Isospin and SU(2) Symmetry

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1 Isospin and SU() Symmetry Javier M. G. Duarte Massachusetts Institute of Technology, MA 04 (Dated: April 8, 009) We describe isospin, an abstract property of elementary particles, which reveals a deep symmetry of the strong force. Particles are assigned a total isospin quantum number, I and an isospin projection in one direction, I 3, 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() and its representations as well as how this symmetry is broken. Finally, with the introduction of a new quantum number, known as strangeness, and selection rules, we utilize isospin to calculate weak decay rates. I. INTRODUCTION AND MOTIVATION When faced with a new problem, a guiding principle of quantum mechanics and physics in general is to exploit its symmetry. Symmetry is a powerful tool, which has allowed physicists to learn about systems, which are otherwise too complicated to investigate. At its most fundamental level, a symmetry is simply the invariance of a system after an operation has been applied. Symmetry can come in a variety of forms, including space-time symmetries, such as rotational invariance, and internal symmetries, which describe the relations between elementary constituents of matter. But how can we as physicists exploit symmetry for problem solving? Emmy Noether provided the critical link. In rough terms, Noether s theorem states that for each symmetry of a system (or equivalently the Lagrangian specifying that system), there is an associated conserved quantity []. This gave physicists an excellent strategy: discover a symmetry of the problem, derive its associated conserved quantity, and use it to drastically simplify calculations. One question that begged for a simple explanation in the early nineteenth century was: Why do the proton and the neutron have such similar masses but differ in electric charge? Heisenberg suggested in 93 that the proton and neutron could be thought of as different states of the same particle: spin up and spin down nucleon. This was the beginning of isospin (originally isotopic spin and sometimes isobaric spin). By analogy with angular momentum, the proton and the neutron can be assigned a vector quantity known as isospin. If we take I I the total isospin vector squared and I 3 the isospin projection in one direction to be our complete set of commuting observables, an isospin state is denoted by II 3. The proton and the neutron are p =, n = () Heisenberg proposed that the strong interaction does Electronic address: woodson@mit.edu n 0 0 p Q= Q=0 S=0 S= S= Q= FIG. : The baryon octet, comprised of the eight lightest baryons. Particles along the same horizontal line have the same strangeness S, while those along the same dotted slanted lines have the same charge Q. not discriminate on the basis of the direction of isospin. In other words, any rotation of a particle s isospin vector will not affect the way it couples to the strong force. The strong force only cares about the magnitude of I. The relationship between isospin and other quantum numbers was worked out empirically later. In the 960s, particle physics was comprised of a grabbag of particles and properties with no rhyme or reason until Murray Gell-Mann proposed his Eightfold Way for classifying the eight lightest baryons. A baryon is an elementary particle containing three quarks protons and neutrons are both baryons. He found that they could be arranged in terms of certain quantum numbers: charge and strangeness, as seen in figure. He generalized Heisenberg s notion by saying that each row in his diagram would be completely degenerate if the strong force was the only force mediating interactions between quarks, was involved. Heisenberg s postulate and Gell-Mann s isospin assignment scheme allow us to perform calculations. Nonetheless, isospin is not an absolute conservation law. Only

