Coleman-Weinberg mechanism
| Erick J Weinberg (2015), Scholarpedia, 10(7):7484. | doi:10.4249/scholarpedia.7484 | revision #150734 [link to/cite this article] |
The Coleman-Weinberg mechanism is a phenomenon by which a theory that
at tree level appears to have a symmetric vacuum actually undergoes
spontaneous symmetry breaking as a result of radiative quantum
corrections.
Contents |
The effective potential
A common example for illustrating spontaneous symmetry breaking is the theory of a real scalar field $\phi$ with a potential of the form \begin{equation} V(\phi) = \frac12 \mu^2 \phi^2 + {\lambda\over 4!} \phi^4 \, . \tag{1} \end{equation} This Lagrangian is symmetric under the transformation $\phi\rightarrow - \phi$. The standard lore is that this is a symmetry of the vacuum if $\mu^2$ is positive, but that the symmetry is spontaneously broken if $\mu^2$ is negative. (In order that the energy be bounded from below, $\lambda$ must be positive.).
The borderline between these two cases, $\mu^2=0$, bears further inspection. Classically, the positivity of the quartic term would be sufficient to guarantee a symmetric vacuum. In a quantum field theory, however, the vacuum energy includes the zero-point energies of the various fields that enter the theory. If these zero-point energies depend upon the value of $\phi$, they can potentially change the situation (Coleman and Weinberg, 1973).
A natural tool for investigating this issue is the effective potential, $V_{\rm eff}$ (Goldstone, Salam and Weinberg, 1962), which can be defined as follows (Jona-Lasinio, 1964). First, we add to the Lagrangian a term $J(x)\phi(x)$ that gives a coupling to a classical source $J(x)$. We then define a quantity $W[J]$ by \begin{equation} e^{iW[J]} = \langle 0^+| 0^-\rangle_J \tag{2} \end{equation} where the quantity on the right-hand side is the amplitude for going from the vacuum in the far past to the vacuum in the far future in the presence of the external source $J$. Perturbatively, $W$ is the generating functional \begin{equation} W[J] = \sum_n {1\over n!}\int d^4x_1 \cdots d^4x_n \,G^{(n)}(x_1, \dots,x_n) J(x_1) \cdots J(x_n) \tag{3} \end{equation} where $G^{(n)}$ is the sum of all connected Feynman graphs with $n$ external $\phi$ lines.
We next define a classical field $\phi_c$ by \begin{equation} \phi_c(x) = {\delta W \over \delta J(x)} \, . \tag{4} \end{equation} A Legendre transformation then defines an effective action \begin{equation} \Gamma[\phi_c(x)] = W[J] - \int d^4x \, J(x)\phi_c(x) \, . \tag{5} \end{equation} The effective potential is then obtained by taking $\phi_c$ to be independent of $x$, and writing \begin{equation} \Gamma[\phi_c(x)] = - \int d^4x \, V_{\rm eff}(\phi_c) \, . \tag{6} \end{equation} A nonzero value of $\phi_c$ in the absence of a source signals spontaneous symmetry breaking. Because \begin{equation} J(x) = - {\delta \Gamma \over \delta \phi_c(x)} \, , \tag{7} \end{equation} this corresponds to a stationary point of the effective action and, assuming spatial homogeneity, a stationary point of the effective potential.
It follows from the above definitions that $V_{\rm eff}$ is given perturbatively as the sum of one-particle-irreducible graphs, with its $n$th derivative being obtained from the sum of all such graphs with $n$ external $\phi$ lines, each carrying zero momentum. More physically, $V_{\rm eff}(\phi_c)$ is the minimum expectation value of the energy density among those states $|\Psi\rangle$ for which the expectation value of the quantum field, $\langle \Psi|\phi({\bf x})|\Psi \rangle$, is equal to $\phi_c$.
