Qmii.pdf

5 The Standard Model of Particle
This chapter will give a brief introduction to the standard model of particle physics.
It covers about 7 − 8 lectures, which are not enough to give a complete picture of allaspects of the standard model but can, hopefully, give an overview of its structure and of some of its most important properties.
The standard model is an extremely successful theory of – with the exception of gravity – all phenomena in the world around us. It is one of the most important intellectual achievements of physics. It show that it is possible to find an extremely compact description of nature based on a few symmetry principles. It has been tested in a large number of high-precision experiments. Nevertheless, the structure of the standard model also suggests that it is not the final answer to the question of how the world around us is build: like many other theories it is an approximation valid for only a certain range of energies.
While we will discuss in this chapter some of the most important building blocks of the standard model, we will not be able to discuss how one can actually calculate, e.g., scattering cross sections. For this further techniques have to be developed which are usually covered in courses on quantum field theory. The reader is encouraged to use this chapter also to recapitulate some basic notions of the previous chapters of this book, including topics like the quantization of light, relativistic quantum mechanics, or the principles of Gauge invariance. We denote such opportunities in the following by a box in the following way: recapitulate: .
Some of the conventions used in the following are taken from the book by Peskin and odier, An introduction to QFT, which gives a much deeper coverage of the topic than we can present here. The book of Nachtmann, Elementary Particle Physics,describes the standard model on a similar level (but much more comprehensive) as 5.1 Lagrange formalism
When writing down the field theory describing the standard model, we will use in the following the Lagrange formalism. We assume that the reader is familiar with this approach from the theory lecture on analytical mechanics. We will, however, recapitulate the basic concepts in the following.
In classical physics, one can replace Newton’s laws by a new postulate, Hamilton’s
variation principle .
recapitulate: Lagrange formulation of classi-
Here one investigates the properties of the action S which is defined as a functional of the path x(t) of a particles, S = S[x(t)]. Hamilton’s variational principle states, that along the physical trajectory the action is extremal (a miminum, a maximum or a saddle point), variations of S therefore vanish when the path is varied, x(t) →x(t) + δx(t) while the endpoints at t = 0 and t = T are kept fixed δS = S[x(t) + δx(t)] − S[R(t)] = 0 with δx(0) = δx(T ) = 0 An action can usually be written as an integral over a Lagrange function L(x(t), ˙x(t), t),
S = 0 dtL(x(t), ˙x(t), t). In this case the variation of the action is given by δS = S[x(t) + δx(t)] − S[x(t)] = From the condition δS = 0 for arbitrary δx(t) we therefore obtain the Euler-
Lagrange equation
written here for several components, i = 1, ., N .
For the Lagrange function L = 12m ˙x2 − V (x), for example, the Euler-Lagrange d (m ˙x) = − ∂ V (x) Newton’s equation are differential equations. The same formalism can, however, also be applied for partial differential equations, i.e. for Schr¨ Maxwell, or Dirac equations. Why do we want to do this? For us, one main reason is that the Lagrange formalism provides a simple and compact way to write down theories and to identify their symmetries. The Lagrange formalism also has the advantage that it allows to choose suitable variables (important for the discussion of the Higgs mechanism). Furthermore, it is also a convenient starting point to formulate a theory in second quantization. Here one can either use the Lagrange formalism to define momenta (pi = ∂L/∂xi) and postulate canonical commutation relations ([xn, pm] = iδnm ). Alternatively, there exists also a formulation of quan- tum mechanics and quantum field theory based on actions S. This is discussed in courses on quantum field theory: it turns out that it is possible to obtain quantum mechanical amplitudes just by summing eiS/ over all possible configurations.
To generalize Hamiltonian’s principle to (classical) field theories, we have to consider actions which are functional of field configurations Φi(r, t), i = 1, 2, .N . In one example, Φi may describe the displacement of a guitar string in another example it could be a wave function of a single particle. Assuming that the action is a local dt L(φ(r, t), ∂t φi, ∇φi(r, t )) d4x L(φi, ∂μφi), μ = 0, 1, 2, 3 Here L is the Lagrange density. As before, we study small variations of the field with fixed boundary conditions at the time t = 0 and t = T φi(r, t = 0) = φ0i(r) , φi(r, T ) = φ1i(r) The boundary conditions in real space depend on the type of problem. For a guitar string one would, for example, consider fixed boundary conditions φ(r = 0) = φ(r = L) = 0. We will use infinite systems in the following thereby ignoring the boundary To derive the Euler-Lagrange equations, we have to consider small variations of the field configuration by a small space- and time-dependent function δφ(r, t) which does vanish at the boundaries t = 0 and t = T .
