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Effects of the UA(1) breaking interaction on baryonic systems
NUCLEAR PHYSICS A
Nuclear physics A!%4 (1993) 635652 No~-~o~l~d
EfIects of the U,(l)
breaking interaction on baryonic systerns Osamu Morimatsu
Institute
forNuciear Study, University of Tokyo, Tanashi, Tokyo 188, .lapan
Makoto Takizawa of physics and ~t~no~y, ~ni~r~i~y of South Carolina, Columbia, SC 29.208, USA’ institute of ~eorer~~al Physics, ~aiuersity of ~ege~~rg D-8400, ~~~sbu~, Germany’ fnstitut fir Ke~pbys~~ Forschungszent~~ .ii&h, L&5170 &iii&, Germany’
~e~~~en~
Received 24 April 1992 (Revised 17 September 1992) Abstract: The effects of the U,( 1) breaking interaction on the baryon number one and two systems are estimated employing the six-quark flavour dete~inantai interaction as the effective interaction of quarks which reproduces the observed mass difference of n and r)’ mesons. This is done by calculating the matrix elements of the U,(l) breaking hamiltonian with respect to unpe~urbed states of the MIT bag model and the nonrelativistic quark model. The dete~inantal interaction induces not only three-body but also two-body interactions of valence quarks. The two-body interaction is attractive, which gives rise to the NA mass difference with the magnitude less than one tenth of the observed one and attraction of two octet baryons at short distances whose magnitude is in the range 20 - 80 MeV depending on the flavour channels and the choice of parameters. The three-body interaction is repulsive, which gives about 10 - 20 MeV repulsion in the H-dibaryon channel and somewhat weaker repulsion in the gavour SU(3) octet and antidecup~et channels of two octet baryons at short distances. We also compare our results with those obtained by using the instanton-induced interaction.
1. Intr~u~tion It is well known that the QCD action has an approximate U,(3) x U,(3) chiral symmetry. It is also known, however, that its subsymmetry, UA(l) symmetry, is badly broken in nature. n’ has a mass much larger than other pseudosealar mesons and the decay process q + 7r0~+~- has a large amplitude. These facts contradict the U,( 1) symmetry. This is the well-known U,( 1) problem I). The key step to solve the U,,( 1) problem was to realize that there is an anomaly in the U,( 1) channel. Namely the UA(l) symmet~ in the classical theory, i.e. in the action, is broken by quantum effects. It was further conjectured that the instanton configuration plays Correspondence to: Dr. 0. Ma~matsu, Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan. ’ Work supported in part by the National Science Foundation, USA. 2 Work supported in part by BMFT grant 06 OR 762. 3 present address. 0375-9474/93/$~.~
@ 1993 - Efsevier Science Publishers B.V. At1 rights reserved
636
0. ~orimat~,
h4. Takizawa j Buryonic systems
a crucial role in the breakdown of the U,( 1) symmetry 2)_ In this interpretation massless quarks coupled to the zero-mode excitation of the gauge geld around the instanton configuration give rise to an effective six-quark interaction in the case of SU(3), which breaks the U,(l) symmetry. This interpretation, however, was questioned by some people ‘). Another scenario for the solution of the U,( 1) problem is in terms of the l/Arc expansion of the QCD “). According to this scenario the U,(l) symmetry, which is preserved in the leading order of the l/N, expansion, is broken in the next-to-leading order suggesting the importance of the nonperturbative effects in QCD. Though this scenario provides us with an effective lagrangian, which is written in terms of pseudoscalar mesons, the underlying dynamical mechanism in terms of quarks and gluons is not really clear. Even though the mechanism of the U,(l) breaking is not really understood yet, it is certainly true that an effective action in terms of quarks should include a term which breaks the U,( 1) symmetry. Actually, an effective six-quark interaction similar to the instanton-induced interaction had been phenomenologically introduced to take account of the U,( 1) breaking even before the discovery of the instanton. We would also like to note that the effect of such an interaction is quite large in the hadronic scale. Namely, the mass difference of n’ and n mesons is as large as several hundreds of MeV, which is the same order as hadron masses. Since it is such a large effect in the pseudoscalar meson sector, it is natural to ask if one can see some effects also in other sectors. Recently, Shuryak and Rosner ‘) pointed out that the instanton configu~tion can play an important role in the description of spin-spin forces, particularly for light baryons. The pattern of these effects can be very hard to disentangle from those more familiar ones due to chromomagnetic interactions arising from one-gluon exchange. Oka and Takeuchi 6*7) studied the effects of the instanton-induced interaction in baryon number B = 2 systems. It was shown that attraction between two nucleons is obtained by the two-body instanton-induced interaction, while the three-body interaction is strongly repulsive in the H-dibaryon channel and makes the H-dibaryon almost unbound. In the former work they are concerned only with the pattern of the effects of such an interaction but not with the strength of the interaction. In the latter it is assumed that the instanton-induced interaction gives $ to $ of the NA mass difference. Our opinion, however, is that given that the strength of the U,(l) breaking has to be determined phenomenologically, it should be determined by the mass difference of the singlet-octet pseudoscalar mesons since it is the quantity most sensitive to the U,(l) breaking term. The purpose of this paper is to give an order estimation of the effects of the U,( 1) breaking interaction on the B = 1 and B = 2 systems by employing the phenomenological six-quark interaction whose strength was determined so as to reproduce the observed pseudoscalar meson spectrum in the context of the Nambu-Jona-Lasinio (NJL) model 9-14).
0. ~or~rn~t~u, M. Takizawa / Baryonic systems
637
This paper is organized as follows. We explain an effective U,( 1) breaking interaction to be used in this paper in sect. 2 and the baryonic states in sect. 3. In sect. 4, we study the effects of the U,( 1) breaking interaction on the baryonic systems both in the flavour SU(3) symmetric case and the flavour SU(3) breaking case. We also compare our results and those obtained by using the instanton-induced interaction there. Finally, we give a brief summary in sect. 5. 2. Effective U,( 1) breaking interaction As the U,(l) breaking interaction in the three-flavour case, we take the following effective lagrangian density given by a dete~inant form in the flavour space ‘):
= -6iK{det ( &cz/L) 4 (h.c.)j ,
(2.1)
where the quark field J, is a column vector in colour, flavour and Dirac spaces, while I,@ stands for its component in flavour (a) and colour-Dirac (A) spaces, I,&= $( 1 f ‘y&j and & = t( 1 - y5) # are the right- and left-handed Dirac fields. -r;Pis the simplest quark interaction which breaks U,( 1) but not SU,(3) x SU,(3) symmetry. By using this lagrangian density, the effects of the U,(l) breaking interaction on meson properties have been extensively studied in the context of the Nambu-Jona-Lasinio (NJL) model 9-*4). We remark that the three-flavour NJL model which includes the U,( 1) breaking interaction .Z&with an appropriate positive coupling constant K reproduces reasonably well the nonet pseudoscalar meson spectrum. However, the interaction, Zk, itself can be used not only in the NJL model but also in a more general context. This interaction can be expressed in terms of the three-body antisymmetrization operator in the flavour space &&: &= -6K{~a”J;h”‘~h”‘~~~~~~‘~~‘~~‘~(h.c.)}, ~~~,=b(l-~~-p?ff-~*-p~*3+~**),
(2.2)
(2.3)
where P$ =;+$A, *)c2, K2, = P-&P& and (n)i (i = 1,. . . ,8) are SU(3) generators (Geil-Mann matrices) for the ffavour space. Here we use the following notation: (2.4) One can express 2$ by using the operators in the colour and spin spaces as 71) ljlR72) +bR 73) s 6 = __+jKqbR [1+~(h~~h~+perm.)+~(a,~o,+perm.) +~(Xf.A~~,.a,+perm.)+$(xT.ASfperm.)(a,.u~fperm.)
+ (h.c.) .
(2.5)
0. Morimutsu, hf. Takizawa / Batyonic systems
638
Here, (A’)j (i = 1,. . . , 8) are SU(3) generators (Gell-Mann matrices) for the colour space and (T’Sare usual Pauli matrices for the spin space. DiZ3 = &(A~)~(A$(A~)~ are symmetric and antisymmetric colour singlet and Ff*3’~j~(AT)i(Aqr(AS)k operators, respectively. The lagrangian density, &, induces not only three-body but also two-body interactions of valence quarks when the vacuum has a nonvanishing condensate ($$). The latter can be described by using an effective four-fermion lagrangian density, &. In the case of the SU(3) symmetric vacuum 26 = -4~{@&$&&~‘~~~+ where &{, is the two-body antisymmetrization
(h.c.)} ,
(2.6)
operator in flavour space:
A{, = f( 1 - p’;,)
(2.7)
and k = K(iiu) = I((&)
= K(Ss) .
cw
This lagrangian density Z& can also be expressed in terms of the operators acting on the colour and spin spaces, ~a=-~~lj;~)~(R2)[1+~~;‘X~+~Cli’~2+~(Xf*)C~)((T*‘~*)]~(LZ)~fi(L1) + (h.c.) .
(2.9)
In the case of the SU(2) symmetric vacuum .& = -za,J;‘,“@(fP,
* P* -$7, *T*)$(Lz)JI(L1)
_ &u~p~k” (
-~=~~(A,)“.(A~).+Q~.~*+~,.Q~)~~)~~)
(2.10)
+ (he.) ,
where 7 is the Pauli matrix acting on the flavour space, P = g-t-v’$ A)’ is the projection operator to the nonstrange sector, Q = 1 -P is the projection operator to the strange sector and K,=K(tiu)=K(&),
(2.11)
Ks = K(n).
