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An extension of the 13P0 model for qq̄ annihilation
Volume 172, number 2
PHYSICS LETTERS B
15 May 1986
AN E X T E N S I O N OF THE t3Po M O D E L FOR q~! ANNIHILATION A.M. G R E E N , J.A. N I S K A N E N ~ and S. W Y C E C H 2,3 Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20 C, 00170 Helsinki, Finland
Received 28 January 1986; revised manuscript received 6 March 1986
It is shown that a natural extension of the 13po model for qCt annihilation introduces effects that are essentially indistinguishable from those usually attributed to the 13S1 component of the gluon exchange model e.g. NN(S-wave) ---,~ro.
In the description of NN annihilation into two mesons there are two popular alternatives for the basic q~ annihilation vertex - the 13S 1 and 13p 0 models [1 ]. Both o f these have varying degrees of success and failure. For example, the process p~ (S-wave) ~ zrp which is observed to have a ~--5% branching ratio is forbidden in lowest order by the 13p 0 model but not by the 13S 1 model. On the other hand, the opposite is true for an understanding of meson decays (M 1 M2M3) and baryon meson vertices (B 1 ~ B2M3), where the 13p 0 model has been rather successful. However, this black-white situation is not the whole story. As shown recently in refs. [2,3], a consistent reduction o f the one-gluon exchange process in fig. 1 leads to a 3S 1 operator of the form 0(3S1) = 01(3S1) + 02(3S1) = (41r/8m3)~tRCR8(r23)[a(1) X o (23)]
• V(rl2)6(rl2 ) + i(4rt/4m3)O~RCRf(r12)o(23) • V(rl)~(r23 ) , ( l )
where the term 01 (3 S i) is the one frequently used in the literature when discussing NN annihilation via a gluon [4]. However, the term 0 2( 3 S 1) arises natural1 On leave of absence from the Department of Theoretical Physics, University of Helsinki, 00170 Helsinki, Finland. 2 Permanent address: Institute for Nuclear Studies, 00681 Warsaw, Poland. 3 Partially supported by the Polish-US Maria Sklodowska-Curie fund under grant number P-F7F037P. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Fig. 1. Quark-antiquark pair annihilation into a gluon with one external quark line.
ly and can not be ignored. In particular it has a character that in some respects is quite different from 01 (3S1). For example in the process p p ~ lr+n - , 01(3S1) contributes in T = S = 1 states such as 33S1, 33D 1 ,'..., whereas 02(3S1) contributes only in T = 0, S = 1 states such as 13P0, 13.P2. In fact 02(3S1) behaves in some ways like the l a P 0 mechanism. It is therefore possible that the best of both q~ annihilation models, 13S 1 and 13P0, may be explained by simply the extended form of 0(3S 1) in eq. (1). This dual character of one-gluon exchange should, of course, not be a surprise. It is simply a reflection o f the fact that a virtual gluon, like a photon, has both 1- (transverse) and 0 ÷ (longitudinal) components. The above state of affairs may sound rather promising. However, with the 13 S1 model there is the drawback that it is based on the one-gluon exchange mechanism being used in a situation where soft gluons are expected to dominate. So, at best, the mechanism must be simulating multigluon processes and, as such, is an "effective" perturbative approach. In this context it has been conjectured that this ob171
PHYSICS LETTERS B
Volume 172, number 2
15May 1986
sation of 01 (13Po ) is the overlap
rt MI~
~M25
0-(13P0) cc f d r 1 ... dr 6 u+(rl)u+(r2) X o+(r 4 ) oa (r 5 ) u+ (r 6) 70 (36) ~ (r 3 - r6 )
X u(rl)u(r2)u(r3)o(r4)o(r5) bl
N 1 23
X 6(½(r l +"4) -1(/'2 + rs) - r')
6/,5 r
i,_
×8(½(rl + i ' 2 + r 3 ) - ½ ( r 4 + r 5 + r 6 ) - r ).
a)
Fig. 2. Quark-antiquark pair annihilation into vacuum. (a) With one external quark line. (b) Embedded in an environment appropriate to NN annihilation.
jection does not apply to situations which involve the creation of heavy quarks [5] e.g. NN ~ YY. Another point against the 3S 1 model is that in more basic approaches, such as the flux-tube form of QCD in a lattice, it is the 13p 0 vertex which seems to be more natural [6]. The question then arises concerning the possibility of extending the usual 13p 0 vertex (see fig. 2a) 01(13P0) = 71136 " V(r36)6(r36)
(2)
to include another term that has some of the features of the 13S 1 vertex 01(13S1). A mechanism for generating such a term is suggested, when it is noted that the form of eq. (2) arises from the overlap
X ~(r 3 -- r6) ,
/ 1 )
u(r3) = N -i11" V(r3)
exp(iq3 • r3) ,
E+m v(r6) = N
( i11" •(r6) ) E +m exp(-iq6 • r6).
