Advanced Chapter 4 Radiative Corrections

Jorge C. Rom˜ao Instituto Superior T´ecnico, Departamento de F´ısica & CFTP A. Rovisco Pais 1, 1049-001 Lisboa, Portugal

Fall 2013 Lecture 6

QED

Vacuum Polarization

Self-energy

The Vertex

Lecture 6

Jorge C. Rom˜ao TCA-2012 – 2 QED Renormalization at one-loop

❐ We will consider the theory described by the Lagrangian Lecture 6 QED Renormalization 1 1 L = − F F µν − (∂ · A)2 + ψ(i∂/ + eA/ − m)ψ . Vacuum Polarization QED 4 µν 2ξ Self-energy

The Vertex ❐ The free are

i β α ≡ S0 (p) p/ − m + iε F βα p  βα

g k k µ ν − i µν + (1 − ξ) µ ν k2 + iε (k2 + iε)2 k  

k k 1 k k = − i g − µ ν + ξ µ ν µν k2 k2 + iε k4   

0 ≡GFµν(k)

Jorge C. Rom˜ao TCA-2012 – 3 QED Renormalization at one-loop

Lecture 6 ❐ The only vertex is given by QED Renormalization

Vacuum Polarization

Self-energy The Vertex e− γ +ie(γµ)βα e = |e| > 0

e−

❐ We will now consider the one-loop corrections to the propagators and to the vertex. We will work in the Feynman gauge (ξ = 1).

Jorge C. Rom˜ao TCA-2012 – 4 Vacuum Polarization

❐ In first order the contribution to the is given by the Lecture 6 diagram QED Renormalization Vacuum Polarization p Self-energy

The Vertex k k

p + k

❐ We write it in the form

′ ′ (1) 0 µ ν 0 Gµν (k) ≡ Gµµ′ (k) i Π (k)Gν′ν (k)

where d4p Tr[γ (p/ + m)γ (p/ + k/ + m)] i Π (k) = − (+ie)2 µ ν µν (2π)4 (p2 − m2 + iε)((p + k)2 − m2 + iε) Z d4p [2p p + p k + p k − g (p2 + p · k − m2) = − 4e2 µ ν µ ν ν µ µν (2π)4 (p2 − m2 + iε)((p + k)2 − m2 + iε) Z

Jorge C. Rom˜ao TCA-2012 – 5 Vacuum Polarization

❐ Simple power counting indicates that this integral is quadratically divergent Lecture 6 for large values of the internal loop momenta. In fact the divergence is QED Renormalization

Vacuum Polarization milder, only logarithmic. Self-energy ❐ The integral being divergent we have first to regularize it and then to define The Vertex a renormalization procedure to cancel the infinities. For this purpose we will use the method of dimensional regularization. ❐ For a value of d small enough the integral converges. If we define ǫ = 4 − d, in the end we will have a divergent result in the limit ǫ → 0. ❐ We note that µ is a parameter with dimensions of a that is introduced to ensure the correct dimensions of the coupling constant in dimension d, − ǫ 4 d ǫ 2 that is, [e] = 2 = 2 . We take then e → eµ ❐ We get therefore

ddp [2p p + p k + p k − g (p2 + p · k − m2)] i Π (k, ǫ) = − 4e2 µǫ µ ν µ ν ν µ µν µν (2π)d (p2 − m2 + iε)((p + k)2 − m2 + iε) Z ddp N (p, k) = − 4e2 µǫ µν (2π)d (p2 − m2 + iε)((p + k)2 − m2 + iε) Z Jorge C. Rom˜ao TCA-2012 – 6 Vacuum Polarization

❐ We have defined a shorthand for the numerator Lecture 6 QED Renormalization 2 2 Nµν (p, k) = 2pµpν + pµkν + pν kµ − gµν (p + p · k − m ) Vacuum Polarization

Self-energy The Vertex ❐ To evaluate this integral we first use the Feynman parameterization to rewrite the denominator as a single term. For that we use (see Appendix)

1 1 dx = 2 ab 0 [ax + b(1 − x)] Z to get

1 d 2 ǫ d p Nµν (p, k) i Πµν(k, ǫ)= − 4e µ dx d 2 2 2 2 2 0 (2π) [x(p + k) − xm + (1 − x)(p − m ) + iε] Z Z 1 ddp N (p, k) = − 4e2 µǫ dx µν d 2 2 2 2 0 (2π) [p + 2k · px + xk − m + iε] Z Z 1 ddp N (p, k) = − 4e2 µǫ dx µν (2π)d 2 2 2 2 Z0 Z [(p + kx) + k x(1 − x) − m + iε]

Jorge C. Rom˜ao TCA-2012 – 7 Vacuum Polarization

❐ For dimension d sufficiently small this integral converges and we can change Lecture 6 variables QED Renormalization

Vacuum Polarization p → p − kx Self-energy The Vertex ❐ We then get

1 d 2 ǫ d p Nµν(p − kx,k) i Πµν(k, ǫ) = −4e µ dx d 2 2 0 (2π) [p − C + iǫ] Z Z where

C = m2 − k2x(1 − x)

