Why physics works: decoupling
The best example: Fermi's theory of beta decay
What "renormalisable" really means
Why high-energy physics is hard
Can a theory work at every energy?
Naturalness: why relevant couplings are a puzzle
Where to go next
Reading list
Exercises

Renormalisation Group VIII: Effective Field Theory

Part 8 of eight. previous: Running couplings · Course home · Glossary

We now have the machinery. This last part is about what it means. In short: the RG explains why we can do physics at all without knowing the final theory and it tells us which questions about very short distances the everyday world can and can't answer.

Why physics works: decoupling

Go back to the power counting of Part 5. A coupling $g$ with negative mass dimension, $[g]=-p<0$, is irrelevant. Its effect at energy $E$ is suppressed by $(E/\Lambda)^p$, where $\Lambda$ is the scale where it came from. Run the flow down from very high energy and every irrelevant coupling fades away. What's left at low energy is a short list, the relevant and marginal couplings, no matter what the high-energy theory was.

Picture it Tong puts it neatly: big things don't affect little things, which is why universities have no astrology departments. And little things mostly affect only slightly bigger things. To set a pot of mishti doi you need milk, sugar and a spoon of yesterday's doi, not quarks. The quarks matter, but only through a few numbers (the masses of protons and neutrons, the strength of chemical bonds) that were fixed long before anyone reached the sweet shop.

That's why chemistry doesn't need quarks and why the Navier-Stokes equation has two parameters instead of $10^{23}$. It's also why heavy particles decouple: a particle of mass $M$ affects physics at $E\ll M$ only by shifting the low-energy couplings, plus corrections suppressed by powers of $E/M$. This is the Appelquist-Carazzone theorem. The ladder of theories in Part 1 is the RG flow drawn as a picture.

🧠 Defn An effective field theory (EFT) is the most general Lagrangian allowed by the symmetries of the problem, written as a series in how big each term's dimension is:

\[ \mathcal L_{\rm eff} = \mathcal L_{d\le4} + \sum_i\frac{c_i}{\Lambda^{\Delta_i-4}}\,\mathcal O_i, \]

with pure numbers $c_i$ of size about one. It's valid for $E\ll\Lambda$ and to any given accuracy you only need to keep a few terms.

Picture it You already know an effective theory: the multipole expansion. From far away, any lump of charge looks like a point charge. Get a bit closer and you notice a small dipole correction, then a smaller quadrupole one, each fading faster with distance. You never need the exact shape of the lump to predict the field far away. You just keep as many terms as your accuracy needs. An EFT does the same thing for a whole theory, with energy playing the role of distance.

The best example: Fermi's theory of beta decay

In 1933 Fermi described a neutron decaying as a contact interaction: four particles meeting at one point, with strength $G_F$. Four fermion fields have dimension $4\times\tfrac32=6$, so $[G_F]=-2$. That's an irrelevant coupling. The measured value is $G_F=1.1664\times10^{-5}\ \mathrm{GeV}^{-2}$.

An irrelevant coupling carries a hidden message: its size tells you where the theory must break down. Fermi's collision rates grow like $G_FE^2$ and would break the rule that probabilities can't exceed one at around $1/\sqrt{G_F}\approx300$ GeV. So something new had to appear before then. It did: the $W$ boson, at $80$ GeV.

Picture it It's like the discovery of Neptune. Astronomers saw that Uranus wobbled slightly off its predicted path and from the size of the wobble they worked out that an unseen planet had to exist and roughly where. The size of $G_F$ was the "wobble" that pointed to the $W$ boson decades before it was found.

$W exchange shrinking to Fermi's contact interaction$

At energies far below $m_W$, the $W$ propagator $g^2/(q^2-m_W^2)$ is basically a constant and the exchange looks like a point interaction. Fermi's theory is the full theory with the $W$ integrated out: one step of Wilson's RG.

Picture it Why does a heavy $W$ look like a contact force? Think of two people passing a very heavy medicine ball. They can only do it standing almost touching, because the ball can't be thrown far. From across the room it looks as if they simply push each other directly. A heavy particle can only travel a tiny distance, about $1/m_W$, so from far away its exchange looks like a push at a single point.

Integrating out the $W$ gives $\frac{G_F}{\sqrt2}=\frac{g^2}{8m_W^2}$. With the weak coupling $g=0.652$ and $m_W=80.377$ GeV, this gives $G_F=1.163\times10^{-5}\ \mathrm{GeV}^{-2}$, within 0.3% of the measured value (loop corrections account for the rest). Fermi's theory wasn't wrong. It was the correct effective theory below $m_W$, with its one number fixed by the physics above.

What "renormalisable" really means

Older textbooks treat renormalisability as a requirement: a sensible theory must only have couplings of zero or positive dimension. The Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) theorem sorts theories into three kinds:

super-renormalisable: every coupling has positive dimension (only finitely many diagrams diverge);
renormalisable: none has negative dimension (only finitely many kinds of infinity, so finitely many counterterms);
non-renormalisable: at least one has negative dimension (you would need infinitely many counterterms to remove the cutoff completely).

