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The effect of η-η′ mixing in the bound state version of the Skyrme model
NUCLEAR P H Y SICS A ELSEVIER
Nuclear Physics A 593 (1995) 281-294
The effect of mixing in the bound state version of the Skyrme model* Yu Cai CCAST (World Laboratory), PO. Box 8730, Beijing 100080, People's Republic of China and Institute of Theoretical Physics, Academia Sinica, PO. Box 2735, Beijing 100080, People's Republic of China I
Received 12 April 1995; revised 2 June 1995
Abstract The T-r/' mixing is incorporated in the symmetry breaking term in the extended Skyrme model Lagrangian. Besides r/-soliton bound states, an s-wave and a p-wave r/'-soliton bound states are found. After fixing the value of the strength parameter X of the "alternative term" to fit the s-wave r/'-soliton bound state to the N(1535) negative-parity nucleon resonance, the r/-soliton bound states disappear. Then only r/'-soliton bound states are identified with nucleon resonances (I = ½) and delta resonances (I = 23-). The predicted resonance masses agree well with experimental values. The decay widths FN* ~N+,7 of the relevant nucleon resonances are also calculated to explain why these particles have large branching ratios in the r/N channel.
1. Introduction Despite its simple structure, the Skyrme model [ 1 ] works well in describing nonstrange baryons [2]. However, attempts to apply it to strange baryons run into some difficulties on account o f breaking of S U ( 3 ) flavor symmetry [ 3 ]. In order to extend the Skyrme model beyond the two flavor sector, several approaches are developed. Among them, the bound state approach proposed by Callan and Klebanov [4] seem to be viable, and has aroused a series of works along this direction [ 5 - 9 ] . In this approach, the strangeness degrees o f freedom are considered as vibrational modes about the zero modes corresponding to isospin rotations o f the classical soliton. After expanding the strange * Work supported in part by the National Natural Science Foundation of China, CCAST (World Laboratory). t Mailing address. 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(95) 00319-3
282
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fields to second order, the problem reduces to the motion of kaons in the background of skyrmions. Bound states of kaons and skyrmions are identified with strange baryons. This scheme works reasonably well in describing the structure of strangeness-flavored hyperons such as mass spectra [4-6] and magnetic moments [7,8]. An interesting problem in the bound state model is the role of the r/ meson which, together with the pions and the kaons, belongs to the pseudoscalar octet. Its comparable mass (mn = 549MeV) to that of the kaons (mK = 495MeV) suggests that it should also be treated as a heavy degree of freedom relative to the pions (m,r = 138MeV), i.e. as a vibrational perturbation about the SU(2) soliton background. This problem was first investigated in Ref. [9]. However, as noticed by the author, the original Skyrme model (with stabilization term being the usual Skyrme term) does not contain much of the dynamics of the eta and skyrmion system, except for an interaction, which is too weak to support a bound state, in the r/mass term. To provide a non-trivial potential for the r/meson, another independent fourth-order derivative term, the so-called "alternative term", was added to the Skyrme term under the requirement that the relevant terms should not contain time derivatives higher than second order. It was shown that this alternative term reduces to the usual Skyrme term in SU(2) sector, thus has no effects to the SU(2) skyrmion. However, it gives different contribution in SU(3) sector. In fact, it was shown in Ref. [ 12] that the increase in the strength of the alternative term relative to the Skyrme term leads to a much better description of the properties of hyperons. The Lagrangian density for SU(3) extended Skyrme model then takes the form Z; = +
Tr(L~L ~) + £mass 1 - X {(TrL~,Lp )2 - (TrL~L~)2} Tr[L~z,L,,] 2 + 1--i-~-e2
( 1)
where Lu = uto~,U with U E SU(3) and the mass term which breaks the SU(3) symmetry can be found, for example, in Ref. [5]. Note that although the WZW term plays a crucial role in the original Callan-Kiebanov approach [4], it makes no contributions to the dynamics of r / - rf-skyrmion system, so it is neglected in Eq. ( 1 ). The last term is the so-called "alternative term". In addition to the pion decay constant F~ and Skyrme parameter e, there is a new parameter X which controls the strength of the "alternative term". The combination of X and ( 1 - X) appearing in front of the fourth-order terms has been made such that the sum of these two terms becomes exactly the usual Skyrme term for U C SU(2). Here it should be emphasized that as r/ is viewed as an octet, the z In mass of 7/ is precisely given by the Gell-Mann-Okubo relation, 3m~ = 4m 2 - m~r. Ref. [9], following the scheme analogous to that of treating the kaons, a bound state located just below the r/-nucleon threshold was found, and was then identified with the N(1535) negative parity $11 nucleon resonance. The motivation for this work is based on the following observations: In the first place, the model mentioned above does not seem to be much convincing since the energy of the r/-skyrmion bound state is always below the r/-nucleon threshold whereas N(1535) lies slightly above the threshold. Therefore one has to leave the parameter X free throughout,
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or, in other words, the model actually unable to make any definite prediction. On the other hand, it is difficult to understand why r/' does not play any role as r / d o e s in the bound state model. This difficulty results from the fact that so far we have adopted an implicit assumption that ~7 is a pure octet. In fact, neither 7/ is an octet nor r / , which has the same quantum numbers of (I, j e ) as those of ~7, is a singlet. Instead, they are superpositions of the octet r/8 and the singlet ~/1, i.e. r/ = ~Tscos 0e - ~71 sin 0e, 7/' ~78 sin 0e + r/1 cos Or,
(2)
where 0t, is mixing angle, empirically, 01, ~ -10.1% In order to overcome these difficulties, we present in this paper an improved model which modifies the Lagrangian (1) to take into account the r/-r/t mixing. As a result, more meson-soliton bound states--not only r/-soliton bound states, but also r/'-soliton o n e s - - a r e found: There exist two bound states, corresponding to 1 = 0, l, in r/sector and r/' sector respectively. The crucial point is that, here it is the l = 0 rl~-soliton bound state rather than r/soliton one that is going to be identified with the N(1535) negative parity Sll nucleon resonance. Once this picture is accepted, the difficulty to explain why N(1535) lies above the r/-nucleon threshold disappears immediately, and the value of the parameter X can be fixed accordingly. Thereupon, as there is no longer any free parameter in the model now, the relevant baryon resonances become predictable in the frame of this improved model. A new feature of the model is the appearance of the coupling term of r/ and r/' at the level of second order in meson fields in the Lagrangian, which originates from the screening effect of the soliton background on the mixing of r/ and r/~. As this term is relevant to the process N* ~ N + r/, its appearance makes it very easy to understand why all the nucleon resonances described in the model have large branching ratios in the r/ N channel. It should be mentioned that several attempts were made to investigate the nucleon and A resonances within the framework of the SU(2) Skyrme model a decade ago [ 17-19]. The basic idea of all these works is interpreting fluctuations around the SU(2) hedgehog configuration as pion-nucleon scattering. The authors of Refs. [ 17,18] calculated the phase shifts in the "breathing mode" of the skyrmion and looked for a resonance in this channel by seeing if and when the phase shift crossed 90 °, nevertheless, a much more reliable criterion for the existence of a resonance was adopted in Ref. [ 19], where resonances were determined by looking for well-defined peaks in the speed [dS/dE] in those regions where the amplitude is curving counterclockwise in the Argand diagram. For comparison the results in Ref. [ 19] are listed in the Table 2 in Section 3 below. This paper is organized as follows. In Section 2 we derive the Lagrangian which modifies the extended Skyrme model Lagrangian to take into account the r / - r / m i x i n g in its symmetry breaking term. In Section 3 the energy levels and wave functions of the meson-soliton bound states as well as the hyperfine splitting constants are calculated. Then the predicted resonance masses are compared with experimental values. In Section 4 we calculate the decay widths /'N*---~N-~-n of the nucleon resonances appearing in our
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model, as compared with experimental values again. Section 5 contains a summarizing discussion.
