The Distribution of Energy Within. Baryonic Sub-Atomic Particles.

Transcription

1 The Distribution of Within Baryonic Sub-Atoic Particles. [2] Peter G.Bass. P.G.Bass P10 Var Noveber 2016

2 ABSTRACT. The total energy within a Baryonic sub-atoic particle consists of three varieties, (i) atter energy - the atter energy of the three constituent uarks, (ii) the energy associated with the intrinsic angular oentu of the particle, and (iii) the energy associated with uark confineent. The purpose of this paper is to deterine the level of each type of energy within all Baryonic subatoic particles that possess intrinsic angular oenta of J 1/2and J 3/2. P.G.Bass i

3 CONTENTS. 1.0 Introduction. 2.0 The Deterination of Represented by one Unit of Intrinsic Angular Moentu. 3.0 The Deterination of the and Confineent Energies for All Baryonic Sub- Atoic Particles Possessing Intrinsic Angular Moenta of J 1/2and J 3/ An Epirical Law for Confineent. 5.0 Conclusions. APPENDICES. A. A plot of by Against the Su of Constituent es. B. A Plot of Confineent Against the Su of Constituent es. C. Epirical Law Relationships for and Confineent Energies. REFERENCES. P.G.Bass ii

4 1.0 Introduction. The total energy content of all 75 Baryonic sub-atoic particles listed in [1] is ade up of three varieties, (i) the atter energy of the three constituent uarks, (ii) the energy associated with intrinsic angular oentu - resonance energy and (iii) the energy associated with uark confineent. The total energy of the ajority these particles is known, firstly by the experiental easureent of ass as reported in [1] and [2], and secondly, for those particles whose ass is reported as unknown in [1] and [2], by virtue of the epirical deterination of their ass in [3]. Also, the level of uark atter energy has been estiated sufficiently in [3] to enable [3] to be prepared, and is therefore considered acceptable for use here. Conseuently, in order to ascertain the apportionent of particle energy between (i), (ii) and (iii), it is only necessary to deterine the level of either resonance energy or uark confineent energy, for each particle. With regard to the latter, the precise nature and source of this energy is unknown, and conseuently there is no eans by which the level of uark confineent energy within a Baryon can be independently deterined. However, because Baryons exist with various levels of intrinsic angular oentu, this enables the levels of resonance energy within all Baryons to be easily obtained. This together with the particles total energy, and its uark atter energy, then enables the levels of uark confineent energy to also be obtained. It should be noted that in this paper, energy will be represented as euivalent ass via the units MeV/c The Deterination of the Represented by One Unit of Intrinsic Angular Moentu. There are 30 Baryonic Particles with an intrinsic angular oentu of J 1/2for which there also exists a resonance energy variant with intrinsic angular oentu of J 3/2 It is reported, [1] and [2], that these particles and their variants contain the sae constituent uarks. Because of this the uark atter energy level in each particle and its resonance energy variant will be identical. Also, because of this sae uark constituency it is believed that their uark confineent energy levels will also be identical, an assuption which is later verified. Conseuently, the siple subtraction of the total energy of each particle fro that of its resonant energy variant, will produce a resonance energy level associated with one unit of intrinsic angular oentu for each particle/variant. This has been effected in the following table for the 60 subject particles. J 1/2 Particle Particle J 3/2 Particle Particle J 1 J 1/2 J 3/2 p uud n udd Σ Σ uus Σ Σ uds Σ Σ dds Ξ Ξ uss Ξ Ξ dss Σ ++ c Σ* ++ c uuc Σ + c Σ* + c udc Σ 0 c Σ* 0 c ddc Ξ /+ c Ξ + c usc Ξ /0 c Ξ 0 c dsc Ω 0 c Ω 0 c ssc Ξ ++ cc Ξ ++ cc ucc Ξ + cc Ξ + cc dcc Ω + cc Ω + cc scc Σ* + b uub Σ + b P.G.Bass 1

