Volume 70B, number 3 PHYSICS LETTERS 10 October 1977

S U M R U L E S F O R T H E M A S S E S O F T H E

2P- A N D H I G H E R L E V E L S O F C H A R M O N I U M

Virendra GUPTA Tara Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

and

Avinash KHARE Tara Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

and Institute of Physics ~, Bhubaneswar 75100 7, India

Received 5 July 1977

We derive a sum rule relating the masses of the four charmonium levels, for each L i> 1, which is valid for a wide class of potentials. The 2P-level sum rule predicts the 1p1 mass to be 3561 -+ 16 MeV. A phenomenological quark con- fining potential is suggested on the basis of the 3pj splittings.

The charmonium model [1] of the narrow vector mesons [2] 4(3.105) and t f (3 .684) as bound states of the charmed quark c and its antiquark ~ predicts a spectrum of (c~)-states. In particular four P-levels are predicted to lie between the ff and 4 ' and it is gratify- ing that the three of the four X-states detected [3] in ~ ' ~ X + 7 at 3413 -+ 5, 3510 + 4 and 3554-+ 5 MeV can be interpreted [4], in the model as the 3P0(0++ ), 3p1(1++ ) and 3P2(2++ ) levels respectively. However, the specific potentials [5, 6] used to calculate the spectrum of the non-relativistic (c~)-system, have not been able to account satisfactorily for these observed splittings between the three 3pj ( j = 0, 1 ,2) levels. The fourth P-level 1 P I ( I + - ) has not been detected so far and generally ignored theoretically since it is not so easily accessible as the 3pj levels. Nevertheless, a prior knowledge of its precise mass may be crucial in its experimental detection. With this in view we show that for a wide class of potentials it is possible to de- rive a sum rule between the masses of the four P-levels which predicts the 1 P1 mass once the masses of the 3pj levels are known. The extension and application of the sum rule to higher orbital levels and other quark- antiquark systems is also briefly discussed. Finally, we suggest a phenomenological potential which may help in understanding the observed splitting of the 3pj levels.

* Permanent address.

I. The Potential: Most of the specific potentials used to obtain the charmonium spectrum are particu- lar cases of the potential

(1) --]--1 [ -U" ( r ) + 1 U'( , ) ] T12 + 2-2-

+ 12m2 3m 2 S1 "S 2 [V2U(r)]

where m c is the mass of the charmed quark, T12 = - 4 S 1 "S 2 + 12(S 1 "~:)(S2"tZ)is the tensor operator, L and S = S 1 + S 2 are the orbital angular momentum and total spin operators. The spin-independent part Vo(r ) (which may contain orbital dependence) gives the common mass while the last three spin-dependent terms treated as a perturbation, split the degenerate states to give the levels of ortho- and para-charmonium. Further, in general Vo(r ) may contain other terms in addition to the spin-dependent generating potential U(r). It is clear that Vo(r ) must contain the quark con- timing potential though U(r) may or may not have such a term. For example, the early calculations [5] used Vo(r ) = -a s / r + ar with U(r) = -as~r, so that the spin-dependence came only from the gluon exchange potential and not the quark confining linear potential. Later it was suggested [6] that the quark confining potential also contributes to the spin-dependence in (1). In fact Schnitzer [6] used Vo(r ) = U(r) = (ar

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- as/r + constant) to obtain bounds on the ratios of the mass splittings of the 2P-states, which are not obeyed experimentally. We thus consider the general potential in (1) and show that a sum rule exists among the masses of the four levels for each L ~> 1 for arbi- trary Vo(r ) and U(r)provided they behave as < 1/r 2 and <~ 1/r respectively at the origin. These conditions ensure that the S-state wave functions at the origin and the expectation value of V2U(r) for S-states are finite. We first derive the sum rule for the 2P-levels.

2. Sum Rule. Denote the 3pj masses by Mj (J = 0, 1,2) and the IP 1 mass byM~ and let mv be the average mass of the 2P-levels. Then treating the spin dependent terms to first order, we have

1 ( 2 a ) M 0 =mp - 2ap - 4bp +~Cp,

' ( 2 b ) M 1 = m p - a p + 2 b p + ~ C p ,

2 t ( 2 c ) M 2 = mp +ap - ~bp +~Cp,

, 3 ( 2 d ) M 1 =mp - zep,

where the expectation values ap etc. are given by

ap = ( 3/2m2)(r - 1 U'(r))p

bp = (1/12m2)(-U"(r) + r- lU'(r))p

Cp = ( 2]3m2c)( V2 U(r))p.

