The properties of scalar quark bound states

Volume 158B, number 2

THE PROPERTIES

PHYSICS LETTERS

8 August 1985

OF SCALAR QUARK BOUND STATES

P. M O X H A Y , Y.J. N G 1 Institute of FieM Physics, Department of Physics and Astronomy, University"of North Carolina, Chapel Hill, NC 27514, USA and S.-H.H. T Y E 2 Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853, USA Received 5 April 1985

We discuss the (static) potential for scalar quark bound states, their leptonic and hadronic decays and their radiative transitions.

1. Introduction. Scalar quarks (spin zero colored particles) are predicted to exist in technicolor models 4:1, supersymmetric models 4:2 and preon models ,3. They have been invoked in the framework o f supersymmetry [4] to explain the " m o n o j e t " events [5] recently observed at CERN. But independent o f any particular model the study o f the phenomenology o f scalar quarks is o f interest in its own right 4:4 Such studies have been carried out b y Nappi [7] and others [8]. In this letter we present results o f the scalar quarkonium system which generalize and improve those previously published. These results will certainly help in the search for scalar quarkonium states. The parity and charge conjugation properties for scalar quarkonlum [P --- ( - 1 ) l, C = ( - 1 ) t] are differt Supported in part by the US Department of Energy and by the Alfred P. Sloan Foundation. z Supported in part by the US National Science Foundation. 4:1 For a review see ref. [ 1 ]. ,2 For a review see ref. [2]. 4:a For a review see ref. [3]. 4:4 If scalar quarks exist they will also add to an understanding of the spin-dependent forces between quarks [6] by providing the spinless case. 170

ent from those for quarkonium [P = ( - 1 ) t+l , C = ( - 1 ) l+S]. F o r scalar quarkonium j P C = 0++, 1 - - , 2 ++ .... for I = 0, 1 , 2 etc. It follows that the S-wave states, havingJ Pc = 0 ++ cannot be produced directly in e+e - annihilation , s . The P-wave states (jPC = 1 - - ) could be produced directly 4:6. Their observability depends on the size o f their leptonic- and radiativedecay widths; their signature will be a small bump in the R ratio accompanied b y a monochromatic photon o f energy of a few hundred MeV to one GeV. To discuss the bound states o f scalar quarks it is crucial to understand their potential. Since such bound states have not yet been observed, their potential can only be inferred by a comparison with the phenomenologically well-known potential for the heavy quarkonium. However, we know o f no a priori convincing reasons why the QCD radiatively corrected (static) potential for the scalar quarks should be 4:5 However, S-wave states can be produced in two-photon or two-gluon processes, see ref. [9]. 4:6 Scalar quarks are therefore produced in P waves in e*e - annihilation, making their thresholds hard to observe. Moreover, even for IPt ~" m the change in the (hadrons to/flu-) R ratio is only one quarter of that for a quark of the same charge. 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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the same as that for the quarks. We take up this issue in section 2 where we show that to one-loop order the two potentials are in fact the same. The rest of this letter is organized as follows. In section 3 we consider the leptonic decay of the Pwave states, in particular we give a formula for F(P -+ e+e - ) including the first order QCD radiative correction. We study the hadronic decays in section 4. In section 5 we discuss radiative transitions. Numerical results are contained in section 6.

2. Potential. The potential between a heavy scalar quark-and-antiquark pair in the color singlet state is expected to have a form similar to the potential proposed by Richardson [I0] or Buchmiiller-GrunbergTye (BGT) [11] for a heavy quark-antiquark pair V(q 2) = -4zrC2(R)as(q2)/q 2 ,

(1)

where as(q2 ) becomes the coupling in perturbative QCD for large q2. But a priori there is no reason why the scale parameters in the respective effective charges as(q 2) should be the same. To settle the issue we have to calculate the complete one-loop contributions to the static (scalar) quark-(scalar) antiquark potential. The result for the spin-l/2 case is well known [11, 12]. Here we report on the calculation for the scalar case. The Feynman diagrams which contribute to the potential (H) between a static scalar quark-antiquark pair are shown in fig. 1. The results for the individual diagrams given below refer to the Feynman gauge. We have used dimensional regularization [13] for ultraviolet divergences, and a small gluon mass X to regularize infrared divergences. The result reads ,7 H(a) = H 0 = - 47ra~0)C2(R )/q 2,

(2)

n ( b ) = (a~O)/aTr)H0 { [s C2(G ) _ _~T(R)Nf]

(Q)

?-?

