_{1}

^{*}

An assumption that
*all* the six flavour quarks are attributed to be the components of
*a same, a*
*common* isospin multiplets space named
STS is proposed. Base on
Pauli Exclusion Principle, every quark is assigned to different flavour marks in STS. Every flavour quark possesses
*its own colour spectral line array* specially appointed. The collection of colour spectral line arrays of the six flavour quarks constructs together the
CSDF, Colour Spectrum Diagram of Flavour, further baryons and mesons could be constructed from
CSDF. STS, Spin Topological Space is a math frame with infinite dimensional matrix representation for spin angular momentum. Flavours is an isospin angular momentum coupling phenomena of the three-colour-quarks.

In isotopic spin space of Standard Model, SM, Gell-Mann M [_{3}, for flavour quarks u, d are 1/2, +1/2 and 1/2, −1/2 respectively, for flavour quarks s , c , b , t are 0, 0. u ( I 3 = + 1 / 2 ) and d ( I 3 = − 1 / 2 ) quarks are assigned to an isodoublet with I = 1 / 2 , the dimension of matrix representation of the isodoublet is equal to 2 × 1 / 2 + 1 = 2 . This matrix representation is an analogy with ordinary angular momentum σ → / 2 , σ → is Pauli matrix. And the remaining four flavour quarks s ( I 3 = 0 ) , c ( I 3 = 0 ) , b ( I 3 = 0 ) , t ( I 3 = 0 ) are assigned to four isosinglets with I = 0, respectively, the dimension of matrix representation of each isosinglet is equal to 2 × 0 + 1 = 1 . Further, there is an isodoublet and there are four isosinglets in isospin scheme for flavour quarks [

It is a curious question, what will happen? if the above six flavour quarks are all put into a common multiplet, that is, if these flavours are treated equally in one isotopic spin space. According to Pauli Exclusion Principle, PEP, each of those values of I 3 ( q i ) of six flavour quarks should not be the same each other. Following the unified math symmetry picture, all eigenvalues of I 3 ( q i ) of flavour q i quark are proposed to be half-integers, to be +5/2, +3/2, +1/2, −1/2, −3/2, −5/2. q i = q t , q c , q u , q d , q s , q b , i = t , c , u , d , s , b respectively are shown in

Flavour | I | I 3 ( q i ) | matrix | PEP | I | I 3 ( q i ) | matrix |
---|---|---|---|---|---|---|---|

Quark | SM | ⇒ | STS | infinite dimension | |||

t | 0 | I 3 ( t ) = 0 | 1 dimension | 1/2 | I 3 ( t ) = + 5 / 2 ^{♦} | infinite dimension | |

c | 0 | I 3 ( c ) = 0 | 1 dimension | 1/2 | I 3 ( c ) = + 3 / 2 ^{♦} | infinite dimension | |

u | 1/2 | I 3 ( u ) = + 1 / 2 | 2 dimension | 1/2 | I 3 ( u ) = + 1 / 2 | infinite dimension | |

d | 1/2 | I 3 ( d ) = − 1 / 2 | 2 dimension | 1/2 | I 3 ( d ) = − 1 / 2 | infinite dimension | |

s | 0 | I 3 ( s ) = 0 | 1 dimension | 1/2 | I 3 ( s ) = − 3 / 2 ^{♦} | infinite dimension | |

b | 0 | I 3 ( b ) = 0 | 1 dimension | 1/2 | I 3 ( b ) = − 5 / 2 ^{♦} | infinite dimension |

In

Spin angular momentum π → of a spin particle in STS math frame, is labelled by two subscripts j , k (if in real region):

π → j , k = ( π 1 ; j , k , π 2 ; j , k , π 3 ; j , k ) (1)

π → j , k satisfy angular momentum commutation rule (2)

π → j , k × π → j , k = i π → j , k (2)

π 1 ; j , k = 1 2 ( π j + + π k − ) (3.1)

π 2 ; j , k = 1 2 i ( π j + − π k − ) (3.2)