2. The model As r/-r/~ mixing is concerned, the Lagrangian density of our model takes the same form as Eq. (1) except that the symmetry breaking term given in Ref. [5] is now replaced by b-2 F_2~ /~mass = ~-:'~-m2Tr(U + U t - 2) - ~ dm2Tr[ x/3,t8 (U + U t ) ] z4 10 2
±F~tm2 - m T192~ +
1
2)[Tr(U-
Ut)]2
Am2)Tr(U - Ut)Tr[v/3As(U-
- ,r (m128+ 96v~
Ut)]
(3)
and the chiral field U takes its value in U ( 3 ) , but not SU(3). As H 3 ( U ( 3 ) ) , the third homotopy group of U ( 3 ) , is exactly the same as / / 3 ( S U ( 3 ) ) , namely, / / 3 ( U ( 3 ) ) = H 3 ( S U ( 3 ) ) = Z, this will not bring us any problem. Concretely, the static energy configurations still exist and the baryon current still be conserved in this case. In Eq. (3), two mass parameters ~ and Am are imposed for sake of conciseness. They are defined as ~2 _
2 2m~ m~ + 3 '
2 Am 2 = mk2 -- m~
(4)
with m,r = 138 MeV and mk = 495 MeV being the masses of pion and kaon respectively. Moreover, the mass matrix elements m 2, m 2 and m2s are introduced to describe r/-rl' mixing. Among them, m 2 is determined by the Gell-Mann-Okubo formula 2
m82= 4m2 - m~r 3 '
(5)
and m~ and m~8 are determined under the requirement that the masses of r/ and 7/' should be fixed to their physical values, i.e. mn = 5 4 9 M e V and mn, = 958MeV. Thus we have m 2 I - m , 12
+mZn, _msa,
m~8 = - w / ( m 2 , -
(6)
m2)(m~- m~).
Consequently, the mixing angle elements as O p = t a n -1
( m~-m2n~-~m-~l -]
8e
(7)
in Eq. (2) can be expressed by the mass matrix
.
Numerically, m8 = 567MeV, ml = 947 MeV, ml8 = 331 MeV and Op = -10.1%
(8)
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In order to describe the r/-r/t-K-skyrmion system we introduce a generalized CallanKlebanov ansatz for the chiral field as follows: U=~
(9)
Ux+n+,7' V / ~ ,
where Ux+o+n' = exp
~
~i~i 4- AoCko 4- a8~b8
,
(10)
i=4
U~=exp ( ~2 e Li=l /~i~i )
( 0 )"n" 01
=
'
(11)
Here, F~ is the pion decay constant (= 186 MeV empirically), ,~i (i = 1. . . . . 8) are the Geil-Mann matrices for SU(3), A0 = V/21, and ~i represent the pseudoscalar mesons. Using the hedgehog ansatz for the SU(2) soliton field u~r = exp ( i r . ~ F ( r ) } ,
(12)
and expanding U up to second order in K, r/, and r/', we get the effective Lagrangian L = -M~I + fd3xCn_n,_so, + • •.,
(13)
where MH is the classical mass of the SU(2) hedgehog soliton. It should be noticed that on the RHS of Eq. (13), "..." stands not only for the terms higher than second order, but also for the second order terms involving kaon fields, because to this order the resulting Lagrangian has neither r/-K coupling term nor r/'-K coupling term, allowing us to study the issue of r/-r/'-skyrmion dynamics without making further reference to the strangeness degrees of freedom. Moreover, £n-n'-sol takes the form £n-n'-sol = ½( a'f/2 - fli.i air/a/r/- y,lr/2)
+½(a'q '2 - fluairl'ajrf - yn,r/'2) + V ( r ) r l r l ' ,
(14)
where c~=1+4(l-X) e2F~
+
e2F~
2sin2F"
[(dF) 2
L,,~)
Lkdr]
+
r--~-- ' (Bij - -xi£j) + ~ ( 6 i j
(15) +£i£j)
],
(16)
yn = 2&2 ( cos Fa2 + b2 ) - ~ Am2 ( cos Fa2 - 2b2 ) - 2 ( fn2 - m2) ( x/2 cos Fa - b ) 2
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Y,' = 2rn2( cos Fb2 + a2) -- 4Am2( cosFb2 - 2a2)
--
2(~2
_
m2) (v,~cos Fb + a) 2
+ ( 2~32m28 + 8 Arn2) (2cos2 Fb2 - v/2cos Fab - 2a2) ,
(18)
V ( r ) = 2th2( 1 - cos F ) a b + 4Am2(cos F + 2 )ab + ~ ( r h z - m 2) [ (2 cos 2 F - 1)ab + x/2cos F ( a 2 - b 2) ]
--
m28 + 8Am2
)I
cos
2(cos 2 F + 1)ab - - - - ~ - ( a 2 - b 2)
]
,
(19)
with a=
cos Op - x/2 sin Op x/~ ,
b=
sin Oe + x/2cos Oe x/~
It should be noticed that due to the screening effect of soliton background, T, and Yn' are varied as r goes from zero to infinity. Asymptotically, lim yn = m 2, , r--+oo
lim Y,' = m 2 , .
(20)
r----+oo
Furthermore, this screening effect also affects on the mixing of r / a n d ~7', as is reflected by V(r) which couples 77 and r/' with each other. Since free r/ and r/' are eigenstates of the mass matrix, we have lim V ( r ) = 0.
(21)
r - - - + o~3
The existence of this V(r) coupling term results in a transition between rf-soliton state and r/-soliton state, and make the problem more complicated. However, this complication will not bring us much difficulties in numerical calculations, because we may treat V(r) as a perturbation owing to the fact that the mixing of r/ and r/' is small. As a first step, we will solve the eigenfrequencies as well as the corresponding wave functions of r/ and r/' in the absence of V(r) term (the lowest order approximation), while the contribution of V(r) will be investigated in the section thereinafter.
3. Bound-state mechanics
By neglecting V(r) term in Eq. (14), and using the standard partial- wave decomposition of the r / a n d r/' fields
rlnL ( r, t) = rlnL ( r ) YLM( O, q~) e -i°~",,d , r/,',L(r, t) = rl,l~(r)YLM(O,d?)e ' -iw,/,,Lt ,
(22) (23)
we obtain the eigenvalue equations of r/ and r f
1 d [r2fldrlnL(r)] [ r 2 dr L --G--r] + ,,o,,,,,.-r,
L ( L + 1)3]
r7
~nL(r) = 0 ,
(24)
Y. Cai/Nuclear Physics A 593 (1995) 281-294 %
. . . .
I
. . . .
i
287
. . . .
_ _ i . . . . . . .
i m~'2
~,(0 (fu,,)
%
7.,(r) V(r)
(doshed) (dot-doshed)
% r g %
II
ii I rn 2
% x cq
o 0
1
2
4
5 r (fin)
Fig. I. 7,7, Y,7', and V as functions ofr. F~ = 108MeV, e = 4.84.
# dr
[ ~J
+
awn'"a-
Yn'
r2
a r/',L(r)=O,
(25)
where
t6 = 2i16ij.fi,
~ = 1+
e2F2~
~
+
r---~j .
