Scalar gluonium and the non-skyrme term

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7 July 1988

SCALAR G L U O N I U M A N D THE N O N - S K Y R M E T E R M M. SPALIlqSKI Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-O0681 Warsaw, Poland Received 26 February 1988; revised manuscript received 16 May 1988

The four derivative terms in the Skyrme model are considered as arising from the decoupling of heavy degrees of freedom of QCD. It is shown that scalar gluonium gives an unambiguous contribution to the non-Skyrme term, its size being given in terms of the gluon condensate.

Some interest has in recent times been devoted to interpreting the four derivative terms in the chiral effective lagrangian in terms of heavy degrees o f freed o m of QCD. This is due to the fact that these terms play a key role in stabilising the Skyrme soliton [ 1 ], and thus it would be of interest if one could understand their magnitude and origin in some way. Here this problem is taken up in the spirit of refs. [ 2,3 ]. In what follows the SU (2) × SU (2) model will be discussed, the lagrangian (in the chiral limit) being E = }f~ Tr(0~ U & U*) + ( 1/32e 2) Tr( [0~ UU*, a. UU*] ) + ( y / 8 e 2) [Tr (a,, U0~ U*) ]2,

( 1)

and since

U=exp(2ill/f~),

/ / = ½~aza,

(2)

where z a are the SU (2) generators (Pauli matrices) normalised to satisfy Tr (zaz b) = 2~ aa, one has =Tr(O~HOUH)

+ ( 1 / e 2 f 4 ) Tr[ (0uH0~H) 2 - (0uH) 4 ] + (2y/e2f~) [ T r ( 0 u H & H ) ]2+ ....

(3)

Here f~ = 93 MeV is the pion decay constant, while e 2 and 7 are the parameters which normalise the Skyrme and non-Skyrme terms respectively. In this note these four derivative terms are considered as arising from the decoupling o f heavy particles from a meson effective lagrangian. This approach has in the past been pursued in refs. [2,3 ], reproducing in particular the 120

"model independent" expression for e 2 obtained in ref. [4]:

1/e2=2f~/m~.

(4)

The coefficient 7 which appears in the Skyrme lagrangian ( 1 ), and measures the strength of the nonSkyrme term was in ref. [2 ] linked with the mass of the ~ resonance using the linear sigma model. The main idea of this letter is to point out that regardless of how one decides to treat the problem of the ~, which involves some arbitrariness, a non-Skyrme term arises from decoupling scalar gluonium, and since the relevant vertices are determined unambiguously by the scale anomaly, the coefficient 7 can be expressed in terms of the gluon condensate which parametrises Q C D sum rules [ 5 ]. In particular, this coefficient turns out to be independent of the mass of scalar gluonium, which as has been claimed in ref. [6 ], is unlikely to be determined experimentally in any direct way. Since the coefficient of the nonSkyrme term cannot be too large if the skyrmion is to be stable, the gluonium contribution should not be very big - the sigma contribution estimated in ref. [ 2 ] already requires y to be quite sizeable. As mentioned above, tree level decoupling in an extended chiral lagrangian has been considered quite explicitly in ref. [3 ]. In what follows a somewhat simpler procedure is employed. Let L denote collectively a set of "light" fields, which below will mean pion fields, and H a set of " h e a v y " fields. Starting from a field theory described by a classical action S ( L ,

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H) the effective lagrangian for the light fields can be defined by

exp(i;d4x~fr(L))=fDHexp[iS(L,H)].

(5)

Using condensed notation which suppresses spacetime dependence and any indices on the fields, the action S can be expanded in powers of the heavy fields as follows:

S ( L , H ) = ~ ~.S~(L)H ~,

(6)

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reason for the "universality" of the quartic terms commented on in ref. [ 3 ]. The meson effective lagrangian on which the decoupling is to be performed has to include the lightest heavy fields expected to contribute to the terms sought: here the vector meson 9 and scalar gluonium are considered. The procedure of constructing such effective lagrangians has been described many times [ 7 ], so here just the relevant results are given. The lagrangian describing the vector mesons interacting with the pions [ 8 ] contains the terms ~p~ = Tr (3,/70"H) - ~Tr[ (a,p, -O,p, )2]

where

+ ½m~Tr (p,p')

S~ (L) = OS(L, H)/3HI . = o .

(7)

For the purposes of a tree level calculation it is sufficient to keep terms of up to second order in H:

S=So-½S1S£ I& + ½(H+ S, SF 1)S2(H+ Sy~ S~ ).

(8)

Performing the gaussian path integral one obtains

exp(ifd4x~rf(L)) =exp[iSo(L ) - ½iS, (L )S£ ~(L )S, ( L ) ] Xdet-~/z[S2(L) ] ,

(9)

thus arriving at the tree level formula for the effective action: f d4x

£P~n-(L)=So (L) - ½S~(L)S~' (L)SI ( L ) . (10)

Since it is only the four derivative terms that are needed here, the propagator S2(L) has to be expanded in powers of derivatives. At this point it is worthwhile to stress that the only vertices in the original action which contribute to terms quartic in L in the effective theory are those of the type HL 2. In the context considered here this means that the couplings which are relevant to this calculation are those quadratic in the pion field and linear in the heavy field. From this it follows that only the vector mesons and scalar particles like the ~ or scalar gluonium can contribute. In particular, the axial vector mesons are not relevant here, as seen in the results of ref. [ 3 ]. One also suspects that this is the