❐ Nµν is a polynomial of second degree in the loop momenta. However as the denominator in only depends on p2 is it easy to show that

ddp pµ = 0 (2π)d [p2 − C + iǫ]2 Z ddp pµpν 1 ddp p2 = gµν (2π)d [p2 − C + iǫ]2 d (2π)d [p2 − C + iǫ]2 Z Z Jorge C. Rom˜ao TCA-2012 – 8 Vacuum Polarization

❐ This means that we only have to calculate integrals of the form Lecture 6 QED Renormalization ddp (p2)r Vacuum Polarization Ir,m = d 2 m Self-energy (2π) [p − C + iǫ] Z − The Vertex dd 1p (p2)r = dp0 (2π)d [p2 − C + iǫ]m Z Z ❐ To make this integration we will use integration in the plane of the complex variable p0 as described in the figure. The deformation of the contour corresponds to the so called Wick rotation,

Im p0

x

0 x Re p

Jorge C. Rom˜ao TCA-2012 – 9 Vacuum Polarization

❐ This means the replacement Lecture 6 QED Renormalization +∞ +∞ 0 0 0 Vacuum Polarization p → ipE ; → i dpE −∞ −∞ Self-energy Z Z The Vertex 2 0 2 2 0 2 2 2 0 and p =(p ) −|~p| = −(pE) −|~p| ≡ −pE, where pE =(pE, ~p) is an euclidean vector, that is

2 0 2 2 pE =(pE) + |~p|

❐ We can then write (see the Appendix for more details),

r d 2 r−m d pE pE Ir,m = i(−1) (2π)d [p2 + C]m Z E where we do not need the iǫ anymore because the denominator is positive definite (C > 0) ❐ The case when C < 0 is obtained by analytical continuation of the final result

Jorge C. Rom˜ao TCA-2012 – 10 Vacuum Polarization

❐ To proceed with the evaluation of Ir,m we write, Lecture 6

QED Renormalization d d−1 Vacuum Polarization d pE = dp p dΩd−1 Self-energy Z Z The Vertex 0 2 2 where p = (pE) + |~p| is the length of of vector pE in the euclidean space with d dimensions and dΩ − is the solid angle p d 1 ❐ We can show (see Appendix) that

d π 2 dΩd−1 = 2 Γ( d ) Z 2 ❐ The p integral is done using the result,

∞ p p+1 x − − Γ( ) dx = π(−1)q 1 ap+1 nq n (xn + an)q n sin(π p+1 )Γ( p+1 − q + 1) Z0 n 2 and we finally get r−m d d d (−1) Γ(r + ) Γ(m − r − ) r−m+ 2 2 2 Ir,m = iC d d 2 Γ(m) (4π) Γ( 2 ) Jorge C. Rom˜ao TCA-2012 – 11 Vacuum Polarization

❐ Note that the integral representation of Ir,m is only valid for d < 2(m − r) Lecture 6 to ensure the convergence of the integral when p → ∞. QED Renormalization Vacuum Polarization ❐ However the final form of can be analytically continued for all the values of d Self-energy

The Vertex except for those where the function Γ(m − r − d/2) has poles, which are, d m − r − 6= 0, −1, −2,... 2

❐ For the application to dimensional regularization it is convenient to write Ir,m after making the substitution d = 4 − ǫ. We get

ǫ r−m 2 ǫ ǫ (−1) 4π − Γ(2 + r − ) Γ(m − r − 2 + ) I = i C2+r m 2 2 r,m (4π)2 C Γ(2 − ǫ ) Γ(m)   2 that has poles for m − r − 2 ≤ 0 (see Appendix).

❐ We now go back to calculate Πµν. First we notice that after the change of variables we get, neglecting linear terms that vanish,

2 2 2 2 2 2 Nµν (p−kx,k) = 2pµpν +2x kµkν −2xkµkν −gµν p + x k − xk − m

Jorge C. Rom˜ao TCA-2012 – 12 Vacuum Polarization

❐ Therefore Lecture 6 d ǫ d p Nµν(p − kx,k) QED Renormalization Nµν ≡µ (2π)d [p2 − C + iǫ]2 Vacuum Polarization Z Self-energy 2 ǫ 2 2 ǫ The Vertex = −1 g µ I + − 2x(1 − x)k k +x(1 − x)k g + g m µ I d µν 1,2 µ ν µν µν 0,2   " # ❐ Using now the results for Irm we can write

ǫ i 4πµ2 2 Γ( ǫ ) µǫI = 2 0,2 16π2 C Γ(2)   i C = ∆ − ln + O(ǫ) 16π2 ǫ µ2   ❐ In doing this we have used the expansion of the Γ function ǫ 2 Γ = − γ + O(ǫ) 2 ǫ   γ being the Euler constant and we have defined 2 ∆ ≡ − γ +ln4π ǫ ǫ

Jorge C. Rom˜ao TCA-2012 – 13 Vacuum Polarization

Lecture 6 ❐ In a similar way QED Renormalization ǫ Vacuum Polarization i 4πµ2 2 Γ(3 − ǫ ) Γ(−1 + ǫ ) Self-energy ǫ 2 2 µ I1,2 = − 2 C ǫ The Vertex 16π C Γ(2 − ) Γ(2)   2 i C = C 1 + 2∆ − 2 ln + O(ǫ) 16π2 ǫ µ2   ❐ Due to the existence of a pole in 1/ǫ in the previous equations we have to expand all quantities up to O(ǫ). This means for instance, that 2 2 1 1 − 1 = − 1 = − + ǫ + O(ǫ2) d 4 − ǫ 2 8

