Building Effective Lagrangians from Low-Energy Physics

The Construction Principle: "Everything Not Forbidden is Compulsory"

We will primarily adopt the "bottom-up" approach we discussed last time. We assume we don't know the ultimate high-energy theory. Instead, we look at the world at our energy scale, $E$, identify the particles and excitations that exist there, and write down every possible interaction between them that is not forbidden by fundamental principles like Lorentz invariance and gauge symmetry. This approach is sometimes called Gell-Mann's "Totalitarian Principle".

The result is a Lagrangian containing an infinite series of terms (operators), organized by their increasing complexity and suppressed by powers of the high-energy cutoff scale, $\Lambda$.

Here, ${\mathcal{L}}_{\text{leading}}$ contains the lowest-dimension operators (typically dimension 4, like the Standard Model Lagrangian), and the sums are over operators ${\mathcal{O}}^{(D)}$ of increasing mass dimension $D>4$. The dimensionless numbers $c_i^{(D)}$ are the Wilson coefficients, which encode the effects of the high-energy physics we are ignorant of. Our task today is to learn how to write down the possible operators ${\mathcal{O}}^{(D)}$.

Step 1: Identify the Degrees of Freedom

The first step is to list all the fields corresponding to particles or excitations with masses much smaller than the cutoff scale $\Lambda$. These are the "light" degrees of freedom that can be created and propagate in our low-energy experiments.

Let's start with a familiar example: the Standard Model itself, viewed as an effective field theory (SMEFT). We assume there is some new physics at a scale $\Lambda$ far above the electroweak scale. Below $\Lambda$, the degrees of freedom are simply all the known Standard Model particles.

The following video segment provides a quick and clear review of these degrees of freedom and their properties, which form the building blocks of our effective theory.

The key takeaway is that constructing an EFT begins with a concrete list of fields and their properties (spin, mass, charges). For SMEFT, these are the gluon, W/Z/photon, Higgs, and three generations of quarks and leptons.

However, the degrees of freedom are not always fundamental particles. In many systems, particularly in condensed matter, they are emergent collective excitations. A classic particle physics example is Chiral Perturbation Theory ($\chi \text{PT}$), the EFT for the strong interaction at very low energies.

This example is profound. It shows that the "degrees of freedom" in an EFT are not necessarily the microscopic constituents, but rather the lightest excitations available at the energy scale of interest.

Step 2: Impose the Symmetries

Once we have our list of fields, we can start writing down interaction terms. An interaction term is a "local operator"—a product of fields and their derivatives evaluated at the same spacetime point. Without any constraints, we could write down infinitely many such terms. Symmetries are the powerful constraints that cut this list down to a manageable, structured set.

The primary symmetries we impose are:

1. Spacetime Symmetries: We require the Lagrangian to be a scalar under Lorentz transformations. This dictates how spacetime indices on fields ($A_{\mu }$) and derivatives (${\partial }_{\mu }$) must be contracted.
2. Internal (Gauge) Symmetries: All terms in the Lagrangian must be invariant under the relevant gauge group transformations. For SMEFT, this is the Standard Model group, $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$. This is an extremely powerful constraint.

Case Study: Constructing the Weinberg Operator

Let's put these two steps into practice by constructing the most important—and simplest—operator one can add to the Standard Model. This is the famous dimension-5 "Weinberg operator," which provides a natural explanation for the smallness of neutrino masses.

Goal: Construct the lowest-dimension operator using only SM fields that is Lorentz invariant, gauge invariant, and violates lepton number.

1. Degrees of Freedom: We need fields that carry lepton number. The obvious choice is the left-handed lepton doublet $L$, which has gauge quantum numbers $(1,2{)}_{-1/2}$ under $SU(3{)}_C\times SU(2{)}_L\times U(1{)}_Y$. We also have the Higgs doublet $H$, with quantum numbers $(1,2{)}_{+1/2}$.

2. Symmetries: We need to combine these fields into a Lorentz scalar and a gauge singlet.

   • Lorentz Invariance: Fermion fields are spinors. A product of an even number of fermion fields can form a Lorentz scalar. The simplest choice is two. So our operator will have the schematic form $LLHH$.
   • Gauge Invariance: Let's focus on the $SU(2{)}_L\times U(1{)}_Y$ part. Both $L$ and $H$ are $SU(2)$ doublets. To make an $SU(2)$ singlet from two doublets, we must use the antisymmetric tensor ${ϵ}_{ij}$. The combination $L_i{H}_j{ϵ}^{ij}$ is an $SU(2)$ singlet.
   • Let's check the hypercharge. $L$ has $Y=-1/2$ and $H$ has $Y=+1/2$. So $LH$ has total hypercharge $Y=0$. Thus, the combination $(LH)\equiv L_i{H}_j{ϵ}^{ij}$ is a complete gauge singlet.
   • To build our full operator, we need two $L$ fields to violate lepton number by two units. The simplest way to make a gauge-invariant Lorentz scalar is to just square this combination.

