Quantum Field Theory: Two-Point Function & Renormalization in Minkowski & Euclidean Spaces, Study notes of Physics

Download Quantum Field Theory: Two-Point Function & Renormalization in Minkowski & Euclidean Spaces and more Study notes Physics in PDF only on Docsity! 11 Perturbation Theory and Feynman Diagrams We now turn our attention to interacting quantum field theories. All of the results that we will derive in this section apply equally to both relativistic and non-relativistic theories with only minor changes. Here we will use the path integrals approach we developed in previous chapters. The properties of any field theory can be understood if the N -point Green functions are known GN (x1, . . . , xN ) = 〈0|Tφ(x1) . . . φ(xN )|0〉 (1) Much of what we will do below can be adapted to any field theory of interest. We will discuss in detail the simplest case, the relativistic self-interacting scalar field theory. It is straightforward to generalize this to other theories of interest. We will only give a summary of results for the other cases. 11.1 The Generating Functional in Perturbation Theory The N -point function of a scalar field theory, GN (x1, . . . , xN ) = 〈0|Tφ(x1) . . . φ(xN )|0〉, (2) can be computed from the generating functional Z[J ] Z[J ] = 〈0|Te i ∫ dDxJ(x)φ(x) |0〉 (3) In D = d + 1-dimensional Minkowski space-time Z[J ] is given by the path integral Z[J ] = ∫ Dφ e iS[φ] + i ∫ dDxJ(x)φ(x) (4) where the action S[φ] is the action for a relativistic scalar field. The N -point function, Eq.(1), is obtained by functional differentiation, i.e., GN (x1, . . . , xN ) = (−i) N 1 Z[J ] δN δJ(x1) . . . δJ(xN ) Z[J ] ∣ ∣ ∣ J=0 (5) Similarly, the Feynman propagator GF (x1−x2), which is essentially the 2-point function, is given by GF (x1 − x2) = −i〈0|Tφ(x1)φ(x2)|0〉 = i 1 Z[J ] δ2 δJ(x1)δJ(x2) Z[J ] ∣ ∣ ∣ J=0 (6) Thus, all we need to find is to compute Z[J ]. We will derive an expression for Z[J ] in the simplest theory, the relativistic real scalar field with a φ4 interaction, but the methods are very general. We 1 will work in Euclidean space-time (i.e., in imaginary time) where the generating function takes the form Z[J ] = ∫ Dφ e −S[φ] + ∫ dDxJ(x)φ(x) (7) where S[φ] now is S[φ] = ∫ dDx [ 1 2 (∂φ)2 + m2 2 φ2 + λ 4! φ4 ] (8) In the Euclidean theory the N -point functions are GN (x1, . . . , xN ) = 〈φ(x1) . . . φ(xN )〉 = 1 Z[J ] δN δJ(x1) . . . δJ(xN ) Z[J ] ∣ ∣ ∣ J=0 (9) Let us denote by Z0[J ] the generating action for the free scalar field, with action S0[φ]. Then Z0[J ] = ∫ Dφ e −S0[φ] + ∫ dDxJ(x)φ(x) = [ Det ( −∂2 + m2 )]−1/2 e 1 2 ∫ dDx ∫ dDyJ(x)G0(x − y)J(y) (10) where ∂2 is the Laplacian operator in D-dimensional Euclidean space, and G0(x− y) is the free field Euclidean propagator (i.e., the Green function) G0(x − y) = 〈φ(x)φ(y)〉0 = 〈x| 1 −∂2 + m2 |y〉 (11) where the sub-index label 0 denotes a free field expectation value. We can write the full generating function Z[J ] in terms of the free field generating function Z0[J ] by noting that the interaction part of the action con- tributes with a weight of the path-integral that, upon expanding in powers of the coupling constant λ takes the form e −Sint[φ] = e − ∫ dDx λ 4! φ4(x) = ∞ ∑ n=0 (−1)n n! ( λ 4! )n ∫ dDx1 . . . ∫ dDxn φ 4(x1) . . . φ 4(xn) (12) 2 rotation, the analytic continuation to imaginary time ix0 → xD, the two-point function in D-dimensional Minkowski space-time, 