Quantizing the four-baryon skyrmion

Nuclear Physics A547 (1992) 423-446 North-Holland

P

LE YSICS

uantizing the four-baryon skyrmio T.S . Walhout'

Institutfür Kernphysik-, KFA Jülich D-5170 Jülich, Germany Received 19 February 1992 (Revised 11 May 1992) Abstract: The zero- and low-energy collective modes about the B = 4 topological soliton of th~- Skyrme model ai:e quantized in a semiclassical scheme . The state of lowest energy corre,-ip ,,)--,Js to the alpha particle, and the next lowest state to its first monopole excitation. The energies and radii are calculated .

1. Introduction Research during the past decade has demonstrated the utility of the simple and elegant picture of the nucleon and nucleon-pion interactions provided by effective meson theories. This view has been further bolstered by recent work. In particular, Verschelde ') has shown that a careful nonrigid quantization using the Dirac formalism for constrained systems yields a correct soft-pion limit in the one-baryon sector and solves some problems which were once thought to be shortcomings of the Skyrme madel. Another supposed shortcoming of the Skyrme model was that, despite yielding a simple and generally qualitatively appealing derivation of the nucleon-nucleon interaction, calculations using the product ansatz, an approximate solution for the two-baryon system, found that the intermediate range attraction in the central channel was missing 2). A recent studY 3), however, indicates that this was merely a shortcoming of the product ansatz and that when exact numerical solutions for the two-baryon system are used, there is an attraction . Since in conventional models this attraction is responsible for the binding of nuclei, one might hope to be able to construct nuclei also in the Skyrme model. This is attempted here. In a previous paper), a calculational scheme was presented for the semiclassical quantization of the topological solitons of effective meson theories such as the Skyrme model. Although these static, minimum-energy, classical solutions for a given baryon number are in general very collective objects in which the identification of individual baryons is impossible, it is nevertheless useful when introducing Correspondence to: Dr. T.S. Walhout . Dept. of Physics, The Ohio State University, 174 West 18th Avenue, Columbus, OH 43210, USA . 1 Present address : Physics Department, The Ohio State University, 174 West 18th Avenue, Columbus, OH 43210, USA. 0375-9474/92/$05 .00 @) 1992 - E!sevier Science Publishers B.V. All rights reserved September 1992

TS. Walhout / Four-baoyon skyrmions

424

collective degrees of freedom for quantization to consider the multisolitons as being built by bringing together distinct solitons of unit baryon number. Then an analogy can be made between the B-baryon soliton and the N-atom molecule, and a group decomposition of the symmetry transforms of the baryon number density provides a classification ofthe vibrational modes according to the symmetries ofthe minimumenergy solution for a given B. This is helpful in identifying the important collective coordinates for semiclassical quantization in the same way that an analysis of the transformation symmetries of a molecule is helpful for the study of its vibrational modes; namely, it allows an identification of the coordinates which diagonalize the hamiltonian. The case of the baryon-number-four sector of the Skyrme model is particularly interesting since semiclassical quantization, while relatively simple, still provides an illustration of how the general problem can be handled. In ref. 4), quantization of the B = 4 skyrmion was dealt with on only a very simple level. espite neglecting all non-zero modes except the monopole, qualitatively reasonable results were obtained for the states of lowest energy, the alpha particle and its monopole excitation . Here we analyze this system in more detail and show how a consideration of the collective modes corresponding to quadrupole vibrations improves the description of the B = 4 nuclei in the Skyrme model. The usual Skyrme model with a stabilizing term of fourth order in derivatives and a pion mass term is used here [for reviews see ref. ')]. In terms of the pion field (X, 0 = 0~-Xp [Ulf_.. )T - a(x, 1)], the lagrangian is f2

Tr OP 1i a1A U') + i ;~ Tr [aP UU a" Uut]2 + 1 2

2

M2 Tr (U

and the conserved, topological baryon number current is 247T'

P

uu

(2)

The parameters are taken to be f,, = 54 MeV and e = 4.84 so as to fit the B = I sector (the nucleon and delta masses) with the pion mass m,, = 138 MeV [ref. ')] . The static, classical, minimum-energy solution in this sector has the hedgehog form U"(x) = exp I iT - x^ F(r)J . In fact, there is a class of solutions with the same energy, defined to be the hedgehog mass M" . These solutions are related to UF] (x) by global translations in space and global rotations in isospace (which, because of the specific hedgehog form, are equivalent to rotations in space). For B = 4, in the asymptotic region, where the system consists of well-separated B = I objects, the field solution is well approximated by the product ansatz 4

Up(X ;1rn1-)1CnD = H CnUH(X - rn)C'n n=1

(3)

ere the C,, are SU(2) matrices defining the orientations ofthe hedgehogs in isospace (each representing three Euler angles), and the r,, are spatial vectors defining the

TS. Walhout / Four-baryon skyrmions

425

hedgehog centers . Since these centers are all well separated, it follows from the asymptotic form of the hedgehog solution F(r) that the total static energy corresponding to (3) is equal to four hedgehog masses plus a sum of two-body potentials of the one-pion -exchange form') V,(r, C) = -!K Tr (,rkC7jC") 2 K12[3(c -

0 a (e 61

rk

61 r,

A)2 _ C 2](I +

r

r

m.,r)

+[ C2_ C2 + 2(c - rA)2] M27rr 21 0

-3

9

(4)

where r = r,, - rm is the distance between the two hedgehog centers, C _= co + i1r - c C'n C,,, is the relative isospatial orientation of the two hedgehogs, and K >0. For the present parameter set, K = 220 MeV - fm 3 [ref. ')]. The asymptotic interaction between two skyrmions is therefore most attractive when one is rotated with respect to the other by 7T about an axis perpendicular to the axis on which their centers lie - that is, when co = 0 and c - r = 0. Taking the centers to be in a tetrahedral arrangement with = l ri(0) = /îj (Pl _ Pq -P) 9 P2 'V/3 11 ( - P Pl -P ) -j

(0)

P3

='A3 (-P -Pl P) ,

r4

j (Pq Pl P)

9

the configuration with maximum attraction between each pair of skyrmions has C"= j where the ri are the Pauli matrices, and C"4 ) '= A, with A arbitrary. i The quantity p, the r.m.s. radius of the baryon number density, is then a measure of the separation between individual baryons. As p --> oo, the product ansatz becomes an exact solution of the equations of motion. For finite p, (3) is only approximate; and - as is evident from exact numerical studies - the approximation becomes increasingly poor as p is decreased. The symmetries of the asymptotic solution with centers and orientations fixed as above are also symmetries of the lagrangian and - since configurations of finite separation can be obtained from the adiabatic time evolution of initially wellseparated skyrmions - are also symmetries of the exact solutions for all separations. These symmetries are combinations of A(i7 ),

AI U(z, x, y)A"1 = U(x, y, z) , with A, = exp (I3 iITT * a'I), d',^ I

=

4'(1, 1, 1), and

A2U( -Y-) X.

