Introduction to Renormalization
A lecture given by Prof. Wayne Repko Notes: Michael Flossdorf
April 4th 2007
Contents
1 Introduction 1
2 RenormalizationoftheQEDLagrangian 2
3 One Loop Correction to the Fermion Propagator 3
4 Dimensional Regularization 6 4.1 SuperficialDegreeofDivergence...... 6 4.2 TheProcedureofDimensionalRegularization ...... 8 4.3 AUsefulGeneralIntegral...... 9 4.4 Dirac Matrices in n Dimensions ...... 11 4.5 Dimensional Regularization of Σ2 ...... 11
5 Renormalization 12 5.1 Summary ofLastSection and OutlineoftheNextSteps ...... 12 5.2 MomentumSubtractionSchemes...... 13 5.3 BacktoRenormalizationinQED...... 14 5.4 Leading Order Expression of Σ2 for Small ǫ ...... 17
6 Outline of the Following Procedure 19
7 The Vertex Correction 19
8 Vacuum Polarization 22 CONTENTS 2
9 The Beta Function of QED 25
Appendix 27
References 30 1 INTRODUCTION 1
First Part of the Lecture 1 Introduction
Starting with the Lagrangian of any Quantum Field Theory, we have seen in some of the previous lectures how to obtain the Feynman rules, which sufficiently de- scribe how to do pertubation theory in that particular theory. But beyond tree level, the naive calculation of diagrams involving loops will often yield infinity, since the integrals have to be performed over the whole momentum space. Renormal- ization Theory deals with the systematic isolation and removing of these infinities from physical observables. The first important insight is, that it is not the fields or the coupling constants which represent measurable quantities. Measured are cross sections, decay width, etc.. As long as we make sure that this observables are finite in the end and can be unambiguously derived from the Lagrangian, we are free to introduce new quantities, called renormalized quantities for every, so called, bare quantity. We can then go a step further and consider the bare quantities to be infinite in a way which would just cancel the infinities coming from our loop calculation. In other words, if we would be able to arrange the infinities of the bare quantities and the infinities coming from the divergent integrals to cancel each other systematically, we would be left with a finite, physically meaningful theory. This is, roughly speaking, what Renormalization Theory does. The general procedure is now done in two different steps. At first, to give the above men- tioned cancelation of infinities mathematically meaning, we need to regularize the divergent integrals. That is, we first have to make the integrals finite, e.g. by im- posing a momentum cut-off. Then we are free to manipulate them. Secondly, the divergent parts of the integrals have to be absorbed into the bare quantities of the Lagrangian. We have to expect that there are different ways to treat the finite parts of the integrals, since a finite quantity can always be absorbed into an infi- nite one, and also that the way the divergences reside in the regularized integrals will depend on the Regularization procedure we choose. As we will work out, the freedom in the treatment of the finite parts is reflected in the existence of differ- ent renormalization schemes and also in the occurrence of scale quantities, which will have mass dimension. The fact that one choice of scale should be as good as any other and should not affect measurable quantities, will lead to the so called Renormalization Group consisting of transformations between different scales. If there exists a consistent way to perform this procedure for a particular the- 2 RENORMALIZATION OF THE QED LAGRANGIAN 2 ory, this theory is said to be renormalizable. QED was historically the first impor- tant theory which could be showed to be renormalizable and we shall see how this can be done explicitly in next to leading order.
2 Renormalization of the QED Lagrangian
In the following we will denote the bare, unrenormalized quantities m0, ψ0, etc. to distinguish them from the renormalized ones. 1 = ψ¯ i∂/ m ψ e ψ¯ γµψ A F 0µνF + LQED 0 − 0 0 − 0 0 0 0µ − 4 0µν LG.F. µ 1 µν Z ψi¯ ∂ψ/ Z′ mψψ¯ Z eψγ¯ ψA Z F F + (1) ≡ 2 − 2 − 1 µ − 3 4 µν LG.F. 1 µ Here G.F. = 2α (∂ Aµ) denotes the Faddeev and Popov Gauge Fixing term. In the followingL we− will not take into account this term in our Renormalization pro- cedure. For a renormalization including this term, see for example [3]. Whenever we will have to choose a gauge, we will pick the Feynman gauge, i.e. set α =1. From the first and last term we see
ψ0 = Z2ψ, A0µ = Z3Aµ. Similarly from the mass term:p p
¯ Z2′ ¯ Z2′ mψψ = mψ0ψ0 Z2 Z2 m = m0. ⇒ Z2′ We now define m = m δm and get 0 − m0 m δm δm Z′ = Z = Z − = 1 Z . 2 2 m 2 m − m 2 The remaining term yields: e Z1 ψ¯ γµψ A = e ψ¯ γµψ A Z2√Z3 0 0 0µ 0 0 0 0µ (2) e = Z2 √Z e . ⇒ Z1 3 0 Note that the covariant derivative now reads ren = ∂ ie Z1 A . In order to Dµ µ − Z2 µ have gauge invariance preserved, Z1 = Z2 should hold. We will find this relation in the next lecture. 3 ONE LOOP CORRECTION TO THE FERMION PROPAGATOR 3
Summarizing the results from above, we have: δm Z′ = 1 Z , (3) 2 − m 2 Z2 e = Z3e0, (4) Z1 p ψ0 = Z2ψ, (5)
A0µ = pZ3Aµ. (6) We can now rewrite the Lagrangian in thep form = + , L Lren Lcounter where 1 ψ¯ i∂/ m ψ eψγ¯ µψA F µνF (7) Lren ≡ − − µ − 4 µν (Z 1) ψ¯ i∂/ m ψ + δmZ ψψ¯ (8) Lcounter ≡ 2 − − 2 1 (Z 1) eψγ¯ µψA (Z 1) F µνF (9) − 1 − µ − 3 − 4 µν (10) We now introduce the interacting part of the hamiltonian:
µ H′ eψγ¯ ψA . (11) ≡ µ −Lcounter Note that the whole counter-term Lagrangian is treated as a perturbation. Remark: Divergences of processes which do not have a tree level correspon- dence, as the photon-photon scattering (see Fig. 1) would also have to be renor- malized by counter. By naive power counting, see section 4, this diagram seems to be logarithmicL divergent. It would now be hardly imaginable to absorb this divergence into a counter term, since there is no corresponding tree level process. But as it turns out when one actually does the calculation, this diagram is not divergent at all.
