experimentally favoured potential for mesons

Z. Phys. C - Particles and Fields 27, 315-319 (1985) Zeitschnlt P a r t i c ~ ~ir Physik C

and �9 Springer-Verlag 1985

Experimentally Favoured Potential for Mesons

D. Olivier*

CERN, CH-1211 Geneva 23, Switzerland

Received 28 June 1984; in revised form 24 August 1984

Abstract. We show first that a natural potential to be used in a semirelativistic wave equation for the S = 1, J = Jmax light and heavy mesons is, for r > 1 Fermi, V ( r ) = a r + b / r with b ~ - - n / 2 , for r < l Fermi, V(r) = cln(r/ro). Next we show that this particular b value favours in fact a QCD-like potential for r < 1 Fermi so that a logarithmic potential is only an effective one in a small r region.

Several years ago, Quigg and Rosner [1], motivated by the apparent equal spacing M ( Y ' ) - M ( F ) = M(~0')- M(O) showed that a natural potential to be used in a Schr6dinger equation describing heavy quarkonia must be V(r) = c ln(r/ro) in a suitable range of r. Here we will somehow extend their analysis by showing that a natural parametrization of a potential to be used in a semirelativistic wave equation [2], which is suitable to describe light and heavy mesons, is

r > l f V ( r ) = a r + b / r

r < l f V ( r ) = e l n ( r / r o )

where the three constants a, b and c have a direct experimental interpretation, namely a is, as is well known, related to the common slope of the Regge trajectories of the p, K* and ~b mesons, b is related to the intercept of the p trajectory and c is, as recalled above, related to the spacing of radial excitations in the charmonium and bellonium spectra, r o, on its side, provides a dimensionless argument for the logarithm and may simply be adjusted to lead a smooth behaved V(r) for r around 1 fermi. Moreover, a and b will both be seen to have a theoretical interpretation which, in

* Present address: LPTHE, Universit6 P. et M. Curie, 4 place Jussieu, F-75230 Paris, France

turn, will suggest a QCD-like potential at short distance so that the logarithmic behaviour for r < 1 f will finally be considered only as a convenient effective one.

The paper is organized as follows: firstly, we recall the well-known structure of the spectra of light and heavy mesons with S = 1 and J = Jmax; secondly, using light quarkonia we derive the potential for large r; then, using heavy quarkonia, we recover the Quigg and Rosner result in the context of a semi-relativistic equation and finally we discuss our results in the light of a model recently proposed.

I. Structure of the Mesonic Spectral Data

We will concentrate on the mesonic states with spin S = l and total kinetic momentum J = J . . . . i.e., J = l + 1 where l is the orbital momentum.

We recall in Fig. 1 the Regge trajectories of the p, K* and ~b mesons. These trajectories appear to be linear to a good approximation, to share approxi- mately the same slope and to be nearly equally spaced. From Fig. 1 we deduce the value of the common slope to be 7 = 1.2 GeV z, the intercept of the p trajectory to be tip = - 0.45, the spacing between the trajectories to be 6 = -0 .21 . We wrote E 2 = 7 ( / - f l - n 6 ) n = 1 or 2 so that 7 = 1/~' and fl = lp(0) in the usual notat ion where I(E) = o~' E 2 + l(0) + n3.

In Fig. 2 we recall the approximate equal spacing between the three first radial excitations of the charmonium ~k and of the bellonium F. Note that we have scaled the r and 1 c spectra to coincide with the first two levels of the non-relativistic Hamiltonian - (d2/dx 2) + In Ixl, For that Hamiltonian we have E 2 - E 1 = 0.703 [3].

These two graphs are all we need to derive the parametrization of V(r) mentioned in the introduction as we shall see now.

316 D. Olivier: Experimentally Favoured Potential for Mesons

6 ( 2 4 5 0 ~ ~ e,s. " k NEAR LE_AST SCI.~RFS FIT ..... ,. ~ r ,x "

...... CHOEEN RECE.,E TRAJECTORY 9u~.~u ; t ~ - _ .~\ ,~ , ~

1, o|

3 ~ ? )

q' K*;| I I I I I [ I I l .t-

O 1 2 3 4 $ 6 H z, GeV 2

Fig. 1. Experimental results drawn on this figure are from the Rosenfeld Tables [14]. The first daughter trajectory is seen to emerge around 2.7 GeV 2

,,d t..030 . . . . . .

