Volume 102B, number 5 PHYSICS LETTERS 25 June 1981

MAGIC MOMENTS IN QCD 2

Andrew BRADLEY, Caroline S. LANGENSIEPEN University of Manchester, UK

and

Graham SHAW NIKHEF-H, Amsterdam, The Netherlands

Received 13 April 1981

The SVZ moment method of calculating resonance masses is studied in the context of two-dimensional QCD. It is shown to underestimate the appropriate parameter by about a factor of three.

1. Introduction. The success o f the moment meth- ods of Shifman, Vainshtein and Zakharov (hereafter referred to as SVZ) in calculating the masses of low-ly- ing resonances from QCD parameters has generated a good deal of interest, and an expanding literature [ I - 3] . However, while their success is undeniable, their physical basis is obscure. For this reason, they have been referred to as "magic moments" by Bell and Bertlmann [3], who have investigated some of the approximations used in the context of non-relativistic potential theory. In this paper, we present a similar in- vestigation in the context o f two-dimensional QCD to leading order o f the 1 [N c expansion [4] . This model is well-known to have many desirable features, combin- ing quark confinement with asymptotic freedom in a fully calculable framework * 1. It offers an ideal oppor- tunity to test approximation schemes against the re- sults o f a complete calculation, in a model whose fea- tures bear a marked resemblance to those in the labo- ratory. In a previous paper [8] we have exploited this to investigate local Q2 duality, which is the starting point for SVZ moment methods, showing that it works

i On leave of absence from the University of Manchester, UK.

#1 For an extensive review of early work on this model, see Ellis [5 ]. Results for e+e - annihilation are given in refs. [6,7 l.

extremely well in QCD 2 . In this paper, we turn to the SVZ moments themselves.

2. S V Z moments [1]. First we summarise the ap- proximations to be investigated, as they apply to the charmonium system in their simplest form [1 ]. The starting point is the spectral representation of the pho- ton propagator

lI(s) = 1 f ds' I?_~(S') , (2.1)

from which the rapidly convergent SVZ moments

_ 1 dlIn_(s.) = 1 f d s Im FI(s) (2.2) Mn - n! ds n s=0 sn+ l

are easily obtained. In the narrow-width approxima- tion

o ~

k~=O 2, 2n+2 (2.3) M N = gk /mk ,

where mk, gk are the mass and coupling of the kth vector meson. For large n,this sum will be dominated increasingly by the ground state, so that the ratio

R n =Mn/Mn+ 1 (2.4)

tends quickly towards the ground-state mass m 2, Thus,

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Volume 102B, number 5 PHYSICS LETTERS 25 June 1981

if the M n can be independently calculated for suffi- ciently large n, the mass m 0 can be determined. This argument can be extended to other quantum numbers by considering the spectral representations of other cur- rents.

To calculate the Mn, SVZ use a very strong and rather novel extension of duality. Specifically, they assume that the weighted averages eq. (2.2) can be ade- quately approximated by the QCD expression

M n = [2n(n + i ) ! / n ( 4 m 2 ) n + l ( Z n + 3)!!]

× [1 + asa n - (n + 3)!~b/(2n + 5) (n - 1 ) ! ] , (2 .5 )

where m c is the quark mass, a s the strong coupling con- stant, and ~ a confinement parameter related to the gluonic field:

~9 = [47ras/q (4m2) 2 ] (0 IGuvaG auu 10). (2.6)

The first term in eq. (2.5) arises from the quark-loop diagram fig. la, while the second term arises from the leading logarithm correction o f perturbative QCD. The constants a n are easily calculated from Schwingers inter- polation formula for the first-order correction, and are relatively slowly varying with n. The last term is the leading power correction from the non-perturbative confining forces, and increases rapidly with large n. Non-leading power contributions would presumably vary even more rapidly, but cannot be calculated.

The usefulness of these ideas for predicting masses depends upon the existence of a range of intermediate n values which are small enough for eq. (2.5) to be a good approximation, but big enough for the moments to be dominated by the ground state alone. The ground-state mass m 0 can then be read offdirectly from the ratio R n . SVZ argue that such a range indeed exists.

(A ) ( B )

( C ) (O~

Fig. 1. Diagrams contributing to e+e - annihilation in QCD 2 to order m 2 = fgZNc/*r).

Using empirical estimates for the charm contribution to e+e - annihilation, they fred that the first four mo- ments are compatible with the perturbative contribu- tion only for the parameter values m c = 1.26 GeV, as(mc) = 0.2. For the next few moments, a discrep- ancy appears which is well accounted for by the lead- ing power correction for a value of the confinement parameter 4) = 1.35 × 10 - 3 . For very large n > 8, agree- ment breaks down, presumably due to the neglect of non-leading contributions in eq. (2.5).

