Lepto-mesons, leptoquarkonium and the QCD potential

23 July 1998

Physics Letters B 432 Ž1998. 167–174

Lepto-mesons, leptoquarkonium and the QCD potential David Bowser-Chao 1, Tom D. Imbo 2 , B. Alex King 3, Eric C. Martell

4

Department of Physics, UniÕersity of Illinois at Chicago, 845 W. Taylor St., Chicago, IL 60607-7049, USA Received 19 March 1998; revised 8 May 1998 Editor: H. Georgi

Abstract We consider bound states of heavy leptoquark-antiquark pairs Žlepto-mesons. as well as leptoquark-antileptoquark pairs Žleptoquarkonium.. Unlike the situation for top quarks, leptoquarks Žif they exist. may live long enough for these hadrons to form. We study the spectra and decay widths of these states in the context of a nonrelativistic potential model which matches the recently calculated two-loop QCD potential at short distances to a successful phenomenological quarkonium potential at intermediate distances. We also compute the expected number of events for these states at future colliders. q 1998 Elsevier Science B.V. All rights reserved.

Particle physicists often lament the fact that the top quark is so short-lived. A long-lived top quark would have allowed us to study various aspects of the Standard Model ŽSM. that are otherwise inaccessible. For instance, had top quarks lived long enough to form bound states, they would have provided an exciting window into the nature of the strong interactions. But the SM top quark width, Gt ™ W b , grows as m3t for large m t , and for m t s 175 GeV is about 1.55 GeV. This corresponds to a lifetime of approximately 4 = 10y2 5 s – very short-lived indeed. However, there may still be a possibility of addressing these questions – not with top quarks, but with leptoquarks ŽLQs.. These hypothetical particles car-

1

E-mail: E-mail: 3 E-mail: 4 E-mail: 2

[email protected]. [email protected]. [email protected]. [email protected].

rying both lepton and baryon number appear naturally in many extensions of the SM. Some LQs may even be relatively light, with masses on the order of a few hundred GeV. ŽThere has been a recent flurry of interest in these models due to data from HERA w1x which initially seemed to hint at the existence of a 200 GeV leptoquark, although new results from the Fermilab Tevatron w2x make this scenario unlikely.. In what follows, we will restrict ourselves to models in which each leptoquark has at least one trilinear coupling with some ordinary quark-lepton pair. 5 Such leptoquarks will be color triplets having a spin

5 It is also usually assumed that a given leptoquark can only couple to quarks and leptons from a single generation – hence the terminology first generation leptoquark, etc. We instead make the less restrictive assumption that a given leptoquark couples to quarks from only one generation and leptons from only one generation, but the two need not be the same.

0370-2693r98r$ – see frontmatter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 6 3 6 - 4

168

D. Bowser-Chao et al.r Physics Letters B 432 (1998) 167–174

of either zero or one. Their widths, which only grow linearly with their masses, will be proportional to the square of the above unknown coupling strengths. These couplings cannot be too large otherwise leptoquarks would have already been seen. Indeed, for reasonably large masses and small couplings, leptoquarks will have widths between about 1 and 100 MeV. As we will see below, this leaves plenty of room for the formation and observation of leptoquark bound states, either with another leptoquark Žleptoquarkonium, LQ- LQ. or with a u, d, s, c or b quark Žlepto-mesons, LQ-q .. 6 It may even be reasonable to consider LQ-t states. The first step in identifying signals of these bound states is understanding their spectra. As with quarkonia, this is most easily done in a nonrelativistic potential model approach. Since leptoquarks are color triplets, the static QCD potential between an LQ and an LQ, or between an LQ and a q, is the same as that between a q and a q Žalthough since leptoquarks have integral spin, relativistic corrections such as hyperfine splittings will be different.. However, the region of the potential probed by our new bound states will be somewhat different than that probed by the known quarkonium states. Thus, we cannot blindly use a generic phenomenological potential which is tailored only to the latter, but must attempt to construct one more suited to our needs. Of course, this potential should be consistent with the known results from QCD as well as what we have learned from quarkonium studies. We will construct such a potential below Žin the spirit of w5x. and then use it to evaluate the energy splitting and the square of the wavefunction at the origin, < c Ž0.< 2 , for the 1S and 2S states of leptoquarkonium and lepto-mesons. This will allow us to calculate approximate widths and production cross sections, and estimate the expected number of events Žon resonance. for these states. What do we know about the global structure of the static QCD potential V Ž r .? At ‘‘short distances’’ it is quasi-Coulombic, with a perturbative expansion in the running coupling a s . The ‘‘long distance’’ behavior, describing quarks bound by tubes of