〈0|Tφ(x1)φ(x2)|0〉 maps onto the two-field correlation function of D-dimensional Euclidean space-time, 〈0|Tφ(x1)φ(x2)|0〉 ←→ 〈φ(x1)φ(x2)〉 (16) Let us formally write the two point function G(2)(x1 − x2) as a power series in the coupling constant λ, G(2)(x1 − x2) = ∞ ∑ n=0 λn n! G(2)n (x1 − x2) (17) However, using the generating functional Z[J ] we can write G(2)(x1 − x2) = 1 Z[J ] δ2 δJ(x1)δJ(x2) Z[J ] ∣ ∣ ∣ J=0 (18) where Z[J ] = e −Sint ( δ δJ ) Z0[J ] (19) Hence, the two-point function can be expressed as a ratio of two series ex- pansions in powers of the coupling constant. The numerator is given by the expansion of δ 2 δJ(x1)δJ(x2) Z[J ] ∣ ∣ ∣ J=0 , δ2 δJ(x1)δJ(x2) Z[J ] ∣ ∣ ∣ J=0 = ∞ ∑ n=0 (−1)n n! ( λ 4! )n ∫ dDy1 . . . ∫ dDyn δ2 δJ(x1)δJ(x2) δ4 δJ(y1)4 . . . δ4 δJ(yn)4 Z0[J ] ∣ ∣ ∣ J=0 (20) and the denominator by the expansion of Z[0], which leads to a similar ex- pression but without a contribution due to the external legs, corresponding to the functional derivatives with respect to the source at the external points x1 and x2. The equivalent expression in Minkowski space-time is obtained by the replacement, −λ←→ iλ (21) at every order in the expansion. We will now look at the form of the first few terms of the expansion of the two-point function in perturbation theory. 11.2.1 Zeroth Order in λ. To zeroth order in λ (i.e., O(λ0)), the numerator reduces to δ δJ(x1) δ δJ(x2) Z[J ] ∣ ∣ ∣ J=0 = δ δJ(x1) δ δJ(x2) Z0[J ] ∣ ∣ ∣ J=0 + O(λ) = G0(x1 − x2) + O(λ) (22) 5 while the denominator is simply equal to one Z[0] = Z0[0] + O(λ) = 1 + 0(λ) (23) Hence, G(2)(x1 − x2) = G0(x1 − x2) + O(λ) (24) 11.2.2 First Order in λ. To first order in λ, the denominator Z[0] is given by Z[0] = 1 + −1 1! ( λ 4! ) ∫ dy δ4 δJ(y)4 Z0[J ]|J=0 + O(λ 2) (25) The expression inside the integrand can be calculated from the Taylor expansion of Z0[J ] in powers of J(x). To find a non-zero contribution we need to bring down from the exponent enough factors of J so that they can be cancelled by the functional derivatives. Since the argument of the exponential factor in Z0[J ] is bilinear in J(x), Z0[J ] = [ Det ( −∂2 + m2 )]−1/2 e 1 2 ∫ dDx ∫ dDyJ(x)G0(x− y)J(y) , (26) only an even number of derivatives in J(x) can be cancelled to a given order. In particular, to first order in λ, we have to cancel four derivatives. This means that we need to expand the exponential in Z0[J ] to second order in its argument to obtain the only non-vanishing contribution to first order in λ to Z[J ] at J = 0, Z[0] = 1 + (−1) 1! ( λ 4! ) ∫ dDx δ4 δJ(x)4 1 2! ( 1 2 ∫ dDy1 ∫ dDy2J(y1)G0(y1 − y2)J(y2) )2 ∣ ∣ ∣ J=0 +O(λ2) (27) The derivatives yield a set of δ-functions δ4 δJ(x)4 [J(y1) · · · J(y4)] ∣ ∣ ∣ J=0 = ∑ P 4 ∏ j=1 δ(ypj − xy) (28) where P runs over the 4! permutations of the four arguments y1, y2, y3 and y4. We can now write the first order correction to Z[0], in the form Z[0] = 1 + ( −1 1! )( λ 4! ) 1 2! ( 1 2 )2 ∫ dDx ∫ dDy1 . . . d Dy4G0(y1 − y2)G0(y3 − y4) ∑ P 4 ∏ j=1 δ(ypj − x) +O(λ2) = 1 + (−1) 1! ( λ 4! ) S ∫ dDx G0(x, x) G0(x, x) + O(λ 2) (29) 6 where S = 4! 2! 