INTRODUCTION

As the Gluon Emission Model (GEM) was first developed by Close (1979) as a description of the formation and decay of vector mesons, perhaps the best place to start to understand the basic features of the GEM is to quote directly (in reduced font within curly brackets) the entire introduction from White (2008), which recounts the basic precepts of Close’s model in some detail, again, as associated with vector mesons. Subsequent to the quotation from White (2008), we will put forth the requisite substitutions designed to make the GEM applicable to the scalar mesons which are the subject of the present work. The introduction from White (2008) follows:

{The details of the Gluon Emission Model (GEM) are explored in detail in Section 4 of ... D. White, International Journal of Theoretical Physics 24 (2), 201 – 216 (1985), but let us now look at some salient features of its development:

In all quantum systems in which natural decay occurs between an excited level and the ground state, the absorption cross-section goes as

σ(ω) = Kα|V|² (1/m)²(1/ω)L(ω),

where K is a constant, ω represents photon frequency, |V|² represents the square of the matrix element descriptive of the photon emission process, the system has mass m, L(ω) is a Lorentz Amplitude with a peak at ω = ω₀ and with a width Γ, and α = (1/137.036) represents the fine structure constant.

Assuming “asymptotic freedom”, i.e., that we may ignore the masses of the decay products (light hadron pairs) in relation to the total energy involved in the system under investigation, we may employ Eq. 1 to predict the width of vector mesons by making the following substitutions to take us from a general quantum electrodynamics (QED) to a specific quantum chromodynamics (QCD) process:

1. We substitute for the photon frequency ω the gluon energy Q₀.
2. We evaluate the right hand side of Eq. 1 at a specific vector meson mass, m_v, i.e., Q₀ = m = m_v. (Hence, the associated Lorentz Amplitude equals unity.)
3. We require |V| ² to be proportional to Σ_i(q_i)⁴, where q_i = quark charge (in units of electron charge magnitude) associated with the quarks comprising the relevant vector meson. (The above criterion is consistent with spin-spin interaction … proportional to q_i², where i denotes quark flavor, giving rise to spin-flip transitions, and the sum is required only in the case of the ρ, as it comprises both the up quark (u) of charge q_u = 2/3 and the down quark (d) of charge q_d = -1/3.)
4. We postulate |V| ² to be proportional to only Σ_i(q_i)⁴, i.e., the precise form of the interaction is universal to all vector mesons in their ground states, except for quark charge differences.
5. We replace α by α_s, the strong coupling parameter, which has the well-known form from QCD gauge invariance theories (see … S. Gasiorowicz and J. L. Rosher, American Journal of Physics 49, 954 & ff (1981)) of:

α_s = B[ln(Q₀/Λ)]^-1, (2)

where B is a constant and Λ is a parameter to be determined. Again, we emphasize that commensurate with the above replacements is that we must assume that the initial energy involved in the formation of a given vector meson is extremely high, i.e., in the “asymptotically free” region of energy space, where the masses of emerging hadron pairs as decay products can be neglected.

At the present juncture it would be helpful to illustrate the main precepts of the Gluon Emission Model of F. E. Close (see … F. E. Close, An Introduction to Quarks and Partons, Academic (1979)). Pertaining to inelastic collisions between electrons and positrons, for example, eventuating in light hadron pair production (i.e., pion pairs and kaon pairs), from page 209 … we read: During hadronization one of the quarks making up one of the pions (or kaons) in the pair production process flips its spin at certain preferred energies (the rest energies of the vector mesons). In the spin-flip process the associated quark emits a gluon …. In this picture the gluon (G) couples directly to [the virtual photon] with unit probability. In the present context the virtual photon referred to in the quote would correspond to one propagating from the collision vertex in a simple Feynman Diagram representative of the colliding beams situation at hand.

Accordingly, then, we find in terms of the above ansatz (normalizing to the ρ)

/Γ_v = A(m_ρ/m_v)³(Σ_i(q_i)⁴)[ln(m_vΛ)]^-1, (3)

where Γ_v represents the width of a given vector meson, v, and A is a constant to be determined.}

Step 2 above is equivalent, of course, to integrating the differential cross-section (see, for example, Merzbacher (1970), p.486), presented as Equation 1 in the direct quotation, over dQ₀δ(Q₀– m_v), thus producing the integrated cross-section, which is then subsequently renormalized into a “stand-alone” width of a given vector meson, v. As a spin-flip of a constituent quark comprising a scalar meson does not occur, the form for the “|V|²” in the afore-mentioned Equation 1 will have to be altered with respect to the assertions made in Step 3, thus making Step 4 moot for the considerations of the present work. As to Step 5, we will employ α_s as derived via the GEM (see White (2008), Equation 7 in Section III), viz.,