2 momentum. Since we are only able to specify the magnitude and a single component of each summand, the magnitude of the sum ranges from I () I () to I () + I (). Clebsch-Gordan coefficients describe the possible states, in the bassis of total isospin, and their probabilities, as seen in figure. A derivation of the Clebsch-Gordan coefficients is carried out in Appendix C of reference []. In particular, consider the state of a neutral pion and a proton. We may use the Clebsch-Gordan table to decompose the two single-particle isospin states into total isospin states. π 0 + p : 0 = (4) FIG. : The Clebsch-Gordan coefficients for and. Note that for each cell of the table there is an implied square root so that / should be read as p /. Taken from [3] III. RELATIVE DECAY RATES AND CROSS SECTIONS the strong force is invariant under rotations in abstract isospin space. However, the weak force respects isospin via selection rules that allow for the derivation of weak decay rates. II. ASSIGNMENT AND ADDITION OF ISOSPIN The assignment of isospin proceeds as follows: a multiplet of I + particles is given a total isospin I and each member is given a projection isospin I 3, which takes on the values I, I +,..., +I in order of increasing charge. We can skip some of the difficulty of assignment by using the empirically-determined Gell-Mann Nishijima formula: I 3 = Q (A + S) () where Q is charge in units of elementary charge, A is baryon number and S is strangeness. Equation implies that the lines of constant I 3 in figure are horizontal. As a simple exercise, the pions form a triplet (π, π 0, π + ) with A = S = 0 so their isospin assignments can be inferred from equation. π =, π 0 = 0, π + = (3) In order to add two isospin vectors, we invoke the formalism from the quantum mechanical addition of angular An important application of isospin conservation in the strong interaction is the calculation of relative decay rates and cross sections (the probability for a decay or a scatter to occur, respectively). An essential concept is that of a matrix element M for a particular interaction, which is related to its cross section. The matrix element connecting the initial and final states, ψ i and ψ f is given by M if = ψ f A if ψ i, (5) where A if is an isospin operator (not to be confused with the Hamiltonian) which only depends on the total isospin, so that A = A / for initial and final states of I = / and A = A 3/ for states of I = 3/. Further, conservation of isospin demands that A = 0 for initial and final states of varying isospin. Knowing this, we define an amplitude for each total isospin I, M I = ψ(i) A I ψ(i) (6) Regardless of the type of interaction (decay or scatter), the probability is always proportional to the absolute square of the matrix element []. σ M if = ψ f A if ψ i, (7) The idea is that since the strong force treats all total isospin I particles the same, we need only specify the parameter M I for each total isospin in order to complete the calculation of relative cross sections. A. Nπ Nπ Scattering These are quantum numbers from particle physics: A is the number of baryons (or three times the number of quarks) and S is the number of strange particles. One of the simplest calculations is nucleon-pion scattering. To proceed, we first write down the possible channels through which the interaction may occur. There are six elastic processes (same particles come out that went

3 3 in) and four charge-exchange processes. (a)π + + p π + + p (b)π 0 + p π 0 + p (c)π + p π + p (d)π + + n π + + n (e)π 0 + n π 0 + n (f)π + n π + n (g)π + + n π 0 + p (h)π 0 + p π + + n (i)π 0 + n π + p (j)π + + p π 0 + n (8) Since the pion has I = and the nucleon has I =, the total isospin of the composite system must be or 3. We use figure to compute the isospin decompositions: π + + p : = 3 3 π 0 + p : 0 = π + + n : = π 0 + n : 0 = π + p : = π + n : = 3 3 (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 elastic processes reveals that two of them are pure I = 3/ and thus can be described by M a = M f = M 3/, where M α is the scattering amplitude for process (α). The rest of the elastic processes are very simple, M b = 3 M 3/ + 3 M /, M c = 3 M 3/ + 3 M / M d = 3 M 3/ + 3 M /, M e = 3 M 3/ + 3 M /. (0) Due to the symmetry of the problem, the chargeexchange processes turn out to all be the same M g = M h = M i = M j = 3 M 3/ 3 M /, () Further, the cross sections must then be in the ratio σ a : σ b : σ c : σ g = 9 M 3/ : M 3/ + M / : M 3/ + M / : M 3/ M /, () where we have left out redundant processes. In certain experimental settings, a single isospin channel dominates. There is a phenomenon known as a resonance particle, which means there is a peak in the scattering cross section as a function of center of mass (CM) energy corresponding exactly to a short-lived particle of certain mass. Concretely, the first resonance in πn scattering is