δS = S[φi + δφi] − S[φi] = By demanding that δS vanishes for arbitrary δφ(r, t), we obtain the Euler-Lagrange
equations for a field theory
As a first example, we will consider the Lagrange density for a real field φ(r, t) = φ(x) ∈ ❘. which has the property to be Lorentz invariant. As d4x = c dt d3r isLorentz invariant, a Lorentz invariant Lagrange density implies a Lorentz invariant action S. Considering only terms quadratic in φ, a natural candidate is L = (∂μφ) (∂μφ) − c2m2φ2 = 0 = 1c ∂t the Euler-Lagrange equations are given by 2 ∂tφ + ∇ −2∇φ = 2 We therefore obtain directly the Klein-Gordon equation
∂μ∂μ + c2m2 φ = 0 Similarly, one can find a Lagrange density describing the Dirac equation formu-
lated for a spinor field, ψ(r, t) ∈ ❈4. For spinors, the following Lagrange density isLorentz invariant recapitulate: Dirac equation and γ matrices
To calculate the corresponding Euler-Lagrange equation, one can express Ψ and Ψ in terms of 8 real fields Re(ψ) and Imψ. Much more convenient is, however, to remember that one can choose arbitrary coordinates when investigating the extrema of the action S. Therefore one can just consider the four component of ψ and the ψ as 8 fields independent from each other. One obtains in total 8 Euler Lagrange equations, the first 4 from varying the action by changing ¯ This already gives the Dirac equation in its usual form.
⇒ ∂μ(iψγμ) = −mψ This equation looks less familiar but we can easily check that this is the Dirac Ψ = γ0ψ∗. To check this, consider the Hermitian conjugate of the ∂μ ψiγμ † = −i (γμ)† ∂μψ = −i (γμ)† γ0 ∂μψ = −mγ0ψ Multiplying the equation from the left with γ0 and using that (γ0)2 = ✶ andγ0 (γμ)† γ0 one obtains again the standard Dirac equation.
5.2 Lagrange function of electrodynamics
Before searching for a Lagrange function to describe quantum electrodynamics, QED, we recapitulate a few facts on Maxell’s equations (see also chapter 2.1) using units where c = 1. As we have seen there, any quantum theory of electromagnetism builds on the concept of a vector potential. Originally this has been introduced so solve two of the four Maxwell’s equations.
The first two equations are solved by introducing the scalar potential φ and the Importantly, these potentials are not unique. All physical properties are invariant φ + ∂tΓ(r, t) ; A → It is useful to rewrite these equations and also the current in terms of contravariant such that the Gauge transformation takes the form recapitulate: 4-vector notation, Lorentz trans-
The electric and magnetic fields are then described by the field strength tensor μ ν = ∂μAν − ∂ν Aμ = ⎜ The field strength tensor has two important properties: First, it is manifestly gauge invariant as under a Gauge transformation Fμν → Fμν + ∂μ∂νΓ − ∂ν∂μΓ = Fμν. Second, it transforms as a tensor under Lorentz-transformation We will not try to find an action S as a functional of Aμ build such that its Euler- Lagrange equations reproduce Maxwells equations. More precisely, we will pretend that we do not know Maxwell’s equation. We will instead try to guess the correct
action S (and therefore Maxwell’s theory) guided by symmetries. We demand 3. simplicity (low powers of the fields) The field strength tensor is Gauge invariant, a natural way to build something Lorentz invariant from it, is to consider Fμ ν F μ ν. If we want to include the electric current jμ in the action, we have to multiply it by another 4-vector to obtain some- thing Lorentz invariant: therefore one naturally considers the combination jμAμ.
We can therefore expect that the action should have the form d4x c1 (Fμ ν F μ ν) + c2Aμjμ where c1 and c2 are unknown constants. To confirm Lorentz invariance, note that = det Λ = 1. We should also check Gauge invariance S[Aμ + ∂μ Γ] − S[Aμ] = d4x c2 (∂μΓ)jμ = d4xc2Γ ∂μjμ + surface terms where we used a partial integration for the last equality. Using charge conservation, ∂μjμ = 0 we see that at least up to surface terms everything is Gauge invariant.