α_s = 1.2 [ln(Q₀/50 Mev)]^-1. (1)

In a recently published work (White (2007)) the f₀(600) is looked upon not as a particle, but instead as a distribution of virtual positronium, called “base balls”, assumed to arise by virtue of the zero momentum energy involved in the head-on collisions of particle beams in accelerators, said energy absorbed by the vacuum comprising an infinite lattice structure of negative energy electrons, as would constitute the Dirac Sea of old. In such view the base ball has a mass of 2m_e, where m_e represents the mass of an electron; also, in such view the “f₀(600)” is representative of a collective phenomenon, not of one describeable by a simple “point A to point B” Feynman Diagram involving a destruction/creation operator ansatz. Basically, the “f₀(600)” of White (2007) is looked upon as the “background” upon which the light, unflavored scalar mesons are laid, they being representative of distributions of base balls themselves, but with special characteristics, viz., their comprising spherical or quasi-spherical collections of the afore-mentioned base balls. Specifically, the η(548) is well represented as a spherical collection of “vacuum shock” base balls such that from the base ball at its center there are five equally spaced base balls along each of three mutually orthogonal coordinate axes on out to its perimeter. Assuming the lattice spacing of the base balls to be uniform in all three dimensions, it is easy to show that the mass of the η(548) in accord with such a model is approximately 536 Mev, i.e., to within ~ 2.2% of the experimentally-determined mass of same. Similarly, the η’(958) and η(1475) are each well-represented by a spherical collection of base balls, the η’(958) with six equally spaced base balls from center to perimeter and the η(1475) with seven, their masses approximated by the model to within ~ 3.3% and ~ 0.5%, respectively. Denoting “N” as the relevant number of base balls along a given coordinate axis (from center to perimeter) as associated with the η(548), the η’(958), or the η(1475), in the work which follows we will assume that “|V|²” from Equation 1 of the direct quotation above is a function of N, ultimately leading to a width formula as a function of N applicable to each of the three scalar mesons mentioned above.

DEVELOPMENT OF THE WIDTH FORMULA FOR THE η– SERIES MESONS

To begin, we let m_s represent the mass of a given scalar meson, “m_s”, then, replacing “m_v” in Equation 3 of the direct quote from White (2008) above, as a first step. Secondly, we index “m_s” via “N”, introduced above, such that:

if N = 5, m₅≈ m_s = mass of the η(548) = 548 Mev; (2 a)

if N = 6, m₆≈ m_s = mass of the η’(958) = 958 Mev; (2 b)

if N = 7, m₇≈ m_s = mass of the η(1475) = 1476 Mev. (2 c)

Introducing Γ_N as representative of the width of the associated scalar meson, we then recast Equation 3 from Ref. 2 as

Γ_N≈ A |f(N)|²(m_ρ/m_N)³α_s, (3 a)

which from Equation 1 may be rewritten as

Γ_N≈ 1.2A|f(N)|²(m_ρ/m_N)³ [ln(m_N / 50 Mev)]^-1. (3 b)

For simplicity we now define G(N) = 1.2 A |f(N)|², (4)

so that we can express Γ_N as Γ_N≈ G(N) (m_ρ/m_N)³ [ln(m_N / 50 Mev)]^-1. (5)

Utilizing experimental data from PDG (2004), pp. 2, 5 and 13, we may obtain numerical evaluations for G(N) for N = 5, 6, and 7 as follows:

For N = 5 the experimentally determined width of the η(548) according to the Particle Data Group (PDG) is Γ₅(PDG) = 0.00129 Mev. Accordingly, then, we set 0.00129 Mev = G(5) (776 / 548)³ [ln(548 / 50)]^-1 = G(5) {1.1860}, from which

G(5) = 0.001088 Mev. (6 a)

For N = 6 the experimentally determined width of the η’(958) according to the PDG is Γ₆(PDG) = 0.202 Mev. In like manner as immediately above we set 0.202 Mev = G(6) (776 / 958)³ [ln(958 / 50)]^-1 = G(6) {0.1800}, from which

G(6) = 1.1223 Mev. (6 b)

Lastly, for N = 7 the experimentally determined width of the η(1475) according to the PDG is Γ₇(PDG) = 87 Mev. Therefore, we set 87 Mev = G(7) (776 / 1476)³ [ln(1476 / 50)]^-1 = G(7) {0.042930}, from which

