Embed Size (px)
Citation preview
Volume 102B, number 2,3 PItYSICS LETTERS 11 June 1981
Q2 DUALITY 1N TWO AND FOUR DIMENSIONS
Andrew BRADLEY and Caroline S. LANGENSIEPEN Department of Theoretical Physics. The University, Manchester M13 9PL, UK
and
Graham SItAW 1 NIKHEt", Amsterdam, The Netherlands
Received 16 .March 1981
Local Q2 duality is investigated in the context of two-dimension',d QCD, and shown to be remarkably successful, for both light and heavy quark masses. The corresponding width predictions in four dimensions for the ff ', T' and T" resonances agree well with experiment.
1. Introduction. The notion of a duality relation between the simplest quark parton diagram for e+e- annihilation (fig. 1 a) and the observed resonance spec- trum has been a topic of interest for many years [ 1 - 5 ] . In addit ion, it forms the basis of the generalis- ed vector dominance model of diffractive photopro- duction on nucleons and nuclei [6] ; and is the starting point for the "SVZ moment" methods of Shifman et al. [7] ,which have led to many interesting results for low-lying resonances in the context of QCD *
In view of this, it is interesting to investigate this duality relation in solvable models. In particular, it has been shown to work well in the context of non- relativistic potential models of the type often used to describe charmonium [4]. In this paper, we investigate whether, and in what way, such Q2-duality relations
I On leave from the University of Manchester, UK. ,1 See for example: Shffman et al. [7], and Reinders et al. [8].
(a) (D)
Fig. 1. Patton model (a) and resonance (b) contributions to ~r(s).
are realised in the context of quantum chromodyna- mics in one space and one time dimension (QCD2). As usual, we work to leading order in N c l , where N c is the number of colours. This model [9] has many desirable features. In particular, it combines asymp- totic scaling with confinement in a fully calculable framework. The mass spectrum consists of an infinite sequence of bound states (n = 0, 1,2 .... ) which be- come equally spaced in mass squared (m 2) at large n, reminiscent of Regge trajectories. At high energies, "sof t" processes like elastic scattering are dominated by "Regge" exchanges with factorisation and duality relations similar to those observed in four dimensions* 2 [10]. Finally, appropriate scaling laws are obtained for deep inelastic processes. In particular, for e+e - annihilation, both the total and single-particle inclu- sive cross sections scale asymptotically [11,12], and for sufficiently high energies, a power-law growth in particle multiplicity is obtained [ 13].
QCD 2 seems, therefore, an almost ideal theoretic- al laboratory in which to investigate Q2 duality. In doing this, we will concentrate on whether it holds locally enough, and for low enough Q2, to yield re- liable results for individual low-lying resonances.
,2 Except that in two dimensions, there is no "diffractive" telm, or pomeron.
180 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North-Holland Publishing Company
Volume 102B, number 2,3 PHYSICS LETTERS 11 June 1981
Finally, encouraged by the results, we re-examine the empirical status of similar relations in four dimensions.
2. E l ec t ron -pos i t ron annihilation in QCD 2 . As- suming Zweig's rule, the vacuum polarization tensor nuv( s = Q2) can be written as a sum over each quark species
nuv( s = Q2) = (quq v - guvq2) ~ e2Ncna(q2) , (2.1) a
where e a is the charge of quark a = u, d, s, c, t ..., and the normalisation factor e a N c has been introduced for later convenience.
In QCD2, to leading order in 1/iV c, Zweig's rule is exact, and each term is given by a set of bound-state contributions [ 11,1 2]
hn na(s ) = n~0 (g~n)2C5 (s - (ma)2) , (2.2)
corresponding to fig. lb. The real part is then given by
1 ~ ds ' lm na(S' ) Re %(s) s ' - ' ( 2 . 3 ) J s
So
a and couplingsg a are determined and the masses m n by the quark masses mr , and quark-gluon coupling constant g. Whatever the values chosen, the couplings gn a satisfy [11,12]
g O = 0 n o d d , (2.4)
which is a consequence of parity, and
n=~o(gna) 2= 1 , (2.5)
which is a completeness relation for the meson wave functions. This implies the asymptotic behaviour
lirn na(S ) = - l f ds Imna(s ) = - 1 / n s , (2.6)
SO
along any direction other than the positive real axis. This is precisely the asymptotic behaviour of the par- ton model diagram fig. la (see below).
