21 Renormalization group

Size: px
Start display at page:

Download "21 Renormalization group"

Transcription

1 Renormalization group. Renormalization and interpolation Probably because they describe point particles, quantum field theories are divergent. Unknown physics at very short distance scales, removes these infinities. Since these infinities really are absent, we can cancel them consistently in renormalizable theories by a procedure called renormalization. One starts with an action that contains infinite, unknown charges and masses, such as e = e e and m = m m, inwhich e is the charge of the electron and m is its mass. One then uses e and m to cancel unwanted infinite terms as they appear in perturbation theory. Because the underlying theory is finite, the value of a divergent scattering amplitude may change by a finite amount when we compute it at two different sets of initial and final momenta. This happens, for example, in the theory of a scalar field with action density L i m g 4. (.) 4! The amplitude for the elastic scattering of two bosons of initial four-momenta p and p into two of final momenta p and p is A = g g 6 k 3 dk +[k + m tx( x)] n dx [k + m sx( x)] +[k + m ux( x)] o (.) to one-loop order (Weinberg, 995, section.). In this formula, s, t, and u are the Mandelstam variables s = (p + p ), t = (p p ), and u = (p p ), and after a Wick rotation k = k + k + k + k 3.

2 756 Renormalization group The amplitude A(s, t, u) diverges logarithmically, but the di erence between it and its value A = A(s,t,u ) at some point (s,t,u )isfinite g A(s, t, u) A = 6 k 3 dk dx x( x)(s s )[k +m (s + s )x( x)] [k + m sx( x)] [k + m s x( x)] +(s, s! t, t )+ (s, s! u, u ). (.3) The second and third terms within the big curly brackets are the same as the first term but with s and s replaced by t and t in the second term and by u and u in the third. The k integral is g apple m s x( x) A(s, t, u) A = dx ln +ln 3 apple m t x( x) m tx( x) +ln m sx( x) apple m u x( x) m ux( x). (.4) If we choose as the renormalization point s = t = u = 4µ /3, then we get the usual result (Weinberg, 995, 996, sections., 8. ).. Renormalization group in quantum field theory We can use the scattering amplitude (.4) to define a running coupling constant g µ at energy scale µ as the experimentally measured, finite, physical amplitude A at s = t = u = µ.thenwithg µ A(s,t,u ), the scattering amplitude (.4) is finite to order g g ( apple +(µ/m) x( x) A(s, t, u) =g µ 3 dx ln s/m (.5) x( x) apple +(µ/m) apple ) x( x) +(µ/m) x( x) +ln t/m +ln x( x) u/m. x( x) Callan (Callan, 97) and Symanzik (Symanzik, 97) noticed that this scattering amplitude, like any physical quantity, is independent of the sliding scale µ. Thus its derivative with respect to µ vanishes t, 3g 3 dx ln +(µ/m) x( dx (µ/m )x( x) +(µ/m) x( x). x) (.6)

3 . Renormalization group in quantum field theory 757 For µ m, the integral is /µ. So at high energies, the running coupling constant obeys the di erential equation (g µ)= 3g 6 = 3g µ 6 (.7) in which the last equality holds to second order in g µ. Integrating the beta function (g µ ), we get ln E E M = ge M µ = dg µ g M (g µ ) = 6 ge dg µ 3 g M gµ = 6. 3 g M g E (.8) So the running coupling constant g µ at energy µ = E is g M g E = 3 g M ln(e/m)/6. (.9) As the energy E = p s rises above M, while staying below the singular value E = M exp(6 /3g M ), the running coupling constant g E slowly increases, as does the scattering amplitude, A g E. Example. (Quantum electrodynamics) Vacuum polarization makes the one-loop amplitude for the scattering of two electrons proportional to A(q )=e + (q ) rather than to e (Gell-Mann and Low, 954), (Weinberg, 995, section.). Here e is the renormalized charge, q = p p is the four-momentum transferred to the first electron, and apple (q )= e x( x)ln + q x( x) m dx (.) represents the polarization of the vacuum. One defines the square of the running coupling constant e µ to be the amplitude A(q ) at q = µ e µ = A(µ )=e + (µ ). (.) For q = µ m, the vacuum-polarization term (µ ) is (exercise.) The amplitude then is apple (µ ) e 6 ln µ m A(q )=e µ and since it must be independent of µ, wehave = d A(q ) + (q ) = d 5 6. (.) + (q ) + (µ ), (.3) e µ + (µ ) d e µ (µ ). (.4)

4 758 Renormalization group So by di erentiating e µ and the vacuum-polarization term (.), we find deµ =e µ (µ ) e d (µ ) deµ µ =e µ (µ ) e e µ 6 µ. (.5) But by (.) the vacuum-polarization term (µ ) is of order e,whichis the same as e µ to lowest order in e µ. Thus we arrive at the Callan-Symanzik equation which we can integrate to ln E M = E M µ = ee µ de µ (e µ)= e3 µ (.6) e M de µ (e µ ) = ee e M de µ e 3 µ =6 e M e E e E = e M e M ln(e/m)/6. (.7) The fine-structure constant e µ/4 slowly rises from =/37.36 at m e to e (45.5GeV) 4 = ln(45.5/.5)/3 = 34.6 (.8) at p s = 9 GeV. When all light charged particles are included, one finds that the fine-structure constant rises to =/8.87 at E = 9 GeV. Example. (Quantum chromodynamics) The beta functions of scalar field theories and of quantum electrodynamics are positive, and so interactions in these theories become stronger at higher energy scales. But Yang- Mills theories have beta functions that can be negative because of the cubic interactions of the gauge fields and the ghost fields (.79). If the gauge group is SU(3), then the beta function is µ dg µ (g µ)= to lowest order in g µ. Integrating, we find ln E E M = ge M µ = dg µ g M (g µ ) = 6 ge dg µ g M gµ 3 g 3 6 = g 3 µ 6 (.9) = 8 g M g E (.)