π 3 ; j , k = 1 2 ( π j + π k − − π k − π j + ) (3.3)

j , k ⊂ STS . π 1 ; j , k and π 2 ; j , k are two infinite dimensional Non-Hermitian Matrices. π 3 ; j , k is an infinite dimensional Hermitian Matrix [

π j , k 2 = π 1 ; j , k 2 + π 2 ; j , k 2 + π 3 ; j , k 2 = 1 4 { ( j − k ) 2 − 1 } I 0 (4)

π 3 ; j , k = π 0 ( 0 ) + 1 2 ( j + k + 1 ) I 0 (5)

π 0 ( 0 ) = diag { ⋯ , 5 , 4 , 3 , 2 , 1 , 0 _ , − 1 , − 2 , − 3 , − 4 , − 5 , ⋯ } (6)

I 0 = diag { ⋯ , 1 , 1 , 1 , 1 , 1 , 1 _ , 1 , 1 , 1 , 1 , 1 , ⋯ } (7)

formulas (4), (5) show π j , k 2 and π 3 ; j , k are diagonal infinite dimensional matrices. Here π 0 ( 0 ) is the vacuum background spin angular momentum of π 3 ; j , k . If in case of no confusion, it is convenient to instead of (5) to use (9) to deal with I 3 , then obtain following expressions

π j , k 2 = 1 4 ( S j , k 2 − 1 ) (8)

π 3 ; j , k = 1 2 ( A j , k + 1 ) (9)

S j , k = j − k , A j , k = j + k (10)

( j , k ) = ( 1 2 ( A j , k + S j , k ) , 1 2 ( A j , k − S j , k ) ) (11)

Call ( j , k ) , STC, Spin Topological Coordinate of spin particle in STS.

Addition of π → j , k and π → r , s , Π → j , k ; r , s is given below

π → j , k × π → j , k = i π → j , k , π → r , s × π → r , s = i π → r , s (12)

Π → j , k ; r , s × Π → j , k ; r , s = i Π → j , k ; r , s (13)

Π → j , k ; r , s = 1 2 ( π → j , k + π → r , s ) (14)

Π j , k ; r , s 2 = 1 16 ( ( S j , k + S r , s ) 2 − 4 ) = 1 4 ( ( S j , k / 2 + S r , s / 2 ) 2 − 1 ) (15)

Π 3 ; j , k ; r , s = 1 2 ( π 3 ; j , k + π 3 ; r , s ) (16)

Π j , k ; r , s 2 and Π 3 ; j , k ; r , s are Casimir operator and the third component of spin particle Π → j , k ; r , s in STS.

Now we continue

component eigenvalues I 3 ( q i ) (19) of flavour q i quarks (fermion I ( q i ) = 1 2 (17)). Details are shown in

I 3 ; j , k ( q i ) | = 1 2 ( A j , k ( q i ) + 1 ) | A j , k ( q i ) | S j , k ( q i ) | ( j , k ) q i |
---|---|---|---|---|

I 3 ; + 3 , + 1 ( t ) = + 5 / 2 ♦ | 1 2 ( + 4 + 1 ) | A + 3 , + 1 ( t ) = + 4 | S + 3 , + 1 ( t ) = + 2 | ( + 3 , + 1 ) t |

I 3 ; + 2 , 0 ( c ) = + 3 / 2 ♦ | 1 2 ( + 2 + 1 ) | A + 2 , 0 ( c ) = + 2 | S + 2 , 0 ( c ) = + 2 | ( + 2 , 0 ) c |

I 3 ; + 1 , − 1 ( u ) = + 1 / 2 | 1 2 ( 0 + 1 ) | A + 1 , − 1 ( u ) = 0 | S + 1 , − 1 ( u ) = + 2 | ( + 1 , − 1 ) u |

I 3 ; 0 , − 2 ( d ) = − 1 / 2 | 1 2 ( − 2 + 1 ) | A 0 , − 2 ( d ) = − 2 | S 0 , − 2 ( d ) = + 2 | ( 0 , − 2 ) d |