For more detailed considerations, numerical values of the masses m a and couplingsg n are required. These can be obtained by numerical solution of the 't Hooft wave equation [9] for the assumed couplingg and
Table 1 A comparison of e*e- couplings estimated using duality [gn(A),gn(B)] with the exact couplings (gn) for the lowest three contributing mesons. Three quark masses are considered corresponding to light (u, d), charm and bottom flavours.
mq n m n gn gn(A) gn(B)
0.270 0 0.769 0.957 0.952 0.952 0.270 2 1.636 0.212 0.233 0.232 0.270 4 2.196 0.118 0.121 0.121
1.442 0 3.097 0.748 0.740 0.729 1.442 2 3.688 0.344 0.357 0.375 1.442 4 4.095 0.248 0.251 0.255
4.645 0 9.459 0.543 0.537 0.526 4.645 2 9.871 0.290 0.296 0.312 4.645 4 10.160 0.230 0.232 0.235
bare quark mass m a. We have done this for a wide range of quark masses, using a method that has notable advantages for low-lying states, and which is reported elsewhere [14]. As a check, we have also used the method of Visnjic and Hildebrandt [ 15]. Some results are given in table 1, corresponding to parameter values
rn = (g2Nc /n ) l /2 = 0.333 GeV, (2.7a)
m u = 0 . 2 7 0 , m c = 1.442, m b = 4 . 6 4 5 , (2.7b)
chosen so that the ground state masses correspond to the vector mesons p, ff and T, and so that for ~ , the first excited state m 1 = 3.445 GeV lies among the even parity x states, and the second m 2 = 3.688 is close to the observed ~b' mass.
3. H o w g o o d is duali ty? In QCD 2 , the naive quark/ parton-model loop diagram fig. la leads to an absorp- tive part
lm n(0)(s)= 2 m 2 0 ( s - 4m2a)/S3/2(s - 4m2) 1/2, (3.1)
which on substituting into eq. (2.3) leads to
lira nCa°)(s) - 1 / ds Im n(0)(s) = - l / n s . (3.2) S--+ oo 7TS
4ma ~
Comparison to eq. (2.6) then gives the global duality relation
f d s Im n(0)(s) = / d s l m T r a ( S ) , (3.3)
So 4mr2
181
Volume 102B, number 2,3 PtlYSICS LETTERS 11 June 1981
between the quark/patton-model absorptive part eq. (3.1), corresponding to free q?::l pairs, and the true ab- sorptive part eq. (2.2), corresponding to narrow reso- nance poles and confined quarks. The interesting question is whether this equality holds for more local averages. In particular, for resonances studies, it must hold over intervals comparable with the resonance spacing; and it must hold for the lowest lying reso- nances * 3
To test this, we estimate the non-vanishing couplings g2n from duality, and compare them with the exact numerical results of table 1. For the ground state, we integrate the parton-model absorptive part from threshold to a point s I midway between the first two contributing poles, and equate it to the ground-state pole. This gives
$!
(g~)2 = f ds Im na(0)(s) = K ( S l ) , (3.4a)
4 ma 2
where
K ( s l ) = sin [tan -1 [(s I - 4 m 2 ) / 4 m 2 ] l / 2 } . (3.4b)
For excited states, integrating over each resonance in turn gives
(g~n) 2 = K(s2n+l ) - K(s2n -1)" (3.5)
For the intermediate points S2n+l , we consider two choices. In the first (hypothesis A) we identify them with the even parity states which do not contribute to e+e - annihilation
S2n+l = (m2n~ , l ) 2 . (3.6a)
In the second (hypothesis B), we simply take the mid- point
SZn+l = } [(mzn+2) 2 + (m2n) 2 ] , (3.6b)
which is a widely used procedure in four dimensions. At large n, where the trajectories are linear, the two prescriptions are equivalent.
To test these duality relations, we assume the mass spectrum to be known, and use eqs. (3 .4)- (3 .6) to predict the resonance couplings. The resulting values are then compared with the exact numerical values, obtained (like the mass spectrum) in section 2. This
, s That it holds in the asymptotic limit n, m2n -, oo has already been shown [12,16].
is done for the ground state (n = 0) and first two non- vanishing excited states (n = 2, 4) in table 1. As can be seen, the agreement is striking for all quark masses considered. For higher states, n ~> 6, the agreement is even better, and becomes exact in the limit n ~ oo as already noted [12,16].