(2.12)
It should be noted that these lagrangian densities =Y$and J& differ from those induced by instantons coupled to the massless quarks, ZrNs and $,,, [ref. “)I. We will discuss the differences in sect. 4.3. 3. Baryonic states Since we use the six-quark interaction whose strength has been determined so as to reproduce the observed pseudoscalar meson spectrum in the NJL model, the
0. Morimatsu, M. Takizawa / Baryonic systems
most consistent systems
way of evaluating
would
unnecessary
be to construct
baryonic
interaction
of the interaction tonian
the effects of the interaction
and too demanding
the U,( 1) breaking
states
in the baryon
to simple
the matrix baryonic
on the baryonic
in the NJL model.
for our purpose
by calculating
with respect
639
of giving
sector. Instead, elements
states based
However,
it is
an order estimation we evaluate
of the U,(l)
of
the effects
breaking
on the valence-quark
hamilpicture.
If the matrix elements are small compared with the baryonic energy scale, the U,( 1) breaking interaction can be regarded as a perturbation to the ordinary U,(l) conserving part. Indeed, we will show later that they are less than 10% of the baryon masses. We assume (six) valence
that
unperturbed
quarks,
states
of baryons
(dibaryons)
are made
of three
which occupy the lowest energy orbit, on the quark condensed
vacuum:
(B= 1) =
C Cj,i,i,a~,UgUXlVaC.)
,
(3.1)
i,i*i)
IB = 2) =
1
Ci,izi,isisisa~,U~~U~~UZaZa;t,lvac.)
,
(3.2)
i,i,i,i,i,i,
where i refers to a spin-flavour-colour quantum number, C is the weight function and ai creates a quark whose wave function is given by 4i =
y;, (
with xi denoting the spin-flavour-colour function is assumed for the spin-flavour for the colour part. We consider B = 1 system and all the possible
(wave)
(3.3)
xi, )
wave function. The ordinary SU(6) wave part and the SU(3) singlet wave function
the octet and decuplet flavour channels for the channels which are made of two octet baryons
for the B = 2 system (8 x 8 = 1 + Ss + 8As + 10+ lO* + 27). It should be noted, however, that one of the two octet channels, &, which is symmetric under the exchange of the flavour
of two baryons,
in eq. (3.2) represents
is forbidden
the situation
by the Pauli principle.
where two baryons
The state described
are located
roughly
on top
of each other. Therefore, the matrix element with respect to such a state gives a measure of the contribution of the U,( 1) breaking interaction either to the dibaryon or to the short-range part of the interaction between two baryons. Explicit forms of u and v are also needed for later numerical evaluation of the orbital integration. We consider two particular cases, i.e. the MIT bag model 16)* and the nonrelativistic quark model (NRQM) “). In the former (3.4) l Strictly speaking, the MIT bag model assumes that the vacuum inside the bag is perturbative, i.e. (k) = 0, which leads to the absence of the two-body U,( 1) breaking interaction. In this paper we do not use the MIT bag model in the strict sense. Instead, we just regard the MIT bag model as a typical example of the valence-quark wave function, while we assume (I&) = const everywhere.
0. Morimatsu, M. Takizawa / Baryonic sysiems
640
(3.5)
where R is the bag radius, m is a quark mass, x is a dimensionless determined by the boundary condition at the bag surface:
parameter
x
(3.6)
tanx=~-mR_[x2+(mR)2]‘/2’ E = ( 1/R)[x2+
( mR)2]1’2 and the normalisation
constant is
2E(E-l/R)+m/R (3.7)
E(E-m)
While in the latter (3-S)
(3.9)
where (3.10)
Q(r) = (rrb2)-3’4exp
and b is the size parameter. In the ordinary de~vation of the Fermi-Breit interaction E is taken to be the on-shell energy and is expanded with respect to p/m. Here we replace E by m where ( ) denotes the expectation value with respect to Q.
4. Evaluation of the U,(l) 4.1. SU(3)
SYMMETRIC
breaking interaction
CASE
We are now ready to evaluate the effect of the UA(l) breaking interaction on the baryonic systems. We first consider the SU(3) limit in order to see qualitative features. The contribution of the three-body term is given by
(v(3)),
-( J d3xZ6(x) ) =9~((o~‘)~-f(~~‘s3’)~)
,
(4.1)
where (4.2) i
(4.3)
0. Morimafsy
M. Takizawa / Barymic systems
641
and F=
I
(U2-$)3d3X,
v*)(~uu)”
(4.4)
d3x.
(4.5)
The operator 02) counts the number of possible combinations of three quarks which are totally antisymmet~c in the flavour space and Q(83)the same combination with the weight -3 (3) for spin doublet (quartet). In the three- or six-quark system of totally symmetric in the orbital space, however, such a three-quark combination can have only spin 4, since the product of the colour and spin wave function of the three quarks must be totally s~met~c, which is possible only when both the colour and spin wave function are mixed symmetric. F and G are positive, since u > v, and they depend on the hadron size, R, roughly as Re6. The three body-term is, thus, repulsive and its magnitude is 9K (F + G) for each combination of flavourantisymmetric three quarks, which is extremely sensitive to the hadron size. The contribution of the two-body term is given by
-(j d3X$(X))
(V(2))=
-f(ag’>.r),
= 4&(0~“)f
(4.6)
where @j’= c ,&.=: >-: (&..$y~,) II i
i
i
i
J
9
(4.7) (4.8)
and I =
I J =
(u2-v2)*d3x,
(4.9)
(2~)~ d3x,
(4.10)
Now the operator S’,” counts the number of pairs which are antisymmetric in the flavour space and 5’g’ the same pairs with the weight -3 (1) for spin singlet (triplet). In the six-quark system of totally symmetric in the orbital space, a flavour-antisymmetric pair can have either spin 0 or 1. In the three-quark system, however, such a pair can have only spin 0, since now the colour wave function of the quark pair is necessarily antisymmet~c and therefore its spin wave function must be also antisymmetric. I and .I are positive and I-J. They depend on the hadron size roughly as Re3. The two-body term is, thus, attractive and its magnitude is 42( I -t J) for spin
0. Marimatsu, M. Takizuwa / Baryonic systems
642
singlet and 4k(Z -4.Z) for spin triplet, which is sensitive to the hadron size but not so extreme as the three-body term. For comparison we also calculate the contribution of the U,( 1) breaking term to the mass of the singlet and octet pseudoscalar mesons in the same approximation. Of course, we do not expect a simple valence picture to be a good approximation for the pseudoscalar mesons. However, it still gives us a qualitative idea about the selectivity of the U,,(l) breaking interaction. In the case of the meson, only the two-body term contributes: (v’2’)=4K{(O~))(;Z+$Z)+(0(82))(;z+$9)},
(4.11)
where
sk”=(f-fx,.x,),
(4.12)
o’,“= (f-$X,*X4)u4-uI.
(4.13)
The flavour-spin matrix elements of the operators 0$‘, @‘, CTp’and Oc,” for various hadrons are given in table 1. We get the following relations:
Singlet-octet
pseudoscalar
meson mass difference, (4.14)
( v(2’)qq,,-( V’2))qrf,8 = -&42ZM +42.Z,) . Decuplet-octet
baryon mass difference, (V(2))& (v(*9q3.10-
Octet-octet
= -X(6Z,s-6&)
TABLE
(4.16)
,
I
matrix elements of the operators O!$, #), 02’ and 0:) for various hadrons in the SU(3) symmetric case 94
(c$‘) &?I Wa3’1 0%‘)
(4.15)
baryon interaction, - 2( V’2’),3+8= 24gZa ( v(2’)q6,1
Ftavour-spin
.
9’
1
8
-1 -3 0 0
f 1 x 0
8 z 3 9 * 0 0
q6 10
1
8
10
10*
21
0 0 0 0
9 -9 4 -12
9 -y S -9
6 -4 0 0
6 -4 2 -6
5 -5 0 0
0. Morimatsu, M. Takizawa / Baryonic systems
(4.17)
(V’2))g~,8-22(V’2’)q3,g=~(18Z~-~~B), ( v(*~}~4*,~ - 2( vt2))& = Zz(121,
-
643
p.Z,>,
( v(*))g6,,0*- 2( Vfz))43,8= Zz(lZZ, +&)
(4.18) (4.19)
,
(V(2))q~,27-2(V(2))q~,8=Z?(8ZB-~.ZB),
(4.20)
( V(3’),6,1= K(36FB+36G,),
(4.21)
( V3))$,s = K(?F,+$G,),
(4.22)
(V~Tf3))&** = K(18F,+186,).
(4.23)
Qualitative co_mpa$son of these quantities can be done by noting that very roughly, KF-KG-KZ-KZ, since F-G-0(1/R6), Z-.Z-0(1/R3) and (qq)-(250 MeV)3 - 0( l/R-‘). The contribution to the octet-decuplet baryon mass difference is one order smaller than that to the singlet-octet pseudo-scalar meson mass difference. In the SU(3)-singlet channel the two-body term is attractive and in the SU(3)-octet, decuplet, antidecuplet and 27-plet channels the cont~bution of the two-body term is smaller than that in the SU(3)-singlet channel. Especially in the case of the MIT bag model, the contribution of the two-body term is small because of the cancellation of the integral Z and J. The three-body term has nonzero matrix elements not only in the SU(3)-singlet channel but also in the SU(3)-octet and antidecuplet channels. The contributions of the three-body term in all these channels are repulsive. 4.2. BROKEN
W(3)
CASE
When the SU(3) breaking is taken into account, the calculation becomes complicated but the essential features above do not change. We summarise here the results. The contribution of the three-body term is (V’3~)=9K((O’,3’)F,-~(8’~3’)G,-f(0~3’)G, +(~~‘)~~-~(~~3’)G
-+Of3))G
336
4,
)
(4.24)
where 0(13’=
C
{Q{&$,Qi + cyclic perm.} ,
(4.25)
{Qi&$kQiUj* Uk + cyclic perm.} ,
(4.26)
i
0i3) =
C i
@Y’=
C i
{Q&&Qi(
(lk
* (fj
+
vi’
OF’= c {Qjd
Qj
@j)
+
+
cyclic per-m%},
(4.27) (4.28)
cyclic perm.} ,
i
og3) = c i
{( Qjd$Qk+ Q/&$kQj)C)‘j' ok i- Cyclic
pWm.} ,
(4.29)
0. Morimatsu,
644
M. Takizawa / Baryonic systems Uk*
Ui
+
Ui’
Uj) + cyclic
perm.} , (4.30)
and F,=
(u’-~‘)‘(~,Z-z~,2)d~x,
(4.31)
I F,=
(uu,-vv,)*(u~-~~)
d3x,
(4.32)
I G, =
(us - 215(2uv)~ d3x,
(4.33)
(~*-~*)(2~z1)(2~,~l,)d~x,
(4.34)
(u*-u*)(u~),+u,~)*d~x,
(4.35)
I G2= I G,= I G4 =
(uu, - VU,)(~UU)( w,+
u,v) d3x.