(b)

(d)

(c)

(e)

Fig. 1. Contributions to the scalar quark-antiquark potential.

H(c) = (0t~0)12rr)H 0 C 2 (G) × {~ [21e - 3' + In 4rr + In 0221m2)] + ~ },

(4)

H(d) = 0x~0)/27r)H0 C2(G ) × {1 [2]e - 3' + In 4rr + ha (/.tZ/m2)] -- ha (XZ/m 2) - ~ },

H(e) = 2C2(R)C2(G)a(sO)2 q-2 In (q2/X2),

(5) (6)

where ot(°) is the unrenormalized strong interaction fine-structure constant (a s = g2]47r), m is the scalar quark mass, e = 4 - D for D space-time dimensions, 12 is the renormalization scale, q the modulus of the spacelike m o m e n t u m transfer, 3' = 0.5772 ... the Euler constant, N f the number of massless quark flavors, and in QCD the group factors read T(R) = 1/2, C2(G ) = 3, C2(R ) = 4]3. Individual terms are quite different from the spin-l/2 case. Summing eqs. ( 2 ) - ( 6 ) we obtain H = H 0 { 1 + t¢a(0)/4rr~ s J tfrla ~ ' g C 2 ( G ) - -~T(R)Nf)

X [2]e - 3' + In 47r - In (q2]/a2)] + -~ C2(G ) - -~ T(R)Nf},

8 August 1985

X ( 2 / e - 3 ' + In 4~)

(3) - In (q2//22) ( ~ C2(G ) - -~ T(R)Nf)

+ (-~ C2(G ) - ~ T(R)Nf)] }, ,7 The ultraviolet divergence of fig. le has been canceled by inserting a counter term having the form of a four-point contact term. Such a term has been considered for scalar QED in ref. [ 14].

(7)

which is identical to the (static) potential between a quark-antiquark pair. Therefore we conclude that the scale parameter for the cases of quarks and scalar 171

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8 August 1985

quarks are the same. Consequently we can apply the Richardson or BGT-like potential to the scalar quark bound states. In particular, given the scalar quark mass, we can obtain the scalar quarkonium spectrum by using the same potential as for quarkonium.

radiative correction factor for the leptonic width as (1 - 2asC3/rr ). This is the analog of the (1 - 16oq/ 370 factor for spin-l/2 triplet quarks, which is essential for obtaining precise values of the leptonic width of the T's. It follows that the leptonic width for a P-wave scalar quark bound state to order as is

3. Leptonic decay. The leptonic width for a Pwave scalar quark bound state at the tree level can be easily evaluated. It is given by

P(P ~ e+e - ) = 8G/tx2Q 2 [IR'p(O)I2/M 4 ]

P(P ~ e+e - ) = 8G r ot202 IR'p(O)12/g 4 ,

(8)

where M is the bound-state mass, R~,(0) is the derivative of the radial wave function at r = 0 [the complete wavefunction is ~0p = ~ riRp(r)/r], ~ is the fine structure constant and Q is the charge of the scalar quark, the color factor [15] reads, e.g., G.r = 3, 6, 8 for color 3, 6, 8 representation respectively. The result ofeq. (8) is in agreement with Nappi [7] for the 3 case. The order-a s correction to the P-wave leptonic width is analogous to the Karplus-Klein [16] correction to the annihilation of two spin-l/2 particles in an S-wave. It is obtained by calculating the vertex corrections shown in fig. 2. The form factor is [17]

F(o) = 1 -- (Ots/2rr)C3f(o),

(9)

where o, in the notation of ref. [17], is half the relative velocity of the scalar quark-antiquark, in the nonrelativistic limit f(o)-~ - zr2/2o + 2, the color factor [15] C 3 reads4~, -g, ao 9 respectively for color 3, 6, 8 representations. For the leptonic width we have IF(o)] 2 = (1 + rrOtsC3/2o) (1 - 2C~sC3/rr).

(10)

But the first factor is already incorporated in the bound state wavefunction. Therefore we identify the

X (1 - 2asC3fir ).

(11)

We note that the coefficient in the a s term in the correction factor (1 - 2%C3/r 0 is one-half that in the spin-l/2 case.

