 ) = 0, _

( J = 1; l 1 0 2 1 J = 1 ; l ) - - g ,

4

( J = O; l l ® 2 [ J = O; 1) = + ~ 8 " We have checked these results for the matrix elements of 0 2 by calculating them in explicitly constructed states. Putting these results together and diagonalizing the two J = 1 flavor octet states gives hyperfine contributions to the L = 0 R-baryon masses as follows: J = O, flavor { 1 } : ±~2 J = O, flavor (8}:

- ~7 n / r ,

J = O, 1 flavor (10}: J=l

flavor (1):

K/r,

Sn/r,

- ~K/r, a

J = 1 flavor (8): (--10.4/8) J = 1 flavor (8): (+2.4/8)

n/r,

n/r,

J = 2, flavor (8):-~K/r, where the number x/4T encountered in the diagonalization of the J --- 1 octets has been replaced by 6.4. To complete the evaluation of the masses, we need to specify the value o f Z 0 and determine the value of r for each state. Ref. [3] assumes that the "zero point energy" parameter Z 0 of a bag is independent of whether it contains 3 quarks or a quark-antiquark

(20, 10)

(1, 2, 8) (8, 4, 8), (8, 2, 8) (10, 2, 8)

(1,4, 10) (8, 2, 10) SU(3) C × SU(2) S × SU(3) F

pair. Extending this ansatz to the R-baryon bags would imply Z 0 = 1.84 for our calculation as well. An alternative picture is that the -Zo/r term is a phenomenological representation of the effects of the quark self~nergies and Coulomb interaction energies. To first order in the coupling it would therefore have the form, for ordinary baryons, ~Nq zo/r. Adding the contribution of the ghiino would change ~Nq 4 • 5Nq 4 + 3, and cause the value of the effective Z 0 to be 3.22 rather than 1.84. The problem of the bag "zero point energy" is poorly understood even for ordinary baryons; therefore we have calculated the masses for both of these choices o f Z 0 and for Z 0 = 0 to give an extreme range of mass predictions. The rest of the mass calculation proceeds by finding for each R-baryon state, the value o f t Which minimizes M given by eq. (1), and then evaluatingM at that value o f t . This is very simple when the quark and gluino masses are either zero or large (~> 1 GeV); for the strange quark mass of 279 MeV it is somewhat messy because of the complicated dependence ofw(m,r) on m andr. See ref. [3] for details. For mg-"1 = 0 and Z 0 = 0, 1.84, and 3.22, the masses of the lightest member of each of the L = 0 multiplets is given in table 2. As already noted in ref. [1 ], the lightest R-baryon is the J = 0 flavor singlet, which can be very long-lived or stable. It is quite remarkable that the hyperflne attraction in the flavor-singlet J = 0 configuration is so strong that this state is several hundred MeV lighter than the lightest members of the nearest non-singlet multiplets, even though it contains a strange quark whose "effective mass" is " d 50 MeV. In fact, the singlet R-baryon must be as heavy as the nucleon, since otherwise the AS = 2 reaction nn (1)0 {1}0 would lead to the disintegration of nuclei over cosmological time scales * 1. This argues in favor For footnote see next page.

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Volume 153B, number 4,5

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Table 2 Mass of the singlet R-baryon, and mass-splittings of higher states relative to it, in MeV, for various choices of gluino mass and zero point energy parameter. Masses are given for the strangeness 0 members of flavor octets and decuplets; for non-zero gluino mass they are only given for the R-baryons which are stable under strong interactions.

m~;Zo M(.[1)0) A( .[8)t ) ix(.[8)O) 4('[1) 1) A( .[8)' 1) 4(.[10)o,1) A( .[8)2 )

0; 1.84(3.22, 0)

ms; 1.84(3.22)

~ 1 GeV; 1.84(0)

890(600, 1300) 210(230, 130) 330(360, 240) 560(610,480) 640(690,530) 720(770,600) 780(840,660)

1010(720) 190(220) 310(340)

(280, 740) + mg--l 250(330) 380(440)

either o f Z 0 less than or comparable to the standard bag-model value o f Z 0 = 1.84, or else a gluino o f mass ~> 1 GeV, as can be seen from table 2, which also gives the singlet R-baryon mass for mg-"1 = m s and for rag-"1 >~ 1 GeV, large enough to apply the non-relativistic approximation. F r o m table 2 we see that the only other states which cannot decay strongly, and which are therefore amenable to observation, are the R-nucleons, the strangeness = 0 members o f the J = 0 and lighter J = 1 octets, which decay weakly to the S = - 1 flavor singlet plus a pion. Fortunately, the Q-value o f these decays is insensitive (see table 2) to the value o f r n ~ and Z 0 , 2 . ,1 This argument is due to E. Witten; see ref. [ 1 ]. It is not excluded that the singlet R-baryon is absolutely stable so long as it is heavier than the proton. ,2 Except for the .[8)1 for Zo ~ 0. This case requires separate analysis.

314

Their lifetimes can be estimated [1] ,2 to be ~ 10 -10 s, in analogy to A decay. For a discussion of the phenomenology and methods for detection o f baryons containing such a light gluino, see ref. [1 ]. G R F wishes to acknowledge the hospitality o f the Universities o f Naples and Rome where some o f this research was performed, as well as helpful discussions with J. Breit, R. Jaffe, K. Johnson and A. Terrano.

References [1] G.R. Farrar, Phys. Rev. Lett. 53B (1984) 13, 1029. [2] M. Chanowitz and S. Sharpe, Phys. Lett. 126B (1983) 225, [3] T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060. [4] N. Cabibbo and H. Ruegg, Phys. Lett. 22 (1966) 85.