In a previous paper[l] we have reconsidered the derivation of a quark- antiquark ... developing a systematic method for deriving a quark-antiquark Hamiltonian in.
IL NUOVO CIMENTO
VOL. 103 A, N. 1
Gennaio 1990
Relativistic Corrections to the Quark-Antiquark Potential and the Quarkonium Spectrum (*). A. BARCHIELLI, N. BRAMBILLA and G. M. PROSPERI Dipartimento di Fisica dell'Universitd - Milano I N F N , Sezione di Milano - Via Celoria 16, 20133 Milano
(ricevuto il 27 Febbraio 1989)
Summary. - - The theoretical derivation from QCD of the q~l potential with velocity-dependent corrections is reconsidered. In particular the problem of ordering between momenta and functions of position in the Hamiltonian is discussed and exact relations among the various potentials are derived. Explicit expressions for the long-range behaviour of the potentials are given, the effect of the velocity-dependent corrections on the quarkonium spectrum is evaluated and compared with the results obtained by different proposals based on more heuristic considerations. PACS 12.40.Bb - Composite models of the structure of hadrons (general models, dynamics, schemes for confinement). PACS 12.40.Qq - Potential models. PACS ll.10.St- Bound and unstable states; Bether-Salpeter equations.
1. -
Introduction.
In a previous paper[l] we have reconsidered the derivation of a quarkantiquark potential from QCD. Starting from a Foldy-Wouthuysen transformation of the one-particle propagator in an external gauge field and the corresponding phase space Feynman integral representation, we succeeded in developing a systematic method for deriving a quark-antiquark Hamiltonian in the form of an inverse quark mass expansion. The main purpose of our work was to give a conceptually well-founded basis to the use of Schr6dinger equation in the computations of the heavy-meson spectrum. However, our method allowed also to obtain from first principles a new velocity-dependent potential Vvd of electrodynamic type never obtained before. Explicit forms for the various (*) To speed up publication, the authors of this paper have agreed to not receive the proofs for correction. 59
60
A. BARCHIELLI, N. BRAMBILLA
and
G. M. PROSPERI
functions Va(r), Vb(r), .... Ve(r) occurring in Vvd were given under the usual assumption of short and long range forces controlled by the weak and the strong interaction limits respectively. Some errors present in the final expressions of ref.[1] were corrected in ref. [2], where some identities involving V(r), Vb(r), ..., Ve(r) were also reported without proof. Such identities are of a similar type of the one established by Gromes [3, 4] for the functions V(r), Vl(r), V2(r) occurring in the spin-dependent part of the potential. In this paper we give the explicit proof of the above-mentioned identities, we reconsider the problem of ordering between momenta and configurational functions in Vvd and other questions that deserve a more complete discussion, and finally we study the effects of the new Vvd term on the spectrum of the mesons. In particular we have enclosed a ,,perimeter term, beside the ,,area term, in the large r expression of the Wilson loop. This amounts to introducing an additional parameter C in the expression of the potential. In the static part of the potential, as it is well known, such a parameter appears as an additive constant (1.1)
V(r) = _x_ + C + zr. r
We shall see that, if we take into account also the first-order relativistic corrections, the structure of the terms involving C is such that this quantity can be absorbed in a redefinition of the quark masses m~---, m~ + C/2, as soon as C/2 is small with respect to m~. Consequently, in our case, the spectrum becomes largely independent of the value of C. Let us notice also that in order to introduce consistently the terms containing C, we have to adopt a limiting procedure in the definition of the functional integral which brings to a not obvious ordering prescription in Vvd, different from that considered in ref. [1]. In trying to understand the effects of our velocity dependent potential on the quarkonium spectrum we have found natural to use as a term of comparison the so-called naive model of ref. [5]. Taking x=0.50, C = - 0 . 8 0 0 G e V and z = = 0.189 GeV2 (a small modification of the values adopted in ref. [5]), treating the whole relativistic correction as a first-order perturbation and choosing mc and ms in order to reproduce correctly the masses of J/~ and 1~ (me= 1.757GeV, mb = 5.170GeV), we obtain a reasonable good agreement with the c~ and bb data in the entire known spectrum. We find that the spin-independent relativistic corrections are generally small (they do not exceed 40 MeV in c~ and 25 MeV in bb); this fact explains the success of the nonrelativistic model. However, for the highest excited states the improvement of our results with respect to those of ref. [5] is clear. In order to test the sensivity of the calculated spectrum to the value of C, we have repeated the calculations for C = 0 with the same value of • and ~ and with mc and mb modified according with the rule given above (m~ = 1.357GeV; m~ = 4.770 GeV). We find that the results are essentially the same for bb, where
RELATIVISTIC CORRECTIONS TO T H E Q U A R K - A N T I Q U A R K
POTENTIAL ETC.
61
C/2mb----0.08, while, as expected, they change appreciably and become definitely worse for c~, where C/2mr-- 0.23. It is worthwhile to mention that for C = 0 the spin-independent relativistic corrections become very large for c~ (up to 275 MeV) questioning the significance of the perturbative treatment. Other proposals of velocity-dependent potentials exist in the literature, which are usually based on some hypotheses of scalar and vector exchange. For what concerns the short-range part of the potential, there is a general agreement on the Breit-Fermi Hamiltonian (and the same holds for us), but for what concerns the long-range part of the velocity-dependent potential, different forms have been proposed, due to some ambiguities in the derivation procedure. We have found particularly significant to compare our Vvd potential with that one proposed in ref. [6, 7], which is structurally different from ours and produces rather large relativistic corrections. We give also a numerical comparison among our model, that of ref. [6, 7] and that of ref. [8], where the long-range part of the Vvd potential is completely neglected, but a much more sophisticated expression is used instead of a pure Coulomb potential in the short-range part of the complete potential. The plan of the paper is the following one: in sect. 2 the form of the q~ potential is discussed, it is shown that the parameter C can be absorbed in the definition of the masses of the quarks and a comparison is made with the expressions proposed in ref. [6, 7] and [8]. In sect. 3 the technique used in the theoretical derivation of the q~ potential is briefly recalled. In sect. 4 the explicit proof of the exact relations among the potentials is given and the perimeter term is also taken into account. Section 5 deals with the relationship between the discretization of the Feynman path integral and the ordering of the operators in the Hamiltonian. Finally in sect. 6 the spectrum of the c~ and bb systems are calculated and the contribution of the velocity-dependent potential is discussed.
