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Home , Coupling constant

Quantum Theory-II Problem Set n. 8 UZH and ETH, FS-2016 Prof. G. Isidori Assistants: A. Greljo, D. Marzocca, J. Shapiro Due: 20-05-2016 http://www.physik.uzh.ch/lectures/qft/

1 Running and matching of the QED constant

1.1 Definition of the effective coupling 1.1.1 Summation of self-energy corrections Diagrammatically, the exact photon may be written as an infinite sum of products of one-particle-irreducible (1PI) corrections as in ¡ = ¡ + ¡1PI + ¡1PI 1PI + ··· . (1) This is a geometrical series of ratio

−i  qρqν  1PI ≡ iΠˆ µρ(q2) gρν − (1 − ξ) , (2) ¡ q2 q2 where the photon propagator is in generic covariant gauge. The hat over Π denotes that the 1PI blob already includes the photon self-energy counterterm, i.e. that the result is already renormalized. The QED Ward identity asserts that corrections to the photon propagator are transverse to all orders in , so that we can write Πˆ µρ(q2) = (q2gµρ − qµqρ)Π(ˆ q2). (3) This yields

 qµqν  1PI = Π(ˆ q2) gµν − , (4) ¡ q2 and since the tensor structure P µν = gµν − qµqν/q2 is a projector (P µρP ρν = P µν) we find   qµqν n h in  qµqν  Π(ˆ q2) gµν − = Π(ˆ q2) gµν − , (5) q2 q2 for each n 6= 0. The 0-th term may be written as qµqν h i0  qµqν  gµν = + Π(ˆ q2) gµν − ; (6) q2 q2 this allows us to sum the geometrical series obtaining

" # −i  qµqρ  1  qρqν  qρqν = gµρ − (1 − ξ) gρν − + = ¡ q2 q2 1 − Π(ˆ q2) q2 q2 −i  qµqν  −i qµqρ = gµν − + ξ . (7) q2[1 − Π(ˆ q2)] q2 q2 q2

1 1.1.2 Effective coupling Consider now the interaction between two conserved currents, in the approximation where this is mediated by a single photon exchange. The conservation law requires that µ µ qµJ1 (q) = 0, qµJ2 (q) = 0; (8) which means that in momentum space

−iα = J µ(q)J µ(−q) . (9) ¡ q2[1 − Π(ˆ q2)] 1 2 The correction to the photon propagator can clearly be reabsorbed into the definition of the coupling constant, so that

−iα = eff J µ(q)J µ(−q) = , (10) ¡ q2 1 2 ¡ where 2 α αeff(q ) = . (11) 1 − Π(ˆ q2) Using this “effective” coupling gives the result for this process at all orders by considering only the tree-level photon propagator.

1.1.3 Π(ˆ q2) in the on-shell scheme

One possibility to fix the photon field constant ZA is requiring the exact photon propagator to have a pole at the physical photon with residue 1 (up to conventional factors of i which can be deduced from the tree-level expression). This choice is partly motivated by the observation that it simplifies considerably the LSZ formula, and is practical only as long as the fermion mass does not vanish so that IR divergences are automatically regulated. The tensor structure of the propagator would require a careful treatment of photon physical states just to write this renormalization condition. In order to avoid a long digression, let us argue that we are interested in that are contracted with a source J µ(q) which is a conserved current. We can therefore neglect all terms in the propagator that contain uncontracted momenta qµ and write the exact photon propagator as

−igµν ∼ . (12) ¡ q2[1 − Π(ˆ q2)] The first requirement that the pole be at q2 = 0 is automatically satisfied: this is a consequence of gauge invariance and ultimately related to the fact that no mass renormalization is needed for the photon. Thus we only need to impose 1 =! 1, (13) 1 − Πˆ OS(0) ˆ OS ˆ 2 2 that is Π (0) = 0. Since Π(q ) = Π(q ) + δZA we immediately find OS δZA = −Π(0). (14) The scale at which we imposed the renormalization condition is completely fixed (it is the physical, ), thus the coupling α does not run in this scheme. Under these conditions α = αeff(0), and it can be directly measured by studying electromagnetic interactions at low momentum transfer.

