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Home , Isospin

Isospin and SU(2)

Javier M. G. Duarte∗ Massachusetts Institute of Technology, MA 02142 (Dated: April 28, 2009) We describe isospin, an abstract property of elementary particles, which reveals a deep symmetry of the strong force. Particles are assigned a total isospin , I and an isospin projec- tion in one direction, I3, in analogy with angular momentum. Exploiting the fact that the strong force is invariant under any rotation in ‘isospin space,’ that is, the strong force treats all particles with the same total isospin I equally, we may derive relations between the cross sections for various interactions mediated by the strong force. Specifically, we calculate the relative cross sections for Nπ scattering, ∆ → Nπ decays, and Kp scattering. We discuss the underlying symmetry group SU(2) and its representations as well as how this symmetry is ‘broken.’ Finally, with the introduction of a new quantum number, known as , and selection rules, we utilize isospin to calculate weak decay rates.

I. INTRODUCTION AND MOTIVATION n p S=0 When faced with a new problem, a guiding principle of and physics in general is to ex- ploit its symmetry. Symmetry is a powerful tool, which 0  has allowed physicists to learn about systems, which are −  S=−1  otherwise too complicated to investigate. At its most fun- damental level, a symmetry is simply the invariance of a system after an operation has been applied. Symmetry − 0 can come in a variety of forms, including space-time sym- S=−2 metries, such as rotational invariance, and internal sym- metries, which describe the relations between elementary constituents of matter. But how can we as physicists Q=−1 Q=0 Q=1 exploit symmetry for problem solving? Emmy Noether provided the critical link. In rough terms, Noether’s the- orem states that for each symmetry of a system (or equiv- FIG. 1: The octet, comprised of the eight lightest alently the Lagrangian specifying that system), there is . Particles along the same horizontal line have the an associated conserved quantity [1]. This gave physicists same strangeness S, while those along the same dotted slanted an excellent strategy: discover a symmetry of the prob- lines have the same Q. lem, derive its associated conserved quantity, and use it to drastically simplify calculations. One question that begged for a simple explanation in not discriminate on the basis of the direction of isospin. the early nineteenth century was: Why do the In other words, any rotation of a particle’s isospin vector and the have such similar but differ in will not affect the way it couples to the strong force. The ? Heisenberg suggested in 1932 that the strong force only cares about the magnitude of I. The proton and neutron could be thought of as different states relationship between isospin and other quantum numbers of the same particle: ‘ up’ and ‘spin down’ . was worked out empirically later. This was the beginning of isospin (originally isotopic spin In the 1960s, was comprised of a grab- and sometimes isobaric spin). By analogy with angular bag of particles and properties with no rhyme or rea- momentum, the proton and the neutron can be assigned son until Murray Gell-Mann proposed his a vector quantity known as isospin. If we take I · I the for classifying the eight lightest baryons. A baryon is total isospin vector squared and I3 the isospin projection an elementary particle containing three – in one direction to be our complete set of commuting and are both baryons. He found that they observables, an isospin state is denoted by |II3i. The could be arranged in terms of certain quantum numbers: proton and the neutron are charge and strangeness, as seen in figure 1. He general-

1 1 1 1 ized Heisenberg’s notion by saying that each row in his p = | 2 2 i, n = | 2 − 2 i (1) diagram would be completely degenerate if the strong Heisenberg proposed that the strong interaction does force was the only force mediating interactions between quarks, was involved. Heisenberg’s postulate and Gell-Mann’s isospin assign- ment scheme allow us to perform calculations. Nonethe- ∗Electronic address: [email protected] less, isospin is not an absolute conservation law. Only 2

momentum. Since we are only able to specify the magni- tude and a single component of each summand, the mag- nitude of the sum ranges from |I(1) − I(2)| to |I(1) + I(2)|. Clebsch-Gordan coefficients describe the possible states, in the bassis of total isospin, and their probabilities, as seen in figure 2. A derivation of the Clebsch-Gordan co- efficients is carried out in Appendix C of reference [2]. In particular, consider the state of a neutral and a proton. We may use the Clebsch-Gordan table to de- compose the two single-particle isospin states into total isospin states. q π0 + p : |10i| 1 1 i = 2 | 3 1 i − √1 | 1 1 i (4) 2 2 3 2 2 3 2 2

1 1 1 FIG. 2: The Clebsch-Gordan coefficients for 2 × 2 and 1 × 2 . Note that for each cell of the table there is an implied square III. RELATIVE DECAY RATES AND CROSS root so that −1/2 should be read as −p1/2. Taken from [3] SECTIONS