22 = 3 (30) x yy yy1 2 3 4 Figure 1: A vertex and its four contractions. It is useful to introduce a picture or diagram to represent this contribution. Let us mark four points y1, . . . , y4 and an additional point at x (which we will call a vertex) with four legs coming out of it. Let us join y1 and y2 by a line and y3 with y4 by another line. To each line we assign a factor of G0(y1 − y2) and G0(y3 − y4) respectively, as in the figure: G0(x − y) = x y Next, because of the δ-functions, we have to identify each of the points y1, . . . , y4 with each one of the legs attached to y in all possible ways (as shown in figure 1) The result has to be integrated over all values of the the coordinates and of x. The result is Z[0] = 1− ( λ 8 ) ∫ dx (G0(x, x)) 2 + O(λ2) (31) Physically, the first order contribution represents corrections to the ground state energy due to vacuum fluctuations. This expression can be represented more simply by the Feynman diagram shown in Fig.2. Here, and below, we denote by a full line the bare propagator G0(x− y). Let us now compute the first order corrections to δ2 δJ(x1)δJ(x2) Z[J ] ∣ ∣ ∣ J=0 . 7 G(2) = + ( ( () ) ) 1+ 1+ +O(λ2) +O(λ2)+O(λ2) Figure 5: Factorization of Feynman diagrams for the two-point function to first order in λ. G(2) = ++ . . . Figure 6: The two-point function to first order in λ. bosonic or fermionic, provided the fields satisfy local canonical commutation (or anti-commutation) relations. The expansion of the two point function has the form 〈φ(x1)φ(x2)〉 = 1 Z[0] ∞ ∑ n=0 ∫ dDy1 . . . d Dyn (−1)n n! 〈φ(x1)φ(x2) n ∏ j=1 Lint (φ(yj))〉0 (37) The denominator factor Z[0] has a similar expansion Z[0] = ∞ ∑ n=0 (−1)n n! ∫ dDy1 . . . d Dyn 〈 N ∏ j=1 Lint (φ(yj))〉0 (38) where 〈A(φ)〉0 denotes an expectation value of the operator A(φ) in the free field theory. Let us consider first the numerator. Each expectation value involves a sum of products of pairwise contractions. If we assign a Feynman diagram to each contribution, it is clear that we can classify these terms into two classes: (a) linked and (b) unlinked diagrams. A diagram is said to be unlinked if it contains a sub-diagram in which a set of internal vertices are linked with each other but not to an external vertex. The linked diagrams satisfy the opposite property. Since the vacuum diagrams by definition do not contain any external vertices, they are unlinked. All the expectation values that appear in the numerator can be written as a 10 sum of terms, each of the form of a linked diagram times a vacuum graph, i.e., 〈φ(x1)φ(x2)Lint (φ(y1)) . . .Lint (φ(yn))〉0 = n ∑ k=0 ( n k ) 〈φ(x1)φ(x2) k ∏ j=1 Lint (φ(yj))〉 ℓ 0 〈 n ∏ j=k+1 Lint (φ(yj))〉0 (39) where the super-index ℓ denotes a linked factor, i.e., a factor that does not contain any vacuum sub-diagram. Thus, the numerator has the form ∞ ∑ n=0 n ∑ k=0 (−1)n n! ( n k ) 〈φ(x1)φ(x2) k ∏ j=1 Lint(yk)〉 ℓ 0〈 N ∏ j=k+1 Lint (φ(yj))〉0 (40) which factorizes into ( ∞ ∑ k=0 (−1)k k! ∫ dDy1 . . . d Dyk 〈φ(x1)φ(x2) k ∏ j=1 Lint (φ(yj))〉 ℓ 0 ) × ( ∞ ∑ n=0 (−1)n n! ∫ dDy1 . . . d Dyn〈 n ∏ j=1 Lint (φ(yj))〉0 (41) We can clearly recognize that the second factor is exactly equal to the denomi- nator Z[0]. Hence we find that we can write the two-point function as a sum of linked Feynman diagrams: 〈φ(x1)φ(x2)〉 = ∞ ∑ n=0 (−1)n n! ∫ dDy1 . . . d Dyn 〈φ(x1)φ(x2) n ∏ j=1 Lint (φ(yn))〉 ℓ 0 (42) This result is known as the linked-cluster theorem. This theorem, which proves that the vacuum diagrams cancel exactly out to all orders in perturbation theory, is valid for all the N -point functions (not just for two-point function) and for any local theory. It also holds in Minkowski space-time upon the replacement (−1)n ↔ in. It holds for all theories with a local