Solutions for PHY-396 K/L: Superficial Degree of Divergence in Quantum Field Theory, Assignments of Physics

Download Solutions for PHY-396 K/L: Superficial Degree of Divergence in Quantum Field Theory and more Assignments Physics in PDF only on Docsity! PHY–396 K/L. Solutions for homework set #18. Problem 2(a): Let us start with the superficial degree of divergence D. At large momenta, bosonic propagators behave as 1/q2 while fermionic propagators behave as 1/q, hence in 4 dimensions D = 4L − 2PB − PF . (S.1) As in the λφ4 theory, we can relate this expression to the numbers of external legs using the vertex valences. Naively, the Yukawa theory has only one vertex type — which connects one bosonic line and two fermionic lines — but we shall see that renormalization requires an additional four-boson vertex of the λφ4 type. Denoting the respected numbers of the two vertex types VY and Vλ, we have 2PF + EF = 2VY , 2PB + EB = VY + 4Vλ , (S.2) while the Euler formula says L − P + V ≡ L − PB − PF + VY + Vλ = 1, (S.3) Combining these three equations, we obtain D = 4L − 2PB − PF = 4(L − PB − PF ) + 3PF + 2PB = 4(1 − VY − Vλ) + 3 2(2VY − EF ) + (VY + 4Vλ − EB) = 4 − 32EF − EB . (S.4) Thus, the external legs of a diagram completely determine its superficial degree of divergence. Consequently, for any number of loops, there are only seven superficially divergent amplitudes, 1 namely (a) D = 4 (b) D = 3 (c) D = 2 (d) D = 1 (e) D = 0 (f) D = 1 (g) D = 0 Furthermore, the amplitude (a) here is the vacuum energy while the amplitudes (b) and (d) van- ish because of the parity symmetry. Indeed, the pseudo-scalar field Φ is parity-odd, hence the amplitudes involving odd number of pseudoscalar particles and no fermions must have parity-odd dependence on the particles’ momenta. But to construct a parity-odd Lorentz-invariant combina- tion of the Lorentz vectors pα1 , p β 2 , . . ., one needs ² tensors, e.g. ²αβγδp α 1 p β 2p γ 3p δ 4, which requires at least 4 linearly independent momenta (in d = 4 spacetime) and hence n ≥ 5 external legs. For the amplitudes (b) and (d) involving one or three pseudoscalars only and no fermions, such construction is not available and the amplitudes vanish identically. Unlike QED, the Yukawa theory does not give rise to Ward identities, so any 1PI amplitude that can diverge generally does. Hence, expanding the 1PI amplitudes (c), (e), (f), and (g) in powers of relevant momenta we find the following independent divergences: (c) Σφ(p 2) = O(Λ2) × const + O(log Λ) × p2 + finite; (e) M(s, t, u) = O(log Λ) × const + finite; (f) Σψ(6p) = O(Λ 1) × const + O(log Λ)×6p + finite; (g) Γ5(p′, p) = γ5 × O(log Λ) × const + finite. To cancel all these divergences in situ in the renormalized perturbation theory, we need four 2 Problem 2(b): At this stage we are ready to calculate the counterterms, beginning with the δλ. At the one-loop level of analysis, the four-boson amplitude comprises the following Feynman diagrams: iM1 loop(k1, k2, k3, k4) = + + + two similar + + five similar. (S.10) The last diagram here yields − ∫ d4p1 (2π)4 Tr { (−gγ5) i 6p1 − M + i0 (−gγ5) i 6p2 − M + i0 (−gγ5) i 6p3 − M + i0 (−gγ5) i 6p4 − M + i0 } (S.11) where p2 = p1 + k1 , p3 = p2 + k2, p4 = p3 + k3, and p1 = p4 + k4 ; there are five similar diagrams related by permutations of the external momenta k1, k2, k3, k4. For generic values of these momenta, the integral (S.11) is quite complicated, but its divergence is k- independent and hence may be evaluated for any particular choice of ki we find convenient. Clearly, the simplest set of ki is k1 = k2 = k3 = k4 = 0; this is off-shell, but that’s OK. Consequently, the 5 integral (S.11) becomes iVψ loop(0, 0, 0, 0) = − ∫ d4p1 (2π)4 tr ( (−gγ5) i 6p − M + i0 )4 = −g4 ∫ d4p1 (2π)4 tr[γ5(6p + M)]4 (p2 − M2 + i0)4 = ∫ d4p1 (2π)4 −4g4 (p2 − M2 + i0)2 (S.12) where the last equality follows from [γ5(6p + M)]2 = γ5(6p + M)γ5(6p + M) = (−6p + M)(6p + M) = −p2 + M2 (S.13) and hence tr[γ5(6p+M)]4 = 4(p2 −M2)2. Evaluating the integral on the last line of eq. (S.12) using dimensional regularization, we obtain Vψ loop(k1 = k2 = k3 = k4 = 0) = −4g4 16π2 ( 1 ²̄ + log µ2 M2 ) (S.14) where 1 ²̄ def = 1 ² − γE + log(4π). (S.15) It remains to multiply the amplitude (S.14) by 6 (for six similar diagrams) and add contributions of the other diagrams (S.10). The latter diagrams have been evaluated in class in the context of the scalar λΦ4 theory, thus to order O(λ2 or g4), V(k1 = k2 = k3 = k4 = 0) = −λ − δλ + 3λ2 32π2 ( 1 ²̄ + log µ2 m2 ) − 24g4 16π2 ( 1 ²̄ + log µ2 M2 ) . (S.16) The renormalization condition for the physical λ coupling is the on-shell four-particle amplitude M(threshold) = −λ, or in other words V = −λ when all external momenta are on shell and at the 6 threshold (s = 4m2, t = u = 0). At other values of external momenta, we should have V(k1, k2, k3, k4) = −λ − λ2 32π2 × finite − 4g4 16π2 × finite + higher loop orders. (S.17) Comparing this formula with eq. (S.16) gives us δ1 loopλ = 3λ2 32π2 ( 1 ²̄ + log µ2 m2 + finite ) − 24g4 16π2 ( 1 ²̄ + log µ2 M2 + finite ) . (S.18) As promised, fermionic loops provide for δλ 6= 0 even if were to start from λ = 0. ? ? ? Next, we want to calculate the δg counterterm, so let us consider the ΦΨγ 5Ψ vertex correction. By analogy with the QED vertex, we denote Γ(5)(p′, p) the 1PI amplitude for two fermions of respective momenta p and p′ and one pseudoscalar of momentum k = p′ − p. At the one-loop level of analysis, −Γ(5)(p′, p) = + + (S.19) = −gγ5 − δgγ 5 + ∫ d4q (2π)4 i q2 − m2 + i0 × (−gγ5) i 6p′+ 6q − M + i0 (−gγ5) i 6p+ 6q − M + i0 (−gγ5). As in the previous calculation, the loop integral here diverges logarithmically, and the divergent part does not depend on the external momenta. Consequently, we may calculate this divergence for any values of p, p′, and k − p′ − p we like, for example p = p′ = k = 0 which makes for a much simpler integral. Indeed, for zero external momenta, the fermionic line becomes (−gγ5) i 6q′+ 6q − M + i0 (−gγ5) i 6q+ 6q − M + i0 (−gγ5) = g3 γ5(6q + M)γ5(6Q + M)γ5 (q2 − M2 + i0)2 = g3 −γ5 (q2 − M2 + i0)2 (S.20) where the second equality follows from eq. (S.13). Consequently, the loop integral in eq. (S.19) 7 the ` → −` symmetry, thus ∫ d4` (2π)4 6` [`2 − ∆ + i0]2 = 0. (S.31) Altogether, this gives us the following expression for the loop integral in eq. (S.25): = −g2 1 ∫ 0 dx [M − (1 − x) 6p ] ∫ d4` (2π)4 1 (`2 − ∆ + i0)2 . (S.32) Curiously, this loop has superficial degree of divergence D = +1 but the actual momentum inte- gral here diverges logarithmically rather than linearly. Evaluating this integral using dimensional regularization, we obtain ∫ d4k (2π)4 1 (`2 − ∆ + i0)2 = i 16π2 ( 1 ²̄ + log µ2 ∆ ) , (S.33) and therefore Σ1 loopψ (6p) = δ ψ M − δ ψ Z 6p + g2 16π2 1 ∫ 0 dx [M −(1−x) 6p ] ( 1 ²̄ + log µ2 (1 − x)m2 + xM2 − x(1 − x)p2 ) . (S.34) The renormalization conditions for the fermion’s propagator correction Σψ(6p) are Σ ∣ ∣ ∣ ∣ 6p = M = 0 and dΣ d 6p ∣ ∣ ∣ ∣ 6p = M = 0. (S.35) In light of eq. (S.34), the second condition (S.35) becomes δψZ [1 loop] = g2 16π2 ∂ ∂ 6p 1 ∫ 0 dx [M − (1 − x) 6p ] ( 1 ²̄ + log µ2 (1 − x)m2 + xM2 − x(1 − x)p2 ) ∣ ∣ ∣ ∣ ∣ ∣ 6p = M = g2 16π2 1 ∫ 0 dx [ (x − 1) ( 1 ²̄ + log µ2 x2M2 + (1 − x)m2 ) + 2x2(1 − x)M2 x2M2 + (1 − x)m2 ] = − g2 32π2 ( 1 ²̄ + log µ2 M2 + finite ) . (S.36) 10 At the same time, the first condition (S.35) implies δψM [1 loop] − Mδ ψ Z [1 loop] = − g2 16π2 1 ∫ 0 dx [M − (1 − x) 6p ] ( 1 ²̄ + log µ2 (1 − x)m2 + xM2 − x(1 − x)p2 ) ∣ ∣ ∣ ∣ ∣ ∣ 6p = M = − g2 16π2 1 ∫ 0 dx xM × ( 1 ²̄ + log µ2 x2M2 + (1 − x)m2 ) = − g2M 32π2 ( 1 ²̄ + log µ2 M2 + finite ) (S.37) and consequently δψM [1 loop] = − g2M 16π2 ( 1 ²̄ + log µ2 M2 + finite ) . (S.38) Note that similar to QED, the fermionic mass counterterm in the Yukawa theory is propor- tional to the mass itself and diverges logarithmically rather than linearly in the UV cutoff (cf. integral (S.32) prior to dimensional regularization). As in QED, this behavior is due an additional symmetry the Yukawa theory acquires when the fermion mass vanishes. Specifically, for M = 0 we have a discrete chiral symmetry Ψ(x) → γ5Ψ(x), Ψ(x) → −Ψ(x)γ5, Φ(x) → −Φ(x). (S.39) Unlike the gauge coupling in QED, the pseudoscalar Yukawa coupling does not respect continuous chiral transforms Ψ(x) → exp(iαγ5)Ψ(x), but the discrete symmetry is sufficient for preventing the massless Yukawa theory from developing a mass shift via loop corrections. 11 ? ? ? Finally, consider the boson’s mass and kinetic energy counterterms δφM and δ φ Z . At the one-loop level of analysis, the pseudoscalar field’s 1PI two-point Green’s function is −iΣ1 loopφ (k 2) = + + = −iδφm + iδ φ Z k 2 + iλm2 32π2 ( 1 ²̄ + 1 + log µ2 m2 ) − ∫ d4p (2π)4 tr ( i 6p − M + i0 (−gγ5) i 6p+ 6k − M + i0 (−gγ5) ) . (S.40) Again, we re-write the fermionic loop integral as + g2 ∫ d4p (2π)4 N D (S.41) where the denominator is the usual D = ( p2 − M2 + i0 ) × ( (p + k)2 − M2 + i0 ) (S.42) and hence 1 D = 1 ∫ 0 dx 1 [`2 − ∆ + i0]2 for ` = p + kx and ∆ = M2 − x(1 − x)k2, (S.43) and the numerator is N = tr [ (6p + M)γ5(6p+ 6k + M)γ5 ] = tr [ (M+ 6p)(M−6p−6k) ] = 4M2 − 4p(p + k) = 4M2 − 4(` − xk)(` + k − xk) = 4M2 − 4`2 + 4x(1 − x)k2 − 4(1 − 2x)` · k. (S.44) Again, the last term here is odd with respect to ` → −` and hence does not contribute to the ∫ d4` 12