BARYON OCTETS

               
 

               
Name Quark composition S B  Y/2

= (B+S) / 2

Iτ Q=Iτ + Y/2 spin mass*e lifetime: τ

in sec

Σ+ uus -1 1 0 1 1 1/2 2328 0.8*10-10
Σ0 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
Ξ0 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  
n0 udd 0 1 1/2 -1/2 0 1/2 1836.6 960
Λ0 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 anti-quark.

* S for strangeness, B for Baryon number, Iτ for iso-spin projection, Y for hyper charge and Q for charge. Q=Iτ + Y/2 is the Gell-Mann Nishijima formula for strong interaction.

* Up quark -> ( 1         down quark -> ( 0       strange quark -> ( 0

                         0                                     1                                     0

                         0 )                                   0 )                                   1 )

in 3-D 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 8-dimensional 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 33 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 Gell-Mann matrices are represented as λi, in su(3) algebra, the generators are gi = λi  /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(iθj gj) where 8  θj are real numbers.

* In math, orthonormality means a norm which is unity. Gell-Mann matrices are however normalized to a value of 2. Thus the trace of the pair of products results in orthonormalization condition ->Tr(λi λj ) =2δij

* In representing baryon octets, We can migrate from I3 -Y basis to U3-Yu basis where U is a new mathematical concept called U- spin and corresponding hypercharge is Yu. The z-axis projection of U-spin is U3.

U3=-(1/2)I3 +(3/4)Y and

Yu =-Q

*Below we consider two applications of the U-spin: SU(3) predictions for the magnetic moments of the octet and the transition magnetic moments of the anti-decuplet. It follows from the assumption of the U-spin conservation that the magnetic moments (and electric charges) of all members of the same U-spin multiplet are equal. From the left panel of Fig. 4, one then immediately obtains that Σ− = Ξ− , Ξ0 = n , p = Σ+

* Gellmann-Okubo G-O formula for mass of baryons : link :1

 M =a1 +a2Y +a3[I(I+1)-Y2 /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  a-b=-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 I3  and b is Y/2.

               

 

 

MESON OCTETS

(Pseudo Scalar Mesons)

               
 

               
Name Quark composition S B   Y/2=

(S+B) / 2

Iτ Q=Iτ + Y/2 spin mass*e lifetime: τ

in sec

π+ ud' 0 0 0 1 1 0 273 2.6*10-8
π0 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
K0 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
K0' d's -1 0 -1/2 1/2 0 0 974  
η0 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 anti-quarks.

* Only 2 quarks / anti-quarks involved where as in baryons , the number is 3. In mesons, each particle has 1 quark and 1 anti-quark.

*Every meson is the bound state of a quark and anti-quark unlike baryons where anti-quarks are not involved in octet. Hence the blue marked combinations excluded. Moreover, in the Baryon octet, there are no anti-baryons whereas in the meson octet, there are anti mesons.(π+-) ,(K+,K- ), (K0 ,K0') are anti-particles of each other. Thus in the meson octet, there are only 6 particles- 3 having their anti-particles and 3 (π0 ,η0 ,η')having no separate anti-particles

* η'  is called the eta prime meson.

 

               
 

BARYON DECUPLET

               
 

               
Name

Quark Composition

S B Y/2=(S+B)/2 Iτ Q=Iτ + 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    
Δ0 ddu (similar to n0 ) 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 Ξ0 ) -2 1 -1/2 1/2 0 3/2    
Σ*0 uds  (similar to Σ0  ) -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

n0 (K0 ) p+ (K+ )
udd (ds') duu(us')
Ξ0 (K0' ) Ξ-(K- )
 uss (d's) dss(u's)
 Σ+( π+  )  Σ-  ( π- )
uus(u'd) dds(d'u)
Σ0  ( π0)  Λ0 ( η0 )
uds(uu') dus(dd')
η'(ss') η'(ss')
* in baryon octet, no anti-quark is involved

*consists of 3 quarks

*Strange quarks are absent in nucleons.

 

* All 8 are particles and there is no anti-particle 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 Σ0 &Λ0 . Both consist of u+d+s. However, their masses are different.Σ0  is heavier than Λ0 by 151me- =  77 Mev & decays into the latter. WHY MASS DIFFERENCE WITH SAME QUARKS??

Σ0  -> Λ0 + γ

* particles transform by interchange of u-d

 

* total u -8, d-8,s-8 sum 24

* 5u-> 5d

   3d->3u

   4s-4s

 

 

* 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 anti-quark.

* 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+,K0) and 3 antiparticles, total=6

   and 3 particles which have no separate anti-particle, 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 anti-quarks.

