BARYON OCTETS 

When we rotate the baryon hexagon so as to align Y with horizontal axis, the rotation is by angle x clockwise about xaxis. Then the coordinates are p (√5/2,0) , n (3/2√5 , 2/√5) ,Σ^{+}(1/√5, 2/√5) ,Σ^{}(1/√5, 2/√5),Ξ^{} (√5/2, 0) , Ξ^{0}(3/2√5 ,2/√5) 

Name  Quark composition  S  B  Y/2 = (B+S) / 2 
I_{τ}  Q=I_{τ} + Y/2  spin  mass*e  lifetime:
τ
in sec 

Σ^{+}  uus  1  1  0  1  1  1/2  2328  0.8*10^{10}  
Σ^{0}  uds  1  1  0  0  0  1/2  2334  0.8*10^{10}  
Σ^{}  dds  1  1  0  1  1  1/2  2343  10^{14}  
Ξ^{}  dss  2  1  1/2  1/2  1  1/2  2586  1.7*10^{10}  
Ξ^{0}  uss  2  1  1/2  1/2  0  1/2  2573  3*10^{10}  
p^{+}  udu  0  1  1/2  1/2  1  1/2  1836.1  
n^{0}  udd  0  1  1/2  1/2  0  1/2  1836.6  960  
Λ^{0}  uds  1  1  0  0  0  1/2  2183  2.5*10^{10}  
no. of possible configurations : 10
[ uuu,sss,ddd], [uus,uss,uud,udd,dds,dss,uds]
First set marked blue do not exist in octets , but in decuplets; Δ^{++}=uuu, Δ^{}=ddd; Ω^{}=sss ; * 3 quarks involved. No antiquark. * S for strangeness, B for Baryon number, I_{τ} for isospin projection, Y for hyper charge and Q for charge. Q=I_{τ} + Y/2 is the GellMann Nishijima formula for strong interaction. * Up quark > ( 1 down quark > ( 0 strange quark > ( 0 0 1 0 0 ) 0 ) 1 ) in 3D Vector Space. * The laws of Physics are invariant under application of unitary transformation to this space i.e ( x (x y = A y z ) z ) Where A is a 3x3 unitary matrix under SU(3) * if one takes A = ( 0 1 0 1 0 0 0 0 1 ) and applies the transformation on up quark, it becomes a down quark and vice versa. This is known as flavour rotation. * when a proton is transformed by every possible flavour rotation A, it turns out that it moves around in an 8dimensional vector space. These 8 dimensions correspond to 8 particles in the so called baryon octet. * Every Lie Group has a corresponding Lie Algebra and each group representation of the Lie group can be mapped to a corresponding representation of the Lie Algebra in the same vector space. The Lie Algebra su(3) can be written as the set of 3x3 traceless hermitian matrices. We normally discuss the representation theory of Lie Algebra su(3) in stead of Lie Group SU(3) since the former is simpler and both are equivalent. * The abstract group SU(3) is represented by a set of eight 3×3 matrices of complex elements which have determinant of unity. These elements of the group can be generated by eight special matrices. These matrices must be Hermitian; i.e., the transpose of their complex conjugates is the same as the matrix. These matrices do not have determinants of unity; instead all have traces (sums of elements on the principal diagonal) of zero. * If GellMann matrices are represented as λ_{i}, in su(3) algebra, the generators are g_{i} = λ_{i} /2 . These matrices are traceless, hermitian & can generate unitary matrix group elements of SU(3) through exponentiation. They obey the rule of trace orthonormality where the orthonormality is 2 instead of 1. Any element of SU(3) can be written as exp(iθ^{j} g_{j}) where 8 θ^{j } are real numbers. * In math, orthonormality means a norm which is unity. GellMann matrices are however normalized to a value of 2. Thus the trace of