Triviality of four dimensional phi^4 theory on the lattice

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Ulli Wolff (2014), Scholarpedia, 9(10):7367. doi:10.4249/scholarpedia.7367 revision #186755 [link to/cite this article]
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Curator: Ulli Wolff

Quantum field theories are only formally defined by their Lagrange density. To extract physical predictions, the theory is regularized (modified) at short distance, an ultraviolet cutoff is imposed. Observable values emerge from suitable renormalized correlation functions in the limit where this cutoff is removed again. As a very simple analog the reader may think of the intermediate consideration of finite differences to define a derivative by a limiting procedure. In both cases the resulting limit is independent of the choice of regularization.

A quantum field theory is called trivial, if the above limit invariably leads to a non-interacting renormalized theory despite the presence of interaction terms in the bare Lagrangian.

It is possible that such a theory can still describe Nature if there is a range of its free parameters such that

  • interactions are strong enough to match experiment,
  • effects of the finite cutoff are unobservably small at accessible momenta.

In other words, the cutoff is not fully removed, but its effects are invisible on a certain limited range of scales to a certain precision. Such an effective field theory is expected to be finally replaced by a more comprehensive but still unknown theory valid beyond the restricted range. Here an analog model could be the Fermi model of weak interactions that has been superseded and extended by the Weinberg-Salam theory.


Quantum field theory of a scalar field

In the focus here is the simplest quantum field theory of a single real scalar field $\phi \left( x \right)$ attached to each point of $D$-dimensional Euclidean space-time. (See the articles Lattice quantum field theory and Critical Phenomena: field theoretical approach). The bridge to particle physics is crossed by setting $D = 4$ and, for many observables, analytically continuing to Minkowski space, which will, however, play no role in this article. A definition of correlation functions as the most important class of observables is given by functional integrals \[ \langle \phi \left( x_1 \right) \phi \left( x_2 \right) \cdots \phi \left( x_n \right) \rangle = \frac{1}{\mathcal{Z}} \int D \phi \, \mathrm{e}^{- S \left[ \phi \right]} \phi \left( x_1 \right) \phi \left( x_2 \right) \cdots \phi \left( x_n \right) \] with the action \[ S = \int d^D x \left\{ \frac{1}{2} \left[ \partial_{\mu} \phi(x) \right]^2 + \frac{\mu_0^2}{2} \phi^2(x) + \frac{g_0}{4!} \phi^4(x) \right\}, \] where $\mathcal{Z}$ is the normalizing partition function achieving $\langle 1 \rangle = 1$. Such a definition is highly formal and a regularization is needed. There are two main strategies to define and evaluate functional integrals:

A: Perturbative approach: the weight $\mathrm{e}^{- S \left[ \phi \right]}$ is expanded in powers of $g_0$ and then the integral becomes a sum of Gaussian (quadratic in the exponent) expectation values of polynomials of $\phi$. Such integrals can be evaluated formally and lead to the famous Feynman diagrams. The individual terms diverge and still need regularization. There are many ways to do this within perturbation theory, as for example continuing to non-integer $D$, or by modifying the action to suppress the large momentum components in $\phi$. Note the unsatisfactory aspect of mixing up approximation and definition in this approach instead of approximating an object that has been well defined before.

B: Nonperturbative lattice approach (see Lattice quantum field theory and text books referenced there): The infinite space-time continuum is replaced by a finite lattice that has to be large and dense enough. Then the functional integral can be rigorously defined as a limit of a sequence of ordinary integrals. In the lattice truncation, various expansions as well as other numerical methods to approximate the high dimensional integrals may be applied with characteristic domains of validity. Most notably this includes Monte Carlo simulation.

Here it is attempted to discuss structural properties of the $\phi^4$ field theory at arbitrary values of its free parameters $\mu_0$ and $g_0$ including such values where the series expansion in $g_0$ may not resemble the true answer. For this reason the focus is mainly on the lattice definition, but inspiration from perturbation theory is still essential to design definitions and parameterizations.