2 1.2 Approximate calculation of the QED effective coupling 1.2.1 Photon wavefunction at one loop The photon self-energy’s expansion in powers of the coupling reads

∞ 2 X 2 ` Π(q ) = Π2`(q ), Π2` ∼ α . (15) `=1 We regulate the result by analytically continuing the number of dimensions to the complex value d = 4 − 2ε. This implies that the bare coupling α0 is not a pure number and a scale µ0 can be singled out setting

2ε α0 = αµ0 . (16) The one-loop corrections to the photon propagator are given by ¡ = ¡ + ¡ + ¡ + O(α2) , (17) and the amputated one-loop diagram has the explicit structure

2 µν µ ν 2 ¡ = i(q g − q q )Π2(q ). (18) The transversality of this correction is guaranteed by the QED Ward identity. The correction is found to be Z 1  −ε 2 α ∆ Π2(q ) = − Γ(ε) dx 2x(1 − x) 2 . (19) π 0 4πµ0 with ∆ = m2 − x(1 − x)q2. No approximation was done so far in this equation. We set 2 2 γe ˜ εγe µ˜0 ≡ 4πµ0e and Γ(ε) ≡ Γ(ε)e (here γe is Euler’s constant) so that Z 1  2 2 −ε 2 α ˜ m q Π2(q ) = − Γ(ε) dx 2x(1 − x) 2 − x(1 − x) 2 . (20) π 0 µ˜0 µ˜0

2 1.2.2 Approximate calculation of Π2(q ) for one fermion 2 The physical part of Π2(q ) is the one that is finite for ε → 0, which means that all terms that vanish in that limit need not be calculated. Expanding in ε we obtain  Z 1   2 2   2 α 1 m q Π2(q ) = − − dx 2x(1 − x) log 2 − x(1 − x) 2 + O(ε) . (21) π 3ε 0 µ˜0 µ˜0 Collecting m2/µ˜2 in the logarithm we get

 2 Z 1   2   2 α 1 1 m q Π2(q ) = − − log 2 − dx 2x(1 − x) log 1 − x(1 − x) 2 + O(ε) , (22) π 3ε 3 µ˜0 0 m and expanding in q2/m2 gives

 2 2 Z 1  2 α 1 1 m q  2 2 2 2 Π2(q ) = − − log 2 + 2 2 dx x (1 − x) + O(ε, q /m ) (23) π 3ε 3 µ˜0 m 0  2 2  α 1 1 m q 2 2 = − − log 2 + 2 + O(ε, q /m ) . (24) π 3ε 3 µ˜0 15m

3 2 2 2 2 In the q > m case we can collect −q /µ˜0 in the logarithm to find  2 Z 1   2   2 α 1 1 −q m Π2(q ) = − − log 2 − dx 2x(1 − x) log x(1 − x) − 2 + O(ε) , (25) π 3ε 3 µ˜0 0 q and neglect the second term in the logarithm getting  2  2 α 1 1 −q 5 2 2 Π2(q ) = − − log 2 + + O(ε, m /q ) . (26) π 3ε 3 µ˜0 9 We will ignore all higher order contributions and set

( 1 1 m2 q2 2 2 α − log 2 + 2 if q  m , 2 3ε 3 µ˜0 15m Π2(q ) = − × 1 1 −q2 5 2 2 (27) π − log 2 + if q  m . 3ε 3 µ˜0 9 Note that for q2 > 4m2 the self-energy develops an imaginary part as required by the optical theorem. The renormalized self-energy in the on-shell scheme is obtained by subtraction of α 1 m2  Π2(0) = − − log 2 , (28) 3π ε µ˜0 which gives

( q2 2 2 OS 2 α 5m2 if q  m , Πˆ (q ) = − × 2 (29) 2 3π 5 −q 2 2 3 − log m2 if q  m .

We observe that the dependence on the arbitrary scaleµ ˜0 has dropped out of the physical result. Let us also stress that this result is ill-defined close to q2 ∼ m2, and in fact discontinuous if equation (29) is interpreted as a piecewise definition.

2 1.2.3 Approximate αeff (q ) for two fermions In order to extend this result to the case of two fermions, we just need to remember that Feynman diagrams are additive and therefore self-energies are too. We thus find

 1 h q2 q2 i 2 2  2 + 2 if q  m1,2,  5 m1 m2 α  2 2 Πˆ OS(q2) = − × 5 −q q 2 2 2 (30) 2 3 − log 2 + 2 if m1  q  m2, 3π m1 5m2  10 −q2 −q2 2 2  − log 2 − log 2 if q  m1,2. 3 m1 m2 2 2 Strictly speaking the existence of the intermediate region implies m1  m2, which means that this result could be approximated further. The effective coupling is then obtained by equation (11).