I 3 ; − 1 , − 3 ( s ) = − 3 / 2 ♦ | 1 2 ( − 4 + 1 ) | A − 1 , − 3 ( s ) = − 4 | S − 1 , − 3 ( s ) = + 2 | ( − 1 , − 3 ) s |

I 3 ; − 2 , − 4 ( b ) = − 5 / 2 ♦ | 1 2 ( − 6 + 1 ) | A − 2 , − 4 ( b ) = − 6 | S − 2 , − 4 ( b ) = + 2 | ( − 2 , − 4 ) b |

Note: A j , k ( q i ) is named as flavour quantum number of quarks, which are even numbers.

I ( q i ) = 1 2 (17)

I j , k 2 ( q i ) = diag { ⋯ , + 3 4 , + 3 4 , + 3 4 , + 3 4 _ , + 3 4 , + 3 4 , + 3 4 , ⋯ } (18)

I 3 ; j , k ( q i ) = diag { ⋯ , + 7 2 ♦ , + 5 2 ♦ , + 3 2 ♦ , + 1 2 _ , − 1 2 , − 3 2 ♦ , − 5 2 ♦ , ⋯ } , i = t , c , u , d , s , b (19)

Now Suppose that except flavour quantum number marked A j , k ( q i ) in STS (_{RGB} (20), which is an array comprised of three colour quantum numbers, marked q_{R}, q_{G} and q_{B}, they are third-fractions.

q RGB ≡ ( q R , q G , q B ) ≡ ( A ( q R ) , A ( q G ) , A ( q B ) ) (20)

Different flavour quark possesses its own colour spectral line array, for example, array u RGB ≡ ( u R , u G , u B ) = ( + 2 3 , + 5 3 , + 11 3 ) is the colour spectral line array of flavour u quark, and array d RGB ≡ ( d R , d G , d B ) = ( − 16 3 , − 13 3 , − 7 3 ) is colour spectral line array of flavour d quark. Flavour A j , k ( q i ) and colour q_{RGB} are the identities of quark particles.

Definition CSDF, Colour Spectrum Diagram of Flavour is composed of six (or more) colour spectral line arrays of flavour quarks, somewhat similar to the Gene diagram of chromosomes. Explicit scheme of CSDF is given below.

Flavours are coupling phenomena of isospin angular momenta of three- colour-quarks.

To track the idea, we make use of CSDF, and of angular momentum formulae (21) and (22) of three spin particles below. Further obtain STC array of flavour quarks in

I 2 ( 3 q ) = 1 36 ( ( S ( q R ) + S ( q R ) + S ( q B ) ) 2 − 9 ) = 3 4 (21)

I 3 ( 3 q ) = 1 3 ( I 3 ( q R ) + I 3 ( q G ) + I 3 ( q B ) ) = 1 2 ( A ( 3 q ) 3 + 1 ) (22.1)

I 3 ( q ) = I 3 ( 3 q ) ¯ = 1 3 I 3 ( 3 q ) (22.2)

( j , k ) q i | ( j , k ) t | ( j , k ) c | ( j , k ) u | ( j , k ) d | ( j , k ) s | ( j , k ) b |
---|---|---|---|---|---|---|

( j , k ) q R | ( j , k ) t R ( + 44 6 , + 32 6 ) t R | ( j , k ) c R ( + 26 6 , + 14 6 ) c R | ( j , k ) u R ( + 8 6 , − 4 6 ) u R | ( j , k ) d R ( − 10 6 , − 22 6 ) d R | ( j , k ) s R ( − 28 6 , − 40 6 ) s R | ( j , k ) b R ( − 46 6 , − 58 6 ) b R |

( j , k ) q G | ( j , k ) t G ( + 47 6 , + 35 6 ) t G | ( j , k ) u G ( + 29 6 , + 17 6 ) c G | ( j , k ) u G ( + 11 6 , − 1 6 ) u G | ( j , k ) u G ( − 7 6 , − 19 6 ) d G | ( j , k ) s G ( − 25 6 , − 37 6 ) s G | ( j , k ) u G ( − 43 6 , − 55 6 ) b G |

