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Spontaneous Symmetry Breaking PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 44 Wednesday, April 8, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topics: Spontaneous Symmetry Breaking. Goldstone’s Theorem. Spontaneous Breaking of Chiral Symmetry. Spontaneous Symmetry Breaking In one of the earlier lectures, we encountered the formal relation between quantum field theory and statistical mechanics. The closest formal analogue of a scalar field theory was seen to be the continuum description of a ferromagnet, or some other system, that allows a second-order phase transition. This analogy raises the possibility that in quantum field theory as well it may be possible for the field to take a nonzero global value. As in a magnet, this global field might have a directional character, and thus violate the symmetry of the Lagrangian. In such a case, we say that the field theory has a hidden or spontaneously broken symmetry. Spontaneous symmetry braking (SSB) is a central concept in the study of quantum field theory. It is an essential ingredient in the theory of the weak interactions. SSB also finds applications in the theory of the strong interactions, and in search for unified models of fundamental physics. Let us analyze the mechanism of this type of symmetry violation. Spontaneous Breaking of Discrete Symmetries Let us begin by analyzing SSB in classical field theory. Consider the φ4 theory Lagrangian, 1 1 λ L = (@ φ)2 − m2φ2 − φ4; (1) 2 µ 2 4! but with m2 replaced by a negative parameter, −µ2: 1 1 λ L = (@ φ)2 + µ2φ2 − φ4: (2) 2 µ 2 4! PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 This Lagrangian has a discrete symmetry (Z2): It is invariant under the operation φ ! −φ. The corresponding Hamiltonian is Z 1 1 1 λ H = d3x π2 + (rφ)2 − µ2φ2 + φ4 : (3) 2 2 2 4! The minimum energy classical configuration is a uniform field φ(x) = φ0, with φ0 chosen to minimize the potential 1 λ V (φ) = − µ2φ2 + φ4 (4) 2 4! (see Fig. 1). This potential has two minima, given by r 6 φ = ±v = ± µ. (5) 0 λ The constant v is called the vacuum expectation value (or vev) of φ. Figure 1: Potential for spontaneous symmetry breaking. The spontaneously broken symmetry is a discrete symmetry, φ ! −φ. To interpret this theory, suppose that the system is near one of the minima (say the positive one). Then it is convenient to define φ(x) = v + σ(x); (6) and rewrite L in terms of the field σ(x). Plugging (6) into (2), we find that the term linear in σ vanishes (as it must, since the minimum of the potential is at σ = 0). Dropping the constant term as well, we obtain the Lagrangian r 1 1 λ λ L = (@ σ)2 − (2µ)2σ2 − µσ3 − σ4: (7) 2 µ 2 6 4! p This Lagrangian describes a simple scalar field of mass 2µ, with σ3 and σ4 interactions. The symmetry φ ! −φ is no longer apparent; its only manifestation is in the relations among the three 2 / 7 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 coefficients in (7), which depend in a special way on only two parameters. This is the simplest example of a spontaneously broken symmetry. Thinking in terms of vacuum expectation values, let us understand what happens to a symmetry when it is spontaneously broken. The original Lagrangian we considered above was invariant under p 2 the Z2 symmetry φ ! −φ. Since hΩjφjΩi = ± 6µ /λ are both minima, there must be two p 2 p 2 different vacua: jΩ+i with hΩ+jφjΩ+i = 6µ /λ and jΩ−i with hΩ−jφjΩ−i = − 6µ /λ. Since the Z2 symmetry takes φ ! −φ, it must take jΩ+i $ jΩ−i as well. The two possible vacua for the theory are equivalent, but one has to be chosen. The new Lagrangian is not invariant under σ ! −σ, so it seems the Z2 symmetry might have disappeared altogether. Actually, the Lagrangian is still invariant under the original φ ! −φ symmetry, because it acts on σ as σ ! −σ − 2v. So the symmetry is still there, it is just realized in a rather unfamiliar way. This is the general feature of spontaneously broken symmetries: the vacuum breaks them, but they are not actually broken in the Lagrangian. Spontaneous Breaking of Global Continuous Symmetries A more interesting theory arises when the broken symmetry is continuous, rather than discrete. The most important example is a generalization of the preceding theory, called the linear sigma model. The Lagrangian of the linear sigma-model involves a set of N real scalar fields