9 Renormalisation

We have encountered already three examples of divergent loop integrals discussing the λφ4

theory. In these cases, it was possible to subtract the infinities in such a way that we obtainedfinite and physically sensible observables. Main aim of this chapter is to obtain a betterphysical understanding of this renormalisation procedure. We will learn that the λφ4 theoryas well as the electroweak and strong interactions of the SM are examples for renormalisabletheories: For such theories, the renormalisation of the finite number of parameters containedin the initial classical Lagrangian is sufficient to make all observables finite in any orderperturbation theory. While most of our discussion stays within our standard perturbativetreatement, we introduce in the last section a non-perturbative approach based on ideasdevelopped in solid-state physics and the renormalisation group.

Why renormalisation at all? We are using perturbation theory with free, non-interactingparticles as asymptotic states as tool to evaluate non-linear quantum field theories. Interac-tions change however the parameters of the free theory, as we know already both from classicalelectrodynamics and quantum mechanics. In the former case, Lorentz studied 1904 the con-nection between the observed electron mass mphy, its mechanical or inertial mass m0 and itselectromagnetic self-energy mel in a toy model. He described the electron as a sphericallysymmetric uniform charge distribution with radius re, obtaining

mphy = m0 +mel = m0 +4e2

5re. (9.1)

Special relativity forces us to describe the electron as a point particle: Taking thus the limitre → 0, classical electrodynamics implies an infinite “renormalisation” of the “bare” electronmass m0 by its electromagnetic self-energy mel.

Another familiar example for renormalisation appears in quantum mechanics. Perturbationtheory is possible, if the Hamilton operator H can be split into a solvable part H(0) and aninteraction λV ,

H = H(0) + λV , (9.2)

and the parameter λ is small. Using then as starting point the normalised solutions |n(0)〉 ofH(0),

H(0)|n(0)〉 = E(0)n |n(0)〉 , (9.3)

we can find the eigenstates |n〉 of the complete Hamiltonian H as a power-series in λ,

|n〉 = |n(0)〉 + λ|n(1)〉 + λ2|n(2)〉 + . . . (9.4)

Since we started with normalised states, 〈n(0)|n(0)〉 = 1, the new states |n〉 are not longercorrectly normalised. Thus going from free (or “bare”) to interacting states requires to re-normalise the states,

R 〈n|n〉R = 1 ⇒ |n〉R ≡ Z1/2 |n〉 . (9.5)

109

9 Renormalisation

q

p

p′

=q

p

p′

+ kq

p

p′

1

Figure 9.1: The general vertex for the interaction of a fermion with a photon and its pertur-bative expansion up to O(e3) within QED.

A very similar problem we encountered introducing the LSZ formalism. In the parlanceof field theory, we continue often to call this procedure wave-function renormalisation, al-though Z renormalises field operators. Note also that in non-degenerate perturbation theory,〈n(0)|n(1)〉 = 0 , so we see that Z = 1 + O(λ2).

The familiar process of renormalisation in quantum mechanics becomes for quantum fieldtheories more mysterious by the fact that the renormalisation constants are infinite. Thus wehave to regularise as first step, i.e. employing a method which makes our expressions finite,before we perform the renormalisation.

9.1 Anomalous magnetic moment of the electron

We start the chapter with the calculation of the anomalous magnetic moment of the electron.Apart from being the first successful loop calculation in the history of particle physics, thisprocess illustrates the case that quantum corrections introduce new types of interactions:While an electron on the classical level interacts via Lint = eψγµψAµ with an electromagneticfield, loop graphs add an (e3/m)ψσµνψFµν interaction. Characteristic for renormalisabletheories is that the loop integrals associated with these new interactions are finite.

Prediction of the Dirac equation We first show that the Dirac current ψγµψ containsalready on the classical level in addition to the “orbital” part—which is identical to the spinlesscase—a new term created by the interaction of the electron spin with an electromagnetic field.

Squaring the Dirac equation,

(iD/ +m)(iD/ −m)ψ = −(D/D/ +m2)ψ = −(

DµDµ −

e

2σµνF

µν +m2)

ψ = 0 , (9.6)

and comparing to the Klein-Gordon equation, we see that the spin of the electron leads to theadditional interaction term σµνF

µν . Evaluating this expression for a homogeneous magneticfield B and a non-relativistic electron, we can derive Hint = (L+2s)eB, i.e. the gs = 2 factorof the electron (problem 9.1). This was the first crucial test that the Dirac equation describescorrectly electrons.

Vertex function We want to write down the most general form Λµ of the coupling termbetween an external electromagnetic field and an on-shell Dirac fermion, consistent with

110

9.1 Anomalous magnetic moment of the electron

Lorentz invariance, current conservation and parity. Since p2 = p′2 = m2, the only non-trivialscalar variable in the problem is p ·p′. We choose to use the equivalent quantity q2 = (p−p′)2

as the variable on which the arbitrary scalar functions in our ansatz for Λµ depend. Imposingparity conservation forbids the use of γ5. Hence the most general ansatz compatible withLorentz invariance and parity is

Λµ(p, p′) = A(q2)γµ +B(q2)pµ + C(q2)p′µ +D(q2)σµνpν +E(q2)σµνp′ν . (9.7)

Current conservation requires qµΛµ(p, p′) = 0 and leads to C = B and E = −D. Hence

Λµ(p, p′) = A(q2)γµ +B(q2)(pµ + p′µ) +D(q2)σµνqν . (9.8)

Hermeticity finally requires that A,B are real and D is purely imaginary.

Gordon decomposition We derive now an identity that allows us to eliminate one of thethree terms in Eq. (9.8), if we sandwich Λµ between two spinors which are on-shell. Weevaluate

Fµ = u(p′)[p/′γµ + γµp/

]u(p) (9.9)

first using the Dirac equation for the two on-shell spinors, finding

Fµ = 2mu(p′)γµu(p) . (9.10)

Secondly, we can use γµγν = ηµν − iσµν , obtaining

Fµ = u(p′)[(p′ + p)µ + iσµν(p′ − p)ν

]u(p) . (9.11)

Equating (9.10) and (9.11) gives the Gordon identity: It allows us to separate the Diraccurrent into a part proportional to (p+ p′)µ, i.e. with the same structure as a scalar current,and a part vanishing for q = p′ − p→ 0 which couples to the spin of the fermion as Eq. (9.6)suggests,

u(p′)γµu(p) = u(p′)

[(p′ + p)µ

2m+

iσµν(p′ − p)ν2m

]

u(p) . (9.12)

Moreover, the Gordon identity shows that the three terms in Eq. (9.8) are not independent.Depending on the context, we can eliminate therefore the most annoying term. We followconventions and introduce the (real) form-factors F1(q

2) and F2(q2) by

Λµ(p, p′) = F1(q2)γµ + F2(q

2)iσµνqν

2m= (9.13)

= F1(q2)

(p′ + p)µ

2m+ [F1(q

2) + F2(q2)]

iσµνqν2m

. (9.14)

The form-factor F1 is the coefficient of the electric charge, eF1(q2)γµ, and should thus go to

one for small momentum transfer, F1(0) = 1. Therefore the magnetic moment of an electronis shifted by 1 + F2(0) from the tree-level value g = 2. The deviation a ≡ (g − 2)/2 is calledanomalous magnetic moment, the two form-factors are often called electric and magneticform-factors.

Note the usefulness of the procedure to express the vertex function using only generalsymmetry requirements but not a specific theory for the interaction: Equation (9.13) allowsexperimentalists to present their measurements using only two scalar functions which in turncan be easily compared to predictions of specific theories.

111

9 Renormalisation

Anomalous magnetic moment After having discussed the general structure of the electro-magnetic vertex function, we turn now to its calculation in perturbation theory for the caseof QED. The Feynman diagrams contributing to the matrix element at O(e3) with wave-functions as external lines are shown in Fig. 9.1. We separate the matrix element into thetree-level part and the one-loop correction, −ieu(p′) [γµ + Γµ]u(p). Using the Feynman gaugefor the photon propagator, we obtain

Γµ(p, p′) =

∫d4k

(2π)4−i

k2 + iε(−ieγν)

i

p/′ − k/ −m+ iεγµ i

p/− k/ −m+ iε(−ieγν) . (9.15)

This integral is logarithmically divergent for large k,

∫ Λ

dkk3

k2k2∝ ln Λ . (9.16)

Before we perform the explicit calculation, we want to understand if this divergence is con-nected to a specific kinematical configuration of the momenta. We split therefore the vertexcorrection into an on-shell and an off-shell part,

Γµ(p, p′) = Γµ(p, p) + [Γµ(p, p′) − Γµ(p, p)] ≡ Γµ(p, p) + Γµoff(p, p′) . (9.17)

Next we rewrite the first fermion propagator for small p′ − p as

1

p/′ − k/ −m=

1

p/− k/ −m+ (p/′ − p/)= (9.18)

=1

p/− k/ −m−

1

p/− k/ −m(p/′ − p/)

1

p/ − k/ −m+ . . . (9.19)

The first term of this expansion leads to the logarithmic divergence of the loop integral forlarge k. In contrast, the reminder of the expansion that vanishes for p′ − p = q → 0 containsadditional powers of 1/k and is thus convergent. Hence the UV divergence is contained solelyin the on-shell part of the vertex correction, while the function Γµ

off(p, p′) = Γµ(p, p′)−Γµ(p, p)is well-behaved. Moreover, we learn from Eq. (9.13) that the divergence is confined to F1(0),while F2(0) is finite. This is good news: The divergence is only connected to a quantityalready present in the classical Lagrangian, the electric charge. Thus we can predict thefunction Γµ(p, p′) for all values p′ 6= p, after we have renormalised the electric charge in thelimit of zero momentum transfer.

We now calculate the vertex function (9.15) explicitly. We set

Γµ(q) = −ie2∫

d4k

(2π)4N µ(k)

D(9.20)

withN µ = γν(p/′ + k/ +m)γµ(p/+ k/ +m)γν (9.21)

andD =

[(p′ + k)2 −m2)

] [(p+ k)2 −m2)

]k2 . (9.22)

Then we combine the propagators introducing Feynman parameter integrals,

Γµ(q) = −ie2∫

d4k

(2π)4

∫ 1

0dα

∫ 1−α

0dβ

2N µ(k)

k2 + α[(p′ + k)2 −m2 − k2] + β[(p + k)2 −m2 − k2]3.

(9.23)

112

9.1 Anomalous magnetic moment of the electron

The complete calculation of the vertex function (9.15) for arbitrary off-shell momenta isalready quite cumbersome. In order to shorten the calculation, we restrict ourselves thereforeto the part contributing to the magnetic moment F2(0). Because of

Λµ(p, p′) =[F1(q

2 + F2(q2)

]γµ − F2(q

2)(p′ + p)µ

2m(9.24)

we can simplify the calculation of N µ(k), throwing away all terms proportional to γµ whichdo not contribute to the magnetic moment. Moreover, we can consider the limit that theelectrons are on-shell and the momentum transfer to the photon vanishes.

Using the on-shell condition, p2 = p′2 = m2, the two square brackets simplify to 2p′ · k and2p · k, respectively,

D =k2 + 2k · (αp′ + βp)

3. (9.25)

Next we eliminate the term linear in k completing the square,

D =

[k + (αp′ + βp)︸ ︷︷ ︸

l

]2 − (αp′ + βp)23

=[l2 − (α2m2 + β2m2 + 2αβp′ · p)

3. (9.26)

Since the momentum transfer to the photon vanishes, q2 = 2m2 − 2p′ · p→ 0, we can replacep′ · p→ m2 and obtain as final result for the denominator

D =l2 − (α+ β)2m2

3. (9.27)

Now we move on to the evaluation of the nominator N µ(k). Performing the change of ourintegration variable from k = l − (αp′ + βp) to l, the nominator becomes

N µ(l) = γν(P/ ′ + l/+m)γµ(P/ + l/+m)γν (9.28)

with P/ ′ ≡ (1 − α)p/′ − βp/ and P/ ≡ (1 − β)p/− αp/′.