4. Hadronic decay. Theoretical estimates of the branching ratios of the bound states depend critically on the values of their hadronic widths. The hadronic width for an S-wave scalar bound state is given by I'(S -* 2g) = 2G2gO~2IRs(O)I2/M 2,

(12)

where R s(r) is the radial wavefunction [the complete wavefunetion is ffs = ~ Rs(r)] • The color factor G2g reads -~, ~, ~ respectively for the color 3, 6, and 8 representations ,s. Our result agrees with Nappi [7] for the 3 case. For the hadronic width of the P-wave bound states we have to consider the three-gluon decay mode shown in fig. 3. Fig. 3c is forbidden by charge conjugation invariance. As explained by Nappi [7], in the radiation gauge, the dominant contributions come from fig. 3a. The scattering matrix is

,~ / ~.~(P - p)~ c/~=--ie 3 e~e2e I [ (if-2--p~-~l

_p) (P-+

qJp/VA

¢

+ (cyclic permutations),

(13)

,8 [12~]defined.here is one-third of the G2g+3,defined in ref.

(a)

(b)

Fig. 2. Order-~s correction to scalar quark electromagnetic form factor. 172

(a)

(b)

(c)

Fig. 3. Three-gluondecay of a P-wavescalar quarkonium state.

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Vo!ume 158B, number 2

where e = ( x / ~ + p2,0) and p = (0, p), (cyclic permutations) stand for the other two terms from permutating the three gluons. Using ~bp = ~ riRp(r)/r for the P wavefunction we have •

]

t

e axpa~p = -lX/~exRp(O)/x/r~.

(14)

To sum over the polarization states it is convenient to average over the three spatial directions/':

8 August 1985

for the hadronic decay of the scalar quarkonium P state•

5. Radiative transitions• The radiative transitions from P to S or D states are given by F(P ~ 75) = ~ aQ2co 3 IOifl 2 , P(P -+ 7D) = ~

~(x~,/e/xe~J)=~.

(15)

Following Barbieri et al. [18] and Novikov et al. [19], we make the logarithmic approximation in the phase space integration f d k l dk 2 dk 3

(19)

aa2co 3 IOifl 2 ,

(20)

where o~ is the photon energy and the nonrelativistic dipole matrix element Dif , including finite size corrections, is Dif = (6/co)(S or

Dlfl(cor/2)lP).

(21)

In practice the finite size corrections are small, so we can use the approximation of the spherical Bessel function jl(cor/2) ~-- cot~6 to write

Jlk113 Ik2l Ik3[ ~(3)(kl ÷ k 2 + k 3) X 8(Ik11 + Ik2l+ I k 3 l - 2m) _ r dlk 1 [

2

= 8neJl--~l-> 8n In (M/A),

(16)

where A is a phenomenological parameter taken to be ~ (r) -1 ((r) is the average radius of the state). Putting in the color factor G3g , we obtain P(P -+ 3g) = (64fir)G3g a3

[[R,p(O)12/m4] In (M/A).

(17) The color factor G3g is ~s, ~s X ~ , 0 for color 3, 6, 8 representations [15] respectively. The result for the triplet case is smaller by a factor of 4n than the value in published literature [7]. Due to the crudeness of the logarithmic approximation, numerical estimates using eq. (17) will have large uncertainties. As a result the various branching ratios may have sizeable uncertainties. The logarithmic factor in eq. (16) is due to a soft gluon in the final state. The same kind of divergence is also found in the annihilation of a 3P 1 quarkonium bound state into hadrons. In the logarithmic approximarion the hadronic width• o f a 3P 1 state is [18] F(3P1 ~ hadrons)

"~ (128/3zr)(~/M4)lRp(O)12In(M/A).

(18)

If a quarkonium system lies near a scalar quarkonium system, one can apply eq. (18) to the quarkonium decay to extract the phenomenological parameter/x

Dif= f 0

drRs, D rRp.

(22)

Probably one of the best ways to detect scalar quark bound states is to look for monochromatic photons in the radiative transitions. This is certainly the case for heavy scalar quarks. Numerical estimates show that the radiative transitions dominate over the threegluon decay mode for heavy enough scalar quarks.

