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Advanced Quantum Field Theory

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Advanced Quantum Field Theory Advanced Quantum Field Theory Doctoral School in Physics, EPFL Prof. Claudio Scrucca Contents 1 Basic formalism for interacting theories 4 1.1 Perturbativeapproach . 4 1.2 Correlationfunctions. ... 5 1.3 Diagrammatics .................................. 6 1.4 AsymptoticstatesandS-matrix. .... 6 1.5 Renormalization ................................. 9 1.6 Vacuum amplitude and generating functional . ....... 11 1.7 Vacuum energy and connected generating functional . ......... 12 1.8 Effective action and 1PI generating functional . ........ 13 1.9 Path-integral representation . ...... 14 2 Path integral and quantum effective action 16 2.1 Saddle-point evaluation of the effective action . ......... 16 2.2 Effective vertices and effective potential . ...... 19 2.3 Symmetry breaking and Goldstone theorem . ..... 19 2.4 Leading quantum corrections and determinants . ........ 21 2.5 World-lineformalism. 23 3 Renormalization group and running couplings 28 3.1 Renormalization at an arbitrary scale . ...... 28 3.2 Dimensionlesscouplings . 29 3.3 Computation of the renormalization group functions . .......... 31 3.4 Runningcouplings ................................ 33 3.5 Schemedependence................................ 36 3.6 Effectofmassparameters . .. .. .. .. .. .. .. 37 3.7 Minimalsubtractionschemes . 38 3.8 Resummation of leading logarithms . ..... 39 3.9 Effectiveaction .................................. 42 1 4 Symmetry breaking and quantum corrections 43 4.1 The O(N)sigmamodel ............................. 43 4.2 Diagrammatic computation of β, γ and γm .................. 44 4.3 Effectivepotential ................................ 45 4.4 Renormalization and counter-terms . ...... 46 4.5 Renormalization group analysis . ..... 47 4.6 Radiativesymmetrybreaking . 48 5 Yang-Mills gauge theories 51 5.1 Gauge-fixing, ghosts and Feynman rules . ..... 51 5.2 BRSTsymmetry ................................. 56 5.3 Diagrammatic computation of β. ........................ 58 5.4 Effectiveaction .................................. 61 6 Effective theories 67 6.1 Low-energy effective theories . 67 6.2 Matching of parameters in a simple example . ...... 69 6.3 Thresholdeffectsandrunning. 73 6.4 Effective action for scalar electrodynamics . ........ 75 6.5 Effective action for spinor electrodynamics . ........ 80 6.6 Pair-production by constant electromagnetic fields . ........... 84 7 Symmetries and anomalies 87 7.1 Classical symmetries and N¨other’s theorem . ........ 87 7.2 Quantum symmetries and Ward identities . ..... 90 7.3 Regularization and anomalies . 93 7.4 Anomalies in global chiral symmetries . ...... 94 7.5 Generalizations ................................. 100 7.6 Anomalies in local gauge symmetries . 101 7.7 Generalstructureofanomalies . 104 7.8 Anomaliesinthestandardmodel . 106 7.9 Scale anomaly and renormalization group functions . ..........109 8 Thermal field theory 116 8.1 Quantum statistical mechanics . 116 8.2 Finite temperature and Euclidean space . 117 8.3 Matsubaramodes.................................119 8.4 Freeenergyoffreefields .. .. .. .. .. .. .. .. 120 8.5 Finite temperature effective potential . .......123 2 9 Tunneling and vacuum decay 127 9.1 Semi-classical tunneling in quantum mechanics . .........127 9.2 Path-integral description of tunneling processes . ............128 9.3 Tunnelinginquantumfieldtheory . 131 9.4 Vacuumdecayandbubbles . 133 3 1 Basic formalism for interacting theories 1.1 Perturbative approach Consider an interacting theory with fields φ(x) and split the Hamiltonian into a free part and an interaction part depending on some small coupling λ: H = H0 + Hint . (1.1) Suppose then that Hint can be treated as a perturbation with respect to H0, in the sense that the spectrum of H is qualitatively similar to that of H0. Call then: 0 : vacuum of H0 with E0 = 0 , | i (1.2) ( n : particle states of H0 with En > 0 , | i and Ω : vacuum of H with EΩ = 0 , | i 6 (1.3) ( N : particle states of H with EN > EΩ . | i Starting from the free theory and switching on the interactions, there can be transitions occurring between states with non-zero probability. However, also the states themselves are changed by the interactions. Perturbation theory is set up in the interaction picture, which is defined from the Heisenberg picture through a time-dependent unitary transformation: iH0(t−t0) −iH(t−t0) U(t,t0)= e e . (1.4) At the reference time t , the two pictures coincide, but