PHY304 Particle Physics Dr C N Booth

Properties of Quarks


In the earlier part of this course, we have discussed three families of leptons but principally concentrated on one doublet of quarks, the u and d.  We will now introduce other types of quarks, along with the new quantum numbers which characterise them.


Isospin

It was noticed that many groupings of particles of similar mass and properties fitted in to common patterns.  One way to characterise these is using isotopic spin or isospin, I.  This quantity has nothing to do with the real spin of the particle, but obeys the same addition laws as the quantum mechanical rules for adding angular momentum or spin.  When the orientation of an isospin vector is considered, it is in some hypothetical space, not in terms of the x, y and z axes of normal co-ordinates.

Nucleons (p, n), pi mesons (π+, π0, π) and the baryons known as Δ (Δ++, Δ+, Δ0, Δ) are three examples of groups of similar mass particles differing in charge by one unit.  The charge Q in each case can be considered as due to the orientation of an “isospin vector” in some hypothetical space, such that Q depends on the third component I3.
Thus the nucleons belong to an isospin doublet: p ≡ |I, I3 = |½, ½; n |½, −½.
Similarly the pions form an isospin triplet, π+ = |1, 1; π0 = |1, 0; π = |1, −1.
The Δ forms a quadruplet with I = .
The rule for electric charge can then be written

Q = eB + I3),
where B is the baryon number which is 1 for nucleons and the Δ and 0 for mesons such as the π.

In terms of quarks, the u and d form an isospin doublet, u = |½, ½; d |½, −½ (both with B = ).

Three quarks with I = can combine to form Itot = or . Itot = gives the nucleons while Itot = forms the Δ. It is useful to consider the symmetry of the quarks inside these baryons. The internal wavefunction can be written as a product of terms, Ψ = ψspinψspaceψisospinψcolour, and must be antisymmetric overall (as quarks are fermions). ψcolour is always antisymmetric (as hadrons are colourless); the symmetry of ψspace is given by (−1)l and l the orbital angular momentum is zero for the long-lived hadrons we consider, so ψspace is symmetric. Thus the product ψspinψisospin must be symmetric, implying either both must be symmetric or both must be antisymmetric. This explains the correlation between allowed spin and isospin states for the baryons: the Δ has I = and s = (both symmetric), while the nucleons have I = and s = (both antisymmetric).

In strong interactions, the total isospin vector (as well as I3) is conserved.  This is not true in electromagnetic or weak interactions. The conservation of isospin has observable effects on the relative rates of strong interactions. For example, the reaction of two protons to form a deuteron and a pion has twice the cross section of the reaction between a proton and neutron to form a similar final state. This will be explained in the lectures.


Strangeness

It was observed that some unstable particles produced in strong interactions had a long lifetime.  This unusual stability for strongly interacting particles led to the term of strangeness.  Such particles are always produced in pairs (associated production), and the quantum number of strangeness, S, was introduced, which is conserved in strong interactions.  Thus in the interaction
π p → Λ0 K0,
the Λ is assigned S = −1 and the K S = +1.  The strange particles can only decay by the weak interaction, which does not conserve strangeness (as we will discuss later).

Isospin multiplet  B    S    I  <Q>/e Y = B + S
π+ π0 π
0
0
1
0
0
p n
1
0
½
½
1
Δ++ Δ+ Δ0 Δ
1
0
½
1
Λ
1
−1
0
0
0
Σ+ Σ0 Σ
1
−1
1
0
0
K+ K0
0
1
½
½
1
K
0
−1
½
−½
−1
Ξ0 Ξ
1
−2
½
−½
−1
Ω
1
−3
0
−1
−2
A selection of strange and non-strange baryon and meson multiplets.

The formula for electric charge must know be modified to read

Q = e(I3 + ½B + ½S) = e(I3 + ½Y)
where Y = B + S is known as the hypercharge. (This formula is known as the Gell-Mann Nishijima relation.) Families of particles with similar properties (e.g. same spin and parity) can be plotted in terms of Y versus I3, and form regular geometrical figures, as shown below.

In terms of quarks we can introduce a new flavour of quark, the strange quark s.  This has charge − and baryon number  (like a d quark) but I = 0 and S = −1.  It is also somewhat heavier than the u and d quarks.  Since baryons consist of qqq, it is clear why no positive baryons exist with |S| > 1, while negative baryons are found with S = −2 or −3.


The lowest-lying pseudoscalar-meson states (JP = 0), with quark assignments indicated.
(The states at the origin are displaced slightly for clarity.)


The vector-meson nonet (JP = 0).
(Quark assignments are the same as above.)


The baryon octet of spin-parity JP = + (with masses in MeV/c2).


The baryon decuplet with spin-parity JP+.


Further quarks

Other, still heavier quarks also exist.  The charm quark, c, has a charge of , like the u, and can be considered as a partner to the s.  In 3 dimensions, as illustrated below, particles containing c quarks can be plotted, and again show regular patterns.  We thus have 2 doublets or generations of quarks - (d, u) and (s, c).  Since there are 3 doublets of leptons, there are theoretical reasons for expecting a third doublet of quarks too.  Particles containing b quarks (bottom or beauty) were discovered in 1977.  The b is an even heavier version of the d.  Its partner, the t (top or truth) was first seen in 1994, and it has the greatest mass of any known fundamental particle at 174 GeV/c2.

Multiplets of hadrons containing up, down, strange and charm quarks.  The slices through these figures where Charm = 0 correspond to the plane figures already shown in the previous diagrams, though containing new particles in the case of the mesons, composed of charm-anticharm quarks.

One of these particles was only discovered recently! See the announcement from CERN.


