Musings

Adams et al study the leading higher-dimension operators which, generically, occur in effective field theory. These appear in the effective lagrangian in the form $$\frac{c_i}{M^{d_i-4}} O_i$$ where $M$ is the cutoff on the effective field theory, $d_i$ is the engineering dimension of the Lorentz-invariant operator, $O_i$. $c_i$ is a dimensionless coupling constant, about which we typically assume that not much can be said, except that we expect it to be $\sim \mathcal{O}(1)$.

The claim of Adams et al is that more, in fact, can be said about the $c_i$. They claim that the coefficients of the leading irrelevant operators are such that the classical stress-energy tensor computed including the contribution of these terms satisfies the dominant energy condition. Only effective theories satisfying this constraint can be completed to local quantum field theories (with or without gravity) in the UV.

They don’t use the phrase “dominant energy condition.” Instead, they talk about superluminal propagation of signals in the background of a nontrivial classical solution. These are, however, really the same condition.

Propagation in a nontrivial translationally-invariant background, given by summing a geometric series in 4-point forward scattering amplitudes.

They then go on to argue that the same positivity constraints on the coefficients $c_i$ follow from looking at the dispersion relations satisfied by the tree-level amplitude for forward scattering. The connection between these two points of view is illustrated in the diagram at left. The propagator in a nontrivial background, which determines the speed of signal propagation, is determined by summing a geometric series of forward-scattering diagrams off the background. (Here, we’re specializing to a translationally-invariant background, so that the momentum is conserved.)

The thing that has confused me is why it is legitimate to treat these leading irrelevant operators as part of the classical action. In the effective field theory, the contribution of nonrenormalizable interactions to scattering occurs at the same order in the derivative expansion as loops involving renormalizable interactions. In a generic effective field theory, it would be incorrect, to this order, to ignore the loop corrections to the forward scattering amplitude. Equally, it would be incorrect to include the nonrenormalizable operator as a part of the classical action, for the purpose of computing the speed of signal propagation.

Of course, in very particular theories, this may not be true. Consider Maxwell theory,

In this theory, the renormalizable part of the theory is free. The leading nontrivial scattering effects come from treating $c_1,c_2$ at tree level.

Another example is the effective Lagrangian for the Goldstone boson of a spontaneously-broken $U(1)$ global symmetry,

(They also consider a slightly more general Lagrangian, without a $\pi\to -\pi$ symmetry, which contains a $(\partial\pi)^2 □ \pi$ interaction. More on that below.)

In both of these theories, the leading interactions come from the nonrenormalizable terms we’ve highlighted. Studying such examples is interesting, but may not give an accurate picture of the state of affairs in a general effective field theory.

That was pretty much the state of my understanding when I started talking to Ben. It became clear that a good model to test these ideas was the interacting version of (2), the chiral Lagrangian for the spontaneous breaking of $SU(2)\times SU(2)\to SU(2)$.

where $$\pi = \frac{1}{2}\left(\array{\arrayopts{\colalign{center}} \pi_0 & \pi_1-i\pi_2\\ \pi_1+i\pi_2& -\pi_0}\right),\qquad \Sigma=e^{2i\pi/f}$$

and $l_1,l_2$ are the leading 4-derivative couplings, whose sign Adams et al would like to bound.

Note that we’ve turned on a nonzero pion mass. This is actually necessary to be able to apply dispersion relations to the forward scattering amplitude. As soon as you turn on interactions, there are 2-particle cuts (in both the $s$- and $u$-channels), which extend all the way down to zero when the particles are massless. At the end, we can try to extrapolate our answers down to zero pion mass (which is where a constant pion gradient, the configuration of interest to Adams et al, is a solution to the equations of motion).

The forward scattering amplitude for, say, $2\pi_0\to 2\pi_0$, is an even function of $\tilde{s}= s-2m^2$.

where $$J(y)= 2+x\log\left(\frac{x-1}{x+1}\right),\qquad x=\sqrt{1-4m^2/y}$$ and $$C = 96\pi^2 \left(l_1(\mu) + l_2(\mu) - \frac{1}{32\pi^2}\log(m^2/\mu^2) \right)$$ is an RG-invariant combination of the couplings. To obtain this forward scattering amplitude to order-$s^2$, we had to sum a set of graphs in which the 4-derivative terms in (2) enter at tree-level, while the other terms appear to 1-loop.