2.596 lO.5(19

z.z9f~ to.3tn

3.68_.~.__ . . . . . . t.8~ 7 lo.o16

q0 : [harmonium Htog I : Bellonium

t.o~ 9.~s~ ~ch.1.36 1.43, 1.00, , 1.4~1

Fig. 2. Experimental results are from [14]. We used H L o g =

-(d2/dx2)+lnlxl and A(Y)• d indicates the energy difference between the first radial excitation and the ground state. The numbers written above the horizontal axis (which has no meaning) are the respective values of the scale parameter Ech for each case

II. Behaviour of the Potent ia l for Large r

The semirelativistic equation for the light mesons we are interested in writes

[x/p 2 + m~+ x /p2+ m2 2 + VL(r)]O(r)=EO(r) (1)

where the m~ are the constituent quark masses and p is the momentum of quark 1 in the centre-of-mass frame.

We will consider first the rn 1 =m2 = 0 case. In that case, imposing linear Regge trajectories leads heuristically to V (r) = ar + b/r.

Indeed, if we postulate first V(r) = ar ~ and approxi-

mate [thus neglecting terms of order O(1/xfi)]*

J l ( l + l ) 1 / x / ~ 1 ) 2 x / p 2 = 2 P~+ r ~ " r

for large enough l, thus neglecting the radial kinetic term [p2] as compared to the centrifugal barrier, we are led to

<r>=r,. L av-

where r,, realizes the minimum of the effective potential Veff = [ 2 x ~ ( / + 1)]/r + ar ~ and then to

which in turn gives linear leading trajectories for v = 1. Such an argument to obtain the long-range potential was used by Grosse and Martin in [4]. Explicitly we have E2..~8a(t+�89 for VL(r)=ar. We expect the slope 8a to be in good agreement with the exact slope as the approximations used should be accurate for high enough l, but the particular intercept - � 8 9 which involves low l may be only a crude estimate. Next it is clear that adding a term + b/r to ar just translates the Regge trajectories by b/2 leading to the most general linear relation

E~ ~- 8a(l + b/2 +�89

in the same approximation. The above heuristic reasoning may be completely justified by explicitly computing the eigenvalue of (1) with V(r)= ar which

* This was noticed by A. Martin

D. Olivier: Experimentally Favoured Potential for Mesons

Table 1. Eigenvalues table of the Hamiltonian (p + l/2rrr)~p(r)= EN,L~b(r)

E Eigen-energies

N L 0 1 2 3

1 0.8908(0.79) 1.1917(1.42) 1.322(2.05) 1.639(2.69) 2 1.3287(1.77) 1.5393(2.37) 1.729 (2.99) 1.902(3.62) 3 1.6615(2.76) 1.8286(3.34) 1.988(3.95) 2.138(4.57) 4 1.9386(3.76) 2.0803(4.33) 2.219(4.92) 2.352(5.53) 5 2.18l (4.76) 2.306 (5.32) 2.43 (5.90) 2.55 (6.50)

~ N.,.k(NI L+~.p(N) L ~ N+~-+3/4 N=0

3 -. BKW / / o AR,EXACT / / /

/

/

o

N=I N=2 N=3 N=~

o, o' �9

~ 5 6

/ A [ ~',gl= 0,28 GeV 2 Ez' GeV2

-1 tl./15 AExP[ 9"g)= 0.30 GeV l

/

Fig. 3. Exact the BKW eigenvalues of the Hamiltonian

(p + 1/27tr)lp(r) = EN,Lff(r )

in turn will provide us with the true value of the intercept.

The results are summarized in Table 1 and Fig. 3 where we have also reported the BKW Regge trajectories [5].

This Table and this graph correspond to the scaled equation:

Ip+~--Tr]~b=Eg, (2)

which was numerically solved by ttie following method:

1) we Fourier-transformed (2); 2) we used the Multhopp method [6] to transform the singular integral equation thus obtained to a set of linear equations; 3) we then solved numerically this linear eigenvalue problem.

Many more details are available [7]. From Table 1 we deduce the following very good

approximate formula for the exact eigenvalues of the Hamiltonian (1) with m 1 = m2 = 0 and V(r) = ar

E2n,t=4r~a(n+ 2(n)l/2 + 3 +l~(n)) n>O l>__0

Table 2. 2(N) and/~(N) are obtained from Table 1

Least Squared Linear Approximation Parameters

N 1 2 3 4 5

0.633 0.617 0.603 0.590 0.58 2 1.27 1.23 1.21 1.18 /t 4.1 x 10 -2 9 • 10 -3 9 • 10 - 4

317

where the first values of 2(n) and #(n) are listed in Table 2.

