Evidence for Color-by-Color Disengagement from the Process of Lepton Production Associated with the ψ-Series and Y-Series Mesons

I. Introduction

In White (2010-f) and White (2010-y) it is shown that form factors (f_i), which represent the fraction of quark (Q)/anti-quark (Q*) states, originally produced in the formation of a given vector meson, that make a transition to a QQ* state comprising quarks of the next lowest mass, figure prominently in the width calculations of said vector mesons via the Gluon Emission Model (GEM) theoretical structure describing the formation and decay of vector mesons. Specifically, it is found that for the ψ-series mesons, characterized by i = 1,

f₁ = (1-q_s²) = (8/9) , (1a)

where q_s = -1/3 represents the charge of the strange (s) quark, and for the Y-series mesons more massive than the Y(1S), characterized by i = 2,

f₂ = (1-q_c²) = (5/9) , (1b)

where q_c = 2/3 represents the charge of the charm (c) quark;

for the Y(1S) f₃ = 1 . (1c)

In the present work we will be almost exclusively concerned with the electron/positron (ee) partial widths (from which the above form factors are readily obtainable) associated with various vector mesons in the ψ- and Y-series. The role that form factors play in the ee partial width calculations (see White (2010-f) & White (2010-y) for details) is exhibited below. Denoting the ee partial width associated with vector meson “X” of mass “M_X” by “Γ_ee”, we find from White (2010-y), for example, where “f_x” represents the appropriate form factor:

Γ_ee = f_x [ (α /2π)(10,042)(2m_e)(m_ρ/M_X)³(q_z)⁴ + (1-f_x){(α /2π)(10,042)(2m_e)(m_ρ/M_X)³(q_x)⁴}] . (2)

In Eq. 2 α = (1/137.036) represents the fine structure constant, m_e = 0.511 Mev represents the mass of the electron, m_ρ = 776 Mev represents the mass of the ρ-meson, q_x represents the charge associated with the original QQ* state comprising X, and q_z represents the charge associated with the quark of next lowest mass relative to the original QQ* structure. Specifically, then, for the ψ-series mesons

Γ_ee(ψN)=f₁[(α /2π)(10,042)(2m_e)(mρ/M_ψN)³(q_s)⁴+(1-f₁){(α/2π)(10,042)(2m_e)(m_ρ/M_ψN)³(q_c)⁴}], (3a)

where N represents the series designate in terms of an identification as per “ψ(NS)”. Similarly,

Γ_ee(YN)=f₂[(α/2π)(10,042)(2m_e)(m_ρ/M_YN)³(q_c)⁴+(1-f₂){(α/2π)(10,042)(2m_e)(m_ρ/M_YN)³(q_b)⁴}], (3b)

where q_b = -1/3 represents the charge of the bottom (b) quark, describes the ee partial width associated with the Y-series mesons. Hadronic partial widths may also be determined for the ψ-series mesons by replacing “α” by α_s = 1.2[ln(M_ψN/50 Mev)]^-1, the strong coupling parameter derived via the GEM (see White (2010-f), White (2010-y), White (2009-a)), and carrying out the relevant calculations associated with appropriate Feynman Diagrams. For example (see White (2010-y)), the hadronic width of the ψ(1S) is determined via the GEM to be 82.0 Kev, which represents nearly an exact match to the most recent report of the Particle Data Group (PDG) of (81.7 ± 1.8) Kev (see PDG (2009-M)). The Y(1S) hadronic width is calculated via the GEM (White (2010-y) again nearly exactly in accord with experiment assuming f_i = f₃ = 1 and the existence of a two-gluon route of decay, as the GEM yields 50.09 Kev compared to the PDG (2009-M) report of (49.99 ± 1.16) Kev. Hence, the GEM is, in general, extremely successful in determining the hadronic widths of the ψ-series and Y-series mesons. The form factors, f_i, however, are derived from observation of the relevant leptonic partial widths. We may calculate f₁ and f₂ directly, in fact, through utilization of Eq. 3a and 3b, respectively, by inserting the appropriate experimental data into said equations and solving for f_i. In each case a quadratic equation is obtained whose solutions are exhibited below:

f₁ = (1/32C)[17C + {289C² – 64CΓ_ee(PDG)}^1/2 ] ; (4a)

f₂ = (1/2C) [17C – {289C² – 4CΓ_ee(PDG)}^1/2] . (4b)