5 Σ 0 b Σ b Ξ /0 b Ξ / b Ω b Ξ /+ cb Ξ /0 cb Ω /0 cb Ω + ccb Ξ 0 bb Ξ bb Ω bb Ω 0 cbb Σ* 0 b Σ* - b Ξ 0 b Ξ b Ω b Ξ + cb Ξ 0 cb Ω 0 cb Ω + ccb Ξ 0 bb Ξ bb Ω bb Ω 0 cbb udb ddb usb dsb ssb ucb dcb scb ccb ubb dbb sbb cbb Table Deterination of the Associated with One Unit of Intrinsic Angular Moentu. In this table the particle ass for Ξ + cc as recorded in [1] and [2] is believed to be low, and so the calculated value fro the epirical law of [3] has been used. The value in the table is shown in red. 3.0 The Deterination of the and Confineent Energies for All Baryonic Sub-Atoic Particles Possessing Intrinsic Angular Moenta of J 1/2 and J 3/2. The final two coluns of table 2.1 provide the levels of resonance energy associated with intrinsic angular oenta of particles with J 1/2and J 3/2. These values could then, as discussed above, be used to deterine the energy associated with uark confineent for the 60 particles listed in Table 2.1. However, prior to this, it is necessary to also cover the 15 particles not contained in this table. To effect this the values so obtained in Table 2.1 ust accordingly be both suitably interpolated and extrapolated, by deterination of a resonance energy epirical law. This is not possible for the coplete range as in Table 2.1 because it is clear fro this table, that resonance energy is not a straight function of either the particle ass, nor the su of the uark asses. However, there is such a correlation apparent when considered in ters of uark content. This is shown in Appendix A, where the resonance energy euivalent to one unit of intrinsic angular oentu, (colun 6 in Table 2.1), has been plotted against the su of the applicable uark asses according to uark content, (i.e. uu + d, s, c, b etc). Curve fitting these plots enables preparation of Table 3.1, which shows the resulting epirical laws, (i.e. Weibull and MMF Models), and also the separate tables for each group of uark content in turn showing resonance energy, (J 1 ), for their su uark ass. uu Series Extrapolated Su ( ) dd Series Interpolated Su ( ) ss Series Interpolated Su ( ) u u u d d d s s s c c c b b b Weibull Model: ya-b*exp(-c*x^d) Weibull Model: ya-b*exp(-c*x^d) Weibull Model: ya-b*exp(-c*x^d) Coefficient Data: Coefficient Data: Coefficient Data: a a a b b b c c c d d d P.G.Bass 2

6 Conten t cc Series Interpolated Su ( ) bb Series Extrapolated Su ( ) Three Series Su ( ) u u uds d d udc s s usc c c dsc b b udb usb MMF Model: MMF Model: y(a*b+c*x^d)/(b+x^d) dsb y(a*b+c*x^d)/(b+x^d) Coefficient Data: Coefficient Data: ucb a a dcb b b scb c c d d th Degree Polynoial Fit: ya+bx+cx^2+dx^3... Coefficient Data: a b c d e f g h 9.88E E E E E E-20 Table Interpolated and Extrapolated Values of for an Intrinsic Angular Moentu of J 1 Incorporated in these tables, (shown in red), are the interpolated/extrapolated values for the five particles with three identical uarks, not included in Table 2.1, fro which the resonance energy for the applicable J 3/2 Baryons is obtained, (i.e. ++,, Ω, Ω ++ ccc, and Ω bbb). The ten particles identified in [3] as low confineent energy particles, are also not present in Table 2.1. Because resonance energy is, as shown above a function of uark content, the resonance energy in these particles will be the sae as their higher energy counterparts. Putting all the above together, the resonance energy of all 75 Baryons has now been deterined and can be used to obtain the level of their uark confineent energy according to the siple relationship where c M (3.1) c confineent energy. p Total particle energy. M Su of the uark content asses. r energy. p This has been effected in Table 3.2 below for particles with J 1/2and Table 3.3 for particles with J 3/2. r P.G.Bass 3

7 Particle Particle Confineent p uud n udd Σ uus Λ uds Σ uds Σ dds Ξ uss Ξ dss Σ ++ c uuc Λ + c udc Σ + c udc Σ 0 c ddc Ξ + c usc Ξ /+ c usc Ξ 0 c dsc Ξ /0 c dsc Ω 0 c ssc Ξ ++ cc ucc Ξ + cc dcc Ω + cc scc Σ + b uub Λ 0 b udb Σ 0 b udb Σ b ddb Ξ 0 b usb Ξ /0 b usb Ξ b dsb Ξ / b dsb Ω b ssb Ξ + cb ucb Ξ /+ cb ucb Ξ 0 cb dcb Ξ /0 cb dcb Ω 0 cb scb Ω /0 cb scb Ω + ccb ccb Ξ 0 bb ubb Ξ bb dbb Ω bb sbb Ω 0 cbb cbb Table 3.2 and Confineent for Particles with Intrinsic Angular Moentu J 1/2 P.G.Bass 4

