April 6, 2020 version 2.0

Quantum Field Theory II Lectures Notes

Part II: Renormalization

Prof. Dr. Gino Isidori ETH & UZH, Spring Semester 2020

i ii Contents

2 Renormalization 1 2.1 Systematics of Renormalization ...... 1 2.1.1 General aspects of ultraviolet divergences ...... 2 2.1.2 Renormalization of the λφ4 theory at the one-loop level ...... 5 2.1.3 Renormalised perturbation theory and counterterms ...... 10 2.2 The Renormalization Group ...... 12 2.2.1 Wilson’s approach to renormalization ...... 12 2.2.2 The renormalization group (Wilsonian approach) ...... 17 2.2.3 Renormalization scale and the Callan-Symanzik equation ...... 19 2.2.4 Solutions of the Callan-Symanzik equation ...... 22 2.2.5 Running coupling ...... 25 2.2.6 Callan-Symanzik equation with local operators and mass terms . . . . . 27

iii iv Chapter 2

Renormalization

2.1 Systematics of Renormalization

Within QFT, when computing correlation functions in perturbation theory beyond the tree- level, one often encounters divergent integrals. As you have already seen in QFT-I, the type of divergences one can encounter are of two types: infrared (IR) and ultraviolet (UV) ones. The divergences appear because the system we are considering has an inﬁnite number of degrees of freedom. In particular, the infrared divergences are due to the inﬁnite volume limit, that implies we have to consider ﬁeld excitations with arbitrary long wave length (unsuppressed in the case of massless ﬁelds). The ultraviolet divergences are related to the continuum limit, that implies we have to consider ﬁeld excitations with arbitrary short wave length. However, both IR and UV divergences are an artefact of perturbation theory: they do not show up in well-deﬁned physical observables. The problem arises because of the non-trivial relation between physical observables (and asymptotic states of the theory) in terms of free ﬁelds and Lagrangian parameters. Dealing with these artefacts requires, in general, a three- step procedure:

i. regularization of the divergences by means of an appropriate IR and UV cut-oﬀ (such as a ﬁnite volume for the IR divergences, and a ﬁnite lattice spacing or a maximal momentum for the UV divergences, or the more abstract dimensional regularization which can cure both type of divergences);

ii. correct identiﬁcation of physical observables;

iii. renormalization of the Lagrangian parameters.

In the theories we are interested in, the ﬁrst two steps are suﬃcient to deal with IR divergences. In orther words, IR divergneces (or better the dependence of correlation functions from the IR cut-oﬀ) cancels automatically when considering realistic observables (in particular when taking into account that we cannot distinguish processes which diﬀer by the emission of quanta with inﬁnitesimal small energy). On the other hand, taming UV divergences usually requires also the last step, i.e. the renormalization of the Lagrangian parameters. In this section we discuss how to deal with this problem in a systematic way.

1 2.1.1 General aspects of ultraviolet divergences

As we will see explicitly in the case of the λφ4 theory and in the case of QED (→ exercise class), the ultraviolet divergences generated at the one-loop level can be “cured” by a re- deﬁnition of the so-called “bare” parameters, i.e via a re-deﬁnition of the parameters appearing in the Lagrangian. More explicitly, in the λφ4 theory we need to renormalize the coupling (λ), the mass (m), and the ﬁeld (i.e. we deﬁne a renormalized ﬁeld, in terms of the “bare” ﬁeld appearing in the Lagrangian). Two questions naturally arise: • Is it procedure possible in any QFT? • What happens when going to higher order loops? To address these questions in general terms, it is convenient to introduce the concept of super- ﬁcial degree of divergence (D) of a Feynman diagram in d dimension, which is deﬁned as

D = d · L − 2P = d · L − 2Pφ − Pψ − 2Pγ (2.1) where d is the space-time dimensions, L is the number of loops and P is the number of propa- gators of a given type (for scalar, fermion, and gauge ﬁelds). This relation follows from the fact that each loop brings an integral over ddk and each propagator brings k (1 power for fermions, 2 for bosons) in the denominator. Let us consider an example: a scalar theory with Lagrangian, 1 1 λ L = (∂ φ)2 − m2φ2 − n φn . (2.2) 2 µ 2 n! Generally for a fully connected diagram with N external lines and V vertices, each with n lines, we have

L = P − Vn + 1 (2.3)

nVn = N + 2P (2.4)

The ﬁrst equation tells us that the number of independent momenta over which integration takes place (L) is equal to the number of propagators (P ) minus the number of relations between momenta, which is equal to Vn (momentum conservation at each vertex) minus one (otherwise overall momentum conservation is counted twice). The second equation tells us that the total number of lines attached to the vertices (n for each) is the sum of the number of external lines and twice the number of propagators. If we take the above two relations into (2.1), we get

d d D = d − N − 1 + V n − 1 − d . (2.5) 2 n 2

• For n = 4 and d = 4, i.e. for the φ4 interaction in ordinary space-time, the above result implies D = 4 − N (2.6) which is independent from the number of vertices. Hence independently of the number of vertices (or loops) we consider, the superﬁcial degree of divergence of the Feynman

2 diagrams is

= 2 for N = 2 quadratic divergence D = = 0 for N = 4 log divergence < 0 for N > 4 “na¨ıve convergence” • For n = 6, d = 4

D = 4 − N + 2V6 , (2.7) hence each insertion of a vertex, with a φ6 interaction, enhances the degree of divergence. • For n = 3, d = 4

D = 4 − N − V3 , (2.8) hence each insertion of a vertex, with a φ3 interaction, reduces the degree of divergence.

Canonical dimension

The coeﬃcient multiplying V in D is the canonical dimension of the coupling λ . To deﬁne Z n d this concept, note that in natural units (~, c = 1) the action S = d x L is dimensionless. Since the mass/energy dimension of x is d[x] = −1, this imply that the mass/energy dimension of the Lagrangian density is d[L] = d (2.9) Using this property, starting from the kinetic terms (i.e. the quadratic derivative terms in the Lagrangian) of free scalar, fermion and gauge ﬁelds, we can derive the canonical mass/energy dimension of each ﬁeld as

(d = 4) d − 2 ∂ φ∂µφ → 2d + 2 = d ⇒ d = = 1 µ φ φ 2 d − 1 3 ψ¯∂ψ/ → 2d + 1 = d ⇒ d = = ψ ψ 2 2 d − 2 F F µν → 2d + 2 = d ⇒ d = = 1 µν A A 2 Using these results, we can then derive the canonical dimension of each coupling in any inter- action term (note that the mass itself can be considered an interaction):

(d = 4) d m2 φ2 → 2d + 2d = d ⇒ d = − d = 1 φ φ mφ mφ 2 φ ¯ mψψψ → 2dψ + dmψ = d ⇒ dmψ = d − 2dψ = 1 4 λ4φ → 4dφ + 2dλ4 = d ⇒ dλ4 = d − 4dφ = 0 ¯ µ eψγ ψAµ → 2dψ + dA + de = d ⇒ de = d − 2dψ − dA = 0 n d λnφ → ndφ + dλn = d ⇒ dλn = d − n 2 − 1 = 4 − n It is now clear that the result in (2.5), generalised to the case where we consider diﬀerent n interactions of the type λnφ , can be re-written as d X D = d − N − 1 + V · d (2.10) 2 n Vn n

3 where dVn denotes the canonical dimension of the coupling λn. Proceeding in a similar manner, in QED one ﬁnds d d − 1 D = d − − 1 N − N + V · d (2.11) 2 γ 2 ψ e e with de = (d − 2)/2 and where Nγ(ψ) denotes the number of external photon (fermion) lines. A general classiﬁcation of QFTs can thus be inferred by looking at the canonical dimension of the interaction terms:

• dV > 0: super-renormalizable theories (ﬁnite number of divergent diagrams) • dV = 0: renormalizable theories (ﬁnite number of divergent basic amplitudes) • dV < 0: “non-renormalizable” theories (inﬁnite number of divergent basic amplitudes) The λφ4 theory and QED, in d = 4, belongs to the class of renormalizable theories. As can be easily understood: only vey few interaction terms belong two the ﬁrst to cases.

