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S-wave π-nucleon scattering in the skyrme model
Nuclear Physics A501 (1989) 621-636 North-Holland. Amsterdam
S-WAVE
n-NUCLEON
SCATTERING
IN THE
SKYRME
MODEL
B.K. JENNINGS
0.V.
MAXWELL
Received 13 September 1988 (Revised 25 April lY89) Abstract:
Although the Skyrme model has enjoyed considerable success in ~-nucleon scattering, the s-waves have remained a problem. In this paper we examine r-wa\c v-nucleon scattering using approximations that, unlike the usual approaches. preserve the 5oft-pion theor-ems. When recoil, a large effect, is included, we find no resonancec helo~v I GeV in either the Sl I or S31partial wabe and obtain very poor agreement with the experimental data in hoth partial waves. The Skyrme model calculations are compared with cloudy bag model calculation5 and, ;tt least for the lowencrgq s-wave scattering, found to be inferior. This comparison suggests additional terms which \cill help cure the problems with the Skjrme model calculations.
1. Introduction
The Skyrme model Much work has been as an efiective model al. ‘.‘), many nuclear with observed nucleon
‘) has enjoyed considerable popularity over the last few years. done on it, since it appears that such a model might arise ‘) of QCD at low energy. Beginning with the work of Adkins et properties have been calculated and reasonable agreement properties obtained. In particular x-nucleon scattering has
been extensively studied (see, for example, refs. ‘-“)). Despite the many claims of success for the Skyrme model in rr-nucleon scattering, the low partial waves have remained a problem ‘.‘,‘). In most calculations the S11 and S31 scattering lengths are equal, in disagreement with soft-pion theorems and experiment. This must be due to approximations introduced in actual calculations with the Skyrme model since the lagrangian has chiral symmetry (or more precisely PCAC) which should guarantee the soft-pion theorems. An exception to this poor s-wave result has been obtained by Holzwarth and collaborators ‘) by decoupling the spin-isospin G: state. Reasonable scattering lengths have also been obtained in ref. “‘). In the approach presented here, we make sure that our approximations respect PCAC and that our results obey the soft-pion theorems. However, we still do not obtain even qualitative agreement with the data. It has been suggested “) that the Skyrme model has features in common with the cloudy bag model, at least as far as pion-nucleon scattering is concerned. We shall see that this is indeed the case and that one of the terms that arises in the Skyrme 0375.9474/89/SO.1..50 ;o Elsevier Science Publishcra (North-Holland Physics Publishing DiGsion)
B.V
622
B. K. Jennings,
0. V. Ma.uwell / S-wave
TN scattering
model is identical, except for a form factor, to that obtained in the cloudy bag model. The Skyrme model, however, has additional terms, which appear to be required by the topological nature of the skyrmion rather than by chiral symmetry. These terms give rise to the resonances we see in the Sll partial wave and perhaps responsible for the resonances seen in previous r-nucleon scattering calculations. In the present calculation we do not find any resonances in the s-waves; however, the model could yield resonances in the higher partial waves. The additional terms in the Skyrme model not present in the cloudy bag model diminish agreement with experiment at low energy. In this sense, the Skyrme model is inferior to the cloudy bag model. On the other hand, comparison with the cloudy bag model suggests ways the calculation could be improved. In particular, there are strong indications that the Skyrme model must include higher order terms before it can be meaningfully compared to experiment. Our approach is rather different from that in previous calculations. First, we use a different nonlinear realization of chiral symmetry, following Weinberg ‘I), rather then using the exponential form of the unitary matrix. This should have no effect on the results but makes the calculations easier. Second, we introduce pion field fluctuations in a manner suggested by Schnitzer “) (see also refs. ‘“,‘4,‘5)), which assists greatly in the maintenance of the soft-pion theorems. In principle, the form of the fluctuations should not affect the results; in practice, however, this cannot be guaranteed due to the various truncations that must be made in an actual calculation. In fact, the ditference in the results obtained with the two forms for the fluctuations provides an estimate of the importance of higher order terms. Third, we project onto states of good spin and isospin before solving the differential equation. This is necessary to satisfy the soft-pion theorems and in this respect, makes our approach superior to that of previous calculations. Finally, we solve the differential equation by Green function techniques in momentum space. While this makes no difference in the results, it does allow us to make explicit connections with field theories and to include recoil effects. We find that recoil is very important. Thus, it appears that calculations done without recoil are not a good guide to the location of resonances in a complete calculation. In the next section we introduce the Skyrme lagrangian and discuss our approach to it. In sect. 3 our quantization procedure is presented and in sect. 4 the fluctuations introduced. Sect. 5 contains a discussion of the procedure adopted for the calculation of s-wave scattering,
while in sect. 6 we present 2. The skyrmion
our results
and conclusions.
lagrangian
Skyrme’s soliton model for baryons interacting with pions is based upon a chirally invariant lagrangian density of the general form ‘)
B. K. Jennings,
0. V. Ma.~wdl
/ S-naoe
623
TN .vcattrring
where ,f, = 93 MeV is the pion decay constant, and U is any SU(2) matrix that transforms like U + LUR--’ under SU(2) x SU(2) chiral transformations. The second term in this expression, involving the dimensionless coupling parameter e, must be included to stabilize the soliton against collapse. The most common
form adopted
for I/ in the literature
U,=exp
[
is the form
p(r) i7.p .L I
(2.2)
This form was used by, among others, Skyrme ‘), Adkins, Nappi, and Witten ‘), and the Siegen group 5m7). We prefer the nonlinear realization of chiral symmetry introduced by Weinberg “): u
=
1 -(v(r)/2,f,)'+i7.
2
((~(r)/.fi)
1+ (dr)/2,fr)2
(2.3)
.
The major advantage of this form is that the covariant derivative simple. This is important because it is more convenient to work with expressed in terms of covariant derivatives rather than in terms matrices U. The covariant derivative operator can be generally defined
is particularly the lagrangian of the unitary by the relation (2.4)
where the U,? is defined by I/ = U,, Us?. In terms of this covariant lagrangian density takes the form
with g = covariant
(e’f t) ‘. For the Weinberg derivative
for an arbitrary
realization
of chiral
symmetry,
derivative,
the
eq. (2.3), the
pion field, C,O,is simply
Duv =
i’,cp 1+ (q3/2f,)’
(2.6)
The connection between the two forms for U[eq. (2.2) or eq. (2.3)] can be expressed as a relationship between the corresponding radial functions (the directions are the same in the two cases) defined by cp(r) = d(r)@ where 4 is a unit vector in the direction of p. In particular, we find
(2.7) with F(r)=+,(r)/f,. In order to take the finite mass of the pion into account, the chirally lagrangian (2.5) must be supplemented by an explicit symmetry-breaking
invariant term. A
5. K. Jennings,
624
common
choice
0. V.
Maxwell / S-waverrN
,sca~terirtg
(see e.g. ref. ‘)) is (2.X)
which we adopt
here. The complete
lagrangian
density
is then
Y=Zsi?Ys,. The simplest
ansatz
(2.9)
for P(Y) with the required q(r)
which yields the static “hedgehog”
properties
is the radial
one “), (2.10)
= 6(r);,
soliton
‘) with mass
(2.11) where (2.12) One could obtain the radial function 4(r) by performing an Euler-Lagrange variation on this expression and solving the resulting nonlinear differential equation directly{. Due to the nature of the boundary conditions, however, it proves to be more convenient to solve the corresponding equation for the quantity F, F
+Isin2F+ 4
sin’
where i= (2fTe)r and pF= m,r, and then to use eq. (2.7) to obtain interpretation of the topological winding number,
(2.13) 4(r).