-z)A 2 = U(x, Y, I

(6)

Z) ,

A with A2 = "P (2 iITT 42 a2 40l -1, 0) . A generic feature of the Skyrme model, which treats baryon structure and interactiuns on the same footing, is that as B > I skyrmions are brought together new symmetries appear, the configuration changes drastically from that of B distinct hedgehogsl and the energy reaches a minimum 7) at some finite value of the r.m.s. radius po . Braaten, Townsend, and Carson first

426

TS. Wathout / Foisr-bai~yon skyrnzions

found the minimum-energy B = 4 solution and identified its symmetries* . It has an octahedral shape and has - in addition to the symmetries (6) and (7) - the symmetry U'(-x, -y, -z)A ; = U(x, y,

Z) ,

where A I = exp(I2ir-r - a,) . The path in conhgura6on space parameterized by the r.m.s. radius p thus describes the construction ofthe global minimum-energy solution from a tetrahedral arrangement of well-separated hedgehogs, the asymptotic at p = A, solution at P -* co. In the present analysis this path, which describes the monopole breathing mode, will be singled out since it corresponds to large-amplitude fluctuations about the minimum-energy solution. For computational convenience, the smaller amplitude vibrations - namely, quadrupole modes and isospatial oscillations - will only be dealt with approximately . The paper is organized as follows. In the next section, semiclassical quantization and the present calculational scheme are discussed . In sect. 3, the effective hamiltonian for the reievant collective coordinates is derived. This is first demonstrated for the asymptotic region, where recourse ca'a be made to the computmionally simpler product ansatz, before proceeding to the more general case. In sect . 4, the quantum states are constructed, and static properties of the alpha particle are calculated . A discussion of the results is given in the final section . 2. Semiclassical quantization Semiclassical quari .ization involves the introduction of collective coordinates which describe quantum fluc,uations about the classical, minimum-energy solution . Those coordinates corresponding to zero and near-zero modes describe low-energy, large-amplitude fluctuations which should be dealt with explicitly. On the other hand, higher modes may be treated as harmonic, and their ground-state energies can be thought to be absorbed through a renormalization of the parameters of the lagrangian - that is, as a first approximation they are ignored . In the asymptotic region, the lower modes are easy to identify: they are given in (3) by the 6B = 24 coordinates r,, and C,,, which when promoted to dynamical variables determine asymptotic zero modes. It is then natural to express the system at finite separation in terms of an effective hamiltonian which contains masses, moments of inertia, and a potential which depend on these variables. In other words, the time dependence of the field U is described via 24 degrees of freedom which reduce to some combination of the above coordinates describing zero modes in the asymptotic limit P - cc. In lieu of any accurate ansatz, the determination of the dependence of the system upon these collective coordinates must be done numerically . The effective lagrangian which results from assuming the system can be parameterized by the above 24 collective variables is obtained by substituting the pion field U(A jq(t)j, JCjt)j) into the Skyrme lagrangian and integrating over three-space . * The symmetry (6), was incorrectly defined in ref. 7) .

Walhout / Four-baryon skyrmions

TS.

42 7

For F

what follows*, it is convenient to express the pion field via a euclidean unit four-vector 0,, as U = 04+iir - 40, with 02a = 1 . Then the lagrangian (1) separates into a "kinetic"' term of second order in the time derivative of the field Yi = 2ll9iOaSiabCliÇbb -) ~f2 7r

.W.b =

2 (ai

«. )2

t5ab - -) ailba ai0b e-

9

and a "potential" term containing only & and its spatial derivatives a j O,, : S

jf2 (ai )2_ 2 7r 40a

1 4e

2

(4900a

4e

The promotion of r,, and C,, _= exp (iT OtOa

7--_

1

)2(ajOh)2+

4

1 n=1

I

-Y n )

2 (aiOaClil0a

)2 _f2 M 2 (I ir 7r -404)-

( 10 )

to dynamical variables implies 190a

&i 9.0.+ ar.i

a Y"i

I

(11)

After substitution in (9) and (10) and integration over three-space one obtains T=j d 3 X

,j4mn &j+2imiAmjj"~, ;j+~,,,jjmn~ t ii njj~ ii

ni'n

V=-j d 3X

('12)

(13)

4Mul,

where MH is the hedgehog masS 6), = «W ni. fi

.90-

(14)

r.j f &X armi 9.b aa0b,

and so on. The motion is here assumed to be adiabatic ; that is, 0,, does not depend explicitly on the time derivatives of the collective coordi-nates . The numerical evahiation of the effective mass X(Jrj, Jyj), the moments of inertia A(frj, Iyj) and J Q r,, 1, J-1,, 1), and the potential VQ r,, 1, 1 y,, 1) is straightforward for the coordinates corresponding to zero modes. Explicitly, given a static solution U(x), there are nine zero modes corresponding to global spatial translations R, global spatial rotations B = exp (-ir - P), and global isospatial rotations A = exp (Pr - cL) of this solution . In other words, the classical energy is invariant under (15)

UW -> A U( (B) - x - R)A' w1jere the spatial rotation parameterized by the SU(2) matrix B is Rij (B) == -12 Tr ( ,riBrjB'') = R ji`(B) T17 len in the I-,ody-fixed frame (where 41 --> 0 and .90. aRj

:--

a(P. = aai

Cijk(Pj Ska

(16)

--> 0), one has a0a a13,

- GijkXjak0a

(17)

* In the present notation, the indices a, b, c run from I to 4; the indices i, j, k run from I to 3; and the indices m, n have variable limits which should be clear from the context.

TS. Walhout / Four-baryon skyrmions

42 8

In a similar fashion, for well-separated hedgehogs, where all 24 coordinates in (3) describe asymptotic zero modes, after a bit of algebra one finds inn Y

inn j, -> 5mr1AH 2ki(Cn)-%-i(Cn)

where

AH

A mn Y -0,

MH45ij18mn

V ->

Y, Vm(Irm - rnliC~~CJ9 m>n

(18)

is moment of inertia of the hedgehog') and a

2ij(cti) =__ 12Eik,R _'km (CJ (ayj R,,,,(C,,)

(19)

