2012 Matthew Schwartz III-2:

1 Introduction

In the previous lecture, we calculated the Casimir effect. We found that the energy of a system involving two plates was infinite; however the observable, namely the force on the plates, was finite. At an intermediate step in calculating the force we needed to model the inability of the plates to restrict ultra-high frequency radiation. We found that the force was independent of the model and only determined by radiation with wavelengths of the plate separation, exactly as physical intuition would suggest. More precisely, we proved the force was independent of how we modeled the interactions of the fields with the plates as long as the very short wavelength modes were effectively removed and the longest wavelength modes were not affected. Some of our models were inspired by physical arguments, as in a step-function cutoff representing an atomic spacing; others, such as the ζ-function regulator, were not. That the calculated force is indepen- dent of the model is very satisfying: macroscopic physics (the force) is independent of micro- scopic physics (the atoms). Indeed, for the Casimir calculation, it doesn’t matter if the plates are made of atoms, aether, phlogiston or little green aliens. The program of systematically making testable predictions about long-distance physics in spite of formally infinite short distance fluctuations is known as . Because physics at short- and long-distance decouples, we can deform the theory at short distance any way we like to get finite answers – we are unconstrained by physically justifiable models. In fact, our most calculationally efficient deformation will be analytic continuation to d = 4 ε dimen- sions with ε 0. In some quantum field theory applications, such as condensed matter− systems → or string theory, there is a real physical cutoff. The beauty of renormalization, however, is that the existence of a physical cutoff is totally irrelevant: quantitative predictions about long dis- tance physics do not care what the short-distance cutoff really is, or even whether or not it exists. The basic principle of renormalization in quantum field theory is Observables, such as S-matrix elements, are finite and (in principle) calculable • functions of other observables.

One can think of general correlation functions Ω T φ( x1 ) φ( xn) Ω as a useful proxy for observables. Most of the conceptual confusion, bothh | { historically and}| amongi students learning the subject, stems from trying to express observables in terms of non-observable quantities, such as coupling constants in a Lagrangian. In practice: Infinite results associated with high-energy divergences may appear in intermediate steps • of calculations, such as in loop graphs. Infinities are tamed by a deformation procedure called regularization. The regulator • dependence must drop out of physical predictions. Coefficients of terms in the Lagrangian, such as coupling constants, are not observable. • They can be solved for in terms of the regulator and will drop out of physical predictions. We will find that loops can produce behavior different from anything possible at tree-level. In particular, Non-analytic behavior, such as ln s , is characteristic of loop effects. • s 0 Tree-level amplitudes are always rational polynomials in external momenta and never involve logarithms. In many cases the non-analytic behavior will comprise the entire physical prediction associated with the loop.

1 2 Section 2

In the previous lecture we gave an example of renormalization in φ4 theory. We found that 4 the expression for a correlation function in terms of the coupling constant λ was infinite: φ s = λ 2 s h i

λ s 16 π 2 ln Λ 2 + = . However, expressing the correlation function at the scale in terms of − ∞ 4 4 the correlation function at a different scale s 0 gave a finite prediction: φ s = φ s 0 1 4 2  s 4 h i h i − 2 φ s 0 ln + . Although φ theory was just a toy example, renormalization in QED, 16 π h i s 0 which we begin in this lecture, is conceptually identical. 2 V r e Recall that the Coulomb potential ( ) = 4πr is given by the exchange of a single e2 = (1) p2

1 ¡ Indeed 4πr is just the Fourier transform of the (cf. Lecture I-3). A one-loop correc- tion to Coulomb’s law comes from a e+ e − loop inside the photon line:

(2) ¡ This will give us a correction to V( r) proportional to e4. We will show that while the charge e is infinite it can be replaced by a finite renormalized charge order-by-order in perturbation theory. The physical effect will be a measurable correction to Coulomb’s law predicted by quantum field theory with logarithmic scale dependence, as in the φ4 toy model. The process represented by the in Eq. (2) is known as vacuum polariza- tion. The diagram shows the creation of virtual e+ e − pairs which act like a virtual dipole. In the same way that a material such as water would get polarized if we put it in a elec- tric field, this vacuum polarization tells us how the vacuum itself gets polarized when it interacts with electromagnetic radiation. Since the renormalization of the graph is no different than it was in φ4 theory, the only diffi- cult part of calculating vacuum polarization is in the evaluation of the loop. Indeed, the loop in Eq. (2) is complicated, involving and spinors, but we can do it exploiting some tricks developed through the hard work of our predecessors. Our approach will be to build up the QED vacuum polarization graph in pieces, starting with φ3 theory, then scalar QED, and finally real QED. For convenience, some of the more mathematical aspects of regularization are com- bined into one place in Appendix B, which is meant to provide a general reference. In the fol- lowing, we assume familiarity with the results from that appendix.