(4.36)
I The contribution ( P’)
of the two-body
term is
= 4K,(( 01291, - ;( O:“)J,)
l
(4.37)
~~:,((~~‘)~~~~(~~*~)~~~(~~*~)~3~~(~~*~)~3),
where ol” = 1 P&sz$yi
= 2 (iPi
i
C
PiPj&d{cPiUi’Uj
= 1
icj
Oi*‘=
C
c
Tj) )
(4.38)
($Pie-&i’Tj)Ui’Uj,
(4.39)
i
(QidiQj+Qjd$Qi)=
C (-a
(Oi~~Qj+Qj~~Oi)ui.uj=
C
(oi~~Oi+Qj~~oj)=
C
)
(4.40)
(-t
i4
(Ai)“(h
C f(QiS+PiOj),
ui’“j9
C4e41)
(4.42)
i
icj Ok”=
(*i)a(*j)a)
a=4
c icj
icj S:“=
;t
i
i
op=
-&*
i
(Qi~~Qi+Qj~~Qj)Ui’Uj=
C +(QiPj+PiQj)Ui*Uj,
(4.43)
i
i
and I,=
(u2-v2)*d3x, I
(4.44)
0. ~orirna#s4 M Takizawa f Baryonic systems TABLE
645
2
Baryon component, SU(3) multiplet, spin, isospin and strangeness of the eight channels of the two octet baryons Baryon component
Spin
SU(3) multiplet
Isospin
Strangeness 0
I
NN
10*
II
NN
27
NE
27 27
-1
NH-NA NE-NA
10
-1 -1 -1
III IV V VI VII
NH
0
-1
10*
NJ-NA
8
H
VIII
1 0
0
1
I2 = I, = J1 = Jz = J3 =
0
-2
J J(u”-u’)(uf-v:) J J J (4.45)
(uu, - uv,)*d3x,
(4.46)
d3x,
(22~)~ d3x,
(4.47)
(~n,+~,n)~d~x,
(4.48)
(2uu)(Zu,v,)
(4.49)
d3x.
Here we have assumed isospin symmetry and u and v are the radial wave functions for the u- and d-quarks while u, and v, are those for the s-quark. We consider the octet and decuplet B = 1 channels as well as the eight channels listed in table 2 which can be made of two octet baryons. The matrix elements of the operators B(‘) and QC3)are listed in table 3. We now explicitly evaluate the orbital integrals TABLE
Flavour-spin
3a
matrix elements of the operators S~“-IY~ (” for the octet and decuplet baryons N
X
S
A
A
z*
g*
R
3
0
0
(sy) (B$*‘)
-5 0 0
0 t -3
0 _f
1 -3 1;
0 0 0 0
0 0 -1 -1
0 0 -1 -1
0 0 0 0
(OS”‘, (t%2’)
0 0
1 -2
1 -2
1 0
0 0
1 1
1 1
0 0
WY)
(W)
0. Morimatsu, M. Takizawa / Baryonic systems
646
TABLE 3b Flavour-spin
matrix
elements
I
(P’) (BY’)
of the operators 0,‘*I- 0’*’ 6 for the eight channels baryons listed in table 2 II
III
5
6 -4
3
-5
(W)
0
0
(W’)
0
0
(@2’)
0
0
(@?)
0
0
IV
-5
VI
4
2 __&7
4 -4
-J
3z
-4
t
-f
0 -1 3 3
V
VII
VIII
4 _f
-3
1
-2
3
1
-2 52
2 -;
of the two octet
2
-2 z
-3 4
1 -2
65
-3
TABLE 3c Flavour-spin (VIII)
matrix elements and strangenessIII
employing
of the operators 0:3’-0i3’ for the H-dibaryon 1 two octet baryon channels (III-VII) IV
the MIT bag model
V
and NRQM
VI
VII
wave functions,
VIII
which
are combined
with the above matrix elements to estimate the effects of the U,(l) breaking interaction on various baryonic systems. The parameters included in the effective lagrangian &, and L&, are taken from ref. 13) and listed in table 4. We consider two sets of parameters, A and B, which correspond to a small and large mixing of the n1 and n8, respectively. The parameters for the unperturbed states are taken from ref. 16) in the case of the MIT bag model, i.e., m, = md = 0,m, = 0.279 GeV and R = 5.0GeV-‘. Those in the case of the NRQM are m, = md = 0.35 GeV, m, = 0.5 GeV and b = 0.6 fm.
TABLE 4 Parameters
included in the effective lagrangian Ye and L?e corresponding to case A and B. These values are taken from ref. 13) K [GeVm5]
case A case B
96.0 33.4
-(Gu)“~
[GeV]
0.247 0.277
-(SS)~‘~ [GeV] 0.264 0.285
0. Morimatsu,
M. Takizawa / Baryonic systems TABLE
Contributions
MIT NRQM
case case case case
of the two-body
A B A B
641
5a
term V@) to octet and decuplet
baryons.
N
z
Z
A
A
-21.6 -9.5 -20.1 -8.8
-19.0 -9.3 -16.9 -8.3
-19.0 -9.3 -16.9 -8.3
-20.7 -9.4 -19.0 -8.6
0 0 0 0
All the entries are in units of MeV Z* 5x 3x 3x 2x
1o-2 lo-* 1o-2 1oP
s*
R
5XlOP 3x10-2 3x10-* 2x1o-2
0 0 0 0
We first discuss the case of J3 = 1. Table 5a shows the contribution of the two-body term. One sees that the octet baryons get the attraction as small as about 20 (10) MeV in the case A (B) and the decuplet baryons have effects much smaller than 1 MeV in all cases. The three-body term has no effect in the B = 1 states. Therefore the NA mass difference due to the UA(1) breaking interaction is less than 10% of the observed one. The effects of the flavour SU(3) breaking on the octet baryons are about 10% in the case of MIT bag model and somewhat smaller in the case of NRQM. We next discuss the case of B = 2. Table 5b shows the contribution of the two-body term. The channel VIII gets the strongest attraction, about 80 (40) MeV in the case A (B), and then the channel VII gets the second strongest attraction. The channels I to VI have similar attraction 40 - 60 (20 - 30) MeV in the case A (B). This order is the same as in the SU(3) symmetric case. Comparing the contribution of the two-body term in channels which belong to the same SU(3) multiplet but have different baryon components, e.g. I and VI or II, III and IV, the effects of the flavour SU(3) breaking turn out to be at most several percent. The contribution of the three-body term to the H-dibaryon and strangeness -1 channels are given in table 5c. It should be noted that the three-body term has no effect on the NN channels. The strengths of repulsion in channels VI, VII and VIII are approximately proportional to the number of combinations of three quarks which are totally antisymmetric in the flavour space. The strongest repulsion in the channel VIII is still less than 20 MeV in all cases with the present choice of parameters. In the channels III, IV and V the nonvanishing contribution of the three-body term comes from the SU(3) breaking effects and is much smaller than that in other channels. TABLE
Contributions
of the two-body
I MIT case case NRQM case case
A B A B
-44.3 -19.4 -59.2 -25.9
5b
term V(z) to the eight channels of two octet baryons the entries are in units of MeV II
III
-41.9 -18.4 -51.9 -22.7
-42.4 -19.7 -48.9 -22.6
IV -42.0 -18.8 -50.6 -22.6
V -45.2 -20.6 -57.1 -25.9
listed in table 2. All
VI
VII
VIII
-45.6 -21.5 -55.4 -25.9
-57.9 -26.9 -70.8 -32.7
-76.7 -36.3 -86.3 -40.7
0. Morimatsu, M. Takizawa j Barycmic sysrems
648
TABLE 5c Contributions of the three-body term Vc3’ to the H-dibaryon (VIII) and strangeness -1 two octet baryon channels (III-VII). All the entries are in units of MeV III MIT case case NRQM case case
A B A B
-2 -1 -1 -5
IV
x 10-z x lo-* x lo-* x 1o-3
V
-2 x lo+ -1 x 10-z -1x 10-z -5 x 10-3
-2 -1 -2 -6
x 10-z x 10-z x 1o-2 x 10-3
VI
VII
VIII
6.4 2.2 8.8 3.0
7.8 2.7 10.8 3.8
12.6 4.4 17.4 6.1
The total effects of the U,( 1) breaking interaction on the B = 2 states can be roughly estimated by the quantity, ( V),,, = ( Vt3’& + ( Vf2))DB - ( V’2’)B - ( V’*&. . Here the subscript DB represents ane of the B=2 channels listed in table 2 and the subscripts B and B’ denote a pair of baryons which consist of the asymptotic state with the lowest energy for the channel DB. (V),, for channels I to VIII are summarized in table 6. The total contribution is attractive in all channels in the NRQM while it is attractive or very weakly repulsive in the MIT bag model. This is because of the large cancellation of the radial integrals I and J in the MIT bag model. In particular it should be noted that the total contribution of the U,(l) breaking interaction is attractive in the H-dibaryon channel VIII because of the strong attraction of the two-body term. However, this does not mean that the introduction of the U,(l) breaking interaction brings more attraction to the Hdibaryon. When the U,( 1) interaction is introduced the strength of the one-gluon exchange has to be reduced since the U,( 1) interaction also contributes to the NA mass difference. Therefore, if one compares the total contribution of the one-gluon exchange and the U,(l) breaking interaction the H-dibaryon gets less attraction than before the introduction of the U,(l) breaking term.