FlavourCharge/e  B     I    I3    S     c    b    t     Mass (GeV/c2)  
d
½
−½
0
0
0
0
0.005
u
+
½
0
0
0
0
0.002
s
0
0
−1
0
0
0
0.095
c
+
0
0
0
+1
0
0
1.3
b
0
0
0
0
−1
0
4.2
t
+
0
0
0
0
0
+1
174
Quark quantum numbers and masses.
Note the convention that quarks with a negative electric charge carry a negative flavour quantum number.
The masses quoted are "bare masses" - when bound in hadrons the effective masses differ, especially for the lightest quarks. (Binding and kinetic energies mean that u and d quarks can be treated as effectively equal in mass.)


The particle content of the Standard Model of Particle Physics (including the Higgs boson which is not covered in this course).


Quark Flavour and the Weak Interaction

As we have already seen, the strong and electromagnetic interactions conserve quark flavour, whereas the weak interaction may change it.  In many weak decays, the changes are within a generation, e.g. in beta decay the W couples a u to a d quark; in the decay
D+K0 π+
it couples a c to an s. However, this is not always the case, e.g. in the decay
K → π0 e ν̅e
the W couples an s to a u quark, and it was observed that such strangeness-changing decays were slightly weaker than strangeness-conserving weak decays.

Cabibbo explained this by proposing that the eigenstates of the weak interaction are different from those of the strong interaction.  The strong interaction eigenstates are the u, d, s, c, b and t quarks, with well-defined isospin, strangeness etc.  The eigenstates of the weak interaction, which does not conserve I, S etc., are said to be those of “weak isospin” T.  For simplicity, let us start by considering the first 2 generations alone.  The weak eigenstates are the leptons and orthogonal linear combinations of the familiar quarks

with dc = α d + β s
sc = −β d + α s (normalisation α2 + β2 = 1)

α is usually known as cos θc, where θc is the Cabibbo angle.  A value of sin θc = 0.25 is consistent with the observed apparent variation of weak coupling constant with reaction type. (Note that by convention it is only the T3 = −½ member of the doublet which is mixed.)

The relationship between weak and strong eigenstates in 2 generations can also be expressed as
 

  = 
weak e-states mixing matrix strong e-states

The same formalism can be used for 3 generations, and the mixing matrix, known as the Cabibbo-Kabayashi-Maskawa or CKM matrix, can be parametrised in a number of ways.

The magnitudes of the matrix elements have been determined experimentally, and are given by

Note that the values along the leading diagonal are quite close to one, those adjacent to it are significantly smaller, and the elements in the top-right and bottom-left corners are much smaller.  This means that the mixing results in states which contain a small admixture of the quark from the next generation, while mixing between 1st and 3rd generation quarks is extremely small. 

Flavour-changing weak interactions always occur via the charged current. That is, they always involve transitions between the two members of the same weak isospin doublet, e.g. between c and s' (in either direction), or between u and d'. The mixing of the negative quarks plays a role for both initial and final state quarks. For example, the decay of a c quark is always to an s' weak eigenstate, which will be bound in a hadron as one of the strong eigenstates of which it can be considered a mixture. On the other hand, when a hadron containing an s quark decays, the s must be considered a mixture of d', s' and b' weak eigenstates, and these decay to u, c and t respectively.

Physically, the relative probability of producing hadrons containing the respective quarks in a weak decay is determined by the elements of the CKM matrix. For example, when a top quark decays it produces a b' quark.  This is bound in a hadron by the strong interaction, so must be revealed as a strong eigenstate.  The b' is most likely to result in a particle containing a b quark, with a smaller probability of an s quark and almost negligible likelihood of producing a d quark.  Therefore, the diagonal structure of the CKM matrix means that weak decays are most likely to be within a generation if allowed by conservation of energy (a particle cannot decay into one that is heavier) or to the next generation below if this is not allowed.  The most likely overall decay chain of a b quark is therefore b → c → s → u.

When the charged-current weak decays are considered along with binding into strong eigenstates in the hadrons, the elements of M can be interpreted as giving the effective transition strengths between quarks as follows:

.

(Again, physical decays must always be to the lighter quark.)

For two generations, one parameter was required to describe the mixing.  This was the Cabibbo angle.  With three generations, 4 independent parameters are needed to define a general unitary matrix, and the individual matrix elements may have imaginary parts.   One possible  parametrisation of the CKM matrix is given below.  Note that the following material is provided for completeness only, and is not examinable!  (Further details are provided in the text books.)

where cij = cos θij and sij = sin θij, with i and j being generation labels {i,j = 1,2,3}.  In the limit θ23 = θ13 = 0, the third generation decouples, and the situation reduces to Cabibbo mixing of the first two generations, with θ12 identified with the Cabibbo angle. The real angles θ12, θ23, θ13 can all be chosen to lie in the first quadrant. c23 is known to differ from unity only in the sixth decimal place.

If the parameter δ is non-zero, then the matrix is complex, and the small degree of CP violation present in the weak interaction can be explained naturally. It has not yet been conclusively proven that this is the explanation for all the observed CP violation!
 

[The above parametrisation and values are taken from the Particle Physics Data Booklet, from "Review of Particle Physics", Chinese Physics C38, July 2014, by the Particle Data Group.]


Supplementary material on the properties of quarks, mainly of a popular or non-technical nature, can be obtained from a number of sources.  You may wish to consult some of the following information on the Web:


Supplementary Reading Material
For further non-technical discussion of the properties of quarks, you might like to consult The Ideas of Particle Physics, as follows:

(The above chapter references are for edition 3. For edition 2, consult chapters 7.6, 8, 10, 36.3-36.5 and 37.)

More information on the CKM matrix and CP violation is available from the following sources.


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