Massive gluinos

Volume 78B, number 4

PHYSICS LETTERS

9 October 1978

MASSIVE GLUINOS ~ P. FAYET 1 California Institute of Technology, Pasadena, CA 91125, USA

Received 7 July 1978

We show how the fermionic partners of the gluons under supersymmetry (gluinos), although massless at the classical level, can acquire masses through quantum effects. This is important for the phenomenological consequences of supersymmetric theories.

Quantum chromodynamics describes the interaction of spin-1 gluons with spin-l/2 quarks. Under supersymmetry these fields are associated, respectively, with spin-l/2 gluons, called gluinos, and spin-0 quarks [1]. After the spontaneous breaking of supersymmetry, spin-0 quarks acquire large masses [2]. If the color gauge group is unbroken, gluons are massless; so are the gluinos, at the classical level. For studying phenomenological consequences of supersymmetric theories embedding QCD, it is important to know whether ghiinos can acquire a mass at higher orders. Gluinos are described by a color-octet of spin-l/2 fields, which do not carry flavor. They may combine with quarks, antiquarks and gluons to form new colorsinglet hadronic states called R-hadrons. These include in particular R-mesons (fermions constructed from a quark, an antiquark and a gluino) and R-baryons (bosons constructed from three quarks and a gluino). R-hadrons are unstable and decay quickly into ordinary hadrons by emitting a neutrino-like particle: the Goldstone fermion (Goldstino) or the fermionic partner of the photon under supersymmetry (photino), collectively called nuinos [31. If gluinos are massless, Rhadrons are naively expected to be relatively light ( - 1-1½ GeV/c 2) [4,5]. Experiments recently performed, designed for charm Work supported in part by the U.S. Department of Energy under Contract EY76-C-03-0068. 1 On leave of absence from Laboratoire de Physique Th~orique de l'Ecole Supdrieure, Paris, France.

detection, give information on this new type of hadron [4]. No experimental evidence for their existence has yet been found, however upper limits on the pair production cross-section of R-hadrons in hadronic collisions can be derived. The limit obtained from CERN beam dump experiments cannot be considered as an absolute upper bound, since it makes use of a theoretical estimate of the reinteraction cross-section of the emitted neutrino-like particles we called nuinos. But experiments looking for missing energy in hadronic collisions are less sensitive to theoretical hypotheses as to the behavior of R-hadrons and nuinos. They lead to an upper bound on the pair production cross-section of R-hadrons much smaller than, e.g., the p~ production cross-section [6]. Unless there is some unforeseen dynamical mechanism inhibiting their production, this probably implies that R-hadrons are more massive than the 1-1½ GeV/c 2 naively expected if constructed from massless ghiinos. We certainly need a better understanding of the QCD of ghiinos and color confinement before ruling out the possibility that massless ghiinos can lead to R-hadron masses larger than 1-1½ GeV/c 2. However our present purpose is not to discuss these questions, but to explore the possibility that massive gluinos can be responsible for larger values of the R-hadron masses, as already mentioned in ref. [3]. In a theory with an unbroken color SU(3) gauge group, gluinos are massless at the classical level: there is no direct mass term, and no mass can be generated through the Higgs fields since all Yukawa couplings o f gluinos involve colored boson fields which 417

