QQ G hybrid mesons in the MIT bag model

Nuclear Physics B224 (1983) 241-264 ~3 North-Holland Pubhshlng C o m p a n y

QQG HYBRID MESONS IN THE MIT BAG MODEL T BARNES and F E CLOSE

Rutherford Appleton Laboratory, Chdton, Dtdcot, OXON OXl l OQX, England F de V I R O N

Dept de Phystque Theortque, Umverstty Cathohque de Louvam, B-1348 Louvam-La-Neuve, Belgtum Recewed 19 October 1982 (Revised 3 February 1983)

We suggest that hybrid (qV:tg) mesons could exast wxth rather light masses The spectrum of the ground state nonets, jPc = (0, 1,2) +; 1 is calculated m the M I T bag model including O(a~) energy shafts We dxscuss hadromc transltaons among these states, consider their possible productaon at L E A R and SPEAR and indicate some interesting decay signatures

I. Introduction In QCD coloured gluons necessarily exist m addition to the well-established coloured quarks. If quarks are permanently confined in colour singlet clusters, then it is to be expected that gluons will also be confined, and hence cannot be isolated and studied individually. Instead we may hope to refer their presence indirectly by observing colour singlet systems containing gluons (gluonia or glueballs) in a manner analogous to that whereby quarks were originally inferred from the observation of the famlhar hadrons. "['he importance of establishing QCD, confirming that gluons exist and of studying their interactions has generated much recent theoretical and experimental interest in glueballs. Various models (potentials [1], bags [2-5], QCD on a lattice [6]) differ m fine details but all agree that the hghtest glueballs will have jPC= 0++,0 +,2++ and masses between 1 and 2 GeV. These are in the very region where qq, or even qqqq [7], mesons occur with these quantum numbers and thus it is possible that glueballs will not exist as relatively pure states but will mix signiflcantl 3 with quark systems. This may make it hard to establish the existence of glueballs. Indeed there are two possible candidates already, the i(1440)0 -+ and 8(1640)2 ÷+, whose masses and j e c are in the right region to be glueballs and whose production and decay are not inconsistent with expectation for glueballs [4, 8]. Even so, these states are far from being established as glueballs; radially excited qcl for i(1440) [9] or qqqq for 241

242

7". Barnes et al / QQG hybrtd mesons

8(1640) [10] remain possible assignments. Furthermore the role of glue in the ,/-,/' system has been controversial for many years. All of this raises the question, at least in the short term, of whether constnuent glue can practically be established by searching for pure glue states. In QCD there exist other simple colour singlet systems which have not yet received much detailed study. These are hybrid states [11,24] which contain both constituent quarks and gluons: qqg, qqqg. The q~lg spectroscopy m particular is expected to exhibit some interesting features. There is a QCD sum rule prediction [12] that a j e c = 1 -+ spin-parity exotic exists at about 1 GeV in mass. If this is correct then we may hope to unambiguously detect constituent glue by identifying such a state in ~b~ vX, ~k~ hadrons or ~p annihilation. j p c = 1 -+ cannot be formed from qcl nor from two transverse gluons. It can be formed from three gluons or qqqq in P-wave, but as such 1 -+ states are expected to be near or above 2 GeV, they should not mix significantly with the qqg state. If the qclg is of order 1 GeV in mass then it may be hard for it to decay hadronically (for example 7rA 1, and KQ may be the only two body hadromc channels open for the I = 0 state). Thus its width should not be anomalously broad and, prima facie, it appears to be the best candidate state for establishing the existence of constituent glue. The purpose of the present paper is to evaluate the prospects of identifying such a state and. to study the spectroscopy of the tightest qqg states. We gwe a general argument that the j P c = 1 -+ (and other q¢tg states) could be rather light, in line with the QCD sum rule result. We shall then present detailed calculations of the hghtest qqg multiplet spectroscopy within the spherical cavity approximation to the MIT bag. This model is known to give a good qualitative, and even quantitative, guide to the spectroscopy of low-spin hadrons made of quarks and possibly even of gluons [2-5]. The literature now contains detailed studies of simple colour singlet systems such as 2 body:

qq [13], gg [2-5, 14],

3 body:

qqq [13], ggg [3, 14],

>/4 body:

qqqq [7], dibaryons [15], q9 and qt5 [16].

Independent of the current interest in establishing qqg states and gluon states in general, it is desirable to complete the study of colour singlet 2 and 3 body states in

T Barnes et a l /

QQG hybrid mesons

243

the M I T bag model. Preliminary results of this calculation were given in ref. [17]. As they tend to confirm the results of the Q C D sum rule calculation and are the result of much detailed computation, the essential details are presented here so that interested readers may verify the results for themselves. In sect. 2 we present a heuristic discussion and zeroth-order estimates of the masses of the lightest hybrid states. In sect. 3 are detailed calculations of the spin-dependent O(as) contributions to the energy of these states, using time-ordered perturbation theory [17, 18] (more details are in appendices). Attention is drawn to spin-independent effects that are still controversial and could lead to an overall shift m the masses but which do not affect the multiplet splittings. In particular ~t is possible that gluons gain a positive self-energy when confined in a cavity. Chanowltz and Sharpe [33] have independently studied these questions and claim that the bag model spectra may compare better with data if the gluon self-energy is some hundreds of MeV. Ways of searching for hybrids are discussed in sect. 4. Finally, in sect. 5 we discuss phenomenological implications of the spectrum and compare our results with the independent work of Chanowltz and Sharpe [33].