It is readily checked that 2(1) _~ 8/2rc thus confirming our heuristic argument for the leading trajectory. We also obtain the intercept of the leading Regge trajectory to be B = 1.25.

Now it is easy to see that no more information is contained in the Regge trajectories of the light mesons as their particular pattern follows from the use of the relativistic kinematics associated with first order perturbation theory. Indeed we may write

m 2 +

so that (1) becomes

[ < +ml +, J

q- ~pz + m 2 + ~ 2 + Vr(r) ~,(r) = E~b(r)

It is then justified to use first order perturbation theory because for a fixed l, the energy differences between the various families (p, K*, 4>) are small as compared to the energy difference in each family between the fundamental and the first radial excitation and that for all I. Therefore we can approximate

E,(m l, m2) ~- Et(O, O) + (m2 + m~) ( 2 j I

But

2 < 3 ) e,(0,0) for large enough I as can be seen from the Feynmann- Hellmann theorem applied to I. Here it is useful to note that apart from providing a coherent kinetic energy for the light quarks, the relativistic kinematics is also important to understand the parallelism of light meson Regge trajectories.

From the experimental data it follows that

a = y/8 = 0.15 GeV 2

and

b = - 2 o + B + = - - 2 0.80+

where VL(r) is written as ar + b/r.

318

2 2 = 2 a 6 = 0.062 G e V 2 which We also have m s - rnu

is quite reasonable. We expect this parametrization to be adequate for V(r) in the range of r tested by the light mesons families, that is for r ~> 1 f.

IlL Behaviour of the Potential for Small r

We start again with (1) which writes for heavy quarkonia in which we are interested, namely charmonium and bellonium,

[2x/p 2 + rn ~ + Vn(r)] ~k(r)= E~b(r) (3)

Now it is clear that the only thing we have to show in order to recover the Quigg and Rosner result is that, in spite of the emerging relativistic behaviour of the charmonium, which is well known (see for instance [1] and [3J), the characteristic mass-independent excitation spacing of the log potential is not altered, at least for the first levels, by the use of relativistic kinematics in the mass region of the c quark. But this last point immediately follows from first-order perturbation theory applied to (3).

Indeed we have

p2 x / p 2 + m 2 =m-t 2m

1 ( p 2 3 2 1

which leads for Vn(r ) = c ln(r/ro) to

c 2 1

A E(rel . . . . . I) 16 m t

We have used the virial theorem to obtain (p2/2mq) = c/4 (assuming a distribution of momentum p sharp enough) which thus appears to be level independent.

To evaluate d E(rel-non rel) we use the c value given by the experimental spacing E ( Y ' ) - E ( Y ) ~ 0.5460GeV. We have c = 0 . 7 G e V and for mc~ 1.5 GeV A E(rel-non rel)_~ 0.02 GeV, which is indeed small as compared to the relevant excitation energies for the first few radially excited levels of the log potential.

Therefore, assuming the approximate equal spacing exhibited in ~ and lc spectra will remain for all heavier qc7 families, we conclude as in [1] that a natural potential to be used in the r region tested by these mesons is Vn(r) = c ln(r/ro) with c = 0.7 GeV.

In fact, we know from [8] that without any extra assumption, VH is determined to be logarithmic only in the range 0.1 f < r < 1 f, r L = 1 f being therefore the upper limit of the small r region.

r o is adjusted so as to provide the smooth universal shape for V(r) in the one-Fermi region. We found c In r o = 0.800 GeV to be a good value.


IV. Discussion

So far we have shown on experimental grounds only that the potential to be used to reproduce the observed spectra of light and heavy S = 1 and J = Jmax mesons must be parametrized as

V L(r) = ar + b/r r > 1 f

Vn(r) = c ln(r/ro) r <~ 1 f

where each relevant parameter (a, b, and c) has a straightforward physical interpretation; but a natural question arises: what could be their theoretical meaning?