Massless scalar quantum electrodynamics
Let us enlarge the theory of Eq. (1) by making $\phi$ complex, with real and imaginary parts $\phi_1$ and $\phi_2$, and coupling it to electromagnetism. Setting $\mu^2 =0$ gives the Lagrangian \begin{equation} {\cal L} = -\frac14 F_{\mu\nu}^2 +\frac12 |D_\mu\phi|^2 -{\lambda \over 4!}|\phi|^4 + {\rm counterterms} \, . \tag{8} \end{equation} Classically, this describes massless scalar electrodynamics. To explore the quantum theory we evaluate the effective potential. This calculation can be simplified by noting that $V_{\rm eff}$ can only depend on $\phi_c^2 = \phi_{1c}^2 + \phi_{2c}^2$. Hence, it is sufficient to calculate the graphs with only external $\phi_1$-lines and replace $\phi_{1c}^2$ by $\phi_c^2$ at the end of the calculation.
The leading approximation to the effective potential is obtained by summing the contributions from the tree and one-loop diagrams. In Landau gauge the latter divide into two classes — those with only internal $\phi$-lines and those with only internal $A_\mu$-lines. We obtain \begin{equation} V_{\rm eff} = {\lambda \over 4!}\phi_c^4 + I(\tfrac12 \lambda\phi^2_c) +I(\tfrac16 \lambda\phi^2_c) + 3 I( e^2 \phi_c^2) -\frac12 B \phi_c^2 - {1 \over 4!} C \phi_c^4 \tag{9} \end{equation} where \begin{equation} I(a^2) = i \int{d^4k \over (2\pi)^4 }\sum_{n=1}^\infty {1 \over 2n} \left( {a^2 \over k^2 + i \epsilon} \right)^n \, . \tag{10} \end{equation} In Eq. (9) the first term arises from the single tree diagram, the next three from the graphs with internal $\phi_1$-loops, $\phi_2$-loops, and photon loops, respectively, and the last from counterterms that must be determined. Note that the loop graphs with $n$ vertices have $2n$ external $\phi$-lines and thus factors of $\phi_c^{2n}$.
The graphs with two and four external $\phi$'s are quadratically and logarithmically divergent, respectively. These divergences will be canceled by the counterterms. Also, the loop graphs with $n\ge 4$ vertices have infrared divergences that become increasingly severe as $n$ gets larger. Indeed, it is just because of these divergences that these graphs must be included even though they would appear to be suppressed by increasing powers of the small couplings. As we will see, these infrared divergences at small $k^2$ combine to give a divergence at small $\phi_c$. Evaluating the sum in Eq. (10), we obtain \begin{equation} I(a^2) = -{i \over 2} \int{d^4k \over (2\pi)^4 } \, \ln\left(1 -{a^2 \over k^2+i\epsilon}\right) \, . \tag{11} \end{equation} The evaluation of the integral is relatively straightforward. Performing a Wick rotation and using a simple momentum-space cutoff $\Lambda$ (which is sufficient for our purposes here), we obtain \begin{equation} I(a^2) = {1\over 32\pi^2}\, a^2 \Lambda^2 + {1\over 64\pi^2}\, a^4 \left(\ln {a^2 \over \Lambda^2} - \frac12 \right) \, , \tag{12} \end{equation} where terms that vanish as $\Lambda^2 \rightarrow \infty$ have been omitted.