(3a)

(3b)

(3c)

Since U(r) is no more singular than 1/r, (V2U(r)~, = (lf '(r) + r-2U'(r))p. This holds for all levels with L >I 1 as their wave functions vanish at the origin as r L . Eliminating the two unknown expectation values (r -1 U'(r)) and (U"(r)) one obtains

3c e = 4(a e - 6be). (4)

From eq. (2) since four masses are now given in terms of only three unknowns by virtue of eq. (4), one finds the sum rule

9M~ = 2 1 M I - 5 M 2 - 7 M 0, (5)

for the 2P-levels. Using the values [4] M 0 = 3413 +-- 5, M 1 = 3510 +4 andM 2 = 3554 + 5 we predict M{ = 3561 -+ 10 MeV for any potential of the type in eq. (1). Knowledge of its mass may help experiment to de- tect the 1Pl'State through its dominant decay modes,

+ t - - - - • •

namely, ~c ~', r/c + "r, Pp and KK1r. It is Important to identify the 1Pl-level since the sum rule (5) would

then provide a test of whether the underlying effective poter~tial is of type (1) without going into a morass of numerical computation.

It should be noted that for some special choices of U(r), namely U(r) = rn/a n+l (n > / - 1 , n 4= 0) there is only one unknown expectation value so in addition to (4) one has n(2 - n) ap = 18 bp which gives in addi- tion to (5) the sum rule for the 3pj masses given by Pumplin et al. [6] which is not satisfied by the experi- mental masses for n >t - 1 .

It is easy to generalize the sum rule (5) for arbitrary L >~ 1. Denote byML,ML± 1 the masses of the three orthocharmonium states and by M~ the mass of the paracharmonium level for a given L, then using eqs. (1), (3) and (4) we obtain the sum rule

3L(L + 1)(2L + 1)(M~ - M L ) (6)

= 2(6L 3 +9L 2 - L - 2)M L - ( 6 L 3 +7L 2 - 3 L ) M L _ 1

- ( 6 L 3 + I lL 2 + L - 4)M L_I .

Specializing to the 3D-states, we obtain

9M(1D2) = 25M(3D2) -7M(3D3) -9M(3D1) . (7)

This sum rule may be of interest in view of Harari's conjecture [7] that the even charge conjugation state detected at 3.45 GeV may not be rT'c but is the 31D2 and further that 33D 1 is expected to lie between 3.75 and 3.80 GeV [8]. If so then the sum rule (7) tells us that the 3D 2 level cannot lie above both the 3D 1 and 3D 3 levels.

One may consider the application to other quark- antiquark systems since our sum rules are independent of the quark mass, the strength of the quark-gluon cou- pling constant and the form of the confining potential. A potential approach to the (SS) bound states has been considered by various authors [9]. It is amusing to compare the sum rules (5) and (7) with the calcula- tions of Lichtenberg and Wills [9] whose potential is of type (1), even though they treat the L .S coupling non-perturbatively. For the 2P-levels, with the identifi- cation of 3P 0, 3P 1 and 3P 2 with S*(993 ± 5), D(1286 + 10) and f'(1516 +- 3) respectively, (5) would predict the IP 1 mass to be 1.388 -+ 30 MeV while they obtain a mass of 1.414 MeV. Further, using their calculated masses of the 3D-levels we find that the sum rule (7) checks within a few per cent.

3. Splitting o f the 3pj levels: The splitting of the

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2P-levels will clearly provide a clue to underlying ef- fective potential and with this in view Schnitzer [6]

1 ~<0 obtained bounds 0.8 ~<R 1 ~< 1.4 and - 3~<R 2 where the mass splitting ratios R 1 and R 2 are given by

M 2 - M 1 M 1 - M] R 1 - M I _ M 0 , R 2 = M I _ M 0 . (8)

However with the identification of the 3pj levels men- tioned earlier, experimentally R 1 = 0.45 which is be- low the bound. If we use value M~ = 3.561 GeV pre- dicted by sum rule (5) then R 2 = - 0 . 5 2 which is also below the bound. It should be remarked that the Schnitzer bounds on R 1 and R 2 are for the particular potential

Vo(r ) = U(r) = c + ar - O~s/r (9)

where c is a constant and a and as are positive. The rea- son why the bounds based on this potential are not satisfied is that ap, bp and Cp are positive definite for this potential and further 6bp ~< ap ~< 18bp indepen- dent of the values of c~ s and a; while experimentally, from (3a)-(3c) , one finds ap = 34.5 MeV, bp = 10.4 MeV and use o f (4 ) yields Cp = -37 .2 MeV for the po- tentials of the type (1). To achieve agreement with ex- periment one has to modify the potential in (9). Phys- ically the quark-antiquark potential is expected to have two parts (i) a short range Coulomb-like part together with spin-dependent terms generated by it and (ii) a part containing the quark confining potential. The choice of the short-range part as arising from quark- gluon interaction is on a rather firm basis but it is not clear w/tat the correct confining potential should be. The earlier attempts [5] used a linear potential with no spin-dependent terms as suggested by lattice gauge theory. Later it was argued [6] that the confining po- tential also generated spin-dependent terms in a man- ner similar to the Coulombic part thus motivating the choice (9) in (1). Though this resulted in larger split- ting among 2P-levels it still does not agree with experi- ment. It is clear therefore that the confining potential needs modification. So on phenomenological grounds we suggest that in (1) we choose

Vo(r ) = U(r) = c - o q / r - a r + ½ b r 2 , a , b > 0. (10)

Note the negative sign of the linear term in contrast to the potentials used so far. This is necessary as a ~< 0 will give ap t> 6bp contrary to experiment. The last two terms will act like confining potential for r > 2a/b .

