Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany

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Introduction to Effective Quantum Field Theories Thomas Mannel Theoretical Physics I (Particle Physics) University of Siegen, Siegen, Germany

2nd Autumn School on High Energy Physics and Quantum Field Theory Yerevan, Armenia, 6-10 October, 2014

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Overview

Lecture 1: Basics of Quantum Field Theory Generating Functionals Functional Integration Perturbation Theory Renormalization Lecture 2: Effective Field Thoeries Effective Actions Effective Lagrangians Identifying relevant degrees of freedom Renormalization and Renormalization Group

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization

Lecture 3: Examples @ work From Standard Model to Fermi Theory From QCD to Heavy Quark Effective Theory From QCD to Chiral Perturbation Theory From New Physics to the Standard Model Lecture 4: Limitations: When Effective Field Theories become ineffective Dispersion theory and effective ﬁeld theory Bound Systems of Quarks and anomalous thresholds When quarks are needed in QCD É.

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization

Lecture 1: Basics of Quantum Field Theory

Thomas Mannel

Theoretische Physik I, Universität Siegen

f q f et Yerevan, October 2014

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Contents

1 Generating Functionals Green’s Functions Connected Green’s Functions Vertex Function, Effective Action 2 Functional Integration Quantum Mechanics Quantum Field Theory Perturbation Theory 3 Renormalization One Loop Renormalization Power Counting Renormalization Theory

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization Introduction

Quantum Mechanics ⊕ Special Relativity = Quantum Field Theory Standard Approach: Canonical Quantization Field Operators Φ(x), conjugate momenta Π(x) with

3 ~ ~ [Φ(x), Π(y)]x0=y0 = i~δ (x − y)

Use this for a (perturbative) calculation of Correlation Functions (or Green’s functions)

Gn(x1, x2, ..., xn) = h0|T [Φ(x1)Φ(x2)...Φ(xn)]|0i

with the ground state |0i and time ordering T .

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Functional Integration Renormalization

Scattering amplitudes S are related to Gn via the LSZ formalism Usual time-dependent perturbation theory to generate a perturbative expansion ... I assume you know this part of the story

However, if we knew all the Gn, we would know everything about this particular ﬁeld theory! For our purpose a different Ansatz will be pursued:

Functional Integrals to generate the Gn

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Green’s Functions Functional Integration Connected Green’s Functions Renormalization Vertex Function, Effective Action Generating Functionals

A Functional is a map from a space of functions into (real) numbers

J(x) → F = F[J]

Generating functional for a set Gn(x1, ..., xn) ∞ X 1 Z Z [J] = d 4x d 4x ...d 4x G (x , ..., x )J(x )J(x )...J(x ) n! 1 2 n n 1 n 1 2 n n=0

Recover any Gn by functional differentiation δ δ Gn(x1, ..., xn) = ... Z [J] (1) δJ(x1) δJ(xn) J=0

Show (1) use the following properties of the functional derivative: δ J(y) = δ4(x − y) δJ(x) δ δF[J] δG[J] (F[J] G[J]) = G[J] + F[J] δJ(x) δJ(x) δJ(x)

For Quantum Field Theory: If we knew Z[J], we would know everything about this theory! There are other generating functionals that can be derived form Z [J]

W [J] = exp(Z [J]) generates the Connected Green’s Functions, e.g.

h0|T [Φ(x1)Φ(x2)Φ(x3)Φ(x4)]|0i

= h0|T [Φ(x1)Φ(x2)Φ(x3)Φ(x4)]|0iconnected

+h0|T [Φ(x1)Φ(x2)]|0i h0|T [Φ(x3)Φ(x4)]|0i

+permutations of x1, x2.x3.x4

Usually we deﬁne

h0|Φ(x)|0i = h0|T [Φ(x)]|0i = 0

h0|T [Φ(x)Φ(y)]|0i = h0|T [Φ(x)Φ(y)]|0iconnected

Still another generating functional: Irreducible Vertex functions or Effective Action: Legendre Transformation Z iΓ[φ] = W [J] − i d 4x J(x)φ(x) with δ φ(x) = (−i) W [J](to be solved forJ) δJ(x)

Show that δ Γ[φ] = J(x) δφ(x) Hint: use the deﬁnition of φ δ φ(x) = (−i) W [J] δJ(x)

Γ[Φ] is a very useful quantity:

Its individual contributions Γn X 1 Z Γ[Φ] = d 4x ...d 4x Γ (x , ...x )Φ(x )...Φ(x ) n! 1 n n 1 n 1 n n are the One Particle Irreducible Vertex Functions. One-Particle-Irreducible: A Diagram is called One Particle Reducible, if it can be split into two pieces by cutting one internal line.

