Lecture 8 - Duke Universitywebhome.phy.duke.edu/~goshaw/Lecture8_2017.pdf3 A. Goshaw Physics 346 Reading Chapter 3 in the text reviews the Lagrangian formalism and use of the Euler-Lagrange

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A. Goshaw Physics 346

Lecture 8 September 21, 2017

Today General plan for construction of Standard Model theory

Properties of SU(n) transformations (review)

Choice of gauge symmetries for the Standard Model

Use of Lagrangian density as the “container” for SM theory Ø  General properties of Lagrangian densities Ø  Free particle Lagrangians and wave equations

L

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Plans to Fall break

●  Abstracts: I still need two of these Essay

●  Essays due Oct. 6 5:00pm. Send me a pdf file by email. ●  Guidelines: Ø  Length ~10 pages Ø  Limit the scope to a specific topic (do not try to cover too much). Ø  Direct presentation to non-experts on topic (others in class)

Homework ●  HW2 due Sept. 23 ●  HW3 will be posted after fall break

Lectures ●  L9 and 10 : development of QED ●  L11 and 12 : development of QCD

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Reading

●  Chapter 3 in the text reviews the Lagrangian formalism and use of the Euler-Lagrange equation. ●  Chapter 4 in the text reviews electromagnetism

●  Chapter 5-6 discusses the Dirac equation. You can skip this if you like. I will review in class and use a different convention for the gamma matrices.

●  Chapter 7-8 discusses QED and related topic.

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Questions from last lecture

●  Why are some meson vector resonances broad and others narrow?

M⇢ = 775 MeV

�⇢ = 148 MeV

(�/M)⇢ = 0.19 (broad)

M� = 1020 MeV

�� = 4.2 MeV

(�/M)� = 4 x 10

�3(narrow)

MJ = 3097 MeV

�J = 0.093 MeV

(�/M)J = 3 x 10

�5

(very narrow)

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Moving forward to SM theory

●  Up to this point in the course, we have developed all the Standard Model infrastructure and used it to calculate some cross sections and decay rates given the transition matrix element for the process.

●  We also established the elementary particle spectrum of the Standard Model, and the constants that must be measured in order to make quantitative predictions.

●  Everything presented was self-contained and did not require any background in quantum field theory, just some basic special relativity and quantum mechanics.

●  The next step is to introduce the dynamic structure of the Standard Model. This will require using the relativistic quantum field equations for fermions and bosons.

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 DONE

 DONE

 DONE  (calculation of

 cross section and  decay widths)

 DONE  (conservation laws

 and kinematics)

 Not Done: dynamic symmetries and experimental constraints  that determine the structure of the Standard Model theory

Moving forward to the SM

 Experimental  constraints

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Introduction to Dynamic Symmetries

●  One of the most profound discoveries in theoretical physics has been that the structure of interactions (forces) can be obtained from assumed symmetries of Nature.

●  The requirement that the basic dynamic equation (say the Lagrangian) be invariant under certain symmetry transformations: 1. Leads to specification of the interaction forces among the particles 2. Predicts conserved dynamic quantities (charges).

●  This is analogous to the requirement of invariance under Poincare symmetry transformations leads to relativistic kinematics and conservation laws:

Ø Invariance under spatial rotations => angular momentum conservation Ø Invariance under spatial translations => linear momentum conservation Ø Invariance under time translations => energy conservation

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●  Classical General relativity is perhaps the first example of an assumed invariance of Nature leading to a force law

Ø  Demanding the invariance under general coordinate transformations (plus the equivalence principle) lead Einstein to write down the classical formulation of the gravitational force.

●  Quantum Electro Dynamics is another example where an assumed symmetry lead to the form of the interaction force

Ø  Demanding the invariance under a local gauge transformation UQ(1) leads to the determination of the interaction of the EM field with particles and requires the conservation of electric charge. See next lecture for details.

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●  The dynamic symmetries governing the forces contained in the Standard Model were discovered by a combination of experimental atomic, nuclear and particle data (suggesting certain symmetries) and theoretical insights. The latter have been recognized with Nobel prizes:

Ø  The SUL(2)xUY(1) gauge symmetry of the electroweak interaction (Nobel Prize (1979) Glashow, Salem and Weinberg)

Ø  The SUc(3) gauge symmetry of the strong interaction (Nobel Prize (2004) Gross, Politzer and Wilczek)


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  Properties of SU(n) transformations

 (a brief review)

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●  The dynamic symmetries of the Standard Model are expressed in terms of invariance under certain SU(n) transformations.

