Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 325–330 c Chinese Physical Society and IOP Publishing Ltd
Vol. 53, No. 2, February 15, 2010
Ground State Meson Spectrum in a Relativistic Model with Instanton Induced Interaction Antony Prakash Monteiro and K.B. Vijaya Kumar∗ Department of Physics, Mangalore University, Mangalagangothri, Mangalore 574199, India
(Received February 19, 2009; revised manuscript received October 9, 2009)
Abstract The mass spectrum of the S-wave mesons is considered in the frame work of relativistic harmonic model (RHM). The full Hamiltonian used in the investigation has the Lorentz scalar plus a vector harmonic-oscillator potential, the confined-one-gluon-exchange potential (COGEP) and the instanton-induced quark-antiquark interaction (III). A good description of the mass spectrum is obtained. The respective role of III and COGEP in the S-wave meson spectrum is discussed. PACS numbers: 14.40.-n, 14.40.Aq, 14.40.Ev, 12.39.-x, 12.39.Ki
Key words: quark model, confined-one-gluon-exchange potential, instanton induced interaction, S-wave spectra
1 Introduction It is accepted that quantum chromodynamics (QCD) is the theory of strong interactions. The QCD is not exactly solvable in the non-perturbative regime which is required to obtain the physical properties of the hadrons. Hence various approximation methods have been employed to solve QCD in the non-perturbative regime. The most promising of these is through lattice gauge theories.[1−3] The lattice gauge theories involve gigantic computation hence the progress has been slow and detailed predictions of the hadron properties have not been made. As a consequence, our understanding of hadrons continues to rely on insights obtained from the experiments and QCD motivated models in addition to lattice QCD results. The phenomenological models developed to explain observed properties of hadrons are either non-relativistic quark models (NRQM) with suitably chosen potential or relativistic models where the interaction is treated perturbatively. The NRQM have been proven to be very successful in describing hadronic properties.[4−12] In most of these works, it is assumed that the quark interaction is dominated by a linear or quadratic confinement potential and is supplemented by a short range potential stemming from the one-gluon exchange mechanism. The Hamiltonian of these quark models usually contains three main ingredients: the kinetic energy, the confinement potential, and a hyperfine interaction term, which has often been taken as an effective one-gluon-exchange potential (OGEP).[13] Other types of hyperfine interaction have also been introduced in the literature. For example the InstantonInduced Interaction (III), deduced by a non-relativistic reduction of the ‘t Hooft interaction[14−15] has already been successfully applied in several studies of the hadron spectra. The main achievement of III in hadron spectroscopy is the resolution of the UA (1) problem, which ∗ Corresponding
author, E-mail:
[email protected]
leads to a good prediction of the masses of η and η ′ mesons. The Goldstone–Boson–Exchange interaction introduced by Glozman and Riska furnishes another example of hyperfine interaction; it allows a good description of the baryon spectrum, and yields, in particular, a correct ordering for the positive and negative parity states.[16−19] However, the model of Glozman and Riska can only be applied to study baryons and is thus unable to provide a unified description of the spectrum of hadrons. Very recently, the light cone harmonic oscillator models have been employed to study meson spectra and have been found to be reasonably successful.[20−22] The success of the NRQM in describing the hadron spectrum is some what paradoxical, as light quarks should in principle not obey a non-relativistic dynamics. This paradox has been avoided in many works based on the constituent quark models by using for the kinetic energy term of the Hamiltonian a semi-relativistic or relativistic expression.[18−19,23] Even in the existing relativistic models though the effect of confinement of quarks has been taken into account, the effect of confinement of gluons has not been taken into account.[24] In our previous works,[24−26] we had investigated ground state masses of mesons in the frame work of NRQM and relativistic harmonic model (RHM) using III. Our conclusion from our previous works is, in addition to the perturbative conventional OGEP derived from QCD, III interactions are needed to keep strong coupling constant (αs ) < 1. If OGEP is taken as the only source of hyperfine interaction the value of αs necessary to reproduce the hadron spectrum is generally much larger than one; this leads to a large spin-orbit interaction, which destroys the overall fit of the spectrum. The inclusion of III diminishes the relative importance of OGEP for the hyperfine splittings. The aim of our earlier investigations
