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 L as the container for SM theory Ø General properties of Lagrangian densities Ø Free particle Lagrangians and wave equations 1
Plans to Fall break Essay Abstracts: I still need two of these 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 2
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. 3
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) M J = 3097 MeV J = 0.093 MeV ( /M ) J = 3 x 10 5 (very narrow) 4
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. 5
Moving forward to the SM DONE DONE DONE (calculation of cross section and decay widths) DONE (conservation laws and kinematics) Experimental constraints Not Done: dynamic symmetries and experimental constraints that determine the structure of the Standard Model theory 6
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 7 Ø Invariance under time translations => energy conservation
Introduction to Dynamic Symmetries 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 U Q (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. 8
Introduction to Dynamic Symmetries 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 SU L (2)xU Y (1) gauge symmetry of the electroweak interaction (Nobel Prize (1979) Glashow, Salem and Weinberg) Ø The SU c (3) gauge symmetry of the strong interaction (Nobel Prize (2004) Gross, Politzer and Wilczek) 9
Properties of SU(n) transformations (a brief review) 10
Comments about SU(n) transformations The dynamic symmetries of the Standard Model are expressed in terms of invariance under certain SU(n) transformations. First a reminder and some notation: Hermetian conjugate (or adjoint) of matrix A = A =(A T ) 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 The elements of SU(n) can be complex, therefore there are 2n 2 parameters. However the unitary and uni-modular constraints reduce the number of independent real parameters to n 2-1. 11
Comments about SU(n) transformations Any unitary, uni-modular matrix can be written in the form: SU(n) =exp[-i P n 2 1 j=1 jt j ] for n 2 where the T j are n x n Hermitian, traceless matrices and the θ j are real functions. T j = T j and T jj =0 The T j are called the generators of the SU(n) transformations. The commutation between the generators define the structure constants f ijk of the SU(n) transformation: [T i,t j ]=2i P n 2 1 k=1 f ijkt k ( i makes the f ijk real numbers and the 2 is just a convention choice). 12
Comments about SU(n) transformations For the dynamics of the SM, we need only the simplest SU(n) transformations. SU(1) = exp[-i Q] SU(2) = exp[-i( 1 1 + 2 2 + 3 3 )] SU(3) = exp[-i P 8 j=1 j j] The generators are a scalar operator Q, the 2x2 Pauli spin matrices i and the 3x3 Gell-Mann matrices i. The, and are real numbers, but can in general be functions of space and time. 13
Pauli matrices Comments about SU(n) transformations Gell-Mann matrices hese matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, ey occur [σ i, in σthe j ] Pauli = 2 i equation f ijk σ k which takes into account the interaction of the spin of particle with an external electromagnetic field. where f ijk = ε ijk ach Pauli (ε 123 matrix = - ε 132 is Hermitian, = 1, ε 113 and = 0 together etc.) with the identity matrix I (sometimes [λ onsidered as the zeroth Pauli matrix σ 0 ), the Pauli matrices a, λ b ] = 2 i f (multiplied abc λ by real c oefficients) form a basis for the vector space of 2 (see 2 page 234 in text for the f Hermitian matrices. abc ) (sum over repeated indices implied) Wolfgang Pauli (1900 1924. Pauli received th in physics in 1945, nom Albert Einstein, for the exclusion principle. ermitian operators represent observables, so the Pauli matrices span the space of observables of the 2-dimens 14 omplex Hilbert space. In the context of Pauli's work, σ k represents the observable corresponding to spin along 3
Comments about SU(n) transformations Examples of some SU(n) symmetries are: Ø SU Q (1) = U Q (1) = a local phase transformation (real space-time) An exact symmetry of the EM interaction Ø SU J (2) = rotation symmetry of angular momentum (real space) An exact symmetry of angular momentum Ø SU L (2) = rotation symmetry for left-handed fermions (internal weak iso-spin space) An exact symmetry of the weak interaction 15
Comments about SU(n) transformations More examples of SU(n) symmetries are: Ø SU c (3) = the color symmetry of the strong interaction (internal color space) An exact symmetry of the strong force Ø SU f (3) = the flavor symmetry of u,d,s quarks forming hadrons (internal flavor space) An approximate symmetry of light mesons and baryons 16
The gauge symmetries of the Standard Model 17
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). 18