5 .. Renormalization group in quantum field theory 759 Asymptotic freedom Figure. The strong-structure constant s (E) as given by the one-loop formula (.5) (thin curve) and by a three-loop formula (thick curve) with = 3 MeV and n f = 5 is plotted for m b E m t. This chapter s Matlab scripts are in Renormalization group at github.com/kevinecahill. and g E = g M apple + g M 8 ln E M (.) which shows that as the energy E of a scattering process increases, the running coupling slowly decreases, going to zero at infinite energy, an e ect called asymptotic freedom (Gross and Wilczek, 973; Politzer, 973). If the gauge group is SU(N), and the theory has n f flavors of quarks with masses below µ, then the beta function is (g µ )= g3 µ N 4 n f 6 (.) which is negative as long as n f < N/. Using this beta-function with N = 3 and again integrating, we get instead of (.) g E = g M apple + ( n f /3)g M 6 ln E M. (.3)

6 76 Renormalization group So with we find (exercise.) M 6 exp ( n f /3)gM s (E) g (E) 4 = (33 n f )ln(e / ) (.4) (.5) which expresses the dimensionless QCD coupling constant s (E) appropriate to energy E in terms of a parameter that has the dimension of energy. Sidney Coleman called this dimensional transmutation. For = 3 MeV and n f = 5, Fig.. displays s (E) in the range 4.9 = m b E m t = 7 GeV. The thin curve is the one-loop formula (.5), and the thick curve is a three-loop formula (Weinberg, 996, p. 56)..3 Renormalization group in lattice field theory Let us consider a quantum field theory on a lattice (Gattringer and Lang,, chap. 3) in which the strength of the nonlinear interactions depends upon a single dimensionless coupling constant g. The spacing a of the lattice regulates the infinities, which return as a!. The value of an observable P computed on this lattice will depend upon the lattice spacing a and on the coupling constant g, and so will be a function P (a, g) of these two parameters. The right value of the coupling constant is the value that makes the result of the computation be as close as possible to the physical value P.So the correct coupling constant is not a constant at all, but rather a function g(a) that varies with the lattice spacing or cuto a. Thus as we vary the lattice spacing and go to the continuum limit a!, we must adjust the coupling function g(a) so that what we compute, P (a, g(a)), is equal to the physical value P. That is, g(a) must vary with a so as to keep P (a, g(a)) constant at P (a, g(a)) = P dp (a, g(a)) da Writing this condition as a dimensionless derivative a dp (a, g(a)) da =. (.6) = da dp (a, g(a)) dp (a, g(a)) = = (.7) d ln a da d ln a we arrive at the Callan-Symanzik equation dp (a, = = d ln ln a + P (a, g(a)). (.8) d ln

7 .3 Renormalization group in lattice field theory 76 The coe cient of the second partial derivative with a minus sign is the lattice -function L(g) dg d ln a. (.9) Since the lattice spacing a and the energy scale µ are inversely related, the lattice -function di ers from the continuum beta-function by a minus sign. In SU(N) gauge theory, the first two terms of the lattice -function for small g are L(g) = g 3 g 5 where for n f flavors of light quarks = (4 ) 3 N 3 n f = (4 ) N 3 Nn f N (.3) N n f and N = 3 in quantum chromodynamics. Combining the definition (.9) of the -function with its expansion L(g) = g 3 g 5 for small g, one gets the di erential equation which one may integrate d ln a =lna ln c = dg d ln a = g 3 + g 5 (.3) dg g 3 + g 5 = g + ln to find that the lattice spacing has an essential singularity at g = a(g) =c + g / e /( g ) g + g g (.3) in which c is a constant of integration. The term g is of higher order in g, and if one drops it and absorbs a power of into a new constant of integration, then one finds a(g) = g / e /( g ). (.33) As g!, the lattice spacing a(g) goes to zero very fast (as long as n f < 7 for N = 3). The inverse of this relation (.33) g(a) ln(a )+( / )ln ln(a ) / (.34) shows that the coupling constant slowly goes to zero with a, whichisa lattice version of asymptotic freedom.

8 76 Renormalization group.4 Renormalization group in condensed-matter physics In classical statistical mechanics, the partition function ( ) isasumover all configurations c of exp( H(c)) in which H(c) is the energy of the configuration c. A configuration might be a d-dimensional array of spins or a field in d-dimensional space (not spacetime) (x) = e ik x (k) d d k (.35) in which the integral is over Fourier modes k with a cuto k< at an inverse lattice spacing. The hamiltonian might be H[ ]= (r ) + g + g g d d x. (.36) The partition function ( ) is an integral over all such configurations ( )= exp [ H[ ]] D. (.37) We change variables from (x) toastretchedfield (x/`) with`> `(x) =a` (x/`) =a` e ik x/` (k) d d k (.38) in which a` is a factor that keeps the partition function ( ) invariant (Fisher, 974, 998; Kosterlitz et al., 976; Kadano, 9; Wilson, 97, 975). To have H[ `] =H[ ], we need each derivative term H i to remain (x/`) H i [ `] = d d x = d d x = `da (x/`) `@x i /` a i d d (x/`) = `d a (x ) d d (x )=H i [ ]. That is, we need a` = `( d)/. We also need the potential terms to remain invariant. They will if the coe cients g n vary suitably with the stretch ` H n [ `] = g n ǹ (x) d d x = g n a ǹ n (x/`) d d x = g n a ǹ n (x/`) `d d d (x/`) = g n `d a ǹ n (x ) d d x = H n [ ], that is, if g n (`) =`d+n( d)/ g n. Similar reasoning shows that a coupling constant g n,p of a term with n factors of the field and p spatial derivative of should vary as g n,p (`) =`d+n( d)/ p g n,p. In quantum statistical mechanics, the partition function (.37) i

9 Exercises 763 a path integral over fields (,x) that are periodic in the inverse temperature = /kt, and the hamiltonian is replaced by a euclidian action. If one stretches space leaving inverse temperature invariant, then the scaling formulas g n (`) =`d+n( d)/ g n and g n,p (`) =`d+n( d)/ p g n,p still apply. Coupling constants with positive exponents d + n(d )/ p > become more important at greater spatial scales and are said to be relevant. Those with negative exponents become less important at greater spatial scales and are said to be irrelevant. Those with vanishing exponents are marginal. Further reading Quantum Field Theory in a Nutshell (ee,, chapters III & VI), An Introduction to Quantum Field Theory (Peskin and Schroeder, 995, chapter ), and The Quantum Theory of Fields (Weinberg, 995, 996, sections. & 8. ). Exercises. Show that for µ m, the vacuum polarization term (.) reduces to (.). Hint: Use ln ab=lna +lnb when integrating.. Show that by choosing the energy scale according to (.4), one can derive (.5) from (.3).

