Advanced Particle Physics 1FA355: Brief notes on gauge theory Rikard Enberg April 1, 2014

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The Standard Model

The Standard Model (SM) is a QFT that describes what the world is made of and how these fundamental constituents interact with each other. The “what” was described in the first lecture: a number of different matter particles (fermions) that build up matter, and a number of force-carrying particles (gauge bosons) that mediated interactions. The “how” of how these interactions happen is described by the following gauge theory Lagrangian: LSM

=

1 a aµν / − Fµν F + iψ Dψ {z } | 4

+

ψ i λij ψj h + h.c. {z } |

gauge sector

flavor sector

+

|Dµ h|2 − V (h) | {z } Higgs sector

with gauge group SU(3)×SU(2)×U(1). That’s all! The first group of terms describes the gauge sector of the SM. It’s a nice and simple gauge theory where all particles are massless. The second group of terms describes the flavor sector of the SM, which after electroweak symmetry breaking describes the fermion masses. The third group of terms describes the Higgs sector of the SM, which tells us about the gauge boson masses and the Higgs dynamics. Partly this course is about unpacking the above equation to understand what it is telling us. To do that, we have to learn a few things about quantum field theories and in particular about gauge theories.

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Quantum field theory

Why quantum field theory? The answer is that in high energy physics we must combine quantum mechanics (QM) with special relativity (SR). QM treats the physics of the very small, such as electrons in an atom or in metals, when special relativity is not needed. SR on the other hand treats things moving very fast when we don’t need quantum mechanics. In high energy physics, we need both—we are dealing with very small things that are moving very fast. Now in QM we have the uncertainty principle that states that the energy of a state can fluctuate. In SR we have E = mc2 , so that particles with mass can be created out of energy. This means that when we combine QM and SR, the number of particles is not a 1

constant, but particles can be created and annihilated. This gives us Quantum Field Theory. QFT is very different from relativistic quantum mechanics (the Dirac equation) where the number of particles is fixed. In quantum mechanics, the electromagnetic field is treated as a field, with the associated particle being the photon. Now we want to treat also matter particles such as the electron as a field, e.g. the electron field.

2.1

Conventions

Before we start, we must take care of some preliminaries and definitions. I am going to use the conventions used in the book by Peskin and Schroeder (which are different from Srednicki). • To start with, we will use natural units, meaning that we put c = ~ = 1. This means that mass, momentum, and energy are measured in GeV. Length and time are measured in GeV−1 . A useful conversion factor is ~c = 1 = 197 MeV · fm. • Relativity: Four-vectors (vectors with index up, or contravariant vectors) are written as xµ = (x0 , x1 , x2 , x3 ) = (t, x, y, z) = (t, ~r) = (t, ~x) and pµ = (E, p~). The metric tensor is g µν = diag(1, −1, −1, −1) = gµν , i.e. we are using the “mostly minus” convention. We are always the Einstein summation convention, summing over repeated indices: P using µ µ xµ x ≡ xµ x . Four-vectors with index down (dual vectors or covariant vectors) are given by lowering the index: xµ = gµν xµ = (x0 , x1 , x2 , x3 ) = (t, −~x) and pµ = (E, p~).

∂ ∂ • Derivatives are written as ∂µ ≡ ∂x∂ µ = ∂t , ∇ and ∂ µ ≡ ∂x∂ µ = ∂t , −∇ . The derivative comes with the index “naturally lowered,” while the vector has it “naturally raised.” In other words, the vector xµ is a vector of the manifold we are considering (Minkowski space), while xµ is a vector in the dual space. • Quantum mechanics and relativity: It’s well-known that in QM, we have the operator ∂ ∂ relation E → i~ ∂t for energy and px → −i~ ∂x or p~ → −i~∇ for momentum. In µ µ relativistic QM we have (with ~ = 1) p → i∂ . You can check that this holds from the QM relations, and that for a plane wave e−ik·x this gives pˆµ e−ik·x = k µ e−ik·x . • Finally, the covariant form of electromagnetism is used. Maxwell’s equations in rationalized Heaviside-Lorentz units are ~ =ρ ∇·E ~ =0 ∇·B ~ ∂B ∂t ~ ~ = ~j + ∂ E . ∇×B ∂t ~ =− ∇×E

~ are defined through E ~ = − ∂ A~ − ∇Φ, B ~ = ∇ × A. ~ Then we The potentials Φ and A ∂t µ ~ We also extract the charge from the can define the four-vector potential A = (Φ, A).