G(7) = 2027 Mev. (6 c)

Having obtained numerical results for G(N), N = 5, 6, and 7, we next plotted x = ln(N!) as the independent variable along a horizontal axis versus y = ln{G(N) / 1 Mev} as the dependent variable, x expressed to five significant figures and y expressed to four decimal places (five significant figures for N = 5 and 7, four significant figures for N = 6), in Microsoft Excel. We then asked Excel for the best linear fit of the data, the equation of the line, and the square of the correlation coefficient between x and y. To five significant figures we obtained the following: y = 3.8626 x – 25.309 with a correlation coefficient, R, equal to 0.999999. Hence, R = 1 to within about one part in one million! Thus,

ln{G(N) / 1 Mev} = 3.8626 ln{N!} – 25.309 essentially exactly. Solving for G(N) yields:

G(N) = (1.0196 x 10^-11)(N!)^3.8626 Mev. (7)

Now, since G(N) = 1.2 A |f(N)|² (Equation 4), we may identify |f(N)|² as A(N!)^3.8626, in which case A = 8.4969 x 10^-12 Mev, and, since α_s = 1.2 [ln(m_N / 50 Mev)]^-1 (Equation 1 applied to the situation at hand), we obtain for Γ_N the following:

Γ_N≈ (8.4969 x 10^-12) α_s (N!)^3.8626 (m_ρ / m_N)3 Mev. (8)

In terms of the electron mass and the speed of light in vacuum, c, (m_ec² = 0.511 Mev) we may express the widths of the η(548) (N = 5), the η’(958) (N = 6), and the η(1475) (N = 7) formally as:

Γ_N≈ (8.3140 x 10^-12) (2m_ec²) α_s (N!)^3.8626 (m_ρ / m_N)³. (9)

APPLICATION OF THE GEM / CTMM WIDTH FORMULA

Utilizing Equation 1, we express α_s in Equation 9 as a function of N as evaluated at the relevant m_N (in units of Mev) as follows: α_s(N) = 1.2 [ln(m_N / 50 Mev)]^-1. In Table 1 below we present for N = 5, 6, and 7 the meson designation, its mass, the associated α_s(N), Γ_N(PDG), and Γ_N resulting from Equation 9.

Table 1: Γ_N as compared to Γ_N(PDG)

As can be seen from Table 1, each Γ_N is well within the uncertainty expressed in the associated Γ_N(PDG), and, as such, constitutes a statistical match thereof, remarkable considering the huge variation in Γ_N from N = 5 to N = 7. The larger the width of a meson, the more unstable it is, of course. From a look at Equation 9 it is evident that the factor “(N!)^3.8626” is the term which leads to the “runaway” width behavior of the η-series scalar mesons considered herein, as such term represents a very rapidly increasing function of complexity in terms of the assumed base ball construction underlying the η-series mesons. What appears to be evident, then, is that, at least as it would pertain to the η-series mesons, the more base balls required to form a given (quasi-spherical) scalar meson, the more unstable it is … to the point that if N = 7, the associated meson has a width of ~ 100 Mev and the presumption, based on the behavior of a factor such as (N!)^{~ 4}, that the N = 8 meson in the η-series would possess a width so large as to be unidentifiable. The latter presumption may, in fact, be tested. The PDG (2004b) has cataloged the η(2190), which corresponds almost exactly to the CTMM’s case of N = 8 (the CTMM predicts a mass of 2192 Mev for N = 8); for said entry the PDG considers its width to be too large for its inclusion in the Meson Table proper. Based upon Equation 9 we may estimate Γ₈ by assuming m₈ = 2192 Mev as follows:

Γ₈≈ (8.3140 x 10^-12) (1.022 Mev) (0.3174) (8!)^3.8626 (776 / 2192)³≈ 73,657 Mev. (10)

To conclusively identify a meson with a width to mass ratio of ~ 34 would indeed be difficult. Of interest as to the above result is that the GEM / CTMM theoretical structure provides a reason why the light, unflavored η-series mesons number only three as identifiable experimentally, though it does indicate an infinite series of η-series mesons does in principle exist, albeit elements in the series of width to mass ratios >> 1 for N ≥ 8.

