6 Perturbative Renormalization

Home , Feynman diagram, Propagator, Quartic interaction, Renormalization, Vacuum polarization

6.1 One–loop renormalization of λφ4 theory Consider the scalar theory

1 1 λ S [φ]= (∂φ)2 + m2φ2 + φ4 d4x (6.1) Λ0 2 2 4! ! " # with initial couplings m2 and λ, deﬁned for momentum modes Λ . (The mass coupling ≤ 0 is dimensionful here.) From the analysis of the previous chapter, we expect that near the Gaussian ﬁxed point, the mass parameter is relevant, while the quartic coupling is marginally irrelevant. Let’s see how these expectations are borne out in perturbative calculations. Firstly, the mass term receives a correction from the Feynman diagram

If we’re integrating out all momentum modes p Λ then this diagram is given by the | |≤ 0 Euclidean signature loop integral

1 d4p 1 Vol(S3) Λ0 p3 dp λ 4 2 2 = λ0 4 2 2 −2 (2π) p + m − 2(2π) 0 p + m ! 2 ! 2 λm2 Λ0/m x dx = (6.2) −32π2 1+x !0 λ Λ2 = Λ2 m2 ln 1+ 0 , 32π2 0 − m2 " $ %# where the factor of 1/2 is the symmetry factor of the diagram, and we have used the fact that Vol(S3) = 2π2. As expected, this result shows that the mass parameter is relevant: There’s a quadratic divergence as we try to take the continuum limit Λ (as well as 0 →∞ a subleading logarithmic one). If we wish to obtain ﬁnite results in the continuum limit, then we must tune the 2 couplings (m ,λ) in our scale-Λ0 action so as to make (6.2) ﬁnite in the limit. Achieving such a tuning directly from (6.2) is complicated, but fortunately we don’t need to do this. Recall from section 5.2.2 that we tune by modifying the action to include counterterms:

CT SΛ0 [φ] SΛ0 [φ]+!S [φ, Λ0] (6.3) → where in this case the counterterm action is 1 1 1 SCT[φ, Λ ]= δZ(∂φ)2 + δm2φ2 + δλ φ4 d4x (6.4) 0 2 2 4! ! " # with (δZ,δm2,δλ) representing our ability to adjust the couplings in the original action (including the coupling to the kinetic term — or wavefunction renormalization). These

– 69 – counterterm couplings will depend explicitly on Λ0, as they represent the tuning that must be performed starting from the Λ0 cut–oﬀtheory. The fact that the counterterm action is proportional to ! means that classical contri- CT butions from S contribute to the same order in ! as 1-loop diagrams from SΛ0 [φ]. Thus the mass term also receives a correction at order !0 from the tree diagram

where the cross represents the insertion of the counterterm δm2 treated as a vertex. The full quantum contribution to the mass term is thus

λ Λ2 δm2 + Λ2 m2 ln 1+ 0 (6.5) 32π2 0 − m2 " $ %# to 1-loop accuracy.

6.1.1 The on–shell renormalization scheme The raison d’etre of counterterms is to ensure (6.5) has a finite continuum limit, so that they must cancel the part of (6.2) that diverges as Λ . This still leaves us a lot of 0 →∞ freedom in choosing how much of the finite part of the loop integral can also be absorbed by the counterterms. There’s no preferred way to do this, and any such choice is called a renormalization scheme. Ultimately, all physically measurable quantities (such as cross– sections, branching ratios, particle lifetimes etc.) should be independent of the choice of renormalization scheme. One physically motivated choice is called the on-shell scheme. In this scheme, we fix the mass counterterm δm2 by asking that, once we take the continuum limit, the pole in the exact propagator d4x eip x φ(x)φ(0) in momentum space occurs at some experimentally · % & measured value. (Recall that the cross-section σ has a peak at the location of poles (in the & complex momentum plane) in the S-matrix, so this location is an experimentally measurable quantity.) For example, it would be natural to try to set up our original action so that the coupling m2 is indeed this experimentally measured value. If we denote

M 2(p2) = sum of all 1PI contributions to the mass2 term (6.6)

then the exact propagator can be written as a geometric series

– 70 – 1 = . p2 + m2 + M 2(p2) Taking into account both the loop integral and the counterterm, to 1-loop accuracy we’d ﬁnd the propagator

λ Λ2 = p2 + m2 + δm2 + Λ2 m2 ln 1+ 0 (6.7) 32π2 0 − m2 " $ %# Consequently, in this scheme we should choose

λ Λ2 δm2 = Λ2 m2 ln 1+ 0 1 (6.8) −32π2 0 − m2 − " " $ % ## so as to completely cancel the 1-loop contribution to the mass term. Notice that we cannot sensibly take the limit Λ in either the 1-loop correction or 0 →∞ the counterterm separately, reflecting the fact that the path integral measure Dφ over all modes in the continuum does not exist. However, we can take the continuum limit of the correlation function φ(x)φ(y) (or it’s momentum space equivalent) after having computed % & the path integral. The combined contribution of the 1-loop diagram using the initial action and the tree–level diagram involving the counterterm (6.8) remains finite in the continuum. Of course, while our tuning (??) is good to 1-loop accuracy, if we computed the path integral to higher order in perturbation theory, including the contribution of higher–loop Feynman diagrams, we would have to make further tunings in δm2 (proportional to higher powers of the coupling λ) so as to still retain a finite limiting result.