2. Mass scales and jPC of lightest hybrids 2 1 HEURISTIC ESTIMATES OF HYBRID MASSES In hadron spectroscopy the up and down quarks act as if they had energies ("constituent masses" Eq) of about 350 MeV. Thus the lightest qqq states have a mass of about 1050 MeV which is indeed the mean of the observed nucleon and A. Analogously one can conceive of a constituent mass Eg for a gluon which will be of the order of ½ of the lightest gluonium, gg, state. The lightest qF:lg or qqqg states will then be expected to have masses of order 2Eq + Eg and 3Eq + Eg respectively. There are several independent reasons to suspect that Eg = 500-1000 MeV. (i) It is possible that the lightest gluonia have masses 1-1.5 GeV. The 1(1440) and 0(1640) are candidates for 0 -+ and 2 ++ gluonia [8]. These masses are consistent with bag model and lattice calculations for 0 -+ and 2 ++ masses, both these models predicting that 0 ++ gluonium has a mass of = 1 GeV. These 0 ++ and 2 ++ masses are also expected from Q C D sum rules [20] though these predict a rather larger mass for the 0 -+ state. If we discount the 0 as a 2 ++ glueball candidate then the 0 ++ and 0 -+ masses of order 1-1-5 GeV will be a measure of confined gluon energies after hyperfine forces have been included. As J = 0 states are pulled down by these forces, the true measure of confined gluon energies could be up to 500 MeV more than these 0-++ mass scales naively suggested. Thus Eg could be as high as 1 GeV, a result which is also found in ref. [33]. (li) Bernard [21] has argued that the energy to break a string joining colour-8 sources yields Eg > 520 MeV. Non-perturbative continuum studies [22] yield Eg =

244

T Barnes et al / QQ_Ghybrid mesons

500 + 200 MeV. Finally perturbative QCD applied to ~k decays may work better if E~ = 700 MeV [231. Thus we anticipate that the tightest states with both quarks and gluons as constituents will have masses of approximately qqg - 1200-1700MeV,

qqqg - 1600-2100MeV.

(2.1)

Qualitative support for such fight states comes from the observed large hyperfine mass shifts in light quark spectroscopy; for both qq and qqq these are several hundreds of MeV. These originate from exchange of a transverse gluon, which gives a perturbative energy shift

Aghfs(X (gqq-- gqqg) -1 .

(2.2)

Substantial energy shifts require that the energy of the intermediate q~lg state is not large; this can be illustrated directly in bag models [18]. 2 2 jec

OF q~lg

In a spherical basis tranversely polarised gluons can be classified in TE or TM modes [2, 3] with j e in the sequence TE(1 +, 2 - . . . ); TM(1 -, 2 + . . . ). Thus a 0 -÷ gg state will be gg(O- +) = ( T E ) ( T M ) ,

(2.3)

whereas the lightest 0 ++, 2 ++ gluonia will be (TE) 2 or (TM) / depending upon whether TE or T M ghions have the lowest energy or "constituent mass". If the gluons are confined inside a sphere, with the usual MIT boundary conditions

n~G~Is =

0,

(2.4)

where n is the outward normal, then the TE(I+) is the lowest eigenmode. For a spherical cavity of 1 fm radius the TM(1-) mode is more energetic: T M ( 1 - ) = TE(1 +) +0.5 GeV.

(2.5)

Thus if the qqg states are reasonably well approximated by the MIT bag model we expect that the lightest states will be (qFt),-0 ® g , + ~ 0 - + , 1 - + , 2 - + ; 1 - - ,

(2.6)

which contain the particularly interesting exotic jec= 1-+ that cannot be made from qFt in a colour singlet. QCD sum rules have predicted [12] that such a state

T Barnes et a l /

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245

should occur " a t about 1 G e V " which is in agreement with our heuristic estimates above. To make quantitative predictions for the mass spectrum, its flavour and jec dependence, we will explicitly compute the O(ets) energy shifts for qqg states in the spherical cavity approximation to the M I T bag. The qq~ system is simpler than qqqq with which it shares several similarities. Jaffe noticed that low-spin qqqq states can be anomalously light [7]. Heuristically, the cost m energy of adding an extra qq is reduced by the large attractive colour magnetic force ansing between (qq)s= 1,8c substates. Analogously in hybrid systems, the cost of adding a gluon tends to be reduced (for low J ) by the corresponding large colour magnetic force generated between (qq)s = 1.8o and (g)s= 1,8o subsystems. As we shall show in sect. 3 explicit calculations of O(cts) effects in a spherical bag confirms that large downward energy shifts occur for the jec = 0-+, 1 -+ qqg states made from (qq)J=,,8c ® (g)'rE ---' ( 0 - + , 1-+)lc,

(2.7)

supporting the Q C D sum rule result for the exotic 1- ÷ states. 2 3 BAG MODEL MASSES AT ZEROTH ORDER IN a s In a spherical cavity the g(1 +) is the lowest energy eigenmode for a gluon whereas qq(1 +) requires P-wave quarks. Thus, for a 1 fm radius cavity, E(qq)t+ - E(g)t+ + 0.5 GeV,

(2.8)

so that the jec = 1 -+ qqg state should be lighter than other jec exotic candidates. This is seen explicitly in the bag as follows. The lowest gluon eigenmodes are TE ( j e = 1 +, Xg = toga = 2.744) and T M ( J P = 1 , X'g = 4.493). These may be compared with quarks, for which the analogous qq combinations are given below.

JP~_ 1 + 1-

g

qq

2.744(TE) 4.493(TM)

5.855 4.086

Thus, adding qq in a JP= 1- state to some system will have sinular effects energetically to adding gTM. Hence, distinguishing qq(qq)j- from qqgTM may be difficult and some mixing may be anticipated. The addition of a TE gluon is much more distinctive: adding (qq)l + costs over twice as much energy and there is hope that qClgTE will not mix much with qqqcl. As the exotic j e c = 1-+ is formed from (qq)) gTE it should be quite distinct from heavier qqqq states.

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T Barnes et al / Q Q G hybrtd mesons

Initially to c o m p a r e with the conventional valence quark h a d r o n spectrum we will adopt the parameters used earlier by the M I T group [3, 7, 13]. The h a d r o n energy is given by

Eo(a ) = E(0) + ( ] T r ) B 0 a3 - Zo a-1 qqg

(2.9)

where

E q~) = Y~,x,a-1 qg

'

Xu.d = 2.043,

XgTE = 2.744,

X' = 4.493 --arM

'

l

B 1/4

=

146 MeV,

Z o = 1.84.