The Euler-Lagrange equations are given by ⇒ 4 c1 ∂νFν μ = c2 jμ Looking at these equations, we see that only the ratio of the two constants, c2/c1, enters. Fixing this ratio is, however, only a convention (it can be reabsorbed into a redefinition of the charge). One uses c2 = 4c1 as a convention. An overall prefactor of the action obviously cannot be obtained by just considering the Euler Lagrange equations. We choose it in such a way that when switching from the Lagrange- to the Hamilton formalism, the Hamiltonfunction can be identified with the energy.
From this condition, one finds c2 = −1 and therefore L = −1Fμ νF μ ν − Aμjμ Amazingly, just from a few simple arguments based on gauge invariance and symme- tries, we were able to derive an action describing all aspects of Maxwell’s equation.
Postulating Gauge invariance also for the Dirac equation and the associated action, we will even be able to get QED, quantum electrodynamics.
5.3 U(1) gauge invariance and Quantum
Electrodynamics (QED)
recapitulate: Gauge invariance, Sec. 3.4
As discussed in Sec. 3.4, the coupling of matter to electromagnetism follows from one powerful postulate, the invariance under local phase transformations of the
quantum fields describing charged particles. As we will see later, a very similar principle can be used to describe also the strong and weak interactions. Therefore we repeat the essential argument here as it allows us to construct the action de- scribing QED. Related concepts are also used in general relativity – from this field the expression ‘Gauge invariance’ (in German: Eichinvarianz) arises (the invariance under changing, e.g., the way how length is measured).
The basic idea of Gauge invariance is to postulate of a invariance of fields under the U(1) transformation Ψ(r, t) → eiφ(r,t)Ψ(r, t). This can, however, be only be achieved,if one considers simultaneously a the transformation of the vector potential All physical observables are invariant under the gauge transformation , Aμ(x) → Aμ(x) − ∂μϕ(x) In this case, the combination of derivative and vector potential (‘minimal coupling’) as we have shown in chapter 3.4. Here q is the elementary charge, i.e. the electron charge for cases when Ψ describes electrons.
To obtain the Lagrange density of QED, we just have to combine the Lagrange den- sity of the electromagnetic fields, Eq. (5.5), with the Lagrange density corresponding to the Dirac equation, Eq. (5.3) replacing / ∂ = γμ∂μ by / D = γμDμ using Eq. (5.7).
D − m)ψ − 1 (F μ νFμ ν) This simple Lagrange density encodes all of QED and therefore most of the physics in the world around us. From this one can derive Maxwell’s equations, the Dirac odinger equation and describe – at least in principle – everything recapitulate: Quantization of the electromag-
By comparing Eq. (5.8) to Eq. (5.5), one can also directly a formula for the current jμ = − d (ψ(i / where q can be identified with the electron charge e. This is up to a prefactor 5.4 Regularization and renormalization of QED
It turns out, that the Lagrange density (5.8) of QED (together with the correspond- ing quantization rules) is not sufficient to define properly the theory of quantum electrodynamics. This can be seen when trying to calculate physical quantities us- ing perturbation theory: typical calculations give divergent results. Therefore one also need a way to treat these divergencies. The reader should study chapter 3.10 recapitulate: Renormalization and regulariza-
Two main results of chapter 3.10 are important for our discussion. First, QED (and the whole standard model) is only well defined when one defines a cutoff Λ, i.e., a maximally allowed energy or momentum (this step is called regularization). For finite Λ (and finite energy) one obtains finite results from perturbation theory, which do, however, diverge for Λ → ∞. Second, by expressing the results of the calculation
in terms of the measured electron mass and the measured fine structure constants
one obtains results for all physical observables which are finite and independent
of Λ for Λ → ∞ (see chapter 3.10). This property is called renormalizability and is a feature of all Gauge theories of the standard model.
While the standard model is renormalizable, this is not true for many other com- peting theories. For example, a local interaction of Fermions described by a term (Ψ†(x)Ψ(x))2 in the Lagrange function (or the Hamilton operator in second quan- tization language) turns out to be not renormalizable, implying that in general
(unknown) details of the physics at some high-energy scale influence the physics at low energies, which drastically reduces the predictive power of a theory. In high- energy physics such non-renormalizable theories have therefore been viewed as less suitable theories to describe nature.
