Some Examples
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
External Fields and Measuring Radiative Corrections
Quantum Electrodynamics (QED), Winter 2015/16 Radiative corrections External fields and measuring radiative corrections We now briefly describe how the radiative corrections Πc(k), Σc(p) and Λc(p1; p2), the finite parts of the loop diagrams which enter in renormalization, lead to observable effects. We will consider the two classic cases: the correction to the electron magnetic moment and the Lamb shift. Both are measured using a background electric or magnetic field. Calculating amplitudes in this situation requires a modified perturbative expansion of QED from the one we have considered thus far. Feynman rules with external fields An important approximation in QED is where one has a large background electromagnetic field. This external field can effectively be treated as a classical background in which we quantise the fermions (in analogy to including an electromagnetic field in the Schrodinger¨ equation). More formally we can always expand the field operators around a non-zero value (e) Aµ = aµ + Aµ (e) where Aµ is a c-number giving the external field and aµ is the field operator describing the fluctua- tions. The S-matrix is then given by R 4 ¯ (e) S = T eie d x : (a=+A= ) : For a static external field (independent of time) we have the Fourier transform representation Z (e) 3 ik·x (e) Aµ (x) = d= k e Aµ (k) We can now consider scattering processes such as an electron interacting with the background field. We find a non-zero amplitude even at first order in the perturbation expansion. (Without a background field such terms are excluded by energy-momentum conservation.) We have Z (1) 0 0 4 (e) hfj S jii = ie p ; s d x : ¯A= : jp; si Z 4 (e) 0 0 ¯− µ + = ie d x Aµ (x) p ; s (x)γ (x) jp; si Z Z 4 3 (e) 0 µ ik·x+ip0·x−ip·x = ie d x d= k Aµ (k)u ¯s0 (p )γ us(p) e ≡ 2πδ(E0 − E)M where 0 (e) 0 M = ieu¯s0 (p )A= (p − p)us(p) We see that energy is conserved in the scattering (since the external field is static) but in general the initial and final 3-momenta of the electron can be different. -
Lecture Notes Particle Physics II Quantum Chromo Dynamics 6
Lecture notes Particle Physics II Quantum Chromo Dynamics 6. Feynman rules and colour factors Michiel Botje Nikhef, Science Park, Amsterdam December 7, 2012 Colour space We have seen that quarks come in three colours i = (r; g; b) so • that the wave function can be written as (s) µ ci uf (p ) incoming quark 8 (s) µ ciy u¯f (p ) outgoing quark i = > > (s) µ > cy v¯ (p ) incoming antiquark <> i f c v(s)(pµ) outgoing antiquark > i f > Expressions for the> 4-component spinors u and v can be found in :> Griffiths p.233{4. We have here explicitly indicated the Lorentz index µ = (0; 1; 2; 3), the spin index s = (1; 2) = (up; down) and the flavour index f = (d; u; s; c; b; t). To not overburden the notation we will suppress these indices in the following. The colour index i is taken care of by defining the following basis • vectors in colour space 1 0 0 c r = 0 ; c g = 1 ; c b = 0 ; 0 1 0 1 0 1 0 0 1 @ A @ A @ A for red, green and blue, respectively. The Hermitian conjugates ciy are just the corresponding row vectors. A colour transition like r g can now be described as an • ! SU(3) matrix operation in colour space. Recalling the SU(3) step operators (page 1{25 and Exercise 1.8d) we may write 0 0 0 0 1 cg = (λ1 iλ2) cr or, in full, 1 = 1 0 0 0 − 0 1 0 1 0 1 0 0 0 0 0 @ A @ A @ A 6{3 From the Lagrangian to Feynman graphs Here is QCD Lagrangian with all colour indices shown.29 • µ 1 µν a a µ a QCD = (iγ @µ m) i F F gs λ j γ A L i − − 4 a µν − i ij µ µν µ ν ν µ µ ν F = @ A @ A 2gs fabcA A a a − a − b c We have introduced here a second colour index a = (1;:::; 8) to label the gluon fields and the corresponding SU(3) generators. -
Path Integral and Asset Pricing