(26)
In our numerical calculations, we fix the parameters F~ and e at values often used in skyrmion physics in which pion mass takes its empirical value of 138 MeV, i.e. F~ = 108 MeV and e = 4.84. In the first place, we study the dependence of binding energies of the bound states on the parameter X by solving Eqs. (24) and (25). To this end, we leave the parameter X free temporarily. We find two bound states, of which one is an s-wave state and the other is a p-wave state, in ~7 and r/~ sector respectively. The dependence of binding energies of these states on X is shown in Fig. 2 and Fig. 3 respectively. As we see, all these energies decrease monotonically as X increases. For r/-soliton bound states, the binding energy of p-wave one is very small and decrease to zero when X ~ 0.5, that of s-wave one is fairly large, however, as X increases, it decreases more and more rapidly and eventually reaches zero when X approaches to 1. This feature of binding energies of ~7-soliton bound states reflects the weakness of the interaction between ~7 and soliton. On the other hand, for ~7'-soliton bound states, the situation is different. As long as the value of X is not very close to 1, the binding energies of both s-wave and p-wave states remain very large. As x approaches to 1, they
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I=0 0ul0 I=1 (dashed)
o
c~
I
o
o 0.2
0.4
0.6
0.8
X
Fig. 2. Binding energies of the ~/-soliton bound states, m, 7 - oJ (in M e V ) , as functions of the parameter X. Full curve is for 1 = 0 state, dashed one is for 1 = I state. F~r = 108 M e V , e = 4.84.
i
I = 0 (full)
I=1 (doshed)
v
E
°
o12
oi,
o16
o18
X
Fig. 3. Binding energies o f the ~/~-soliton bound states, m,y - ¢o (in M e V ) , as functions o f the parameter X. Full curve is for l = 0 state, dashed one is for l = 1 state. F~r = 1 0 8 M e V , e = 4.84.
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decrease rapidly to fairly small values, leaving us room for adjusting X to fit s-wave state to N(1535). In order to obtain the correct isospin, spin and parity quantum numbers as well as energy splittings of these states, we now take into account the quantization of the collective coordinates associated with rotations of the SU(2) soliton. The SU(2) soliton field is rotated according to
uTr
>A(t)uzrAt(t),
(27)
where A(t) is a time-dependent matrix in SU(2). Because r/ and 77' are isoscalar fields, this transformation leaves them unaffected. The time dependence of A(t) then introduces extra time derivative terms to the Lagrangian, which to lowest order in the angular velocity /~i (given as g i = -~Tr(~'iAtA)) is 6L=2S2g2+8(1-X)
e2F~
f
Jd
3 sin2F"
x~K.r
x (VTIil+ Vrl'(7')
=2f2K 2 - 2cnK. L, 1 - 2c,,K. L , , .
(28)
Here, O is the soliton moment of inertia, L, 7 and L, 7, are orbit angular momenta of r/-soliton bound state and r/'-soliton bound state respectively. The hyperfine splitting constants c~ and c,~, have the forms OO
c,7 =
4( 1 - X)
e2F2
f
oJn j d r s i n 2 Frl(r) 2 ,
(29)
o 00
cn , -
4( 1 - X)
e2F2
o~,
j dr sin 2 Frl'(r) 2
(30)
o
Taking into account the rotational correction to the energy obtained from Eq. (28), the bound state spectrum is finally 1
E=Mtt+w+~-~[cJ(J+l)+(1-c)l(l+l)+c(c-1)L(L+l)],
(31)
with I being the isospin quantum number of the soliton and J the total angular momentum quantum number. This spectrum formula holds for both r/-soliton bound states and r/'-soliton bound states, the quantities (~o, c, L) in it stands for either (o~,1, c n, L,7) or (w,,, cn,, L,7, ), depending on whether the meson bound by the soliton is r / o r r/'. Hitherto, the parameter X still remains undetermined. Now we turn to discuss this issue. In Ref. [9], the r / - N bound state was explained as the N(1535) negative parity Sll nucleon resonance. However, this explanation does not seem convincing since N(1535) lies above the r / - N threshold while the energy of ~/-N bound state can never beyond this threshold. From our point of view, the N(1535) resonance should rather be identified with the lowest ~7'-N bound state, which has the same quantum numbers of I ( J p) as those of N(1535) too. Upon this consideration, we can immediately fix the value of X
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Table 1 Eigenfrequencies (in M e V ) and hyperfine splitting constants of the bound states. Parameters are as follows: F~r = 108 MeV, e = 4.84, X = 0.89 r / - N bound states
L = 0 L = 1
~/~-N bound states
(-or/
Cr/
(.Or/,'
Cr/;
520
0.025
597 809
0.062 0.039
by requiring the energy of the lowest r/t-N bound state to be 1535 MeV. In view of the fact that this state is a s-wave state, which implies that no hyperfine energy shift is devoted to its energy, this amounts to requiring its eigenfrequency to to be the mass difference between N(1535) and nucleon. The corresponding value of X is found to be 0.89 using the dependence of to on X. By setting X = 0.89, we then determine the eigenfrequencies of all the bound states and calculate their hyperfine splitting constants as well. The results are shown in Table 1. In comparison of our results with experiment [15], we first pay our attention to the r/-soliton bound states. From Table 1 we observe that the p-wave r/-soliton state disappears due to the fact that for this state, X¢, the critical value of X for which the state lies just at the threshold, is less than 0.89. On the other hand, the s-wave one still exists with to = 520 MeV, lying very close to the r/-soliton threshold (549 MeV). Unfortunately, this leads to disagreement with experiment since no N* resonance with I ( J P ) = ½( ½- ) is observed lying below the r/-N threshold. We argue that, if a suitable set of higher-derivative terms were added to the Skyrme Lagrangian, this disaster would not happen. The point is that the inclusion of higher-derivative terms will increase the interactions between soliton and mesons, while the increasing extent of the r/I-soliton interaction will be larger than that of the r/-soliton interaction, for these higher-derivative terms mainly account for the short-range behavior of the interactions, and m~, is fairly larger than mn. So, in view of the fact that the r/-soliton bound state lies very close to the r/-soliton threshold, once we take these terms into account the fitting of the lowest r//-soliton bound state to N(1535) will lead to the disappearance of this state. Therefore, we suggest that this state be neglected, and that the relevant states to be identified with low-lying baryon resonances be only r/~-soliton bound states. After performing the collective coordinate quantization of the SU(2) soliton, these r / ' soliton bound states get their isospin and spin quantum numbers I = J = ½, 3. Because of the pseudoscalar nature of r/~, the parity of the s-wave bound state is negative while that of p-wave bound state is positive. We then identify them with nucleon resonances ( I = ½) and delta resonances (1 = 3). The masses of the predicted resonances are shown in Table 2, in comparison with the available experimental data [ 15]. We observe that as a whole our model works fairly well. In fact, the predicted masses of N(1710) Pll and N(1720) PI3 nucleon resonances agree very well with their empirical values, with relative errors of no more
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291
Table 2 The calculated spectrum of the baryon resonances (in MeV) as compared to empirical ones. Parameters are the same as in Table 1. For comparison, the results in Ref. 119] are also listed
1
JP
Mass
± 2
i7+
i2 ± 2
-2 ~+ 2
Ref. [ 19 ]
Particle
1535(Fi~ed)
1478
N(1535)
1739
1427
N(1710)
1751
1982
N(1720)
2
~ 7+-
1829
1737
A(1700)
2
2021
1982
A(1910)
2
-2 ~+
2033
1946
A(1920)
2
2+ 2
2053
1931
A(2000)
than 2%. Nevertheless, the predicted delta resonances come out 50 ~ 130 MeV higher than the empirical values. Even so, the corresponding relative errors of no more than 8% are still satisfactory as compared with the general accuracy of the topological soliton model of about 30%. For comparison, the masses of the relevant resonances predicted in Ref. [ 19] are also listed in Table 2. Here we stress that the masses of the nucleon resonances predicted by our approach are much better than those in Ref. [ 19]. Apart from the fact that, as will be discussed in Section 4, these nucleon resonances have large branching ratios in the tIN channel, the above fact seems to provide us another support to the viewpoint that these resonances should be regarded as rf-skyrmion bound states.