The
(2.14) as the baryon number in the soliton picture and the requirement that the hedgehog corresponds to a single baryon imposes the condition F(F= 0) = IT. Asymptotica~Iy, as U-tco, (2.1s) where c is a constant determined by the solution of the differential equation and related to the nNN coupling constant. Fig. 1 displays our result for 4(r) for the choice e = 5.471. Note that C$is a smooth function of r and that it goes to infinity as r + 0.
625
Fig.
1. The radia1
function
d, as a functinn
3. Quantization
of r. It diverges
at r = 0.
procedure
The static hedgehog soliton is neither a spin nor an isospin eigenstate and thus, cannot directly represent a physical baryon. To describe physical baryons, it is necessary to perform a gIobal rotation in isospin space, as discussed, for example, by Walliser and Eckart “)+ We can accomplish this by replacing the simple hedgehog ansatz, (2.10), by the expression, SPO-I= +b(rz:* where angles
&apy)l :
the rotation effected by the matrix L? is characterized cy, p, y. In terms of the matrix elements of 6, 5&,,, = e^$,’* ii.
which are just the Wigner
with g,,,, denoting
a spherical
D-functions,
i?,,,,
(3.1) by the three
Euler
(3.2)
eq. (3.1) becomes
unit vector.
Q~antization of the soliton requires that the Euler angles be made time dependent ‘)_ If we then introduce the momenta conjugate to cy, & y through the relations,
(3.4)
R. K. Jennings, 0. V. Mnxweli / S-wme TN scartering
626 we
can quantize
according
to the prescription
“)
(3
Ph(Yj
pct+-iA
-
,
),
L+:cotp
pv+-iih
a
(> -
,
aY
where A is the associated moment of inertia. time derivative of L+Y takes the form
involving
With this prescription
the symmetrized
an operator f=
--
1
’ a’ a-+-+2 sin’@ [ &x’ 3-y’
which is just the squared
angular
cos p-
momentum
&$,, = J(J The mass of the quantized
soliton
i)”
-) -a--cot &Yay I ap’
@A ap ’
(3.7)
operator:
+ l)L?$M
can be expressed
M,, = Mstati, + ( TM,-M, /
where
(3.5)
.
d’x ( -dYQy) / 77&M,)
A&,,,, is the mass of the static soliton
(3.8)
as ,
(3.9)
given by (2.11), and
(3.10) is the contribution to the lagrangian density (2.5) from terms involving time derivatives. The matrix element here is evaluated between S = T baryon states with spin, isospin projections MS, M,: ITM+~)=(-I)T+M~
J
2T+1 -plr
9; I,,,.. ;Ff,(NPY) .
(3.11)
Eq. (3.6) does not uniquely specify how to quantize the Iagrangian of eq. (3.10) since there is still the question of the order of the operators associated with different q. We adopt the prescription here that all orderings of the alqO’s and other p’s are to be included with equal weights except that no external operators are permitted between two D-matrices arising from the same dot product 9. 9. Thus, for example, the contraction of the D-matrices implicit in the definition eq. (2.12) for d will not be disturbed. Inserting eqs (3.3) and (3.6) for 9 and its time derivative into (3.10) and noting that (3.12)
0. V.
B. K. Jennings,
Ma.xwell/ S-wave TAHLE
The various
M \I.I,IC
c~tltrjbutions
mass as given
627
in eq. (3.13) 1
3
2A
scatrering
1
to the nucleon
T(T+Il
TN
Total
2h
yields the result
MB =
Mst;r7ic
+
T(T+I) 2h
I 3 I 2rg y dr 2 ‘I 4A A’ I 0 ’ 0d
(3.13)
with (3.14) The second term on the right-hand side of this expression is the standard result for the rotational kinetic energy that appears in all versions of the quantized soliton. The last two terms, however, which play the role of a zero-point energy in the baryon masses, are a special feature of the quantization scheme employed in this work and are not present in previous calculations (see, for example, ref. ‘)). Their presence here is a sign of the large ambiguity inherent in quantizing the classical soliton. Such ambiguities are always present when quantizing a classical system and in general, different ordering algorithms will yield different zero point energies. In other words, the complete specification of a model requires not only that the lagrangian be specified but the quantization procedure as well. Because we are interested specifically in the rr-nucleon s-wave interaction, we have chosen to fix,f, at the empirical value of93 MeV (since this fixes the isospin-odd s-wave scattering lengths through the soft pion theorems). With this value of.f7, we cannot get the empirical nucleon mass for any value of the coupling parameter e, so we just minimize the mass as given by eq. (3.13) with respect to e. This procedure yields the nucleon mass contributions displayed is 800 MeV too high. Note that the zero-point two zero-point terms is in excess of 300 MeV. sensitive to the quantization procedure and the properties. We stick to it because it is easy to is the main emphasis of this paper.