In general, however, the derivatives with respect to the 15 collective coordinates which do not define zero modes will not have such a simple form, and the evaluation of these tensors is numerically rather involved . The calculational scheme outlined in ref. 4) will be followed here. The relative isospatial variables are initially fixed at the orientations which produce the minimumenergy configuration (the C,," above), and the dependence of the system on the spatial variables is then calculated. The dependence of the potential, masses and moments of inertia on the relative isospatial angles may be added perturbatively. The zero modes are then quantized exactly for the "unperturbed" effective hamiltonian, which depends explicitly on only the relative spatial variables . Also the relative isospatial variables can be simply quantized since the free quantum states are the eigenstates of the unperturbed hamiltonian . (There are nine independent relative isospatial angles which can be represented by the matrices D, == C4, C' , D3 = C"*C,, 4 - D =- C4'C3 .) The remaining vibrational modes may be classified by an analysis of the group transformation symmetries of the baryon number density [see, for example, ref. 9 )] . Such an analysis (see ref. 4)) shows that the B = 4 solution for general p transforms as the tetrahedral symmetry group Td and decomposes into the representations A I + E + T, + 2 T. The modes may be interpreted as three rotations (TI ), three translations (T,), one monopole vibration (A,), and five quadrupole vibrations (E + T2 ) . The extra symmetry (8) of the minimum-energy solution is just inve:sion i, which means the total B = 4 state transforms as the full octahedral group Oh = Td x i - that is, as the above representations with either even or odd parity. Pictorially, the total state can be viewed as a superposition of the two states corresponding to pulling baryons out from four equally separated corners of the octahedral solution . One sees that without the extra symmetry of the minimum-energy solution the quantum states would not be parity definite. A simple and quick way of quantizing the spatial non-zero modes would be to assume that the monopole and quadrupole oscillations about the classical soliton are harmonic . The frequencies of the oscillations can be determined by solving the time-dependent classical equations of motions") for a small initial monopole or quadrupole distortion of the classical solution, which may be obtained by solving the time-independent classical equations of motion with Lagrange constraints") .

TS. Walhout / Four-baryon skyrmions

42 9

This can be easily done by modifying existing codes, and the following frequencies are found for the present parameter set*. One monopole mode at W, = 168 MeV, two quadrupole modes at (02 =102 MeV, and three quadrupole modes at W, 118 MeV. Then the energy of the ground state is & = &I + 1w, + 2X 2l102+ 3 X 2!(03= 50 2

MeV,

(20)

where the classical binding energy E I = -313 MeV is the static energy of the classical solution minus four hedgehog masses. There is no bound state. Of course, this analysis is oversimplified ; one cannot expect the monopole vibration, at least, to be of small amplitude. So indeed, the analysis of the monopole mode in ref. 4), where no harmonic approximation was made (but where quadrupole modes were ignored), found two bound states at E,) = - 176 MeV and E, = -40 MeV. The lowest monopole excited state has energy EI -E,) =136MeV 20)1 ; the harmonic approximation does not work. In what follows, the analysis of ref. 4) will be expanded upon by including also the five quadrupole degrees of freedom . 3. The effective hamiltonian 3.1 . THE COLLECTIVE COORDINATES

The dependence of the B=4 skyrmion system on the six spatial collective coordinates corresponding to non-zero modes can be studied numerically by fixing via Lagrange constraints 11,12) the mean-square radius P 2 = 41 f d'x X 2~(yj jx) and the d 3 X X 2 y(2)( M XA )'0;qjX)' with Y (,,,' a spherical harmonic. quadrupole moments Q,,, These coordinates are linearly independent combinations of the moments qij = d'x x-x 04,jx) . From the group analysis above we expect the effective hamiltonian to diagonalize when expressed via these coordinates . Small quadrupole oscillations separate into the doubly degenerate representation E, which transforms as the Z 2 _ X 2 _ 2) , v3 r-:;( X 2 _ Y2)) combinations !((2 1(2q33 - qII - q22, 0 (q, I - q22)), and 2 Y 2 the triply degenerate representation T2 , which transforms as '13_ ((yz), (xz), (xy~) ,13- ( q23, qI .3, q12) . Thus if we keep terms of only second order in the moments Q,,, (which are defined in the body-fixed frame), the effective potential for fixed relative isospatial angles may be written as V(P'

I Q. 1)

5 2 2 2 = V(,(P) + 81r kjp)[3qI2+3qI .4+3q231 +

5 8

k2(P)I I4 (2q33 - qI 1 - q22) 2 + 41 ( qI 1 - q22)21

Vjp) +~'k I (p)[ Q*,

QI + Q* - I Q- 1 - 2-21 (Q2 - Q-2

)2] + k,(p)[Ç02 +1 O 2(Q-I+Q-2 2 * For comparison, the monopole mode of the hedgehog is 240 MeV [ref. "'fl .

)2]

(21)

431)

T S.

IValhatit / Four-bar~von sk-yrmions

the numerical calcultions is the classical field solution fact, "at one Rnds Q; A 10 1) for fixe values of the relative isospatial orientations, and one need are small. Solving the Schri5dinger equation, however, is not assume that the considerably simpler for a potential of the form (21); and one can expect that anharmonic corrections will be small. ecalling the discussion of the preceding section, ti ie kinetic part (12) of the e ctive lagrangian should be re-expressed in terms of he cordinates, R, A, B, Di, p, an ,, . In the body-fixed frame, the spatial coordina es are related to the rn via

-

he vect

hS.

rs P,,

Qc -- R)2,

4

2 I P =4

-

4

n =I

( r hT)2V2QC) . H n 13

(22)

above are restricted by the condition hT

0C 1: r hT X ih .r. = 0 ,

(23)

a ely, the total orbital angular momentum in the body-fixed frame is zero . si ly corresponds to spatial translations, which can be ignored by working in the ce ter-of mass frame. One can dehne three additional independent combinations f the n, the Jacobi vectors, as (24)

The relevant collective variables can then be expressed in terms of the Jacobi variaLles via I 4

13 052

n-1

11

n y2y ^ bW) . q~ = 1 9Cf ni 4n

(25)

n-1

ese relations will be useful later for censtructing the quantum states. nder global isospatial rotations, G - AG ; while under global spatial rotations, h" Y B) and Cn --> Cft Invariance with respect to A = exp (ér - et) means the total isospin is conserved, whereas invariance with respect to B = exp (- i-r - 13) means the total angular momentum (orbital plus spin) is conserved . The relative o6entations D, = (Q&J" - exp Ur - -%) are then expressed in the body-fixed frame and are fixed at D, " = iri in first approximation. The corresponding masses and moments of inertia follow from substituting

(

. .90. n=1 0 nni 3

rn ; 16 ale, = 1 A~ ap + 1. Qn - + OliEijl0j 161. + eiEijkXjakl0a + M=1

19QM

a

(26)

into (9) . The conjugate momenta and the effective hamiltonian can then be obtained by the usual methods, This requires one to Ifind the field solutions for many values of p and Qni (and D-,), a numerically tedious task. Instead, some approximations will be made here.

TS. Walhout / Four-baryon skyrmions

431

3.2 . THE ASYMPTOTIC REGION

It is perhaps instructive to first perform this rearrangement of variables for the asymptotic case of well-separated hedgehogs in the tetrahedral arrangement given by (3) and (5). Substituting (18) into (12), one finds i;2 +'kji i2j ' T =211:1M (27) "

n

n

n

where i i.i = 2EijkR,_-j'(C,,)a,Rjk(C,,)

= 9ij Wj~nj .

(28)

Using C,, = AD,,B, D4 -= 1, and r,, = R - '(B) - rh,,-", and working in the body-fixed frame a --* 0 and 8 ---> 0, after a fair amount of work one finds T = 1(4MH )É2 + '(4MH )fi2 + X

+

l6p

)2 102 _ 0 2 (02+ 0-2 (02 _

AHA

!