the pair of products results in orthonormalization condition >Tr(λ_{i} λ_{j} ) =2δ_{ij} * In representing baryon octets, We can migrate from I_{3} Y basis to U_{3}Y_{u} basis where U is a new mathematical concept called U spin or magnetic spin quantum number and corresponding hypercharge is Y_{u}. The zaxis projection of Uspin is U_{3}. U_{τ}=(1/2)I_{τ} +(3/4)Y or 4U_{τ} = 2I_{τ} + 3Y and Y_{u} =Q which simplifies to U_{τ} = Y + Y_{u} /2 analogous to Q= I_{τ} + Y/2 Similar to conservation of charge, there is conservation of U spin corresponding to magnetic moment. Q= I_{τ} + Y/2 Q= 2Y  2U_{τ} Q =4/3 * (I_{τ }+ U_{τ}/2) Migration from Q to U_{τ} Q= I_{τ} + Y/2 U_{τ}= (1/2)I_{τ} +(3/ 4)Y Here , the coefficient of I_{τ} has decreased by 150% and the coefficient of Y has increased by 150%. +Baryon Octets : U Spin
* You shall observe that u = Y + Y_{u}/ 2 Q = ( Y' Y_{u}' ) / √2 where Y' = R * Y Y_{u}' Y_{u} and R= cos 45 sin 45 sin 45 cos 45 *Below we consider two applications of the Uspin: SU(3) predictions for the magnetic moments of the octet and the transition magnetic moments of the antidecuplet. It follows from the assumption of the Uspin conservation that the magnetic moments (and electric charges) of all members of the same Uspin multiplet are equal. From the left panel of Fig. 4, one then immediately obtains that µΣ− = µΞ− , µΞ0 = µn , µp = µΣ+ * GellmannOkubo GO formula for mass of baryons : link :1 M =a1 +a2Y +a3[I(I+1)Y^{2} /4] where a1,a2, a3 are experimentally determined. The formula works for baryons within 0.5% of measured value. * We observe that charge of u quark is 2/3 whereas charge of d quark is 1/3. Question arises why there is so much asymmetry between charges of both quarks while p,e have same magnitude of charge. It may be that charge is not an indivisible attribute and consists of multiple more fundamental attributes which are symmetric. Supposing that there are 2 attributes a , b such that a+b=2/3 and ab=1/3 . solving them, a=1/6, b=1/2 . So now u,d are more symmetric in their charges since a is same for u and d and same is the case for b attribute. In fact, it seems b is I_{3} and a is Y/2. * It is quite likely that u is fractional because of rotation of charge q which is distributed on the surface of a sphere with radius r=1 which gives effective charge as (2/3)qr^{2} =2q/3 similar to the moment of inertia of a hollow sphere about an axis passing through the center. 
MESON OCTETS (Pseudo Scalar Mesons) 



Name  Quark composition  S  B  Y/2= (S+B) / 2 
I_{τ}  Q=I_{τ} + Y/2  spin  mass*e  lifetime:
τ in sec 

π^{+}  ud'  0  0  0  1  1  0  273  2.6*10^{8}  
π^{0}  uu'  0  0  0  0  0  0  264  0.8*10^{16}  
π^{}  u'd  0  0  0  1  1  0  273  2.6*10^{8}  
K^{}  u's  1  0  1/2  1/2  1  0  966  1.2*10^{8}  
K^{0}  ds'  1  0  1/2  1/2  0  0  974  
K^{+}  us'  1  0  1/2  1/2  1  0  966  1.2*10^{8}  
K^{0'}  d's  1  0  1/2  1/2  0  0  974  
η^{0}  dd'  0  0  0  0  0  0  1074  10^{17}  
η^{'}  ss'  0  0  0  0  0  0  
