Skyrmions and nuclear forces

Nuclear Physics A463 (1987) 169c - 180c North-Holland, Amsterdam 169c

SKYRMIONS AND NUCLEAR FORCES

Sakae SAITO

Department of Physics, Nagoya Universi ty, Nagoya 464, Japan

I t is shown that the Skyrme model underlied by the large-N c l i m i t of the QCD has a qua l i t a t i ve success in describing the s ta t i c properties of nucleon and the NN and the ~N in teract ions. Despite th is , no medium-range at- t rac t ion in the central NN potent ia l and no Yukawa ~N in teract ion term are serious. I w i l l ta lk about how they are.

I . INTRODUCTION

The Skyrme model I has recently been revived as a low-energy e f fec t i ve

theory of the QCD. The large-N c l i m i t of the l a t t e r is known to become a

theory of mesons interact ing weakly, and baryons may emerge as sol i tons of the

mesonic f i e l ds . 2 At low energies, such a lagrangian must be a ch i r a l l y in-

var iant one described by Goldstone bosons(pions). The Skyrme model is con-

sidered to be such a candidate and possesses a so l i t on i c solut ion (Skyrmion).

The nucleon and A isobar are iden t i f i ed as i ts spinning states. The s ta t ic

propert ies of nucleon are reproduced wi th in error up to 30%. 3 The predicted

NN in teract ion 4 has the features character is t ic to the r e a l i s t i c one: ( I ) the

inner repulsive core about the order of the nucleon mass, (2) the one-pion

exchange t a i l , and (3) the r ight order of the p-meson contr ibut ion. The ~N

scatter ing is also in good agreement with experiment for the higher par t ia l 5 waves.

Despite the considerable success, shortcomings of the model have been

reported. The most serious one is the missing of the medium-range a t t rac t ion

in the NN central potent ia l . Is i t possible to get such an a t t rac t ion by

introducing ( ch i r a l l y invar iant ) higher der ivat ive terms into the Skyrme

lagrangian? This is a question raised recently by many authors. Another is

that the A state cannot decay into a nucleon and a pion in the classical

leve l ; namely, there is no Yukawa in teract ion term in the pion-Skyrmion system.

The so l i ton model does not al low such a term, because of the s t a b i l i t y of the

so l i ton .

In th is ta lk , I w i l l review the present status of the study on the

Skyrmion-Skyrmion (SS) in te rac t ion , and discuss fur ther about the d i f f i c u l t y

of no Yukawa coupling term in the pion-Skyrmion in teract ion. The f i r s t part

of my ta lk is based on the work with H. Kanada, T. Kurihara, T. Otofuj i and

0375--9474/87/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

170c S. Saito / Skyrmions and nuclear forces

M. Yasuno, and the second part with T. Otofu j i and M. Yasuno. In section 2,

the Skyrme model is b r i e f l y reviewed fo r the case of two f lavors(SU(2)). The

SS in terac t ion is discussed in section 3. The ~N in terac t ion is discussed in

section 4. Conclusion of my ta lk is given in section 5.

2. SKYRME MODEL

The lagrangian of the Skyrme model is given by

L=f~/16 Tr[3 ut3~u]+I/32e 2 Tr { [ (3 u)ut,(~vu)ut]2}+f2m2/16 ~ ~ Tr[U+Ut-2], ( I )

where the f i r s t term, L 2, is that of the lowest-order nonl inear sigma model,

the second, L 4, the Skyrme term, and the t h i r d , L×S B, the chira l symmetry

breaking term, respect ively. In Eq.(1), U denotes an SU(2) matrix describing

the pion f i e l d : U = t 0 + IT" t with T the Pauli matrices. The Skyrme term

plays a role of a s t a b i l i z e r of the so l i t on i c so lu t ion.

The f i e l d U approaches a constant (we take i t to be un i ty) as r-~, since

the energy must be f i n i t e . Consequently, the points at r=~ can be i den t i f i ed

as the same point , which means that the conf igurat ion space is compactified to

S 3. U is an element of SU(2) which is isomorphic to S 3, and the mapping given

by U is , thus, from S 3 to S 3. Such a continuous mapping is c lass i f i ed by the

winding number( topological charge ): We note that the fo l lowing B ~ is a

conserved current(3 B~=O) i r respect ive of the equation of motion:

B ~ = (I/24~2)~ ~vaB Tr[(3 u)ut(3 u)ut (3~u)ut ] . (2)

The winding number is given by the spat ia l integral of the time-component B O.

