- Email: [email protected]
Mixed scalar glueballs and mesons
cm .__
__ B
ELSEVIER
25 September 1997 PHYSICS LETTERS B
Physics Letters B 410 (1997) I-5
Mixed scalar glueballs and mesons L.S. Kisslinger, J. Gardner, C. Vanderstraeten Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Received 18 April 1997;revised 2 July 1997 Editor: J.-P. Blaizot
Abstract We find QCD sum rule solutions for scalar meson-glueball mixed states consistent with the f,(15OO) being largely a glueball and the f,(1370) being a largely meson state, consistent with recent experiments. Pure scalar gluebail solutions are in the 300-600 MeV region and pure 0 ++ Fjq meson solutions are found near 1 GeV. 0 1997 Published by Elsevier Science B.V. PACS:
12.38.Lg; 14.4O.C~
1. Introduction The detection of glueballs and the measurement of their decay properties is important for the study of strong interactions. Recently, the Crystal Barrel Collaboration [l] has discovered a scalar CO++) state, the f&500), and
established from its branching ratios that it is likely the state of a scalar glueball admixed with a scalar meson [1,2]. This mass is consistent with recent lattice gauge calculations 131which find a scalar glueball in the region of 1500-1700 MeV. A K-matrix analysis of the Crystal Barrel and other data in the 500-1900 MeV region [4] finds suggests that there is a scalar glueball mixed with two L = 1 O++ meson states in the 1300-1500 MeV region. A recent quark model calculation [5] also finds a mixed glueball/meson state in this region. The coupled channel analysis by the Crystal Barrel Collaboration of the isoscalar f&1370) and isovector a&450) O++ resonances are consistent with their being largely 44 meson states. This analysis of the f&370) also involves
the f,(980) state, whose nature is unclear. Soon after the method of QCD sum rules was formulated it was applied to the search for scalar glueballs [6], as well as scalar mesons [7]. In recent years there have been a number of detailed sum rule studies of the lowest-lying scalar glueball [8], which all find a light, narrow glueball. A typical solution is 500 MeV, in conflict with experiment and lattice gauge calculations [3]. A solution for the scalar meson was found at about 1.0 GeV [7] and was interpreted as the f,(980), with the I = 1 scalar degenerate. The scalar mesons have also been treated in an instanton picture [9]. In this model scalar solutions are difficult to find, but also in the instanton model of the scalar qq mesons the masses seem to be near the f,(980) and the a,(980), which the
authors interpret as states not well-described as qFj mesons. In the present work we explore the possibility of mixed glueball-meson solutions within the method of QCD Sum Rules. It is known that there must be strong coupling between scalar glueballs and scalar mesons in the 0370-2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00964-7
2
L.S. Kisslinger
et al/Physics
Letters B 410 (1997) 1-5
same mass region. We show that the QCD sum rule for a current that is an admixture of a scalar meson with a scalar glueball current, consistent with known mixing [see Section 31, does not have the solutions for a light mass found [8] with a pure glueball current. Indeed, a solution with the mixed current is found near the f,(1500) that is about 80 % glueball. Moreover, with a current for a state that is 80 % scalar meson with a scalar glueball admixed the solutions for the isoscalar and isovector are found near the f&1370) and the a&450). In Section 2 we review the QCD sum rules for pure scalar mesons and pure scalar glueballs. In Section 3 we derive sum rules for mixed scalar glueballs and mesons, and discuss our results in Section 4.
2. Review of QCD sum rules for scalar glueballs mesons The starting point of the QCD sum rule method, as well as the lattice gauge calculations, is the correlator n( p’) = i/&e’“”
(OIT(J(++(O))IOL
(1)
a two point function with J(X) a composite field operator, the current. The currents are J&)=@)q(x),
Jo(x)=~G(x)G(x),
(2)
for the scalar meson and glueball, respectively. By meson we mean a Cjq state. GPy is the gauge field tensor and we have used the normalization of Novikov et al. [6] for the glueball. Equating the once-subtracted dispersion relation to the operator product expansion of the correlator for the glueball one finds [6,8]
w~Gwl = - -17,(O)-
2$p21n(pl)
4a; + pl(G.G)+~T’6)+~~@),.