❐ We now substitute these expressions to obtain Nµν

Jorge C. Rom˜ao TCA-2012 – 14 Vacuum Polarization

❐ We obtain Lecture 6

QED Renormalization Vacuum Polarization 1 1 i C N =g − + ǫ + O(ǫ2) C 1 + 2∆ − 2 ln + O(ǫ) Self-energy µν µν 2 8 16π2 ǫ µ2 The Vertex       i C + − 2x(1 − x)k k +x(1 − x)k2g + g m2 ∆ − ln + O(ǫ) µ ν µν µν 16π2 ǫ µ2 " #     i C = − k k ∆ − ln 2x(1 − x) 16π2 µ ν ǫ µ2    i 2 C + 2 gµν k ∆ǫ x(1 − x) + x(1 − x) + ln 2 − x(1 − x) − x(1 − x) 16π " ! µ ! 1 1 + x(1 − x) − 2 2  # i C 1 1 + g m2 ∆ (−1+1)+ln (1 − 1)+(− + ) 16π2 µν ǫ µ2 2 2  

Jorge C. Rom˜ao TCA-2012 – 15 Vacuum Polarization

❐ Finally Lecture 6

QED Renormalization i C 2 Vacuum Polarization N = ∆ − ln g k − k k 2x(1 − x) µν 16π2 ǫ µ2 µν µ ν Self-energy   The Vertex  ❐ Substituting in the expression for Πµν we get

1 1 C Π (k) = − 4e2 g k2 − k k dx 2x(1 − x) ∆ − ln µν 16π2 µν µ ν ǫ µ2 Z0    2 2 = − gµνk − kµkν Π(k , ǫ)

where  2α 1 m2 − x(1 − x)k2 Π(k2, ǫ) ≡ dx x(1 − x) ∆ − ln π ǫ µ2 Z0   ❐ This expression clearly diverges as ǫ → 0. Before we show how to renormalize it let us discuss the meaning of Πµν(k).

Jorge C. Rom˜ao TCA-2012 – 16 Vacuum Polarization

Lecture 6 ❐ The full photon propagator is given by the series QED Renormalization

Vacuum Polarization

Self-energy

The Vertex Gµν =

= + +

+ + . . .

❐ Where we have defined

≡ i Πµν(k) = sum of all one-particle irreducible (proper) diagrams to all orders

Jorge C. Rom˜ao TCA-2012 – 17 Vacuum Polarization

❐ In lowest order we have the contribution Lecture 6

QED Renormalization Vacuum Polarization = Self-energy

The Vertex which is what we have just calculated. ❐ To continue it is convenient to rewrite the free propagator of the photon (in an arbitrary gauge ξ) in the following form

k k 1 k k 1 k k iG0 = g − µ ν + ξ µ ν = P T + ξ µ ν µν µν k2 k2 k4 µν k2 k4   0T 0L ≡iGµν + iGµν

T where we have introduced the transversal projector Pµν defined by

k k P T = g − µ ν µν µν k2  

Jorge C. Rom˜ao TCA-2012 – 18 Vacuum Polarization

❐ This projector satisfies the relations, Lecture 6 QED Renormalization µ T k Pµν = 0 Vacuum Polarization Self-energy  P T νP T = P T The Vertex  µ νρ µρ

❐ The full photon propagator can also in general be written separating its transversal an longitudinal parts

T L Gµν = Gµν + Gµν

T where Gµν satisfies

T T Gµν = PµνGµν

❐ The result we got at one loop means that, to first order, the vacuum polarization tensor is transversal, that is

2 T i Πµν(k) = −ik Pµν Π(k)

Jorge C. Rom˜ao TCA-2012 – 19 Vacuum Polarization

❐ This result is in fact valid to all orders of perturbation theory, a result that Lecture 6 can be shown using the Ward-Takahashi identities. This means that the QED Renormalization

Vacuum Polarization longitudinal part of the photon propagator is not renormalized,

Self-energy L 0L The Vertex Gµν = Gµν

❐ For the transversal part we obtain from the series expansion

′ ′ T T 1 T 1 2 Tµ ν 2 T 1 iG =P + P ′ (−i)k P Π(k )(−i)P ′ µν µν k2 µµ k2 ν ν k2 1 1 1 + P T (−i)k2P Tρλ Π(k2)(−i)P T (−i)k2P Tτσ Π(k2)(−i)P T + ··· µρ k2 λτ k2 σν k2 1 =P T 1 − Π(k2)+Π2(k2) + ··· µν k2   ❐ This gives, after summing the geometric series, 1 iGT = P T µν µν k2 1+Π(k2)  

Jorge C. Rom˜ao TCA-2012 – 20 Vacuum Polarization

❐ All that we have done up to this point is formal because the function Π(k) Lecture 6 diverges. QED Renormalization Vacuum Polarization ❐ The most satisfying way to solve this problem is the following. The initial Self-energy