6.1.2 Dimensional regularization

While the idea of integrating out momenta only up to a cut–off Λ0 is very intuitive, in more complicated examples it becomes very cumbersome to perform the loop integrals over p with a finite upper limit. In studying the local potential approximation in different | | dimensions (such as for the Wilson–Fisher fixed point), we saw that different couplings change their behaviour, becoming either relevant, marginal or irrelevant, as the dimension of the space is changed. Since the irrelevant couplings always have vanishing continuum limits, whereas (untuned) relevant ones diverge in the continuum, this suggests that we can regularize our loop integrals by analytically continuing the dimension d of our space. I stress that this is purely a convenient device for regularizing loop integrals — there is no suggestion that Nature ‘really’ lives in non–integer dimensions. Furthermore, dimensional regularization is only a perturbative regularization scheme: whilst we shall see that it does allow us to regulate individual loop integrals over the full range p [0, ), unlike impos- | |∈ ∞ ing a cut–offor a lattice regularization, dimensional regularization does not provide any definition of a finite–dimensional path integral measure. Despite these conceptual short- comings, its practical convenience makes it an essential tool in perturbative calculations, especially in gauge theories as we shall see later. In d–dimensions, the quartic coupling λ has non–zero mass dimension 4 d, so we − replace λ µ4 dλ where the new λ is dimesionless, and µ is an arbitrary mass scale. → −

– 71 – Thus, in dimensional regularization, we obtain the 1-loop correction to the mass coupling

d d 1 d 1 1 4 d d p 1 4 d Vol(S − ) ∞ p − dp λµ − = λµ − , (6.9) 2 (2π)d p2 + m2 2(2π)d p2 + m2 ! !0 d 1 where Vol(S − ) is the surface volume of a unit sphere in d dimensions. To compute this, note that d (x,x) d d 1 ∞ r2 d 1 π 2 = e− d x = Vol(S − ) e− r − dr Rd 0 ! d 1 ! (6.10) Vol(S − ) ∞ u d 1 = e− u 2 − du 2 !0 and thus d d 1 2π 2 Vol(S − )= d . (6.11) Γ 2 The remaining integral is ' ( d 1 2 d/2 1 2 4 d ∞ p − dp 1 4 d ∞ (p ) − d(p ) µ − = µ − p2 + m2 2 p2 + m2 !0 !0 2 4 d 1 m µ − d 1 d = (1 u) 2 − u− 2 du (6.12) 2 m 0 − ) * ! m2 µ 4 d Γ( d )Γ(1 d ) = − 2 − 2 , 2 m Γ(1) ) * where u := m2/(p2 + m2) and we have used the deﬁntion of the Euler beta–function

1 Γ(s)Γ(t) s 1 t 1 B(s, t)= = u − (1 u) − du. (6.13) Γ(s + t) − !0 Combining the pieces the 1-loop contribution to the mass–shift is

m2λ Vol(Sd 1) µ 4 d Γ( d )Γ(1 d ) m2λ µ 4 d d − − 2 − 2 = − Γ 1 (6.14) 2 2(2π)d m Γ( d ) 2(4π)d/2 m − 2 ) * 2 ) * $ % Expanding around d = 4 by setting d =4 ', this is − m2λ 2 4πµ2 γ + log + (') (6.15) − 32π2 ' − m2 O " $ %# as ' 0, where we’ve used the basic properties → Γ(z + 1) 1 Γ(z)= and Γ(z) γ + (') z ∼ z − O of the Gamma function, with γ 0.577 being the Euler–Mascheroni constant. The di- ≈ vergence we saw as Λ in the cut–oﬀregularization has become a pole in d = 4 in 0 →∞ dimensional regularization. The simplest renormalization scheme is minimal subtraction (MS) — one simply chooses the counterterm m2λ δm2 = MS (6.16) −16π2'

– 72 – so as to remove the purely divergent parts of the loop diagrams. A more common scheme is modiﬁed minimal subtraction (MS) in which one also removes the Euler–Mascheroni constant and the log 4π term, so that

m2λ 2 δm2 = γ + log 4π MS . (6.17) −32π2 ' − $ % We shall see later that these minimal subtraction schemes are rather diﬀerent in character from the on–shell renormalization scheme. Nonetheless, it has become the most frequently used renormalization scheme in the literature.

6.1.3 Renormalization of the quartic coupling The 1-loop correction to the quartic vertex is given by the three Feynman diagrams

which lead to the momentum space integrals

λ2µ4 d ddp 1 1 − + other channels (6.18) 2 (2π)d p2 + m2 (p + k + k )2 + m2 ! 1 2 in dimensional regularization. Because these Feynman diagrams involve two separate ver- tices, they will lead to non–local contributions; indeed, expanding (6.18) in powers of the momenta k generates new derivative interactions such as φ2(∂φ)2. You should check i ∼ that all such derivative terms are irrelevant as expected — they remain ﬁnite in the limit d 4 in dimensional regularization, and would in fact vanish in the continuum limit → Λ had we worked with a hard cut–oﬀ. 0 →∞ The k-independent part of each of the three loop diagrams involves the integral

d 1 2 (d 2)/2 4 d ∞ p − dp 1 ∞ (p ) − 2 µ − = d(p ) (p2 + m2)2 2 (p2 + m2)2 !0 !0 4 d 1 1 µ − 1 d d 1 = u − 2 (1 u) 2 − du (6.19) 2 m 0 − ) * ! 1 µ 4 d Γ 2 d Γ d = − − 2 2 2 m Γ(2) ) * ' ( ' ( Combining all three diagrams, we have a total 1-loop contribution