in table 1 with a nucleon mass that energy is not small-the sum of the This indicates that the mass is very present one is not the best for static use in the scattering problem which
4. Fluctuations There exist several schemes for incorporating quantum fluctuations within the skyrmion model. The most common scheme is simply to replace +D by cp-+ 6~ [refs. “.“.‘)I, where 69 represents the fluctuating field. An alternative prescription,
628
introduced
B. K. Jentzjt7g~, 0. K :Musrwll / S-waw xN
by Schnitzer
“f and
employed
mttering
in the present
work,
is to make
the
replacement
u -+11w2uu,,*,
(4.1)
with 1+ ir ’ (@/2f,) U”!‘=~1+(sa,2~*)‘l’iLr where again 69 represents
the fluctuating
field. This is equivalent
(4.2) to the replacement
(4.3) in eq. (2.5) for the chirally
invariant &
d
:
lagrangian
and
bP++d’ 41 +(~~/2.L)21 ’
(4.4)
in the symmetry-breaking term, (2.x), with d given by (2.12) and the covariant derivative c?, defined as previously. The resulting fagrangian can be expanded as a power series in &,o. The linear term in this series contains the p-wave rr-nucleon vertex and hence, is sufficient for the discussion of the p-wave rr-nucleon resonances. There are no s-wave contributions to the linear term, however, so that a study of s-wave rr-nucleon scattering and the associated resonances requires consideration of the quadratic term in 9’. The chirally invariant contribution to the quadratic term may be extracted from (2.5) by incorporating the replacement (4.3) with the covariant derivative of 6~ replaced by the ordinary derivative. Combining the result with the second-order contribution to A& yields the expression
(4.5) with
The time derivatives occurring in these expressions have to be carried out symmetrically according to some prescription for the ordering of the rotation matrices. We have chosen
the same prescription
here as in the mass calculation
with the zeroth-
order lagrangian; i.e., we have included all orderings that do not require the disruption of a D-matrix contraction appearing in the original expression. Let us now consider eq. (4.5) in more detail. First we note that the only terms without derivatives acting on 6~ arise from the pion mass contributions. The terms with one derivative
on 6cp can be rewritten
as
where I^(+o)” is the vector current of the soliton. The strength of this contribution is fixed by PCAC and is the strength required to give the correct scattering lengths in the Born approximation. A term of this form with the same strength also occurs in the cloudy bag model, although in that model the vector current is due to the quarks. The cloudy bag model contains no additional terms; the Skyrme model, on the other hand contains several terms with two derivatives acting on the fluctuating fields. These terms are not required by chiral symmetry. but are connected with the topological nature of the skyrmion. Consider, for example, the term proportional to ii’”(6~) * 3, (6~ f. Far from the soliton, the coefficient of this term approaches one half as it should for a free pion. it is also one half at the origin, but it must vanish at some point in between because the factor of [ 1 - (+0/2,#,)‘] in the coefficient of this term passes through zero for some finite value of Y. This is a consequence of the topological nature of the soliton, i.e., the identification of the topological winding number as the baryon number, which requires that the radial function (b(r) decrease from infinity to zero as r ranges from zero to infinity [see fig. 1 and the discussion following eq. (2.13) in sect. 21. Thus there is, in fact, no choice of +Dthat can make the additional baryon number resonances.
terms in the potential one. Presumably,
vanish
and still yield a topological
it is these terms that are responsible
soliton
with
for the baryon
5. Skyrmion s-wave scattering
The usual procedure at this point is to perform a variation with respect to 6cp on YZ to obtain the classical equation of motion for iiq. This yields a coordinate space differential equation and is usually solved directly in coordinate space. We will not use this technique, however, but instead employ a Green function technique in momentum space. This simplifies the calculation and permits the inclusion of recoil, which has a large efTect.
B.K. Jennings,
630
0. V. A4n.wvell/
S-wave nN .scattering
We begin by separating the second-order skyrmion lagrangian previous section into a term associated with the free pion field, Y= =&J&D)
discussed
. tP(&o) - m’,(Sp ’ S$o)] )
in the
(5.1)
- -rP,. Isolation of the latter term permits and a r-baryon interaction term, ~ipi,,= LX?> us to define an elastic n-nucleon s-wave potential in momentum space as the matrix element
of the integrated
interaction V(ki,
lagrangian
k,-,w)= (f/
density: (5.2)
d”x (-.Yi,,t)li)
where ki and kr are the initial and final momenta in the centre-of-mass, and w is the centre-of-mass pion energy. The hadronic states between which this matrix element is evaluated consist of a nucleon and a pion. For the initial state, we write /i>= j( T=;)M,Ms)/mi)
(5.3)
and similarly for the final state, where the first factor on the right-hand side of (5.3) represents the nucleon state given by eq. (3.11) and the second factor denotes the pion state with isospin projection mi. An alternative potential, defined with respect the n-nucleon s-channel,
~(ki, k,., can be obtained
w)=
((lt)Z’M’,/
from (5.2) by a simple
~(ki, kf, W) =
C
I
to states of total isospin
,
d’x (-Yi~~/(l~)JMI)
recoupling
I = i, s in
(5.4)
of the isospins:
(lmi~M,lfM,)(lm~~M:II’M:)V(ki,
kr, w).
(5.5)
Vl,l?l,M,M;
This latter potential is actually diagonal in both the total isospin I and its projection, and hence, is more convenient than V for a study of n-nucleon scattering. To construct either form of the potential, we have to integrate the lagrangian density and then take matrix elements between the appropriate states. After inserting the expansions of q and its derivatives, eqs. (3.3), (3.6) and (3.12), into eqs. (4.5)(4.7), it is found that the fagrangian density involves scalar and vector products of the fluctuating pion field and its derivatives. This field can be expanded in momentum eigenstates in the usual manner, yielding
SF=
d'k (271.)“l?&
~~(k)exp~(~f-k.~)+u~~k)exp(-~(~~-k.~)].
(5.6)
The second quantized operators here, a(k) and a.‘(k), which respectively destroy and create a pion with momentum k, have to be contracted with analogous operators in the initial and final pion states. One then obtains (f~S~~S~~i>=exp[i(k,--ki)~x]-t-exp[-i(k~-ki)~x], for the matrix element of 6~. 6p and similar involving the space and time derivatives of 6~.
expressions
(5.7) for matrix
elements
B.K.
We now expand interest
Jennings, 0. V. Ma.w~Il
the exponentials
in this work to s-wave
expansions
need be retained;
in partial
r-nucleon
to I = 0 terms
rh!
st‘nflering
waves. Since we have confined
scattering,
i.e., we can replace
(f/SF. This restriction of 8~ radial,
/ S-waw
our
only the I= 0 terms in these
(5.7) by
&o/i)= 2~~~(~i~),~~,(~~~) .
has the added
631
advantage
(5.8)
that it makes the gradient
which considerably simplifies those terms in the lagrangian density that contain vector products of 6~ and its derivatives. In fact, these terms just involve the familiar matrix elements,
Combining these matrix elements and those represented by eq. (5.8) with the various radial functions appearing in the lagrangian density, we finally obtain for the two forms of the skyrmion r-nucleon potential,
and ci(ki, kr, w)=[A(ki, where the isospin
kr, w)+ B(ki, kr,
independent
and isospin
w)(26,~-6,~)16,,,6n,,ar~3
dependent
r2 dr.~:,(kir).i:,(k,r)[p,(r)
(5.13)
pieces are given by
-P4(r)11
-[kijb(kir)j,,(k,-r) (5.14)
632
B. K. Jennings,
0. V Maxwvll / S-waue TN .scaffering
with
(5.15) and jh( z) = dj,( z)/dz. Using the potential equation in the form
of eqs.
(5.12)-(5.15)
we write
the
Lippmann-Schwinger
Note in this equation that it is the on-shell energy, W, that occurs in the potential. In the original work on scattering in the cloudy bag model “,I’), it was the energy k, that occurred. This choice caused corresponding to the off-shell momentum, considerable problems associated with the violation of the soft-pion theorems. These problems were ameliorated in ref. “) by taking into account terms that arise from the conversion of a lagrangian to a hamiltonian in the presence of derivative coupling. The point was considered more generally in ref. I’)) in the context of relativistic two body propagators. There it was shown that using the on-shell energy leads to better properties for the two-body r-matrix and takes into account higher order graphs. This effect is included automatically in the Skyrme model calculation. In ref. I’), it was also shown how to include recoil. In the current problem their prescription for including recoil amounts to multiplying the propagator of eq. (5.16) N w h ere M, is the nucleon mass. As we shall by wN/(wN+w) with w,=Jk’+M’ see in the next section, recoil included
according
to this prescription
has a large effect.