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current, j µ → ej µ and define the field strength tensor F µν = Dµ Aν − ∂ ν Aµ . Then the relativistic form of Maxwell’s equations is Dµ F µν = ej µ µνρσ



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∂ν Fρσ = 0.

(1) (2)

Gauge theory

A gauge symmetry is a continuous, local, internal symmetry. A gauge theory is a quantum field theory where the Lagrangian (L) is invariant under some gauge symmetry. The set of possible gauge transformations form a group, known as the gauge group. This is a Lie group, and its generators form a Lie algebra. In the Standard Model, we have the groups U(1), SU(2), and SU(3). The group SU(N ) consists of all special unitary N × N -matrices U , i.e., they fulfill U † U = U U † = 1 and det U = 1.

3.1

U(1) gauge theory

Let us start by considering a U(1) gauge theory. U(1) transformations consist of all unitary 1 × 1-matrices, i.e., the set of phase factors eiα . The Dirac Lagrangian is given by L = ψ(x) (iγ µ ∂µ − m) ψ(x).

(3)

We demand invariance under the U(1) gauge transformation ψ(x) → ψ 0 (x) = eiα(x) ψ(x) ≡ U (x)ψ(x).

(4)

If α is independent of x, then it’s easy to see that L is invariant, but this does not hold if α = α(x). Then we instead get L → L0 = ψ(x) (iγ µ ∂µ − m) ψ(x) + ψiγ µ ψ i∂µ α(x).

(5)

The additional term can be compensated for by introducing the covariant derivative ∂µ → Dµ ≡ ∂µ − igAµ

(6)

where g is called the the gauge coupling and Aµ is called the gauge field or the connection. The gauge field must have the simultaneous gauge transformation 1 Aµ (x) → Aµ (x) + ∂µ α(x), g

(7)

because then, under the gauge transformation (4), Dµ ψ(x) → eiα(x) Dµ ψ(x) = U (x)Dµ ψ(x),

(8)

so now Dµ ψ transforms in the same way as the field ψ itself, meaning that the Lagrangian will be invariant. To get invariance under the gauge transformation, we are forced to introduce the new field Aµ (x). This is a physical new field, but what is it? To answer this question we write out the new Lagrangian with the covariant derivative: L = ψ(x) (iγ µ Dµ − m) ψ(x) = ψ(x) (iγ µ ∂µ − m) ψ(x) + gψγ µ ψAµ . 3

(9)

This is the original Dirac Lagrangian plus an additional term. The additional term is clearly an interaction term since it contains a product of three fields. It is in fact the interaction term of QED, if g is replaced by the electric charge −e, and the field Aµ is interpreted as the photon field. The gauge field of QED is therefore the photon field. Gauge fields are the force carriers of gauge theories, and in fact all force-carrying fields of particle physics are gauge fields. We have seen how the requirement of gauge invariance on the non-interacting Dirac Lagrangian dictates the form of the interaction. The field Aµ must also have a kinetic term. This is given by the Maxwell Lagrangian

where in U(1) theory, Fµν

1 − F µν Fµν 4 ≡ ∂µ Aν − ∂ν Aµ . We also have [Dµ , Dν ] = −igFµν .