1

1

2

( !5~_ 'Ir)

- 0-2

)2+

4

0*1 01 + 0*- 1 0- 1 1 + 2!('J4MHP2 + 4À A 2i

4

1 n=1

2

E (ài+ini n~1

)2 ~

(29)

where 2 ijk o9 j Rjj (Dj&k '(Dj

i i ni = E

= Rij(Dn) a~k (Dn) ~nk

(30)

(note that i4i = 0) . Here IQ .. I << p has been assumed so that the orbital moment of inertia simplifies to A L = 344M" )P2 . If now one fixes p and keeps only terms of order Q2m in the potential (18) -that is, if one allows only small quadrupole deviations from the positions r" n given in (5) - one finds V(PIIQ.I)--

a 5 41rKM a 2

10p a

2 2 I + + 2) e-a a a e-a

J(

3 5 6 6) +_+ a a2+ a3+ a4 )2]

X

[Q*]Ql + Q*_ I Q- I - 12(Q2 - Q-2

+

1( 1+ 1 3 6 + 6)[Q2 )2]1, + + 0 + !2 (Q2 + Q-2 2 a a 2 a3 a 4

(31)

where a = N/~3 m,p. At large p, the coefficients kn (P) of (2 1) - although very small become negative, and it is obviously then a poor approximation to suppose that the Qm are small. A state consisting, for example, of two well-separated B=2 skyrmions with the same value of p is energetically preferred to the tetrahedral arrangement of isolated hedgehogs studied here, which must therefore be in weakiy unstable equilibrium at large p. Configurations with sucIA1, a large p are far removed 11

432

TS. Walhout / Four-haj~wn skyrinsons

from the minimum-energy solution, however, and should not make a large contribution to the quantum wave function . At smaller p, the assumption that the quadrupole distortions are small seems reasonable . ne may now construct the effective hamiltonian in the asymptotic limit. This requires that T be re-expressed in terms of the canonical momenta: 4

aT

aài

aT -

afl,

fî --

1

tg

=1

4

=(A 1- +4A t1 )#j -A 1j I R,_-j'(DJ(d;+,q,,j), q 0 OR-A

RjJDJ~j , ~

lj = - R,,(DP,)Y,,j = -A 11 afi

Pl'

~i +

A .4 R P-,'(DJ(cij + &j ) ,

= 4M.6

aT Q.

7rmul 1OP2

Vi"', is the total isospin, and J j"- is the total angular momentum, both in the body-fixed frame. The ~h,, correspond to right shifts of D,,, and Y-,, i to left shifts of D,, (with ~e , Substituting (32) into (29), and working in the c.m . frame, one finds 2

Oî

T

2 I if + -1 OP 2(4Mti ) 20WO

P*P

11

fi

2

3

fi

1 Ylli 1

)

3

+

E

n

1

1 2AL

(Ylli )

,

J h.r. + ihf. -

Pli + R 1

1-

11i

J

The hamiltonian is then H = T + V, with T in (33) and V given in (3 1) . Quantization proceeds by introducing the usual commutation relations between the coordinates and their conjugate momenta. It is natural to interpret -T,,, and R, in (32) as the isospin and spin, respectively, f. of the nth baryon (where n = 1, 2, 3). Then ~ I b,f- Y, YP, i and -R _= Sib - Ell d a2ni i are the isospin and spin of the fourth baryon, where the total angular momentum has been decomposed into total spin and orbital angular momentum Ji = Li + Si Since YP, + ~h,,i = 0 for a hedgehog (isorotations are equivalent to spatial rotations), the thhd term in (33) is just L 2 124 - the usual contribution from the orbital angular momentum . Y4i

4i

TS. Walhout / Four-baryon skyrmions

433

Of course, to construct bound states one must study the behavior of tse system near the rninimum-energy solution . This will be undertaken next. 3.3. GENERAL SEPARATION

For general separation, p, the masses and moments of inertia follow from substitution of (26) into (9). Thus . =

d 3X EWCSCEimnOm -)

JY

J -13

=I

sij',a17 = A Crp i =

J

d 3X EW0kSY1aEimnXm d 3X EMOksila

f

d3X Eiik0j'aka

f

d3 X 190a ap

a-0. (9 71'j -

49

n0a i

1)

0a I ap

9.

a-0. = d3 X Eijk0jSika 9 aQ. J A i.

aQ

XPP

4ah

190h

(9p

I

(34)

and so on. The field 0a is found numerically by discretizing the equations of motion (with la grange constraints) on a 26 x 26 x 52 lattize "), where because the symmetries (6) and (7) are imposed the calculation is reduced to one-fourth of space. Thus 0,, is found as a function of xi or given values of p and Q,,, with the relative isospatial orientations fixed at the values which minimize the energy. The tensors labelled with just a and P are then simple to calculate. Those labelled with p and Q involve derivatives with respect to these coordinates, which requires findin g many different Lagrange-constrained field solutions . This difficulty will be avoided here by assuming that the A and ft are equal to their asymptotic values, which is motivated by a study of the effective mass Mpp for the two-skyrmion system that found a difference from the asymptotic value 2M" only at small separations 13) . Since small separations involve large potential energies, as can be seen from fig. 1, such configurations will not contribute greatly to the quantum bound state. Note that this approximation also means that A = 0. Riska and Nyman 14), using the product ansatz, have shown that for the B = 2 system such terms are non-zero at finite separation and give rise to a spin-orbit interaction. For the ground state and monopole mode of the alpha particle, where the main contribution to the wave function comes from relative s-states, one can expect that even if A 0 0 such terms will not contribute to the energy.

TS. Walhout / Four-baryon skyrmitms

LLJ

6

-

CL

rms radius (frn) Fig. 1 . Numerical determination of the four-baryon potential Vo as a function of the r.m .s. radius p (squares) . The thin curve is the asymptotic form (31) determined from the product ansatz.