no. of possible configurations : 21 [ (uu,ss,dd),(ud,us,ds),(u'u',d'd',s's'),(u'd',u's',d's')], [(uu',dd',ss'),(ud',u'd,us'.u's,ds',d's)] . First set numbering 12 marked blue do not exist because they shall result in fractional charges and they are either all quarks or all antiquarks. *From(u,d,s) , the combination of 2 quarks is 3c2 =3 (ud,us,ds) *From(u',d',s') , the combination of 2 quarks is 3c2 =3 (u'd',u's',d's') *From (u,d,s) to (u',d',s'), combination of 2 is 3 x3 =9 [(uu',dd',ss'),(ud',u'd,us'.u's,ds',d's)] *With repetition of (u,d,s) , combination is 3 (uu,ss,dd) *With repetition of (u',d',s') , combination is 3 (u'u',d'd',s's') First 2 sets do not occur , because the charges are not integers. 3rd set of 9 is fully exhausted in meson octet with spin 0 and vector meson octet of spin 1. 4th and 5th sets again do not occur because charges are not integers. If we consider the combination of non integer charges, total is 12 out of which 6 are negative of another 6. The frequency distribution of fractional charges is 4/3  1 1/3 2 2/3 3 weighted average= [(4/3)*1+(1/3)*2 +(2/3)*3] / 6 = 0/6 =0 Same is true for 4/3,1/3.2/3 . * Only 2 quarks / antiquarks involved in mesons whereas in baryons , the number is 3. In mesons, each particle has 1 quark and 1 antiquark. *Every meson is the bound state of a quark and antiquark unlike baryons where antiquarks are not involved in octet. Hence the blue marked combinations excluded. Moreover, in the Baryon octet, there are no antibaryons whereas in the meson octet, there are anti mesons.(π^{+},π^{}) ,(K^{+},K^{} ), (K^{0} ,K^{0'}) are antiparticles of each other. Thus in the meson octet, there are only 6 particles 3 having their antiparticles and 3 (π^{0} ,η^{0} ,η^{'})having no separate antiparticles * η^{' } is called the eta prime meson. *Only Gauge invariant objects are observable. Since quarks and gluons are not gauge invariant, they are not observable. Hadrons, being gauge invariant, are observable. * Meson nonet can be put in matrix form uu' du' su' ud' dd' sd' us' ds' ss' In terms of charge Q, the matrix is as under 0 1 1 1 0 0 1 0 0 Since 2 rows are identical, this is singular, traceless , skew symmetric matrix. * In a st. line along I_{τ} , if we take the maximum absolute value of I_{τ} and denote it by n, then total no. of particles in that line will be 2n+1. Exa If n=1, then no. of slots in that line =2*1+1=3 Taking only 1st generation quarks into consideration, total no. of arrangement is 10 out of which 4 have integer charge(π^{+}.π^{}.π^{0}.η^{0}) and 6 fractional charge . The ratio of fractional to integer charge is 3:2: (u,u',d,d') no. of combination  4 C 2 =6 uu' Q =0 dd' Q=0 ud' Q=1 u'd Q=1 ud Q= 1/3 (this is equal to the charge of s' quarks with I_{τ}=0 but spin 1/0) u'd' Q=1/3 (this is equal to the charge of s quarks with I_{τ}=0 but spin 1/0) Then with repetition (no. of arrangements=4) uu Q=4/3 u'u' Q=4/3 dd Q=2/3 d'd' Q= 2/3 Combinations with blue and red color are either all quarks or all anti quarks and have fractional charges. we can arrange the charges as 0/3. 1/3. 2/3,3/3,4/3 with plus and minus sign and 4 combinations (2 each) of black color have integer charges. Below combination (4 in number in red) will have only spin zero and spin 1 is ruled out because all quantum no. of fermions cannot be identical. Of course, this paved the way for a new quantum no. called color quarks.
