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arXiv:hep-ph/0403186 v1 16 Mar 2004
International Journal of Modern Physics D c World Scientific Publishing Company
GLUEBALL-GLUEBALL INTERACTION IN THE CONTEXT OF AN EFFECTIVE THEORY
MARIO L. L. DA SILVA Instituto de F´ısica, Universidade Federal do Rio Grande do Sul, Av. Bento Gon¸calves, 9500 Porto Alegre, Rio Grande do Sul, CEP 91501-970, Brazil
[email protected] DIMITER HADJIMICHEF Instituto de F´ısica e Matem´ atica, Universidade Federal de Pelotas, Campus Universit´ ario, Pelotas, Rio Grande do Sul, CEP 96010-900, Brazil
[email protected]
Received (received date) Revised (revised date) In this work we use a mapping technique to derive in the context of a constituent gluon model an effective Hamiltonian that involves explicit gluon degrees of freedom. We study glueballs with two gluons using the Fock-Tani formalism. Keywords: Constituent Models; Glueballs; Fock-Tani.
1. Introduction The gluon self-coupling in QCD implies the existence of bound states of pure gauge fields known as glueballs. Numerous technical difficulties have so far been present in our understanding of their properties in experiments, largely because glueball states can mix strongly with nearby q q¯ resonances. However recent experimental and lattice studies of 0++ , 2++ and 0−+ glueballs seem to be convergent. In the present we follow a different approach by applying the Fock-Tani formalism in order to obtain an effective interaction between glueballs 1 . A glueball-glueball crosssection can be obtained and compared with usual meson-meson cross-sections. 2. Fock-Tani Formalism for Glueballs The starting point is the creation operator of a glueball formed by two constituent gluons 1 † † G†α = √ Φµν α aµ aν 2 where gluons obey the following commutation relations [aµ , aν ] = 0 ; [aµ , a†ν ] = δµν 1
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The composite glueball operator satisfy non-canonical commutation relations [Gα , Gβ ] = 0 ; [Gα , G†β ] = δαβ + ∆αβ where γρ δαβ = Φ⋆ργ α Φβ
;
γρ † ∆αβ = 2Φ⋆µγ α Φβ a ρ a µ
The Fock-Tani formalism introduces “ideal particles” which obey canonical relations, in our case they are ideal glueballs [gα , gβ ] = 0 ; [gα , gβ† ] = δαβ This way one can transform the composite glueball state |αi into an ideal state |α ) by π |α ) = U −1 (− ) G†α |0i = gα† |0i 2 where U = exp(tF ) and F is the generator of the glueball transformation given by ˜α − G ˜ † gα F = gα† G α
(1)
with ˜ α = Gα − 1 ∆αβ Gβ − 1 G† [∆βγ , Gα ]Gγ G 2 2 β In order to obtain the effective glueball-glueball potential one has to use (1) in a ˜ a set of Heisenberg-like equations for the basic operators g, G, dgα (t) ˜α ; = [gα , F ] = G dt
˜ α (t) dG ˜ α (t), F ] = −gα . = [G dt
The simplicity of these equations are not present in the equations for a √ √ 2 µν † daµ (t) † a g + = [aµ , F ] = − 2Φµν Φ a ∆βα gβ ν β β dt 2 β ν ′
† † γµ + Φ⋆µγ α Φβ (Gβ aµ′ gβ − gβ aµ′ Gβ ) ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ √ ⋆γ ρ µ γ − 2(Φµρ )G†γ a†µ′ Gβ gβ Φγ⋆γ ρ + Φµα ρ Φµγ ρ Φγ α Φρ
The solution for these equation can be found order by order in the wavefunctions. (0) So, for zero order one has aµ = aµ (0)