Path Integral and Asset Pricing Zura Kakushadze§†1 § Quantigicr Solutions LLC 1127 High Ridge Road #135, Stamford, CT 06905 2 † Free University of Tbilisi, Business School & School of Physics 240, David Agmashenebeli Alley, Tbilisi, 0159, Georgia (October 6, 2014; revised: February 20, 2015) Abstract We give a pragmatic/pedagogical discussion of using Euclidean path in- tegral in asset pricing. We then illustrate the path integral approach on short-rate models. By understanding the change of path integral measure in the Vasicek/Hull-White model, we can apply the same techniques to “less- tractable” models such as the Black-Karasinski model. We give explicit for- mulas for computing the bond pricing function in such models in the analog of quantum mechanical “semiclassical” approximation. We also outline how to apply perturbative quantum mechanical techniques beyond the “semiclas- sical” approximation, which are facilitated by Feynman diagrams. arXiv:1410.1611v4 [q-fin.MF] 10 Aug 2016 1 Zura Kakushadze, Ph.D., is the President of Quantigicr Solutions LLC, and a Full Professor at Free University of Tbilisi. Email: [email protected] 2 DISCLAIMER: This address is used by the corresponding author for no purpose other than to indicate his professional affiliation as is customary in publications. In particular, the contents of this paper are not intended as an investment, legal, tax or any other such advice, and in no way represent views of Quantigic Solutions LLC, the website www.quantigic.com or any of their other affiliates. 1 Introduction In his seminal paper on path integral formulation of quantum mechanics, Feynman (1948) humbly states: “The formulation is mathematically equivalent to the more usual formulations. -
The Weak Charge of the Proton Via Parity Violating Electron Scattering
The Weak Charge of the Proton via Parity Violating Electron Scattering Dave “Dawei” Mack (TJNAF) SPIN2014 Beijing, China Oct 20, 2014 DOE, NSF, NSERC SPIN2014 All Spin Measurements Single Spin Asymmetries PV You are here … … where experiments are unusually difficult, but we don’t annoy everyone by publishing frequently. 2 Motivation 3 The Standard Model (a great achievement, but not a theory of everything) Too many free parameters (masses, mixing angles, etc.). No explanation for the 3 generations of leptons, etc. Not enough CP violation to get from the Big Bang to today’s world No gravity. (dominates dynamics at planetary scales) No dark matter. (essential for understanding galactic-scale dynamics) No dark energy. (essential for understanding expansion of the universe) What we call the SM is only +gravity part of a larger model. +dark matter +dark energy The astrophysical observations are compelling, but only hint at the nature of dark matter and energy. We can look but not touch! To extend the SM, we need more BSM evidence (or tight constraints) from controlled experiments4 . The Quark Weak Vector Charges p Qw is the neutral-weak analog of the proton’s electric charge Note the traditional roles of the proton and neutron are almost reversed: ie, neutron weak charge is dominant, proton weak charge is almost zero. This suppression of the proton weak charge in the SM makes it a sensitive way to: 2 •measure sin θW at low energies, and •search for evidence of new PV interactions between electrons and light quarks. 5 2 Running of sin θW 2 But sin θW is determined much better at the Z pole. -
1 Drawing Feynman Diagrams
1 Drawing Feynman Diagrams 1. A fermion (quark, lepton, neutrino) is drawn by a straight line with an arrow pointing to the left: f f 2. An antifermion is drawn by a straight line with an arrow pointing to the right: f f 3. A photon or W ±, Z0 boson is drawn by a wavy line: γ W ±;Z0 4. A gluon is drawn by a curled line: g 5. The emission of a photon from a lepton or a quark doesn’t change the fermion: γ l; q l; q But a photon cannot be emitted from a neutrino: γ ν ν 6. The emission of a W ± from a fermion changes the flavour of the fermion in the following way: − − − 2 Q = −1 e µ τ u c t Q = + 3 1 Q = 0 νe νµ ντ d s b Q = − 3 But for quarks, we have an additional mixing between families: u c t d s b This means that when emitting a W ±, an u quark for example will mostly change into a d quark, but it has a small chance to change into a s quark instead, and an even smaller chance to change into a b quark. Similarly, a c will mostly change into a s quark, but has small chances of changing into an u or b. Note that there is no horizontal mixing, i.e. an u never changes into a c quark! In practice, we will limit ourselves to the light quarks (u; d; s): 1 DRAWING FEYNMAN DIAGRAMS 2 u d s Some examples for diagrams emitting a W ±: W − W + e− νe u d And using quark mixing: W + u s To know the sign of the W -boson, we use charge conservation: the sum of the charges at the left hand side must equal the sum of the charges at the right hand side. -