Vol(Sd 1) 1 µ 4 d Γ 2 d Γ d 3λ2 µ 4 d d 3λ2 − − − 2 2 = − Γ 2 (6.20) 2(2π)d 2 m Γ(2) 2(4π)d/2 m − 2 ) * ' ( ' ( ) * $ % using our result (6.11) for Vol(Sd 1). Setting d =4 ', this becomes − − 3λ2 2 4πµ2 γ + log + (') . (6.21) 32π2 ' − m2 O $ %

– 73 – Consequently, in the MS scheme we choose our counterterm δλ to be

3λ2 2 δλ = γ + log 4π (6.22) 32π2 ' − $ % removing both the pole as ' 0 and the γ log 4π terms. → − The remaining 1-loop correction to the quartic vertex leads to a β-function

∂λ 3λ2 β = µ = (6.23) ∂µ 16π2 agreeing (to this order) with what we found in (??) for the local potential approximation. The fact that the β-function is positive shows that the quartic coupling is (marginally) irrelevant in d = 4. Thus, no matter how small we choose the interaction to be at some scale µ, if it is non–zero, then there is a scale µ# >µin the UV at which the coupling diverges. Of course, our perturbative treatment is not powerful enough to really say what occurs as this happens, but more sophisticated treatments indeed show that λφ4 does not exist as a continuum QFT. Notice that there are no 1-loop diagrams that can contribute to the wavefunction 2 renormalization factor ZΛ here — to obtain a correction to the kinetic term (∂φ) we would need a momentum space diagram involving precisely two external φ fields. With a purely quartic interaction, the only such 1-loop diagram is the one relevant for the mass shift. However, the momentum running around the loop in this diagram does not involve any momentum being brought in by the external fields, and therefore cannot contribute the factor of k2 necessary to be interpreted as a correction to the kinetic term. Put differently, this loop diagram is a purely local contribution, so does not affect any derivative terms. Thus ZΛ = 1 to 1-loop accuracy. There are non-trivial wavefunction renormalization factors beginning at 2-loops.

6.2 One–loop renormalization of QED The action for QED describes a massive charged Dirac spinor coupled to the electromagnetic ﬁeld is 1 S [A,ψ]= ddx F µνF + ψ¯D/ψ + mψψ¯ (6.24) QED 4e2 µν ! " # µ where the covariant derivative in the fermion kinetic term is D/ψ = γ (∂µ +iAµ)ψ, and the Dirac matrices γ obey γ ,γ = +2δ in Euclidean signature. (We’ll understand more µ { µ ν} µν about covariant derivatives when we look at Yang–Mills theory.) In order for the covariant derivative Dµ = ∂µ +iAµ to make sense, the gauge ﬁeld must have mass dimension 1 even in d dimensions. Thus, the electric charge e has dimensions (4 d)/2, so is relevant when − d<4, irrelevant in d>4 and marginal in d = 4, at least to leading order. Introducing an arbitrary mass scale µ as before we introduce a dimensionless coupling g(µ) as

2 4 d 2 e = µ − g (µ) . (6.25)

– 74 – To do perturbation theory, we’d like the kinetic terms to be canonically normalized, so we new old introduce a rescaled photon ﬁeld Aµ = eAµ . In terms of this rescaled ﬁeld the action becomes

new d 1 µν 2 d S [A ,ψ]= d x F F + ψ¯(∂/ + m)ψ +iµ − 2 g ψ¯A/ψ (6.26) QED 4 µν ! " # with g(µ) appearing only in the electron–photon vertex, as befits a coupling. Notice that the new photon field has mass dimension (d 2)/2, just like a scalar field. − 6.2.1 Vacuum polarization: loop calculation Rules Feynman Rules Naive 6.4.1 Feynman Naive 6.4.1 s h ue are rules the ns, are fermio We’ll rules For the now takens, a theory. look fermio this at For the for simplest, rules theory. and this probably for Feynman also the rules the most important determine Feynman to the 1-loop want graphWe determine to want We in QED; the photon self–energy graph, also known as vacuum polarization. h aea hs ie nScin5 h e icsare: pieces new are: The 5. pieces new Section in The 5. given those Section as in same given the those as same the Electromagnetic forces between charged particles are mediated by photon exchange. Quantum corrections modify the form of this propagator, for example by the 1-loop graph = j i, an with = j comes i, an line the with of comes end line Each the of line. end wavy a Each by line. photon wavy the a by denote photon We the denote We • • ecluae h rnvrephoton transverse the photon calculated transverse We the . ! A of calculated We . component ! A of the us telling component the index us 3 , 2 , 1 telling index 3 , 2 , 1 Aρ A j p i p i j p i p tr i tr ij δ = ij ij δ D = contributes ij D and contributes is and it (6.33): in is it propagator (6.33): in propagator 2 ! p − " i + 2 2 ! p p − " i + 2 p q " | | ! " | | !