In Eq. 4 Γ_ee(PDG) represents the ee partial width as reported in PDG (2009-M) and

C = (α /2π)(10,042)(2m_e)(m_ρ/M_X)³(1/3)⁴ . (5)

Thus, for i = 1 M_X = M_ψN , and for i = 2 or 3 M_X = M_YN .

In Section II we will calculate f₁ as associated with the ψ(1S), the ψ(2S), the ψ(4040), the ψ(4160), and the ψ(4415) assuming, at first, for each one all three quark colors are participating in lepton production. We shall find, however, that f₁ = (8/9) is realized in the case of the ψ(2S) only if two quark colors are operative in lepton production; in the cases of the ψ(4040), the ψ(4160), and the ψ(4415), f₁ = (8/9) is realized only if one quark color is operative in lepton production. In Section III we will calculate f₂ as associated with the Y(2S), the Y(3S), the Y(4S), the Y(10860), and the Y(11020). The leptonic widths of the Y-series mesons are not as precisely determined as those of the ψ-series, but we believe we can demonstrate that the same sequential color disengagement from lepton production as seen in the ψ-series is at work in the Y-series. All experimental data in Sections II and III (and Section IV) comes from PDG (2009-M).

II. Calculation of f₁ , the Form Factor of the Ψ-Series

A. The Ψ(1S)

Assuming that all quark colors are operative in the leptonic decay mode of the Ψ (1S), as the GEM as presented so far does, and inserting appropriate experimental quantities (with M_Ψ1 = 3097 Mev) into Eq. 3a yields

Γ_ee(GEM) = 5.72 Kev,

where Γ_ee(GEM), here and onward, represents the relevant ee partial width as determined via the GEM. The PDG (2009-M) reports

Γ_ee(PDG) = (5.55 ± 0.16) Kev.

Thus, Γ_ee(GEM) is just out of the experimental range of the currently accepted ee partial width of the Ψ(1S). Turning now to Eq. 4a, in which C = 2.3149, we find:

f₁(Ψ1) = 0.8951 = (8/9){1.0070} . (6)

Hence, the form factor associated with the Ψ(1S), viz., f₁(Ψ1), is seen to be calculated as very nearly the GEM-theoretical value of f₁ = (8/9), thus indicating that, indeed, all quark colors do participate in the leptonic decay mode of the Ψ(1S).

B. The Ψ(2S)

Again, assuming that all quark colors are operative in the leptonic decay of the Ψ(2S) and inserting the appropriate experimental quantities into Eq. 3a (with M_Ψ2 = 3686 Mev), we obtain

Γ_ee(GEM) = 3.39 Kev,

whereas Γ_ee(PDG) = (2.36 ± 0.04) Kev.

We first note that if we were to assume that each quark color contributes the same amount to Γee and that one color has disengaged from the process of lepton production, then

Γ_ee(GEM) → (2/3)3.39 Kev = 2.26 Kev,

which, again, lies just outside the PDG’s range. Now, again assuming a two color contribution to Γee , C = (2/3)[1.3731] = 0.9154, from which we find

f₁(Ψ2) = 0.8792 = (8/9){0.9891} . (7)

Basing our confidence level on the relative closeness of f₁(Ψ2) to “(8/9)”, we are ~ 99% sure that

f₁(Ψ2) = f₁ = (8/9)

and, in turn, ~99% sure that one quark color has disengaged from lepton production as associated with the Ψ(2S).

C. The Ψ(4040) [Our Designation: Ψ(3S)]

Again, assuming that all quark colors are operative in the leptonic decay of the Ψ(3S) and inserting the appropriate experimental quantities into Eq. 3a (with M_Ψ3 = 4039 Mev), we obtain

Γ_ee(GEM) = 2.58 Kev,

whereas Γ_ee(PDG) = (0.86 ± 0.07) Kev.