The Skyrmion is given by the hedgehog ansatz UO(~) = exp[ i~ .RF(r ) ] , where

the ch i ra l angle F(r) fol lows the boundary condit ion F(r)÷O as r-~o and F(O)=n~.

This mapping has the winding number n. Skyrme conjectured fo r n to be the

baryon number. The topological current B ~ has been shown 6 to be an induced

quark current when the Skyrmion exists as a background. For the case of three

f lavors , the Wess-Zumino term, FWZ, must be included into the Skyrme lagran-

gian, in order to correctlYNdescribe the symmetry of the QCD. Witten showed 7

that rWZ gives a factor (-) c for the 2~ ro tat ion of the Skyrmion. This means

that it must be fermion for the odd number of colors. For the case of two

f lavors , the Wess-Zumino term vanishes i d e n t i c a l l y , so that we cannot apply the

same argument, but the SU(2) Skyrmion can be quantized 8 as fermion by the

resu l t of the homotopy group ~4(SU(2))=Z2.

The Skyrme lagrangian is invar ian t under the vector ro ta t ion U+AUA t , where

A is a constant SU(2) matr ix. To remove the degeneracy, we consider A as a

S. Saito / Skyrmions and nuclear forces 171 c

collective coordinate which is time-dependent but spatial-independent. Using

a canonical quantization method we obtain the Hamiltonian

H = M 0 + J~212~ , (3)

where M 0 is the s ta t i c sol i ton mass, ~ the moment of i ne r t ia and J the spin

operator. Due to the hedgehog ansatz, there appear only l=J states, where I

is the magnitude of the isospin. The l=J=I/2 state is iden t i f ied as the

nucleon, and the 3/2 state as the A isobar. The s ta t i c properties of nucleon

have been calculated by Adkins et a l . , 3 and i t s agreement with data is con-

sidered rather good (roughly 30%). The th i rd column in Table 1 shows the cal-

culated values.

3. SKYRMION-SKYRMION INTERACTION

3.1. Product ansatz

The invest igat ions on the Skyrmion-Skyrmion(SS) interact ion are mainly

based on the product ansatz of the chiral f i e l d

UB=2(x;rI,AI;r2,A 2) = AIUo(x-rl)AIA2Uo(x-r2)A 2 z UIU 2 , (4)

where Uo(~-~i) fo r i=1,2 is the hedgehog solution located at ~ i ' and A i the

co l lec t ive coordinate to describe the rotat ion. The SS potential is defined

by

VSS(~) = - i d3x {L(UIU2) - L(UI) - L(U2)} ' (5)

where ~ is the re la t i ve coordinate between two Skyrmions(~=~l-~2). The

product ansatz has reasonable properties: ( I ) the baryon number is addi t ive,

(2) i t gives a correct behavior at the asymtotic region, i . e . , the one-pion

exchange potent ia l , and (3) at ~=0 the product form is very close to the

hedgehog solution of the baryon number 2.

3.2. Asymptotic form of the SS potential

For the massive pion the chiral angle has the asymptotic form F(r) ÷

~exp[-m r]/m r as r ~ . Compared with a pseudoscalar coupling theory we obtain

= (3/4~)(m~/MN)(g~NN/f ~) , (6)

where g~NN is the pseudoscalar coupling constant. Using the asymptotic form

of F(r) one obtains the SS potential at large r

VSS(~) ÷ (9/4~)(g~NN/2MN)2DiaDja~i3j(e-n~Tr/r), (7)

where Dia = Tr[A~iAtTa]/2. The matrix element of Dia between the projected

states is


< BIDialB'> = A(BB')SiT a ,

where A(NN)=-I/3, A(NA)=I/¢~, A(&&)=-I/15, etc. , and S i (generalized) spin and isospin operators, respectively. into Eq.(7) we get the Sugawara-von Hippel form

< BIB21Vss(~)iB~B ~ >÷(mJl2~)(m~/2MN)2g~BIB ~ g~B2B ~

x ~l.#2[~l.S2Y(m~r) + Sl2Z(m ~r ) ] ,

where Y(x)=exp(x)/x and Z(x)=(l+3/x+3/x2)y(x). The coupling constants sat is fy the strong coupling relations

(8)

and T a are the

Substituting Eq.(8)

(9)

g~NA/g~NN = 3/ /3 and g A&/g~NN = I /5.

