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Home , Charm (quantum number), Hypercharge, Isospin, Strangeness

Properties of Many groupings of of similar and properties fitted in to common patterns. So far, we have discussed three families of leptons but isotopic or principally concentrated on one doublet of quarks , the One way to characterise these is using isospin , I. u and d. This quantity has nothing to do with the real spin of the , but obeys the same addition laws as the We will now introduce other types of quarks, along quantum mechanical rules for adding angular with the new quantum numbers which characterise momentum or spin. them. ‰ When the orientation of an isospin vector is considered, it is in some hypothetical space, not in terms of the , y and z axes of normal co-ordinates. 1 2

Nucleons (p, n) , pi ( π+, π0, π–) and the known as ∆ (∆++ , ∆+, ∆0, ∆–) are three examples of groups of similar mass particles differing in by one unit. The rule for can then be written, The charge Q in each case can be considered as due to =(1 + ) Q e2 B I 3 the orientation of an “isospin vector” in some where B is the number which is 1 for hypothetical space, such that Q depends on the third and the ∆ and 0 for mesons such as the π. component I3. Thus the nucleons belong to an isospin doublet

‰ ≡ = 1 1 =1 − 1 In terms of quarks, the u and d form an isospin doublet, pI , I 3 2 , 2 n2 , 2 u= 1 , 1 =1 − 1 Similarly the form an isospin triplet , 2 2 d2 , 2 + 0 − 1 π = 1,1 π = 1, 0 π =1, − 1 (both with B = /3).

3 The ∆ forms a quadruplet with I = /2. 4 6 1 Three quarks with I = /2 can combine to form It was observed that some unstable particles produced in 1 3 Itot = /2 or /2. strong interactions had a long lifetime. This unusual stability for strongly interacting particles led to the term of strangeness . I = 1/ gives the nucleons while I = 3/ forms the ∆. tot 2 tot 2 Such particles are always produced in pairs (associated ‰ production), and the of strangeness , S, was introduced, which is conserved in strong In strong interactions, the total isospin vector (as well as interactions. Thus in the interaction π– p → Λ0 K0, the Λ is assigned I3) is conserved. S = –1 and the K S = +1. ‰ The strange particles can only decay by the weak This is not true in electromagnetic or weak interactions. interaction, which does not conserve strangeness (as we ‰ ‰ 7 will discuss later). 8 (Table)

Hadrons, quarks and quantum numbers: The formula for electric charge must now be modified to read the story so far =( ++1 1) =( + 1 ) QeI32 B 2 S eI 3 2 Y where Y = B + S is known as the . ° Groups of particles with similar properties are characterised by same quantum numbers, (This formula is known as the Gell-Mann Nishijima e.g. isospin I and strangeness S relation.) ° Electric charge determined by 3 rd component of

isospin, I3 Families of particles with similar properties (e.g. same ° Up and down form isospin doublet, I = ½ spin and parity) can be plotted in terms of Y versus I3, and form regular geometrical figures (see plots). =( + 1 ) Q eI3 2 Y

where Y = B + S is known as the hypercharge .

9 10 Y = S Y = S + + K 0 (ds) +1 K (us ) K*0 +1 K*

− η + − φ + π (du) π (ud ) ρ ω ρ 0 0 –1 − 1 π 1 +1 I 3 –1 − 1 ρ 1 +1 I 3 2 2 2 2

− ( ) 0 *− *0 K su –1 K (sd) K –1 K

The lowest-lying pseudoscalar- states The vector-meson nonet ( JP = 1 –). (JP = 0 –), with quark assignments indicated. 12 13

Y Y − ∆0 ∆+ ++ p N( 939 ) ∆ ∆ n +1 I = 3 (1232) 2 +1

Σ− Σ0 Σ+ = I 1 I 3 (1384) –1 − 1 1 +1 Σ− Σ0 Σ+ Σ (1193 ) 2 2

I 3 –1 − 1 Λ 1 +1 Λ (1116 ) 2 2

= 1 Ξ− Ξ0 I 2 –1 (1533)

Ξ− Ξ0 –1 Ξ (1318 ) − –2 (1672) I = 0 Ω P + The baryon octet of spin-parity ( J ) ½ P 3 + The baryon decuplet of spin-parity ( J ) /2 14 15 (with in MeV/ c2). (with masses in MeV/ c2). Quark Quantum Numbers In terms of quarks we can introduce a new flavour of quark, the s . Flavour charge/ e BII3 S ... This has charge –1/ and 1/ (like a 3 3 d –1/ 1/ 1/ –1/ 0 d quark) but I = 0 and S = –1. 3 3 2 2 u +2/ 1/ 1/ +1/ 0 It is also somewhat heavier than the u and d quarks. 3 3 2 2 s –1/ 1/ 0 0 –1 Since baryons consist of qqq , it is clear why no positive 3 3 baryons exist with | S| > 1, while negative baryons are . found with S = –2 or –3. Note that the type of quark is known as its flavour . Quarks. carry flavour and colour , and each flavour of quark exists in three colours. .