φi(x): 1 1 λ L = (@ φi)2 + µ2(φi)2 − [(φ )2]2; (8) 2 µ 2 4 i with an implicit sum over i in each factor (φi)2. The Lagrangian (8) is invariant under the symmetry φi ! Rijφj; (9) for any N × N orthogonal matrix R. The group of transformations (9) is just the rotation group in N dimensions, also called the N-dimensional orthogonal group or simply O(N). i Again, the lowest-energy classical configuration is a constant field φ0, whose value is chosen to minimize the potential 1 λ V (φi) = − µ2(φi)2 + [(φi)2]2 (10) 2 4 i (see Fig. 2). This potential is minimized for any φ0 that satisfies µ2 (φi )2 = : (11) 0 λ i This condition determines only the length of the vector φ0; its direction is arbitrary. It is i conventional to choose coordinates so that φ0 points in the N-th direction: µ φi = (0; 0; ··· ; 0; v); where v = p : (12) 0 λ 3 / 7 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Figure 2: Potential for spontaneous breaking of a continuous O(N) symmetry, drawn for the case N = 2. Oscillations along the trough in the potential correspond to the massless π fields. We can now define a set of shifted fields by writing φi(x) = (πk(x); v + σ(x)); k = 1; 2; ··· ;N − 1: (13) (Note: Due to historic reasons, this notation comes from the application of this formalism to the effective field theory of pions, in the case N = 4.) It is now straightforward to rewrite the Lagrangian (8) in terms of the π and σ fields. The result is 1 1 1 L = (@ πk)2 + (@ σ)2 − (2µ2)σ2 2 µ 2 µ 2 p p λ λ λ − λµσ3 − λµ(πk)2σ − σ4 − (πk)2σ2 − [(πk)2]2: (14) 4 2 4 We obtain a massive σ field just as in (7) and also a set of N − 1 massless π fields. The original O(N) symmetry is hidden, leaving only the subgroup O(N − 1), which rotates the π fields among themselves. Referring to Fig. 2, we note that the massive σ field describes oscillations of φi in the radial direction, in which the potential has a nonvanishing second derivative. The massless π fields describe oscillations of φi in the tangential directions, along the trough of the potential. The trough is an (N − 1)-dimensional surface, and all N − 1 directions are equivalent, reflecting the unbroken O(N − 1) symmetry. Goldstone’s Theorem The appearance of massless particles when a continuous symmetry is spontaneously broken is a general result, known as Goldstone’s theorem. To state the theorem precisely, we must count the number of linearly independent continuous symmetry transformations. In the linear sigma model, there are no continuous symmetries for N = 1, while for N = 2 there is a single direction of 4 / 7 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 rotation. A rotation in N dimensions can be in any one of N(N − 1)=2 planes, so the O(N)- symmetric theory has N(N − 1)=2 continuous symmetries. After SSB there are (N − 1)(N − 2)=2 remaining symmetries, corresponding to rotations of the (N − 1) π fields. The number of broken symmetries is the difference, N − 1. Goldstone’s theorem states that for every spontaneously broken continuous symmetry, the the- ory must contain a massless particle. The massless fields that arise through SSB are called Goldstone bosons. Many light bosons seen in physics, such as the pions, may be interpreted (at least approxi- mately) as Goldstone bosons. More generally we can consider a continuous global symmetry G broken down to a subgroup H. The vacuum is then invariant under H, but not under the remaining elements of G, which are denoted as a coset, and written as G=H. The coset is not a subgroup of G. The number of Goldstone bosons is the dimension of the coset space, which is the number of generators of G that are not also generators of H. This result does not depend on what representation of G the fields belong to not on what form the potential V takes: the number of Goldstone bosons is simply the dimension of G=H. Spontaneous Breaking of Chiral Symmetry Let us consider the QCD Lagrangian with two light flavors of quarks, u and d: 1 L = − (F a )2 + iu¯Du= + id¯Dd= − m uu¯ − m dd:¯ (15) 4 µν u d If the quark masses were equal, this theory would have a global SU(2) symmetry that rotates the up and down quarks into each other. In reality, the masses of the up and down quarks are close but not equal. More importantly, they are very small compared to the QCD scale, ΛQCD. So let us set the masses to zero for now.
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