The induced cubic interaction is a manifestation that the discrete Z2 symmetry of L is spontaneously broken by the ground state (classical) solution, and has the effect that particle number modulo 2 is no longer conserved. The last term is a quartic self-interaction that is unaffected by the symmetry breaking.
For the intermediate caseµ2 = 0 it is not sufficient to consider the theory classically.
One must extend the discussion to consider the effective potential, which reduces to the potential at tree-level in an expansion in the number of loops. The model considered here is difficult to study; the apparent minimum is at nonzeroν but occurs outside of the range of validity of the expansion. However, in more realistic theories, such as the standard model, the symmetry is spontaneously broken for µ2 = 0 (Coleman and Weinberg, 1973). The effective potential is extremely useful for incorporating higher-order effects in the study of symmetry breaking and for considering field theory at finite temperature (see, e.g., Dolan and Jackiw, 1974; Weinberg, 1995; Quiros, 1999).
One can perturb the potential in (3.133) by linear or cubic terms that explicitly break theZ2symmetry, as in (3.129). For definiteness, we will consider the lineartadpoleoperator
−aφwitha >0,
V (φ) =µ2φ2
2 −aφ+λφ4
4 . (3.141)
An immediate consequence is that theZ2 symmetry must be violated in the ground state as well, i.e., hφi 6= 0. Forµ2 >0 the VEV is induced by the explicit breaking,ν =hφi= a/µ2+O(a3). The most important consequence is the presence of a small cubic term for the shifted fieldφ0=φ−ν,
V (φ0) = µ2
2 φ02+λνφ03+λ
4φ04+O(a2), (3.142) similar to the (larger) one in (3.140). For µ2 < 0 the potential is shifted, as seen in the dotted curve inFigure 3.4. The deepest (globalor true) minimum is at
ν =ν0+ a
2ν20 +O(a2), (3.143)
where ν0 = p
−µ2/λ is the unperturbed minimum. For small enough a there is another metastablelocal orfalse minimum forν <0. Such metastable local minima are frequently encountered in field theories, and in some cases it makes sense to quantize around them rather than around the global minimum (e.g., Kusenko et al., 1996; Intriligator et al., 2006; Degrassi et al., 2012; Camargo-Molina et al., 2014). The relevant issues for a realistic theory with a metastable vacuum are: (a) Is the lifetime of the metastable vacuum due to tunneling (Linde, 1983) long compared to the 1010 year age of the observed Universe?
(b) Which vacuum would have been occupied initially as the Universe cooled?
3.3.2 A Digression on Topological Defects
There are also more energetic classical solutions to (3.134). For example, the staticdomain wall solution is an infinite wall perpendicular, e.g., to the x direction, with φ(x) varying from φ(−∞) =−ν toφ(+∞) = +ν. This is illustrated inFigure 3.4, where the center of the wall is at x= 0 and the wall is parallel to the y and z directions. Energy is stored in the wall in the transition region near x= 0. The thickness d of the transition region can be estimated by minimizing the sum of the kinetic energy density∼(ν/d)2=|µ|2/λd2and the potential energy density (with respect to the minimum) ∼µ4/λ, leading tod∼1/|µ|. Thus, the energy density per unit area is ∼ dµ4/λ ∼ |µ|3/λ. Since the wall is infinite in extent, it would take infinite energy to tunnel to one of the ground states φ(x) = ±ν, so the wall is stable. (Such objects are known astopological defects.)
This simple model illustrates a generic difficulty with spontaneously broken discrete symmetries. Such walls would presumably have formed in the early Universe as it cooled from a temperatureT much larger than|µ|because causally disconnected regions would have fallen randomly into either of the two minima, somewhat like the formation of ferromagnetic domains. Both walls and anti-walls (making transitions from +νto−ν) would have formed.
Most would presumably have been annihilated, but one would expect at least one to survive in a volumeV ∼R3of the size of our observable Universe, contributing to the energy density and anisotropy of our Universe. To get an idea of the magnitude, let us assume the average energy per unit volume due to a single domain wall is bounded by the observed average energy density ρtot,
energy volume ∼ |µ|3
λR < ρtot∼ 3×10−3eV4
. (3.144)
Of course, this underestimates the constraint since the observed energy density is ex-tremely isotropic, unlike a domain wall. UsingR∼1.4×1010yr∼4×1017sec, this yields
|µ|/λ1/3<30 MeV. Discrete symmetries spontaneously broken at a larger scale are therefore cosmologically dangerous.18
Spontaneous symmetry breaking in particle physics can lead to other defects of possible cosmological relevance, which may involve classical values for gauge fields as well as scalars.
