PHY 396 K. Solutions for problem set #3. The one-loop diagram ) yields amplitude F δk) iλ d 4 p π) 4 p m + i p δk p ) m + i, S.) but the momentum integral here diverges logarithmically as p, so it needs to be regularized. In the Pauli Villars regularization scheme, one subtracts from ) a similar diagram where internal lines belong to ghost fields of extremely large mass Λ m. The subtraction is done before the momentum integration, F PV δk) iλ d 4 p { π) 4 p m + i p δk p ) m + i p Λ + i p δk p ) Λ + i, S.) so for p > Λ the net integrand behaves as OΛ /p 6 ) instead of /p 4 and the integral converges. Out task is to evaluate the integral S.), so let s start with the Feynman s trick for simplifying the propagator product. As discussed in class, p m + i p δk p ) m + i dx dx where q p x δk. [ x)p m + i) + xp m + i)] [q + x x) δk m + i] S.3) Similarly, p Λ + i p δk p ) Λ + i dx [q + x x) δk Λ + i] S.4) for exactly same q p xδk. Hence, we plug both propagator products into eq. S.),
change the order of integration, and than change the momentum variable from p to q, F PV δk t) dx F PV t, x) S.5) where F PV t, x) iλ iλ d 4 p { π) 4 [q p xδk) + tx x) m + i] [q p xδk) + tx x) Λ + i] d 4 q { π) 4 [q + tx x) m + i] [q + tx x) Λ + i]. S.6) Next, we analytically continue the momentum integral from the Minkowski momentum q µ to the Euclidean Momentum q µ E, thus d 4 q id 4 q E, q q E, S.7) and hence F PV t, x) λ d 4 { q E π) 4 [qe + m x x)t] [qe + Λ x x)t]. S.8) At this point, we go to spherical coordinates in the 4D Euclidean space and focus on the radial coordinate q E. This gives us d 4 q E π q E 3 d q E π q E dq E S.9) and hence F PV t, x) λ 3π dq E { qe [qe + m x x)t] qe [qe + Λ x x)t]. S.)
The last integral here has form ) y y + A) y y + ) S.) which evaluates to log/a). Indeed, ) y y + A) y y + ) y + A y + log y + A y + + A y + A log A + + log A ) log A ) ) A y + A) + y + ) ) y + A + A ) A + ) ) ) log A. S.) Hence, F PV t, x) λ 3π log Λ x x)t m x x)t λ 3π log Λ m x x)t S.3) since we assume not only Λ m but also Λ t, u, s. Integrating this formula over x, we arrive at the Pauli Villars regularized amplitude, F PV t) λ 3π λ 3π dx log log Λ m + I Λ m x x)t )) t m S.4) where I ) t m def dx log m m x x)t. S.5) In class, we have calculated the same one-loop amplitude using the hard-edge cutoff; our 3
result was F hard edge λ 3π )) log Λ t m + I m S.6) where It/m ) is exactly as in eq. S.5). omparing eqs. S.6) and S.4), we immediately see that the only difference is the term inside the parentheses in eq. S.6). In other words, the two regularization schemes produce similar amplitudes except for a constant term λ /3π ) O). Moreover, this constant term may be eliminated by adjusting the cutoff scales of the two regulators. Indeed, the cutoff scale Λ HE of the hard-edge regulator the maximal value of the Euclidean momenta allowed in that scheme does not have to be exactly the same as the mass Λ PV of the ghost fields in the Pauli Villars regularization scheme. To produce a similar physical effect, the two scales should have similar orders of magnitude, but this generally means Λ HE Λ PV c ) for some O) constant c rather than naive identification Λ HE Λ PV. onsequently, in the Pauli Villars regularization scheme F t) λ 3π log Λ PV m + I )) t m S.7) while in the hard-edge regularization scheme F t) λ 3π log Λ HE m + I )) t m, S.8) and log Λ PV may be different from log Λ HE by some O) constant log c according to eq. ). Thus, the one-loop amplitudes S.7) and S.8) are in perfect agreement with each other, provided we identify log Λ PV Λ HE, S.9) i.e., Λ PV Λ HE exp). 4
