Chapter 2.4 by Manu Verma

Chapter 2 Elementary particles

2.1 Classification of Particles 2.2 Leptons 2.3 Quarks 2.4 Hadrons 2.5 Interactive Exercise

2.4 Hadrons

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Hadrons Brief history of hadron discoveries First known hadrons were proton and neutron The lightest are pions π (pi-mesons). There are charged pions π+, π- with mass of 0.140 GeV/c2, and neutral ones π0, mass 0.135 GeV/c2. Pions and nucleons are the lightest particles containing u- and d-quarks only. Pions were discovered in 1947 in cosmic rays, using photo emulsions to detect particles. Some reactions induced by cosmic rays primaries: p + p → p + n + π+ → p + p + π0 → p + p + π+ + πSame reactions can be reproduced in accelerators, with higher rates, although cosmic rays may provide higher energies.

Fig 1: First observed pions: a π+ stops in the emulsion and decays to a μ+ and νμ, followed by the decay of μ+

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In emulsions, pions were identified by much more dense ionization along the track, as compared to electrons Figure 1 shows examples of the reaction π+ → μ+ + νμ

(1)

where pion comes to the rest, producing muons having equal energies, which in turn decay by the reaction μ+ → e+νeνμ. Charged pions decay mainly to the muon-neutrino pair (branching ratio about 99.99%), having lifetimes of 2.6 ✕ 10-8 s. In quark terms: (ud) → μ+ + νμ Neutral pions decay mostly by the electromagnetic interaction, having shorter lifetime of 0.8 ✕ 10-16 s: π0→ γ + γ Discovered pions were fitting very well into Yukawa’s theory - they were supposed to be responsible for the nuclear forces:

Fig 2: Yukawa model of direct (a) and exchange (b,c) nuclear forces

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The resulting potential for this kind of exchange is of Yukawa type (6), and at the longest range reproduces observed nuclear forces very well, including even spin effects. However, at the ranges comparable with the size of nucleons, this description fails, and the internal structure of hadrons must be taken into account. Strange mesons and baryons were called so because, being produced in strong interactions, had quite long lifetimes and decayed weakly rather than strongly. The most light particles containing s-quark are: mesons K+, K- and K0, K0:”kaons”, lifetime of K+ is 1.2 ✕ 10-8 s baryon Λ , lifetime of 2.6 ✕ 10-10 s Principal decay modes of strange hadrons: K+ → μ+ + νμ

(B=0.64)

K+ → π + + π 0

(B=0.21)

Λ → π- + p (B=0.64) 0 Λ→π +n (B=0.36) While the first decay in the list is clearly a weak one, decays of Λ can be very well described as strong ones, if not the long lifetime: (udd) → (du) + (uud) must have a lifetime of order 10-23 s, thus Λ can not be another sort of neutron... Dayalbagh Educational Institute

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Solution: To invent a new “strange” quark, bearing a new quark number --“strangeness”, which does not have to be conserved in weak interactions S=1

S= -1 Λ (1116) = uds

K+ (494) = us

K- (494) = su

K0 (498) = ds

K0 (498) = sd

In strong interactions, strange particles have to be produced in pairs in order to conserve total strangeness (“associated production”): π- + p → K0 + Λ

(2)

In 1952, bubble chambers were invented as particle detectors, and also worked as targets, providing, for instance, the proton target for reaction (2).

Liquid Particles

Fig 3: A bubble chamber picture of the reaction (2)

Magnetic coils

Camera

Piston

Magnetic field Dayalbagh Educational Institute

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How does a bubble chamber work: − It is filled with a liquid under pressure (hydrogen) − Particles ionize the liquid along their passage − When pressure drops, liquid boils preferentially along the ionization trails Bubble chambers were great tools of particle discovery, providing physicists with numerous hadrons, all of them fitting u-d-s quark scheme until 1974. In 1974, a new particle was discovered, which demanded a new flavour to be introduced. Since it was detected simultaneously by two rival groups in Brookhaven (BNL) and Stanford (SLAC), it received a double name: “dzei-psai” J/ψ (3097) = cc The new quark was called “charmed”, and the corresponding quark number is charm, C. Since J/ψ itself has C = 0, it is said to contain “hidden charm”. Shortly after that particles with “naked charm” were discovered as well: D+(1869) = c d, D0(1865) = cu D-(1869) = dc, D0(1865) = u c Λc+ (2285) = u dc Even heavier charmed mesons were found – those which contained strange quark as well: Ds+(1969) = cs, Ds- (1969) = sc Lifetimes of the lightest charmed particles are of order 10-13 s, well in the expected range of weak decays. Dayalbagh Educational Institute