It is the general believe that the standard model in its present form is only a low- energy approximation to some other, presently unknown theory.
the standard model is renormalizable means that one can nevertheless predict all experiments with high precision without any knowledge of the high-energy theory.
While this helps a lot to understand the laws of nature around us, it also implies that it is very difficult to access the physics beyond the standard model in present-day experiment. From this point of view, the success of the standard model is actually the main obstacle to understand what it behind the standard model.
5.5 Strong interactions: quarks and gluons
5.5.1 Quarks
Protons and neutrons are not elementary particles. High-energy scattering exper- iment have revealed that they are actually made of so-called quarks. Quarks are
fermionic spin 1/2 particles described by a Dirac equation (see below).
Bound states of 3 quarks are called baryons – the most important examples
are the proton and the neutron – while meson is the name for bound states
A pion is, for example, a meson. A single free quark has never been observed and – as we will discuss in Sec. 5.5.3 – cannot even exist at low energies: only certain types of bound states are allowed. As quarks are fermions, this implies that baryons are fermions while pions act as bosons.
There are, in total, 6 different types of quarks: One speaks of 6 different flavors of quarks. As we will see later when discussing
electroweak interactions, the u/d, the c/s and the t/b quarks each form a pair. The quantum numbers of the quarks, e.g. their charge, will be discussed later in Sec.5.8, after we have worked out the electroweak theory.
The quarks have all different masses1. With the two lightest quarks, the up and the down quark, one can form a stable bound states of 3 quarks which cannot decay (at least not within the standard model) as one cannot form states with lower energy The proton is made from two up and one down quark (uud). The an-
tiproton is therefore build from two three antiquarks (¯ anti-particles by a bar). The neutron, in contrast, is made from two down
As a single particle in vacuum it decays after about 15 min (see Sec. 5.6). Also an d) exist. All other bound states of three quarks decay 1As no free single quarks exist, the precise definition of the mass of a quark is a tricky business.
Note that the mass of the proton is not just the sum of the masses of the quarks it is madefrom as according to E = mc2 also the binding energy contributes to the mass.
The quark picture was developed in the early 80th by people like Murray Gell-Mann which tried to understand a large zoo of different baryons and mesons by assuming that they are bound states of some unknown elementary particles. It was realized early that just two quarks (the u and d) are not sufficient but that at least a third one was needed. Indeed the c, t and b quarks are much heavier than the u, d and s quarks and therefore all ‘lighter’ baryons and mesons are made from the latter three.
Ignoring all of the heavier quarks, one can as a first crude approximation consider the mass of the three remaining lighter quarks u, d and s as being approximately equal (or even zero). Considering only the strong interactions, u, d and s also feel the same forces. Therefore an approximate SU(3) symmetry exist: just changing theu, d and s wave function by an SU(3) matrix, (u, d, s) → U(u, d, s) with U ∈ SU(3)should not strongly affect the mass of a bound state.2 This approximate symmetry is called the SU(3)-flavor symmetry, which should not be confused with the SU(3)
gauge symmetry (acting on the color instead of flavor) discussed in Sec. 5.5.2. The usefulness of the SU(3)-flavor symmetry relies on the fact that it allows to classify, sort and even predict the zoo of barionic and mesonic states by using the theory of We will not explain the representation theory of SU(3) or how it is precisely related to the baryons and mesons but just remind the reader that one can classify for example atoms by the total angular moment J = S + L. While a singlet state, j = 0, is unique, one obtains two degenerate states for j = 1/2, three states for j = 1 and so on. Each j = 0, 1/2, 1, 3/2, . labels one representation of SU(2) and gives rise to a 2j + 1-fold degenerate state.
recapitulate: total angular moment (Sec. 3.6),
Similarly, SU(3) has various representations and one expects that the bound states of quarks can be labeled by these. The approximate SU(3)-flavor symmetry and the group theory for SU(3) therefore predicts that there is a group of 8 light mesons (the origin of the number 8 will become clear in Sec. 5.5.2) of similar mass (3 pions, 4 K- mesons and 1 eta meson) and, similarly, that is a group of 8 light baryons (an octet) to which the neutron and proton and 6 more baryons belong. Another group of 10 baryons (a decuplet) can also be viewed as belonging to one representation of the approximate SU(3) flavor symmetry. Long before the strong interactions had been understood and long before any bound states of quarks could be calculated (and before even quarks have been introduced), it was therefore possible to understand 2SU(3) the is the group of all unitary 3 × 3 matrices U with det U = 1.
the basic structure of baryons and mesons based on the quark model and some symmetry considerations. This approach was coined the ‘eightfold way’ by Murray Gell-Mann (refering to the ‘Noble Eightfold Path’ of Buddhism).