Consider the small r behaviour of V(r) first. As stated above, the log parametrization appears, from the current spectroscopy, to be strictly valid in a fairly restricted range of r; moreover it does not correspond to any common theoretical expectation, so that altogether it sounds much more like a convenient form of an effective potential than anything fundamental. On the contrary, the ar part of the large r behaviour of VL(r ) is well known to reflect the string dynamics of light mesons so that we may wonder if the b/r term could not also have such an interpretation. In fact it does! Indeed, we expect from the string model an additional contribution to the large distance potential coming from the quantum fluctuations of the string, a contribution which has been shown to write x / r [9, 10] where x = - rc for the Nambu string and x = - 7r/2 in the Neveu-Schwartz model. Comparing with our b value we see that for a sensible u constituent quark mass fin,-~0.1 GeV), b is quite near the Neveu- Schwartz value, which is noticeable. Moreover, using a preliminary model recently proposed in [11], we can reach a complete agreement with it. Indeed in that model, based on the coupled dynamics of a string and of a U(1) gauge field, the potential to be used in a semirelativistic equation is shown to be for large r: V(r) = ar + ~c/r - (M2/r ) where (M2o/r) comes from the coupling with the U(1) field and extends to short distances as well and where Mo 2 remains to be determined. From the b value we get

m 2

2m~ M g = 0.03 + - -

a

so for 0 .1< m, __< 0.2 GeV, 0.16<M2o<0.57 which, remarkably, is seen to include the particular values used in the early calculation of Eichten et al. [12]. Such a result may suggest that the true short distances potential is in fact QCD-like*.

* We should not worry too much about how to define the potential in the 1 Fermi region. Indeed from [10] it is clear that the form a r + ~c/r is not an accurate approximation to the string potential in that region (its exact form is for

r >~ V ( r ) = x / ( a r ) 2 - 2 x a


Let us emphasize finally that by using the trans- parent b/r trick, there is no need for any negative constant to be added to the linear term. We think that removes a long-lived embarrassment (see for instance 1-133).

Acknowledoements . We are grateful J.L. Basdevant for suggesting to look at the meson spectroscopy in the context ofa semirelativistic Schr6dinger equation and for informing us about the Multhopp method. We also thank him for discussions, and thank A. Martin for a careful reading of the manuscript.

References

1. C. Quigg, J. Rosner: Phys. Lett. 71B, 153 (1977) 2. Let us mention the earliest use of a semirelativistic equation to

describe mesons by D. Stanley and D. Robson: Phys. Rev. D23, 2776 (1981) and the subsequent appearance of such an equation

319

in J.L. Basdevant and G. Preparata: Nuovo Cimento 67A, 19 (1982) and J.L Basdevant, P. Colangelo and G. Preparata: Nuovo Cimento 71A, 445 (1982). After completion of this paper, we were informed by H.J. Schnitzer that he and J.S. Kang already used it in 1975 (Phys. Rev. D12, 84l (1975))

3. C. Quigg, J. Rosner: Phys. Rep. 56C, 168 (1979) 4. H. Grosse, A. Martin: Phys. Rep. 60C, 6 (1980) 5. P. Cea et al.: Phys. Rev. D26, 1157 (1982) 6. A.J. Hanson, R.D. Peccei, M.K. Prasad: Nucl. Phys. B121, 477

(1977); A. Robinson, M. Laurmann: Wing Theory, 193 (1958) 7. D. Olivier: Th6se de 3e Cycle, Universit6 de Paris VI, Jussieu

(1983) 8. C. Quigg, J. Rosner, H.B. Thacker: Phys. Rev. D23, 2625 (1981) 9. M. Lfischer, K. Symanzik, P. Weisz: Nucl. Phys. B173, 365

(1980); L. Brink, H. Nielsen: Phys. Lett. 45B, 4 (1973) 10. J.F. Arvis: Phys. Lett. 127B, 106 (1983) 11. D. Olivier: preprint LPTHE 83-15 (1983, LPTHE 84, 19 (1984);

LPTHE 84-29 (1984)); see also [7] 12. E. Eichten et al.: Phys. Rev. D21, 203 (1980) 13. D. Gromes: Phys. C--Particles and Fields 11, 147 (1981) 14. Particle Data Group: Phys. Lett. l l lB , 1 (1982)

so that we should do if we want to use the potential

K Mo 2 r~>l f V ( r ) = a r . . . .

1" r

Mo ~ r~<l f V(r) = - - -

r

is to join the r ~> 1 f curve and the r ~< 1 f curve with a smooth line

experimentally favoured potential for mesons

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