6. Results and discussion In figs. 2 and 3 we show results of the full calculation including recoil corrections for the Sll and S31 r-nucleon phase shifts, together with the data. As one can see, not even the general trend of the empirical curve is reproduced by the model. There are two possible reasons for this: either the Skyrme model is simply not suitable for the analysis of s-wave r-nucleon scattering, or the approximations made are not valid. To explore the second possibility, we have shown in figs. 4 and 5 the results obtained with and without recoil and the results with only the term YChlrLrl given by eq. (4.8) as the potential. The latter results are very similar to those obtained in the cloudy bag model “) and while not in good agreement with the data, are
R. K. Jennings,
1078
0.
1278
v.
Ma.well / S-woe
1478
nM
1678
.scatrering
1878
633
2078
Em&W Fig. 2. The Sl
I
phase shift from threshold
to 1 GeV
above threshold.
model result while the dashed curve is the experiment
Fig. 3. The S31 phase shift from threshold
to
1GeV
uhove threshold.
model result while the dashed curve is the experiment
definitely
better than the full calculation.
The solid curve i’l the Skyrme
phase-shift
The wlid
phase-shift
This indicates
analysis from ref. “‘1.
curve is the Skyrme
analysis from ref. “‘).
that the low-order
Skyrme
model is not as goad a starting point for the description of low-energy rr-nucleon scattering as the cloudy bag model. However, the comparison with the cloudy bag model calculations suggests that the Skyrme model can be brought into better agreement with the data by including higher order graphs I’.‘“) involving pion rescattering and vertices with three pions and the soliton. These graphs, which were found to be essential in the cloudy bag model to obtain the correct energy dependence at low energy, also occur in the Skyrme model as terms of higher order in Sq. There
634
Fig. 4. The various approximations for the Sll phase shift. The solid line is the full calculation while the dash-dotted line is the result without recoil. The dashed and dotted curves are the chiral term with and without recoil. 100
I
1
I
1 _-_/--
,’
70
.’
s 2
40 -
z .6 (u :: &
10
-20
-
-50 1078
‘.-Y-7. --._y__ --...:----____ . . . . . .._.
! 1278
I 1478
‘....____ ----___ ------. . . . . . ..__._. ‘..“........_.______,_ I 1678
, 1878
2078
Ecm(MeVf Fig. 5. The various approximations for the S31 phase shift. The solid line is the full calculation while the dash-dotted line is the result without recoil. The dashed and dotted curves are the chiral term with and without recoil.
will also be additional graphs of the same order in Sy, not present in the cloudy bag model that may be important. These higher terms may be included as in ref. I’). Thus, it appears likely that at least some of the problems encountered in the present calculations are due to a truncation of the expansion at too low an order. There is no reason to expect these higher order graphs to be small in higher partial waves. As seen in fig. 4, it is clear that recoil has an appreciable effect in both the S.11 and S31 partial waves. Similar effects will occur in higher partial waves and indicate
B.lS. .lenning.x, 0. V. Muswell
that recoil must be included comparison
/ S-watw xN .seartering
at least to the extent discussed
635
here before a meaningful
can be made with experiment.
Our calculations are completely elastic, which may partially account for the lack of resonances in the S31 partial wave, where experimentally the observed resonance couples strongly to the n-4 channel. This channel could be inc’tuded in our calculation as a coupled channel. In the Sll partial wave the first resonance couples mainly to the +nucleon and the n-nucleon channels, so to include the effects of inelasticity in this partial wave, one would have to include the n meson in the calculation. The second St I resonance is up to 65”/0 elastic, so it might be expected to appear even without coupled channels, while our phase shift in this channel does increase to above 90”, it is not the rapid increase that is expected for a good well behaved resonance. Notice that the phase shift no longer increases above 90” if we keep only the terms forced by chiral symmetry. In table2 we display the scattering lengths obtained in the present calculation and in the cloudy bag model caIcuiation of ref. “1 As for scattering at finite energy, the scattering lengths are not in good agreement with the data. The Born limit of the term associated with chiral symmetry is the same in both models, as it is fixed by PCAC, and taken alone agrees with experiment. Solving the multiple scattering equation makes the scattering lengths more attractive in both cases but gives a larger effect for the cloudy bag model. This is presumably due to the softer form factors in the skyrmion model. The other terms included in the two models are quite diiferent and, as can be seen in the table, change the scattering lengths in opposite directions. In particular, the extra terms in the Skyrme lagrangian are attractive, while the
A comparison of the scartering lengths as obtained from the Skyrme model and the cloudy bag model. Results are given with and without recoil as well as from using just the chiral term, Y‘,,,,,,,. The cloudy bag results are from ref. “)
Experiment
SIl
s31
SII
0.24
-0.14
0.24
CBM (with higher Born (recoil) Full (recoil) Born (no recoilj Full (no recoil)
0.29 0.22 0.34 0.22 CBM ichiral
Born Full Born Full
(recoil) (recoil) (no recoil) (no recoil 1
0.22
0.47
order
terms) -0.13 -0.04 -0.15 -0.01
term only) -0.11 -0.09
Skyrme
s3.1 -0.14
(with topological
0.27 0.46 0.3 1 O.hl Skyrme 0.22
0.35 0.25 0.45
terms -0.06 -0.06 -0.07 -0.06
ichiral
term only) -0. I 1 -0.09 -0.13 -0.11
1
3. K. Jetw~ing.s. 0. V. Mawelf
636
/ S-ware
rN
scariering
higher order terms included in the cloudy bag model are repulsive. Thus, as noted above, the inclusion of terms in the Skyrme model similar to those in the cloudy bag model should result in an improvement in the results obtained. While our calculations are somewhat different from previous ones, we expect that the validity
of our main conclusions,
that recoil is important
and that higher order
effects must be included before one can meaningfully compare to experiment, will not depend on details of either the model employed or the manner in which the calculation is carried out. Clearly, much work remains to be done before it can be decided whether the Skyrme model is capable of describing z-nucleon scattering.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
T.H.R. Skyrme, Proc. Roy. Sot. A260 (1961) 127; Nucl. Phys. 31 (1962) 556 E. Witten. Nucl. Phys. B160 (1979) 57 C.S. Adkins, C.R. Nappi and E. Witten, Nucl. Phys. 8228 (1953) 552 G.S. Adkins and C.R. Nappi, Nucl. Phys. R233 (1984) 109 G. tiolzwarth and 8. Schwesinger, Rep. Prog. Phys. 49 i 1986) 825 H. Walliser and G. Eckart, Nucl. Phys. A429 (1984) 514 C. Eckart, A. Hayashi and G. Holzwarth, Nucl. Phys. A448 (1986)732 M.P. Mattis and M. Karliner, Phys. Rev. D31 (1985) 2833 M.P. Mattis and M. Karliner. Phys. Rev. D32 (1985) 5X M. Uehara and H. Kondo, Prog. Theor. Phys. 75 ( 1986) 981 E.D. Cooper, B.K. Jennings, PAM. Guichon and A.%‘. Thomas, Nucl. Phys. A469 (1987) 717 S. Weinberg, Phys. Re%. 166 (1968) 1568 H.J. Schnitzer, Phys. Lat. Bl39 (1984) 217 M. Uehnra, Prog. Theor. Phys. 75 (1986) 212 S. Klimt and W. Weise, Regcnsburg Preprint TPR-88-13. ED. Cooper and B.K. Jennings, Phys. Rev. D33 (1986! 1509 E.A. Veit, B.K. Jennings and A.W. Thomas. Phys. Rev. D33 t tY8hf 1859 EA. Veit, B.K. Jennings, A.W. Thomas and R.C. Barrett, Phys. Rev. D31 (1986) 1033 E.D. Cooper and B.K. Jennings, Nucl. Phys. A483 (1988) 601 R.A. Arndt and L.D. Roper, Scattering analysis dial in (SAID program) from VP1 8: SU, solution WI87
S-WAVE
n-NUCLEON
SCATTERING
IN THE
SKYRME
MODEL
B.K. JENNINGS
0.V.