The first thing to notice, of course, is that there’s no meaning to trying to bound the sign of the couplings, $l_i$, themselves. They are additively renormalized, and can have either sign, depending on what scale you evaluate them. What we can bound is the RG-invariant quantity, $C$. Applying standard dispersion-relation arguments, one finds

Repeating the same analysis for $A_{\pi_+\pi_0\to\pi_+\pi_0}(t=0)$, one obtains

where $$C' = 48\pi^2\left(l_2(\mu) - \frac{1}{48\pi^2}\log(m^2/\mu^2)\right)$$ is RG-invariant. From the sign of the $\beta$-functions, we see that the running couplings increase as we flow to the IR. The “bare” couplings, $l_i(\mu=\Lambda_{\chi SB})$, could perfectly well start out negative, and still satisfy the bounds on $C,C'$ by flowing to suitable positive values, $l_i(\mu= m)$, in the IR. (Note that $l_1(\mu=m)$ can be negative, and indeed is likely negative in the real world.)

Note that, in order to get these results, we had to include loop-contributions from the two-derivative terms in the action. In other words, the computation is not a classical one. We may, in the end, obtain a bound on some RG-invariant quantities. But we do not expect (and do not obtain) a constraint on the signs of the “bare” couplings in the classical action.

The same remarks, obviously, hold in more general theories. The leading higher-derivative terms in the action should not be thought-of as part of the classical action. Rather, they contribute to physical processes at the same order as loop corrections. This applies, whether one is computing scattering amplitudes, or studying propagation is a nontrivial background.

As another example of this, consider, instead of Maxwell Theory (1), QED with $F^4$ interactions. The contributions to light-by-light scattering (equivalently, the speed of light in a constant elecromagnetic field background) include not just the “classical” $F^4$ terms, but also electron loops. Even at weak coupling, the latter dominate over the former, for $m_e\ll \Lambda$.

One of the most interesting applications of their work is an attempt to kill the Dvali-Gabadadze-Porrati model, a plausible-sounding braneworld model. In the DGP model, there is both Einstein gravity on the brane and in the 5D bulk.

In a certain decoupling limit, the theory reduces to a single 4D scalar, with Lagrangian $$L = 3 (\partial \pi)^2 - \frac{(\partial\pi)^2 □ \pi}{\Lambda^3} + \mathcal{O} (\partial^m(\partial^2\pi)^n)$$

where $\Lambda\sim M_5^2/M_4$ is held fixed, as we send the 4D and 5D Planck masses to $\infty$.

This theory looks nontrivial, but all the interaction terms vanish by the equations of motion. So they do not contribute to on-shell scattering. The S-matrix is trivial, and one does not obtain any constraint on the theory by considering this limit. (That’s not to say that the DGP model makes sense; just that you can’t kill it this way.)

The $□ \pi$ factor in the above interaction vanishes when applied to an external line in any scattering amplitude. However, as they compute in their paper, there are nontrivial contributions to scattering, where the $□$ hits only internal lines in the graph. Normally, such interaction terms can be shifted away by field redefinitions, but — in this case — not in a way compatible with the shift symmetry, $\partial_\mu\pi\to \partial_\mu\pi+c_\mu$, which they wish to maintain. At tree-level, they get a nonvanishing 4-point scattering amplitude, $$A(s,t) = \frac{1}{\Lambda^6}(s^3+t^3+u^3)+\dots$$ This tree-level amplitude is nontrivial, but vanishes in the forward direction, not just to $O(s^2)$, which already causes problems for the dispersion relation, but to all orders $$s^3+t^3+u^3 = 3 s t u + (s+t+u)(s^2+t^2 +u^2 - s t - t u - u s) = - 3 s t (s+t)$$ (A little thought shows that all tree diagrams will have this kind of structure.)

Having said this much, there is still a loophole in this argument. To apply dispersion relations, we need to turn on a small mass, so as to separate the cuts which, in the massless limit, extend all the way down to $s=0$. When you do this, you regenerates a nonzero contribution to the forward scattering amplitude $$A(s, t=0) \propto \frac{m^2}{\Lambda^6}(s-2m^2)^2 +\dots$$ Presumably, everything now goes through. We can compute the dispersion relation. The coefficient of $s^2$ is positive, as it must be, but vanishes in the $m^2\to 0$ limit, which still violates unitarity.

It sure looks, then that the DGP model is ruled out.

These look just like the constraints (5),(6) above, except that the latter:

• involve the renormalized couplings, $l_i(\mu=m)$,
• have a finite shift on the right-hand side of the inequality.

“Bollocks!” you say, “I can absorb a finite shift like that in a change in renormalization scheme for these couplings.” Up to an astute choice of such a renormalization scheme, all I seem to have done is to replace the classical couplings, by their renormalized counterparts.

But that’s because I’ve been holding out on you. There’s actually a third constraint

following from dispersion relations for $\pi^+\pi^+$ scattering, that isn’t reproduced by the classical superluminality analysis. The true shape of the allowed region is shown in the figure below.

The region allowed by the dispersion relation analysis is above all the lines. In Gasser & Leutwyler’s notation, $\overline{l}_1=C-C'$ and $\overline{l_2}=C'$.

The extra disallowed region is depicted in burnt-orange. Some wags have suggested that the original disallowed region should have been drawn in crimson, instead of pale blue.