(3)
1
Here m is the mass of the quark and N the normalisation constant. In the case of the particular multiquark process shown in fig. 2b, the natural general.i172
In eqs. (3) ant (4) the proportionality factor has dimensions and carries the relic of the underlying dynamics. If the quark wave functions in eqs. (4) are now taken to be th~ solutions of the single-particle Dirac equation containing the oscillator potential ½(1 + 70)tx2r 2, then they have the form, for example,
u(r, ~) = u,~
1)
-i11" V(r)
exp(--~c~'2).
(5)
E,~ +m This results in an extended 13p 0 vertex with the structure 0-(13P0) = 0-1(13P0) + 02(13P0) = -i1136 " r
+ [af3/(Ea 4 m)(E# + m)]-~(a +/~)-1 X [1 --~(t~ +/3)r 2] X 1136" [(( ~1 - 114) - (112 - 115)) X r ' ] ,
01(13Po) cc f d r 3 dr6 v+(r6)70(36)u(r3)
where
(4)
b)
(6)
where, in the notation of ref. [1 ] , a and 13refer to the oscillator paraaaeters in_the N or N and final mesons respectively. The term 0 2 arises as a recoil effect and it should be erlphasized that this is not a spurious CM motion since t his calculation was carried out using throughout relative coordinates for describing the N, and meson wave functions. The term 0-1 illustrates, through the presence of the r factor, the feature that the basic 13p 0 model o f e q . (2) acts in NN P-waves and is forbidden in S-waves. On the other hand, the term 0-2 is reminiscent of the 01(13S1) term in eq. (1). It is interesting that both the terms in eq. (6) apparently arise from pure kinematics without any explicit dynamic input. However, it should be noted
Volume 172, number 2
PItYSICS LETTERS B
that the bound states are solutions of homogeneous IAppmann--Sghwinger equations actually involving the confining interaction - presumably of gluonic origin - to all orders. Since O1 and 02 in eq. (6) act in different partial waves their relative magnitudes cannot be compared directly in a model independent way. However, a measure of the importance of the small components in the wave function of eq. (5) - the origin of 0 2 can to some extent be judged by inspecting the normalisation integral N2
f [1 + a2r2/(E + m) 2] exp(-,m "2) = 0r/a) 3/2 [1 + -~a/(E + m) 2 ] = 0r/c03/2(1.32),
(7)
using the parameters ofref. [7] a = 2472 MeV2; E = 438 MeV and m = 94.3 MeV. This shows 230% effects are introduced by the advent of small components in the quark wave functions. However, this finding should not be surprising since for light quarks (P/m) ~ 1 and as shown in ref. [8], their presence can account for decays that would otherwise be forbidden e.g. r ~ AlV r. In NN annihilation a more interesting comparison is that between the NN (S-wave)~ M IM 2 branching ratios calculated both with 01 (13 S 1) in eq. (1) and 02 in eq. (6). The most detailed and explicit use of 01(13S 1) is in refs. [3,9]. These works contain many refinements and so the comparison can be carried out at different levels, the most simple being to calculate the corresponding spin-isospin dependent matrix elements - in the notation of fig. 2b
~ C(TSS'MI4M~SM~6) MI4M~s
[NRIrS=
M'36S' X I[MIaM:~5] rS'M~6] rs,
(9)
where M'I4 , M~5 are s-wave meson wave functions (rr, 7?, p and ~o) with the third meson always being an ~o-meson. The C can be derived as in ref. [1 ]. In eq. (8) the final wave function is simply two S-wave mesons in a relative p-wave. The operator 09 annihilates the to and converts [M~4M~5 ]TS' into [x~/14M25 ] TS'. It might be thought that the w does not have the correct internal angular momentum to annihilate into the vacuum. However, this is only an illusion since the original operator in eq. (4) did need the annihilating q3q6 to be in a p-wave. The latter p-wave is then neutralized to be replaced by a p-wave in the final state as seen when comparing 01,2 in eq. (6). The net effect of this is that the ratios of the M(TS) in eq. (8) turn out to be precisely the same as those obtained from the spin-isospin structure of 01 (3 S 1) the usual one-gluon exchange operator. The actual values of M are shown in table 1 and agree (upto an overall multiplicative factor)with the results of refs. [3,9]. The above findings should not be surprising since it can be shown by directly comparing the operators 01 (3S1) and 02(13P0) that the angular momentum structures are indeed the same. However, the radial dependences will be different. For example, 01(3S1) leads to
Table 1
M(TS) of eq. (8) normalised so that M(00, tow) = 1
M(TS, M 14M25)
= C(TSS 'MI4M~5co) X ([ [M14M25 ] TS" X Y1 (r')] TS i1~-2 U[[M~4M~5 ] TS'~o] TS).