We make a table of quarks of 1st generation with fractional charges


BARYON DECUPLET 



Name 
Quark Composition 
S  B  Y/2=(S+B)/2  I_{τ}  Q=I_{τ} + Y/2  spin  mass*e  lifetime:
τ in sec 

Δ^{++}  uuu  0  1  1/2  3/2  2  3/2  
Δ^{}  ddd  0  1  1/2  3/2  1  3/2  
Δ^{+}  uud (similar to p^{+} )  0  1  1/2  1/2  1  3/2  
Δ^{0}  ddu (similar to n^{0} )  0  1  1/2  1/2  0  3/2  
Σ^{*+}  uus (similar to Σ^{+} )  1  1  0  1  1  3/2  
Σ^{*}  dds (similar to Σ^{} )  1  1  0  1  1  3/2  
Ξ^{*}  dss (similar to Ξ^{} )  2  1  1/2  1/2  1  3/2  
Ξ^{*0}  uss (similar to Ξ^{0} )  2  1  1/2  1/2  0  3/2  
Σ^{*0}  uds (similar to Σ^{0} )  1  1  0  0  0  3/2  
Ω^{}  sss  3  1  1  0  1  3/2  8.21*10^{11}  
Transformation from one Type to Another
n^{0} (K^{0} )  p^{+} (K^{+} ) 
udd (ds')  duu(us') 
Ξ^{0} (K^{0'} )  Ξ^{}(K^{} ) 
uss (d's)  dss(u's) 
Σ^{+}( π^{+} )  Σ^{} ( π^{} ) 
uus(u'd)  dds(d'u) 
Σ^{0} ( π^{0})  Λ^{0} ( η^{0} ) 
uds(uu')  dus(dd') 
η^{'}(ss')  η^{'}(ss') 
* in baryon octet, no
antiquark is involved *consists of 3 quarks *Strange quarks are absent in nucleons.
* All 8 are particles and there is no antiparticle in the octet.
* out of 10 possible combinations, 7 find their place in octet; 3 in ducuplet of resonance baryons sss for Ω^{} , uuu for Δ^{++} , ddd for Δ^{} . * Baryon no. is 1 * there is degeneracy of quarks in Σ^{0} &Λ^{0} . Both consist of u+d+s. However, their masses are different.Σ^{0} is heavier than Λ^{0} by 151m_{e} = 77 Mev & decays into the latter. WHY MASS DIFFERENCE WITH SAME QUARKS?? Σ^{0} > Λ^{0 }+ γ * particles transform by interchange of ud
* total u 8, d8,s8 sum 24 * 5u> 5d 3d>3u 4s4s
* No. of Baryons with 0 strange quark : 2 No. of Baryons with 1 strange quark : 4 No. of Baryons with 2 strange quark : 2 * all spin 1/2 particles with exception of Ω^{} which has 3/2 spin. 
* In meson octet, in every
particle, there is one quark and one antiquark. * consist of 2 particles , 1 quark & 1 anti quark * strange quark/anti quark absent in pions, neutral eta meson. * there are 3 particles(K^{+},π^{+},K^{0}) and 3 antiparticles, total=6 and 3 particles which have no separate antiparticle, total=3
* out of 21 possible combinations, 9 find their place in meson octet. Other 12 ruled out because they have fractional charges and moreover they are made up of either only quarks or only antiquarks. * Baryon no. is 0 * There is no such degeneracy in mesons.
* particles transform by interchange of ud and u'd'. * ' represents antiquarks. *total u3,u'3,d3,d'3,s2,s'2 sum 16 3u>3d 3u'>3d' 1s1s 1s'1s' η' is excluded . * if η' is included, total u3,u'3,d3,d'3,s3,s'3 sum 18 * No. of Mesons with 0 strange quark : 4 No. of Mesons with 1 strange quark : 4 No. of Mesons with 2 strange quark : 1 * all spin 0 particles but vector mesons have spin 1 with odd parity. 
