Effective Field Theories

This note introduces effective field theories (EFTs) within the framework developed in On theories, symmetries and gauge. The guiding idea is that EFTs do not require a new notion of theory: they are theories in exactly the same sense, but with a scale‑dependent selection mechanism.
Roughly speaking, an EFT is a theory whose criteria $E$ are designed to distinguish fields only up to a given resolution, ignoring or averaging over finer details.

EFTs as Scale‑Restricted Theories

Recall the working definition:

A theory is a mechanism that selects a subset $S \subset S = Γ (M, E)$ by means of a criterion $E$ .

An effective field theory is specified by:

the same (or a reduced) bundle of fields $E \to M$ ,
together with a family of criteria ${E_{Λ}}$ , indexed by a scale $Λ$ .

Here $Λ$ represents an energy, momentum, or length scale. The defining property is:

$E_{Λ}$ is insensitive to variations of the fields at scales shorter than $Λ^{- 1}$ .

Accordingly, the selected set

S_{Λ} = {Φ \in Γ (M, E) ∣ δ E_{Λ} [Φ] = 0}

characterizes fields only up to coarse‑graining. Different microscopic configurations may be indistinguishable at the level of the EFT.

the selection mechanism is modified,
the field space may be reduced,
but the notion of theory itself is unchanged.

Example 1 Revisited: Zero‑Dimensional EFT

Recall Example 1 in on theories, symmetries and gauge, where

S ≅ R^{n}, E [x] = \frac{1}{2} | x |^{2} - ⟨ b, x ⟩,

and the selected field is $x = b$ .

Splitting Scales

Decompose the field as

x = (x_{L}, x_{H}) \in R^{k} \oplus R^{n - k},

where $x_{H}$ represents “heavy” components. Consider a modified criterion

E_{Λ} [x_{L}, x_{H}] = \frac{1}{2} | x_{L} |^{2} + \frac{1}{2} Λ^{2} | x_{H} |^{2} - ⟨ b_{L}, x_{L} ⟩ .

For large $Λ$ , variations in $x_{H}$ are strongly suppressed. The Euler–Lagrange equations give

x_{H} = 0, x_{L} = b_{L} .

Interpretation

This result should not be read as an arbitrary truncation of the solution, nor as a voluntary decision to “keep only the first components.” Rather, it reflects a genuine change in the selection mechanism.
The effective criterion $E_{Λ}$ defines a theory that is dynamically insensitive to variations in the heavy directions:

The equations of motion force $x_{H}$ to a fixed, trivial configuration.
This occurs independently of any microscopic value it may have had.
In this sense, information about the heavy components (the last ones) is lost at the level of the criterion, not discarded at the level of solutions. The EFT therefore remains a fully predictive theory, but only for the light variables $x_{L}$ ; it is structurally incapable of resolving or distinguishing different microscopic values of $x_{H}$ .

Here is a specific, concrete realization of Example 1:

The Scenario: Optimizing a Chemical Reactor

Imagine you are an engineer trying to configure a chemical reactor to achieve a specific production target.

The Space ( $M = {p}$ ): The "spacetime" is just the single reactor unit itself. There is no spatial variation (everything happens in this one pot).
The Field ( $x \in R^{n}$ ): The vector $x$ represents the deviation from the factory default settings.
- $x = 0$ means "all dials at default."
- $x \neq 0$ means we are tweaking the machine.
The Vector $b$ (The Source): This represents the market demand or external requirement. The market demands a specific mix of outputs (Yield, Purity, etc.) that corresponds to a machine configuration $b$ .
- We want to set $x = b$ to perfectly satisfy the market.
The Cost Function ( $E$ ): The variational principle represents the Energy Cost or "Effort" required to maintain the settings, minus the "Profit" from satisfying the demand.
$E [x] = \underset{Cost of tweaking}{\underset{⏟}{\frac{1}{2} ∥ x ∥^{2}}} - \underset{Gain from matching demand}{\underset{⏟}{⟨ b, x ⟩}}$
Minimizing $E$ balances the difficulty of changing the settings against the need to satisfy the external source $b$ .

The Full Theory (Microscopic View)

In the "full" theory, we assume we can tweak every knob on the machine.

Let's say our state vector $x$ has two components:

x = (\begin{matrix} x_{L} \\ x_{H} \end{matrix})

$x_{L}$ (Light Field): The Input Valve. It is a lightweight plastic dial. It is very easy (low energy cost) to turn.
$x_{H}$ (Heavy Field): The Reactor Wall Thickness. To change this, you have to physically forge a new steel vessel. It is incredibly "stiff" or expensive (high energy cost).

If the market demands a configuration $b = (b_{L}, b_{H})$ (e.g., "Open the valve AND make the walls thicker"), the full theory says:

Mechanism: Minimize $E [x]$ .
Selected Field: $x_{L} = b_{L}$ and $x_{H} = b_{H}$ .
Result: You turn the valve and you spend millions rebuilding the reactor walls.

