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Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism - Scholarpedia

# Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism

Post-publication activity

Curator: Tom W B Kibble

Figure 1: The sombrero potential with an unstable state at $$\phi=0$$ and minima around a circle.

The Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism provides the means by which gauge vector bosons can acquire nonzero masses in the process of spontaneous symmetry breaking. It is a key element of the electroweak theory that forms part of the standard model of particle physics, and of many models, such as Grand Unified Theories, that go beyond it.

## Role of the mechanism

The discovery of the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism effectively removed a major obstacle to constructing a unified theory of weak and electromagnetic interactions. Since the weak interactions are of short range and very weak at low energies, it was clear that if they were mediated by an intermediate vector boson $$W\ ,$$ it must have a large mass -- an apparent problem, since gauge bosons were believed to be by nature massless, like the photon. Some mechanism was required to give mass to the $$W$$ while leaving the photon massless.

Initially, invoking spontaneous symmetry breaking was thought to introduce yet another problem, the appearance of the massless scalar Nambu-Goldstone bosons, as seemingly required by the Goldstone theorem (for reasons explained below). It turns out, however, that these two problems in a sense "cancel out".

The mechanism is essentially a relativistic version of one that operates in a superconductor. For brevity, it is often called the "Higgs mechanism", although strictly speaking that terminology is historically inaccurate, not least because it ignores the contribution of Englert & Brout (1964), published just before those of Higgs (1964a,b), but also because it omits any mention of the slightly later, but independent, contribution of Guralnik, Hagen & Kibble (1964), as well as of several earlier ones, notably those of Nambu (1960) and of Anderson (1963). The history of the development of the idea is discussed in the article Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism (history).

## Spontaneous symmetry breaking

Figure 2: (a) A ground state of the ferromagnet, with all spins aligned; (b) another ground state, with all spins rotated; (c) a low-energy spin-wave excitation.

Spontaneous symmetry breaking occurs when the ground state of a system does not share the full symmetry of the underlying theory. The clearest example is that of an isotropic ferromagnet, composed of an array of spins attached to the sites of a lattice, and described by the Hamiltonian $\tag{1} H = -\frac{1}{2}\sum_{\mathbf{x},\mathbf{y}} J_{\mathbf{x}-\mathbf{y}}\mathbf{S}_{\mathbf{x}}\cdot\mathbf{S}_{\mathbf{y}},$

where $$\mathbf{S}_{\mathbf{x}}$$ denotes the spin at the lattice site $$\mathbf{x}\ .$$ The strength of the interaction $$J$$ depends only on the distance between the lattice sites $$\mathbf{x}$$ and $$\mathbf{y}\ ,$$ and is assumed to fall off rapidly as the distance becomes large. (This assumption is important for reasons explained below.)

This model is invariant under a simultaneous rotation of all the spins, and correspondingly the total spin $\tag{2} \mathbf{S}=\sum_{\mathbf{x}}\mathbf{S}_{\mathbf{x}}$

is conserved. But if $$J$$ is positive, then clearly the lowest energy state is one with all the spins aligned (see Figure 2(a)). The direction in which they point is arbitrary. Thus there is a degenerate ground state. Making a simultaneous rotation of all the spins yields another ground state, with exactly the same energy (see Figure 2(b)).

A very important consequence of spontaneous symmetry breaking of a continuous symmetry like this one is that there are excitations whose energy goes to zero in the long wavelength limit. These are the Goldstone bosons or Nambu-Goldstone bosons. In this case these are spin waves, in which a periodic space-dependent rotation is applied to the spins (see Figure 2(c)). Since it costs no energy to rotate all the spins, from (a) to (b), it costs very little energy to make a long-wavelength periodic change. This is the content of the Goldstone theorem.

### Goldstone's U(1) model

The simplest relativistic field theory (Goldstone 1961) that exhibits such spontaneous symmetry breaking is one of a complex scalar field $$\phi$$ with the Lagrangian density function $\tag{3} L=\partial_\mu\phi^*\partial^\mu\phi-V(\phi),$

where $\tag{4} V(\phi)=m^2\phi^*\phi+\tfrac{1}{2}\lambda(\phi^*\phi)^2.$

Here units are chosen so that $$c=\hbar=1\ ,$$ the metric is $$(1,-1,-1,-1)$$ and $$m$$ and $$\lambda$$ are the mass and self-interaction coupling constant of the scalar field. The model is invariant under a global change of phase $\tag{5} \phi(x)\to\phi(x)e^{i\alpha}.$

This set of transformations defines the Abelian symmetry group U(1).