MAXWELL
Received 13 September 1988 (Revised 25 April lY89) Abstract:
Although the Skyrme model has enjoyed considerable success in ~-nucleon scattering, the s-waves have remained a problem. In this paper we examine r-wa\c v-nucleon scattering using approximations that, unlike the usual approaches. preserve the 5oft-pion theor-ems. When recoil, a large effect, is included, we find no resonancec helo~v I GeV in either the Sl I or S31partial wabe and obtain very poor agreement with the experimental data in hoth partial waves. The Skyrme model calculations are compared with cloudy bag model calculation5 and, ;tt least for the lowencrgq s-wave scattering, found to be inferior. This comparison suggests additional terms which \cill help cure the problems with the Skjrme model calculations.
1. Introduction
The Skyrme model Much work has been as an efiective model al. ‘.‘), many nuclear with observed nucleon
‘) has enjoyed considerable popularity over the last few years. done on it, since it appears that such a model might arise ‘) of QCD at low energy. Beginning with the work of Adkins et properties have been calculated and reasonable agreement properties obtained. In particular x-nucleon scattering has
been extensively studied (see, for example, refs. ‘-“)). Despite the many claims of success for the Skyrme model in rr-nucleon scattering, the low partial waves have remained a problem ‘.‘,‘). In most calculations the S11 and S31 scattering lengths are equal, in disagreement with soft-pion theorems and experiment. This must be due to approximations introduced in actual calculations with the Skyrme model since the lagrangian has chiral symmetry (or more precisely PCAC) which should guarantee the soft-pion theorems. An exception to this poor s-wave result has been obtained by Holzwarth and collaborators ‘) by decoupling the spin-isospin G: state. Reasonable scattering lengths have also been obtained in ref. “‘). In the approach presented here, we make sure that our approximations respect PCAC and that our results obey the soft-pion theorems. However, we still do not obtain even qualitative agreement with the data. It has been suggested “) that the Skyrme model has features in common with the cloudy bag model, at least as far as pion-nucleon scattering is concerned. We shall see that this is indeed the case and that one of the terms that arises in the Skyrme 0375.9474/89/SO.1..50 ;o Elsevier Science Publishcra (North-Holland Physics Publishing DiGsion)
B.V
622
B. K. Jennings,
0. V. Ma.uwell / S-wave
TN scattering
model is identical, except for a form factor, to that obtained in the cloudy bag model. The Skyrme model, however, has additional terms, which appear to be required by the topological nature of the skyrmion rather than by chiral symmetry. These terms give rise to the resonances we see in the Sll partial wave and perhaps responsible for the resonances seen in previous r-nucleon scattering calculations. In the present calculation we do not find any resonances in the s-waves; however, the model could yield resonances in the higher partial waves. The additional terms in the Skyrme model not present in the cloudy bag model diminish agreement with experiment at low energy. In this sense, the Skyrme model is inferior to the cloudy bag model. On the other hand, comparison with the cloudy bag model suggests ways the calculation could be improved. In particular, there are strong indications that the Skyrme model must include higher order terms before it can be meaningfully compared to experiment. Our approach is rather different from that in previous calculations. First, we use a different nonlinear realization of chiral symmetry, following Weinberg ‘I), rather then using the exponential form of the unitary matrix. This should have no effect on the results but makes the calculations easier. Second, we introduce pion field fluctuations in a manner suggested by Schnitzer “) (see also refs. ‘“,‘4,‘5)), which assists greatly in the maintenance of the soft-pion theorems. In principle, the form of the fluctuations should not affect the results; in practice, however, this cannot be guaranteed due to the various truncations that must be made in an actual calculation. In fact, the ditference in the results obtained with the two forms for the fluctuations provides an estimate of the importance of higher order terms. Third, we project onto states of good spin and isospin before solving the differential equation. This is necessary to satisfy the soft-pion theorems and in this respect, makes our approach superior to that of previous calculations. Finally, we solve the differential equation by Green function techniques in momentum space. While this makes no difference in the results, it does allow us to make explicit connections with field theories and to include recoil effects. We find that recoil is very important. Thus, it appears that calculations done without recoil are not a good guide to the location of resonances in a complete calculation. In the next section we introduce the Skyrme lagrangian and discuss our approach to it. In sect. 3 our quantization procedure is presented and in sect. 4 the fluctuations introduced. Sect. 5 contains a discussion of the procedure adopted for the calculation of s-wave scattering,
while in sect. 6 we present 2. The skyrmion
our results
and conclusions.
lagrangian
Skyrme’s soliton model for baryons interacting with pions is based upon a chirally invariant lagrangian density of the general form ‘)
B. K. Jennings,
0. V. Ma.~wdl
/ S-naoe
623
TN .vcattrring
where ,f, = 93 MeV is the pion decay constant, and U is any SU(2) matrix that transforms like U + LUR--’ under SU(2) x SU(2) chiral transformations. The second term in this expression, involving the dimensionless coupling parameter e, must be included to stabilize the soliton against collapse. The most common
form adopted
for I/ in the literature
U,=exp
[
is the form
p(r) i7.p .L I
(2.2)
This form was used by, among others, Skyrme ‘), Adkins, Nappi, and Witten ‘), and the Siegen group 5m7). We prefer the nonlinear realization of chiral symmetry introduced by Weinberg “): u
=
1 -(v(r)/2,f,)'+i7.
2
((~(r)/.fi)
1+ (dr)/2,fr)2
(2.3)
.