3.3. General form of the SS potential The SS potential has the following form at any r

VSS(~) = Vc(r) + TI.T2[SI.S2Vss(r) + SI2VT(r)]

(i0)

+ second rank spin-isospin tensor terms, ( I I )

where V c, Vss and V T denote the central, spin-spin and tensor terms, respec- t i ve ly . The G-parity structure can be seen by comparing with the Skyrmion- anti-Skyrmion (S~) potent ia l , where the G-parity transformation is given by

le t t ing U+U + which is associated with that of the pion f ie ld 7 . ÷ - 7 . . 1 1 Calculated results 4 show that (1) Vss and V T are in good agreement with those

of the Paris potential at large distances, (2) the spin-dependent terms, V ss and V T, have both G-parities, and the p-meson contribution is of the r ight

order (the contribution is simulated by mp ~ 500MeV and f2pNN/4 ml.2~l.8), and

(3) the central potential, Vc, has a repulsive core of about IGeV. These show

that the quality of the agreement with nuclear forces is remarkably high.

However, the repulsive part of V c is long-ranged and gives no attraction in the

medium range. Further, i t has positive G-parity; that is, the potential is

l ike a a-exchange but has wrong sign.

3.4. Extended Skyrme model

I t is known that the Skyrme term is not a unique quartic term in the chiral 9 lagrangian. There exists another term in the chiral l im i t

L~ = (y/ge2){Tr[3 U 3~ut]} 2 . (12)

Donoghue et al. 9 showed that the low energy data of the D-wave ~ scattering

S. Saito / Skyrmions and nuclear forces 173c

gives ¥=0.12-0.20. On the other hand, Pham et. al 9 obtained #=0.28-0.34

by f i t t i n g the S-wave ~ scatter ing. Since Y>O, L~ gives a negative contr ibu-

t ion to the so l i ton energy, so that there is an expectation of gett ing an

a t t rac t i ve part in the central potent ia l . lO However, the term works as a

des tab i l i ze r . To prevent the system from col lapsing, an w-meson coupling term

is usual ly introduced. This is quite natural : In the SU(3) level the

Wess-Zumino term plays an important role in the Skyrme model. I f we require

a local gauge invariance, then the vector mesons such as the P, A I , w mesons

come into play as the gauge bosons. Now, l e t us reduce the SU(3) gauge group

into the SU(2), then the Wess-Zumino term survives as being the w-meson

coupling term, where the coupling constant gm is given by gNc/2 with g the

gauge coupling constant determined usually by the p-meson decay width. I I

At the large mm-limit, the m-meson coupling term is equivalent to L 6

introduced by Jackson et a l . 12

2 2 B ~ (13) L 6 = -(gJ2m ) B .

Here, B is the baryon current in Eq.(4)

Thus, i t is natural to extend the Skyrme model to the fo l lowing:

L = L 2 + L 4 + L~ + L 6 + LXS B . (14)

13 I show a recent study with the above lagrangian by Otofuj i et a l . The

parameters are chosen to be #=0.I , f~=125MeV, e=12 and g = I I .2 , which are

within the experimental errors of the low energy data of the ~ scatter ing.

¥=0.I is almost the c r i t i c a l value above which no solut ion of the classical

Table I . Predict ion of s ta t ic properties of nucleon

Quantit ies Present Model Skyrme Model Exp.

f 125MeV 108MeV 186MeV TF

2 I /2 <r >I=0 O.74fm O.68fm O.72fm

2 I /2 <r > I : I 1.04fm 1.04fm O.88fm

2 I /2 0.94fm O.96fm 0.81 fm <r >M,I=O Up 2.12 1.97 2.79

-1.33 -1.24 -1.91 n

g~NN 12.9 I I . 8 13.5

gA O. 80 O. 63 1.23

174c ,.7. Saito / Skyrmions and nuclear forces

1.2

.8

.4

0

~ otal VC

',_ vi

vi

FIGURE 1 The c a l c u l a t e d c e n t r a l po- t e n t i a l in the Sky rm ion-