Comments about SU(n) transformations

●  First a reminder and some notation:

●  The elements of SU(n) can be complex, therefore there are 2n2 parameters. However the unitary and uni-modular constraints reduce the number of independent real parameters to n2-1 .

• Hermetian conjugate (or adjoint) of matrix A = A† = (AT )⇤

• A Hermetian matrix has H = H†

• A unitary matrix U has an inverse U�1 = U†

• SU(n) transformation can be represented by n dimensional matrices that

are both unitary U�1= U†

and uni-modular |U| = 1

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where the Tj are n x n Hermitian, traceless matrices and the θj are real functions.

●  Any unitary, uni-modular matrix can be written in the form:

●  The Tj are called the generators of the SU(n) transformations.

T†j = Tj and Tjj = 0

●  The commutation between the generators define the structure constants fijk of the SU(n) transformation:

( “i” makes the fijk real numbers and the 2 is just a convention choice).

SU(n) = exp[-i

Pn2�1j=1 ✓jTj ] for n � 2

[Ti, Tj ] = 2iPn2�1

k=1 fijkTk

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●  For the dynamics of the SM, we need only the simplest SU(n) transformations.

• The ✓,↵ and � are real numbers, but can in general

be functions of space and time.

SU(1) = exp[-i✓Q]

SU(2) = exp[-i(↵1�1 + ↵2�2 + ↵3�3)]

SU(3) = exp[-i

P8j=1 �j�j ]

• The generators are a scalar operator Q, the 2x2 Pauli spin

matrices �i and the 3x3 Gell-Mann matrices �i.

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Gell-Mann matricesFrom Wikipedia, the free encyclopedia

The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3x3traceless Hermitian matrices used in the study of the strong interaction in particle physics. They span theLie algebra of the SU(3) group in the defining representation.

Contents1 Matrices2 Properties

2.1 Trace orthonormality2.2 Commutation relations2.3 Fierz completeness relations

3 Representation theory3.1 Casimir operators and invariants

4 Application to quantum chromodynamics5 See also6 References

Matrices

and .

PropertiesThese matrices are traceless, Hermitian (so they can generate unitary matrix group elements throughexponentiation), and obey the extra trace orthonormality relation. These properties were chosen by Gell-Mann because they then naturally generalize the Pauli matrices for SU(2) to SU(3), which formed the basisfor Gell-Mann's quark model. Gell-Mann's generalization further extends to general SU(n). For their

Gell-Mann matrices - Wikipedia https://en.wikipedia.org/wiki/Gell-Mann_matrices#Matrices

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Pauli matricesFrom Wikipedia, the free encyclopedia

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2complex matrices which are Hermitian and unitary.[1] Usually indicated by the Greekletter sigma (σ), they are occasionally denoted by tau (τ) when used in connection withisospin symmetries. They are

These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics,they occur in the Pauli equation which takes into account the interaction of the spin ofa particle with an external electromagnetic field.

Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimesconsidered as the zeroth Pauli matrix σ0), the Pauli matrices (multiplied by realcoefficients) form a basis for the vector space of 2 × 2 Hermitian matrices.

Hermitian operators represent observables, so the Pauli matrices span the space of observables of the 2-dimensionalcomplex Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kthcoordinate axis in three-dimensional Euclidean space ℝ3.

The Pauli matrices (after multiplication by i to make them anti-Hermitian), also generate transformations in the sense ofLie algebras: the matrices iσ1, iσ2, iσ3 form a basis for su(2), which exponentiates to the special unitary groupSU(2).[nb 1] The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of ℝ3, called thealgebra of physical space.

Contents1 Algebraic properties

1.1 Eigenvectors and eigenvalues1.2 Pauli vector1.3 Commutation relations1.4 Relation to dot and cross product1.5 Some trace relations1.6 Exponential of a Pauli vector

1.6.1 The group composition law of SU(2)1.6.2 Adjoint action

1.7 Completeness relation1.8 Relation with the permutation operator

2 SU(2)2.1 SO(3)2.2 Quaternions

3 Physics3.1 Classical mechanics3.2 Quantum mechanics

Wolfgang Pauli (1900–1958), ca.1924. Pauli received the Nobel Prizein physics in 1945, nominated byAlbert Einstein, for the Pauliexclusion principle.