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory Feynmans Approach to Quantum Theory

A quantum mechanical amplitude is constructed by the “sum” over all trajectories a particle can go from point q1 to q2, weighted by a phase proportional to the action of the trajectory

X −i A(q1, q2) ∼ exp S[q] all Paths ~

S[q]: Action functional depending on the trajectory q(t) with q(t1) = q1 and q(t2) = q2

Z t2 S[q] = dt L(q(t), q˙ (t), t) t1

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory

Mathematically speaking: The sum is a functional integral (assigning a number to a functional) X −i Z −i exp S[q] = [dq] exp S[q] all Paths ~ ~ [dq] Integration measure in the space of functions, in this case the trajectories q(t). It can be shown that for quantum mechanics, the integration measure −i [dq] exp S[q] ~ is well deﬁned for all smooth potentials, and thus this formulation is equivalent to the canonical one.

We just transfer this program to QFT The classical ﬁeld theory be deﬁned by its Lagrangian Density Z 3 L = d ~x L(Φ, ∂µΦ)

The Action Functional is a function of the ﬁeld Z 4 S[Φ] = d x L(Φ, ∂µΦ)

Quantization is done via Z −i A ∼ [dΦ] exp S[Φ] ~

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory

To be more precise: The generating functional Z [J] of QFT can be written as a functional integral: Z Z Z[J] = [dΦ] exp −iS[Φ] + i d 4x Φ(x)J(x)

Remarks Normalization: Z[0] = R [dΦ] exp (−iS[Φ]) = 1 One point function:

(−i) δ Z[J] = R [dΦ] Φ(x) exp (−iS[Φ]) = 0 δJ(x) J=0 Two point function: (−i) δ δ Z[J] = δJ(x) δJ(y) J=0 R [dΦ] Φ(x)Φ(y) exp (−iS[Φ]) h0|T [Φ(x)Φ(y)]|0i = D−1(x, y)

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory

Functional Integrals in QFT are more subtle: The general integral measure is mathematically ill deﬁned ... only Gaussian measures are well deﬁned: Z S[Φ] = d 4x d 4y Φ(x)D(x, y)Φ(y)

with some (symmetric) distribution D(x, y) The Gaussian measures correspond to (generalized) free ﬁelds! More general measures are ill deﬁned, may be related to the necessity of renormalization.

Show that S[Φ] = R d 4x d 4y Φ(x)D(x, y)Φ(y) yields the generating functional for generalized free ﬁelds. Use the following assumptions D has an inverse: R d 4z D(x, z)D−1(z, y) = δ4(x − y) Perform the functional integration using quadratic completion, together with an appropriate shift of the integration variables. Why is the result Z Z [J] = exp d 4x d 4y J(x)D−1(x, y)J(y)

a (generalized) free ﬁeld?

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory

Local Quantum Field Theory: S is the integral over a (local) Lagrangian density Z 4 S[Φ] = d x L(Φ(x), ∂µΦ(x))

The Lagrangian Density is usually split up as

1 1 L(Φ, ∂ Φ) = (∂ Φ)(∂µΦ)− m2Φ2−V (Φ) = L −V (Φ) µ 2 µ 2 0 the potential is usually written as a power series in Φ g g V (Φ) = 3 Φ3 + 4 Φ4 + ... 3! 4!

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory

The generalization to more than one field is straightforward → For every field we introduce a source J → Complex fields require also complex sources Including fermions is nontrivial, since we need to introduce anticommuting variables as filelds → Anticommuting source terms → Anticommuting differentiation with respect to the anticommuting sources Gauge fields introduce a few more subtleties...

Simple and closed formulation of perturbation theory: A functional differentiation generates a ﬁeld

Z Z Z [J] = [dΦ] exp −i d 4x [L − Φ(x)J(x)]

Z Z δ = [dΦ] exp −i d 4xV −i δJ(x) Z 4 exp −i d x [L0 − Φ(x)J(x)] Z δ = exp −i d 4xV −i Z [J] δJ(x) 0

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory

Z0[J] is a Gaussian integral

1 Z Z [J] = exp d 4x d 4y J(x)D (x, y)J(y) 0 2 F with the Feynman Propagator

Z d 4p −i D (x, y) = e−ip(x−y) F (2π)4 p2 − m2 + i

Perturbative expansion in powers of the gi from the expansion of the exponential with the potential Feynman Diagrams That would be it, if there were not divergent expressions