Substituting this result into Eq. (9) gives an expression with quadratic and logarithmic divergences that must be canceled by the counterterms. Requiring that \begin{equation} \left. {d^2 V_{\rm eff} \over d\phi_c^2} \right|_{\phi_c=M} = 0 \tag{13} \end{equation} determines $B$ and eliminates the quadratic divergence. We can fix $C$ by imposing a condition on the fourth derivative of $V_{\rm eff}$. However, this cannot be imposed at $\phi_c=0$, because there is a logarithmic infrared singularity. Our Lagrangian has only dimensionless parameters, so there is no natural alternative choice. Instead, we must choose some arbitrary point $\phi_c=M$ and require that \begin{equation} \left. {d^4 V_{\rm eff} \over d\phi_c^4} \right|_{\phi_c=M} = \lambda \, . \tag{14} \end{equation} Different choices of $M$ correspond to different definitions of $\lambda$, but these are different parameterizations of the same theory, related by the renormalization group. Imposing these renormalization conditions, we obtain \begin{equation} V_{\rm eff} = {\lambda \over 4!} \, \phi_c^4 + \left({5\lambda^2 \over 1152 \pi^2 } + {3e^4 \over 64\pi^2}\right) \phi_c^4 \left(\ln {\phi_c^2 \over M^2} - \frac{25}{6} \right) \, . \tag{15} \end{equation}
Let us examine this result. Because the logarithm of a small number is large and negative, the minimum of the tree-level potential at $\phi=0$ has become a local maximum, indicating that there is a minimum at some nonzero value $\langle\phi\rangle$. The suggests that we set $M = \langle\phi\rangle$ in Eq. (15). Requiring for consistency that the derivative of $V_{\rm eff}$ actually vanish at this point gives the relation \begin{equation} \lambda = {11 \over 8 \pi^2} \left( 3e^4 + {5 \over 18}\lambda^2 \right) \, . \tag{16} \end{equation} Consistency of our perturbative calculation requires that $\lambda$ be small, which in turn means that we can drop the $\lambda^2$ term on the right-hand side and set \begin{equation} \lambda = {33 e^4\over 8 \pi^2} \, . \tag{17} \end{equation} We then obtain our final expression for the one-loop effective potential, \begin{equation} V_{\rm eff} = {3e^4 \over 64\pi^2}\, \phi_c^4 \left(\ln {\phi_c^2\over \langle \phi \rangle^2} -\frac12 \right) \, , \tag{18} \end{equation} which is shown in Figure 1.
Because $\lambda$ is of the same order as $e^4$, the one-photon-loop contributions are comparable in size to the tree-level potential, and have given rise to spontaneous symmetry breaking. Expanding about the asymmetric vacuum in the usual manner, we find that instead of a massless photon and a massless complex scalar, as suggested by the tree-level analysis, we have a massive vector and a massive neutral scalar. The ratio of their masses is \begin{equation} {m^2(S) \over m^2(V)} = {3 e^2 \over 8 \pi^2} \, . \tag{19} \end{equation}
Note that $V_{\rm eff}$ is rather flat around the maximum at $\phi=0$. It was for this reason that the one-bubble new inflationary cosmology was first proposed in the context of a Coleman-Weinberg type potential (Linde, 1982; Albrecht and Steinhardt 1982).
Zero-point energy and the effective potential
The connection with the zero-point energies of the quantum fields is somewhat obscured in the covariant calculation outlined above. It can be made clearer by rewriting the integral in Eq. (11) as \begin{equation} I(a^2) = \int {d^3{\bf k} \over (2\pi)^3 }K(a^2) \tag{20} \end{equation} where \begin{eqnarray} K(a^2) &=& -{i \over 2}\int_{-\infty}^\infty {d k_0 \over (2\pi)} \ln \left(1 -{a^2 \over k_0^2 -{\bf k}^2 + i \epsilon} \right) \cr\cr &=&-{i \over 2}\int_{-\infty}^\infty {d k_0 \over (2\pi)} \left\{ {d \over dk_0} \left[k_0 \, \ln \left(1 - {a^2 \over k_0^2 -{\bf k}^2 + i \epsilon} \right) \right] - {2k_0^2 \over k_0^2 -{\bf k}^2 -a^2+ i \epsilon} +{2k_0^2 \over k_0^2 -{\bf k}^2 + i \epsilon} \right\} \cr\cr &=& \frac12\sqrt{ {\bf k}^2 + a^2 } - \frac12\sqrt{ {\bf k}^2 } \, . \tag{21} \end{eqnarray} The third line is obtained from the second by dropping the surface terms from the integration by parts and then evaluating the remaining integral by contour integration. Thus, we see that the symmetry breaking in "massless" scalar electrodynamics can be traced to the zero-point energies of the three polarizations of the now-massive vector meson.