Pauli matrices - Wikipedia https://en.wikipedia.org/wiki/Pauli_matrices

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 Pauli matrices  Gell-Mann matrices

[σi , σj ] = 2 i fijk σk where fijk = εijk (ε123 = - ε132 = 1, ε113 = 0 etc.)

[λa , λb ] = 2 i fabc λc (see page 234 in text for the fabc )

 (sum over repeated indices implied)

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●  Examples of some SU(n) symmetries are:

Ø  SUQ(1) = UQ(1) = a local phase transformation (real space-time)

An exact symmetry of the EM interaction

Ø  SUL(2) = rotation symmetry for left-handed fermions (internal weak iso-spin space)

An exact symmetry of the weak interaction

Ø  SUJ(2) = rotation symmetry of angular momentum (real space)

An exact symmetry of angular momentum


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●  More examples of SU(n) symmetries are:

Ø  SUc(3) = the color symmetry of the strong interaction (internal color space)

An exact symmetry of the strong force

Ø  SUf(3) = the flavor symmetry of u,d,s quarks forming hadrons (internal flavor space)

An approximate symmetry of light mesons and baryons


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 The gauge symmetries  of the Standard Model

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Dynamic Symmetries of the Standard Model

●  The dynamic symmetries determining the structure of the Standard Model are specified by SU(n) transformations in a space of internal particle coordinates (e.g. weak isotopic spin space or color space).

●  The procedure is to write down the Relativistic Quantum Field equation for the non-interacting fields and particles, and then demand that the equation remains invariant under the assumed SU(n) transformations of the particle and field wave functions.

●  This exercise determines the form of the interaction of the particles with the fields, and requires the conservation of the charges responsible for the interactions (electric, weak or color charges).

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The concept of gauge transformations

●  Consider the change of a wave function caused by a transformation:

where the sum over repeated indices j is implied, g is a real constant and the αj(x) are real functions.

●  This is referred to as a gauge transformation of ψ :

Ø  global gauge transformations are those with αj(x) = constant Ø  local gauge transformations are those with with αj(x) varying in space-time (“x” => ct, x,y,z)

! exp[-i g ↵j(x)Tj ]

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●  As described above, in the Standard model we consider Tj that are the generators of the SU(n) symmetry transformations:

Ø  n = 1 T = a scalar operator Ø  n = 2 Tj = 2x2 matrix operators (three Pauli matrices) Ø  n = 3 Tj = 3x3 matrix operators (eight Gell-mann matrices)

●  The theoretical postulate (guess) is that the dynamic equations describing the Standard Model particles and fields are invariant under these gauge transformations.

●  The Standard model is therefore called a “gauge theory” and the bosons representing the force fields are called gauge bosons.

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●  The constant g will turn out to be the coupling strength of the boson force fields (electromagnetic, weak or strong) to the fermions.

●  The αj(x) are the components of a vector in an n dimensional space and αj(x)Tj performs a rotation of ψ in this n dimensional space.

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●  For small changes in ψ generated by the real parameters g αj(x):

●  For the SU(n) transformations finite changes can be generated by integrating over infinitesimal changes so it is sufficient to consider the changes to ψ generated by [1 - i g αj(x)Tj ] .


0= exp[-i g ↵j(x)Tj ] ⇡ [ 1 - i g ↵j(x)Tj ]

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Abelian and non-abelian gauge transformations

●  If the Tj commute (fijk = 0) the field theory is abelian and there is no self coupling of the field bosons. An example is pure electromagnetism where the photon is the gauge boson.

●  If the Ti do not commute then the field theory is non-abelien and there will be self-coupling between the boson fields. An example is quantum chromo-dynamics where there the gluon is the gauge boson.

●  Recall the definition of structure constants on page 9:

[Ti, Tj ] = 2ifijkTk

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●  Start with the free particle Lagrangian densities for: Ø  the force field bosons [ γ , Zo, W+, gluons] Ø  the particle fermions [leptons and quarks]

●  Postulate that these free particle equations must be modified in such a way that they remain invariant under an SU(n) local gauge transformations:

where g will specify the coupling strength Tj are the generators of the SU(n) transformation αj(x) are arbitrary functions of x = (ct,x,y,z)

●  The speculation is that the equations that satisfy the desired SU(n) invariance will express the dynamics of the interactions between the boson fields and the lepton/quark fermions.

THE PLAN: Generating dynamics from

gauge symmetries

!