P2
s(s +p2)
P
(3) P
where (G . G) is the usual gluon condensate, r@) = ( g,f,,,G;vG,b GC > and r(*) = (14(f,b,G:vG&)2 (&,,G,“yG,6,>2). Treating the continuum in the usual way, having an .r’ fpo’l;m with a threshold s0 and carrying out the Bore1 transform one finds the sum rule for A&, the lightest scalar glueball mass I&(0)(
f?-“b/MZ-
2a,2 1.0) + TEl(so)
KY =~;;~M”+~u:(C.C)-~~(~‘-
SCY
4?rcY3 -r(s), M4
(4)
with M the Bore1 mass and E,(s,) = M4(1 + s,,/M2)exp(-- s0/M2). In deriving Eq. (4) we use the low energy theorem discussed in Refs. 1681, giving h4& I&(O) = (I CX,G]GB>~. For the scalar meson the QCD sum rule, keeping terms up to D = 6, is [7]
In obtaining Eq. (5) the factorization approximation of the four-quark vacuum matrix elements given in Ref. [6] has been used. The constant g, is defined as (]J,lm(O”)> = g,. The value of the parameters appearing in Eqs. (41, (5) in the early work [6] cxJ?i-= 0.1 ( a,G . G) = 0.0377 GeV4 r@) = 0.01 14GeV6 r(s) = 0.56(( cr,G . G))2 = 0.0081 GeV* no(O) = 3.5(G. G),
(6) with the expression for I&(O) derived from a low energy theorem discussed by Novikov et al. in Ref. [61. To the extent that factorization is a good approximation, the greatest uncertainty in the parameters is the value of
L.S. Kisslinger et al./Physics
3
Letters B 410 (1997) l-5
r@). In Refs. [8] the range of values from lattice gauge calculations and other work was discussed and used in the sum rules. Here we take the range of - 0.01 to 0.05 GeV6. We also explore larger values of (G . G), increasing the value of the gluon condensate by up to a factor of two times the standard value given in Eq. (6). Our results are that we find a solution for the scalar glueball mass in the range of 300 MeV 5 MGB< 600 MeV, in agreement with the results of the most recent calculations Ref. [8]. For the calculation of the lightest scalar meson mass, M,,, we eliminate the parameter g, by using the derivative sum rule obtained by taking d/&/M’> of both sides of Eq. (5). We fimd a stable solution: Mm = 980 MeV + 10 %, as in Ref. [7].
3. QCD sum rules for mixed scalar glueballs and mesons Although it is generally expected that gluonic and & modes do not mix strongly, the scalar glueball and meson currents do couple rather strongly, and there is a low-energy theorem [see, e.g., Novikov et al., Ref. [6]] which we can use to obtain a sum rule for a mixed scalar glueball/meson resonance. We introduce the current, J 0++ for the mixed system: J o++=PMoJm+(l-
IPI)J,,
(7)
where MO is a constant that we take to be 1 GeV. This M, constant effectively modifies the parameter g,, defined after Eq. (5). For mainly glueball or mainly meson solutions to the sum rules this parameter is eliminated, as discussed in Section 2 and below. For solutions which are not predominately meson or glueball our analysis would not be satisfactory and the M,g, parameter must be treated more carefully. Solutions are sought in the range 0 < I p I < 1, with p = 1 a scalar meson and p = 0 a scalar glueball. Using analyses similar to those used in the two calculations discussed in Section 2, which we describe below, we find stable solutions for /3 = 0.3 and j3 = -0.7, corresponding after normalization to a glueball with 20% meson admixture and a meson with 20% glueball admixture. We treat the former using the QCD sum rule approach for a glueball and the latter using the techniques developed for the meson. For I j3 I s 0.3 we assume that the pole term in the dispersion relation for n( p2> can be treated as a glueball for which we use the low energy theorem for n(O) given in Eq. (6). After the Bore1 transform the resulting sum rule is e-“:++/M2 + 0.0475&( so) = (1 - I p I)’
0.0077 1.876 1 - 0.0475M4 - 0.1125 - M4 - or@)
(
-1.83M2+---
0.039
0.033
M2
M4
-P(l-
lPl)O.235,
(8)
where we have used the low energy theorem [6] / d4x(OtT( J,( x) J,(O))iO> = F(4q).
