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EFT-hedron: the landscape within the EFT quagmire

Physicists have clearly worked more intensely during the Christmas and New Year break than the moderators. Consequently, the hep-th archive overwhelmed us with 168 new preprints hours ago (whose identifiers still start with the year '20), including 70 new hep-th-primary papers, 35 new papers cross-listed to hep-th, and 63 replacements. That's some 200% above the daily average.

You may ask which of the new papers is going to be most influential. Well, you have to be cautious, we don't really know. But you may also ask which of them is the longest one. A good idea is to search for -Hamed on the page and you will quickly find this new paper:

The EFT-Hedron
by Nima Arkani-Hamed, Cchu-Čen Huang, Ju-tin Huang (IAS, Caltech, Taiwan)
If you care whether you should write it with a hyphen, it is spelled 3 times with and 61 times without. ;-) Note that even though this paper appeared on January 1st in the morning, it's the 66th paper on that day and is still counted in 2020, unlike the Weak Gravity Conjecture that appeared as the first hep-th paper of 2006.

Well, two days ago, I had a fun two-hour-long chat with one of the authors whose name isn't Huang (not only about the Millenials' lack of independence and ambitions) and he prepared me for that paper. Sometime in 2021, a paper of mine could appear and you might like it. Well, it's January 1st in the morning and it's here. ;-) Unless he meant a different paper that will appear later, of course.

They describe a huge new class of very explicit inequalities constraining the coefficients of higher-dimensional operators \(a_{\Delta, q}\) in any consistent effective field theory (EFT). While the words "string" and "gravity" appear 60 and 27 times in the paper, respectively, they discuss general enough EFTs. The first focus is on the two-to-two scattering in spinless EFTs where only the \(s\)-channel (and no \(u\)-channel) contributes. The geometric effect of all these inequalities is to cut an allowed polytope in the space of the (properly reparameterized) coefficients of the operators, the EFT-hedron.

Note that in 2013, Arkani-Hedron and Trnka introduced the amplituhedron and Nima and pals also later dragged an older mathematical concept of an associahedron to particle physics. I chose a more technical terminology for the authors; note that a hedron is a shape while hamed is a praiser in Arabic. ;-) Now, one of the fathers co-introduced this EFT-hedron today while the other one, Trnka (CZ-US), released a 72-page-long paper on the gravituhedron, one about the gravitational NMHV amplitudes.

But let's return to the EFT-hedron. We are dealing with the effective field theories (where we start by not caring about renormalizability because the expectation is that they are only used at low enough energies – but with the desire to get an arbitrary precision). The general 4-particle scattering amplitude is\[ {\mathcal A} (s,t) = \sum_{\Delta, q} a_{\Delta, q} s^{\Delta - q} t^q \] which is a Taylor expansion organized according to the total degree \(\Delta\) and the distribution of these powers-of-energies between the \(s,t\) Mandelstam variables, given by the \(t\)'s exponent called \(q\). These \(a_{\Delta,q}\) coefficients are the coefficients of the higher-dimensional operators and are being shown to be inside the EFT-hedron.