Lattice formulation, symmetric phase

A cubic lattice with spacing $a$ and periodicity $L$ in each direction (hypertorus) is visualized in the article on Lattice quantum field theory. There are $\left( L / a \right)^D$ points or sites $x$, and the functional integral $\int D \phi \ldots$ is defined to mean independent integrations over the entire real line of $\phi \left( x \right)$ for each $x$. In the definition of the action $S$ we replace \[ \int d^D x \ldots \rightarrow a^D \sum_x \ldots \hspace{1em} {\rm and} \hspace{1em} \partial_{\mu} \phi(x) \rightarrow \frac{1}{a} [\phi(x+a\hat{\mu})-\phi(x)] \] where $\hat{\mu}$ is a unit vector that $x+a\hat{\mu}$ is the nearest neighbor of $x$ in the $\mu$-direction.

An important observable is the two-point correlation $\left\langle \phi \left( x \right) \phi \left( y \right) \right\rangle = G \left( x - y \right)$ where $x, y$ must be lattice sites, of course, and translation invariance on the torus makes $G$ depend on $x - y$ only. We can pass to momentum space \[ \tilde{G} \left( p \right) = a^D \sum_x \mathrm{e}^{- i p x} G \left( x \right), \hspace{1em} p \in \left( \frac{2 \pi}{L} \mathbb{Z} \right)^D \] where the components $p_{\mu}$ on the torus are integer multiples of $2 \pi / L$. In the free theory ($g_0 = 0$), one can now perform the finite dimensional Gauss integrals and obtains the free propagator \[ \tilde{G} \left( p \right) = \frac{1}{\hat{p}^2 + \mu_0^2}, \hspace{1em} \hat{p}_{\mu} = \frac{2}{a} \sin \left( a p_{\mu} / 2 \right) \] which, in position space, corresponds to an asymptotic decay of $G \left( x \right)$ proportional to $\exp \left( - \mu_0 \left| x \left| \right) \right. \right.$ up to power corrections. The physical scale that is thus encoded in the correlations is the correlation length $\xi = 1 / \mu_0$. In this case the lattice is considered large and dense (sufficiently fine resolution) if $a \ll \xi \ll L$ holds. One may now consider the

  • thermodynamic limit $L / \xi \rightarrow \infty$,
  • continuum limit $a / \xi = a \mu_0 \rightarrow 0$.

They may be taken in either order (or even simultaneously) and in any case the extrapolation leads to a diverging number of lattice sites. Renormalization theory suggests, that limits of the type just discussed also exist in the interacting theory with $g_0 > 0$, although in a considerably more complicated fashion as far as the relation between the bare parameters $\mu_0, g_0$ and physical properties of the correlations are concerned.

For example, the two-point function becomes more complicated. However, all experiences from perturbation theory suggest that in the limit of small momenta (only) one can still match \[ \tilde{G} \left( p \right) = \frac{Z}{\hat{p}^2 + m_R^2} \hspace{1em} \left( \hat{p}^2 \rightarrow 0 \right) \] where the dimensionless quantities $Z , a m_R$, extracted from the limiting behavior of $\tilde{G} \left( p \right)$ at small $p$ [or $G \left( x \right)$ at large $\left| x \left| \right. \right.$], are non-trivial but well-defined functions of $a \mu_0$ and $g_0$. The decay of correlations is now characterized by $\xi = 1 / m_R$. The (renormalized) mass of the scalar particles described by this quantum field theory is essentially given by $m_R$, while the parameter $\mu_0$ no longer has a direct physical meaning. A physical measure of interaction is related with scattering cross-sections of the particles. For certain processes they are determined by four-point correlations. This leads to a possible definition of a renormalized coupling constant (interaction strength) $g_R$ by \[ g_R = \frac{3 \langle M^2 \rangle^2 - \langle M^4 \rangle}{\langle M^2 \rangle^2} \left( m_R L \right)^D, \hspace{1em} M = a^D \sum_x \phi \left( x \right) . \] This $g_R$ has a number of important properties:

  • $g_R$ is dimensionless and non-negative (Lebowitz inequality),
  • $g_0 = 0 \Rightarrow g_R = 0$ as the numerator vanishes for Gaussian measures,
  • the thermodynamic limit exists smoothly and is assumed to have been taken from now on with the situation hence being: $0 < a m_R \ll 1, m_R L = \infty$.