Old-Fashioned Perturbation Theory
2012 Matthew Schwartz I-4: Old-fashioned perturbation theory 1 Perturbative quantum field theory The slickest way to perform a perturbation expansion in quantum field theory is with Feynman diagrams. These diagrams will be the main tool we’ll use in this course, and we’ll derive the dia- grammatic expansion very soon. Feynman diagrams, while having advantages like producing manifestly Lorentz invariant results, give a very unintuitive picture of what is going on, with particles that can’t exist appearing from nowhere. Technically, Feynman diagrams introduce the idea that a particle can be off-shell, meaning not satisfying its classical equations of motion, for 2 2 example, with p m . They trade on-shellness for exact 4-momentum conservation. This con- ceptual shift was critical in allowing the efficient calculation of amplitudes in quantum field theory. In this lecture, we explain where off-shellness comes from, why you don’t need it, but why you want it anyway. To motivate the introduction of the concept of off-shellness, we begin by using our second quantized formalism to compute amplitudes in perturbation theory, just like in quantum mechanics. Since we have seen that quantum field theory is just quantum mechanics with an infinite number of harmonic oscillators, the tools of quantum mechanics such as perturbation theory (time-dependent or time-independent) should not have changed. We’ll just have to be careful about using integrals instead of sums and the Dirac δ-function instead of the Kronecker δ. So, we will begin by reviewing these tools and applying them to our second quantized photon. -
'Diagramology' Types of Feynman Diagram
`Diagramology' Types of Feynman Diagram Tim Evans (2nd January 2018) 1. Pieces of Diagrams Feynman diagrams1 have four types of element:- • Internal Vertices represented by a dot with some legs coming out. Each type of vertex represents one term in Hint. • Internal Edges (or legs) represented by lines between two internal vertices. Each line represents a non-zero contraction of two fields, a Feynman propagator. • External Vertices. An external vertex represents a coordinate/momentum variable which is not integrated out. Whatever the diagram represents (matrix element M, Green function G or sometimes some other mathematical object) the diagram is a function of the values linked to external vertices. Sometimes external vertices are represented by a dot or a cross. However often no explicit notation is used for an external vertex so an edge ending in space and not at a vertex with one or more other legs will normally represent an external vertex. • External Legs are ones which have at least one end at an external vertex. Note that external vertices may not have an explicit symbol so these are then legs which just end in the middle of no where. Still represents a Feynman propagator. 2. Subdiagrams A subdiagram is a subset of vertices V and all the edges between them. You may also want to include the edges connected at one end to a vertex in our chosen subset, V, but at the other end connected to another vertex or an external leg. Sometimes these edges connecting the subdiagram to the rest of the diagram are not included, in which case we have \amputated the legs" and we are working with a \truncated diagram". -
TRIUMF & Canadian Scientists Help Measure Proton's Weak Charge