1.3 Exact calculation of the QED effective coupling (optional) 1.3.1 Integral representation of the hypergeometric function The integral may be explicitly solved by using the x ↔ (1 − x) to write

Z 1  q2 −ε Z 1/2  q2 −ε dx 2x(1 − x) 1 − x(1 − x) 2 = dx 4x(1 − x) 1 − 4x(1 − x) 2 , (31) 0 m 0 4m and changing variables to 1 √ 1 t = 4x(1 − x), x = 1 − 1 − t , dx = (1 − t)−1/2, (32) 2 4

4 finding

Z 1  2 −ε Z 1  2 −ε q 1 −1/2 q dx 2x(1 − x) 1 − x(1 − x) 2 = dt t(1 − t) 1 − t 2 . (33) 0 m 4 0 4m This is an integral representation of the hypergeometric function Z 1 b−1 c−b−1 −a Γ(b)Γ(c − b) dt t (1 − t) (1 − tz) = 2F1(a, b; c; z), (34) 0 Γ(c) so we identify a = ε, b = 2, c = 5/2 and set z ≡ q2/4m2 to get

 2 −ε   2 α ˜ m 5 Π2(q ) = − Γ(ε) 2 2F1 ε, 2; ; z . (35) 3π µ˜0 2

The renormalized self-energy is found using 2F1(a, b; c; 0) = 1 to compute Π(0), which gives

 2 −ε     ˆ OS 2 α ˜ m 5 Π2 (q ) = − Γ(ε) 2 2F1 ε, 2; ; z − 1 . (36) 3π µ˜0 2

1.3.2 Solution as an expansion in q2/m2 In order to expand in the ratio q2/m2 we use

∞ X rn (1 − r)−ε = (ε) , (37) n n! n=0 where (x)n ≡ Γ(x + n)/Γ(x) is the Pochhammer symbol, which implies

−ε ∞ n 2α m2  X 1  q2  Z 1 Π (q2) = − Γ(˜ ε) (ε) dx xn+1(1 − x)n+1 . (38) 2 π µ˜2 n n! m2 0 n=0 0 The integral may be carried out in closed form since it’s of Euler type (first kind)

−ε ∞ n 2α m2  X Γ(2 + n)2 1  q2  Π (q2) = − Γ(˜ ε) (ε) . (39) 2 π µ˜2 Γ(4 + 2n) n n! m2 0 n=0 Using the duplication formula

− 1 2k−1 Γ(2k) = Γ(k)Γ(k + 1/2) π 2 2 , (40) with k = 2 + n one gets

− 1 3+2n Γ(4 + 2n) = Γ(2 + n)Γ(5/2 + n) π 2 2 , (41) √ which together with Γ(5/2) = 3 π/4 gives

n Γ(4 + 2n) = 6Γ(2 + n)(5/2)n4 . (42) Inserting this identity we find

 2 −ε ∞  2 n α m X (2 + n)n(ε)n 1 q Π (q2) = − Γ(˜ ε) ; (43) 2 3π µ˜2 (5/2) n! 4m2 0 n=0 n this is precisely the series representation of the hypergeometric function we found before.

5 2 1.3.3 Analytic properties of Π2(q ) The dependence on q2 of the photon self-energy is entirely contained in the hypergeometric function. This is defined by its series representation (43) for |z| < 1 and in the whole complex plane via analytic continuation. Its branch cut is conventionally positioned on the real axis for z > 1. A good series expansion for |z| > 1 can be obtained via the identity  Γ(b − a)  1  F (a, b; c; z) = Γ(c) (−z)−a F a, 1 − c + a; 1 − b + a; + (a ↔ b) , (44) 2 1 Γ(b)Γ(c − a) 2 1 z which allows to recover the limit q2  m2. We observe that the region corresponding to a photon exchange (as it happens e.g. when a is thrown against a target at rest) actually corresponds to a space-like separation 2 2 q < 0. To make this explicit, the effective coupling should be computed as αeff(−Q ). In this region there are no ambiguities due to the resolution of the branch: the effective coupling is real and regular on the negative real axis. For timelike momentum transfer q2 > 0 the examined process can involve charged fermion . As mentioned before, this implies that Π(ˆ q2) acquire a non-vanishing imaginary part according the optical theorem for q2 > 4m2, and the i0+ prescription needs to be used to choose the branch.