A generic interaction terms with high powers of ﬁelds or derivatives naturally leads to dV < 0. The so-called non-renormalizable theories (or better theories which are not renormalizable in the classical sense) are not necessarily problematic. However, they are theories that cannot be extrapolated to arbitrary high energies. We will discuss this in more detail in the next section, but we can have a ﬁrst understanding of the what happens by the following argument. Consider the n-point correlation function Z iS G(x1, . . . , xn) = h0|T {φ(x1) ··· φ(xn)}|0i = N Dφ φ(x1) ··· φ(xn)e (2.12) expanding the interaction term in perturbation theory brings down powers of λn. Assuming the correlation function corresponds to a process characterised by an energy scale E, then dimensional analysis implies

X p −p·dV G ∼ κn cp × (λn) × E (2.13) p where κn is a (dimensional) proportionality factor, which depends on the number of external ﬁelds but is independent from the perturbative expansion, cp are dimensionless coeﬃcients, and p is the order of the perturbative expansion. If we now look at the evaluation of Feynman diagrams, in addition to the physical energy (E) we have an extra (unphysical) energy scale associated to the maximal momentum/energy in the loops (i.e. the UV cut-oﬀ, ΛUV). If dV < 0, this implies we have more and more divergent integrals the more we expand in the couplings: this is why the theory is said to be non-renormalizable. For concreteness, let us consider the 5 λ5φ theory (where dλ5 = −1) and let’s concentrate on the 4-point function:

Z 2 1 Z A ∼ d4k λ2 ∼ λ2 d2k ∼ λ2[Λ2 + ...] 1 5 (k2)3 5 5 UV | {z } d=0

Z 5 1 Z A ∼ d4k λ4 ∼ λ4 d4k ∼ λ4[Λ4 + ...] 2 5 (k2)8 5 5 UV | {z } d=0

4 The problem of increasing number of divergent terms seems to be dramatic. However, it is g not so if we limit ourself to analyse the amplitude at small energies. Deﬁning λ = , where 5 M now dg = 0, we get something of the type

g2Λ2 g4Λ4 “inﬁnite” g2 g4 A(φ φ → φ φ ) = UV + UV + ··· + × E2 + O × E4 . 1 2 3 4 M 2 M 2 but E-indep. M 2 M 4 | {z } λeﬀ (2.14)

Paying the price of introducing a new coupling (λeﬀ ) and limiting ourself to the small energy g region, deﬁned by the condition E M, we ﬁnd a well-behaved expansion in powers of . M This illustrates that also most of the so-called non-renormalizabile theoreis can be handled (yielding physically interesting results) with a proper renormalization procedure.

2.1.2 Renormalization of the λφ4 theory at the one-loop level

In this section we discuss in detail the explicit example of UV renormalization at the one-loop level in the λφ4 theory. We anlyse how to regularize the divergences, and how they can be removed via a redeﬁnition (renormalization) of ﬁeld and couplings.

Lagrangian and Feynman rules

The Lagrangian density of the λφ4 theory is 1 m2 λ L = ∂ φ ∂µφ − φ2 − φ4 (2.15) 2 µ 2 4! | {z } | {z } Lfree Lint From this we deduce the following Feynman rules in momentum space:

i = p2 − m2 + i 1. For each propagator

= −iλ

2. For each vertex

3. Impose momentum conservation at each vertex Z d4k 4. For each closed loop, integrate over internal momentum (2π)4 5. Divide by symmetry factors

Corrections to the 4-point correlation function

Let us start by calculating the one particle irreducible (1PI) 4-point correlation function up to O(λ2), i.e. including one-loop corrections:

5 p1 p3 p1 p3 p1 p3 p1 p3 p1 p3

= + + +

p2 p4 p2 p4 p p p p p2 p4 2 4 2 4

The amplitude for the second diagram is

p1 k p3 (−iλ)2 Z d4k i i = 2 (2π)4 k2 − m2 + i (k + p)2 − m2 + i = (−iλ)2iV (p2) (2.16)

p2 k + p p4 where the prefactor 1/2 is the symmetry factor.

Introducing the Mandelstam variables,

2 2 2 (p1 + p2) = s, (p3 − p1) = t, (p4 − p1) = u , (2.17) it is easy to realise that p2 in Eq. (2.16) is nothing else but the Mandelstan variable s, and that the amplitudes for the third and the fourth graph are obtained setting p2 = t and p2 = u, respectively. The complete 1PI 4-point correlation function is then

2 iM12→34(s, t, u) = −iλ + (−iλ) [iV (s) + iV (t) + iV (u)] (2.18)

Feynman parametrization to compute the loop integral

Let us proceed to the explicit calculation of V (p2), which is deﬁned in equation (2.16)

1 Z d4k i i iV (p2) = (2.19) 2 (2π)4 k2 − m2 + i (k + p)2 − m2 + i

Using the Feynman parametrisation

1 Z 1 dx = 2 , (2.20) AB 0 [xA + (1 − x)B)] our denominator reads

(k2 − m2)(1 − x) + (k + p)2 − m2 x2 = k2 + p2x − m2 + 2kpx2 . (2.21)

Applying the shift of variable l = k + px to remove the terms linear in k and deﬁning ∆ = m2 − p2x(1 − x), we ﬁnally obtain

Z 4 Z 1 Z Λ 4 Z Λ 3 2 i d l 1 d l dl l V (p ) = 4 dx 2 2 ∼ 4 ∼ 4 ∼ log Λ (2.22) 2 (2π) 0 [l − ∆] l l leading to a logarithmic divergence.

6 Dimensional regularization

Regularizing the divergence by means of an explicit momentum cut-oﬀ, as done in Eq. (2.22), is not always possible (it is not a well-deﬁned procedure when going to higher loops). A rather convenient regularization procedure which can easily be applied to all orders in most theories is the so-called dimensional regularization. This consists in computing the integral in d < 4 dimensions, where the integral converges and where we can perform the Wick rotation from Minkowski to Euclidean space. The parameter ε = 4 − d (not to be confused with the i in the propagator!) becomes the regulator of UV divergences. After rotating tor Euclidean space the amplitude has no poles in momentum space and the divergence appears as a pole in d → 4 (or ε → 0). Applying this to our integral (2.22), we get

Z d Z d Z Z ∞ d−1 d l 1 d lE 1 dΩd lE I ≡ d 2 2 = i d 2 2 = i d dlE 2 2 (2.23) (2π) [l − ∆ + i] (2π) [lE + ∆] (2π) 0 [lE + ∆]

0 0 ~ ~ where in the ﬁrst equality we applied the Wick rotation l = ilE and l = lE, and dΩd is the solid angle in d dimensions. The area of a unit sphere in d dimensions is

Z d Z √ d −x2 − P x2 ( π) = dx e = dx1 ··· dxd e i Z Z ∞ Z Z ∞ d−1 −x2 1 d −1 −t = dΩd dx x e = dΩd dt t 2 e (2.24) 0 2 0 | {z } d Γ( 2 )

Z ∞ where we have used the integral form Γ(x) = dt tx−1e−t. Thus we get 0

d Z 2π 2 dΩd = d . (2.25) Γ 2 ∆ The radial part of the integral can be computed with the change of variables t = 2 with lE + ∆ which we obtain

d d ∞ d−1 1 2 −1 −2 1 Z l Z dt ∆ ∆ 2 Z d d E 1− 2 2 −1 dlE 2 2 = − ∆ = dt t (1 − t) . (2.26) 0 [lE + ∆] 0 2∆ t 2 0 Using Z 1 Γ(α)Γ(β) dx xα−1(1 − x)β−1 = , (2.27) 0 Γ(α + β) we ﬁnd ∞ d−1 d d d Z 2 −2 lE ∆ Γ(2 − 2 )Γ( 2 ) dlE 2 2 = , (2.28) 0 [lE + ∆] 2 Γ(2) which ﬁnally implies

Z d d ε d l 1 i 1 Γ(2 − 2 ) i Γ 2 I = = = ε ε . (2.29) d 2 2 d 2− d 2− (2π) [l − ∆] (4π) 2 ∆ 2 Γ(2) (4π) 2 ∆ 2

7 Using Γ(n) = (n − 1)! , expanding the Gamma function in O(ε) ε 2 Γ = − γ + O(ε) (2.30) 2 ε EM where γEM ≈ 0.5772 is the Euler-Mascheroni constant and expanding

ε 2 4π ε ln 4π ε 4π 2 = e 2 ∆ = 1 + ln + O(ε ) , (2.31) ∆ 2 ∆ we then ﬁnd for small ε i 2 I = − γ + ln 4π − ln ∆ + O(ε) (2.32) (4π)2 ε EM which ﬁnally yields Z 1 2 1 2 2 2 V (p ) = − 2 dx − γEM + ln 4π − ln(m − p x(1 − x)) + O(ε) (2.33) 32π 0 ε For future purposes, here are general formulae useful to compute generic loop integrals:

Z d d d lE 1 1 Γ(n − ) d 2 2 −n d 2 n = d ∆ (2.34) (2π) [lE + ∆] (4π) 2 Γ(n) Z d 2 d d lE l 1 d Γ(n − − 1) d E 2 2 −n+1 d 2 n = d ∆ (2.35) (2π) [lE + ∆] (4π) 2 2 Γ(n) Z d 4 2 d d lE l 1 3d Γ(n − − 2) d E 2 2 −n+2 d 2 n = d ∆ (2.36) (2π) [lE + ∆] (4π) 2 4 Γ(n)

The integral over an odd number of `E vanishes since we are integrating from −∞ to +∞. Note that all integrals above are in Euclean space. In order to compute the corresponding 0 0 integrals in Minkowski space one must insert an overall i from the measure (given lE = −il ) and a (−1)n+p where p is the power of l2 in the numerator. Note also that in d dimension µν µ ν gµνg = d hence for integrals of the type l l we have Z ddl gµν Z ddl f(l2)lµlν = f(l2)l2 . (2.37) (2π)d d (2π)d

Renormalization

The key observation that allows us renormalize the theory is the fact that the divergence in

M12→34(s, t, u) is a local term, i.e. is a constant (momentum independent) term, exactly as the tree-level contribution:

2 M12→34(s, t, u) = −λ − λ [V (s) + V (t) + V (u)] = −λ − 3λ2 × V (0) −λ2 [V (s) + V (t) + V (u) − 3V (0)] . (2.38) | {z } | {z } UV divergent UV regular This imply we can reabsorb it with a redeﬁnition of the Lagrangian parameter λ, which is not a physical observable. In particular, we can impose the following renormalization condition

2 M12→34(s = 4m , t = u = 0) = −λren , (2.39) which from a diagramatic point of view reads

8 2 = −iλren at s = 4m , t = u = 0.

amputated

The renormalized coupling (λren) is what it can be extracted from a hypothetical measurement (in this case the 2 → 2 scattering at threshold). It must be stressed that the speciﬁc renor- malization condition we have imposed is an arbitrary choice: we decided to deﬁne λren from the magnitude of the 2 → 2 scattering amplitude at threshold, but we could have chosen a diﬀerent processes and/or the same process at a diﬀerent energy scale. As we shall see in the next section, this freedom leads to very important physical consequences. The relation between renormalized and “bare” coupling (i.e. the coupling appearing in the Lagrangian) reads

2 2 − iλren = −iλ + (−iλ) i[V (4m ) + 2V (0)] (2.40) 2 Z 1 λren 2 2 2 2 λ = λren + 2 dx 3 − γE + ln(4π) − ln m − 4m x(1 − x) − 2 ln m 32π 0 ε

As can be seen, we can keep λren ﬁnite provided λ → ∞ for → ∞. Notice that to get λ as 2 2 3 function of λren we used the fact that at λ = λren + O(λren). Given our renormalization condition, the 4-point correlation function expressed in terms of the renormalized coupling reads

2 2 M12→34(s, t, u) = −λren − λren V (s) + V (t) + V (u) − V (4m ) − 2V (0) 2 Z 1 2 λren m − x(1 − x)s = −λren − 2 dx ln 2 2 + 32π 0 m − 4m x(1 − x) m2 − x(1 − x)t m2 − x(1 − x)u ln + ln , (2.41) m2 m2 which is a ﬁnite non-trivial result. Note in particular that once we ﬁx the renormalized coupling from the threshold amplitude, then we can predict the scattering amplitude at other energies without introducing extra free parameters. We can repeat the same procedure with the 1PI 2-point correlation function imposing the renormalization condition

iZ 2 2 = 2 2 + (terms regular at p = m ) p − mren which speciﬁes the location of the pole and the residue. The latter is such that if we compute the 2-point correlation function in terms of the renormalized ﬁeld 1 φren = √ φ0 (2.42) Z

2 2 we obtain a free-ﬁeld result (with pole at p = mren).

9 The two point function up to O(λ2) is given by

= +

As can easily be checked, at this order Z = 1 (corrections to Z appear only at the two-loop level in the λφ4 theory). On the other hand, resumming the one-loop diagrams (as you learned in QFT-I) one obtains d 2 2 λ Γ 1 − 2 m = mren − d 2 1−d/2 . (2.43) 2(4π) 2 (m )

2 The fact that Z = 1 + O(λ ) ensures the result in Eq. (2.41), with m = mren, is the correct 3 1 expression of the 2 → 2 scattering amplitude up to O(λren) corrections.

2.1.3 Renormalised perturbation theory and counterterms

Let us summarise what we have seen in the analysis of the λφ4 theory. Adopting a standard notation, we denote with a subscript “0” bare parameters (both couplings and ﬁelds). Using this notation, the Lagrangian density of the theory reads

1 1 λ L = ∂ φ ∂µφ − m2φ2 − 0 φ4 (2.44) 2 µ 0 0 2 0 0 4! 0 where m0 and λ0 are the bare quantities and φ0 the bar ﬁeld. In this theory, UV divergences 1/2 can be canceled by redeﬁnition of m0 and λ0, as well as the ﬁeld φ0 = Z φr. We imposed renormalization conditions considering the two-point correlation function in momentum space, Z 2 4 ipx iM2(p ) = d x h0|T {φ0(x)φ0(0)|0i e (2.45)

and the 1PI four-point correlation function in momentum space, M4(s, t, u). We imposed in particular the following three conditions

2 −1 1. M2(p ) = 0 −→ m (pole of the 2-point function) p2=m2

2 2 2 2.( p − m )M2(p ) = Z −→ Z (residue of the pole ) p2=m2

2 3. M4(s = 4m , t = u = 0) = −λ −→ λ (renormalized coupling)

The two parameters m and λ are “physical” parameters: they are deﬁned from physical quantities, and are independent from the choice of the regularization procedure. The intro- duction of these parameters do not necessitate any modiﬁcation of the original Lagrangian. √ 1 phys −4 1PI According to the LSZ reduction formulae, M12→34(s, t, u) = ( Z) M12→34(s, t, u).

10 However, it is useful to re-write the original Lagrangian in terms of these parameters (as well as the renomalized ﬁeld). This leads to split the original Lagrangian into

1 1 λ L = Z ∂ φ ∂µφ − m2Zφ2 − 0 Z2φ4 (2.46) 2 µ r r 2 0 r 4! r 1 1 λ 1 1 δ = ∂ φ ∂µφ − m2φ2 − φ4 + δ (∂ φ )2 − δ φ2 − λ φ4 , (2.47) 2 µ r r 2 r 4! r 2 Z µ r 2 m r 4! r | {z } | {z } ﬁnite inﬁnite terms that cancel the local divergence of loop diagrams where 2 2 2 δZ = Z − 1 , δm = m0Z − m , δλ = λ0Z − λ . (2.48) The terms among square brackets in Eq. (2.47) are known as counterterms. It looks like we have “added” this term to the original Lagrangian, written in terms of physical couplings, but this is not the case: we do not add any term! We do not modify the original Lagrangian, but we re-arrange it into a part depending on the physical couplings (with the same functional form of the original Lagrangian) and a part which explicitly depends on how the theory has been regularised (i.e. the counterterms). We can then make calculations using directly the renormalised couplings (and ﬁelds) that, in practice, has some advantages. To do so, we must consider as free Lagrangian 1 1 Lr = ∂ φ ∂µφ − m2φ2 (2.49) free 2 µ r r 2 r and treat all the rest (including the counterterms) as interaction terms. Proceeding this way we are allowed to interpret m and λ in the Feynamn rules listed below Eq. (2.15) as the renormalised mass and coupling; however, we must add the following additional Feynman rules to describe the eﬀect of the counterterms

2 = i(p δZ − δm)

= −iδλ

This procedure (i.e. avoiding bare couplings and adding Feynaman rules to describe the eﬀfect of the counterterms) is known as renormalised perturbation. This procedure is somehow less intuitive: we loose the physical intuition that couplings and ﬁelds must be renormalised, choosing speciﬁc physical processes. However, it is technically more advantageous, especially if one imposes generic renormalization conditions such as the so-called minimal subtraction scheme (examples of this technique will be provided in the exercise classes).