2 Scalar φ3 theory

As a warm-up for the vacuum polarization calculation in QED, we will start with scalar φ3 theory with Lagrangian 1 g = φ(  + m2 ) φ + φ3 (3) L − 2 3! Now we want to compute

4 p − k 1 d k i i i ( p) = = ( ig) 2 (4) loop p p 4 2 2 2 2 M k 2 (2π) ( p k) m + iε k m + iε Z − − − which will tell¡ us the 1-loop correction to the Yukawa potential. We will allow the initial and 2 2 final line to be off-shell: p  m , since we are calculating the correction to the initial φ propa- i 2 2 p  m gator p2 − m 2 which also must have to make any sense, and since we will be embedding this graph into a correction to Coulomb’s law (see Eq. (14) below). First, we can use the Feynman parameter trick from Appendix B

1 1 1 = dx (5) AB [ A + ( B A) x] 2 Z0 − Scalar φ3 theory 3 with A = ( p k) 2 m2 + iε and B = k2 m2 + iε. Then we complete the square − − − A + [ B A] x = ( p k) 2 m2 + iε + [ k2 ( p k) 2 ] x − − − − − (6) = ( k p(1 x)) 2 + p2 x(1 x) m2 + iε which gives − − − − g2 d4k 1 1 i ( p) = dx (7) M loop 2 (2π) 4 [( k p(1 x)) 2 + p2 x(1 x) m2 + iε] 2 Z Z0 − − − − Now shift k µ k µ + pµ(1 x) in the integral. The measure is unchanged, and we get → − g2 d4k 1 1 i ( p) = dx (8) M loop 2 (2π) 4 [ k2 ( m2 p2 x(1 x)) + iε] 2 Z Z0 − − − At this point, we need to introduce a regulator. We will use Pauli-Villars regularization (see Appendix B), which adds a fictitious scalar of Λ with fermionic statistics. This particle is an unphysical ghost particle. We can use the Pauli-Villars formula from Appendix B: d4k 1 i ∆ = ln (9) (2π) 4 ( k2 ∆ + iε) 2 − 16π2 Λ2 Z − Comparing to Eq. (8), we have ∆ = m2 p2 x(1 x) so that − − ig2 1 m2 p2 x(1 x) i loop ( p) = dx ln − − (10) M − 32π2 Λ2 Z0   This integral can be done – the integrand is perfectly well behaved between x = 0 and x = 1 . For m = 0 it has the simple form g2 p2 loop( p) = 2 ln − (11) M 32π2 − Λ2   Note that the 2 cannot be physical, because we can remove it by redefining Λ2 Λ2 e − 2 . Also note that when this diagram contributes to the Coulomb potential (as in Eq. (14)→ below), the − 2 pµ p2 < p virtual momentum is spacelike ( 0) so ln Λ 2 is real. Then, g2 Q 2 ( p) = ln (12) M loop − 32π2 Λ2

An important point is that the regulator scale Λ has to be just a number, independent of any external momenta. With the Pauli-Villars regulator we are using here, Λ is the mass of some heavy fictitious particle. It corresponds to a deformation of the theory at very high energies/short distances, like the modeling of the wall in the Casimir force. On the other hand, Q is a physical scale like the plate separation in the Casimir force. Thus Λ cannot depend on Q. In particular, the ln Q 2 dependence cannot be removed by a redefinition of Λ like the 2 in Eq. (11) was. This point is so important it’s worth repeating: the short distance deformation ( Λ) can’t depend on long-distance physical quantities ( Q). This separation of scales is critical to being able to take Λ to make predictions by relating observables at different long-distance → ∞ 2 scales such as Q and Q 0. The coefficient of ln Q is in fact regulator independent and will gen- erate the physical prediction from the loop.