3 One Loop Correction to the Fermion Propagator
In this section we will look at the one loop correction to the fermion propaga- tor. The relation to the counter-term Lagrangian will be worked out in the later sections. 3 ONE LOOP CORRECTION TO THE FERMION PROPAGATOR 4
Figure 1: Box diagram of photon-photon scattering.
The scattering amplitude is given by
4 S = T exp i d xH′(x) − Z 2 4 ( i) 4 4 =1 i d xH′(x)+ − d x d x′ T [H′(x)H′(x)] + ... . − 2! Z Z Z We will use Wick’s theorem to evaluate the time ordered products. See (A.1- A.4) in the appendix for the definition of the field contractions. We will introduce a photon mass λ to regularize the infrared divergence in the loop integral. Fig. 2 shows the process for which we will calculate the amplitude now. To use Wicks theorem we now have to take into account all possible contrac- tions which correspond to this process. The factor of two in the calculation below is due to the fact that, in our case, there are actually two equivalent ways to do the contractions.
k
p q = p + k p′ = p
Figure 2: One loop correction to the fermion propagator. 3 ONE LOOP CORRECTION TO THE FERMION PROPAGATOR 5
p′,s′ S 1 p,s h | − | i 2 ( ie) 4 4 µ ν =2 − p′,s′ d x d x′ψ¯(x)γ ψ(x)A (x) ψ¯(x′)γ ψ(x′)A (x′) p,s 2! h | µ ν | i Z 2 4 4 ip′x s′ µ ν ipx′ s = e d x d x′ e u¯ (p′)γ iS (x x′)iD (x x′)γ e− u (p) − Fαβ − Fµν − Z Z 2 4 4 s′ µ /q + m 1 ν s = e (i)( i) d qd k u¯ (p′)γ g γ u (p) − − q2 m2 + iǫ µν k2 λ2 + iǫ Z − − 1 4 4 ip′x ipx′ iq(x x′) i( k)(x x′) d x d x′ e e− e− − e− − − ×(2π)8 Z Z
This gives us two δ-functions: δ(q p k) and δ(p′ q + k). Note that the sign of k depends on how we choose the− momentum− direction− in the loop and that we choose k to be consistent with our assignments in Fig. 2. Now we perform the q-integration:−
2 s′ 4 µ /p+k/+m 1 s = e u¯ (p )δ(p p) d kγ 2 2 2 2 γ u (p) ′ ′ (p+k) m +iǫ k λ +iǫ µ − − ′ 4 4 − s − s = (2π) δ(p′ p) iM (2π) δ(p′ p) u¯ (p′)( iΣ ) u (p) . − ≡R − − 2 Where M denotes the matrix element which we could have obtained directly by using the Feynman rules:
iM = sum of all connected, amputated diagrams. At this point we should stress one subtlety: We introduced a counter-term La- grangian which will lead to additional Feynman rules. Also, we did not take into account the counter-terms in the interacting Hamiltonian we used to calculate the scattering amplitude. We shall come back to that in section 5. Note that we in general want the calculated diagram to be part of a bigger one, so we don’t include the Dirac spinors from the external lines in our definition of Σ2. Thus we have:
2 µ e 4 γ /p + k/ + m γµ 1 iΣ2(p) − d k . (12) − ≡ (2π)4 (p + k)2 m2 + iǫ k2 λ2 + iǫ Z − − A standard way to evaluate integrals like this one is to first combine the two de- nominators using the Feynman trick:
1 1 1 dx dx ...dx n =(n 1)! . . . 1 2 n δ 1 x , a a ...a − (a x + a x + . . . + a x )n − i 1 2 n 0 0 1 1 2 2 n n i=1 ! Z Z X 4 DIMENSIONAL REGULARIZATION 6 in our case the following form is sufficient:
1 1 dx = 2 . (13) a b 0 [a +(b a) x] Z − Now let a = k2 λ2 + iǫ and b = k2 +2pk + p2 m2 + iǫ, then − − a +(b a) x = k2 λ2 + iǫ + 2pk + λ2 + p2 m2 x. − − − Thus using the Feynman trick, we can rewrite (12):
2 1 e 4 µ Σ2(p) = i dx d kγ /p + k/ + m γµ − (2π)4 Z0 Z 1 ×[k2 +2xpk + p2x m2x + λ2 (x 1) + iǫ]2 − − 2 1 µ e γ /p + k/ + m γµ = i dx d4k , (14) 4 2 2 − (2π) 0 (k + xp) ∆+ iǫ Z Z − where we have defined
∆ p2x (1 x)+ m2x + λ2 (1 x) . ≡ − − − 4 Dimensional Regularization
4.1 Superficial Degree of Divergence We are following [1] here. In order to figure out the degree of divergence of a certain Feynman diagram, it is useful to have a rule of thumb at hand. Here we will consider a more general Lagrangian than the QED-Lagrangian, with a scalar field, a fermion field and a massless vector gauge boson field in n-dimensional space-time. Suppose we have a diagram with
- B external boson lines,
- F external fermion lines,
- L loops,
- ni ith type vertices, 4 DIMENSIONAL REGULARIZATION 7
- bi boson lines in the ith type vertex,
- fi fermion lines in the ith type vertex,
- di number of derivatives in the ith type vertex, - IB internal boson lines, - IF internal fermion lines. The structure of the graph gives us the relations
B +2IB = nibi, (15) i X F +2IF = nifi and (16) i X L = IB + IF n +1. (17) − i i X The so called superficial degree of divergence D is then given by
D nL 2IB IF + n d ≡ − − i i i X (18) = n IB + IF n +1 2IB IF + n d , − i − − i i i ! i X X where we used (17) in the second step. Plugging in (15) and (16) now yields:
n 2 n 1 D = − n b B + − n f F 2 i i − 2 i i − " i # " i # X X + n (d n)+ n i i − i X n 2 n 1 = n − B − F + n δ , (19) − 2 − 2 i i i X where d 2 d 1 δ d + − b + − f n (20) i ≡ i 2 i 2 i − is called the index of divergence. When D =0 the integral is said to be superficial logarithmic divergent, linear divergent if D =1 and quadratic divergent if D =2. 4 DIMENSIONAL REGULARIZATION 8
Note that this is really just a rule of thumb, which for example in case of the photon-photon scattering turns out to give the wrong answer. Nevertheless the real divergence of a diagram can not be worse than the hereby estimated divergence.