So long as $$m^2>0\ ,$$ this model is just a self-interacting scalar field, whose quanta are particles and antiparticles of mass $$m\ .$$

Things are different, however, if $$m^2<0\ .$$ Then $$\phi=0$$ is a maximum of the potential $$V\ ,$$ representing an unstable equilibrium. Here $\tag{6} V(\phi)=-\tfrac{1}{2}\lambda v^2\phi^*\phi +\tfrac{1}{2}\lambda(\phi^*\phi)^2,$

where $$v^2=-2m^2/\lambda\ .$$ This is often called the sombrero potential (see Figure 1 above). Its minima now lie on the circle $$|\phi|^2 = v^2/2 \ .$$ So in the ground state, or vacuum state, one expects the value of $$\phi$$ to be non-zero, with a magnitude close to $$v/\sqrt{2}\ ,$$ but arbitrary phase. There will be a degenerate family of vacuum states $$|0_\alpha\rangle\ ,$$ labelled by the phase angle $$\alpha\ .$$

For a ferromagnet, the symmetry is broken even in finite volume. For the Goldstone model, the situation is different. If this system were confined to a finite volume, then the true vacuum state would be a symmetric linear superposition of all these "ground states", in which the expectation value of $$\phi$$ would still vanish. But this is not the case for a field theory (in infinite volume). Then the states $$|0_\alpha\rangle$$ are mutually orthogonal, and all matrix elements between distinct vacua involving a finite number of fields vanish. Each $$|0_\alpha\rangle$$ is the vacuum state of a distinct Hilbert space constructed by applying the field operators to it. (Note that the representation of the canonical commutation relations on this Hilbert space is inequivalent to the usual Fock-space representation -- in contrast to the situation in ordinary quantum mechanics where all irreducible representations of the Heisenberg commutation relations are unitarily equivalent.)

Figure 3: Definition of the shifted fields, representing fluctuations about the stable equilibrium state (red dot, on right), instead of the unstable point (green dot, centre).

Nambu-Goldstone bosons appear here too. One can see this by choosing one particular minimum, say the one where $$\phi$$ is real and positive, and expanding about that point (see Figure 3), defining shifted real fields $$\varphi_{1,2}$$ by $\tag{7} \phi=\frac{1}{\surd2}(v+\varphi_1+i\varphi_2).$

Then one finds $\tag{8} L = \tfrac{1}{2}[(\partial_\mu\varphi_1)^2 +(\partial_\mu\varphi_2)^2] - V,$

with $\tag{9} V=-\tfrac{1}{8}\lambda v^4+\tfrac{1}{2}\lambda v^2\varphi_1^2 +\tfrac{1}{2}\lambda v\varphi_1(\varphi_1^2+\varphi_2^2) +\tfrac{1}{8}\lambda(\varphi_1^2+\varphi_2^2)^2.$

The first term here is merely an unimportant constant. By construction, there is no linear term, and, importantly, nor is there a term in $$\varphi_2^2\ .$$

Now canonical quantization can proceed as normal in terms of the fields $$\varphi_{1,2}\ .$$ The model describes two kinds of particles, the $$\varphi_1$$ quanta of mass $$\sqrt\lambda v$$ and the massless $$\varphi_2$$ quanta, with cubic and quartic couplings. The $$\varphi_2$$ quanta, corresponding to a spatial variation of the phase angle, are the Nambu-Goldstone bosons. Their presence is required by the Goldstone theorem in any manifestly Lorentz covariant theory in which a continuous symmetry is spontaneously broken. (The theorem is also true for non-relativistic theories, provided that an additional assumption is satisfied. This is discussed further below.)