(2.10)

The strange quark mass is 279 MeV. In addition the energy shifts from transverse gluon ("colour magnetic") interaction are calculated to O(as) using " b a g g e d Q C D " (sect. 3). These yield an energy shift A E M. Thus finally

E = E o ( a ) + A E M,

(2.11)

and the total energy is minimized with respect to the cavity radius ddEa a=ao = 0 .

(2.12)

The hadronlc mass is then gwen by E(a = ao). For a s = 0 this yields E = 1 2 1 5 M e V (uUg, dag, u d g ) , 1365MeV(u~g,d~g), 1 5 1 0 M e V (s~g).

(2.13)

That u~g is some 10% higher than the mean NA b a r y o n mass is expected because one quark has been replaced by a single gluon whose individual contribution is roughly 30% up on a single quark. The proximity of qclg to the NA mass scale causes the prediction to be rather insensitive to model details (such as the magnitude and source of Z0) once parameters are adjusted to give the correct NA masses. To illustrate this we can consider two alternative but similar approaches. (1) Ftxed spherwal cavtty: the Bogohubov model. Here Z 0 = B = 0. The energy is E = EX a- l directly without minirmsation. If one fixes E N = E,a = 1100 MeV (at O(as°)) then aeff = 5.5 GeV i. Then with this aef f we obtain Eua ~ = 1260 M e V ,

Euudg = 1773 M e V .

(2.14)

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T Barnes et al / Q Q G hybrid mesons

Centreof mass mo&fiedMIT bag.

(ii) It has been argued [26] that the parameter Z 0 includes, m part, a c.m. correction. The energy in the standard MIT model is the pnce for localising N constituents in a fixed sphere. Thus we have localised the sphere as well and hence overpaid in energy. The authors of ref. [26] propose that this be taken into account as follows: remove the individual E p 2 from the computed E 2 thus

M 2=

~_,x,a-')2 -~.,(x,a- I ).2 l

(2.15)

1

So for N identical eigenvalues, as in the qqq proton,

M=NEa ~/1 N1 _NE--a 2 E_NEa

al ,

(2.16)

and hence contributes 1 to the phenomenologlcal parameter Z 0. To fit the pA mean mass at 1100 MeV requires aeff = 4.5 G e V - ~. For hybrids this approach yields Mu~g = 1220 MeV,

Muudg= 1685 MeV.

(2.17)

These estimates ignore any gluon self-mass contnbutmns. It ~s possible that perturbatlon theory in as wdl generate a large positive self-energy for the gluon, thereby raising the overall mass scales of this section and affecting the mixing among qqg and qqqq states. Such computations are beyond the scope of the present paper. When calculations of gluon self-energies have been completed, the remarks in this section might need to be revised. The pattern of supermultiplet sphttings discussed hereon is essenUally independent of this question.

3. Hyperfine splittings 3 1 STATESAND QUANTUM NUMBERS TO derive the hyperfine structure of bag model qqg states, we must explicitly obtain their colour and spin structure. They are overall colour smglets, formed from a (qq) colour octet state and a colour octet transverse gluon. 8

[qCtg> = ~ ~/~]qCt>~[g)a,

(3.1)

a=l

3 ] q q ) a = f ~ - Y~ h~jlq),]ct)j. t,J=l

(3.2)

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7". Barnes et al. / QQG hybrid mesons

The ~ - in the Iqq)a colour octet state is required for normalisation.The (q~l) and g angular momenta are combined as usual to construct these j p c qqg states; as an example, the j p c = 2-+ state with Jz = 2 is explicitly

g=(+)/ Xaj~----~.

Jqqg2-+,Jz=2>=E q,($)

(3.3)

In calculating hyperfine shifts for these qqg states, we will sometimes display the multiplet shifts as an array with entries representing the various states

{2+ 1 +

/

1

(3.4)

0 +

(Sq~= 1)

(Sq~ = o)

For example, the spin-spin term 8 E = cSg . Sq~asa-1,

(3.5)

can be conveniently displayed as 1

8E = casa- 1 -1 -2

(3.6)

3 2 t-CHANNEL DIAGRAMS The most important hyperfine diagrams are t-channel gluon exchange. Here we shall d~scuss the qg diagrams explicitly and just quote the F:lg and qq results. There are two time orderings of qg t-channel transverse gluon exchange amplitudes which yield identical contribunons:

a a' i

t" = ~-~V'E[tlfx"'"Sg½ga

3 ¢,

J'

' ] [ - t b , S q ~ X , , g a - ' ] ~j/.

(3.7)

Taking the colour singlet matrix element of this amplitude, as in (3.5), gives its contribution to the O(as) energy shift in colour single qqg states.

T Barneset a l / QQG hybridmesons

249

The clg diagrams with spectator quarks (2~ ~ - A T) give the same result, but with Sq ~ Sq. The sum of t-channel qg and qg gluon exchange diagrams is thus

8E = ( 24~rb't' ) asa- 'Sg "( M

+S~).

(3.8)

.J Y

(0.7324) There is also t-channel transverse gluon exchange between the q and ~l, with the valence gluon acting as a spectator. This gives [ 167rb2 ) B E = - 1 3Xg

(3.9)

asa-'Sq" $4'

(0.1178) which is an order of magnitude smaller than qg and qg diagrams, primarily due to the small X. ~ matrix element for a colour octet qq pair. The total t-channel transverse gluon exchange contribution to the hyperfine sphttings is thus

3E t - c h a n n e l

transverse

=as a-1

gluon exchange

0.7324). - 1 -2 "W

qg + qg

0

)(

4 I 4

+ 0.1178

3

.

I 4

J \

..y

J

qCt (3.10) 3 3 COMPTON DIAGRAMS An interesting feature of the qqg system is the presence of internal Compton scattering of the valence gluon off the valence quark and antiquark. These give contributions to hyperfme splittings which are almost as important here as t-channel transverse gluon exchange. The Compton diagrams which give the following rE matrix elements between colour smglet qqg states after summing over q and q (Sq~ ~ Sq q'-STq), are given below: a

('

a'

.\/
a J~

(I) s-channel

I

a'

= ( ~bZl )(Otsa-l)Tr[XaX~'Xa'Xa]fll - Sg. Sqv~), (3.11)

8Xg

~6

T Barnes et a l /

250

,X

(II)

QQG hybrid mesons

u-channel

= -1[~rb28Xg) ( a s a -

1)Tr[~a~ '~a?~'](1 + Sg. Sqra).