4.2 The Procedure of Dimensional Regularization In most parts we are following [2] in this section. Mathematically this section is necessarily a bit sloppy. A mathematically more profound treatment can be found in [1] and especially in [3]. The procedure of Dimensional Regularization can be summarized in the following way: Compute the Feynman diagram as an analytic function of space-time. Which is tricky, because therefore the integral expression has to be analytically continued, since the divergence will reside as a pole (as we will see) for n = 4 (note that in Dimensional Regularization n is a complex number). See [3, p. 115] for details here. If now n is small enough, any loop integral will finally converge. For QED, Dimensional Regularization is especially important, because it preserves the Ward Identity (in contrast to imposing a direct momentum cut-off, see [2, p. 248]). After regularization and renormalization the expressions for observables will have a well-defined limit as n 4. The first thing we do is to go to Euclidean space by performing→ a Wick rota- 0 0 4 4 tion, i.e. let k = ikE. We thus have d kE = id k. As in three dimensions, we can introduce angle coordinates and generalize− our four dimensional space-time integrals in the following way:
4 n d kE d kE dΩn n 1 n = n dkE k − . (2π)4 → (2π) (2π) E
In this notation Ωn = dΩ is the surface of the unit sphere in n dimensions. To obtain an explicit expression for this, we recall the Gaussian integral R
∞ x2 dx e− = √π Z−∞ 4 DIMENSIONAL REGULARIZATION 9 to derive:
n n x2 π 2 = dx e− Z n P x2 = d x e− i Z ∞ n 1 x2 = dΩn dx x − e− Z Z0 1 ∞ n 1 y = dΩ dy y 2 − e− n 2 Z Z0 1 n = dΩ Γ . (21) n 2 2 Z Here we used the Euler form of the Gamma function
∞ x 1 t Γ(x)= dt t − e− . Z0 See the appendix for a summary of its most important properties. Using equation (21) we find
n 2π 2 dΩ = . (22) n Γ n Z 2 4.3 A Useful General Integral
Before we proceed with our calculation of Σ2, we will study the general integral of the form dnk E , (23) (k2 + ∆ iǫ)β Z E − where β is an integer. At first we introduce the following trick:
1 ∞ α k2 +∆ iǫ = dα e− ( E − ). k2 + ∆ iǫ E − Z0 Secondly we note that
β 1 β 1 1 ( 1) − ∂ − 1 = − . 2 β 2 (k + ∆ iǫ) (β 1)! ∂∆ kE + ∆ iǫ E − − − 4 DIMENSIONAL REGULARIZATION 10
Using this we obtain
β 1 β 1 n ( 1) − ∂ − ∞ 2 d kE n α(kE +∆ iǫ) β = − dα d kE e− − (k2 +∆ iǫ) Γ(β) ∂∆ E − Z0 Z β 1 β 1 R 2 ( 1) − ∂ − ∞ ∞ n 1 α(k +∆ iǫ) = − dα dΩ dk k − e− E − . Γ(β) ∂∆ n E E Z0 Z Z0 Now we perform the variable substitution t = αk2 dk = dt on the last E E 2αkE integral: ⇒
n 1 αk2 1 ∞ n 2 t ∞dk k − e E = dt k − e− 0 E E − 2α E Z0 R n 2 1 1 − ∞ n 1 t = dt t 2 − e− 2α √α Z0 1 n n = α− 2 Γ 2 2 Using this result and (22) we can go on with our calculation
β 1 n β 1 n ( 1) − 2π 2 ∂ − ∞ 1 n n d kE α(∆ iǫ) 2 β = − dα e− − α− Γ (k2+∆ iǫ) Γ(β) Γ n ∂∆ 2 2 − 2 Z0 R β 1 β 1 ( 1) − n ∂ − ∞ n α(∆ iǫ) = − π 2 dα α− 2 e− − Γ(β) ∂∆ 0 n Z 2 π ∞ β n 1 α(∆ iǫ) = dα α − 2 − e− − Γ(β) 0 Z ′ dα Now substitute α′ = α (∆ iǫ) dα = ∆ iǫ . n − ⇒ − π 2 dα′ 1 ′ n ′ β 2 1 α = β n 1 α − − e− Γ(β) (∆ iǫ) (∆ iǫ) − 2 − Zn − − 2 n π Γ β 2 = β n − (24) (∆ iǫ) − 2 Γ(β) − This result could also have been obtained more elegantly (but not as straight for- 2 ward) by substituting x = ∆/ (kE + ∆) in our starting expression for the integral in (23) to massage it into a form where we can use a relation of the so called beta function 1 α 1 β 1 Γ(α)Γ(β) B(α, β) dx x − (1 x) − = . ≡ − Γ(α + β) Z0 4 DIMENSIONAL REGULARIZATION 11
Compare [2, p. 250]. We should note that integrals like that belong to a general class of functions, called Passarino-Veltman functions, which can be found in the literature.