11 2.2 The Renormalization Group

What we have seen so far is how the renormalization procedure works, at the technical level, in perturbation theory. As already stressed, the key feature that allow us to renormalise the theory is the fact that UV divergences generate a local correction, which is reabsorbed into a redeﬁnition of the couplings. Beside the technical aspects, this is the consequence of a very important physical property of QFT: physics at a given energy scale is insensitive to what happens at higher energy scales, modulo a redeﬁnition of the couplings. This observation is at the basis of the so-called Renormalization Group, i.e. the fact that in any QFT there is a well-deﬁned ﬂow of the couplings with the energy scale (or the length scale). To better illustrate the physical consequence of the renormalization procedure, it is conve- nient to adopt a physical cut-oﬀ and analyse the problem at the level of functional integrals. This is what we will discuss in the next two sections. We then proceed identifying the diﬀeren- tial equations that describe the energy-ﬂow of the couplings, and discuss the possible general form of their solutions.

2.2.1 Wilson’s approach to renormalization

Let us consider a scalar QFT, with an explicit cut-oﬀ Λ on the high-frequency modes of our functional integral. The explicit cut-oﬀ remove (=regularize) all the UV divergences of the theory. To simplify the discussion, let us also consider the theory in Euclidean space-time (avoiding the problems of physical poles in the amplitudes). The generating functional is given by Z Z 1 1 λ W [J] = [Dφ] exp − ddx (∂ φ)2 + m2φ2 + φ4 + Jφ (2.50) Λ 2 µ 2 4!

Y R 4 −ikx where [Dφ]Λ = dφ(k) and φ(k) = d x φ(x)e . |k|<Λ

Let us now split [Dφ]Λ into two parts: |k| < bΛ and bΛ ≤ |k| < Λ, with b < 1. To do so, we deﬁne two sets of (orthogonal) ﬁelds,

( ( φ(k) |k| < bΛ 0 |k| < bΛ φ˜(k) = , φˆ(k) = (2.51) 0 bΛ ≤ |k| ≤ Λ φ(k) bΛ ≤ |k| ≤ Λ

Since the external currents are arbitrary, we choose external currents that only couples to the low-frequency modes φ˜. To simplify the notation, from now φ˜ → φ. Separating the integrals on the two sets of ﬁelds in (2.50), one gets

Z R d Z Z 1 1 W [J] = Dφ e− d x[L(φ)+Jφ] Dφˆ exp − ddx (∂ φˆ)2 + m2φˆ2 2 µ 2 ! (2.52) λ + (φˆ4 + 4φˆ3φ + 6φˆ2φ2 + 4φφˆ 3) 4!

Note that the φˆ-dependent part of the action has no mixed quadratic terms, i.e. quadratic terms

12 ˆ µ ˆ of the type φφ, ∂µφ∂ φ vanish. This is a consequence of the orthogonality of the two ﬁelds

Z Z Z ddk Z ddp ddx φ(x)φˆ(x) = ddx e−ikxφ(k) e−ipxφˆ(p) (2π)d (2π)d Z Z = (2π)−d ddk ddp δd(k + p)φ(k)φˆ(p) Z = (2π)−d ddk φ(k)φˆ(−k) (2.53) | {z } no overlap =0

On the other hand, the triple term φ2φˆ (and other type of “interaction” terms) do not vanish:

Z Z Z ddk Z ddp Z ddl ddx φ2(x)φˆ(x) = ddx e−ikxφ(k)e−ipxφ(p)e−ilxφˆ(l) (2π)d (2π)d (2π)d Z Z Z = (2π)−2d ddk ddp ddl δ(k + p + l)φ(k)φ(p)φˆ(l) Z Z = (2π)−2d ddk ddp φ(k)φ(p)φˆ(−k − p) (2.54) | {z } can be nonzero at the same time 6=0

We can now proceed by performing explicitly the integral over φˆ. We expect to ﬁnd an expres- sion of the form Z Z 4 W [J] = [Dφ]bΛ exp − d x [Leﬀ (φ) + Jφ] (2.55) where Leﬀ (φ) is an eﬀective Lagrangian where the heavy degrees of freedom have been integrated out. According to the modern point of view, all quantum ﬁeld theories are eﬀective theories, i.e. the low-energy limit of more “fundamental” theories with heavy modes that we have not identiﬁed yet, and whose eﬀect amounts only to a redeﬁnition of the interaction terms among the low-energy modes. Although it may not appear obvious a ﬁrst sight, what we are doing is renormalizing the theory (in a way that is conceptually very similar to what we have seen in the previous section): we are removing the sensitivity to the original cut-oﬀ Λ, writing a new theory with a (lower) cut-oﬀ bΛ. The correction terms in the eﬀective Lagrangian will result from the removal of the large momentum components φˆ of the ﬁelds. Let us compute explicitly the integral over the high-momentum ﬁelds

Z Z 1 1 λ Dφˆexp − ddx (∂ φˆ)2 + m2φˆ2 + φˆ4 + 4φˆ3φ + 6φˆ2φ2 + 4φφˆ 3 (2.56) 2 µ 2 4! 1 We treat L ≡ ∂ φ∂ˆ µφˆ as “free term” and all the rest as an interaction term, that we evaluate 0 2 µ in perturbation theory. We also treat the mass term as a perturbation, since we are interested in the limit m2 Λ2. In the absence of a frequency cut-oﬀ, the propagator is

d 0 Z d k 1 0 hφ(x)φ(y)i = eik (x−y) (2.57) (2π)d k02 and in momentum space

d d d 0 d Z d x d y d k 1 0 Z 1 (2π) hφ(k)φ(p)i = eik (x−y)e−ikxe−ipy = ddy e−i(k+p)y = δd(k + p) (2.58) (2π)d k02 k2 k2

13 In the case of high-frequency modes, we have ( (2π)d 1 bΛ ≤ |k| ≤ Λ hφˆ(k)φˆ(p)i = δd(k + p)θ(k), θ(k) = (2.59) k2 0 otherwise This result allows us to identify the Feynman rule for the propagator of the high-frequency modes. Taking into account the various interaction terms between low- and high-frequency modes we deduce the following set of Feynman rules:

ˆ φ 1 λ θ(k) − k2 4

λ λ − − 6 24

λ − 6

Using these tools we can proceed evaluating the various contributions in perturbation theory.

The eﬀective mass term Integrating out the heavy modes on the φ2φˆ2 interaction term, leads to

φ φ

Z λ 1 Z − ddx φ2φˆφˆ =. − ∆m2 ddx φ2 (2.60) 4 2 with Z d Z Λ 2 λ d k 1 λ Ωd d−3 ∆m = d 2 = d |k| d|k| 2 bΛ≤|k|≤Λ (2π) k 2 (2π) bΛ d−2 2 (2.61) λ Λ d−2 d=4 λ Λ 2 = d (1 − b ) −−→ 2 (1 − b ) (4π) 2 Γ(d/2) d − 2 16π 2 d d−1 d where we used d k = |k| d|k| dΩd and Ωd = 2π 2 /Γ(d/2). This term, as well as its powers, sum up to reconstruct an exponential term in Leﬀ (φ): 1 1 m2 φ2 → (m2 + ∆m2)φ2 (2.62) 2 Λ 2 Λ This eﬀective mass 2 2 2 meﬀ = mΛ + ∆m (2.63) is the (tree-level) mass of the low-frequency modes, in a theory where the cut-oﬀ is bΛ. This is not a physical mass, but if we choose bΛ ≈ mphys, in such theory the one-loop corrections between the Lagrangian mass and the physical mass are small, if λ 1:

meﬀ = mphys + small corrections (2.64)

14 The eﬀective coupling term At order λ2 we have terms due to the contractions of two interaction terms of the type φ2φˆ2. There are two possible contractions, but only one is fully connected and give rise to a correction to the φ4 interaction,

1 Z λ 2 1 Z − ddx φ2φˆφˆ =. − ∆λ ddx φ4 . (2.65) 2 4 4!

φ φ

φ φ

We can compute ∆λ evaluating the four-point function above in the limit of external mo- 2 2 2 menta pi satisfying (pi + pj) b Λ for any i, j. In this limit all the external momenta can be neglected and we obtain

1 λ2 Z ddk 1 2 −∆λ = − (4! × 2) d 2 , (2.66) 2 4 bΛ≤|k|≤Λ (2π) k where the multiplicity factor 4! × 2 is due to the possible contractions (4!) of the four φ and the exchange of the internal lines. The explicit calculation yileds λ2 Z Λ ∆λ = −3 |k|d−5d|k| d d 2 Γ( 2 )(4π) bΛ λ2 Λd−4 3λ2 = −3 (1 − bd−4) −−→d=4 ln b (2.67) d d 2 2 d − 4 16π Γ( 2 )(4π) where we have used bd−4 = e(d−4) log b = 1 + (d − 4) log b. Proceeding as in the case of the mass, in the eﬀective theory of the low-frequency modes, with cut-oﬀ bΛ, we thus ﬁnd 3λ2 1 λ = λ + ∆λ = λ − Λ ln < λ (2.68) eﬀ Λ Λ 16π2 b Λ | {z } >0

Note the great diﬀerence between the mass correction ∆m2, which is quadratically sensitive to Λ, and the coupling constant correction ∆λ, which is only logarithmically sensitive to the cut-oﬀ. Deﬁning Λlow = bΛ we can indeed write 2 2 2 λλ 2 2 3λΛ Λ ∆m = 2 (Λ − Λlow) , ∆λ = − 2 ln 2 . (2.69) 32π 16π Λlow Note also that, beside diﬀerences in ﬁnite terms (i.e. in cut-oﬀ independent terms), the calcula- tion of ∆λ is exactly the same we did in the previous section to renormalize λ: not by chance, 2 2 the coeﬃcient of ln(Λ /Λlow) in (2.69) is the same coeﬃcient of the 2/ε term in (2.40).