6

Previous mention of bound states containing leptoquarks can be found in w3,4x.

chromo-electric flux, is expected to be linear. There is currently no known analytic method to describe V Ž r . between these regimes. However, many phenomenological potential models have been developed to fit the cc and bb spectra which basically probe this region, all of which have a quasi-logarithmic form at these distance scales. Finally, we note that V Ž r . is known to be concave downward for all r; that is, V X Ž r . ) 0 and V XX Ž r . - 0 w6x. We first discuss the short range potential aV Ž m.

Vshort Ž r . s y 43 , where a V can be expanded r in terms of a s . Until recently, this expansion was only known to one loop in the presence of n f massless flavors of quarks w7x, although it was known to two loops in the absence of fermions w8x. However, the full two loop potential has recently been computed w9x. The result, of course, is dependent on both the renormalization scale Ž m s m Ž r .. and scheme. We will use the BLM method w10x for which Vshort Ž r . 16p

žž

sy 3r

qc2

ž

aMS Ž m ) .

/ ž q c1

4p

aMS Ž m ) ) ) . 4p

aMS Ž m ) ) . 4p

2

/

3

/

/

q... .

Ž 1.

Here the computation is performed in the MS scheme, and the scale at each order is chosen so that the coefficients c i are n f-independent. Combining this with the general results of w9x yields c1 s y8 and c 2 s 14r3 q 54p 2 y 9p 4r4 y 220 z 3 , as well as

m) Ž r . s

1 r

m) ) Ž r . s

ž

exp y g q 56 y 1 r

exp y

ž

p 2b 0 6

49 y 52 z 3 16

/

ž

aMS Ž m ˜. 4p

//

;

Ž 2.

for the first two BLM scales Žwhere g , 0.57722 and z 3 , 1.20206.. To two loops, the BLM method does not determine the scales m ) ) ) in 1 and m ˜ in 2. For

D. Bowser-Chao et al.r Physics Letters B 432 (1998) 167–174

simplicity, we choose both of these scales to be m ) ) . An approximate value for a MS at any scale m can be found from a reference value a MS Ž q . Žwe use a MS Ž m Z . s 0.118. and the following formula w9x:

aMS Ž m . s aMS Ž q . 1 q

q

q

ž ž 1 2

aMS Ž q .

m2

aMS Ž q .

ž

/ ž / b 0 ln

4p

2

q2

m2

/ ž / Ž . /ž ž / b 1 ln

4p

3

aMS q

m2

b 2 ln

4p

y b 0 b 1 ln

q2

2

m2

q2

y1

ž // q2

,

Ž 3.

which, like the potential, is complete to third order. Here

b 0 s 11 y 23 n f ;

b 1 s 102 y 383 n f ;

5033 325 2 b 2 s 2857 2 y 18 n f q 54 n f .

Ž 4.

