Traditionally, in weak interaction, I 3, Y and four flavour quantum numbers are not conserved but several empirical selection rules work well. Recently, it was found that, in weak interaction, there are three levels of conservation of additive quantum numbers, and fermion quantum number F is conserved in all kinds of interactions. It is known that weak interaction has three types: fermionic, pure hadronic and pure leptonic, corresponding to the first and the second level of conservation of additive quantum numbers respectively. It is demonstrated in this paper that the selection rules in all types of weak interaction can be interpreted by conservation of F, and the formula of relation between Q/e, F and F 0 is more general than Gell-Mann-Nishijima formula. Description of weak interaction becomes simpler, If only we take Q, F 0 and F, based on the conserved physical quantities.
At the most elementary level, at present, fermions include quarks and leptons. Traditionally, isospin (I) with isospin projection (I3) is assigned to u and d quarks, but the other four quarks have the flavour quantum numbers:S, C, B* and T for s, c, b and t quarks respectively. All quarks have the baryon quantum number B and electric charge Q. Hypercharge Y is defined from I3 andQ. Leptons, much simpler than quarks, have Q and lepton quantum number L. Q, B, L, I, I3, Y and four flavour quantum numbers are additive quantum numbers. Q, B and L are conserved in strong interaction, electromagnetic interaction and weak interaction. I3, Y and four flavour quantum numbers are conserved in strong interaction and electromagnetic interaction, but not conserved in weak interaction. I is conserved in strong interaction, but not conserved in either electromagnetic interaction or weak interaction. On the other hand, in weak interaction, the systematic way in which symmetry related to conservation is broken leads to several empirical selection rules which may serve to check observations and to put constraint on models. It is conventional to refer to weak interaction events as being pure leptonic, pure hadronic (or nonleptonic) or fermionic (or semi-leptonic, semi-hadronic) depending on whether they involve leptons only, hadrons only, or both leptons and hadrons. Knowledge above is described in textbooks, e.g., [
Recently, it was found that, in weak interaction, there are three levels of conservation of additive quantum numbers [
Characteristics of additive quantum numbers are described in [
I 3 = ( D + U ) / 2 . (1)
H is sum of all six flavor quantum numbers:
H = D + U + S + C + B * + T = { − 1 : quark is ‘ d ’ type + 1 : quark is ‘ u ’ type (2)
Antiquarks have additive quantum numbers with the same absolute values as quarks but the opposite sign of quarks, except that antiquarks have same I as quarks. H of each lepton is zero.
HL of leptons is similar with H of quarks:
H L = { − 1 : lepton is ‘ d ’ type + 1 : lepton is ‘ u ’ type (3)
Antileptons have additive quantum numbers with the same absolute values as leptons but opposite sign of leptons. HL of each quark is zero.
F for all fermions is combined from H and HL:
F = H + H L = { − 1 : fermion is ‘ d ’ type + 1 : fermion is ‘ u ’ type (4)
The formula of relation between electronic charge Q/e, F and F0 is
Q / e = ( F 0 + F ) / 2 , (5)
where F0 is:
F 0 = { B = + 1 / 3 : fermion is quark − L = − 1 : fermion is lepton (6)
Values of additive quantum numbers of quarks and leptons are listed in
Q / e = I 3 + Y / 2 , (7)
we can see that Equation (5) only includes conserved additive quantum numbers so as to express more general and more profound connotation than Equation (7) including not-conserved additive quantum numbers in weak interaction.
In the following paragraphs we check all three types of weak interaction one by one to verify if the selection rules are related to conservation of F. Firstly, hadrons composed by only u, d and s quarks are considered, so from Equation (2),
H = D + U + S . (8)
After it, the relation is generalized to heavier quarks (Section 3.4).
From Equation (1) and (8),
| quark | Q/e | B | I | I3 | D | U | S | C | B* | T | Y | H | F |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| d | −1/3 | +1/3 | +1/2 | −1/2 | −1 | 0 | 0 | 0 | 0 | 0 | +1/3 | −1 | −1 |
| u | +2/3 | +1/3 | +1/2 | +1/2 | 0 | +1 | 0 | 0 | 0 | 0 | +1/3 | +1 | +1 |
| s | −1/3 | +1/3 | 0 | 0 | 0 | 0 | −1 | 0 | 0 | 0 | −2/3 | −1 | −1 |
| c | +2/3 | +1/3 | 0 | 0 | 0 | 0 | 0 | +1 | 0 | 0 | +4/3 | +1 | +1 |
| b | −1/3 | +1/3 | 0 | 0 | 0 | 0 | 0 | 0 | −1 | 0 | −2/3 | −1 | −1 |
| t | +2/3 | +1/3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | +1 | +4/3 | +1 | +1 |
| Lepton | Q/e | L | HL | F |
|---|---|---|---|---|
| e− | −1 | +1 | −1 | −1 |
| νe | 0 | +1 | +1 | +1 |
| μ− | −1 | +1 | −1 | −1 |
| νμ | 0 | +1 | +1 | +1 |
| τ− | −1 | +1 | −1 | −1 |
| ντ | 0 | +1 | +1 | +1 |
Δ H = Δ ( S + D + U ) = Δ S + 2 Δ I 3 . (9)
The pure hadronic weak interaction obeys selection rule
Δ S = ∓ 1 , Δ I = ± 1 / 2 , Δ I 3 = ± 1 / 2 . (10)
Obviously, the selection rule (Equation (10)) can be deduced from ΔH = 0 (Equation (9)), i.e., conservation of H as well as conservation of F because in pure hadronic weak interaction HL = 0 and then F = HL (Equation (4)). For example, in K0 decay
K 0 → π + + π − , (11)
or in quark terms
s ¯ d → d ¯ u + u ¯ d , (12)
we can see that ΔH = 0 as well as ΔF = 0 which leads to ΔS = −1 and ΔI3 = 1/2.