Multiplying out the two brackets and ordering the result according to powers of m, weobserve first that the term ∝ m2 leads to ∝ γµ and thus does not contribute. Next we splitfurther the term linear in m according to powers of l: The term linear in l vanishes afterintegration, while the term ml/0 results in

m(γν(P/ ′γµγν + γνγµP/γν) = 4m(P ′µ + Pµ) = 4m[(1 − 2α)p′µ + (1 − 2β)pµ] . (9.29)

Using the symmetry in the integration variables α and β, we can rewrite this expression as

→ 4m[(1 − α− β)(p′µ + pµ)] . (9.30)

We split the m0 term in the same way according to the powers of l/. The m0 l/2 term gives aγµ term, the m0 l/ vanishes after integration, and the m0 l/0 gives after some work

γνP/ ′γµP/γν → 2m[α(1 − α) + β(1 − β)](p′ + p)µ . (9.31)

Finally, the m0 term contributes to the anomalous magnetic moment

→ −2m(p′ + p)µ[2(1 − α)(1 − β)] . (9.32)

113

9 Renormalisation

Combining all terms, we find

N µ = 4m(1 − α− β)(p′ + p)µ + 2m[α(1 − α) + β(1 − β)](p′ + p)µ (9.33)

−4m(1 − α)(1 − β)(p′ + p)µ = (9.34)

= 2m[(1 − α− β)(α+ β)](p′ + p)µ . (9.35)

Thus

Γµ2 (0) = −2ie2

∫

dαdβ

∫d2ωl

(2π)2ω

N µ

[l2 − (α+ β)2m2]3, (9.36)

where the subscript 2 indicates that we account only for the contribution to the anomalousmagnetic moment.

We can perform now the l integral using the formula for I(ω, a) with ω = 2 and a = 3,

I(2, 3) =i

32π2

1

(α+ β)2m2 + iε, (9.37)

obtaining as expected a finite result. As last step, we do the integrals over the Feynmanparameters α and β,

∫ 1

0dα

∫ 1−α

0dβ

1 − α− β

α+ β=

1

2(9.38)

and find thus

Γµ2 (0) =

e2

8π2

1

2m(p′ + p)µ . (9.39)

We have reproduced the result of the first successful calculation of a loop correction in aquantum field theory, performed by Schwinger 1948, F2(0) = α/2π. Together with Bethe’sprevious estimate of the Lamb shift in the hydrogen energy spectrum, this stimulated theview that a consistent renormalisation of QED is possible.

The currently most precise experimental value for the electron anomalous magnetic momentae ≡ F2(0) is

aexpe = 0.001 159 652 180 73(28)[0.24 ppb] . (9.40)

The calculation of the universal (i.e. common to all charged leptons) QED contribution hasbeen completed up to fourth order. There exists also an estimate of the dominant fifth ordercontribution,

auniℓ = 0.5

(α

π

)

− 0.328 478 965 579 193 78 . . .(α

π

)2

+1.181 241 456 587 . . .(α

π

)3− 1.9144(35)

(α

π

)4+ 0.0(4.6)

(α

π

)5

= 0.001 159 652 176 30(43)(10)(31) · · · (9.41)

The three errors given in round brackets are: The error from the uncertainty in α, thenumerical uncertainty of the α4 coefficient and the error estimated for the missing higherorder terms [Jeg07].

Comparing the measured value and the prediction using QED, we find an extremely goodagreement. First of all, this is strong support that the methods of perturbative quantum fieldtheory we developed so far can be successfully applied to weakly coupled theories as QED.Secondly, it means that additional contributions to the anomalous magnetic moment of theelectron have to be tiny.

114

9.1 Anomalous magnetic moment of the electron

Electroweak and other corrections The lowest order electroweak corrections to the anoma-lous magnetic moment contain in the loop virtual gauge bosons (W±, Z) or a higgs boson hand are shown in Fig. 9.2. We will consider the electroweak theory describing these diagramsonly later; for the present discussion it is sufficient to know that the weak coupling constant isg ∼ 0.6 and that the scalar and weak gauge bosons are much heavier than leptons, M ≫ m.

The second diagram corresponds schematically to the expression

∼ g2

∫d4k

(2π)41

k2 −M2

A(m2, k)

[(p − k) −m2]2. (9.42)

As in QED, this integral has to be finite and we expect that it is dominated by momentaup to the mass M of the gauge bosons, k <∼M . Therefore its value should be proportionalto g2m2/M2 (times a possible logarithm ln(M2/m2)) and electroweak corrections to theanomalous magnetic moment of the electron are suppressed by a factor (m/M)2 ∼ 10−10

compared to the QED contribution. The property that the contribution of virtual heavyparticles to loop processes is suppressed in the limit |q2| ≪M2 is called “decoupling”. Notethe difference to the case of the mass of a scalar particle or the cosmological constant: In theseexamples, the loop corrections are infinite and we cannot predict these quantities. However,we consider it as unnatural that the measured value ρvac is so much smaller than the expectedminimal value Λ4 >∼ (TeV)4. In contrast, the anomalous magnetic moment is finite but, aswe include loop momenta up to infinity, depends in principal on all particles coupling tothe electron, even if they are arbitrarily heavy. Only if these heavy particles “decouple,” wecan calculate ae without knowing e.g. the physics at the Planck scale. Thus the decouplingproperty is a necessary ingredient of any theory of high energy physics, otherwise nothing likechemistry or solid state physics would be possible before knowing the “theory of everything”.

Clearly, the contribution of heavy particles (either electroweak gauge and higgs bosonsor other not yet discovered particles) is more visible in the anomalous magnetic momentof the muon than of the electron. Moreover, a relativistic muon lives long enough that ameasurement of its magnetic moment is feasible. This is one example how radiative corrections(here evaluated at q2 = 0) are sensitive to physics at higher scales M : If an observable canbe measured and calculated with high enough precision, one can be sensitive to suppressedcorrections of order g2m2/M2. In the case of the anomalous magnetic moment of the muon,the achieved precision is high enough to probe generically scales of M ∼ 100 TeV, i.e. muchhigher than the mass scales that can be probed directly at current accelerators as LHC. Foran overview see [Jeg07].

W W

νµ Z Hµ

γa) b) c)

Figure 9.2: Lowest order electroweak corrections to the anomalous magnetic moment offermions.

115

9 Renormalisation

Finite versus divergent parts of loop corrections We found that the vertex correction couldbe split into two parts

Λµ(p, p′) = F1(q2)γµ + F2(q

2)iσµνqν

2m, (9.43)

where the form factor F2(q2) is finite for all q2, while the form factor F1(q

2) diverges forq2 → 0. The important observation is that F2(q

2) corresponds to a Lorentz structure thatis not present in the original Lagrangian of QED. This suggests that we can require from a“nice” theory that

• all UV divergences are connected to structures contained in the original Lagrangian,all new structures are finite. The basic divergent structures are also called “primitive”divergent graphs.

• If there are no anomalies, then loop corrections respect the original (classical) symme-tries. Thus, e.g., the photon propagator should be at all orders transverse, respectinggauge invariance. We will see that as consequence the high-energy behaviour of thetheory improves.

In such a case, we are able to hide all UV divergences in a renormalisation of the originalparameters of the Lagrange density.

9.2 Power counting and renormalisability

We try to make the requirements on a “nice” theory a bit more precise. Let us consider theset of λφn theories and check which graphs are divergent. We define the superficial degree Dof divergence of a Feynman graph as

D = 4L− 2I , (9.44)

where L is the number of independent loop momenta and I the number of internal lines.The former contributes a factor d4p, while the latter corresponds to a scalar propagator with1/(p2 − m2) ∼ 1/p2 for p → ∞. We can restrict our analysis to those diagrams called 1Pirreducible (1PI) which cannot be disconnected by cutting an internal line: All 1P reduciblediagrams can be decomposed into 1PI diagrams which do not contain common loop integralsand can be therefore analysed separately.

Momentum conservation at each vertex leads for an 1PI-diagram to

L = I − (V − 1) , (9.45)

where V is the number of vertices and the −1 takes into account the delta function leadingto overall momentum conservation. Thus

D = 2I − 4V + 4 . (9.46)

Each vertex connects n lines and any internal line reduces the number of external lines bytwo. Therefore the number E of external lines is given by

E = nV − 2I . (9.47)

As result, we can express the superficial degree D by the order of perturbation theory (V ),the number of external lines E and the degree n of the interaction polynomial φn,

D = (n− 4)V + 4 − E . (9.48)

From this expression, we see that

116

9.2 Power counting and renormalisability

• for n = 3, the coefficient of V is negative. Therefore only a finite number of terms inthe perturbative expansion is infinite. Such a theory is called super-renormalisable, thecorresponding terms in the Hamiltonian are also called relevant.

• For n = 4, we find D = 4 − E. Thus the degree of divergence is independent of theorder of perturbation theory being only determined by the number of external lines.Such theories contain an infinite number of divergent graphs, but they all correspond toa finite number of divergent structures—the so-called primitive divergent graphs. Suchinteractions are also called marginal and are candidates for a renormalisable theory.

• Finally, for n > 4 the degree of divergence increases with the order of perturbation the-ory. As result, there exists an infinite number of divergent structures, and increasing theorder of perturbation theory requires more and more input parameter to be determinedexperimentally. Such a theory is called non-renormalisable, the interaction irrelevant.

In particular, the λφ4 theory as an example for a renormalisable theory has only three diver-gent structures: i) the case E = 0 and D = 4 corresponding a contribution to the cosmologicalconstant, ii) the case E = 2 and D = 2 corresponding to the self-energy, and iii) the caseE = 2 and D = 0, i.e. logarithmically divergence, to the four-point function. As we will dis-cuss in the next section, the three primitive divergent diagrams of the λφ4 theory correspondto the following physical effects: Vacuum bubbles renormalise the cosmological constant. Theeffect of self-energy insertions is twofold: Inserted in external lines it renormalises the field,while self-energy corrections in internal propagators lead to a renormalisation of its mass.The vertex correction finally renormalises the coupling strength λ.

Let us move to the case of QED. Repeating the discussion, we obtain the analogue toEq. (9.49), but accounting now for the different dimension of fermion and bose fields,

D = 4 −B −3

2F , (9.49)

where B and F count the number of external bosonic and fermionic lines, respectively. Thereare six different superficially divergent primitive graphs in QED: The photon and the fermioniccontribution to the cosmological constant (D = 4), the vacuum polarisation (D = 2), thefermion self-energy (D = 1), the vertex correction (D = 0) and light-by-light scattering(D = 0).

In a gauge theory, the true degree of divergence can be smaller than the superficial one.For instance, light-by-light scattering corresponds to a term L ∼ A4 that violates gaugeinvariance. Thus either the gauge symmetry is violated by quantum corrections or such aterm is finite.

Because of the correspondence of the dimension of a field and the power of its propagator,we can connect the superficial degree of divergence of a graph to the dimension of the couplingconstants at its vertices. The superficial degree D(G) of divergence of a graph is connectedto the one of its vertices Dv by

D(G) − 4 =∑

v

(Dv − 4) (9.50)

which in turn depends as

Dv = δv +3

2fv + bv = 4 − [gv] (9.51)

117

9 Renormalisation

on the dimension of the coupling constant g at the vertex v. Here, fv and bv are the numberof fermion and boson fields at the vertex, while δv counts the number of derivatives.

Thus the dimension of the coupling constant plays a crucial role deciding if a certain theoryis “nice” in the naive sense defined above. Clearly D = 0 or [g] = 0 is the border-line case:

• If at least one coupling constant has a negative mass dimension, [g] < 0 and Dv > 4,the theory is non-normalisable. Examples are the Fermi theory of weak interactions,[GF ] = −2, and gravitation, [GN ] = −2.

• If all coupling constants have positive mass dimension, [g] > 0 and Dv < 4, the theoryis super-normalisable. An example is the λφ3 theory in D = 4 with [λ] = 1.

• The remaining cases, with all [gi] = 0, are candidates for renormalisable theories. Ex-amples are Yukawa interactions, λφ4, Yang-Mills theories that are unbroken (QED andQCD) or broken by the Higgs mechanism (electroweak interactions).

Theories with massive gauge bosons We have assumed that the propagators of bosonsbehave as 1/(p2 − m2) ∼ 1/p2 for p → ∞. This is true for scalars and for massless spin-1particles like the photon and gluons. In contrast, the longitudinal part in the propagator ofa massive spin-1 particles leads to a worse asymptotic behaviour. Including a mass term forgauge bosons breaks gauge invariance and leads to a non-renormalisable theory. A solutionto this problem is the introduction of gauge boson masses via interactions that respect thegauge symmetry as in the Higgs mechanism.