15 May 1986
(8)
Here the initial wave function is related to the original S-wave NN wave function by the rearrangement
S
T
S"
Mesons
M(ST,Mt4JI=s)
0 0 0 0
0 0 1 1
I 1 1 1
toto pp pto prr
+1 - ! &/3 +2~/2/3 -x/2
1 I
0 0
1
,on
l
wO
- 11~/2/6 - l/x/6
1
1
0
rrtr
+2/x/3
1
1
0
pp
+2/3
1
1
1
lrw
1
1
I
np
1
I
2
pp
-7/3,,,/6 - 1/3~/6 -x/5/3 173
Volume 172, number 2
PHYSICS LETTERS B
Rs(r"r)=r'exp(-r'22
4a+3fl5ct + 4fl
_ r2 3 7 ~ 2 + 18t~fl + 9fl 2
8 --r'r
4 a + 3fl
,3
~ ~T~-]
r ' e x p ( - 0 . 6 5 00`'2 - 1.82 a~r2 - 0.43 r • r ' ) , (10) for ct = fl -- the result not bein~ strongly dependent on ft. On the other hand, O2(1~P0) becomes simply
Rp(r', r)
= r ' [ 1 - -~(a + fl)r 2 ]
X exp [-lc~r '2 - ~ ( o t +
3fl)r2l
= r ' [ 1 - - ~ a r 2 ] e x p [ - ½ a r '2 - ~ a r 2 ] ,
(11)
which is very similar to the form for NN annihilation into two mesons - one being a p-wave meson - (ref. [10] eq. (2.5)). It is seen that the main qualitative difference between eqs. (10) and (11) is the appearance in the latter o f a node at -~-0.5 fm. This could have a large effect for a given initial-state wave function and lead to significant differences between the separate NN partial waves. However, for a given partial wave it naturally does n o t affect the relative branching ratios. This is not so for the r' dependence in eqs. (10) and (11). There it is seen that R s has a somewhat shorter range ( 1 / ~ ) compared with the extended 3P 0 model form (Rp) and its range o f 1 / ~ . As shown in table 2, this range difference greatly enhan-
Table 2 NN ~ MtM 2 phase-space integrals from eqs. (10) and (11) for a = 2.78 fm -2 at NN threshold. Numbers in parentheses correspond to E(p, lab) = 300 MeV [units 109MeV4 ] Mesons
pp pro p~r n~/ lr~r 174
Integrals eq. (10)
eq. (11)
ratio
5.75 7.20 2.91 3.34 0.81
3.16 3.99 0.97 0.96 0.13
1.82 1.80 2.98 3.46 6.36
(2.16) (z26) (3.56) (4.73) (8.64)
15 May 1986
ces the branching ratios for the lighter mesons e.g. a factor o f 8.64/2.16 ~ 4 for 7rzr compared with pp at E(ff, lab) = 30£~ MeV. The possibility o f distinguishing between th~ 3S 1 and extended 3P 0 models therefore reduces to their relative differences in the phasespace integrals and not to some symmetry in the basic operators o f eq. (1) and (6). The situation could well be worse than this, since eq. (1) for convenience considers the one-gluon exchange interaction to be ofz~.~o range (the 6(r12 ) factor). It is this additional 6-function in eq. (1) that then forces R s in eq. (10) to t e shorter range in r' t h a n R p in eq. (11) and so leads to the larger branching ratios in table 2 for light mesons. Another difference between an explicit gluon exchange eq. (10) and the present approach is that the former involves also a non-separable term ~ e x p ( - ~ - a r • r ' ) ( i f a = fl) acting in all partial waves aad n o t appearing in eq. (11). The conclu,qon to be reached by the above arguments is that b'N annihilation into two s-wave mesons is not contrary to the spirit o f the 13P0-model. A natural extension to this model then leads to branching ratk,s, which are hard to distinguish from those generated b y one-gluon exchange.