If we now assume that two colors are now inoperative, i.e., only one color participates in lepton production associated with the Ψ(3S),

Γ_ee(GEM) →(1/3) 2.58 Kev = 0.86 Kev,

which represents an exact match to Γ_ee(PDG). Commensurately, C = (1/3)[1.0437] = 0.3479, which translates to

f₁(Ψ3) = 0.8886 = (8/9){0.9997}. (8)

We are essentially 100% confident, therefore, that now two quark colors have disengaged from lepton production by the time we reach the (assumed) “3S” level.

D. The Ψ(4160) [Our designation: Ψ(4S)]

Again, assuming that all quark colors are operative in the leptonic decay of the Ψ(4S) and inserting the appropriate experimental quantities into Eq. 3a (with M_Ψ4 = 4153 Mev), we obtain

Γ_ee(GEM) = 2.37 Kev,

whereas Γ_ee(PDG) = (0.83 ± 0.07) Kev.

If we again assume that only one color participates in lepton production associated with the Ψ(4S),

Γ_ee(GEM) →(1/3) 2.37 Kev = 0.79 Kev,

which represents a statistical match to Γ_ee(PDG). Commensurately, C = (1/3)[0.9600] = 0.3200, which translates to

f₁(Ψ4) = 0.8778 = (8/9){0.9876} . (9)

We are therefore ~99% confident that only one color participates in lepton production as associated with the Ψ(4S).

E. The Ψ(4415) [Our designation: Ψ(5S)]

Again, assuming that all quark colors are operative in the leptonic decay of the Ψ(5S) and inserting the appropriate experimental quantities into Eq. 3a (with M_Ψ5 = 4421 Mev), we obtain

Γ_ee(GEM) = 1.96 Kev,

whereas Γ_ee(PDG) = (0.58 ± 0.07) Kev.

If we again assume that only one color participates in lepton production associated with the Ψ(5S),

Γ_ee(GEM) →(1/3) 1.96 Kev = 0.65 Kev,

which again represents a statistical match to Γ_ee(PDG). Commensurately, C = (1/3)[0.7958] = 0.2653, which translates to

f₁(Ψ5) = 0.9128 = (8/9){1.0269} . (10)

We are therefore ~97% confident that only one color participates in lepton production as associated with the Ψ(5S).

F. Short Summary of Section II

Of great interest is the fact that the average number multiplying “(8/9)” in Eq. 6 – 10 is given by (to three significant figures)

〈n〉= 1.00 ± 0.01 , (11)

which suggests with very high confidence that (1) f₁ = (8/9) for all Ψ-series objects and that (2) sequential quark color disengagement from the process of lepton production takes place with increasing energy in the Ψ-series mesons. Chart 1 below illustrates the phenomenon as per Section II.

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III. Calculation of f₂ , the Form Factor of the Y-Series

For the exposition in Section III to follow, it proves to be convenient to add a subscript, j = 1, 2, or 3, to certain previously-defined variables, where “j” indicates the number of quark colors assumed to be operative in lepton decay. For example, “Γ_ee1(GEM)” indicates a leptonic width per the GEM theory assuming only one color participates in ee production, and “f₂₁” indicates the associated form factor, f₂ , under such condition. All calculations follow along the lines seen in Section II with Eq. 3a and 4a supplanted by Eq. 3b and 4b, respectively. In addition, as the methodology for determining leptonic widths and form factors is by now clear, our presentation in the present section will not feature the entire set of detailed information put forth in Section II.

A. The Y(2S)

Inserting M_Y2 = 10023 Mev into Eq. 3b, we obtain

Γ_ee3(GEM) = 0.624 Kev,

a figure just outside of the experimental range, given by

Γ_ee(PDG) = (0.612 ± 0.011) Kev.

The resulting form factor is found from Eq. 4b to be

f₂₃ = 0.5446 = (5/9){0.9803}. (12)

We are thus ~98% sure that three colors are operative in the leptonic decay of the Y(2S) and that f₂ = (5/9).