An important application of isospin conservation in the the strong force is invariant under rotations in abstract strong interaction is the calculation of relative decay rates isospin space. However, the weak force respects isospin and cross sections (the probability for a decay or a scatter via selection rules that allow for the derivation of weak to occur, respectively). An essential concept is that of a decay rates. matrix element M for a particular interaction, which is related to its cross section. The matrix element connecting the initial and final states, ψ and ψ is given by II. ASSIGNMENT AND ADDITION OF i f ISOSPIN Mif = hψf |Aif |ψii, (5) The assignment of isospin proceeds as follows: a mul- where Aif is an isospin (not to be confused tiplet of 2I + 1 particles is given a total isospin I and with the Hamiltonian) which only depends on the to- each member is given a projection isospin I , which takes 3 tal isospin, so that A = A1/2 for initial and final states on the values −I, −I + 1, ..., +I in order of increasing of I = 1/2 and A = A3/2 for states of I = 3/2. Further, charge. We can skip some of the difficulty of assign- conservation of isospin demands that A = 0 for initial ment by using the empirically-determined Gell-Mann– and final states of varying isospin. Knowing this, we de- Nishijima formula: fine an amplitude for each total isospin I, 1 I = Q − (A + S) (2) MI = hψ(I)|AI |ψ(I)i (6) 3 2 Regardless of the type of interaction (decay or scatter), where Q is charge in units of elementary charge, A is the probability is always proportional to the absolute and S is strangeness1. Equation 2 implies square of the matrix element [2]. that the lines of constant I3 in figure 1 are horizontal. As a simple exercise, the form a triplet 2 2 σ ∝ |Mif | = |hψf |Aif |ψii| , (7) (π−, π0, π+) with A = S = 0 so their isospin assignments can be inferred from equation 2. The idea is that since the strong force treats all total isospin I particles the same, we need only specify the parameter for each total isospin in order to complete − 0 + MI π = |1 − 1i, π = |10i, π = |11i (3) the calculation of relative cross sections.

In order to add two isospin vectors, we invoke the for- malism from the quantum mechanical addition of angular A. Nπ → Nπ Scattering

One of the simplest calculations is nucleon-pion scat- 1 These are quantum numbers from particle physics: A is the num- tering. To proceed, we first write down the possible chan- ber of baryons (or three times the number of quarks) and S is nels through which the interaction may occur. There are the number of ‘strange’ particles. six elastic processes (same particles come out that went 3 in) and four charge-exchange processes. (a)π+ + p → π+ + p (b)π0 + p → π0 + p (c)π− + p → π− + p (d)π+ + n → π+ + n (e)π0 + n → π0 + n (f)π− + n → π− + n (8) (g)π+ + n → π0 + p (h)π0 + p → π+ + n (i)π0 + n → π− + p (j)π+ + p → π0 + n

1 Since the pion has I = 1 and the nucleon has I = 2 , 1 3 the total isospin of the composite system must be 2 or 2 . We use figure 2 to compute the isospin decompositions:

π+ + p : |11i| 1 1 i = | 3 3 i 2 2 q 2 2 π0 + p : |10i| 1 1 i = 2 | 3 1 i − √1 | 1 1 i 2 2 3 2 2 3 2 2 q π+ + n : |11i| 1 − 1 i = √1 | 3 1 i + 2 | 1 1 i 2 2 3 2 2 3 2 2 q π0 + n : |10i| 1 − 1 i = 2 | 3 − 1 i + √1 | 1 − 1 i 2 2 3 2 2 3 2 2 q π− + p : |1 − 1i| 1 1 i = √1 | 3 − 1 i + 2 | 1 − 1 i 2 2 3 2 2 3 2 2 − 1 1 3 3 π + n : |1 − 1i| 2 − 2 i = | 2 − 2 i (9) As prescribed by equation 5, we take the inner product of the initial and final state to find the coefficient in front of the scattering amplitude. A quick inspection of the FIG. 3: Total cross sections for π+p and π−p, shown with elastic processes reveals that two of them are pure I = solid and dashed lines, respectively. Figure taken from 3/2 and thus can be described by Ma = Mf = M3/2, [1] (original source: S. Gasiorowicz. Elementary Particle where Mα is the scattering amplitude for process (α). Physics, New York: Wiley, 1966, page 294). The rest of the elastic processes are very simple, 2 1 1 2 Mb = 3 M3/2 + 3 M1/2, Mc = 3 M3/2 + 3 M1/2 1 2 2 1 B. ∆ Baryon Decays Md = 3 M3/2 + 3 M1/2, Me = 3 M3/2 + 3 M1/2. (10) ++ + 0 − Due to the symmetry of the problem, the charge- All ∆ particles (∆ , ∆ , ∆ , ∆ ) decay quickly to a exchange processes turn out to all be the same combination of a pion and nucleon [1]. Conserving charge √ √ leads us to the possible interactions, 2 2 Mg = Mh = Mi = Mj = 3 M3/2 − 3 M1/2, (11) (a)∆++ → π+p Further, the cross sections must then be in the ratio (b)∆+ → π0p (c)∆+ → π+n 2 2 0 0 0 − (14) σa : σb : σc : σg = 9|M3/2| : |2M3/2 + M1/2| (d)∆ → π n (e)∆ → π p 2 − − : |M3/2 + 2M1/2| (f)∆ → π n 2 : 2|M3/2 − M1/2| , (12) The left-hand side of each of the processes is pure where we have left out redundant processes. I = 3/2. Thus we find that the coefficient in front of In certain experimental settings, a single isospin chan- the I = 3/2 state on the right-hand side of each pro- nel dominates. There is a phenomenon known as a ‘reso- cess provides the square root of the decay rate ratio in nance particle,’ which means there is a peak in the scat- each case. The decompositions of the right-hand sides of tering cross section as a function of center of (CM) the interactions are exactly the same as those listed in energy corresponding exactly to a short-lived particle of equation 9. Following through, certain mass. Concretely, the first resonance in πN scat- tering is the ∆ baryon at m = 1232 MeV with I = 3/2, Γa :Γb :Γc :Γd :Γe :Γf =3:2:1:2:1:3, (15) as seen prominently in figure 3. When the CM energy reaches 1232 MeV, the probability of the p and the N where Γα is the decay rate of process (α). converting into a ∆ is much larger than that of any other interaction. So we say that the isospin 3/2 channel dom- inates and thus the scattering amplitude for I = 3/2 is C. Kp → Σπ Scattering much larger than for I = 1/2. In this limit, we can sim- plify the results of equation 12 to A slightly more mathematically involved, but no more σ : σ : σ : σ = 9 : 4 : 1 : 2 (13) conceptually difficult example is as follows. Two a b c g − ¯ 0 1 (K , K ) form a doublet with isospin- 2 , while the three 4 sigma baryons (Σ−, Σ0, Σ+) form a triplet with isospin- group representation. A rigorous treatment can be in [4]. 1, much like the pions. Suppose we are investigating the For our purposes it will suffice to state that a represen- processes: tation of a group is a map which preserves multiplication and sends a member of the group to a linear operator − 0 0 − + − (a)K + p → Σ + π (b)K + p → Σ + π on a . The dimension of the representation ¯ 0 + 0 ¯ 0 0 + (16) (c)K + p → Σ + π (d)K + p → Σ + π is just the dimension of the vector space. Not all repre- sentations yield new information, though: some can be From the isospin decompositions, we find that (a) may decomposed into the direct product of two smaller di- only proceed via the I = 0 channel, (c) and (d) only via mension representations. An irreducible representation the I = 1 channel, and (b) may proceed via both. Thus is one that cannot be broken up into smaller dimension √ representations. The π and the ∆ baryons con- σ : σ : σ : σ = | |2 : | 6 + |2 : 3| |2 : 3| |2 a b c d M0 2 M1 M0 M1 M1 stitute the three and four dimensional representations of (17) SU(2), respectively [5]. If, for example, we satisfy conditions such that M1 >> M0 in the previous example, then we may take the limit It is widely recognized that the internal isospin sym- in order to simplify the expression above: metry of the actual particles is not exactly SU(2) and that this is due to the mass splittings between the quarks σa : σb : σc : σd = 0 : 1 : 2 : 2 (18) [1]. However, the mass splittings of the lightest quarks (the up and the down) are at most 2 or 3%, which is These derivations freely use a conservation law (and why SU(2) is a relatively ‘good’ symmetry of the light- thus a symmetry) of the strong force. We now state what est baryons. symmetry group this corresponds to and explain qualita- tively what is meant by a representation of a group. Isospin continues to be a useful concept even as one moves away from the strong sector, where it is conserved. By introducing the quantum number of strangeness, IV. SU(2) SYMMETRY and using the relationship between quantum numbers in equation 2, we may partially recover the machinery of The of dimension 2, or SU(2) is isospin and Clebsch-Gordan coefficients in the weak sec- a group of matrices which follow the rule that: tor and use it to calculate relative weak decay rates.

