Yang-Mills Lagrangian & Gauge Boson Self-Interactions

Constructing the Yang-Mills Lagrangian

Our goal is to build a kinetic term for the gauge fields, ${\mathcal{L}}_{YM}$, that is both Lorentz invariant and gauge invariant. We have the field strength tensor $F_{\mu \nu }$, which is an anti-symmetric Lorentz tensor and transforms covariantly under a gauge transformation.

How can we construct a quantity that is a Lorentz scalar and also a gauge singlet (i.e., fully invariant) from $F_{\mu \nu }$?

• Lorentz Invariance: The simplest Lorentz scalar we can form is the contraction $F_{\mu \nu }{F}^{\mu \nu }$.
• Gauge Invariance: However, this quantity is still a matrix in the Lie algebra. Under a gauge transformation, it transforms as: $$F_{\mu \nu }{F}^{\mu \nu }\to (U{F}_{\mu \nu }{U}^{†})(U{F}^{\mu \nu }{U}^{†})=U{F}_{\mu \nu }{F}^{\mu \nu }{U}^{†}$$ This is not yet gauge invariant. To get a scalar number from this matrix that is invariant under such a "similarity transformation," we can take the trace. The cyclic property of the trace ($\text{Tr}(ABC)=\text{Tr}(CAB)$) is the key: $$\text{Tr}(F_{\mu \nu }^{\prime }{F}^{\prime \mu \nu })=\text{Tr}(U{F}_{\mu \nu }{F}^{\mu \nu }{U}^{†})=\text{Tr}(U^{†}U{F}_{\mu \nu }{F}^{\mu \nu })=\text{Tr}(F_{\mu \nu }{F}^{\mu \nu })$$ We have found our gauge-invariant and Lorentz-invariant object! We can now define the Yang-Mills Lagrangian.

The following video segment lucidly explains this final step of the construction.

As shown in the video, the standard form of the Yang-Mills Lagrangian is:

The factor of $1/g^2$ is a convention, which we'll discuss later. Using the normalization for the SU(N) generators, $\text{Tr}(T^a{T}^b)=\frac{1}{2}{\delta }^{ab}$, this can be rewritten in component form:

By absorbing the coupling constant into the fields ($A_{\mu }\to g{A}_{\mu }$), we can write the Lagrangian in the form you will most often see:

where the field strength is now defined as $F_{\mu \nu }^a={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+g{f}^{abc}{A}_{\mu }^b{A}_{\nu }^c$. This form makes the role of $g$ as a coupling strength for the interaction vertices more apparent.

For a more formal and detailed presentation of this construction, the following text is an excellent resource.

Unveiling the Gauge Boson Self-Interactions

The compact form ${\mathcal{L}}_{YM}=-\frac{1}{4}{F}_{\mu \nu }^a{F}_a^{\mu \nu }$ hides a wealth of physics. Let's reveal it by substituting the full expression for $F_{\mu \nu }^a$:

Expanding this product reveals terms with different powers of the gauge field $A_{\mu }$:

1. Quadratic Terms ($\sim A^2$):
   $-\frac{1}{4}({\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a)({\partial }^{\mu }{A}_a^{\nu }-{\partial }^{\nu }{A}_a^{\mu })$
   This part looks just like a sum of $N^2-1$ copies of the Maxwell Lagrangian. It is the kinetic term for the free propagation of the gauge bosons.

2. Cubic Terms ($\sim A^3$):
   $-\frac{1}{2}g{f}^{abc}({\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a)A_b^{\mu }{A}_c^{\nu }$
   This term involves three gauge fields. It is an interaction term. In the language of Feynman diagrams, this corresponds to a vertex where three gauge bosons meet.

3. Quartic Terms ($\sim A^4$):
   $-\frac{1}{4}{g}^2{f}^{abc}{f}_{ade}{A}_{\mu }^b{A}_{\nu }^c{A}_d^{\mu }{A}_e^{\nu }$
   This term involves four gauge fields and describes a four-gauge-boson interaction vertex.

The Ramond notes provide the explicit expansion of the Lagrangian, clearly showing these terms.

This is a monumental result. Unlike photons in QED, which do not interact with each other (the Maxwell Lagrangian is purely quadratic in $A_{\mu }$), the gauge bosons of a non-Abelian theory like QCD (the gluons) do interact with themselves. This self-interaction is a direct consequence of the non-zero commutator in the field strength tensor, rooted in the non-Abelian nature of the symmetry group. These interactions are responsible for many of the unique properties of the strong force, such as asymptotic freedom and confinement.

The image below provides a powerful visual summary, connecting the terms in the QCD Lagrangian directly to the Feynman rules for gluon propagators and their self-interaction vertices.

Conclusion

In this lesson, we have constructed the dynamical Lagrangian for non-Abelian gauge fields and uncovered its profound physical meaning.

Key Takeaways:

• The Yang-Mills Lagrangian, ${\mathcal{L}}_{YM}=-\frac{1}{4}{F}_{\mu \nu }^a{F}_a^{\mu \nu }$, is the unique gauge-invariant and Lorentz-invariant kinetic term for SU(N) gauge fields that leads to a unitary quantum theory.
• Gauge invariance of the Lagrangian is guaranteed by the structure of the field strength tensor $F_{\mu \nu }$ and the cyclic property of the trace.
• Expanding the Lagrangian reveals not only a standard kinetic term (quadratic in $A_{\mu }$), but also cubic and quartic self-interaction terms.
• These self-interactions are a hallmark of non-Abelian theories and mean that the force carriers (like gluons) interact with each other. This is fundamentally different from QED, where photons do not self-interact.

Preview of the Next Lesson:

We now have the Lagrangian for the matter fields (from the previous lesson) and the Lagrangian for the gauge fields. In the next lesson, we will put them together to formulate the complete Lagrangian for Quantum Chromodynamics (QCD), the theory of the strong nuclear force. We will introduce the concept of "color charge" and see how quarks (the matter fields) interact via the exchange of gluons (the gauge fields).