Sky rm ion i n t e r a c t i o n .

equation of motion exists. We obtain a

very good agreement in the stat ic properties

of nucleon( See Table l ) . We found that

the spin-dependent part of the calculated

SS potential is almost in agreement with

that of the previous calculations, in spite

of the small Skyrme term. For the central

potential, i t di f fers from that of the pure

Skyrme model. Fig.l shows the calculated +

central potential Vc, where V 6 and V 6 denote

the contributions from L 6 with the positive

and negative G-parity components, re-

spectively. We see that the negative

component, V 6, is strong and short range.

This contribution is written as the folding

of the baryon densities of two Skyrmions

V6(~ ) = 292/m 2 [ d3x BO(~-~ I)BO(~-~2), CJ CO •

(15) where Bo(~-~i) denotes the baryon density

of the i - t h Skyrmion located at r i . In

F i g . l , we note that the cont r ibut ion from i L 4, V~, is a t t r ac t i ve , but i t s magnitude is not s u f f i c i e n t to overcome the

repulsion from L 6. The net resu l t of V c is , thus, en t i r e l y repulsive.

3.5. I n s t a b i l i t y due to the L~ term

Can we increase the coupling constant ¥ to get the medium-range at t ract ion?

We note that L~ gives a cont r ibut ion proport ional to F '4 with a negative coef-

f i c i e n t to the mass, whi le L 6 and the other terms give contr ibut ions of F '2,

so that we have only a l oca l l y stable so lut ion even i f there exists a so lu t ion

of the Euler-Lagrange equation. Such a local s t a b i l i t y , however, disappears

when the baryon number is increased. 13 This can be seen with a l i t t l e calcu-

l a t i on ; the c r i t i c a l coupling constant Yc decreases l i ke I /n 2 when n, the

baryon number, is increased: for the same values of the parameters in the

above, we found that ¥c is 0 . I , 0.02 and 0.005 fo r n=l, 2 and 3, respect ively.

Consequently, the product ansatz fo r the chi ra l f i e l d U of the two baryons is

not j u s t i f i e d : the system f a l l s in to a negative energy. The inc lus ion of the

L~ term seems not accepted as an e f fec t i ve lagrangian of the Skyrmion physics,

i f the lagrangian should also describe multi-Skyrmion systems.

Several attempts 14 to derive the e f fec t ive lagrangian from the QCD or i t s

S. Saito I Skyrmions and nuclear forces 175c

substi tutes show that the eventual lagrangian involves the L~ term, supporting

the phenomenological lagrangian for the ~ scatter ing. However, there ex is t

a large number of higher der ivat ive terms, since such an e f fec t ive lagrangian

is given by a fermi on determinant. Ripka et a l . showed that a l l the terms of

the determinant gives a stable sol i ton. Therefore, higher der ivat ive terms

play an important role in i t s s t a b i l i t y . In viewing th is , l e t us consider the

fol lowing in place of L~:

L ° = ( I / 2 ) ~ o ~o - ( I /2) m 2 o 02 - X~4 + go~ Tr[~ U ~VUt]. (16)

Such a scalar (ch i ra l ly singlet) meson term without a self-coupling term was

f i r s t introduced by the Paris group, lO and i ts large-mass l im i t brings about

y/e=4g~/m~. We found that such a model does not change the L~ with low-energy v v

f i t of the ~ scattering and possesses also stable multiple-Skyrmions, i f the

strength of the self-coupling term is suitably chosen. Such a feature is very

desirable for the Skyrmion physics. However, we could obtain no medium-range

attraction in the central NN potential, even i f the strength of the scalar

coupling y is increased.

3.6. Quantum correction?

We have shown that it is very difficult to get the medium-range attraction

in V c by an extended model without spoiling the stability of multi-Skyrmians.