( j , k ) q B | ( j , k ) u B ( + 53 6 , + 41 6 ) t B | ( j , k ) u B ( + 35 6 , + 23 6 ) c B | ( j , k ) u B ( + 17 6 , + 5 6 ) u B | ( j , k ) u B ( − 1 6 , − 13 6 ) d B | ( j , k ) s B ( − 19 6 , − 31 6 ) s B | ( j , k ) u B ( − 37 6 , − 49 6 ) b B |

Due to baryons all are “white colour” particles, which are made of colourful quarks. To help with CSDF, picking up three colour quantum numbers: q R 1 ⊆ q RGB 1 , q G 2 ⊆ q RGB 2 and q B 3 ⊆ q RGB 3 , respectively from any three quarks q^{1}, q^{2} and q^{3} (q^{1}, q^{2} and q^{3} can be any flavour), then various visible baryons could be produced. In this way, for example, baryon decuplet is constituted as shown in

According to SM, the “colourless phenomena” of all mesons could be satisfied by blending with a quark with colour and antiquark with anti-colour, that is to say, q R q ¯ R ¯ , or q G q ¯ G ¯ , or q B q ¯ B ¯ . Contrary to SM, in STS a meson is similar to a baryon, a meson also is a three-body system that comprises a quark (colour), an antiquark (anti-colour) and a gluon (white). This gluon plays the role of mediator to fasten quark and antiquark together in a meson.

Colour spectral line array of meson is symbolized with q i q ¯ j ¯ g k = ( q i , q ¯ j ¯ , g k ) . The mentioned above is the case of i = j , k = 0 . The discussion about i ≠ j , k ≠ 0 will be given later.

In what follows base on CSDF, we list the weight diagram

This paragraph suggests some ideas, similar to spin-spin S i → ⋅ S j → interaction in spin space [^{1}, q^{2} and q^{3}. Each baryon exists in one of three possible states, labelled with cases: I, II and III in case (23.1) and case (23.2) respectively. The results in

I 3 ( Δ − ) = − 3 2 , − 4 ( d R , d G , d B ) ( − 16 3 , − 13 3 , − 7 3 ) | I 3 ( Δ 0 ) = − 1 2 , − 2 ( u R , d G , d B ) ( + 2 3 , − 13 3 , − 7 3 ) | I 3 ( Δ + ) = + 1 2 , 0 ( u R , d G , u B ) ( + 2 3 , − 13 3 , + 11 3 ) | I 3 ( Δ + + ) = + 3 2 , + 2 ( u R , u G , u B ) ( + 2 3 , + 5 3 , + 11 3 ) | |||
---|---|---|---|---|---|---|

I 3 ( Σ ∗ − ) = − 5 2 , − 6 ( d R , s G , d B ) ( − 16 3 , − 31 3 , − 7 3 ) | I 3 ( Σ ∗ 0 ) = − 3 2 , − 4 ( u R , s G , u B ) ( + 2 3 , − 31 3 , − 7 3 ) | I 3 ( Σ ∗ + ) = − 1 2 , − 2 ( u R , s G , u B ) ( + 2 3 , − 31 3 , + 11 3 ) | ||||

I 3 ( Ξ ∗ − ) = − 7 2 , − 8 ( d R , s G , s B ) ( − 16 3 , − 31 3 , − 25 3 ) | I 3 ( Ξ ∗ 0 ) = − 5 2 , − 6 ( u R , s G , s B ) ( + 2 3 , − 31 3 , − 25 3 ) | |||||

I 3 ( Ω − ) = − 9 2 , − 10 ( s R , s G , s B ) ( − 34 3 , − 31 3 , − 25 3 ) |

Note: in _{3} of every baryon all is half integer, contrary to those what the I_{3} might take both half integer and integer (include zero) in case of J P = 3 / 2 + in SM.