canonical structure, relativistic or not. 11.4 Summary of Feynman Rules for φ4 theory 11.4.1 Position Space The general rules to construct the diagrams for the N -point function 〈0|Tφ(x1) . . . φ(xN )|0〉 in φ 4 theory in Minkowski space and 〈φ(x1) . . . φ(xN )〉 in Euclidean space, in position space are 11 1. A general graph for the N -point function has N external points (ore ver- tices) and n interaction vertices, where n is the order in perturbation theory. Each vertex is a point with a coordinate label and 4 lines (for a φ4 theory) coming out of it. 2. Draw all topologically distinct graphs by connecting the external points and the internal vertices in all possible ways. Discard all graphs which contain sub-diagrams not linked to at least one external point. 3. The following weight is assigned to each graph: (a) For every vertex a factor of −i λ4! in Minkowski space and − λ 4! in Euclidean space . (b) For every line connecting a pair of points z1 and z2, a factor of 〈0|Tφ(z1)φ(z2)|0〉0 = −iG (2) 0 (z1, z2) in Minkowski space, or 〈φ(z1)φ(z2)〉0 = G0(z1 − z2) in Euclidean space. (c) An overall factor of 1n! . (d) A multiplicity factor which counts the number of ways in which the lives can be joined without altering the topology of the graph. (e) Integrate over all internal coordinates. For example, the 4-point function G(4)(x1, x2, x3, x4) = 〈φ(x1)φ(x2)φ(x3)φ(x4)〉 (43) has the contribution at order λ2 shown in the diagram of figure 7. These two x1x1 x2x2 x3 x3 x4x4 y1y1 y2y2 + Figure 7: Two contributions to the four-point function to order λ2. diagrams have exactly the same weight (if G (2) 0 (1, 2) = G (2) 0 (2, 1)), and their total contribution to the 4-point function is 1 2! ( −λ 4! )2 S ∫ dDy1 ∫ dDy2 G (2) 0 (x1, y1) G (2) 0 (x2, y1) [ G (2) 0 (y1, y2) ]2 G (2) 0 (x3, y2) G (2) 0 (x4, y2) (44) but are topologically distinct and, thus, count as separate contributions. The multiplicity factor S is S = (4× 3)2 × 2. 12 += Figure 10: The Dyson equation for the two-point function. Here the thick lines are the full propagator, the thin line is the bare prop- agator, and the shaded blob represents the irreducible diagrams, i.e., diagrams with amputated external legs. We represent the blob by the self-energy operator Σ(p), shown in Fig.11 Thus, the total sum satisfies the Dyson equation G(2)(p) = G (2) 0 (p) + G (2) 0 (p) Σ(p) G (2)(p) (50) The inverse of G(2)(p), Γ(2)(p), satisfies Γ(2)(p) = G (2) 0 (p) −1 − Σ(p) = p2 + m2 − Σ(p) (51) To first order in λ, Σ(p) is just the tadpole term Σ(p) = − λ 2 ∫ dDq (2π)D 1 q2 + m2 + O(λ2) (52) which happens to be independent of the external momentum pµ. Of course, the higher order terms in general will be functions of pµ. In terms of the renormalized mass µ2, to order one-loop (i.e., O(λ)) we get µ2 = m2 − Σ(p) = m2 + λ 2 ∫ dDq (2π)D 1 q2 + m2 + O(λ2) (53) Thus, we conclude that vacuum fluctuations renormalize the mass. However, a quick look at Eq.(53) reveals that this is very large renormalization. Indeed, fluctuations of all momenta, ranging from long wave-lengths (and low energies) with q ∼ 0, to short wave-lengths (or high energies) contribute to the mass renormalization. In fact, the high-energy fluctuations, with q2 ≫ m2, yield the largest contributions to Eq.