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Discovery of “charmed” particles was a triumph for the electro-weak theory, which demanded number of quarks and leptons to be equal. In 1977, “beautiful” mesons were discovered: Υ(9460) = bb B+ (5279) = ub, B0(5279) = db B- (5279) = bu, B0(5279) = bd and the lightest b-baryon: Λb0 (5461) = udb And this is the limit: top-quark is too unstable to form observable hadrons.

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Baryons Each baryon is assigned a baryon number B = +1, and each antibaryon has B = -1. This number is conserved in all instructions i.e. baryons can only be created in pairs (of B – B). Any particle which is not a baryon has baryon number zero so all the leptons and mesons have B = 0. Baryon number of a particle made from quarks is B=(nq - nq )

B can only take the values -1, 0 or +1 Quarks

Example

Name

qqq

p (uud), n (udd)

Baryon

qqq

-1

p (ūūd), n (ūd d)

Anti-Baryon

π+ (u d), K- (sū)

(Anti-) Meson

Baryon number is conserved in (almost) all interactions

p+n→p+n+p+p 1 + 1 → 1 + 1 + 1 + -1 Allowed

p + n → p + μ+ + μ1+1→1+0 +0 Forbidden

Note, there is no meson number; π- → μ- + υμ Dayalbagh Educational Institute

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Example: The process p+p→p+p+p+p has

1 + 1 → 1 + 1 + 1 + (-1)

is allowed, provided there is sufficient energy. The process p + p → p + p + π+ + π+ B:

1 + 1 → 1 + (-1)+ 0 + 0

is energetically possible, but forbidden by conservation of baryon number.

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Isospin The strong interaction does not distinguish between the behavior of neutron and proton, then we regard nucleon to have an internal degree of freedom described by an Isospin vector I in a fictitious space. For I = ½, we have two possible orientation of the vector with I3 = ± ½ along the quantization axis corresponding respectively.

the

proton

and

neutron

The Isospin I is a general property of the strong interactions, hence this concept extends to all particles which can interact strongly i.e. the hadrons. Hence all hadrons belong to Isospin multiplets. Similar to the ordinary spin S with 2s + 1 different orientations of spin vector, isospin I has 2I + 1 different orientations of isospin vector corresponding to the different particles in the Isospin multiplet. General reaction between charge Q (expressed in units of|e|) and I3 is: Q = I3 + B/2 Strong Isospin is a quantum number that is conserved in the strong interaction In an analogous way to spin, particles in an isospin family are seen as different projections of the isospin family;

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Example: The p and n represent an isospin family; I=ď š (Notice the change of 1 in Iz values) Examples of isospin families

Examples of isospin families

In strong interactions, isospin is vectorially conserved

Forbidden Dayalbagh Educational Institute

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Strangeness Some particles observed (like Λ0, π-) had a strange behavior: production via the strong interaction and decay via the weak. Gell-Mann, Nishijima introduce ‘strangeness’ quantum number S which was assumed to be conserved in the strong interaction but not in the weak. Thus above equation was modified to I3 + B+S /2 B + S = Y is called hypercharge. Meson I K+ K0 π+ π0 πη0 K0 K-

½ ½ 1 1 1 0 ½ ½

Baryon

+½ -½ +1 0 -1 0 +½ -½

+1 +1 0 0 0 0 -1 -1

0 0 0 0 0 0 0 0

+1 +1 0 0 0 0 -1 -1

p n

½ ½ 1 1 1 0 ½ ½

+½ -½ +1 0 -1 0 +½ -½

0 0 -1 -1 -1 -1 -2 -2

1 +1 1 +1 1 0 1 0 1 0 1 0 1 -1 1 -1

Σ+ Σ0 ΣΛ0 Ξo Ξ-

Table 2: Summary of Jp = 0- mesons and the Jp = ½+ baryons.

