H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4-Theories November 22, 2016 (/home/kleinert/kleinert/books/kleischu/regulkl.tex) 8 Regularization of Feynman Integrals
For dimensions close to D = 4, the Feynman integrals in momentum space derived in Chapter 4 do not converge since their integrands fall off too slowly at large momenta. Divergences arising from this short-wavelength region of the integrals are called ultraviolet (UV)-divergences. For massive fields, these are the only divergences of the integrals. In the zero-mass limit relevant for critical phenomena, there exists further divergences at small momenta and long wavelength. These so-called infrared (IR)-divergences will be discussed in Chapter 12. In this chapter we shall consider only UV-divergences. We shall therefore often omit speci- fying their UV character. The divergences can be controlled by various mathematical methods whose advantages and disadvantages will be pointed out, and from which we shall select the best method for our purposes. In principle, all masses and coupling constants occurring here ought to carry a subscript B indicating that the perturbative calculations are done starting from the bare energy functional EB[φB] with bare mass mB and bare field φB, introduced earlier in Section 7.3.1:
2 D 1 2 mB 2 EB[φB]= d x (∂φB) + φB , (8.1) Z "2 2 # and perturbing it with the bare interaction
int D λB 4 EB [φB] d x φB(x). (8.2) ≡ Z 4! However, in Chapter 9 we shall see that the renormalized quantities can eventually be calculated from the same Feynman integrals with the experimentally observable mass m and coupling constant λ. For this reason, the subscripts B will be omitted in all integrals.
8.1 Regularization
In four dimensions, the integrals of two- and four-point functions diverge. With the help of so-called regularization procedures they can be made finite. A regularization parameter is introduced, so that all divergences of the integrals appear as singularities in this parameter. There are various possible regularization procedures:
(a) Momentum cutoff Λ regularization In field descriptions of condensed matter systems, Feynman diagrams are regularized naturally at length scales a, where the field description breaks down. All momentum integrals are limited naturally to a region p < Λ = π/a, so that no UV-divergences | | 102 8.1 Regularization 103
can occur. Examples of Feynman integrals in a four-dimensional φ4-theory with cutoff regularization are
1 λ d4p 1 gm2 Λ2 Λ2 0 ✎☞ −1 = 4 2 2 = 2 2 log 2 + Λ , 2 Λ − 2 Λ (2π) p + m −32π "m − m # O ✍✌r Z 4 2 2 . . 0 3 ...... 3λ d p 1 3λ Λ ....✓✏.... −1 ...... r r. = 2 2 2 2 2 = 2 log 2 + Λ . 2 Λ − 2 Λ (2π) [(p k) + m ](p + m ) 32π " m # O ✒✑ Z − The subscript Λ of the diagrams emphasizes that the momentum integrals are carried out only up to p2 = Λ2. Both integrals are divergent for Λ . The first behaves for −→ ∞ large Λ like Λ2, and is called quadratically divergent. The second behaves like log Λ and is called logarithmically divergent. Phase transitions do not depend on the properties of the system at short distances and should therefore not depend on the cutoff. If a field theory is to give a correct description of the phase transition, it must be possible to go to the limit Λ at the end without → ∞ changing the critical behavior. The cutoff regularization has an undesirable feature of destroying the translational in- variance. Methods that do not suffer from this are
(b) Pauli-Villars regularization [1] In this case, convergence is enforced by changing the propagator in such a way that it decreases for p faster than before. This is done by the replacement | | → ∞ (p2 + m2)−1 (p2 + m2)−1 (p2 + M 2)−1 . (8.3) → − in which M 2 plays the role of a cutoff.
(c) Analytic regularization [2] The propagator is substituted by (p2 + m2)−z where z is a complex number with Re(z) large enough to make the integrals convergent. The result is then continued analytically to a region around the physical value z = 1. All divergences manifest themselves as simple poles for z = 1. Finite physical quantities for z = 1 can be defined by subtracting the pole terms.