The Vertex lagrangian from which we started has been obtained from the classical theory and nothing tell us that it should be exactly the same in quantum theory. In fact, as we have just seen, the normalization of the wave functions is changed when we calculate one-loop corrections, and the same happens to the physical parameters of the theory, the charge and the mass. ❐ Therefore we can think that the correct lagrangian is obtained by adding corrections to the classical lagrangian, order by order in perturbation theory, so that we keep the definitions of charge and mass as well as the normalization of the wave functions. The terms that we add to the lagrangian are called counterterms ❐ This interpretation in terms of quantum corrections makes sense. In fact we can show that an expansion in powers of the coupling constant can be interpreted as an expansion in ¯hL, where L is the number of the loops in the expansion term

Jorge C. Rom˜ao TCA-2012 – 21 Vacuum Polarization

❐ The total lagrangian is then, Lecture 6

QED Renormalization Ltotal = L(e,m,...) + ∆L Vacuum Polarization

Self-energy The Vertex ❐ Counterterms are defined from the normalization conditions that we impose on the fields and other parameters of the theory. In QED we have at our disposal the normalization of the and photon fields and of the two physical parameters, the electric charge and the electron mass. ❐ The normalization conditions are, to a large extent, arbitrary. It is however convenient to keep the expressions as close as possible to the free field case, that is, without radiative corrections. ❐ We define therefore the normalization of the photon field as,

2 RT T lim k iGµν = 1 · Pµν k→0

RT where Gµν is the renormalized propagator (the transversal part) obtained from the lagrangian Ltotal.

Jorge C. Rom˜ao TCA-2012 – 22 Vacuum Polarization

❐ The justification for this definition comes from the following argument. Lecture 6 Consider the Coulomb scattering to all orders of perturbation theory. We QED Renormalization

Vacuum Polarization have then the situation described in the figure

Self-energy

The Vertex

=

+ + +

❐ Using the Ward-Takahashi identities one can show that the last three diagrams cancel in the limit q = p′ − p → 0. Then our normalization condition, means that the experimental value of the electric charge is determined in the limit q → 0 of the Coulomb scattering.

Jorge C. Rom˜ao TCA-2012 – 23 Vacuum Polarization

❐ We have then the situation described in Lecture 6

QED Renormalization

Vacuum Polarization

Self-energy lim = The Vertex q→0

❐ The counterterm lagrangian has to have the same form as the classical lagrangian to respect the symmetries of the theory. For the photon field it is traditional to write 1 1 ∆L = − (Z − 1)F F µν = − δZ F F µν 4 3 µν 4 3 µν

corresponding to the following Feynman rule

k k k k µ ν − i δZ k2 g − µ ν 3 µν k2  

Jorge C. Rom˜ao TCA-2012 – 24 Vacuum Polarization

❐ We have then Lecture 6 QED Renormalization k k iΠ =iΠloop − i δZ k2 g − µ ν Vacuum Polarization µν µν 3 µν k2 Self-energy   2 T The Vertex = − i (Π(k, ǫ) + δZ3) k Pµν

❐ Therefore we should make the substitution

Π(k, ǫ) → Π(k, ǫ) + δZ3

in the photon propagator. We obtain,

T T 1 1 iGµν = Pµν 2 k 1+Π(k, ǫ) + δZ3

❐ The normalization condition implies

Π(0, ǫ) + δZ3 = 0

from which one determines the constant δZ3.

Jorge C. Rom˜ao TCA-2012 – 25 Vacuum Polarization

Lecture 6 ❐ We get QED Renormalization Vacuum Polarization 2α 1 m2 Self-energy δZ = − Π(0, ǫ) = − dx x(1 − x) ∆ − ln 3 π ǫ µ2 The Vertex Z0   α m2 = − ∆ − ln 3π ǫ µ2   ❐ The renormalized photon propagator can then be written as

P T iG (k) = µν + iGL µν k2[1+Π(k, ǫ) − Π(0, ǫ)] µν

❐ Notice that the photon mass is not renormalized, that is the pole of the photon propagator remains at k2 = 0.

Jorge C. Rom˜ao TCA-2012 – 26 Vacuum Polarization

❐ The finite radiative corrections are given through the function Lecture 6 QED Renormalization ΠR(k2) ≡Π(k2, ǫ) − Π(0, ǫ) Vacuum Polarization 1 2 2 Self-energy 2α m − x(1 − x)k = − dx x(1 − x) ln The Vertex π m2 Z0   2 2 1/2 2 1/2 α 1 2m 4m − 4m = − + 2 1 + − 1 cot 1 − 1 − 1 3π 3 k2 k2 k2 (   "    #) where the last equation is valid for k2 < 4m2. ❐ For values k2 > 4m2 the result for ΠR(k2) can be obtained by analytical continuation. Using (k2 > 4m2)

− − iπ cot 1 iz = i − tanh 1 z + 2   and

4m2 1/2 4m2 − 1 → i 1 − k2 k2   r Jorge C. Rom˜ao TCA-2012 – 27 Vacuum Polarization

❐ We get then Lecture 6

QED Renormalization 2 2 2 1/2 Vacuum Polarization R 2 α 1 2m 4m −1 4m Π (k ) = − + 2 1 + 2 −1 + 1 − 2 tanh 1 − 2 Self-energy 3π 3 k k k (   " r   The Vertex π 4m2 −i 1 − 2 2 r k #)

❐ The imaginary part of ΠR is given by

α 2m2 4m2 4m2 Im ΠR(k2) = 1 + 1 − θ 1 − 3 k2 k2 k2   r   and it is related to the pair production that can occur for k2 > 4m2. ❐ In fact, for k2 > 4m2 there is the possibility of producing one pair e+e−. Therefore on top of a virtual process (vacuum polarization) there is a real process (pair production).