Introduction to Quantum Chromodynamics (QCD)

Introduction to Quantum Chromodynamics (QCD) Introduction to Quantum Chromodynamics (QCD) Jianwei Qiu Theory Center, Jefferson Lab May 29 June 15, 2018 Lecture One The plan for my four lectures q The Goal: To understand the strong interaction dynamics

More information

REVIEW REVIEW. Quantum Field Theory II

REVIEW REVIEW. Quantum Field Theory II Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,

More information

Quantum Field Theory II

Quantum Field Theory II Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,

More information

Review of scalar field theory. Srednicki 5, 9, 10

Review of scalar field theory. Srednicki 5, 9, 10 Review of scalar field theory Srednicki 5, 9, 10 2 The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate

More information

Lecture notes Particle Physics II. Quantum Chromo Dynamics. 6. Asymptotic Freedom. Michiel Botje Nikhef, Science Park, Amsterdam

Lecture notes Particle Physics II. Quantum Chromo Dynamics. 6. Asymptotic Freedom. Michiel Botje Nikhef, Science Park, Amsterdam Lecture notes Particle Physics II Quantum Chromo Dynamics 6. Asymptotic Freedom Michiel Botje Nikhef, Science Park, Amsterdam December, 03 Charge screening in QED In QED, a charged particle like the electron

More information

Theory of Elementary Particles homework VIII (June 04)

Theory of Elementary Particles homework VIII (June 04) Theory of Elementary Particles homework VIII June 4) At the head of your report, please write your name, student ID number and a list of problems that you worked on in a report like II-1, II-3, IV- ).

More information

Effective Field Theory

Effective Field Theory Effective Field Theory Iain Stewart MIT The 19 th Taiwan Spring School on Particles and Fields April, 2006 Physics compartmentalized Quantum Field Theory String Theory? General Relativity short distance

More information

Quantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University

Quantum Field Theory. and the Standard Model. !H Cambridge UNIVERSITY PRESS MATTHEW D. SCHWARTZ. Harvard University Quantum Field Theory and the Standard Model MATTHEW D. Harvard University SCHWARTZ!H Cambridge UNIVERSITY PRESS t Contents v Preface page xv Part I Field theory 1 1 Microscopic theory of radiation 3 1.1

More information

A Renormalization Group Primer

A Renormalization Group Primer A Renormalization Group Primer Physics 295 2010. Independent Study. Topics in Quantum Field Theory Michael Dine Department of Physics University of California, Santa Cruz May 2010 Introduction: Some Simple

More information

By following steps analogous to those that led to (20.177), one may show (exercise 20.30) that in Feynman s gauge, = 1, the photon propagator is

By following steps analogous to those that led to (20.177), one may show (exercise 20.30) that in Feynman s gauge, = 1, the photon propagator is 20.2 Fermionic path integrals 74 factor, which cancels. But if before integrating over all gauge transformations, we shift so that 4 changes to 4 A 0, then the exponential factor is exp[ i 2 R ( A 0 4

More information

Unitarity, Dispersion Relations, Cutkosky s Cutting Rules

Unitarity, Dispersion Relations, Cutkosky s Cutting Rules Unitarity, Dispersion Relations, Cutkosky s Cutting Rules 04.06.0 For more information about unitarity, dispersion relations, and Cutkosky s cutting rules, consult Peskin& Schröder, or rather Le Bellac.

More information

An Introduction to. Michael E. Peskin. Stanford Linear Accelerator Center. Daniel V. Schroeder. Weber State University. Advanced Book Program

An Introduction to. Michael E. Peskin. Stanford Linear Accelerator Center. Daniel V. Schroeder. Weber State University. Advanced Book Program An Introduction to Quantum Field Theory Michael E. Peskin Stanford Linear Accelerator Center Daniel V. Schroeder Weber State University 4B Advanced Book Program TT Addison-Wesley Publishing Company Reading,

More information

TENTATIVE SYLLABUS INTRODUCTION

TENTATIVE SYLLABUS INTRODUCTION Physics 615: Overview of QFT Fall 2010 TENTATIVE SYLLABUS This is a tentative schedule of what we will cover in the course. It is subject to change, often without notice. These will occur in response to

More information

VI.D Self Duality in the Two Dimensional Ising Model

VI.D Self Duality in the Two Dimensional Ising Model VI.D Self Duality in the Two Dimensional Ising Model Kramers and Wannier discovered a hidden symmetry that relates the properties of the Ising model on the square lattice at low and high temperatures.

More information

VI.D Self Duality in the Two Dimensional Ising Model

VI.D Self Duality in the Two Dimensional Ising Model VI.D Self Duality in the Two Dimensional Ising Model Kramers and Wannier discovered a hidden symmetry that relates the properties of the Ising model on the square lattice at low and high temperatures.