4.3. COMPARISON
WITH INSTANTON-INDUCED
INTERACTION
We now compare our results with those obtained by using the instanton-induced interaction. The instanton-induced interaction is described by the following TABLE 6 Total contributions
of the U,(l) breaking interaction to the eight channels of the two octet baryons listed in table 2. All entries are in units of MeV I
MIT case A
case B NRQM case A case B
-1.1 -0.4 -19.0 -8.3
II 1.3 0.6 -11.7 -5.1
III -1.8 -0.9 -11.9 -5.5
IV 0.3 0.1 -11.5 -5.2
V -2.9 -1.7 -18.0 -8.5
VI 1.4 -0.5 -9.6 -5.8
VII
VIII
-7.8 -5.3 -20.9 -11.5
-22.6 -13.1 -30.8 -17.4
0. Morimatsu, M. Takizawa / Baryonic systems
649
lagrangian densities:
= -$‘$~‘@‘~+@[l
+~(Xf~A~+perm.)+&(X~~X~u,~uz+perm.)
-&Df23+&Df23(u1 x
‘u2+perd
-&Ff23(u1
x u2)
‘031
(4.50)
(h.c.) ,
@‘@JIz”+
and 2rNS = -fZ?&‘@‘&{,(
1 -:a,
- u2)$f2’&‘+
(h.c.)
= -3a’~k”3k”[l+BA~.A;+B(AT.A5)(u,.u2)]JI’L2’~(L” (4.51)
+ (h.c.) , with Z?=K’(Iiu)=K’(&f)=K’(&).
(4.52)
For comparison, we calculate ( VC2’)and ( V”‘) for the above lagrangian densities in the SU(3) symmetric case. The contribution of the three-body term (4.50) to baryonic systems is given by
-( j-d’xP,,,(x))
(V”‘>=
=~K’((@)(F++G)-(CT$‘)(+F+fG)),
(4.53)
and that of the two-body term (4.51) by (V”‘>=
-( j-d3&,,,(x))
=~k:‘((6’,2’)(z+&q -(oc,“)(fZ+&r)).
(4.54)
To the meson only the two-body term contributes, which gives (v”‘)=4Z?{(6~‘)(Z+.Z)+(6’,2’)(Z+.Z)}. Therefore the quantities corresponding Singlet-octet
pseudoscalar
to (4.14)-(4.23) become as follows:
meson mass difference,
( V(2))4aI- (V(2))4q8= , -Z?(24Z Decuplet-octet
(4.55)
M
+24&,).
(4.56)
baryon mass difference, ( v(2))$Jo - ( V2))& = _Zz’(6ZB+ 65,) .
(4.57)
650
Octet-octet
0. Mwimatsu, M. Takizawu ,! Batyonic systems
baryon interaction, ( v(2))s6,*- 2( v(2))& = i:‘( 191,-t 7Jn) )
(4.58)
( V2))& - 2( v(Z))y3,8= #,($Z, +@n) )
(4.59)
( V2))$,o - 2( P))&
= ?(9Z,
-&)
,
(4.60)
( v(z)}~6,~~* - 2( v(*))$,* = B’(9Z8 -:.&> ,
(4.61)
( v(2))&7 - 2( V2’) $,8 = K’(7z,+JJJ3) ,
(4.62)
( VC3’),6,i= K’(36&+36G,),
(4.63)
( V’3’),, 8 = K’(FF,
(4.64)
+ FG B) 9
( v(3))q6*,o*= K’(18F,+
18Gu).
(4.65)
One sees that the relative contribution of the UA(l) breaking interaction within the baryonic sector or within the mesonic sector is similar for two different choices of the interaction. However, the ratio of those in the baryonic sector to those in the mesonic sector is about + different. Namely, if one fixes the strength of the interaction so as to give the same mass difference of singlet and octet pseudoscalar mesons in the above approximation, the effects of the instanton-induced interaction would be about 5 stronger than those of the determinantal interaction in the baryonic sector. Oka and Takeuchi 6,7)performed more involved calculations than the above. They studied the AA, NE and XX coupled-channels scattering problem in the framework of the quark-cluster model ‘8Z’9)including the instanton-induced interaction as well as one-gluon exchange and confinement potentials between quarks and (n-, K, C> meson exchange potentials between baryons. By choosing the strength of the interaction to give $ to $ of the NA mass difference, they concluded that appreciable attraction between two nucleons is obtained by the two-body instanton-induced interaction, while the three-body interaction is strongly repulsive in the H-dibaryon channel and makes the H-dibaryon almost unbound. Qualitatively our results are similar to theirs. Quantitatively, however, their results seem to be much bigger than ours. In addition to the difference of the interaction, which could account for about a factor s difference as has been seen above, there are some more possible origins of the difference. In their calculation they completely ignored the contributions coming from the lower components of valence quarks f&r the instanton-induced interaction even though they included those for one-gluon exchange potential in the form of the Fermi-Breit interaction. In this sense their calculation is not fully consistent. We would also like to mention that those contributions are far from being able to be ignored in our calculation. Another point is that they have chosen the size parameter which is a little smaller than ours, It gives rise to stronger effects since roughly ( V’*‘) - 0( I/ R3) and ( Vc3’) - 0( l/ R’) as has been seen before, where R is the hadron size.
0. Morimatsu,
M. Takizawa / Baryonic systems
651
Before closing this section we would like to make a short comment on theoretical as well as phenomenological grounds for the choice of the interaction. Theoretically, though the instanton-induced interaction might sound nicer, it is not at all clear that the form of the interaction derived with the dilute-gas approximation survives after including all the nonperturbative effects. Moreover, since the strength of the interaction has to be determined phenomenologically anyway, the interaction itself should be regarded as a phenomenological one. Then, from the phenomenological viewpoint, both choices of the interaction work equally well in the pseudoscalarmeson sector, which is the only place in our opinion where one can single out the effect of the U,(l) breaking interaction. In this sense, the possibility of different choices of the interaction should be regarded as the ambiguity of the calculation at the present stage.
5. Summary
In this paper, we have studied the effects of the U,(l) breaking interaction on the B = 1 and B = 2 systems. We have employed a flavour determinantal six-quark interaction as the U,( 1) breaking interaction for quarks which has been extensively used to study the effects of the U,(l) breaking interaction on meson properties in the context of the Nambu-Jona-Lasinio (NJL) model. In particular we would like to emphasize that the masses of 71 and 7’ are correctly reproduced. We evaluated the effects of the U,( 1) breaking interaction on the baryonic systems by calculating the matrix elements of the interaction hamiltonian with respect to the unperturbed wave functions for the B = 1 and B = 2 systems. We first studied qualitative features in the flavour SU(3) limit. In the B = 1 systems only the two-body term gives nonvanishing contributions. It gives NA mass difference which is about one order smaller than the observed one. In the B = 2 systems the two body term gives attraction of two flavour-octet baryons in all flavour channels while the three-body term gives repulsion in flavour-singlet, octet, anti-decuplet channels but no effects in other channels. We then used the MIT bag model and nonrelativistic quark model (NRQM) wave functions in order to make quantititative study. Here SU(3) breaking effects are fully taken into account. The contributions of the U,( 1) breaking interaction to the single octet baryons are attractive and as small as about 20 MeV and those to the single decuplet baryons are repulsive and much smaller than 1 MeV, which vanish in the flavour SU(3) symmetric limit. The effects of the U,(l) breaking interaction on the NN system are attractive in the NRQM both for T = 1 and T = 0 channels whose strength ranges about -5 to -20 MeV depending on the parameters. However, in the MIT bag model the effects are small (either repulsive or attractive). In the H-dibaryon channel the two-body term gives strong attraction (-36 - -86 MeV) while the three-body term gives rise to considerable repulsion (4- 17 MeV). The total contribution of the U,(l) term is attractive (-13 - -30 MeV) in this channel.