Jorge C. Rom˜ao TCA-2012 – 28 Self-energy of the electron

❐ The electron full propagator is given by the diagrammatic series Lecture 6

QED Renormalization = + + Vacuum Polarization

Self-energy

The Vertex + + . . .

❐ This of can be written as,

S(p) =S0(p) + S0(p) − i Σ(p) S0(p) + ···  

=S0(p) 1 − i Σ(p)S(p)   where we have identified

≡ −i Σ(p)

Jorge C. Rom˜ao TCA-2012 – 29 Self-energy of the electron

− ❐ Multiplying on the left with S 1(p) and on the right with S−1(p) we get Lecture 6 0 QED Renormalization −1 −1 Vacuum Polarization S0 (p) = S (p) − i Σ(p)

Self-energy

The Vertex which we can rewrite as

−1 −1 S (p) = S0 (p) + i Σ(p)

❐ Using the expression for the free field propagator,

i − S (p) = =⇒ S 1(p) = −i(p/ − m) 0 p/ − m 0

we can then write

−1 −1 S (p) =S0 (p) + iΣ(p) = −i p/ − (m + Σ(p)) h i ❐ We conclude that it is enough to calculate Σ(p) to all orders of perturbation theory to obtain the full electron propagator. The name self-energy given to Σ(p) comes from the fact that it comes as an additional (momentum dependent) contribution to the mass.

Jorge C. Rom˜ao TCA-2012 – 30 Self-energy of the electron

❐ In lowest order there is only the diagram Lecture 6 QED Renormalization k Vacuum Polarization

Self-energy The Vertex p p + k p

❐ Therefore we get,

d4k g i −iΣ(p)=(+ie)2 (−i) µν γµ γν (2π)4 k2 − λ2 + iε p/ + k/ − m + iε Z where we have chosen the Feynman gauge (ξ = 1) for the photon propagator and we have introduced a small mass for the photon λ, in order to control the infrared divergences (IR) that will appear when k2 → 0 (see below). ❐ Using dimensional regularization and the results of the Dirac algebra in dimension d,

µ µ γµ(p/ + k/)γ = − (p/ + k/)γµγ + 2(p/ + k/) = −(d − 2)(p/ + k/) µ mγµγ =m d

Jorge C. Rom˜ao TCA-2012 – 31 Self-energy of the electron

❐ We get Lecture 6

QED Renormalization Vacuum Polarization d ǫ 2 d k 1 p/ + k/ + m µ Self-energy −i Σ(p)=− µ e γ γ (2π)d k2 − λ2 + iε µ (p + k)2 − m2 + iε The Vertex Z ddk −(d − 2)(p/ + k/) + m d =− µǫe2 (2π)d [k2 − λ2 + iε][(p + k)2 − m2 + iε] Z 1 ddk −(d − 2)(p/ + k/) + m d =− µǫe2 dx (2π)d 2 2 2 2 2 Z0 Z [(k − λ ) (1 − x) + x(p + k) − xm + iε] 1 ddk −(d − 2)(p/ + k/) + m d =− µǫe2 dx d 2 2 2 2 2 0 (2π) [(k + px) + p x(1 − x) − λ (1 − x) − xm + iε] Z Z 1 ddk −(d − 2) [p/(1 − x) + k/] + m d =− µǫe2 dx d 2 2 2 2 2 0 (2π) [k + p x(1 − x) − λ (1 − x) − xm + iε] Z Z 1 ǫ 2 =− µ e dx − (d − 2)p/(1 − x) + m d I0,2 Z0  

Jorge C. Rom˜ao TCA-2012 – 32 Self-energy of the electron

❐ Where Lecture 6 QED Renormalization i I = ∆ − ln −p2x(1 − x) + m2x + λ2(1 − x) Vacuum Polarization 0,2 16π2 ǫ Self-energy    The Vertex ❐ The contribution from the loop to the electron self-energy Σ(p) can then be written in the form,

Σ(p)loop = A(p2) + B(p2) p/

with

1 1 A =e2µǫ(4 − ǫ)m dx ∆ − ln −p2x(1 − x) + m2x + λ2(1 − x) 16π2 ǫ Z0    1 1 B = − e2µǫ(2 − ǫ) dx (1 − x) ∆ 16π2 ǫ Z0 "

− ln −p2x(1 − x) + m2x + λ2(1 − x) #   Jorge C. Rom˜ao TCA-2012 – 33 Self-energy of the electron

❐ Using now the expansions Lecture 6

QED Renormalization 1 Vacuum Polarization µǫ(4 − ǫ) =4 1 + ǫ ln µ − + O(ǫ2) Self-energy 4 "   # The Vertex 1 µǫ(4 − ǫ)∆ =4 ∆ + 2 ln µ − + O(ǫ) ǫ ǫ 4 "   #