1.3.4 Expansion in ε The expansion in ε can be performed equivalently before or after the integration over Feynman parameters. It is anyway a bit involved to carry out analytically because it requires either the integration over a logarithm with non-trivial argument or the expansion of a hypergeometric function in one of its parameters. Once this difficulties have been dealt with one finds " # α 5 1  1 r1 − z r z Πˆ OS(z) = − + − 2 + arctan + O(ε) . (45) 2 3π 3 z z z 1 − z

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Figure 1: Plots of the real and imaginary part of 3πΠˆ 2(z)/α.

1.4 Running of the QED coupling α 1.4.1 β function in QED In general, renormalization requires an extra unphysical scale µ to be introduced. We define the β function as the logarithmic derivative of the renormalized coupling α with respect to this scale: ∂ log α(µ) ∂ log e(µ) β ≡ = . (46) ∂ log µ2 ∂ log µ

6 Note that this definition differs from the problem sheet by a factor of e, due to the fact that we take the derivative of log e rather than e. Since in QED

Ze e0 = 1/2 e, (47) ZA Zψ and the bare coupling does not depend on any scale, we get ∂ log e 1 ∂ log Z ∂ log Z ∂ log Z 0 = 0 ⇒ β = A + ψ − e . (48) ∂ log µ 2 ∂ log µ ∂ log µ ∂ log µ The Ward identity guarantees that in suitable renormalization schemes the last two terms cancel each other, thus we conclude 1 ∂ log Z β = A . (49) 2 ∂ log µ

1.4.2 Computation of β in the MS scheme A good renormalization condition fixes the renormalization constant for the photon field strength 2 ZA so that it includes the regulated UV poles of the bare photon self-energy Π(q ) plus arbi- trary finite terms. This determines a renormalization scheme. A simple choice when adopting dimensional regularization in d = 4 − 2ε is to include no finite terms at all; expanding the photon self-energy in powers of the coupling and the regulator parameter ε, ∞ ∞ 2 X X (n) 2 n Π(q ) = Π2` (q )ε , (50) `=0 n=−` this corresponds to setting

∞ −1 ∞ ∞ X X (n) n ˆ MS 2 X X (n) 2 n δZA = − Π2` ε , Π (q ) = Π2` (q )ε . (51) `=0 n=−` `=0 n=0

(n) 2 (n) ` where we made it explicit that Π2` does not depend on q for negative n. Since Π2` ∼ α its logarithmic derivative with respect to the unphysical regularization scale µ =µ ˜0 is

∂ ` ∂ ` 2`ε ` 2`ε ∂ (n) 2 (n) 2 α = α0µ0 = 2`εα0µ0 ⇒ Π2` (q ) = 2`εΠ2` (q ). (52) ∂ logµ ˜0 ∂ logµ ˜0 ∂ logµ ˜0 Thus the logarithmic derivative of the counterterm is simply

∞ −1 ∂δZA X X (n) = −2ε `Π εn, (53) ∂ log µ 2` `=0 n=−` which at one loop translates simply into1

1 ∂δZA (−1) β = = −Π2 . (54) 2 ∂ log µ ε=0 2 From any of the expression giving Π2(q ) above we read immediately α β = . (55) 3π Note that in order to compute this result only the pole of the photon self-energy is needed, and that taking into account fermion is not required.

1This is just a formal generalization of the argument that logarithmic derivatives with respect to µ of renormalization constants are equal to −2 times the pole coefficient.

7 1.4.3 Running of α in mass-independent schemes In general one defines ∞ α X  α n β(α) = − β , (56) 4π n 4π n=0 (the minus sign is conventional) so that at one loop α β(α) = − β . (57) 4π 0

At this order, if β0 does not depend on µ, the differential equation for α can be easily solved by separation of variables: d log α β − = 0 d log µ2, (58) α 4π −1 − log α integrating from µi to µf (remember that α = e ) gives

2 1 1 β0 µf − = log 2 . (59) α(µf) α(µi) 4π µi

Solving for α(µf) we obtain

α(µi) α(µf) = 2 . (60) α(µi) µf 1 + β0 log 2 4π µi

1.5 The running coupling in the effective field theory framework 1.5.1 1-loop running with two fermion families

2 The Appelquist-Carazzone theorem states that at scales µ  mi a particle of mass mi de- couples and can therefore be integrated out. On the other hand we know that for µ & mi the associated loop corrections become important. We will therefore work with a theory that con- tains the particle i (with i = 1, 2 standing for e.g. the electron and the muon) for µ ≥ µi ∼ mi and with an effective theory where the field i is not dynamical for µ ≤ µi. The boundary condition is determined by requiring that the calculations in the two different theories give the same physical result at µi: this procedure is called matching.