Higher-dimensional interaction terms We also have higher dimensional interaction terms (i.e. operators with more ﬁelds and/or more derivatives) generated by integrating out φˆ. For example, an eﬀective interaction of the type (k/6!)φ6 is generated at the tree-level by the following diagram

15 and it receives cubic corrections in λ by the following one-loop diagram:

φ φ

φ φ

φ φ

Focusing the attention only on the latter diagram, and deﬁning

1 Z λ 3 1 Z − ddx φ2φˆφˆ = − ∆k ddx φ6 (2.70) 3! 4 6! we ﬁnd

3 2 Z d 3 3 Z Λ λ 6! × 2 d k 1 15λ d−7 ∆k = − − d 2 = d |k| d|k| 4 3! (2π) k (4π) 2 Γ(d/2) bΛ (2.71) 3 d−6 3 −2 15λ Λ d−6 d=4 15λ Λ 1 = d 1 − b −−→ 2 2 − 1 (4π) 2 Γ(d/2) d − 6 16π 2 b

Proceeding in a similar manner one ﬁnds also derivative interactions, that arise when we no longer neglect the external momenta. In general, all possible interactions of the ﬁelds φ and their derivatives (with arbitrary powers of ﬁelds and derivatives) ere generated.

Collecting all the terms We ﬁnd an eﬀective action of the type

Z Z 1 1 1 S [φ] = ddxL (φ) = ddx (1 + ∆Z)(∂ φ)2 + (m2 + ∆m2)φ2 + (λ + ∆λ)φ4 eﬀ eﬀ 2 µ 2 4! 1 1 + (k + ∆k)φ6 + (ξ + ∆ξ)(∂ φ)4 + ··· (2.72) 6! 4! µ

With these eﬀective Lagrangian we are able to compute correlation functions of the φ ﬁelds (low-frequencey modes) at small energies. The loop diagrams must be integrated only up to the cut-oﬀ Λlow = bΛ: choosing this scale close the scale of the physical processes we are interested in, the quantum corrections are small if the eﬀective couplings are small. What we have achieved with this procedure is to re-absorb the eﬀects of the heavy modes into the eﬀective couplings. This inevitably leads also to non-renormalizable terms in the Lagrangian, might seem disturbing. However, as we shall see next, their eﬀects on physical processes is small once we take into account the fact that the eﬀective theory as a smaller Λ cut-oﬀ.

16 2.2.2 The renormalization group (Wilsonian approach)

Let us now consider the following shift of variables: k x0 = bx, k0 = =⇒ 0 < |k0| < Λ(|k| now integrated up to Λ) (2.73) b We do this rescaling on the whole system, hence all the physical momenta, including the external 0 momenta, are rescaled as pi = pi/b: we basically “zoom-in” in the low-energy region. After rescaling, the eﬀective action looks like

Z 1 1 1 S [φ] = ddx0b−d (1 + ∆Z)b2(∂0 φ)2 + (m2 + ∆m2)φ2 + (λ + ∆λ)φ4 eﬀ 2 µ 2 4! ∆k ∆ξ + φ6 + b4(∂0 φ)4 + ··· (2.74) 6! 4! µ

We can now redeﬁne the ﬁeld, 1 φ0 = b2−d(1 + ∆Z) 2 φ (2.75) 1 such that the kinetic term is canonical i.e. (∂0 φ0)2. With this ﬁnal change of variables the 2 µ eﬀective action becomes Z 1 1 0 0 1 0 S [φ0] = ddx0 (∂0 φ0)2 + m 2φ 2 + λ0φ 4 + ··· (2.76) eﬀ 2 µ 2 4!

The theory we have obtained with this procedure is very similar to the original one, but for p • scaled space-time (momentum) coordinates [p → p0 = ] b • modiﬁed couplings [λ → λ0]:

(m0)2 = (m2 + ∆m2)(1 + ∆Z)−1b−2 λ0 = (λ + ∆λ)(1 + ∆Z)−2bd−4 k0 = (k + ∆k)(1 + ∆Z)−3b2d−6 . . (2.77)

The scaling with b is the normal scaling (dictated by naive dimensional analysis) while the factor in front is the anomalous scaling due to quantum ﬂuctuations. This procedure can be iterated

Λ → bΛ → b2Λ → · · · (2.78)

For b ∼ 1 the shells in momentum space are inﬁnitesimal and we get a continuous transforma- tion:

x → x0 {λ, m, · · · } → {λ0, m0, ···}. (2.79)

The procedure of integrating over the high-momentum degrees of freedom in a QFT obtaining a new Lagrangian with a lower UV cut-oﬀ can be described as a ﬂow in the space of all

17 possible Lagrangians. Despite this set of Lagrangians (or couplings) does not form a group in the mathematical sense, this ﬂow is called the renormalization group (RG) ﬂow of the theory. Physics must be invariant under this transformation. If we ﬂow to the regime where the UV cut-oﬀ is close (i.e. just above) the scale of the processes we are interested in, simply looking a the value of a given coupling in the Lagrangian we understand its role at that given energy. To clarify this concept, lit is convenient to make a simple example. Let’s consider the following adimensional amplitude p2 A = λ log (2.80) m2 At the classical level (or in our QFT at the tree-level), the scaling of variables imply

(p0)2 (p/b)2 p2 A −→class λ log = log = λ log (2.81) (m0)2 (m/b)2 m2 As expected, at the classical level an adimensional amplitude is unchanged by the scale trans- formation (while a dimentional one gets a trival scaling of the type bn dictated by the canonical dimension of the amplitude). The non-trivial fact is that we can deﬁne the scaling also beyond the tree-level, considering also the UV cut-oﬀ and the anomalous scaling of the couplings. Sup- 2 pose for instance the amplitude depends only on (pi + pj) , and let’s assume the masses are negligible, then we have

(p + p )2 (p + p )2 A = f λ; i j RGﬂow−→ f λ0; i j (2.82) Λ2 b2Λ2

2 2 If the amplitude is adimensional, the dependence on the variable (pi + pj) /Λ is typically a logarithmic dependence (as in the tree-level example above). Then choosing the appropriate value of b we can ﬂow to a region where this log is small and all quantum eﬀects are encoded in the scaled coupling λ0. In other words, using the renormalization group we can renormalize the couplings to the speciﬁc energy relevant to our process: the couplings thus renromalized immediately indicate their relevance at that physical energy. We can thus classify couplings in a QFT Lagrangian according to their behavior under the RG ﬂow: • Couplings with positive mass dimension are called relevant: their value grow, in units of the reference energy, when ﬂowing to lower energies. For example (m0)2 = m2b−2 + ··· (the dots indicate the quantum correction terms that, in a weakly interacting theory, do not change this general behaviour). • Couplings with negative mass dimension are called irrelevant: their value decrease, in units of the reference energy, when ﬂowing to lower energies. For example k0 = kb2 + ··· (d = 4). • Couplings with zero mass dimension are called marginal: here quantum corrections play a key role and change them into relevant or irrelevant couplings depending on the theory. Let’s consider for example the coupling λ in the φ4 theory. Here for d = 4 we ﬁnd

3λ2 3λ λ0 = λ + ln b ≈ λbε , ε = (2.83) 16π2 16π2 we then deduce that λ is irrelevant for d = 4, i.e. if we start from a given value at high energies, the interaction become weaker and weaker at low energies. Note that if we “start” with a