The concept of gauge transformations Consider the change of a wave function caused by a transformation:! exp[-i g j (x)t j ] where the sum over repeated indices j is implied, g is a real constant and the α j (x) are real functions. This is referred to as a gauge transformation of ψ : Ø global gauge transformations are those with α j (x) = constant Ø local gauge transformations are those with with α j (x) varying in space-time ( x => ct, x,y,z) 19
The concept of gauge transformations As described above, in the Standard model we consider T j that are the generators of the SU(n) symmetry transformations: Ø n = 1 T = a scalar operator Ø n = 2 T j = 2x2 matrix operators (three Pauli matrices) Ø n = 3 T j = 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. 20
The concept of gauge transformations The constant g will turn out to be the coupling strength of the boson force fields (electromagnetic, weak or strong) to the fermions. The α j (x) are the components of a vector in an n dimensional space and α j (x)t j performs a rotation of ψ in this n dimensional space. 21
The concept of gauge transformations For small changes in ψ generated by the real parameters g α j (x): 0 =exp[-ig j (x)t j ] [1-ig j (x)t j ] 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 α j (x)t j ]. 22
Abelian and non-abelian gauge transformations Recall the definition of structure constants on page 9: [T i,t j ]=2if ijk T k If the T j commute (f ijk = 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 T i 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. 23
THE PLAN: Generating dynamics from gauge symmetries Start with the free particle Lagrangian densities for: Ø the force field bosons [ γ, Z o, 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:! 0 =exp[-ig j (x)t j ] where g will specify the coupling strength T j are the generators of the SU(n) transformation α j (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. 24
Review of Lagrangian densities 25
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 L(q, q) =K(q, q) V (q) where q is a generalized coordinate and q 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). 26
General Properties of Lagrangian 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: L(, @ µ ) @ @ µ 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). Other properties of L : Ø The dimensions are energy density = E/L 3 Ø It must be invariant under Lorentz transformations. L(, @ µ )=L( 0, @ µ 0 0 ) where the primes indicate the Lorentz transformed quantities evaluated at the same space-time point. 27
General Properties of Lagrangian 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. 28
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. 29
Free particle Lagrangian densities 1. A spin 0 boson of mass mc 2 described by a real field φ : L = 1 2 @ µ @ µ - 1 2 (mc/~)2 2 Using the Euler-Grange equation this leads to the Klein-Gordon equation: (see HW2 problem): (~c) 2 @ µ @ µ +(mc 2 ) 2 =0 30
Free particle Lagrangian densities 2. A spin 0 boson of mass mc 2 described by a complex scalar field φ = (φ 1 + i φ 2 )/ 2 : L = 1 2 @ µ 1@ µ 1-1 2 (mc/~)2 2 1 + 1 2 @ µ 2@ µ 2-1 2 (mc/~)2 2 2 or if expressed directly in terms of the complex field φ : L = @ µ @ µ -(mc/~) 2 Using the Euler-Grange equation, and varying φ * and φ independently (a procedure that is equivalent to varying φ 1 and φ 2 separately): (~c) 2 @ µ @ µ +(mc 2 ) 2 =0 This is needed for the Higgs boson, the only spin 0 SM particle. 31
Free particle Lagrangian densities 3. An abelian (non-self-interacting) spin 1 boson of mass mc 2 described by a vector field A µ (the so-called Proca field). L = 1 4 F µ F µ + 1 2 (mc/~)2 A A where F µ = @ µ A @ A µ Using the Euler-Grange equation this leads to the field equation: (~c) 2 @ µ F µ +(mc 2 ) 2 A =0 This is needed for the electromagnetic field where m = 0 and the 4-vector of the spin 1 field A µ =(c, A)with ~ and A ~ the electric scalar and vector potentials. 32
Free particle Lagrangian densities 4. A spin ½ fermion of mass mc 2 described by a 4 dimensional column spinor ψ : L = i(~c) µ @ µ -(mc 2 ) where = 0 and µ are 4x4 Dirac matrices. Using the Euler-Grange equation this leads to the Dirac equation. Treat and as independent, and apply the Euler-Lagrange equation to one of them.) i(~c) µ @ µ -(mc 2 ) =0 This is needed for all quarks and leptons. 33
Free particle Lagrangian densities 5. The last Lagrangian we will need is for a non-abelian (self-interacting) spin 1 massive boson described by a vector field W µ 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: G µ = @ µ W @ W µ gw µ W where g introduces the self-coupling of the fields. We will leave the details for when we study the weak interaction. 34
Next Lecture: Quantum ElectroDynamics 35