Hypercharge (Y) mapping [Bijective]
Baryon  Mesons
particles  Y  mapping  particles  Y  
p^{+}, n^{0}  1  K^{+}, K^{0}  1  
Σ^{+}, Σ^{0} , Σ^{}, Λ^{0}  0  π^{+},π^{0} ,π^{} ,η^{0}  0  
Ξ^{0} ,Ξ^{}  1  K^{0'}, K^{}  1  
Strangeness (S) mapping [Bijective]
Baryon  Mesons
particles  S  mapping  particles  S  
p^{+}, n^{0}  0  K^{+}, K^{0}  1  
Σ^{+}, Σ^{0} , Σ^{}, Λ^{0}  1  π^{+},π^{0} ,π^{} ,η^{0}  0  
Ξ^{0} ,Ξ^{}  2  K^{0'}, K^{}  1  
Quark Triplet
particle  spin I_{3}  type  B  S  Y=B+S  Y/2  I_{τ}  Q=I_{τ}+Y/2  U spin=I_{τ} /2 + 3Y/4  
u  1/2  fermion  1/3  0  1/3  1/6  1/2  2/3  0  
u'  1/2  antifermion  1/3  0  1/3  1/6  1/2  2/3  0  
d  1/2  fermion  1/3  0  1/3  1/6  1/2  1/3  1  
d'  1/2  anti fermion  1/3  0  1/3  1/6  1/2  1/3  1  
s  1/2  fermion  1/3  1  2/3  1/3  0  1/3  1  
s'  1/2  anti fermion  1/3  1  2/3  1/3  0  1/3  1 
1+5+2+3+2+5+1=19
AB=AC=DF=EF=√13/6
BC=DE=1
BD=CE=2/3
AE=AD=BF=CF=√5/2
BE=CD=√13/3
AF=4/3
sin DAF =1/√5
DAF=26.5650 °
ADE=AED=63.4349 °
ADC=CDE=AEB=DEB=31.71745 °
AOD=121.7175 °
Contribution (%) of Hypercharge (Y) & IsoSpin Projection (I_{τ})
to
the Charge Q of particles
particle  type  Y (%) contribution  I_{τ} (%) contribution  (%) total contribution 
u quark  fermion  25  75  100 
d quark  fermion  25  75  100 
s quark  fermion  100  0  100 
e^{}  lepton (fermion)  50  50  100 
μ^{}  lepton (fermion)  50  50  100 
τ^{}  lepton (fermion)  50  50  100 
(K^{+}) p^{+} ^{(us')udu} 
nucleon/baryon/hadron (meson/hadron) 
50 (+)  50 (+)  100 
( π^{+})
Σ^{+} ^{ (ud')uus} 
hyperon/baryon/hadron (meson/hadron) 
0  100 (+)  100 
( π^{})Σ^{} ^{(u'd)dds} 
hyperon/baryon/hadron (meson/hadron) 
100 ()  0  100 
(K^{}) Ξ^{} ^{(u's)dss} 
hyperon/baryon/hadron (meson/hadron) 
50 ()  50()  100 
(K^{0}) n^{0} ^{(u's)dud} 
nucleon/baryon/hadron (meson/hadron) 
50 (+)  50 ()  0 
(K^{0'}) Ξ^{0}
^{(d's)uss} 
hyperon/baryon/hadron  50 ()  50 (+)  0 
( π^{0}) Σ^{0} ^{(uu')uds} 
hyperon/baryon/hadron (meson/hadron) 
0  0  0 
(η^{0})Λ^{0} ^{(dd')uds} 
hyperon/baryon/hadron (meson/hadron) 
0  0  0 
Δ^{++} uuu 
baryon/hadron  25  75  100 
Δ^{} ddd 
baryon/hadron  50  150  100 
Ω^{} sss 
hyperon/baryon/hadron  100  0  100 
Vector Mesons (nonet)
Vector mesons are mesons with total spin 1 and of odd parity. Pseudo vector mesons
have total spin 1 and are of even parity.