* Baryon no. is 0

* There is no such degeneracy in mesons.

 

* particles transform by interchange of  u-d and u'-d'.

* ' represents anti-quarks.

*total u-3,u'-3,d-3,d'-3,s-2,s'-2 sum 16

3u->3d

3u'->3d'

1s-1s

1s'-1s'

η' is excluded .

* if η' is included,

  total u-3,u'-3,d-3,d'-3,s-3,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+, n0 1    K+, K0 1  
  Σ+, Σ0 , Σ-, Λ0 0    π+,π0 ,π- ,η0 0  
  Ξ0 ,Ξ- -1    K0', K- -1  
             

Strangeness (S) mapping [Bijective]

 Baryon - Mesons

  particles S mapping particles S  
   p+, n0 0    K+, K0 1  
  Σ+, Σ0 , Σ-, Λ0 -1    π+,π0 ,π- ,η0 0  
  Ξ0 ,Ξ- -2    K0', K- -1  
             

Quark Triplet

 

Contribution (%) of Hypercharge (Y) & Iso-Spin Projection (Iτ)

to

the Charge Q of particles

particle type Y  (%) contribution Iτ (%) 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
 (K0) n0

(u's)dud

nucleon/baryon/hadron

(meson/hadron)

50 (+) -50 (-) 0
(K0') Ξ0

(d's)uss

hyperon/baryon/hadron -50 (-) 50 (+) 0
( π0) Σ0

(uu')uds

hyperon/baryon/hadron

(meson/hadron)

0 0 0
00

(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

iso-spin(I) 3rd component

of

iso-spin (I3)

weak iso-spin(T) 3rd component of

weak iso-spin (T3

Hypercharge(Y) weak

hypercharge

(Yw)

Left handed particles

mass

Mev

B-L 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' anti-fermion + -1/2 -1/2 -1/2 -1/2(RH) -1/3 -1/3(RH)     -1(RH)
d' anti-fermion + -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) 90-95   +1(LH)
s' anti-fermion + 0 0 -1/2 +1/2(RH) +2/3 -1/3(RH)     -1(RH)
                       
  Leptons                 B-L X charge
e- fermion - 0 0 1/2 -1/2(LH) 0 -1(LH) 0.511 -1 -3(LH)
νe- 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)
νμ- 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)
ντ- fermion - 0 0 1/2 1/2(LH)

0 (RH)

0 -1(LH)

0 (RH)

  -1 -3(LH)
                       

* X-charge 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 X-charge +1 where as a right handed quark has X-charge -1(up,charm,top quarks) or -3(down,strange,bottom quarks)

X=5(B-L) -2Yw

* X-Charge 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.

p+ → e+ + π0

This form of decay violates the conservation of both baryon number and lepton number, however the X-charge is conserved. Similarly, all experimentally confirmed forms of decay also conserve the X-charge value.

 

* For Strong Interaction, Q=I3 + Y/2  where Y=(B+S + C+t+b')/2

* For Weak Interaction, Q=T3  +  Yw /2 where  Yw =(5/2)(B-L) -X/2 Where B is baryon no., L lepton no. and X is a conserved quantity under GUT.

or

X +2 Yw = 5(B-L) . Pl. note that B-L 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.

Iso-spin is similar to but should not be confused with weak iso-spin.

* Weak iso-spin is the gauge symmetry of the weak interaction which connects quark & lepton doublets of left handed particles, (u,d), (t,b),(e- ,νe- )

*Iso-spin 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

* iso-spin is conserved in only in strong interaction whereas weak iso-spin & weak hypercharge are  conserved in electroweak interactions.  It is conserved (only terms that are overall weak-hypercharge 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 T3). Only a specific combination of them, Q = T3 +  YW /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 iso-spin angular momentum is

* anti-parallel to direction of motion when it is receding and the iso-spin rotation is anti-clockwise.

* parallel to direction of motion when it is approaching  and iso-spin rotation is clockwise.

     
                       

Comparison of iso-spin vrs weak iso-spin 

&

hypercharge vrs Weak hypercharge

Q=   I3+ Y /2 = T3+ YW/2 2YW

=

5(B-L)  -   X            
  I3 T3   Y YW   X   B L B-L
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
νe- 0 -1/2   0 -1   -3   0 1 -1
μ- 0 -1/2   0 -1   -3   0 1 -1
νμ- 0 1/2   0 -1   -3   0 1 -1
τ- 0 1/2   0 -1   -3   0 1 -1
ντ- 0 1/2   0 -1   -3   0 1 -1