The continuum limit for this model amounts to tuning the bare parameters $a \mu_0, g_0$ to arbitrary values (where the integrals exist, however) such that $a m_R = a / \xi$ goes to zero. Generically, we might expect this to happen on a line in the $\left( a \mu_0, g_0 \right)$ plane [including the free point $(0,0)$], and different points on this 'continuum line' may correspond to different physical interaction strengths $g_R$ in the continuum theory. Note that renormalized continuum correlations emerging in the limit may still depend on $m_R$ which thus can survive as the physical scale. Our naive generic picture is essentially correct for $D = 2, 3$, but in $D \geqslant 4$ we encounter the phenomenon of triviality: we find that $g_R \searrow 0$ whenever we tune for the continuum limit $a m_R \rightarrow 0$. Thus only free non-interacting theories are reached in the strict continuum limit.

Triviality of lattice $\phi^4$ theory in this sense has been rigorously proven for $D > 4$ while for the most interesting borderline case $D = 4$ we have only partial results but very strong evidence from numerical simulations. It is interesting to note what perturbation theory has to say in this context. One way to set up the Callan-Symanzik renormalization group equation for the $\phi^4$ field theory is to define the $\beta$ function by \[ \frac{d g_R}{d \ln \left( a m_R \right)} \left|_{g_0} = \beta \left( a m_R, g_R \right) . \right. \] The rate of change refers to varying $a \mu_0$ at arbitrary fixed $g_0$-values, but the result is written as a function of $g_R$ and $a m_R$. In perturbation theory one may now show that as a power series in $g_R$ \[ \beta \left( a m_R, g_R \right) = \sum_{n = 1}^{\infty} b_n \left( a m_R \right) g_R^{n + 1}, \hspace{1em} b_1 \left( 0 \right) = \frac{3}{\left( 4 \pi \right)^2}, \hspace{1em} b_2 \left( 0 \right) = - \frac{17}{3 \left( 4 \pi \right)^4}, \hspace{1em} \ldots . \] $\beta$ exists to all orders and the limits $b_n \left( 0 \right)$ of the coefficients are smooth and finite. If we integrate this differential equation to leading order we obtain the asymptotic behavior \[ g_R \simeq \frac{1}{- b_1 \ln \left( a m_R \right) } \] that predicts a (slow!) logarithmic vanishing of $g_R$ in the continuum limit. In this sense, perturbation theory predicts triviality as long as the truncated expansion in $g_R$ is justified. Note that this is entirely possible without $g_0$ being small since we argue within renormalized perturbation theory. However, there remains the logical possibility that a non-trivial continuum limit exists with some $g_R$ where perturbation theory is simply irrelevant. It is this gap which one tries to close by numerical experiments.