For B = 4 configurations with general relative orientations, the D', relate the field 0,, in different regions of space. For the asymptotic tetrahedral configuration (3) with (5), this is evident as the field in the volume 'V,,, with x E 'V,, if Ix - r n r(O'l M Ch.r. for all m 0 n, is just C UH (x - r,,(') ) n ' and is related to the field in volume V.. by anFor isopatial rotation D'Dn, = (C C", finite separations, the field for given Dn can be calculated by employing the symmetry (O) J < Ix -

-I

,, -rb

x E Vn

-> x'= R(Bnn ) -

x c Y,,,,

U(x) -+ D,,,B,,,,,D il U(x)D,,B"Mil D n-'

9

(35)

where R(B,,,,,) rotates In to 'V,,, . According to the present scheme, in first approximation the Dn are fim-d at D,"' = i7i , with D4 = D4( 0) = 1 . With the above discussion and the results of sect. 3.2 in mind, we will assume that the moments of inertia depend on the Dn as follows: -0 "jin .in

[,0 "i,k~ I 'Vn2,kj Dn

n

ii

~,k~" ik

] VnR kI Dn ) gly ( Dn

ilptiti 79'7 == 8 Pnn '~~ki (Dtj)[-O'kl y

J,O(, = 4 ii

E

[ 9 'i'k]V,,RJ(Dn),

(36)

where [ 9 ] Vn indicates that the integral in the defining relation (34) is to be performed only over the volume Vn . The tensors are evaluated for (~,, at different values of p

TS. Walhout / Four-baryon skyrtnions

435

but with the Q,,, set to zerv (since I Qm I < p is assumed, and the contribution to the energy from quantization of the rotational modes is already small) and the D,,, fixed at D(nO . Then the lattice calculations show that

' y"]'V"=AJ(P)8jj' V ['9

]'V"

AIJ (P) Bij , 4

,O'jii'j "= Y, [,O'j'j"]'V,,=Aj(p)sij, ii n=l

(37)

within numerical uncertainty. After the above assumptions, the kinetic part of the effective lagrangian for the general case is not greatly modified from (29). Calculating the conjugate momentum as above, then, one arrives at the following expression for the effective hamiltonian : H=

2 5 1 172+ lop Y_ P*Pn 2(4M H) 2(ITMH) n = , n

1 Ib .r. J b.r. + +_ E (Yni + eni il=, 2AL(P) AI(P) ( ' 3

1 '

1 + i 2A I (p) [(

3

E Ylli

2

n=l

3

1

»

]2

+ E ( Y,,, f- +V(p,fQ,,,I), n=l

where V(p, f Qm 1) is given in (13) and (21), and 4A 2Ij (P ) ,A L(P) :_ AJ (P) - t A (p)

(38)

(39)

I b.f._13 =1 From the arguments in the previous subsection, (Yni + Ini) in (38) is just n Sb .f. . the total spin in the body-fixed frame i The coefficients kn(P) in (21) may be extracted from the static energy of the Lagrange-constrained solutions OJx ; p, JQm I, ID((')I). For the region where these coefficients are positive, it is numerically convenient to instead use time-dependent simulations '0), as described in sect. 2, to extract the frequencies of the quadrupole oscillations for a given p, which can be fixed by a Lagrange constraint. From these and the masses in (38) the oscillator constants k n can be found as functions of p. At large p, expressions flor these quantities are given in (31). In figs. I and 2 are shown the terms VO(p) and ^,Jp) in the potential (21) as extracted from the numerical calculations, along with the corresponding asymptotic expressions from (31) . In fig. 3 are shown the moments of inertia A, (p), A lj (p), andIAL(P) from (37) and (34). As can be seen, these agree with the product ansatz results (29) at large p. The stage is now set for quantization of the collective inodes .

TS. Walhomm/ Four-baryon x0 yr m omo

436

rms radius (fm) Fig. 2. Numerical determination of the coefficients k, (circles) and k2 (diamonds) for harmonic quadas functions of the cmy . radius p. The thin curves show the corresponding qoaox6tes from (31) determined using the product ansatz approximation . The quantities plotted are ^,Jp) . rupole distortions of the four-baryon skyrmion

E 3000

o -~ m ~ o M

1000

1

2

rms radius Qm)

3

Fig.l The monoeomo f inertia &AL (p)(uguureu) A u(p)(d iam nodu) A / (p)(cirdca) .Tbc thin curve is the asymptotic orbital moment 4AL = :Mu z, and the thin line is the asymptotic isospin moment AH 195 MeV '6n z .

TS. Walhout / Four-baryon skyrmions

437

4. Construction of the quantum states 4.1 . ANGULAR MOMENTUM AND ISOSPIN

The rotational and isorotational coordinates A, B, and D,,, can be dealt with exactly. That the hamiltonian (38) is invariant with respect to A and B means that total isospin and angular momentum are conserved. Invariance with respect to the D,, is of course only good to zeroth order in a perturbative expansion in these variables. The Wigner rotation matrices are eigenstates of the operators in j b. )j b.f. i f* and J = -R,-j '(B j - the total angular momentum in the body-fixed and lab I frames, respectively - with total angular momentum J = (J b- f-) " =j j + 1) and third components - m' in the body-fixed frame and m in the lab frame '). Similar relations hold for and Ii = -Rij(A)I .j"-r* - the isospin in the body-fixed and lab frames, respectively - and for each pair _02, and -T,,i = -Rq (A)R,# . Rotational and isorotational eigenstates X(A, B, JDJ) of the hamiltonian (38) can be constructed then from products of !Vikk') (B), !VI;*,'(A), and 2 (S") k,,I,, ( Dn allowed, however. The values of i, j, and s', are Not all quantum numbers are restricted by the condition that the B = 4 state be composed of four fermions, so that the wave function should pick up a phase -1 for each fermion that is rotated by 27r under a given symmetry transformation . This relates to the well-known Finkelstein- Rubinstein constraints for quantization of solitons ' 6), which require that closed paths in configuration space are either contractible or non-contractible depending on whether an even or odd number of solitons are rotated by 27r. Thus sending B --> - B corresponds to a global spatial rotation of four baryons by 27r, which means the wave function should pick up a phase (_1)4= + I and hence the restriction thatj be integral . Similarly, A ---> -A rotates four baryons by 2 7r in isospace, and so i is also an integer. On the other hand, sending D,, --> -D,, rotates only the field in the volume 'V,, - namely, one baryon - by 27r. The correspGnding eigenstates should thus pick up a phase -1, which means the s,, are half-integral . Finally, there are constraints corresponding to the symmetries (6) and (40) U( -x, -Y, Z) = U(x, Y, Z), which is just of (7) applied twice [the path associated with (7) alone is not closed]. This has been described in detail by Carson for the quantization of the zero modes of the B = 3 minimum-energy solution 17) . The transformation symmetries are nearly identical for the present case. First, eq. (6) applied three times rotates four baryons by 2 ir in space and isospace, thus giving a. phase + 1 . The closed path corresponding to (6) must therefore also be contractable, since it would then be inconsistent for the wave function to pick up a phase -1 under (6) . Since eq. (40) followed by eq. (6), applied three times, is the identity transformation, it follows that the path associated with (40) is also contractible. Expressing the symmetry transformations (6) and (40) in terms of the operators I bi - '- and J hi -'- , the wave function is thus

TS. Walhout / Four-bajyon skyrmions

438

restriete

by the conditions exp

W'-" +J t)-f )

A.A " I -

+

exp [ i7rj 3h .C.] 'V = +'V .