More information

Quantum Field Theory 2 nd Edition

Quantum Field Theory 2 nd Edition Quantum Field Theory 2 nd Edition FRANZ MANDL and GRAHAM SHAW School of Physics & Astromony, The University of Manchester, Manchester, UK WILEY A John Wiley and Sons, Ltd., Publication Contents Preface

More information

QUANTUM FIELD THEORY. A Modern Introduction MICHIO KAKU. Department of Physics City College of the City University of New York

QUANTUM FIELD THEORY. A Modern Introduction MICHIO KAKU. Department of Physics City College of the City University of New York QUANTUM FIELD THEORY A Modern Introduction MICHIO KAKU Department of Physics City College of the City University of New York New York Oxford OXFORD UNIVERSITY PRESS 1993 Contents Quantum Fields and Renormalization

More information

The 1/N expansion method in quantum field theory

The 1/N expansion method in quantum field theory III d International School Symmetry in Integrable Systems and Nuclear Physics Tsakhkadzor, Armenia, 3-13 July 2013 The 1/N expansion method in quantum field theory Hagop Sazdjian IPN, Université Paris-Sud,

More information

Finite Temperature Field Theory

Finite Temperature Field Theory Finite Temperature Field Theory Dietrich Bödeker, Universität Bielefeld 1. Thermodynamics (better: thermo-statics) (a) Imaginary time formalism (b) free energy: scalar particles, resummation i. pedestrian

More information

Summary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization:

Summary of free theory: one particle state: vacuum state is annihilated by all a s: then, one particle state has normalization: The LSZ reduction formula based on S-5 In order to describe scattering experiments we need to construct appropriate initial and final states and calculate scattering amplitude. Summary of free theory:

More information

Vacuum Energy and Effective Potentials

Vacuum Energy and Effective Potentials Vacuum Energy and Effective Potentials Quantum field theories have badly divergent vacuum energies. In perturbation theory, the leading term is the net zero-point energy E zeropoint = particle species

More information

6.1 Quadratic Casimir Invariants

6.1 Quadratic Casimir Invariants 7 Version of May 6, 5 CHAPTER 6. QUANTUM CHROMODYNAMICS Mesons, then are described by a wavefunction and baryons by Φ = q a q a, (6.3) Ψ = ǫ abc q a q b q c. (6.4) This resolves the old paradox that ground

More information

Effective Field Theory in Cosmology

Effective Field Theory in Cosmology C.P. Burgess Effective Field Theory in Cosmology Clues for cosmology from fundamental physics Outline Motivation and Overview Effective field theories throughout physics Decoupling and naturalness issues

More information

Quantum Field Theory II

Quantum Field Theory II Quantum Field Theory II PHYS-P 622 Radovan Dermisek, Indiana University Notes based on: M. Srednicki, Quantum Field Theory Chapters: 13, 14, 16-21, 26-28, 51, 52, 61-68, 44, 53, 69-74, 30-32, 84-86, 75,

More information

The Strong Interaction and LHC phenomenology

The Strong Interaction and LHC phenomenology The Strong Interaction and LHC phenomenology Juan Rojo STFC Rutherford Fellow University of Oxford Theoretical Physics Graduate School course Introduction and motivation: QCD and modern high-energy physics

More information

Introduction to perturbative QCD and factorization

Introduction to perturbative QCD and factorization Introduction to perturbative QCD and factorization Part 1 M. Diehl Deutsches Elektronen-Synchroton DESY Ecole Joliot Curie 2018 DESY Plan of lectures 0. Brief introduction 1. Renormalisation, running coupling,

More information

Wilsonian renormalization

Wilsonian renormalization Wilsonian renormalization Sourendu Gupta Mini School 2016, IACS Kolkata, India Effective Field Theories 29 February 4 March, 2016 Outline Outline Spurious divergences in Quantum Field Theory Wilsonian

More information

Bare Perturbation Theory, MOM schemes, finite volume schemes (lecture II)

Bare Perturbation Theory, MOM schemes, finite volume schemes (lecture II) Bare Perturbation Theory, MOM schemes, finite volume schemes (lecture II) Stefan Sint Trinity College Dublin INT Summer School Lattice QCD and its applications Seattle, August 16, 2007 Stefan Sint Bare

More information

Renormalization Group: non perturbative aspects and applications in statistical and solid state physics.

Renormalization Group: non perturbative aspects and applications in statistical and solid state physics. Renormalization Group: non perturbative aspects and applications in statistical and solid state physics. Bertrand Delamotte Saclay, march 3, 2009 Introduction Field theory: - infinitely many degrees of

More information

AN INTRODUCTION TO QCD

AN INTRODUCTION TO QCD AN INTRODUCTION TO QCD Frank Petriello Northwestern U. & ANL TASI 2013: The Higgs Boson and Beyond June 3-7, 2013 1 Outline We ll begin with motivation for the continued study of QCD, especially in the

More information

QCD in the light quark (up & down) sector (QCD-light) has two mass scales M(GeV)

QCD in the light quark (up & down) sector (QCD-light) has two mass scales M(GeV) QCD in the light quark (up & down) sector (QCD-light) has two mass scales M(GeV) 1 m N m ρ Λ QCD 0 m π m u,d In a generic physical system, there are often many scales involved. However, for a specific

More information

Asymptotic safety of gravity and the Higgs boson mass. Mikhail Shaposhnikov Quarks 2010, Kolomna, June 6-12, 2010

Asymptotic safety of gravity and the Higgs boson mass. Mikhail Shaposhnikov Quarks 2010, Kolomna, June 6-12, 2010 Asymptotic safety of gravity and the Higgs boson mass Mikhail Shaposhnikov Quarks 2010, Kolomna, June 6-12, 2010 Based on: C. Wetterich, M. S., Phys. Lett. B683 (2010) 196 Quarks-2010, June 8, 2010 p.

More information

Dimensional Reduction in the Renormalisation Group:

Dimensional Reduction in the Renormalisation Group: Dimensional Reduction in the Renormalisation Group: From Scalar Fields to Quantum Einstein Gravity Natália Alkofer Advisor: Daniel Litim Co-advisor: Bernd-Jochen Schaefer 11/01/12 PhD Seminar 1 Outline

More information

Effective Field Theory for Nuclear Physics! Akshay Vaghani! Mississippi State University!