## Spontaneous breaking of a local gauge symmetry

The global symmetry (5) may be promoted to a local symmetry provided a corresponding gauge field is introduced, with potential $$A_\mu\ ,$$ say. The Lagrangian is now $\tag{10} L=D_\mu\phi^*D^\mu\phi -\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-V(\phi),$

where the covariant derivative and gauge field are given by $D_\mu\phi=\partial_\mu\phi+ieA_\mu\phi,\qquad F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.$ This Lagrangian is invariant under the local gauge transformations $\tag{11} \phi(x)\to\phi(x)e^{i\alpha(x)},\qquad A_\mu(x)\to A_\mu(x)-\frac{1}{e}\partial_\mu\alpha(x).$

This is the simplest of all gauge theories, with an Abelian U(1) gauge group.

So long as $$m^2>0\ ,$$ this is ordinary scalar electrodynamics. It describes a charged particle of mass $$m\ ,$$ and its antiparticle, interacting with massless photons. But what happens if $$m^2$$ is again taken negative, say $$m^2=-\tfrac{1}{2}\lambda v^2<0$$ as before?

Here too one can make the substitution (7), and find that $$V$$ takes the form (9). Now, however, there are additional quadratic terms arising from the first, kinetic term in (10). The Lagrangian now takes the form $\tag{12} L=\tfrac{1}{2}(\partial_\mu\varphi_1)^2 +\tfrac{1}{2}(\partial_\mu\varphi_2+evA_\mu)^2 -\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu} -\tfrac{1}{2}\lambda v^2\varphi_1^2+\dots,$

where the dots denote cubic and quartic interaction terms (and the constant term). It is now clear that the second term gives the gauge field a mass. Indeed, one can introduce a new field $B_\mu=A_\mu+\frac{1}{ev}\partial_\mu\varphi_2,$ so that the second term becomes a mass term for $$B_\mu\ ,$$ with mass $$ev\ ,$$ and no kinetic term for $$\varphi_2$$ remains.

A rather neater way of describing the situation is to note that one may choose a gauge such that $$\varphi_2=0\ ,$$ so that $$A_\mu$$ and $$B_\mu$$ coincide (though care is needed in dealing with points where $$\phi=0\ ;$$ these are associated with topological defects).

This is then a model that describes only massive fields, a vector field $$B_\mu$$ with mass $$ev\ ,$$ interacting with a scalar field $$\varphi_1\ ,$$ whose mass is again $$\sqrt\lambda v\ .$$ The apparently massless gauge field $$A_\mu$$ and the apparently massless scalar field $$\varphi_2$$ have combined to give the massive vector field $$B_\mu\ .$$ Note that the number of helicity states is unchanged: the two polarization states of $$A_\mu$$ and the single one of $$\varphi_2$$ combine to give the three of $$B_\mu\ .$$ This is the essence of the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism.

How was it possible to arrive at a theory with no massless particles, despite the Goldstone theorem which might seem to demand their presence?

### Proof of the theorem

To answer this question it may be helpful to start by examining one of the proofs of Goldstone's theorem. It is based on three hypotheses.

Firstly, the theory has a continuous symmetry, and correspondingly a Noether current $$j_\mu(x)$$ which is conserved, satisfying $$\partial_\mu j^\mu=0\ .$$ In the case of a scalar field with U(1) symmetry, this current is $\tag{13} j_\mu=-i\phi^*D_\mu\phi+i\phi D_\mu\phi^*.$

Secondly, this symmetry is spontaneously broken; there is no invariant vacuum state, but rather a degenerate family of non-invariant vacuum states. Specifically, for any vacuum state $$|0\rangle\ ,$$ there exists some field $$\phi$$ (possibly composite) whose vacuum expectation value $$\langle0|\phi|0\rangle$$ is not invariant. Now the generator $$Q$$ of the transformation is the spatial integral of $$j^0\ :$$ $\tag{14} Q(t)=\int d^3\mathbf{x}\, j^0(t,\mathbf{x}),$

so what this means is $\tag{15} -i\int d^3\mathbf{x}\, \langle0|[j^0(0,\mathbf{x}),\phi(0)]|0\rangle\ne0.$

Finally, the theory is manifestly Lorentz covariant.