(d) Dimensional regularization [3, 4, 5] Instead of changing the power of the propagator, the measure of momentum integration is changed by allowing the dimension D in the integrals to be an arbitrary complex number. This regularization will be introduced in detail in the next section and used in all our calculations. It was invented by ’t Hooft and Veltman to regularize nonabelian gauge theories where all previous cutoff methods failed. There are several attractive features of dimensional regularization. First, it preserves all symmetries of the theory, in particular gauge symmetry. Second, it allows an easy identification of the divergences. Third, it suggests in a natural way a minimal subtraction scheme (MS scheme), that greatly simplifies the calculations. Fourth, it regularizes at the same time IR-divergences in massless theories, as will be discussed in Section 12.3. A difficulty with dimensional regularization is the treatment of certain tensors whose definition does not permit an analytic extrapolation to an arbitrary complex number 104 8 Regularization of Feynman Integrals
of spatial dimensions, the most prominent example being the completely antisymmetric tensor εαβγδ. Fortunately, this tensor does not appear in the theories to be discussed in this text, so that dimensional regularization can be applied without problem.
8.2 Dimensional Regularization
−2 The integrand in the diagram ❧ behaves for large loop momenta p like p , and the ...... q −4 | | integrand in the diagram ...... like p . The momentum integrals are therefore defined only q q❧ | | for dimensions D < 2 and D < 4 respectively. The idea is to calculate a Feynman integral for a continuous-valued number of dimensions D for which convergence is assured. For this the Feynman integrals for integer D must be extrapolated analytically to complex D. The concept of a continuous dimension was introduced by Wilson and Fisher [6], who first calculated physical quantities in D = 4 ε dimensions with Re ε > 0, and expanded − them in powers of the deviation ε from the dimension D = 4. This concept was subsequently incorporated into quantum field theory [7], giving rise to many applications in statistical physics. The dimensional regularization by ’t Hooft and Veltman was the appropriate mathematical tool for such expansions. We shall first derive all formulas needed for the upcoming calculations. The analytic extra- polation to noninteger dimension will be based on the extrapolation of the factorial of integer numbers to real numbers by the Gamma function. In Subsection 8.2.2 we shall describe another approach in which D-dimensional Gaussian integrals are used to arrive at the same formulas. For completeness, the original procedure of ’t Hooft and Veltman will be reviewed in Sub- section 8.2.4, to be followed in Subsection 8.2.5 by a slightly different method of Collins [8], who introduced integrals in continuous dimensions D via a certain subtraction method.
8.2.1 Calculation in Dimensional Regularization To explain the method of dimensional regularization, consider first the simplest Feynman inte- gral dDp 1 I(D)= D 2 2 . (8.4) Z (2π) p + m It is UV-divergent for D 2, and IR-divergent for D 0. After introducing polar coordinates ≥ ≤ as explained in Appendix 8A, it can be rewritten as
D D−2 π ∞ d p 1 2π k D−1 1 I(D) = = sin ϑk dϑk dp p (8.5) (2π)D p2 + m2 (2π)D 0 0 p2 + m2 Z kY=1 Z Z ∞ SD D−1 1 = D dp p 2 2 , (8.6) (2π) Z0 p + m wherey 2πD/2 S = (8.7) D Γ(D/2) is the surface of a unit sphere in D dimensions [recall (1.98)]. The resulting one-dimensional integral can, after the substitution p2 = ym2, be cast into the form of an integral for the Beta function Γ(α)Γ(γ) ∞ B(α, γ) = dy yα−1(1 + y)−α−γ, (8.8) ≡ Γ(α + γ) Z0