18 small value for λ (i.e. λ ∼ 1), then we go to lower and lower values, hence higher-order loop corrections cannot change this behaviour: we can thus conclude that λ = 0 is an IR ﬁxed point of the RG ﬂow. What is less obvious is what happens at high energies: form the one-loop result it seems there is not ﬁnite value for λ at arbitrary large energies (i.e. for arbitrary large UV cut oﬀ), but we cannot trust this extrapolation, since for large values of λ higher-order loop corrections can be come relevant. Most important, new high-energy modes (i.e. new heavy particles) could change this behaviour. Let’s not consider the φ4 theory but for for d < 4. In this case we ﬁnd 3λ2 bd−4 − 1 λ0 = λ − Λd−4 bd−4 (2.84) (4π)d/2Γ(d/2) 4 − d This implies that there is a value of λ at which the increase due to (classical) rescaling is compensated by the decrease caused by the nonlinear eﬀect due to quantum corrections. This causes a second ﬁxed point of Lagrangian. If the scale of the Λ cut-oﬀ is much higher than that of experiments, only the relevant and marginal terms appear in the eﬀective Lagrangian: the coeﬃcients of the other terms ﬂow to zero after integrating out over many momentum shells. This is the reason why at low energies the non-renormalizable terms have a negligible eﬀect. All theories “ﬂow” towards renormalizable theories at low energies. It could well be that all QFT are non-renormalizable theories with an intrinsic UV cut-oﬀ (i.e. theories that make no sense without such cut-oﬀ . . . ): at suﬃciently low energies we cannot distinguish them from renormalizable theories.

2.2.3 Renormalization scale and the Callan-Symanzik equation

The properties of the RG in more general theories can be derived using a more formal procedure related to the explicit renormalization of the couplings (or of the Green functions) of the theory. The key observation is that we can renormalize the theory choosing an arbitrary renormalization scale, or imposing renormalization conditions at an arbitrary reference energy scale. In Wilson’s approach, we are free to choose the scaling parameter of lengths and energies as we prefer. However, the physical amplitudes must be independent of this choice. This fact enables us to predict the RG ﬂow of the parameters of a renormalized quantum ﬁeld theory. In Sect. 2.1.2 we have seen how to renormalize the φ4 theory imposing speciﬁc conditions on the two- and four-point Green’s functions. There we have imposed speciﬁc renormalization conditions related to speciﬁc processes at speciﬁc energies (such as the 2 → 2 scattering at threshold). However, as we discuss below, we could have chosen diﬀerent conditions, related to diﬀerent energies. For simplicity, let’s consider ﬁrst the case of a massless φ4 theory. In this case we can impose for instance the following generic renromalization conditions

i 1. h0|T {φ(p)φ(−p)} |0i = at p2 = −µ2 p2

2. M4(φφ → φφ) s=t=u=−µ2 = −λ where µ is an arbitrary renormalization scale (if m 6= 0, we can still impose conditions of this type provided µ2 m2). These conditions make use of the two- and four-point Green’s

19 functions at a reference energy to deﬁne the renormalized coupling, λ, and the renormalized −1/2 ﬁeld φ = Z φ0. Let’s now consider a generic n-point un-renormalised Green function:

n n (n) h0|T {φ0(x1) ··· φ0(xn)} |0i = Z 2 h0|T {φ(x1) ··· φ(xn)} |0i = Z 2 G ({xi}; λ, µ) (2.85)

The l.h.s. of this expression contains bare ﬁelds and bare couplings, and is thus manifestly independent of µ, while the r.h.s., where the renormalized Green function and the renormalized coupling appear, has an apparent µ dependence: the latter must cancel once all eﬀects are taken into account. Performing the derivative on both sides of (2.85) with respect to µ we obtain

d n (n) 0 = µ Z 2 G ({x }; λ, µ) dµ i √ n 1 d (n) n dλ ∂ (n) ∂ (n) = nZ 2 √ µ Z G + Z 2 µ G + µ G , (2.86) Z dµ dµ ∂λ ∂µ | {z } |{z} γ β that implies

∂ ∂ nγ + µ + β G(n) ({x }; λ, µ) = 0 , (2.87) ∂µ ∂λ i with dλ 1 d √ β = µ and γ = √ µ ( Z) . (2.88) dµ Z dµ (n) The functions β and γ are the same for every G and therefore cannot depend on {xi} (or µ {p }). Being dimensionless, they can only depend on the dimensionless quantities λ and . i Λ However, since the Green function G(n) is renormalized, β and γ cannot depend on the cut-oﬀ Λ, hence we deduce they are only functions of the (renormalized) coupling λ. Showing explicitly such dependence we re-write Eq. (2.87) as

∂ ∂ µ + β(λ) + nγ(λ) G(n)({x }; λ, µ) = 0 (2.89) ∂µ ∂λ i which is known as the Callan-Symanzik equation for a theory with a single adimensional cou- pling λ. As it appear from their deﬁnition in Eq. (2.88), β(λ) and γ(λ) tell us how the renormalized coupling and the ﬁeld strength renormalization change under a change of renormalization scale, they thus allows to identify the RG ﬂux discussed in the previous section. Note also that we can use Callan-Symanzik on any pair of Green functions at our choice to extract β(λ) and γ(λ): we are not obliged to use the two- and four-point functions as done so far (although in the φ4 case this is clearly the simplest choice).

Evaluating β(λ) in the φ4 theory at one-loop As we have already seen, in the φ4 theory the ﬁeld strength renormalization does not receives contributions at the one-loop level, namely Z = 1 + O(λ2). On the other hand, the 1PI four-point function, obtained by the following diagrams

20 + + permutations + counterterm reads 2 M4(s, t, u) = −λ − λ [V (s) + V (t) + V (u)] − δλ (2.90)

2 where δλ is the vertex counterterm. The renormalization condition M4(s = t = u − µ ) = −λ then implies

2 2 δλ = −3λ V (−µ ) 2 Z 1 3λ 2 2 2 = + 2 dx − γEM + ln 4π − ln(m + µ x(1 − x)) 32π 0 ε 3λ2 2 = + − ln(µ2) + const. (2.91) 32π2 ε where we have used the expression for V (p2) derived in Eq. (2.33), and the last line follows 2 2 (4) 2 under the assumption µ m . Taking into account that G ∝ M4, and that Z = 1 + O(λ ), (4) applying the Callan-Symanzik equation on G (i.e. requiring M4 to be independent of µ), we deduce that d d 3λ2 β(λ) = µ λ = −µ δ = + O(λ3). (2.92) dµ dµ λ 16π2

β functions in more general theories In more general theories it is less obvious how to deﬁne renormalization conditions on the coupling. However, as anticipated, we can use the Callan-Symanzik equation to determine β. The general strategy is the following:

1. Compute the γ’s from 2-point functions of all the independent ﬁelds

2. Compute β(gi) selecting a suﬃcient number of independent Green functions and apply the Callan-Symanzik equation

" a b # ∂ X ∂ X µ + β(g ) + n γ (g ) G(n1, ··· , nb)({x }; g , µ) = 0 (2.93) ∂µ i ∂g j j j i i i=1 i j=1

to reconstruct the evolution of the couplings (na indicates the number of independent couplings and nb the number of independent ﬁelds on the given Green function).