The simplest way to prove the theorem is to consider the Fourier transform $\tag{16} f^\mu(k)=-i\int d^4x\,e^{ik\cdot x}\langle0|[j^\mu(x),\phi(0)]|0\rangle$

The broken symmetry condition (15) then becomes $\tag{17} \int_{-\infty}^\infty dk^0\,f^0(k^0,\mathbf{0}) \ne0.$

On the other hand, because the current is conserved, we have $\tag{18} k_\mu f^\mu(k)=0.$

This would seem to require in particular that $\tag{19} k^0f^0(k^0,\mathbf{0})=0.$

It follows from Eqs. (17) and (19) that $$f^0(k^0,\mathbf{0})$$ contains a non-zero multiple of $$\delta(k^0)\ .$$ If we insert a complete set of intermediate states into (16), we see that this implies that there must be states that couple to the vacuum through $$\phi$$ for which $$k^0\to0$$ as $$\mathbf{k}\to\mathbf{0}\ ,$$ i.e., massless particle states.

### The case of gauge symmetry

So how does this argument fail in the case of a local, gauge symmetry?

The key lies in the third hypothesis, of manifest covariance. To quantize a gauge theory, one must choose a gauge, and impose a gauge-fixing condition of some kind. For the U(1) gauge theory, if one wants a formulation in terms of a Hilbert space containing only physical states, this cannot be done in a manifestly covariant way. The simplest choice is the Coulomb or radiation gauge, defined by the condition $$\partial_kA^k=0\ ,$$ which is non-covariant since it depends on the choice of time axis. In this case, commutators are not required to vanish outside the light cone, and in fact fall off only like an inverse power. Consequently, the continuity equation $$\partial_\mu j^\mu=0$$ no longer requires that the charge operator (14) be time-independent, because when we integrate by parts the contribution from the surface term at infinity need not vanish. (In fact, $$Q$$ is not really well defined, except inside commutators.) So it no longer follows that $$k_\mu f^\mu(k)=0\ .$$ (In the non-relativistic case, the third hypothesis of the Goldstone theorem is that there are no long-range forces involved, because that ensures that commutators fall off sufficiently rapidly at infinity that the surface integral can be dropped.)

Of course, there is an alternative, manifestly covariant way of quantizing the electromagnetic field, namely to use the Lorentz gauge, defined by $$\partial_\mu A^\mu=0\ .$$ In that case, however, the Hilbert space necessarily contains states of unphysical scalar and longitudinal photons. Here the Goldstone theorem definitely does apply, and there do exist massless Nambu-Goldstone bosons. But it turns out that they are purely gauge modes, uncoupled to the physical particles of the theory.

This mechanism is often said to exhibit "spontaneously broken gauge symmetry". That is a convenient shorthand description, but the terminology is potentially somewhat misleading. The process of quantization requires a choice of gauge, i.e., an explicit breaking of the gauge symmetry. However, the resulting theory does retain a global phase symmetry that is broken spontaneously by the choice of the phase of $$\langle\phi\rangle\ .$$

It is, however, very important that the model derives from an initial fully gauge-invariant one. In general, models involving massive vector fields are non-renormalizable. But this theory is renormalizable, as proved by 't Hooft (1971) (see gauge theories), primarily because it retains the Ward identities that play a key role in the renormalizability of quantum electrodynamics.

## Non-Abelian symmetry breaking

Essentially the same mechanism can apply in the case of a non-Abelian gauge theory.

The simplest example is that of a triplet of scalar fields, $$\vec\phi=(\phi_j)\ ,$$ transforming according to the three-dimensional adjoint representation of the group SO(3). The Lagrangian here is $\tag{20} L=\tfrac{1}{2} \overrightarrow{D_\mu\phi}\cdot\overrightarrow{D^\mu\phi} -\tfrac{1}{4}\vec F_{\mu\nu}\cdot\vec F^{\mu\nu}-V(|\vec\phi|),$

where $\overrightarrow{D_\mu\phi}=\partial_\mu\vec\phi+g\vec A_{\mu}\times\vec\phi,$ $\vec F_{\mu\nu}=\partial_\mu\vec A_{\nu}-\partial_\nu\vec A_{\mu} +g\vec A_{\mu}\times\vec A_\nu,$ and $$g$$ is the gauge coupling constant.

This Lagrangian is invariant under SO(3) gauge transformations; in infinitesimal form these are $\delta\vec\phi=\overrightarrow{\delta\omega}\times\vec\phi$ and $\delta\vec A_\mu=\overrightarrow{\delta\omega}\times\vec A-\frac{1}{g}\partial_\mu\overrightarrow{\delta\omega},$ where $$\overrightarrow{\delta\omega}=(\delta\omega_j)$$ are three infinitesimal parameters.

