Volume 208 number l PHYSICS LETTERS B 7 July 1988

SCALAR G L U O N I U M AND 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 ef- fective lagrangian in terms of heavy degrees o f free- dom 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 under- stand 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 /8e 2) [Tr (a,, U0~ U*) ]2, ( 1 )

and since

U = e x p ( 2 i l l / 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 /e2 f 4 ) Tr[ (0uH0~H) 2 - (0uH) 4 ]

+ (2y /e2 f~) [ 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 effec- tive lagrangian. This approach has in the past been pursued in refs. [2,3 ], reproducing in particular the

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

1 / e 2 = 2 f ~ / m ~ . (4)

The coefficient 7 which appears in the Skyrme la- grangian ( 1 ), and measures the strength of the non- Skyrme 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 unambig- uously by the scale anomaly, the coefficient 7 can be expressed in terms of the gluon condensate which parametrises QCD 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 non- Skyrme 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 collec- tively a set of "light" fields, which below will mean pion fields, and H a set of "heavy" 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

e x p ( i ; d 4 x ~ f r ( L ) ) = f D H e x p [ i S ( L , H ) ] . (5)

Using condensed notation which suppresses space- time 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)

where

S~ (L) = OS(L, H)/3HI . = o . ( 7 )

For the purposes of a tree level calculation it is suffi- cient 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

e x p ( i f d 4 x ~ r f ( 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 ex- panded 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 ax- ial vector mesons are not relevant here, as seen in the results of ref. [ 3 ]. One also suspects that this is the

reason for the "universality" of the quartic terms commented on in ref. [ 3 ].

The meson effective lagrangian on which the de- coupling 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]

+ ½m ~ Tr (p,p') + ½ igp~ 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 cal- culation at hand, but in general are necessary to en- sure 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 effec- tive 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

- 1 a [ l n ( G / < G > ) - 1 ] , (12)

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

G=<G)+Zh, (13)

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

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canonical normal iza t ion o f the h kinetic term and ex- pressing the constant Z by the g luonium mass mh [ def ined by the coefficient of the term quadra t ic in h in the lagrangian (12) after the subst i tut ion (13) ]. This gives the t r i l inear couplings needed for this cal- culat ion as

~ 2 ~ ~ ½igo~ ~ T r ( p " H ~ , H )

+ ( m h / ( G ) '/2)h T r ( 0 , H S " H ) . (14)

To make use of formula (10) one needs scalar and vector meson propagators expanded in powers of de- rivatives; to the required accuracy they are given by

( h ) a y l (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~ and m~ are the g luonium and vector meson masses squared. Using this one obta ins

f d4x ~(o) eft"

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

× T r ( r ~ H 0 . H ) (17)

for the four der ivat ive p cont r ibut ion to the effective theory. Making use of the ident i ty

T r ( r ~ M ) T r ( r ~ M ) = 2 T r ( M 2) ( 18 )

val id for a traceless hermi t ian matr ix M, and inte- grating by parts one can bring the 9 contr ibut ion ( 17 ) to the form

~[~(p ) 2 ~fr : (gp~=/2m4) Tr [ (O.H3,H) 2 - (OuH) 4] (19)

if the pion equat ions of mot ion are used and only quartic, four der ivat ive terms are kept. One recog- nises that the P h a m - T r u o n g relat ion (4 ) for e 2 fol- lows from ( 19 ) and (3) i f the K S F R relat ion

2 2 2 go~=mp/ f~ (20)

is used to e l iminate go~. The gluonium cont r ibut ion reads

P.~") = ½ ( G) -'[Tr(~.H3~'H) ]2 (21) eft

which is the advert ised 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 ropor t iona l to mh. This is a con- sequence of the pat tern of d i la ta t ion symmetry breaking in QCD and is not in any sense model de- pendent . The formula for 7 which follows from (21) and (3) ,

I 2 f 2 / 7--gparcl(G) , (22)

can be used to get a bound on the gluon condensate from the requirement that 7 should not exceed 0.21 to ensure skyrmion stabil i ty [2 ]. This would require ( G ) >t0.003 GeV 4, which is a very weak bound. I f ( G ) is assumed to have the SVZ [5] value 0.0135 GeV4, 7 is de te rmined by (22) as 0.043. Thus gluon- ium 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 inspir ing discussions at the Zakopane School of Theoret ical Physics.

References

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