Fine, now you try to reorganize the amplitude by fixing \(t\) while changing \(s\) and therefore \(u\). You will notice that the amplitude gets poles as a function of \(s\) and \(u\). These poles are "somewhere" because they come from terms of the type\[ \frac{1}{s-M^2}, \qquad \frac{1}{u-M^2} \] and this \(M^2\) has to be integrated over. Now, and you should look at the equation (1.3) to understand what I am saying, before you integrate over \(M^2\), these poles are also multiplied by a particular function of \(M^2\), the probabilities \(p_l(M^2)\), which must be positive, as well as the Gegenbauer polynomials \(G_l(x)\) where \(x=1+2t/M^2\). These polynomials have various shapes which are labeled by \(l\) and you must sum over \(l\). I should probably reproduce the equation (1.3) here; it's pretty similar to a pole-like expansion of the Veneziano amplitude in basic chapters of string theory textbooks.\[ \begin{align} {\mathcal A} (s,t)&={\mathcal A}_0 (t)+{\mathcal A}_1 (t)s+\\ &+\!\!\int dM^2 \sum_l p_l(M^2) G_l(1\!+\! \frac{2t}{M^2}\!) \zav{\! \frac{1}{s\!-\!M^2} + \frac{1}{u\!-\!M^2} \! } \end{align} \] Excellent. The paper focuses on inequalities and how they restrict the allowed space of the coefficients. The three principles or conditions are referred to as
vacuum stability; causality; unitarity.
This is what all the constraints come from. In the conventions spoken not only around the Arkani-Hedron, these words mean
positivity of the mass (of what propagates); the subluminal speed of what propagates; the non-negativity of probabilities.
The word "unitarity" has other meanings; the unitary matrix primarily has to obey \(U g U^\dagger = g\) where \(g\) defines some metric on the space and may be indefinite when the generalization of the matrix is called "pseudounitary" (negative entries appear with the bad ghosts in the spectrum). OK, Nima and others don't really care that some general identity like that holds; they mean the absense of bad ghosts (when we look at the spectrum through the virtual particles that influence other particles' scattering) i.e. the very rule that \(g\) is positively definite. Fine. The equation for the amplitude above is a wonderful place to translate the somewhat vague "inequality-like conditions" to something like mathematics. The vacuum stability, causality, and unitarity translate to\[ \begin{align} M^2 & \gt 0\\ x & \gt 0\\ p_l(M^2) & \gt 0 \end{align} \] The condition \(x\gt 0\) means \(1+2t/M^2 \gt 0\); the Guggenheim Museum polynomial is a multiplicative correction that could change the speed to a superluminal one. The negative \(M^2\) would mean that tachyons are running in the channel and they may destabilize the vacuum, too; the negative \(p_l\), the main factor in the residues, would mean that the thing running inside is a negative-probability bad ghost. Great. What's left is to do some 122 pages of reasoning and calculations and determine what these conditions – imposed on many functions inside an expansion – mean for the coefficients of the higher-dimensional operators. And the answer is cool and you will like it if you have a positive emotional attachment to concepts such as minors, Hankel matrix, and a Vandermonde matrix in the linear algebra.

In particular, the "positive probability" (unitarity) conditions will tell you that if you write down a matrix composed of entries called \(f\) which are just some properly integrated functions \(p_l(M^2)\) that we mentioned above, all the minors of this matrix have to be positive. Minors are some determinants of a submatrix. A matrix whose minors are universally positive is called "a totally positive matrix". (Similarly, when all minors are educated to be positive towards all politically correct falsehoods, what you get is a "positively totalitarian civilization living in the Matrix". I made this analogy myself.)

Geometrically, the intersection of all the regions defined by these inequalities is the EFT-hedron. A damn polytope that looks very analogous to other A-hedrons of Dr Arkani-Hedron and his collaborators, followers, and imitators. At some level, it must be a shocking similarity to Nima others. At another level, I find this similarity both ill-defined and unsurprising. It's a bit ill-defined because the previous hedrons worked with rather different variables so what was left was just some determinant-like products and inequalities (but in the EFT-hedron paper, there are no spinorial or twistorial variables); and it's a bit unsurprising because in both cases, they were dealing with similar conditions restricting similar theories.

Don't get me wrong. I do find something very surprising, especially in this EFT-hedron case. And it's the very fact that you get a polytope – that in the variables that are rather easy to write down, the boundaries are "flat" or linear spaces. Note that a determinant is an \(N\)-th degree expression in the entries of an \(N\times N\) matrix in general; but it is at most linear in each entry because of the antisymmetrization that is included in the definition of a determinant. The higher-order factors in the same variable seem to prohibited. Do they clearly understand why this ends up being the case? Why aren't the boundaries of the EFT-hedron curved? Is that some old consistency condition that has a physical meaning, a new physical one, or a completely mysterious new mathematical condition that hasn't been translated to a straightforward intuition? (Update: Nima reminded me that in general, the boundaries are curvy.)