2.1 Renormalization The diagram we computed is a correction to the tree level φ propagator. To see this, observe that the propagator is essentially the same as the t-channel scattering diagram

p1 p3 i i 0( p) = p = ( ig) 2 (13) M p2

p2 p4 ¡ 4 Section 3

If we insert our scalar bubble in the middle, we get

p1 p3 p 2 2 1 2 i i 2 1 g p 1 i ( p) = = ( ig) i loop ( p) = ig ln − (14) M p2 M p2 p2 − 32π2 Λ2 p2 p  

¡p2 p4

Since p2 < 0, let us write Q 2 = p2 with Q > 0. Then, − g2 1 g2 Q 2 ( Q) = 0( Q) + 1 ( Q) = 1 ln + ( g4) (15) M M M Q 2 − 32π2 Q 2 Λ2 O   Note that g is not a number in φ3 theory but has dimensions of mass. This actually makes φ3 a little more confusing than QED, but not insurmountably so. Let’s just simplify things by 2 2 g writing the whole thing in terms of a new Q-dependent variable g˜ 2 which is dimensionless. ≡ Q Then 1 Q 2 ( Q) = g˜ 2 g˜ 4 ln + ( g˜ 6) (16) M − 32π2 Λ2 O

Then we can define a renormalized coupling g˜R at some fixed scale Q 0 by

g˜ 2 ( Q ) (17) R ≡ M 0 This is called a renormalization condition. It is a definition, and by definition, it holds to all orders in perturbation theory. The renormalization condition defines the coupling in terms of an observable. Therefore, you can only have one renormalization condition for each parameter in the theory . This is critical to the predictive power of quantum field theory. 2 It follows that g˜R is a formal power series in g~

1 Q 2 g˜ 2 = ( Q ) = g˜ 2 g˜ 4ln 0 + ( g˜ 6) (18) R M 0 − 32π2 Λ2 O which can be inverted to give g˜ as a power series in g˜R

1 Q 2 g˜ 2 = g˜ 2 + g˜ 4 ln 0 + ( g˜ 6 ) (19) R 32π2 R Λ2 O R Substituting into Eq.(16) produces a prediction for the matrix element at the scale Q in terms of the matrix element at the scale Q 0

1 Q 2 1 Q 2 ( Q) = g˜ 2 g˜ 4 ln + ( g˜ 6 ) = g˜ 2 + g˜ 4 ln 0 + ( g˜ 6 ) (20) M − 32π2 Λ2 O R 32π2 R Q 2 O R

Thus we can measure at one Q and then make a nontrivial prediction at another value of Q. M

3 Vacuum polarization in QED

In φ3 theory, we found

ig2 1 m2 p2 x(1 x) = dx ln − − (21) − 32π2 Λ2 Z0

The integral in QED¡ is quite similar. We will first evaluate the vacuum polarization graph in scalar QED, and then in spinor QED. Vacuum polarization in QED 5

3.1 Scalar QED In scalar QED the vacuum polarization diagram is

k p − d4k i i = ( ie) 2 (2k µ pµ)(2kν pν ) p p − (2π) 4 ( p k) 2 m2 + iε k2 m2 + iε − − k Z − − −

For external¡ photons, we could contract the µ and ν indices with polarization vectors, but instead we keep them free so that this diagram can be embedded in a Coulomb exchange dia- gram as in Eq. (14). This integral is the same as in φ3 theory, except for the numerator factors. In scalar QED there is also another diagram