Lattice formulation, broken-symmetry phase

So far it was tacitly assumed that $G \left( x \right)$ decays to zero at large distance. If we imagine a negative value of $\mu_0^2 < 0$, then, for constant fields, $S$ is an integral over a double-well shaped density with minima at $\phi \left( x \right) \equiv \pm \overline{\phi}$ with $\overline{\phi}^2 = - 6 \mu_0^2 / g_0$. Then in perturbation theory, for $D>1$, the dominant contributions to the path integral come from fluctuations around one of these minima (rather than the zero field as before) which is picked by the spontaneous breaking of the symmetry $\phi \rightarrow - \phi$. This leads to a non-vanishing constant one-point function $\langle \phi \left( x \right) \rangle$ and particles are associated with the small fluctuations around it. From the subtracted two-point function \[ G \left( x \right) = \left\langle \phi \left( x \right) \phi \left( 0 \right) \right\rangle - \left\langle \phi \right\rangle^2, \] one may now extract $m_R$ and $Z$ in the same way as before. Now perturbation theory is set up after shifting the integrations by $\overline{\phi}$. From the quadratic part of the shifted action one reads off the leading order result $m_R^2 = - 2 \mu_0^2 + \ldots$

For a discussion beyond perturbation theory, it is useful to change to a different but equivalent parameterization of the action on the lattice \[ S = \sum_x \left[ \varphi \left( x \right)^2 + \lambda \left( \varphi \left( x \right)^2 - 1 \right)^2 \right] - 2 \kappa \sum_{\langle x y \rangle} \varphi \left( x \right) \varphi \left( y \right) . \] The second sum runs over all nearest-neighbor pairs on the lattice (links) and comes from the discretized $\partial_{\mu}$. By simple algebra, this form is seen to be entirely equivalent to the previously given one with the following identifications (for $D=4$) \[ a \phi = \sqrt{2 \kappa} \varphi, \hspace{1em} a^2 \mu_0^2 = \frac{1 - 2 \lambda}{\kappa} - 8, \hspace{1em} g_0 = \frac{6 \lambda}{\kappa^2} . \]

Figure 1: Phase diagram of the $\phi^4$ theory in the lattice parameterization (qualitative, except end points)

The parameters ($\left. a \mu_0, g_0 \right)$ have been traded for ($\kappa, \lambda$). Symmetry breaking is caused by the interaction and, in this phase, there is an alternative and natural definition of a renormalized coupling given by \[ g_R' = 3 Z m_R^2 / \left\langle \phi \right\rangle^2 \] with similar properties as $g_R$ above.

From various reasonings and simulations, the phase diagram is expected to look as shown in the figure. Quantitatively correct (for $D = 4$) are here the points $\lambda = 0, \kappa = 1 / 8$ and the point at $\lambda = \infty .$ The rest of the line is meant to be qualitative only.

If the critical line is approached from below the symmetric continuum limit is taken, from above the broken one is reached (for $\lambda > 0$). In the limit $\lambda = \infty$ the phase diagram looks perfectly regular. For simulations, this is in fact a very convenient case. By the steeply rising action $S$, the path integral is restricted to the vicinity of $\varphi \left( x \right) = \pm 1$ on all sites and it thus reduces to the Ising model, which facilitates Monte Carlo simulations. At the same time, assuming a monotonic relation between bare and renormalized coupling, this is then the most relevant case to establish triviality.

Trivial theories may be effective

Trivial theories can still be useful as effective theories, if there are parameter values with enough interaction to match experiments and at the same time small enough $a m_R$ that effects of the unremoved lattice (or other UV cutoff) are so small as to not contradict experiment. This can then hold true only up to momenta where $a \left| p \left| \ll 1 \right. \right.$ is so small that the lattice cannot be resolved. Effective theories can thus only represent physics at low enough energies, as one does not expect any space-time granularity to really be given by one of the standard cutoffs such as the lattice. In $D = 4$ one can argue that the coupling dies out only logarithmically, $g_R \propto c / |\ln (a m_R)|$ and the effective scenario is quite viable. Presumably the status of being an effective theory in the above sense is also true for the Standard Model of particle physics, which contains the scalar Higgs sector. Then a bound on $g_R'$ translates into a bound on the Higgs mass $m_R^2$ because the proportionality factor $\langle \varphi \rangle^2 / Z$ is fixed by the phenomenological scale generated, the raison d'être of the Higgs field.