This form of the potential is only valid at energy scales far above quark pair creation thresholds, where all n f quark flavors can be considered ‘‘light’’. Since this will be far from true in the applications that follow, we need a way to incorporate the effect of quark masses. A naive method would be to make n f a discontinuous function of m , increasing by one unit as each threshold is crossed. However, this is not very realistic. In the absence of a full two loop calculation with massive quarks, we shall instead model these effects in the manner suggested in w11x, letting n f be an analytic function of m : n f Ž m . s 2 Ý q Ž m2 mq m˜ 2q . . Here, the sum is over all quark flavors and the m ˜ q ’s are the current quark masses Žnot to be confused with the constituent masses used later.. As in w11x we choose m ˜ u s 0.004 GeV, m˜ d s 0.008 GeV, m ˜ s s 0.2 GeV, m˜ c s 1.5 GeV, m˜ b s 4.5 GeV and m ˜ t s 175 GeV. ŽOur final results are rather insensitive to these numerical choices.. We believe

169

this definition of Vshort Ž r . to be valid out to r , 0.3 GeVy1 . The ‘‘intermediate distance’’ range is the least understood Žfrom first principles. portion of the QCD potential. Running from the edge of the perturbative short distance region out to the long distance linear regime, it includes the quasi-logarithmic range Ž r , 0.9–5 GeVy1 . where the low-lying cc and bb states live. Since the properties of certain bound states containing leptoquarks will be sensitive to the transition between the perturbative and logarithmic regions, we want to connect them as smoothly as possible. To this end we would like an intermediate potential which attaches to our perturbative potential at some reasonable distance scale r a in such a way that V, V X and V XX are continuous at r s r a . We also require that V Ž r . does a reasonable job of fitting the known Žspin-averaged. cc Ž1S, 2S and 1 P . and bb Ž1S, 2S, 3S, 1 P and 2 P . states. We choose to match at r a s 0.2 GeVy1 , which is safely Žbut not too far. within the perturbative region. Since the first two derivatives of V will be continuous here, our results are somewhat insensitive to this choice. We choose the form of the intermediate potential to be a slightly modified version of the standard log potential w12x: Vint Ž r . s Aln

r

ž / r0

q aeyb Ž ryr a . .

Ž 5.

As in w12x we take A s 0.733 GeV. The constants r 0 , a and b are determined by our continuity criteria to be r 0 , 5.501 GeVy1 , a , y0.013 GeV and b , 49.30 GeV. The second term in 5 is non-negligible only in the region r , 0.2–0.4 GeVy1 , so we do not expect it to have much of an effect on the quarkonium states. As it stands, our potential does not have the appropriate linear behavior at ‘‘large’’ distances. We can, of course, fix this in any number of ways. However, we feel that this is unnecessary for our purposes since most of the states in which we are interested are not sensitive to this linear regime. The few states that we consider which have somewhat small reduced masses ŽQ 0.5 GeV. still lie mostly in the region well-described by the log potential Žat least in the 1S case – see, for example, w13x.. In any event, the whole nonrelativistic approach of this paper is suspect for these states. We only include

170

D. Bowser-Chao et al.r Physics Letters B 432 (1998) 167–174

Fig. 1. Our version of the static QCD potential.

them in order to show certain qualitative trends. 7 Our full potential ŽFig. 1. is analytic except at r s r a , and is everywhere concave downward. Using the above potential and the nonrelativistic Schrodinger equation, we have numerically com¨ puted the bound state energies and wavefunctions at the origin for the 1S and 2S states of lepto-mesons and leptoquarkonium ŽTable 1., for various constituent leptoquark masses between 300 GeV and 1 TeV. ŽWe also checked that V Ž r . fits the low-lying cc and bb states with approximately the same x 2 as the standard log potential alone.. This analysis also required choices for the constituent quark masses. For the c and b quarks we used those from w12x: m c s 1.5 GeV and m b s 4.906 GeV. For the top quark we used m t s 175 GeV, while for the lighter quarks we chose m s s 0.5 GeV and m u s m d s 0.35 GeV. The states LQ-urd and LQ-s are only presented for a leptoquark mass of 300 GeV, since the reduced mass of the system is basically independent of the leptoquark mass in the range of interest. The radii of the 1S heavy-heavy states in Table 1 ŽLQ- LQ and LQ-t . lie well within the perturbative regime of V Ž r ., ranging from approximately 0.02 to 0.05 GeVy1 as the reduced mass decreases. The 2S heavy-heavy states have radii from about 0.05 to