+ ρσ i where a virtual i e e−i pair is formed andi then reabsorbed. If we let Πloop(q) denote the otat ihthe with the contracts γ 1PI with on contributions index contracts The γ to. the on γ photonie index self–energy, The . γ contributes thenie at one loop ordercontributes the vertex onlyThe contribution vertex The ρσ − − • • to Πloop is from the diagram above. The Feynman rules following from (6.26) give ne ntepoo line. photon the on line. index photon the on index d ρ σ ρσ 4 d 2 d p tr ( ip/ + m)γ ( i(p/ /q)+m)γ Π (q)=µ − g − − − , (6.27) 1 loop (2π)d (p2 + m2)((p q)2 + m2) y x y x ' ( h o-oa neato hc,i oiinsae sgvnby given is by space, given is position in ! space, which, position in interaction − which, non-local The interaction non-local The and we must now evaluate this integral. • • 0 0 2 0 0 0 2 0 To begin, note that ) y x ( δ ) γ e ) ( i y x ( δ ) γ e ( i − of − factor a of contributes factor a contributes 1 y ! 1 x ! 1 π 4 1 1y ! x ! dπ x4 = | − | = | − | (6.28) AB B A A − B [(1 x)A + xB]2 okn in working r fo in paid working we’ve r fo price paid the is we’ve This − price " the messy. is # ! This rather 0 are − messy. rules rather are Feynman rules These Feynman These sit something into ns so something that expressio we into can these ns combine massage the expressio two can these propagators we that massage in( show 6.27 can ) we now as that We’ll show gauge. now We’ll Coulomb gauge. Coulomb instantaneous g endin ff o 1 the instantaneous g with endin ff start o dx the Let’s with start invariant. Let’s 1 Lorentz and invariant. dx simple Lorentz more and much simple more much 2 = 2 2 2 2 0 [(p2 + m2)(1 x) + ((p q)2 + m2)x] 0 [p + m 2xp q + q x] opnn fteguefil,w ol r to try could we to try field, ! could gauge we the of field, − gauge −component the 0 of A the ! from component 0 A comes it − the · from Since comes it (6.29)interaction. Since interaction. 1 dx ic hc ilcpueti em nfc,it fact, In term. it this fact, In capture term. will this which capture piece 00 will D a which = include piece to 00 D a 2 2 propagator include 2 to the 2 . redefine propagator the redefine 0 [(p qx) + m + q x(1 x)] t ut ieyi hsfr:i elo nmmnu pc,w have we space, have we momentum in space, look we if ! momentum in form: − look this we in if nicely form: − quite this in fits nicely quite fits If we now change variables p p = p + qx then (6.27) becomes (dropping the prime) → # ) y x ( ip 4 ) y x ( ip 0 4 0 0 ρ 0 σ − · e p d ddp − 1 · ) tre y ( p i(d p/x ( +δ /qx)+m) )γy ( i(p/x ( /qδ (1 x)) + m)γ ρσ 4 d 2 − − − − (6.83) (6.83) Π1 loop(q)=µ − g d = dx − = 2 − 2 , 2 ! p 4 ) π (2(2 π) 02 ! p y ! '4 ) x ! π π (2 4 [y ! p +∆x ! ] π 4 ( | | ! # ! | | | − | # | − | (6.30) tnpoaao by propagator by oton ph propagator transverse oton the ph with transverse the interaction with non-local the interaction combine non-local can the we so combine can we so enn e htnpropagator photon new a propagator photon defining new a defining – 75 – i i =0 ν µ, =0 ν µ, + + 2 ! p 2 ! p | | | | p p i j p i p  i  (6.84) (6.84) j i )= p ( D )= p ( ν µ D =0 j = ν , =0 i = =0 µ j = ν , =0 i = µ δ  ij δ ν µ  " " " "2 ! p − ij " i + 2 2 ! p p   − " i + 2 p   " | | ! " | |   !   otherwise otherwise 0 0       ne,with index, 3 , 2 , 1 , with =0 ν index, µ, 3 a , 2 , 1 , carries =0 ν now µ, a line carries photon now wavy line the   photon wavy propagator, the   this With propagator, this With need w no We need w no interaction. We instantaneous interaction. the of care instantaneous taking the of care component 0 taking = µ extra component the 0 = µ extra the hc correctly which µ γ ie correctly by which µ γ replaced ie gets by above i γ replaced ie gets the above i γ slightly: ie the vertex our slightly: change to vertex our change to − − − − ic nteisatnosinteraction. instantaneous interaction. the in piece 2 ) instantaneous 0 γ e the ( in the for piece 2 ) 0 γ e ( accounts the for accounts

4 – 141 – – 141 – where ∆:= m2 + q2x(1 x). − The next step is to perform the traces over the Dirac matrices. We’ll do this treating the Dirac spinors as having 4 components29 as appropriate for our ﬁnal goal of d = 4. Thus tr(γργσ) = 4 δρσ (6.31) tr(γµγργνγσ) = 4(δµρδνσ δµνδρσ + δµσδνρ) − in Euclidean signature, so that tr ( i(p/ + /qx)+m)γρ( i(p/ /q(1 x)) + m)γσ − − − − = 4 [ (p + qx)ρ(p q(1 x))σ +(p + qx) (p q(1 x))δρσ (6.32) ' − − − (· − − (p + qx)σ(p q(1 x))ρ + m2δρσ . − − − Thus the loop integral becomes + d 1 ρσ 4 d 2 d p 1 Π1 loop(q)=4µ − g d dx 2 2 (2π) 0 [p +∆] ! ! (6.33) [ (p + qx)ρ(p q(1 x))σ +(p + qx) (p q(1 x))δρσ × − − − · − − (p + qx)σ(p q(1 x))ρ + m2δρσ , − − − which would be quadratically divergent in d = 4. + We’re now ready to perform the loop integral. Observing that whenever d N, any ∈ term involving an odd number of powers of momentum would vanish, we drop these terms. For the same reason, we replace 1 (p2)2 pµpν δµνp2 and pµpνpρpσ [δµν δρσ + δµρ δνσ + δµσ δνρ] → d → d(d + 2) where the tensor structure is ﬁxed by Lorentz invariance and permutation symmetry, and the numerical factors are determined by contracting both sides with metrics. Finally, since the integrand now depends only on p2, the angular integrals may be performed trivially to obtain d d 1 d p d 1 p − dp 1 2 d 1 2 = Vol(S − ) = (p ) 2 − d(p ) (6.34) (2π)d (2π)d (4π)d/2 Γ(d/2) as in section 6.1.2. Thus (6.33) becomes 2 ρσ 4 d g Π (q)=4µ − 1 loop d d (4π) 2 Γ( ) 2 (6.35) 1 2 2 ρσ ρ σ 2 2 ρσ ∞ 2 2 d 1 p (1 d ) δ + (2q q q )x(1 x)+m δ dx d(p )(p ) 2 − − − − . × [p2 +∆]2 !0 !0 , - To go further, we use the integrals d 2 d 1 2 d d (p ) 2 1 − 2 Γ(2 )Γ( ) ∞ d(p2) − = − 2 2 (p2 +∆)2 ∆ Γ(2) 0 $ % ! d (6.36) 2 d 1 d d (p ) 2 1 − 2 Γ(1 + )Γ(1 ) ∞ d(p)2 = 2 − 2 (p2 +∆)2 ∆ Γ(2) !0 $ % 29In certain supersymmetric theories, it is often convenient to work instead with d–dimensional spinors, which is known as dimensional reduction, rather than dimensional regularization.