4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 8.2 Dimensional Regularization 105 where Γ(z) is the Gamma function with the integral representation
∞ Γ(z)= dt tz−1e−t. (8.9) Z0 We then find
∞ ∞ SD D−1 1 SD 2 D/2−1 D/2−1 −1 I(D) = D dp p 2 2 = D (m ) dy y (1 y) (2π) Z0 p + m 2(2π) Z0 − 1 Γ(D/2)Γ(1 D/2) (m2)D/2−1 = (m2)D/2−1 − = Γ(1 D/2). (8.10) (4π)D/2Γ(D/2) Γ(1) (4π)D/2 −
The Gamma function provides us with an analytical extrapolation of the integrals in integer dimensions D to any complex D. In general, Feynman integrals contain more complicated denominators than just p2 + m2. They may, for instance, contain another momentum q:
dDp 1 q I(D; )= D 2 2 . (8.11) Z (2π) p +2pq + m This integral can be reduced to the previous one in (8.4) by completing the squares in the denominator, yielding
dDp 1 dDp 1 1 q 2 q2 D/2−1 I(D; )= D 2 2 = D 2 2 2 = D/2 (m ) Γ(1 D/2). Z (2π) p +2pq+m Z (2π) p +m q (4π) − − − (8.12) By differentiating this with respect to the mass, we get a formula for arbitrary integer powers of such propagators:
dDp 1 1 Γ(a D/2) q 2 q2 D/2−a I(D, a; )= D 2 2 a = D/2 − (m ) , (8.13) Z (2π) (p +2pq + m ) (4π) Γ(a) − which can be extended analytically to arbitrary complex powers a. We may differentiate this equation with respect to the external momentum q, and obtain a further formula dDp p 1 Γ(a D/2) q Iµ(D, a; q) µ = − µ . (8.14) ≡ (2π)D (p2 +2pq + m2)a (4π)D/2 Γ(a) (m2 q2)a−D/2 Z − More differentiations with respect to q yield formulas with higher tensors qµ1 qµn in the · · · integrand. For products of different propagators, the integrals are reduced to the above form with the help of Feynman’s parametric integral formula:
1 Γ(a + b) 1 xa−1 (1 x)b−1 = dx − , (8.15) Aa Bb Γ(a)Γ(b) 0 [Ax + B(1 x)]a+b Z − which is a straightforward generalization of the obvious identity
1 1 1 1 1 B 1 1 1 = = dz = dx . (8.16) AB B A A − B B A A z2 0 [Ax + B(1 x)]2 − − Z Z − 106 8 Regularization of Feynman Integrals
Differentiation with respect to A and B yields (8.15) for integer values of the powers a and b, and the resulting equation can be extrapolated analytically to arbitrary complex powers. More generally, Feynman’s formula reads:
1 1 1 δ(1 x1 + + xn) = Γ(n) dx1 dxn − · · · n , (8.17) A A 0 · · · 0 (x A + ... + x A ) 1 · · · n Z Z 1 1 n n as can easily be proved by induction [10]. By differentiating both sides ai times with respect to Ai, one finds
1 1 a1−1 a2−1 an−1 1 Γ(a1 + ... + an) δ(1 x1 + + xn)x1 x2 xn = dx1 dxn − · · · · · · · , a1 an a1+...+an A1 An Γ(a1) Γ(an) Z0 · · · Z0 (x1A1 + ... + xnAn) · · · · · · (8.18) valid for integer a1,...,an. By analytic extrapolation, this formula remains valid for complex values of ai. The formula is, of course, only true as long as the integrals over xi converge. As an example for the use of the Feynman parametrization, take the D-dimensional mo- mentum integral containing two propagators of arbitrary power. This reduces to the single integral dDp 1 Γ(a + b) 1 dDp (1 x)a−1 xb−1 = dx − (2π)D (p2 + m2)a[(p k)2 + m2]b Γ(a)Γ(b) 0 (2π)D [p2 + m2 2pk x + k2 x]a+b Z − Z Z − 1 Γ(a + b) 1 (1 x)a−1 xb−1 = dx − . (8.19) (4π)D/2 Γ(a)Γ(b) 0 [m2 + k2 x (1 x)]a+b−D/2 Z − 8.2.2 Dimensional Regularization via Proper Time Representation Schwinger observed that all propagators may be rewritten as Gaussian integrals by using a so- called proper time representation (see Appendix 8C) of Feynman integrals, also referred to as parametric representation. This permits defining momentum integrals in D complex dimensions via a generalization of the Gaussian integral to D complex dimensions [4, 11, 12, 13], since the latter possesses a rather straightforward analytic continuation with the help of Gamma functions. This method is closely related to what is called analytic regularization, and many properties are common to the two approaches. In Schwinger’s proper time representation, each scalar propagator is rewritten as an integral