B. The Y(3S)

Inserting M_Y3 = 10355 Mev into Eq. 3b, we obtain

Γ_ee3(GEM) = 0.566 Kev,

a figure well outside of the experimental range, given by

Γ_ee(PDG) = (0.443 ± 0.008) Kev.

We find Γ_ee2(GEM) = 0.377 Kev, which is well below the above experimental range of values. However, we do note that

〈Γ_ee3+2(GEM)〉= (1/2)[ Γ_ee3(GEM) + Γ_ee2(GEM)] = 0.471 Kev, a figure which is

only ~6% discrepant with experiment. Furthermore, we find

〈f₂〉= (1/2)[f₂₃ + f₂₂] = (1/2)[0.4217 + 0.6565] = 0.5441 = (5/9){0.9794} (13)

Hence, we obtain the same confidence level as to f₂ from the Y(3S) data as we did from the Y(2S) data if it is assumed that the leptonic decay of the Y(3S) takes place with an even mix of two and three color participation. Viewed in such manner, the Y(3S) would represent the very threshold for the onset of color non-participation in lepton decay associated with the Y-series mesons.

C. The Y(4S)

Inserting M_Y4 = 10579 Mev into Eq. 3b, we obtain

Γ_ee3(GEM) = 0.531 Kev,

a figure well outside of the experimental range, given by

Γ_ee(PDG) = (0.272 ± 0.029) Kev.

Assuming only two colors operative, we obtain Γ_ee2(GEM) = 0.354 Kev, and assuming one operative color, we obtain Γ_ee1(GEM) = 0.177 Kev. We now note that

〈Γ_ee2+1(GEM)〉 = (1/2)[ Γ_ee2(GEM) + Γ_ee1(GEM)] = 0.266 Kev,

which represents a statistical match to Γ_ee(PDG) above. Thus, by assuming f₂ = (5/9) and that lepton decay of the Y(4S) takes place with an even mix of one and two color participation, we realize excellent agreement with experiment.

D. The Y(10860) [Our Designation: Y(5S)]

Inserting M_Y5 = 10860 Mev into Eq. 3b, we obtain

Γ_ee3(GEM) = 0.490 Kev,

a figure well outside of the experimental range, given by

Γ_ee(PDG) = (0.31 ± 0.07) Kev.

We note, however, that Γ_ee2(GEM) = 0.33 Kev with an associated f₂₂ = 0.5271 = (5/9){0.9488}, thus indicating ~95% confidence that two colors are operating in the leptonic decay of the Y(5S) and that f₂ = (5/9).

E. The Y(11020) [Our Designation: Y(6S)]

Inserting M_Y6 = 11019 Mev into Eq. 3b, we obtain

Γ_ee3(GEM) = 0.470 Kev,

a figure well outside of the experimental range, given by

Γ_ee(PDG) = (0.130 ± 0.030) Kev.

We note, however, that Γ_ee1(GEM) = 0.157 Kev, which is in the experimental range of Γ_ee(PDG) above, thus indicating that only one color is operating in the leptonic decay of the Y(6S) and that f₂ = (5/9).

F. Short Summary of Section III

There is clear indication of color disengagement from lepton production in conjunction with f₂ = (5/9) as associated with the Y-series mesons. Assuming the color participation outlined below in Chart 2 along with assuming that f₂ = (5/9) throughout, the average relative discrepancy between Γee(GEM) and Γee(PDG) associated with the Y-series mesons is only 7.5%, whereas the data set itself has an average imprecision of 12.0%.

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Concluding Remarks

We find the phenomenon of “color shutdown” as to the process of lepton production in the ψ-series and Y-series mesons extremely interesting. The confidence level is seen to be extremely high (at least regarding the ψ-series mesons) that the phenomenon exists, but what causes it? Because of the color-by-color sequential nature of the color turn-off with increasing mass within the subset of integer number of colors operative in lepton decay, it is unlikely that the phenomenon has its basis in color screening effects. The phenomenon we have explored above looks much too quantized for that. As interesting and, frankly, mysterious, as the color-by-color turn-off phenomenon is, the half-integer turn-offs are far more so. There is even an example of possibly a “three-quarter color” turn-off not yet explored in the present work, to which we turn presently:

The ψ(3770) (M_ψ(3770) = 3773 Mev) has an anomalously small ee partial width, viz.,

Γ_ee(PDG) = (0.265 ± 0.018) Kev.

