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10 Particle Physics – II
The sum of the squares of the quark charges refer to those quark pairs
which can possibly contribute to the reactions at a given CMS energy (
√
s).
The factor 3 in (1) arises due to 3 colors. At the low s-values, below the c c
threshold, only u, d, s quarks are involved, the expected ratio being
R(
√
s < 3 GeV) = 3 × [(2
3)
2
+ (1
3)
2
+ (1
3)
2 ] = 2
R(3.2 <
√
s < 9.2 GeV) = 3 × [(2
3)
2
+ (1
3)
2
+ (1
3)
2
+ (2
3)
2 ] = 10
3
R(9.2 <
√
s < 350 GeV) = 3 × [(2
3)
2
+ (1
3)
2
+ (1
3)
2
+ (2
3)
2
+ (1
3)
2 ] = 11
3
R(
√
s > 350 GeV) = 3 × [(2
3)
2
+ (1
3)
2
+ (1
3)
2
+ (2
3)
2
+ (1
3)
2
+ (2
3)
2 ] = 5
10.48 In the quark model Σ
−
= dds, P = uud , n = udd , K
−
= su, π
−
= du
σ (Σ
− n) = 9σ (qq)
σ ( pp) = 9σ (qq)
σ (K
− p) = 3σ (qq) + 3σ (qq)
σ (π
− p) = 3σ (qq) + 3σ (qq)
It follows that
σ (Σ
− n) = σ ( pp) + σ (K
− p) − σ (π
− p)
10.49 In the quark model, Λ = uds, p = uud , n = udd , K
−
= su, π
+
= ud
σ (Λ p) = σ (uds)(uud)
= σ (uu + uu + ud + du + du + dd + su + su + sd) = 9σ (qq)
Similarly
σ ( pp) = 9σ (qq)
σ (K
− n) = 3σ (qq) + 3σ (qq)
σ (π
+ p) = 3σ (qq) + 3σ (qq)
Therefore
σ (Λ p) = σ ( pp) + σ (K
− n) − σ(π
+ p)
where we have assumed that σ (qq) = σ (qq)
10.50 Substract 1 MeV from the rest energies of the charged particles and take the
difference in the masses.
n(940) − p(938 − 1) = 3 MeV
= udd − uud = d − u
Σ
− (1,197 − 1) − Σ
0 (1,192) = 4 MeV
= dds − uds = d − u
Σ
0 (1192) − Σ
+ (1189 − 1) = 4 MeV
= uds − uus = d − u
K
0 (498) − K
+ (494 − 1) = 5 MeV
= ds − us = d − u
Thus the mean difference in the masses of d-quark and u-quark is 4 MeV.