Volume 78B, number 4

PHYSICS LETTERS

~ /

xc

i

f

sq ~

\

/

~

;r

xr

I

/

~N

\

\

;u

Fig. 1. One-loop diagrams generating a h-g"mass term (X and are Majorana spinors, so that h R = (hL)*, fR = (fL)*). cannot be translated if color is to be conserved. But a gluino mass may arise from quantum corrections which involve other fields of the theory. In other words, quantum effects may lift the gluon-gluino mass degeneracy which at the classical level persisted even after the spontaneous breaking of the supersymmetry. In the simplest versions of the models the generation of a gluino mass, which is in principle allowed since supersymmetry is spontaneously broken, is in fact forbidden by the existence of a set of R-transformations [7] acting as ?5-transformations on gluinos fields. A mass term will be allowed for gluinos as soon as R-invariance is broken, spontaneously or explicitly *a. Models of the latter type can be obtained easily from those given in ref. [2]" Explicit discussion of the gluino mass in these nroddls leads to the examination of several-loop diagrams #2. Since our purpose here is to demonstrate as simply as possible that ghiinos can be massive, we prefer to use a slightly different mechanism, apparently more complicated, but requiring the evaluation of simpler diagrams. In addition to the ghiinos we have already considered, now to be called orthogluinos, X, we introduce a second color-octet of spinorial particles which we call paragluinos, ~', with opposite R-transformation properties. Both X and ~"are octets of Majorana spinors. The construction of a massive Dirac gluino field from X and ~" is compatible with R-invariance. We just show how the relevant mass term connecting k to ~"emerges at the one-loop level. For this purpose we shall consider ,1 We shall not considere here the breaking of R-invariance due to gravitation, although it may also generate a gluino mass [8]. ,2 The breaking of both.supersymmetry and R-invariance is needed for the generation of a gluino mass. Therefore diagrams responsible for such a mass should involve at least " (i) particles for which the mass degeneracy due to supersymmetry is spontaneously broken at the classical level and (ii) masses or vertices breaking R-invariance. 418

9 October 1978

loops involving a heavy quark q with its spin-0 partners sq and tq (see fig. 1). We shall not identify q with one o t the usual quarks, owing in particular to their different transformation properties under the weakand-electromagnetic gauge group. Before evaluating the above diagrams we shall specify, for the reader familiar with the techniques of supersymmetry, the terms of the Lagrangian density which are relevant for our analysis. All the superfields involved here have already been introduced in ref. [2], except the (left-handed) superfield G describing the paragluinos ~"and their spin-0 partners. The gluons and the orthogluinos k are described by the (real) superfield V, the h e a w quark q and its spin-0 partners Sq, tq by the superfields Sq (left-handed) and Tq (right-handed). The definition of R-invariance given in ref. [2] is extended to the new superfield G in such a way that k and f have opposite R-transformation properties. Therefore under an R-transformation we have

V(x, O,-0) ~ V(x, 0e -ic~, 0eia), G(x, O, O) ~ G(x, 0e-i% 0eia),

(1)

Sq(x, O, O) -~ eiaSq(x, 0e -ia, 0e ia) , Tq(X, O, O) "-,"e-iC~Tq(X, 0e - i a , 0eic~) , so that the R-transformation properties of the fields appearing in the one-loop diagrams of fig. 1 are: X -~ e~S~X

Sq ~ e~Sq q-+ q

~"-+ e-TSa~"

(2) tq -+ e-i~tq

The Lagrangian density, constrained by R-invariance, can be, for example, the one constructed in ref. [2] but two more terms must be added owing to the existence of the new superfield G. The first one describes its kinetic energy and gauge interactions, the second one its super-Yukawa interaction with Sq and Tq. Let us write two terms in the Lagrangian density which are important for us;

.~= ... + [2mqT~qSq + 4hcT~qSqG]F.component.

(3)

The summation over color indices is understood, mq is the mass of the heavy quark q, and h c is the (super) Yukawa coupling constant which will appear in fig. 1 at the vertex of the paragluino f. In order to generate a mass term through the diagrams of fig. 1 we also need to break the mass degene-

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PHYSICS LETTERS

racy between q, Sq and tq. This will arise from the interactions of Sq and Tq with weak-and-electromagnetic gauge superfields. The auxiliary D-component of at least one o f the neutral gauge superfields (for example the one denoted V" in ref. [2]) has a non-vanishing vacuum expectation value, so that the mass 2 of Sq and 2 The existence o f the tq are shifted from the value mq. direct mass term 2mq [Tq~Sq] F-component in the Lagrangian density implies that Sq and Tq have the same gauge transformation properties. After the spontaneous breaking of the supersymmetry, this will result in the following mass relation (cf. ref. [9]) m2(Sq) + m2(tq) = 2 m 2 ,

(4)

implying in particular that Sq and tq have different masses ,3. After this digression defining more precisely the theory, we return to the loop diagrams o f fig. 1. We find the mass coefficient / . d4k

mq

m x f ~ gchc J (~g)4 k 2 + m 2

,(1 ×

k2+m2(tq)