Effective Field Theory for Nuclear Physics! Akshay Vaghani! Mississippi State University! Effective Field Theory for Nuclear Physics! Akshay Vaghani! Mississippi State University! Overview! Introduction! Basic ideas of EFT! Basic Examples of EFT! Algorithm of EFT! Review NN scattering! NN scattering

More information

Baroion CHIRAL DYNAMICS

Baroion CHIRAL DYNAMICS Baroion CHIRAL DYNAMICS Baryons 2002 @ JLab Thomas Becher, SLAC Feb. 2002 Overview Chiral dynamics with nucleons Higher, faster, stronger, Formulation of the effective Theory Full one loop results: O(q

More information

Introduction to High Energy Nuclear Collisions I (QCD at high gluon density) Jamal Jalilian-Marian Baruch College, City University of New York

Introduction to High Energy Nuclear Collisions I (QCD at high gluon density) Jamal Jalilian-Marian Baruch College, City University of New York Introduction to High Energy Nuclear Collisions I (QCD at high gluon density) Jamal Jalilian-Marian Baruch College, City University of New York Many thanks to my colleagues, A. Deshpande, F. Gelis, B. Surrow

More information

SM, EWSB & Higgs. MITP Summer School 2017 Joint Challenges for Cosmology and Colliders. Homework & Exercises

SM, EWSB & Higgs. MITP Summer School 2017 Joint Challenges for Cosmology and Colliders. Homework & Exercises SM, EWSB & Higgs MITP Summer School 017 Joint Challenges for Cosmology and Colliders Homework & Exercises Ch!"ophe Grojean Ch!"ophe Grojean DESY (Hamburg) Humboldt University (Berlin) ( christophe.grojean@desy.de

More information

July 2, SISSA Entrance Examination. PhD in Theoretical Particle Physics Academic Year 2018/2019. olve two among the three problems presented.

July 2, SISSA Entrance Examination. PhD in Theoretical Particle Physics Academic Year 2018/2019. olve two among the three problems presented. July, 018 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 018/019 S olve two among the three problems presented. Problem I Consider a theory described by the Lagrangian density

More information

MASS GAP IN QUANTUM CHROMODYNAMICS

MASS GAP IN QUANTUM CHROMODYNAMICS arxiv:hep-th/0611220v2 28 Nov 2006 MASS GAP IN QUANTUM CHROMODYNAMICS R. Acharya Department of Physics & Astronomy Arizona State University Tempe, AZ 85287-1504 November 2006 Abstract We present a heuristic

More information

Light-Cone Quantization of Electrodynamics

Light-Cone Quantization of Electrodynamics Light-Cone Quantization of Electrodynamics David G. Robertson Department of Physics, The Ohio State University Columbus, OH 43210 Abstract Light-cone quantization of (3+1)-dimensional electrodynamics is

More information

Scale symmetry a link from quantum gravity to cosmology

Scale symmetry a link from quantum gravity to cosmology Scale symmetry a link from quantum gravity to cosmology scale symmetry fluctuations induce running couplings violation of scale symmetry well known in QCD or standard model Fixed Points Quantum scale symmetry

More information

Introduction to particle physics Lecture 7

Introduction to particle physics Lecture 7 Introduction to particle physics Lecture 7 Frank Krauss IPPP Durham U Durham, Epiphany term 2009 Outline 1 Deep-inelastic scattering and the structure of protons 2 Elastic scattering Scattering on extended

More information

Zhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016

Zhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016 Zhong-Zhi Xianyu (CMSA Harvard) Tsinghua June 30, 2016 We are directly observing the history of the universe as we look deeply into the sky. JUN 30, 2016 ZZXianyu (CMSA) 2 At ~10 4 yrs the universe becomes

More information

Quantum Field Theory. Kerson Huang. Second, Revised, and Enlarged Edition WILEY- VCH. From Operators to Path Integrals

Quantum Field Theory. Kerson Huang. Second, Revised, and Enlarged Edition WILEY- VCH. From Operators to Path Integrals Kerson Huang Quantum Field Theory From Operators to Path Integrals Second, Revised, and Enlarged Edition WILEY- VCH WILEY-VCH Verlag GmbH & Co. KGaA I vh Contents Preface XIII 1 Introducing Quantum Fields

More information

Reφ = 1 2. h ff λ. = λ f

Reφ = 1 2. h ff λ. = λ f I. THE FINE-TUNING PROBLEM A. Quadratic divergence We illustrate the problem of the quadratic divergence in the Higgs sector of the SM through an explicit calculation. The example studied is that of the

More information

Anomaly. Kenichi KONISHI University of Pisa. College de France, 14 February 2006

Anomaly. Kenichi KONISHI University of Pisa. College de France, 14 February 2006 Anomaly Kenichi KONISHI University of Pisa College de France, 14 February 2006 Abstract Symmetry and quantization U A (1) anomaly and π 0 decay Origin of anomalies Chiral and nonabelian anomaly Anomally

More information

Effective Field Theory for Cold Atoms I

Effective Field Theory for Cold Atoms I Effective Field Theory for Cold Atoms I H.-W. Hammer Institut für Kernphysik, TU Darmstadt and Extreme Matter Institute EMMI School on Effective Field Theory across Length Scales, ICTP-SAIFR, Sao Paulo,

More information

NTNU Trondheim, Institutt for fysikk

NTNU Trondheim, Institutt for fysikk NTNU Trondheim, Institutt for fysikk Examination for FY3464 Quantum Field Theory I Contact: Michael Kachelrieß, tel. 99890701 Allowed tools: mathematical tables Some formulas can be found on p.2. 1. Concepts.