The major advantage of this form is that the covariant derivative simple. This is important because it is more convenient to work with expressed in terms of covariant derivatives rather than in terms matrices U. The covariant derivative operator can be generally defined
is particularly the lagrangian of the unitary by the relation (2.4)
where the U,? is defined by I/ = U,, Us?. In terms of this covariant lagrangian density takes the form
with g = covariant
(e’f t) ‘. For the Weinberg derivative
for an arbitrary
realization
of chiral
symmetry,
derivative,
the
eq. (2.3), the
pion field, C,O,is simply
Duv =
i’,cp 1+ (q3/2f,)’
(2.6)
The connection between the two forms for U[eq. (2.2) or eq. (2.3)] can be expressed as a relationship between the corresponding radial functions (the directions are the same in the two cases) defined by cp(r) = d(r)@ where 4 is a unit vector in the direction of p. In particular, we find
(2.7) with F(r)=+,(r)/f,. In order to take the finite mass of the pion into account, the chirally lagrangian (2.5) must be supplemented by an explicit symmetry-breaking
invariant term. A
5. K. Jennings,
624
common
choice
0. V.
Maxwell / S-waverrN
,sca~terirtg
(see e.g. ref. ‘)) is (2.X)
which we adopt
here. The complete
lagrangian
density
is then
Y=Zsi?Ys,. The simplest
ansatz
(2.9)
for P(Y) with the required q(r)
which yields the static “hedgehog”
properties
is the radial
one “), (2.10)
= 6(r);,
soliton
‘) with mass
(2.11) where (2.12) One could obtain the radial function 4(r) by performing an Euler-Lagrange variation on this expression and solving the resulting nonlinear differential equation directly{. Due to the nature of the boundary conditions, however, it proves to be more convenient to solve the corresponding equation for the quantity F, F
+Isin2F+ 4
sin’
where i= (2fTe)r and pF= m,r, and then to use eq. (2.7) to obtain interpretation of the topological winding number,
(2.13) 4(r).
The
(2.14) as the baryon number in the soliton picture and the requirement that the hedgehog corresponds to a single baryon imposes the condition F(F= 0) = IT. Asymptotica~Iy, as U-tco, (2.1s) where c is a constant determined by the solution of the differential equation and related to the nNN coupling constant. Fig. 1 displays our result for 4(r) for the choice e = 5.471. Note that C$is a smooth function of r and that it goes to infinity as r + 0.
625
Fig.
1. The radia1
function
d, as a functinn
3. Quantization
of r. It diverges
at r = 0.
procedure
The static hedgehog soliton is neither a spin nor an isospin eigenstate and thus, cannot directly represent a physical baryon. To describe physical baryons, it is necessary to perform a gIobal rotation in isospin space, as discussed, for example, by Walliser and Eckart “)+ We can accomplish this by replacing the simple hedgehog ansatz, (2.10), by the expression, SPO-I= +b(rz:* where angles
&apy)l :
the rotation effected by the matrix L? is characterized cy, p, y. In terms of the matrix elements of 6, 5&,,, = e^$,’* ii.
which are just the Wigner
with g,,,, denoting
a spherical
D-functions,
i?,,,,
(3.1) by the three
Euler
(3.2)
eq. (3.1) becomes
unit vector.
Q~antization of the soliton requires that the Euler angles be made time dependent ‘)_ If we then introduce the momenta conjugate to cy, & y through the relations,
(3.4)
R. K. Jennings, 0. V. Mnxweli / S-wme TN scartering
626 we
can quantize
according
to the prescription
“)
(3
Ph(Yj
pct+-iA
-
,
),
L+:cotp
pv+-iih
a
(> -
,
aY
where A is the associated moment of inertia. time derivative of L+Y takes the form
involving
With this prescription
the symmetrized
an operator f=
--
1
’ a’ a-+-+2 sin’@ [ &x’ 3-y’
which is just the squared
angular
cos p-
momentum
&$,, = J(J The mass of the quantized
soliton
i)”
-) -a--cot &Yay I ap’
@A ap ’
(3.7)
operator:
+ l)L?$M
can be expressed
M,, = Mstati, + ( TM,-M, /
where
(3.5)
.
d’x ( -dYQy) / 77&M,)
A&,,,, is the mass of the static soliton
(3.8)
as ,
(3.9)
given by (2.11), and
(3.10) is the contribution to the lagrangian density (2.5) from terms involving time derivatives. The matrix element here is evaluated between S = T baryon states with spin, isospin projections MS, M,: ITM+~)=(-I)T+M~
J
2T+1 -plr
9; I,,,.. ;Ff,(NPY) .
(3.11)
Eq. (3.6) does not uniquely specify how to quantize the Iagrangian of eq. (3.10) since there is still the question of the order of the operators associated with different q. We adopt the prescription here that all orderings of the alqO’s and other p’s are to be included with equal weights except that no external operators are permitted between two D-matrices arising from the same dot product 9. 9. Thus, for example, the contraction of the D-matrices implicit in the definition eq. (2.12) for d will not be disturbed. Inserting eqs (3.3) and (3.6) for 9 and its time derivative into (3.10) and noting that (3.12)
0. V.
B. K. Jennings,
Ma.xwell/ S-wave TAHLE
The various
M \I.I,IC
c~tltrjbutions
mass as given
627
in eq. (3.13) 1
3
2A
scatrering
1
to the nucleon
T(T+Il
TN
Total
2h
yields the result
MB =
Mst;r7ic
+
T(T+I) 2h
I 3 I 2rg y dr 2 ‘I 4A A’ I 0 ’ 0d
(3.13)
with (3.14) The second term on the right-hand side of this expression is the standard result for the rotational kinetic energy that appears in all versions of the quantized soliton. The last two terms, however, which play the role of a zero-point energy in the baryon masses, are a special feature of the quantization scheme employed in this work and are not present in previous calculations (see, for example, ref. ‘)). Their presence here is a sign of the large ambiguity inherent in quantizing the classical soliton. Such ambiguities are always present when quantizing a classical system and in general, different ordering algorithms will yield different zero point energies. In other words, the complete specification of a model requires not only that the lagrangian be specified but the quantization procedure as well. Because we are interested specifically in the rr-nucleon s-wave interaction, we have chosen to fix,f, at the empirical value of93 MeV (since this fixes the isospin-odd s-wave scattering lengths through the soft pion theorems). With this value of.f7, we cannot get the empirical nucleon mass for any value of the coupling parameter e, so we just minimize the mass as given by eq. (3.13) with respect to e. This procedure yields the nucleon mass contributions displayed is 800 MeV too high. Note that the zero-point two zero-point terms is in excess of 300 MeV. sensitive to the quantization procedure and the properties. We stick to it because it is easy to is the main emphasis of this paper.