∞ 1 −τ(p2+2pq+m2) 2 2 = dτ e . (8.20) p +2pq + m Z0 The variable τ is called proper time for reasons which are irrelevant to the present development. With the integral representation of the Gamma function (8.9), this can be generalized to
∞ 1 1 a−1 −τ(p2+2pq+m2) 2 2 a = dτ τ e , (8.21) (p +2pq + m ) Γ(a) Z0 valid for a> 0. We now assume that in a typical D-dimensional Feynman integral
D D ∞ d p 1 1 d p a−1 −τ(p2+2pq+m2) D 2 2 a = D dτ τ e , (8.22) Z (2π) (p +2pq + m ) Γ(a) Z (2π) Z0 the integral over the proper time can be exchanged with the momentum integral, leading to
∞ D ∞ D 1 a−1 d p −τ(p2+2pq+m2) 1 a−1 −τ(m2−q2) d p −τp2 dτ τ D e = dτ τ e D e . (8.23) Γ(a) Z0 Z (2π) Γ(a) Z0 Z (2π)
4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 8.2 Dimensional Regularization 107
The D-dimensional Gaussian integral on the right-hand side can easily be done and yields:
D D D/2 d p −τp2 dp −τp2 1 D e = e = . (8.24) Z (2π) "Z 2π # 4πτ The important point is now that this integral can easily be generalized from integer values of D to any complex number D of dimensions. While the left-hand side of (8.24) makes initially sense only if D is an integer number, the right-hand side exists for any complex D, so that Eq. (8.24) can be used as a definition of the Gaussian integral on the left-hand side for complex dimensions D. Inserting (8.24) into (8.23), we are left with a proper-time integral which can be evaluated using the integral representation (8.9) for the Gamma function:
∞ 2 2 1 dτ τ a−D/2−1e−τ(m −q ) = Γ(a D/2) . (8.25) 0 − (m2 q2)a−D/2 Z − The left-hand side is defined only for D < 2a, but the Gamma function possesses a unique analytic continuation to larger D. In this way we find for the Feynman integral (8.22): dDp 1 1 Γ(a D/2) 1 = − . (8.26) (2π)D (p2 +2pq + m2)a (4π)D/2 Γ(a) (m2 q2)a−D/2 Z − By expanding this integral in powers of a up to order (a) and comparing the coefficients of a we find a further important integral which will be neededO for the calculation of vacuum energies in Eqs. (8.116) and (10.131): dDp 1 2 k2 kq 2 2 q2 D/2 D log( +2 + m ) = D/2 Γ(1 D/2) (m ) , (8.27) Z (2π) (4π) D − − Strictly speaking, the logarithm on the left-hand side does not make sense since its argument has the dimension of a square mass. It should therefore always be written as log[(k2 +2kq+m2)/µ2] with some auxiliary mass µ. If m = 0, the auxiliary mass µ can, of course, be taken to be m 6 itself. In dimensional regularization, however, this proper way of writing the logarithm does not change the integral at all, as we shall soon demonstrate. The reason is that it merely adds an integral over a constant, and this vanishes by the so-called Veltman formula, to be derived in Eq. (8.33). A proper-time representation exists for any Feynman integral. Each propagator is replaced by an integral over τi. As a result, all momentum integrals are of the type (8.26), (8.14), and their straightforward generalizations. The UV-divergences arise then from the integration region τ 0, which is regulated by the analytic continuation of the Gamma function. i ≈ Massless Tadpole Integrals Let us now derive a result which will turn out to be important for the evaluation of massless Feynman integrals associated with so-called tadpole diagrams. Tadpole diagrams are quadrati- cally divergent diagrams which have only one external vertex. They contain therefore no exter- 4 nal momenta. In the φ -theory, the simplest tadpole diagram is ❤. The detailed discussion of these diagrams will take place in Section 11.4, in particular theirq role in the renormalization process. Consider the following Feynman integral: dDp m2 dDp 1 1 D 2 2 2 = D 2 2 2 . (8.28) Z (2π) p (p + m ) Z (2π) "p − (p + m )# 108 8 Regularization of Feynman Integrals