5.5.2 Color and SU(3) gauge theory
Quarks have an important extra quantum number, their so-called color. Each quark
comes in three different colors, often referred to a red, green and blue. We will also label them by i = 1, 2, 3. To describe all 6 flavors of quarks we need 72 = 6 × 3 × 4quantum fields as each quark (for example a red up-quark) is described by a 4- component Dirac spinor: the 4 spinor components describe the spin-up and spin- down states of the quark and its antiparticle. The 12 up-quark fields, for example, where i = 1, 2, 3 describes the color and the second index of uiα is the spinor index.
The Lagrange density of the up-quarks is given for vanishing quark mass by the recapitulate: Dirac equation and its interpre-
To discuss that color index we group the three colors in a 3-component vector ⎝q2⎠, where q = u, d, s, c, t, b for each of the 6 quarks and each qi stands for a 4-component spinor as discussed above for q = u.
Let us recall that the basic idea behind the theory of electromagnetism is the U(1) gauge invariance: physical laws and observables do not change under a transforma- tion Ψ → eiϕ(t,r)Ψ. The basic idea behind the strong interaction is exactly the same:here one postulates that the physical laws do not change under an arbitrary local q(x) = ⎝q2(x)⎠ → q (x) = ⎝q2(x)⎠ = U(x) ⎝q2(x)⎠ , U(x) ∈ SU(3) where x = (t, r) and U (x) as an element of SU (3) is an arbitary unitary 3 ×3 matrix
with det U = 1. The SU(3) Gauge theory of the color-index is called quantum
chromodynamics, or shorter, QCD.
Before learning how to cope with such a situation, we have to study a view basic properties of SU (3). We start by counting the number of independent real param- eters needed to describe a SU(3) matrix. A complex 3 × 3 matrix is described by18 real fields but the 9 equations U †U = ✶ and the extra equation det U = 1 reducethe number of independent components to 8 = 18 − 9 − 1. We need a convenientway to parametrize the matrices with 8 real fields. Before doing so, let us recall how this is accomplished in the SU (2) case which probably more familiar to the reader.
An arbitrary SU(2) matrix can be written as ei 3i=1 φiσi/2 where φi are three angles parametrizing the matrix and σi are the three Pauli matrices which form a basis of the hermitian, traceless 2 × 2 matrices with tr σiσj = 2δij.
Similarily, a SU(3) matrix can be written as U (x) = ei 8α=1 ϕα(x) λα The eight functions ϕα(x) parametrize U (x) and the eight 3 × 3 matrices λα arethe natural generalizations of the Pauli matrices to SU(3). They are a basis of all traceless, hermitian 3 × 3 matrices normalized to tr λiλj = 2δij. A possibleparametrization is, for example, λ1 = ⎝ 1 0 0 ⎠ , λ2 = ⎝ i 0 ⎠ , λ3 = ⎝ 0 −1 0 ⎠ λ4 = ⎝ 0 0 0 ⎠ , λ5 = ⎝ 0 0 0 ⎠ λ6 = ⎝ 0 0 1 ⎠ , λ7 = ⎝ 0 0 −i ⎠ , λ8 = √ ⎝ 0 1 The λi are the 8 generators of SU(3). Similar to QED, we face the problem that when μ(U (x)q) = ∂μ(ei 8α=1 ϕα(x) λα 2 q) eight terms U (x)∂μϕα α are generated. For QED the solution of this problem was to introduce a vector potential to cancel the corresponding term. This is precisely what we are forced to do also in the SU(3) case. Local SU(3) invariance is only possible if for each ϕα we also introduce a corresponding vector potential.