Is the product ansatz too bad? I t has been shown that the deformation effect

of the Skyrmions gives an additional attraction but is not large. 15

Non-adiabatic effects such as the time-dependence of the relative or the

col lect ive(rotat ional) coordinates yield no serious attractive terms. 16

Further, a la t t ice calculation without the product ansatz also shows no such attraction. 17

These suggest that classical calculations on the Skyrme model may give no

correct answer for such a problem. Can any quantum correctiop give the

answer? Such a calculation has been recently carried out by Zahed et al.,18

where a one-loop correction of the f luctuating f ie ld was included. This is

based on Weinberg's idea on phenomenological lagrangians: since S-matrix

elements are not unitary in the tree graph approximation, one has to include

graphs with loops. 19 The Skyrme model, essentially the nonlinear u-model,

is not renormalizable. Therefore, one needs i n f i n i t e l y many counter terms.

However, the n loop graphs are suppressed by (p2)n in comparison to the tree

graphs, so that the counter terms could be included as higher derivative terms

of the lagrangian, since the theory must be ch i ra l ly invariant. Zahed et al.

obtained in this way that the soft-pion correction of one-loop diagrams pushes 20 down the Skyrmion mass about 20%. Encouraged by this, Jackson et al.


evaluated the NN in terac t ion and showed that the one-loop corrected lagrant ian

can recover the medium-range a t t rac t ion . This indicates that quantum correc-

t ions are important. However, there seems d i f f i c u l t to obtain a de f i n i t e

answer because of ca lcu la t iona l ambiguity. We need a fu r ther study on th is

problem.

4. PIONIC FLUCTUATION AROUND SKYRMION

4.1. Chiral f l uc tua t i on

The ch i ra l f l uc tua t ion around the Skyrmion describes the pion-Skyrmion +

system, Following Schnitzer 21 we wr i te U = U U(t)U with U = exp[i~.@/f ] ,

where U(t) is given by AUoAt with the hedghog ansatz U O, and ~ the p icn ic

f i e l d . Subst i tu t ing U in to the Skyrme lagrangian we obtain

L(U U(t)U ) = L B + L M + L I , (17)

where LB=L(U(t)) is the baryonic part , and LM=L(U U ~) the mesonic part. L I is

the in terac t ion part described by the sum of the sof t and hard pion terms.

The sof t pion term is wr i t ten as

LS ( l / f ) ( 3 j ) . ~ O (1/ f~)(3 ~ ) x T . ~ 2 +2 = + + (mJ4) t T r [ l - Uo], (18)

where ~u and ~ are the ax ia l -vec tor and vector currents of the Skyrmion,

respect ively. Here, we have used the equation of motion for the c lassical

so l i ton solut ion

3iAi = m2F2/4 T r [ i ~ U( t ) ] , (19)

and dropped surface terms. As i t should be, there exists no spat ia l part of

the Yukawa coupling term in Eq.(18): the vanishing of the l inear terms in

is equivalent to the s t a b i l i t y of the classical so lut ion. Nonvanishing of the

t ime-der ivat ive term is because the Skyrmion is defined as the ro ta t ing s ta t i c 22 solution. As stressed by Uehara, nonexistence of the Yukawa coupling term

is crucial in describing the P wave interaction. On the other hand, the

vector coupling and the ~ terms are cor rec t ly reproduced in Eq.(18).

The Siegen and the SLAC groups analyzed 5 the ~N scatter ing on the basis

of an adiabat ic approximation to neglect the time dependence of the co l lec t i ve

coordinate describing the ro ta t ion. A very in te res t ing resu l t is the l inear

re la t ion of the S-matrix elements of the ~N scat ter ing. A comparison with

experiment shows a subs tan t ia l l y qua l i t a t i ve agreement fo r L>3. This is one

of the remarkable success of the Skyrme model. However, disappoint ing resul ts

are obtained fo r the lower par t ia l waves, especia l ly for the P wave: there

ex is ts no P33 resonance at the A state, since the A pole does not appear.