The left hand side is calculated with the help of Schwinger’s proper-time integral as D 2 ∞ 2 2 ∞ 2 2 d p −(τ1+τ2)p −τ2m m −D/2 −τ2m m D dτ1 dτ2 e = D/2 dτ1 dτ2 (τ1 + τ2) e Z (2π) Z0 (4π) Z0 2 2 −D/2+1 ∞ ∞ 2 ∞ 2 m −D/2 −τ2m m τ2 −τ2m = D/2 dτ2 dτ12 τ12 e = D/2 dτ2 − e (4π) 0 τ2 (4π) 0 1 D/2 Z Z Z − (m2)D/2−1 = Γ(1 D/2). (8.29) − (4π)D/2 − The same result is obtained by applying Eq. (8.26) to the second term on the right-hand side of (8.28): dDp 1 (m2)D/2−1 D 2 2 = D/2 Γ(1 D/2). (8.30) − Z (2π) p + m − (4π) − This implies that for any complex D, the first term on the right-hand side of (8.28) must vanish: dDp 1 D 2 =0. (8.31) Z (2π) p After a polar decomposition of the measure of integration as in Eq. (8.6), the integral (8.31) reads ∞ SD D−3 D dk k =0. (8.32) (2π) Z0 The vanishing of this for all D implies, again via a polar decomposition, the vanishing of the more general integral dDp p2 κ D ( ) =0 for κ, D complex. (8.33) Z (2π) This is known as Veltman’s formula [9]. Actually, it could have been derived before from Eq. (8.26) by taking the limit a 0, which is well defined for all D =2, 4,... : → 6 dDk =0. (8.34) Z By analytic interpolation, it is expected to hold for all D. The consistency of (8.33) has been discussed in various ways [8, 9]. It is nontrivial owing to the absence of any regime of convergence as a function of D, so that no proper mathematical analytic continuation can be invoked. Leibbrandt introduced an extended Gaussian integral containing an auxiliary mass term which permits taking simultaneously the limits m2 0 and D 4. Below we shall give further arguments for the validity of (8.33) on the basis of→ Collins’ → subtraction method in Subsections 8.2.4 and 8.2.5. The vanishing of the integrals over simple powers of the momentum has the pleasant con- sequence that in the course of renormalization via the so-called minimal subtraction procedure it will be superfluous to calculate diagrams which contain a tadpole, as will be explained in Sections 11.4 and 11.7–11.8 (see also on page 208).
8.2.3 Tensor Structures The generalization of tensors from integer space dimension D to complex values of D pro- ceeds by replacing four-vectors by D-vectors. The integrations are performed and the results interpolated analytically. Consider for instance the integral
D −τp2 d ppµe =0, Z
4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 8.2 Dimensional Regularization 109 which vanishes in integer dimension. This integral is defined to be zero for all D. As another example, we evaluate
dDp p 1 µ (8.35) (2π)D (p2 + m2)a [(p k)2 + m2]b Z − k Γ(a + b D/2) 1 D/2−a−b = µ − dτ τ b(1 τ)a−1 k2τ(1 τ)+ m2 . (4π)D/2 Γ(a)Γ(b) 0 − − Z h i When dealing with Feynman integrals of second or higher order we frequently encounter the unit tensor. For consistency, its trace which is defined initially only for integer values of D must be assumed to have a continuous value D for arbitrary complex D:
δµµ = D. (8.36) µ X This relation can be used to simplify integrals over tensors in momentum space:
dDp 1 dDp p2 p2 p2 D pµpν f( )= D δµν f( ). (8.37) Z (2π) D Z (2π)
The correctness of this can be verified by contraction with δµν and using the previous integration rules in a complex number D of dimensions.