¡p p d4k i = 2ie2 gµν (22) (2π) 4 k2 m2 + iε Z −

So we can add them together to get

4 µ ν µ ν ν µ µ ν µν 2 2 µν d k 4k k + 2 p k + 2 p k p p + 2 g (( p k) m ) iΠ = e2 − − − − (23) 2 − (2π) 4 [( p k) 2 m2 + iε][ k2 m2 + iε] Z − − − Fortunately, we do not need to evaluate the entire integral. By looking at what possible form it could have, we can isolate the part which will contribute to a correction to Coulomb’s law and µν just calculate that part. By Lorentz invariance, the most general form that Π2 could have is µν 2 2 2 µν 2 2 µ ν Π2 = ∆1 ( p , m ) p g + ∆2 ( p , m ) p p (24) µν µ µ for some form factors ∆1 and ∆ 2 . Note that Π2 cannot depend on k , since k is integrated over. As a correction to Coulomb’s law, this vacuum polarization graph will contribute to the same process that the photon propagator does. Let us define the photon propagator in momentum space by 4 d p − Ω T A µ( x) Aν ( y) Ω = eip ( x y) iG µν ( p) (25) h | { }| i (2π) 4 Z Note that this expression only depends on x y by translation invariance. This is the all-orders non-perturbative definition of the propagator− G µν( p) , which is sometimes called the dressed propagator. At leading order, in Feynman gauge, the dressed propagator reduces to the free propagator igµν iG µν( p) = − + ( e2 ) (26) p2 + iε O Including the 1-loop correction, with the parametrization in Eq. (24), the propagator is (sup- pressing the iε pieces)

µν µα βν µν ig ig 2 ig 4 iG ( p) = − + − iΠαβ − + ( e ) p2 p2 p2 O µν µ ν ig i µν p p 4 = − + − ∆1 g + ∆ 2 + ( e ) (27) p2 p2 p2 O  µ ν  µν p p (1+∆1 ) g + ∆ 2 2 = i p − p2 + iε Note that we are calculating loop corrections to a Green’s function, not an S-matrix element, so we do not truncate the external and add polarization vectors. One point of using a µν dressed propagator is that once we calculate ∆1 and ∆ 2 we can just use G ( p) instead of the tree level propagator in QED calculations to include the loop effect. Next note that the ∆ 2 term is just a change of gauge – it gives a correction to the unphysical gauge parameter ξ in covariant gauges. Since ξ drops out of any physical process, by gauge invariance, so will ∆2 . Thus we only need to compute ∆ 1 . This means extracting the term pro- portional to gµν in Π µν. 6 Section 3

Most of the terms in the amplitude in Eq. (23) cannot give gµν. For example, the pµpν term µ ν must be proportional to p p and can therefore only contribute to ∆ 2 , so we can ignore it. For the pµkν term, we can pull pµ out of the integral, so whatever the remaining integral gives, it must provide a pν by Lorentz invariance. So these terms can be ignored too. The k µkν term is important – it may give a pµpν piece, but may also give a gµν piece, which is what we’re looking for. So we only need to consider 4 µ ν µν 2 2 µν d k 4k k + 2 g (( p k) m ) Π = ie2 − − − (28) 2 (2π) 4 [( p k) 2 m2 + iε][ k2 m2 + iε] Z − − − Now we need to compute the integral. The denominator can be manipulated using Feynman parameters just as with the φ3 theory

4 1 µ ν µν 2 2 µν d k 4k k + 2 g (( p k) m ) Π = ie2 dx − − − (29) 2 (2π) 4 [( k p(1 x)) 2 + p2 x(1 x) m2 + iε] 2 Z Z0 − − − − However, now when we shift k µ k µ + pµ(1 x) we get a correction to the numerator. We get → − 4 1 µ µ ν ν µν 2 2 µν d k 4( k + p (1 x))( k + p (1 x))+2 g (( xp k) m ) Π = ie2 dx − − − − − (30) 2 (2π) 4 [ k2 + p2 x(1 x) m2 + iε] 2 Z Z0 − − As we have said, we don’t care about pµpν pieces, or pieces linear in pν. Also, pieces like p k are odd under k k while the rest of the integrand, including the measure, is even. So these· terms must give zero→ − by symmetry. All that is left is

4 1 µ ν µν 2 2 2 2 µν d k 2k k + g ( k + x p m ) Π = 2ie2 dx − − (31) 2 (2π) 4 [ k2 + p2 x(1 x) m2 + iε] 2 Z Z0 − − It looks like this integral is much more badly divergent than the φ3 theory – it is now quadratically instead of logarithmically divergent. That is, if we cut off at k = Λ we will get something proportional to Λ2 due to the k µkν and k2 terms. Quadratic divergences are not tech- nically a problem for renormalization. However, in Lecture III-8 we will see, on very general grounds, that in gauge theories like scalar QED, all divergences should be logarithmic. In this case, the quadratic divergence from the k µkν term and the k2 term precisely cancel due to gauge invariance. This cancellation can only be seen using a regulator which respects gauge invariance, d k µkν 1 k2 gµν such as dimensional regularization. In dimensions (using d from Appendix B), the integral becomes → 2 1 ddk (1 ) k2 + x2 p2 m2 Πµν = 2ie2 µ4 − dgµν dx − d − (32) 2 (2π) d [ k2 + p2 x(1 x) m2 + iε] 2 Z0 Z − − Using the formulas from Appendix B:

ddk k2 d i 1 2 d = Γ (33) d 2 2 d/2 − d − (2π) ( k ∆ + iε) − 2 (4π) 1 2 2 Z − ∆   ddk 1 i 1 4 d = Γ (34) d 2 2 d/2 − d − (2π) ( k ∆ + iε) (4π) 2 2 2 Z − ∆   with ∆ = m2 p2 x(1 x) we find − − 2 1 − d − d e − d d 1 1 2 d 1 2 2 µν gµνµ4 d dx x2 p2 m2 Π2 = 2 d/2 1 Γ(1 ) + ( )Γ(2 ) − (4π) 0 − 2 − 2 ∆ − − 2 ∆ Z        Using Γ(2 d ) = 1 d Γ(1 d ) this simplifies to − 2 − 2 − 2   2 1 − d e d − 1 2 2 Πµν = 2 p2 gµνΓ(2 ) µ4 d dxx(2x 1) (35) 2 d/2 − (4π) − 2 0 − ∆ Z   For completeness, we also give the result including also the pµpν terms

2 1 − d 2e d − 1 2 2 µν p2 gµν pµpν µ4 d dxx x Π2 = − d/2 ( )Γ(2 ) (2 1) 2 2 (4π) − − 2 0 − m p x(1 x) Z  − −  Vacuum polarization in QED 7

You should verify this through direct calculation (see Problem 1), but it is the unique result µν consistent with Eq. (35) which satisfies the Ward identity pµΠ2 = 0. Expanding d = 4 ε we get, in the ǫ 0 limit, − → − e2 1 2 4πe γE µ2 Π µν = ( p2 gµν pµpν ) dxx(2x 1) + ln + ( ε) (36) 2 − 8π2 − − ε m2 p2 x(1 x) O Z0   − −   1 µ 2 The ε gives the infinite, regulator-dependent constant. It is also standard to define ˜ = − − 4πe γE µ2 which removes the ln (4π) and e γE factors. For Q 2 = p2 > 0 and m Q, the inte- gral over x is easy to do and we find − ≪ e2 2 µ˜ 2 8 Πµν = ( p2 gµν pµpν ) + ln + , m Q (37) 2 − 48π2 − ε Q 2 3 ≪   At this point, rather than continue with the scalar QED calculation, let’s calculate the loop in QED, as it’s almost exactly the same.

3.2 Spinor QED In spinor QED, the loop is

k p − d4k i i = ( ie) 2 Tr[ γµ( k p + m) γν( k + m)] p p − − (2π) 4 ( p k) 2 m2 k2 m2 − k Z − − − where¡ the 1 in front comes from the fermion loop. Note that there is only 1 diagram in this case. − Using our trace formulas (see Lecture II-6), we find Tr[ γµ( k p + m) γν ( k + m)] = 4[ pµkν k µpν + 2k µkν + gµν( k2 + p k + m2 )] (38) − − − − · We can drop the pµ and pν terms as before giving

4 µ ν µν 2 2 µν d k 2k k + g ( k + p k + m ) iΠ = 4 e2 − · (39) 2 − (2π) 4 [( p k) 2 m2 + iε][ k2 m2 + iε] Z − − − Introducing Feynman parameters and changing k µ k µ + pµ(1 x) and again dropping the pµ and pν terms we get → − 4 1 µ ν µν 2 2 2 µν d k 2k k g ( k x(1 x) p m ) Π = 4ie2 dx − − − − (40) 2 (2π) 4 [ k2 + p2 x(1 x) m2 ] 2 Z Z0 − − This integral is quite similar to the one for scalar QED, Eq. (31). The result is

