Quantum Field 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 coupling constant 1.1 Definition of the effective coupling 1.1.1 Summation of photon self-energy corrections Diagrammatically, the exact photon propagator 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 ≡ iΠ^ 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 perturbation theory, so that we can write Π^ µρ(q2) = (q2gµρ − qµqρ)Π(^ q2): (3) This yields qµqν 1PI = Π(^ 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(^ q2) gµν − = Π(^ q2) gµν − ; (5) q2 q2 for each n 6= 0. The 0-th term may be written as qµqν h i0 qµqν gµν = + Π(^ 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(^ q2) q2 q2 −i qµqν −i qµqρ = gµν − + ξ : (7) q2[1 − Π(^ 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(^ 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(^ 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(^ q2) in the on-shell scheme One possibility to fix the photon field renormalization constant ZA is requiring the exact photon propagator to have a pole at the physical photon mass 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 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 qµ and write the exact photon propagator as −igµν ∼ : (12) ¡ q2[1 − Π(^ 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 − Π^ OS(0) ^ OS ^ 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, pole mass), 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 1 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 − 2". This implies that the bare coupling α0 is not a pure number and a regularization scale µ0 can be singled out setting 2" 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 −" 2 α ∆ Π2(q ) = − Γ(") 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(") ≡ Γ(")e (here γe is Euler's constant) so that Z 1 2 2 −" 2 α ~ m q Π2(q ) = − Γ(") 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 " ! 0, which means that all terms that vanish in that limit need not be calculated. Expanding in " we obtain Z 1 2 2 2 α 1 m q Π2(q ) = − − dx 2x(1 − x) log 2 − x(1 − x) 2 + O(") : (21) π 3" 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(") ; (22) π 3" 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("; q =m ) (23) π 3" 3 µ~0 m 0 2 2 α 1 1 m q 2 2 = − − log 2 + 2 + O("; q =m ) : (24) π 3" 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(") ; (25) π 3" 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("; m =q ) : (26) π 3" 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 3" 3 µ~0 15m Π2(q ) = − × 1 1 −q2 5 2 2 (27) π − log 2 + if q m : 3" 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 " µ~0 which gives ( q2 2 2 OS 2 α 5m2 if q m ; Π^ (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 8 1 h q2 q2 i 2 2 > 2 + 2 if q m1;2; > 5 m1 m2 α < 2 2 Π^ 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) symmetry to write Z 1 q2 −" Z 1=2 q2 −" 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 p 1 t = 4x(1 − x); x = 1 − 1 − t ; dx = (1 − t)−1=2; (32) 2 4 4 finding Z 1 2 −" Z 1 2 −" 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 = ", b = 2, c = 5=2 and set z ≡ q2=4m2 to get 2 −" 2 α ~ m 5 Π2(q ) = − Γ(") 2 2F1 "; 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 −" ^ OS 2 α ~ m 5 Π2 (q ) = − Γ(") 2 2F1 "; 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 1 X rn (1 − r)−" = (") ; (37) n n! n=0 where (x)n ≡ Γ(x + n)=Γ(x) is the Pochhammer symbol, which implies −" 1 n 2α m2 X 1 q2 Z 1 Π (q2) = − Γ(~ ") (") 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) −" 1 n 2α m2 X Γ(2 + n)2 1 q2 Π (q2) = − Γ(~ ") (") : (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) p 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 −" 1 2 n α m X (2 + n)n(")n 1 q Π (q2) = − Γ(~ ") ; (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.