These include monopoles, which occur when non-abelian symmetries are broken down to a subgroup containing aU(1) factor, andcosmic strings, associated withU(1) symmetries (seeProblem 4.1).In another class aretextures, which are not topologically stable but may be long-lived. These matters are discussed in much more detail in (Coleman, 1985; Vilenkin, 1985; Kolb and Turner, 1990; Brandenberger, 2013).
3.3.3 A Complex Scalar: Explicit and Spontaneous Symmetry Breaking Consider a complex scalarφwith
L0= (∂µφ)†∂µφ−V (φ), V(φ) =µ2φ†φ+λ φ†φ2
, (3.145)
whereλ >0 so thatV is bounded from below. As discussed inSection 2.4.1,L0is invariant under theU(1) phase transformationsφ→eiβφ, with a conserved charge corresponding to particle number. It is convenient to go to a Hermitian basis,φ= (φ1+iφ2)/√
2, as in (2.91),
18They can be avoided if the reheating temperature of the Universe after a period of inflation is smaller than µ. Also, the addition of a small explicit symmetry breaking term −κφ3 or −aφ to (3.133) would eliminate the problem; the energy difference would lead to an attractive force between domain walls and anti-walls and therefore more rapid annihilation.
so that As shown in (3.75), theU(1) transformation takes the form of a rotation19 in this basis
φ1
(In fact, U(1) and SO(2) are equivalent.) Again, the vacuum corresponds to a classical solution of the field equations
The minimum condition requires that the eigenvaluesm21,2of
∂2V must be non-negative. (These are interpreted as the mass-squares of the physical mass eigenstate particles when one expands around the minimum.)
Forµ2 >0, the minimum is at ν1 =ν2 = 0, as seen in Figure 3.6. One can quantize around this point, obtaining degenerateφ1,2with mass-squareµ2and the relations between the quartic couplings unbroken (or, equivalently, degenerate φ and φ†, with a conserved particle number). Thus, there is an unbroken symmetry in both the equations of motion and the ground state. This is known as theWigner-Weylrealization of the symmetry.
There are two ways in which a Lagrangian symmetry can be broken. One is to add small explicit breaking terms, as we did forSU(3) in (3.113). For example, withµ2>0 and
L=L0−
2φ22 (3.151)
theU(1)∼SO(2) symmetry would be broken, with nondegenerate masses
m21=µ2, m22=µ2+, (3.152)
corresponding to the Hermitian mass eigenstates φ1 and φ2. In this example, the quartic relations are not modified at tree-level, though there would be finite corrections induced at loop-level.20However, there would no longer be a conserved particle number (Problem 3.29).
The other possibility isspontaneous symmetry breaking(SSB), also known as the Nambu-Goldstonerealization of the symmetry (Nambu, 1960; Nambu and Jona-Lasinio, 1961; Gold-stone, 1961). Forµ2<0 theMexican hat(orwine bottle) potential in (3.146) has its minima
19The full symmetry ofL0 isO(2). The extraφ2→ −φ2 reflection symmetry corresponds toφ↔φ†in the complex basis.
20They are finite because the symmetry breaking issoft, i.e., the coefficient has a positive power in mass units.
φ1
φ2
V(φ)
φ1
φ2
V(φ)
Figure 3.6
Left: the potential in (3.146) for
µ2>0. Right: potential for
µ2<0.
away from the origin, as seen in Figure 3.6. The rotational symmetry leads to a circle of degenerate minima at
φ21+φ22=ν2≡ −µ2
λ >0, (3.153)
as illustrated inFigure 3.7. The true ground state must pick a specific point on this circle, spontaneously breaking the rotational symmetry in much the same way that the spins in a ferromagnetic domain line up in a specific direction. Since the classical field carries aU(1) charge, the symmetry is broken and the charge is not conserved.
φ1
φ2
ν
Figure 3.7