μ(x), α = 1, ., 8, μ = 0, 1, 2, 3 are needed to The quantized vector potential Aμ describes photons.
recapitulate: Quantization of the electromag-
Siimilarly, in quantized form the fields Gα μ(x) describe eight bosonic particles, the gluons. While electromagnetic forces are described by the photon field Aμ, the
gluons are the particles responsible for the strong forces. They glue the quarks It is useful to combine the 8 gluon fields in one Hermitian and traceless 3 × 3 matrixGμ using the definition The formulas for non-abelian gauge groups like SU(3) or SU(2) are slighty more complicated than for the abelian U(1) gauge group. Gauge invariance is achieved by the following combined transformation of quark fields and gluon fields q2(x)⎠ → U(x) ⎝q2(x)⎠ , Gμ(x) → U(x) Gμ(x) U(x)† − i U(x)∂μU†(x) The equations above have been constructed such that the combination Dμ = ∂μ + igsGμ is allows to formulate a SU(3) gauge invariant Lagrange density L = i ¯ us check this claim by studying the Gauge transformation of Dμq (∂μ + igsGμ) ⎝q2⎠ → ∂μ + igsUGμU† + U(∂μU†) U ⎝q2⎠ μ + igsGμ + U †(∂μU ) + U †U (∂μU †)U ⎝q2⎠ = UDμ ⎝q2⎠ where we used that U (∂μU †) = −(∂μU)U† as 0 = ∂μ✶ = ∂μ(UU†) = (∂μU)U† +U (∂μU †). From Eq. 5.12 it follows directly that the Lagrange density does not change under an SU(3) gauge transformation.
Note that in the U (1) case where U (x) = eigsϕ(x) the complicated looking formu- las (5.10) reduce to the simpler formulas (5.6) when identifying gs = q/ c as, for example, − ig U∂μU† = −∂μϕ in this case.
After we have found that the postulate of Gauge invariance enforces the existence of gluons and fixes the form of their coupling to the quarks, we still have to find the SU(3) generalization of the electric and magnetic fields and of the Maxwell equations. To solve this question, we have to repeat just the arguments of Sec. 5.2, where we constructed the action underlying Maxwell’s equations. We first need the the tensor describing the SU(3) version of electric and magnetic fields. It is defined = ∂λGρ − ∂ρGλ + igs[Gλ, Gρ] where we used that the commutator of ∂λ with an arbitrary function f (x) is given by ∂λf (x) as [∂λ, f ]g = ∂λ(f g) − f ∂λg = (∂λf )g for arbitrary f (x) and g(x). Notethat each Gλρ is a 3 × 3 matrix in color space.
We have to check how Gλρ transforms under a SU(3) Gauge transformation. We can use that we have shown above, that Dλq → UDλq = UDλU†Uq. Therefore, Gλρ → 1 [UDλU†, UDλU†]) = UGλρU† This implies directly that the combination tr GλρGλρ is both invariant under a gauge transformation and a Lorentz transformation (the trace runs over the color indices while the summation over λ and ρ is part of the usual Einstein convention).
tr GλρGλρ is therefore the natural candidate to describe in the Lagrange density the gluon analog of Maxwell’s equations.3 action (5.13), we conclude that Lagrange density of QCD which describes the strong LQCD = −1tr GλρGλρ + qj γμ(∂μ + igsGμ) q j Here the index j runs over the 6 quark flavors, with q1,.,6 = u, d, s, t, b, c. Each of these fields has two indices, qν,i, where ν = 1, 2, 3, 4 is the spinor index whilei = 1, 2, 3 is the color index. The 4 × 4 Dirac matrices γμ act on the spinor indexwhile the 3×3 matrices Gμ act on the color index. The factor 1/2 in front of the firstterm is just a convention (it can be absorbed in a redefinition of the fields and gs.
Therefore the only free parameter of this theory (as long as we neglect the masses of the quarks as above) is the coupling constant of the strong interaction, gs.
LQCD describes all aspectes of the strong interaction, the force which holds, forexample, the quarks in the nucleus together. It is parametrized by one coupling constant gs, which takes over the r The most important qualitative aspect which distinguishes LQCD from QED arisesfrom the last term in Eq. (5.14) which is quadratic, not linear in the fields Gα Therefore the first term in LQCD has contributions of the form tr((∂λGρ)[Gλ, Gρ])and tr([Gλ, Gρ] · [Gλ, Gρ]). Gαμ. These terms describe interactions of gluons.
Gluons interact with each other, as described by terms cubic and quartic μ. The coupling constant is given by gs.
While photons do not interact with each other directly, gluons have strong interac- tions. The consequences of this are discussed in Sec. 5.5.3.
3It is possible to add another term, the so-called θ term which does, however, violate time- reversal symmetry. According to experiment this term is either absent or has an extremlysmall prefactor.

Source: http://www.thp.uni-koeln.de/~schuette/advqm/scriptRoschWeek13.pdf