2 1 − d e d − 1 2 2 µν p2 gµν µ4 d dxx x Π2 = 8 d/2 Γ(2 ) (1 ) 2 2 (41) − (4π) − 2 0 − m p x(1 x) Z  − −  e2 1 2 µ˜ 2 = p2 gµν dxx(1 x) + ln + ( ε) − 2π2 − ε m2 p2 x(1 x) O Z0   − −   So we find (for large Q 2 = p2 m2 ) − ≫ e2 2 µ˜ 2 5 Π µν = p2 gµν + ln + + ( ε) , m Q 2 − 12π2 ε Q 2 3 O ≪   µ˜ 2 We see that the loop gives the same pole and ln Q 2 terms as a scalar loop, multiplied by a factor of 4. It is not hard to compute the pµpν pieces as well (see Problem 1). The full result is

2 1 − d 8e d − 1 2 2 Π µν = ( p2 gµν pµpν)Γ(2 ) µ4 d dxx(1 x) (42) 2 − d/2 2 2 (4π) − − 2 0 − m p x(1 x) Z  − −  which, as in the scalar QED case, automatically satisfies the Ward identity. 8 Section 4

4 Physics of vacuum polarization

We have found that vacuum polarization loop in QED gives i Πµν = i( p2 gµν + pµpν) e2 Π ( p2 ) (43) 2 − 2 where 1 1 2 µ˜ 2 Π ( p2 ) = dxx(1 x) + ln (44) 2 2π2 − ε m2 p2 x(1 x) Z0   − −  Thus the dressed photon propagator at 1-loop in Feynman gauge is

k p − µν µν g i µν i µ ν iG = b + = i + − iΠ − + p p terms (45) p p − p2 p2 2 p2 k

¡2 2 µν (1 e Π2 ( p )) g = i − + pµpν terms − p2 This directly gives the Fourier transform of the corrected Coulomb potential: 2 2 1 e Π2 ( p ) V˜ ( p) = e2 − (46) p2 Now we need to renormalize. A natural renormalization condition is that the potential between two particles at some ref- 2 e R erence scale r should be V( r ) which would define a renormalized eR. It is easier to 0 0 4πr0 ≡ − ˜ 2 2 − 2 continue working in momentum space and to define the renormalized charge as V ( p0 ) eRp0 exactly. So ≡ 2 2 ˜ 2 2 4 2 e p V ( p ) = e e Π ( p ) + (47) R ≡ 0 0 − 2 0 4 Solving for the bare coupling e as a function of eR to order eR gives

2 2 4 2 e = eR + eRΠ2 ( p0) + (48) 2 Since Π2 ( p0) is infinite, e is infinite as well, but that is ok since e is not observable. The potential at another scale p, which is measurable, is

2 ˜ 2 4 2 2 4 2 2 p V ( p) = e e Π ( p ) + = e e [Π ( p ) Π ( p )] + (49) − 2 R − R 2 − 2 0 For concreteness, let us take p0 = 0, corresponding to r = so that the renormalized electric charge agrees with the macroscopically measured electric charge.∞ Then

1 2 2 1 p Π2 ( p ) Π2 (0) = 2 dxx(1 x) ln 1 2 x(1 x) (50) − − 2π 0 − − m − Thus, we have Z   e2 e2 1 p2 V˜ ( p2 ) = R 1 + R dxx(1 x) ln 1 x(1 x) + ( e4 ) (51) p2 2π2 − − m2 − O R  Z0    which is a totally finite correction to the Coulomb potential. It is also a well-defined perturba- tion expansion in terms of a small parameter eR which is also finite. We will now study some of the physical implications of this potential

4.1 Small momentum – First, let’s look at the small-momentum, large-distance limit. For p2 m | | ≪ 1 p2 1 p2 p2 dxx(1 x) ln 1 x(1 x) dxx(1 x) x(1 x) = (52) − − m2 − ≈ − − m2 − − 30m2 Z0   Z0   implying 2 4 ˜ eR eR V ( p) = + (53) p2 − 60π2 m2 Physics of vacuum polarization 9