4. Duality in four dimensions. Q2 duality in the real world has been discussed by many authors [ 1-5 ]. In particular, considerations similar to those of section 3, based on a parton-model sum rule of the form eq. (3.3), and incorporating q~l threshold effects explicitly, were proposed long ago by Sakurai [2]. Most discus- sions, like the recent paper by Gounaris [5], tend to concentrate on the ground-state contribution, linking it to other parameters like the quark mass, and, if first-order QCD effects are included, the scale parameter A. However, Gounaris [5] also estimated the ~b' width with reasonable success, and estimated tile T ' ( I0 .02) width to be 0.6 keV if the T " was as- sumed to be at 10.4 keV. This was compared to the then experimental value of 0.37 +-0.16 keV [17]. More recent results give 0.55 -+ 0.09 keV [ 18]. Further now the masses of four T states are known
rn = 9 .46 , m ' = 10.02 ,
m ' '= 10.35, m ' " = 10.57. (4.1)
We therefore reexamine the T case. The appropriate absorptive part, including perturbative QCD effects in leading log approximation **, is [19]
10( 3 02)[ I 4 Im ~'(O)(s) = ae 2 ~ - + ~ asf (V)] , (4.2a)
where
f ( v ) = 7r/2v - ~ (3 + v) (~ n - 3/47r), (4.2b)
and
v(s) = (1 -- 4 m 2 /s) 1/2 . (4.2c)
In contrast, the narrow resonance approximation gives
I m ~'(s) = 3n" ~ rnrFrS(S _ m 2 ) . (4.3) O~ r
Using local duality, one then easily derives the analo- gues of eqs. (3.4), (5), adopting hypothesis (B) to de- fine the duality interval. For a given as, the experi-
~4 There are no log corrections to scaling in QCD 2.
182
Volume 102B, number 2,3 PHYSICS LETTERS 11 June 1981
mental ground-state width 1.29 -+ 0.20 keV [18] leads
to an effect ive quark mass, which can then be used
to predict the exci ted-s ta te widths. The pure par ton
model a s = 0 gives
m b = 3.93 -+0.12 GeV, I"(10.02)=0.44_+0.02 keV,
(4.4) i"(10.35) = 0.29 +0 .02 k e V ,
compared to the exper imcnta l values [181
1"(10.02) = 0.55 -+0.09, I ' (10 .35) = 0.41 -+ 0.06 keV,
whereas wi th a s = 0.2 (cor respondmg 1o an effect ive
A = 0.16), the values
I ' (10 .02) = 0.58 k e V , (4.5)
F(10.35) = 0.37 keV (mb = 4.30 GeV),
in good agreement with experiment. Finally, we note that using the same arguments,
and effect ive A value, for the ~ sequence m = 3.097,
tn' = 3.688, m" = 4 .028, leads to
m c = 1 . 2 7 G e V , m F ( 3 . 6 8 8 ) = 9 . 8 5 ( × l O - 6 G e V 2 ) ,
compared to the exper imenta l value 7.56 +- 0.85.
However , this ignores the ~kD(3.77), and including
this in the same duali ty interval increases the effect ive
m F(expt . ) to 8.50 _+ 0.86, in reasonable agreement
with the above.
5. Conclusions. We have shown that local Q2 dual-
ity works ex t remely well in QCD2, for bo th light and
heavy quark masses. Similar results have been reported
by o ther authors [4] in the con tex t o f non-relativistic
potent ia l theory . Taken together with the empir ical
success summarised in section 4 for T and ~k mesons,
this const i tu tes strong evidence for the relevance o f
dual i ty to even the lowest lying resonance widths. To
obtain detailed results on resonance masses, however ,
more sophist icated me thods [7,8] are required.
QCD 2 should prove an interest ing testir/g ground for these me thods as well.
C.S.L. grateful ly acknowledges financial support
f rom the SRC.
R cferetl ces
[ 1 ] A. Bramon, E. Etim and M. Greco, Phys. Lett. 41B (1972) 609; M. Greco, Nucl. Phys. 1363 (1973) 398.
[2] J.J. Sakurai, Phys. Left. 46B (1973) 207; Proc. Intern. School of Sub-nuclear physics (Erice, Sicily, 1973), ed. A. Zichichi (Periodici Scientifici, Milan, 1975) Part A, p. 290.