(9)
Using the values of r@) in the range discussed above we the sum rule of Eq. (8) has stable solutions for p = 0.3 and the threshold parameter s,, in the range 2.5-2.8 and yields values for the mass of this scalar system M,, + (80% glueball) = 1450 + 100 MeV.
(10)
By a stable solution we mean that it satisfies the criteria of satisfying Eq. (8) over a range of Bore1 masses in the region of the value we obtain for M,, +. Also, the continuum contributes less that 50 %. For values of p less than about 0.2 or greater than about 0.4 the solutions are much less stable. Therefore it is consistent to identify this solution as a mainly-gluonic hadron in the region of the f&500).
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L.S. Kisslinger
et al./Physics
Letters B 410 (1997) l-5
Another set of solutions is found for a largely scalar meson. For values for I p I 2 0.7 we use the analysis described in Section 2 for the scalar meson. The sum rule is 3 7 + 0.039 - 0.329 I 2.48 0.01 0.0628M4-0.149---&=x .
M~++&-M:++/M= = ~2
1.43( 1 - E,( So)/@
1
(11)
Taking the ratio of the derivative of Eq. (11) to the equation itself one obtains the sum rule 0.023 2(1-E,(%))+~+
Ml++ = M2 l-E,(s,)/M4+~-x-
0.0272
0.023
’ (12)
with E,(s,) = (1 + so/M2 + si/2M4)exp(-s,/M2). Using the values of r@) in the range discussed above, we find that the sum rule of Eq. (12) has stable solutions for p = - 0.7 and the threshold parameter s0 = 2.3; and that it yields values for the mass of this scalar system M,++(8O%meson) = 1400 + 1OOMeV. (13) For larger or smaller values of p in this region of /3 = - 0.7 the solutions are less stable. The Z = 0 and Z = 1 solutions are degenerate in this treatment and therefore it is consistent to identify these solutions seem with the f&370) and a&1450). Since these resonances are quite broad, and the widths were not included in our sum rules, the error given in Eq. (13) is probably an underestimate. 4. Conclusions
The main result of this work is that it is essential to explicitly take into account the mixing of the scalar meson with the scalar glueball in the current used for predicting the masses via the QCD sum rule method. For the scalar glueball there is a dramatic effect: the low-mass solution obtained with a purely gluonic current is not present with a mixed current and the most stable solutions, found in the region of about 1450 MeV, are about 80% glueball. Although not so dramatic, the corresponding solutions, about 80% scalar mesons, move from the region of the f,(980) to about 1400 MeV. Although the accuracy of this work is not sufficient for a definitive spectroscopy, it is consistent to identify the largely glueball solution with the f,(1500), and the largely meson solutions with the f&370) and a&450). Gur solutions are also consistent with the f,(980) having a small ijq content. The pure scalar glueball solution at about 500 MeV is far from known ijq states.We are studying this solution at present. In future work we shall use our recently developed three-pont method [lo] to obtain branching ratios for decays of these resonances for experimental tests of the nature of these states. Acknowledgements
We would like to acknowledge helpful discussions with Curtis Meyer. This work is supported in part by National Science Foundation grant PHY-93 19641, and in part was a CMU undergraduate research project of J.G. and C.V. References [l] Crystal Barrel Collaboration, Phys. Lett. B 355 (1995) 425. [2] C. Amsler, F.E. Close, Phys. Lett. B 353 (1995) 385; Phys. Rev. D 53 (1996) 295.