Furry’s theorem We have not included in our list of primitive divergent graphs of QEDloops with an odd number of fermions, i.e. the tadpole (B = 1 and D = 3) and “photonsplitting” graph (B = 3 and D = 1). Such diagrams vanish in vacuum, as fact known asFurry’s theorem. Since the momentum flow in a loop graph plays no rule, we can write aloop as “1/2(clockwise+ anti-clockwise)”. Then we insert CC−1 = 1 between all factors inthe trace, use CγµC−1 = −γµT and CSF (−x)C−1 = ST

F (x). Hence for an odd number ofpropagators, the two contributions cancel.

Alternatively, we can use a symmetry argument to convince us that all diagrams withan odd number of photons are zero: Because the QED Langrangian is invariant under C,ψ → ψc = CψT and Aµ =→ −Aµ, all Green functions with an odd number of photonsvanish.

9.3 Renormalisation of the λφ4 theory

We have argued that a theory with dimensionless coupling constant is renormalisable, i.e.that a multiplicative shift of the parameters contained in the classical Lagrangian is sufficientto obtain finite Green functions. The simplest theory of this type in D = 4 is the λφ4 theoryfor which we will discuss now the renormalisation procedure at one loop level. As starter, weexamine the general structure of the divergences and review different regularisation options.

9.3.1 Renormalisation and regularisation

Structure of the divergences We learnt that the degree of divergence decreases increasingthe number of external lines, since the number of propagators increases. The same effect has

118

9.3 Renormalisation of the λφ4 theory

taking derivatives wrt external momenta p,

∂

∂p/

1

k/ + p/−m= −

1

(k/ + p/−m)2.

This means that

1. we can Taylor expand loop integrals, confining the divergences in the lowest order terms.Choosing e.g. p = 0 as expansion point in the fermion self-energy,

Σ(p) = A0 +A1p/+A2p2 + . . .

with

An =1

n!

∂n

∂p/n Σ(p) ,

we know that A0 is (superficially) linear divergent. Thus A1 can be maximally loga-rithmic divergent, while all other An are finite.

2. We could choose a different expansion point, leading to different renormalisation con-ditions (within the same regularisation scheme).

3. The divergences can be subtracted by local operators, i.e. polynomials of the fields andtheir derivatives.

The last statement requires that non-local terms as e.g. ln(p2/µ2) which can be generated by sub-loops in a diagram of order gn are cancelled by counter-terms of order n′ < n. A sketch why thisshould be true goes as follows:Green functions become singular for coinciding points, i.e. when the convergence factor e−kx in theEuclidean Green function becomes one. In the simplest cases (separate divergences in one diagram)as

〈0|φ(x′)φ(x)|0〉x′→x =

∫d3k

(2π)32ωke−ik(x′−x)

∣∣∣∣x′→x

=

∫d3k

(2π)32ωk, (9.52)

the infinities are eliminated by normal ordering. More complicated are overlapping divergences wheretwo or more divergent loops share a propagator. Wilson suggested to expand the product of two fieldsas the sum of local operator Oi times coefficient functions Ci(x− y) as

φ(x)φ(y) =∑

i

Ci(x− y)Oi(x) ,

where the whole spatial dependence is carried by the coefficients. For a massless scalar field, dimen-sional analysis dictates that Ci(x) ∝ x−2+di , if the local operator Oi has dimension di. Note thatonly the unity operator has a singular coefficient function 1/x2 corresponding to the massless scalarpropagator. Similarly we can expand product of operators,

On(x)Om(y) =∑

i

Cinm(x− y)Oi(x) ,

where now Cinm(x) ∝ x−dn−dm+di . Thus we can use this operator product expansion (or briefly

“OPE”) to rewrite the overlapping divergences in terms of coefficient functions and finite local oper-

ators. Moreover, the sub-divergence occurring at order k, when p < k points coincide, have the form

found at order p. We conclude that non-local terms due to overlapping divergencies are cancelled by

the counter terms found at lower order.

119

9 Renormalisation

Regularisation schemes Mathematical manipulations as shifting the integration variable indivergent loop integral are only well-defined, if we convert first these integrals into convergentones. This process is called regularisation and can be done in a number of ways. In general,one reparametrises the integral in terms of a parameter Λ (or ε) called regulator such thatthe integral becomes finite for finite regulator while the limit Λ → ∞ (or ε → 0) returns tothe original integral.

• We can avoid UV divergences evaluating loop-integrals introducing an (Euclidean) mo-mentum cutoff,

I → IΛ ≡

∫ Λ

d4k F (k) = A (Λ) +B + C

(1

Λ

)

.

Somewhat more sophisticated, we could introduce instead of a hard cutoff a smoothfunction which suppresses large momenta. Using Schwinger’s proper-time representation

1

p2 +m2=

∫ ∞

0ds e−s(p2+m2) (9.53)

we can cut-off large momenta setting

1

p2 +m2→

e−s(p2+m2)/Λ2

p2 +m2=

∫ ∞

Λ−2

ds e−s(p2+m2) . (9.54)

Although conceptual very easy, both regularisation schemes violate generically all sym-metries of our theory. This is not a principal flaw, since we should recover these sym-metries in the limit Λ → ∞. But calculations become much thougher since we cannotuse the symmetries at intermediate steps, and therefore these schemes are in practisenot useful except for the simplest cases.

• Pauli-Villars regularisation is a scheme respecting gauge invariance of QED. Its basicidea is to add heavy particles coupled gauge invariantly to the photon,

1

k2 −m2 + iε→

1

k2 −m2 + iε+

∑

i

ai

k2 −M2 + iε.

For k2 ≪ M2, physics is unchanged, while for k2 ≫ M2 and ai < 0 the combinedpropagators scale as M2/k4 and the convergence of loop integrals improves. Since theheavy particles enter with the wrong sign, they are unphysical and serve only as amathematical tool to regularise loop diagrams.

• Dimensional Regularisation (DR) is one of the most useful and least intuitive regulators.The reason for its usefulness is that it preserves gauge invariance. In DR, we replaceour integrals with

I → Id ≡

∫ ∞

0ddkF (k)

where d is the dimension of our measure. As the example∫

ddk k−2 = 0 which wewill discuss later shows the “measure” we implement by physical requirements with DRis not a positive measure—as a mathematician would require. Nevertheless, we can

120

9.3 Renormalisation of the λφ4 theory

perform integrals in d 6= 4 space-time dimensions. Then we can replace d → 4 − ε inthe result and take the limit ε→ 0, splitting the result into poles and finite parts.

Evaluating fermion traces in the denominator using DR, we have to extend the Cliffordalgebra to d dimensions. A natural choice is trγµ, γν = dηµν . Problematic is howeverthe treatment of γ5 ≡ iγ0γ1γ2γ3 relying heavily on d = 4. As a result DR breaks chiralsymmetry. The same is true for supersymmetry.

• In lattice regularisation one replaces the continuous space-time by a discrete lattice.The finite lattice spacing a introduces a momentum cutoff, eliminating all UV diver-gences. Moreover the (Euclidean) path-integral becomes well-defined and can be calcu-lated numerically without the need to do perturbation theory. Thus this approach isparticularly useful in the strong-coupling regime of QCD. It has been very successful inpredicting many results of low-energy QCD, including hadron masses and form-factors,which otherwise are only calculable in low-energy approximations to QCD. Note thatlattice regularisation for finite a respects the gauge symmetry, but spoils translationand Lorentz symmetry of the underlying QFT. Nevertheless, one recovers in the limita → 0 a relativistic QFT. A longstanding problem of lattice theory was how to imple-ment correctly chiral fermions. This question has been recently solved and thus the SMcan be now in a mathematically consistent, non-perturbative way defined as a latticetheory.

• Various other regularisation schemes as e.g. zeta function regularisation or point split-ting methods exists.

Even fixing a regularisation scheme, e.g. DR, we can choose various renormalisation condi-tions. Three popular choices are

• on-shell renormalisation. In this scheme, we choose the subtraction such that the on-shell masses and couplings coincide with the corresponding values measured in processeswith zero momentum transfer. For instance, we define the renormalised electric chargevia the Thomson limit of the Compton scattering amplitude. While very intuitive, it isnot practical for QCD.

• the minimal subtraction (MS) scheme, where we subtract only the 1/ε poles.

• the modified minimal subtraction MS (read em-es-bar) scheme, where we subtract ad-ditionally the ln(γE) + 4π term appearing frequently. This scheme gives compacterexpressions than the others and is most often used in theoretical calculations. As draw-back, quantities like mMS

e have to be translated into the physical mass me.

9.3.2 The λφ4 theory at one-loop

There are two equivalent ways to perform perturbative renormalisation. In the one we usefirst called often “conventional” perturbation theory we use the “bare” (unrenormalised)parameters in the Lagrangian,

L = L0 + Lint =1

2(∂µφ0)

2 −1

2m2

0φ20 +

λ0

4!φ4

0 . (9.55)

Then we introduce a renormalised field φR = Z−1/2φ φ0 and choose the parameters Zφ,m0 and

λ0 as function of the regularisation parameter (ε, Λ, . . .) such that the field φR has finite

121

9 Renormalisation

Green functions. In the following we discuss the renormalisation procedure at one-loop levelfor the Green functions of the λφ4 theory in this scheme. Since any 1P reducible diagramcan be decomposed into 1PI diagrams which do not contain common loop integrals, we canrestrict our analysis again to 1PI Green functions.

Mass and wave-function renormalisation The Feynman propagator in momentum space is

i∆F (p) =

∫

d4x eipx 〈0|Tφ0(x)φ0(0) |0〉 =i

p2 −m20 + iε

(9.56)

at lowest order perturbation theory. The full propagator i∆F (p) is the sum of an infinitechain of self-energy insertions,

i∆F (p) =i

p2 −m20 + iε

+i

p2 −m20 + iε

(−iΣ(p2)

) i

p2 −m20 + iε

+ . . . (9.57)

This expression looks like an infinite tower of independently propagating particle with differentmasses. But if we sum up the terms of this binomial series, we obtain

i∆F (p) =i

p2 −m20 + iε

1

1 + iΣ(p2) ip2−m2

0+iε

=i

p2 −m20 − Σ(p2) + iε

. (9.58)

Now we see that the effect of the interactions results in a shift of the particle mass, m20 →

m20 + Σ(p2). Next we have to show that Σ(p2) is finite after renormalisation. The 1-loop

expression is

−iΣ(p2) =−iλ0

2

∫d4k

(2π)4i

k2 −m20 + iε

, (9.59)

i.e. quadratically divergent. As a particularity of the φ4 theory, the p2 dependence of theself-energy Σ shows up only at the 2-loop level.

We perform a Taylor expansion of Σ(p2) around the arbitrary point µ,

Σ(p2) = Σ(µ2) + (p2 − µ2)Σ′(µ2) + Σ(p2) , (9.60)

where Σ(µ2) ∝ Λ2, Σ′(µ2) ∝ ln Λ and Σ(p2) is the finite reminder. A term linear in Λ isabsent, since we cannot construct a Lorentz scalar out of pµ. Note also Σ′(µ2) = Σ(µ2) = 0.

Now we insert (9.60) into (9.58),

i

p2 −m20 − Σ(p2) + iε

=i

p2 −m20 − Σ(µ2)

︸ ︷︷ ︸

p2−µ2

−(p2 − µ2)Σ′(µ2) − Σ(p2) + iε, (9.61)

where we see that we can identify µ with the physical mass given by the pole of the propagator.

We aim at rewriting the remaining effect for p2 → µ2 of the self-energy insertion, Σ(µ2), asa multiplicative rescaling. In this way, we could remove the divergence from the propagatorby a rescaling of the field. At leading order in λ, we can write

Σ(p2) =[1 − Σ′(µ2)

]Σ(p2) + O(λ2

0) (9.62)

122

9.3 Renormalisation of the λφ4 theory

and thus

i∆F (p) =1

1 − Σ′(µ2)

i

p2 − µ2 − Σ(p2) + iε=

iZφ

p2 − µ2 − Σ(p2) + iε(9.63)

with the wave-function renormalisation constant

Zφ =1

1 − Σ′(µ2)= 1 + Σ′(µ2) . (9.64)

Close to the pole, the propagator is the one of a free particle with mass µ,

i∆F (p) =iZφ

p2 − µ2 + iε+ O(p2 − µ2) . (9.65)

Thus the renormalisation constant Zφ is the same Z appearing in the LSZ formula. As faras S-matrix elements are concerned, the renormalisation of external lines has therefore thesole effect of cancelling the Z−1 factor from the LSZ formula. Therefore we can obtain S-matrix elements replacing propagators on external lines directly with on-shell wave-functions,excluding at the same time all diagrams where external lines are renormalised.