6. Numerical results. Some numerical estimates of the leptonic, hadronic, and radiative decays of scalar quark bound states are given in tables 1-3, based on eqs. (11), (17), and (19), assuming color triplets. The wavefunctions and bound state masses were taken from the potential model of ref. [20]. More precise-

Table 1 Leptonic and hadronic widths of 1P scalar quark bound states; m is the scalar quark mass and Q = 2/3. m (GeV)

Ri~(O)(GeVs/2)

re+e- (eV)

rha d (keV)

5 10 20 30 40 50

1.3 3.0 7.2 11.9 17.1 22.6

73 28 9.7 5.5 3.6 2.6

19.5 4.8 0.39 0.20 0.13 0.09 173

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Table 2 Hadronic widths of 1S scalar quark bound states. m (GeV)

RS(0) (GeV3/2)

rha d (keV)

5 10 20 30 40 50

2.6 4.5 7.6 10.3 12.8 15.2

3600 1900 1000 730 570 470

Table 3 Radiative widths of 1P scalar quark bound states; Q = 2/3. m (GeV)

Dif (GeV-1 )

F(P ~ 7S) (keV)

5 10 20 30 40 50

1.07 0.76 0.53 0.44 0.38 0.34

150 76 37 25 19 15

ly, we took the values which this model gave at the T mass (m = 4.9 GeV) and scaled them to higher masses assuming an effective power-law potential r v with v ~ 0. To calculate Phad, we took cts = 0.2 at the T mass and scaled it logarithmically, assuming A = 0.2 GeV. We took A = 1/(r), where
174

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We thank R. Robinett and J. Rosner for useful discussions. [1] See, e.g., E. Farhi and L. Susskind, Phys. Rep. 74 (1981) 277. [2] See, e.g., H.P. Nilles, Phys. Rep. 110 (1984) 1; see also K. Yamamoto, P. Moxhay, Y.J. Ng and S.-H.H. Tye, University of North Carolina preprint IFP-243/ UNC (1985). [3] See, e.g., R.D. Peccei, Max-Planck-Institut preprint MPIPAE/Pth 42/84; see also Y.J. Ng and B. Ovrut, Phys. Lett. 125B (1983) 147; Nucl. Phys. B233 (1984) 144. [4] See, e.g., J. Ellis and H. Kowalski, Phys. Lett. 142B (1984) 441. [5] UA1 Collab.,G. Arnison et al., Phys. Lett. 139B (1984) 115. [6] See J.L. Rosner, Enrico Fermi Institute preprint EFI84/33 (1984); see also Y.J. Ng, J. Pantaleone and S.-H.H. Tye, University of North Carolina preprint IFP-239/UNC (1985), and references therein. [7] C. Nappi, Phys. Rev. D25 (1982) 84. [8] G. Barbiellini, G. Bonneaud, G. Coignet, J. Ellis, M.K. GaiUard, C. Mateuzzi and B.H. Wiik, DESY report No. 7967, unpublished; S.-H.H. Tye and C. Rosenfeld, Phys. Rev. Lett. 53 (1984) 2215. [9] P. Moxhay and R.W. Robinett, University of Massachusetts preprint UMHEP-207,Phys. Rev. D, to be pubfished. [10] J i . Richardson, Phys. Lett. 82B (1979) 272. [11] W. Buchmiiller, G. Grunberg and S.-H.H. Tye, Phys. Rev. Lett. 45 (1980) 103,598(E); W. Buchm/iller and S.-H.H. Tye, Phys. Rev. D24 (1981) 132. [12] W. Fischler, Nucl. Phys. B129 (1977) 157. [13] G. 't Hooft, Nucl. Phys. B61 (1973) 455; G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1981) 189. [14] F. Rohrlich, Phys. Rev. 80 (1950) 666; L.L. DeRaad Jr., Phys. Rev. D l l (1975) 282. [15] Y.J. Ng and S.-H.H. Tye, Phys. Rev. Lett. 41 (1978) 6; H. Fritsch, Phys. Lett. 78B (1978) 611; H. Georgi and S.L. Glashow, Nucl. Phys. B159 (1979) 29. [16] R. Karplus and A. Klein, Phys. Rev. 87 (1952) 848. [17] J. Schwinger, Particles, sources and fields, Vol. II (Addison-Wesley,New York, 1973) pp. 390,427,428. [18] R. Barbieri, R. Gatto and E. Remiddi, Phys. Lett. 61B (1976) 465. [19] V.A. Novikov et al., Phys. Rep. 41C (1978) 1. [20] P. Moxhay and J.L. Rosner, Phys. Rev. D28 (1983) 1132.