CONCLUDING REMARKS

That the masses of m₅, m₆, m₇, and m₈ should correspond so closely to the masses of the η(548), the η’(958), the η(1475), and the η(2190), respectively, via (White, 2007, Equation 5) m_N≈ [(4π/3)N³ + 1](2m_ec²) and that the widths of at least the first three can be quantitatively so well represented by a very simple function of N, certainly must be representative of circumstances which have roots in something far beyond mere coincidence. Not only are the readily measureable quantitative aspects of the elements in the η-series mesons easily understood in terms of the functional dependences of m_N and Γ_N on N, but the qualitative aspects of said series are as well determined by the behavior of both m_N and Γ_N, they developed in accord with both the CTMM of Ref. 6 and the GEM. We have already seen, for example, why the η(2190) should be very difficult to identify, thus obtaining a firm understanding as to why the η(1475) should be at the high energy end of the easily identifiable elements in the η-series. At the other end, i.e., the low energy end, one can easily show that m₄ = 275 Mev ≈ (1/2)[2m_π⁰ + 2m_π^±], where m_π⁰ represents the mass of the neutral pion and m_π^± represents the mass of the charged pion, thus showing that the first available (quasi-) spherical distribution of base balls would be of mass m₅, the mass of the η(548).

Central to both the CTMM and the GEM as it is applied to the η-series mesons is, of course, the assumed existence of a cubic lattice structure of negative energy electrons, each of energy -m_ec², a collection of which such that each absorbs energy ~ +2m_ec² in the vicinity of colliding beams, thus leading to a collection of virtual positronium “base balls”, sub-collections of same of quasi- or pure spherical form describing the η-series mesons. In short, the CTMM and subsequent GEM analysis assumes the existence of the Dirac Sea. Contrarily, the “Physics Establishment” (meaning the prevailing view in the literature) has deemed the Dirac Sea as “obsolete”. What that means really, in our view, is that it has become “more convenient” in recent years to work simply with “creation” and “destruction” operators than it is to try to provide manifestly some kind of theoretical framework which would underlie and make more rationally understandable a given physical process in which an “x” is destroyed, leading to the creation of an associated “y”. What led to the CTMM, in fact, was a desire to make understandable a situation which has yet to be understood in terms of the methods employed by the “Establishment” over more than 50 years – the problem of the f₀(600). Since the 1950’s, a controversy has persisted over whether the f₀(600) is a “particle” or not. If the f₀(600) is a particle, then its characteristics can be described by assuming that when two particle beams collide, a virtual photon (or gluon) can be created which “resonates” at the f₀(600)’s mass, is destroyed upon creation of the f₀(600), which then disappears upon creation of a gluon, which then disappears upon coupling to a quark /anti-quark pair, which then undergo “hadronization”, leading to the creation of two emerging pions (primarily). That’s it: destruction of the original particle beams’ identities … creation of a virtual photon or gluon … destruction of the virtual photon or gluon … creation of the f₀(600) … destruction of the f₀(600) … creation of a gluon … destruction of the gluon … creation of a pion pair, i.e., point to point to point interactions involving a series of “particle” entities. The problem remains: no one has been able to show conclusively that the f₀(600) is actually a “particle” entity. Without quoting extensively from White (2007), it is sufficient to say that if no one over the last five decades has been able to prove conclusively that the f₀(600) is a particle, it (1) probably isn’t and (2) it doesn’t have to be for it’s apparent existence to be reasonably well understood, as long as one is willing to use a different mode of description from the creation / destruction operator, point to point interaction approach.

The “different mode of description” mentioned above involves, of course, the Dirac Sea. Since m₄ has the mass, essentially, of two pions, all the features of f₀(600) “creation” and “destruction” are contained within any collection of base balls large enough to constitute enough rest energy subsequently observed historically as the f₀(600), and various light, unflavored scalar mesons would show themselves as unstable (quasi-) spherical collections of base balls characterized by N = 5 to 8 at the highest (see White, 2007 for refinements and exact limits). In the CTMM framework, then, the f₀(600) is simply representative of the virtual photon mass distribution not associated with the known light, unflavored scalar mesons; it constitutes the background for such entities as the resulting virtual photon mass distribution associated with a given collision of original colliding particle beams.