in table 1 with a nucleon mass that energy is not small-the sum of the This indicates that the mass is very present one is not the best for static use in the scattering problem which
4. Fluctuations There exist several schemes for incorporating quantum fluctuations within the skyrmion model. The most common scheme is simply to replace +D by cp-+ 6~ [refs. “.“.‘)I, where 69 represents the fluctuating field. An alternative prescription,
628
introduced
B. K. Jentzjt7g~, 0. K :Musrwll / S-waw xN
by Schnitzer
“f and
employed
mttering
in the present
work,
is to make
the
replacement
u -+11w2uu,,*,
(4.1)
with 1+ ir ’ (@/2f,) U”!‘=~1+(sa,2~*)‘l’iLr where again 69 represents
the fluctuating
field. This is equivalent
(4.2) to the replacement
(4.3) in eq. (2.5) for the chirally
invariant &
d
:
lagrangian
and
bP++d’ 41 +(~~/2.L)21 ’
(4.4)
in the symmetry-breaking term, (2.x), with d given by (2.12) and the covariant derivative c?, defined as previously. The resulting fagrangian can be expanded as a power series in &,o. The linear term in this series contains the p-wave rr-nucleon vertex and hence, is sufficient for the discussion of the p-wave rr-nucleon resonances. There are no s-wave contributions to the linear term, however, so that a study of s-wave rr-nucleon scattering and the associated resonances requires consideration of the quadratic term in 9’. The chirally invariant contribution to the quadratic term may be extracted from (2.5) by incorporating the replacement (4.3) with the covariant derivative of 6~ replaced by the ordinary derivative. Combining the result with the second-order contribution to A& yields the expression
(4.5) with
The time derivatives occurring in these expressions have to be carried out symmetrically according to some prescription for the ordering of the rotation matrices. We have chosen
the same prescription
here as in the mass calculation
with the zeroth-
order lagrangian; i.e., we have included all orderings that do not require the disruption of a D-matrix contraction appearing in the original expression. Let us now consider eq. (4.5) in more detail. First we note that the only terms without derivatives acting on 6~ arise from the pion mass contributions. The terms with one derivative
on 6cp can be rewritten
as
where I^(+o)” is the vector current of the soliton. The strength of this contribution is fixed by PCAC and is the strength required to give the correct scattering lengths in the Born approximation. A term of this form with the same strength also occurs in the cloudy bag model, although in that model the vector current is due to the quarks. The cloudy bag model contains no additional terms; the Skyrme model, on the other hand contains several terms with two derivatives acting on the fluctuating fields. These terms are not required by chiral symmetry. but are connected with the topological nature of the skyrmion. Consider, for example, the term proportional to ii’”(6~) * 3, (6~ f. Far from the soliton, the coefficient of this term approaches one half as it should for a free pion. it is also one half at the origin, but it must vanish at some point in between because the factor of [ 1 - (+0/2,#,)‘] in the coefficient of this term passes through zero for some finite value of Y. This is a consequence of the topological nature of the soliton, i.e., the identification of the topological winding number as the baryon number, which requires that the radial function (b(r) decrease from infinity to zero as r ranges from zero to infinity [see fig. 1 and the discussion following eq. (2.13) in sect. 21. Thus there is, in fact, no choice of +Dthat can make the additional baryon number resonances.
terms in the potential one. Presumably,
vanish
and still yield a topological
it is these terms that are responsible
soliton
with
for the baryon
5. Skyrmion s-wave scattering
The usual procedure at this point is to perform a variation with respect to 6cp on YZ to obtain the classical equation of motion for iiq. This yields a coordinate space differential equation and is usually solved directly in coordinate space. We will not use this technique, however, but instead employ a Green function technique in momentum space. This simplifies the calculation and permits the inclusion of recoil, which has a large efTect.
B.K. Jennings,
630
0. V. A4n.wvell/
S-wave nN .scattering
We begin by separating the second-order skyrmion lagrangian previous section into a term associated with the free pion field, Y= =&J&D)
discussed
. tP(&o) - m’,(Sp ’ S$o)] )
in the
(5.1)
- -rP,. Isolation of the latter term permits and a r-baryon interaction term, ~ipi,,= LX?> us to define an elastic n-nucleon s-wave potential in momentum space as the matrix element
of the integrated
interaction V(ki,
lagrangian
k,-,w)= (f/
density: (5.2)
d”x (-.Yi,,t)li)
where ki and kr are the initial and final momenta in the centre-of-mass, and w is the centre-of-mass pion energy. The hadronic states between which this matrix element is evaluated consist of a nucleon and a pion. For the initial state, we write /i>= j( T=;)M,Ms)/mi)
(5.3)
and similarly for the final state, where the first factor on the right-hand side of (5.3) represents the nucleon state given by eq. (3.11) and the second factor denotes the pion state with isospin projection mi. An alternative potential, defined with respect the n-nucleon s-channel,
~(ki, k,., can be obtained
w)=
((lt)Z’M’,/
from (5.2) by a simple
~(ki, kf, W) =
C
I
to states of total isospin
,
d’x (-Yi~~/(l~)JMI)
recoupling
I = i, s in
(5.4)
of the isospins:
(lmi~M,lfM,)(lm~~M:II’M:)V(ki,
kr, w).
(5.5)
Vl,l?l,M,M;
This latter potential is actually diagonal in both the total isospin I and its projection, and hence, is more convenient than V for a study of n-nucleon scattering. To construct either form of the potential, we have to integrate the lagrangian density and then take matrix elements between the appropriate states. After inserting the expansions of q and its derivatives, eqs. (3.3), (3.6) and (3.12), into eqs. (4.5)(4.7), it is found that the fagrangian density involves scalar and vector products of the fluctuating pion field and its derivatives. This field can be expanded in momentum eigenstates in the usual manner, yielding
SF=
d'k (271.)“l?&
~~(k)exp~(~f-k.~)+u~~k)exp(-~(~~-k.~)].
(5.6)
The second quantized operators here, a(k) and a.‘(k), which respectively destroy and create a pion with momentum k, have to be contracted with analogous operators in the initial and final pion states. One then obtains (f~S~~S~~i>=exp[i(k,--ki)~x]-t-exp[-i(k~-ki)~x], for the matrix element of 6~. 6p and similar involving the space and time derivatives of 6~.
expressions
(5.7) for matrix
elements
B.K.
We now expand interest
Jennings, 0. V. Ma.w~Il
the exponentials
in this work to s-wave
expansions
need be retained;
in partial
r-nucleon
to I = 0 terms
rh!
st‘nflering
waves. Since we have confined
scattering,
i.e., we can replace
(f/SF. This restriction of 8~ radial,
/ S-waw
our
only the I= 0 terms in these
(5.7) by
&o/i)= 2~~~(~i~),~~,(~~~) .
has the added
631
advantage
(5.8)
that it makes the gradient
which considerably simplifies those terms in the lagrangian density that contain vector products of 6~ and its derivatives. In fact, these terms just involve the familiar matrix elements,
Combining these matrix elements and those represented by eq. (5.8) with the various radial functions appearing in the lagrangian density, we finally obtain for the two forms of the skyrmion r-nucleon potential,
and ci(ki, kr, w)=[A(ki, where the isospin
kr, w)+ B(ki, kr,
independent
and isospin
w)(26,~-6,~)16,,,6n,,ar~3
dependent
r2 dr.~:,(kir).i:,(k,r)[p,(r)
(5.13)
pieces are given by
-P4(r)11
-[kijb(kir)j,,(k,-r) (5.14)
632
B. K. Jennings,
0. V Maxwvll / S-waue TN .scaffering
with
(5.15) and jh( z) = dj,( z)/dz. Using the potential equation in the form
of eqs.