References [1] A.M. Greer and J.A. Niskanen, International Review of Nuclear Ph esics, Vol. 1, ed. W. Weise (World Scientific, Singapore, 1984) p. 569; A.M. Greer and J.A. Niskanen, Helsinki preprint HUTFT-85-60. in: Progress in Nuclear and Particle Physics, ed. A. Faessler (Pergamon, Oxford, 1986), to be published. [2] M. Kohno and W. Weise, Phys. Lett. B t52 (t985) 303; Regensburj, preprint TPR-85- 25. [3] E.M. Henley, T. Oka and J. Vergados, Phys. Lett. B 166 (1986) 214. [4] A. Faessler, G. Li~beck and K. Shimizu, Phys. Rev. D26 (1982) 32[ 0; M. Alberg ct al., Phys. Rev. D27 (1983) 536. [5] H. Rubinst~in and H. SneUman, Phys. Lett. B 165 (1985) 18~. [6] N. Isgur and J. Paten, Phys. Lett. B 124 (1983) 247; Phys. Rev. D31 (1985) 2910. [7] R. Tegen, R. Brockmann and W. Weise, Z. Phys. A307 (1982) 339. [8] C. Hayne and N. Isgur, Phys. Rev. 25D (1982) 1944. [9] M. Maruyarna, Ph.D. thesis O~aka University (1984), unpublished. [10] A.M. Green, V. Kuikka and J.A. Niskanen, Nuct. Phys. A446 (1985) 543.
PHYSICS LETTERS B
15 May 1986
AN E X T E N S I O N OF THE t3Po M O D E L FOR q~! ANNIHILATION A.M. G R E E N , J.A. N I S K A N E N ~ and S. W Y C E C H 2,3 Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20 C, 00170 Helsinki, Finland
Received 28 January 1986; revised manuscript received 6 March 1986
It is shown that a natural extension of the 13po model for qCt annihilation introduces effects that are essentially indistinguishable from those usually attributed to the 13S1 component of the gluon exchange model e.g. NN(S-wave) ---,~ro.
In the description of NN annihilation into two mesons there are two popular alternatives for the basic q~ annihilation vertex - the 13S 1 and 13p 0 models [1 ]. Both o f these have varying degrees of success and failure. For example, the process p~ (S-wave) ~ zrp which is observed to have a ~--5% branching ratio is forbidden in lowest order by the 13p 0 model but not by the 13S 1 model. On the other hand, the opposite is true for an understanding of meson decays (M 1 M2M3) and baryon meson vertices (B 1 ~ B2M3), where the 13p 0 model has been rather successful. However, this black-white situation is not the whole story. As shown recently in refs. [2,3], a consistent reduction o f the one-gluon exchange process in fig. 1 leads to a 3S 1 operator of the form 0(3S1) = 01(3S1) + 02(3S1) = (41r/8m3)~tRCR8(r23)[a(1) X o (23)]
• V(rl2)6(rl2 ) + i(4rt/4m3)O~RCRf(r12)o(23) • V(rl)~(r23 ) , ( l )
where the term 01 (3 S i) is the one frequently used in the literature when discussing NN annihilation via a gluon [4]. However, the term 0 2( 3 S 1) arises natural1 On leave of absence from the Department of Theoretical Physics, University of Helsinki, 00170 Helsinki, Finland. 2 Permanent address: Institute for Nuclear Studies, 00681 Warsaw, Poland. 3 Partially supported by the Polish-US Maria Sklodowska-Curie fund under grant number P-F7F037P. 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Fig. 1. Quark-antiquark pair annihilation into a gluon with one external quark line.