We define the renormalised field φ = Z−1/2φ φ0 such that the renormalised propagator

i∆R(p) =

∫

d4x eipx 〈0|Tφ(x)φ(0) |0〉 = Z−1φ i∆(p) =

i

p2 − µ2 − Σ(p2) + iε(9.66)

is finite. Similarly, we define renormalised n-point functions by

G(n)R (x1, . . . , xn) = 〈0|Tφ(x1) · · ·φn(xn) |0〉 = Z

−n/2φ G

(n)0 (x1, . . . , xn) . (9.67)

Coupling constant renormalisation We can choose an arbitrary point inside the kinematicalregion, s + t + u = 4µ2 and s ≥ 4µ2, to define the coupling. For our convenience and lesswriting work, we choose instead the symmetric point

s0 = t0 = u0 =4µ2

3.

The bare four-point function is (see section 7.5)

Γ(4)0 (s, t, u) = −iλ0 + Γ(s) + Γ(t) + Γ(u) , (9.68)

the renormalised four-point function at (s0, t0, u0) is

Γ(4)R (s0, t0, u0) = −iλ . (9.69)

Next we expand the bare 4-point function around s0, t0, u0,

Γ(4)0 (s, t, u) = −iλ0 + 3Γ(s0) + Γ(s) + Γ(t) + Γ(u) (9.70)

where the Γ(x) are finite and zero at x0. Now we define a vertex (or coupling constant)renormalisation constant by

−iZ−1λ λ0 = −iλ0 + 3Γ(s0) (9.71)

123

9 Renormalisation

Inserting this definition in (9.70) we obtain

Γ(4)0 (s, t, u) = −iZ−1

λ λ0 + Γ(s) + Γ(t) + Γ(u) (9.72)

which simplifies at the renormalisation point to

Γ(4)0 (s, t, u) = −iZ−1

λ λ0 . (9.73)

We use now the connection between renormalised and bare Green functions,

Γ(4)R (s, t, u) = Z2

φΓ(4)0 (s, t, u) (9.74)

and thus−iλ = Z2

φZ−1λ (−iλ0) . (9.75)

The relation between the renormalised and bare coupling in the λφ4 theory is thus

λ = Z2φZ

−1λ λ0 . (9.76)

Now we have to show that Γ(4)R (s, t, u) is finite.

Γ(4)R (s, t, u) = Z2

φΓ(4)0 (s, t, u)

= −iZ2φZ

−1λ λ0 + Z2

φ[Γ(s) + Γ(t) + Γ(u)]

= −iλ+ Z2φ[Γ(s) + Γ(t) + Γ(u)] (9.77)

Since Zφ = 1 + O(λ2), this is equivalent to

Γ(4)R (s, t, u) = −iλ+ [Γ(s) + Γ(t) + Γ(u)] + O(λ3) (9.78)

consisting only of finite expressions. This completes the proof that at one-loop order allGreen functions in the λφ4 theory are finite, renormalising the field φ, its mass and couplingconstant as

φ = Z−1/2φ φ0 (9.79a)

λ = Z2φZ

−1λ λ0 , (9.79b)

µ2 = m20 + Σ(µ2) = m2

0 + δm2φ2 . (9.79c)

Renormalised perturbation theory We can use the conditions (9.79) in order to formulateperturbation theory using directly only renormalised quantities. We set L0 = L +Lct wherehas the same structure as L but is expressed through renormalised quantities,

L =1

2(∂µφ)2 −

1

2m2φ2 +

λ

4!φ4 (9.80)

and thus

Lct =1

2(Zφ − 1)(∂µφ)2 −

1

2δm2φ2 + (Zλ − 1)

λ

4!φ4 (9.81)

The term Lct is called counter term Lagrangian and contains the divergent renormalisationconstants. The latter are all of order λ and thus we can treat Lct as a perturbation. Applyingrenormalised perturbation theory consists of the following steps:

124

9.3 Renormalisation of the λφ4 theory

1. Starting from (9.80), one derives propagator and vertices.

2. One calculates 1-loop 1PI diagrams, finds the divergent parts which determine the

counter terms in L(1)ct at order O(λ).

3. The new Lagrangian L + L(1)ct is used to generate 2-loop 1PI diagrams and L

(2)ct .

4. The procedure is iterated moving to higher orders.

Renormalisation group equations Consider two renormalisation schemes R and R′. In

the two schemes, the renormalised field will in general differ, being φR = Z−1/2φ (R)φ0 and

φR′ = Z−1/2φ (R′)φ0, respectively. Hence the connection between the two renormalised fields

is

φR′ =Z

−1/2φ (R′)

Z−1/2φ (R)

φR ≡ Z−1/2φ (R′, R)φR . (9.82)

As both φR and φR′ are finite, also Zφ(R′, R) is finite. The transformations Z−1/2φ (R′, R)

form a group, called the renormalisation group.

If we consider

G(n)0 (x1, . . . , xn) = Z

−n/2φ G

(n)R (x1, . . . , xn) , (9.83)

we know that the bare Green function is independent of the renormalisation scale µ. Takingthe derivative with respect to µ, the LHS thus vanishes. For the simplest case of a masslesstheory, we find thus

0 =d

d lnµ

[

Zn/2φ G

(n)R (x1, . . . , xn)

]

=

[∂

∂ lnµ+

∂λ

∂ lnµ

∂

∂λ−n

2

∂ lnZφ

∂ lnµ

]

G(n)R (x1, . . . , xn)

≡

[∂

∂ lnµ+ β

∂

∂λ−n

2γ

]

G(n)R (x1, . . . , xn) . (9.84)

Here we introduced in the last step the beta function

β(λ) = µ∂λ(µ)

∂µ, (9.85)

which determines the logarithmic change of the coupling constant and the anomalous dimen-sion

γ(µ) = µ∂ lnZφ(µ)

∂µ. (9.86)

of the field φ. The physical meaning of γ(µ) will be the topic of problem 9.6.

Knowing these two functions, we can calculate the change of any Green function undera change of the renormalisation scale µ. The general solution of (9.84) can be found bythe method of characteristics or by the analogy of d/d lnµ with a convective derivative, cf.problem 9.7.

Equations of the type (9.84) are called generically renormalisation group equations orbriefly RGE. They come in various flavors, carrying the name of their inventors: Stuckelberg–Petermann, Callan–Symanzik, Gell-Man–Low, . . .

125

9 Renormalisation

g

β(g)

gc g

β(g)

gc

IR

UV

UV

IR

Figure 9.3: Left: Example of a beta function with a perturbative IR and an UV fixed point.Right: General classification of UV and IR stable fixed points of the RGE flow.

Asymptotic behaviour of the beta-function The behaviour of the beta-function β(µ) in thelimit µ → 0 and µ → ∞ provides a useful classification of quantum field theories. Considere.g. the example shown in the left panel of Fig. 9.3: This beta function has a trivial zero atzero coupling, as we expect it in any perturbative theory, and an additional zero at gc. Howdoes the beta-function β(µ) evolve in the UV limit µ→ ∞?

• Starting in the range 0 ≤ g(µ) < gc implies β > 0 and thus dg/dµ > 0. Therefore ggrows with increasing µ and the coupling is driven towards gc.

• Starting with g(µ) > gc implies β < 0 and thus dg/dµ < 0. Therefore g decreases forincreasing µ and we are driven again towards gc.

Fixed points gc approached in the limit µ → ∞ are called UV fixed points, while IR fixedpoints are reached for decreasing µ. The range of values [g1 : g2] which is mapped by theRGE flow on the fixed point is called its basin of attraction.

To see what happens for µ → 0, we have only to reverse the RGE flow, dµ → −dµ, andare thus driven away from gc: If we started in 0 ≤ g(µ) < gc, we are driven to zero, whilethe coupling goes to infinity for g(µ) > gc. Thus g = 0 is an IR fixed point. The distinctionbetween IR and UV fixed points is sketched in the right panel of Fig. 9.3. If the beta functionhas several zeros, the theory consists of different phases which are not connected by the RGflow.

Working through problem 9.5, you should show that the beta-function of the λφ4 theoryat one loop is given by

β(λ) = b1λ2 + b2λ

3 + . . . =3λ2

16π2+ O(λ3) , (9.87)

so that the running coupling satisfies

λ(µ) =λ0

1 − 3λ0/16π2 ln(µ2/µ20). (9.88)

Thus the theory has λ = 0 as an IR fixed point and the use of free particles as asymptoticstates is sensible (but requires renormalisation). On the other hand, the coupling increases

126

9.4 Vacuum polarisation

for µ → ∞ formally as λ → ∞. Clearly, we cannot predict the behaviour of λ(µ) in thestrong-coupling limit, because perturbation theory breaks down. The solution (9.88) suggestshowever that the coupling explodes already for a finite value of µ: The beta function has apole for a finite value of µ called Landau pole where the denominator of (9.88) becomes zero.

The large logs Although we derived the beta functions only at O(λ), the running cou-pling contains arbitrary powers of [λ ln(µ2/µ2

0)]n, since λ(µ) = λ0

∑

n[b1λ ln(µ2/µ20]

n/n!. Thisshould be contrasted with our original definition of the running coupling in Eq. (7.131). Ifthe expansion parameter b1λ is small, we managed to sum up the leading terms from higherorder corrections. If on the other hand the expansion parameter b1λ is large, our perturbativeapproach fails anyways in this regime.

9.4 Vacuum polarisation

We now turn from the λφ4 theory to the case of unbroken gauge theories as QED and QCD.There are several complications compared to the simpler case of a scalar theory: First, thecomplexity of the calculations will increase together with the number of vertices. Second,we have to ensure that the gauge symmetry is not spoiled by loop corrections. From a moretechnical point of view, we note that in processes involving gauge bosons small errors canhave drastic consequences, because the leading terms in a gauge-invariant set of Feynmandiagrams often cancel. Finally, theories containing massless spin 1 or 2 particles have IRdivergencies additional to UV divergencies, showing up in the emission of real or virtual softphotons, gluons or gravitons. Our discussion aims not to be complete. Instead we pick out asingle process, the vacuum polarisation from which we can deduce the salient point of gaugetheories: asymptotic freedom.

Using “conventional” perturbation theory, we define wave-function renormalisation con-stants for the electron and photon as

ψ0 ≡ Z1/22 ψ (9.89)

Aµ0 ≡ Z

1/23 Aµ (9.90)

Analogous to Eq. (9.76), we expect that the electric coupling is renormalised by

e(µ) =Z2Z

1/23

Z1e0 , (9.91)

where Z1 is the charge renormalisation constant, and Z2 and Z1/23 take into account that two

electron fields and one photon field enter the 3-point function.As it stands, the renormalisation condition (9.91) creates two major problems: First, the

factor Z1/22 will vary from fermion to fermion. For instance, the wave-function renormalisation

constant of a proton includes strong interactions while the one of the electron does not. Asa result, it is difficult to understand why the electric charge of an electron and an protonare renormalised such that they have the same value for q2 → 0, while they would disagreefor q2 6= 0. In particular, we would expect that the universe is not electrically neutral, evenassuming the same number density of both particles, ne = np. Second, we see that therenormalised covariant derivative

Dµ = ∂µ + iZ1

Z2e(µ)Aµ (9.92)

127

9 Renormalisation

remains only gauge-invariant, if Z1 = Z2. Clearly, this condition would also ensure theuniversality of the electric charge. We postpone the proof that Z1 = Z2 holds in suitablerenormalisation schemes to section 9.5.2, where we will derive the Ward-Takahashi identities.In a non-abelian theory as QCD, where we have to ensure that the gauge coupling in all termsof Eq. (8.56) remains after renormalisation the same, several constraints of the type Z1 = Z2

arise.