– 76 – that can be evaluated using the substition u =∆/(p2 +∆) and the deﬁnition of the Euler B–function, just as we did in section 6.1.2. Using these integrals to evaluate (6.35), altogether one ﬁnds that the 1-loop contribution to vacuum polarization is given by

4g2µ4 d d Πρσ (q)= − Γ 2 1 loop − (4π)d/2 − 2 $ % 1 δρσ( m2 + x(1 x)q2)+δρσ(m2 + x(1 x)q2) 2x(1 x)qρqσ dx − − − − − × 2 d !0 " ∆ − 2 # := (q2 δρσ qρqσ) π (q2) , − 1 loop (6.37) 2 where in the last line we have deﬁned π1 loop(q ) to be the dimensionless quantity

2 d 1 2 d/2 8g (µ)Γ 2 µ2 − π (q2)= − 2 dxx(1 x) . (6.38) 1 loop − (4π)d/2 − ∆ ' ( !0 $ % and we recall that ∆= m2 + q2x(1 x) (and that d<4). − 6.2.2 Counterterms in QED The ﬁrst thing to notice about our result (6.37) is that it, if all couplings remain constant, it will diverge in the physically interesting dimension because Γ(2 d ) has a pole when − 2 d = 4. To obtain a ﬁnite continuum result we must tune the initial couplings in the action, which as always we do by introducing counterterms. For QED the counterterms are

1 SCT[A,ψ,']= ddx δZ F µνF + δZ ψ¯D/ψ + δm ψψ¯ . (6.39) 3 4 µν 2 ! " # Adding these to the QED action (6.26) allows us to tune the initial values of the photon and electron wavefunction renormalizations, and the electron mass. The labels (δZ3,δZ2) for the photon and electron wavefunction renormalization factors are conventional. The fact that the entire kinetic term for the electron, including the gauge covariant derivative operator D/ = ∂/+ieA/, receives only one counterterm assumes that the regularized path integral preserves gauge invariance: Provided our regularized path integral is indeed gauge invariant, then ∂ψ/ and iA/ψ cannot appear independently. Gauge invariance is main- tained in lattice regularization, but would fail if one simply imposed a cut–off Λ0, because the requirement that fields only contain Fourier modes with p Λ is not preserved under | |≤ 0 the gauge transformation ψ eiχ(x)ψ, even if it is true of ψ and χ separately.30 The desire → to maintain manifest gauge invariance was one of the main motivation to use dimensional regularization in the first place, and our result (6.37) vindicates this decision: we see that the 1-loop correction Πρσ (q2) is proportional to (q2δρσ qρqσ), so 1 loop − ρσ qρΠ1 loop(q) = 0 . (6.40)

30In fact, the conceptually simple idea of integrating over modes only up to a cut–oﬀ can be done in gauge theory, but requires the introduction of a fair amount of technology beyond the scope of this course; see e.g. K. Costello’s book cited in the introduction.

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To fix the counterterm δZ3, note that the loop diagram (6.37) diverges in the physical dimension since Γ(2 d/2) has a pole as d 4 . Indeed, setting d =4 ' we have − → − − 2 1 2 = j i, an with comes line the of end d 4 Each g (µ) line. 2 wavy a 4π by µ photon the denote We 2 → π1 loop(q ) 2 dxx(1 x) γ + ln + (') (6.41) −→ − 2π 0 − ' − ∆ O • ! ! $ % ecluae h rnvrephoton transverse the where again calculated γ is We the. Euler–MascheroniA of constant. The contribution component the us telling index 3 , 2 , 1 p p i j i δ = tr D contributes and is it (6.33): in propagator 2 ! p − ij " i + 2 p ij " | | ! from 1 δZ F ρσF must remove this pole, and in the MS scheme we’d set − 4 3 ρσ g2(µ) 2 δZ = γ + ln 4π (6.42) i i 3 − 12π2 ' − otat ihthe with contracts γ on index The . γ ie $ % contributes vertex The so as also to remove the contribution ( γ + ln 4π). (To check that this counterterm does − 1 ∝ − • indeed cancel the pole, note that dxx(1 x)= 1 .) Thus the total contribution to the 0 − 6 effective photon self–energy at one loop is & line. photon the on index Πρσ(q)= q2δρσ qρqσ π(q2) (6.43) − where ' ( g2(µ) 1 m2 + x(1 x)q2 π(q2) = + dxx(1 x) ln − (6.44) y x 2π2 − µ2 !0 " # h o-oa neato hc,i oiinsae sgvnby given is in the MS scheme. space, position in which, interaction non-local The Strikingly, the loop correction to the photon propagator has created the logarithm31 • 0 0 2 0 ) y ln m2 + x (1( δ x) q) 2 γ e ( i − − of factor a contributes in momentum space. This is quite unlike. y ! anythingx ! you’veπ +4 seen at tree–level, where Feynman diagrams are always rational functions of momenta, but it is very similar to the logarithms we obtained from integrating out fields| in lower− dimensional| examples. In the present case, the logarithm has a branch cut in the region m2 + x(1 x)q2 < 0, or in other words when okn in working r fo paid we’ve price the is This messy. − rather are rules Feynman These x(1 x) q2 >m2 . (6.45) − Lorentz sit something into ns expressio back in these Lorentzian signature. massage Since can x(1 x we ) 1/4 for that x [0, 1], the show smallest now value of q2 We’ll gauge. Coulomb − ≤ ∈ instantaneous g endin ff o the at which with this branch start cut is reached Let’s is invariant. Lorentz and simple more much 4 2 qLorentz =4m (6.46) 31Actually, it has produced a certain integral of a logarithm involving x(1 x). This integral can be opnn fteguefil,w ol r to try could we field, gauge the of component 0 A the − from comes it Since interaction. explicitly computed in terms of dilogarithms, but we won’t need to know the result. ic hc ilcpueti em nfc,it fact, In term. this capture will which piece 00 D a include to propagator the redefine t ut ieyi hsfr:i elo nmmnu pc,w have we space, momentum in look we if form: this in nicely quite fits – 78 –