ly and can not be ignored. In particular it has a character that in some respects is quite different from 01 (3S1). For example in the process p p ~ lr+n - , 01(3S1) contributes in T = S = 1 states such as 33S1, 33D 1 ,'..., whereas 02(3S1) contributes only in T = 0, S = 1 states such as 13P0, 13.P2. In fact 02(3S1) behaves in some ways like the l a P 0 mechanism. It is therefore possible that the best of both q~ annihilation models, 13S 1 and 13P0, may be explained by simply the extended form of 0(3S 1) in eq. (1). This dual character of one-gluon exchange should, of course, not be a surprise. It is simply a reflection o f the fact that a virtual gluon, like a photon, has both 1- (transverse) and 0 ÷ (longitudinal) components. The above state of affairs may sound rather promising. However, with the 13 S1 model there is the drawback that it is based on the one-gluon exchange mechanism being used in a situation where soft gluons are expected to dominate. So, at best, the mechanism must be simulating multigluon processes and, as such, is an "effective" perturbative approach. In this context it has been conjectured that this ob171
PHYSICS LETTERS B
Volume 172, number 2
15May 1986
sation of 01 (13Po ) is the overlap
rt MI~
~M25
0-(13P0) cc f d r 1 ... dr 6 u+(rl)u+(r2) X o+(r 4 ) oa (r 5 ) u+ (r 6) 70 (36) ~ (r 3 - r6 )
X u(rl)u(r2)u(r3)o(r4)o(r5) bl
N 1 23
X 6(½(r l +"4) -1(/'2 + rs) - r')
6/,5 r
i,_
×8(½(rl + i ' 2 + r 3 ) - ½ ( r 4 + r 5 + r 6 ) - r ).
a)
Fig. 2. Quark-antiquark pair annihilation into vacuum. (a) With one external quark line. (b) Embedded in an environment appropriate to NN annihilation.
jection does not apply to situations which involve the creation of heavy quarks [5] e.g. NN ~ YY. Another point against the 3S 1 model is that in more basic approaches, such as the flux-tube form of QCD in a lattice, it is the 13p 0 vertex which seems to be more natural [6]. The question then arises concerning the possibility of extending the usual 13p 0 vertex (see fig. 2a) 01(13P0) = 71136 " V(r36)6(r36)
(2)
to include another term that has some of the features of the 13S 1 vertex 01(13S1). A mechanism for generating such a term is suggested, when it is noted that the form of eq. (2) arises from the overlap
X ~(r 3 -- r6) ,
/ 1 )
u(r3) = N -i11" V(r3)
exp(iq3 • r3) ,
E+m v(r6) = N
( i11" •(r6) ) E +m exp(-iq6 • r6).
(3)
1
Here m is the mass of the quark and N the normalisation constant. In the case of the particular multiquark process shown in fig. 2b, the natural general.i172
In eqs. (3) ant (4) the proportionality factor has dimensions and carries the relic of the underlying dynamics. If the quark wave functions in eqs. (4) are now taken to be th~ solutions of the single-particle Dirac equation containing the oscillator potential ½(1 + 70)tx2r 2, then they have the form, for example,
u(r, ~) = u,~
1)
-i11" V(r)
exp(--~c~'2).