9.4.1 Vacuum polarisation in QED

We calculate next the one-loop correction to the photon propagator, the so called vacuumpolarisation tensor. Using the Feynman rules for QED, we obtain for the contribution of onefermion species with mass m

iΠµν(q) = −(−ie)2∫

d4k

(2π)4tr[γµi(k/ +m)γν i(k/ + q/+m)]

(k2 −m2)[(k + q)2 −m2]= −e2

∫d4k

(2π)4N µν

D. (9.93)

Our first task is to show that the vacuum polarisation tensor respects gauge invariance, i.e. hasthe structure Πµν(q) = (q2gµν−qµqν)Π(q2). For this it is sufficient to show that qµΠµν(q) = 0.We write

q/ = (q/+ k/ −m) − (k/ −m) (9.94)

and obtain

qµNµν = tr[(q/+ k/ −m) − (k/ −m)](k/ +m)γν(k/ + q/+m) (9.95)

= [(q + k)2 −m2]tr(k/ +m)γν − (k2 −m2)trγν(k/ + q/+m) . (9.96)

Employing dimensional regularisation (DR) with d = 4 − 2ε in order to obtain well-definedintegrals,

qµiΠµν(q) = −e2µ2ε

∫ddk

(2π)d

tr[(k/ +m)γν ]

(k2 −m2)−

tr[γν(k/ + q/+m)]

(k + q)2 −m2

, (9.97)

we are allowed to shift the integration variable in one of the two terms. Thus qµΠµν(q) = 0and hence the vacuum polarisation tensor at order O(e2) is transverse as required by gaugeinvariance.

Let us pause a moment and summarise before we start with the evaluation of Π(q2): In ourpower-counting analysis we found as superficial degree of divergence D = 2. This result wasbased on the assumption that the numerator N behaves as a constant. But the only constantavailable is m2 which would lead to a mass term of the photon, e2AµΠµνAν ∼ e2m2Aµg

µνAν .Thus the transversality of Πµν implies that the m2 term in the numerator will disappear atsome step of our calculation. Thereby the convergence of the polarisation tensor improves,becoming a “mild logarithmic” one.

We proceed with the explicit evaluation of Πµν . Taking the trace of (9.93) and using itstransversality, we find in d dimensions

Πµµ(q) = (q2δµ

µ − q2)Π(q2) = (d− 1)q2Π(q2)

and

(d− 1)q2iΠ(q2) = e2d

∫ddk

(2π)dtr[γµ(k/ +m)γµ(k/ + q/+m)]

(k2 −m2)[(k + q)2 −m2]= e2d

∫ddk

(2π)dN

D. (9.98)

128

9.4 Vacuum polarisation

We combine the two propagators introducing the Feynman parameter integral

1

ab=

∫ 1

0

dx

[ax+ b(1 − x)]2(9.99)

with a = (k − q)2 −m2 and b = k2 −m2. Thus the denominator becomes

D = az + b(1 − z) = k2 − 2kqx+ q2x−m2 = (k − qx)2 + q2x(1 − x) −m2 . (9.100)

Next we introduce as new integration variable K = k − qx,

D = K2 + q2x(1 − x) −m2 = K2 − a , (9.101)

and a as short-cut.Evaluating the trace in the denominator using DR, we have to extend the Clifford algebra

to d dimensions. A natural choice is trγµ, γν = dηµν , giving with γµγµ = d and γµq/γµ =(2 − d)q/ as result for the trace

N = N µµ = d[(2 − d)k · (k + q) + dm2] = d(2 − d)[K2 − q2x(1 − x)] + dm2 . (9.102)

In the last step, we performed the shift k → K + qx omitting linear terms in K that vanishafter integration. Finally, we want to perform a Wick rotation which is allowed as long aswe do not pass a singularity. Since the prefactor x(1 − x) of q2 has as maximum 1/4, thisrequires q2 < 4m2,

(d− 1)q2Π(q2) = e2dd

∫ 1

0dx

∫ddK

(2π)d(2 − d)[K2 + q2x(1 − x)] − dm2

(K2 + a)2. (9.103)

The integrand depends only on K2 and we can therefore write

µ2ε

∫d4−2εK

(2π)4−2ε= µ2ε(2π)−4+2ε

∫

dΩd

∫ ∞

0dKK3−2ε . (9.104)

We consider first the K integrals,

I1 +I2 +I3 =

∫ ∞

0dK

(2 − d)K2(3−ε)−1

(K2 + a)2− dm2 K

2(2−ε)−1

(K2 + a)2− (2 − d)q2x(1 − x)

K3(3−ε)−1

(K2 + a)2

.

(9.105)We evaluate the integrals rewriting the definition of the Euler beta function, Eq. (A.21), as

∫ ∞

0dK

K2x−1

(K2 + a)y= ax−y Γ(x)Γ(y − x)

2Γ(y). (9.106)

We start to look for the first two terms where we expect a cancellation of the m2 term in thenominator,

I1 + I2 = (2 − d)a1−εΓ(3 − ε)Γ(−1 + ε)/2 − dm2a−εΓ(2 − ε)Γ(ε)/2 . (9.107)

Next we use zΓ(z) = Γ(z + 1) to combine the two terms, obtaining

I1 + I2 = Γ(ε)Γ(3 − ε)(a1−ε −m2a−ε

)= −Γ(ε)Γ(3 − ε)

q2x(1 − x)

[m2 − q2x(1 − x)]ε. (9.108)

129

9 Renormalisation

Hence the m2 term dropped indeed out of the nominator. Evaluating I3 in the same way andadding then a factor (2π)d/2Γ(2/d) = (2π)2 from the angular integral dΩd, we find

(d− 1)q2Π(q2) =e2

2π2q2d

∫ 1

0dx x(1 − x)

[4πµ2

m2 − q2x(1 − x)

]ε

Γ(ε)(3 − 2ε) . (9.109)

The result is proportional to q2, as it should be to match the q2 on the LHS. As last step wehave to expand this expression around1 d = 4 − 2ε using (A.20) and aε = 1 + ε ln a+ O(ε2),

Π(q2) =e202π2

1

ε+ γ + ln(4π) − 3

∫ 1

0dx x(1 − x) ln

[µ2

m2 − q2x(1 − x)

]

. (9.110)

The prefactor x(1 − x) has its maximum 1/4 for x = 0.5. Thus the branch cut of thelogarithm starts at q2 = 4m2, i.e. when the virtuallity of the photon is large enough that iscan decay into a fermion pair of mass 2m. This is a nice illustration of the optical theorem:The polarisation tensor is real below the pair creation threshold, and acquires an imaginarypart above (which equals the pair creation cross section of a photon with mass m2 = q2, cf.problem 9.8).

The x integral can be integrated by elementary function, but we display the result only forthe limiting case |q2|/m2 → 0,

Π(q2) =e20

12π2

[1

ε+ γ + ln(4π) + ln(µ2/m2) +

q2

5m2+ . . .

]

. (9.111)

In the MS scheme, we obtain the renormalisation constant Z3 for the photon field as thecoefficient of the pole term,

Z3 = 1 −e20

12π2ε. (9.112)

More often the on-shell renormalisation scheme is used in QED. Here we require that quantumcorrections to the electric charge vanish for q2 = 0, i.e. we choose Z3 such that Πon(q2 = 0) =0. This is obviously achieved setting

Πon(q2) = Π(q2) − Π(0) =e20

60π2

q2

m2+ . . . , (9.113)

for |q2| ≪ m2. This q2 dependence leads to a modification of the Coulomb potential, whichcan be measured e.g. in the Lamb shift.

Beta function We can derive the scale dependence of the renormalised electric charge from

e0 =µε

Z1/23

e , (9.114)

where we used Z1 = Z2. Then the beta function is given as

β(e) ≡ µ∂e

∂µ= µ

∂

∂µ

(

µ−εZ1/23 e0

)

. (9.115)

1Note we use both conventions d = 4 − 2ε and d = 4 − ε, depending on the calculation.

130

9.4 Vacuum polarisation

Since the bare charge e0 is independent of µ, we have to differentiate only µ and Z3,

β(e) = µ∂e

∂µ= −εµ−εZ

1/23 e0 + µµ−ε 1

2Z

−1/23

∂Z3

∂µe0 (9.116)

= −εe+µ

2Z3

∂Z3

∂µe . (9.117)

Inserting Z3 = 1 − e2/(12π2ε) and thus

∂Z3

∂µ= −

1

12π2ε

2e∂e

∂µ(9.118)

gives

β(e) = −εe−µ

12π2εZ3

∂e

∂µe2 = −εe−

1

12π2εZ3β(e) e2 . (9.119)

Note that Z3 is scheme-dependent, while the beta-function remains scheme independent upto two loop (problem 9.5). Solving for β and neglecting higher order terms in e2, we find inthe limit ε→ 0

β(e) = −εe

(

1 −e2

12π2ε+ O(e4)

)

=e3

12π2. (9.120)

Thus the beta function contains indeed the renormalised charge on the RHS, as announcedafter Eq. (??). Finally we note that the beta function can be re-expressed as

β(e2) ≡ µ2 ∂e2

∂µ2= eβ(e) =

e4

12π2. (9.121)

Its solution,

e2(µ) =e2(µ0)

1 − e2(µ0)6π2 ln

(µµ0

) (9.122)

shows not only explicitly the increase of e2 with µ, but moreover that the electric couplingdiverges for a finite value of µ. This singularity called Landau pole happens at

µ = µ0 exp(6π2/e2(µ0)) = me exp(3π/2α(me)) ∼ 1056 GeV ≫MPl (9.123)

and has therefore no physical relevance.

9.4.2 Vacuum polarisation in QCD

We only sketch the calculations in QCD, stressing the new or different points compared toQED. In Fig. 9.4, we show the various 1-loop diagrams contributing to the vacuum polarisationtensor in a non-abelian theory as QCD. Most importantly, the three-gluon vertex allows inaddition to the quark loop now also a gluon loop. Since a fermion loop has an additional minussign, we expect that the gluon loop gives a negative contribution to the beta function. Thisopens the possibility that non-abelian gauge theories are in contrast to QED asymptoticallyfree, if the number of fermion species is small enough.

131

9 Renormalisation

1

Figure 9.4: Feynman diagrams describing vacuum polarisation in QCD at one-loop.

Quark loop The vertex changes from −ieγµ in QED to −igsTaγµ in QCD. Since the quark

propagator is diagonal in the group index, a quark loop contains for each flavor additionallythe factor

trT aT a =1

2δaa = 4 . (9.124)

Thus we have only to replace e → 4nfgs in the QED result, where nf counts the number ofquark flavors. For the three light quarks, u, d and s, it is an excellent approximation to setm = 0. In contrast, the masses of the other three quarks (c, b and t) can not be neglected.As the calculation for massive quarks shows is the effect of particle masses well approximatedincluding in the loop only particles with mass 4m2 < µ2, making nf scale dependent.

Loop with three-gluon vertex Since the three-gluon vertex connects identical particles, wehave to take into account symmetry factors similar as in the case of the λφ4 theory. Welearnt that the imaginary part of a Feynman diagram corresponds to the propagation of realparticles. Thus the imaginary part of the gluon vacuum polarisation can be connected to thetotal cross section of g → gg scattering. This cross section contains a symmetry factor 1/2!to account for two identical particles in the final state. Therefore the same symmetry factorshould be associated to the vacuum polarisation with a gluon loop2. Applying the Feynmanrule for the three-gluon and using the Feynman-t’Hooft gauge, we find

iΠµνab 2(q

2) =1

2(−ig)2

∫d4k

(2π)4N µν

ab

(k2 + iε)[(k + q)2 + iε](9.125)

with

N µνab = fbcd[−η

νρ(q + k)σ + ηρσ(2k − q)ν + ησν(2q − k)ρ] (9.126)

× facd[−ηµρ(q + k)σ + ηρσ(q − 2k)µ + ηµ

σ(k − 2q)ρ] . (9.127)

Evaluating the colour trace,

facdf cdb = Ncδab (9.128)

2This argument does not apply to the quark loop, since cutting leads in this case to a distinguishable qq state

132

9.4 Vacuum polarisation

and extracting in the usual way the pole part using DR one obtains for d = 4 − ε

iΠµνab 2(q

2) = −iNcg

2

16π2ε

(µ2

q2

)ε/2 (11

3qµqν −

19

6ηµνq2

)

+ O(ε0) . (9.129)

Thus the contribution from the three-gluon vertex alone is not transverse—demonstratingagain that a covariant gauge requires the introduction of ghost particles.