I 3 ( k 0 ) = + 1 , A ( k 0 ) = + 1 ( d R , s ¯ R ¯ , g 0 ) ( − 16 3 , + 28 3 , − 3 3 ) | I 3 ( k + ) = + 2 , A ( k + ) = + 3 ( u R , s ¯ R ¯ , g 0 ) ( + 2 3 , + 28 3 , − 3 3 ) | |||
---|---|---|---|---|

I 3 ( π − ) = − 1 , A ( π − ) = − 3 ( d R , u ¯ R ¯ , g 0 ) ( − 16 3 , − 8 3 , − 3 3 ) | I 3 ( π 0 , η ) = 0 , A ( π 0 , η ) = − 1 ( u R , u ¯ R ¯ , g 0 ) ( + 2 3 , − 8 3 , − 3 3 ) ( d R , d ¯ R ¯ , g 0 ) ( − 16 3 , + 10 3 , − 3 3 ) | I 3 ( π + ) = + 1 , A ( π + ) = + 1 ( u R , d ¯ R ¯ , g 0 ) ( + 2 3 , + 10 3 , − 3 3 ) | ||

octet mesons | I 3 ( k − ) = − 2 , A ( k − ) = − 5 ( s R , u ¯ R ¯ , g 0 ) ( − 34 3 , − 8 3 , − 3 3 ) | I 3 ( k ¯ 0 ) = − 1 , A ( k ¯ 0 ) = − 3 ( s R , d ¯ R ¯ , g 0 ) ( − 34 3 , + 10 3 , − 3 3 ) |

Note: in _{3} of every meson all is integer, contrary to those what the I_{3} might take both half integer and integer in case of J P = 0 − in SM.

There is an amusing equality (23) below among q^{1},q^{2} and q^{3} that is obtained from q RGB ≡ ( q R , q G , q B )

IV ( r - h ) . q 1 + q 2 + q 3 ≡ I . q R 1 + q G 2 + q B 3 = II . q R 2 + q G 3 + q B 1 = III . q R 3 + q G 1 + q B 2 , A = 1 3 ( q 1 + q 2 + q 3 ) (23.1)

IV ( l - h ) . q 2 + q 1 + q 3 ≡ I . q R 2 + q G 1 + q B 3 = II . q R 3 + q G 2 + q B 1 = III . q R 1 + q G 3 + q B 2 , A = 1 3 ( q 2 + q 1 + q 3 ) (23.2)

example of p^{+}

IV ( r - h ) . u 1 + d 2 + u 3 ≡ I . u R + d G + u B = II . d R + u G + u B = III . u R + u G + d B , A = 1 3 ( u 1 + d 2 + u 3 ) = 0 3 = 0 (24.1)

= + 2 3 + − 13 3 + + 11 3 = − 16 3 + + 5 3 + + 11 3 = + 2 3 + + 5 3 + − 7 3 , (24.2)

example of n^{0}

IV ( r - h ) . u 1 + d 2 + d 3 ≡ I . u R + d G + d B = II . d R + d G + u B = III . d R + u G + d B , A = 1 3 ( u 1 + d 2 + d 3 ) = − 18 3 = − 6 (25.1)

= + 2 3 + − 13 3 + − 7 3 = − 16 3 + − 13 3 + + 11 3 = − 16 3 + + 5 3 + − 7 3 , (25.2)

If array q RGB is defined as a vector (26)

q RGB ≡ ( q R , q G , q B ) = A → ( q i ) (26)

Then the next two tables are constructed from CSDF, which may offer some heuristic search for classification of particle mass.