More information

Renormalization of the fermion self energy

Renormalization of the fermion self energy Part I Renormalization of the fermion self energy Electron self energy in general gauge The self energy in n = 4 Z i 0 = ( ie 0 ) d n k () n (! dimensions is i k )[g ( a 0 ) k k k ] i /p + /k m 0 use Z

More information

Quark Structure of the Pion

Quark Structure of the Pion Quark Structure of the Pion Hyun-Chul Kim RCNP, Osaka University & Department of Physics, Inha University Collaborators: H.D. Son, S.i. Nam Progress of J-PARC Hadron Physics, Nov. 30-Dec. 01, 2014 Interpretation

More information

The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local Gauge Transformations Dynamics of Field Ten

The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local Gauge Transformations Dynamics of Field Ten Lecture 4 QCD as a Gauge Theory Adnan Bashir, IFM, UMSNH, Mexico August 2013 Hermosillo Sonora The Gauge Principle Contents Quantum Electrodynamics SU(N) Gauge Theory Global Gauge Transformations Local

More information

1 Equal-time and Time-ordered Green Functions

1 Equal-time and Time-ordered Green Functions 1 Equal-time and Time-ordered Green Functions Predictions for observables in quantum field theories are made by computing expectation values of products of field operators, which are called Green functions

More information

National Nuclear Physics Summer School Lectures on Effective Field Theory. Brian Tiburzi. RIKEN BNL Research Center

National Nuclear Physics Summer School Lectures on Effective Field Theory. Brian Tiburzi. RIKEN BNL Research Center 2014 National Nuclear Physics Summer School Lectures on Effective Field Theory I. Removing heavy particles II. Removing large scales III. Describing Goldstone bosons IV. Interacting with Goldstone bosons

More information

CFT approach to multi-channel SU(N) Kondo effect

CFT approach to multi-channel SU(N) Kondo effect CFT approach to multi-channel SU(N) Kondo effect Sho Ozaki (Keio Univ.) In collaboration with Taro Kimura (Keio Univ.) Seminar @ Chiba Institute of Technology, 2017 July 8 Contents I) Introduction II)

More information

lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab

lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab a black box? QCD lattice QCD observables (scattering amplitudes?) in these lectures, hope to give you a look inside the box 2 these lectures how

More information

Introduction to Quantum Chromodynamics (QCD)

Introduction to Quantum Chromodynamics (QCD) Introduction to Quantum Chromodynamics (QCD) Jianwei Qiu August 16 19, 018 Four Lectures The 3 rd WHEPS, August 16-4, 018, Weihai, Shandong q The Goal: The plan for my four lectures To understand the strong

More information

QCD Phases with Functional Methods

QCD Phases with Functional Methods QCD Phases with Mario PhD-Advisors: Bernd-Jochen Schaefer Reinhard Alkofer Karl-Franzens-Universität Graz Institut für Physik Fachbereich Theoretische Physik Rab, September 2010 QCD Phases with Table of

More information

Loop corrections in Yukawa theory based on S-51

Loop corrections in Yukawa theory based on S-51 Loop corrections in Yukawa theory based on S-51 Similarly, the exact Dirac propagator can be written as: Let s consider the theory of a pseudoscalar field and a Dirac field: the only couplings allowed

More information

Quantum Gravity and the Renormalization Group

Quantum Gravity and the Renormalization Group Nicolai Christiansen (ITP Heidelberg) Schladming Winter School 2013 Quantum Gravity and the Renormalization Group Partially based on: arxiv:1209.4038 [hep-th] (NC,Litim,Pawlowski,Rodigast) and work in

More information

Quarks, Leptons and Gauge Fields Downloaded from by on 03/13/18. For personal use only.

Quarks, Leptons and Gauge Fields Downloaded from  by on 03/13/18. For personal use only. QUARKS, LEPTONS & GAUGE FIELDS 2nd edition Kerson Huang Professor of Physics Mussuchusetts Institute qf Technology Y 8 World Scientific Singapore New Jersey London Hong Kong Publirhed by World Scientific

More information

STANDARD MODEL and BEYOND: SUCCESSES and FAILURES of QFT. (Two lectures)

STANDARD MODEL and BEYOND: SUCCESSES and FAILURES of QFT. (Two lectures) STANDARD MODEL and BEYOND: SUCCESSES and FAILURES of QFT (Two lectures) Lecture 1: Mass scales in particle physics - naturalness in QFT Lecture 2: Renormalisable or non-renormalisable effective electroweak

More information

Finite-temperature Field Theory

Finite-temperature Field Theory Finite-temperature Field Theory Aleksi Vuorinen CERN Initial Conditions in Heavy Ion Collisions Goa, India, September 2008 Outline Further tools for equilibrium thermodynamics Gauge symmetry Faddeev-Popov

More information

Theory toolbox. Chapter Chiral effective field theories

Theory toolbox. Chapter Chiral effective field theories Chapter 3 Theory toolbox 3.1 Chiral effective field theories The near chiral symmetry of the QCD Lagrangian and its spontaneous breaking can be exploited to construct low-energy effective theories of QCD

More information

Two particle elastic scattering at 1-loop

Two particle elastic scattering at 1-loop Two particle elastic scattering at 1-loop based on S-20 Let s use our rules to calculate two-particle elastic scattering amplitude in, theory in 6 dimensions including all one-loop corrections: For the

More information

Quantum Field Theory III

Quantum Field Theory III Quantum Field Theory III Prof. Erick Weinberg March 9, 0 Lecture 5 Let s say something about SO(0. We know that in SU(5 the standard model fits into 5 0(. In SO(0 we know that it contains SU(5, in two

More information

Lecture II: Owe Philipsen. The ideal gas on the lattice. QCD in the static and chiral limit. The strong coupling expansion at finite temperature

Lecture II: Owe Philipsen. The ideal gas on the lattice. QCD in the static and chiral limit. The strong coupling expansion at finite temperature Lattice QCD, Hadron Structure and Hadronic Matter Dubna, August/September 2014 Lecture II: Owe Philipsen The ideal gas on the lattice QCD in the static and chiral limit The strong coupling expansion at

More information

lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab

lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab lattice QCD and the hadron spectrum Jozef Dudek ODU/JLab the light meson spectrum relatively simple models of hadrons: bound states of constituent quarks and antiquarks the quark model empirical meson