1 µǫ(2 − ǫ) =2 1 + ǫ ln µ − + O(ǫ2) 2 "   #

1 µǫ(2 − ǫ)∆ =2 ∆ + 2 ln µ − + O(ǫ) ǫ ǫ 2 "   # we can finally write,

4 e2m 1 1 −p2x(1 − x) + m2x + λ2(1 − x) A(p2) = dx ∆ − − ln 16π2 ǫ 2 µ2 Z0   

Jorge C. Rom˜ao TCA-2012 – 34 Self-energy of the electron

❐ And for B Lecture 6

QED Renormalization 2 1 2 2 2 2 2 e −p x(1 − x) + m x + λ (1 − x) Vacuum Polarization B(p ) = − 2 dx (1−x) ∆ǫ − 1 − ln 2 16π 0 µ Self-energy Z    The Vertex ❐ To continue with the renormalization program we have to introduce the counterterm lagrangian and define the normalization conditions. We have

µ µ ∆L = i (Z2 − 1) ψγ ∂µψ−(Z2 − 1) m ψψ+Z2δm ψψ+(Z1 −1)e ψγ ψAµ

and therefore we get for the self-energy

loop −i Σ(p) = −i Σ (p) + i (p/ − m) δZ2 + iδm

❐ Contrary to the case of the photon we see that we have two constants to determine. In the on-shell renormalization scheme that is normally used in QED the two constants are obtained by requiring that the pole of the propagator corresponds to the physical mass (hence the name of on-shell renormalization), and that the residue of the pole of the renormalized electron propagator has the same value as the free field propagator.

Jorge C. Rom˜ao TCA-2012 – 35 Self-energy of the electron

❐ This implies, Lecture 6 QED Renormalization Σ(p/ = m) = 0 ⇒ δm =Σloop(p/ = m) Vacuum Polarization

Self-energy ∂Σ ∂Σloop The Vertex = 0 ⇒ δZ = ∂p/ 2 ∂p/ p/=m p/=m

❐ We then get for δm (δm is not IR divergent, so λ → 0 ) δm =A(m2) + mB(m2) 2 me2 1 m2x2 + λ2(1 − x) = dx 2∆ − 1 − 2 ln 16π2 ǫ µ2 Z0    m2x2 + λ2(1 − x) −(1 − x) ∆ − 1 − ln ǫ µ2    2 m e2 3 1 1 m2x2 + λ2(1 − x) = ∆ − − dx (1 + x) ln 16π2 2 ǫ 2 µ2  Z0   3αm 1 2 1 m2x2 = ∆ − − dx (1 + x) ln 4π ǫ 3 3 µ2  Z0  

Jorge C. Rom˜ao TCA-2012 – 36 Self-energy of the electron

❐ In a similar way we get for δZ2, Lecture 6 QED Renormalization ∂Σloop ∂A ∂B Vacuum Polarization δZ = = + B + m 2 ∂p/ ∂p/ ∂p/ Self-energy p/=m p/=m p/=m

The Vertex where

∂A 4 e2 m2 1 2(1 − x)x = dx ∂p/ 16π2 −m2x(1 − x) + m2x + λ2(1 − x) p/=m Z0

2 αm2 1 (1 − x)x = dx π m2x2 + λ2(1 − x) Z0

α 1 m2x2 + λ2(1 − x) B = − dx (1 − x) ∆ − 1 − ln 2π ǫ µ2 Z0   

∂B α 1 2x(1 − x)2 m = − m2 dx ∂p/ 2π m2x2 + λ2(1 − x) p/=m Z0

Jorge C. Rom˜ao TCA-2012 – 37

Self-energy of the electron

❐ Substituting we get, Lecture 6 QED Renormalization α 1 1 1 m2x2 1 (1+x)(1−x)xm2 Vacuum Polarization δZ =− ∆ − − dx(1−x)ln −2 dx 2 2π 2 ǫ 2 µ2 m2x2 +λ2(1−x) Self-energy  Z0   Z0  The Vertex α m2 λ2 = −∆ − 4+ln − 2 ln 4π ǫ µ2 m2   where we have taken the λ → 0 limit in all cases that was possible.

❐ It is clear that the final result diverges in that limit, therefore implying that Z2 is IR divergent. This is not a problem for the theory because δZ2 is not a physical parameter. We will see that the IR diverges cancel for real processes. ❐ If we had taken a general gauge (ξ 6= 1) we would find out that δm would not be changed but that Z2 would show a gauge dependence. Again, in physical processes this should cancel in the end.