Alternatively, this can be reexpressed in terms of the ratio R n , which tends to rn 2 as n becomes large, so that the ground state dominates. If this is plotted against n, then the parton model contribution alone decreases steadily towards the quark threshold value 4m 2 < m02, as is seen from eqs. (2.2,2.4, 2.5). On add- ing the other terms in eq. (2.5), and in particular the power correction term, R n flattens out to a minimum, and eventually begins to rise, presumably reflecting the neglected higher-power corrections. For the parameter values cited above

m c = l . 2 6 G e V , a s ( m c ) = 0 . 2 , ~ b = 1 . 3 5 × 1 0 - 3 , (2.7)

this minimum occurs for n values of 6•7. For these n values, the lowest resonance does dominate the mo- ments, so that R n at the minimum gives a good pre- diction for m 0 .

As we shall see immediately, in QCD 2 a very simi- lar situation is found, except that in this case the basic parameters corresponding to eq. (2.7) are precisely known. One can thus test whether the parameter values assumed to obtain good mass predictions are the true values, or not.

3. S V Z momen t s in QCD 2 . In QCD 2 to leading or- der in 1/Nc, the absorptive part of the photon propaga- tor

Iluv(S = q2) = (quq v _ g u v q 2 ) ~ e Z N c i i a ( S ) (3.1) 12

is given by a set of narrow resonance contributions [6]

Im II(s) = k~0__ (gk)26 (s - mk2). (3.2)

The factor e a 2N c (where e a is the quark charge, a

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Table 1 Masses and couplings of "charmonium" states resulting from a numerical solution of the 't Hooft wave equation of QCD 2 with the parameters eq. (3.3).

k rn k g k

0 3.0974 0.7482 1 3.4446 0 2 3.6877 0.3439 3 3.9049 0 4 4.0950 0.2482 5 4.2727 0 6 4.4360 0.1988 7 4.5915 0 8 4.7377 0.1674

= u, d, s, c, b..) has been abstracted for convenience, and in eq. (3.2) we have specialised to a single quark species II(s) - l-lc(S ). The masses m k and couplings gk depend on the quark mass m c and the quark-gluon coupling constant g. Their values can be determined by numerical solution of the 't Hooft wave equation [4]. For the parameter values

m e = 1.442 GeV, m 2 = (g2Nc/Tr) = 0.111 , (3.3)

the results o f table 1 are obtained ,2 in which the low- lying mass values correspond closely to those of the charmonium system. The vanishing of the couplings gk for odd k is a general property reflecting the alternat- ing parity of the states.

The moments M n and ratios R n can now be defin- ed precisely as in eqs. (2.2, 2.4), and evaluated using the numerical results of table 1 , 3 . To complete the analysis, we need the analogue of eq. (2.5). To obtain this, we expand both l-l(s) and the corresponding mo- ments M n in terms of the coupling parameter m 2 = (g2Xc/v:)

II(s) = ll(0)(s) + m2II(1)(s) + .... (3.4)

~2~ , r (1 ) ( 3 . 5 ) Mn =M(n O) + . . . . . . n + . . . .

Asymptotically, II (n)(s) ~ s - (n+2) , the leading term II(0)(s) corresponding to the parton model diagram fig. 1 a. On evaluating its contribution to the moments Mn,

we obtain

,2 For further details, see refs. [8,9]. ,3 The small contributions for n >~ 10 can be estimated very

accurately using the duality results o f ref. [8 ].

Mn (0) = 2n+l(n + 1 ) ! [ ( 4 m 2 ) n + l ( 2 n + 3 ) ! ! , (3.6)

similar to that result in four dimensions, eq..(2.5). There are no log corrections in QCD 2, and the leading power corrections II(1)(s) arise from figs. lb, c, d. In two dimensions, the single-gluon exchange potential of diagram 1 b is already confining, whilst diagrams lc, d partially renormalise the bare quark mass me. On evaluating their contributions to Mn, we find

[1 m 2 ( ( 2 n + 3 ) , , Mn = M(O) + ( n + l )

177 c

n (3.7) X Z) 1)!!(2X + 1)!!/J k=0

with m 2 playing the role of the confinement parameter ¢ in four dimensions.

This expression can now be used to calculate the moments M n and ratios R n for the parameter values eq. (3.3). The results for R n are plotted in fig. 2a. Again, while the parton contribution tends towards 4m 2 for large n, on adding the power correction term, R n flattens out to give a min~num, and then rises rapid- ly as the approximation eq. (3.7) breaks down. Assum- ing R n = m2o at the minimum then gives

m 0 = 3 . 1 9 G e V , E 0 = m 0 - 2 m c = 0 . 3 1 G e V (3.7')

for the mass and excitation energy o f the ground state, compared to the true values (table 1)

m 0 = 3 . 0 7 9 G e V , E 0 = m 0 - 2 m e = 0 . 2 1 3 G e V . (3.8)

Since the excitation energy represents the non-trivial part of the mass value, this is not very impressive, and indicates that there is no range o f n in which eq. (3.7) and the ground-state saturation hypothesis R n = m 2

are both good approximations. This can be seen explic- itly in fig. 2b.