Dimensional transmutation
At the beginning of our analysis, the theory was described by two dimensionless parameters, $\lambda$ and $e$, and no manifest dimensionful ones. However, there was also a hidden quantity with dimensions of mass, namely the renormalization point $M$. This doesn't really add an extra parameter, because any change in the value of $M$ can be compensated by changes in the values of $\lambda$ and $e$. However, it offers the possibility of exchanging the dimensionless parameter $\lambda$ for a dimensionful one, $\langle\phi\rangle$, that can be viewed as defining the unit of mass. This phenomenon is known as dimensional transmutation. In this example the net result is that a theory that at first sight appears to depend on two arbitrary parameters actually depends on only one. Perhaps more dramatic is the case of quantum chromodynamics with massless quarks, where the dimensionless gauge coupling constant can be exchanged for, e.g., the nucleon mass, leaving a theory with no free parameters at all.
Adding a scalar mass term
Although the original calculation was for a theory with a superficially massless scalar, radiative corrections can also drive spontaneous symmetry breaking when a small positive mass term is present. If a mass term \begin{equation} \mu^2 |\phi|^2 \equiv \beta \, {3e^4 \over 64\pi^2} \, \langle \phi\rangle^2 \,|\phi|^2 \tag{22} \end{equation} is added to the Lagrangian of Eq. (8), with $\lambda$ still much less than $e^2$ so that $\phi$-loops can be ignored, the one-loop effective potential takes the form \begin{equation} V_{\rm eff} = {3e^4 \over 64\pi^2} \left\{\frac12 \beta \langle \phi\rangle^2 \phi_c^2 + \phi_c^4 \left[ \ln {\phi_c^2\over \langle \phi \rangle^2} - {(2 +\beta) \over 4} \right] \right\} \tag{23} \end{equation}
With $\beta$, and thus $\mu^2$, positive, there is a symmetric minimum at the origin, $\phi_c=0$. However, there is also an asymmetric minimum at $\phi_c = \langle \phi \rangle \ne 0$. For $0 < \beta < 2$, the asymmetric minimum is lower, and thus represents a stable symmetry-breaking vacuum. At $\beta=2$ the two vacua are degenerate, and for $2 < \beta <4$ the symmetric vacuum is lower while the asymmetric one is metastable and can decay by the nucleation of bubbles of the symmetric vacuum. [If $\beta > 4$, then $\langle \phi\rangle$ becomes a local maximum rather than a local minimum, with the minimum located at a larger value of $\phi_c$. The expression in Eq. (23) is then simply a reparameterization of one with $\beta < 4$.] In the asymmetric vacuum the masses of the scalar and vector are related by \begin{equation} {m^2(S) \over m^2(V)} = \left(1-{\beta \over 4}\right){3 e^2 \over 8 \pi^2} \, . \tag{24} \end{equation}
Before taking the radiative corrections into account, it seemed that the theory with a symmetric vacuum went over smoothly to the symmetry-breaking one as $\mu^2$ went from positive to negative. This behavior is reminiscent of a continuous second-order phase transition. We see here that the effect of the radiative corrections is to replace this by a discontinuity similar that which characterizes a first-order transition.
Complexity and convexity
In the discussion above it was assumed that the scalar self-coupling was small enough that the scalar-loop contribution to the effective potential could be neglected. If this is not the case, then one must also include a term of the form \begin{equation} V_{\rm scalar~loop} = {1 \over 64\pi^2} [V''(\phi_c)]^2 \ln[V''(\phi_c)/M^2] \tag{25} \end{equation} where the value of $M^2$ is determined by the renormalization conditions. If the tree-level potential $V$ displays spontaneous symmetry breaking, there will be a range of $\phi_c$ for which $V''$ is negative. The logarithm will then have an imaginary part, rendering $V_{\rm eff}$ complex. This is in conflict with the statement that $V_{\rm eff}(\phi_c)$ is the (manifestly real) minimum expectation value of the energy density among states for which the expectation value of the quantum field $\phi$ is $\phi_c$.