) y x ( ip 4 0 0 − · e p d ) y x ( δ (6.83) = − 2 ! p 4 ) π (2 y ! x ! π 4 | | # | − | tnpoaao by propagator oton ph transverse the with interaction non-local the combine can we so enn e htnpropagator photon new a deﬁning i =0 ν µ, + 2 ! p | | j p i p i  (6.84) =0 j = ν , =0 i = µ δ  )= p ( ν µ D " " 2 ! p − ij " i + 2 p   " | | !   otherwise 0    ne,with index, 3 , 2 , 1 , =0 ν µ, a carries now line photon wavy the   propagator, this With need w no We interaction. instantaneous the of care taking component 0 = µ extra the hc correctly which µ γ ie by replaced gets above i γ ie the slightly: vertex our change to − − ic nteisatnosinteraction. instantaneous the in piece 2 ) 0 γ e ( the for accounts

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integrals weighted by e− . y x y x Rules Feynman Naive 6.4.1 h o-oa neato hc,i oiinsae sgvnby given is We’vey by space, justx given found is position that by in space, given is which, position in space, interaction which, position in non-local interaction which, The non-local interaction The non-local The ρ • • • 0 0 2 0 ρ 0 20 ρ 2 0 q0 qσ 0 2 2 0 ) y x ( δ ) Π ) γ σe =( y i q x δ( δ σ ) ) γ e ( y i π(x q( δ )) γ e ( i (6.48) − − of − factor a q2 − of factor contributes a of factor contributes a contributes y ! x ! π 4 y ! $ x ! π 4 %y ! x ! π 4 where, to 1-loop accuracy,| − π| (q2) is given| − by| (6.44)| in− the| MS scheme. In fact, one can okn in working r fo in paid working showr we’ve fo in that price paid (6.48 the working is r ) we’ve holds fo This price to paid all the orders is messy. we’ve in This price the rather the electromagnetic are is messy. rules This rather are coupling messy. 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Therefore the exact photon propagator may be written opnn fteguefil,w ol r to try could to we try field, could gauge to we the try of field, could gauge we the component of field, 0 A the gauge the from component of 0 A comes the it from component Since 0 A comes the it from interaction. Since comes it interaction. 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(2 π 4 y ! x ! π 4 | | # | − | by summing this geometric| | series. # | | | − | # | − | tnpoaao by propagator by oton ph propagator by Let’soton transverse think ph the about propagator with whatoton transverse ph the this interaction exact with transverse propagator the non-local interaction the with tells us combine non-local about interaction can the the we eso ffective combine non-local can the action we so for combine the can we so photon that we’d obtain after performing propagator the path photon integral new propagator a over photon thedefining electron new propagator a field. photon defining new The a defining classical Maxwell action is i i i =0 ν µ, 1 1 + =0 ν µν µ, d =0 ν 2 µ, µ ˜ν 2 ν+ ˜µ + ˜ ˜ d F Fµν d x = ! p i q A ( q) ! p q A ( q) 2 i ! p qµAν(q) qνAµ(q) d q 4 4 | −| − | −|  − − ! j p i p ! j p i i p p i p | |  i  (6.50) (6.84) (6.84) (6.84) ) µ ν )= p ( j ν i µ *D ) )= p ( ν µ D *)= p ( D =0 j = ν , =0 i =0 = j µ = ν , =0 i =0 1= j µ = ν ij , δ =0 i q= qµ  ij δ  δ  ν µ =2 q2 δµν 2 2   A˜ ( q2 )A˜ 2 (q)d ij dq 2  " " " " " ! p − "" i + ! p 2p −  µ " i + ! p ν p −   " i + p  " | 2 | ! " −| q|  ! −" | |  !   otherwise otherwise ! 0 $ %0 otherwise 0 0 when written in momentum space. Note that  ∆µν is indeed the inverse of this momentum       ne,with index, 3 , 2 , with 1 , =0 ν index, space3 µ, , 2 a , with 1 kinetic, =0 carries termν index, 3 µ, , now 2 a for, 1 , line polarizations=0 carries ν µ, photon now a transverse line wavy carries the  photon now to qµ line . wavy Thus, propagator, the   photon the this exact wavy With photon propagator, the   this propagatorWith propagator, this With need w no We need w no would interaction. We need followw no from interaction. We instantaneous a momentum the of care interaction. instantaneous space the taking of quadratic care instantaneous term component the taking 0 of = µ care extra component taking 0 the = µ extra component 0 the = µ extra the µ µ i i µ ν hc correctly which γ ie correctly by which γ ie replaced correctly by gets (2)which ˜µ γ above ie replaced 1γ ie by gets the 2above replaced γ 2 ie slightly: gets µν the q vertex above qi γ our ie ˜ slightly: the change ˜ vertex to our d slightly: change vertex to our change to Seff [A]= [1 + π(q )] q δ Aµ( q)Aν(q)dq (6.51) − − 4− − 2 − 0 − q2 2 − 0 − 2 0 ic nteisatnosinteraction. interaction. ! instantaneous the in interaction. piece $ instantaneous ) the γ e ( in the %piece instantaneous for ) the γ e ( in accounts the piece for ) γ e ( accounts the for accounts