More information

Ginsparg-Wilson Fermions and the Chiral Gross-Neveu Model

Ginsparg-Wilson Fermions and the Chiral Gross-Neveu Model Ginsparg-Wilson Fermions and the DESY Zeuthen 14th September 2004 Ginsparg-Wilson Fermions and the QCD predictions Perturbative QCD only applicable at high energy ( 1 GeV) At low energies (100 MeV - 1

More information

Introduction to chiral perturbation theory II Higher orders, loops, applications

Introduction to chiral perturbation theory II Higher orders, loops, applications Introduction to chiral perturbation theory II Higher orders, loops, applications Gilberto Colangelo Zuoz 18. July 06 Outline Introduction Why loops? Loops and unitarity Renormalization of loops Applications

More information

Units, dimensions and regularization

Units, dimensions and regularization Chapter Units, dimensions and regularization Instructor: Sudipta Mukherji, Institute of Physics, Bhubaneswar. Natural Units Like other courses in physics, we start with units and dimensions by first listing

More information

Introduction to the Parton Model and Pertrubative QCD

Introduction to the Parton Model and Pertrubative QCD Introduction to the Parton Model and Pertrubative QCD George Sterman, YITP, Stony Brook CTEQ summer school, July 0, 20 U. of Wisconsin, Madison II. From the Parton Model to QCD A. Color and QCD B. Field

More information

An Introduction to Quantum Field Theory (Peskin and Schroeder) Solutions

An Introduction to Quantum Field Theory (Peskin and Schroeder) Solutions An Introduction to Quantum Field Theory (Peskin and Schroeder) Solutions Andrzej Pokraka February 5, 07 Contents 4 Interacting Fields and Feynman Diagrams 4. Creation of Klein-Gordon particles from a classical

More information

The Strong Interaction and LHC phenomenology

The Strong Interaction and LHC phenomenology The Strong Interaction and LHC phenomenology Juan Rojo STFC Rutherford Fellow University of Oxford Theoretical Physics Graduate School course Lecture 2: The QCD Lagrangian, Symmetries and Feynman Rules

More information

4. The Standard Model

4. The Standard Model 4. The Standard Model Particle and Nuclear Physics Dr. Tina Potter Dr. Tina Potter 4. The Standard Model 1 In this section... Standard Model particle content Klein-Gordon equation Antimatter Interaction

More information

Luciano Maiani: Lezione Fermi 11. Quantum Electro Dynamics, QED. Renormalization

Luciano Maiani: Lezione Fermi 11. Quantum Electro Dynamics, QED. Renormalization Luciano Maiani: Lezione Fermi 11. Quantum Electro Dynamics, QED. Renormalization 1. Divergences in classical ED, the UV cutoff 2. Renormalization! 3. The electron anomaly 4. The muon anomaly 5. Conclusions

More information

Acknowledgements An introduction to unitary symmetry The search for higher symmetries p. 1 The eight-baryon puzzle p. 1 The elimination of

Acknowledgements An introduction to unitary symmetry The search for higher symmetries p. 1 The eight-baryon puzzle p. 1 The elimination of Preface p. xiii Acknowledgements p. xiv An introduction to unitary symmetry The search for higher symmetries p. 1 The eight-baryon puzzle p. 1 The elimination of G[subscript 0] p. 4 SU(3) and its representations

More information

arxiv:quant-ph/ v1 12 Feb 2002

arxiv:quant-ph/ v1 12 Feb 2002 Ultraviolet analysis of one dimensional quantum systems Marco Frasca Via Erasmo Gattamelata, 3, 0076 Roma (Italy) (February, 008) arxiv:quant-ph/00067v Feb 00 Abstract Starting from the study of one-dimensional

More information

19 Entanglement Entropy in Quantum Field Theory

19 Entanglement Entropy in Quantum Field Theory 19 Entanglement Entropy in Quantum Field Theory So far we have discussed entanglement in ordinary quantum mechanics, where the Hilbert space of a finite region is finite dimensional. Now we will discuss

More information

Regularization Physics 230A, Spring 2007, Hitoshi Murayama

Regularization Physics 230A, Spring 2007, Hitoshi Murayama Regularization Physics 3A, Spring 7, Hitoshi Murayama Introduction In quantum field theories, we encounter many apparent divergences. Of course all physical quantities are finite, and therefore divergences

More information

Seiberg Duality: SUSY QCD

Seiberg Duality: SUSY QCD Seiberg Duality: SUSY QCD Outline: SYM for F N In this lecture we begin the study of SUSY SU(N) Yang-Mills theory with F N flavors. This setting is very rich! Higlights of the next few lectures: The IR

More information

Precision (B)SM Higgs future colliders

Precision (B)SM Higgs future colliders Flavor and top physics @ 100 TeV Workshop, IHEP/CAS, MARCH 5, 2015 Seung J. Lee (KAIST) Precision (B)SM Higgs Studies @ future colliders 1. Study of SM Higgs boson partial widths and branching fractions

More information

Asymptotically free nonabelian Higgs models. Holger Gies

Asymptotically free nonabelian Higgs models. Holger Gies Asymptotically free nonabelian Higgs models Holger Gies Friedrich-Schiller-Universität Jena & Helmholtz-Institut Jena & Luca Zambelli, PRD 92, 025016 (2015) [arxiv:1502.05907], arxiv:16xx.yyyyy Prologue:

More information

The mass of the Higgs boson

The mass of the Higgs boson The mass of the Higgs boson LHC : Higgs particle observation CMS 2011/12 ATLAS 2011/12 a prediction Higgs boson found standard model Higgs boson T.Plehn, M.Rauch Spontaneous symmetry breaking confirmed

More information

Perturbative Static Four-Quark Potentials

Perturbative Static Four-Quark Potentials arxiv:hep-ph/9508315v1 17 Aug 1995 Perturbative Static Four-Quark Potentials J.T.A.Lang, J.E.Paton, Department of Physics (Theoretical Physics) 1 Keble Road, Oxford OX1 3NP, UK A.M. Green Research Institute