Although lower probability than ΔS = ±1, there exists the pure hadronic weak interaction with ΔS = ±2. For example, in the decay
Ξ − → n + π − , (13)
or in quark terms
ssd → ddu + u ¯ d , (14)
the selection ΔS = 2 and ΔI3 = −1 can still be deduced from ΔH = 0 (Equation (9)) as well as ΔF = 0.
In consequence, in the pure hadronic weak interaction including u, d and s quarks, the selection rules can be deduced from conservation of H as well as conservation of F.
In the pure leptonic weak interaction, because S = 0 and H = 0, there is no selection rule, but HL as well as F is conserved. For example, in muon decay
μ − → e − + ν ¯ e + ν μ , (15)
ΔHL = 0 as well as ΔF = 0 is satisfied.
The fermionic weak interaction, in which both hadrons and leptons are involved, obeys two selection rules:
1) The first selection rule is
Δ S = 0 , Δ I = ± 1 , Δ I 3 = ± 1 . (16)
From Equation (4) and Equation (9),
Δ F = Δ H L + 2 Δ I 3 . (17)
For example, in β decay
n → p + e − + ν ¯ e , (18)
or in quark terms
ddu → uud + e − + ν ¯ e , (19)
where s quark does not appear, we can see that ΔF = 0 (Equation (17)) which leads to ΔI3 = 1 and ΔHL = −2.
Another example is that in Σ+ decay
Σ + → Λ 0 + e + + ν e , (20)
or in quark terms
suu → sud + e + + ν e , (21)
where S is not changed as s quark so called “spectator” does not participate in the reaction, we can see that ΔF = 0 (Equation (17)) which causes ΔI3 = −1 and ΔHL = 2.
2) The second selection rule is
Δ S = Δ Q h = ± 1 , Δ I = ± 1 / 2 , Δ I 3 = ± 1 / 2 , (22)
where ΔQh is change of electric charge of hadrons. From Equation (4) and Equation (9),
Δ F = Δ H L + Δ S + 2 Δ I 3 . (23)
For example, in decay
Σ − → n + e − + ν ¯ e , (24)
or in quark terms
sdd → ddu + e − + ν ¯ e , (25)
ΔS = ΔQh = 1, ΔI3 = 1/2, and ΔHL = −2. The selection can be deduced from ΔF = 0 (Equation (23)).
A reverse case is that a reaction with ΔS = −ΔQh is consistent with Gell-Mann-Nishijima formula Equation (7), but has not been observed. For example,
Σ + → n + e + + ν ¯ e , (26)
or in quark terms
suu → ddu + e + + ν ¯ e , (27)
where ΔS = −ΔQh = 1 and ΔI3 = −3/2, but ΔHL = 0, so that ΔF = −2. So, the reason why a reaction with selection ΔS = −ΔQh has not been observed is clear: F is not conserved in the case.
In consequence, in the fermionic weak interaction including u, d and s quarks, the selection rules can be deduced from conservation of F.
Relation between selection rules in weak interaction and F conservation can be extended to the other quarks including c, b and t quarks. For example, including c quark, from Equation (2), H = U + D + S + C and then from Equation (1),
Δ F = Δ S + 2 Δ I 3 + Δ C + Δ H L (28)
D+ decay is a fermionic weak interaction:
| type of weak interaction | conserved additive quantum number | selection rule in weak interaction |
|---|---|---|
| pure leptonic | F, HL | null |
| pure hadronic | F, H | 1) ΔS = ∓1, ΔI = ±1/2, ΔI3 = ±1/2. |
| 2) ΔS = ∓2, ΔI3 = ±1 | ||
| fermionic | F | 1) ΔS = ΔQh = ±1, ΔI = ±1/2, ΔI3 = ±1/2 |
| 2) ΔS = 0, ΔI = ±1, ΔI3 = ±1 |
D + → Κ ¯ 0 + e + + ν e , (29)
or in quark terms
c d ¯ → d ¯ s + e + + ν e , (30)
we can see that ΔF = 0 (conservation of F) leads to selection ΔS = −1, ΔC = −1, ΔI3 = 0 (Equation (28)). In consequence, conservation of F determines the selection rules of weak interaction including heavier quarks.
This Conservation of F determines the selection rules in all types of weak interaction. Especially, conservation of H, as the hadronic part of F, determines the selection rules in pure hadronic weak interaction, and conservation of HL, as the leptonic part of F, determines the selection rules in pure leptonic weak interaction (
Conservation of F indicates that both hadrons and leptons must be considered together in weak interaction, but not separately. In fact, F due to its conservation gives a unified concept about both hadrons and leptons in weak interaction. If we only take Q, F0 and F, description of weak interaction becomes simpler, only based on the conserved physical quantities. Conservation of F in weak interaction, strong interaction and electromagnetic interaction and a tight correlation among Q, F0 and F should be utilized in quantum field theory.
This research was supported by National Natural Science Foundation (NSFC) grant number U2031103 in China.
The author declares no conflicts of interest regarding the publication of this paper.
Ma, X. H. (2022) Selection Rules in Weak Interaction and Conservation of Fermion Quantum Number. Journal of Modern Physics, 13, 700-706. https://doi.org/10.4236/jmp.2022.135039