4.2. Quantization of the f l uc tua t i ng f i e l d

S. Saito / Skyrrnions and nuclear forces 177c

The above suggests that the adiabatic treatment may be inval id in the lower

part ial waves. This is apparently so in the S wave scattering, since we must manipulate the time-component of the vector coupling term which was neglected in the adiabatic calculations. Now, is i t possible to get nonzero decay width of the A state without using adiabatic approximation? We have attempted a col lect ive coordinate quantization of the f luctuat ing f ie ld . 23 Let us write

the integrated lagrangian

Id3x L =-Mo + 2X a2 +~ I d3x ~aMa ~ + ½1 d3x {~aKab~b - ~a~ab~b }, (20)

where M 0 is the s tat ic mass, and X the moment of iner t ia . In the lowest order of the I/N c expansion the coupling term between the f luctuat ing f ie ld and the

col lect ive coordinate is given by the time-component of the axial current

Ma a uSMaiO i = (1/ f )A~ , (21)

where we have changed the coordinates a into three angle variables 0 i on S 3, 's are not independent (Z a 2 = ~). = since a v Then, we have ~2~ OiBijO j . - .

Now, le t us denote the momenta conjugate to 0 i and @a by ~i and ~a " We readily note that they are not independent

~i I d3x ~a(K-1)ab~bi : O. (22)

After defining the new momenta ~ perpendicular to the col lect ive motions, we a note that the following ~i are conjugate to 0i:

~i = ~i + (4X)-I ~j I d3x 3~ i - [ (B- l ) jk~ak] ~a " (23)

The Hamiltonian of this system is given by

H = M 0 + ( 8x ) - l ~ i (B - l ) i j ~j + ½ i d3x {~a(K-l)ab ~b + 9a ~ab @b } " (24)

4.4. Yukawa coupling term

Substituting Eq.(23) into Eq.(24) we obtain coupling terms between the col lect ive modes and the pionic f ie lds . We estimate the coupling term by

expanding the pion f ie ld @a in the plane waves. The matrix element gives

v BB (0) m u(k) v~ S.k T v B (25) jk : i (4~) I /2 (fBB'~ / ~) j ' '

where B(B') is the state of baryon, VB(VB, ) i ts spinor, and u(k) the form factor normalized to unit at k=O. The effect ive coupling constant f(O! is BB proportional to the rotat ional energy. Therefore, we have

f(O) : (0) and f f (O) I f ( 0 ) ) 2 NN~ : "AA~ ' NAT" NN~ = 9 (3/12) 2 . (26)


For the parameters of Adkins et a l . we have obtained ( f )2=0.003 and ~'NA~'

=0.124. These are compared with the renormalized values of the cloudy bag

2 =(0) is very small, but th is is model: 24 (fNN~)2=O.061 and (fNA~) = 0.176. "NN~

desirable to the so l i ton picture of nucleon because the self-energy correct ion

(0) is close to so that we may have a is small. On the other hand, fNA~ fNA~'

decay width comparable to experiment. However, our ca lcu la t ion shows that the

form fac tor u(k) decreases rapid ly : 0.33 at k=225MeV(the A resonance).

Although the resu l t is not yet conclusive, the decay width seems to be too

small.

The above ca lcu la t ion indicates that the tree diagrams are not s u f f i c i e n t

to reproduce the decay width of A. I f we include the higher order terms in the

f l uc tua t i ng f i e l ds , the decay of the A state is expressed by the diagrams with

loops. Such loop diagrams were considered to be important in the NN

in te rac t ion . I t is in te res t ing to invest igate what an e f fec t the loop correc-

t ions bring about in the pion-Skyrmion in te rac t ion .

5. CONCLUSION

In conclusion, the Skyrme model has a qua l i t a t i ve success in describing the

s ta t i c propert ies of nucleon, the NN in terac t ion and the ~N in te rac t ion . The

lagrangian is that of pions, and the terms are consistent with the ~

scat ter ing. This is a l i t t l e surpr is ing, since the model has no matter f i e l ds .

The baryons are i den t i f i ed as the topological so l i tons of the mesonic(boson)

f i e l d . Thus, the Skyrme model real izes a scenario of the large-N c l i m i t of the

QCD. However, we have shown that the Skyrme model in the classical level has

serious shortcomings: the missing of the medium range a t t rac t ion in the NN

po ten t ia l , and no decay width of the A state. There ex is t no convincing ex-

planations of them. Quantum correct ions with loops of the f l uc tua t i ng f i e lds

may play an important role in resolving the d i f f i c u l t y .

ACKNOWLEDGEMENT

I would l i ke to thank H. Kanada, T. Kurihara, T. Otofuj i and M. Yasuno fo r

f r u i t f u l discussions and col laborat ions.

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Skyrmions and nuclear forces