4.4 Dirac Matrices in n Dimensions Before we begin to work n dimensional space-time, we have to appropriately generalize our Dirac matrix identities. We define the n Dirac matrices by
γµ,γν =2gµν, { } µν where g gµν = n in n dimensions. We will now use this definition to show:
µ γ γµ = n, γµγνγ = (2 n) γν, (25) µ − γµγνγργ =4gνρ (4 n) γνγρ. µ − − The first identity can easily be obtained by writing
γµ,γ =2γµγ =2gµ =2n. { µ} µ µ Using this we can also show the second identity:
µ ν µ ν ν γ ,γ γµ = γ γ γµ + γ n { } µν ν =2g γµ =2γ .
And finally we use our last result to show the third identity:
γµ,γν γργ = γµγνγργ + (2 n) γνγρ { } µ µ − =2gµνγργ =2γργν =4gρν 2γνγρ. µ −
4.5 Dimensional Regularization of Σ2 After all the preparatory work we can now go ahead and transform equation (14) for Σ2 into n dimensions:
2 1 µ ie 4 n n γ /p + k/ + m γµ Σ (p)= − µ − dx d k (26) 2 n 2 2 (2π) 0 (k + xp) ∆+ iǫ Z Z − 5 RENORMALIZATION 12
Here we introduced the scale µ, a quantity which has mass dimension and must be introduced so that e remains dimensionless also in n dimensions:
2 n e eµ − 2 . →
In the following calculation of (26), we first shift the integration variable k′ = k + xp dk′ = dk, k/ = k/′ x/p and then drop ’ for notational simplicity. Then we perform⇒ a Wick rotation,− use our general expression (24) and note that the k/-term is zero by symmetry.
2 1 µ ie 4 n n γ k/ + (1 x) /p + m γµ Σ2 (p) = − µ − dx d k − n 2 2 (2π) 0 (k ∆+ iǫ) Z Z − 2 1 µ ie 4 n n γ (1 x) /p + m γµ = − µ − dx i d k − n E 2 2 (2π) 0 (k + ∆ iǫ) Z Z E 2 1 µ µ − e 4 n (1 x) γ /pγµ + γ γµm = n µ − dx − n (2π) 2 2 Z0 (∆ iǫ) − 2 −1 e n n (1 x)(2 n) /p + nm 4 n 2 = n µ − π Γ 2 dx − − n (27) 2 2 (2π) − 2 0 (∆ iǫ) − Z − Where in the last step we used (25). This result is usually written in the following way: 2 2 Σ2(p)= A p + B p p/ m , (28) − where
2 1 e n n 2 x (2 n) 2 4 n 2 A(p ) n µ − π Γ 2 dx − − n m and (29) ≡ (π) − 2 2 2 Z0 (∆ iǫ) − 2 1− e n n 1 x 2 4 n 2 B(p ) n µ − π Γ 2 (2 n) dx − n . (30) 2 2 ≡ (π) − 2 − 0 (∆ iǫ) − Z − 5 Renormalization
5.1 Summary of Last Section and Outline of the Next Steps We should now pause for a minute and ask ourselves what we have accomplished so far. We regularized the loop integral: To cure the infra-red divergence we 5 RENORMALIZATION 13 introduced a photon mass λ and dimensional regularization took care of the UV- divergences. Note that there exist alternative regularization procedures, too, like the Pauli-Villar regularization (discussed for example in [1, p. 45]). In the beginning of the lecture we introduced a counter-term Lagrangian. But in our derivation of the one loop correction to the fermion propagator, we did not take into account its contribution to the interacting Hamiltonian H′. Thinking back on our motivation to introduce it, we clearly would want this contribution to cancel the divergent parts of our calculation. We therefore will in this section reinterpret our result to also contain the Feynman diagrams resulting from the full interacting Lagrangian, including also the counter term pieces. In that context it is important to note that the factors of Z in the complete Lagrangian are not defined at all, unless we exactly define how much of the finite parts of the loop diagrams we want them to absorb. There are different ways to do that and to choose one, means to choose a particular renormalization scheme. That is, renormalization schemes differ in the treatment of the finite parts of the loop integrals. In this lecture we choose a momentum subtraction scheme (another choice would have been the Minimum Subtraction Scheme, for example).