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Antony Prakash Monteiro and K.B. Vijaya Kumar
was also to test whether αs can be treated as a perturbative effect and to understand the role played by the III in meson spectra. Though the results obtained for masses of ground state mesons were satisfactory in both NRQM and RHM with III and OGEP, the effect of confinement of gluons in these models have not been taken into account. In our present work, we have investigated the effect of exchange of confinement of gluons on the masses of light mesons and their radially excited states in the frame work of RHM with III.[25,27] The essential new ingredient in our investigation of the mesonic states is to take into account the confinement of gluons in addition to the confinement of quarks. In the existing quark models, Fermi–Breit interaction which gives rise to π–ρ and N-∆ splitting is treated as perturbation. The OGEP being attractive for π, and for a nucleon a naive perturbative treatment of one gluon hyperfine interaction is incorrect and hence one obtains a high value for the pion mass. This leads to further renormalization of strength of interaction for a better fit. Also, the most prominent flaw of NRQM is the neglect of relativistic effects and gluon dynamics. In our present work, for the confinement of quarks we are making use of the RHM which has been successful in explaining the properties of light hadrons. For the confinement of gluons we have made use of the current confinement model (CCM) which was developed in the spirit of the RHM.[28−29] The CCM has been quite successful in describing the glue-ball spectra. The confined gluon propagators (CGP) are derived in CCM. Using CGP we have obtained confined one gluon exchange potential (COGEP). The full Hamiltonian used in the investigation has Lorentz scalar plus a vector harmonic-oscillator potential, in addition to two-body COGEP and III. In our earlier work, it was shown that the terms in COGEP arising out of confinement of gluons give the required intermediate range attraction in the nucleon-nucleon interaction. Also, we had obtained good n-p and p-p differential cross sections.[30] In our present work, the total mass of the meson is obtained by calculating the energy eigenvalues of the Hamiltonian in the harmonic oscillator basis spanned over a space extending upto the radial quantum number nmax = 5. The masses of the ground state mesons are obtained after diagonalising for various values of nmax . In Sec. 2, we review the RHM and CCM models and give a brief description of CGP. The COGEP is obtained using CGP. The central parts of III used in the interaction are explained. We also discuss the parameters involved in our model. The results of the calculations are presented in Sec. 3. Conclusions are given in Sec. 4.
2 Relativistic Harmonic Model In RHM,[25,27] quarks in a hadron are confined through the action of a Lorentz scalar plus a vector harmonic-
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oscillator potential 1 (1 + γ0 ) A2 r2 + M , (1) 2 where γ0 is the Dirac matrix: 1 0 γ0 = , (2) 0 −1 M is the quark mass and A2 is the confinement strength. They have a different value for each quark flavour. In RHM, the confined single quark wave function (ψ ) is given by: ! φ ψ=N , (3) σ·P φ E+M with the normalization r 2(E + M ) N= , (4) 3E + M where E is an eigenvalue of the single particle Dirac equation with the interaction potential given in (1). The lower component is eliminated by performing the similarity transformation, Vconf (r) =
Uψ = φ , where U is given by, 1
1
(5) σ·P E+M . 1
(6) i σ·P P2 − E+M (E + M )2 Here, U is a momentum and state (E) dependent transformation operator. With this transformation, the upper component φ satisfies the harmonic oscillator wave equation. h P2 i + A2 r2 φ = (E − M )φ , (7) E+M which is like the three dimensional harmonic oscillator equation with an energy-dependent parameter Ω2n : Ωn = A(En + M )1/2 . (8) h N 1+