Furthermore, depending on the parameters in (10), ap, bp , Cp are no longer positive definite and in fact the ratios R 1 and R 2 are not constrained any more. For purpose of illustration, we choose m c = 1.6 GeV, a s = 0.2, a = 0.2 (GeV) 2 and further estimate (I/r) and (1/r 3) using three-dimensional harmonic oscillator

• 1 2 wave functions for a potentlal-ffKr . Then the harmon- ic oscillator parameter ~ = ( ½ m c K ) l / 4 and b can be determined from the experimental values of ap and bp. Forap = 34.5 MeV and bp = 10.4 MeV one obtains

1.015 GeV and b = 0.054 (GeV) 3. It is interesting to compare these numbers with

Harari's suggestion [7] that the 3P 1 state may be at 3.455 GeV instead of 3.510 GeV and 3P 0 and 3P 2 masses as before. With this choice sum rule (5) pre- dicts the 1 P1 to lie at 3.433 GeV i.e. below the level 3P 1 . Furthermore for this choice R 1 = 2.357,R 2 = +0.52 which lie above the Schnitzer bound in con- trast to the case discussed above where they lay below the bound. The experimental values of ap = 48.25 MeV, bp = - 1 MeV and Cp = 72.3 MeV for this choice with m c, a s and a as above, lead to ~ = 0.25 GeV and b = 0.1176 (GeV) 3 .

The above discussion shows that with the potential given by (10) it is possible to explain the splitting of the 2P-levels. Whether it can simultaneously explain tp, if' etc. requires a numerical solution of the Schr6dinger equation and will be reported elsewhere. However, what we wish to emphasize here is that the importance of the experimental detection of the 1P 1 state since this together with the knowledge of the 3pj states can provide an important clue to the nature of the spin-dependence of the quark confining poten- tial.

One of us (A.K.) is grateful to Prof. V. Singh and other members of the T.I.F.R. Theory Group for warm and cordial hospitality.

References

[1] T. Appelquist and H.D. Politzer, Phys. Rev. Lett. 34 (1975) 43; Phys. Rev. D12 (1975) 1404; A. De Rujula and S.L. Glashow, Phys. Rev. Lett. 34 (1975) 46; S. Borchardt et al., Phys. Rev. Lett. 34 (1975) 38.

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[2] J.J. Aubert et al., Phys. Rev. Lett. 33 (1974) 1404; J. Augustin et al., Phys. Rev. Lett. 33 (1974) 1406; C. Bacci et al., Phys. Rey. Lett. 33 (1974) 1408; G. Abrams et al., Phys. Rev. Lett. 33 (1975) 1453; J. Augustin et al., Phys. Rev. Lett. 34 (1975) 764.

[3] W. Braunschweig et al., Phys. Lett. 57B (1975) 407; G.J. Feldman et al., Phys. Rev. Lett. 35 (1975) 84; W. Tannenbaum et al., Phys. Rev. Lett. 35 (1975) 1323.

[4] Particle Data Group, Phys. Lett. 68B (1977) 1. [5] For an excellent review and other references see

J.D. Jackson, LBL-5500 (1976) to be published in the Proc. Summer Institute on Particle Physics, SLAC, Stanford, California; E. Eichten, K. Gottfried, T. Kinosbita, J. Kogut, K.D. Lane, and T.M. Yah, Phys. Rev. Lett. 34 (1975) 369;

R. Barbieri, R. Gatto, R. KiSgerler and Z. Kunszt, Nucl. Phys. B105 (1976) 125.

[6] J. Pumplin, W. Repko and A. Sato, Phys. Rev. Lett. 35 (1975) 1538; H.J: Schnitzer, Phys. Rev. Lett. 35 (1975) 1540.

[7] H. Harari, Phys. Lett. 64B (1976) 469. [8] A.B. Henriques, B.H. Kellett and R.G. Moorhouse, Phys.

Lett. 64B (1976) 85; H.J. Schnitzer, Phys. Rev. D13 (1976) 74.

[9] J.F. Gunion and R.S. Willey, Phys. Rev. D12 (1975) 174; D. Lichtenberg and J.G. Wills, Phys. Rev. Lett. 35 (1975) 1055; A. De Rujula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147.

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