5.2 Momentum Subtraction Schemes In this subsection we will again follow [1]. The general idea of Momentum Sub- traction Schemes is the observation that divergences of loop integrals will occur only in the first few terms of a Taylor expansion in external momenta of the Feyn- man diagrams. Note that by divergences we here mean properly regularized (so cut-off or dimensional dependent) quantities. An easy example shall illustrate that. Suppose we wanted to calculate the following integral: d4l 1 1 Γ p2 = . (2π)4 (l p)2 m2 + iǫ l2 m2 + iǫ Z − − − We then observe that its first derivative
2 ∂Γ(p ) 1 ∂ 2 ∂p2 = 2 pµ Γ p 2p ∂pµ 1 d4l (l p) p 1 = 4 − 2 , p2 (2π) (l p)2 m2 + iǫ l2 m2 + iǫ Z − − − is finite! And all further terms of the expansion will be finite, too. Which terms of the expansion diverge depend on how badly divergent the integral is. 5 RENORMALIZATION 14
Clearly there is a freedom related to the expansion point (in that context some- times called subtraction point) here. This is were naturally a scale, let’s call it M, comes in: To pick a certain expansion point means, loosely speaking, to define the theory at the scale M. Later we will come back to that question when we pick the so called renormalization conditions. Since every expansion point is as good as every other, we can also ask how different choices of M are connected to each other. It turns out that transformations from one scale to another form a group, called the Renormalization Group. For details see [2, Chapter 12.2].
5.3 Back to Renormalization in QED We would like to get a renormalized propagator of the form: i + (terms regular at p2 =+m, finite). (31) /p m − p Note that this form fixes the mass to be on shell (since it is precisely the pole of the propagator) and also fixes the residue to be one. Now the identity p/ + m /p + m 1 = = p2 m2 /p + m /p m /p m − − − motivates to write /p = m shortly for (the position of the pole at) p2 = +m. Further we note that also ∂ is well-defined: ∂/p p
∂ µ 1 ∂ ∂ =(γ )− = γµ , ∂p/ ∂pµ ∂pµ where we used (25). This will allow as a more readable notation. We will now see that (31) for the form of our propagator exactly defines two of our renormalization conditions: We are doing on (mass) shell renormalization. To see that, we expand Σ2 around /p = m:
∂Σ2 Σ2(p) = Σ2 /p = m + /p m + Σf (p) . (32) ∂p/ /p=m − Where Σ (p) should be finite when n 4 by the above argument. By construc- f → tion (higher order term of a Taylor expansion) Σf (p) further satisfies:
Σf /p = m =0 ∂Σ (33) f =0 . ∂p/ /p=m
5 RENORMALIZATION 15
In a few moments we will find the renormalized propagator in one loop approxi- mation to be i . (34) /p m Σf (p) − − Using (33) we immediately find by expanding Σf (p) around /p = m
2 i 1 ∂ Σf = 1+ 2 /p m + . . . . /p m Σf (p) /p m ∂p/ /p=m − − − − So indeed, the propagator has a pole at /p = m and its residue is 1. In order to see explicitly how the counter-terms have to be chosen to cancel the divergent parts of Σ2 in the on shell renormalization scheme (defined by (33)), we rewrite our Counter-Lagrangian for the last time: 1 = δ (F µν)2 + ψ¯ iδ ∂/ δ ψ eδ ψγ¯ µψA , (35) Lcounter −4 3 2 − m − 1 µ where δ = Z 1, δ = Z 1, δ = Z 1 and δ = Z m m (don’t confuse 1 1 − 2 2 − 3 3 − m 2 0 − δm and δm). Now we read off the Feynman rules. This is at first sight not straight forward to 1 µν 2 do for 4 (F ) , but by integration by parts (and throwing away the surface term − 1 2 µν µ ν as usual) one obtains 2 Aµ ( ∂ g + ∂ ∂ ) Aν , where one can readily read off the corresponding Feynman− Rule.− All Feynman rules are shown in Figure 3. Now
µν 2 µ ν = i g q q q δ3 − − = i pδ/ 2 δm −
µ = ieγ δ1 −
Figure 3: Feynman rules of the counter-terms. we see that for the process we calculated there is another diagram contributing: 5 RENORMALIZATION 16
The second one in Fig. 3. For our final result of the one loop correction of the fermion propagator we thus have to include that diagram, too: The amplitude for the one loop correction is the sum of the two diagrams, which has to be finite and which we already denoted as Σf , whereas Σ2 on the other hand only contains the loop diagram:
iΣf = + −
= iΣ2 ( i)Σ2 p/ = m +( i)Σ′2 /p=m /p m , − − − − | −
where we used the Taylor expansion from eqn. (32). Note that the right hand side of this equation is only finite, because the infinite parts (and also some of the finite parts) of the loop diagram amplitude is canceled by Σ2 /p = m and Σ′ . In the following we will have to figure out the connection between this 2 /p=m two quantities and the constants occurring in the Feynman rules of the counter terms. Therefore we will plug in /p = m in the last equation, use (33) and the defini- tion of the counter-term amplitude in Fig. 3 to figure out how we have to define δ2 and δm to get the desired cancelation:
0= iΣ (m)+ i (mδ δ ) . − 2 2 − m Now we use our second renormalization condition in (33):
0= iΣ′ (m)+ iδ . − 2 2 The two last equations now give us
mδ δ = Σ (m) and (36) 2 − m 2 δ = ∂Σ2 , (37) 2 ∂/p p/=m
which exactly defines how the counter-terms have to look like in our on-shell scheme. Since we now have the Feynman rules, we can justify the form of the propagator already given in (34). It is the geometric sum over the one loop dia- 5 RENORMALIZATION 17 grams and the corresponding counter term diagrams displayed in Fig. 4: i i i + ( iΣf ) + . . . /p m /p m − /p m − − − i Σ Σ 2 = 1+ f + f + . . . /p m /p m /p m − " − − # i = . /p m Σf − −
++ + ... + + + ...
Figure 4: The one loop approximation for the renormalized propagator.