+ ½igp~ Tr (p"H~,I1)...,

( 11 )

where P is the rho meson field, and gp~ is the x-9 coupling constant. The terms not explicitly written out in the equation above are not relevant for the calculation at hand, but in general are necessary to ensure correct symmetry behaviour of the action. Scalar gluonium is brought in so as to reproduce the pattern of scale symmetry breaking in QCD [ 6,9 ]. This is done by first ensuring that all terms in ( 11 ) are scale invariant by multiplying each term of scale dimension d by a factor of G / ( G ) to the power - d/4, where G is the field representing gluonium and ( G ) is the gluon condensate. Then two extra terms have to be added: a scale invariant kinetic term for G (which has scale dimension four), and a term which ensures that the anomalous conservation law of the dilatation current of QCD is reproduced at the effective lagrangian level. This results in

P.~o~= (G/ ( G) )1/2 Tr(O.HO'll) - ~ (G/(G)

)

Tr[ (O,,p. -O.p. )2]

+ ½(G/(G) )1/2m2 Tr(p.p") + ½igo~(G/(G) )'/~ Tr(p¢'H~.H) -~aG-3/2(O,G)2 - 1a[ln(G/<

G>)

- 1],

(12)

where a is a constant determined later on. The physical gluonium field h (of dimension 1 ) is introduced by

G=
(13)

as in ref. [ 9 ], and one now repeats the procedure explained there, which consists in fixing a by requiring 121

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canonical n o r m a l i z a t i o n o f the h kinetic term a n d expressing the constant Z by the gluonium mass mh [ defined by the coefficient o f the term q u a d r a t i c in h in the lagrangian ( 1 2 ) after the substitution ( 1 3 ) ]. This gives the trilinear couplings n e e d e d for this calculation as ~ 2 ~ ~ ½igo~~ T r ( p " H ~ , H )

+ ( m h / ( G ) '/2)h T r ( 0 , H S " H ) .

(X, y) "~ - - ( 1 / m ~ ) d ( x - - y )

(15)

for gluonium, and

~"SG~ (x, y) = ( 1 / m ~ ) X[g,~(1/rn~)(gu~2-~,3~)]d(x-y)

(16)

for the P, where m~ a n d m~ are the gluonium a n d vector meson masses squared. Using this one obtains f d4x ~(o) eft"

-

(17)

for the four derivative p c o n t r i b u t i o n to the effective theory. Making use o f the identity T r ( r ~ M ) T r ( r ~ M ) = 2 T r ( M 2)

( 18 )

valid for a traceless h e r m i t i a n matrix M, and integrating by parts one can bring the 9 contribution ( 17 ) to the form 2 : (gp~=/2m4) T r [ (O.H3,H) 2 - (OuH) 4]

(19)

if the pion equations o f m o t i o n are used and only quartic, four derivative terms are kept. One recognises that the P h a m - T r u o n g relation ( 4 ) for e 2 follows from ( 19 ) a n d ( 3 ) if the K S F R relation

g o2 ~ = m p2/ f ~2

(20)

is used to eliminate go~. The gluonium c o n t r i b u t i o n reads

P.~") eft = ½( G) -'[Tr(~.H3~'H) ]2

122

I 2f2 / 7--gparcl(G) ,

(22)

can be used to get a b o u n d on the gluon condensate from the requirement that 7 should not exceed 0.21 to ensure skyrmion stability [2 ]. This would require ( G ) >t0.003 GeV 4, which is a very weak bound. If ( G ) is assumed to have the SVZ [5] value 0.0135 GeV4, 7 is d e t e r m i n e d by (22) as 0.043. Thus gluonium gives a negligible contribution to the non-Skyrme term coefficient. I would like to thank Professor I.J.R. Aitchison for getting me interested in these ideas, and for some very inspiring discussions at the Z a k o p a n e School o f Theoretical Physics.

(go~Jm 2 4o) I d4xTr(r~HO.H)(O2g,,.-3.3.)

×Tr(r~H0.H)

~[~(p ~fr)

which is the advertised non-Skyrme term. Its strength is fixed by the gluon condensate rather than by the gluonium mass, which is due to the fact that the p i o n gluonium vertex is p r o p o r t i o n a l to mh. This is a consequence of the pattern o f dilatation s y m m e t r y breaking in Q C D and is not in any sense m o d e l dependent. The formula for 7 which follows from (21) and (3),

(14)

To make use o f formula (10) one needs scalar and vector meson propagators e x p a n d e d in powers o f derivatives; to the required accuracy they are given by (h)ayl

7 July 1988

(21)

References

[1 ] I. Zahed and G.E. Brown, Phys. Rep. 142 (1986) 3. [2] M. Mashall, T.N. Pham and T.N. Truong, Phys. Rev. D 34 (1986) 3484. [3] I.J.R. Aitchison, C.M. Fraser and P.J. Miron, Phys. Rev. D 33 (1986) 1994; I.J.R. Aitchison, Lectures at the Zakopane School of Theoretical Physics ( 1986 ). [4] T.N. Pham and T.N. Truong, Phys. Rev. D 31 ( 1985 ) 3027. [5 ] M. Shifman, A. Vainstein and V. Sacharov, Nucl. Phys. B 149 (1979) 385. [6] J. Lanik, Phys. Lett. B 144 (1984) 439; J. Ellis and J. Lanik, Phys. Len. B 150 ( 1985 ) 289. [7] S. Pokorski, Gauge field theories (Cambridge U.P., Cambridge, 1987 ). [8] O. Kaymackalan and J. Schechter, Phys. Rev. D 31 (I985) 1109.

[ 9 ] H. Gomm, P. JaJn and J. Schechter, Phys. Rev. D 33 ( 1985 ) 801; A. Solomone, J. Schechter and T. Tudron, Phys. Rev. D 23 (1981) 1143.