As such, the f₀(600) is symbolic … or representative … of a collective phenomenon, not representative of a point to point phenomenon. Obviously, however, (1) the CTMM (even inclusive of the GEM application to the η-series mesons) cannot offer a complete treatment of the decay of the f₀(600) and/or other light, unflavored mesons on its own, and (2) the point to point interaction approach has shown itself to be an extremely valuable tool for the understanding of a plethora of high energy phenomena. Thus, what we are here advocating is not something like the supplanting of one mode of description over another; rather, we are attempting to make a case for a new kind of complementarity … one in which “collective-ism” complements “particle-ism”, so to speak, in words coined with hopefully well-understood meanings in the present context. With such complementarity, for example, the f₀(600) is seen as identical to the virtual photon propagating from the colliding beams vertex to the pion pair vertex in a simple Feynman Diagram, the virtual photon mass distribution quantized in units of 2m_e, therefore. From the “collective-ism” end, the pion pairs come from the break-up of the m₄“embryo” of mass ~ 2m_π⁰ or ~ 2m_π^±. What about the quarks making up the pions? For the answer to that we may appeal to the routinely used, though still mysterious process from the “particle-ism” end, none other than “hadronization”– a word which means no more than “a destruction operator terminates the f₀(600) and simultaneously creates all quarks necessary and their proper binding modes to create an emerging pion pair”. In the language of “collective-ism”“hadronization” means “the m₄ reconfigures into all quarks necessary and their proper binding to create an emerging pion pair”… essentially the same thing.

What, now, about the Standard Model, which puts forth that (not the f₀(600), because the Standard Model only deals with known particles) the η-series mesons, for example, comprise certain definite quarks? There is no question that the quark assignments for the light, unflavored mesons and beyond are self-consistent in the main and make quite understandable the experimental results of decays of given “particles” in terms of developing a catalogue of reactions, interactions, spontaneous decays, etc. Therein is a point to be made, viz., the Standard Model is heavy on cataloging but light on providing a detailed, underlying view of the physical processes involved in high energy physics. Accepting as complementary to the Standard Model the CTMM with its, in turn, assumption of the existence of the Dirac Sea, we have, we think, something far more dynamic … far more fluid … far more interesting to contemplate than what two quarks make up the η(548), for example.

In any event, the η(548), as is any meson, by definition, an intermediate state between a (hopefully) known initial state and a (hopefully) known final state. The properties of all intermediate states must be inferred; none can be directly observed. What one infers depends on what one’s universe of terms is, in terms of what one develops as one’s descriptive tools. With complementarity one’s set of descriptive tools is not “right”, as opposed to another’s “wrong” set of same. Hence, acceptant of the complementarity we are proposing herein, we are served just as well by considering a given η-series meson as comprising a particular quark structure, which then decays and undergoes hadronization to create emerging pion pairs, as we are by considering that a given unstable m_N (N = 5, 6, or 7) making up a particular virtual photon distribution undergoes hadronization to create emerging pion pairs. If one is interested in a tidy, undetailed, qualitative picture of the process, the Standard Model will suffice. If one is interested in a quantitative picture as to what may underlie the masses of the m_N mentioned above and their associated widths – a full, quantitative picture, therefore – then the CTMM / GEM will suffice, but it in turn must be recognized that the CTMM and, indeed, the GEM were born of the Standard Model and all of its precepts. Even in representing the decay of the f₀(600) or, as well, any of the considered η-series mesons via the descriptive tools of the CTMM / GEM, at the hadronization stage the destruction / creation operator concept is still an indispensible element, just as it is in a purely qualitative sense using the language of “pure” QCD, i.e., language devoid in considerations involving the Dirac Sea. At any rate it seems to us that until hadronization is more fully understood – if indeed it is a phenomenon which will ever lend itself to being more fully understood – the Dirac Sea must be reinstated as part of a viable descriptive framework to better understand the f₀(600) and certain of the light, unflavored scalar mesons at the very least.

REFERENCES

1) Close, F. (1979) An Introduction to Quarks and Partons, Academic Press.

2) White, D. (2008) “The Gluon Emission Model for Hadron Production Revisited”, Journal of Interdisciplinary Mathematics, 11 (4), pp. 543 - 551.

3) White, D. (1985) “Calculation of the Strong Coupling Constant, α_s, from Considerations of Virtual Synchrotron Radiation Resulting in Hadron Pair Emission”, International Journal of Theoretical Physics, 24 (2), p. 210.

4) Gasiorowicz, S. and Rosher, J. L. (1981) “Hadron Spectra and Quarks”, American Journal of Physics, 49, p. 954.

5) Merzbacher, E. (1970) Quantum Mechanics, Wiley, p. 486.

6) White, D. (2007) “A Mathematical Model for the f₀(600) based on Chaos Theory”, Journal of Interdisciplinary Mathematics, 10 (5), pp. 625 – 635.

7) PDG (2004) “Mesons”, accessed online November 7, 2008, http://pdg.lbl.gov/2004/tables/mxxx.pdf.

8) PDG (2004b) “Further States”, accessed online November 7, 2008, http://pdg.lbl.gov/2004/listings/m300.pdf.

This article was originally published in the Journal of Applied Global Research. It is reprinted here courtesy of Dr. David King and the Intellectbase International Consortium.