4. N*
, N + r/decay width
FN.--.N+,!
As we have mentioned in Section 2, in order to simplify our numerical calculations we regard the V(r) term in the Lagrangian (14) as a perturbation, and, as lowest order approximation, neglect it in the course of calculating the eigenfrequencies and wave functions of the meson-soliton bound states. Nevertheless, provided higher order corrections are included, this term results in a transition between r/P-soliton bound state and r/-soliton bound state, and provides us a simple, intuitive picture in explaining why experimentally the nucleon resonances described in our model have large branching ratios in the r/N channel. In this section, we calculate the contributions of this term to first order, the only relevant process is N* ~ N + r/. Here, we regard the rotating soliton (the nucleon) just as a background field in which the mesons (r/, r/~) is moving. As usual in the Skyrme model [ 14,16], the mass of the nucleon is regarded as being infinitely large, then the recoil effect is neglected. So, studying the above process amounts to calculating matrix elements of/~int = V ( r ) ~ 7 ~ t between the initial state (bound ~7~ s t a t e ]~TtnLM)) and the final state (scattered 7/ state ]7/~,LM)):
292
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2~'8( E f
-
Ei)'Zfi = f d 4X(rl~oLMl£intlrlnLg), t J
(32)
where Ei = to,,, E f = o) are initial and final energies respectively. The transition amplitude 7- is related to the S-matrix by Sfi = 8fi - 2~'iS( Ef - Ei)7-fi. By expanding the r/and r/~ fields in terms of their eigenmodes: (~o
r l = ~ - ~ f do~[rlo,t.(r)YLi~(O, da)e-i,Ota,7,oLU+rto,L(r)y?.M(O, dp)e io,ta,lo~LM t ]
,
(33)
LM 0 L M t"'~ ~ , q ~-t'e-i'o"t) U r l, n L M "-l -- t-l 'n L t ',r ' Y ) L* M 't,0 , q~"-i'°"tat ))e .q,nLMI,
r/= Z[r/.L(rlE
(34)
nLM
and requiring to have the standard form for the Hamiltonian operator in the second quantization scheme [4], the wavefunctions are normalized as (>o
f r2dr(w + o)')tr(r)rla, L(r)rl~o,L(r ) = 8(to -- m t) , o
(35)
oo
f
r2 dr( o), + ~on,)a( r )rl.L ( r)rl.,L ( r ) = 6~n, .
'
0
(36)
,
After performing the time integral and the angle part of the spatial integral in Eq. (32), we obtain oo
~ f i = ~LL' 6 M M '
/ rZ drrl,,,L ( r ) V ( r ) rllnL( r )
(37)
,
o
where the appearance of the factor t~LL, ~ M M ' is due to conservation of the angular momentum. The decay width FN*---,N+n is then easily expressed as 2
77" llq*_ ----~N+rI = - (.0
[fo r2drrl,oL (r) V(r) r/. L (r)
,
(38)
where rl',L(r) is the radial wavefunction of the bound ~/~ state to which N* corresponds, w = mN* - mu is the energy of the outgoing r/meson, with mN- given by Eq. (31). The numerical results of the decay width from Eq. (38) are listed in Table 3. We observe that as a whole our results are larger to different extents as compared with the experimental values. In view of the inherent accuracy of the Skyrme model, the decay width for N(1535) is satisfactory, that for N(1710) is still acceptable, however, that for N(1720) is too large by a factor as much as five. This is due to the fact that the picture presented here is too simple to account for why the difference between FN(1710)---*N+~ and FN(1720)~N+n is SO large that the former is several times larger than the latter.
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Table 3 The calculated decay widths FN*~s+,7 (in MeV) as comparedto empirical ones. Parameters are the same as in Table 1
5.
N*
n
L
J
FN*--,N+n(MeV)
Exp. (MeV)
N(1535)
1
0
N(1710)
1
1
l
87
~ 75
l
41
~
N(1720)
1
1
~2
38
~ 8
28
Conclusion
In this paper, we have investigated the effects of the ~?-rf mixing in the bound state version of the Skyrme model. To this end, we have improved the extended Skyrme model by incorporating the r/-r/~ mixing in the symmetry breaking term in the Lagrangian of the model. Up to the second order in pseudoscalar meson fields, neither r/-K nor r/~-K coupling terms arise, so the strange baryon spectrum predicted by the original model remains unchanged. However, this improvement is helpful to our understanding of nonstrange baryon resonances. As our calculation has demonstrated, there exist a number of rl-soliton and ~'-soliton bound states. For ~ - s o l i t o n bound states, by identifying their quantum numbers with those of the relevant baryon resonances, they are explained as these resonances. The calculated mass spectrum agrees very well with experiment, with relative errors of no more than 8% which are much small as compared with the general accuracy of the topological soliton model of about 30%. Unfortunately, an exceptional state predicted by the l = 0 r/-soliton bound state exists. We see this not as a problem with our approach, but rather a problem with the mesonic action that has been adopted. In fact, the large-Nc QCD Lagrangian, to which the Skyrme Lagrangian is just an approximation, of course contains a multitude of higher-derivative terms and there is no prior reason to neglect them. They have been neglected only for reasons of computational simplicity. If a suitable set of such terms were added to the Skyrme Lagrangian, this exceptional state would disappear. An interesting co-product of our approach is the appearance of the coupling term of r/ and r/~ at the level of second order in the meson fields in the Lagrangian, which originates from the screening effect of the soliton background on the mixing of r/ and r/~. This coupling term results in a transition between r/~-soliton bound state and r/soliton bound state, and provides us with a simple, intuitive picture of explaining why experimentally the nucleon resonances described in our model have large branching ratios in the r/N channel. Some other problems concerning this approach, such as the calculation of the magnetic moments of the relevant baryon resonances, and the model that contains heavy flavors, deserve further investigation. We will discuss these problems in detail elsewhere.
294
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Acknowledgements T h e author w o u l d like to thank Dr. J.H. Dai, and Dr. J. Z h a n g for helpful discussions. The w o r k is finished during the S u m m e r Institute sponsored by C C A S T and c o - s p o n s o r e d by ITP and supported by N S F C . T h e author is also supported in part by Local Natural Foundation of Xinjiang.
References [1] [2] [31 [4] [51 [61 [71 [8] [9] [101 Jill [12] [13] [14] [151 [16] 1171
[18] [19]
T.H.R. Skyrme, Proc. Roy. Soc. (London) A 260 (1961) 127. G. Adkins, C. Nappi and E. Wiuen, Nucl. Phys. B 228 (1983) 552. D. Masak, Phys. Rev. D 39 (1989) 305, and references therein. C.G. Cailan and I. Klebanov, Nucl. Phys. B 262 (1985) 365; C.G. Callan, I. Klebanov and K. Hombostel, Phys. Lett. B 202 (1988) 269. U. Biom, K. Dannbom and D.O. Riska, Nucl. Phys. A 493 (1989) 384. D.O. Riska and N.N. Scoccola, Phys. Lett. B 265 (1991) 188. J. Kunz and P.J. Mulders, Phys. Rev. D 41 (1990) 1578. E.M. Nyman and D.O. Riska, Nucl. Phys. B 325 (1989) 593. G. Pail, Phys. Lett. B 261 (1991) 347. A. Hosaka, Phys. Lett. B 293 (1992) 23. G. Pail, B. Schwesinger and H. Walliser, Phys. Lett. B 255 (1991) 1. G. Pail and N.N. Scoccola, Phys. Lett. B 296 (1992) 391. C. Gobbi and N.N. Scoccola, Phys. Lett. B 318 (1993) 382. C. Gobbi, A. Hosaka and N.N. Scoccola, Nucl. Phys. A 562 (1993) 461. Particle Data Group, Phys. Lett. B 239 (1990). M. Rho, D.O. Riska and N.N. Scoccola, Z. Phys. A 341 (1992) 343. I. Zahed, U.-G. Meissner and U.B. Kaulfuss, Nucl. Phys. A 426 (1984) 525. J.D. Breit and C.R. Nappi, Phys. Rev. Lett. 53 (1984) 889. M.P. Mattis and M. Kaliner, Phys. Rev. D 31 (1985) 2833; I. Zahed, Excited Baryons in the Skyrme Model, in: Excited Baryons (1988), eds. N.C. Mukhopadhyay and P. Stoler (World Scientific, Singapore, 1989).