(10)

(11)

Note that from the gauge transformation of the field Aµ in Eq. (7), it is easy to see that the product Aµ Aµ is not gauge invariant. It is therefore not possible to add a term such as m2 Aµ Aµ to the Lagrangian, and gauge fields must therefore be massless if the gauge symmetry is exact. (This is the whole reason for introducing the Higgs mechanism to be discussed later.) A product such as ψψ is however gauge invariant, and mass terms for fermions are thus allowed in both QED and QCD. We will have a problem later when we consider the electroweak theory, where mass terms are not allowed at all (we will look into why later in the course). Since all fermions in the Standard Model carry electroweak charges, fermions must also be massless in the Standard Model. This will be discussed later in the course. So why did this happen? The reason we need to include the covariant derivative is that when we take the partial derivative ∂µ ψ we are in effect trying to compare the field at two different points in space-time, ψ(x) and ψ(x + dx), but because the gauge transformation os local, i.e., it depends on the space-time point, the field at these two points have completely different gauge transformation properties. The covariant derivative compensates for this through the gauge field or connection. This is similar to the situation in general relativity where space-time is curved and we introduce parallel transport.

3.2

Non-abelian Lie groups and Lie algebras

The above discussion can be generalized to more complicated gauge groups than U(1). All Lie groups can be represented by matrices, and except U(1) they are all non-Abelian, meaning that the commutator of two elements of the group is non-zero. The gauge groups of the Standard Model are U(1), SU(2) and SU(3), and the gauge groups used in theories beyond the Standard Model are mostly SU(N ) and SO(N ). I will from now on discuss only SU(N ) groups. A general group element of SU(N ) is an N × N -matrix with determinant equal to one. It can be written as U = exp (iαa T a ) where αa is a set of N 2 − 1 parameters and T a is the set of N 2 − 1 generators of the group. The generators are matrices and fulfill the commutation relation h i T a , T b = if abc T c (12) where f abc are called structure constants. The commutation relation defines the Lie algebra associated with the group and can be obtained by writing, for infinitesimal parameters αa  1, U ' 1 + iαa T a . 4

(13)

Thus the generators form a sort of basis for infinitesimal unitary transformations. Any group element can then be obtained by exponentiation. To be clear, there are infinitely many group elements U in SU(N ): they are all possible matrices fulfilling the requirements of being unitary and having unit determinant. There is a finite number of generators T a , a = 1, . . . , N 2 − 1, which fulfill the Lie algebra. Since the group elements are unitary, the generators are Hermitian, i.e. T a† = T a . A representation of a group consists of a set of matrices that fulfill the same multiplication law as the elements of the more abstract group, or in the case of the generators of the group, they fulfill the same commutation relations. Any set of N 2 − 1 matrices T a that fulfills the Lie algebra is a viable representation, and those matrices do not have to be N × N matrices. There is however always a representation in terms of N × N matrices. This is called the defining representation, or the fundamental representation. I.e., for a SU(N ) group, the fundamental representation consists of all N × N -matrices with unit determinant. The fundamental representation of the corresponding SU(N ) Lie algebra then consists of a set of N 2 − 1 different N × N matrices that fulfill the commutation relations. Since the gauge transformations are matrices acting on the fields, the fields must be matrices too. Example. SU(2) is the group of all unitary 2 × 2-matrices with unit determinant. The gauge transformations are then in the fundamental representation the matrices U ∈ SU(2), such that ψ(x) → U (x)ψ(x). Then the field in the fundamental representation must be a column vector   ψ1 (x) ψ(x) = ψ2 (x)

(14)

(15)

We say that ψ is a doublet of SU(2), or more precisely that it transforms as a doublet, or that it transforms as a spinor under the spin-1/2 representation of SU(2). It is also possible to have triplet fields. Then the generators are 3 × 3-matrices. This corresponds to spin 1.

