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SKYRMION PHYSICS

Tran N. TRUONG Centre de Physique Theorique de l'Ecole Polytechnique Plateau de Palaiseau - 91128 Palaiseau - Cedex - France

AB STRACT

Sum rules are given for evaluating the strength of the two derivative quartic terms in the pion chiral Lagrangian. The Skyrme term which stab ilises the soliton and the non-Skyrme term which destabilises the soliton are shown , respec­ tively, inversely proportional to m2 and m2 . The general approaches to Skyrmion physics are discussed. p (J 338

( ) In 1960 , in a very original paper, Skyrme I proposed a unified field theory of mesons and baryons by considering baryons as soliton solutions of a chiral meson field theory. His main motivation is to give a proper treatment of an extended obj ect. Skyrme ' s idea is recently revised by a numb er of people <2 ,3) The real impetus in this direction is due to Witten who showed in the limit of the large number of color N , Quantum Chromodynamics becomes an effective theory of c (4) mesons , and baryons could emerge as soliton solutions of the effective theory . If this idea was correct, the static properties of the baryons could be determined by the low energy effective meson theory wh ose strength is not arbitrary but is determined by the low energy phenomenology .

The purpose of this communication which is done in the collaboration with T.N. Pham is two fold. We show how to determine the strength of the low ener­ gy effective field theory and then clarify the prevailing confusing question of how to stab ilize the baryon solitons (Skyrmions) : we first show that the Skyrme term which is used in the litterature to stabilise the soliton is related to the field. (In the limit its mass tends to infinity, the strength of the Skyrme p term vanishes) . In addition to this term there exists another term which destabi­ lises the soliton; its strength is inversely proportional to the effective scalar meson a mass. Low energy pion pion scattering phenomenology shows that the con­ tributions of these two quartic terms are not sufficient to stabilise the solitons . It is then argued that , on the general ground , the contribution (and possibly w A ) which gives rise to a six power of the derivatives of the pion field in the 1 Lagrangian mus t be taken into account.

We write down the effective Lagrangian for meson meson interaction as

£ £ £ = + Q 0 with £ is the standard minimal term in the non linear a-model 0 2 11 + 2 2 £ f Tr ( a M a M ) I m f (2 - Tr M) ( I ) 0 8 'Jr \1 - 4 'Jr 'Jr where M exp ih �. 1i/f and f 1 35 Mev and 'Jr 'Jr

(2)

The first term on the RHS is identified as the Skyrme term the second is called non-Skyrme term. 339

2 To calculate e and we evaluate the contribution of £ to the y Q elastic pion pion scattering and identify its contribution with those evaluated from the forward dispersion relation to obtain two separate sum rules. The method (5) is discussed in details in our recent publication . We give the main results : o Tr+Tf ds s(s 4m ) o (s) �TI tot (3) I 2 3 2e (s-2m ) TI

2 0TI 0 ds /sr---::.(s-4m ) oTI (s) tot TI (4) .J_2 e

Eqs . (3) and (4) clearly show that high energy contribution are strongly suppres­ 2 sed and that e 0 and O. In the sum rule (3) only isospin I= ! and I=2 > y > contribute and sum rule (4) only isospin I=O and I=2 contribute. It is an 1 experimental fact that at low energy the I=2 cross section is negligible and therefore can be neglected in these sum rules. Instead of using the experimental data in these sum rules to get a number , it is more transparent to parametrise experimental data such that the Weinberg low energy expansion is satisfied . After <5) some manipu. 1 ations. , we h ave : 2 f I TI 2e m2 p (5) 2 f :r..2 I TI e 32 m 0 2 2 2 2 where m "" 30 m and m = 25 m which fits the low energy and more TI s p phase shiftsp (up to about0 900 Mev)TI; is the usual vector meson and is an p 0 effective isoscalar scalar meson. Hence I 30 and "" 0.35. 2""e y

It can be shown, on general grounds, that the soliton energy is nega­ (5) tive for Numerical integration of the differential equation derived y > � from the chiral Lagrangian £ with the usual Skyrme ansatz and boundary condition ( restricts 0.10 6 • 7). Our value of given by Eq. (5) violates this inequa­ y � y lity, hence no soliton solution exists with £Q given by Eq. (2) .

A large value of as given by Eq. (5) is unavoidable. It is related y to the discrepancies between current algebra predictions and experimental data on (S) S wave I=O pion pion scattering length, TITieV and 3TI rate . K + n + 340

Let us try to see whether it is possible to change this situation. One can argue that the effective Lagrangian given above is not sufficiently general. It essentially takes into account the indirect effect of the and field p a (in other words a chiral Lagrangian obtained after eliminating the heavy and p fields) . But this is not sufficient . To be consistent we have to take into a account of the effect of the field whose mass is almost degenerate with w p

The addition of the - contribution in the general effective Lagrangian induces w a term which is 6th power of the derivative of the pion fields (6-point function) . Knowing the experimental width of 3rr , the strength of this term can be calcu­ w + lated.

Numerical integration of the Euler equations obtained from Eqs. and (I) (2) together with the contribution yield a stable soliton solution with (6) w 0.27 , which is roughly consistent with Eq. (5) , allowing some uncertainties y < in the determination of m The baryon mass is however a factor of 2 too large. It remains to be seen whethera the A contribution can change the situation. 1

(9) Our approach differs from that of Adkins and Nappi who took into account of the contribution independently of the quartic terms . They have w carefully pointed out, to be consistent , the contribution must be taken into p account which, we show here, is the Skyrme term.

In conclusion, I would like to point out two possible general approa­ ches to the Skyrmion problem. The first method consists in extending the non ( O) linear model to include , A contributions I ; the non-Skyrme term has a p, w 1 to be put in by hand . The second method is to write down the linear model with a , , A contributions . In either approach one should verify the contributions p w 1 of other heavy fields to the soliton calculation are small. (We have explicitly calculated the f contribution to the strength of the non Skyrme term Eq. ( 4) and found to be smal0 l, of the order of 10% of the contribution) . a

REFERENCES

( I ) T.H.R. Skyrme Proc . Roy . Soc. London, Der. A260 , 127 (1961). (2) A.P. Balachandran et al , Phys. Rev. Lett. 49, 1124 (1982) . (3) E. Witten, Nucl. Phys. B223, 422, 433 (1983) . (4) For review see for example talk given by J.P. Blaizot at this conference. (5) T.N. Pham and Tran N. Truong , Phys. Rev. D31, 3027 (1985) . The formulae for in this publication are too large by a factor of 2. y (6) M. Mashaal , T.N. Pham and Tran N. Truong (to be published) . (7) R. Vinh Mau et al, I.P.N. Orsay preprint (1985) . 341

(8) For a review see Tran N. Truong , Acta Polonica B_!2 , 633 (1984) . (9) G.S. Adkins and C.R. Nappi, Phys . Lett . .!2Z_B , 251 (1984) . ( I O) R. Vinh Mau et al (Private Communication) .