652
0. Morimatsu, M. Takizawa / Baryonic systems
We also compared our results with those obtained by using the instanton-induced interaction. Some origins of differences between our results and those by Oka and Takeuchi are discussed. An interesting result of the present work is that the three-body term gives rise to repulsion also in some S = - 1 channels, which include NZ spin-triplet and Nx - NA spin-triplet channels. Since S = - 1 channels are easier to study experimentally than S = -2 channels, it would be interesting and important to see the implication of the present result in the S = - 1 channels. We would like to express our sincere thanks to Kiyotaka Shimizu for letting us use his Fortran code for the calculation of the flavour-spin matrix elements. We would also like to thank Makoto Oka and Sachiko Takeuchi for useful discussions and critical comments. One of us (M.T.) wishes to acknowledge useful discussions with Kuniharu Kubodera, Kensuke Kusaka, Fred Myhrer, Woffram Weise and Hiroyuki Yabu and the other (O.M.) to Bob Jaffe. Most of the numerical works were performed on VAX Station 3100 of Theory Division, Institute for Nuclear Study, University of Tokyo, References 1) S. Weinberg, Phys. Rev. D11 (1975) 3583 2) 3) 4) 5) 6) 7) 8) 9)
10) 11) 12) 13) 14) 15) 16) 17) 18) 19)
G. ‘t Hooft, Phys. Reports 142 (1986) 357 R.J. Crewther, Phys. Lett. B70 (1977) 349 E. Witten, NucI. Phys. B149 (1979) 285; Bl56 (1979) 269 E.V. Shuryak and J.L. Rosner, Phys. Lett. 8218 (1989) 72 M. Oka and S. Takeuchi, Phys. Rev. Lett. 63 (1989) 1780; Nucl. Phys. A524 (1991) 649 S. Takeuchi and M. Oka, Phys. Rev. Lett. 66 (1991) 1271 M. Kobayashi, H. Kondo and T. Maskawa, Prog. Theor. Phys. 45 (1971) 1955 T. Kunihiro and T. Hatsuda, Phys. Lett. B206 (1988) 385; T. Kunihiro, Prog. Theor. Phys. 80 (1988) 34; T. Hatsuda and T. Kunihiro, Z. Phys. C51 (1991) 49 V. Bernard, R.L. Jaffe and U.-G. Meissner, Nucl. Phys. B308 (1988) 753 H. Reinhardt and R. Alkofer, Phys. Lett. B207 (1988) 482; R. Alkofer and H. Reinhardt, Z. Phys. C45 (1989) 275 Y. Kohyama, K. Kubodera and M. Takizawa, Phys. Lett. B208 (1988) 165; M. Takizawa, K. Tsushima, Y. Kohyama ;nd K. Kubodera, Prog. Theor. Phys. 82 (1989) 481 M. Takizawa, K. Tsushima, Y. Kohyama and K. Kubodera, Nucl. Phys. A507 (1990) 611 S. Klimt, M. Lutz, U. Vogl and W. Weise, Nucl. Phys. A516 (1990) 429; U. Vogi, S. Klimt, M. Lutz and W. Weise, Nucl. Phys. A516 (1990) 469 G. ‘t Hooft, Phys. Rev. D14 (1976) 3432; M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B163 (1980) 46 T. DeGrand, R.L. Jaffe, K. Johnson and J. Kiskis, Phys. Rev. D12 (1975) 2060 A. De Rtijula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147 M. Oka and K. Yazaki, Quarks and nuclei, ed. W. Weise (World Scientific, Singapore, 1984) p. 489 K. Shimizu, Rep. Prog. Phys. 52 (1989) 1
Nuclear physics A!%4 (1993) 635652 No~-~o~l~d
EfIects of the U,(l)
breaking interaction on baryonic systerns Osamu Morimatsu
Institute
forNuciear Study, University of Tokyo, Tanashi, Tokyo 188, .lapan
Makoto Takizawa of physics and ~t~no~y, ~ni~r~i~y of South Carolina, Columbia, SC 29.208, USA’ institute of ~eorer~~al Physics, ~aiuersity of ~ege~~rg D-8400, ~~~sbu~, Germany’ fnstitut fir Ke~pbys~~ Forschungszent~~ .ii&h, L&5170 &iii&, Germany’
~e~~~en~
Received 24 April 1992 (Revised 17 September 1992) Abstract: The effects of the U,( 1) breaking interaction on the baryon number one and two systems are estimated employing the six-quark flavour dete~inantai interaction as the effective interaction of quarks which reproduces the observed mass difference of n and r)’ mesons. This is done by calculating the matrix elements of the U,(l) breaking hamiltonian with respect to unpe~urbed states of the MIT bag model and the nonrelativistic quark model. The dete~inantal interaction induces not only three-body but also two-body interactions of valence quarks. The two-body interaction is attractive, which gives rise to the NA mass difference with the magnitude less than one tenth of the observed one and attraction of two octet baryons at short distances whose magnitude is in the range 20 - 80 MeV depending on the flavour channels and the choice of parameters. The three-body interaction is repulsive, which gives about 10 - 20 MeV repulsion in the H-dibaryon channel and somewhat weaker repulsion in the gavour SU(3) octet and antidecup~et channels of two octet baryons at short distances. We also compare our results with those obtained by using the instanton-induced interaction.
1. Intr~u~tion It is well known that the QCD action has an approximate U,(3) x U,(3) chiral symmetry. It is also known, however, that its subsymmetry, UA(l) symmetry, is badly broken in nature. n’ has a mass much larger than other pseudosealar mesons and the decay process q + 7r0~+~- has a large amplitude. These facts contradict the U,( 1) symmetry. This is the well-known U,( 1) problem I). The key step to solve the U,,( 1) problem was to realize that there is an anomaly in the U,( 1) channel. Namely the UA(l) symmet~ in the classical theory, i.e. in the action, is broken by quantum effects. It was further conjectured that the instanton configuration plays Correspondence to: Dr. 0. Ma~matsu, Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan. ’ Work supported in part by the National Science Foundation, USA. 2 Work supported in part by BMFT grant 06 OR 762. 3 present address. 0375-9474/93/$~.~
@ 1993 - Efsevier Science Publishers B.V. At1 rights reserved
636
0. ~orimat~,
h4. Takizawa j Buryonic systems
a crucial role in the breakdown of the U,( 1) symmetry 2)_ In this interpretation massless quarks coupled to the zero-mode excitation of the gauge geld around the instanton configuration give rise to an effective six-quark interaction in the case of SU(3), which breaks the U,(l) symmetry. This interpretation, however, was questioned by some people ‘). Another scenario for the solution of the U,( 1) problem is in terms of the l/Arc expansion of the QCD “). According to this scenario the U,(l) symmetry, which is preserved in the leading order of the l/N, expansion, is broken in the next-to-leading order suggesting the importance of the nonperturbative effects in QCD. Though this scenario provides us with an effective lagrangian, which is written in terms of pseudoscalar mesons, the underlying dynamical mechanism in terms of quarks and gluons is not really clear. Even though the mechanism of the U,(l) breaking is not really understood yet, it is certainly true that an effective action in terms of quarks should include a term which breaks the U,( 1) symmetry. Actually, an effective six-quark interaction similar to the instanton-induced interaction had been phenomenologically introduced to take account of the U,( 1) breaking even before the discovery of the instanton. We would also like to note that the effect of such an interaction is quite large in the hadronic scale. Namely, the mass difference of n’ and n mesons is as large as several hundreds of MeV, which is the same order as hadron masses. Since it is such a large effect in the pseudoscalar meson sector, it is natural to ask if one can see some effects also in other sectors. Recently, Shuryak and Rosner ‘) pointed out that the instanton configu~tion can play an important role in the description of spin-spin forces, particularly for light baryons. The pattern of these effects can be very hard to disentangle from those more familiar ones due to chromomagnetic interactions arising from one-gluon exchange. Oka and Takeuchi 6*7) studied the effects of the instanton-induced interaction in baryon number B = 2 systems. It was shown that attraction between two nucleons is obtained by the two-body instanton-induced interaction, while the three-body interaction is strongly repulsive in the H-dibaryon channel and makes the H-dibaryon almost unbound. In the former work they are concerned only with the pattern of the effects of such an interaction but not with the strength of the interaction. In the latter it is assumed that the instanton-induced interaction gives $ to $ of the NA mass difference. Our opinion, however, is that given that the strength of the U,(l) breaking has to be determined phenomenologically, it should be determined by the mass difference of the singlet-octet pseudoscalar mesons since it is the quantity most sensitive to the U,(l) breaking term. The purpose of this paper is to give an order estimation of the effects of the U,( 1) breaking interaction on the B = 1 and B = 2 systems by employing the phenomenological six-quark interaction whose strength was determined so as to reproduce the observed pseudoscalar meson spectrum in the context of the Nambu-Jona-Lasinio (NJL) model 9-14).
0. ~or~rn~t~u, M. Takizawa / Baryonic systems
637
This paper is organized as follows. We explain an effective U,( 1) breaking interaction to be used in this paper in sect. 2 and the baryonic states in sect. 3. In sect. 4, we study the effects of the U,( 1) breaking interaction on the baryonic systems both in the flavour SU(3) symmetric case and the flavour SU(3) breaking case. We also compare our results and those obtained by using the instanton-induced interaction there. Finally, we give a brief summary in sect. 5. 2. Effective U,( 1) breaking interaction As the U,(l) breaking interaction in the three-flavour case, we take the following effective lagrangian density given by a dete~inant form in the flavour space ‘):
= -6iK{det ( &cz/L) 4 (h.c.)j ,
(2.1)
where the quark field J, is a column vector in colour, flavour and Dirac spaces, while I,@ stands for its component in flavour (a) and colour-Dirac (A) spaces, I,&= $( 1 f ‘y&j and & = t( 1 - y5) # are the right- and left-handed Dirac fields. -r;Pis the simplest quark interaction which breaks U,( 1) but not SU,(3) x SU,(3) symmetry. By using this lagrangian density, the effects of the U,(l) breaking interaction on meson properties have been extensively studied in the context of the Nambu-Jona-Lasinio (NJL) model 9-*4). We remark that the three-flavour NJL model which includes the U,( 1) breaking interaction .Z&with an appropriate positive coupling constant K reproduces reasonably well the nonet pseudoscalar meson spectrum. However, the interaction, Zk, itself can be used not only in the NJL model but also in a more general context. This interaction can be expressed in terms of the three-body antisymmetrization operator in the flavour space &&: &= -6K{~a”J;h”‘~h”‘~~~~~~‘~~‘~~‘~(h.c.)}, ~~~,=b(l-~~-p?ff-~*-p~*3+~**),
(2.2)
(2.3)
where P$ =;+$A, *)c2, K2, = P-&P& and (n)i (i = 1,. . . ,8) are SU(3) generators (Geil-Mann matrices) for the ffavour space. Here we use the following notation: (2.4) One can express 2$ by using the operators in the colour and spin spaces as 71) ljlR72) +bR 73) s 6 = __+jKqbR [1+~(h~~h~+perm.)+~(a,~o,+perm.) +~(Xf.A~~,.a,+perm.)+$(xT.ASfperm.)(a,.u~fperm.)
+ (h.c.) .
(2.5)
0. Morimutsu, hf. Takizawa / Batyonic systems
638
Here, (A’)j (i = 1,. . . , 8) are SU(3) generators (Gell-Mann matrices) for the colour space and (T’Sare usual Pauli matrices for the spin space. DiZ3 = &(A~)~(A$(A~)~ are symmetric and antisymmetric colour singlet and Ff*3’~j~(AT)i(Aqr(AS)k operators, respectively. The lagrangian density, &, induces not only three-body but also two-body interactions of valence quarks when the vacuum has a nonvanishing condensate ($$). The latter can be described by using an effective four-fermion lagrangian density, &. In the case of the SU(3) symmetric vacuum 26 = -4~{@&$&&~‘~~~+ where &{, is the two-body antisymmetrization
(h.c.)} ,
(2.6)
operator in flavour space:
A{, = f( 1 - p’;,)
(2.7)
and k = K(iiu) = I((&)
= K(Ss) .
cw
This lagrangian density Z& can also be expressed in terms of the operators acting on the colour and spin spaces, ~a=-~~lj;~)~(R2)[1+~~;‘X~+~Cli’~2+~(Xf*)C~)((T*‘~*)]~(LZ)~fi(L1) + (h.c.) .