[3] M. Bohm, It. Joos and M. Krammer, Acta Phys. Austriaca 38 (1973) 122; G.J. Gounaris, Nucl. Phys. B68 (1974) 574; B88 (1975) 451;Phys. Lett. 72B (1977) 91; G.J. Gounaris, E.K. Manesis and A. Verganelakis, Phys. Lett. 56B (1975) 457; V. Barger, W.F. Long and M. Olsson, Phys. Lett. 57B (1975) 452; E.C. Poggio, II.R. Quinn and S. Weinberg, Phys. Rev. D13 (1976) 1958; M. Kramer, P. Leal-Ferreira, Rev. Bras. Fis. 6 (1976) 7; t..E. Close, D.M. Scott and D. Sivers, Phys. Lett. 62B (1976) 213; Nucl. Phys. Bl17 (1976) 134; M.Bohm and M. Krammer, Nucl. Phys. B I20 (1977) 113; V.A. Novikov et al., Phys. Rev. Lett. 38 (1977) 626,791 (E); Phys. Lett. 67B (1977) 409; Phys. Rep. 41C (1978) 1; C. Quigg and J.L. Rosner, Phys. Rev. DI7 (1978) 2364; M. Greco, Phys. Lett. 77B (1978) 84; J.S. Bell and J. Pasupathy, Phys. Lett. 83B (1979) 389; E. Tainov, Phys. Lett. 97B ~1980) 283.
[4] K. ishikawa and J.J. Sakurai, Z. Phys. CI (1979) 117; J.S. Bell and R.A. Bertlmann, Z. Phys. C4 (1980) 11.
[51 G.J. Gounaris, Z. Physik CI (1979) 399. [6] For reviews and references, see: A. Donnachie and G.
Shaw in: Electromagnetic interactions of hadrons, Vol. II, eds. A. Donnachie and G. Shaw (Plenum, 1978); . G. Grammer and J.D. Sullivan, in: Electromagnetic inter- actions of hadrons, Vol. II, eds. A. Donnachie and G. Shaw (Plenum, 1978).
[7] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448,519;Phys. Lett. 76B (1978) 471; Phys. Rev. Lett.42 (1979) 297.
[8] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. 94B (1980) 203;95B (1980) 103;97B (1980) 257.
[9] G. 't tlooft, Nucl. Phys. B75 (1974) 461. [ 10] R.C. Brower, J. Ellis, M.G. Schmidt and J.H. Weis, Phys.
Lett. 65B (1976) 249; Nucl. Phys. B128 (1977) 131,175; M.B. Einhorn, S. Nussinov and E. Rabinovici, Phys. Rev. D15 (1977) 2282; M.B. Eirthorn and E. Rabinovici, Nucl. Phys. B128 (1977) 421.
[ 11 ] C.G. Callan, N. Coote and D.J. Gross, Phys. Rev. D 13 (1976) 1649.
[12] M.B. Einhorn, Phys. Rev. DI4 (1976) 3451; D15 (1977) 3O37.
[13] J. Randa and G. Shaw, Plays. Rev. D19 (1979) 3037. [ 14] A. Bradley, C.S. Langensiepen and G. Shaw, Manchester
University preprint, in preparation.
183
Volume 102B, number 2,3 PHYSICS LETTERS 11 June 1981
[15] S. Hildebrandt and V. Visnjic, Phys. Rev. DI7 (1978) 1618.
[16] R.C. Brower, W.L. Spence and J.H. Weiss, Phys. Rev. D19 (1979) 3024.
[17] J.K. Bienlein et al., Phys. Lett. 78B (1978) 360; C.W. Darden et al., Phys. Lett. 78B (1978) 364.
[18] D. Andrews et al., Phys. Rev. Lett. 44 (1980) 1108; Phys. Rev. Lett. 45 (1980) 219; T. Bohringer et al., Phys. Rev. Lett. 44 (1980) 1111 ; H. Schroder et al., DESY 80/61; B. Niezyporuk et al., DESY 80/58 and DESY 80/81; Ch. Berger et al., Phys. Lett. 93B (1980) 497.
[ 191 T. Appelquist and H.D. Politzer, Phys. Rev. DI 2 (1975) 1404.
184