As an exercise, it is worth computing these functions in QED. There one ﬁnds

e2 γ (e) = (in the Feynman gauge) ψ 16π2

e2 γ (e) = (transverse component, gauge independent) A 12π2

e2 β(e) = (gauge independent) 12π2 21 2.2.4 Solutions of the Callan-Symanzik equation

Let us consider a 2-point function G(2)(p2; λ, µ). By dimensional arguments, it has to be of the form i p2 G(2)(p2; λ, µ) = G˜(2) − ; λ (2.94) p2 µ2 ˜(2) (2) ˜(2) with Gtree = 1. Similarly to G , also the adimensional G satisﬁes the Callan-Symanzik equation: ∂ ∂ p2 µ + β(λ) + 2γ(λ) G˜(2) − ; λ = 0 (2.95) ∂µ ∂λ µ2 Observe that ∂ ∂ |p| G˜(2) = −µ G˜(2) (2.96) ∂|p| ∂µ ∂ ∂ where p2 = −|p|2 (spacelike momentum), so we can trade µ with −|p| . If we now deﬁne ∂µ ∂|p| |p| t = log µ , then ∂ ∂|p| ∂ ∂ ∂ = = µet = |p| (2.97) ∂t ∂t ∂|p| ∂|p| ∂|p| so that our equation becomes ∂ ∂ − + β(λ) + 2γ(λ) G˜(2) (t; λ) = 0 (2.98) ∂t ∂λ The Callan-Symanzik equation tells us that the dependence on t (and hence on the momenta) and λ are related (which is obvious in the limit γ → 0). Let us introduce the function λ¯(t, λ) deﬁned by Z λ¯(t,λ) dx = t (2.99) λ β(x) Diﬀerentiating with respect to t we get ∂λ¯ 1 ∂λ¯ = β(λ¯) 1 = ¯ =⇒ ∂t (2.100) β(λ) ∂t λ¯(0, λ) = λ (boundary condition)

Diﬀerentiating with respect to λ we get 1 ∂λ¯ 1 ∂λ¯ β(λ¯) 0 = − =⇒ = (2.101) β(λ¯) ∂λ β(λ) ∂λ β(λ) thus we ﬁnd that the function λ¯(t, λ) satisﬁes the homogeneous Callan-Symanzik equation (i.e. the equation with γ = 0) ∂ ∂ − + β(λ) λ¯(t, λ) = 0 (2.102) ∂t ∂λ From this result, it also follows that any function F (λ¯(t, λ)) solves the omogeneous Callan- Symanzik equation. Let’s now turn to the complete (non-homogeneous) Callan-Symanzik equation. Given the above ﬁndings, its solution can be conveniently decomposed as

G˜(2)(t, λ) = F (λ¯(t, λ))U(t, λ) (2.103)

22 ∂ ∂ Deﬁning Kˆ ≡ − β , the (non-homogeneous) Callan-Symanzik equation for the two-point ∂t ∂λ function assumes the form h i Kˆ + nγ G(2)(t, λ) = 0 (2.104) where n = 2, but we keep it generic for later convenience. Using the decomposition G(2)(t, λ) = FU we ﬁnd

Kˆ (FU) = ( KFˆ )U + F (KUˆ ) =! nγF U (2.105) |{z} =0(hom.) then

KUˆ = nγU ===Ansatz⇒ U = enW Keˆ nW = n(KWˆ )enW = nγenW =⇒ KWˆ = γ

The function W that solves the above equation is

Z λ¯(t,λ) γ(x) W = dx (2.106) λ β(x) as it can be checked explicitly:

∂ ∂ γ(λ¯) ∂λ¯ γ(λ¯) ∂λ¯ γ(λ) − β(λ) W = − β(x) − ∂t ∂λ β(λ¯) ∂t β(λ¯) ∂λ β(λ) γ(λ¯) ∂ ∂ = − β(x) λ¯ + γ(λ) = γ(λ) (2.107) β(λ¯) ∂t ∂λ

Putting everything together we thus obtain

R λ¯ γ(x) t (2) 2 dx 2 R dt0γ[λ¯(t0)] G˜ (t; λ) = F [λ¯(t, λ)]e λ β(x) = F [λ¯(t, λ)]e 0 (2.108) where we used dt = dλ/β¯ (λ¯). Note that all the momentum dependence is carried by λ¯(t, λ). The above result can easily be generalised to any family of Green’s functions that are characterised by a single scale of momenta, obtaining

(n) 2 R t dt0γ[λ¯(t0)] G˜ (t; λ) = F [λ¯(t, λ)] × U(t, λ) = F [λ¯(t, λ)] × e 0 (2.109)

• F (λ¯(t, λ)) is a function that we can determine looking at the perturbative evaluation of the Green function we are interested in. At the tree level we obtain a function of the bare coupling. At the one-loop level, when we renormalise the coupling, we ﬁnd corrections of the type λ2 ln(p2/µ2), and so on. The general structure in Eq. (2.109) tells us that the dependence from µ and λ is not generic and, beside the ﬁeld-strength rescaling, is simply encoded by λ¯. In particular, we can take into account all the leading terms of the series λn+1 ln(p2/µ2)n if, in the lowest-order result, we replace the bare coupling with λ¯. • U(t, λ) is the accumulated ﬁeld-strength rescaling of the correlation function from µ to the actual momentum p at which G is evaluated. This rescaling depend on the number of external legs of the Green function.

23 Consider, for example, the connected 4-point function of the φ4 theory evaluated at space- 2 2 2 like momenta pi such that pi = −P and pipj = 0 [hence s, t, and u are all of order O(−P )]. ˜(4) ¯ From the tree level result Gtree = −iλ we can determine the leading structure of F (λ) and deduce that ˜(4) ¯ ¯2 Gloop(t; λ) = −iλ + O(λ ) (2.110)

2 2 If we measure λ at the kinematical point t = 0, corresponding to a speciﬁc value P = P0 and determine β(λ) theoretically, we can evolve λ to the relevant kinematical conﬁguration 2 2 P 6= P0 , corresponding to t 6= 0, by solving the RG equation with boundary condition,

∂λ¯ = β(λ¯) ∂t (2.111) λ¯(0) = λ

The “evolved” coupling λ¯(t, λ), i.e. the coupling renormalized at the scale of the process we are interested in, is the coupling controlling the strength of the interaction. The problem is slightly more complicated if we have more physical scales in the system. However, also in that case we can obtain a general solution of the Callan-Symanzik equation:

∂ ∂ µ + β(λ) + nγ(λ) G(n)({p }; λ, µ) = 0 (2.112) ∂µ ∂λ i

Let’s assume there is a scale µ0 where we “know” the value of the coupling (e.g. we measure −t it, or we predict it in terms of speciﬁc hypotheses). We can then write µ = µ0e , such that ∂ ∂ µ = − and the Callan-Symanzik assumes the form ∂µ ∂t

∂ ∂ − + β(λ) + nγ(λ) G(n)({p }; λ, t) = 0 (2.113) ∂t ∂λ i [µ0]

We can proceed as in the previous case, deﬁning λ¯(t) such that

∂λ¯ = β(λ¯) ∂t (2.114) λ¯(0) = λ

Then Z t (n) (n) ¯ ¯ 0 0 G ({pi}; λ, µ) = G ({pi}; λ(t), µ0) exp n γ(λ(t ))dt (2.115) 0

For a complicated Green function with several pi, it is not obvious which is the optimal renor- malization scale. The generic perturbative result (in the λφ4 theory) is

" # X pipj G(n) ∼ λkf(p ) 1 + λ c ln + non-log terms + O(λ2) (2.116) i ij µ2 ij

The µ dependence appears only beyond the tree level and only in logs. In the cases where all the momenta are of the same order we can eliminate these large logs by selecting µ = O(pipj), ﬁxing in this way the relevant scale for λ. Note that these large logs are the real potential problem of renormalization, not the “apparent” inﬁnities.

24 2.2.5 Running coupling

We are interested in solving the equation dλ¯ 3 = β(λ¯) = λ2 + O(λ3), λ¯(0, λ) = λ (2.117) dt 16π2 where λ is the coupling constant renormalized at the scale µ2. Since Z λ¯ dx t = = ln(|p|/µ) (2.118) λ β(x) we get 16π2 Z λ¯ dλ¯0 Z t |p| 16π2 1 1 |p| = dt0 = ln =⇒ − = ln (2.119) ¯02 ¯ 3 λ λ 0 µ 3 λ λ µ and ﬁnally we arrive at λ λ¯ = λ(|p|) = (2.120) 3λ |p| 1 − ln 16π2 µ As already discussed in the context of the Wilsonian approach, we ﬁnd that • the eﬀective coupling λ¯ decreases for small |p|: λ¯(|p|) → 0 as |p|/µ → 0; • the eﬀective coupling diverges at high energies, signaling that the theory is not well-deﬁned if we remove the UV cut-oﬀ. This expression (2.120) partially solves large log problems. Expanding in λ the above result we ﬁnd a series of potentially large logs which are resummed in the Green functions (given the general structure of the solution to the Callan-Symanzik equation) thanks to introduction of λ¯, that is also known as the running coupling: 3λ2 |p| 3 2 |p| λ¯ = λ + log + λ3 log2 + ··· (2.121) 16π2 µ 16π2 µ Suppose we measure (or the model predicts) λ ≈ π at a very high scale µ = 1016 GeV and we want to measure the φφ → φφ scattering at 1 GeV. Apparently, in a naive perturbative expansion using λ ≈ π, the one-loop result is a large as the tree-level one: 3λ2 |p| log ≈ −3 (2.122) 16π2 µ But this is an artifact of using λ at the wrong renormalization scale. If we use λ¯ = λ(1 GeV) ≈ 1 (2.123) we have a perfectly well-behaved perturbation theory for processes around 1 GeV.