 a b  A ∈ SU(2),A = , det A = aa∗ + bb∗ = 1 −b∗ a∗ (19) It is said that SU(2) has ‘dimension’ three, or three de- grees of freedom, because it requires four real numbers V. ISOSPIN AND STRANGENESS IN WEAK (the real and imaginary component of a and b) subject INTERACTIONS to a single constraint. As done in Appendix B of reference [2], τ1, τ2, τ3 the 1 isospin- 2 matrix operators may be introduced (which A new quantum number, known as strangeness, was are related to their respective Pauli spin matrices by proposed by Gell-Mann and Nishijima in the 1950s to ex- 1 τ = 2 σ). For shorthand, we may express the isospin plain why some extremely long-lived particles seemed to wavefunction as a two-component vector, with p and n only be produced in pairs. It was argued that strangeness representing the spin-up and spin-down basis elements. S was conserved in the strong production of these par- ticles with opposite strangeness. However this conserva-  1   0  p = , n = , (20) tion would be violated in the , so that 0 1 single strange particles could decay into non-strange par-

1 ticles. To verify that τ3 admits ± 2 as its eigenvalues, we com- pute, Despite the fact that the weak interaction neither con- serves S nor I , there are selection rules, which can spec-     3 1 1 0 0 1 ify relative amplitudes of weak interactions. In general, τ3n = = − n (21) 2 0 −1 1 2 nonleptonic decays of strange particles are characterized by the rule ∆S = 1, ∆I = 1 , which arises from the ex- and likewise for the proton p. 2 1 change of a strange s(S = −1,I = 0) with a non- It is no coincidence that there are three isospin- ma- 1 2 d(S = 0, ∆I = ). trices (the same number of degrees of freedom in SU(2)). 2 In a sense the three 3 Pauli spin matrices generate the In practice, the way to implement the rule is to intro- group SU(2). duce a hypothetical particle (sometimes called a “spu- 1 SU(2) can be considered to be acting on the two- rion”) with I = 2 to the left-hand side of the reaction. dimensional space spanned by (p, n). This is called a For example, say we want to find a relationship between 5 the amplitudes of several Σ weak decays 2: VI. SUMMARY AND CONCLUSIONS

(a)Σ+ → n + π+ (b)Σ− → n + π− (22) + 0 (c)Σ → p + π . From these basic exercises, it is evident that isospin and symmetry in general are powerful tools. We first Then we may choose the spurion to have isospin ket | 1 − 2 postulated an invariance of the strong force under rota- 1 2 i. We can read off the isospin sum decompositions from tions in isospin space. Noether’s theorem predicts that equation 9 by making the analogy between sigma baryons this leads to a conservation of isospin in strong inter- and pi mesons and between spurion and the neutron. The actions. We assigned isospin according to Gell-Mann’s resulting system of linear equations is Eightfold-Way and imported the framework of addition of angular momentum from quantum mechanics. Using = 1 + 2 Ma 3 M3/2 3 M1/2 the amplitudes corresponding to a particular value of I, = Mb √ M3/√2 (23) total isospin, we calculate the relative cross sections and 2 2 Mc = 3 M3/2 − 3 M1/2, decay rates of a variety of strong processes. We made mention of the underlying group theory in the study of which can be solved to yield, particle physics. In addition, we expanded the reach of √ 1 isospin to the weak sector by adding a caveat (the ∆I = 2 Ma + 2Mc = Mb. (24) selection rule) and compared these results to experiment. 1 Another theoretical exercise using the ∆I = 2 rule, which agrees wonderfully with experiment, is computing Moreover, these calculations are merely the beginning the ratio of decay rates for: in terms of exploiting the symmetry of elementary par- (a)K → π+π−/K → π0π0 ticles. This glimpse of isospin captures its essence: sym- S S (25) (b)Ξ → Λπ−/Ξ → Λπ0, metry leads to vast simplifications, which often make im- portant calculations possible. By the same method as in the previous example, we ar- rive at the ratio of 2:1 for (a) and (b). Table I lists exper- imental values for the branching ratios and decay times of these interactions. The experimentally determined ra- tios are 2.19 for (a) and 1.8 for (b). So we agree with experiment to within 10% solely based on this isospin selection rule of the weak force.

Particle Mean life [s] Decay Mode Fraction −10 + − KS 0.892(2) × 10 π π 68.6 % π0π0 31.3 % Σ+ 0.800(4) × 10−16 pπ0 52 % bπ+ 48 % Σ0 6 × 10−20 Λγ 100% Acknowledgments Σ− 1.48(1) × 10−10 nπ− 100% Ξ0 2.9(1) × 10−10 Λπ0 100% Ξ− 1.64(2) × 10−10 Λπ− 100% The author thanks Sara L. Campbell for peer-editing TABLE I: Summary of experimental decay times and branch- this paper, R. L. Jaffe for suggesting section V, and David ing ratios for common decays of K, Σ and Ξ particles [2]. Guarrera for reviewing the first draft.

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