16 18

The Standard Model of Further Quarks

Fermions Bosons Other, still heavier quarks also exist. 2 The quark , c, has a charge of /3, like the u, and Leptons Quarks can be considered as a partner to the s. ν γ e e d u ± 0 µ νµ s c W , Z In 3 dimensions (see figure) particles containing τ ντ b t g c quarks can be plotted, and again show regular patterns.

‰ H0 19 22 Disputed claim 2002 Discovered 2017! Quark Quantum Numbers

Not yet! Flavour charge/ e BII3 S c

1 1 1 1 d – /3 /3 /2 – /2 0 0

2 1 1 1 u + /3 /3 /2 + /2 0 0

1 1 s – /3 /3 0 0 –1 0

2 1 c + /3 /3 0 0 0 +1 of 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 21 the case of the mesons, composed of cc . 23

Not yet!

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. Multiplets of hadrons containing up, down, strange and charm quarks. ‰ Example decays 24 25 Quark Quantum Numbers We thus have 2 doublets or generations of quarks – (d, u) and (s, c). Flavour charge/ e BII S c b 3 Since there are 3 doublets of leptons, there are

1 1 1 1 theoretical reasons for expecting a third doublet of d – /3 /3 /2 – /2 0 0 0 quarks too. u +2/ 1/ 1/ +1/ 0 0 0 3 3 2 2 Particles containing b quarks (bottom or beauty ) were

1 1 discovered in 1977. The b is an even heavier version s – /3 /3 0 0 –1 0 0 of the d. 2 1 c + /3 /3 0 0 0 +1 0 Its partner, the t (top or truth ) was first seen in 1994, 2 1 1 and its mass has now been measured at 174 GeV/c . b – /3 /3 0 0 0 0 –1

26 27

Quark Quantum Numbers The Standard Model of Particle Physics

Flavour charge/ e BII S c b t 3 Fermions Bosons

1 1 1 1 d – /3 /3 /2 – /2 0 0 0 0 Leptons Quarks

u +2/ 1/ 1/ +1/ 0 0 0 0 ν γ 3 3 2 2 e e d u s –1/ 1/ 0 0 –1 0 0 0 ± 0 3 3 µ νµ s c W , Z c +2/ 1/ 0 0 0 +1 0 0 3 3 τ ντ b t g

1 1 b – /3 /3 0 0 0 0 –1 0

t +2/ 1/ 0 0 0 0 0 +1 0 3 3 H ‰ 28 29 What you should have learned Quark Flavour and the As we have already seen, the strong and electromagnetic • Isospin, multiplets and charge interactions conserve quark flavour , whereas the weak • Isospin and interaction may change it . • Strangeness and the strange quark In many weak decays, the changes are within a generation, • Hypercharge e.g. in the W couples a u to a d quark; +→0 π + • Multiplets in 2D ( Y versus I3) in the decay D K it couples a c to an s. ‰ • Charm – multiplets in 3D However, this is not always the case, e.g. in the decay −→ π0 − ν K e e the W couples an s to a u quark, • b and t quarks ‰ and it was seen that such strangeness-changing decays were slightly weaker than strangeness-conserving weak decays. 30 31

Reaction Coupling Constant Cabibbo explained this by proposing that the eigenstates of the weak interaction are different from those of the . µ→ν+ + ν µ e e G The strong interaction eigenstates are the u, d, s, c, b →+ ν and t quarks, with well-defined isospin, strangeness etc. p n e e 0.97 G The eigenstates of the weak interaction, which does not π→π−0 − ν e e 0.97 G conserve I, S etc, are said to be those of “” T. ‰ −→ π0 − ν K e e 0.25 G For simplicity, let us first consider the first 2 generations alone.