(5)
E,~ +m This results in an extended 13p 0 vertex with the structure 0-(13P0) = 0-1(13P0) + 02(13P0) = -i1136 " r
+ [af3/(Ea 4 m)(E# + m)]-~(a +/~)-1 X [1 --~(t~ +/3)r 2] X 1136" [(( ~1 - 114) - (112 - 115)) X r ' ] ,
01(13Po) cc f d r 3 dr6 v+(r6)70(36)u(r3)
where
(4)
b)
(6)
where, in the notation of ref. [1 ] , a and 13refer to the oscillator paraaaeters in_the N or N and final mesons respectively. The term 0 2 arises as a recoil effect and it should be erlphasized that this is not a spurious CM motion since t his calculation was carried out using throughout relative coordinates for describing the N, and meson wave functions. The term 0-1 illustrates, through the presence of the r factor, the feature that the basic 13p 0 model o f e q . (2) acts in NN P-waves and is forbidden in S-waves. On the other hand, the term 0-2 is reminiscent of the 01(13S1) term in eq. (1). It is interesting that both the terms in eq. (6) apparently arise from pure kinematics without any explicit dynamic input. However, it should be noted
Volume 172, number 2
PItYSICS LETTERS B
that the bound states are solutions of homogeneous IAppmann--Sghwinger equations actually involving the confining interaction - presumably of gluonic origin - to all orders. Since O1 and 02 in eq. (6) act in different partial waves their relative magnitudes cannot be compared directly in a model independent way. However, a measure of the importance of the small components in the wave function of eq. (5) - the origin of 0 2 can to some extent be judged by inspecting the normalisation integral N2
f [1 + a2r2/(E + m) 2] exp(-,m "2) = 0r/a) 3/2 [1 + -~a/(E + m) 2 ] = 0r/c03/2(1.32),
(7)
using the parameters ofref. [7] a = 2472 MeV2; E = 438 MeV and m = 94.3 MeV. This shows 230% effects are introduced by the advent of small components in the quark wave functions. However, this finding should not be surprising since for light quarks (P/m) ~ 1 and as shown in ref. [8], their presence can account for decays that would otherwise be forbidden e.g. r ~ AlV r. In NN annihilation a more interesting comparison is that between the NN (S-wave)~ M IM 2 branching ratios calculated both with 01 (13 S 1) in eq. (1) and 02 in eq. (6). The most detailed and explicit use of 01(13S 1) is in refs. [3,9]. These works contain many refinements and so the comparison can be carried out at different levels, the most simple being to calculate the corresponding spin-isospin dependent matrix elements - in the notation of fig. 2b
~ C(TSS'MI4M~SM~6) MI4M~s
[NRIrS=
M'36S' X I[MIaM:~5] rS'M~6] rs,
(9)
where M'I4 , M~5 are s-wave meson wave functions (rr, 7?, p and ~o) with the third meson always being an ~o-meson. The C can be derived as in ref. [1 ]. In eq. (8) the final wave function is simply two S-wave mesons in a relative p-wave. The operator 09 annihilates the to and converts [M~4M~5 ]TS' into [x~/14M25 ] TS'. It might be thought that the w does not have the correct internal angular momentum to annihilate into the vacuum. However, this is only an illusion since the original operator in eq. (4) did need the annihilating q3q6 to be in a p-wave. The latter p-wave is then neutralized to be replaced by a p-wave in the final state as seen when comparing 01,2 in eq. (6). The net effect of this is that the ratios of the M(TS) in eq. (8) turn out to be precisely the same as those obtained from the spin-isospin structure of 01 (3 S 1) the usual one-gluon exchange operator. The actual values of M are shown in table 1 and agree (upto an overall multiplicative factor)with the results of refs. [3,9]. The above findings should not be surprising since it can be shown by directly comparing the operators 01 (3S1) and 02(13P0) that the angular momentum structures are indeed the same. However, the radial dependences will be different. For example, 01(3S1) leads to
Table 1
M(TS) of eq. (8) normalised so that M(00, tow) = 1
M(TS, M 14M25)
= C(TSS 'MI4M~5co) X ([ [M14M25 ] TS" X Y1 (r')] TS i1~-2 U[[M~4M~5 ] TS'~o] TS).