(2.9)
In the case of the SU(2) symmetric vacuum .& = -za,J;‘,“@(fP,
* P* -$7, *T*)$(Lz)JI(L1)
_ &u~p~k” (
-~=~~(A,)“.(A~).+Q~.~*+~,.Q~)~~)~~)
(2.10)
+ (he.) ,
where 7 is the Pauli matrix acting on the flavour space, P = g-t-v’$ A)’ is the projection operator to the nonstrange sector, Q = 1 -P is the projection operator to the strange sector and K,=K(tiu)=K(&),
(2.11)
Ks = K(n).
(2.12)
It should be noted that these lagrangian densities =Y$and J& differ from those induced by instantons coupled to the massless quarks, ZrNs and $,,, [ref. “)I. We will discuss the differences in sect. 4.3. 3. Baryonic states Since we use the six-quark interaction whose strength has been determined so as to reproduce the observed pseudoscalar meson spectrum in the NJL model, the
0. Morimatsu, M. Takizawa / Baryonic systems
most consistent systems
way of evaluating
would
unnecessary
be to construct
baryonic
interaction
of the interaction tonian
the effects of the interaction
and too demanding
the U,( 1) breaking
states
in the baryon
to simple
the matrix baryonic
on the baryonic
in the NJL model.
for our purpose
by calculating
with respect
639
of giving
sector. Instead, elements
states based
However,
it is
an order estimation we evaluate
of the U,(l)
of
the effects
breaking
on the valence-quark
hamilpicture.
If the matrix elements are small compared with the baryonic energy scale, the U,( 1) breaking interaction can be regarded as a perturbation to the ordinary U,(l) conserving part. Indeed, we will show later that they are less than 10% of the baryon masses. We assume (six) valence
that
unperturbed
quarks,
states
of baryons
(dibaryons)
are made
of three
which occupy the lowest energy orbit, on the quark condensed
vacuum:
(B= 1) =
C Cj,i,i,a~,UgUXlVaC.)
,
(3.1)
i,i*i)
IB = 2) =
1
Ci,izi,isisisa~,U~~U~~UZaZa;t,lvac.)
,
(3.2)
i,i,i,i,i,i,
where i refers to a spin-flavour-colour quantum number, C is the weight function and ai creates a quark whose wave function is given by 4i =
y;, (
with xi denoting the spin-flavour-colour function is assumed for the spin-flavour for the colour part. We consider B = 1 system and all the possible
(wave)
(3.3)
xi, )
wave function. The ordinary SU(6) wave part and the SU(3) singlet wave function
the octet and decuplet flavour channels for the channels which are made of two octet baryons
for the B = 2 system (8 x 8 = 1 + Ss + 8As + 10+ lO* + 27). It should be noted, however, that one of the two octet channels, &, which is symmetric under the exchange of the flavour
of two baryons,
in eq. (3.2) represents
is forbidden
the situation
by the Pauli principle.
where two baryons
The state described
are located
roughly
on top
of each other. Therefore, the matrix element with respect to such a state gives a measure of the contribution of the U,( 1) breaking interaction either to the dibaryon or to the short-range part of the interaction between two baryons. Explicit forms of u and v are also needed for later numerical evaluation of the orbital integration. We consider two particular cases, i.e. the MIT bag model 16)* and the nonrelativistic quark model (NRQM) “). In the former (3.4) l Strictly speaking, the MIT bag model assumes that the vacuum inside the bag is perturbative, i.e. (k) = 0, which leads to the absence of the two-body U,( 1) breaking interaction. In this paper we do not use the MIT bag model in the strict sense. Instead, we just regard the MIT bag model as a typical example of the valence-quark wave function, while we assume (I&) = const everywhere.
0. Morimatsu, M. Takizawa / Baryonic sysiems
640
(3.5)
where R is the bag radius, m is a quark mass, x is a dimensionless determined by the boundary condition at the bag surface:
parameter
x
(3.6)
tanx=~-mR_[x2+(mR)2]‘/2’ E = ( 1/R)[x2+
( mR)2]1’2 and the normalisation
constant is
2E(E-l/R)+m/R (3.7)
E(E-m)
While in the latter (3-S)
(3.9)
where (3.10)
Q(r) = (rrb2)-3’4exp
and b is the size parameter. In the ordinary de~vation of the Fermi-Breit interaction E is taken to be the on-shell energy and is expanded with respect to p/m. Here we replace E by m where ( ) denotes the expectation value with respect to Q.
4. Evaluation of the U,(l) 4.1. SU(3)
SYMMETRIC
breaking interaction
CASE
We are now ready to evaluate the effect of the UA(l) breaking interaction on the baryonic systems. We first consider the SU(3) limit in order to see qualitative features. The contribution of the three-body term is given by
(v(3)),
-( J d3xZ6(x) ) =9~((o~‘)~-f(~~‘s3’)~)
,
(4.1)
where (4.2) i
(4.3)
0. Morimafsy
M. Takizawa / Barymic systems
641
and F=
I
(U2-$)3d3X,
v*)(~uu)”
(4.4)
d3x.
(4.5)
The operator 02) counts the number of possible combinations of three quarks which are totally antisymmet~c in the flavour space and Q(83)the same combination with the weight -3 (3) for spin doublet (quartet). In the three- or six-quark system of totally symmetric in the orbital space, however, such a three-quark combination can have only spin 4, since the product of the colour and spin wave function of the three quarks must be totally s~met~c, which is possible only when both the colour and spin wave function are mixed symmetric. F and G are positive, since u > v, and they depend on the hadron size, R, roughly as Re6. The three body-term is, thus, repulsive and its magnitude is 9K (F + G) for each combination of flavourantisymmetric three quarks, which is extremely sensitive to the hadron size. The contribution of the two-body term is given by
-(j d3X$(X))
(V(2))=
-f(ag’>.r),
= 4&(0~“)f
(4.6)
where @j’= c ,&.=: >-: (&..$y~,) II i
i
i
i
J
9
(4.7) (4.8)
and I =
I J =
(u2-v2)*d3x,
(4.9)
(2~)~ d3x,
(4.10)
Now the operator S’,” counts the number of pairs which are antisymmetric in the flavour space and 5’g’ the same pairs with the weight -3 (1) for spin singlet (triplet). In the six-quark system of totally symmetric in the orbital space, a flavour-antisymmetric pair can have either spin 0 or 1. In the three-quark system, however, such a pair can have only spin 0, since now the colour wave function of the quark pair is necessarily antisymmet~c and therefore its spin wave function must be also antisymmetric. I and .I are positive and I-J. They depend on the hadron size roughly as Re3. The two-body term is, thus, attractive and its magnitude is 42( I -t J) for spin
0. Marimatsu, M. Takizuwa / Baryonic systems
642
singlet and 4k(Z -4.Z) for spin triplet, which is sensitive to the hadron size but not so extreme as the three-body term. For comparison we also calculate the contribution of the U,( 1) breaking term to the mass of the singlet and octet pseudoscalar mesons in the same approximation. Of course, we do not expect a simple valence picture to be a good approximation for the pseudoscalar mesons. However, it still gives us a qualitative idea about the selectivity of the U,,(l) breaking interaction. In the case of the meson, only the two-body term contributes: (v’2’)=4K{(O~))(;Z+$Z)+(0(82))(;z+$9)},
(4.11)
where
sk”=(f-fx,.x,),
(4.12)
o’,“= (f-$X,*X4)u4-uI.
(4.13)
The flavour-spin matrix elements of the operators 0$‘, @‘, CTp’and Oc,” for various hadrons are given in table 1. We get the following relations:
Singlet-octet
pseudoscalar
meson mass difference, (4.14)
( v(2’)qq,,-( V’2))qrf,8 = -&42ZM +42.Z,) . Decuplet-octet
baryon mass difference, (V(2))& (v(*9q3.10-
Octet-octet
= -X(6Z,s-6&)
TABLE
(4.16)
,
I
matrix elements of the operators O!$, #), 02’ and 0:) for various hadrons in the SU(3) symmetric case 94
(c$‘) &?I Wa3’1 0%‘)
(4.15)
baryon interaction, - 2( V’2’),3+8= 24gZa ( v(2’)q6,1
Ftavour-spin
.
9’
1
8
-1 -3 0 0
f 1 x 0
8 z 3 9 * 0 0
q6 10
1
8
10
10*
21
0 0 0 0
9 -9 4 -12
9 -y S -9
6 -4 0 0
6 -4 2 -6
5 -5 0 0
0. Morimatsu, M. Takizawa / Baryonic systems
(4.17)
(V’2))g~,8-22(V’2’)q3,g=~(18Z~-~~B), ( v(*~}~4*,~ - 2( vt2))& = Zz(121,
-
643
p.Z,>,
( v(*))g6,,0*- 2( Vfz))43,8= Zz(lZZ, +&)
(4.18) (4.19)
,
(V(2))q~,27-2(V(2))q~,8=Z?(8ZB-~.ZB),
(4.20)
( V(3’),6,1= K(36FB+36G,),
(4.21)
( V3))$,s = K(?F,+$G,),
(4.22)
(V~Tf3))&** = K(18F,+186,).