Consider the Equation of Motion with source term:

2 0 ( + m )φJ + V (φJ ) = J

with the classical solution φJ (with φJ = 0 for J = 0)

Shift the functional integration variable: Φ → Φ + φJ , thus Z[J] = exp (iSJ [φJ ]) exp (iQ[φJ ]) with Z 1 1 S [φ ] = d 4x (∂ φ )(∂µφ ) − m2φ2 − V (φ ) − Jφ J J 2 µ J J 2 J J J and Q represents quantum corrections

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory

Some intermediate steps:

L(Φ + φJ , ∂µ[Φ + φJ ]) + J[Φ + φJ ] = L(φJ , ∂µφJ ) + JφJ

1 µ 1 2 2 0 + (∂µΦ)(∂ Φ) − m Φ − [V (Φ + φ ) − V (φ ) − V (φ )Φ] 2 2 J J J

Note tat the last term is at least quadratic in the ﬁeld Φ

1 V (Φ + φ ) = V (φ ) + V 0(φ )Φ + V 00(φ )Φ2 + O(Φ3) J J J 2 J

Consequently we have

exp (iQ[φJ ]) Z Z 4 1 µ 1 2 2 0 = [dΦ] exp −i d x[ (∂µΦ)(∂ Φ) − m Φ − [V (Φ + φ ) − V (φ ) − V (φ )Φ]] 2 2 J J J

The quadratic part is again a Gaussian integral, so this can be performed and yields the one loop effective potential.

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals Quantum Mechanics Functional Integration Quantum Field Theory Renormalization Perturbation Theory

Generating functional of connected Green’s functions

(−i)G[J] = (−i) ln Z [J] = SJ [φJ ] + Q[φJ ]

Effective action: Legendre Transformation Z δG[J] 4 δQ[φJ ] δφJ (y) Φ(x) = (−i) = φJ (x)+(−i) d y δJ(x) δφJ (y) δφJ (x) the last term is a quantum correction. Effective Action:

Z Γ[Φ] = (−i)G[J] − d 4x J(x)Φ(x)

= S0[Φ] + Quantum Corrections

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory Renormalization

Power Counting (of mass dimensions): The Action functional S is dimensionless! Conseqences: The dimension of the Lagrangian is dim[L] = 4 The dimension of a scalar field is dim[Φ] = 1 The dimension of a fermion field is dim[ψ] = 3/2 ... The dimension of a field with spin s is dim[ψ] = 1 + s Note 1: This looks strange for higher spins, but it can be derived from the propagator Note 2: Gauge Symmetries reduce the degree of divergence

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

Exercise 4: Dimension of a Vector ﬁeld Vµ

Show that dim[Vµ] = 2 Use the propagator of the massive vector ﬁeld

k k 1 D (k) = (−i) g − µ ν µν µν M2 k 2 − M2

and look at the Fourier transform

F.T . Dµν(k) −→ h0|T [Vµ(x)Vν(0)]|0i

At tree level, all Feynman diagrams are ﬁnite: The Effective Action is just the classical action Γ[Φ] = S[Φ] For the Φ4 theory we have Z 1 1 g S[Φ] = d x (∂ Φ)(∂µΦ) − m2Φ2 − 4 Φ4 2 µ 2 4!

→ only Γ2 6= 0 and Γ4 6= 0 at tree level:

2 4 Γ2(x, y) = −( + m )δ (x − y) 4 4 4 Γ2(x, y, z, w) = g4δ (x − y)δ (x − z)δ (x − w) All other Feynman Diagrams are 1P reducible

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

At one loop level, divergent diagrams appear:

The effective action contains all terms Γn 4 Φ theory: Only Γ2 and Γ4 contain divergent terms The structure of these divergent terms is:

4 Γ2(x, y) = −(a + b)δ (x − y) 4 4 4 Γ2(x, y, z, w) = cδ (x − y)δ (x − z)δ (x − w)

with divergent coefﬁcients a, b, c

At one loop, there is always only a ﬁnite number of Γn plagued with this problem

Renormalization = Redeﬁnition of the “bare parameters“ Bare parameters = Parameters that have been introduced at tree level Idea of Renormalizaton: The sum of the bare parameter and the divergent term yields the observed value of the parameter

Observed mass squared = m2 + b

Observed coupling = g4 + c Normalization of the Field = 1 + a

(this comes more precisely below)

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

At two loops, this systematics continue

The effective action contains all terms Γn 4 Φ theory: Again only Γ2 and Γ4 are divergent However, in general is can be that new structures of divergent terms appear This distinguishes super-renormalizable, renormalizable and non-renormlizable theories (more precisely deﬁned below)