(5.12)-(5.15)
we write
the
Lippmann-Schwinger
Note in this equation that it is the on-shell energy, W, that occurs in the potential. In the original work on scattering in the cloudy bag model “,I’), it was the energy k, that occurred. This choice caused corresponding to the off-shell momentum, considerable problems associated with the violation of the soft-pion theorems. These problems were ameliorated in ref. “) by taking into account terms that arise from the conversion of a lagrangian to a hamiltonian in the presence of derivative coupling. The point was considered more generally in ref. I’)) in the context of relativistic two body propagators. There it was shown that using the on-shell energy leads to better properties for the two-body r-matrix and takes into account higher order graphs. This effect is included automatically in the Skyrme model calculation. In ref. I’), it was also shown how to include recoil. In the current problem their prescription for including recoil amounts to multiplying the propagator of eq. (5.16) N w h ere M, is the nucleon mass. As we shall by wN/(wN+w) with w,=Jk’+M’ see in the next section, recoil included
according
to this prescription
has a large effect.
6. Results and discussion In figs. 2 and 3 we show results of the full calculation including recoil corrections for the Sll and S31 r-nucleon phase shifts, together with the data. As one can see, not even the general trend of the empirical curve is reproduced by the model. There are two possible reasons for this: either the Skyrme model is simply not suitable for the analysis of s-wave r-nucleon scattering, or the approximations made are not valid. To explore the second possibility, we have shown in figs. 4 and 5 the results obtained with and without recoil and the results with only the term YChlrLrl given by eq. (4.8) as the potential. The latter results are very similar to those obtained in the cloudy bag model “) and while not in good agreement with the data, are
R. K. Jennings,
1078
0.
1278
v.
Ma.well / S-woe
1478
nM
1678
.scatrering
1878
633
2078
Em&W Fig. 2. The Sl
I
phase shift from threshold
to 1 GeV
above threshold.
model result while the dashed curve is the experiment
Fig. 3. The S31 phase shift from threshold
to
1GeV
uhove threshold.
model result while the dashed curve is the experiment
definitely
better than the full calculation.
The solid curve i’l the Skyrme
phase-shift
The wlid
phase-shift
This indicates
analysis from ref. “‘1.
curve is the Skyrme
analysis from ref. “‘).
that the low-order
Skyrme
model is not as goad a starting point for the description of low-energy rr-nucleon scattering as the cloudy bag model. However, the comparison with the cloudy bag model calculations suggests that the Skyrme model can be brought into better agreement with the data by including higher order graphs I’.‘“) involving pion rescattering and vertices with three pions and the soliton. These graphs, which were found to be essential in the cloudy bag model to obtain the correct energy dependence at low energy, also occur in the Skyrme model as terms of higher order in Sq. There
634
Fig. 4. The various approximations for the Sll phase shift. The solid line is the full calculation while the dash-dotted line is the result without recoil. The dashed and dotted curves are the chiral term with and without recoil. 100
I
1
I
1 _-_/--
,’
70
.’
s 2
40 -
z .6 (u :: &
10
-20
-
-50 1078
‘.-Y-7. --._y__ --...:----____ . . . . . .._.
! 1278
I 1478
‘....____ ----___ ------. . . . . . ..__._. ‘..“........_.______,_ I 1678
, 1878
2078
Ecm(MeVf Fig. 5. The various approximations for the S31 phase shift. The solid line is the full calculation while the dash-dotted line is the result without recoil. The dashed and dotted curves are the chiral term with and without recoil.
will also be additional graphs of the same order in Sy, not present in the cloudy bag model that may be important. These higher terms may be included as in ref. I’). Thus, it appears likely that at least some of the problems encountered in the present calculations are due to a truncation of the expansion at too low an order. There is no reason to expect these higher order graphs to be small in higher partial waves. As seen in fig. 4, it is clear that recoil has an appreciable effect in both the S.11 and S31 partial waves. Similar effects will occur in higher partial waves and indicate
B.lS. .lenning.x, 0. V. Muswell
that recoil must be included comparison
/ S-watw xN .seartering
at least to the extent discussed
635
here before a meaningful
can be made with experiment.
Our calculations are completely elastic, which may partially account for the lack of resonances in the S31 partial wave, where experimentally the observed resonance couples strongly to the n-4 channel. This channel could be inc’tuded in our calculation as a coupled channel. In the Sll partial wave the first resonance couples mainly to the +nucleon and the n-nucleon channels, so to include the effects of inelasticity in this partial wave, one would have to include the n meson in the calculation. The second St I resonance is up to 65”/0 elastic, so it might be expected to appear even without coupled channels, while our phase shift in this channel does increase to above 90”, it is not the rapid increase that is expected for a good well behaved resonance. Notice that the phase shift no longer increases above 90” if we keep only the terms forced by chiral symmetry. In table2 we display the scattering lengths obtained in the present calculation and in the cloudy bag model caIcuiation of ref. “1 As for scattering at finite energy, the scattering lengths are not in good agreement with the data. The Born limit of the term associated with chiral symmetry is the same in both models, as it is fixed by PCAC, and taken alone agrees with experiment. Solving the multiple scattering equation makes the scattering lengths more attractive in both cases but gives a larger effect for the cloudy bag model. This is presumably due to the softer form factors in the skyrmion model. The other terms included in the two models are quite diiferent and, as can be seen in the table, change the scattering lengths in opposite directions. In particular, the extra terms in the Skyrme lagrangian are attractive, while the
A comparison of the scartering lengths as obtained from the Skyrme model and the cloudy bag model. Results are given with and without recoil as well as from using just the chiral term, Y‘,,,,,,,. The cloudy bag results are from ref. “)
Experiment
SIl
s31
SII
0.24
-0.14
0.24
CBM (with higher Born (recoil) Full (recoil) Born (no recoilj Full (no recoil)
0.29 0.22 0.34 0.22 CBM ichiral
Born Full Born Full
(recoil) (recoil) (no recoil) (no recoil 1
0.22
0.47
order
terms) -0.13 -0.04 -0.15 -0.01
term only) -0.11 -0.09
Skyrme
s3.1 -0.14
(with topological
0.27 0.46 0.3 1 O.hl Skyrme 0.22
0.35 0.25 0.45
terms -0.06 -0.06 -0.07 -0.06
ichiral
term only) -0. I 1 -0.09 -0.13 -0.11
1
3. K. Jetw~ing.s. 0. V. Mawelf
636
/ S-ware
rN
scariering
higher order terms included in the cloudy bag model are repulsive. Thus, as noted above, the inclusion of terms in the Skyrme model similar to those in the cloudy bag model should result in an improvement in the results obtained. While our calculations are somewhat different from previous ones, we expect that the validity
of our main conclusions,
that recoil is important
and that higher order
effects must be included before one can meaningfully compare to experiment, will not depend on details of either the model employed or the manner in which the calculation is carried out. Clearly, much work remains to be done before it can be decided whether the Skyrme model is capable of describing z-nucleon scattering.
References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20)
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