(53), since the mass effectively cuts-off the contri- butions in the infrared, IR q → 0. Moreover, for all dimensions D ≥ 2 the high-energy (or ultraviolet, UV, q →∞) contribution is divergent. If we were to cutoff the integral at a high-momentum scale Λ, in general space-time dimension . . .+++= Figure 11: Feynman diagrams summed by the Dyson equation. 15 D the diagram diverges as ΛD−2. In particular, the tadpole contribution to the mass renormalization is logarithmically divergent for D = 2 (1 + 1) dimensions, and quadratically divergent for D = 4 dimensions. Thus, although it is consistent (and quite physical) to regard the leading effect of fluctuations as a mass renormalization, they amount to a divergent change. The reason for this divergence is that all wavelengths contribute, from the IR to the UV. This happens since space-time is continuous: we assumed that there is no intrinsic short-distance scale below which local field theory would not be valid. There is a way to think about this problem. The problem of how to under- stand the physics of these singular contributions, indeed of how the continuum limit (a theory without cutoff) of quantum field theory is the central purpose of the Renormalization Group (RG). We will study this approach in detail next term. Here we will discuss some qualitative features. From the point of view of the RG the problem is that the continuum theory (i.e., defining a theory with- out a UV cutoff) cannot be done naively. We will see next term that for such a procedure to work it is necessary to be able to define the theory in a regime in which there is no scale, i.e., in a scale-invariant regime. This requirement means that one should look at a regime in which the renormalized mass becomes arbitrarily small, µ2 → 0. As we will see below, this requires to fine tune the bare coupling constant and the bare mass to some determined critical values. It turns out that, near such a critical point a continuum field theory (without a UV cutoff) can be defined. The RG point of view relates the problem of the definition of a Quantum Field Theory to that of finding a continuous phase transition, a central problem in Statistical Physics. However, there are alternative descriptions, such as String Theory, that pos- tulate that local field theory is not the correct description at short distances, typically near the Planck scale, ℓPlanck = √ ~G c3 ∼ 10 −33cm (!), where G is New- ton’s gravitation constant. From this view point, these singular contributions at high energies signal a breakdown of the theory at those scales. Before we try to compute Σ(p), it is worth to mention the Hartree Approxi- mation. It consists in summing up all tadpole diagrams (and only the tadpole diagrams) to all orders in λ. A typical graph is shown in figure 12. The sum of all the tadpole diagrams can be done by means of a very simple trick. Let us modify the expression for the self-energy to make it self consistent, i.e., Σ0(p) = − λ 2 ∫ dDq (2π)D 1 q2 + m2 − Σ0(q) (54) This formula is equivalent to a Dyson equation in which the internal propagator is replaced by the full propagator, as in figure 13. This approximation becomes exact for a theory of an N -component real scalar field φa(x) (a = 1, . . .N), with O(N) symmetry, and interaction Lint[φ] = λ 4! ( ( ~φ )2 )2 = λ 4! ( N ∑ a=1 φa(x)φa(x) )2 (55) 16 Figure 12: A typical tree diagram. in the large N limit, N →∞. Otherwise, the solution of this integral equation just yields the leading correction. Figure 13: The self-energy in the one-loop (Hartree) approximation. Eq.