– 79 – 4 – 141 – – 141 – – 141 – where the electron loop eﬀects are incorporated in the factor [1 π(q2)]. The part π(0) − of π(q2) that is independent of q2 just provides an overall factor multiplying the classical action, and so corresponds straightforwardly to the position space term

2 1 2 (2) 1+π(0) µν 4 1 g m µν 4 Seff [A]= F (z)Fµν(z)d z = 1+ 2 dxx(1 x) ln 2 F Fµν d z 4 4 2π 0 − µ ! " ! # ! (6.52) where the second expression uses our result (6.44) for the 1-loop contribution to π(0) in the MS scheme in d = 4. As expected, this is a contribution to photon wavefunction renormalization. Expanding π(q2) as a power series in q2/m2 shows that the remaining, q2-dependent terms correspond to an infinite series of higher derivative interactions of the n µν n schematic form ∂ F ∂ Fµν. All these higher derivative couplings are irrelevant in d =4 and may be expected to be small at energies much lower than the electron mass. (In particular, the original loop integrals for these terms were finite in d = 4.)

6.2.3 The β-function of QED Knowing the effective action allows us to read offthe β-function for the electric charge. To relate the photon kinetic term – involving wavefunction renormalization – to the β function old new for the electromagnetic coupling e, we first undo our rescaling Aµ = eAµ and work back old in terms of the original gauge field Aµ . Then the quadratic term (6.52) in the effective action becomes 1 1 1 1 m2 S(2)[Aold]= + dxx(1 x) ln F µνF d4z. (6.53) eff 4 g2 2π2 − µ2 µν " !0 # ! In this way, we can view the vacuum polarization as a quantum correction to the value of 1/g2. Therefore ∂ 1 2 1 1 µ = β(g)= dxx(1 x) (6.54) ∂µ g2 −g3 −π2 − $ % !0 so that the β function for g(µ) is

∂g g3 β(g) := µ = . (6.55) ∂µ 12π2

Solving this g(µ) gives 1 1 = C ln µ, (6.56) g2(µ) − 6π2 or equivalently g2(µ ) g2(µ)= # (6.57) 1 g2(µ )/6π2 ln(µ/µ ) − # # which ﬁxes the coupling g(µ) at arbitrary scales in terms of its value at some arbitrary reference scale µ#. For example, we could choose µ# to be the scale of the (physcial) electron mass, at which g2(m )/4π 1/137 is found experimentally. e ≈ As for the quartic coupling of λφ4 theory, the fact that the β-function in QED is positive shows that the electromagnetic coupling is marginally irrelevant in d = 4, at least near the

– 80 – Gaussian critical point g = 0. Consequently, it is believed that pure QED does not exist as a continuum QFT in four dimensions! This fact has no immediate phenomenological consequences, because taking g2(m )/4π 1/137 from experiment, the scale µ at which e ≈ the coupling (6.57) diverges is fixed to be 10286 GeV, well beyond any point at which ∼ we claim to even vaguely trust QFT as a description of Nature. Nonetheless, the lesson from the β–function is that pure QED can only exist as a low–energy effective theory — in our world it unifies with the weak interactions at around 100 GeV, where the physics ∼ of non–Abelian gauge theories comes into play. Finally, I wish to point out a small peculiarity inherent in the MS renormalization scheme. In studying the local potential approximation in section 5.3.2 we discovered that the quantum contributions to the β-functions due to high energy states circulating in the loop became suppressed at scales much lower than the masses of these states. This made good sense: the heavy states decoupled from low–energy physics. However, in the MS scheme the β-function (6.54) shows no suppression at any scale and the coupling (6.57) still runs even at scales µ m . There is nothing wrong with this: as µ 0 there , e → is a balance in the effective action (6.51) between e2(µ) 0 and the growing effect of → the loop contribution ln(1/µ2) so that the actual observable physics does remain ∝ →∞ constant. However, it’s strange to have loop effects dominating the classical ones, so for some purposes it is better to proceed as follows. We consider two different theories. One includes the electron and is valid at scales µ m , while the second does not and is valid ≥ e at scales µ m . Physical quantities in the two theories are matched at µ = m . The ≤ e e effects of the electron (or other heavy particle) are then manually frozen out as we continue on to our other theory at lower scales. In particular, since pure Maxwell theory is free, for any µ m the fine structure constant α(µ)=g2(µ)/4π will remain frozen at its value ≤ e 1/137 at the electron mass. ≈ We could avoid the need to decouple by hand had we used a renormalization scheme, such as on–shell renormalization, that fixes the value of the counterterm δZ3 in terms of 2 2 2 π1 loop(q ) at some definite scale q = µ# . That is, we set g2 1 2 4πµ2 δZ = dxx(1 x) γ + ln (6.58) 3 2 2 2 −2π 0 − ' − m x(1 x)µ ! $ − − # % In this scheme, the β-function instead becomes ∂g g3 1 x(1 x)µ 2 β(g)=µ = dxx(1 x) ln − # (6.59) 2 2 2 ∂µ 2π 0 − m x(1 x)µ ! − − # in four dimensions. This does indeed approach zero when µ m . However, for most # , e purposes (particularly in more complicated theories such as Yang–Mills theory, or the full Standard Model) the MS scheme is so convenient that it’s worth paying the price of having to decouple the electron by hand.