8 Particle Particle Confineent uuu uud udd ddd Σ uus Σ uds Σ dds Ξ uss Ξ dss Ω sss Σ* + c udc Σ* ++ c uuc Σ* 0 c ddc Ξ + c usc Ξ 0 c dsc Ω 0 c ssc Ξ ++ cc ucc Ξ + cc dcc Ω + cc scc Ω ++ ccc ccc Σ* + b uub Σ* 0 b udb Σ* - b ddb Ξ 0 b usb Ξ b dsb Ω b ssb Ξ + cb ucb Ξ 0 cb dcb Ω 0 cb scb Ω + ccb ccb Ξ 0 bb ubb Ξ bb dbb Ω bb sbb Ω 0 cbb cbb Ω bbb bbb Table 3.3 and Confineent for Particles with Intrinsic Angular Moentu J 3/2 Fro these tables it is clear that the level of uark confineent energy is identical for the two groups of particles contained in Table 2.1. For those particles the axiu difference between Tables 3.2 and 3.3 being 0.05MeV/c 2 between Ω 0 cbb and Ω 0 cbb with the ajority of the others being identical to two decial places. This verifies the assuption ade in Section 2.0. It is also apparent that + is a resonance variant of p +, and 0 is a resonance variant of n 0. For the ten particles, (J 1/2that have low confineent energy variants, correlation of their uark confineent energy with their resonance energy variants appears in two groups. This is detailed in the following table. P.G.Bass 5

9 J 1/2 J 3/2 Low Confineent High Confineent Particle Particle Variant uds Λ 0 Σ 0 Σ 0 Σ 0 udc Λ + c Σ + c Σ + c Σ + c usc Ξ + c Ξ /+ c Ξ + c Ξ /+ c dsc Ξ 0 c Ξ /0 c Ξ 0 c Ξ /0 c udb Λ 0 b Σ 0 b Σ 0 b Σ 0 b usb Ξ 0 b Ξ /0 b Ξ 0 b Ξ 0 b dsb Ξ b Ξ / b Ξ b Ξ b ucb Ξ + cb (1) Ξ /+ cb (1) Ξ + cb Ξ + cb dcb Ξ 0 cb (2) Ξ /0 cb (2) Ξ 0 cb Ξ 0 cb scb Ω 0 cb (3) Ω /0 cb (3) Ω 0 cb Ω 0 cb Table Confineent Correlation of Particle Variants. J 3/2 to J 1/2 Confineent Correlation. Thus for the first five particles in Table 2.5, correlation of uark confineent energy occurs between the resonance energy variant and the high confineent energy particle. Whereas for the second five, it occurs with the low confineent energy particle. The reason for this variation is unclear. For particles annotated (1), their ass was reported as unknown in [1] and [2] and was therefore evaluated via the epirical law of [3]. Their ass was therefore calculated to be identical which led to identical uark confineent energies. As the ass and uark confineent energy of the pried particle should be higher, it would be expected that correlation of uark confineent energy would occur between the resonance variant and the un-pried J 1/2particle as has been reflected in the above table. This discussion also applies to particles annotated (2) and (3). 4.0 An Epirical Law for Confineent. The fact that uark confineent energy is identical for particles with J 1/2and J 3/2indicates that it is not a direct function of, nor directly affected by, the particles intrinsic angular oentu. Therefore it is concluded that it ust be a function of soe other aspect of the uark content. Plotting uark confineent energy against the su of the constituent uark ass, fro either Table 2.3 or 2.4 produces the figure shown in Appendix B. However, curve fitting this plot produces an epirical law that exhibits errors which range fro +5.56% to -8.40%. Conseuently it is only useful as an indication or guide. The above errors are priarily due to the use of resonance energy in deterining uark confineent energy via E.(3.1), and it was shown, Table 3.1, that resonance energy is not a direct function of the su of the uark asses but only as dictated by uark content. It is therefore expected that epirical laws for uark confineent energy should be based upon the sae preise. This has been effected and is shown in Table 4.1. P.G.Bass 6

10 uu Series dd Series ss Series Confineent Confineent Confineent % Error % Error % Error Actual Epirical Calculation Actual Epirical Calculation Actual Epirical Calculation E % E % E % E % E % E % E % E % E % E % E % E % E % E % E % 3rd degree Polynoial Fit: ya+bx+cx^2+dx^3 3rd degree Polynoial Fit: ya+bx+cx^2+dx^3 3rd degree Polynoial Fit: ya+bx+cx^2+dx^3 Coefficient Data: Coefficient Data: Coefficient Data: a a a b b b c c c d 3.71E-07 d 3.76E-07 d 1.22E-07 cc Series bb Series Three Series Confineent Confineent % Error Confineent % Error Su % Error ( Epirical Actual Epirical Calculation Actual Epirical Calculation ) Actual Calculation E E-3% E E-4% uds E E-6% E E-3% E E-4% udc E E-4% E E-5% E E-5% usc E E-3% E E-6% E E-7% dsc E E-3% E E-6% E E-7% udb E E-3% usb E E-2% 3rd degree Polynoial Fit: 3rd degree Polynoial Fit: ya+bx+cx^2+dx^3 ya+bx+cx^2+dx^3 dsb E E-2% Coefficient Data: Coefficient Data: ucb E E-2% a a dcb E E-2% b b scb E E-3% c -8.31E-05 c 7.24E-05 d 3.55E-08 d 5.81E-08 6th Degree Polynoial Fit: ya+bx+cx^2+dx^3... Coefficient Data: a b c d -6.62E-06 e 2.12E-09 f -3.12E-13 g 1.73E-17 Table Confineent Plus Epirical Law Calculations and Error Levels. The axiu error range here is in the ss uark series, % to %. The epirical law relationships for both resonance energy and uark confineent energy are shown in Appendix C. 5.0 Conclusions. In view of the uncertainty with which the uark asses are "known", [3], it is concluded that the results derived here exhibit an excellent level of accuracy. This conclusion is based upon the following three factors - (i) The excellent agreeent on the values of uark confineent energy between applicable particles with intrinsic angular oenta of J 1/2and those with J 3/2 ii The regularity and accuracy with which the epirical laws for particle resonance energy have been derived. P.G.Bass 7