0.22 GeVy1 . Here we begin to see some sensitivity to the short-intermediate matching region for lighter leptoquark masses. The 1S LQ-b states all lie around 0.6 GeVy1 , and are also somewhat sensitive to the matching. All other states considered have radii of 1.2 GeVy1 and above, and thus mainly see the logarithmic part of the potential. ŽFor an alternative treatment of LQ-c and LQ-b states, see w4x.. Technically, the results in Table 1 represent spin-aÕeraged quantities since we have not included hyperfine effects in our potential. However, for scalar leptoquarks this statement is trivial since their leptoquarkonium states can only have spin-0 and their lepto-meson states spin-1r2. For vector leptoquarks there are hyperfine splittings. These are negligible for the lepto-mesons containing lighter quarks, but larger for the LQ-t states and for leptoquarkonium. However, they are still substantially smaller than the 1S–2S splittings ŽQ 0.1 D E2S -1S ., justifying our assumption that the static potential alone is a good starting point. For simplicity, in the remainder of this paper we will only consider scalar leptoquarks. Table 1 Values of < cn S Ž0.< 2 and the 1S–2S energy splitting for lepto-mesons and leptoquarkonium. All quantities are in appropriate GeV units m LQ

System

< c 1 S Ž0.< 2

< c 2 S Ž0.< 2

D E2S -1S

300

LQ-ur d LQ-s LQ-c LQ-b LQ-t LQ-LQ

0.020 0.035 0.190 1.285 1985 4574

0.010 0.018 0.097 0.649 308.5 640.8

0.589 0.589 0.589 0.593 1.346 1.751

500

LQ-c LQ-b LQ-t LQ-LQ

0.191 1.299 3072 1.819=10 4

0.097 0.656 445.8 2622

0.589 0.593 1.542 2.725

700

LQ-c LQ-b LQ-t LQ-LQ

0.191 1.305 3791 4.489=10 4

0.097 0.659 538.1 6776

0.589 0.593 1.649 3.604

1000

LQ-c LQ-b LQ-t LQ-LQ

0.191 1.310 4486 1.167=10 5

0.097 0.662 629.2 1.820=10 4

0.589 0.593 1.741 4.808

7

We have also ignored the effects of leptoquark-Higgs interactions in our potential. These very model-dependent couplings, even if large, would only affect the LQ-LQ Žor possibly LQ-t . states – and even then only if the Higgs mass is ‘‘small enough’’. We neglect the model-dependent leptoquark couplings to electroweak gauge bosons as well.

D. Bowser-Chao et al.r Physics Letters B 432 (1998) 167–174

Our analysis so far has only utilized the mass and spin of the leptoquark, along with the assumption that it transforms as an SUŽ3.c triplet. The study of leptoquark bound state production and decay, however, is more model-dependent; for example, gg ™ LQ- LQ involves the leptoquark’s electric charge. We follow w14x in assuming that at the mass scales considered, the SM is extended only by the addition of leptoquarks which respect baryon and lepton number and SUŽ3.c = SUŽ2.L = UŽ1. Y . The leptoquark isospin and hypercharge then determine its coupling to the SM gauge bosons, and to which quark-lepton pairs it may couple. We explicitly consider only one Žpresumably the lightest. scalar leptoquark. A degenerate weak isospin doublet or triplet of leptoquarks would not drastically change our conclusions, but could enrich the phenomenology Žby giving rise to the decay LQ– LQ™Wq Wy, for example.. Except in the case of lq ly™ LQ- LQ, we assume a purely chiral leptoquark coupling; if its trilinear coupling to a lepton of chirality i Ž i s L, R . and quark of opposite chirality is denoted by l i , then we assume that either l L / 0, l R s 0 or l R / 0, l L s 0 in order to satisfy magnetic moment constraints Žsee below.. For convenience, we write l i s e l˜ i , with e 2r4p s a em Ž m Z . s 1r128. To ensure that the bound state widths are substantially smaller than the 1S–2S splitting Žmaking the states easily identifiable., we require weakly coupled leptoquarks Ž l˜ 2i Q 1.. This leads to bound state widths of at most O Ž100 MeV.. Finally, we continue to ignore the model-dependent leptoquark couplings to W, Z and Higgs bosons – including the effects of partial widths such as LQ– LQ™ZZ would not greatly change our results. The production cross-section Žon resonance. of a narrow width bound state B from an initial state i, summed over all decay modes of the resonance w15x, is well-approximated by