More information

Effective Theories are Dimensional Analysis

Effective Theories are Dimensional Analysis Effective Theories are Dimensional Analysis Sourendu Gupta SERC Main School 2014, BITS Pilani Goa, India Effective Field Theories December, 2014 Outline Outline The need for renormalization Three simple

More information

Identical Particles. Bosons and Fermions

Identical Particles. Bosons and Fermions Identical Particles Bosons and Fermions In Quantum Mechanics there is no difference between particles and fields. The objects which we refer to as fields in classical physics (electromagnetic field, field

More information

Manifestly diffeomorphism invariant classical Exact Renormalization Group

Manifestly diffeomorphism invariant classical Exact Renormalization Group Manifestly diffeomorphism invariant classical Exact Renormalization Group Anthony W. H. Preston University of Southampton Supervised by Prof. Tim R. Morris Talk prepared for Asymptotic Safety seminar,

More information

QCD Lagrangian. ψ qi. δ ij. ψ i L QCD. m q. = ψ qi. G α µν = µ G α ν ν G α µ gf αβγ G β µ G γ. G α t α f αβγ. g = 4πα s. (!

QCD Lagrangian. ψ qi. δ ij. ψ i L QCD. m q. = ψ qi. G α µν = µ G α ν ν G α µ gf αβγ G β µ G γ. G α t α f αβγ. g = 4πα s. (! QCD Lagrangian L QCD = ψ qi iγ µ! δ ij µ + ig G α µ t $ "# α %& ψ qj m q ψ qi ψ qi 1 4 G µν q ( ( ) ) ij L QED = ψ e iγ µ!" µ + iea µ # $ ψ e m e ψ e ψ e 1 4 F F µν µν α G α µν G α µν = µ G α ν ν G α µ

More information

Transverse momentum-dependent parton distributions from lattice QCD. Michael Engelhardt New Mexico State University

Transverse momentum-dependent parton distributions from lattice QCD. Michael Engelhardt New Mexico State University Transverse momentum-dependent parton distributions from lattice QCD Michael Engelhardt New Mexico State University In collaboration with: B. Musch P. Hägler J. Negele A. Schäfer Lattice theorists go shopping...

More information

The cosmological constant puzzle

The cosmological constant puzzle The cosmological constant puzzle Steven Bass Cosmological constant puzzle: Accelerating Universe: believed to be driven by energy of nothing (vacuum) Vacuum energy density (cosmological constant or dark

More information

The asymptotic-safety paradigm for quantum gravity

The asymptotic-safety paradigm for quantum gravity The asymptotic-safety paradigm for quantum gravity Astrid Eichhorn Imperial College, London Helsinki workshop on quantum gravity June 1, 2016 Asymptotic safety Generalization of asymptotic freedom (! Standard

More information

Kern- und Teilchenphysik II Lecture 1: QCD

Kern- und Teilchenphysik II Lecture 1: QCD Kern- und Teilchenphysik II Lecture 1: QCD (adapted from the Handout of Prof. Mark Thomson) Prof. Nico Serra Dr. Marcin Chrzaszcz Dr. Annapaola De Cosa (guest lecturer) www.physik.uzh.ch/de/lehre/phy213/fs2017.html

More information

QCD and Rescattering in Nuclear Targets Lecture 2

QCD and Rescattering in Nuclear Targets Lecture 2 QCD and Rescattering in Nuclear Targets Lecture Jianwei Qiu Iowa State University The 1 st Annual Hampton University Graduate Studies Program (HUGS 006) June 5-3, 006 Jefferson Lab, Newport News, Virginia

More information

Quantum ChromoDynamics (Nobel Prize 2004) Chris McLauchlin

Quantum ChromoDynamics (Nobel Prize 2004) Chris McLauchlin Quantum ChromoDynamics (Nobel Prize 2004) Chris McLauchlin Outline The Four Fundamental Forces The Strong Force History of the Strong Force What These People Did Experimental Support 1 Fundamental Forces

More information

Critical exponents in quantum Einstein gravity

Critical exponents in quantum Einstein gravity Critical exponents in quantum Einstein gravity Sándor Nagy Department of Theoretical physics, University of Debrecen MTA-DE Particle Physics Research Group, Debrecen Leibnitz, 28 June Critical exponents

More information

Physique des Particules Avancées 2

Physique des Particules Avancées 2 Physique des Particules Avancées Interactions Fortes et Interactions Faibles Leçon 6 Les collisions p p (http://dpnc.unige.ch/~bravar/ppa/l6) enseignant Alessandro Bravar Alessandro.Bravar@unige.ch tél.:

More information

The QCD equation of state at high temperatures

The QCD equation of state at high temperatures The QCD equation of state at high temperatures Alexei Bazavov (in collaboration with P. Petreczky, J. Weber et al.) Michigan State University Feb 1, 2017 A. Bazavov (MSU) GHP2017 Feb 1, 2017 1 / 16 Introduction

More information

QCD β Function. ǫ C. multiplet

QCD β Function. ǫ C. multiplet QCD β Function In these notes, I shall calculate to 1-loop order the δ counterterm for the gluons and hence the β functions of a non-abelian gauge theory such as QCD. For simplicity, I am going to refer

More information

Evolution of 3D-PDFs at Large-x B and Generalized Loop Space

Evolution of 3D-PDFs at Large-x B and Generalized Loop Space Evolution of 3D-PDFs at Large-x B and Generalized Loop Space Igor O. Cherednikov Universiteit Antwerpen QCD Evolution Workshop Santa Fe (NM), 12-16 May 2014 What we can learn from the study of Wilson loops?

More information

Dynamic Instability of the Standard Model and the Fine-Tuning Problem. Ervin Goldfain

Dynamic Instability of the Standard Model and the Fine-Tuning Problem. Ervin Goldfain Dynamic Instability of the Standard Model and the Fine-Tuning Problem Ervin Goldfain Photonics CoE, Welch Allyn Inc., Skaneateles Falls, NY 13153, USA Email: ervingoldfain@gmail.com Abstract The Standard

More information