A second puzzle is the observation that the effective potential, having been defined via a Legendre transform, should be everywhere convex (Iliopoulos, Itzykson and Martin, 1975). With a negative $V''$ the scalar-loop contribution does not satisfy this requirement. Indeed, even the effective potential of Eq. (18) fails this convexity condition.
These two puzzles have a common resolution (Weinberg and Wu, 1987). In a theory with two degenerate vacua, say at $\phi=\pm \sigma$, a state degenerate with these, but with $-\sigma < \langle \phi({\bf x}) \rangle < \sigma$ can be obtained by taking an appropriate linear combination of the original two vacuum states. This is the state whose (real) energy is given by the true effective potential. The latter takes the form indicated by the solid curve in Figure 2, and is manifestly convex.
However, this is not the state addressed by the perturbative calculation. Instead, that calculation focuses on states $|\Psi\rangle$ that not only have $\langle \Psi| \phi({\bf x})| \Psi\rangle = \phi_c$, but that also satisfy the further requirement that their wave functional be concentrated on configurations with $\phi({\bf x}) \approx \phi_c$. The minimum value of $\langle \Psi| H | \Psi\rangle$ among such states gives the real part of the perturbative effective potential. The imaginary part reflects the instability of these states, even when an external source is applied to maintain the condition $\langle \Psi| \phi({\bf x}) | \Psi\rangle = \phi_c$. Classically, it would be energetically advantageous for a configuration with a spatially uniform field to break up into an inhomogeneous mixture of domains, with the same overall average value of $\phi({\bf x})$, provided that the energy gained by reducing $V(\phi)$ was greater than the cost in gradient energy at the domain boundaries. This is the case if $V''(\phi_c)$ is negative, which is precisely the situation where the one-loop effective potential becomes complex. In fact, one can show that the imaginary part of the perturbative effective potential agrees quantitatively with an independent calculation of the decay rate of the initial state.
The region with negative $V''(\phi_c)$ corresponds to the existence of a classical instability. There is also the possibility of a quantum instability, with the spreading of the initially homogeneous state driven by quantum bubble nucleation, even when $V''(\phi_c) >0$. This gives a nonperturbative contribution to the imaginary part over the entire region between the classical minima, as indicated in Figure 2.
The relation between the exact effective potential and the one addressed by perturbation theory is quite analogous to that between the exact free energy obtained by a Maxwell construction and the analytic continuation of the free energy that describes a metastable phase.
Gauge-dependence of the effective potential
The calculation of the one-loop effective potential of scalar electrodynamics given in Eq. (18) was performed in Landau gauge. It is not obvious that working in another gauge would give the same result (Jackiw, 1974). Indeed, although the leading, $O(e^4)$, approximation is gauge-independent, gauge dependence appears at $O(e^6)$ (Dolan and Jackiw, 1974). Although this may appear troubling at first sight, it should not be, and can be readily understood. The scalar field $\phi(x)$ is itself gauge-dependent, as is even a statement that $\phi$ is spatially uniform. Hence, asking for the value of the effective potential at a given value of $\phi_c$ is not a well-defined question until the gauge is fixed. What is required is that physically measurable quantities be gauge-independent. Thus, the existence of a symmetry-breaking minimum and the difference in energy density between this minimum and the symmetric state should be gauge-independent. Identities that show that physical quantities such as these are indeed gauge-invariant have been derived (Nielsen, 1975; Fukuda and Kugo, 1976). The gauge independence of the scalar-vector mass ratio of Eq. (19) has been verified by explicit calculation (Kang, 1974). Similarly, the rate at which a metastable symmetric vacuum decays by the nucleation of bubbles of asymmetric true vacuum — a calculation that requires some care (Weinberg, 1993) — can be shown to be gauge-invariant (Metaxas and Weinberg, 1996).
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