the baryon at m = 3 MeV with I = 3/, as seen prominently in figure 3. When the CM energy reaches 3 MeV, the probability of the p and the N converting into a is much larger than that of any other interaction. So we say that the isospin 3/ channel dominates and thus the scattering amplitude for I = 3/ is much larger than for I = /. In this limit, we can simplify the results of equation to σ a : σ b : σ c : σ g = 9 : 4 : : (3) FIG. 3: Total cross sections for π + p and π p, shown with solid and dashed lines, respectively. Figure taken from [] (original source: S. Gasiorowicz. Elementary Particle Physics, New York: Wiley, 966, page 94). B. Baryon Decays All particles ( ++, +, 0, ) decay quickly to a combination of a pion and nucleon []. Conserving charge leads us to the possible interactions, (a) ++ π + p (b) + π 0 p (c) + π + n (d) 0 π 0 n (e) 0 π p (f) π n (4) The left-hand side of each of the processes is pure I = 3/. Thus we find that the coefficient in front of the I = 3/ state on the right-hand side of each process provides the square root of the decay rate ratio in each case. The decompositions of the right-hand sides of the interactions are exactly the same as those listed in equation 9. Following through, Γ a : Γ b : Γ c : Γ d : Γ e : Γ f = 3 : : : : : 3, (5) where Γ α is the decay rate of process (α). C. Kp Σπ Scattering A slightly more mathematically involved, but no more conceptually difficult example is as follows. Two kaons (K, K 0 ) form a doublet with isospin-, while the three

4 4 sigma baryons (Σ, Σ 0, Σ + ) form a triplet with isospin-, much like the pions. Suppose we are investigating the processes: (a)k + p Σ 0 + π 0 (b)k + p Σ + + π (c) K 0 + p Σ + + π 0 (d) K 0 + p Σ 0 + π + (6) From the isospin decompositions, we find that (a) may only proceed via the I = 0 channel, (c) and (d) only via the I = channel, and (b) may proceed via both. Thus σ a : σ b : σ c : σ d = M 0 : 6 M +M 0 : 3 M : 3 M (7) If, for example, we satisfy conditions such that M >> M 0 in the previous example, then we may take the limit in order to simplify the expression above: σ a : σ b : σ c : σ d = 0 : : : (8) These derivations freely use a conservation law (and thus a symmetry) of the strong force. We now state what symmetry group this corresponds to and explain qualitatively what is meant by a representation of a group. IV. SU() SYMMETRY The Special Unitary group of dimension, or SU() is a group of matrices which follow the rule that: ( ) a b A SU(), A = b a, det A = aa + bb = (9) It is said that SU() has dimension three, or three degrees of freedom, because it requires four real numbers (the real and imaginary component of a and b) subject to a single constraint. As done in Appendix B of reference [], τ, τ, τ 3 the isospin- matrix operators may be introduced (which are related to their respective Pauli spin matrices by τ = σ). For shorthand, we may express the isospin wavefunction as a two-component vector, with p and n representing the spin-up and spin-down basis elements. ( ) ( ) 0 p =, n =, (0) 0 To verify that τ 3 admits ± as its eigenvalues, we compute, τ 3 n = ( ) ( ) 0 0 = 0 n () and likewise for the proton p. It is no coincidence that there are three isospin- matrices (the same number of degrees of freedom in SU()). In a sense the three 3 Pauli spin matrices generate the group SU(). SU() can be considered to be acting on the twodimensional space spanned by (p, n). This is called a group representation. A rigorous treatment can be in [4]. For our purposes it will suffice to state that a representation of a group is a map which preserves multiplication and sends a member of the group to a linear operator on a vector space. The dimension of the representation is just the dimension of the vector space. Not all representations yield new information, though: some can be decomposed into the direct product of two smaller dimension representations. An irreducible representation is one that cannot be broken up into smaller dimension representations. The π mesons and the baryons constitute the three and four dimensional representations of SU(), respectively [5]. It is widely recognized that the internal isospin symmetry of the actual particles is not exactly SU() and that this is due to the mass splittings between the quarks []. However, the mass splittings of the lightest quarks (the up and the down) are at most or 3%, which is why SU() is a relatively good symmetry of the lightest baryons. 