(3.12)

32 3

Diagrams where one gluon interacts with the quark and the other gluon interacts with the antiquark yield a net vanishing contribution. Combining the two sets, we have finally

~Ecompto n =

which is comparable to these we must add the states. These involve the the qqg energy shifts are

as a - l (0.2651

16

(3.13)

the t-channel transverse gluon exchange result (3.10). To Z-graph topologies with qqqq and qqqqgg intermediate annihilation vertices (A.10), (A.11). Their contributions to respectively a~2

(I) s-channel = ~Z~r

X+Xp+Xg

~ ( I I )

1

(1 - Sg-Sqv~)as a-l

(l + Sg. Sq~)asa-'.

u-channel = - 4~r X+Xp-Xg

(3.14)

(3.15)

The sum of Z and direct graphs is 1.14 × direct (for s-channel) and 0.61 × direct (for u-channel). These Z-topologies have physical significance in the decay amplitudes of qqg. Their contributions to 8E are

6E =

- 0.0232 + 0.0674 +0.113

) + 0.0221

a s a - ~.

(3.16)

This is a small but non-negligible contribution to the 0 -+ and 1-+ hybrid energy shifts. 3 4 QQ A N N I H I L A T I O N INTO A T R A N S V E R S E G L U O N

The internal annihilation of the qq valence pair into a TM (1 -) gluon is a measure of mixing with gg intermediate states. This effect is small for massless quarks but

T Barnes et al / QQG hybrid mesons

251

quite significant for strange quarks.

=

( 4~ra~

--)a~a-'(2S,.Sf). X'g- 2X

_

(3.17)

The time ordering that has an ggqclqcl intermediate state is some 20 times smaller and is ignored. The 2S 1• Sf can be Flerz transformed to

2S"Sf=3+Sq'Srt=

1,

Sqrt= 1

O,

SqFq = 0 .

This gwes the anmhllation hyperfine shift contribution for 0 -+, 1-+, 2 -+ to be 8Ea~n~l,auo, = -asa-1(0.00982) (per ufi or dd) = - a s ' ( O . 1 2 1 ) (for sg).

(3.18)

3 5 COULOMB D I A G R A M S

Naively one would expect the Coulomb diagrams to be spin-independent in this problem, as S-wave bag model quarks give a spherically symmetric charge density which does not couple to their spin. This is indeed the case for t-channel Coulomb diagrams; but the s-channel qq annihilation Coulomb diagram is spin-dependent. It is zero between our qq states unless Sq~ = 1. The reason is that the qq annihilation charge density, unlike the qcl transition charge density, has non-trivial ( / = 1) angular dependence;

([P (x)lq, qj)im=o=-2ot2+Jo(kr)Jl(kr)[~ljo2o a

Tho q and 0 are Pau'i spinors, with

•

I a ~q,]~Xj,.

(3.19)

] andq =O m=[O]. T, e

usual Vq antiquark Pauli spinor is ~ = -io2~. ) It ~s convement to write the entire Coulomb hamiltonian's matrix element as a single effective four-quark Coulomb vertex. Assuming the qq pairs are in colour octets, this vertex ~s l t

X

X

= Cl~sq.~,l~.j,~,/t~s a

--1

•

(3.20)

T Barnes et al. / QQG hybrtd mesons

252

The coefficient is given by an integral over unit spheres;

C 1= ~(a2a3)2ff~=o~dxdy[jo(xrx)jl(xrx)][(rx)---~

( r y ) ] ~ . f a G ( a x , ay) (3.21)

= (0.0750).

(3.22)

This coefficient is related to the CM( = 1.20. 10 -2) quoted in fig. 7 of Maciel and Monaghan [28] by C I = 2~'(1.20- 10 2) = 0.075.

(3.23)

The energy shift between the colour slnglet qclg states is B E = + 0.150C~sa- 18sq~.I.

(3.24)

So, S-wave qcl annihilation through a Coulomb gluon does not affect the ( 1 - - ) qclg state, which is a quark spin singlet (Sqr~ = 0), but shifts the isoscalar (0 -÷, 1-+, 2 -+) qqg states up by 0.15 asa -1. This breaks the I = 0; I = 1 degeneracy by as much as 100 MeV: a distinctive feature of hybrid spectroscopy. 3 6 OTHER SPIN-INDEPENDENT EFFECTS There are several spin-independent contributions to the q~tg state energies which we have not included in our spectrum calculations. We have neglected these for two reasons. First, we are primarily interested in the multiplet splitting pattern as a characteristic feature of the qclg state spectrum; this revolves only spin-dependent effects. Second, although we can calculate the O(as) spin-dependent corrections to qclg state energies, several of the spin-independent effects are not yet well understood. Perhaps they can be adequately treated in future work; for the present we subsume these terms in the multiplet centre of gravity E 0. The important spin-independent contributions to qqg energies are (i) Casimir effects, (ii) colour Coulomb interactions, and Off) self-energy diagrams. The Casimir energy in the bag model certainly requires further work, as there is no consensus regarding either the value of the zeroth order contribution Zoa-1 to the bag energy [37, 38] or the O(as) radiative correction Z l a s to this energy [38]. It is usually either fitted as a free parameter (in which case it can imitate other spin-independent effects) or set equal to zero, although it should be a calculable number.

T Barnes et al / QO_Ghybrid mesons

253

The spin independent t-channel colour Coulomb interaction does not present any difficulues. However there ~s still some controversy as to what extent its effects are cancelled out by self-energy contributions. The self-energy diagrams for quarks and gluons in the bag have only recently begun to be discussed [14,35,39]. O(as) self-energy effects also should be calculable, but this has not yet been done. The approach of two recent bag model papers [14, 35] has been to parametrize self-energy contributions as aE~¢,r-¢,¢rgy =

(G or

-e,)a~a-'.