Now suppose that $$V$$ is again chosen so that its minima are not at the symmetric point $$\vec\phi=\vec 0$$ but on a sphere of non-zero radius, $$|\vec\phi|=v\ .$$ Then there will as before be a degenerate family of vacuum states, characterized by vacuum expectation values lying on the sphere. For example, one such vacuum will have $$\langle0|\vec\phi|0\rangle=(0,0,v)=\vec\phi_0\ ,$$ say. Other points on the sphere will correspond to other vacua.

In this case, the symmetry group SO(3) is not broken completely. There is an unbroken subgroup SO(2), comprising rotations in the $$xy$$ plane, which leave $$\vec\phi_0$$ unchanged. In this case, one of the three gauge bosons, the component $$A_{3\mu}$$ corresponding to this unbroken symmetry, remains massless. The other two $$A_{1\mu}$$ and $$A_{2\mu}$$ acquire masses, equal to $$gv\ .$$ As before the longitudinal components of these massive gauge bosons come from the would-be Nambu-Goldstone bosons $$\phi_1$$ and $$\phi_2\ .$$ Oscillations in the radial direction, described by $$\phi_3-v\ ,$$ are the massive Higgs bosons.

### Case of a general symmetry group

In the general case, suppose that the multiplet $$\phi$$ of real fields $$\phi^\alpha$$ belongs to a representation of the symmetry group $$G$$ with Hermitian (in fact imaginary, antisymmetric) generators $$T_a\ ,$$ and that the symmetry is spontaneously broken down to a subgroup $$H$$ when $$\phi$$ acquires a non-zero expectation value $$\phi_0\ .$$ Then it is easy to see that the mass matrix for the vector fields is $(M^2_v)_{ab}=g^2\tilde\phi_0 T_aT_b\phi_0,$ where the tilde denotes transposition. The generators of $$H$$ are those generators of $$G$$ for which $$T_a\phi_0=0\ .$$ For them, the corresponding elements of $$M^2$$ vanish. So if $$G$$ is of dimension $$n$$ and $$H$$ is of dimension $$m\ ,$$ there will be $$m$$ massless gauge bosons and $$n-m$$ massive ones. For example, in the Weinberg-Salam model, where $$G=SU(2) \times U(1)$$ and $$H=U(1)\ ,$$ we have one massless and three massive gauge bosons.

Once again, the longitudinal components of the massive gauge bosons come from the would-be Nambu-Goldstone bosons, while the remaining components of $$\phi$$ (or rather of $$\phi-\phi_0$$) will acquire masses given by the scalar mass matrix, $(M^2_s)_{\alpha\beta}= \left(\frac{\partial^2V}{\partial\phi^\alpha\partial\phi^\beta}\right)_{\phi=\phi_0}.$ These are the Higgs bosons.

The primary purpose of the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism was to provide masses for the gauge vector bosons. A secondary effect, however, was to give masses to other fundamental particles.

Any fermion that interacts with the scalar field $$\phi$$ via an interaction term of the form $$h\bar\psi'\phi\psi$$ may acquire a mass, of order $$hv\ ,$$ by virtue of the non-zero expectation value $$\langle0|\phi|0\rangle\ .$$ In the standard model, this is the mechanism that gives masses to the leptons and the quarks. These remain arbitrary parameters, however, because the masses are determined by the arbitrary coupling constants $$h\ ;$$ the top quark is heavy, for example, because it interacts relatively strongly with the Higgs field.

It is sometimes said that the Higgs field gives masses to all other particles, but that is not strictly correct. It is important to note that most of the mass of the nucleon in particular does not arise in this way. Only the masses of the quarks come from the Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism. The larger part of the nucleon mass comes from a mechanism along the lines sketched out earlier by Nambu (see Englert-Brout-Higgs-Guralnik-Hagen-Kibble mechanism (history)).

## References

• Goldstone, J (1961). Field theories with superconductor solutions. Nuovo Cim. 19: 154-64.
• Higgs, P W (1964a). Broken symmetries, massless particles and gauge fields. Phys. Lett. 12: 132-3.
• 't Hooft, G (1971). Renormalizable Lagrangians for massive Yang-Mills fields. Nuc. Phys. B 35: 167-88.

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