L.S. Kisslinger et al./Physics Letters B 410 (1997) 1-5 [3] G. Bali et al. (UKQCD), Phys. Lett. B 309 (1993) 378; J. Sexton, A. Vaccarino, D. Weingarten, Phys. Rev. Lett. 75 (1995) 4563. [4] V.V. Anisovich, Yu.D. Prokoshkin, A.V. Saramtsev, Phys. Lett. B 389 (19%) 388; A.V. Anisovich, V.V. Anisovich, A.V. Saramtsev, Phys. Lett. B 395 (1997) 123. [5] F.E. Close, G.R. Farrar, 2. Li, Phys. Rev. D 55 (1997) 5749. [6] V.A. Novikov, M.A. Shifman, AI. Vainstein, V.I. Zakharov, Nucl. Phys. B 165 (1980) 67; B 191 (1981) 301; P. Pascual, R. Tarrah, Phys. Lett. B 113 (1982) 495. [7] LJ. Reinders, S. Yazaki, H.R. Rubinstein, Nucl. Phys. B 196 (1982) 125. [S] C.A. Dominguez, N. Paver, Zeit. Phys. C 31 (1986) 591; J. Bordes, V. Gimenez, J.A. Penarrocha, Phys. Lett. B 223 (1989) 251; J. Liu, D. Liu, J. Phys. G: Nucl. Part. Phys. 19 (1993) 373. [9] E.V. Shuryak, Nucl. Phys. B 319 (1989) 541; E.V. Shuryak, J.J.M. Verbaarschot, Nucl. Phys. B 410 (1993) 55. [lo] M.B. Johnson, L.S. Kisslinger, submitted for publication (1996).
__ B
ELSEVIER
25 September 1997 PHYSICS LETTERS B
Physics Letters B 410 (1997) I-5
Mixed scalar glueballs and mesons L.S. Kisslinger, J. Gardner, C. Vanderstraeten Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA
Received 18 April 1997;revised 2 July 1997 Editor: J.-P. Blaizot
Abstract We find QCD sum rule solutions for scalar meson-glueball mixed states consistent with the f,(15OO) being largely a glueball and the f,(1370) being a largely meson state, consistent with recent experiments. Pure scalar gluebail solutions are in the 300-600 MeV region and pure 0 ++ Fjq meson solutions are found near 1 GeV. 0 1997 Published by Elsevier Science B.V. PACS:
12.38.Lg; 14.4O.C~
1. Introduction The detection of glueballs and the measurement of their decay properties is important for the study of strong interactions. Recently, the Crystal Barrel Collaboration [l] has discovered a scalar CO++) state, the f&500), and
established from its branching ratios that it is likely the state of a scalar glueball admixed with a scalar meson [1,2]. This mass is consistent with recent lattice gauge calculations 131which find a scalar glueball in the region of 1500-1700 MeV. A K-matrix analysis of the Crystal Barrel and other data in the 500-1900 MeV region [4] finds suggests that there is a scalar glueball mixed with two L = 1 O++ meson states in the 1300-1500 MeV region. A recent quark model calculation [5] also finds a mixed glueball/meson state in this region. The coupled channel analysis by the Crystal Barrel Collaboration of the isoscalar f&1370) and isovector a&450) O++ resonances are consistent with their being largely 44 meson states. This analysis of the f&370) also involves
the f,(980) state, whose nature is unclear. Soon after the method of QCD sum rules was formulated it was applied to the search for scalar glueballs [6], as well as scalar mesons [7]. In recent years there have been a number of detailed sum rule studies of the lowest-lying scalar glueball [8], which all find a light, narrow glueball. A typical solution is 500 MeV, in conflict with experiment and lattice gauge calculations [3]. A solution for the scalar meson was found at about 1.0 GeV [7] and was interpreted as the f,(980), with the I = 1 scalar degenerate. The scalar mesons have also been treated in an instanton picture [9]. In this model scalar solutions are difficult to find, but also in the instanton model of the scalar qq mesons the masses seem to be near the f,(980) and the a,(980), which the
authors interpret as states not well-described as qFj mesons. In the present work we explore the possibility of mixed glueball-meson solutions within the method of QCD Sum Rules. It is known that there must be strong coupling between scalar glueballs and scalar mesons in the 0370-2693/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO370-2693(97)00964-7