(3.25)

The actual values of these coefficients used m the two papers differ considerably: Cq = - e q -

0.4 [35], 0.8 [14],

CTE = --eVE -- 1.2--3.6 [35], 0.3 [14],

(3.26) (3.27)

so ~t ~s evident that no agreement on the size of these spin-independent effects has arisen from these papers in their fits to the spectrum. In view of the as yet uncalculated spin-independent shifts due to the Caslmir and self-energy contributions, we have d e o d e d to exclude them from our discussions of the hybrid spectrum for the present, and concentrate only on the relatively wellunderstood multlplet splittings. 3 7 TOTAL O(a,) MULTIPLET SPLITTINGS

The total spin-dependent shift of the q~lg hybrid energies is determined from the sum of the t-channel transverse gluon exchange (3.10), Compton (3.13), (3.14), annihilation (3.17), (3.18), and s-channel qq annihilation into a Coulomb gluon (3.24). The sum of these terms is, for the m q = 0 states with I = 0 and 1 :

6Eqrag(m q = O) (asa -1 )

0.703 + 0.059 - 0.019 6to - 0.023 + 0.300 81o - 0 . 7 6 2 + 0.471 - 0.019 810 + 0.067 + 0.300 6to, - 1.494 + 0.678 - 0.019 8io + 0.113 + 0.300 61o

0.088 + 0.265 + 0 + 0.022 + 0 /

(3.28)

=

I

0.739 -0.224 - 0.0703

0.375

1

+

1 1

0 (0.281) 8zo.

(3.29)

T Barnes et al / Q Q G hybrM mesons

254

Inclusion of a strange-quark mass changes the various contributions somewhat due to changes in energy denominators and vertex strengths. For the ~ states, with rn~a = 1.65, we fred

8Es~(msa = 1.65) (asa

=

')

0.546 + 0.035 + 0.037 - 0.027 + 0.087 - 0 . 5 8 2 + 0.279 + 0.037 + 0.079 + 0.087 1.145 + 0.402 + 0.037 + 0.134 + 0.087

0.052 + 0.157 + 0 + 0.026 + 0

(3.30) 0.678 - 0.100, - 0.485

} 0.235

•

(3.31)

These numbers are quite close to results presented earlier [17]; our expectations for the masses and multlplet sphttings of these states have not changed significantly. The differences of - 0. lasa I between these numbers and the earlier results are due primarily to the inclusion of the effects of C o m p t o n Z-graphs and qq annihilation into a C o u l o m b gluon. These are - 50 MeV effects in a total multlplet splitting of - 5 0 0 MeV, although the latter effect does lead to some interesting and perhaps easdy observable results. Adding the bag energy BV to the splittings (3.29), (3.31) and mlnirmzing with respect to the bag radius a, as usual, will multiply the spllttings by a factor of 4. The resulting spectrum for a = 6 G e V - l and t~s = 2.2 is shown in fig. 1. It ~s worth noting that the masses of the I = 0 tog hybrid mesons are displaced upwards from the I = 1 Og states by

m(pg)--m(tog) = (2) u~ dd

. (4) bag + qq~

(0.15~sa - t )

- 100 M e V ,

(3.32)

C o u l o m b shift

due to the colour C o u l o m b annihilation diagram. This affects only the Sqva = 1 states (0-+, 1 -+, 2 - +); the (Sq~ = 0)1 qqg states (~lq)g and ~rg are predicted to degenerate. A n alternative way of computing qqg multiplet splittmgs is to use the S-matrix formalism derived by Sucher [30, 31] in the context of Q E D . Tbas a p p r o a c h gives the energy shift of the states due to an HI in terms of the admbatic S-matrix constructed from that H x. This is reviewed m [32] and yields results in accord with sect. 3. 4. Hybrid phenomenology The spectroscopy of the low-lying states is dlustrated in fig. 1. These masses are given for the parameters a 0 = 6 GeV o l, as = 2.2. The splittings are independently

T Barnes et al / QQG hybrid m e s o n s 0 -+

I-+

255 2

I--

-÷

2 0 _

_

K~

g

M(qrqg) l (GeV) - -

(ns)g

-

K

-

g

,,,, ,] (nq)g,~rg

q~g

~

~g

1 5

K g

K g ~0g

- - P g

1 0

,

i

0 -÷

,

,

I-+

,

I--

,

i

,

2-÷

Fig 1 H y b n d m e s o n m a s s e s Parameters are a = 6 G e V - i, as = 2 2, thas gwes E 0 = 1 52 GeV

confirmed by Chanowitz and Sharpe [33]. There arises an interesting pattern of allowed and forbidden transitions among these states due to ~r or K emission. If we label the states in an obvious notation then possible transitions are

~r:

pg ~ wg,

K~ ~ K~.

(4.1)

Sq~ = 0 qqg state labels, e.g. (Pg ---' ~rg), may be inserted anywhere in the above list as well. We note that the K emlssmns between 1-+ and 0 -+ may be kinemaUcally forbidden, so q~g(1-+), K~(1-+) will decay to conventional quark matter as will the

256

T Barnes et al / QQG hybrzd mesons

0 - ÷ states. T h e t r a n s i t i o n (4.2)

(pg), +-* ( g)o

if energetically allowed will generate a slow p i o n which m a y be an aid in d e t e c t i n g these states via the ~r° --, ~"r d e c a y [34]. T h e quasi t w o - b o d y decays that m a y be i m p o r t a n t d e p e n d u p o n the a b s o l u t e m a s s scale of the hybrids. T h e most f a v o u r a b l e S- a n d P-wave decays are hsted in table 1. S o m e p a r t i c u l a r l y f a v o u r a b l e m u l t i b o d y decays are also listed: there are, of course, m a n y o t h e r m o d e s n o t listed. 41 l +DECAYS T h e 1 + state is of p a r t i c u l a r interest zn view of its exotic j e c q u a n t u m n u m b e r s . T h e d e c a y to two distznct p s e u d o s c a l a r s is allowed. If E o is low then c o n v e n t i o n a l t w o - b o d y decays could be totally f o r b i d d e n for ~0g(~TrAl) a n d O g ( ~ KQ). T h e f o r m e r state seems most p r o m i s i n g m ~p ---, y ( t % ) --, y(Trqro) ---, V(4*r),

