574 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 cc
threshold, only u, d, s quarks are involved, the expected ratio being
R(√
s<3GeV)= 3 ×[(2/
3)^2 +(1/
3)^2 +(1/
3)^2 ]= 2
R(3. 2 <√
s< 9 .2GeV)= 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 ]= 510.48 In the quark modelΣ−=dds, P=uud,n=udd,K−=su,π−=du
σ(Σ−n)= 9 σ(qq)
σ(pp)= 9 σ(qq)
σ(K−p)= 3 σ(qq)+ 3 σ(qq)
σ(π−p)= 3 σ(qq)+ 3 σ(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(Λp)=σ(uds)(uud)
=σ(uu+uu+ud+du+du+dd+su+su+sd)= 9 σ(qq)
Similarly σ(pp)= 9 σ(qq)
σ(K−n)= 3 σ(qq)+ 3 σ(qq)
σ(π+p)= 3 σ(qq)+ 3 σ(qq)
Therefore σ(Λp)=σ(pp)+σ(K−n)−σ(π+p)
where we have assumed thatσ(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)=3MeV
=udd−uud=d−u
Σ−(1, 197 −1)−Σ^0 (1,192)=4MeV
=dds−uds=d−u
Σ^0 (1192)−Σ+(1189−1)=4MeV
=uds−uus=d−u
K^0 (498)−K+(494−1)=5MeV
=ds−us=d−u
Thus the mean difference in the masses of d-quark and u-quark is 4 MeV.