I 2 + II 2 + III 2 | Prediction | Experiment | ||||
---|---|---|---|---|---|---|

A 2 ( p + ) = A → ( p + ) ⋅ A → ( p + ) | I 2 ∝ ( + 2 ) 2 + ( − 13 ) 2 + ( + 11 ) 2 = 294 | 774 | | | | | ⇔ | ||

II 2 ∝ ( − 16 ) 2 + ( + 5 ) 2 + ( + 11 ) 2 = 402 | 774 + 1098 = 1872 | 938 + 940 = 1878 | ||||

III 2 ∝ ( + 2 ) 2 + ( + 5 ) 2 + ( − 7 ) 2 = 78 | 1872 / 2 = 936 | 1878 / 2 = 939 | ||||

A 2 ( n 0 ) = A → ( n 0 ) ⋅ A → ( n 0 ) | I 2 ∝ ( + 2 ) 2 + ( − 13 ) 2 + ( − 7 ) 2 = 222 | 1098 | | | | | 774 / 936 = 0.827 | ⇔ | 938 / 939 = 0.999 |

II 2 ∝ ( − 16 ) 2 + ( − 13 ) 2 + ( + 11 ) 2 = 546 | 1098 / 936 = 1.173 | 940 / 939 = 1.001 | ||||

III 2 ∝ ( − 16 ) 2 + ( + 5 ) 2 + ( − 7 ) 2 = 330 | 0.827 + 1.173 = 2 | 0.999 + 1.001 = 2 |

A 2 ( q ) = q R q R + q G q G + q B q B | A 2 ( q ) / A 2 ( u ) | Mev / c 2 ( q ) ⇒ M ( q ) / M ( u ) | q |
---|---|---|---|

A 2 ( t ) ∝ ( + 38 ) 2 + ( + 41 ) 2 + ( + 47 ) 2 = 5334 | ⇒ 5334 / 150 = 35.56 | 173 × 10 3 ⇒ 7521.7 | t |

A 2 ( c ) ∝ ( + 20 ) 2 + ( + 23 ) 2 + ( + 29 ) 2 = 1770 | ⇒ 1770 / 150 = 11.8 | 1.275 × 10 3 ⇒ 554.3 | c |

A 2 ( u ) ∝ ( + 2 ) 2 + ( + 5 ) 2 + ( + 11 ) 2 = 150 | ⇒ 150 / 150 = 1 | 2.3 × 10 0 ⇒ 1 | u |

A 2 ( d ) ∝ ( − 16 ) 2 + ( − 13 ) 2 + ( − 7 ) 2 = 474 | ⇒ 474 / 150 = 3.16 | 4.8 × 10 0 ⇒ 2.1 | d |

A 2 ( s ) ∝ ( − 34 ) 2 + ( − 31 ) 2 + ( − 25 ) 2 = 2742 | ⇒ 2742 / 150 = 18.28 | 95.0 × 10 0 ⇒ 41.3 | s |

A 2 ( b ) ∝ ( − 52 ) 2 + ( − 49 ) 2 + ( − 43 ) 2 = 6954 | ⇒ 6954 / 150 = 46.36 | 4.18 × 10 3 ⇒ 1817.4 | b |

In

In this paper we have pointed links between flavour quarks and colour quarks in math frame STS, Spin Topological Space: the flavour viewed as a number, named as flavour quantum number A j , k ( q i ) and the colour viewed as an array, named as colour spectral line array q_{RGB} consist of three colour quantum numbers q_{R}, q_{G} and q_{B} or A ( q R ) , A ( q G ) and A ( q B ) . The former is even number, the latter are third-fractions. When one thinks I 3 ( q R ) , I 3 ( q G ) and I 3 ( q B ) as three distinct angular momentums respectively, using momentum addtion of three-body, one can construct a variety of baryons.

In contrast to SM, mesons only are made of quark and antiquark, it becomes more complex, as now gluon joins into meson mechanism. In account of what happened in colour spectral line array q i q ¯ j ¯ g k when i ≠ j , k ≠ 0 , many efforts are needed, after all, so much is not fully understood.

Perhaps CSDF, Colour Spectrum Diagram of Flavour is an essential conception for us to realize what flavour and colour of quarks are.

The author thanks my intimate friend, Xin Mao, for valuable discussions and encouragements, and grateful to Jia Guan who helpfully supportes this paper Latex version.

The author declares no conflicts of interest regarding the publication of this paper.

Ren, S.X. (2021) Flavour and Colour of Quarks in Spin Topological Space. Journal of Modern Physics, 12, 380-389. https://doi.org/10.4236/jmp.2021.123027