(4.3)

p r o d u c e d b y the colour C o u l o m b i n s t a n t a n e o u s interaction, fig. 2 (note that this enables 1 + q q g to be p r o d u c e d : the p r o d u c t i o n b y two transverse gluons in c o n t r a s t is b e h e v e d to b e s u p p r e s s e d b y Y a n g ' s theorem). It is interesting that a p o s s i b l e ~r~rp e n h a n c e m e n t is o b s e r v e d b e l o w 1400 M e V in ~ ~ "t(47r) (fig. 24 of ref. [35]). This w o u l d c o r r e s p o n d to E 0 = 1300 MeV. The 1 - + ( K ~ ) state will have a genuine TABLE 1

Some low threshold S and P convenUonal decays of hybnds Flavour" spm

pg, %1

S

~re ~ rr(rr~r)s

p

K~, Kgl-

q,g, (~s)gl

Ke ~ K(~r~r)s

KK --* K(rr K) s

~p

~e ~ ~(~r~r)s 3~r ~ qrKK. KK*

IrK*

KK*

S

~rD

~rAt

~rQ

KQ

P

~r~r* ~',KK* ~A z

~rK

~/' KK*

S

~r~/ KK* ~rf

P

~rp

0 +

1

tOg, ('qq)gl

+

~rK**

2 ÷

I-

S P

rr~r q'gto

~rK*

KK*

~rB

~rQ

KQ

~rp

~rK

KK

T Barnes et a l /

257

QQG hybrtd mesons

f

"l

q

Fig 2 Instantaneous colour Coulomb production of 1-+ q~g m heavy QQ radiative decay

t w o - b o d y decay into Kor which could aid its detection in + ~ KKoror via

I

I

" Kor(1 +)

(4.4)

• K~r(1--).

N o t e that ~b---, K ' K * so the recoil spectrum against K* should be studied for evidence of a spin-one state. The 1-+(pg) can decay into orB. Studies of PP --' cr(orB),

(4.5)

at L E A R m a y be worthwhile. The orB combination is particularly interesting in that 0 +÷, 1-+, 2 ÷+ are the d o m i n a n t combinations for masses below 1500 MeV. The 2 +÷ is clean (A2); the 0 ÷+ is in any event interesting and the 1 -+ is the hybrid of interest. 4 2 0-+ DECAYS The qq m qqg are in 8c and so must end up in different mesons in the decay q , q z g ~ (q,q2)8(q3q4)8 ~ ( q l q 4 ) , ( q s q 2 ) t •

(4.6)

This has the following consequences for the ~g and Wg states (both producible in q~ ---, ~,X). The ~g decay will lead to [S~l,~q] final states. Thus B~ror is forbidden whereas KKor is allowed. This behaviour is precisely that observed for the 0-+(i, 1440). However, this state c a n n o t be s~g because ¢0g would then exist at 1100 MeV and be p r o d u c e d at least twice as copiously. N o such state has been seen. The cog has a converse behaviour: it can decay both to Boror and KK~r. Thus the signatures for 0-÷(q~lg) in ~b ~ Y(q~lg) are that of a pair of states with the decay charactenstlcs of i(1440) and &(1800) and can be observed but only with the relative

T. Barnes et a l /

258

Q Q G hybrid mesons

masses reversed [36]. The relative strengths of £(0-+ (0~)-~0)

(4.7)

r(o-+(,o~)--, ~.~) '

provide an interesting test of dynamics (if ~ is P-wave q~l). The magneUc gluon produces qq in a relative P-wave: (1S)(1P) in the cavity energy elgenstates. If the 1P quark preserves its 1P state then ~r~ is produced. In order to yield 7rp the 1P eigenstate must convert into a IS mode and the unit of angular momentum be converted into an orbital P-wave between the rr and p (fig. 3). 43

2 - +STATES

If there is no significant price to pay for colour rearrangement then apparent violatmns of the OZI rule can occur in the 2 -+ system in particular. The S-wave

Fig 3a Conversion of a TE gluon into an S and P qVqpmr

S

}~

~

$ Fig 3b The 2 -+ decay to ~r8

$ Fig 3c Conversion of P mode to S mode and relative P-wave yields ~rp

T Barnes et a l /

259

QQG hybrtd mesons

decay 2-+(ps)~rf

',

(4.8)

produces a final state with I = 1 and carrying manifest hidden strangeness. Thus PP ~ ~r (rrf'),

(4.9)

could be an interesting signature. Chanowitz and Sharpe [33] have noted that a 2-+(1850) appears in the data of ref. [40]. This is anomalously close to the A3(1660 ) and so CS suggest that the 1850 could be a qqg state. If so, then we predict that the 2 - + ( w s ) occurs 150 MeV higher almost degenerate with the 2-+(K~). The anmhilation (eq. (3.24)) effects lead to m ( W g ) - m(Kg), which lS a characteristic signature for identifying hybrid spectroscopy, with the values of m s and c~s used here.

5. Spectroscopy With

B I/4 =

146 MeV, the bag radius becomes a o ( G e V - 1 ) = 5.1E o (GeV) 1/3 ,

(5.1)

where the zeroth-order energy E 0 is in GeV. For u~g states E0(ct s = 0) is naively expected to be - 1.2 GeV. This leads to u n c o m f o r t a b l y low masses for the 0 - + qqg states, and we suspect that the self-energies of gluons m a y yield a positive contribution to the state masses of order some hundreds of MeV. This was assumed to be the case m refs. [17, 33], and the range of masses in these lowest qqg multlplets will cover the 1 - 2 GeV range. For presentation of splittings we will use a constant a 0 = 6.0 G e V - 1, which gives a mean rnq = 0 energy of E 0 = 1.52 GeV. This corresponds to B ~ / 4 = 143 MeV. For the strange quark we use a mass of rosa = 1.65 and hence (w s - Wu)a - 170 MeV. The latter is - 50 MeV less than Chanowltz and Sharpe, who use a large gluon self-energy and a large value of m s. We are in agreement with Chanowitz and Sharpe [33] (henceforward referred to as CS) on the prominent features of the spectroscopy. Specifically, the ordering of the masses 0 + 1 + 1 - 2 -+, with splittings between states of equal flavour being - rn.. F u r t h e r m o r e we agree on the important colour C o u l o m b shift that raises the ~g and wg states relative to K~ and pg, such that ws and K~ are roughly degenerate. (This degeneracy depends crucially on the values of m s and a s assumed). It is interesting to note in this connection that such colour C o u l o m b terms should also be important m q2q2 spectroscopy and lift the I = 0, I = 1 degeneracy. This has been ignored in bag models to date and questions the jusuflcation of using the degeneracy of S*(980) and 8(980) as supportive of a (ufi + dd)sg structure. A near degeneracy of the Kg and Wg