Jorge C. Rom˜ao TCA-2012 – 38 The Vertex

❐ The diagram contributing to the QED vertex at one-loop is Lecture 6 ′ QED Renormalization p

Vacuum Polarization Self-energy k The Vertex µ p ❐ In the Feynman gauge (ξ = 1) this gives a contribution,

d ′ d k g ie µǫ/2Λloop(p ,p) =(ie µǫ/2)3 (−i) ρσ µ (2π)d k2 − λ2 + iε Z i[(p/′ + k/) + m] i[(p/ + k/) + m] γσ γ γρ (p′ + k)2 − m2 + iε µ (p + k)2 − m2 + iε

where Λµ is related to the full vertex Γµ through the relation

loop iΓµ =ie (γµ + Λµ + γµδZ1)

R =ie γµ + Λµ

Jorge C. Rom˜ao  TCA-2012 – 39 The Vertex

Lecture 6 ❐ The integral that defines Λloop(p′,p) is divergent. As before we expect to QED Renormalization µ

Vacuum Polarization solve this problem by regularizing the integral, introducing counterterms and

Self-energy normalization conditions. The counterterm has the same form as the vertex

The Vertex ❐ The normalization constant is determined by requiring that in the limit q = p′ − p → 0 the vertex reproduces the tree level vertex because this is what is consistent with the definition of the electric charge in the q → 0 limit of the Coulomb scattering. ❐ Also this should be defined for on-shell . We have therefore that the normalization condition gives,

u(p) Λloop + γ δZ u(p) = 0 µ µ 1 p/=m 

❐ If we are interested only in calculating δZ1 and in showing that the divergences can be removed with the normalization condition then the problem is simpler. It can be done in two ways.

Jorge C. Rom˜ao TCA-2012 – 40 st The Vertex: Calculating δZ1: 1 method

′ ❐ We use the fact that δZ1 is to be calculated on-shell and for p = p . Then Lecture 6

QED Renormalization

Vacuum Polarization d Self-energy loop 2 ǫ d k 1 1 1 ρ iΛµ (p,p) = e µ γρ γµ γ The Vertex (2π)d k2 − λ2 + iε p/ + k/ − m + iε p/ + k/ − m + iε Z ❐ However we have

1 1 ∂ 1 γ = − p/ + k/ − m + iε µ p/ + k/ − m + iε ∂pµ p/ + k/ − m + iε

and therefore

∂ ddk 1 p/ + k/ + m iΛloop(p,p) = − e2µǫ γ γρ µ ∂pµ (2π)d k2 − λ2 + iε ρ (p + k)2 − m2 + iε Z ∂ = − i Σloop(p) ∂pµ

Jorge C. Rom˜ao TCA-2012 – 41 st The Vertex: Calculating δZ1: 1 method

❐ We conclude then, that Λloop(p,p) is related to the self-energy of the Lecture 6 µ

QED Renormalization electron

Vacuum Polarization loop ∂ loop Self-energy Λµ (p,p) = − µ Σ The Vertex ∂p

❐ This result is one of the forms of the Ward-Takahashi identity ❐ On-shell we have ∂Σloop Λloop(p,p) = − = −δZ γ µ ∂pµ 2 µ p/=m p/=m

and the normalization condition gives

δZ1 = δZ2

❐ As we have already calculated δZ2 then δZ1 is determined.

Jorge C. Rom˜ao TCA-2012 – 42 nd The Vertex: Calculating δZ1: 2 method

❐ In this second method we do not rely in the Ward identity but just calculate Lecture 6 the integrals for the vertex. For the moment we do not put p′ = p but we will QED Renormalization

Vacuum Polarization assume that the vertex form factors are to be evaluated for on-shell spinors. Self-energy ❐ Then we have The Vertex d ′ ρ ′ d k u(p)γ [p/ + k/ + m)] γ [p/ + k/ + m)] γ u(p) i u(p )Λloopu(p) =e2µǫ ρ µ µ (2π)d D D D Z 0 1 2 ddk N =e2µǫ µ (2π)d D D D Z 0 1 2 where

2 ′ ′ Nµ =u(p) (−2 + d)k γµ + 4p · p γµ + 4(p + p ) · kγµ + 4m kµ  ′ − 4k/ (p + p )µ + 2(2 − d)k/kµ u(p)  2 2 D0 =k − λ + iǫ ′ 2 2 D1 =(k + p ) − m + iǫ 2 2 D2 =(k + p) − m + iǫ Jorge C. Rom˜ao TCA-2012 – 43 nd The Vertex: Calculating δZ1: 2 method

❐ Now we use the results of the Appendix to express the ingrals with three Lecture 6 denominators QED Renormalization Vacuum Polarization ′ ′ rµ =p µ ; rµ = pµ ,P µ = x p µ + x pµ Self-energy 1 2 1 2 The Vertex 2 2 2 2 ′ C =(x1 + x2) m − x1x2 q + (1 − x1 − x2)λ , q = p − p ❐ We get, 1 1−x1 ′ α 1 i u(p )Λloopu(p) =i Γ(3) dx dx µ 4π 1 2 2C Z0 Z0 ′ 2 2 2 2 ′ ′ u(p )γµu(p) − (−2 + d)(x1m + x2m + 2x1x2p · p) − 4p · p 2 n ′ h ′ (2 − d) C +4(p + p ) · (x p + x p) + C ∆ − ln 1 2 2 ǫ µ2   ′ ′ + u(p)u(p)m 4(x1p + x2p)µ − 4(p + p)µ(x1 + x2) h ′ − 2(2 − d)(x1 + x2)(x1p + x2p)µ ) i 2 2 ′ = i u(p) G(q ) γµ + H(q )(p + p ) u(p)