1 k 2+m2(sq)

)

(5, '

in which gc (the color gauge coupling constant) and h c come from the X and f vertices, respectively. The relative sign of the contributions o f the two diagrams can be understood from the fact that m M. should vanish when m(Sq) = rn(tq) = mq: no gluon-gluino mass splitting can result from diagrams which do not reflect the spontaneous supersymmetry breaking. The integral, convergent, is easily evaluated. We find

mx ~ ~ gchc(mq/ A m 2) [m2 (Sq) log m2(Sq) + m2(tq)logm2(tq)-

2m2 logm2] ,

(6)

in which we have defined the mass 2 splitting Am 2 = Im 2 - m2(Sq)l = I m q2 -- m 2 ( t q ) l < m 2

(7)

We shall only need the approximate expression

m x f ~ gchc( Am2 /mq) .

(8)

4:3 It follows that parity is broken in the sector of the heavy quark q. Note, however, that the mass relation (4) does not apply for ordinary quarks which get their masses through Yukawa couplings with Higgs bosons.

9 October 1978

The most natural order o f magnitude for ~ m 2 / m q is the mass o f the intermediate boson W e but this is not necessary. In fact, owing to the arbitrary value of the (super) Yukawa coupling constant h c , 4 , the mass coefficient mx~. appears as arbitrary. The Majorana spinors X and f, now connected by an induced mass term, join to form a massive Dirac gluino field. However this is not the end of the story since the paragluino fields ~"are accompanied by two real octets of spin-0 fields which, presently, are also massless at zeroth order. We must consider the stability of the vacuum state we are using. Calculations performed at the one-loop level show that one of the spin-0 octets acquires a positive mass 2 but the other acquires a negative mass 2 , s , reflecting the fact that the vacuum state It is conceivable that theories similar to the one we consider here can be made invariant under an N = 2 extended supersymmetry algebra (hypersymmetry). Then ortho- and paragluinos are related by a global internal symmetry transformation, and the super-Yukawa coupling constant h c is no longer a free parameter, but is fixed by the gauge coupling constant gc. This can already be achieved for the part of the theory which describes the color gauge hypermultiplet (V, G) and the heavy quark hypermultiplet (So, Tq). 2 • The mass2 ma, m b2 of the spin-0 partners of paragluinos are due to loop-diagrams involving the heavy quark q and its spin-0 partners, Sq, tq. They can be evaluated in terms of the integrals fd4k( 1 + 1 J(2rr)4 k2-+m2(sq) k2+m2(tq) 21r)4

[k2+m2(sq)12

2 )

k2+m~t '

[k2+m2(tq)] 2

[k2+m~l 2 "

These are convergent, owing in particular to the mass relation m 2 (Sq) + m 2 (tq) - 2 m~ = 0 (note the similarity with PauliVillars regularization for loop-diagrams). It is remarkable although not surprising that the mass relation needed for the convergence of the above one-loop diagrams is precisely the one which constrains the mass spectrum of the spontaneously broken supersymmetric theory, at the tree approximation. The first integral also appeared in the evaluation of mx~- (see formulas (5) and (6)). We find mg = h 2 [m2 (Sq)log m 2 (Sq) + m 2 (tq) log m 2 (tq) 4n2 - 2rn~logm~] > 0 , 2

rn2a=mb + - ~ 2 - [log m 2 (Sq) + log m 2(tq) - 2 log m~] < 0, mg is ~hc(Am 2 2) 2/mq; 2 so is ma, 2 unless m2(Sq) or m2(tq) is • 2 accidentally small with respect to mq. -