stot s

ž

16p S MB2

/ž

GB ™ i G

/

.

Here, G and GB ™ i are the total and partial widths of the bound state, and the rest-frame total energy E is well within G of the resonance mass MB . In the particular case of unpolarized incoming beams, the spin-multiplicity factor S is given by NBrŽ N1 N2 ., with NB the spin-multiplicity of the resonance and

171

N1 , N2 those of the incoming particles comprising the state i. When the colliding beams are not monochromatic compared to G , we assume a gaussian distribution in E strongly peaked at the resonance mass MB with a spread of sE Ž sE R G .. At the energies considered,

stot s

16p S

p

GB ™ i

MB2

8

sE

ž / ž( / ž /

.

We assume that leptoquarks would first be discovered in non-bound state production, and that m LQ would be sufficiently well-determined to allow a search for the resonances considered here. We first treat lepto-mesons Ž MB , m LQ q m q .. Throughout, our notation will not distinguish between particle and antiparticle so that Žfor example. LQ-q represents LQ–q or LQ–q, and G LQ ™ e q represents G LQ ™ e " q or G LQ™ e " q. The principal lepto-meson decay channels should be the spectator decays of the quark Ž GB ™ ŽLQ. qXqX s Gq ™ qXqX . and leptoquark:

GB ™ l q q s G LQ ™ l q s

a em l˜ 2i 4

m2LQ y m2q

ž

m2LQ

2

/

Ž l s e, m ,t . .

m LQ

Ž 6. ŽWe will ignore leptoquark couplings to neutrinos as well as all lepton masses.. As mentioned earlier, for large m LQ the leptoquark decay width increases only as m LQ . The quark decay width GB ™ ŽLQ. qXqX is negligible except in the case of the top quark, where GB ™ ŽLQ. qXqX s Gt ™ W b . We will also need the very narrow width GB ™ l ag b s Ž1q ab . GB ™ l g for decay into 4

l g with helicities ar2 and b respectively: GB ™ l g s

Nc 16p =

ž

2

Ž 4paem l˜ i . Ql

MB

y

Qq mq

2

/

c Ž 0. MB m LQ

.

2

Ž 7.

Here Nc s 3 and Q is the electric charge Žin units of the proton charge.. The width GB ™ l g is inconsequential as a contribution to the total width, which is

D. Bowser-Chao et al.r Physics Letters B 432 (1998) 167–174

172

well-approximated by G s GB ™ ŽLQ. qXqX q GB ™ l q q . Over the mass range we consider, the LQ-t width GB ™ l t t ranges from l˜ 2i P 255 MeV to l˜ 2i P 1835 MeV, while for lighter quarks the LQ-q widths GB ™ l q q all range from about l˜ 2i P 586 MeV to l˜ 2i P 1953 MeV. We may now consider the production of lepto-mesons at an ey g collider Žfor non-bound state production, see w16x.. With an expected beam spread of sE s 0.05's f 0.05MB w17x, the cross-section for a polarized eyg collider is

stot , a b s

16p

p

GB ™ e ag b

MB2

8

sE

ž / ž( / ž

/

cay channel is not dominant. The two gluon or two photon Žwith helicities a and b . decay modes are more significant:

GB ™ g a g b s

GB ™ g ag b s

.