the evolution to t = t of the 0 6 0 operators and the states is different in the two pictures, although all the expectation values are the same. Denote then the interaction picture fields, whose evolution is controlled by the free Hamiltonian H0, as iH0(t−t0) −iH0(t−t0) φI (t, ~x)= e φ(t0, ~x)e = field depending on free creation and annihilaton operators , (1.5) and the Heisenberg picture fields, whose evolution is controlled by the full Hamiltonian H, as iH(t−t0) −iH(t−t0) φ(t, ~x)= e φ(t0, ~x)e = field depending on the full complication of interactions . (1.6) The relation between these fields is: † φ(t, ~x)= U (t,t0)φI (t, ~x)U(t,t0) . (1.7) The states evolve with the operator U(t,t0) in the interaction picture and are time inde- pendent in the Heisenberg picture. 4 In order to be able to use efficiently the interaction picture, we need in practice to express the evolution operator U(t,t0) in terms of the fields φI (x). To do so, we start from the differential equation that this operator satisfies, which reads: ∂ i U(t,t )= H (t)U(t,t ) , (1.8) ∂t 0 I 0 in terms of the interaction Hamiltonian in the interaction picture, which is a function of the fields φI (x): iH0(t−t0) −iH0(t−t0) HI (t)= e Hinte . (1.9) We can now solve this equation with the boundary condition U(t0,t0) = 1 by first con- verting it to an integral equation and then solving this by infinite iteration. The result can be expressed in a very compact form as a time-ordered product: t U(t,t )= T exp i dt′H (t′) . (1.10) 0 − I Zt0 This can be generalized to U(t2,t1) with arbitrary times, defined as iH0(t2−t0) −iH(t2−t1) −iH0(t1−t0) U(t2,t1)= e e e . (1.11) This satisfies the same differential equation with the boundary conditon U(t,t) = 1, and its form in terms of the fields φI (x) reads: t2 U(t ,t )= T exp i dt′H (t′) . (1.12) 2 1 − I Zt1 1.2 Correlation functions The basic objects that one wants to compute are the correlation functions defined by the vacuum expectation value of time-ordered products of fields: φ(x ) φ(x ) = Ω T φ(x ) φ(x ) Ω . (1.13) h 1 · · · n i h | 1 · · · n | i In order to compute these objects, we have to relate them to expectation values of the fields φ (x) on the free vacuum 0 , which can be easily computed. We need then to I | i express the fields φ(x) in terms of the field φ (x), and find a way to relate the vacuum Ω I | i in terms of the free vacuum 0 . This can be done by using the trick of acting on Ω with | i | i the Heisenberg evolution operator e−iHT at very large times T , in a slightly imaginary direction, in such a way to select its component along the lowest-lying free state 0 . More | i precisely, we take: T + (1 iǫ) . (1.14) → ∞ − Using the completeness of the set of states N and the fact that Ω is among these the | i | i one with lowest energy, one deduces that in this limit e−iHT = e−iEΩT Ω Ω . Applying | ih | this kind of result to 0 , one finds that: | i eiEΩ(t0+T/2) eiEΩ(T/2−t0) Ω = U(t , T/2) 0 , Ω = 0 U(T/2,t ) . (1.15) | i Ω 0 0 − | i h | 0 Ω h | 0 h | i h | i 5 One computes then: eiEΩT Ω Ω = 0 U(T/2, T/2) 0 , (1.16) h | i Ω 0 2 h | − | i |h | i| and eiEΩT Ω T φ(x ) φ(x ) Ω = 0 T φ (x ) φ (x )U(T/2, T/2) 0 . (1.17) h | 1 · · · n | i Ω 0 2 h | I 1 · · · I n − | i |h | i| Using the normalization condition Ω Ω = 1, one finally deduces that the correlation h | i functions can be expressed as: T/2 0 T φI (x1) φI (xn)exp i dt HI(t) 0 h | · · · − −T/2 | i φ(x ) φ(x ) = Z . (1.18) h 1 · · · n i T/2 0 T exp i dt H (t) 0 h | − I | i Z−T/2 Note moreover that the value of the vacuum energy EΩ can be deduced from the T - dependent exponent in the expression for Ω Ω , and takes the form: h | i i T/2 E = log 0 T exp i dt H (t) 0 . (1.19) Ω T h | − I | i Z−T/2 1.3 Diagrammatics Expanding the exponential of the evolution operator and using Wick’s theorem, one can evaluate the correlation functions in a perturbative expansion in powers of the coupling λ, leading to Feynman diagrams. The result is that: T/2 sum of 0 T exp i dt H (t) 0 = exp , (1.20) h | − I | i disc. diag. Z−T/2 T/2 sum of conn. sum of 0 T φ (x ) φ (x )exp i dt H (t) 0 = exp . h | I 1 · · · I n − I | i diag. with x disc. diag. Z−T/2 i It follows then that sum of conn. φ(x ) φ(x ) = , (1.21) h 1 · · · n i diag. with x i and also that i sum of E = . (1.22) Ω T disc. diag. 1.4 Asymptotic states and S-matrix In order to compute cross sections, we need to define asymptotic in and out states cor- responding to isolated particles of definite momentum and polarization in the far past 6 and far future of a collision event.
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