Scalar mesons and the phenomenology of the trace of the energy momentum tensor

280

Nuclear Physics B (Proc. Suppl.) 23B (1991) 280-282 North-Holland

SCALAR MESONS AND THE PHENOMENOLOGY OF THE TRACE OF THE ENERGY MOMENTUM TENSOR

S t e p h a n NARISON LPM*) Universit4 Montpellier II, Sciences et Techniques du Languedoc Place Eugene Bataillon, 34095 - MONTPELLIER Cedex 5 (FRANCE)

Properties of the scalar gluonia and quarkonia derived from QCD spectral sum rules

(QSSR) and

momentum can

some low-energy

tensor are

be orthogonal

angle

theorems related

to the trace of the energy

summarized. The large width ~(0.9) and the narrow fo(0.98) eigenstates of

the qq-gluonium

mixing matrix with a mixing

of about -45". The G(I.6) seen at GAMS appears to be an almost pure gluo-

nium

state. The

I =

1 ao(0.98)

is still compatible with a qq assignement. We

also discuss the effects of these mesons in the decays of the light Higgs boson. i.

Introduction Hadron

ble

physics continues to show a remarka-

degree of

blems

to our

from

the naive

properties

complexity and poses severe pro-

we have obtained from the unsubtracted subtracted

(USR) and

(SSR) sum rules the values in MeV :

basic hadronic schemes originated quark model. In particular,

of the

scalar jPc = O++

the

f

mesons are

~- 546 - 677 769 - 931

ST 2.ST

not quite understood and lead to many open interpretations

(four-quark, KK molecule...). In this

talk,

we

discuss

these

mesons and

the

f ~ 478 - 533 G 544 - 656

gluonium assignement of

the mixing between the qq and

gluonium states.

for

two values

standard order

2.

need

a second

we have estimated the decay constants of the

value

of the

scalar

to

for the

Properties of the scalar gluonia*

gluonia q(0.9)

analysis

the trace

pure

and G(I.6) from the QSSR

of the two-point correlator associated of the

energy-momentum tensor in

at SSR

o 0~ •)

-4 ~(%) Gz G = 4~

gluon condensate we should

(ST =

note that in

two sum rules to be fulfilled we state. This

is due

to the huge

subtraction constant which pushes

higher energy the region of stability of the and then

emphasizes the

important role of

The couplings of the gluonia to the pairs of Goldstone

hosons have

fG M2~ , (I)

Unit4 de Recherche Associ4e au CNRS n ° URA 040768

0920 5632/91/$03.50 © 1991 - Elsevier Science Publishers B.V.

(2)

the G(I.6) in this sum rule.

Yang-Mills theory. By defining these decay

constants as :

of the

value). Here,

ST 2.ST

All rights reserved.

been estimated

from the

S. Narison / The trace

low-energy

behaviour

of the

281

energy momentum tensor

and then :

of the vertex :

F(G ~ n'~) [MeV] ~52 ± 9 44 + 8 where

Gnn

gG~n z sin 8 _~, gG.., =~ the low-energy sum rule (LESR)

r

- 0.26 , (i0)

: where we have used in (8) the experimental bound

--

=

(4)

1

g=n~ 6 9.75 GeV in

M2

4 ~, G

F

one obtains

f --

(9)

F

its derivative is known to be 1 at q2 = O

and in the chiral limit m 2 = O. Then

ST 2.ST

deduced from

q' ~ QE~ data and

(I0) the ~-~' mixing angle 8

~- -18". One

I

should not consider (9) as an absolute bound due Neglecting the G coupling to 44, the LESR gives:

to

the approximations

particular

we

have

used for deriving it. In neglected

the higher mass

state contributions. However, our results in (9) F(cr ~ 4+4-+24 ")[MeV] _~ (798 _+ 429) (410 _+ 220)

ST , (5) 2.ST

which is comparable to the I = O s-wave 44 phase shifts

result.

couples

(4)

also

universally to

and

(I0) compare

and

suggests that the G(1.6) state seen by this

favourably with the GAMS data

group is an almost pure gluonium state.