Ghost loop This diagram has the same dependence on the structure constants as the pre-vious one,

iΠµνab, 3(q

2) = −(−ig)2∫

d4k

(2π)4fbdc(k − q)νfacdkν

(k2 + iε)[(k + q)2 + iε](9.130)

and can thus be combined with the three-gluon loop. Evaluating the integral results in

iΠµνab, 3(q

2) = −iNcg

2

16π2ε

(µ2

q2

)ε/2 (1

3qµqν +

1

6ηµνq2

)

+ O(ε0) (9.131)

and summing the three-gluon and ghost loops gives the expected gauge-invariant expression.Moreover, the sum has the opposite sign as the quark loop and can thus lead to the oppositebehaviour of the beta function as in QED.

Four-gluon loop and tadpole diagrams The loop with the four-gluon vertex contains amassless propagator and does not depend on external momenta,

iΠµνab,4(q

2) ∝

∫ddk

k2 + iε. (9.132)

Our general experience with DR tells us that this graph is zero, as gluons are massless.However, this loop integral is also in DR ambiguous: For any space-time dimension d, theintegral is either UV or IR divergent. To proceed, we split therefore the integral introducingthe arbitrary mass M as

1

k2 + iε=

1

k2 −M2 + iε−

M2

(k2 + iε)(k2 −M2 + iε). (9.133)

Now the IR and UV divergence is separated, and we can use d < 2 in the first term and2 < d < 4 in the second. By dimensional analysis, both terms have to be proportional to apower of the arbitrary mass M . As the LHS is independent of M , the only option is that thetwo terms on the RHS cancel, as an explicit calculation confirms. The two remaining tadpolediagrams (5 and 6 of Fig. 9.4) vanish by the same argument.

Asymptotic freedom Deriving the beta function in QCD, we should evaluate

g(µ) =Z2Z

1/23

Z1g0 , (9.134)

where Z1 is the charge renormalisation constant, and Z2 and Z3 are quark and gluon renor-malisation constants, respectively. Combining all contributions to the 1/ε poles as 1-loopcontribution b1 to the beta function of QCD gives

β = µ2 ∂g2

∂µ2= −

α2s

4π

(b1 + b2αs + b3α

2s + . . .

)(9.135)

133

9 Renormalisation

with αs ≡ g2s/(4π),

b1 =11

3Nc −

2

3nf (9.136)

and nf as number of quark flavors. For nf < 16, the beta function is negative and therunning coupling decreases as µ → ∞. Asymptotic freedom of QCD explains the apparentparadox that protons are interacting strongly at small Q2, while they can be described indeep-inelastic scattering as a collection of independently moving quarks and gluons.

Let us consider now the opposite limit, µ→ 0. The solution of (9.135) at one loop,

1

αs(µ2)=

1

αs(µ20)

+ b1 ln

(µ2

µ20

)

(9.137)

shows that the QCD coupling constant becomes formally infinite for a finite value of µ.We define ΛQCD as the energy scale where the running coupling constant of QCD diverges,α−1

s (Λ2QCD) = 0. Experimentally, the best measurement of the strong coupling constant has

been performed at the Z resonance at LEP, giving αs(m2Z) ∼ 0.1184. Thus at one-loop level,

ΛQCD = mZ exp

(1

2b1αs(m2Z)

)

. (9.138)

ΛQCD depends on the renormalisation scheme and, numerically more importantly, on the

number of flavors used in b1: For instance, ΛMSQCD ∼ 220 MeV for nf = 3. The fact that the

running coupling provides a characteristic energy scale is called dimensional transmutation:Quantum corrections lead to the break-down of scale-invariance of classical QCD with masslessquarks and to the appearence of massive bound-states, the mesons and baryons with masses oforder ΛQCD. Note also that we are able to link exponentially separated scales by dimensionaltransmutation.

Coupling constant unification While the strong coupling αs ≡ α3 decreases with increasingµ2, the electromagnetic coupling αem ≡ α1 increases. Since two lines in a plane meet at onepoint, there is a point with α1(µ∗) = α3(µ∗) and one may speculate that at this point atransition to a “unified theory” happens. Since the running is only logarithmic, unificationhappens at exponentially high scales, µ∗ ∼ 1016 GeV, but interestingly still below the Planckscale MPl. The problems becomes more challenging, if we add to the game the third, theweak coupling α2. The situation in 1991 assuming the validity of the SM is shown in theleft panel of Fig. 9.5. The width of the lines indicates the experimental and theoretical error,and the three couplings clearly do not meet within these errors. The right panel of the samefigure assume the existence of supersymmetric partners to all SM particles with an “averagemass” of around MSUSY ∼ 200 GeV. As a result, the running changes above µ = 200 GeV,and now the three couplings meet for 2 × 1016 GeV.

9.5 Effective action and Ward identities

In this section we introduce first the effective action as the generating functional of 1PI Greenfunctions. Then we will use the developed formalism to derive the Ward identities which implye.g. that the exact photon propagator is transverse and that the renormalisation of the electriccharge is universal. Later on in Sec.11.2 we will see that the discussion of the renormalisationof theories with spontaneous symmetry breaking simplifies using the effective action.

134

9.5 Effective action and Ward identities

60

50

40

30

20

10

103 105 107 109 1011 1013 1015 10170

World average 91

Q (GeV)

α1 (Q)-1

α i (

Q)

-1

α2 (Q)-1

α3 (Q)-1

(a)

60

50

40

30

20

10

0102 104 106 108 1010 1012 1014 1016 1018

α–1 (µ)1

µ [GeV]

α–1 (µ)2

α–1 (µ)3

(b)

Figure 9.5: The measurements of the gauge coupling strengths at LEP do not evolve towardsa unified value in the SM (left), but meet at 2 × 1016 GeV assuming low-scalesupersymmetry (right).

9.5.1 Effective action

We start with our familiar generating functional

Z[J ] =

∫

Dφ expi

∫

d4x(L (x) + J(x)φ(x)) = eiW [J ] , (9.139)

then we define a classical field φc(x) as φc(x) = δW [J ]/δJ(x). Performing the functionalderivative in its definition, we see immediately why this definition makes sense,

φc(x) =δW [J ]

δJ(x)=

1

iZ

δZ[J(y)]

δJ(x)=

1

Z

∫

Dφφ(x) exp i

∫

d4y(L + Jφ)

=〈0|φ(x)|0〉J

〈0|0〉J= 〈φ(x)〉J .

Thus the classical field φc(x) is defined as the vacuum expectation value of the field φ(x) inthe presence of the source J(x).

Now we define the effective action Γ[φc] as the Legendre transform of W [J ],

Γ[φc] = W [J ] −

∫

d4yJ(y)φ(y) . (9.140)

This procedure is completely analogous to the construction of the Hamilton function fromthe Lagrange function in classical mechanics: It will allow us to answer the question “whichsource J(x) produces a given φc(x),” because we can use φc as independent variable in Γ[φc].We compute the functional derivative w.r.t. φc of this new quantity,

δΓ[φc]

δφc(y)=

∫

d4xδJ(x)

δφc(y)

δW

δJ(x)−

∫

d4xδJ(x)

δφc(y)φc(x) − J(y).

Using the definition δWδJ(x) = φc(x), the first and second term cancels and we end up with

δΓ[φc(x)]

δφc(y)= −J(x) . (9.141)

135

9 Renormalisation

This is the desired relation determing the source J(x) producing a given classical field φc(x).To see why Γ[φc] is called effective action, we consider a free scalar field. Now

W0[J ] = −1

2

∫

d4xd4x′ J(x)∆F (x− x′)J(x′), (9.142)

and

φc(x) =δW

δJ(x)= −

∫

d4x′ ∆F (x− x′)J(x′). (9.143)

If we apply the Klein-Gordon operator to the classical field,

( +m2)φc(x) = −

∫

d4x( +m2)∆F (x− x′)J(x′) (9.144)

=

∫

d4x δ(x − x′)J(x′) = J(x) (9.145)

we obtain a solution of the classical field equation. Now we have all we need to write downan explicit expression for the free effective action

Γ0[φc] = W0[J ] −

∫

d4xJ(x)φc(x)

=1

2

∫

d4xd4x′ J(x)∆F (x− x′)J(x′) .

Inserting the expression (9.145) for J(x), and integrating partially, we obtain

Γ0[φc] =1

2

∫

d4xd4x′[( +m2)φc(x)

]∆F (x− x′)

[(′ +m2)φc(x

′)]

=1

2

∫

d4xφc(x)( +m2)φc(x) = S[φc].

That is, the effective action for a free scalar field is just the usual action of the classical field,justifying its name.

To gain some more information of the meaning of Γ[φ], we can evaluate it perturbativelyand expand it in φc, giving us a new set of n-point Green functions,

Γ[φc] =∑

n

1

n!

∫

d4x1 · · · d4xnΓ(n)(x1, ..., xn)φc(x1) · · · φc(xn) . (9.146)

The functions Γ(n) are the one-particle-irreducible Green functions. Hence, the effective actionis the generating functional for 1PI Green’s functions, the proof we postpone to section 11.2.

Example: Show that a) Γ(2) is equal to the inverse propagator or inverse 2-point function, and b)derive the connection of Γ(3) to the connected 3-point function.a.) We write first

δ(x1 − x2) =δφ(x1)

δφ(x2)=

∫

d4xδφ(x1)

δJ(x)

δJ(x)

δφ(x2)

using the chain rule. Next we insert φ(x) = δW/δJ(x) and J(x) = −δΓ/δφ(x) to obtain

δ(x1 − x2) = −

∫

d4xδ2W

δJ(x)δJ(x1)

δ2Γ

δφ(x2)δφ(x2).

136

9.5 Effective action and Ward identities

Setting J = φ = 0, it follows∫

d4x iG(x, x1) Γ(2)(x, x2) = −δ(x1 − x2) (9.147)

or Γ(2)(x1, x2) = iG−1(x, x1).b.) The connected 3-point function G(3)(x1, x2, x3) is obtained by appending propagators to theirreducible 3-point vertex function Γ(3)(x1, x2, x3), one for each external leg,

G(3)(x1, x2, x3) = i

∫

d4x′1d4x′2d

4x′3G(2)(x1, x

′1)G

(2)(x2, x′2)G

(2)(x3, x′3)Γ

(3)(x1, x2, x3) .

This generalises to n > 3 and therefore one calls the Γ(n) also amputated Green functions.

9.5.2 Ward-Takahashi identities

The redundancy implied by gauge invariance leads us to believe that not all of Green func-tions of a gauge theory are independent: Gauge invariance implies relations between Greenfunctions called Ward or Ward-Takahashi identities in QED and Slavnov-Taylor identities inthe non-abelian case. We use now the formalism of the effective action we developed to derivethese for QED: Consider the generating functional

Z[Jµ, η, η] =

∫

DADψDψ expi

∫

d4xLeff, (9.148)

where Jµ is a four-vector source, η and η are Graßmannian sources and the effective La-grangian is composed of a classical term, a source term and a gauge fixing term,

Leff = Lcl + Ls + Lgf

Lcl = −1

4FµνF

µν + iψγµDµψ −mψψ

Ls = JµAµ + ψη + ηψ

Lgf =1

2ξ(∂µA

µ)2

We consider now the renormalised version of the Lagrangian Leff , where the renormalisedcovariant derivative Dµ is given by Eq. (9.92) with Z1 6= Z2 in general. This implies that aninfinitesimal gauge transformation has the form

Aµ → A′µ = Aµ + ∂µΛ (9.149a)

ψ → ψ′ = ψ − ieΛψ with e ≡Z1

Z2e . (9.149b)

As Lcl is gauge invariant by construction, the variation of Leff under an infinitesimal gaugetransformation consists only of

δ

∫

d4x (Lgf + Ls) =

∫

d4x

[

−1

ξ(∂µA

µ)Λ + Jµ∂µΛ + ieΛ(ψη − ηψ)

]

. (9.150)

Now we integrate partially the first term twice and the second term once, to factor out thearbitrary function Λ,

δ

∫

d4xLeff =

∫

d4x

[

−1

ξ(∂µA

µ) − ∂µJµ + ie(ψη − ηψ)

]

Λ . (9.151)

137

9 Renormalisation

Thus the variation of the generating functional Z[Jµ, η, η] is

δZ[Jµ, η, η] =

∫

DADψDψ exp

(

i

∫

d4xLeff

)

iδ

∫

d4xLeff . (9.152)

The fields Aµ, ψ and ψ are however only integration variables in the generating functional.The gauge transformation (9.149) is thus merely a change of variables which cannot affectthe functional Z[Jµ, η, η]. Thus its variation has to vanish, δZ[Jµ, η, η] = 0.