Nuclear Physics A 593 (1995) 281-294
The effect of mixing in the bound state version of the Skyrme model* Yu Cai CCAST (World Laboratory), PO. Box 8730, Beijing 100080, People's Republic of China and Institute of Theoretical Physics, Academia Sinica, PO. Box 2735, Beijing 100080, People's Republic of China I
Received 12 April 1995; revised 2 June 1995
Abstract The T-r/' mixing is incorporated in the symmetry breaking term in the extended Skyrme model Lagrangian. Besides r/-soliton bound states, an s-wave and a p-wave r/'-soliton bound states are found. After fixing the value of the strength parameter X of the "alternative term" to fit the s-wave r/'-soliton bound state to the N(1535) negative-parity nucleon resonance, the r/-soliton bound states disappear. Then only r/'-soliton bound states are identified with nucleon resonances (I = ½) and delta resonances (I = 23-). The predicted resonance masses agree well with experimental values. The decay widths FN* ~N+,7 of the relevant nucleon resonances are also calculated to explain why these particles have large branching ratios in the r/N channel.
1. Introduction Despite its simple structure, the Skyrme model [ 1 ] works well in describing nonstrange baryons [2]. However, attempts to apply it to strange baryons run into some difficulties on account o f breaking of S U ( 3 ) flavor symmetry [ 3 ]. In order to extend the Skyrme model beyond the two flavor sector, several approaches are developed. Among them, the bound state approach proposed by Callan and Klebanov [4] seem to be viable, and has aroused a series of works along this direction [ 5 - 9 ] . In this approach, the strangeness degrees o f freedom are considered as vibrational modes about the zero modes corresponding to isospin rotations o f the classical soliton. After expanding the strange * Work supported in part by the National Natural Science Foundation of China, CCAST (World Laboratory). t Mailing address. 0375-9474/95/$09.50 (~) 1995 Elsevier Science B.V. All rights reserved SSDI 0375-9474(95) 00319-3
282
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fields to second order, the problem reduces to the motion of kaons in the background of skyrmions. Bound states of kaons and skyrmions are identified with strange baryons. This scheme works reasonably well in describing the structure of strangeness-flavored hyperons such as mass spectra [4-6] and magnetic moments [7,8]. An interesting problem in the bound state model is the role of the r/ meson which, together with the pions and the kaons, belongs to the pseudoscalar octet. Its comparable mass (mn = 549MeV) to that of the kaons (mK = 495MeV) suggests that it should also be treated as a heavy degree of freedom relative to the pions (m,r = 138MeV), i.e. as a vibrational perturbation about the SU(2) soliton background. This problem was first investigated in Ref. [9]. However, as noticed by the author, the original Skyrme model (with stabilization term being the usual Skyrme term) does not contain much of the dynamics of the eta and skyrmion system, except for an interaction, which is too weak to support a bound state, in the r/mass term. To provide a non-trivial potential for the r/meson, another independent fourth-order derivative term, the so-called "alternative term", was added to the Skyrme term under the requirement that the relevant terms should not contain time derivatives higher than second order. It was shown that this alternative term reduces to the usual Skyrme term in SU(2) sector, thus has no effects to the SU(2) skyrmion. However, it gives different contribution in SU(3) sector. In fact, it was shown in Ref. [ 12] that the increase in the strength of the alternative term relative to the Skyrme term leads to a much better description of the properties of hyperons. The Lagrangian density for SU(3) extended Skyrme model then takes the form Z; = +
Tr(L~L ~) + £mass 1 - X {(TrL~,Lp )2 - (TrL~L~)2} Tr[L~z,L,,] 2 + 1--i-~-e2
( 1)
where Lu = uto~,U with U E SU(3) and the mass term which breaks the SU(3) symmetry can be found, for example, in Ref. [5]. Note that although the WZW term plays a crucial role in the original Callan-Kiebanov approach [4], it makes no contributions to the dynamics of r / - rf-skyrmion system, so it is neglected in Eq. ( 1 ). The last term is the so-called "alternative term". In addition to the pion decay constant F~ and Skyrme parameter e, there is a new parameter X which controls the strength of the "alternative term". The combination of X and ( 1 - X) appearing in front of the fourth-order terms has been made such that the sum of these two terms becomes exactly the usual Skyrme term for U C SU(2). Here it should be emphasized that as r/ is viewed as an octet, the z In mass of 7/ is precisely given by the Gell-Mann-Okubo relation, 3m~ = 4m 2 - m~r. Ref. [9], following the scheme analogous to that of treating the kaons, a bound state located just below the r/-nucleon threshold was found, and was then identified with the N(1535) negative parity $11 nucleon resonance. The motivation for this work is based on the following observations: In the first place, the model mentioned above does not seem to be much convincing since the energy of the r/-skyrmion bound state is always below the r/-nucleon threshold whereas N(1535) lies slightly above the threshold. Therefore one has to leave the parameter X free throughout,
Y. Cai/Nuclear Physics A 593 (1995) 281-294
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or, in other words, the model actually unable to make any definite prediction. On the other hand, it is difficult to understand why r/' does not play any role as r / d o e s in the bound state model. This difficulty results from the fact that so far we have adopted an implicit assumption that ~7 is a pure octet. In fact, neither 7/ is an octet nor r / , which has the same quantum numbers of (I, j e ) as those of ~7, is a singlet. Instead, they are superpositions of the octet r/8 and the singlet ~/1, i.e. r/ = ~Tscos 0e - ~71 sin 0e, 7/' ~78 sin 0e + r/1 cos Or,
(2)
where 0t, is mixing angle, empirically, 01, ~ -10.1% In order to overcome these difficulties, we present in this paper an improved model which modifies the Lagrangian (1) to take into account the r/-r/t mixing. As a result, more meson-soliton bound states--not only r/-soliton bound states, but also r/'-soliton o n e s - - a r e found: There exist two bound states, corresponding to 1 = 0, l, in r/sector and r/' sector respectively. The crucial point is that, here it is the l = 0 rl~-soliton bound state rather than r/soliton one that is going to be identified with the N(1535) negative parity Sll nucleon resonance. Once this picture is accepted, the difficulty to explain why N(1535) lies above the r/-nucleon threshold disappears immediately, and the value of the parameter X can be fixed accordingly. Thereupon, as there is no longer any free parameter in the model now, the relevant baryon resonances become predictable in the frame of this improved model. A new feature of the model is the appearance of the coupling term of r/ and r/' at the level of second order in meson fields in the Lagrangian, which originates from the screening effect of the soliton background on the mixing of r/ and r/~. As this term is relevant to the process N* ~ N + r/, its appearance makes it very easy to understand why all the nucleon resonances described in the model have large branching ratios in the r/ N channel. It should be mentioned that several attempts were made to investigate the nucleon and A resonances within the framework of the SU(2) Skyrme model a decade ago [ 17-19]. The basic idea of all these works is interpreting fluctuations around the SU(2) hedgehog configuration as pion-nucleon scattering. The authors of Refs. [ 17,18] calculated the phase shifts in the "breathing mode" of the skyrmion and looked for a resonance in this channel by seeing if and when the phase shift crossed 90 °, nevertheless, a much more reliable criterion for the existence of a resonance was adopted in Ref. [ 19], where resonances were determined by looking for well-defined peaks in the speed [dS/dE] in those regions where the amplitude is curving counterclockwise in the Argand diagram. For comparison the results in Ref. [ 19] are listed in the Table 2 in Section 3 below. This paper is organized as follows. In Section 2 we derive the Lagrangian which modifies the extended Skyrme model Lagrangian to take into account the r / - r / m i x i n g in its symmetry breaking term. In Section 3 the energy levels and wave functions of the meson-soliton bound states as well as the hyperfine splitting constants are calculated. Then the predicted resonance masses are compared with experimental values. In Section 4 we calculate the decay widths /'N*---~N-~-n of the nucleon resonances appearing in our
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model, as compared with experimental values again. Section 5 contains a summarizing discussion.