(4.23)
Qualitative co_mpa$son of these quantities can be done by noting that very roughly, KF-KG-KZ-KZ, since F-G-0(1/R6), Z-.Z-0(1/R3) and (qq)-(250 MeV)3 - 0( l/R-‘). The contribution to the octet-decuplet baryon mass difference is one order smaller than that to the singlet-octet pseudo-scalar meson mass difference. In the SU(3)-singlet channel the two-body term is attractive and in the SU(3)-octet, decuplet, antidecuplet and 27-plet channels the cont~bution of the two-body term is smaller than that in the SU(3)-singlet channel. Especially in the case of the MIT bag model, the contribution of the two-body term is small because of the cancellation of the integral Z and J. The three-body term has nonzero matrix elements not only in the SU(3)-singlet channel but also in the SU(3)-octet and antidecuplet channels. The contributions of the three-body term in all these channels are repulsive. 4.2. BROKEN
W(3)
CASE
When the SU(3) breaking is taken into account, the calculation becomes complicated but the essential features above do not change. We summarise here the results. The contribution of the three-body term is (V’3~)=9K((O’,3’)F,-~(8’~3’)G,-f(0~3’)G, +(~~‘)~~-~(~~3’)G
-+Of3))G
336
4,
)
(4.24)
where 0(13’=
C
{Q{&$,Qi + cyclic perm.} ,
(4.25)
{Qi&$kQiUj* Uk + cyclic perm.} ,
(4.26)
i
0i3) =
C i
@Y’=
C i
{Q&&Qi(
(lk
* (fj
+
vi’
OF’= c {Qjd
Qj
@j)
+
+
cyclic per-m%},
(4.27) (4.28)
cyclic perm.} ,
i
og3) = c i
{( Qjd$Qk+ Q/&$kQj)C)‘j' ok i- Cyclic
pWm.} ,
(4.29)
0. Morimatsu,
644
M. Takizawa / Baryonic systems Uk*
Ui
+
Ui’
Uj) + cyclic
perm.} , (4.30)
and F,=
(u’-~‘)‘(~,Z-z~,2)d~x,
(4.31)
I F,=
(uu,-vv,)*(u~-~~)
d3x,
(4.32)
I G, =
(us - 215(2uv)~ d3x,
(4.33)
(~*-~*)(2~z1)(2~,~l,)d~x,
(4.34)
(u*-u*)(u~),+u,~)*d~x,
(4.35)
I G2= I G,= I G4 =
(uu, - VU,)(~UU)( w,+
u,v) d3x.
(4.36)
I The contribution ( P’)
of the two-body
term is
= 4K,(( 01291, - ;( O:“)J,)
l
(4.37)
~~:,((~~‘)~~~~(~~*~)~~~(~~*~)~3~~(~~*~)~3),
where ol” = 1 P&sz$yi
= 2 (iPi
i
C
PiPj&d{cPiUi’Uj
= 1
icj
Oi*‘=
C
c
Tj) )
(4.38)
($Pie-&i’Tj)Ui’Uj,
(4.39)
i
(QidiQj+Qjd$Qi)=
C (-a
(Oi~~Qj+Qj~~Oi)ui.uj=
C
(oi~~Oi+Qj~~oj)=
C
)
(4.40)
(-t
i4
(Ai)“(h
C f(QiS+PiOj),
ui’“j9
C4e41)
(4.42)
i
icj Ok”=
(*i)a(*j)a)
a=4
c icj
icj S:“=
;t
i
i
op=
-&*
i
(Qi~~Qi+Qj~~Qj)Ui’Uj=
C +(QiPj+PiQj)Ui*Uj,
(4.43)
i
i
and I,=
(u2-v2)*d3x, I
(4.44)
0. ~orirna#s4 M Takizawa f Baryonic systems TABLE
645
2
Baryon component, SU(3) multiplet, spin, isospin and strangeness of the eight channels of the two octet baryons Baryon component
Spin
SU(3) multiplet
Isospin
Strangeness 0
I
NN
10*
II
NN
27
NE
27 27
-1
NH-NA NE-NA
10
-1 -1 -1
III IV V VI VII
NH
0
-1
10*
NJ-NA
8
H
VIII
1 0
0
1
I2 = I, = J1 = Jz = J3 =
0
-2
J J(u”-u’)(uf-v:) J J J (4.45)
(uu, - uv,)*d3x,
(4.46)
d3x,
(22~)~ d3x,
(4.47)
(~n,+~,n)~d~x,
(4.48)
(2uu)(Zu,v,)
(4.49)
d3x.
Here we have assumed isospin symmetry and u and v are the radial wave functions for the u- and d-quarks while u, and v, are those for the s-quark. We consider the octet and decuplet B = 1 channels as well as the eight channels listed in table 2 which can be made of two octet baryons. The matrix elements of the operators B(‘) and QC3)are listed in table 3. We now explicitly evaluate the orbital integrals TABLE
Flavour-spin
3a
matrix elements of the operators S~“-IY~ (” for the octet and decuplet baryons N
X
S
A
A
z*
g*
R
3
0
0
(sy) (B$*‘)
-5 0 0
0 t -3
0 _f
1 -3 1;
0 0 0 0
0 0 -1 -1
0 0 -1 -1
0 0 0 0
(OS”‘, (t%2’)
0 0
1 -2
1 -2
1 0
0 0
1 1
1 1
0 0
WY)
(W)
0. Morimatsu, M. Takizawa / Baryonic systems
646
TABLE 3b Flavour-spin
matrix
elements
I
(P’) (BY’)
of the operators 0,‘*I- 0’*’ 6 for the eight channels baryons listed in table 2 II
III
5
6 -4
3
-5
(W)
0
0
(W’)
0
0
(@2’)
0
0
(@?)
0
0
IV
-5
VI
4
2 __&7
4 -4
-J
3z
-4
t
-f
0 -1 3 3
V
VII
VIII
4 _f
-3
1
-2
3
1
-2 52
2 -;
of the two octet
2
-2 z
-3 4
1 -2
65
-3
TABLE 3c Flavour-spin (VIII)
matrix elements and strangenessIII
employing
of the operators 0:3’-0i3’ for the H-dibaryon 1 two octet baryon channels (III-VII) IV
the MIT bag model
V
and NRQM
VI
VII
wave functions,
VIII
which
are combined
with the above matrix elements to estimate the effects of the U,(l) breaking interaction on various baryonic systems. The parameters included in the effective lagrangian &, and L&, are taken from ref. 13) and listed in table 4. We consider two sets of parameters, A and B, which correspond to a small and large mixing of the n1 and n8, respectively. The parameters for the unperturbed states are taken from ref. 16) in the case of the MIT bag model, i.e., m, = md = 0,m, = 0.279 GeV and R = 5.0GeV-‘. Those in the case of the NRQM are m, = md = 0.35 GeV, m, = 0.5 GeV and b = 0.6 fm.
TABLE 4 Parameters
included in the effective lagrangian Ye and L?e corresponding to case A and B. These values are taken from ref. 13) K [GeVm5]
case A case B
96.0 33.4
-(Gu)“~
[GeV]
0.247 0.277
-(SS)~‘~ [GeV] 0.264 0.285
0. Morimatsu,
M. Takizawa / Baryonic systems TABLE
Contributions
MIT NRQM
case case case case
of the two-body
A B A B
641
5a
term V@) to octet and decuplet
baryons.
N
z
Z
A
A
-21.6 -9.5 -20.1 -8.8
-19.0 -9.3 -16.9 -8.3
-19.0 -9.3 -16.9 -8.3
-20.7 -9.4 -19.0 -8.6
0 0 0 0
All the entries are in units of MeV Z* 5x 3x 3x 2x
1o-2 lo-* 1o-2 1oP
s*
R
5XlOP 3x10-2 3x10-* 2x1o-2
0 0 0 0
We first discuss the case of J3 = 1. Table 5a shows the contribution of the two-body term. One sees that the octet baryons get the attraction as small as about 20 (10) MeV in the case A (B) and the decuplet baryons have effects much smaller than 1 MeV in all cases. The three-body term has no effect in the B = 1 states. Therefore the NA mass difference due to the UA(1) breaking interaction is less than 10% of the observed one. The effects of the flavour SU(3) breaking on the octet baryons are about 10% in the case of MIT bag model and somewhat smaller in the case of NRQM. We next discuss the case of B = 2. Table 5b shows the contribution of the two-body term. The channel VIII gets the strongest attraction, about 80 (40) MeV in the case A (B), and then the channel VII gets the second strongest attraction. The channels I to VI have similar attraction 40 - 60 (20 - 30) MeV in the case A (B). This order is the same as in the SU(3) symmetric case. Comparing the contribution of the two-body term in channels which belong to the same SU(3) multiplet but have different baryon components, e.g. I and VI or II, III and IV, the effects of the flavour SU(3) breaking turn out to be at most several percent. The contribution of the three-body term to the H-dibaryon and strangeness -1 channels are given in table 5c. It should be noted that the three-body term has no effect on the NN channels. The strengths of repulsion in channels VI, VII and VIII are approximately proportional to the number of combinations of three quarks which are totally antisymmetric in the flavour space. The strongest repulsion in the channel VIII is still less than 20 MeV in all cases with the present choice of parameters. In the channels III, IV and V the nonvanishing contribution of the three-body term comes from the SU(3) breaking effects and is much smaller than that in other channels. TABLE
Contributions
of the two-body
I MIT case case NRQM case case
A B A B
-44.3 -19.4 -59.2 -25.9
5b
term V(z) to the eight channels of two octet baryons the entries are in units of MeV II
III
-41.9 -18.4 -51.9 -22.7
-42.4 -19.7 -48.9 -22.6
IV -42.0 -18.8 -50.6 -22.6
V -45.2 -20.6 -57.1 -25.9
listed in table 2. All
VI
VII
VIII
-45.6 -21.5 -55.4 -25.9
-57.9 -26.9 -70.8 -32.7
-76.7 -36.3 -86.3 -40.7
0. Morimatsu, M. Takizawa j Barycmic sysrems
648
TABLE 5c Contributions of the three-body term Vc3’ to the H-dibaryon (VIII) and strangeness -1 two octet baryon channels (III-VII). All the entries are in units of MeV III MIT case case NRQM case case
A B A B
-2 -1 -1 -5
IV
x 10-z x lo-* x lo-* x 1o-3
V
-2 x lo+ -1 x 10-z -1x 10-z -5 x 10-3
-2 -1 -2 -6
x 10-z x 10-z x 1o-2 x 10-3
VI
VII
VIII
6.4 2.2 8.8 3.0
7.8 2.7 10.8 3.8
12.6 4.4 17.4 6.1
The total effects of the U,( 1) breaking interaction on the B = 2 states can be roughly estimated by the quantity, ( V),,, = ( Vt3’& + ( Vf2))DB - ( V’2’)B - ( V’*&. . Here the subscript DB represents ane of the B=2 channels listed in table 2 and the subscripts B and B’ denote a pair of baryons which consist of the asymptotic state with the lowest energy for the channel DB. (V),, for channels I to VIII are summarized in table 6. The total contribution is attractive in all channels in the NRQM while it is attractive or very weakly repulsive in the MIT bag model. This is because of the large cancellation of the radial integrals I and J in the MIT bag model. In particular it should be noted that the total contribution of the U,(l) breaking interaction is attractive in the H-dibaryon channel VIII because of the strong attraction of the two-body term. However, this does not mean that the introduction of the U,(l) breaking interaction brings more attraction to the Hdibaryon. When the U,( 1) interaction is introduced the strength of the one-gluon exchange has to be reduced since the U,( 1) interaction also contributes to the NA mass difference. Therefore, if one compares the total contribution of the one-gluon exchange and the U,(l) breaking interaction the H-dibaryon gets less attraction than before the introduction of the U,(l) breaking term.