If we substitute fields by functional derivatives of their sources, the part from Eq. (9.151)can be moved outside the functional integral, leading to

0 =

[

−1

ξ∂µ

δ

δJµ− ∂µJ

µ + ie

(

ηδ

δη− η

δ

δη

)]

expiW

= −1

ξ

(

∂µδW

δJµ

)

− ∂µJµ + ie

(

ηδW

δη− η

δW

δη

)

. (9.153)

Differentiating this equation with respect to Jν(y) and setting then the sources Jµ, η and ηto zero gives us our first result,

−1

ξ

(

∂µδ2W

δJµ(x)δJν(y)

)

= ∂µgµνδ(x− y) . (9.154)

The second derivative of W w.r.t. to the vector sources Jµ is the full photon propagatorDµν(x− y). If we go to momentum space, we have

i

ξk2kµ

Dµν(k) = kν . (9.155)

Splitting the propagator into a transverse part and a longitudinal part as in (8.39), thetransverse part immediately drops out and we find

i

ξk2kνDL(k2) = kν . (9.156)

Thus the longitudinal part of the exact propagator agrees with the longitudinal part of thetree level propagator,

iDL(k2) = iDL(k2) =ξ

k2. (9.157)

This implies that higher order corrections do not affect the longitudinal part of the photonpropagator. Since we can expand all relations as power series in the coupling constant e, thisholds also at any order in perturbation theory.

Let us go back to the constraint for the variation of the generating functional Z undergauge transformations, Eq. (9.153). We aim to derive identities between 1PI Green functionsand want therefore to transform it into a constraint for the effective action Γ. If we Legendretransform W [J, η, η] into Γ[A, ψ, ψ],

Γ[ψ, ψ,Aµ] = W [η, η, Jµ] −

∫

d4x(JµAµ + ψη + ηψ) , (9.158)

138

9.6 Renormalisation and critical phenomena

we can replace the functional derivatives of W with classical fields, and the sources withfunctional derivatives of Γ, i.e.

δW

δJµ= Aµ,

δW

δη= ψ,

δW

δη= ψ

δΓ

δAµ= −Jµ,

δΓ

δψ= −η,

δΓ

δψ= −η

with all the fields classical. This transforms Eq. (9.153) into

1

ξ(∂µA

µ(x)) − ∂µδΓ

δAµ(x)+ ie

(

ψδΓ

δψ(x)− ψ

δΓ

δψ(x)

)

= 0 , (9.159)

a master equation from which we can derive relations between different types of Green frunc-tion. Differentiating with respect to ψ(x1) and ψ(x2) and setting then the fields to zero givesus the most important one of the Ward-Takahashi identities, relating the 1PI 3-point functionΓ(3)(x, x1, x2) to the 2-point function Γ(2)(x1, x2) of fermions,

∂µΓ(3)µ (x, x1, x2) = ie

(Γ(2)(x, x1)δ(x− x2) − Γ(2)(x, x2)δ(x − x1)

)(9.160)

or, after Fourier transforming

kµΓ(3)µ (p, k, p+ k) = eS−1

F (p + k) − eS−1F (p) . (9.161)

The Green functions in this equation are finite, renormalised quantities and thus Z1/Z2 hasto be finite in any consistent renormalisation scheme too. In any scheme where we identifydirectly e(0) with the measured electric charge, also the finite parts of Z1 and Z2 agree, i.e.the measured electric charge is universal.

9.6 Renormalisation and critical phenomena

Overview The behaviour thermodynamical systems exhibit close to the critical points intheir phase diagram is called “critical phenomenon.” For a fixed number of particles, we cancharacterise thermodynamical systems using the free energy F = U − TS. Ehrenfest intro-duced the classification of phase transitions according to the order of the first discontinuousderivative of F with respect to any thermodynamical variable φ. Hence a phase transitionwhere at least one derivative ∂nF/∂φn is discontinuous while all ∂n−1F/∂φn−1 are continuousis called a n.th order phase transition.

Critical phenomena are for a particle physicist interesting by at least three reasons:

• We can learn about symmetry breaking: We should look out for ideas how we cangenerate mass terms without violating gauge invariance. Systems like ferromagnetsshow that symmetries as rotation symmetry can be broken at low energies although theHamiltonian governing the interactions is rotation symmetric.Another example is a plasma: Here the screening of electric charges modifies theCoulomb potential to a Yukawa potential; the photon has three massive degrees offreedom, still satisfying gauge invariance, kµΠµν(ω,k) = 0, but with ω2 − k

2 6= 0.

139

9 Renormalisation

• Experimentally one finds that close a critical point, T → Tc, the correlation length ξdiverges, while otherwise correlations are exponentially suppressed. Thus the 2-pointfunction of a certain order parameter φ scales as

〈φ(x)φ(0)〉 ∝ exp(−|x|/ξ) .

Comparing this to the 2-point function of an Euclidean scalar field φ,

〈φ(x)φ(0)〉 →4π

|x|2exp(−m|x|)

in the limit m|x| ≫ 1, we find the correspondence ξ ∝ m−1. Considering a statisticalsystem on a lattice with spacing a, i.e. m|x| = na/ξ, we see that the continuum limita→ 0 corresponds to ξ → ∞ for finite m.Thus the correlation functions of a statistical system correspond for non-zero a to bareGreen functions and a finite value of the regulator of the corresponding quantum fieldtheory. The connection to renormalised Green functions (a → 0 or Λ → ∞) is onlypossible when the statistical system is at a critical point.

• Near a critical point, T → Tc, thermodynamical systems show a universal behaviour.More precisely, they fall in different universality classes which unify systems with verydifferent microscopic behaviour. The various universality classes can be characterised bycritical exponents, i.e. by the exponents γi with which characteristic thermodynamicalquantities Xi diverge approaching Tc, i.e. Xi = [b(T − Tc)]

−γi .This phenomenon is similar to our realisation that e.g. two λφ4 theories, one withλ = 0.1 and another one with λ = 0.2, are not fundamentally different but connectedby a RGE transformation.

Landau’s mean field theory Landau suggested that close to a second-order phase transitionthe free energy can be expanded as an even series in the order parameter. Considering e.g.the magnetisation M , we can write for zero external field H the free energy as

F = A(T ) +B(T )M2 + C(T )M4 + . . . (9.162)

We can find the possible value of the magnetisation M by solving

0 =∂F

∂M= 2B(T )M + 4C(T )M3 . (9.163)

The variable C(T ) has to be positive in order that F is bounded from below. If also B(T )is positive, only the trivial solution M = 0 exists. If however B(T ) is negative, two solutionwith non-zero magnetisation appear. Let us use a linear approximation, B(T ) ≈ b(T − Tc),and C(T ) ≈ c valid close to Tc. Then

M =

0 for T > Tc,

±[

b2c(T − Tc)

]1/2for T < Tc.

(9.164)

Note also that the ground-state breaks the M → −M symmetry of the free energy for T < Tc.Representing the thermodynamical quantity M as integral of the local spin density,

M =

∫

d3x s(x) , (9.165)

140

9.6 Renormalisation and critical phenomena

we can rewrite the free energy in a way resembling the Hamiltonian of a stationary scalarfield,

F =

∫

d3x[(∇s)2 + b(T − Tc)s

2 + cs4 − H · s]. (9.166)

Here, (∇s)2 is the simplest ansatz leading to an alignment of spins in the continuous language.Minimising F will give us the ground-state of the system for a prescribed external field H(x)and temperature T . For small s, we can ignore the s4 term. The spin correlation function〈s(x)s(0)〉 is found as response to a delta function-like disturbance H0δ(x) as

〈s(x)s(0)〉 =

∫d3k

(2π)3H0e

ik·x

k2 + b(T − Tc)(9.167)

as it follows immediately by analogy with the Yukawa potential, m2 → b(T − Tc). Thus thecorrelation function is

〈s(x)s(0)〉 =H0

4πre−r/ξ (9.168)

with

ξ = [b(T − Tc)]−1/2 . (9.169)

Hence Landau’s theory reproduces the experimentally observed behaviour ξ → ∞ for T → Tc.Moreover, the theory predicts as critical exponent 1/2. Notice that the value of the expo-nent depends only on the polynomial assumed in the free energy, not on the underlyingmicro-physics. Thus another prediction of Landau’s theory is an universal behaviour of ther-modynamical systems close to their critical points in the dependence on T − Tc.

Experiments show that this prediction is too strong: Thermodynamical systems fall intodifferent universality classes, and we should try to include some micro-physics into the de-scription of critical phenomena.

Kadanoff’s block spin transformation Close to a critical point, collective effects play adecisive role even in case of short-range interactions. In d dimension, a particle is coupled bycollective effects to (ξ/a)d particles and standard perturbative methods will certainly fail forξ → ∞.

Kadanoff suggested to remove the short-wave length fluctuations by the following procedure:Each step of a block spin transformation consists of i) dividing the lattice into cells of size(2a)d, ii) assigning a common spin variable to the cell, iii) rescaling 2a→ a.

At each step, the number of strongly correlated spins is reduced. After n transformations,the correlation length decreases as ξn = ξ/(2n). When the correlation length becomes ofthe order of the lattice spacing, collective effects play no role: All the physics can be readoff from the Hamiltonian. If the procedure is not trivial, this implies that in each step theHamiltonian changes. In particular, the coupling constant is changed as

K2 = f(K) , K3 = f(K2) = f(f(K)) , . . . (9.170)

One-dimensional Ising model We illustrate the idea behind Kadanoff’s block spin transfor-mation using the example of the one-dimensional Ising model. This model consists of spinswith value si ± 1 on a line with spacing a, interacting via nearest neighbour interactions.

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9 Renormalisation

sN−1

s0

s1

s2

s3

s4

s5

s′0

s′1

f

s′2

Figure 9.6: One block spin transformation a→ 2a for an one-dimensional lattice model.

We consider only the piece of six spins shown in Fig. 9.6. The corresponding partitionfunction is

Z6 =∑

sN−1,s0,s1,...,s4

exp [K(sN−1s0 + s0s1 + . . . s3s4)] (9.171)

=∑

s′0,s′

1,s′

2

∑

sN−1,s1,s3

exp[K(sN−1s

′0 + s′0s1 + . . . s3s4)

]. (9.172)

The step a → 2a requires to perform the sums over the unprimed spins. Expanding theexponentials using (sisj)

2n = 1 gives terms like

exp[K(s′0s1)] = 1 +Ks′0s1 +K2

2!+K3

3!s′0s1 + . . . (9.173)

= cosh(K) + s′0s1 sinh(K) (9.174)

= cosh(K)[1 + s′0s1 tanh(K)] . (9.175)

The terms linear in s1 cancel in the sum and we obtain∑

s1

exp[K(s′0s1)] exp[K(s1s′1)] = 2 cosh2(K)[1 + tanh2(K) s′0s

′1] . (9.176)

Thus the summation over the unprimed spins changes the strength of the nearest neighbour-hood interaction and generates additionally a new spin-independent interaction term. We trynow to rewrite the last expression in a form similar to the original one,

2 cosh2(K)[1 + tanh2(K) s′0s′1] = exp[g(K) +K ′s′0s

′1)] . (9.177)

Using (9.175) to replace exp(K ′s′0s′1), we find

tanh(K ′) = tanh2(K) . (9.178)

This determines the function g as

g(K) = ln

(2 cosh2K

coshK ′

)

. (9.179)

The summation over the other spins s3, s5, . . . can be performed in the same way. Thusthe partition function on a lattice of size 2a has the same nearest-neighbour interactionswith a new coupling K ′ ≡ K1 determined by (9.178). Iterating this procedure generates arenormalisation flow with

tanh(Kn)) = tanh2n(K) . (9.180)

142

9.6 Renormalisation and critical phenomena

Fixed point behaviour In general, we will not be able to calculate the transformation func-tion f(K). But even without the knowledge of f(K), we can draw some important insightfrom general considerations. First, the exact RGE equation3 is of the type of a heat ordiffusion equation,

∂tX = ∇2X ,

on the space of functionals X = exp(−S). Its flow is therefore a gradient flow (“Fick’s law”)which has only two possible asymptotics: a runaway solution to infinity and the approach toa fixed point defined by Kc = f(Kc).