GB ™ gg s

The spin-multiplicity factor S has been specified implicitly in this case Ž S s 1. as is done below for the other cross-sections computed. For unpolarized incoming beams, S s 1r2. With the exception of LQ-t, for which the top decay width dominates, the 1S–2S splitting is larger than the width of the states for l˜ 2i Q 1r10. ŽFor LQ-t, there may still be the possibility of resolving the 1S–2S peaks for very heavy leptoquarks – see Table 1.. In Table 2 we present the expected number of events per l˜ 2i for the production of 1S states. Given a yearly integrated luminosity of up to 50 fby1 w17x, these resonances could be observable. Resolving these states, however, will be difficult in practice given the broad collider energy distribution and the non-trivial task of precisely reconstructing the lepto-meson given that there is a jet in the final state. Turning to leptoquarkonium Ž MB , 2 m LQ ., the partial width for the spectator decay of either constituent leptoquark is GB ™ l qŽLQ. s 2 G LQ ™ l q . For a very weak trilinear coupling Ž l˜ 2i Q 1r1000. this de-

Table 2 Estimates of the number of eventsr l˜ 2i for 1S lepto-mesons produced approximately on resonance at a polarized eyg collider. Masses are in GeV. We assume sE s 0.05MB and an integrated luminosity of 10 fby1 . The particle-antiparticle assignments shown in the column headings were chosen to maximize production m LQ

LQ-t

LQ-b

LQ-c

LQ-s

LQ-u

LQ-d

300 500 700 1000

72.36 26.63 12.24 4.979

13.58 3.127 1.171 0.410

98.51 21.31 7.769 2.665

40.36 8.765 3.202 1.100

194.5 41.98 15.30 5.246

48.11 10.43 3.807 1.307

da b

c Ž 0.

2p Nc

MB2

da b 2 1

2p

2 2 Ž 4pa s . ,

Ž 8.

GB ™ gg ;

Nc

c Ž 0.

2

MB2

2 4 . Ž 4pa em . Q LQ

Ž 9.

On the other hand, for l˜ 2i R 1r100 we have G , GB ™ l qŽLQ. . We can now discuss the production of leptoquarkonium at a gg collider. Assuming a gg center of mass energy resolution sE of 0.15's f 0.15MB w17x, the cross-section for unpolarized incoming photons is

stot s

16p MB2

(

p GB ™ gg 8

2 sE

.

Ž 10 .

ŽThere is an additional factor of 2 in this equation to compensate for the factor of 1r2 that was included in Eq. 9 to account for the identical photons in the final state.. Eq. 9 shows that production will be greatly enhanced or suppressed depending on the value of Q LQ Ž Q LQ s 1r3,2r3,4r3 or 5r3.. We have taken the most optimistic possibility, Q LQ s 5r3, in Table 3. It is interesting to consider a very small l˜ 2i , say l˜ 2i s 1r10000, where the dominant decay width Ž; 1.7 MeV for m LQ s 300 GeV. is into gluon pairs. In this case the gg decay channel, with a partial width of 0.45 MeV, has a branching ratio of about 1r5. The resulting narrow resonance in gg ™ gg will stand out in a spectacular way over the continuum background w18x, which should be more than an order of magnitude smaller Žat somewhat less than 1 fb within a reasonable mass bin.. Similar results are obtained even for Q LQ as small as 2r3. Finally, for leptoquarkonium production at an eqey or mqmy collider we will need the decay

D. Bowser-Chao et al.r Physics Letters B 432 (1998) 167–174 Table 3 Estimates of the number of events Žon resonance. at polarized mqmy and unpolarized gg colliders for 1S leptoquarkonium. Masses are in GeV and the integrated luminosity is 10 fby1 . In the mqmy case, we assume that sE - G , Q LQ s1r3, and LQ couples to m t with l˜ 2L s l˜ 2R s 0.1. In the gg case, we take sE s 0.15MB and Q LQ s 5r3 m LQ

mqmy ™LQ-LQ

gg ™LQ-LQ

300 500 700 1000

578.0 52.55 12.05 2.568

853.8 264.0 121.1 52.91

width into a pair of charged leptons Ž lq ly . with helicities ar2 and br2:

GB ™ lqa lyb

173

sion, we have constructed a version of the static QCD potential which matches Ža slightly updated form of. the recently calculated two-loop perturbative piece at short distances to a successful phenomenological quarkonium potential at intermediate distance scales. We then used this potential to determine the spectra and decay widths of bound states containing heavy leptoquarks; namely, lepto-mesons and leptoquarkonium. The expected number of events for these states at future colliders were also estimated. For reasonable values of leptoquark masses and couplings, many of these exotic hadrons should be observable. Note added. After the completion of this work, we noticed that a similar version of the static QCD potential has recently been constructed in w22x, although used for different purposes.

2

s d a b 32p Nc ž a em l˜ qa l˜ ya /

2

c Ž 0 . m2q

Ž MB2 q 4 m2q .

2

. Acknowledgements

Ž 11 . ŽFor a review of non-bound state production, see w19x.. The cross-section for a polarized lq ly collider operating at the resonance is then

stot , a b s

16p

GB ™ lqa lyb

MB2

G

ž /ž

/

,

which vanishes unless we relax our condition that the coupling be purely chiral; in fact, it is maximized if we take l˜ L s l˜ R Žas in Table 3.. The maximal cross-section will be for an LQ that couples to l t, which we assume here. Since the next generation eqey colliders are expected to attain at best sE , 0.01's 4 G , and the measured electron magnetic moment requires an extremely small value for l˜ L l˜ R w20x Žand thus a tiny partial width GB ™ eq ey ., production at these machines would be heavily suppressed. By contrast, proposed muon colliders could attain sE , 2.1 = 10y5 's w21x. Furthermore, the constraints from the muon g y 2 are much looser, allowing l˜ 2L s l˜ 2R as large as 0.14 for m LQ G 300 GeV. Results for mq my™ LQ– LQ may be found in Table 3. In this case, the useful decay mode would most likely be B ™ my q mq q Žwith a branching ratio of 0.97 for m LQ s 300 GeV. so that this channel would not only be visible, but might also have the potential of resolving the 1S–2S splitting. In conclu-

We thank Markus Peter for clarification of his results. We also thank Mark Adams, Russell Betts, Adam Falk, Ben Grinstein, Wai-Yee Keung, YoungKee Kim, Julius Solomon and Uday Sukhatme for helpful discussions. This research was supported in part by the US Department of Energy under Grant Number DE-FG02-91ER40676. References w1x H1 Collaboration, C. Adloff et al., Z. Phys. C 74 Ž1997. 191; ZEUS Collaboration, J. Breitweg et al., Z. Phys. C 74 Ž1997. 207. w2x D0 Collaboration, B. Abbott et al., Phys. Rev. Lett. 79 Ž1997. 4321; CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 79 Ž1997. 4327. w3x J. Blumlein, DESY preprint 93-153, contribution to the ¨ Proceedings of the Workshop eq ey Collisions at 500 GeV, Munich, Annecy, Hamburg, 1993; C. Friberg, E. Norrbin, T. Sjostrand, Phys. Lett. B 403 Ž1997. 329; J.L. Hewett, T.G. ¨ Rizzo, hep-phr9703337, version 3. w4x V.V. Kiselev, hep-phr9710432. w5x K. Hagiwara, A.D. Martin, A.W. Peacock, Z. Phys. C 33 Ž1986. 135. w6x C. Borgs and E. Seiler, Comm. Math. Phys. 91 Ž1983. 329; C. Bachas, Phys. Rev. D 33 Ž1986. 2723. w7x A. Billoire, Phys. Lett. B 92 Ž1980. 343. w8x W. Fischler, Nucl. Phys. B 129 Ž1977. 157. w9x M. Peter, Nucl. Phys. B 501 Ž1997. 471; M. Peter, Phys. Rev. Lett. 78 Ž1997. 602.

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