X,

(41)

e wave function V for the dependence on the rotational and isorotational degrees of freedom is then of the Orm I

jal)=E Q921(B).11"(A) f] .9"WA), k

I

(42)

where the C,, must be chosen so that (41) is fulfilled. Here a denotes the set of quantum numbers Q% 1% k, - k3 , 1 1 - 0. The energy corresponding to these states is obtained by sand&chi , te hamiltonian (38) between them, and it is clear that states of lowest energy nni have low values of the quantum numbers i, j, and s n . ConMde6ng (38), the C,, should be chosen to be proportional to Clebsch-Gordan e cients in the indkes k n and 1, such that the X is an eigenstate of definite angular momenta Y-r, Y, Y,,, and ~IA Tj The Finkelstein-Rubinstein constraints, however, in general will not allow the construction of states of good total spin. The full wave function will include, of course, dependence on the vibrational degrees of freedom p and Q,,,. In order to quantize these spatial coordinates, it will be useful to combine with them the coordinates corresponding to orbital rotations, which means separating out from (42) states of total orbital angular momentum L. This can easily be accomplished by transforming from the coordinates (A, B, Dj to (G, B), where G = AaB (with C4 = AB), via the well-known relations [see, for example, ref. "')]

114X) &-!Y""2%~,(A) , (A)

a§!, (A)

L

Y_

L=11 1 -121 M,N=-L

( 1 1 M I ~' M21

LM)

x(1^~-,n2jLN).13~(L) MN (A).

(43)

Then the wave function (42) can be rewritten as X(A,

f

(Il

C(,, E cro -yL6~ MM,(B)q('4) (AB) a4b4 Lo

C

3

n=1

p2U)

(44) ~ 1 Xc,,P-yL(f il MM '(B) , cap

where P denotes the (summed) quantum numbers (M, M', a, - a4 , bj - b4,, S4) and -y denotes the (good) quantum numbers (j, k, 1) 19, S I - S3) . The coefficient C,,,O,, is a generally messy combination of Clebsch-Gordan coefficients. The total wave function may now be written as flfc.l'

Ai fQmD'~ E

Lao

X~Y[3-yL(fcnl)lPa[3-y L(P-) fQmI, B) ,

(45)

TS. Walhout / Four-baryon skyrnlions

439

where the dependence on B - the rotation matrix _q(L) MM (B) - has been absorbed intoOaf3yL Let us now specialize to the states of lowest energy, namely, those with i = j = 0. Clearly, the conditions (41) are satisfied byV in (42) with these values of the total isospin and angular momentum . As mentioned above, the X should be eigenstates of the operators YTj and RTj . This may be accomplished by choosing the C', in (42) - or, equivalently, in (44) - as Ç,, = 1 (s i k I S2 k:! lUbt >(SI Il S2 121 0"9 ' )(OMS3k3l SLk,

' S3 131 SRk

(46)

In fact, the hamiltonian (38) may be diagonalized immediately for i = j = 0 by further multiplying the y obtained from (44) and (46) by (_I)kL-4-M(SL,)

-kL ,SR ,

(47)

L, -M)

kRl

and summing over kL and k R . This simply gives an eigenstate of the angular momentum Shi-1 .

(YTi + -RTO

n

(R,,i +

(48)

'P11j)

where the last equality follows since Ji =O. From (44), (46), and (47), the wave function is Xo(A, B,

I D, 1)

16

C.Y ""-' a4,b4 (AB)

3

fj

n=l

C.,:, '(AD,,B ) _q (L)

mm

a, bn

~ ( B)

~

(49)

where 8 is now (a, - a 4 , bI - b4,, M') and = 1. . ( -,)M'--a4(S aS2a2lui£)(s 1 bS2b2l e 'll ' ) Co gg ,v

X ( 6r US3a3l SL ., - a4)(O" IL ' S3b3l SRv)(sLb4SR el L, -m) .

(50)

This decomposition supports the identificationOf -~PT+-92T in (48) with the orbital angular momentum L. Expressing the total wave function for the collective coordinates as in (45), the state i =j = 0 can be written 4

11

ZOQ Cn 1AP,1QmD = 1Cj3 11 -6~(a,,b),,(Cn)4'LMM'-y(P-)IQrnl,B), n=l is

where

S4 2:` SL .

(51)

The wave function for the spatial variables is IPLMM'-I (Pg IQ,,,I, B) = OyLM(A,

I

Q",

1) g (LM) M

(B)

(52)

where y represents the set of vibrational quantum numbers . The hamiltonian in this state becomes simply 1

A2

1)+

3

2~j-[L(L+l)]+ -1 E Sn(Sn + 1 [SL(SL + H = H'ill+ f1~I 2À , 2 A IAL

(53)

440

TS. 111alhout / Four-baqon skyrndons

where H, j , depends only on the vibrational coordinates . The state of lowest energy has si, = sR = 2 , L = 0, and the s,, = 2I , which corresponds to individual solitons quantized as nucleons . An account of the D,, dependence of the system through perturbation theory will add states of higher s,, - for example, a delta admixture - and states of higher L. 4.2. THE VIBRATIONAL MODES

.,(p, IQ .. 1, B) for the spatial degrees Now we may find the wave function of freedom . From (38), (53), and (54), the hamiltonian to be solved is 2 1 5 ~ff2 + 8P H= 1 1 P* P. "' 2(4M FI ) 2(2MH ) 41r,,,7-

+

3 _ 2 + 7/* (P, 1 Q"' 1 ) L 2(8M"p')

(54)

where the effective potential is I I ( I ) I Y(P' IQ. 1) = V(P' IQ ..D+- --2 Al(p) 'AH I 'A 2 (P ) 3 2( I

+-

2

1J

p

)AL(P)

8 MHP 2

4

S"(S"+0 1-1

L(L+I) .

(55)

Here the asymptotic orbital energy has been added to the kinetic part and subtracted from the effective potential . This is convenient for the basis functions which will be chosen below. Note also that the constant Y_ 4~ 1 s,, (s,, + I )/ A " has been subtracted so that the energy goes to zero at large separation . Since the potential V was defined by subtracting four hedgehog masses from the static energy, for all the s,, = 21 this means that the asymptotic energy is just that of four nucleons . Recall that the parameters have been chosen so thatMH+93AH= 939 MeV, the experimental value of the nucleon mass . Of the vibrational modes, the monopole is to be treated most carefully . A natural way to approach quantization is then the hyperspherical expansion method "'), which expresses the spatial variables as 3 c .m . coordinates (which can be ignored in the .c.m. frame), a hyperradius (the r.m.s. radius p), and 3 B - 4 angles. Three of these angles correspond to the global rotations B which generate the orbital angular momentum L. These and the other five angles may be expressed in terms of the Jacobi variables in the lab frame as El = 2p siri

"/-l Àîrig 1 ,

92 =2peos0sinym^ -), t3 ==

2p cos 0 cos y M3 A

(56)

TS. Wathout / Four-baryon skyrmions

with 0 ----

441

e, Y'--- 127T- Using g,, = R-'(B) . gbn_1 and (25), we have ^ 1 ) + COS2 e [ y (2)( 2y+ y(2)( sin 2 e y (2)( M M m M2)sin m M3 A

)

A

COS2y]

