Even Term 2020 Dr. Anosh Joseph, IISER Mohali
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 + (∇φ)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! √ 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 hΩ|φ|Ωi = ± 6µ /λ are both minima, there must be two p 2 p 2 diﬀerent vacua: |Ω+i with hΩ+|φ|Ω+i = 6µ /λ and |Ω−i with hΩ−|φ|Ω−i = − 6µ /λ. Since the Z2 symmetry takes φ → −φ, it must take |Ω+i ↔ |Ω−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 = √ . (12) 0 λ
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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 √ √ 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.
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. With mu = md = 0, the theory actually has two independent SU(2) symmetries, since the left-handed quarks and the right-handed quarks are completely decoupled. Indeed, writing the right- and left-handed spinors as
1 ψR/L = (1 ± γ )ψ , (16) q 2 5 q the Lagrangian is
1 L = − (F a )2 + iu¯LDu/ L + iu¯RDu/ R + id¯LDd/ L + id¯LDd/ L. (17) 4 µν
This Lagrangian is invariant under two separate rotations ! ! ! ! uL uL uR uR → gL and → gR , (18) dL dL dR dR
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where gL ∈ SU(2)L and gR ∈ SU(2)R. Equivalently, the symmetry can be written as
h a a i q → exp i(θaτ + γ5βaτ ) , (19) where ! u q = d is a ﬂavor doublet of the Dirac spinors u and d. The set of transformations parametrized by θa, with βa = 0 is called isospin. The set of transformations parametrized by βa with θa = 0 are called the axial rotations.
The SU(2)L × SU(2)R symmetry of QCD is called a chiral symmetry, since it acts diﬀerently on left- and right-handed ﬁelds. Actually, the Lagrangian is invariant under
U(2) × U(2) = SU(2)L × SU(2)R × U(1)V × U(1)A, (20) with the two U(1) symmetries called vector and axial. In our Universe, the spontaneous symmetry breaking of SU(2)×SU(2) happened about 14 billion years ago, when the temperature of the universe cooled below a critical temperature, TC ∼ ΛQCD. Below that scale, the thermal energy of quarks dropped below their binding energy, and instead of a big quark gluon plasma, hadrons appeared. Although it has not been proven from QCD itself, the ground state of QCD apparently has a nonzero expectation value for the quark bilinears uu¯ and dd¯
¯ 3 huu¯ i = hddi = ΛQCD. (21)
The above equation holds to a good approximation in QCD. (Note that we do not yet have a clear understanding of how and when spontaneous symmetry breaking occurs in general Yang-Mills theories.) The symmetry breaks as SU(2) × SU(2) → SU(2)isospin. The unbroken symmetry is the diagonal subgroup, which rotates left- and right-handed ﬁelds the same way.
Let us model this spontaneous symmetry breaking with a set of scalar ﬁelds Σij(x) transforming linearly under † † † † Σ → gLΣgR, Σ → gRΣ gL. (22) An eﬀective Lagrangian for this ﬁeld is the linear sigma model
λ L = |∂ Σ|2 + m2|Σ|2 − |Σ|4, (23) µ 4
2 † where |Σ| = ΣijΣji. This Lagrangian is invariant under SU(2) × SU(2) through Eq. (22). Note that only ordinary (not covariant) derivatives are required since we are interested in the global (not local) symmetries. Also, the potential has been chosen so that spontaneous symmetry breaking
6 / 7 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 occurs. The potential is minimized for ! v 1 0 hΣiji = √ , (24) 2 0 1 √ where v = 2m/ λ, which breaks the SU(2) × SU(2) symmetry down to the diagonal SU(2). We expect v ∼ ΛQCD. Let us decompose Σ in terms of a modulus ﬁeld σ(x) and angular ﬁelds π(x)
v + σ(x) πa(x)τ a Σ(x) = √ exp 2i , (25) 2 fπ
a with fπ = v, chosen so that π (x) have canonically normalized kinetic terms. Then according to Goldstone’s theorem there should be 3 massless bosons associated with the spontaneous symmetry breaking. We parametrize them as: π0 = π3 and π± = √1 (π1 ± iπ2). That is, 2 √ " √ !# 2 πaτ a i π0 2π− Σ(x) → U(x) ≡ exp 2i = exp √ . (26) + 0 v fπ fπ 2π −π
The matrix U depends only on the three πa degrees of freedom (not on σ) and has UU † = 1. † Like Σ, it also transforms under SU(2) × SU(2) as U → gLU gR. The constant fπ is called the pion decay constant. Its value is fπ = 92 MeV. This constant can + + be computed, for instance, from the reaction π → µ νµ, which gives relation between fπ, pion lifetime, and the masses of the pion and the muon. Note that in real world, quark masses explicitly break chiral symmetry, and this in turn implies that pions are not massless Goldstone bosons, but massive pseudo-Goldstone bosons.
 M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995).
 L. H. Ryder, Quantum Field Theory, Cambridge University Press (1996).
 M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013).
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