Canada’s national laboratory for particle and nuclear physics Laboratoire national canadien pour la recherche en physique nucléaire et en physique des particules News Release | For Immediate Release | 17 Sep 2013, 5:00 p.m. PDT TRIUMF & Canadian Scientists Help Measure Proton’s Weak Charge (Newport News, VA, USA) --- An international team including Canadian researchers at TRIUMF has reported first results for the proton’s weak charge in Physical Review Letters (to appear in the October 18, 2013 issue) based on precise new data from Jefferson Laboratory, the premier U.S. electron-beam facility for nuclear and particle physics in Newport News, Virginia. The Q-weak experiment used a high-energy electron beam to measure the weak charge of the proton—a fundamental property that sets the scale of its interactions via the weak nuclear force. This is distinct from but analogous to its more familiar electric charge (Q), hence, the experiment’s name: ‘Q-weak.’ Following a decade of design and construction, Q-weak had a successful experimental run in 2010–12 in Hall C at Jefferson Laboratory. Data analysis has been underway ever since. “Nobody has ever attempted a measurement of the proton’s weak charge before,” says Roger Carlini, Q- weak’s spokesperson at Jefferson Laboratory, “due to the extreme technical challenges to reach the required sensitivity. The first 4% of the data have now been fully analyzed and already have an important scientific impact, although the ultimate sensitivity awaits analysis of the complete experiment.” The first result, based on Q-weak’s commissioning data set, is Q_W^p= 0.064 ± 0.012. -
Neutrino Masses-How to Add Them to the Standard Model
he Oscillating Neutrino The Oscillating Neutrino of spatial coordinates) has the property of interchanging the two states eR and eL. Neutrino Masses What about the neutrino? The right-handed neutrino has never been observed, How to add them to the Standard Model and it is not known whether that particle state and the left-handed antineutrino c exist. In the Standard Model, the field ne , which would create those states, is not Stuart Raby and Richard Slansky included. Instead, the neutrino is associated with only two types of ripples (particle states) and is defined by a single field ne: n annihilates a left-handed electron neutrino n or creates a right-handed he Standard Model includes a set of particles—the quarks and leptons e eL electron antineutrino n . —and their interactions. The quarks and leptons are spin-1/2 particles, or weR fermions. They fall into three families that differ only in the masses of the T The left-handed electron neutrino has fermion number N = +1, and the right- member particles. The origin of those masses is one of the greatest unsolved handed electron antineutrino has fermion number N = 21. This description of the mysteries of particle physics. The greatest success of the Standard Model is the neutrino is not invariant under the parity operation. Parity interchanges left-handed description of the forces of nature in terms of local symmetries. The three families and right-handed particles, but we just said that, in the Standard Model, the right- of quarks and leptons transform identically under these local symmetries, and thus handed neutrino does not exist. -
Path Integral in Quantum Field Theory Alexander Belyaev (Course Based on Lectures by Steven King) Contents
Path Integral in Quantum Field Theory Alexander Belyaev (course based on Lectures by Steven King) Contents 1 Preliminaries 5 1.1 Review of Classical Mechanics of Finite System . 5 1.2 Review of Non-Relativistic Quantum Mechanics . 7 1.3 Relativistic Quantum Mechanics . 14 1.3.1 Relativistic Conventions and Notation . 14 1.3.2 TheKlein-GordonEquation . 15 1.4 ProblemsSet1 ........................... 18 2 The Klein-Gordon Field 19 2.1 Introduction............................. 19 2.2 ClassicalScalarFieldTheory . 20 2.3 QuantumScalarFieldTheory . 28 2.4 ProblemsSet2 ........................... 35 3 Interacting Klein-Gordon Fields 37 3.1 Introduction............................. 37 3.2 PerturbationandScatteringTheory. 37 3.3 TheInteractionHamiltonian. 43 3.4 Example: K π+π− ....................... 45 S → 3.5 Wick’s Theorem, Feynman Propagator, Feynman Diagrams . .. 47 3.6 TheLSZReductionFormula. 52 3.7 ProblemsSet3 ........................... 58 4 Transition Rates and Cross-Sections 61 4.1 TransitionRates .......................... 61 4.2 TheNumberofFinalStates . 63 4.3 Lorentz Invariant Phase Space (LIPS) . 63 4.4 CrossSections............................ 64 4.5 Two-bodyScattering . 65 4.6 DecayRates............................. 66 4.7 OpticalTheorem .......................... 66 4.8 ProblemsSet4 ........................... 