260

T Barnes et a l /

Q Q G hybrid mesons

states would be a clear indication of the importance of this colour Coulomb annihilation effect. It will also mix tog and ~g. We have not studied this but CS find that the mixing angle is less than 10 °. The main differences between CS and this work are their inclusion of gluon and quark self-energies as parameters, which raises the overall energy scale Eo; also there are minor differences* in two places. The latter differences are stated below. (1) The relative sign of the two time orderings of the qq ~ g ( T E ) ~ qcl annihilation diagrams in (3d); CS now agree with our results (3.18) [42]. (il) We believe that it is unnecessary to explicitly impose the exclusion principle in intermediate states (and hence the 7 factors in CS (5.4) become 1 in our (3.14), (3.15)) for reasons as outlined in the first footnote of sect. 5 of CS. CS have independently also decided to replace the 7 by 1 [42]. We also note that CS have studied the energy shifts which arise due to P3/2 and D3/2 intermediate quark modes, which we have neglected. These contributions turn out to be rather small. Apart from these minor points the pattern of splitting in CS and this work are in good agreement, which provides a welcome confirmation of the computation of the diagrams considered by both groups, and reinforces the proposed experimental signatures that may aid in ~dentification of this new spectroscopy. If E 0 = 1500 MeV then some interesting phenomenology ensues. The 2-+pg will be at 1850 MeV where, as pointed out by CS, a 2 -+ state is seen [40] and not easily interpreted as qq in view of the nearby qq2-+(1640). The important test of qqg status would be to find the I = 0 and strange partners of the 2-+(1850). The significant Coulomb shift for I = 0 hybrids yields the I = 0 tog at around 2 GeV, almost degenerate with the K~ state, manifestly not degenerate with the pg: a distinctive phenomenon contrasting with conventional spectroscopy where mK. = ½(m~ + m , ) . The 1 -+ states will have the tog at about E o = 1500 MeV. This should be looked for in ~b---,~(47r) as a non-pp enhancement below the 1600 MeV structure observed in ref. [35]. Indeed comparison of the pp and non-pp structures suggest that an enhancement may be present at this very mass. The 1 - - and 0 -+ states are predicted to lie in the same region of mass as the conventional radial excitations. Interesting mixing effects may ensue. An amusing possibility emerges if E o = 1600 MeV. The 0-%r(1300) and K'(1400) [41] are then precisely where the pg and K~qqg states are predicted to be. The K~ - tog states are predicted to be nearly degenerate so the 0-+tog has similar mass to 1(1440). The 0-+fig state is then expected at 1650 MeV and should be produced in + ~ 3 , X at half the rate of the tog. The absence of a prominent ~k~3,KK~r enhancement at 1650 MeV prevents us identifying the (whole of the) i signal as the 0-+tog state. In this scenario the 1-+tog will be contained in the 1650 MeV enhancement in hb ~ y(4~r) and the 2-+pg emerges at 2 GeV, consistent with the constructive interference fit of ref. [40]. * See also a c k n o w l e d g e m e n t s

T Barnes et al /

261

Q Q G hybrtd mesons

In general the near degeneracy of tog with K*, in contrast to to and 0, is perhaps the most clear signal that will enable confirmation of hybrid as against other assignments in non-exotic j e c multiplets. The exotic j e c is, of course, the most dramatic signal. We are indebted to J. Weyers for his interest and suggestions and to M. Chanowitz and S. Sharpe for informing us of their independent work and comparisons of our two calculations. In particular they pointed out a sign error m our original calculation of the Z-graph Compton diagrams. Appendix A VERTICES

Many of the bag model qqg" and ggg vertices have been derived earlier [2, 27, 28]. We label the numerical vertex strength by type; a, = annihilation, b, = bremsstrahlung, C1= Coulomb, t, = three-gluon. a

,

>

~

j =ib,Sq.e~(haj,)ga - l ,

(A.1)

j =-,blSq.eg(~,)ga-' ,

(A.2)

a

, ~

>

a

, - C

"a~'.,

j =-tblS~.e;(-kS;)ga

<

j =lblS~q.eg(-hS;)ga

-l,

(A.3)

a

, ~

',

(A.4)

a = - a l S m . e * (g~\ a j l )ga -1

(A.5)

= -a,S m . eg(X~,)ga-',

(A.6)

= _ a l S f m . e g (~xj~)g a - l ,

(A.7)

= -- a l S f i n • e g* ( h jTU ,)ga -

(A.8)

l

j ~

t J l

a ~

j

~-----i

C

~

A = _ tISg " ~* , ¢abc..- ) ©A 2-J ,5" , B

1,

(A.9)

262

T Barnes et a l /

QQG hybrtd mesons

The transverse (TE) constituent gluon can annihilate into an (S-wave; P-wave) or (PS) quark antiquark pair [27]. The coefficient a2 is of course different from the (TE)(S)(S) vertex;

A

~

~

(A.10)

= - - a 2 i S j k ' e A ~ k g a -1 ,

(p) - k 1

(

S

k"~)

A' = _ a z t S k . e A'*,a'~k,ga - '.

~

(a.11)

(TEl

F o r lowest modes, Xq(S) - X = 2.043, Xq(P) - Xp = 3.812, x g ( T E ) = 2.744, and a z = 0.092,

(A.12)

b, = 0 . 1 3 8 9 ,

(a.13)

1

×

(t3jo(n)2jo

n

Jo

n

\×

/

,

a I = 0.01694,

(A.14)

where a +z a 3 = 0.4100,

a'la = 2.172,

X = 2.043 ,

t 1 = 0.1919.