Now consider the higher-derivative regularization scheme. In this scheme, the scalar field φ has very small higher-derivative terms in its Lagrangian, L HD µφ) m φ λ 4 φ4 Λ φ), S.) which softens the scalar s propagator for very high momenta p > Λ : i p m + i i p m + i Λ p 4 i p m + i Λ p Λ + i. S.) onsequently, in the higher-derivative regularization scheme, the one-loop amplitude ) becomes F HD δk) iλ d 4 p π) 4 p m + i Λ p Λ + i p m + i Λ p Λ + i S.) where p δk p. For all but extremely large momenta p Λ, the integrand here is indistinguishable from the un-regularized loop integral S.), but for p > Λ it becomes softer behaves like Λ 4 /p 8 for p instead of /p so the integral S.)converges. Our task is to evaluate this integral, so let s start by simplifying the propagator product by using the Feynman s trick S.3) and then interchanging the order of dx and d 4 p integrals: F HD δk) dx F HD δk, x) S.3) where F HD δk, x) iλ d 4 p π) 4 [q p xδk) + x x)δk m + i] Λ p Λ + i Λ p δk p ) Λ + i. S.4) Note that we have used the Feynman trick only for the /p m +i) and /p m +i) factors, the remaining Λ dependent factors remain as they are on the second line of eq. S.4). 5
Next, inside the dx integral, we change the momentum integration variable from p to q p xδk, thus F HD δk, x) iλ d 4 q π) 4 [q + x x)δk m + i] Λ p q + xδk) Λ + i Λ p q + x )δk) Λ + i. S.5) The UV cutoff scale Λ must me much larger than the scalar s mass m and also than any component δk µ of the net momentum transfer δk. onsequently, for any q µ we may approximate Λ q + Oδk)) Λ + i Λ q Λ + i : S.6) For q Λ this approximation works because the whole q+oδk)) term in the denominator is negligible compared to the Λ term, while for q Λ or large, Oδk) correction to q becomes negligible because δk q. factors in eq. S.5), we have Applying this approximation to both Λ dependent Λ p q + xδk) Λ + i Λ p q + x )δk) Λ + i Λ 4 [q Λ + i] S.7) and hence F HD δk, x) iλ d 4 q π) 4 [q + x x)δk m + i] Λ 4 [q Λ + i]. S.8) At this point, we analytically continue the momentum integral from the Minkowski momentum q µ to the Euclidean momentum q µ E : d4 q becomes id 4 q E, q becomes q E, and the integral S.8) becomes F HD δk t, x) λ λ 3π d 4 q E π) [qe + m x x)t] Λ 4 [qe + Λ ] dq E q E Λ4 [q E + m x x)t] [q E + Λ ]. S.9) On the second line here, we have integrated over the directions of the q µ E in the 4D Euclidean 6
space. As to the remaining radial integral, it has form y y + A) y + ) S.3) where A m x x)t is much less than Λ. The simplest way to evaluate this integral is to split it at some point which is much bigger than A but much smaller that. Thus y y + A) y + ) y y + A) y + ) + where in the zero to integral y allows us to approximate y y + A) y + ) S.3) y y + A) y + ) y y + A) log A + A + A log A, S.3) while in the to infinity integral y A makes for y + A) y in the denominator and hence Altogether, for A, y y + A) y + ) yy + ) log + { y y + log. + y + ) S.33) y y + A) y + ) log A + log log A. S.34) note that drops out of net result; if it did not, our approximations would be inconsistent. 7
Alternatively, we may evaluate the integral S.3) without using any approximations by expanding the integrand which is a rational function of y into its poles: y y + A) y + ) α y + A) + β y + ) + γ y + A + δ y + S.35) for some constants α, β, γ, δ. The values of α and β follow by matching the coefficients of the double poles at y A and y at both sides, thus α A A), β 3 A). S.36) Subtracting the double poles from both sides of eq. S.35) and matching the residues of the remaining single poles, we obtain γ δ + + A) A) 3. S.37) onsequently, y y + A) y + ) A) A) [ + A A y + A ) y + [ + A A log A ] ] A y + A) y + ) for any > A > S.38) log A for A, in perfect agreement with eq. S.34). Plugging this formula into the momentum integral S.9), we obtain F HD t, x) λ 3π Λ ) log m x x)t S.39) 8
and consequently F HD t) λ 3π λ 3π dx Λ ) log m x x)t log Λ m + I )) t m S.4) where It/m ) is as in eq. S.5). Similar to the Pauli Villars case, the Λ parameter of the higher-derivative regularization scheme does not have to be exactly equal to the hard-edge cutoff Λ HE to produce the same physical effect. Instead, we expect Λ hard edge Λ higherderivative c ) for some O) numerical constant c. onsequently, log Λ in the amplitude S.4) can be shifted by a constant, and in this way the one-loop diagram ) regularized using the higherderivative term become consistent with the other regulators hard-edge and Pauli Villars for the same amplitude. Indeed, eq. S.4) re-written as F t) λ 3π log Λ HD m + I )) t m S.4) becomes identical with eqs. S.8) and S.7) when we identify log Λ HD log Λ PV log Λ HE, S.4) or equivalently Λ hard edge Λ PauliVillars exp+) Λ higher derivative exp). ) 9