6.2.4 Physical interpretation of vacuum polarization When light propagates through a region containing an insulating medium with no relevant degrees of freedom, on general grounds we expect the low–energy eﬀective ﬁeld theory to

– 81 – be governed by an action that modifies the coefficients of the electric and magnetic fields in the usual Maxwell action by terms that respect the microscopic symmetries of the medium. In the present case, the medium is simply the vacuum itself ! Since the vacuum is Lorentz invariant, these modifications must be proportional to the Lorentz invariant combination E2 B2 = F µνF . In (6.52) we see explicitly that this is true. If we place a medium such − µν as water in the presence of an electric field, it will become polarized due to the large dipole moment of the H2O molecules. Likewise, at distances ! 1/m the vacuum itself becomes a dielectric medium in which virtual electron–positron pairs form dipoles, polarizing the vacuum. The first effect is that vacuum polarization leads to a measureable change in the Coulomb potential. Recall that in the non-relativistic limit (in Lorentzian signature), the Fourier transform of the (Feynman gauge) photon propagator δρσ/q2 is the Coulomb potential V (r)=e2/4πr, as I hope is familiar from Rutherford scattering. Let’s compute the 1-loop quantum corrections to this result. We consider a scattering process in which 1 two spin 2 charged particles interact electromagnetically. The Feynman diagrams

make contributions of the form

e1e2 4 2 µ S(1, 2 1#, 2#)=− δ (p + p p p ) 1+π(q ) u¯ γ u u¯ γ u (6.60) → 4π2q2 1 2 − 1! − 2! 1! 1 2! µ 2 . + where u1,2 are the on–shell Dirac wavefunctions of the incoming particles andu ¯1!,2! are the on–shell Dirac wavefunctions for the outgoing particles. The 1–loop diagram modiﬁes the classical answer by the factor [1 + π(q2)]. Non–relativistically, the energy transfer q0 q ,| | and

µ iδm1,m 1! u¯1! γ u1 − , ≈ / 0 0 where the factor of δmm! enforces that the spins of the two particles should be aligned. Thus, in the non–relativistic limit we have

e1e2 4 2 S(1, 2 1#, 2#) − δ (p1 + p2 p1 p2 ) 1+π(q ) δm1m δm2,m . (6.61) → ≈ 4π2q2 − ! − ! 1! 2! . + We can compare this result to the calculation of scattering in non–relativistic quantum mechanics for a potential V (r) in the Born approximation, where

e1e2 4 3 iq r SBorn(1, 2 1#, 2#)=− δ (p1 + p2 p1 p2 ) δm1,m δm2,m d r V (r)e− · (6.62) → 4π2 − ! − ! 1! 2! ! This shows that the 1-loop corrected amplitude looks just like the amplitude we would find from Born level scattering offa modified classical potential V1(r) whose Fourier transform

– 82 – 2 2 is e1e2[1 + π(q) ]/q , or in other words

2 e1e2 3 iq r 1+π(q ) V (r)= d q e · . (6.63) 1 (2π)3 q2 ! To lowest order in π(q2), this is just the potential energy

ρ (x) ρ (y) V ( r )= d3x d3y 1 2 (6.64) | | x y + r ! | − | produced at point r from the electrostatic interaction of two charge distributions ρ1(x) and ρ2(y) deﬁned by 3 e1,2 3 iq r 2 ρ (r)=e δ (r)+ d q e · π(q ) . (6.65) 1,2 1,2 2(2π)3 ! In particular, we see that

3 1 3 3 iq r 2 d r ρ (r)=e 1+ d r d q e · π(q ) 1,2 1,2 2(2π)3 ! " ! # 1 (6.66) = e 1+ π(0) 1,2 2 " # = e1,2 where the last line uses the on–shell renormalization scheme result (??). Thus the total charge seen by the long–range part of the Coulomb potential is the same as the charge governing the interaction in Feynman diagrams. After a contour integration using the on–shell scheme result for π(q2), one ﬁnds

2 1 ρ1,2(r) 3 e mr mr = (1+L)δ (r) 3 3 dxx(1 x) 1+ exp − (6.67) e1,2 − 8π r 0 − , x(1 x)- / x(1 x)0 ! − − where L is the integral 1 1

3 1 e 3 1 mr mr L = 3 d r 3 dxx(1 x) 1+ exp − . (6.68) 8π r , 0 − , x(1 x)- / x(1 x)0- ! ! − − This integral diverges at short distances r 10. The interpretation1 is that the bare point → charge of strength e1(1 + L) sitting at r = 0 polarizes the vacuum, attracting virtual particles of opposite charge towards it and repelling their antiparticles as they circulate around the loop. Thus the bare charge is partially shielded and we see only a ﬁnite charge e1.

– 83 –