Part 2's formula $D=d-\frac{d-2}2N+(d-4)V$ is where this comes from. The effective-theory view, which Melo's notes argue for carefully, changes the verdict on the last kind:

"Non-renormalisable theories have infinities that can't be removed." False. They need infinitely many counterterms, one per term in the Lagrangian, but only if you insist on $\Lambda\to\infty$.
"Non-renormalisable theories can't predict anything." Only if you want them to work at every energy. At $E\ll\Lambda$ you keep a few terms and predict to any fixed accuracy with finitely many numbers. Fermi's theory, the theory of pions and gravity at everyday energies all work this way.
"Renormalisable theories are fundamental." No. They're what any theory looks like far below its cutoff, because everything else is irrelevant. The Standard Model being renormalisable tells you its new-physics scale is far above what we've tested. It doesn't tell you there's no new physics.

Why high-energy physics is hard

Universality is wonderful if you care about long distances, because it lets you ignore the details. Tong points out that it's a curse if you care about short distances.

Picture it Imagine many different rivers all flowing into the same lake. Standing at the lake, you can't tell which river a drop of water came from. That's universality seen from the other side: many different short-distance theories flow to the same long-distance physics, so measuring the long-distance physics can't tell you which one is true. Tong jokes that nature seems to be in on the conspiracy: black-hole horizons hide the most extreme gravity from view and cosmic inflation wiped out traces of whatever came before it.

Otaku corner This is the Attack on Titan situation. Humanity lives inside the walls (low energies) and badly wants to know what's outside (the Planck scale). Universality is the wall: from inside, many different outsides look exactly the same. The only way to find out is to climb higher, which in physics means building bigger colliders.

This is why reaching new physics usually needs higher energy, not just more precision and why an irrelevant coupling with a measurably large effect (like $G_F$) is such a valuable clue.

Can a theory work at every energy?

Can a theory be valid at all energies, so that the cutoff really can go to infinity? Run the RG backwards, towards short distances and ask where it goes (Melo, "The continuum limit").

Picture it It's like tracing a river upstream. Either you reach a source, a spring the river starts from (a fixed point), or the river gets wilder and wilder until the map simply breaks down (a Landau pole). A theory that works at every energy is one whose river has a source.

Trivial. In $\phi^4$ theory in four dimensions the coupling grows towards short distances and hits a Landau pole (Part 7). The only way to push the cutoff to infinity and keep the pole above it is to set the low-energy coupling to zero, which leaves a free theory. The same is believed of QED. Neither is a problem in practice, because the poles are at absurd energies ($10^{277}$ GeV for QED with the electron alone). These are perfectly good effective theories.
Asymptotically free. In QCD the coupling shrinks to zero at short distances, heading into the Gaussian fixed point. That's a river with a source, so the theory makes sense at all energies. Proving this rigorously for Yang-Mills theory is one of the Clay Millennium Prize Problems.
Asymptotically safe. The source is an interacting fixed point. We met one in Part 6: in $d<4$, a theory whose short-distance end is Wilson-Fisher. Weinberg's asymptotic safety idea asks whether gravity, whose Newton constant is irrelevant ($[G_N]=-2$), could have such a fixed point in four dimensions. Nobody knows yet.

Naturalness: why relevant couplings are a puzzle

Irrelevant couplings are harmless because they shrink as you go to lower energies. Relevant ones do the opposite: they grow towards low energy, so a small relevant coupling at low energy needs very careful tuning at high energy. That's the hierarchy problem from exercise 1.4, now in RG language.

Picture it A small relevant coupling is like balancing a full bhanr of cha on one finger inside a moving Kolkata bus. Or a pencil balanced on its tip. It can stand there, but only if everything is arranged just right and the slightest nudge sends it falling. A light Higgs boson with a cutoff near the Planck scale is a pencil balanced to about thirty decimal places.

Apart from the vacuum energy, the Higgs mass term is the Standard Model's only relevant coupling and its quantum corrections grow like $\Lambda^2$. For the Higgs to be light compared with a cutoff near the Planck scale, the bare parameter must cancel the loop to about thirty digits. You saw this yourself in the Part 2 widget. The cosmological constant, a coupling of dimension four, is even worse, off by about 120 powers of ten. Whether these are clues about new physics or just accidents is one of the big open questions. The RG doesn't answer it, but it's the reason it's a question at all.

Where to go next

You now know the core of the renormalisation group. Some directions from here:

The functional RG. Instead of following a few couplings, follow the whole potential or action as it flows. Melo's notes derive the "local potential approximation", an equation for $V(\phi)$ that finds the Wilson-Fisher point directly in $d=3$, with no $\epsilon$ expansion. The modern version is the Wetterich equation.
Conformal field theory. A fixed point doesn't change when you zoom and usually it has an even bigger symmetry called conformal symmetry. That symmetry is so powerful that the conformal bootstrap pins down the 3D Ising exponents to six digits. That's where the $\nu=0.629971$ in Part 6 comes from. Tong's §3.6 is a gentle first look.
Continuous symmetries. Magnets whose spins can point in any direction, the Mermin-Wagner theorem and the Kosterlitz-Thouless transition, where a whole line of fixed points appears: Tong's Chapter 4.
Gauge theories. The QCD beta function, including the ghost fields: Peskin §16.5 to §16.7.