68 1 2 CONTENTS 5 Path Integrals in Quantum Mechanics 69 5.1 Introduction............................. 69 5.2 The Point to Point Transition Amplitude . 70 5.3 ImaginaryTime........................... 74 5.4 Transition Amplitudes With an External Driving Force . ... 77 5.5 Expectation Values of Heisenberg Position Operators . .... 81 5.6 Appendix .............................. 83 5.6.1 GaussianIntegration . 83 5.6.2 Functionals ......................... 85 5.7 ProblemsSet5 ........................... 87 6 Path Integral Quantisation of the Klein-Gordon Field 89 6.1 Introduction............................. 89 6.2 TheFeynmanPropagator(again) . 91 6.3 Green’s Functions in Free Field Theory . -
Feynman Diagrams Particle and Nuclear Physics
5. Feynman Diagrams Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 5. Feynman Diagrams 1 In this section... Introduction to Feynman diagrams. Anatomy of Feynman diagrams. Allowed vertices. General rules Dr. Tina Potter 5. Feynman Diagrams 2 Feynman Diagrams The results of calculations based on a single process in Time-Ordered Perturbation Theory (sometimes called old-fashioned, OFPT) depend on the reference frame. Richard Feynman 1965 Nobel Prize The sum of all time orderings is frame independent and provides the basis for our relativistic theory of Quantum Mechanics. A Feynman diagram represents the sum of all time orderings + = −−!time −−!time −−!time Dr. Tina Potter 5. Feynman Diagrams 3 Feynman Diagrams Each Feynman diagram represents a term in the perturbation theory expansion of the matrix element for an interaction. Normally, a full matrix element contains an infinite number of Feynman diagrams. Total amplitude Mfi = M1 + M2 + M3 + ::: 2 Total rateΓ fi = 2πjM1 + M2 + M3 + :::j ρ(E) Fermi's Golden Rule But each vertex gives a factor of g, so if g is small (i.e. the perturbation is small) only need the first few. (Lowest order = fewest vertices possible) 2 4 g g g 6 p e2 1 Example: QED g = e = 4πα ∼ 0:30, α = 4π ∼ 137 Dr. Tina Potter 5. Feynman Diagrams 4 Feynman Diagrams Perturbation Theory Calculating Matrix Elements from Perturbation Theory from first principles is cumbersome { so we dont usually use it. Need to do time-ordered sums of (on mass shell) particles whose production and decay does not conserve energy and momentum. Feynman Diagrams Represent the maths of Perturbation Theory with Feynman Diagrams in a very simple way (to arbitrary order, if couplings are small enough). -
The Link Between a Vacuum Diagram and the Renormalization Group Equation
Radboud University Nijmegen Department of Theoretical High Energy Physics Master's thesis The link between a vacuum diagram and the renormalization group equation Author: Supervisor: Daphne van den Elzen Prof. dr. Wim Beenakker September 1, 2016 Contents 1 Introduction3 2 Theoretical context4 2.1 The cosmological constant problem.................4 2.2 Previous research on the cosmological constant problem.....5 2.3 The Higgs Lagrangian........................6 2.4 Scalar Yukawa theory........................8 2.5 Renormalization...........................9 3 Research 11 3.1 The 1-loop self-energy........................ 11 3.2 The K¨all´en-Lehmannspectral representation........... 16 3.3 The vacuum bubble......................... 21 3.4 Comparison with the renormalization group equation....... 25 4 Conclusion and outlook 32 A 'Altered' renormalization group equation 34 2 1 Introduction What exactly is the vacuum? It seems a rather simple question, but when zoom- ing in on this subject it actually turns out to be a pretty complex one. The most obvious definition of the vacuum is empty space; the absence of matter and radiation. When looking at it from the angle of quantum field theory however, the notion of empty space is replaced with that of a vacuum state. It is defined to be the lowest energy state, also called the ground state, of all fundamental fields. This energy isn't zero, as expected from classical mechanics, but it has a value due to zero-point fluctuations of the fields made possible by quantum mechanics. Summing all contributions of zero-point energies leads to an infinite vacuum energy. To make sense of a quantum field theory we need to remove this infinity.