2
L.S. Kisslinger
et al/Physics
Letters B 410 (1997) 1-5
same mass region. We show that the QCD sum rule for a current that is an admixture of a scalar meson with a scalar glueball current, consistent with known mixing [see Section 31, does not have the solutions for a light mass found [8] with a pure glueball current. Indeed, a solution with the mixed current is found near the f,(1500) that is about 80 % glueball. Moreover, with a current for a state that is 80 % scalar meson with a scalar glueball admixed the solutions for the isoscalar and isovector are found near the f&1370) and the a&450). In Section 2 we review the QCD sum rules for pure scalar mesons and pure scalar glueballs. In Section 3 we derive sum rules for mixed scalar glueballs and mesons, and discuss our results in Section 4.
2. Review of QCD sum rules for scalar glueballs mesons The starting point of the QCD sum rule method, as well as the lattice gauge calculations, is the correlator n( p’) = i/&e’“”
(OIT(J(++(O))IOL
(1)
a two point function with J(X) a composite field operator, the current. The currents are J&)=@)q(x),
Jo(x)=~G(x)G(x),
(2)
for the scalar meson and glueball, respectively. By meson we mean a Cjq state. GPy is the gauge field tensor and we have used the normalization of Novikov et al. [6] for the glueball. Equating the once-subtracted dispersion relation to the operator product expansion of the correlator for the glueball one finds [6,8]
w~Gwl = - -17,(O)-
2$p21n(pl)
4a; + pl(G.G)+~T’6)+~~@),.
P2
s(s +p2)
P
(3) P
where (G . G) is the usual gluon condensate, r@) = ( g,f,,,G;vG,b GC > and r(*) = (14(f,b,G:vG&)2 (&,,G,“yG,6,>2). Treating the continuum in the usual way, having an .r’ fpo’l;m with a threshold s0 and carrying out the Bore1 transform one finds the sum rule for A&, the lightest scalar glueball mass I&(0)(
f?-“b/MZ-
2a,2 1.0) + TEl(so)
KY =~;;~M”+~u:(C.C)-~~(~‘-
SCY
4?rcY3 -r(s), M4
(4)
with M the Bore1 mass and E,(s,) = M4(1 + s,,/M2)exp(-- s0/M2). In deriving Eq. (4) we use the low energy theorem discussed in Refs. 1681, giving h4& I&(O) = (I CX,G]GB>~. For the scalar meson the QCD sum rule, keeping terms up to D = 6, is [7]
In obtaining Eq. (5) the factorization approximation of the four-quark vacuum matrix elements given in Ref. [6] has been used. The constant g, is defined as (]J,lm(O”)> = g,. The value of the parameters appearing in Eqs. (41, (5) in the early work [6] cxJ?i-= 0.1 ( a,G . G) = 0.0377 GeV4 r@) = 0.01 14GeV6 r(s) = 0.56(( cr,G . G))2 = 0.0081 GeV* no(O) = 3.5(G. G),
(6) with the expression for I&(O) derived from a low energy theorem discussed by Novikov et al. in Ref. [61. To the extent that factorization is a good approximation, the greatest uncertainty in the parameters is the value of
L.S. Kisslinger et al./Physics
3
Letters B 410 (1997) l-5
r@). In Refs. [8] the range of values from lattice gauge calculations and other work was discussed and used in the sum rules. Here we take the range of - 0.01 to 0.05 GeV6. We also explore larger values of (G . G), increasing the value of the gluon condensate by up to a factor of two times the standard value given in Eq. (6). Our results are that we find a solution for the scalar glueball mass in the range of 300 MeV 5 MGB< 600 MeV, in agreement with the results of the most recent calculations Ref. [8]. For the calculation of the lightest scalar meson mass, M,,, we eliminate the parameter g, by using the derivative sum rule obtained by taking d/&/M’> of both sides of Eq. (5). We fimd a stable solution: Mm = 980 MeV + 10 %, as in Ref. [7].