Conservation of charge in Strong / weak interactions
particles  type  chirality + or right handed chirality  or left handed 
isospin(I)  3rd component
of isospin (I_{3}) 
weak isospin(T)  3rd component of
weak isospin (T_{3}) 
Hypercharge(Y)  weak hypercharge (Y_{w}) Left handed particles 
mass Mev 
BL  X charge 
u  fermion    1/2  +1/2  1/2  +1/2(LH)  1/3  +1/3(LH)  1.4  +1(LH)  
d  fermion    1/2  1/2  1/2  1/2(LH)  1/3  +1/3(LH)  4.5  +1(LH)  
u'  antifermion  +  1/2  1/2  1/2  1/2(RH)  1/3  1/3(RH)  1(RH)  
d'  antifermion  +  1/2  +1/2  1/2  +1/2(RH)  1/3  1/3(RH)  1(RH)  
s  fermion    0  0  1/2  1/2(LH)  2/3  +1/3(LH)  9095  +1(LH)  
s'  antifermion  +  0  0  1/2  +1/2(RH)  +2/3  1/3(RH)  1(RH)  
c  fermion  +  0  0  1/2  +1/2(LH))  4/3  1/3(LH)  
c'  antifermion    
Leptons  BL  X charge  
e^{}  fermion    0  0  1/2  1/2(LH)  0  1(LH)  0.511  1  3(LH) 
ν_{e}  fermion    0  0  1/2  1/2(LH) 0 (RH) 
0  1(LH) 0 (RH) 
1  3(LH)  
μ^{}  fermion    0  0  1/2  1/2(LH)  0  1(LH)  52.5  1  3(LH) 
ν_{μ}  fermion    0  0  1/2  1/2(LH) 0 (RH) 
0  1(LH) 0 (RH) 
1  3(LH)  
τ^{}  fermion    0  0  1/2  1/2(LH)  0  1(LH)  1  3(LH)  
ν_{τ}  fermion    0  0  1/2  1/2(LH) 0 (RH) 
0  1(LH) 0 (RH) 
1  3(LH)  
* Xcharge is a conserved quantum number associated with the SO(10) Grand Unification Theory. It is thought to be conserved in electromagnetic, strong, weak, gravitational, Higgs interaction. Because it is associated with weak hypercharge, it varies with the helicity of a particle. A Left handed quark has an Xcharge +1 where as a right handed quark has Xcharge 1(up,charm,top quarks) or 3(down,strange,bottom quarks) X=5(BL) 2Y_{w} * XCharge in proton decay: Proton decay is a hypothetical form of radioactive decay, predicted by many grand unification theories. During proton decay, the common baryonic proton decays into lighter subatomic particles. However, proton decay has never been experimentally observed and is predicted to be mediated by hypothetical X and Y bosons. Many protonic decay modes have been predicted, one of which is shown below.
This form of decay violates the conservation of both baryon number and lepton number, however the Xcharge is conserved. Similarly, all experimentally confirmed forms of decay also conserve the Xcharge value.
* iso spin symmetry for strong interaction involves u,d quarks and it is an approximate symmetry. The exact symmetry is broken by mass (u quark mass is different from d quark mass) and electromagnetism(u,d quarks have different charges). Iso spin symmetry is defined as the invariance of the Hamiltonian of the strong interaction under the action of Lie Group SU(2). * For Strong Interaction, Q=I_{3} + Y/2 where Y=(B+S + C+t+b')/2 * For Weak Interaction, Q=T_{3} + Y_{w} /2 where Y_{w }=(5/2)(BL) X/2 Where B is baryon no., L lepton no. and X is a conserved quantity under GUT.This is applicable to left handed particles. X=1 or X +2 Y_{w} = 5(BL) . Pl. note that BL is a conserved quantity in weak interaction. Weak interaction is one where neutrinos of any of the 3 (or combination/mixing of 3 flavors) different flavors are produced. Isospin (for strong interaction) is similar to but should not be confused with weak isospin. * Weak isospin is the gauge symmetry of the weak interaction which connects quark & lepton doublets of left handed particles, (u,d), (t,b),(e^{} ,ν_{e} ) *Isospin is a symmetry of strong interaction under the action of Lie group SU(2), the 2 states being up flavor and down flavour. .Strong interaction connects only up and down quarks , acts on both chiralities (left, right) and is a global (not gauge) symmetry. * Iso spin corresponding to weak interactions is known as weak iso spin T or T3. A quark of one flavor can transform to a quark of another flavor and the matrix of interaction is called CKM matrix. * Quarks are always in a bound or confined stage but in QGP(quark gluon plasma), they occur in deconfined stage. * For massless particles, chirality and helicity are same thing whereas for massive particles, both must be distinguised. Universe prefers left handed chirality. (why???). Chirality is defined by whether the