1 Running and matching of the QED coupling constant

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1 Quantum Field Theory-II UZH and ETH, FS-6 Assistants: A. Greljo, D. Marzocca, J. Shapiro Problem Set n. 8 Prof. G. Isidori Due: -5-6 Running and matching of the QED coupling constant. Definition of the effective coupling.. Summation of photon self-energy corrections Diagrammatically, the exact photon propagator may be written as an infinite sum of products of one-particle-irreducible PI corrections as in = This is a geometrical series of ratio PI + PI + PI PI +. iˆπ µρ q i q g ρν ξ qρ q ν q, where the photon propagator is in generic covariant gauge. The hat over Π denotes that the PI 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 perturbation theory, so that we can write This yields ˆΠ µρ q = q g µρ q µ q ρ ˆΠq. 3 PI = ˆΠq g µν qµ q ν q, 4 and since the tensor structure P µν = g µν q µ q ν /q is a projector P µρ P ρν = P µν we find [ ] ˆΠq g µν qµ q ν n n = [ˆΠq ] g µν qµ q ν, 5 q for each n. The -th term may be written as g µν = qµ q ν + [ˆΠq ] g µν qµ q ν ; 6 q q this allows us to sum the geometrical series obtaining = i q g [ ] µρ ξ qµ q ρ g q ˆΠq ρν qρ q ν + qρ q ν = q q i = g q [ ˆΠq µν qµ q ν ] q q + i q ξ qµ q ρ q. 7

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 which means that in momentum space q µ J µ q =, q µ J µ q = ; 8 = iα q [ ˆΠq ] J µ qj µ q. 9 The correction to the photon propagator can clearly be reabsorbed into the definition of the coupling constant, so that where = iα eff q J µ qj µ q = α eff q =, α ˆΠq. Using this effective coupling gives the result for this process at all orders by considering only the tree-level photon propagator...3 ˆΠq in the on-shell scheme One possibility to fix the photon field renormalization constant Z A is requiring the exact photon propagator to have a pole at the physical photon mass with residue 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 propagators 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 q µ and write the exact photon propagator as ig µν q [ ˆΠq ]. The first requirement that the pole be at q = 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 ˆΠ OS that is ˆΠ OS =. Since ˆΠq = Πq + δz A we immediately find! =, 3 δz OS A = Π. 4 The scale at which we imposed the renormalization condition is completely fixed it is the physical, pole mass, thus the coupling α does not run in this scheme. Under these conditions α = α eff, and it can be directly measured by studying electromagnetic interactions at low momentum transfer.

3 . Approximate calculation of the QED effective coupling.. Photon wavefunction at one loop The photon self-energy s expansion in powers of the coupling reads Πq = Π l q, Π l α l. 5 l= We regulate the result by analytically continuing the number of dimensions to the complex value d = 4 ε. This implies that the bare coupling α is not a pure number and a regularization scale µ can be singled out setting The one-loop corrections to the photon propagator are given by α = αµ ε. 6 = + and the amputated one-loop diagram has the explicit structure = iq g µν q µ q ν Π q. 8 The transversality of this correction is guaranteed by the QED Ward identity. The correction is found to be Π q = α ε π Γε dx x x. 9 4πµ with = m x xq. No approximation was done so far in this equation. We set 4πµ e γe and Γε Γεe εγe here γ e is Euler s constant so that + + Oα, 7 Π q = α π Γε m dx x x x x q ε... Approximate calculation of Π q for one fermion The physical part of Π q is the one that is finite for ε, which means that all terms that vanish in that limit need not be calculated. Expanding in ε we obtain Π q = α [ ] m π 3ε dx x x log x x q + Oε. Collecting m / µ in the logarithm we get Π q = α π 3ε [ ] m log dx x x log x x q + Oε, 3 m and expanding in q /m gives Π q = α π 3ε 3 = α π log m + q m 3ε m log + q 3 5m + Oε, q /m dx [ x x ] + Oε, q /m