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

Power Counting, continued: Expansion of V (Φ): “Vertices”

g3 g4 X gn V (Φ) = Φ3 + Φ4 + ··· = Φn 3! 4! n! n≥3

Dimension of the monomials: dim[Φn] = n Possible generalization: V could also contain derivatives: dim[∂mΦn] = n + m 2 2 2 Examples: dim[Φ Φ ] = 6, dim[Φ(∂µΦ )(∂µΦ)] = 6 etc. dim[L] = 4 thus the corresponding couplings have to carry dimensions:

dim[gn] = 4 − n

Super-Renormalizable: A QFT is Super-Renormalizable, if all couplings have positive dimension Only a finite number of Feynman diagrams are divergent Renormalizable: A QFT is Renormalizable, if all couplings have non-negative dimension Infinitely many divergent diagrams, but contained in only a finite set of the Γn Only a finite number of inputs needed for renormalization, → predictive

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

Non-Renormalizable: A QFT is Non-Renormalizable, if at least on elf the couplings has negative dimension

infinitely many divergent diagrams, appearing in all Γn At first sight, no predictive power, since infinitely many inputs are needed to fix the Γn However,

The negative dimension of gn in the non-renormalizable term(s) corresponds to a mass scale Expansion in this mass scale: Only a ﬁnite number of insertions of non-renormalizable terms is required! This can be dealt with!

Consider the generating functional Z [J], which depends also on the coupling parameters m and gi

Z [J] = Z[J; m, gi ]

In a renormalizable theory all Green’s functions can be rendered finite by a redefinition of the fields (equivalent to a redefinition of the source term J) and a redefinition of the coupling parameters m and gi :

1 2 2 J = √ Jr m = Zmmr gi = Zgi gi,r ZΦ

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

These redeﬁnitions generate the appropriate counter terms to render the Green’s functions ﬁnite. Example:

Z Z 4 Z [J; m, g4] = [dΦ] exp −i d x 1 µ 2 1 2 1 4 (∂µΦ)(∂ Φ) + m Φ + Φ g − Φ(x)J(x) −→ 2 2 4! 4 Z Z 1 2 4 Z [ √ Jr ; Zmmr , Zg gi,r ] = [dΦ] exp −i d x ZΦ 1 µ 2 1 2 1 4 1 (∂µΦ)(∂ Φ) + Zmmr Φ + Φ Zg g4,r − Φ(x) √ Jr (x) 2 2 4! ZΦ Z Z 4 = [dΦr ] exp −i d x 1 µ 2 1 2 1 4 2 ZΦ (∂µΦr )(∂ Φr ) + ZΦZmm Φ + Φ Z Zg g , − Φr (x)Jr (x) 2 r 2 r 4! r Φ 4 r √ with a redeﬁnition of the integration variable Φ = ZΦΦr

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

This construction generate all necessary counter terms to render all Green’s functions ﬁnite. For Φ4 theory the counter terms needed are: 1 1 1 L = a (∂ Φ)(∂µΦ) + m2b Φ2 + c Φ4 C 2 µ 2 4! and thus at one loop

2 2 2 ZΦ = (1+a) ZΦZmmr = (1+b)m ZΦZg = (1+c) This Program holds in a renormalizable theory to any loop order, i.e. to any order in the perturbative expansion. The Renormalization procedure is deﬁned in perturbation theory.

In a renormalizable theory only a finite number of effective vertices are divergent Once the counter term have removed the divergent parts, the finite part need to be fixed by renormalization conditions 4 Example: The coupling constant g4 of Φ theory: 4 Consider the Φ vertex in the effective action Γ4 Fix the value of the Fourier Transform of Γ4 Z ˆ 4 4 4 4 ip1x1 ip2x2 ip3x3 ip4x4 Γ4(p1, p2, p3, p4) = d x1 d x2 d x3 d x4 Γ4(x1, x2, x3, x4)e e e e

2 2 2 at some normalization point, e.g. pi = 0 and (p1 + p2) = µ ˆ 2 Γ4(p1, p2, p3, p4)normalization point ≡ g4(µ )

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

Remark 1: Renormalization always introduces a (mass) scale: Renormalization scale µ Remark 2: Likewise for the other divergent function Γ2(x1, x2) Remark 3: All the parameters depend on µ: m(µ) and g4(µ) Fourier Transform of the Renormalized vertex functions:

ˆ 2 4 (n/2) ˆ 2 4 Γn(p1, ..., pn; m , g , Λ) = ZΦ (Λ, µ)Γn,r (p1, ..., pn; m (µ), g (µ)) with a cut off Λ to regularize.