3. QCD sum rules for mixed scalar glueballs and mesons Although it is generally expected that gluonic and & modes do not mix strongly, the scalar glueball and meson currents do couple rather strongly, and there is a low-energy theorem [see, e.g., Novikov et al., Ref. [6]] which we can use to obtain a sum rule for a mixed scalar glueball/meson resonance. We introduce the current, J 0++ for the mixed system: J o++=PMoJm+(l-
IPI)J,,
(7)
where MO is a constant that we take to be 1 GeV. This M, constant effectively modifies the parameter g,, defined after Eq. (5). For mainly glueball or mainly meson solutions to the sum rules this parameter is eliminated, as discussed in Section 2 and below. For solutions which are not predominately meson or glueball our analysis would not be satisfactory and the M,g, parameter must be treated more carefully. Solutions are sought in the range 0 < I p I < 1, with p = 1 a scalar meson and p = 0 a scalar glueball. Using analyses similar to those used in the two calculations discussed in Section 2, which we describe below, we find stable solutions for /3 = 0.3 and j3 = -0.7, corresponding after normalization to a glueball with 20% meson admixture and a meson with 20% glueball admixture. We treat the former using the QCD sum rule approach for a glueball and the latter using the techniques developed for the meson. For I j3 I s 0.3 we assume that the pole term in the dispersion relation for n( p2> can be treated as a glueball for which we use the low energy theorem for n(O) given in Eq. (6). After the Bore1 transform the resulting sum rule is e-“:++/M2 + 0.0475&( so) = (1 - I p I)’
0.0077 1.876 1 - 0.0475M4 - 0.1125 - M4 - or@)
(
-1.83M2+---
0.039
0.033
M2
M4
-P(l-
lPl)O.235,
(8)
where we have used the low energy theorem [6] / d4x(OtT( J,( x) J,(O))iO> = F(4q).
(9)
Using the values of r@) in the range discussed above we the sum rule of Eq. (8) has stable solutions for p = 0.3 and the threshold parameter s,, in the range 2.5-2.8 and yields values for the mass of this scalar system M,, + (80% glueball) = 1450 + 100 MeV.
(10)
By a stable solution we mean that it satisfies the criteria of satisfying Eq. (8) over a range of Bore1 masses in the region of the value we obtain for M,, +. Also, the continuum contributes less that 50 %. For values of p less than about 0.2 or greater than about 0.4 the solutions are much less stable. Therefore it is consistent to identify this solution as a mainly-gluonic hadron in the region of the f&500).
4
L.S. Kisslinger
et al./Physics
Letters B 410 (1997) l-5
Another set of solutions is found for a largely scalar meson. For values for I p I 2 0.7 we use the analysis described in Section 2 for the scalar meson. The sum rule is 3 7 + 0.039 - 0.329 I 2.48 0.01 0.0628M4-0.149---&=x .