A closer parallel to the early SVZ analysis of char- monium is obtained by treating the parameters me, m 2 as "unknowns" to be determined from a moment anal- ysis of the "data" of table 1. Following the arguments of section 2, we then find that, as in the four-dimen- sional case, the first few M n values are compatible with the parton model results M~n 0) to high accuracy for a suitably chosen quark mass (specifically, the ratio M(nO)/Mn = 0.99, 0.99, 1.00, 1.02 for n = 0, 1 ,2 ,3 ) .

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Volume 102B, number 5 PHYSICS LETTERS 25 June 1981

R r l

( G e V ~ ) 11

10

2 Mku--

4mc2__

8 I

0

I 1 I I I I I I I I

/

0 0

I

1

X

X X

X X X X X

I I I I I 1 1 I l

2 3 4 5 6 7 8 9 10

R n

(GeV 2)

10

2

11

9

I

0

4m~

8

I I I

O

x

o

x o

X o --,g-

l I I T I I l I

0 0 - - O - - .

x x x x

0 0

X X X

[ I I i I I I I I L

1 2 3 4 5 6 7 8 9 10

o

Fig. 2. (a) The moment ratio R n = M n / M n + l for the " t rue" parameter values m c = 1.442 GeV, m 2 = 0.111 GeV 2 . [Crosses (X) - zeroth-order calculation; circles (o) - first-order calculation; dots (o) - exact numerical results]. (b) As in (a) but for the fitted parameter values rn c = 1.471, m 2 = 0.037.

For the n e x t few m o m e n t s , a d i sc repancy appears

w h i c h is well a c c o u n t e d for by t he leading p o w e r cor- r ec t ion M (1 ) for a su i t ab ly c h o s e n value o f t he con-

f i n e m e n t p a r a m e t e r m 2 . In th is way , we arrive at the

p a r a m e t e r values

m e = 1.471 G e V , m 2 = 0 .037 , (3 .9 )

w h i c h in t u r n lead to the R n values o f fig. 2b . As can

be seen, th is exh ib i t s a shal low m i n i m u m cor respond-

ing closely to the g round-s ta te mass squared .

This d iscuss ion exh ib i t s all the nice fea tures o f the

c h a r m o n i u m analysis o f sec t ion 2. However , whi le in

four d imens ions we had no check o n the quark mass

and c o n f i n e m e n t p a r a m e t e r ~b, in this case we k n o w

the " t r u e " values o f mc , m 2 c o r r e s p o n d i n g to the " d a t a '

used. On compar ing eq. (3 .9) to eq. (3 .3) , we see t h a t

the m o m e n t analysis s l ightly overes t imates the bare

qua rk mass , b u t u n d e r e s t i m a t e s the c o n f i n e m e n t pa ram-

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eter m 2 by about a factor of three. This provides strong support for the conjecture of Bell and Bertlmann [3], based on potential studies, that the SVZ value of the confinement parameter ~, eq. (2.7), may be a sub-

stantial underestimate; and throws further doubt on the physical basis of the ratio method as a means of

studying resonance masses.

One of us (G.S.) would like to thank John Bell for a most helpful discussion at the start of this work.

C.S.L. would like to thank the Science Research Council for financial support.

References

[1] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448, 519; Pis'ma Zh. Eksp. Teor. F~. (USSR) 27 (1978) 60; Phys. Lett. 76B (1978)471; Phys. Rev. Lett. 42 (1979) 297.

[2] M,A. Shifman, A.I. Vainshtein, M. Voloshin and V.I. Zakharov, Phys. Lett. 77B (1978) 80; Yad. Fiz. 28 (1978) 465 ; V.A. Novikov et al., Phys. Rep. 41C (1978) 1 ; L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. 94B (1980)203; 95B (1980) 103; 97B (1980)257; D. Morgan, Rutherford Lab. preprint RL-80-051 (1980); B. Guberine, R. Meckback, R.D. Peccei and R. Riickl, Max Planck Institut preprint, MPI-PAE/PTh 52/80.

[3] J.S. Bell and R.A. Bertlmann, CERN preprints TH2880 (1980); TH2896 (1980); Nucl. Phys. B, to be published.

[4] G. 't Hooft, Nucl. Phys. B75 (1974) 461. [5] J. Ellis, Acta Phys. Pol. B8 (1977) 1019. [6] C.G. Callan, N. Coote and D.J. Gross, Phys. Rev. D13

(1976) 1649; M.B. Einhorn, Phys. Rev. D14 (1976) 3451 ; D15 (1977) 3037.

[7} J. Randa and G. Shaw, Phys. Rev. D19 (1979) 3314. [8] A. Bradley, C.S. Langensiepen and G. Shaw, Manchester

University preprint M/C TH 81/90. 01 A. Bradley, C.S. Langensiepen and G. Shaw, Manchester

University preprint, in preparation.

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