indicates that the

44 and

Let

us finally discuss the couplings of the

gluonia

to 7~ and heavy quarkonia. These can be

deduced

from

KK pairs in the

SU(3) r symmetric limit. the

low-energy

behaviour of the

gg77 box diagram and from the uses of dispersive The singlet

couplings of ~

pairs

the gluonia

can be

to the U(1) A

deduced from the low-

techniques and meson dominance. We obtain :

energy behaviour of the vertex : B(~ ~ 709 B(V ~ g4) ~ (6 - 16) 10 -4

B(~, ~ 7G) B(G ~ ~l'~l + 44") ~- (13 - 25) 10 -4

f d4xld'x,e

i(qlxl+q'xz )
)8~10 >

, (6) r(~ -, 77) -~ (0.03 - 0.08) key

where

Q(x) ~ _~s GG

84

is

the

[~(G ~ 7Y) ~- (0.3 - 0.6) keV

topological charge

(ii)

density. By relating this vertex through a Wardtype identity to the U(1)

two-point correlator A

F~M(q2) = i f d4x

el qX <0 1~

These the

Q(x)(Q(O))+I O> (7)

branching these ting

with

F~M(O) ~ -(180 MeV) 4 ,

one can

predictions are

not accurate but suggest

order of magnitude of these decay rates and ratios. Experimental

quantities would

measurements of

certainly help for tes-

the nature of the ~ and G. However, we are

deduce the aware

LESR :

for the

difficult identification

~'~ and 4~" final states in ~-decays. 1 Z 4 o',G

g*n,"1 ~

fl -~ 1.15 GeV 2

(8)

of the

S. Narison / The trace of the energy momentum tensor

282

3.

qq and gluonium s c e n a r i o 2 f o r t h e s c a l a r

is

mesons below 1 GeV

the

The

ss , the

the

ao(0.98)

having t h e p a r a m e t e r s g i v e n i n

PDG booklet satisfies well the QCD test for

a meson associated to the divergence of the isovector

SU(2)

vector current 3. Moreover, its TT F

width has also been estimated within the hadronic tadpole

framework by using p-~ mixing data. The

obtained value :

about 1.45

f (1.3) is a non-trivial mixing between the o

excitation.

in

reasonable

agreement

(12)

has shown

that a qq-

main features of the scalar mesons phenomenology Improvments

of

progresses

these

of both

we would

of the

predictions

needs

some

the theory and experiments. expect that

one of

the best

quark content of the scalar mesons

can be performed in a %-factory experiment.

with the data. In 4.

this

Our analysis

gluonium minimal scheme can still accomodate the

test

is

~(0.9) and eventually its first radial

Indeed,

['(a° -. ~'7) ~- 0.3 keV

GeV. Then one could imagine that

Effects of the scalar mesons in the light

scenario the K*(I.4) in the K~ data is the Highs decay"

SU(3)~ partner of the a° and satisfies different These

scalar

mesons

can

also

affect the

QCD tests as well. decays The

situation of the I = O isoscalar mesons

K~,

of the

KK and

Higgs is

much more

subtle. We

the fo(0.975)

7T. In

both couple

boson into pairs of

fact, as the scalar and the to 8 ~,

the mesons appear as

develop here a scheme intermediate

where

light Higgs

states in

the Higgs decay. There-

and the v(0.9) is a mixing fore

a scenario

similar to

the case of vector

between the scalar gluonium discussed previously and

the qq

The

hypothetical hadronic and 77 widths of this

isoscalar partner

of the ao(0.98).

meson

dominance for

large

enhancement of

the unmixed qq state can be deduced from the ones of the

a ° from the uses of the good realization of

$U(2)

Higgs mass

the photon appears here. A these width will occur if

is in one of these meson poles.

In this case, the Higgs leptonic branching ratio will

be strongly suppressed making their detec-

symmetry. The TT width of the fo is also tion quite difficult. However, the present limit

known from the data. With these different input, one

can derive the value of the qq-gluonium mi-

xing angle to be about :

given

by the

LEP experiments on the Higgs mass

might

not be

affected by

comes

from the Z ~ ~+~-H process in a inclusive

these effects

as it

way. 8qg

~-- 45"

(13) References

The

narrow hadronic

width of the a

is now uno derstood from a destructive interference between

1

This

Narison the

large

width

gluonium

and

discussion is

based on the paper by S.

and G. Veneziano, Int. Mod. Phys. Lett.

the unmixed qq 4A (1989) 2751.

state. We also predict the ratio of the hadronic 2 couplings : !

gf0,+,which

This

part is based on the work by : A Bramon

and S. Narison, Mod. Phys. Lett. 4A (1989) 1113. l]g~o-+-- ~ 1.5 - 2

3 (14)

is still consistent with the PDG data. In

this qq scheme the expected mass of the ss state

For

Notes

a review, see e.g. : S. Narison, Lecture in

Physics

Vol.

26

(1990).

World

Scientific. 4 This part is based on the papers : $. Narison~ Phys. Lett. 228B (1989) 513 ; ibid. 236B (1990) 474.