36 37 The weak eigenstates are the leptons and orthogonal linear combinations of the familiar quarks Reaction Coupling Constant

= + 1  ν ν + + T3  e  µ  u  c µ→νe ν GG 2        µ e T = − 1 e µ d s 3 2      c c →+ ν θ p n e e 0.97 G G Cos C α β with dc = d + s −0 − β α α2 β2 π→πe ν θ sc = – d + s (normalisation + = 1) e 0.97 G G Cos C

α θ θ −0 − is usually known as cos c, where c is the Cabibbo → π ν θ K e e 0.25 G G Sin C angle. θ A value of sin c = 0.25 is consistent with the observed apparent variation of weak coupling constant with ‰ reaction type. 38 39

The relationship between weak and strong eigenstates in The magnitudes of the matrix elements have been 2 generations can also be expressed as determined experimentally, and are given with 90% confidence limits by d  cosθ sin θ  d  c=  c c     0.9743± 0.0002 0.2254 ± 0.0006 0.0036 ± 0.0002  s− sin θ cos θ c  c c  s  =± ± ±  M 0.2252 0.0006 0.9734 0.0002 0.041 0.001    weak e-states mixing matrix strong e-states  0.0089± 0.0003 0.040 ± 0.001 0.99914 ± 0.00005 

The same formalism can be used for 3 generations, and Note that the values along the leading diagonal are quite the mixing matrix, known as the Cabibbo-Kabayashi- close to one , those adjacent to it are significantly smaller , Maskawa or CKM matrix, can be parametrised in a and the elements in the top-right and bottom-left corners number of ways. are much smaller . d′  d This means that the mixing results in states which   s′ = M s contain a small admixture of the quark from the next   generation, while mixing between 1 st and 3 rd generation b′  b   41 quarks is extremely small. 42 Physically, this is shown in weak decays by the relative probability of producing hadrons containing the respective quarks. For example, when a decays it produces a b′ quark . The most likely overall decay chain of a b quark is therefore This is bound in a 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 . ‰ b → c → s → u. Therefore, the near-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. 43 44

Mixing between 3 generations: For two generations, one parameter was required to δ− 1 0 0  c 0 s e i  c s 0 describe the mixing. This was the Cabibbo angle .   13 13  12 12  M= 0 c s  0 1 0 − s c 0 With three generations, 4 independent parameters are 23 23   12 12  − − iδ   needed to define a general unitary matrix, and the 0 s23 c23  s13 e 0 c13  0 0 1 individual matrix elements may have imaginary parts. One possible parametrisation of the CKM matrix is given Mixing between 2 & 3 Mixing between 1 & 2 below. Mixing between 1 & 3 Note that the following material is provided for but a general 3x3 (unitary) transformation requires 4 completeness only, and is not examinable ! (Further parameters details are provided in the text books.)  cc sc s e−i δ   1213 1213 13  =−−iδ − i δ [The following parametrisation and previous values are taken from M  sc1223 css 122313e cc 1223 sss 122313 e sc 2313  the Particle Properties Data Booklet, from "Review of Particle  −iδ − − i δ   ss1223 ccs 122313e cs 1223 scs 122313 e cc 2313  Physics", Chinese Physics C38, July 2014, by the .] 45 47  cc sc s e−i δ  What you should have learned  1213 1213 13  =−−iδ − i δ M  sc1223 css 122313e cc 1223 sss 122313 e sc 2313   −iδ − − i δ  • Strangeness-changing and strangeness-  ss1223 ccs 122313e cs 1223 scs 122313 e cc 2313  conserving weak decays θ θ where c ij = cos ij and s ij = sin ij , with i and j being generation labels { i,j = 1,2,3}. • Cabibbo theory. θ θ – Weak & strong eigenstates, quark mixing In the limit 23 = 13 = 0, the third generation decouples, and the situation reduces to the usual Cabibbo mixing of the first θ • CKM matrix two generations, with 12 identified with the Cabibbo angle. θ θ θ (The real angles 12 , 23 , 13 can all be chosen to lie in the first • Preference for weak decays:- quadrant.) c13 is known to differ from unity only in the sixth decimal place. – within a generation (if allowed) If the parameter δ is non-zero, then the matrix is complex, and – otherwise to next generation below the small degree of CP violation present in the weak • Another illustration of weak & strong eigenstates next interaction can be explained naturally. This is still the subject 48 lecture! 49 of research!

Reading for next week from “Ideas of Particle Physics” (Edition 3): • Chapters 34, 35.1-2 – “R” • Chapters 25, 26, 27 – Scaling • Section 32.2 – Scaling violation.

From Edition 2: • Chapters 35, 36.1-2 – “R” • Chapters 26, 27, 28 – Scaling • Section 33.2 – Scaling violation.

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