Historically, all our theories seem to be eventually superseded by a next more comprising theory revealing itself beyond a certain domain of validity. Triviality may now even be seen as a bonus rather than a defect. It implies an order of magnitude 'prediction' about the range of applicability of the present model. However, one has to emphasize that this is really only an order of magnitude as one is by construction concerned with non-universal properties which in detail differ from one (unphysical) regularization to another.

Numerical results

In numerical tests the typical goal is to verify, if in a certain range of the phase diagram that is close to the critical line, a renormalized coupling constant $g_R$ evolves under changing the cutoff in the way that is predicted by the perturbative renormalization group. If the answer is affirmative for some more or less convincing range, then one has confidence that for the rest of the trajectory toward the continuum limit with smaller and smaller coupling the agreement with perturbation theory gets even better. Then one checks numerical support for the triviality conjecture in $D = 4$.

The difficulty in such simulations is the presence of the length scales $a$, $\xi$, and $L$. In computer simulations the number of sites $L / a$ in each direction is limited. As triviality is considered an ultraviolet renormalization effect, one may study a finite size scaling continuum limit where one fixes $L / \xi$ to some order one value and thus achieves a larger variation of $\xi / a \gg 1$ to verify renormalization group behavior instead paying for the thermodynamic limit. An additional problem that has to be mastered is critical slowing down of simulations for growing $\xi / a$. Here the Ising limit allows to apply particularly efficient techniques in the form of cluster and worm algorithms. Tests conducted along these lines have supported the triviality scenario up to now.


rigorous proofs:

  • Aizenman, Michael (1981). Proof of the Triviality of phi**4 in D-Dimensions Field Theory and Some Mean Field Features of Ising Models for D > 4. Phys. Rev. Lett. 47: 1.
  • Fröhlich, Jürg (1981). On the Triviality of lambda phi**4 Theories and the Approach to the critical Point in d(-) > 4 Dimensions. Nucl. Phys. B200: 281.

lattice formulation:

  • Lüscher(1987). Scaling Laws and Triviality Bounds in the Lattice phi**4 Theory. 1. One Component Model in the Symmetric Phase. Nucl. Phys. B290: 25.
  • Lüscher(1988). Scaling Laws and Triviality Bounds in the Lattice phi**4 Theory. 2. One Component Model in the Phase with Spontaneous Symmetry Breaking. Nucl. Phys. B295: 65.
  • Heller, Urs. M.; Klomfass, Markus; Neuberger, Herbert and Vranas, Pavlos M. (1983). Regularization dependence of the Higgs mass triviality bound. Nucl.Phys.Proc.Suppl. 30: 685. arXiv:hep-lat/9210026

implications for the standard model:

  • Dashen(1983). How to Get an Upper Bound on the Higgs Mass. Phys.Rev.Lett. 50: 1897.

numerical tests:

  • Montvay, Istvan; Münster, Gernot and Wolff, Ulli (1988). Percolation Cluster Algorithm and Scaling Behavior in the four-dimensional Ising Model. Nucl. Phys. B305: 143.
  • Jansen, Karl; Trappenberg, Thomas; Montvay, Istvan; Münster, Gernot and Wolff, Ulli (1989). Broken phase of the four-dimensional Ising model in a finite volume. Nucl. Phys. B322: 698.

finite size scaling:

  • Wolff, Ulli (2009). Precision check on triviality of $\phi^4$ theory by a new simulation method. Phys. Rev. D79: 105002. arXiv:0902.3100
  • Weisz(2011). Triviality of $\phi^4_4$ theory: small volume expansion and new data. Nucl. Phys. B846: 316. arXiv:1012.0404
  • Hogervorst(2012). Finite size scaling and triviality of $\phi^4$ theory on an antiperiodic torus. Nucl. Phys. B855: 885. arXiv:1109.6186
  • Siefert(2014). Triviality of $\varphi^4$ theory in a finite volume scheme adapted to the broken phase. Phys. Lett. B: to appear. arXiv:1403.2570

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