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

Remark 4: µ is arbitrary; l.h.s. does not depend on µ. d h i 0 = µ Z (n/2)(Λ, µ)Γˆ (p , ..., p ; m2(µ), g (µ), µ) dµ Φ n,r 1 n 4 This leads to the Renormalization Group Equation:

∂ (n/2) 2 (n/2) d 2 µ Z Γˆn,r (p , ..., pn; m (µ), g (µ), µ) + Z µ Γˆn,r (p , ..., pn; m (µ), g (µ), µ) ∂µ Φ 1 4 Φ dµ 1 4 n (n/2) ∂ ˆ 2 = Z µ ln ZΦ Γn,r (p , ..., pn; m , g , µ) 2 Φ ∂µ 1 4 " 2 # (n/2) dm ∂ 2 +Z µ Γn,r (p , ..., pn; m , g , µ) Φ dµ ∂m2 1 4 (n/2) dg4 ∂ 2 +ZΦ µ Γn,r (p1, ..., pn; m , g4, µ) dµ ∂g4

(n/2) ∂ 2 +Z µ Γn,r (p , ..., pn; m , g , µ) = 0 Φ ∂µ 1 4

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

Renormalization Group Equation ∂ ∂ ∂ n µ + β + β + γ Γ ( , ..., ; 2, , µ) = m 2 Φ n,r p1 pn m g4 0 ∂µ ∂m ∂g4 2 Renormalization Group Functions Beta function for the mass: dm2 β (µ) = µ = β (g (µ)) m dµ m 4 Beta function for the coupling: dg β(µ) = µ 4 = β(g (µ)) dµ 4 Anomalous dimension of the ﬁeld: ∂ γ (µ) = µ ln Z = γ (g (µ)) Φ ∂µ Φ Φ 4 (assuming a “mass independent scheme”)

Renormalization necessarily introduces a scale In the case at hand: Renormalization scale µ Large momentum behavior from RG study of the massless case: ∂ ∂ n µ + β + γΦ Γn,r (p1, ..., pn; 0, g4, µ) = 0 ∂µ ∂g4 2 Dimensional analysis: Scale dimensional quantities: d Γn,r (λp1, ..., λpn; 0, g4, λµ) = λ Γn,r (p1, ..., pn; 0, g4, µ)

(d: Mass dimension of Γi ) or (Euler Eq.) " # ∂ X ∂ µ + pν + d Γ (p , ..., p ; 0, g , µ) = 0 ∂µ i ∂pν n,r 1 n 4 i i

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

Insert RGE " # X ν ∂ ∂ n pi ν − β + d − γΦ Γn,r (p1, ..., pn; 0, g4, µ) = 0 ∂p ∂g4 2 i i

Modiﬁed Scaling through renormalization Scale generation through renomalization large momentum behavior and renormalization Rationale for the name “anomalous dimension” A renormlizable theory is scalable to inﬁnitely high scales! Parametrization of ignorance (about the high scales).

One can insert Local Composite Operators Finite number of insertions! Let O be a composite operator, dim[O] = k Z Z Z [J, R] = [dΦ] exp −iS[Φ] + i d 4x [Φ(x)J(x) + R(x)O(x)]

Differentiation with respect to R generates Green’s functions with O insertions

h0|T [Φ(x1)...Φ(xn)O(y)]|0i

However, this is a bit too simple ...

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1 Generating Functionals One Loop Renormalization Functional Integration Power Counting Renormalization Renormalization Theory

Renormalization requires to introduce the full set of all (lin. independent) operators Oi of the same dim. Z Z 4 Z [J, Ri ] = [dΦ] exp −iS[Φ] + i d x [Φ(x)J(x) + Ri (x)Oi (x)]

“Mixing” under renormalization

Ri (x)Oi (x) → Rj (x)Zji Oi (x)

Renormalization group equation (massless case)

∂ ∂ n Oi µ + β + γΦ + γji Γn,r (p1, ..., pn, q; 0, g4, µ) = 0 ∂µ ∂g4 2

Anomalous dimension matrix γij (see below)

Theories with coupling constants with non-negative mass dimension are renomalizable (Caveat: Higher Spin particles) Greens functions with a ﬁnite number of insertions of higher dimensional operators can be renormalized Operators with the same dimension mix under renormalization Renormalization group can be used to investigate the behavior of the theory at high scales.

T. Mannel, Siegen University Effective Quantum Field Theories: Lecture 1