BARYON OCTETS 



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. 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 and corresponding hypercharge is Y_{u}. The zaxis projection of Uspin is U_{3}. U_{3}=(1/2)I_{3} +(3/4)Y and Y_{u} =Q *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 a is I_{3} and b is Y/2. 
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. * Only 2 quarks / antiquarks involved where as 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.


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 
* 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 
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
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 
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)  
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.
* 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. 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 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. * 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 in 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. 

Comparison of isospin vrs weak isospin
&
hypercharge vrs Weak hypercharge
Q=  I_{3}+ Y /2 = T_{3}+ Y_{W}/2  2Y_{W} 
= 
5(BL)   X  
I_{3}  T_{3}  Y  Y_{W}  X  B  L  BL  
u  1/2  1/2  1/3  1/3  1  1/3  0  1/3  
d  1/2  1/2  1/3  1/3  1  1/3  0  1/3  
s  0  1/2  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  