For a theory with nF active fermions in the photon self-energy loop we determine from eq. (55) α 4 β (α) = n ⇒ β = − n . (61) nF F 3π 0 3 F The MS running coupling is the solution of the differential equation  β (α) if µ2 ≤ µ2, ∂ log α(µ)  nF=0 1 = β (α) if µ2 < µ2 < µ2, (62) ∂ log µ2 nF=1 1 2  2 2 βnF=2(α) if µ > µ2, which reads  2 2 α0 if µ ≤ µ ,  1  α1 if µ2 ≤ µ2 ≤ µ2, α1 µ2 1 2 α(µ) = 1− 3π log 2 (63) µ1  α2 if µ2 ≥ µ2.  2α2 µ2 2  1− 3π log 2 µ2 2T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 (1975).

8 1.5.2 Tree-level matching

The constants α1 and α2 can be determined by requiring the current-current one-photon- exchange interaction measured by α(µ) αMS(µ2) = (64) eff 1 − Πˆ MS(−µ2) ˆ to yield a unique result at µ1 and µ2. At tree-level Π = 0, which means that one simply requires

lim α(µ) = lim α(µ) ⇒ α0 = α1, (65) − + µ→µ1 µ→µ1 α lim α(µ) = lim α(µ) ⇒ 1 = α . (66) µ2 2 µ→µ− µ→µ+ α1 2 2 2 1 − log 2 3π µ1 This gives just  α if µ2 ≤ µ2,  0 1  α0 2 2 2  α µ2 if µ1 ≤ µ ≤ µ2, 1− 0 log α(µ) = 3π 2 (67) µ1  α0 2 2    if µ ≥ µ2,  α0 µ2 µ2  1− 3π log 2 +log 2 µ1 µ2 which contains only one free parameter to be fixed by a measurement. In principle one could choose to use the value at low energies, but we know the effective theory to be only an ap- proximation of the real physics in that regime: we shall therefore disregard this option and determine α0 from high energy experiments. A real measurement would give a value that is accurate at all orders in α (within its experimental uncertainty); nevertheless in the absence of actual data we will require the effective coupling to agree with the most accurate prediction available, that is our one loop calculation α (0) α (−µ2) = eff . (68) eff 2 2 αeff(0)  µ µ 10  1 − log 2 + log 2 − 3π m1 m2 3

The dependence on µ drops out because the MS result contains the correct logarithms at all orders in α, and we find α (0) α = eff . (69) 0  2 2  αeff(0) µ1 µ2 10 1 − log 2 + log 2 − 3π m1 m2 3 We see that, provided that the logarithms in the denominator are not large enough to compen- 3 sate αeff(0), the difference in the value at low momentum transfer is formally of higher order. With tree-level matching we have obtained boundary conditions requiring that the running coupling be continuous,4 however in general the requirement that physical observables be con- tinuous – which is much more important – requires to sacrifice the continuity of α(µ) at the matching points. 2 In figure 2 we plot the effective coupling αeff and the running coupling α as a function of Q for QED with two fermions. Note that in the are also charged fermions which enter loop corrections to the photon propagator: in principle their contribution could be included just by appropriately modifying nF (although there are subtleties). For Q ∼ MZ,W

3 One could tweak the values of µ1 and µ2 in such a way that the two agree exactly, but this is not really honest since it requires the knowledge of the full result. 4This actually happens with one-loop matching as well, but it breaks down at two loops.

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Figure 2: One-loop running of the QED coupling constant with an electron of mass 0.5 MeV and a muon of mass 105 MeV. The solid blue line is the effective coupling αeff to all orders and the dashed orange line is the MS running α.

more radical modifications take place since electromagnetic interactions merge with the weak , so the range of scales where the corrections we computed are valid is practically quite small. This fact, together with the small size of the corrections to α, make the model discussed here not particularly relevant for . However let us point out that the running of the strong coupling αs is more pronounced and, notwithstanding the very different resulting behavior, its calculation in MS is not too different from the one that was carried out.

10