8.2.4 Dimensional Regularization of ’t Hooft and Veltman We are now going to review the construction of an analytical continuation in the number of dimensions originally used by ’t Hooft and Veltman [3, 4, 5]. This regularization procedure may be exemplified using the integral (8.4), which is obviously UV-divergent for D 2 and ≥ IR-divergent for D 0. It can be used to define the integration in a continuous number of ≤ dimensions D in the region D < 2. In order to extrapolate the integral to a larger domain, ’t Hooft and Veltman introduced a procedure called “partial p”. This is based on inserting into the integrand the unit differential operator 1 ∂p =1, (8.38) D ∂p and carrying out a partial integration in the region 0 1 D ∂ 1 I(D) = d ppi 2 2 (8.39) −D Z ∂pi p + m 2 SD D−1 2p = dp p 2 2 2 . (8.40) − D Z (p + m ) Inserting m2 m2 and re-expressing the right-hand side by I(D) leads to − 2 S m2 1 I(D)= D dp pD−1 . (8.41) D 2 (p2 + m2)2 − Z The region of convergence of the integral is now extended to 0 ∞ 2 D/2−2 D−1 1 (m ) Γ(D/2)Γ(2 D/2) dp p 2 2 2 = − . (8.42) Z0 (p + m ) 2 Γ(2) The UV-divergence of the integral is reflected in the pole of the Gamma function at D = 4. After including the prefactor of (8.41) and the definition (8.7) of SD, we obtain D 1 D/2 2 D/2−1 I(D)= d p 2 2 = π (m ) Γ(1 D/2). (8.43) Z p + m − The Gamma function is analytic in the entire complex-D plane, except for isolated poles at D = 2, 4,... . Recall that the same analytic expression was obtained by a naive evaluation of (8.6) in (8.10), without an explicit consideration of ranges of convergence of the momentum integrals. The considerations of ’t Hooft and Veltman justify further the simple direct result. The above analytic continuation in the dimension of a specific class of integrals can now easily be extended to more general Feynman integrals. There the integrand will not only depend on the magnitude of the momentum p but also on its direction, which may point in some n- dimensional subspace of the D-dimensional space, where n is an integer number n 4. Let ≤ this space be spanned by unit vectors qi, (i =1,...,n). The integrand is then some function 2 f(p , p q1, p q2,..., p qn). Now ’t Hooft and Veltman split the measure of integration into · · n· D−n an n-dimensional part d pk and a remainder d p⊥, and write the integral as dDp f(p2, p q , p q ,..., p q ) (8.44) · 1 · 2 · n Z ∞ D−n 2 2 = dp1 dpn d p⊥f(pk, p⊥, p q1, p q2,..., p qn). Z−∞ · · · Z · · · The integration over the n-dimensional subspace is an ordinary integration in integer dimen- sions. The remaining D n -dimensional integration is integration in continuous dimensions. − The function f is independent of the direction of p⊥ because the scalar products p q1,..., p qn 2 2 · 2 · 2 depend only on p1,..., pn, such that p⊥ appears only in the argument p = p1 + ... + pn + p⊥. The D n -dimensional integration over p is therefore rotationally invariant. The angular in- − ⊥ tegration can now be carried out in arbitrary dimensions D n, leaving only a one-dimensional − radial integral: dD−np f(p2, p2 , p q , p q ,..., p q ) ⊥ k ⊥ · 1 · 2 · n Z ∞ D−n−1 2 2 = SD−n dp⊥ p⊥ f(pk, p⊥, p q1, p q2,..., p qn), (8.45) Z0 · · · where SD−n is the surface of a unit sphere in D n dimensions [see (8.7)]. Together with the remaining integrations in (8.44)− we have (D−n)/2 ∞ D 2 2π d p f(p , p