Reading list

D. Tong, Statistical Field Theory, Cambridge lecture notes. The best free introduction to the magnet side (Parts 3 to 6 here) and the source of most of this course's intuition.
J. F. Melo, Introduction to Renormalisation, arXiv:1909.11099. Short and conceptual and the backbone of the field theory side here. It has one sign slip in its low-energy QED discussion (flagged in Part 7).
L.-F. Li, Introduction to Renormalization in Field Theory, arXiv:1208.4700. Short and concrete on counterterms, BPH subtraction and divergences inside divergences (used in Part 2).
D. V. Shirkov, "Fifty years of the renormalization group", CERN Courier (2001). The history, told by one of the people who built it.
M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory: Ch. 6.3 (Wick rotation), Ch. 10 (infinities and counterterms) and Ch. 12 (Wilson, Callan-Symanzik, running).
A. Das, Field Theory: A Path Integral Approach, 3rd edition: Ch. 4 (Euclidean rotation), Ch. 10 (effective action and the loop expansion), Ch. 14 and 15 (critical exponents, the Ising model and duality).
M. Kardar, Statistical Physics of Fields, Ch. 3 to 5. The standard textbook version of Parts 4 to 6.
K. G. Wilson and J. Kogut, "The renormalization group and the ε expansion", Physics Reports 12 (1974) 75. Where it all started and still readable.

Exercises

🤔 Problem 8.1. Use $G_F/\sqrt2=g^2/(8m_W^2)$ with $g=0.652$ and $m_W=80.377$ GeV to work out $G_F$. Then estimate the energy at which Fermi's theory, without the $W$, would have to fail.

Show solution

$G_F=\sqrt2\,g^2/(8m_W^2)=\sqrt2(0.4251)/(51683)=1.163\times10^{-5}\ \mathrm{GeV}^{-2}$ (the measured value is $1.1664\times10^{-5}$). Collision rates grow like $G_FE^2$, so they become too large at $E\sim G_F^{-1/2}\approx293$ GeV. More careful bounds move this by a factor of a few. The real theory changed at $m_W=80$ GeV, safely below.

🤔 Problem 8.2. The simplest term you can add to the Standard Model has dimension 5 (the Weinberg operator) and it gives neutrinos a mass $m_\nu\sim v^2/\Lambda$, with $v=246$ GeV. If $m_\nu\approx0.05$ eV, what is $\Lambda$?

Show solution

$\Lambda\sim v^2/m_\nu=(246)^2/(5\times10^{-11})\approx1.2\times10^{15}$ GeV, up to factors of order one. Tiny neutrino masses may be our first sight of an irrelevant term coming from physics near $10^{15}$ GeV. It's exactly the same logic as reading off the $W$ scale from $G_F$.

🤔 Problem 8.3. Newton's constant has $[G_N]=-2$, with $G_N=1/M_{\rm Pl}^2$ and $M_{\rm Pl}=1.22\times10^{19}$ GeV. How much are quantum-gravity effects suppressed at the LHC ($E\sim10^4$ GeV)? Is gravity being "non-renormalisable" a problem for the LHC?

Show solution

The relative size is $\sim E^2/M_{\rm Pl}^2=(10^4/1.22\times10^{19})^2\approx7\times10^{-31}$. General relativity works perfectly well as an effective theory below the Planck scale and people have even calculated its small quantum corrections to Newton's law. Being non-renormalisable only means the theory has to be replaced by something else near $M_{\rm Pl}$. It's a statement about $10^{19}$ GeV, not about the LHC.

🤔 Problem 8.4. (Putting it together.) In a few sentences each, explain in RG language: (a) why the 3D Ising model and a liquid-gas critical point share exponents; (b) why the Higgs mass is "unnatural" but the electron mass isn't; (c) why the fine-structure constant is $1/137$ at low energy and $1/128$ at $m_Z$.

Show solution

(a) Both flow to the same fixed point, Wilson-Fisher with $N=1$ in three dimensions. The exponents are growth rates of the flow at that point and the differences between the two systems are irrelevant directions that fade away. (b) A scalar mass term is relevant and depends on the cutoff like $\Lambda^2$, so a light Higgs needs fine-tuning. The electron mass is protected by a symmetry (chiral symmetry): if $m_e$ were zero the theory would gain a symmetry, so its corrections are proportional to $m_e$ itself and only depend on $\Lambda$ through a logarithm. (c) The QED coupling runs with a positive beta function, because virtual charged particles screen the charge. Probing at shorter distances (higher $Q$) means less screening, so $\alpha$ looks larger. The size of the effect comes from adding up the loops of every charged particle lighter than the $Z$.

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