0= exp[-i g ↵j(x)Tj ]

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 Review of Lagrangian densities

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General Properties of Lagrangian

densities

●  To get started we must chose a formalism for expressing the relativistic quantum field equations. A common way to do this is to use Lagrangian densities L

●  Recall that in classical mechanics for systems of point particles the dynamics of the system can be expressed in terms of a Lagrangian where q is a generalized coordinate and its first time derivative.

●  For distributed systems (a string or other continuous media) this is generalized to a Lagrangian density (see discussion in the text).

L(q, q) = K(q, q)� V (q)q

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densities

●  In relativistic quantum mechanics the system is described by fields “ψ “ that replace the classical mechanics generalized coordinates “q”. ●  The Lagrangian density will therefore be of the form:

Higher order derivatives such as introduce nasty problems with energy flow and are not included in the SM Lagrangian ( just as not needed in classical mechanics).

L( , @µ )

●  Other properties of : Ø  The dimensions are energy density = E/L3 Ø  It must be invariant under Lorentz transformations.

L

@⌫@µ

L( , @µ ) = L( 0, @0µ 0)

where the primes indicate the Lorentz transformed quantities

evaluated at the same space-time point.

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densities

●  The field equations for ψ can be obtained in a manner analogous to classical mechanics by minimizing the action which leads to the condition imposed by the Euler-Lagrange equation (see discussion in the text or a QM book):

@µ[ @L/@(@µ ) ] - @L/@ = 0

●  The field equations can then be used to obtain the matrix elements expressing the dynamics of the system.

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General Properties of

Free particle Lagrangian densities

●  I collect here a few examples of free particle Lagrangian densities for fermions and bosons that we will now need in development of the SM theory.

●  Note that conventions enter here as can be multiplied by constants or functions can be added that cancel out when applying the Euler-Lagrange equations.

L

●  In the end the particle field equations that predict measurable quantities will be unchanged.

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1. A spin 0 boson of mass mc2 described by a real field φ :

L = 1

2@µ�@µ� - 1

2 (mc/~)2�2

(~c)2@µ@µ�+ (mc2)2� = 0

Using the Euler-Grange equation this leads to the Klein-Gordon equation: (see HW2 problem):

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2. A spin 0 boson of mass mc2 described by a complex scalar field φ = (φ1 + i φ2 )/√2 :

L = 12@µ�1@µ�1 - 1

2 (mc/~)2�21 + 1

2@µ�2@µ�2 - 12 (mc/~)2�2

2

or if expressed directly in terms of the complex field φ :

L = @µ�⇤@µ� - (mc/~)2�⇤�

(~c)2@µ@µ�+ (mc2)2� = 0

Using the Euler-Grange equation, and varying φ* and φ independently (a procedure that is equivalent to varying φ1 and φ2 separately):

This is needed for the Higgs boson, the only spin 0 SM particle.

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3. An abelian (non-self-interacting) spin 1 boson of mass mc2 described by a vector field Aµ (the so-called Proca field).

L = � 14F

µ⌫Fµ⌫ + 12 (mc/~)2A⌫A⌫

where Fµ⌫ = @µA⌫ � @⌫Aµ

Using the Euler-Grange equation this leads to the field equation: (~c)2@µFµ⌫ + (mc2)2A⌫ = 0

This is needed for the electromagnetic field where m = 0 and the 4-vector of the spin 1 field Aµ = (c�, ~A) with � and

~A the electric scalar and vector potentials.

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4. A spin ½ fermion of mass mc2 described by a 4 dimensional column spinor ψ :

Using the Euler-Grange equation this leads to the Dirac equation.

Treat as independent, and apply the Euler-Lagrange equation to one of them.)



L = i(~c) �µ@µ - (mc2)

and

This is needed for all quarks and leptons.

i(~c)�µ@µ - (mc2) = 0

where

¯ = †�0 and �µ are 4x4 Dirac matrices.

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5. The last Lagrangian we will need is for a non-abelian (self-interacting) spin 1 massive boson described by a vector field Wµ

We will need this for the description of the W and Z bosons. The basic change from the abelian case described in 3. is that the field tensor is now of the form: where g introduces the self-coupling of the fields.

Gµ⌫ = @µW⌫ � @⌫Wµ � gWµW⌫

We will leave the details for when we study the weak interaction.

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 Next Lecture:  Quantum ElectroDynamics

Documents

Lecture 8 - Duke Universitywebhome.phy.duke.edu/~goshaw/Lecture8_2017.pdf3 A. Goshaw Physics 346 Reading Chapter 3 in the text reviews the Lagrangian formalism and use of the Euler-Lagrange