4 In the q > m case we can collect q / in the logarithm to find Π q = α π 3ε [ ] q log dx x x log x x m 3 q + Oε, 5 and neglect the second term in the logarithm getting Π q = α π 3ε q log Oε, m /q. 6 We will ignore all higher order contributions and set Π q = α π { 3ε 3 3ε 3 log m log q + q 5m if q m, if q m. 7 Note that for q > 4m 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 Π = α [ ] m log, 8 3π ε which gives ˆΠ OS { q = α q 3π if q m, 5m 5 q log if q m. 3 m We observe that the dependence on the arbitrary scale µ has dropped out of the physical result. Let us also stress that this result is ill-defined close to q m, and in fact discontinuous if equation 9 is interpreted as a piecewise definition Approximate α eff 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 [ ] q ˆΠ OS q = α + q if q m 5 m 3π m,, 5 q log + q if m 3 m 5m q m, 3 q log log q if q m 3 m m,. Strictly speaking the existence of the intermediate region implies m m, which means that this result could be approximated further. The effective coupling is then obtained by equation..3 Exact calculation of the QED effective coupling optional.3. Integral representation of the hypergeometric function The integral may be explicitly solved by using the x x symmetry to write ε / ε dx x x x x q = dx 4x x 4x x q, 3 m 4m and changing variables to t = 4x x, x = t, dx = 4 t /, 3 4

5 finding ε dx x x x x q = m 4 This is an integral representation of the hypergeometric function ε dt t t / t q. 33 4m dt t b t c b tz a = ΓbΓc b F a, b; c; z, 34 Γc so we identify a = ε, b =, c = 5/ and set z q /4m to get Π q = α 3π Γε m ε F ε, ; 5 ; z. 35 The renormalized self-energy is found using F a, b; c; = to compute Π, which gives ˆΠ OS q = α 3π Γε m ε [F ε, ; 5 ] ; z Solution as an expansion in q /m In order to expand in the ratio q /m we use r ε = where x n Γx + n/γx is the Pochhammer symbol, which implies n= n= ε n r n n!, 37 Π q = α π Γε m ε q n ε n dx x n+ x n+. 38 n! m The integral may be carried out in closed form since it s of Euler type first kind Using the duplication formula with k = + n one gets Π q = α π Γε m ε Γ + n Γ4 + n ε q n n. 39 n! m which together with Γ5/ = 3 π/4 gives n= Inserting this identity we find Π q = α 3π Γε m ε Γk = ΓkΓk + / π k, 4 Γ4 + n = Γ + nγ5/ + n π 3+n, 4 Γ4 + n = 6Γ + n5/ n 4 n. 4 n= + n n ε n q n ; 43 5/ n n! 4m this is precisely the series representation of the hypergeometric function we found before. 5

6 .3.3 Analytic properties of Π q The dependence on q of the photon self-energy is entirely contained in the hypergeometric function. This is defined by its series representation 43 for z < and in the whole complex plane via analytic continuation. Its branch cut is conventionally positioned on the real axis for z >. A good series expansion for z > can be obtained via the identity F a, b; c; z = Γc [ Γb a ΓbΓc a z a F a, c + a; b + a; z ] + a b, 44 which allows to recover the limit q m. We observe that the region corresponding to a photon exchange as it happens e.g. when a charge is thrown against a target at rest actually corresponds to a space-like separation q <. To make this explicit, the effective coupling should be computed as α eff 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 q > the examined process can involve charged fermion pair production. As mentioned before, this implies that ˆΠq acquire a non-vanishing imaginary part according the optical theorem for q > 4m, and the i + prescription needs to be used to choose the branch..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 ˆΠ OS z = α 3π [ z + z z arctan z z z ] + Oε π α Π π α Π Figure : Plots of the real and imaginary part of 3π ˆΠ z/α..4 Running of the QED coupling α.4. 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: β log αµ log µ = log eµ log µ. 46 6