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, and using the relationship between quantum numbers in equation, we may partially recover the machinery of isospin and Clebsch-Gordan coefficients in the weak sector and use it to calculate relative weak decay rates. V. ISOSPIN AND STRANGENESS IN WEAK INTERACTIONS A new quantum number, known as strangeness, was proposed by Gell-Mann and Nishijima in the 950s to explain why some extremely long-lived particles seemed to only be produced in pairs. It was argued that strangeness S was conserved in the strong production of these particles with opposite strangeness. However this conservation would be violated in the weak interaction, so that single strange particles could decay into non-strange particles. Despite the fact that the weak interaction neither conserves S nor I 3, there are selection rules, which can specify relative amplitudes of weak interactions. In general, nonleptonic decays of strange particles are characterized by the rule S =, I =, which arises from the exchange of a strange quark s(s =, I = 0) with a nonstrange quark d(s = 0, I = ). In practice, the way to implement the rule is to introduce a hypothetical particle (sometimes called a spurion ) with I = to the left-hand side of the reaction. For example, say we want to find a relationship between

5 5 the amplitudes of several Σ weak decays : VI. SUMMARY AND CONCLUSIONS (a)σ + n + π + (b)σ n + π (c)σ + p + π 0. () Then we may choose the spurion to have isospin ket. We can read off the isospin sum decompositions from equation 9 by making the analogy between sigma baryons and pi mesons and between spurion and the neutron. The resulting system of linear equations is M a = 3 M 3/ + 3 M / M b = M 3/ M c = 3 M 3/ 3 M /, which can be solved to yield, (3) M a + M c = M b. (4) Another theoretical exercise using the I = rule, which agrees wonderfully with experiment, is computing the ratio of decay rates for: (a)k S π + π /K S π 0 π 0 (b)ξ Λπ /Ξ Λπ 0, (5) By the same method as in the previous example, we arrive at the ratio of : for (a) and (b). Table I lists experimental values for the branching ratios and decay times of these interactions. The experimentally determined ratios are.9 for (a) and.8 for (b). So we agree with experiment to within 0% solely based on this isospin selection rule of the weak force. Particle Mean life [s] Decay Mode Fraction K S 0.89() 0 0 π + π 68.6 % π 0 π % Σ (4) 0 6 pπ 0 5 % bπ + 48 % Σ Λγ 00% Σ.48() 0 0 nπ 00% Ξ 0.9() 0 0 Λπ 0 00% Ξ.64() 0 0 Λπ 00% TABLE I: Summary of experimental decay times and branching ratios for common decays of K, Σ and Ξ particles []. From these basic exercises, it is evident that isospin and symmetry in general are powerful tools. We first postulated an invariance of the strong force under rotations in isospin space. Noether s theorem predicts that this leads to a conservation of isospin in strong interactions. We assigned isospin according to Gell-Mann s Eightfold-Way and imported the framework of addition of angular momentum from quantum mechanics. Using the amplitudes corresponding to a particular value of I, total isospin, we calculate the relative cross sections and decay rates of a variety of strong processes. We made mention of the underlying group theory in the study of particle physics. In addition, we expanded the reach of isospin to the weak sector by adding a caveat (the I = selection rule) and compared these results to experiment. Moreover, these calculations are merely the beginning in terms of exploiting the symmetry of elementary particles. This glimpse of isospin captures its essence: symmetry leads to vast simplifications, which often make important calculations possible. Acknowledgments The author thanks Sara L. Campbell for peer-editing this paper, R. L. Jaffe for suggesting section V, and David Guarrera for reviewing the first draft. [] D. J. Griffiths. Introduction to Elementary Particles. Wiley-VCH, 008. [] D. H. Perkins. Introduction to High Energy Physics. Addison Wesley, 987. [3] J. Negele Quantum Physics II Course Notes. [4] M. Artin. Algebra. Prentice Hall, 99. [5] C. D Coughlan and J. E. Dodd. The Ideas of Particle Physics. Cambridge University Press, 99. [6] D. J. Griffiths. Introduction to Quantum Mechanics. Prentice Hall, 005. [7] I. S. Hughes. Elementary Particles. Cambridge University Press, 985.