2. The model As r/-r/~ mixing is concerned, the Lagrangian density of our model takes the same form as Eq. (1) except that the symmetry breaking term given in Ref. [5] is now replaced by b-2 F_2~ /~mass = ~-:'~-m2Tr(U + U t - 2) - ~ dm2Tr[ x/3,t8 (U + U t ) ] z4 10 2
±F~tm2 - m T192~ +
1
2)[Tr(U-
Ut)]2
Am2)Tr(U - Ut)Tr[v/3As(U-
- ,r (m128+ 96v~
Ut)]
(3)
and the chiral field U takes its value in U ( 3 ) , but not SU(3). As H 3 ( U ( 3 ) ) , the third homotopy group of U ( 3 ) , is exactly the same as / / 3 ( S U ( 3 ) ) , namely, / / 3 ( U ( 3 ) ) = H 3 ( S U ( 3 ) ) = Z, this will not bring us any problem. Concretely, the static energy configurations still exist and the baryon current still be conserved in this case. In Eq. (3), two mass parameters ~ and Am are imposed for sake of conciseness. They are defined as ~2 _
2 2m~ m~ + 3 '
2 Am 2 = mk2 -- m~
(4)
with m,r = 138 MeV and mk = 495 MeV being the masses of pion and kaon respectively. Moreover, the mass matrix elements m 2, m 2 and m2s are introduced to describe r/-rl' mixing. Among them, m 2 is determined by the Gell-Mann-Okubo formula 2
m82= 4m2 - m~r 3 '
(5)
and m~ and m~8 are determined under the requirement that the masses of r/ and 7/' should be fixed to their physical values, i.e. mn = 5 4 9 M e V and mn, = 958MeV. Thus we have m 2 I - m , 12
+mZn, _msa,
m~8 = - w / ( m 2 , -
(6)
m2)(m~- m~).
Consequently, the mixing angle elements as O p = t a n -1
( m~-m2n~-~m-~l -]
8e
(7)
in Eq. (2) can be expressed by the mass matrix
.
Numerically, m8 = 567MeV, ml = 947 MeV, ml8 = 331 MeV and Op = -10.1%
(8)
Y Cai/Nuclear Physics A 593 (1995) 281-294
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In order to describe the r/-r/t-K-skyrmion system we introduce a generalized CallanKlebanov ansatz for the chiral field as follows: U=~
(9)
Ux+n+,7' V / ~ ,
where Ux+o+n' = exp
~
~i~i 4- AoCko 4- a8~b8
,
(10)
i=4
U~=exp ( ~2 e Li=l /~i~i )
( 0 )"n" 01
=
'
(11)
Here, F~ is the pion decay constant (= 186 MeV empirically), ,~i (i = 1. . . . . 8) are the Geil-Mann matrices for SU(3), A0 = V/21, and ~i represent the pseudoscalar mesons. Using the hedgehog ansatz for the SU(2) soliton field u~r = exp ( i r . ~ F ( r ) } ,
(12)
and expanding U up to second order in K, r/, and r/', we get the effective Lagrangian L = -M~I + fd3xCn_n,_so, + • •.,
(13)
where MH is the classical mass of the SU(2) hedgehog soliton. It should be noticed that on the RHS of Eq. (13), "..." stands not only for the terms higher than second order, but also for the second order terms involving kaon fields, because to this order the resulting Lagrangian has neither r/-K coupling term nor r/'-K coupling term, allowing us to study the issue of r/-r/'-skyrmion dynamics without making further reference to the strangeness degrees of freedom. Moreover, £n-n'-sol takes the form £n-n'-sol = ½( a'f/2 - fli.i air/a/r/- y,lr/2)
+½(a'q '2 - fluairl'ajrf - yn,r/'2) + V ( r ) r l r l ' ,
(14)
where c~=1+4(l-X) e2F~
+
e2F~
2sin2F"
[(dF) 2
L,,~)
Lkdr]
+
r--~-- ' (Bij - -xi£j) + ~ ( 6 i j
(15) +£i£j)
],
(16)
yn = 2&2 ( cos Fa2 + b2 ) - ~ Am2 ( cos Fa2 - 2b2 ) - 2 ( fn2 - m2) ( x/2 cos Fa - b ) 2
Y Cai/Nuclear Physics A 593 (1995) 281-294
286
Y,' = 2rn2( cos Fb2 + a2) -- 4Am2( cosFb2 - 2a2)
--
2(~2
_
m2) (v,~cos Fb + a) 2
+ ( 2~32m28 + 8 Arn2) (2cos2 Fb2 - v/2cos Fab - 2a2) ,
(18)
V ( r ) = 2th2( 1 - cos F ) a b + 4Am2(cos F + 2 )ab + ~ ( r h z - m 2) [ (2 cos 2 F - 1)ab + x/2cos F ( a 2 - b 2) ]
--
m28 + 8Am2
)I
cos
2(cos 2 F + 1)ab - - - - ~ - ( a 2 - b 2)
]
,
(19)
with a=
cos Op - x/2 sin Op x/~ ,
b=
sin Oe + x/2cos Oe x/~
It should be noticed that due to the screening effect of soliton background, T, and Yn' are varied as r goes from zero to infinity. Asymptotically, lim yn = m 2, , r--+oo
lim Y,' = m 2 , .
(20)
r----+oo
Furthermore, this screening effect also affects on the mixing of r / a n d ~7', as is reflected by V(r) which couples 77 and r/' with each other. Since free r/ and r/' are eigenstates of the mass matrix, we have lim V ( r ) = 0.
(21)
r - - - + o~3
The existence of this V(r) coupling term results in a transition between rf-soliton state and r/-soliton state, and make the problem more complicated. However, this complication will not bring us much difficulties in numerical calculations, because we may treat V(r) as a perturbation owing to the fact that the mixing of r/ and r/' is small. As a first step, we will solve the eigenfrequencies as well as the corresponding wave functions of r/ and r/' in the absence of V(r) term (the lowest order approximation), while the contribution of V(r) will be investigated in the section thereinafter.
3. Bound-state mechanics
By neglecting V(r) term in Eq. (14), and using the standard partial- wave decomposition of the r / a n d r/' fields
rlnL ( r, t) = rlnL ( r ) YLM( O, q~) e -i°~",,d , r/,',L(r, t) = rl,l~(r)YLM(O,d?)e ' -iw,/,,Lt ,
(22) (23)
we obtain the eigenvalue equations of r/ and r f
1 d [r2fldrlnL(r)] [ r 2 dr L --G--r] + ,,o,,,,,.-r,
L ( L + 1)3]
r7
~nL(r) = 0 ,
(24)
Y. Cai/Nuclear Physics A 593 (1995) 281-294 %
. . . .
I
. . . .
i
287
. . . .