q1,..., p qn) = dp1 ...dpn · · Γ((D n)/2) −∞ Z − Z D−n−1 2 2 dp⊥p⊥ f(pk, p⊥, p q1,..., p qn). (8.46) × Z · · The splitting of the measure produces artificial IR-divergences in the remaining radial integral D−n−1 via the factor p⊥ . In ’t Hooft and Veltman’s approach, these are eliminated by partial 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 8.2 Dimensional Regularization 111 integration throwing away the surface terms. This procedure is shown in Appendix 8B for the simple integral of Eq. (8.4) in which the splitting of the measure also generates artificial IR- divergences. Only after generating a finite domain of convergence can the procedure “partial p” be used to go to higher D. The results show again that the Gamma function provides us with a universal analytic continuation in D. Note that, for brevity, we have omitted here the typical factor 1/(2π)D in the measure of all momentum integrations since they are irrelevant to the above arguments. This factor will again be present in all subsequent calculations. 8.2.5 Subtraction Method Another procedure for the analytic extrapolation of Feynman integrals in D dimensions was presented by Collins [8]. He differs from ’t Hooft and Veltman by giving an explicit procedure to subtract the artificial IR-divergences which disappear in the previous approach by discarding the surface terms. For simplicity, consider a rotationally invariant integrand f(p2), in which case we may take n = 0 in Eq. (8.44). The integrand is supposed to fall off sufficiently fast for large p to give the one-dimensional integral in Eq. (8.45) a finite region of convergence 0 < D < D′. For the analytic extrapolation to smaller D < 0, the subtraction procedure consists in adding and subtracting the leading orders of an expansion of the integrand around p2 = 0, which contain the IR-divergences. The continuation to 2 < Re D < D′ is given by − dDp 2(4π)−D/2 ∞ p2 D−1 2 D f( ) = dp p f(p ) (8.47) Z (2π) Γ(D/2) ZC C CD + dp pD−1 f(p2) f(0) + f(0) . 0 − D ) Z h i The integral over f(p2) f(0) p2f (1)(0) + (p4) obviously converges at the origin for 2 < − ∼ O − Re D < 0. This formula holds initially only for 0 < D < D′. We now derive the left-hand side by extrapolating the right-hand side analytically to D < 0. There, the limit C can be → ∞ taken since f(0)CD/D 0, leaving → dDp 2(4π)−D/2 ∞ f(p2) = dp pD−1 f(p2) f(0) , (8.48) (2π)D Γ(D/2) 0 − Z Z h i which is a convergent definition for the left-hand side valid for 2 < Re D< 0. Repeated application of the subtraction procedure leads to a− continuation formula for 2l − − 2 < Re D< 2l (l =0, 1,...): − 2(4π)−D/2 ∞ f (l)(0) dDp f(p2)= dp pD−1 f(p2) f(0) p2f ′(0) p2l . (8.49) Γ(D/2) 0 − − −···− l! Z Z h i After a sufficient number of steps, the one-dimensional integral will be convergent and can be evaluated with the help of the integral representation of the Beta function (8.8) and the Gamma function (8.9). This, in turn, can be extrapolated analytically to higher D, thereby exhibiting poles at isolated values of D. This continuation to higher D is found to be consistent with the subtraction formula as all subtraction levels lead to the same Gamma function. We demonstrate this by treating the one-loop integral (8.4): dDp 1 I(D)= D 2 2 , (8.50) Z (2π) (p + m ) 112 8 Regularization of Feynman Integrals which is UV-divergent for D 2 and becomes IR-divergent for D 0 upon splitting the mea- sure of integration in Eq. (8.6).