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Volume 78B, number 4

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9 October 1978

is unstable and color is spontaneously broken. Such an effect, very interesting from the point of view of field theory, c a n n o t be tolerated here if color is to be conserved. Therefore we add to the Lagrangian density a new term (/a[ G 2 ]F-component in superfield n o t a t i o n ) so that it now contains a direct mass term for the paragluinos 1 ( - ~ i t ~ ' ) as well as for their spin-0 partners. This allows us to have positive mass 2 for b o t h of the latter. The v a c u u m state considered, for which color is conserved, is now stable (at least locally) , 6 . The term added to the Lagrangian density breaks R-invariance explicitly b u t softly (i.e., for terms of dimension ~< 3). However, since it has [RI = 2, it still preserves the discrete s y m m e t r y ( - 1 ) R called R-parity [3,4]. R-parity is such that the usual fields appearing in gauge theories are R-even, while the n u i n o and (ortho or para) gluino fields are R-odd. The conservation o f R-parity is i m p o r t a n t for the p h e n o m e n o l o g y of the new particles carrying R. The ghiino mass matrix n o w reads:

In particular, if/~ is large compared to rnx~ , the mass eigenstates are approximately the paragluino ~', with mass/~, and the orthogluino X, with mass m2X~/t~ ,7: the i n t r o d u c t i o n of the massive paragluino field ~ generates a mass for the orthogluino field X. In conclusion, the gluon/gluino mass degeneracy, which is persistent at the classical level w h e n supersymmetry is spontaneously broken, can be lifted by q u a n t u m corrections provided no 3'5"invariance forbids a gluino mass term , 8 . In the example we have described, weak-and-electromagnetic interactions induce a s p o n t a n e o u s breaking of supersymmetry, communicated to the pair gluon/gluino through a pair heavy quark/heavy spin-0 quark. Various other mechanisms based o n the same principles can also be constructed. In any case, the gluino mass is generally calculated in terms of several arbitrary parameters so that it is largely arbitrary. In the phenomenological search for R-hadrons b o t h the possibilities of massless gluinos, and massive gluinos of indeterminate mass, should be considered.

~"

I am very grateful to Dr. G.R. Farrar for m a n y helpful discussions.

+ m x f ( X 7 5 ~"+ ~75X) •

(9)

It leads to two non-vanishing masses for the gluinos:

(¼ 2 + m r)l/2 +

(lo)

,6 The positivity of the mass2 of both spin-0 octets (which requires ta > ~ (h c Arn2/mq) from the one-loop calculations of footnote 5) ensures that the vacuum state is a local minimum for the effective potential, but not necessarily an absolute minimum. In fact, at the tree approximation, the potential does have a minimum which is lower than the one of interest to us, for which some of the colored fields acquire large vacuum expectation values (-mq/hc) and color is spontaneously broken. Thus if the tree approximation can be trusted, the vacuum state we use is only metastable. But the length and height of the potential barrier can be made large enough so that our color-conserving vacuum state is effectively stable, since in many cases they are - mq/h c and

Inf ( la2rn~/h 2c,~m~/ h ~.

4-7 Since rnx~-has been computed at the one-loop level, we should also consider, in principle, the (one-particle irreducible) two-loop contributions to the orthogluino mass. Those, proportional to g~h2c~, cannot cancel the contribu2 22 22 2 tlon rnh~/~t ~ g~hc(~m ) /mqu. They can even be neglected when ~ ~ Am2/mq. •

420

,s The same considerations can be applied to the photon and its fermionic partner, the photino. The mass of the photino will vanish with the electrical charge. The Goldstino must stay massless (before the introduction of gravitation and its absorption in the massive gravitino field) if the Goldstone theorem is to be valid at higher orders.

References [1] P. Fayet, Phys. Lett. 64B (1976) 159; and in: Proc. 18th Int. Conf. on High Energy Physics, 1976 (Dubna, USSR), vol. 2, T8. [2] P. Fayet, Phys. Lett. 69B (1977) 689. [3] P. Fayet, in: Proc. of the Orbis Scientiae, Coral Gables, (FI., USA) Jan. 1978, to be published (Caltech preprint CALT 68-641). [4] G.R. Farrar and P. Fayet, Caltech preprint CALT 68-648, to be published in Phys. Lett. B. [5] T. Goldman, Los Alamos preprint LA-UR-78-1104. [61 G.R. Farrar and P. Fayet, in preparation. [7] P. Fayet, Nucl. Phys. B90 (1975) 104. [8] P. Fayet, Phys. Lett. 70B (1977) 461. [9] P. Fayet and J. Iliopoulos, Phys. Lett. 51B (1974) 461.