There are four quark numbers S≡ -[N(s) – N(s)] B≡ -[N(b) – N(b)]

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C≡ [N(c) – N(c)] T≡ [N(t) – N(t)]

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Particle

π-

(dū)

-1

K-

(sū)

-1

D-

(d c)

-1

B-

(bū)

-1

Table3

Note: We use B to distinguish form baryon number B. As the strong & EM interactions create or destroy like quark/antiquark pairs, these quantum numbers are conserved in these interactions π- (dū) + p(uud) → K° (d s) + Λ(uds) S=

0 →

-1

As the weak interaction produces unlike quark/ antiquark pairs, it does not have to conserve these quantum numbers Λ(uds) → π- (dū) + p(uud) S=

-1

→

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Hadron Quantum Numbers Characteristics of a hadron: 1) Mass 2)Quantum numbers arising from space symmetries : J, P, C. Common notation:

- JP (e.g. for proton: ½+), or - JPC if a particle is also an eigenstate of C-parity (e.g. for π0 : 0-+) Internal quantum numbers: Q and B (always conserved), ˆ T (conserved in e.m. and strong interactions) S, C, B, How do we know what are quantum numbers of a newly discovered hadron? How do we know that mesons consist of a quark-antiquark pair, and baryons - of three quarks? Some a priori knowledge is needed: Particle

Mass (GeV/c2)

Quark Composition

ˆ B

0.938

uud

0.940

udd

K-

0.494

-1

D-

1.869

-1

B-

5.279

-1

Table 4

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Considering the lightest 3 quarks (u, d, s), possible 3quark and 2-quark states will be (qi,j,k are u- or dquarks): sss

ssqi

sqiqj

qiqjqk

-3

-2

-1

0;-1

1;0;-1

2;1;0;-1

sqi

qiqi

qiqj

-1

0;-1

1;0

-1;1

Table 5 (a) and (b)

Hence restrictions arise: for example, mesons with S =-1 and Q=1 are forbidden Particles which fall out of above restrictions are called exotic particles (like ddus , uuuds etc.) From observations of strong interaction processes, quantum numbers of many particles can be deduced: p + p → p + n + π0

p + p → p + n + π+ Q=

p + π- → π 0 + n Q=

-1

Table 6 (a), (b) and (c)

Observations of pions confirm these predictions, ensuring that pions are non-exotic particles. Dayalbagh Educational Institute

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Assuming that K- is a strange meson, one can predict quantum numbers of Λ-baryon: K- + p → π 0 + Λ Q=

-1

Table 7

And further, for K+-meson: π - + p → K+ + π - + Λ Q=

-1

Table 8

All of the more than 200 observed hadrons satisfy this kind of predictions, and no exotic particles have been found so far It confirms validity of the quark model, which suggests that only quark-antiquark and 3-quark (or 3-antiquark) states can exist Even more quantum numbers... It is convenient to introduce some more quantum numbers, which are conserved in strong and e.m. interactions: – Sum of all internal quantum numbers, except of Q, hypercharge Y ≡ B + S + C +

– Instead of Q : I3 ≡ Q - Y/2 Dayalbagh Educational Institute

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which is to be treated as a projection of a new vector:

– Isospin I ≡ (I3)max so that I3 takes 2I+1 values from -I to I It appears that I3 is a good quantum number to denote up- and down- quarks, and it is convenient to use notations for particles as I(JP)or I(JPc) If a baryon has I3=1, the only possibility for isospin is I=1, and we have a triplet: Σ+, Σ0, ΣIndeed, all these particles have been observed:

Masses and quark composition of Σ-baryons are: Σ+(1189) = uus ; Σ0(1193) = uds ; Σ-(1197) = dds It clearly indicates that d-quark is heavier than uquark under following assumptions: a) strong interactions between quarks do not depend on their flavour and give contribution of M0 to the baryon mass

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1/3

2/3

1/2

1/3

-1/3

1/3

-1

-2/3

-1/3

1/3

4/3

2/3

1/3

-1

-2/3

-1/3

1/3

4/3

2/3

Table 9

Hypercharge Y, isospin I and its projection I3 are additive quantum numbers, so that corresponding quantum numbers for hadrons can be deduced from those of quarks:

Proton and neutron both have isospin of very close masses:

½, and also

p(938) = uud ; n(940) = udd : proton and neutron are said to belong to the isospin doublet Other examples of isospin multiplets: K+(494) = us ; K0(498) = ds : π+(140) = ud ; π-(140) = du : π0(135) = (uu-dd)/√2 : Dayalbagh Educational Institute

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Principle of isospin symmetry: it is a good approximation to treat u- and d-quarks as having same masses Particles with I=0 are isosinglets : Λ(1116) = uds, I(J)P = 0(1/2)+ By introducing isospin, we imply new criteria for nonexotic particles: sss

ssqi

sqiqj

qiqjqk

-3

-2

-1

0;-1

1;0;-1

2;1;0;-1

1/2

0;1

3/2; 1/2

Table 10

sqi

qiqi

qiqj

-1

0;-1

1;0

-1;1

1/2

0;1

Table 11

In all observed interactions these criteria are satisfied as well, confirming once again the quark model. This allows predictions of possible multiplet members: suppose we observe production of the Σ+ baryon in a strong interaction: K- + p → π - + Σ + which then decays weakly : Σ+→π+ + n Σ+→π0 + p Dayalbagh Educational Institute

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It follows that Σ+ baryon quantum numbers are: B=1, Q=1, S=-1 and hence Y=0 and I3=1. Since I3>0 ⇒I ≠ 0 and there are more multiplet members!

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Quark diagrams Quark diagrams are convenient way of illustrating strong interaction processes Consider an example: ∆++ → p + π+ The only 3-quark state consistent with ∆++ quantum numbers is (uuu), while p=(uud) and π+=(ud)

Fig : Quark diagram of the reaction ∆++ → p + π+

Analogously to Feynman diagrams: arrow pointing to the right denotes a particle, and to the left – antiparticle time flows from left to right Allowed resonance formation process:

Fig : Formation and decay of ∆++ resonance in π+ p elastic scattering Dayalbagh Educational Institute

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Hypothetical exotic resonance:

Fig : Formation and decay of an exotic resonance Z ++ in K+ p elastic scattering

Quantum numbers of such a particle Z++ are exotic, moreover, there are no resonance peaks in the corresponding cross-section:

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Problem 1 The Σ baryon exists in three charge states +, 0, and -. Show that the Σ has strangeness S=-1.

Solution

The three states corresponds to three possible orientation of the isospin vectors. Therefore I=1, with I3 = +1,0 and +1 corresponding to Σ+, Σ0 and Σ- respectively. From Gell-Mann-Nishijima formula, Q = I3+ +1 = +1 + (1+S)/2 => S= -1 0 = 0 + (1+S)/2 => S=-1 -1 = (-1) + (1+S)/2 => S = -1 We find that for any of the charge states, we have S= -1.

Problem 2 The coulomb self-energy of a hadron with charge +e or −e is about 1 MeV. The quark content and rest energies (in MeV) of some hadrons are n(udd)940, p(uud)938, Σ−(dds)1197, Σ0(uds)1192, Σ+ (uus)1189, K0(ds)498, K+(us)494 The u and d quarks make different contribution to the rest energy. Estimate this difference. Dayalbagh Educational Institute

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Solution 2: Substract 1 MeV from the rest energies of the charged particles and take the difference in the masses. n (940) − p (938 − 1) = 3MeV = udd − uud = d − u Σ−(1,197 − 1) − Σ0(1,192) = 4MeV = dds − uds = d − u Σ0(1192) − Σ+(1189 − 1) = 4MeV = uds − uus = d − u K0(498) − K+(494 − 1) = 5MeV = ds − us = d − u Thus the mean difference in the masses of d-quark and u-quark is 4 MeV.

Problem 3: Below the production threshold of charm particles, the cross-section for the reaction e+e− → μ+μ− is 20 nb. Estimate the cross-section for hadron production.

Solution 3: According to the quark model, the ratio

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where Q i is the charge of the quark and the factor 3 arises due to the three colors in which the quarks can appear. Now the charges of the three quarks: u, d, and s are +2/3, −1/3, and −1/3 Therefore

and R = 3 × 2/3 = 2. Hence σ (e +e − → hadrons) = 3 × 2/3 × σ (e +e − → μ+μ−) = 2 × 20 nb = 40 nb Actually the value of R will be slightly greater than 2 because of an increased value of αs , the running coupling constant.

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