5.4 Leading Order Expression of Σ2 for Small ǫ We now again set ǫ = 4 n. Then using the formulas given in the appendix, the − leading term of (27) for small ǫ is (here λ is set to zero): 1 e2 e2 1 Σ2(p)= p/ +4m + /p (1 + γ) m (1+2γ) ǫ 16π2 8π2 −2 − (38) 1 4πµ2 + dx (1 x) /p +2m log + O(ǫ) − p2x (1 x)+ m2x Z0 − It is interesting to compare this result to the result one gets for regularization using the Pauli-Villars method. This method introduces a cut-off Λ and does the following replacement to regularize the integral: 1 1 1 k2 λ2 + iǫ → k2 λ2 + iǫ − k2 Λ2 + iǫ − − − We then would have obtained a result quite similar to (38): α 1 (1 x)Λ2 Σ2(p)= dx (1 x) /p +2m log − . (39) 2π − p2x (x 1) + m2x Z0 − 5 RENORMALIZATION 18
Compare [2, p. 218]. Here we can see that the logarithmic divergent terms in cut- off regularization are replaced by simple poles in Dimensional Regularization. 6 OUTLINE OF THE FOLLOWING PROCEDURE 19
Second Lecture 6 Outline of the Following Procedure
In the last lecture we introduced dimensional regularization and the momentum subtraction scheme to handle the divergent parts of loop integrals. We also ob- tained new Feynman rules corresponding to the counter-terms in the renormalized Lagrangian. All Feynman rules for renormalized QED are summarized in the ap- pendix. The constants in this counter-diagrams now have to be calculated order by order in perturbation theory. In the first lecture we already worked out how δ2 and therefore Z2 = δ2 +1 have to be defined in our on shell renormalization scheme. Now we will proceed calculating Z1 and Z3. We already stressed the dependence of the finite parts of the Z-variables on the renormalization scheme we use. The details of the renormalization scheme are expressed in the, so called, renormalization conditions. Eqns. (33) already define two of these conditions we have chosen. To evaluate Z1 and Z3 we will have to define two more. Our goal in this lecture is to derive the beta function of QED which expresses the running coupling in QED. We already saw at the beginning of the last lecture, eqn. (2), the connection between the bare and the renormalized coupling constant, which for dimensional regularization we rewrite in the following way:
2 4 n Z2 µ − α = 2 Z3α0. (40) Z1
As we mentioned, we would like to find Z1 = Z2 in order to ensure gauge in- variance. Indeed, we will now start to explicitly show this relation in first order of perturbation theory. Note that this identity holds to all orders of perturbation theory, which can be shown by using the Ward Identity. The running coupling will therefore only be dependend on Z3.
7 The Vertex Correction
We now turn to vertex corrections in QED. On one loop level this means we look at the diagrams displayed in Fig. 5. We can use the Feynman rules given in Fig. 7 in the appendix to see that the sum of the two diagrams is equal to ieγµ ieΛµ, − − 7 THE VERTEX CORRECTION 20 where
4 µ 2 d k ν i µ i i ′ Λ (p ,p) ( ie) 4 γ γ γν 2 −2 ≡ − (2π) p/′ + k/ m + iǫ /p + k/ m + iǫ k λ + iǫ Z − − − 4 2 d k µ 1 1 1 = ie 4 γ γν γµ 2 2 . − (2π) p/′ + k/ m + iǫ /p + k/ m + iǫ k λ + iǫ Z − − − (41)
Note that as before the diagram is supposed to be part of a larger diagram, so
p′ p′
one loop corr. k q −−−−−−−−−→ q p p
Figure 5: One loop correction of the QED vertex. there are no external particle contributions in the amplitude. We could now evaluate (41) with the calculational tools we derived in the first lecture. Indeed, for example for the order-α contribution to the anomalous mag- netic moment of the electron, this would have to be done, since this diagram gives the strongest contribution here. See for example [3] for a detailed calculation. For our purpose we will see that this has not to be done, since we are primarily inter- ested in finding the relation Z1 = Z2. Now in order to show this, we at first need to introduce a new renormalization condition, which will define the finite parts of Z1: µ Λf (p = p′)=0, (42) where Λf denotes the finite amplitude, which we obtain if we also take into ac- count the counter term Feynman diagram:
µ µ µ ieΛ (p,p′) ieΛ (p,p′) ieγ δ , (43) − f ≡ − − 1 so ν ν Λ (p = p′)= γ δ . (44) − 1 7 THE VERTEX CORRECTION 21
In order to proceed, we will need to know what the derivative of an inverse matrix (with respect to a real variable) is. Therefore we perform a (infinitesimal) small 1 variation of A− , where A is a matrix:
1 1 1 1 1 1 δ A− =(A + δA)− A− = A 1+ A− δA − A− − 1 1 1 1 1 1 1 = 1+ A− δA − A− A− = 1 A− δA A− A− 1 1 − − − = A− δA A− . − Using this, we follow the procedure outlined in [5] and perform the following derivative: ∂ 1 1 1 = γµ . (45) ∂pµ /p m −/p m /p m − − − This enables us to calculate the derivative of Σ2 as defined in eqn. (12):
2 ∂Σ2 ie 4 µ 1 1 1 = d kγ γν γµ . (46) ∂pν (2π)4 /p + k/ m + iǫ /p + k m + iǫ k2 λ2 + iǫ Z − − − Comparing this with (41) immediately gives us
∂Σ2 Λ (p = p′)= . (47) ν − ∂pν If we now use (44) and remember what we have already found in (37):
∂Σ2 δ2 = , ∂p/ we can derive: Λ = γ δ ν − ν 1 ∂Σ2 µ ∂Σ2 µ ∂Σ2 ∂Σ2 = = g = γνγ = γν − ∂pν − ν ∂pµ − ∂pµ − ∂p/ = γ δ . − ν 2 So we arrive at the important result:
δ1 = δ2, (48) or Z1 = Z2. (49) 8 VACUUM POLARIZATION 22
Remark About Infrared Divergences We shall now briefly discuss infrared divergences. More details could be found in [2, p. 175], for instance. We had to introduce the photon mass λ in Λµ to cure the infrared divergence. But at first sight this means that the first order correction to physical observable processes like Coulomb scattering (which contain this diagram) would have a dependence on the unphysical photon mass λ. Indeed, a separate cross section correction like that would be divergent as λ 0. The solution to this problem again lies in→ the insight that cross sections like that one are not measurable individually. We have forgotten to take into account another first order correction to processes like Coulomb scattering: Bremsstrahlung. Together with the fact that every photon detector can detect photons only down to some limiting energy, we conclude that observable is only the sum of the cross section for both processes. Now Bremsstrahlung corrections are also infrared di- vergent. But it turns out that the λ-dependence of the sum of the two processes cancels! So we are again left with finite (λ-independent) predictions for observ- ables.