With ξn+1 = ξ/2 and labeling the n dependence implicitly vis ξn = f(Kn) we can write

ξ(f(K)) =1

2ξ(K) . (9.181)

At a fixed point Kc = f(Kc) only two solutions exist,

ξ(Kc) = 0 and ξ(Kc) → ∞ . (9.182)

The second possibility corresponds to the approach of a critical point, allowing the limit a→ 0and thus the continuum limit necessary for the transition to a QFT. This point is called acritical fixed point, while the fixed point with zero correlation length is called trivial.

We can now easily generalise our previous discussion of the fixed point behaviour for thebeta function from one to n-dimensions. The general behaviour of the RGE flow can beunderstood from the two-dimensional example in Fig. 9.7. The dashed lines show surfacesof constant correlation length, including a critical surface ξ = ∞. Also shown are threecritical fixed points (A, B, C) and a trivial one (D). We remember that in each RGE stepthe correlation length decreases. Thus the trivial fixpoint is an attractor, i.e. inside a smallenough neighbourhood all points will flow towards it. On the other hand, the critical line hasat least one unstable direction, the one orthogonal to its surface: Even points infinitesimalclose to the surface will flow away and eventually end in a trivial fixpoint. Moreover, we seethat also inside the critical surface stable and unstable directions exist: The fixpoint B willattract all points in-between A and C (“its basin of attraction”).

We can identify universality classes of QFTs with stable critical fixpoints and their basinof attraction.

Effective action, RGE flow and irrelevant operators The RGE flow generates all kind ofcouplings compatible with the symmetries of the fundamental Hamiltonian. The task ofunderstanding the RGE flow in an infinite-dimensional space of couplings is simplified by thefollowing observation: Relevant and marginal interactions are typically already included inour starting Hamiltonian. For instance, in the case of the λφ4 theory in d = 4, we includea constant term4 (d = 0, leading to a cosmological constant ρΛ), the mass term m2φ2 withd = 2, and the interaction λφ4. Hence, the RGE flow will renormalise the values ρΛ, m2 andλ and introduce an infinite set of irrelevant operator Od,i with dimension d ≥ 6.

It is now time to explain the reason why the non-renormalisable operators are called ir-relevant in this context. Let us start from the Euclidean generating functional restricted to

3A brief derivation for the curious is given in Section ??4If we do not, the normalisation constant of the path integral will do the job.

143

9 Renormalisation

ξ = ∞

ξ = 10

ξ = 1

A

B

C

D

Figure 9.7: A two-dimensional illustration of the RGE flow: three fixed points A, B and Con the critical surface ξ = ∞, where A and C have a stable direction along thecritical surface, while B has two unstable directions. The trivial fixed point D isstable fixed point, attracts all points starting not on the critical surface.

wave-numbers below a cutoff, k ≤ Λ,

Z[J ] =

∫

DφΛ exp−

∫

Ωd4x

(1

2∂µφ∂

µφ+1

2m2φ2 +

λ

4!φ4

)

, (9.183)

where DφΛ = Dφ|k|≤Λ. Now we want to integrate out the fields with wave-numbers betweensΛ ≤ k ≤ Λ from the generating functional. We express φ(x) as Fourier modes, obtaining e.g.

Sint =1

24

∫d4k1

(2π)4· · ·

d4k4

(2π)4(2π)4δ(k1 + . . .+ k4)φ(k1) · · · φ(k4) . (9.184)

Then we introduce the field φ(k) which coincides with the original field φ in the range wewant to integrate out and vanishes otherwise,

φ(k) =

φ(k) for sΛ ≤ |k| ≤ Λ,

0 otherwise.(9.185)

We set now φ = φ + φ in Eq. (9.183) and call afterwords the integration variable φ againφ. Then

Z[J ] =

∫

DφsΛDφ exp

[

−

∫

d4x

(1

2(∂µφ+ ∂µφ)2 +

1

2m2(φ+ φ)2 +

λ

4!(φ+ φ)4

)]

=

∫

DφsΛe−S[φ]Dφ exp

[

−

∫

d4x

(1

2(∂µφ)2 +

1

2m2φ2 +

λ

6φ3φ+

λ

4φ3φ2 +

λ

6φφ3 +

λ

4φ4

)]

=

∫

DφsΛe−Seff [φ] , (9.186)

144

9.6 Renormalisation and critical phenomena

where we used that the mixed quadratic terms vanish (being orthogonal). To proceed, wewould have to expand the exponential and to evaluate it perturbatively. Apart from generatingterms which renormalises the original parameters, we would obtain an infinite number ofhigher-dimension operators Od,i with dimension d ≥ 6. For instance O2n,1 = φ2n, O2n,2 =φ2n−1

φ, etc. Thus the effective Lagrangian including fluctuations up to Λ1 ≡ sΛ can bewritten as

Leff =1

2∂µφ∂

µφ+1

2m2

Λ1φ2 +

λΛ1

4!φ4 +

∑

d≥6

∑

i

Cd,i(Λ1)Od,i . (9.187)

The coefficients C2n,1 are given by a loop integral over n propagators,

C2n,1 ∝ λn

∫ Λ

Λ1

d4k

(2π)4

(1

k2 +m2

)n

∝ λn

(1

Λ2n−4−

1

Λ2n−41

)

. (9.188)

We see now that relevant operators are determined by the high-energy cutoff, marginal op-erators (∝ ln(Λ/Λ1) are influenced in the same way by each decade in k, while irrelevantoperators are determined by the low-energy cutoff.

As a side remark, we note that both directions of the RGE flow—towards the UV or theIR—are useful discussing QFTs. The point of view of a RGE flow towards the IR is useful,if we want to connect a theory at high-scales to a simpler theory at lower energy scales.An example for this approach is chiral perturbation theory where one connects QCD to aneffective theory of mesons and baryons at low energies. In the opposite view, we may looke.g. at the SM as an effective theory known to be valid up to scales around TeV and ask whathappens if we increase the cutoff.

You may have noticed that we have rescaled the effective Lagrangian in Eq. (9.187) suchthat the kinetic term maintained its canonical normalisation. According to step iii) in theKadanoff-Wilson prescription, we have to rescale Z after each step a→ fa (or Λ → Λ/f). Ifwe rescale distances by x→ x′ = x/f , the functional integral is again over modes φ(x′) withx > a. Keeping the kinetic term invariant,

∫

d4x (∂µφ)2 =

∫

d4x′ (∂′µφ′)2 =

1

f2

∫

d4x (∂µφ′)2 (9.189)

requires thus a rescaling of the field as φ′ = fφ. Let us consider now an irrelevant interaction,e.g. g6φ

6. Then

g6

∫

d4xφ6 =g6f2

∫

d4x′ φ′2 (9.190)

shows that the new coupling g′6 is rescaled as g′6 = g6/f2: As f grows and the cutoff scale Λ

decreases, the value of an irrelevant coupling is driven to zero. Clearly, a relevant operator asthe cosmological constant ρ or a mass term m2φ2 shows the opposite behaviour and grows. Asresult, irrelevant couplings are in our low-energy world suppressed and as first approximationa renormalisable theory emerges at low energies.

We can generalise now our earlier discussion of the two-dimensional RGE flow, Fig. 9.7. TheRGE flow stops at fixed points on critical surfaces of low dimension. All initial values on thecritical surface (or basin of attraction) flow to the critical fixed point. Directions perpendicularto the critical surface are controlled by the irrelevant interactions; flows beginning off thesurface are driven to the trivial fixed point. On the way towards ξ → ∞ only the relevant andmarginal interactions survive. Insisting not on ξ → ∞ and keeping a finite cutoff (somewherebetween TeV and MPl, depending on the limit of validity of the theory we assume) we cankeep irrelevant interaction but may require some fine-tuning of their parameters.

145

9 Renormalisation

Summary of chapter

Using a power counting argument for the asymptotic behaviour of the free Green functions,we singled out theories with dimensionless coupling constants: Such theories with marginalinteractions are renormalisable, i.e. are theories with a finite number of primitive divergentdiagrams. Consequently, the multiplicative renormalisation of the finite number of parameterscontained in the classical (effective) Lagrangian is sufficent to obtain finite Green functionsat any order perturbation theory.

The scale dependence of renormalised Green functions can be interpreted as a running ofcoupling constants and masses. The use of a running coupling sums up the leading logarithmsof type lnn(µ2/µ2

0), and a suitable choice of the renormalisation scale in a specific problemsreduces the remaining scale dependence of any perturbative result.

Different interactions can be characterised by the asymptotic behavior of their couplingconstants. Gauge theories with a sufficiently small number of fermions are the only renormal-isable interactions which are asymptotically free, i.e. their running coupling constant goes tozero for µ→ ∞.

The non-perturbative approach of Wilson provides an argument why the SM as describtionof our low-energy world is renormalisable: Integrating out high-energy degrees of freedom,irrelevant couplings are driven to zero and thus it is natural that a renormalisable theoryemerges at low energies.

Further reading

Our discussion of the renormalisation of non-abelian gauge theories left out most details.For instance, we did not introduce the non-abelian analogue of the Ward-Takahshi identifieswhich are easiest derived using the formalism of the BRST symmetry. I recommend thoseinterested to fill the gaps to start with the books of Pokorski and Ramond. Banks gives aextremely concise and lucid discussion of renormalisation.

Exercises

9.1 gs-factor of the electron.

Derive (9.6) and show that the Dirac equationpredicts for the interaction of non-relativistic elec-tron with a static magnetic field Hint = (L +2s)eB, i.e. a gs equal to two for the electron.

9.2 gs-factor of gauge bosons.

Derive the gs-factor of gauge bosons from the non-abelian Maxwell equations.

9.3 Comparison of cutoff and DR.

Recalculate the three basic primitive diagramsof a scalar λφ4 theory using as regularisation acutoff Λ. Find the correspondence between thecoefficients of poles in DR and divergent terms in

Λ.

9.4 Primitive divergent diagrams of scalar

QED.

Find the basic primitive diagrams of scalar QED

L =1

2(Dµφ)†Dµφ−

1

2m2φ†φ−

1

4FµνF

µν

and their superficial degree of divergence.

9.5 β function of the λφ4 theory.

The β function determines the logarithmic changeof the coupling constants. a) Show that the β

146

9.6 Renormalisation and critical phenomena

function can be written in DR as

β(λ) = −ελ−µ

Z

dZ

dµλ

with Z−1 = Z−1λ Z2

φ. b.) Show that Z−1λ =

1−3λ/(16π2ε) in one loop approximation and findthe β-function. c.) Up to which order is the βfunction scheme independent? d.) Solve the dif-ferential equation for λ(µ).

9.6 Anomalous dimension.

.

9.7 General solution of RGE equation.

Find the general solution of the RGE equa-tion (9.84) using the method of characteristics orthe following analogue: Coleman suggested to con-sider the growth rate g(x) of bacteria in an one-dimensional flow as analogue to the RGE.a.) Show that the density ρ(x, t) of bacteria satis-fies [

∂

∂t+ v(x)

∂

∂x− g(x)

]

ρ(x, t) = 0

b.) The position of a fluid element is described byx = x(x, t) with the initial condition x(x, 0) = x.Then x(x, t) satisfies

d

dtx(x, t) = v(x)

Show that for ρ(x, 0) = ρ0(x) at later time ρ(x, t)is given by

ρ(x, t) = ρ0(x(x, t)) exp

(∫ t

0

dt′ g(x(x, t′))

)

.

9.8 Imaginary part of the photon polarisa-

tion tensor.

Derive the imaginary part of the photon polari-sation tensor at one loop and show that it equalsthe pair creation cross probability of a photon withvirtuality q2.

9.9 Choosing the scale in multi-scale prob-

lems .

Show that the optimal choice µ for the scaleof the running coupling constant in a problemwith multiple scales µi is their geometrical mean,µ = (µ1 · · ·µn)1/n.

9.10 A toy model for the effective action

approach.

Consider

Z =

∫

dxdy exp([−(x2 + y2 + λx4 + λx2y2)]

as a toy model for the generating functional oftwo coupled scalar fields. Integrate out the fieldy, assume then that λ is small: Expand first theresult and then rewrite it as an exponential. Showthat this process results in a.) a renormalisationof the mass term x2, b.) a renormalisation of thecoupling term x4, and c.) the appearance of new(“irrelevant”) interactions xn with n ≥ 6.

147