(54) is an integral non-linear equation for Σ0(p). Equations of this type are common in many-body physics. For example, the gap equation of the BCS theory of superconductivity has a similar form. Let us now evaluate the integral in the equation for Σ0(p), Eq.(54). Clearly Σ0(p) is a correction due to virtual fluctuations with momenta qµ ranging from zero to infinity. These fluctuations do not obey the mass shell condition p2 = m2. Notice that, at this level of approximation, Σ(p) is independent of the momentum. This is only correct to order one-loop. Before computing the integral, let us rewrite Eq.(54) in terms of the effective or renormalized mass µ2, µ2 = m2 − Σ(p) = m2 + λ 2 ∫ dDq (2π)D 1 q2 + µ2 (56) Let us denote by m2c the value of the bare mass such that µ 2 = 0: 0 = m2c + λ 2 ∫ dDq (2π)D 1 q2 (57) Clearly, m2c is IR divergent for D ≤ 2 and UV divergent for D ≥ 2. Let us now express the renormalized mass µ2 in terms of m2c , and define δm 2 = m2 −m2c . 17 12 The four point function and the effective cou- pling constant We will now discuss briefly the perturbative contributions to the four point function, G(4)(x1, x2, x3, x4) = 〈φ(x1)φ(x2)φ(x3)φ(x4)〉 (64) which is also known as the two-particle Green function. We will discuss its connection with the effective (or renormalized) coupling constant. To zeroth order in perturbation theory, O(λ0), the four point function fac- torizes into a product of all (three) possible two point functions obtained by pair-wise contractions of the four field operators. G(4)(1, 2, 3, 4) = 1 2 4 3 + 1 3 42 + 1 3 2 4 +O(λ) In other words, G(4)(x1, x2, x3, x4) = G0(x1, x3)G0(x2, x4) + G0(x1, x2)G0(x3, x4) + G0(x1, x4)G0(x2, x3) + O(λ) (65) As it is apparent, to zeroth order in λ, the four point function reduces to just products of bare two-point functions and hence nothing new is learned from it. We will show next semester that to all orders on perturbation theory the four-point function has the following structure: G(4)(x1, x2, x3, x4) = G(2)(x1, x3)G (2)(x2, x4) + G (2)(x1, x2)G (2)(x3, x4) + G (2)(x1, x4)G (2)(x2, x3) + ∫ dDy1 . . . d Dy4 G (2)(x1, y1)G (2)(x2, y2)G (2)(x3, y3)G (2)(x4, y4) Γ (4)(y1, y2, y3, y4) (66) where the factors of G(2)(x, x′) represent the exact two-point function, and the new four-point function, Γ(4)(y1, y2, y3, y4), is known as the four-point vertex function. The vertex function is defined as the set of one-particle irreducible (1PI) Feynman diagrams, i.e., diagrams that cannot be split in two by cutting a single propagator line, with the external lines “amputated” (they are already accounted for in the propagator factors). 20 In momentum space, due to momentum conservation at the vertex, Γ(4) has the form Γ(4)(p1, . . . , p4) = (2π) D δD ( 4 ∑ i=1 pi ) Γ(4)(p1, . . . , p4) (67) The lowest order contribution to Γ(4)(y1, y2, y3, y4) appear at order λ Γ(4)(y1, y2, y3, y4) = λ + O(λ 2) (68) depicted by the tree level diagrams: Γ(4) = +O(λ 2) λ which, in momentum space is Γ(4)(p1, . . . , p4) = λ + O(λ 2) (69) To one-loop order, O(λ2), the four-point vertex function is a sum of (three) Feynman diagrams of the form p1 p3 p2 p4 q p1 + p2 − q The total contribution to the vertex function Γ(4), to order one-loop, is Γ(4)(p1, . . . , p4) = λ− λ2 2 { ∫ dDq (2π)D 1 (q2 + m2) ((p1 + p2 − q)2 + m2) + two permutations } + O(λ3) (70) This expression has a