(57) in terms of the byperspherical angular coordinates D = (e, becomes

Y, I MA

the hamiltonian

(58) i P +'V'P' where the hermitian hyperradial momentum operator O=,91ap+41p has been introduced, and An is the eight-dimensional annular operator which is the generalization of the operator L 2 for simple spherical coordinates. (311-4) The wave function can be expanded in the hyperspherical harmonics 6&K,, which are eigenfunctions of the 3 D - 4 angular operator 2(4MH )

ap ( P" 49P

-K(K+3B _ 5)ùU(3R-4) Ka

~à(38-4)0&OB-4) = fi

Kce

(59)

and form a complete, orthonormal basis on the (3B - 4) -sphere. Here a represents 3 B - 5 other quantum numbers which include L and M. Expanding for B = 4 as 41(p,

P-4

0nKa(P)

K,a

011(Ka( 12 )

(60)

s

the Schr6dinger equation can be rewritten as the hyperradial equation I 8MH

J_

+ K'a' I

d 2 + ( K +3)(K +4)1 OKa (P) 2 dp 2 P C

WKaK'a'(P)OK'a'(P)=EKaOKJP)

where WrKaK'.'(P)

=fM

6UKa(f2

)V(A

f2)6UK'-'(W

(61)

(62)

is the angle-averaged potential. (For the remainder of the paper, the superscript (8) on V will be suppressed .) The differential angular volume is (63) M = dO sin 20 COS5 e dy sin 2,y COS2 y dm', dm^ , dM3 P-4 d P 0 2K, in (60), theO K,, are normalized as Because ofthe explicit factor The hyperspherical harmonics can be written explicitly for B = 4 [ref. ' 5 )1 .. I

I'l3, 1

q1 KLMk

E (1 1 MIM

M, Im I LM) Y'm,(m A

(9) X Y I 2 1 5(M . ., ~ (O)e(6)13(,Y) ~ kj, l"I - M3)eK,I l k

where 1 2 1 3( '2~ Ylm M M3) =

E (LM2 13 1tfll M'M3

(13 ) . I M ) Y (12 ~( M, M2) "' Y M3 (M3)

(64)

(65)

TS. Wathout / Four-baryon skwndons

442

an &-,ah (y)

e "'

y p(a+1/2,h+(ti-5)/2) (cos 2 y) . =W ik-a-bV2 &-ah sin'y cosh

(66)

are Jacobi's polynomials. Now from ere "" I kah is a normalization factor and P'"'" sect . 2 we want to choose the state to preserve the extra symmetry (8) of the minimum energy solution, namely, e

* #h "

e ( - x) = -:E gf (x) ,

(67)

which means we must choose states of definite parity . Since the hyperspherical functions transform under parity as (-I)' [ref. ") ], eq. (67) is ensured by restricting the sum in eq. (60) to either only even or only odd X Finally, according to the analysis in sect. 4. 1, we are interested in only the L = 0 states, for which (64) reduces to I 1,1, -

a 0?1 K,V

V

A/-

(-01,-12+1 3

11

, (do

Y'M~".

(68)

where the matrix is a Wigner 3j symbol. The number of L = 0 states for a given K is (K + 8)!!/(K!!8!!) if K is even, and (K + 5)!!I((K - 3)!!8!!) if K is odd , 5) . Clearly the lowest-energy states will be primarily K = 0, with a small admixture of higher even-K harmonics which depends uponW/'I
621 K Q (B) - tki1) = 611 K Q Ai 1) -

(69)

(70) dfl"-- 611 KJf2b'f- )'V(P-) n h'f.)611 K'a'(f2 b* f. ),) we may easily move to the body-hxed Name, where the potential V is defined. Setting sn = ! and L = 0 in (55), and using (21), (25), and (57), one finds 1 3 1 Y(PI M = vo(p) + 2 ( À 1 (p) À ui

13

e2 j e2 y 1121 Xï n, illki(p)[ ln ;Mn - 1 r (2 + i 2'2)* (t~h Y2 n + k2(P)[ Y( A YVI*(Aîi

f P`n) Y(

1-1.) y ~n

(2

( M n ) + y(2 - 1) :*- ( #fi b.f.) y~2) (lûb.f. ) ", 1 n b f.

y~2

f.

) (ffi

ny y(2 2)* (f~ n; ' W Y~) -1

ri

+

y(2) f. -2 ( t~ 1', »P .

(71)

TS. Walhout / Four-baryon skyrmions

44 3

Evaluating (70) for K = K'= 0 gives ,WOO(P) = VO(P)+ -3 ( 1 - 1 ) +~OP4 [3k,(p)+2k 2 (P)l 2 Aj(p) AH 11W

(72)

Furthermore, the matrix elements IVOK ,,,, are zero for all K'= 2 harmonics. This means the lowest harmonics that can mix with the K = 0 state are K = 4. We will assume here that this mixing is negligible . The equation to be solved, then, from (61) and (72), is the hyperradial Schri5dinger equation d 3 (73) E,, - W,)o(p) +8MH 2 Idp 2MH 2 0', (P) = 0 -

p Il

1

The effective hyperradial potential WOO +3/2M H P2 obtained from the numerical calculations is plotted in fig. 4. Also si..own are the energy levels calculated from (73) with a Numerov algorithm [see, for example, ref. "fl, where at large p the solution 0,, was fit to the asymptotic form which can easily be derived analytically. In agreement with phenomenology, only two bound states were found. They correspond to the alpha-particle ground state and the first 0' excitation, and have energies EO = -79.1 MeV and E, = - 10.2 MeV, respectively . The experimental values are EO = -28 .1 MeV and E, = -8.0 MeV [ref. 20)] . From the solutions to (73), the r.m.s. radius of the ground state is determined to be rr.m .s .

f

P )P 20 0(p ) dp g( o

1

1/2

1 .58 fm,

(74)

800

(D C LLJ

C 0 13-400 1

rms radius (fm)

2

P2 (solid Fig . 4. The potentials Vo(p) (dashed line), 7VOO(p) (dash-dotted line), and W",,(p)+3/2MH line) obtained from the numerical calculations as functions of the r.m .s. radius p. The ground- and excited-state energies are given by the thin lines.

TS. Wallsous / Four-bao~yon skyrndons

444

200

-200

.

I

.

I

.

.

.

.

.

.

I

3

-

-

__

rms radius (fm)

I-

4

.

.

.