_ _ i . . . . . . .
i m~'2
~,(0 (fu,,)
%
7.,(r) V(r)
(doshed) (dot-doshed)
% r g %
II
ii I rn 2
% x cq
o 0
1
2
4
5 r (fin)
Fig. I. 7,7, Y,7', and V as functions ofr. F~ = 108MeV, e = 4.84.
# dr
[ ~J
+
awn'"a-
Yn'
r2
a r/',L(r)=O,
(25)
where
t6 = 2i16ij.fi,
~ = 1+
e2F2~
~
+
r---~j .
(26)
In our numerical calculations, we fix the parameters F~ and e at values often used in skyrmion physics in which pion mass takes its empirical value of 138 MeV, i.e. F~ = 108 MeV and e = 4.84. In the first place, we study the dependence of binding energies of the bound states on the parameter X by solving Eqs. (24) and (25). To this end, we leave the parameter X free temporarily. We find two bound states, of which one is an s-wave state and the other is a p-wave state, in ~7 and r/~ sector respectively. The dependence of binding energies of these states on X is shown in Fig. 2 and Fig. 3 respectively. As we see, all these energies decrease monotonically as X increases. For r/-soliton bound states, the binding energy of p-wave one is very small and decrease to zero when X ~ 0.5, that of s-wave one is fairly large, however, as X increases, it decreases more and more rapidly and eventually reaches zero when X approaches to 1. This feature of binding energies of ~7-soliton bound states reflects the weakness of the interaction between ~7 and soliton. On the other hand, for ~7'-soliton bound states, the situation is different. As long as the value of X is not very close to 1, the binding energies of both s-wave and p-wave states remain very large. As x approaches to 1, they
Y Cai/Nuclear Physics A 593 (1995) 281-294
288
I=0 0ul0 I=1 (dashed)
o
c~
I
o
o 0.2
0.4
0.6
0.8
X
Fig. 2. Binding energies of the ~/-soliton bound states, m, 7 - oJ (in M e V ) , as functions of the parameter X. Full curve is for 1 = 0 state, dashed one is for 1 = I state. F~r = 108 M e V , e = 4.84.
i
I = 0 (full)
I=1 (doshed)
v
E
°
o12
oi,
o16
o18
X
Fig. 3. Binding energies o f the ~/~-soliton bound states, m,y - ¢o (in M e V ) , as functions o f the parameter X. Full curve is for l = 0 state, dashed one is for l = 1 state. F~r = 1 0 8 M e V , e = 4.84.
Y. Cai/Nuclear Physics A 593 (1995) 281-294
289
decrease rapidly to fairly small values, leaving us room for adjusting X to fit s-wave state to N(1535). In order to obtain the correct isospin, spin and parity quantum numbers as well as energy splittings of these states, we now take into account the quantization of the collective coordinates associated with rotations of the SU(2) soliton. The SU(2) soliton field is rotated according to
uTr
>A(t)uzrAt(t),
(27)
where A(t) is a time-dependent matrix in SU(2). Because r/ and 77' are isoscalar fields, this transformation leaves them unaffected. The time dependence of A(t) then introduces extra time derivative terms to the Lagrangian, which to lowest order in the angular velocity /~i (given as g i = -~Tr(~'iAtA)) is 6L=2S2g2+8(1-X)
e2F~
f
Jd
3 sin2F"
x~K.r
x (VTIil+ Vrl'(7')
=2f2K 2 - 2cnK. L, 1 - 2c,,K. L , , .
(28)
Here, O is the soliton moment of inertia, L, 7 and L, 7, are orbit angular momenta of r/-soliton bound state and r/'-soliton bound state respectively. The hyperfine splitting constants c~ and c,~, have the forms OO
c,7 =
4( 1 - X)
e2F2
f
oJn j d r s i n 2 Frl(r) 2 ,
(29)
o 00
cn , -
4( 1 - X)
e2F2
o~,
j dr sin 2 Frl'(r) 2
(30)
o
Taking into account the rotational correction to the energy obtained from Eq. (28), the bound state spectrum is finally 1
E=Mtt+w+~-~[cJ(J+l)+(1-c)l(l+l)+c(c-1)L(L+l)],
(31)
with I being the isospin quantum number of the soliton and J the total angular momentum quantum number. This spectrum formula holds for both r/-soliton bound states and r/'-soliton bound states, the quantities (~o, c, L) in it stands for either (o~,1, c n, L,7) or (w,,, cn,, L,7, ), depending on whether the meson bound by the soliton is r / o r r/'. Hitherto, the parameter X still remains undetermined. Now we turn to discuss this issue. In Ref. [9], the r / - N bound state was explained as the N(1535) negative parity Sll nucleon resonance. However, this explanation does not seem convincing since N(1535) lies above the r / - N threshold while the energy of ~/-N bound state can never beyond this threshold. From our point of view, the N(1535) resonance should rather be identified with the lowest ~7'-N bound state, which has the same quantum numbers of I ( J p) as those of N(1535) too. Upon this consideration, we can immediately fix the value of X
Y Cai/Nuclear Physics A 593 (1995) 281-294
290
Table 1 Eigenfrequencies (in M e V ) and hyperfine splitting constants of the bound states. Parameters are as follows: F~r = 108 MeV, e = 4.84, X = 0.89 r / - N bound states
L = 0 L = 1
~/~-N bound states
(-or/
Cr/
(.Or/,'
Cr/;
520
0.025
597 809
0.062 0.039
by requiring the energy of the lowest r/t-N bound state to be 1535 MeV. In view of the fact that this state is a s-wave state, which implies that no hyperfine energy shift is devoted to its energy, this amounts to requiring its eigenfrequency to to be the mass difference between N(1535) and nucleon. The corresponding value of X is found to be 0.89 using the dependence of to on X. By setting X = 0.89, we then determine the eigenfrequencies of all the bound states and calculate their hyperfine splitting constants as well. The results are shown in Table 1. In comparison of our results with experiment [15], we first pay our attention to the r/-soliton bound states. From Table 1 we observe that the p-wave r/-soliton state disappears due to the fact that for this state, X¢, the critical value of X for which the state lies just at the threshold, is less than 0.89. On the other hand, the s-wave one still exists with to = 520 MeV, lying very close to the r/-soliton threshold (549 MeV). Unfortunately, this leads to disagreement with experiment since no N* resonance with I ( J P ) = ½( ½- ) is observed lying below the r/-N threshold. We argue that, if a suitable set of higher-derivative terms were added to the Skyrme Lagrangian, this disaster would not happen. The point is that the inclusion of higher-derivative terms will increase the interactions between soliton and mesons, while the increasing extent of the r/I-soliton interaction will be larger than that of the r/-soliton interaction, for these higher-derivative terms mainly account for the short-range behavior of the interactions, and m~, is fairly larger than mn. So, in view of the fact that the r/-soliton bound state lies very close to the r/-soliton threshold, once we take these terms into account the fitting of the lowest r//-soliton bound state to N(1535) will lead to the disappearance of this state. Therefore, we suggest that this state be neglected, and that the relevant states to be identified with low-lying baryon resonances be only r/~-soliton bound states. After performing the collective coordinate quantization of the SU(2) soliton, these r / ' soliton bound states get their isospin and spin quantum numbers I = J = ½, 3. Because of the pseudoscalar nature of r/~, the parity of the s-wave bound state is negative while that of p-wave bound state is positive. We then identify them with nucleon resonances ( I = ½) and delta resonances (1 = 3). The masses of the predicted resonances are shown in Table 2, in comparison with the available experimental data [ 15]. We observe that as a whole our model works fairly well. In fact, the predicted masses of N(1710) Pll and N(1720) PI3 nucleon resonances agree very well with their empirical values, with relative errors of no more
Y. Cai/Nuclear Physics A 593 (1995) 281-294
291
Table 2 The calculated spectrum of the baryon resonances (in MeV) as compared to empirical ones. Parameters are the same as in Table 1. For comparison, the results in Ref. 119] are also listed
1
JP
Mass
± 2
i7+
i2 ± 2
-2 ~+ 2
Ref. [ 19 ]
Particle
1535(Fi~ed)
1478
N(1535)
1739
1427
N(1710)
1751
1982
N(1720)
2
~ 7+-
1829
1737
A(1700)
2
2021
1982
A(1910)
2
-2 ~+
2033
1946
A(1920)
2
2+ 2
2053
1931
A(2000)
than 2%. Nevertheless, the predicted delta resonances come out 50 ~ 130 MeV higher than the empirical values. Even so, the corresponding relative errors of no more than 8% are still satisfactory as compared with the general accuracy of the topological soliton model of about 30%. For comparison, the masses of the relevant resonances predicted in Ref. [ 19] are also listed in Table 2. Here we stress that the masses of the nucleon resonances predicted by our approach are much better than those in Ref. [ 19]. Apart from the fact that, as will be discussed in Section 4, these nucleon resonances have large branching ratios in the tIN channel, the above fact seems to provide us another support to the viewpoint that these resonances should be regarded as rf-skyrmion bound states.