3.3

Non-abelian gauge theory

Let us now consider an SU(N ) gauge theory. Then the field transforms as ψ → (1 + iαa T a ) ψ,

(16)

ψi → ψi + iαa (T a )ij ψj ,

(17)

or in component form which shows that the field is a column vector of the same dimension as the generators. This dimension is different for different representations of the generators. This in turn depends on what representation is chosen for the fields. In the SU(2) example given above, the generators are 2 × 2-matrices and the field is a doublet. In QCD, the quark fields are in the triplet representation of SU(3), meaning that they are 3-dimensional column vectors. The generators acting on the quarks are then 3 × 3-matrices. In SUSY theories, there are fermions called gluinos that are in the 8-dimensional, or octet, representation of SU(3). This is known as 5

the adjoint representation. Then the generators are 8 × 8-matrices. (The gluinos are the supersymmetric partners of the gluons.) We need a covariant derivative so that the covariant derivative of the field transforms under the gauge transformations in the same way as the field itself. This turns out to be given by Dµ = ∂µ − igAaµ T a (18) where the sum runs over all the generators, a = 1, . . . N 2 − 1. There are therefore N 2 − 1 different gauge fields. (For example, in QCD, which is an SU(3) gauge theory, there are 8 gluons.) The covariant derivative can be written in component form as (Dµ )ij = δij ∂µ − igAaµ (T a )ij ,

(19)

which shows that the covariant derivative is a matrix of the same dimension as the generators. The gauge fields transform under the gauge transformation as h i 1 Aaµ T a → Aaµ T a + (∂µ αa )T a + i αa T a , Abµ T b , g

(20)

and you can check that simultaneously transforming ψ and Aµ gives the transformation Dµ ψ → (1 + iαa T a ) Dµ ψ,

(21)

just as we wanted. The field-strength tensor of a non-abelian theory is more complicated than for U(1), where we had [Dµ , Dν ] = −igFµν For SU(N ) this is generalized to a [Dµ , Dν ] = −igFµν T a,

(22)

h i a Fµν T a = ∂µ Aaν T a − ∂ν Aaµ T a − ig Aaµ T a , Abν T b ,

(23)

which gives the expression

a T a , so one for the field strength. Note that each term in this expression is a matrix, e.g. Fµν a T a , as could write this in matrix form, with definitions such as Fµν = Fµν

Fµν = ∂µ Aν − ∂ν Aµ − ig [Aµ , Aν ] .

(24)

  a in another useful form we can use the commutation relation T a , T b = if abc T c on To put Fµν the commutator in Eq. (23), which gives a Fµν = ∂µ Aaν − ∂ν Aaµ + gf abc Abµ Acν .

(25)

This form is useful to find the Lagrangian and Feynman rules for Aaµ . a , since these give The most important knowledge from all this are the forms of Dµ and Fµν all the interactions involving the gauge field. The covariant derivative tells us how all other fields interact with the gauge field. In the Standard Model, this means how the quarks interact with gluons, how the quarks and leptons

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interact with the electroweak gauge bosons γ, W ± , Z 0 , and how the Higgs boson interacts with the W ± and Z 0 . For example, the Dirac kinetic term is given by iψγ µ Dµ ψ = iψγ µ ∂µ ψ + gψγ µ ψAaµ T a ,

(26)

which shows that the Feynman rule is that one should include a factor igγ µ T a . Note that γ µ and T a are both matrices, but they act in different spaces and are not multiplied with each other as matrices. They therefore commute with each other. The kinetic term for the gauge bosons tell us how the gauge bosons interact with each other. This is a new phenomenon for non-abelian gauge theories and comes from the commutator in Eq. (23). We have 1 a Lkin = − F aµν Fµν = (kinetic terms for Aµ ) 4 − gf abc (∂µ Aaν )Abµ Acν 1 − g 2 f abc f ade Abµ Acν Adµ Aeν , 4

(27)

where I didn’t write out the kinetic terms involving the partial derivative ∂µ . The last two lines are interaction terms for the gauge field, which arise from the kinetic term. We can immediately see that there are three-particle and four-particle interactions. The Feynman rule for the three-boson interaction includes momenta, since there is a derivative. In fact, if the gauge group is SU(3), this gives exactly the interactions of QCD.

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