gα(0) (t) = Gα sin t + gα cos t ; Gβ (t) = Gβ cos t − gβ sin t (1)
(1)
In the first order gα = 0, Gβ = 0 and √ µν † (0) 2Φβ aν [Gβ − Gβ ] a(1) µ (t) = In the second order we found ′
′
′
†(0)
† † (0) ⋆µγ γµ (0) ⋆µγ γµ ⋆µγ γµ a(2) µ (t) = −2Φα Φβ Gβ aµ′ Gα + Φα Φβ Gβ aµ′ Gα + Φα Φβ Gβ aµ′ Gα
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Glueball-Glueball Interaction in The Context of an Effective Theory
3
(3)
To obtain the third order aµ (t) is straightforward and shall be presented elsewhere. The glueball-glueball potential can be obtained applying in a standard way the Fock-Tani transformed operators to the microscopic Hamiltonian 1 H(µν; σρ) = Taa (µ)a†µ aµ + Vaa (µν; σρ)a†µ a†ν aρ aσ 2 where one obtains for the glueball-gluball potential Vgg Vgg =
4 X
Vi (αγ; δβ)gα† gγ† gδ gβ
(2)
i=1
and ⋆νξ ρξ στ V1 (αγ; δβ) = 2Vaa (µν; σρ)Φ⋆µτ α Φγ Φδ Φβ ⋆νξ ρτ σξ V2 (αγ; δβ) = 2Vaa (µν; σρ)Φ⋆µτ α Φγ Φδ Φβ ⋆λξ σλ ρξ V3 (αγ; δβ) = Vaa (µν; σρ)Φ⋆µν α Φγ Φδ Φβ ⋆νλ λξ ρσ V4 (αγ; δβ) = Vaa (µν; σρ)Φ⋆µξ α Φγ Φδ Φβ .
Fig. 1. Diagrams representing the scattering amplitude hf i for glueball-glueball interaction with constituent gluon interchange.
The next step is to obtain the scattering T -matrix from Eq. (2) T (αβ; γδ) = (αβ|Vgg |γδ) . Due to translational invariance, the T -matrix element is written as a momentum conservation delta-function, times a Born-order matrix element, hf i : T (αβ; γδ) = δ (3) (P~f − P~i ) hf i , where P~f and P~i are the final and initial momenta of the two-
glueball system. This result can be used in order to evaluate the glueball-glueball scattering cross-section Z 0 4π 5 s dt |hf i |2 (3) σgg = 2 2) s − 4MG −(s−4MG
where MG is the glueball mass, s and t are the Mandelstam variables.
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3. The Constituent Gluon Model On theoretical grounds, a simple potential model with massive constituent gluons, namely the model of Cornwall and Soni 2 ,3 has been studied 4 ,5 and the results are consistent with lattice and experiment. In the conventional quark model a 0++ state is considered as q q¯ bound state. The 0++ resonance is a isospin zero state so, in principal, it can be either represented as a q q¯ bound state, a glueball, or a mixture. In particular there is growing evidence in the direction of large s¯ s content with some mixture with the glue sector. It turns out that this resonance is an interesting system, in the theoretical point of view, where one can compare models. In the present work we consider two possibilities for 0++ : (i) a as pure s¯ s and calculate, in the context of a quark interchange picture, the cross-section; (ii ) as a glueball where a new calculation for this cross-section is made, in the context of the constituent gluon model, with gluon interchange. The potential Vaa is determined in the Cornwall and Soni constituent gluon model 2 Vaa (r) = where OGEP V2g (r)
= −λ
1 ace bde OGEP V2g (r) + VS (r) f f 3
e−mr π ω1 + ω2 2 D(r) r m
,
(4)
VS (r) = 2m (1 − e−β m r ) (5)
and D(r) =
k 3 m3 −k2 m2 r2 e π 3/2
,
λ=
3 g2 4π
,
ω1 =
1 1 ~2 + S 4 3
,
5 ~2 ω2 = 1 − S . (6) 6