The eigenvalue of (7) is given by, En2 = M 2 + (2n + 1)Ω2n . (9) Note that Eq. (7) can also be derived by eliminating the lower component of the wave function using the FoldyWouthuysen transformation as it has been done in [27]. Adding the individual contributions of the quarks we obtain the total mass of the hadron. The spurious centre of mass (CM) is corrected[31] by using intrinsic operators P P for the i ri 2 and i ∇2i terms appearing in the Hamiltonian. This amounts to just subtracting the CM motion zero point contribution from the E 2 expression. It should be noted that this method is exact for the 0S-state quarks as the CM motion is also in the 0S state. The quark-antiquark potential is the sum of COGEP and III potential. Vq (r) = VCOGEP (r) + VIII (r) . (10) COGEP is obtained from the scattering amplitude[29] λbj g2 λa ab Mij = s ψ¯i γ µ i ψi Dµν (q)ψ¯j γ ν ψj , (11) 4π 2 2
No. 2
Ground State Meson Spectrum in a Relativistic Model with Instanton Induced Interaction
where ψ¯ = ψ + γ0 , ψi/j are the wave functions of the quarks ab in the RHM, Dµν = ∂ab Dµν are the CCM gluon propagators in momentum representation, gs2 /4π (= αs ) is the quark-gluon coupling constant and λi is the color SU(3)c generator of the i-th quark. In CCM,[28,32] the coupled non-linear terms in the equation of motion of a gluon are simulated by a selfinduced color current jµ = θµν Aν (= m2 Aµ ) or equivalently an effective mass term for all the gluons with m2 = c4 r2 −2c2 δµ0 . The equations of motion: Aµ +m2 Aµ = 0 are easily solved by using harmonic oscillator modes in the gauge ∂ µ Aµ = 0. The consistency of ∂ µ Aµ = 0 and ∂ µ jµ = ∂ µ (m2 Aµ ) = 0 imposes a secondary gauge condition: ∇ · A + c2 r · A = a · A termed “oscillator gauge”, where a is the usual harmonic oscillator annihilation operator. The propagators are then obtained very simply using the properties of harmonic oscillator wave functions as follows: E D c ′ D1 (r, r′ , E = 0) ≡ r r 2a · a+ + 3 X ψ ∗ (r)ψN (r′ ) N =c . (12) 2N + 3 {N }
Transferring the source point (r) to the origin we obtain (r − r′ → r) X ψN (r)ψN (0) D1 (r, 0, E = 0) ≡ D1 = c 2N + 3 {N }
=
Γ3/4 c(cr)−3/2 W0;−1/4 (c2 r2 ). (4π)3/2
(13)
Similarly, D0 (r, 0, E = 0) ≡ D0 = c
X ψN (r)ψN (0) , 2N + 1
{N }
Γ1/4 = c(cr)−3/2 W1/2;−1/4 (c2 r2 ), (14) (4π)3/2 cent VCOGEP (r) =
where the W’s are Whittaker functions (∼ exp[−(rc)2 /2]/r). The complete propagators are given by D00 (r) = 4πD0 (r) , (15) where D0 (r) is given by Eq. (14). The Dik (r) is given by a+ ak Dik (r) = 4π δik − i + D1 (r) , (16) a·a where D1 is given by Eq. (13). It should be noted that these propagators are similar to those given by Feynman et al.[33] apart from the time coordinate which is suppressed here. The closed analytical expression for D0 (r) and D1 (r) were obtained in a translationally invariant ansatz.[28] We perform a similarity transformation on ψ and ψ + and express ψ and ψ + in terms of φ and φ+ . The details can be found in Refs. [28–29, 34]. For example ψi+ ψi = ψi+ U + (U + )−1 U −1 U ψi + −1 −1 = φ+ U φi . i (U )
Similarly, + −1 ψi+ αi ψi = φ+ αi U −1 φi , i (U )
where α1 = 1.035 994, α2 = 2.016 150 fm−1 , c0 = (3.001 453)1/2 fm−1 , γ = 0.863 933 6, and c2 = (4.367 436)1/2 fm−1 .
(18)
where hN andi U are given by Eqs. (4) and (6) respectively. 0 σ αi = σi 0i where the σi ’s are the usual Pauli matrices. With this transformation the two components of ψ are eliminated without any approximation. The scattering amplitude is now expressed in terms of the two-component spinor φ and the momentum dependent operator U . After substituting φ and U in the expression for the scattering amplitude (Eq. (11)), it essentially corresponds to the Born amplitude in the momentum representation 1 +˜ Mij = αs N 4 ϕ+ (19) i ϕj U ϕi ϕj λi · λj . 4 By taking the Fourier transformation of each term in the Born amplitude, we obtain COGEP. The central part of COGEP is[34]
h i αs N 4 1 λi · λj D0 (r) + [4πδ 3 (r) − c4 r2 D1 (r)][1 − 2/3σi · σj ] . 2 4 (E + M )
To calculate the matrix elements (ME) of COGEP, we have fitted the exact expressions of D0 (r) and D1 (r) by Gaussian functions. It is to be noted that the D0 (r) and D1 (r) are different from the usual Coulombic propagators. However, in the asymptotic limit (r → 0) they are similar to Coulombic propagators and in the infra-red limit (r → ∞) they fall like Gaussian. α h −r2 c2 i 1 0 D0 (r) = + α2 exp , r 2 h 2 2i γ −r c2 , D1 (r) = exp r 2
327
(20)
We have used the following form for III potential.[9,18] −8gδ(r)δS,0 δL,0 , for I = 1, −8g ′ δ(r)δ δ , for I = 1/2, S,0 L,0 VIII = (21) √ ′! g 2g 8 √ δ(r)δS,0 δL,0 , for I = 0. 2g ′ 0 The symbols S, L, and I are respectively the spin, the relative angular momentum, and the iso-spin of the system. The g and g ′ are the coupling constants of the interaction. The Dirac delta-function appearing has been regularized and replaced by a Gaussian-like function: 2 1 r δ(r) → √ 3 exp . (22) (Λ π) Λ2 The parameters of RHM are listed in Table 1.