15 May 1986
(8)
Here the initial wave function is related to the original S-wave NN wave function by the rearrangement
S
T
S"
Mesons
M(ST,Mt4JI=s)
0 0 0 0
0 0 1 1
I 1 1 1
toto pp pto prr
+1 - ! &/3 +2~/2/3 -x/2
1 I
0 0
1
,on
l
wO
- 11~/2/6 - l/x/6
1
1
0
rrtr
+2/x/3
1
1
0
pp
+2/3
1
1
1
lrw
1
1
I
np
1
I
2
pp
-7/3,,,/6 - 1/3~/6 -x/5/3 173
Volume 172, number 2
PHYSICS LETTERS B
Rs(r"r)=r'exp(-r'22
4a+3fl5ct + 4fl
_ r2 3 7 ~ 2 + 18t~fl + 9fl 2
8 --r'r
4 a + 3fl
,3
~ ~T~-]
r ' e x p ( - 0 . 6 5 00`'2 - 1.82 a~r2 - 0.43 r • r ' ) , (10) for ct = fl -- the result not bein~ strongly dependent on ft. On the other hand, O2(1~P0) becomes simply
Rp(r', r)
= r ' [ 1 - -~(a + fl)r 2 ]
X exp [-lc~r '2 - ~ ( o t +
3fl)r2l
= r ' [ 1 - - ~ a r 2 ] e x p [ - ½ a r '2 - ~ a r 2 ] ,
(11)
which is very similar to the form for NN annihilation into two mesons - one being a p-wave meson - (ref. [10] eq. (2.5)). It is seen that the main qualitative difference between eqs. (10) and (11) is the appearance in the latter o f a node at -~-0.5 fm. This could have a large effect for a given initial-state wave function and lead to significant differences between the separate NN partial waves. However, for a given partial wave it naturally does n o t affect the relative branching ratios. This is not so for the r' dependence in eqs. (10) and (11). There it is seen that R s has a somewhat shorter range ( 1 / ~ ) compared with the extended 3P 0 model form (Rp) and its range o f 1 / ~ . As shown in table 2, this range difference greatly enhan-
Table 2 NN ~ MtM 2 phase-space integrals from eqs. (10) and (11) for a = 2.78 fm -2 at NN threshold. Numbers in parentheses correspond to E(p, lab) = 300 MeV [units 109MeV4 ] Mesons
pp pro p~r n~/ lr~r 174
Integrals eq. (10)
eq. (11)
ratio
5.75 7.20 2.91 3.34 0.81
3.16 3.99 0.97 0.96 0.13
1.82 1.80 2.98 3.46 6.36
(2.16) (z26) (3.56) (4.73) (8.64)
15 May 1986
ces the branching ratios for the lighter mesons e.g. a factor o f 8.64/2.16 ~ 4 for 7rzr compared with pp at E(ff, lab) = 30£~ MeV. The possibility o f distinguishing between th~ 3S 1 and extended 3P 0 models therefore reduces to their relative differences in the phasespace integrals and not to some symmetry in the basic operators o f eq. (1) and (6). The situation could well be worse than this, since eq. (1) for convenience considers the one-gluon exchange interaction to be ofz~.~o range (the 6(r12 ) factor). It is this additional 6-function in eq. (1) that then forces R s in eq. (10) to t e shorter range in r' t h a n R p in eq. (11) and so leads to the larger branching ratios in table 2 for light mesons. Another difference between an explicit gluon exchange eq. (10) and the present approach is that the former involves also a non-separable term ~ e x p ( - ~ - a r • r ' ) ( i f a = fl) acting in all partial waves aad n o t appearing in eq. (11). The conclu,qon to be reached by the above arguments is that b'N annihilation into two s-wave mesons is not contrary to the spirit o f the 13P0-model. A natural extension to this model then leads to branching ratk,s, which are hard to distinguish from those generated b y one-gluon exchange.
References [1] A.M. Greer and J.A. Niskanen, International Review of Nuclear Ph esics, Vol. 1, ed. W. Weise (World Scientific, Singapore, 1984) p. 569; A.M. Greer and J.A. Niskanen, Helsinki preprint HUTFT-85-60. in: Progress in Nuclear and Particle Physics, ed. A. Faessler (Pergamon, Oxford, 1986), to be published. [2] M. Kohno and W. Weise, Phys. Lett. B t52 (t985) 303; Regensburj, preprint TPR-85- 25. [3] E.M. Henley, T. Oka and J. Vergados, Phys. Lett. B 166 (1986) 214. [4] A. Faessler, G. Li~beck and K. Shimizu, Phys. Rev. D26 (1982) 32[ 0; M. Alberg ct al., Phys. Rev. D27 (1983) 536. [5] H. Rubinst~in and H. SneUman, Phys. Lett. B 165 (1985) 18~. [6] N. Isgur and J. Paten, Phys. Lett. B 124 (1983) 247; Phys. Rev. D31 (1985) 2910. [7] R. Tegen, R. Brockmann and W. Weise, Z. Phys. A307 (1982) 339. [8] C. Hayne and N. Isgur, Phys. Rev. 25D (1982) 1944. [9] M. Maruyarna, Ph.D. thesis O~aka University (1984), unpublished. [10] A.M. Green, V. Kuikka and J.A. Niskanen, Nuct. Phys. A446 (1985) 543.