The Fourier transform of a 1 is δ( r) , so we find e2 e4 V( r) = R R δ( r) (54) − 4πr − 60π2 m2 This agrees with the Coulomb potential up to a correction known the Uehling term [Uehling Phys. Rev. 48, 55-63 (1935)] What is the physical effect of this extra term? One way to find out is to plug this potential into the Schrödinger equation and see how the states of the atom change. Equiva- lently, we can evaluate the effect in time-independent perturbation theory by evaluating the e 4 leading order energy shift ∆E = ψi ∆V ψi using ∆V = 4 2 δ( r) . Since only the L = 0 h | | i − 60 π m atomic orbitals have support at r = 0 this extra term will only affect the S states of the Hydrogen atom. The energy is negative, so their energies will be lowered. You might recall that, at leading order, the energy spectrum of the Hydrogen atom is determined only by the principal atomic number n, so the 2P1/2 and 2S1/2 levels (for example) are degenerate. Thus the Uehling term contributes to the splitting of these levels, known as the Lamb shift. It changes the 2S1/2 state by 27 MHz which is a measurable contribution to the 1028 MHz Lamb shift. More− carefully, you can show in Problem 2 that the 1-loop− potential is

2 2 ∞ 2 e e − 2x + 1 V( r) = 1 + dxe 2 mrx √x2 1 (55) − 4πr 6π2 2x4 −  Z1  This is known as the [Uehling 1935]. For r 1 , ≫ m

α α 1 − 1 V( r) = 1 + e 2 mr , r (56) − r" 4√π ( mr) 3/2 # ≫ m

This shows that the finite correction has extent 1/m = re, the Compton radius of the electron. r L a m Since e is much smaller than the characteristic size of the modes, the Bohr radius 0 α , our δ-function approximation is valid. ∼ By the way, the measurement of the Lamb shift in 1947 by Wallis Lamb [Ref] was one of the key experiments that convinced people to take quantum field theory seriously. Measurements of the hyperfine splitting between the 2S1/2 and 2P1/2 states of the Hydrogen atom had been attempted for many years, but it was only by using microwave technology developed during the Second World War that Lamb was able to provide an accurate measurement. He found ∆E 1000 MHz. Shortly after his measurement, calculated the dominant theoretical con-≃ tribution. His calculation was of a vertex correction which was infrared divergent. Now we know that the infrared divergence is canceled when all the relevant contributions are included, but Bethe simply cut off the divergence by hand at what he argued was a natural physical scale, the 4 5 electron mass. His result was that ∆E = Z α m e ln( α4Z 4) 1000 MHz in excellent agreement − 12 π ≈ − with Lamb’s value. The next year, Feynman, Schwinger and Tomonaga all independently pro- vided the complete calculation, including the Uehling term and the -orbit coupling. Due to a subtlety regarding gauge invariance, only Tomonaga got it right the first time. The full 1-loop result gives E(2S1/2 ) E(2P1/2 ) = 1051MHz. The current best measurement of this shift is 1054 MHz. −

4.2 Large momentum – logarithms of p r 1 In the small distance limit m it is easier to consider the potential in momentum space. Then we have from Eq. (51) ≪ e2 e4 1 1 p2 V˜ ( p2 ) = R + R dxx(1 x) ln 1 x(1 x) + ( e6 ) (57) p2 p2 2π2 − − m2 − O R Z0   e2 e4 1 p2 1 R + R ln dxx(1 x) + ( e6 ) (58) ≈ p2 p2 2π2 − m2 − O R   Z0 2 2 2 eR eR p 6 = 1 + ln − + ( eR) (59) p2 12π2 m2 O   10 Section 4

Recall that for t-channel exchange, Q 2 = p2 > 0 so this logarithm is real. If we compare the potential at two high− energy scales Q m and Q m we find ≫ 0 ≫ e4 Q 2 Q 2 V˜ ( Q 2 ) Q 2 V( Q 2 ) = R ln 0 (60) − 0 0 12π2 Q 2 which is independent of m. Note however, that setting m = 0 directly in Eq. (57) results in a divergence. This kind of divergence is known as a mass singularity, which is a type of infrared