I find the apparent ban on the higher-order "not anti-symmetrized" factors to be mathematically analogous to the presence of \(p\)-forms in the spectrum of a supergravity theory. At the end, regardless of the value of \(p\), the \(p\)-form fields carry the spin one. They can't have a higher spin along an axis exactly because the index for the direction/axis can't be repeated (due to the antisymmetry). Only the graviton, a single one, is allowed to have \(j=2\). The rest must have a lower spin. So what I see is some analogy to the "problems with the higher spin" in field theories except that we must discuss "some kind of a spin within the space of coefficients" of the EFT. Sounds very intriguing, especially because I have believed for 25 years that it would be very profound when we learn how to treat world volumes, spacetimes, and configuration and moduli spaces on a more equal footing as different perspectives on a general theory.

The general point that the coefficients of EFTs are highly constrained by inequalities seems obvious to me. Most EFTs must produce inconsistencies. They imply ghosts or tachyons as ingredients of the "necessary completions" at higher energies. If you guarantee that an elementary particle in an EFT seems positively-probabilistic and subluminal, it doesn't necessarily prevent some implied bound states from being ghosts or tachyons (the implication is valid in UV-complete theories, however, because the absence of ghosts and tachyons may be proved "fundamentally" from the infinitely short distance scales). The class of EFTs seems "larger" than the class of UV-complete quantum field theories and the EFTs simply don't guarantee that "everything is OK at the shorter distances" (although it ultimately should be, even though the approximation, an EFT, may seem to tell you "anything goes"). But even with all the consistency conditions, the space of UV-complete, consistent quantum field theories is rather large and partially understood, right?

They did initial steps to fully translate the conditions into equations for more general theories, more general scattering involving a greater number of particles. Special structures and perhaps stronger conditions should exist when you include gravity (because the graviton has a higher spin, some of the exponents will be shifted and either much more constraining or much less constraining). If there is at least some health left in the research community, there will be a large number of followups that will gradually clarify a large number of these questions and more.

Some of the figures in the paper show the (flat) boundaries of the new EFT-hedron. You may see that the allowed EFT landscape is sometimes a really tiny, slim slice of the surrounding forbidden space that I called a quagmire to be separated from the analogous swampland (the latter is a set of nontrivially sick effective quantum theories with gravity).

The authors have noticed that the actual restrictions on the coefficients might actually be much more stringent than what has been found so far. In particular, they observe that the EFTs constructed from the fully consistent UV-complete quantum field theories don't just produce the coefficients inside the EFT-hedron. In fact, they seem to produce the coefficients at a point that is inside but very close to a boundary of the EFT-hedron. There might be a reason for that, perhaps a very explicit and very beautiful reason. It wouldn't be the first time when you may derive an inequality; but Nature and deeper mathematical logic actually wants to saturate (or nearly saturate) the inequality. For example, you may prove an inequality like \(D\leq 26\) or \(D\geq 26\) from the absence of ghosts in the spectrum of bosonic string theory but at the end, only \(D=26\) is the critical string theory that sort of works (except for the tachyons).

It's a bit ironic because they generally tend to assume naturalness which involves a "comparable magnitude of analogous enough coefficients". However, the functions measuring "the distance from the boundary of the EFT-hedron" seem to be much smaller than the distance of the generic points inside the EFT-hedron. Doesn't it mean that they have noticed some naturally generated unnaturalness? ;-)

Professionals who know how to do such things should get back to work and analyze the regions in the parameter space of EFTs' coefficients more fully. This is clearly another step in the "paradigm shift" of recent decades that I have repeatedly mentioned in recent months, namely the shift of physics from randomly guessing some hypotheses or theories taken from an "uncontrollably large and mysterious land of fantasies and possibilities" (and praying that the guess will be good enough) to systematically analyzing and grading the whole spaces and sets of hypotheses or theories that are mathematically possible. Here the authors are already clearly analyzing regions in a potentially infinite-dimensional space labeled by coefficients in a Lagrangian. When people really master quantum field theories, EFT, and especially quantum gravity and string theory, I think that they will consider many statements (including some very new statements) about these infinite-dimensional spaces and regions to be "intuitively obvious".

Stay tuned. Or, even better, try to tune other people's antennas to yours yourself.

If some equation above didn't fit into the green column, it was intentional. Now you may read the blog post for the second time but in the mobile template where the equations are short enough to be seen in their entirety.

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