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Antony Prakash Monteiro and K.B. Vijaya Kumar
Table 1 Values of the parameters used in our model. b
0 .77 fm
Mu,d
251.3 MeV
Ms
427 MeV
αs.
0.2
Λ
0.35 fm
c
1.74 fm−1
g
0.1425 × 10−4 MeV−2
g′
0.095 85 × 10−4 MeV−2
3 Results and Discussion In the present study, the product of quark-antiquark oscillator wave functions is expressed in terms of oscillator wave functions corresponding to the relative and CM coordinates using the Moshinsky transformations.[35] The oscillator size parameter b is fixed by minimizing the expectation value of the Hamiltonian for the vector mesons. We constructed 5 × 5 Hamiltonian matrix for both pseudoscalar and vector mesons in the harmonic oscillator basis. To fit αs , c, and Mu (mass of u quark), we started with a set of reasonable values of the above parameters and diagonalized the 5 × 5 matrix for ρ meson. Having fitted αs , c, and Mu , we fitted g, g ′ , and Λ to the experimental mass of the π meson. The additional term in COGEP (c4 r2 D1 (r)) is arising exclusively from the effect of confinement of gluons. For 1 S0 state, the color magnetic part (CMP) of the above term is repulsive and the overall contribution of the CMP of COGEP is also repulsive. If we carry out the diagonalisation for pseudoscalar mesons with nmax = 5, the physical masses of pseudoscalar mesons agree with the experimental masses. In our previous work, using NRQM,[26] it was required to diagonalise 11 × 11 matrix to obtain the physical mass of the pseudoscalar mesons. Hence, the additional term in COGEP arising out of confinement of gluons plays a significant role in restricting the harmonic oscillator basis to radial quantum number nmax = 5. Also, further increase of the oscillator basis does not lead to any significant change in the masses for pseudoscalar mesons. It is to be noted that the contribution of III is attractive for pseudoscalar mesons. Hence the attractive contribution of III and the CMP of δ 3 (r) term is partly canceling the CMP of c4 r2 D1 (r) term. The masses of the pseudoscalar and vector mesons after diagonalisation for successive values of nmax are listed in Tables 2 and 3 respectively. Respective diagonal contributions of III, chromo-electric, and chromo-magnetic parts of COGEP are listed in Table 4. The successive diagonalisation in the space of radially excited light meson states brings down the value of meson mass to its physical mass. For example: with nmax = 1 the naive mass of Kmeson is 733.88 MeV. After diagonalising 5 × 5 matrix the
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physical mass of the meson turns out to be 495.42 MeV which is in good agreement with the experimental mass of the K meson. If we carry out the diagonalisation for K meson without the III, the na¨ıve mass of K meson turns out to be 903.67 MeV for the parameters specified in Table 1. In order to reproduce experimentally observed masses without the III contribution, we need to choose αs = 1.65 which is contrary to our proposition that the value of αs should be consistent with the theory of perturbation. It is to be noted that III does not contribute to the masses of the vector mesons. The CMP of δ 3 (r) and c4 r2 D1 (r) are repulsive and attractive respectively and almost cancel each other (Table 4). Hence the masses of vector mesons do not change appreciably when diagonalisation is carried out in a larger basis. Hence the convergence can be achieved in a smaller basis for ρ mesons (Table 3). Table 2 The pseudoscalar meson masses (in MeV) for successive values of nmax . nmax
π
K
1
516.88
733.88
2
327.34
611.73
3
164.65
511.99
4
142.56
498.28
5
138.15
495.42
Expt.
139.57
493.67
Table 3 The vector meson masses (in MeV) for successive values of nmax . nmax
ρ
K∗
φ
1
772.12
898.71
1023.63
2
770.03
896.03
1020.4
3
769.65
895.51
1019.79
4
769.58
895.43
1019.71
5
769.57
895.42
1019.69
Expt.