8 Vacuum Polarization
In this section we will calculate the last important one loop diagram: The vac- uum polarization diagram, which is a correction to the photon propagator due to fermion loops:
q + k
µ ν iΠµν(k) k k ≡ q
The only Lorentz tensor in Πµν(q) can be gµν and kµkν. On the other hand the µν Ward identity tells us that kµΠ (k)=0. So we expect the following structure:
Πµν (k)= k2gµν kµkν Π k2 . − We will soon find exactly this structure (without using the Ward identity). 8 VACUUM POLARIZATION 23
Using again the Feynman rules in the appendix and noting that the closed fermion loop gives a factor of 1 and a trace over the product the Dirac matrices, we get: −
d4q 1 1 iΠ(k)µν = e2 Tr γµ γν − (4π)4 /q m + iǫ /q + k/ m + iǫ Z − − d4q γµ /q + m γν /q + k/ + m = e2 Tr − (2π)4 (q2 m2 + iǫ) (q + k)2 m2 + iǫ Z − − We will now use the Feynman trick in the form of eqn. (13) to combine the de- nominators: B A =2qk + k2 A =(B A−) x = q2 m2 + (2qk + k2) x + iǫ − − 2 =(q + kx) xk2 (x 1) m2 + iǫ 2 − − − (q + xk) ∆+ iǫ, ≡ − where we defined ∆ xk2 (x 1) + m2. ≡ − Noting that the trace of an odd number of Dirac matrices vanishes and going to n dimensions, we thus get:
1 n µ ν µ ν 2 µν 2 4 n d q γ /qγ /q + k/ + γ γ m iΠ(k) = e µ − dx Tr n 2 2 − 0 (2π) (q + xk) ∆+ iǫ Z Z − 1 n µ / ν / 2 µ ν 2 4 n d q γ /q xk γ /q + k(1 x) + m γ γ = e µ − dx n Tr − − − (2π) (q2 ∆+ iǫ)2 Z0 Z 1 n µ ν − µ ν 2 µ ν 2 4 n d q γ /qγ /q x (1 x) γ kγ/ k/ + m γ γ = e µ − dx Tr − − , n 2 2 − 0 (2π) (q ∆+ iǫ) Z Z − where we shifted the integration variable q q + xk in the first step and used the → fact that because of the antisymmetry, terms linear in q vanish in the second step. We now need to evaluate the three traces:
µ ν µ ρ ν σ µ ν 2 µν Tr γ /qγ /q = qρqσ Tr γ γ γ γ = 4(2q q q g ) − Tr γµkγ/ νk/ = 4(2kµkν k2gµν) Tr γµγν =4gµν. − 8 VACUUM POLARIZATION 24
Using this identities we arrive at:
1 n µν µν 2 4 n d q 2 2 2 g iΠ(k) = e µ − dx 4 m + k (1 x) q n 2 2 − 0 (2π) − − (q ∆+ iǫ) Z Z − kµkν qµqν 8x (1 x) +8 . − − (q2 ∆+ iǫ)2 (q2 ∆+ iǫ)2 − − To calculate the remaining integrals, we first note
dnq qµqν 1 dnq q2 = gµν , (50) (2π)n D n (2π)n D Z Z where D denotes a symmetric denominator. The derivation of that identity is straight forward: For µ = ν the integral is zero because of the antisymmetry. 6 Because of the Lorentz structure the result must be proportional to gµν. The factor µν 1/n can be checked by multiplying both sides with gµν and using gµνg = n. Using our general integral from eqn. (24) we can also show:
n 2 n dnq q2 d qE q + ∆ d qE 1 n 2 = i + i∆ (2π) (q2 ∆+iǫ) n 2 2 n 2 2 − − (2π) (q + ∆ iǫ) (2π) (q + ∆ iǫ) Z E − Z E − R dnq 1 dnq 1 = i E + i∆ E n 2 n 2 2 − (2π) (qE + ∆ iǫ) (2π) (qE + ∆ iǫ) Z n − Z − iπ 2 1 n n = − n 1 n Γ 1 Γ 2 (2π) (∆ iǫ) − 2 − 2 − − 2 n − h i iπ 2 1 n n = − n 1 n Γ 1 (51) (2π) (∆ iǫ) − 2 − 2 2 − Using this integrals we can finish our calculation:
1 n µν 2 4 n 1 Γ 2 2 iΠ(k) = 4ie µ − dx n −n − 2 ∆2 2 Z0 (4π) − gµν m2 + x (1 x) k2 + gµν m2 + x (1 x) q2 2x (1 x) kµkν × − − − − − = k2gµν kµkν iΠ k2 , − where
1 2 8α n x (1 x) Π k = − n Γ 2 dx − n . (52) 2 1 − 2 2 2 2 2 (4π) − Z0 [k x (x 1) + m ] − − 9 THE BETA FUNCTION OF QED 25
9 The Beta Functionof QED
So far we have not said how we are going to renormalize our result from the last chapter. It is logarithmic divergent, so only the zero order term of a Taylor expan- sion is infinte. Therefore the following renormalization condition is sufficient: Π k2 = M 2 =0. (53) f − Thus Π k2 = Π k2 Π M 2 . (54) f − − 2 2 Meaning we make Π(k ) finite by subtracting at (the space-like momentum) k = M 2. Like in the last chapters, this defines our counter term to be − δ = Π M 2 , (55) 3 − or 2 Z3 =1+Π M − 1 8α n x (1 x) (56) =1 n Γ 2 dx − n 2 1 2 2 − (4π) − − 2 0 [ M 2x (x 1) + m2] − Z − − Here we introduced the scale M (compare to section 5.2). This condition now 1 defines us the renormalized coupling in the lagangian. Note that α = 137 would only hold if we chose M 2 =0. We now take the derivative of (40) with respect to the scale M and use that Z1 = Z2:
dα (4 n) dZ3 = µ− − α dM dM 0 dZ = 3 α dM 8α n n 1 x2 (1 x)2 = − n Γ 2 2 2M dx − n 2 1 3 2 (4π) − − 2 − − 2 0 [M 2x (1 x)+ m2] − Z − 4Mα 1 x2 (1 x)2 = dx − n (57) 3 2 π 0 [M 2x (1 x)+ m2] − Z − 4 n where in the second step we used that µ − α α since Z is already of order α. ≈ 0 3 For M much bigger than m we can now read off the beta function of QED: dα 2α2 β(α) M = . (58) ≡ dM 3π 9 THE BETA FUNCTION OF QED 26
This equation can be integrated:
2 2 α(M0 ) α M = 2 2 . (59) α(M0 ) M 1 3π ln 2 − M0 This defines the running of the renormalized coupling constant in our renormal- ization scheme. Note that for example in the Minimum Subtraction (MS) scheme we get exactly the same first order approximation for the beta function. Instead of the subtraction point (which we don’t have in that scheme, because there only the 1 part proportional to ǫ of the expansion for small ǫ in the dimensional regularized loop integrals is subtracted), the renormalized coupling constant is dependent on the scale µ that you have to introduce in order to get the dimensions right, which in the MS scheme does not cancel in the expression for the beta function as it did in our case. See for example [3] for details here. 9 THE BETA FUNCTION OF QED 27
APPENDIX
Contraction Rules The relation between contractions and propagators is given by:
′ 4 q + m e iq(x x ) d q / αβ − − iSF (x x′) = ψα(x)ψ¯β(x′)= i , (A.1) αβ − (2π)4 q2 m2 + iǫ Z − 4 ik(x x′) µν d q e− − iDF (x x′) = Aµ(x)Aν(x′)= ig , (A.2) µν − − (2π)4 k2 λ2 + iǫ Z − ipx′ s ψ(x′) p,s = e− u (p) 0 , (A.3) | i | i ip′x s′ p′,s′ ψ¯ = 0 e u¯ (p′). (A.4) h | h | Note that we use the Feynman gauge and that we introduced a photon mass λ to regularize infrared divergences.
The Gamma Function The Gamma function Γ(z) has the following important properties:
Γ(n +1) = n! n IN, x Γ(x) =Γ(x +∀ 1)∈ x IR>0. ∀ ∈ Also useful for expansions of the Gamma function is the following product repre- sentation ∞ 1 γz z z = ze 1+ e− n , (A.5) Γ(z) n n=1 Y where γ 0.5772 is the Euler-Mascheroni constant. ≈ Fig. 6 displays the Gamma function. Note that it has a poles at x = 0 and at every negative integer. 9 THE BETA FUNCTION OF QED 28
GHxL
4
2
x -4 -2 2 4
-2
-4
Figure 6: Plot of the Gamma function.
Leading Order Expansions From (A.5) one can derive the leading order expansion of Γ(x): 1 Γ(ǫ)= γ + O(ǫ) (A.6) ǫ − near x =0, and ( 1)n 1 1 Γ( n + ǫ)= − γ +1+ . . . + + O(ǫ) (A.7) − n! ǫ − n near x = n. Another expression which will occur in regularized integrals is − 1 ǫ =1 ǫ log∆+ .... (A.8) ∆ − Particularly useful is now the following expansion, which can be directly derived from that:
n n 2 2 Γ 2 2 1 − 1 2 − n = log ∆ γ + log(4π)+O(ǫ) , (A.9) 2 ∆ 2 ǫ − − (4π) (4π) with ǫ =4 n. − 9 THE BETA FUNCTION OF QED 29
Feynman rules for QED
− µν 2 µ ν igµν = i g q q q δ3 = 2+ − − q iǫ = i = i /pδ2 δm /p−m+iǫ −
µ µ = ieγ δ1 = ieγ − −
Figure 7: Feynman rules for QED in renormalized perturbation theory. Note that a closed fermion loop additionally gives a factor of -1 and a trace of a product of Dirac matrices. See [2, p. 120]. REFERENCES 30
References
[1] Ta-Pei Cheng and Ling-Fong Li, Gauge Theory of Elementary Particle Physics, Oxford University Press, 1984.
[2] Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Westview Press, 1995.
[3] Pierre Ramond, Field Theory: A Modern Primer, Westview Press, 1990.
[4] Franz Schwabl, Quantenmechanik fuer Fortgeschrittene (Advanced Quan- tum Mechanics), Springer, 2005.
[5] James D. Bjorken and Sidney D. Drell, Relativistic Quantum Mechanics, McGraw-Hill, 1964.