M~++&-M:++/M= = ~2
1.43( 1 - E,( So)/@
1
(11)
Taking the ratio of the derivative of Eq. (11) to the equation itself one obtains the sum rule 0.023 2(1-E,(%))+~+
Ml++ = M2 l-E,(s,)/M4+~-x-
0.0272
0.023
’ (12)
with E,(s,) = (1 + so/M2 + si/2M4)exp(-s,/M2). Using the values of r@) in the range discussed above, we find that the sum rule of Eq. (12) has stable solutions for p = - 0.7 and the threshold parameter s0 = 2.3; and that it yields values for the mass of this scalar system M,++(8O%meson) = 1400 + 1OOMeV. (13) For larger or smaller values of p in this region of /3 = - 0.7 the solutions are less stable. The Z = 0 and Z = 1 solutions are degenerate in this treatment and therefore it is consistent to identify these solutions seem with the f&370) and a&1450). Since these resonances are quite broad, and the widths were not included in our sum rules, the error given in Eq. (13) is probably an underestimate. 4. Conclusions
The main result of this work is that it is essential to explicitly take into account the mixing of the scalar meson with the scalar glueball in the current used for predicting the masses via the QCD sum rule method. For the scalar glueball there is a dramatic effect: the low-mass solution obtained with a purely gluonic current is not present with a mixed current and the most stable solutions, found in the region of about 1450 MeV, are about 80% glueball. Although not so dramatic, the corresponding solutions, about 80% scalar mesons, move from the region of the f,(980) to about 1400 MeV. Although the accuracy of this work is not sufficient for a definitive spectroscopy, it is consistent to identify the largely glueball solution with the f,(1500), and the largely meson solutions with the f&370) and a&450). Gur solutions are also consistent with the f,(980) having a small ijq content. The pure scalar glueball solution at about 500 MeV is far from known ijq states.We are studying this solution at present. In future work we shall use our recently developed three-pont method [lo] to obtain branching ratios for decays of these resonances for experimental tests of the nature of these states. Acknowledgements
We would like to acknowledge helpful discussions with Curtis Meyer. This work is supported in part by National Science Foundation grant PHY-93 19641, and in part was a CMU undergraduate research project of J.G. and C.V. References [l] Crystal Barrel Collaboration, Phys. Lett. B 355 (1995) 425. [2] C. Amsler, F.E. Close, Phys. Lett. B 353 (1995) 385; Phys. Rev. D 53 (1996) 295.
L.S. Kisslinger et al./Physics Letters B 410 (1997) 1-5 [3] G. Bali et al. (UKQCD), Phys. Lett. B 309 (1993) 378; J. Sexton, A. Vaccarino, D. Weingarten, Phys. Rev. Lett. 75 (1995) 4563. [4] V.V. Anisovich, Yu.D. Prokoshkin, A.V. Saramtsev, Phys. Lett. B 389 (19%) 388; A.V. Anisovich, V.V. Anisovich, A.V. Saramtsev, Phys. Lett. B 395 (1997) 123. [5] F.E. Close, G.R. Farrar, 2. Li, Phys. Rev. D 55 (1997) 5749. [6] V.A. Novikov, M.A. Shifman, AI. Vainstein, V.I. Zakharov, Nucl. Phys. B 165 (1980) 67; B 191 (1981) 301; P. Pascual, R. Tarrah, Phys. Lett. B 113 (1982) 495. [7] LJ. Reinders, S. Yazaki, H.R. Rubinstein, Nucl. Phys. B 196 (1982) 125. [S] C.A. Dominguez, N. Paver, Zeit. Phys. C 31 (1986) 591; J. Bordes, V. Gimenez, J.A. Penarrocha, Phys. Lett. B 223 (1989) 251; J. Liu, D. Liu, J. Phys. G: Nucl. Part. Phys. 19 (1993) 373. [9] E.V. Shuryak, Nucl. Phys. B 319 (1989) 541; E.V. Shuryak, J.J.M. Verbaarschot, Nucl. Phys. B 410 (1993) 55. [lo] M.B. Johnson, L.S. Kisslinger, submitted for publication (1996).