The parameters λ, m, k and β assume known values3,4,5 while the wave function 6 Φµν α is given in . The glueball-glueball scattering amplitude hf i is given by hf i (s, t) =
6 X 3 Ri (s, t) R0 (s) 8 i=1
(7)
where R0 R1 R2 R3 R4
1 s 4 2 exp − 2 = − MG 2b 4 (2π)3/2 b3 √ √ Z (2) √ λω1 4 2π ∞ q t q u q2 q2 = J + J exp − dq 2 0 0 2 2 2 3 q + m 2b 2b 2b2 0 √ (2) tk 2 m2 uk 2 m2 λω2 2 2πb3 k 3 m exp − + exp − = 4(b4 + 2b2 k 2 m2 ) 4(b4 + 2b2 k 2 m2 ) 3(b2 + 2k 2 m2 )3/2 √ Z ∞ √ √ q t q u q2 32 2π q 2 βm2 J + J exp − = dq 2 0 0 2 m2 )2 2 2 3 (q + β 2b 2b 2b2 0 √ r Z ∞ (3) q s 3q 2 q λω1 16 2π b2 2 q dq 2 exp − 2 sinh − MG =− 3 q + m2 8b 2b2 4 s 2 0 − M G 4
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" # 2 k 2 m2 4s − MG 16πb3 k 3 m exp − 2(2b4 + 3b2 k 2 m2 ) (2b2 + 3k 2 m2 )3/2 √ r 2 Z ∞ 128 2πb2 s qβm2 q 3q 2 R6 = − q dq 2 sinh exp − MG (q + β 2 m2 )2 8b2 2b2 4 2 0 3 4s − MG
5
(3)
λω R5 = − 2 3
here b =
√ √ 3 2 r0
(8)
where r0 is the glueball’s rms radius and J0 (x) = sin x/x. In (8) (i)
(i)
one finds the following notation ω1 and ω2 , where the index i correponds to the number of the evaluated diagram in figure (1) The cross-section is obtained inserting (7) in (3). From reference 6 one obtains the corresponding cross-section for a 0++ meson with a s¯ s content − 4bξ2 2 2 1 − e 2 4b ξ 5ξ ξ ξ 4b 64 4πα s 128 − − − − s e 12b2 − e 24b2 σfs¯si = + e 6b2 + e 8b2 + √ 81m4q ξ 27 3 3 ξ 2 with ξ = s−4MG . The comparison between the cross-sections in the glueball picture and the quark picture for the 0++ meson is given in figure (2).
40 ++
glueball 0 r0=0.58 fm meson ssbar r0=0.58 fm meson ssbar r0=0.71 fm meson ssbar r0=0.8 fm meson ssbar r0=1.0 fm
σ (mb)
30
20
10
0 3.2
3.6
4 1/2
s
4.4
(GeV)
Fig. 2. Cross-section comparison for 0++ with the following parameters β = 0.1, λ = 1.8, k = 0.21, gluon mass m = 0.6GeV. The s¯ s quark model parameters: mq = 0.55 GeV, αs = 0.6.
4. Conclusions In this work we have extended the Fock-Tani Formalism to a hadronic model in which the bound state is composed by bosons. The Cornwall-Soni constituent gluon model has been successful in describing low mass glueballs, in particular the 0++ resonance, which is a isospin zero state. This state can be either represented as a
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q q¯ bound state, a glueball, or a mixture. In the present work we have considered two possibilities for 0++ : a as pure s¯ s and as a glueball. A comparison of the cross-sections reveals that a quark composition for the 0++ implies in a larger rms radius than in the constituent gluon picture. This could represent a criterion for distinguishing between pictures. Acknowledgements The author (M.L.L.S.) was supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ ogico (CNPq). References 1. 2. 3. 4. 5. 6.
Hadjimichef D, Krein G, Szpigel S and Veiga J S da Ann. of Phys. 268 105 (1998). Cornwall J M and Soni A Phys. Lett. 120B, 431 (1983). Hou W S and Soni A Phys. Rev. D29, 101 (1984). Hou W S, Luo C S and Wong G G Phys. Rev. D64, 014028 (2001). Hou W S and Wong G G Phys. Rev. D67, 034003 (2003). Szpigel S Intera¸c˜ ao M´eson-M´eson no Formalismo Fock-Tani. PhD thesis (Doutorado em Ciˆencias) - Instituto de F´ısica, Universidade de S˜ ao Paulo, S˜ ao Paulo, 1995.