771.1
893.14
1019.42
Table 4 The contributions of III, the chromo-electric (CE) and chromo-magnetic (CM) parts of COGEP to the 0S state in MeV. Meson
III
CE-COGEP
CM-COGEP
π
–260.02
–30.88
3.58
K
–174.89
–30.83
7.55
η
–138.8
–97.54
33.58
η′
270.54
–97.58
33.58
ρ
–30.88
–1.19
K∗
–30.83
–2.52
φ
–35.66
–0.62
No. 2
Ground State Meson Spectrum in a Relativistic Model with Instanton Induced Interaction
In Tables 5 and 6 the values of αs required to obtain the experimental masses of mesons for various values of nmax are given, where rest of the parameters are taken from Table 1. Here we have noted that all the masses converge reasonably well to respective experimental values for αs = 0.2. For smaller configuration space, the value of αs needs to be increased. This also indicates that perturbative techniques are fully adequate and are justified. It is to be noted that COGEP acts both on spin singlet and triplet systems whereas III acts only on the spin singlet systems. Table 5 The masses of pseudoscalar mesons (in MeV) when diagonalisation is carried out for different values of αs. . The rest of the parameters are taken from Table 1. nmax
αs.
π
K
1
2.97
138.87
411.51
2
0.96
138.1
419.77
3
0.288
138.1
483.85
4
0.21
139.41
494.92
5
0.2
138.15
495.42
Table 6 The masses of vector mesons (in MeV) when diagonalisation is carried out for different values of αs. . The rest of the parameters are taken from Table 1. nmax
αs.
ρ
K∗
φ 1019.09
1
0.225
768.11
894.54
2
0.22
766.41
892.15
1016.12
3
0.215
766.86
892.51
1016.49
4
0.21
767.73
893.43
1017.5
5
0.2
769.58
895.41
1019.69
Table 7 Values of the parameters used in our model for η and η ′ .
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To obtain the masses of η and η ′ mesons, we calculated the ME for η mesons for 0S state. To obtain the physical mass of η and η ′ mesons, III is added by diagonalising Eq. (21). Equation (21) mixes pseudo-scalar iso-singlet η0 and pseudo-scalar iso-octet η8 . The III lowers the mass of the η0 state while it pushes up the mass of the η8 . The calculated ground state masses for η and η ′ using the parameters of Table 7 are 547.24 MeV and 956.58 MeV respectively. The III contributions are listed in Table 4.
4 Conclusions In this work, we have investigated the effect of III and the effect of confinement of gluons on the ground state masses of light mesons in the frame work of RHM. In our earlier works, we had obtained the masses of ground state mesons using NRQM[26] and RHM with OGEP and III.[25] In NRQM, to obtain the experimental masses of the observed pseudoscalar mesons it was necessary to diagonalise 11 × 11 matrix keeping αs < 1. In both NRQM and RHM the CMP of OGEP was attractive. In our present model, we have an additional term in one gluon interaction which arises solely due to using CGP. For pseudoscalar mesons, the CMP of COGEP arising from the confinement of gluons is repulsive which compensates to a large extent the attractive contribution of III and hence it is sufficient to diagonalise in a smaller basis. For vector mesons, the contribution from the CMP of c4 r2 D1 (r) almost cancels the contribution from δ 3 (r) term. Hence, to obtain the masses of vector mesons COGEP is sufficient. Hence, it is justified to use a combination of COGEP and III potentials to obtain the ground state meson masses of both pseudoscalar and vector mesons. However, the convergence is achieved in a smaller basis for vector mesons. To obtain the physical masses of η and η ′ mesons it is necessary to include III. Hence, it is justified to use a combination of COGEP with a smaller strength (compatible with perturbative treatment) and III to obtain the physical masses of the mesons. The calculations clearly show that the computation of mesonic masses using only COGEP is inadequate in the case of pseudoscalar mesons. The above work could be extended to investigate the masses of P - and Dwave mesons. The work in this direction is in progress.
b
0.77 fm
Mu,d
251.3 MeV
Ms
427 MeV
αs
0.7
Acknowledgement
Λ
0.35 fm
c
1.74 fm−1
g
0.0361 × 10−4 MeV−2
g′
0.0531 × 10−4 MeV−2
One of the authors (APM) is grateful to the DST, India, for granting the JRF. The other author (KBV) acknowledges the DST for funding the project (Sanction No. SR/S2/HEP-14/2006).
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