4. N*
, N + r/decay width
FN.--.N+,!
As we have mentioned in Section 2, in order to simplify our numerical calculations we regard the V(r) term in the Lagrangian (14) as a perturbation, and, as lowest order approximation, neglect it in the course of calculating the eigenfrequencies and wave functions of the meson-soliton bound states. Nevertheless, provided higher order corrections are included, this term results in a transition between r/P-soliton bound state and r/-soliton bound state, and provides us a simple, intuitive picture in explaining why experimentally the nucleon resonances described in our model have large branching ratios in the r/N channel. In this section, we calculate the contributions of this term to first order, the only relevant process is N* ~ N + r/. Here, we regard the rotating soliton (the nucleon) just as a background field in which the mesons (r/, r/~) is moving. As usual in the Skyrme model [ 14,16], the mass of the nucleon is regarded as being infinitely large, then the recoil effect is neglected. So, studying the above process amounts to calculating matrix elements of/~int = V ( r ) ~ 7 ~ t between the initial state (bound ~7~ s t a t e ]~TtnLM)) and the final state (scattered 7/ state ]7/~,LM)):
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Y C a i / N u c l e a r Physics A 593 (1995) 2 8 1 - 2 9 4
2~'8( E f
-
Ei)'Zfi = f d 4X(rl~oLMl£intlrlnLg), t J
(32)
where Ei = to,,, E f = o) are initial and final energies respectively. The transition amplitude 7- is related to the S-matrix by Sfi = 8fi - 2~'iS( Ef - Ei)7-fi. By expanding the r/and r/~ fields in terms of their eigenmodes: (~o
r l = ~ - ~ f do~[rlo,t.(r)YLi~(O, da)e-i,Ota,7,oLU+rto,L(r)y?.M(O, dp)e io,ta,lo~LM t ]
,
(33)
LM 0 L M t"'~ ~ , q ~-t'e-i'o"t) U r l, n L M "-l -- t-l 'n L t ',r ' Y ) L* M 't,0 , q~"-i'°"tat ))e .q,nLMI,
r/= Z[r/.L(rlE
(34)
nLM
and requiring to have the standard form for the Hamiltonian operator in the second quantization scheme [4], the wavefunctions are normalized as (>o
f r2dr(w + o)')tr(r)rla, L(r)rl~o,L(r ) = 8(to -- m t) , o
(35)
oo
f
r2 dr( o), + ~on,)a( r )rl.L ( r)rl.,L ( r ) = 6~n, .
'
0
(36)
,
After performing the time integral and the angle part of the spatial integral in Eq. (32), we obtain oo
~ f i = ~LL' 6 M M '
/ rZ drrl,,,L ( r ) V ( r ) rllnL( r )
(37)
,
o
where the appearance of the factor t~LL, ~ M M ' is due to conservation of the angular momentum. The decay width FN*---,N+n is then easily expressed as 2
77" llq*_ ----~N+rI = - (.0
[fo r2drrl,oL (r) V(r) r/. L (r)
,
(38)
where rl',L(r) is the radial wavefunction of the bound ~/~ state to which N* corresponds, w = mN* - mu is the energy of the outgoing r/meson, with mN- given by Eq. (31). The numerical results of the decay width from Eq. (38) are listed in Table 3. We observe that as a whole our results are larger to different extents as compared with the experimental values. In view of the inherent accuracy of the Skyrme model, the decay width for N(1535) is satisfactory, that for N(1710) is still acceptable, however, that for N(1720) is too large by a factor as much as five. This is due to the fact that the picture presented here is too simple to account for why the difference between FN(1710)---*N+~ and FN(1720)~N+n is SO large that the former is several times larger than the latter.
Y Cai/Nuclear Physics A 593 (1995) 281-294
293
Table 3 The calculated decay widths FN*~s+,7 (in MeV) as comparedto empirical ones. Parameters are the same as in Table 1
5.
N*
n
L
J
FN*--,N+n(MeV)
Exp. (MeV)
N(1535)
1
0
N(1710)
1
1
l
87
~ 75
l
41
~
N(1720)
1
1
~2
38
~ 8
28
Conclusion
In this paper, we have investigated the effects of the ~?-rf mixing in the bound state version of the Skyrme model. To this end, we have improved the extended Skyrme model by incorporating the r/-r/~ mixing in the symmetry breaking term in the Lagrangian of the model. Up to the second order in pseudoscalar meson fields, neither r/-K nor r/~-K coupling terms arise, so the strange baryon spectrum predicted by the original model remains unchanged. However, this improvement is helpful to our understanding of nonstrange baryon resonances. As our calculation has demonstrated, there exist a number of rl-soliton and ~'-soliton bound states. For ~ - s o l i t o n bound states, by identifying their quantum numbers with those of the relevant baryon resonances, they are explained as these resonances. The calculated mass spectrum agrees very well with experiment, with relative errors of no more than 8% which are much small as compared with the general accuracy of the topological soliton model of about 30%. Unfortunately, an exceptional state predicted by the l = 0 r/-soliton bound state exists. We see this not as a problem with our approach, but rather a problem with the mesonic action that has been adopted. In fact, the large-Nc QCD Lagrangian, to which the Skyrme Lagrangian is just an approximation, of course contains a multitude of higher-derivative terms and there is no prior reason to neglect them. They have been neglected only for reasons of computational simplicity. If a suitable set of such terms were added to the Skyrme Lagrangian, this exceptional state would disappear. An interesting co-product of our approach is the appearance of the coupling term of r/ and r/~ at the level of second order in the meson fields in the Lagrangian, which originates from the screening effect of the soliton background on the mixing of r/ and r/~. This coupling term results in a transition between r/~-soliton bound state and r/soliton bound state, and provides us with a simple, intuitive picture of explaining why experimentally the nucleon resonances described in our model have large branching ratios in the r/N channel. Some other problems concerning this approach, such as the calculation of the magnetic moments of the relevant baryon resonances, and the model that contains heavy flavors, deserve further investigation. We will discuss these problems in detail elsewhere.
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Y Cai/Nuclear Physics A 593 (1995) 281-294
Acknowledgements T h e author w o u l d like to thank Dr. J.H. Dai, and Dr. J. Z h a n g for helpful discussions. The w o r k is finished during the S u m m e r Institute sponsored by C C A S T and c o - s p o n s o r e d by ITP and supported by N S F C . T h e author is also supported in part by Local Natural Foundation of Xinjiang.
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