divergence. In this case, the divergence is naturally regulated by m  0. On other occasions we will have to introduce an artificial infrared regulator (like a photon mass) to produce finite answers. Infrared divergences are the subject of Lecture III-6. One way to write the radiative correction to the potential is e2 ( Q) V˜ ( Q 2 ) = eff (61) p2 where e2 Q 2 e2 ( Q) = e2 1 + R ln (62) eff R 12π2 m2    In this case, for simplicity, we have defined the renormalized charge eR e ( m) at Q = m rather ≡ eff than at Q = 0 . (One could also define eR at Q = 0 as with the Uehling potential; however then one would need to include the full m dependence to regulate the m = 0 singularity.) Eq. (62) is to be interpreted as an effective charge in QED which grows as the distance gets smaller (momentum gets larger). Near any particular fixed value of the momentum transfer µ 2 p , the potential looks like a Coulomb potential with a charge eeff( p ) instead of eR. This is a useful concept because the charge depends only weakly on p2 , through a logarithm. Thus, for small variations of p around a reference scale, the same effective charge can be used. Eq. (62) only comes into play when one compares the charge at very different momentum transfers. Q The sign of the coefficient of the ln m term is very important; this sign implies the effective charge gets larger at short distances. At large distances, the charge is increasingly screened by the virtual electron- dipole pairs. At smaller distances, there is less room for the screening and the effective charge increases. However, the effective charge only increases at small 2 α e R 1 e distances very slowly. In fact taking R = 4π = 137 so that R = 0.303 we get an effective fine structure constant of the form 2 2 1 p αeff( p ) = 1 + 0.00077ln − (63) − 137 m2    Because the coefficient of the logarithm is numerically small, one has to measure the potential at extremely high energies to see its effect. In fact, only very few high precision measurements are sensitive to this logarithm. Despite the difficulty of probing extremely high energies in QED experimentally, one can at least ask what would happen if we attempted scattering at t = p2 m2 . From Eq. (63) we can − ≫ see that at some extraordinarily high energies p 10286 eV, the loop correction, the logarithm, is as important as the tree-level value, the 1. Thus∼ perturbation theory is breaking down. At these scales, the 2-loop value will also be as large as the 1-loop and tree level values, and so on. The scale where this happens is known as a . So, QED has a Landau Pole: perturbation theory breaks down at short distances. • This means that QED is not a complete theory in the sense that it does not tell us how to com- pute scattering amplitudes at all energies.

4.3 Running coupling It is not difficult to include certain higher-order corrections to the effective electric charge. Adding more loops in series, we can sum a set of graphs to all orders in the coupling constant:

µν

G ( p) = b + + +

¡ ¡ Physics of vacuum polarization 11

These corrections to the propagator immediately translate into corrections to the momentum- space potential: 2 2 2 2 2 2 ˜ eR eR Q eR Q V ( Q) = 1 + ln + ln + (64) − Q 2 12π2 m2 12π2 m2     2 1 eR = 2 2 2 (65) − Q  e R Q  1 2 ln 2 − 12 π m   So now the momentum-dependent electric charge becomes

2 2 eR eeff( Q) = 2 2 (66) e R Q 1 2 ln 2 − 12 π m which is known as a running coupling. Note we have defined this running coupling to have 2 2 the same renormalization condition as the one-loop effective charge: eeff = eR at p = m . Although the running coupling includes contributions from all orders in perturbation theory,− it still has a Landau pole at p 10286 eV. Running couplings will∼ play an increasingly important role as we study more complicated problems in quantum field theory. They are best understood through the renormalization group. As a preview of how the renormalization group works, note that Eq. (66) can be written as 1 1 1 Q 2 2 = 2 2 ln 2 (67) eeff( Q) eR − 12π m The renormalization group comes from the simple observation that there is nothing special about the renormalization point. Here we have defined eR = eeff( m) , but we could have renor- malized at any other point µ2 instead of m2 , and the results would be the same. Then we would have 1 1 1 Q 2 2 = 2 2 ln 2 (68) eeff( Q) eeff( µ) − 12π µ The left hand side is independent of µ. So taking the µ derivative gives 2 d 1 2 0 = 3 eeff + 2 (69) − eeff dµ 12π µ Or de e3 µ eff = eff (70) dµ 12π2 This is known as a renormalization group equation. We even have a special name for the right hand side of this particular equation, the β-function. In general, de µ β( e) (71) dµ ≡ 3 β e e and we have derived that ( ) = 12 π 2 at 1-loop. The renormalization group is the subject of Lec- ture Lecture III-9.