7 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 e = and the bare coupling does not depend on any scale, we get Z e Z / A Z e, 47 ψ log e log µ = β = log Z A log µ + log Z ψ log µ log Z e log µ. 48 The Ward identity guarantees that in suitable renormalization schemes the last two terms cancel each other, thus we conclude.4. Computation of β in the MS scheme β = log Z A log µ. 49 A good renormalization condition fixes the renormalization constant for the photon field strength Z A so that it includes the regulated UV poles of the bare photon self-energy Πq plus arbitrary finite terms. This determines a renormalization scheme. A simple choice when adopting dimensional regularization in d = 4 ε is to include no finite terms at all; expanding the photon self-energy in powers of the coupling and the regulator parameter ε, this corresponds to setting δz A = l= n= l Πq = l= n= l Π n l εn, ˆΠMS q = Π n l q ε n, 5 l= n= Π n l q ε n. 5 where we made it explicit that Π n l does not depend on q for negative n. Since Π n l logarithmic derivative with respect to the unphysical regularization scale µ = µ is log µ α l = log µ α l µ lε = lεα l µ lε Thus the logarithmic derivative of the counterterm is simply δz A log µ = ε l= n= l α l its log µ Π n l q = lεπ n l q. 5 lπ n l εn, 53 which at one loop translates simply into β = δz A log µ = Π. 54 ε= From any of the expression giving Π q above we read immediately β = α 3π. 55 Note that in order to compute this result only the pole of the photon self-energy is needed, and that taking into account fermion masses is not required. This is just a formal generalization of the argument that logarithmic derivatives with respect to µ of renormalization constants are equal to times the pole coefficient. 7

8 .4.3 Running of α in mass-independent schemes In general one defines βα = α 4π α n β n, 56 4π n= the minus sign is conventional so that at one loop βα = α 4π β. 57 At this order, if β does not depend on µ, the differential equation for α can be easily solved by separation of variables: d log α α integrating from µ i to µ f remember that α = e log α gives Solving for αµ f we obtain = β 4π d log µ, 58 αµ f αµ i = β 4π log µ f. 59 µ i αµ f = αµ i + β αµ i 4π log µ f µ i.5 The running coupling in the effective field theory framework.5. -loop running with two fermion families. 6 The Appelquist-Carazzone theorem states that at scales µ m i a particle of mass m i decouples and can therefore be integrated out. On the other hand we know that for µ m i the associated loop corrections become important. We will therefore work with a theory that contains the particle i with i =, standing for e.g. the electron and the muon for µ µ i m i and with an effective theory where the field i is not dynamical for µ µ i. The boundary condition is determined by requiring that the calculations in the two different theories give the same physical result at µ i : this procedure is called matching. For a theory with n F active fermions in the photon self-energy loop we determine from eq. 55 β nf α = n F α 3π The MS running coupling is the solution of the differential equation log αµ β nf =α if µ µ, = β log µ nf =α if µ < µ < µ, β nf =α if µ > µ, β = 4 3 n F. 6 6 which reads α if µ µ, α if µ αµ = α µ µ µ, log 3π µ α if µ µ. α 3π log µ µ 63 T. Appelquist and J. Carazzone, Phys. Rev. D,

9 .5. Tree-level matching The constants α and α can be determined by requiring the current-current one-photonexchange interaction measured by α MS eff µ = αµ ˆΠ MS µ 64 to yield a unique result at µ and µ. At tree-level ˆΠ =, which means that one simply requires lim µ µ lim µ µ αµ = lim αµ α = α, 65 µ µ + αµ = lim αµ µ µ + α α 3π log µ µ = α. 66 This gives just α if µ µ, α if µ αµ = α µ µ µ, log 3π µ α if µ µ, α 3π log µ µ +log µ µ 67 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 approximation of the real physics in that regime: we shall therefore disregard this option and determine α 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 eff µ = α eff 3π α eff. 68 log µ + log µ m m 3 The dependence on µ drops out because the MS result contains the correct logarithms at all orders in α, and we find α = α eff 3π α eff. 69 log µ + log µ m m 3 We see that, provided that the logarithms in the denominator are not large enough to compensate α eff, the difference in the value at low momentum transfer is formally of higher order. 3 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 continuous which is much more important requires to sacrifice the continuity of αµ at the matching points. In figure we plot the effective coupling α eff and the running coupling α as a function of Q for QED with two fermions. Note that in the Standard Model quarks are also charged fermions which enter loop corrections to the photon propagator: in principle their contribution could be included just by appropriately modifying n F although there are subtleties. For Q M Z,W 3 One could tweak the values of µ and µ in such a way that the two agree exactly, but this is not really honest since it requires the knowledge of the full result. 4 This actually happens with one-loop matching as well, but it breaks down at two loops. 9

10 α /α - - Figure : One-loop running of the QED coupling constant with an electron of mass.5 MeV and a muon of mass 5 MeV. The solid blue line is the effective coupling α eff 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 force, 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 phenomenology. However let us point out that the running of the strong coupling α s 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.