The Effective Field Theory (EFT View)

Now, suppose we are operating at a "low energy scale." We have a limited budget, or perhaps we need to make changes on a timescale of seconds, not months. We cannot afford to forge new steel walls.

Here, the parameter $Λ$ represents the Stiffness or Cost Multiplier of the heavy component.

####### 1. The Separation of Scales

We rewrite the cost function (the selection mechanism) to reflect reality: changing the wall thickness ( $x_{H}$ ) is exponentially harder than turning the valve ( $x_{L}$ ).

E_{Λ} [x_{L}, x_{H}] = \underset{Easy dial}{\underset{⏟}{\frac{1}{2} (x_{L})^{2}}} + \underset{Expensive rebuild}{\underset{⏟}{\frac{1}{2} Λ^{2} (x_{H})^{2}}} - \underset{Valve demand}{\underset{⏟}{⟨ b_{L}, x_{L} ⟩}}

Note specifically what changed:

The Stiffness ( $Λ^{2}$ ): A huge penalty factor $Λ^{2}$ (where $Λ ≫ 1$ ) is attached to the heavy field $x_{H}$ .
The Source Filter: The term $- ⟨ b_{H}, x_{H} ⟩$ has been dropped (or is negligible compared to $Λ^{2}$ ). This is the crucial EFT step: The selection mechanism stops "listening" to demands that require high-energy responses. The theory effectively says, "I don't care if the market wants thicker walls ( $b_{H}$ ); at this budget scale, that demand is invisible to us."

####### 2. The Solution (Selected Fields)

We solve for the minimum ( $δ E_{Λ} = 0$ ):

For the Light Field ( $x_{L}$ ):
$\frac{\partial E}{\partial x_{L}} = x_{L} - b_{L} = 0 ⟹ x_{L} = b_{L}$
The valve is set exactly as requested.
For the Heavy Field ( $x_{H}$ ):
$\frac{\partial E}{\partial x_{H}} = Λ^{2} x_{H} = 0 ⟹ x_{H} = 0$
(Even if we had kept a small source term $b_{H}$ , the solution would be $x_{H} = b_{H} / Λ^{2}$ , which vanishes as $Λ \to \infty$ ).

Why This Matters

In this specific example:

What is $b$ ? $b$ is the ideal configuration requested by the external world (the source).
Why the split? We split $x$ into $x_{L}$ and $x_{H}$ because the physics (or economics) of the machine imposes vastly different costs on them.
The EFT Consequence: The EFT tells us that for all practical purposes at low energy, the variable $x_{H}$ does not exist as a degree of freedom. It is "frozen out."

The "Theory" (selection mechanism) has changed from:

"Find the best settings for everything."

To:

"Find the best settings for the valve, assuming the walls are fixed."

We did not just "ignore" $x_{H}$ ; the mechanism $E_{Λ}$ forced it to zero dynamically due to the scale $Λ$ . This is accurate to the physics of the machine: if you push on the reactor wall with your hand (low energy probe), it doesn't move. It effectively isn't a variable you can play with.

Example 2 Revisited: Coarse‑Grained Localization

Recall Example 2, where the criterion singles out a field supported sharply on the line $x = 2$ .

Sharp vs Effective Localization

The original functional involves a Dirac delta:

δ (x - 2) .

This corresponds to infinite resolution in the $x$ -direction.

An EFT description replaces this by a smeared distribution

δ_{Λ} (x - 2),

localized within a width $Λ^{- 1}$ . The effective criterion becomes

E_{Λ} [f] = \iint (x - 2)^{2} f^{2} d x d y + \iint δ_{Λ} (x - 2) (f - 1)^{2} d x d y .

Selected Fields

The solutions are no longer sharply supported on $x = 2$ , but rather fields that:

are approximately $1$ in a tubular neighborhood of the line,
decay away from it over a scale $Λ^{- 1}$ .

Different microscopic profiles correspond to the same effective solution.

Thus:

the EFT selects an equivalence class of fields,
sharp localization is replaced by scale‑dependent indistinguishability.

EFTs and Symmetry

A central principle:

An effective criterion must respect the active symmetries observed at the given scale.

Gauge symmetries and exact spacetime symmetries therefore constrain the allowed terms in $E_{Λ}$ . Higher‑order terms are permitted only if they are invariant under the same symmetry group.

This fits naturally with the discussion in On theories, symmetries and gauge:

symmetries constrain admissible criteria,
EFTs encode this constraint approximately but systematically.

Conceptual Summary

Within this framework:

An EFT is a perfectly ordinary theory in the sense of field selection.
What is new is that the criterion $E$ is scale‑dependent and deliberately incomplete.
EFTs describe not single fields, but coarse‑grained equivalence classes of fields.

Seen this way, effective field theories are not a compromise—they are the natural expression of the fact that physical distinction is always made at finite resolution.