Jorge C. Rom˜ao   TCA-2012 – 44 nd The Vertex: Calculating δZ1: 2 method

❐ In the last equation we have defined Lecture 6

QED Renormalization Vacuum Polarization 1 1−x1 2 2 2 2 2 α (x1 + x2) m − x1x2q + (1 − x1 − x2)λ Self-energy G(q ) ≡ ∆ −2−2 dx dx ln 4π ǫ 1 2 µ2 The Vertex  Z0 Z0 − 1 1 x1 −2(x + x )2m2 − x x q2 − 4m2 + 2q2 + dx dx 1 2 1 2 1 2 (x + x )2m2 − x x q2 + (1 − x − x )λ2 Z0 Z0 1 2 1 2 1 2 2(x + x )(4m2 − q2) + 1 2 (x + x )2m2 − x x q2 + (1 − x − x )λ2 1 2 1 2 1 2  − α 1 1 x1 −2m (x + x ) + 2m (x + x )2 H(q2) ≡ dx dx 1 2 1 2 4π 1 2 (x + x )2m2 − x x q2 + (1 − x − x )λ2 Z0 Z0 1 2 1 2 1 2 

❐ Finally we get for the renormalized vertex,

′ R ′ ′ 2 2 ′ u(p )Λµ (p ,p)u(p) = u(p ) G(q ) + δZ1 γµ + H(q )(p + p )µ u(p)    Jorge C. Rom˜ao TCA-2012 – 45 nd The Vertex: Calculating δZ1: 2 method

′ ❐ As δZ1 is calculated in the limit of q = p − p → 0 it is convenient to use the Lecture 6 Gordon identity to get rid of the (p′ + p)µ term. We have QED Renormalization

Vacuum Polarization ′ ′ ′ Self-energy ν u(p )(p + p)µ u(p) = u(p ) 2mγµ − iσµν q u(p) The Vertex   and therefore,

′ R ′ ′ 2 2 2 ν u(p )Λµ (p ,p)u(p) =u(p ) G(q ) + 2mH(q ) + δZ1 γµ − iH(q ) σµν q u(p)   

′ i =u(p ) γ F (q2) + σ qνF (q2) u(p) µ 1 2m µν 2   ❐ In the last expression we have introduced the usual notation for the vertex form factors,

2 2 2 F1(q ) ≡G(q ) + 2mH(q ) + δZ1

2 2 F2(q ) ≡ − 2mH(q )

Jorge C. Rom˜ao TCA-2012 – 46 nd The Vertex: Calculating δZ1: 2 method

❐ The normalization condition implies F1(0) = 0, that is, Lecture 6

QED Renormalization δZ1 = −G(0) − 2mH(0) Vacuum Polarization

Self-energy The Vertex ❐ We have therefore to calculate G(0) and H(0). In this limit the integrals are much simpler. We get (we change variables x1 + x2 → y),

α 1 1 y2m2 + (1 − y)λ2 G(0) = ∆ − 2 − 2 dx dy ln 4π ǫ 1 µ2  Z0 Zx1 1 1 −2y2m2 − 4m2 + 8ym2 + dx dy 1 y2m2 + (1 − y)λ2 Z0 Zx1  α 1 1 −2m y + 2m y2 H(0) = dx dy 4π 1 y2m2 + (1 − y)λ2 Z0 Zx1 ❐ Now using

1 1 y2m2 + (1 − y)λ2 1 m2 dx dy ln = ln − 1 1 µ2 2 µ2 Z0 Zx1  

Jorge C. Rom˜ao TCA-2012 – 47 nd The Vertex: Calculating δZ1: 2 method

❐ And Lecture 6 QED Renormalization 1 1 −2y2m2 − 4m2 + 8ym2 λ2 Vacuum Polarization dx1 dy 2 2 2 =7+2ln 2 0 x1 y m + (1 − y)λ m Self-energy Z Z The Vertex 1 1 −2m y + 2m y2 1 dx dy = − 1 y2m2 + (1 − y)λ2 m Z0 Zx1 (where we took the limit λ → 0 if possible) we get,

α m2 λ2 G(0) = ∆ + 6 − ln +2ln 4π ǫ µ2 m2   α 1 H(0) = − 4π m

❐ Substituting the previous expressions we get finally,

α m2 λ2 δZ = −∆ − 4+ln − 2 ln 1 4π ǫ µ2 m2   in agreement with the previous result

Jorge C. Rom˜ao TCA-2012 – 48 The Vertex: Form factors

2 2 ❐ The general form of the form factors Fi(q ), for q 6= 0, is quite Lecture 6 2 QED Renormalization complicated. We give here only the result for q < 0

Vacuum Polarization 2 θ/2 Self-energy 2 α λ θ F1(q ) = 2 ln + 4 (θ coth θ − 1) − θ tanh − 8 coth θ β tanh βdβ The Vertex 4π m2 2 (  Z0 ) α θ F (q2) = 2 2π sinh θ where θ q2 sinh2 = − · 2 4m2

❐ In the limit of zero transferred momenta (q = p′ − p = 0) we get

F1(0) = 0 α  F (0) =  2 2π a result that we will use while discussing the anomalous magnetic moment of the electron. Jorge C. Rom˜ao TCA-2012 – 49