11 (iii) The regularity and accuracy with which the epirical laws for uark confineent energy have been derived. Despite this, the ethod adopted here cannot, as it stands, be applied to Baryons in general. This is because there are any other variants in [1] and [2], with various levels of resonance energy, to which the epirical laws derived here do not exactly apply. It is believed that this is because there are any other levels of uark confineent energy which Baryons can exhibit, and the precise nature of the variation of this paraeter needs to be deterined, in order that the ethod used here can be extended such that it can be applied globally to all Baryons, whatever their resonance and uark confineent energy levels. Finally, for the particular particles studied here, the results will now enable the deterination of the distribution of the three varieties of particle energy, between their three constituent uarks, which will in turn illustrate the energy transitions that occur during their decay. P.G.Bass 8

12 APPENDIX A - Fig.A bb+ Series, Mev/c ss+ Series cc+ Series u++ and d++ Series ud+ Series 0 uu+ and 0 dd+ Series Su ( ), MeV/c 2 Fig.A.1 - Plots of Particle as a Function of Su es by. P.G.Bass 9

13 APPENDIX B - Fig.B Confineent Enery, (MeV/c 2 ) Su (Quar k M as s e s ), (M e V /c 2 ) Fig.B.1 - Plot of Confineent as a Function of Su es. P.G.Bass 10

14 Appendix C. Epirical Law Relationships for and Confineent Energies. In the relationships below, the following definitions apply. p Total Particle, (Euivalent ). r Particle, (Euivalent ). c Particle Confineent, (Euivalent ). M Su of the of Applicable Constituent s, (Euivalent ). C.1 Epirical Laws, (Fro Table 3.1). These euations are for one unit of intrinsic angular oentu. For particles with J 1/2 ultiply the RHS by 1/2. For particles with J 3/2 ultiply the RHS by 3/2. (i) The uu Series, (Weibull Model). r [ ] 0.. M EXP (C.1) (ii) The dd Series, (Weibull Model). r [ ] 0.. M EXP (C.2) (iii) The ss Series, (Weibull Model). r [ M ] EXP (C.3) (iv) The cc Series, (MMF Model). r x M M (C.4) (v) The bb Series, (MMF Model). r x M M (C.5) (vi) The Three Series, (7 th Degree Polynoial Model). r ( 3. 25E 20) M + ( 7. 54E 16) M ( 6. 98E 12) M + ( 3. 27E 8) 3 2 ( 8. 06E 5) M + ( E 2) M M M 4 (C.6) P.G.Bass 11

15 C.2 Particle Confineent Epirical Laws, (Fro Table 4.1). (i) The uu Series, (3 rd Degree Polynoial Model). 3 2 ( 3. 71E 7) ( E 3) c (C.7) (ii) The dd Series, (3 rd Degree Polynoial Model). 3 2 ( 3. 76E 7) ( E 3) c (C.8) (iii) The ss Series, (3 rd Degree Polynoial Model). 3 2 ( 1. 22E 7) ( E 3) c (C.9) (iv) The cc Series, (3 rd Degree Polynoial Model). 3 2 ( 3. 55E 8) ( 8. 31E 5) c (C.10) (v) The bb Series, (3 rd Degree Polynoial Model). 3 2 ( 5. 81E 8) + ( 7. 24E 5) c (C.11) (vi) The Three Series, (6 th Degree Polynoial Model). c ( 1. 73E 17) M ( 3. 12E 13) M + ( 2. 12E 9) M ( 6. 62E 6) 2 + ( E 3) M M M 3 (C.12) REFERENCES. [1] Wikipedia, List of Baryons, en.wikipedia.org. [2] Particle Data Group, Particle Listings, pdg.lbl.gov. [3] P.G.Bass, Derivation of Epirical Laws for the of Sub-Atoic Baryonic Particles, P.G.Bass 12