particle transforms in a right handed or left handed representation of the Poincare Group * isospin is conserved only in strong interaction whereas weak isospin & weak hypercharge are conserved in electroweak interactions. It is conserved (only terms that are overall weakhypercharge neutral are allowed in the Lagrangian). However, one of the interactions is with the Higgs field. Since the Higgs field vacuum expectation value is nonzero, particles interact with this field all the time even in vacuum. This changes their weak hypercharge (and weak isospin T_{3}). Only a specific combination of them, Q = T_{3} + Y_{W} /2(electric charge), is conserved. * Weak hypercharge corresponds to Gauge Symmetry U(1). Hypercharge is the eigen value of the charge operator. * Gauge transformation is one where transformation of the potential leaves the field invariant. * In nature, so far we have not come across right handed neutrinos. When the left handed neutrino is a) receding from the observer or b) approaching the observer , its spin axis is always parallel to the direction of motion. However isospin angular momentum is * antiparallel to direction of motion when it is receding and the isospin rotation is anticlockwise. * parallel to direction of motion when it is approaching and isospin rotation is clockwise. * A quark cannot decay in weak interaction, into another quark having the same 3rd component of weak iso spin i.e. T3. In case of leptons, something similar happens for left handed leptons (+ chirality) such as electron, muon, tao particle and corresponding right handed neutrinos. * Fermions with + chirality (LH fernions) and anti fermions with  chirality(RH anti fermions)) form singlets with T=T3=0 and do not undergo charged weak interaction. *Strange particles can only decay through weak interaction and strangeness is not conserved in weak interaction. * Iso spin and 3rd component of isospin is not conserved in electromagnetic and weak interaction. Hence a separate term weak iso spin, weak hypercharge is coined which is conserved in weak interaction. * We consider left handed particles as spinning anticlockwise while the particle is receding from the observer. Chirality of such particles is termed positive. * There are 8 generators of SU(3) out of which 6 correspond to 6 flavors of quarks, 1 corresponds to isospin and 1 corresponds to hypercharge. Cabibbo Angle and Mixing : There are 3 families {(u,d) , (c,s) , (t,b) }of quarks and 3 families {(e,ν_{e}),(μ,νμ),(τ,ντ)} of leptons. The coupling constant g between members of the same family is same . Coupling constant universality is a characteristic feature of quarks and leptons. The transition amplitude L is proportional to g/ 2√π. The transition probability α_{w} is proportional to square of L. The transition probability from u to d should be same as that of e to ν_{e} . But in reality, it is slightly less. The difference is the transition probability from u to s. This is explained by Cabibbo angle. We define flavor generators T for first 2 generations of quarks as under T(u) = T(c) = 2/3 0 0 2/3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 T(d) = T(s) = 1/3 0 0 1/3 0 0 0 0 0 0 0 0 0 0 1 0 0 0 sinθ_{c} =Tr [T(d)T(c) ] =2/9 and angle is about 12.7 degree. Mixing is between flavor mass eigen states(u,c,t,d,s,b) and flavor interaction states(u',c',t',d's',b'). u' = u c' c t' t
d' =V_{ckm} * d s' s b' b 

Comparison of isospin vrs weak isospin
&
hypercharge vrs Weak hypercharge
Q=  I_{3}+ Y /2 = T_{3}+ Y_{W}/2  (LH)2Y_{W} 
= 
5(BL)   X  
(RH)2Y_{W}  +5(BL)for(u,c,t) 5(BL)for(d,s,b) 
+X  
I_{3}  T3(RH)  (LH)T_{3}  Y  (RH)Y_{W}  (LH)Y_{W}  X  B  L  BL  
u  1/2  0  1/2  1/3  4/3  1/3  1  1/3  0  1/3  
d  1/2  0  1/2  1/3  2/3  1/3  1  1/3  0  1/3  
s  0  0  1/2  2/3  2/3  1/3  1  1/3  0  1/3  
c  0  0  1/2  4/3  4/3  1/3  1  1/3  0  1/3  
t  0  0  1/2  4/3  4/3  1/3  1  1/3  0  1/3  
b  0  0  1/2  2/3  2/3  1/3  1  1/3  0  1/3  
e^{}  0  1/2  0  1  3  0  1  1  
ν_{e}  0  1/2  0  1  3  0  1  1  
μ^{}  0  1/2  0  1  3  0  1  1  
ν_{μ}  0  1/2  0  1  3  0  1  1  
τ^{}  0  1/2  0  1  3  0  1  1  
ν_{τ}  0  1/2  0  1  3  0  1  1  