Fig. 5. The ground- (full line) and first-excited-state (dashed line) solutions to the Schr6dinger eq. (73). The wave functions are multiplied by a factor 100, and the effective hyperradial potential of fig . 4 is also shown (thin line).

and the transition matrix ehement M=2

f

)P200(p) = dp 0*(P I

1 .12 fm 2

(75)

Experimentally, r,.m .,. = 1 .63 :1-- 0.04 fm and M = I - 10 -:1-- 0. 16 fm2 [ref. 2')]. The wave functions 0,)(p) and -0,(p) are shown in fig. 5 . One can see that the quantum states contain large contributions from configurations corresponding to well-separated skyrmions, where the dependence on the relative orientation C is small . Since the r.m .s. radius of the excited state is 2 .64 fin, the skyrmions in this state are on average much further apart than those in the ground state; and so one can expect that corrections due to dependence on C will be much smaller for this state - presumably small enough to keep it bound. 5. Conclusions The static minimum-energy classical solution to the Skyrme lagrangian in the 4 sector has the shape of a cube with holes in its sides. It is bound by 313 MeV with respect to four free skyrmions, and its nni.s. radius is 1 . 12 fin. Ifthe semiclassical expansion is valid, and if the Skyrn-,e model is a reasonable low-energy approximation to QCD, then this object is the leading-order contribution to the alpha particle. A proper quantum-theoretical treatment would include fluctuations about this static soliton, but the semiclassical approximation allows the truncation of this infinite

TS. Walhout / Four-baryon skyrmions

445

number of degrees of freed_~m to some finitt: number corresponding to zero and nearly-zero modes. So as to be consistent with the B = I sector quantization of the hedgehog as a nucleon, these collective degrees of freedom were chosen to be those which correspond to separate translations and isorotations of the four individual B = I objects in the asymptotic region when all skyrmions are well separated. When the B = 4 object is quantized, then, these 24 variables are promoted to dynamical degrees of freedom. Naturally, since only a finite number of degrees of freedom are kept, the present calculation is sensitive to how the collective coordinates are chosen. The particular choice employed here was brought about by calculational convenience, but identifying the fluctuations about the soliton via the moments of the baryon number density does seem natural. In particular, it is easy to see how higher-order fluctuations may be included . Moreover, this choice is motivated by a study of the group transformation symmetries of the baryon number density of the minimum-energy solution . Such a study implies that the hamiltonian will be diagonal when expressed in variables corresponding to one monopole and five quadrupole modes. These are low-energy modes which in the limit of large separation become spatial zero modes of the four individual solitons. There are also non-zero modes corresponding to the relative isospatial orientations of the solitons. These orientations should be localized about those particular orientations which produce the maximum attraction between the skyrmions. So the potential and moments of inertia were calculated for these specific orientations, with the assumption that corrections due to changes in the relative orientations could be added perturbatively. Then, to lowest order, the system depends only on the spatial collective coordinates ; and it was possible to treat those collective coordinates corresponding to the relative orientations as free. Higher-order perturbative effects have not been considered here. So after this truncation of the degrees of freedom, this definition of the collective coordinates, this perturbative treatment of the relative isospatial degrees of freedom, and the assumption that the quadrupole oscillations are small, the quantum states corresponding to the alpha particle and its lowest 0' excitation were found to have energies -79.1 and - 10.2 MeV. There is a significant contribution to the energy from these modes, and there is a qualitative agreement with experiment . The r.m.s . radius of the ground state and the transition matrix element M are in even better agreement with experiment . Recall that if the dependence on the quadrupole coordinates was ignored, the energies were found to be - 176 and -40 MeV. So including the monopole vibrations gives the largest effect, and the quadrupole vibrations give a smaller contribution, as one would expect . It seems reasonable that an even smaller effect will come from considering the dependence on the relative orientations . Such a contribution will necessarily be smaller for the excited state, where the skyrmions are mostly well separated, where the dependence on the relative orientations will be small.

446

P.S. Without / Four-haj~yoti skyrinions

his -work was supported in part by the Alexander von Humboldt Foundation and by NSF grants PHY-91-02922 and PHY-88-58250. I thank E. Migli and I ambach for helpful discussions. eferences 1) H . Verschelde, Phys. Lett. B209 (1988) 34; B215 (1988) 444; H. Verschelde and H. Verbeke, Nucl. Phys. A495 (1989) 523 2) A. Jackson, A.D. Jackson and V. Pasquier, Nucl. Phys. A432 (1985) 567 ; R. Vinh Mau, M. Lacombe, B. Loiseau, W.N. Cottingham and P. Lisboa, Phys. Lett . 13150 (1985) 259 3) T.S. Walhout and J. Wambach, Phys. Rev. Lett. 67 (1991) 314 4) T.S. Walhout, Nucl. Phys. A531 (1991) 596 5) 1. Zahed and G.E . Brown, Phys. Reports 142 (1986) 1 ; G. Holzwarth and B. Schwesinger, Rep. Prog. Phys. 49 (1986) 825 6) G.S. Adkins and C.R. Nappi, Nucl. Phys. B233 (1984) 109 7) E. Braaten, S. Townsend and L. Carson, Phys. Lett. B235 (1990) 147 8) T.H.R. Skyrme, Nucl. Phys. 31 (1962) 556 9) M. Tinkham, Group theory and quantum mechanics (McGraw-Hill, New York, 1964) pp. 50-64, 323-330; L.D. Landau and E.M. Lifshiftz, Quantum mechanics, 2nd ed . (Pergamon, Oxford, (965) 332 10) J.J.M. Verbaarschot, T.S. Walhout, J. Wambach and H.W. Wyld, Nucl. Phys. A461 (1986) 603 ; T.S. Walhout, Nucl. Phys. A484 (1988) 397 11) J.J.M. Verbaarschot, T.S. Walhout, I Wambach and H.W. Wyld, Nucl. Phys. A468 (1987) 520 12) 1 Wambach, H.W. Wyld, and M.S. Sommermann, Phys. Lett. B186 (1987) 272 13) T.S. Walhout and J. Wambach, in preparation 14) D.O. Riska and E.M. Nyman, Phys. Lett. B183 (1987) 7 15) Yu.A. Simonov, in The nuclear many-body problem, ed. F. Calogero and C. Ciofi Degli Atti (Editrice Compositori, Bologna) 1973; A.I. Bai, Yu.T. Grifi, V.F. Demin and M.V. Zhukov, Sov. J . Part. Nucl. 3 (1972) 137 ; V.D. tfros, Sov. J. Nucl. Phys. 15 (1972) 128 16) D. Finkelstein and J. Rubinstein, J. Math. Phys. 9 (1968) 1762; J.G. Williams, J. Math. Phys. 11 (1970) 2611 ; E. Braaten and L. Carson, in Workshop on nuclear chromodynamics, ed S. Brodsky and E. Moniz (World Scientific, Singapore, 1986) p. 454 17) L. Carson, Phys . Rev. Lett. 66 (1991) 1406; 'Static properties of 3 He and 3 H in the Skyrme model', Alabama Prepint #UAHEP912 18) K. Gottfried, Quantum mechanics (Benjamin, New York, 1974) p. 264 19) S.E . Koonin and D.C. Meredith, Computational physics (Addison-Wesley, Reading, 1990) 55 20) S. Fiarman and W .E. Meyerhof, Nucl. Phys. A206 (1973) 1 21) H. Theissen, Springer tracts in modern physics vol. 65 (1972) p. 1 ; M.W. Kirson, Nucl . Phys. A257 (1976) 58