The resulting operator is:

This operator has mass dimension $2\times \frac{3}{2}(\text{for }L)+2\times 1(\text{for }H)=5$. It is the unique operator of dimension 5. The full effective Lagrangian is then:

After electroweak symmetry breaking, the Higgs gets a vacuum expectation value $⟨H⟩$, and this operator generates a Majorana mass for the neutrinos of the form $m_{\nu }\sim c_5\frac{v^2}{\Lambda }$. This is the famous seesaw mechanism. The observed smallness of neutrino masses ($\sim 0.1$ eV) implies that if the dimensionless coupling $c_5\sim 1$, the scale of new physics $\Lambda$ must be enormous, $\sim 10^{14}$ GeV.

This example is a crowning achievement of the EFT method. By simply asking "What is allowed by symmetry?", we discover a natural explanation for a profound mystery of particle physics.

For a more detailed walkthrough of this construction, the following reading is excellent.

The Subtlety: Redundant Operators and Operator Bases

A crucial point in constructing an effective Lagrangian is that not all operators you can write down are physically independent. Some operators can be transformed into others using:

1. Integration by parts (IBP): Total derivatives in the Lagrangian do not affect the equations of motion and can be dropped. This means, for example, that an operator like $\varphi ({\partial }_{\mu }{\partial }^{\mu }\varphi )$ is equivalent to $-({\partial }_{\mu }\varphi )({\partial }^{\mu }\varphi )$.
2. Equations of Motion (EOM): The leading-order Lagrangian implies classical equations of motion for the fields. For example, for a scalar field $\mathcal{L}=\frac{1}{2}(\partial \varphi {)}^2-\frac{m^2}{2}{\varphi }^2-\frac{\lambda }{4!}{\varphi }^4$, the EOM is $\square \varphi =-m^2\varphi -\frac{\lambda }{6}{\varphi }^3$. We can use this relation to eliminate operators containing $\square \varphi$ in favor of operators with more fields but fewer derivatives.

This means that a naive list of operators will be "redundant". The goal is to find a complete and non-redundant set, known as a basis. Any other valid operator can be written as a linear combination of the operators in the basis. For the Standard Model EFT, the most commonly used basis is the "Warsaw basis".

The process of identifying and removing redundant operators can be complex, but it is a systematic procedure that ensures you are not double-counting physical effects.

This idea of a basis is what makes the EFT construction rigorous. It guarantees that we have a well-defined and unambiguous way to parametrize all possible low-energy effects of new physics.

Conclusion

In this lesson, we have moved from the philosophy of EFT to its practical application. We have seen that constructing an effective Lagrangian is a systematic process rooted in identifying the active degrees of freedom and the inviolable symmetries of the system.

Key Takeaways:

• Construction Principle: The effective Lagrangian includes all local operators consistent with the chosen degrees of freedom and symmetries.
• Step 1: Degrees of Freedom: These are the light fields or excitations relevant at the energy scale of interest. They can be fundamental particles (like in SMEFT) or emergent excitations (like the pions in $\chi \text{PT}$).
• Step 2: Symmetries: Lorentz invariance and gauge symmetries are powerful constraints that drastically limit the set of allowed operators.
• A Concrete Process: By listing fields and their quantum numbers, you can systematically build gauge- and Lorentz-invariant operators. The Weinberg operator for neutrino masses is a prime example of this bottom-up construction yielding deep physical insight.
• Operator Bases: The set of all possible operators is redundant. Using integration by parts and the equations of motion, one can define a minimal, complete basis of independent operators (e.g., the Warsaw basis for SMEFT).

We now have a method to write down the form of our effective theory, resulting in an infinite tower of operators with unknown Wilson coefficients. This seems daunting. How can a theory with infinite terms be predictive? The answer lies in the third and final step of the EFT recipe: power counting. In our next lesson, we will explore how dimensional analysis provides a natural hierarchy, allowing us to organize these operators by their importance and systematically truncate the theory to make predictions at any desired level of precision.