Possible values of β and their meaning In view of the above considerations, we can classify QFT theories according to the value of the β-function: > 0 theory potentially UV unstable dλ β(λ) = µ = = 0 ﬁxed point (2.124) dµ < 0 theory potentially IR unstable Examples of the ﬁrst case β(λ) > 0 are QED and λφ4 in their perturbative regime.

25 β(λ) β(λ) UV ﬂux (p ↑) IR ﬂux (p ↓)

• • λ λ

UV stable IR stable ﬁxed point ﬁxed point

Dimensional transmutation In (2.117) we have seen that for the λφ4 theory we have dλ¯ 3 = β(λ) = β λ2 + O(λ3) with β = (2.125) dt 0 0 16π2 which leads to λ λ¯(µ) λ¯(Q) = = (2.126) 1-loop ¯ 1 − β0λ log (Q/µ) 1 − β0λ(µ) log (Q/µ) Taken at face value, this result implies the existence of a so-called Landau pole, i.e. a divergent behaviour of the coupling constant in the UV:

λ(Q)

Landau pole •

µ Λ0 Q

If, on the other hand, we had dλ¯ = β(λ) = −β λ¯2 , (2.127) dt 0 with β0 > 0, we would have found

λ¯(µ) Qµ 1 λ¯(Q) = −−−→ (2.128) 1-loop ¯ 1 + β0λ(µ) log (Q/µ) β0 log (Q/µ) which implies a divergence of the coupling at low energies. In both cases there appears a speciﬁc energy scale in the theory, i.e. the scale Λ0 where the coupling diverges: such scale was not there in the classical theory (in the limit where we neglect masses, the classical theory is scale invariant). Looking at the above expressions of λ(Q), which are representative of one-loop results, we can deﬁne this dynamical generated scale as

( 1 1 , β0 > 0,Q Λ0, ¯ |β0| log(Λ0/Q) λ(Q) = = 1 (2.129) β log(Λ /Q) , β0 < 0,Q Λ0, 0 0 |β0| log(Q/Λ0)

26 where 1 ¯ β0λ(Q) Λ0|1-loop = Qe (2.130) The appearance of this scale is a non-perturbative phenomenon which goes beyond perturbation theory. A general expression for the dynamically generated scale, beyond the one-loop result, is Z λ0 1 Λ0 = µ exp dx , (2.131) λ(µ) β(x) where λ0 is a generic (large) value of the coupling constant, independet of µ, satisfying λ0 λ(µ). As it can easily be checked by explicit derivation, dΛ0/dµ = 0, while using the perturba- tive expansion of β(λ) and considering λ0 → ∞ we recover the result in Eq. (2.129).

2.2.6 Callan-Symanzik equation with local operators and mass terms

So far we have only considered quantum ﬁeld theories with dimensionless parameters, or renor- malizable theories in the massless limit. What happens if we have mass terms and dimensionful coeﬃcients? The key point is that the renormalization scale µ is mass-independent, and it is often convenient to keep it as a generic scale independent of any other scale possibly appearing in the Lagrangian. To do so we treat mass terms, and in general any additional local operator, as new interaction terms in the Lagrangian. For the mass term in the λφ4 theory we thus have. 1 λ 1 L = ∂ φ∂µφ − φ4 − m2φ2 (2.132) 2 µ 4! 2 | {z } | {z } L0 Lint

More generally X L = L0 + CiOi(x) (2.133) i where Oi(x) are local operators of arbitrary mass dimension and Ci are their coeﬃcient. For ¯ µ 5 2 2 example in the Fermi theory we have GF ψ(x)γ (1 − γ )ψ(x) where GF has dimension 1/M . Similarly to ﬁelds and adimensional couplings, the operators can be renormalized by imposing speciﬁc renormalization conditions at the scale µ, using a Green-function which involves them. For instance, in the case of the mass term, we can impose a condition such as 1 2 i i 2 2 2 2 h0|T φ(p)r φ(k) φ(q)r |0i = 2 2 at p = q = k = −µ , (2.134) 2 r p q

1 2 which corresponds to impose the tree-level matrix element for the operator 2 φ on a correlation function with two φ ﬁelds.

The renormalized operator Or diﬀers from the bare operator O0 for a scale parameter ZO

O0(x) = ZO(µ)Or(x) . (2.135)

Separating the eﬀect related to the renormalization of the (n) ﬁelds appearing in the operator, we can decompose ZO as n/2 −1 ZO = ZﬁeldsZvertex (2.136)

27 ¯ µ For operators corresponding to conserved currents (such as ψγ ψ in QED) we have ZO = 1, 4 2 but in general ZO 6= 1. In the φ theory, where Zφ = 1 + O(λ ), at the one-loop level the renormalization of the operator Since bare operators do not depend on µ

∂ ∂ h n i µ h0|T {φ (x ) ··· φ (x )O (x)} |0i = µ Z 2 Z G(n,1)({x }, x; λ, µ) = 0 (2.137) ∂µ 0 1 0 n 0 ∂µ φ O i we can derive the following Callan-Symanzik equation for the (renormalized) Green functions with the insertion of n ﬁelds and one operator O:

∂ ∂ µ + β(λ) + nγ (λ) + γ (λ) G(n,1) = 0 (2.138) ∂µ ∂λ φ O

∂ where γ = µ ln Z . Generalising to G(n,k) = h0|{φ(x ) ··· φ(x )[CO]k}|0i we obtain O ∂µ O 1 n

∂ ∂ µ + β(λ) + nγ (λ) + kγ (λ) G(n,k) = 0 (2.139) ∂µ ∂λ φ O

∂ Since acting with C on the Green function returns the power k, we can rewrite the above ∂C equation as ∂ ∂ ∂ µ + β(λ) + nγ (λ) + C γ (λ) G(n,k) = 0 (2.140) ∂µ ∂λ φ ∂C O For example, for a mass term ∂ 1 2 ∂ 2 ∂ γ C = γ 2 m = γ 2 m (2.141) O m 1 2 m 2 ∂C 2 ∂( 2 m ) ∂m In general " # ∂ ∂ X ∂ µ + β(λ) + nγ (λ) + γ (λ)C G(n)({x }; λ, C , µ) = 0 (2.142) ∂µ ∂λ φ Oi i ∂C i i i i

It is important to notice that there is no problem in having a large number of Ci or having Ci corresponding to non-renormalizable terms, provided that

• the number of Ci is ﬁnite at a given renromalization scale µ (initial scale) , • we are interested in processes at E µ (i.e. we are interested only in the low-energy regime of the theory) .

To see the last point more clearly, we redeﬁne the (dimensional) coupling of the operator (Ci) as X 4−di CiOi = ρi µ Oi(x) (2.143) i where ρi is a dimensionless parameter, µ is the initial reference scale, and di is the canonical n dimension of Oi (e.g. d = n for φ ). Expressing Ci in terms of ρi, the derivatives in the Callan-Symanzik equation gets modiﬁed as follows

∂ ∂ ∂ ∂ ∂ Ci −→ ρi , µ −→ µ − (4 − di)ρi , (2.144) ∂Ci ∂ρi ∂µ ∂µ ∂ρi

28 where the last term is necessary to remove the extra dependence in µ introduced by the redef- inition of the Ci. The Callan-Symanzik equation then becomes " # ∂ ∂ X ∂ µ + β + nγ + [γ (λ) + d − 4]ρ G(n)({p }; µ, λ, {ρ }) = 0 (2.145) ∂µ ∂λ Oi i i ∂ρ i i i i

and deﬁning βi(λ) = [γOi (λ) + di − 4]ρi we ﬁnd " # ∂ ∂ X ∂ µ + β + nγ + β G(n)({p }; µ, λ, {ρ }) = 0 (2.146) ∂µ λ ∂λ i ∂ρ i i i i

For λ and ρi small we can easily solve the equation for the running parameters corresponding to di 6= 4. For small γOi we have ∂ρ¯ i = β (¯ρ , λ) = (d − 4 + γ )¯ρ ≈ (d − 4)¯ρ ∂t i i i Oi i i i (2.147) ρ¯i(0) = ρi

|p| where t = log µ and |p| is the generic momentum of the Green function. From the diﬀerential equation above we obtain ( |p|di−4 d > 4 (di−4)t i ρ¯i = ρi e = ρi ρi for (2.148) µ |p| µ

We thus recover the structure of the energy ﬂow discussed in the context of Wilson’s approach to the RG.

29