2. - S t r u c t u r e o f t h e q ~ p o t e n t i a l .
As shown in ref. [1] the Hamiltonian of the quark-antiquark system can be written as (2.1)
(mj-4 2my p~ 8mP~'~} _+ v(r) + Vvd + vsd,
H = ~=,~
where ml and m2 are the quark and antiquark masses, V is the static potential, Vsa and Vvd are the spin velocity-dependent contributions to the potential. The static potential is given by the famous expression derived by Wilson [9] (2.2)
V(r)=lim~l~
r
A. BARCHIELLI, N. BRAMBILLA and G. M. PROSPERI
62
w h e r e / ' , is a rectangular loop with a temporal side of length ~ and a spatial side of length r and the symbol (...) denotes the expectation with respect to the measure determined by the Yang-Mills action (in Minkowski space). The spin-dependent part of the potential has been obtained by Eichten and Feinberg[10] and it is given by (2.3)
V~d =
LI" $1 +
L2" $2 ~ r
+ (L~. $2 + L2" S~)
1
[V(r) + 2Vl(r)] +
1 -h/rhr k mlm2r2b'l[--fi
1 ,hk\ 5~ )S~V3(r)+
d V2(r) + - -
m~ m2 r dr
+.
1
6ml m2
SI"S2V4(r).
with L~ = r • p~, L2 = - r x P2; the summation from 1 to 3 over repeated indices is understood. The velocity-dependent part of the potential has been obtained in ref. [1] and it is given by (2.4)
Vvd = -~
+
A[V(r) + Va(r)] + +
1
ml m2
2
1
{{p~,p,z,S }}~,1~+
{{P),Pj, T hk} j_ l
.
where the symbol {{..., ..., ...}}z~,~ denotes the ordering prescription (2.5)
1~~p)h, {p~,X}) ~--~ . 1 1-~, ~pj~p,~,~t, , t ={ {p], p~ , X ~] ]~3.,~ :=-~2 -~
=_~tzp)p~A+p]Xp~ + p~k Xp~h + 2Xp~p]) and (2.6a)
S~=~Vb(r)+(3~--r:;k)Vr
(2.6b)
Thk= ~hkVd(r) + ~
[1,,hk -- rhr~-rk )~Y e ( r ) .
The potentials V i - V,, V a - Vd are complicated functional integrals containing the Wilson loop with cromoelectric and cromomagnetic insertions (see ref. [1], eqs. (1.4) and (1.9)). Let us stress that in eq. (2.4) we have changed the ordering of momenta and functions of position with respect to that given in eq. (1.8) of ref. [1]. This is related to the limiting procedure adopted in the definition of the functional integrals. We shall discuss this point in sect. 5.
RELATIVISTIC CORRECTIONS TO THE QUARK-ANTIQUARK POTENTIAL ETC.
63
The potentials V, V1, V~ satisfy the exact relation, found by Gromes [3, 4], (2.7)
d IV(r) + Vl(r) - V2(r)] = 0. dr
In sect. 4 we shall show that the potentials V, Vb, Vo, Vd and Ve satisfy the somewhat analogous relations (cf. [2]) (2.8a)
(2.8b)
Vd(r)+
Vb(r)+
Ve(r) +
1
r dV(r) 12 dr = 0 '
V(r)
r dV(r) Vc(r) +-~ d---~-- 0,
or, equivalently, (2.9)
rhr k d V 3 hkV+ 2r dr"
S hk=-2T hk-
It is usually believed that the expression (2.2) is well represented for small r by its perturbation expansion and for large r by the area law for the Wilson loop (coming from lattice QCD) plus a perimeter correction. In this way one obtains the Cornell potential (1.1). Following some conjectures of Eichten and Feinberg[10] and a suggestion by Gromes [11] on the applicability of that area law to more general situations, in ref. [1] all the potentials have been estimated (the corrections given in the Erratum [2] have to be taken into account). In ref. [1] the ,~perimeter~ terms had not been considered; in sect. 4 we shall show how to treat them. The final result is (2.10a)
(2.10b)
Vl(r)
V3(r) =
= -
C -
3~
(2.12b)
V2(r) = - ~
r'
V4 = 2 hV2(r)
r'
(2.11) (2.12a)
or,
= 8r:•
V~(r) = O, Vb(r) = 2x
3r
r 9 '
V~(r) = - 4 C
Vc(r) =
r9 '
• 2r
Ve(r) =
zr 6 ' zr 6 "
The potentials V b - Ve have been estimated without using eqs. (2.8), but they turn out to satisfy them.
64
A. BARCHIELLI, N. BRAMBILLA
and G. M. PROSPERI
Let us now collect from H the mass terms, the nonrelativistic kinetic energy terms and all the terms containing the parameter C; we get the expression mj + P3
P~ C~
But H has been obtained as an expansion in 1/m up to the order 1/m2. Up to this 2
order, if C