4.3. COMPARISON
WITH INSTANTON-INDUCED
INTERACTION
We now compare our results with those obtained by using the instanton-induced interaction. The instanton-induced interaction is described by the following TABLE 6 Total contributions
of the U,(l) breaking interaction to the eight channels of the two octet baryons listed in table 2. All entries are in units of MeV I
MIT case A
case B NRQM case A case B
-1.1 -0.4 -19.0 -8.3
II 1.3 0.6 -11.7 -5.1
III -1.8 -0.9 -11.9 -5.5
IV 0.3 0.1 -11.5 -5.2
V -2.9 -1.7 -18.0 -8.5
VI 1.4 -0.5 -9.6 -5.8
VII
VIII
-7.8 -5.3 -20.9 -11.5
-22.6 -13.1 -30.8 -17.4
0. Morimatsu, M. Takizawa / Baryonic systems
649
lagrangian densities:
= -$‘$~‘@‘~+@[l
+~(Xf~A~+perm.)+&(X~~X~u,~uz+perm.)
-&Df23+&Df23(u1 x
‘u2+perd
-&Ff23(u1
x u2)
‘031
(4.50)
(h.c.) ,
@‘@JIz”+
and 2rNS = -fZ?&‘@‘&{,(
1 -:a,
- u2)$f2’&‘+
(h.c.)
= -3a’~k”3k”[l+BA~.A;+B(AT.A5)(u,.u2)]JI’L2’~(L” (4.51)
+ (h.c.) , with Z?=K’(Iiu)=K’(&f)=K’(&).
(4.52)
For comparison, we calculate ( VC2’)and ( V”‘) for the above lagrangian densities in the SU(3) symmetric case. The contribution of the three-body term (4.50) to baryonic systems is given by
-( j-d’xP,,,(x))
(V”‘>=
=~K’((@)(F++G)-(CT$‘)(+F+fG)),
(4.53)
and that of the two-body term (4.51) by (V”‘>=
-( j-d3&,,,(x))
=~k:‘((6’,2’)(z+&q -(oc,“)(fZ+&r)).
(4.54)
To the meson only the two-body term contributes, which gives (v”‘)=4Z?{(6~‘)(Z+.Z)+(6’,2’)(Z+.Z)}. Therefore the quantities corresponding Singlet-octet
pseudoscalar
to (4.14)-(4.23) become as follows:
meson mass difference,
( V(2))4aI- (V(2))4q8= , -Z?(24Z Decuplet-octet
(4.55)
M
+24&,).
(4.56)
baryon mass difference, ( v(2))$Jo - ( V2))& = _Zz’(6ZB+ 65,) .
(4.57)
650
Octet-octet
0. Mwimatsu, M. Takizawu ,! Batyonic systems
baryon interaction, ( v(2))s6,*- 2( v(2))& = i:‘( 191,-t 7Jn) )
(4.58)
( V2))& - 2( v(Z))y3,8= #,($Z, +@n) )
(4.59)
( V2))$,o - 2( P))&
= ?(9Z,
-&)
,
(4.60)
( v(z)}~6,~~* - 2( v(*))$,* = B’(9Z8 -:.&> ,
(4.61)
( v(2))&7 - 2( V2’) $,8 = K’(7z,+JJJ3) ,
(4.62)
( VC3’),6,i= K’(36&+36G,),
(4.63)
( V’3’),, 8 = K’(FF,
(4.64)
+ FG B) 9
( v(3))q6*,o*= K’(18F,+
18Gu).
(4.65)
One sees that the relative contribution of the UA(l) breaking interaction within the baryonic sector or within the mesonic sector is similar for two different choices of the interaction. However, the ratio of those in the baryonic sector to those in the mesonic sector is about + different. Namely, if one fixes the strength of the interaction so as to give the same mass difference of singlet and octet pseudoscalar mesons in the above approximation, the effects of the instanton-induced interaction would be about 5 stronger than those of the determinantal interaction in the baryonic sector. Oka and Takeuchi 6,7)performed more involved calculations than the above. They studied the AA, NE and XX coupled-channels scattering problem in the framework of the quark-cluster model ‘8Z’9)including the instanton-induced interaction as well as one-gluon exchange and confinement potentials between quarks and (n-, K, C> meson exchange potentials between baryons. By choosing the strength of the interaction to give $ to $ of the NA mass difference, they concluded that appreciable attraction between two nucleons is obtained by the two-body instanton-induced interaction, while the three-body interaction is strongly repulsive in the H-dibaryon channel and makes the H-dibaryon almost unbound. Qualitatively our results are similar to theirs. Quantitatively, however, their results seem to be much bigger than ours. In addition to the difference of the interaction, which could account for about a factor s difference as has been seen above, there are some more possible origins of the difference. In their calculation they completely ignored the contributions coming from the lower components of valence quarks f&r the instanton-induced interaction even though they included those for one-gluon exchange potential in the form of the Fermi-Breit interaction. In this sense their calculation is not fully consistent. We would also like to mention that those contributions are far from being able to be ignored in our calculation. Another point is that they have chosen the size parameter which is a little smaller than ours, It gives rise to stronger effects since roughly ( V’*‘) - 0( I/ R3) and ( Vc3’) - 0( l/ R’) as has been seen before, where R is the hadron size.
0. Morimatsu,
M. Takizawa / Baryonic systems
651
Before closing this section we would like to make a short comment on theoretical as well as phenomenological grounds for the choice of the interaction. Theoretically, though the instanton-induced interaction might sound nicer, it is not at all clear that the form of the interaction derived with the dilute-gas approximation survives after including all the nonperturbative effects. Moreover, since the strength of the interaction has to be determined phenomenologically anyway, the interaction itself should be regarded as a phenomenological one. Then, from the phenomenological viewpoint, both choices of the interaction work equally well in the pseudoscalarmeson sector, which is the only place in our opinion where one can single out the effect of the U,(l) breaking interaction. In this sense, the possibility of different choices of the interaction should be regarded as the ambiguity of the calculation at the present stage.
5. Summary
In this paper, we have studied the effects of the U,(l) breaking interaction on the B = 1 and B = 2 systems. We have employed a flavour determinantal six-quark interaction as the U,( 1) breaking interaction for quarks which has been extensively used to study the effects of the U,(l) breaking interaction on meson properties in the context of the Nambu-Jona-Lasinio (NJL) model. In particular we would like to emphasize that the masses of 71 and 7’ are correctly reproduced. We evaluated the effects of the U,( 1) breaking interaction on the baryonic systems by calculating the matrix elements of the interaction hamiltonian with respect to the unperturbed wave functions for the B = 1 and B = 2 systems. We first studied qualitative features in the flavour SU(3) limit. In the B = 1 systems only the two-body term gives nonvanishing contributions. It gives NA mass difference which is about one order smaller than the observed one. In the B = 2 systems the two body term gives attraction of two flavour-octet baryons in all flavour channels while the three-body term gives repulsion in flavour-singlet, octet, anti-decuplet channels but no effects in other channels. We then used the MIT bag model and nonrelativistic quark model (NRQM) wave functions in order to make quantititative study. Here SU(3) breaking effects are fully taken into account. The contributions of the U,( 1) breaking interaction to the single octet baryons are attractive and as small as about 20 MeV and those to the single decuplet baryons are repulsive and much smaller than 1 MeV, which vanish in the flavour SU(3) symmetric limit. The effects of the U,(l) breaking interaction on the NN system are attractive in the NRQM both for T = 1 and T = 0 channels whose strength ranges about -5 to -20 MeV depending on the parameters. However, in the MIT bag model the effects are small (either repulsive or attractive). In the H-dibaryon channel the two-body term gives strong attraction (-36 - -86 MeV) while the three-body term gives rise to considerable repulsion (4- 17 MeV). The total contribution of the U,(l) term is attractive (-13 - -30 MeV) in this channel.
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We also compared our results with those obtained by using the instanton-induced interaction. Some origins of differences between our results and those by Oka and Takeuchi are discussed. An interesting result of the present work is that the three-body term gives rise to repulsion also in some S = - 1 channels, which include NZ spin-triplet and Nx - NA spin-triplet channels. Since S = - 1 channels are easier to study experimentally than S = -2 channels, it would be interesting and important to see the implication of the present result in the S = - 1 channels. We would like to express our sincere thanks to Kiyotaka Shimizu for letting us use his Fortran code for the calculation of the flavour-spin matrix elements. We would also like to thank Makoto Oka and Sachiko Takeuchi for useful discussions and critical comments. One of us (M.T.) wishes to acknowledge useful discussions with Kuniharu Kubodera, Kensuke Kusaka, Fred Myhrer, Woffram Weise and Hiroyuki Yabu and the other (O.M.) to Bob Jaffe. Most of the numerical works were performed on VAX Station 3100 of Theory Division, Institute for Nuclear Study, University of Tokyo, References 1) S. Weinberg, Phys. Rev. D11 (1975) 3583 2) 3) 4) 5) 6) 7) 8) 9)
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