Assuming f₁ = (8/9) and that Γ_ee is due to an even mix of zero color participation in the ψ(3770) lepton decay and one-half color participation in same, the GEM yields via Eq. 4a Γ_ee(GEM) = 0.264 Kev, a figure representing nearly an exact match to experiment. On average, then, the ψ(3770) has only ¼ of a color participating in lepton production. We have characterized the half-integer color participations in Section III as representations of even mixes in contribution to lepton decay from “k” colors operative and “k-1” colors operative. If such representation is accurate, then in some instances it must be the case that one quark of the di-quark meson system has “k” colors operative in lepton decay, while the other has “k-1” colors operative. Such would explain the half-integer color contributions exhibited in Section III. The case of the ψ(3770), however, suggests something else entirely, as its leptonic width result must be explained by an even mix of zero color contribution to lepton production and one-half color contribution to same. Unless the case of the ψ(3770) represents merely a numerical coincidence, an alternative explanation would have to be that we have uncovered a new feature associated with quarks … that of “shade”, in that each color must possess two “shades” (“light”? and “dark”?) in order to explain the quarter-integer color contribution to lepton production of the ψ(3770).

In any event we may estimate the threshold for the complete turn-off of lepton production in the ψ-series and in the Y-series by plotting the number of operative colors versus (M_ψN / M_ψ1) and (M_YN / M_Y2), respectively, omitting the ψ-(3770) outlier case, and estimating where on said plots N_C = 0.5 (our definition for the above-mentioned threshold), where N_C represents the number of operative colors from Charts 1 and 2. From Fig. 1 we note that

N_C ≈ 3 x₁^-3.5 , where x₁ = (M_ψN / M_ψ1). (14a)

Hence, x₁ ≈ (5/3) when N_c = ½ , thus indicating a lepton shut-down threshold for the ψ-series at about 5170 Mev. From Fig. 2 we note that

N_C ≈ 3 x₂^-9.75 , where x₂ = (M_YN / M_Y2). (14b)

Hence, x₂ ≈ (6/5) when N_c = ½ , thus indicating a lepton shut-down threshold for the ψ-series at about 12050 Mev. Comparing the color shut-down behaviors, as exhibited by Eqs. 14a and 14b, it is apparent that the Y-series shutdown with increasing energy is much more rapid, relatively speaking, than is that of the ψ-series. As a final word, we remark that whether quark colors each can be either “light” or “dark” … or not … the color shut-down phenomenon is extremely interesting and should prove to be the subject of future research in high energy physics theoretical work.

Figures

Used only with express written permission

Used only with express written permission

D. White (2010), “ Evidence for Color-by-Color Disengagement from the Process of Lepton Production Associated with the ψ-Series and Y-Series Mesons”, Communications in Mathematics and Applications, Vol. 1, No. 3, pp. 183 – 193.

References

1. D. White (2010-f), “Form Factor Analysis Derived from the Gluon Emission Model Applied to the ψ(2S) and the Y(2S)”, Communications in Mathematics and Applications. Vol. 1, No. 3, pp. 165 – 181.
2. D. White (2010-y), “GEM and the Y(1S)”, The Journal of Informatics and Mathematical Sciences, Vol. 2, Nos. 2 & 3, pp. 71 – 93.
3. D. White (2009-a), “Mathematical Modeling of the Strong Coupling Parameter Based upon the Gluon Emission Model for Hadron Production Associated with Vector Meson Decay”, Journal of Interdisciplinary Mathematics, Vol. 12, No. 6, pp.825 – 838.
4. PDG (2009-M), pdg.lbl.gov, “Meson Table”.

See also: http://www.intechopen.com/articles/show/title/vector-mesons