logarithmic UV divergence in D = 4, and more severe divergencies for D > 4. To address this problem let us proceed by analogy 21 with the mass renormalization and define the physical or renormalized coupling constant g by the value of Γ(4)(p1, . . . , p4) at zero external momenta, p1 = . . . = p4 = 0. (It is up to us to define it at any momentum scale we wish.) g ≡ lim pi→0 Γ(4)(p1, . . . , p4) = Γ(4)(0, . . . , 0) (71) This definition is convenient and simple but it is problematic if the renormalized mass µ2 vanishes (i.e., in the massless or critical theory). To order one-loop, the renormalized coupling constant g is g = λ− 3 λ2 2 ∫ dDq (2π)D 1 (q2 + m2) 2 + O(λ 3) (72) Using the same line of argument we used to the dine the self energy, γ(2), we will now sum all the one loop diagrams, as shown in the figure: + + + . . . q q1 q2 q1 q2 q3 This “bubble” sum is a geometric series, and it is equivalent to the replacement of Eq.(72) by g = λ− 3 g2 2 ∫ dDq (2π)D 1 (q2 + m2) 2 + O(λ 3) (73) or, alternatively, to write the bare coupling constant λ in terms of the renor- malized coupling g as λ = g + 3 2 g2 ∫ dDq (2π)D 1 (q2 + µ2) 2 + O(g 3) (74) 22 A Integrals We introduce a momentum cutoff Λ and to suppress the contributions at large momenta, q ≫ Λ of integrals of the form ID ( µ2 Λ2 ) = ∫ dDq (2π)D 1 q2 + µ2 e − q2 Λ2 (80) where we used a Gaussian cutoff function (or regulator). We will only be inter- ested in the regime µ2 ≪ Λ2. Using a Feynman-Schwinger parametrization we can write ID ( µ2 Λ2 ) = ∫ ∞ 0 dα ∫ dDq (2π)D e − q2 Λ2 − α(q2 + µ2) = ∫ ∞ 0 dα e−αµ 2 ∫ dDq (2π)D e − ( 1 Λ2 + α ) q2 = e µ2 Λ2 ( µ2 ) D 2 −1 (4π)D/2 ∫ ∞ µ2/Λ2 dt t− D 2 e−t (81) Hence ID ( µ2 Λ2 ) = ( µ2 ) D 2 −1 (4π)D/2 Γ ( 1− D 2 , µ2 Λ2 ) e µ2 Λ2 (82) where Γ(ν, z) is the incomplete gamma function, with z = µ 2 Λ2 and ν = 1− D 2 , Γ(ν, z) = ∫ ∞ z dt tν−1e−t (83) and Γ(ν, 0) = Γ(ν) is the Gamma function Γ(ν) = ∫ ∞ 0 dt tν−1e−t (84) If the regulator Λ is removed (i.e., if we take the limit Λ→∞), ID(µ 2) formally becomes: ID(µ 2) = ( µ2 ) D 2 −1 (4π)D/2 Γ ( 1− D 2 ) (85) For general D, Γ(1 −D/2) is a meromorphic function of the complex variable D, and has simple poles for ν = 0 or any negative integer, Γ(1−D/2) has poles for D = 2, 4, 6, . . . In D = 4 dimensions, ν = −1 where Γ(ν) has a pole, the incomplete Gamma function at ν = −1 is (as z → 0) Γ(−1, z) = ( 1 z − ln 1 z ) e−z + γ (86) 25 where γ is the Euler-Mascheroni constant γ = − ∫ ∞ 0 dt e−t ln t = 0.5772 . . . (87) Hence, for µ2 ≪ Λ2, I4 is I4 ( µ2 Λ2 ) = Λ2 16π2 − µ2 8π2 ln ( Λ µ ) + γ 16π2 µ2 (88) Here we see that the leading singularity is quadratic in the regulator Λ, with a sub-leading logarithmic piece. In two dimensions I2 has instead a logarithmic singularity for µ 2 ≪ Λ2 I2 ( µ2 Λ2 ) = 1 4π Γ ( 0, µ2 Λ2 ) = 1 2π ln ( Λ µ ) − γ 4π (89) A second integral of interest is JD ( µ2 Λ2 ) = ∫ dDq (2π)D 1 q2 (q2 + µ2) = 1 µ2 ( ID(0)− ID ( µ2 Λ2 )) (90) JD(µ 2/Λ2) is UV finite if D < 4, where it is given by JD ( µ2 ) = ( µ2 ) D 2 −2 (4π)D/2 Γ ( D 2 − 2 ) D 2 − 1 (91) In four dimensions, J4, has a logarithmic divergence J4 ( µ2 Λ2 ) = 1 8π2 ln ( Λ µ ) − γ 16π2 (92) A third useful integral is I ′D( µ2 Λ2 ) = ∫ dDq (2π)D 1 (q2 + µ2) 2 e − q2 Λ2 = − ∂ID ∂µ2 (93) In the massless limit it becomes I ′D(0) = 1 (4π)D/2 4 (D − 2)(D − 4) ΛD−4 (94) In four dimensions it becomes I ′4( µ2 Λ2 ) = 1 8π2 ln ( Λ µ ) − ( γ + 1 16π2 ) (95) 26