Xg' = 4.493 ,

(A.15) (A.16)

All quarks are in the lowest X = 2.043 mode. The gluons are in the lowest TE(1 +) mode, Xg = 2.744, except in the annihilanon diagrams where this m o d e gives no contribution. The gluon m o d e in the annihilation diagrams is the lowest TM(1 ) state, Xgt = 4.493.

Appendix B There is an interesting check on the rather complicated C o m p t o n result (3.13). R e m o v e the non-essential colour degree of freedom, replace the b~X in the vertices

T Barnes et a l /

QQG hybrid mesons

263

by b 1 and explicitly evaluated the energy shift due to the four sets of diagrams,

including the complicated "twisted" diagrams (second and fourth sets) which we eliminated in the text because they cancel.

\/

\/ '/ X The energy shift is

i

=

(b,ga - I )2/~og

}

0-1+°

- l + 0 + l

+3+½_3

Replacing b by b I and multiplying by the colour factors ~ ; 3-, J6. _ 2; 716, we do indeed recover the full qqg Compton result (3.13). As a second check, note that the sum of the first and second sets gives the total contrlbuUon due to qq intermediate states

2

(b,ga)

=

/~0g nt. . . .

1

.

d,ates

This is zero unless j P C = 0-+ or 1 -, as it must be; only these channels have qq states to max with if we keep the quarks m the x = 2.043 mode.

References [I] T Barnes, Z Phys CI0 (1981) 275 [2] T Barnes, F E Close and S Monaghan. Phys Lett ll0B (1981)159. Nucl Phys B198. 380 (1982) [3] R L Jaffe and K Johnson. Phys Lett 60B (1976) 301, J Donoghue. K Johnson and B L . Phys Lett 99B (1981) 416 [4] J Donoghue. Expectations for glueballs, Amherst preprlnt (1982) [5] C B Thorn, unpublished [6] B Berg, Phys Lett 97B (1980)401, G Bhanot. Phys Lett 101B (1981) 95, G Bhanot and C Rebbl, Nucl Phys BI80 (1981) 469; H Hamer and G Pans1, Phys Rev Lett 25 (1981) 1792, K Ishtkawa et al, DESY-81-089 [7] R L Jaffe, Phys Rev D15 (1977)267

264

T Barnes et a l /

Q Q G hybrtd mesons

[8] K Ishakawa, Phys Rev Lett 46 (1981)978, M Chanowltz, Phys Rev Lett. 46 (1981) 981, D Scharre, SLAC-PUB-2880 (1981), H J LIplon, Phys Lett 109B (1982) 326; J Donoghue and H Gomm, Phys Lett 112B (1982) 409 [9] I Cohen and H J Lxpkin, Nucl Phys B151 (1979) 16 [10] M Chanowltz, Particles and fields 1981, eds C Hensch and W Kirk (AIP Conf Proc no 81) p 85 [11] T Barnes, PhD thesis, Cahech 1977, p 103, Nucl Phys B158 (1979) 171; F E Close, Proc of Kupan Summer School 1980, ed J Elhs, RL-81-066 (1981 EPS Int Conf on H:gh energy physics, Lisbon, 1981) [12] I I Bahtsky, D Dyakanov and A Yung, Phys Lett l12B (1982)71 [13] T de Grand, R Jaffe, K Johnson and J Kaskls, Phys Rev DI2 (1975) 2060 [14] R Konophtch and M Schepkan, Nuovo Cam 67A (1982) 211, C E Carlson, T H Hansson and C Peterson, MIT prepnnt CTP 1020 (1982) [15] R L Jaffe, Phys Rev Lett. 38 (1977) 195, E 617 [16] G Bhamati et al, Madras M U T P / 8 2 / I [17] T Barnes and F E Close, Phys Lett, in press, RL-82-037 [18] F E Close and R R Horgan, Nucl Phys B164 (1980) 413 [19] F de Vlron and J Weyers, Nucl Phys B185 (1981) 391 [20] V Nonkov et al Phys Lett 86B (1979)347 [21] C Bernard, Phys Lett. 108B (1982) 431 [22] J M Cornwall and A Sore, UCLA 82/TEP3 (1982) [23] G Parlsl and R Petronzio, Phys Lett 94B (1980) 51 [24] D Horn and J Mandula, Phys Rev D17 (1978)898, P Hasenfratz, R R Horgan, J Kut~ and J M Richard, Phys Lett 95B (1980) 299 [25] P N Bogohubov, Ann Inst Henrt Poancar~, 8 (1968) 163 [26] J Donoghue and K Johnson, Phys Rev D21 (1980) 1975 [27] F E Close and S Monaghan, Phys Rev D23 (1978) 2098, A Macael and J Piton, Nucl Phys B197 (1982) 201 [28] A Macael and S Monaghan, Oxford TP 25/82 (1982) [29] T D Lee, Phys Rev DI9 (1979) 1802 [30] J Sucher, Phys Rev 107 (1957) 1448 [31] S Schweber, Introduction to relativistic quantum field theory (Harper and Row) [32] Preprmt version of present paper RL-82-088 [33] M Chanowatz and S Sharpe, Nucl Phys B222 (1983) 211 [34] A Odian, private commumcatlon; Mark III collaboration at SPEAR [35] D Burke and G Tnlhng et al, Phys Rev Lett 49 (1982) 632 [36] E Bloom, Proc 21st Int Conf on High energy physics, Pans 1982 [37] K Milton, Phys Rev D22 (1980) 1441, 1444, J Baacke, unpubhshed and private commumcataon [38] K Johnson, SLAC-PUB-2436 (1979) [39] J Baacke, Y Iganshx and G Kaspendus, Dortmund DO-TH 82/13 (1982), H Hansson and R Jaffe, unpubhshed [40] C Daum et al Phys Lett 89B (1980) 285 [41] Part:de Data Group, Phys Lett 111B (1982) 1 [42] S Sharpe, private commumcatxon