≥ Expressing the integral directly in≤ terms of Gamma functions as in (8.10) gives dDp 1 (m2)D/2−1 I(D)= D 2 2 = D/2 Γ(1 D/2). (8.51) Z (2π) p + m (4π) − To find a definition for the integral for 2 < Re D< 0, we use the subtraction formula (8.48). − We derive D d p 1 SD D−1 1 1 I(D) = D 2 2 = D dp p 2 2 2 Z (2π) (p + m ) (2π) Z p + m − m ! 2 2 D/2−1 ∞ SD D−1 p (m ) D/2 −1 = D dp p 2 −2 2 = − D/2 dy y (1 + y) (2π) Z m (p + m ) (4π) Γ(D/2) Z0 (m2)D/2−1 (m2)D/2−1 Γ(1 D/2) = − B(D/2+1, D/2) = − . (8.52) (4π)D/2Γ(D/2) − (4π)D/2 Γ(1) This is the same result as for the unsubtracted formula (8.51). The same is true for higher subtractions like (p2/m2)2, implying that the integration over any pure power of p2 gives zero in this regularization. These manipulations show again that the naive integration in Subsection 8.2.1 performed without any subtractions provides us with the desired analytic extrapolation to any complex dimension D. 8.3 Calculation of One-Particle-Irreducible Diagrams up to Two Loops in Dimensional Regularization In order to illustrate dimensional regularization, we shall now calculate explicitly the diagrams (2) (4) (0) in Γ (k) and Γ (ki), and Γ up to two loops: 1 1 ✎☞ 1 Γ(2)(k) = k2 + m2 ✎☞+ + ✓✏, (8.53) − 2 4 ✎☞✍✌r 6 r r ✍✌r ✒✑ ✍✌r ...... 3 ...... 3 ...... 3 (4) ...... ✓✏...... ❥ ...... ✓✏r ...... Γ (k ) = ...... 3 .....✓✏r ...... (8.54) i r♠ r r♠ ...... − r − 2 r r − r r − 4 − 2 .. .. ✒✑ r r ✒✑ ✒✑ (0) 1 ✓✏ 1 ✓✏✓✏ Γ = + s . (8.55) 2 ✒✑ 8 ✒✑✒✑ In Subsection 8.3.4, we shall also do the calculation for the vacuum diagrams. The results will be given a power series in ε, with the divergences appearing as pole terms in ε. 8.3.1 One-Loop Diagrams On the one-loop level, there are only two divergent diagrams. The Feynman integral associated with the diagram ❧ is divergent for D 2. q ≥ dDp 1 ❧ = λ D 2 2 . (8.56) q − Z (2π) p + m 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 8.3 Calculation of One-Particle-Irreducible Diagrams 113 Using Eq. (8.27), we find (m2)D/2−1 ❧ = λ D/2 Γ(1 D/2). (8.57) q − (4π) − The Feynman integral is UV-divergent in two, four, six, ... dimensions, which is reflected by poles in the Gamma function at D = 2, 4, 6,... . The poles can be subtracted in various ways, parametrized by an arbitrary mass parameter µ. Introducing the dimensionless coupling constant g [similar to the dimensionless λˆ = λmD−4 of Eq. (7.28)] g λµD−4 = λµ−ε, (8.58) ≡ the integral reads in terms of g and ε 2 ε/2 2 g 4πµ ♠ = m Γ(ε/2 1) . (8.59) q − (4π)2 m2 ! − The arbitrary mass parameter µ appears in a dimensionless ratio with the mass. It is this kind of terms which contains IR-divergences in the limit m2 0. They are expanded in powers of ε: → 4πµ2 ε/2 ε 4πµ2 =1+ log + (ε2) . (8.60) m2 ! 2 m2 ! O The ε-expansion of the Gamma function in (8.59) reads (see Appendix 8D): ( 1)n 1 ε π2 Γ( n + ε)= − + ψ(n +1)+ + ψ2(n + 1) ψ′(n + 1) + (ε2) , (8.61) − n! ( ε 2 " 3 − # O ) where ψ(z) Γ′(z)/Γ(z) is the Euler Digamma function. Inserting these into (8.59) we find ≡ the Laurent expansion in ε: 2 2 g 2 4πµ ♠ = m + ψ(2) + log + (ε) . (8.62) q (4π)2 " ε m2 ! O # The residue of the pole is proportional to m2 and independent of µ...... The integration over two propagators in .. q q❧.. is convergent for D< 4: D ...... d p 1 1 ...... 2 q q❧ = λ D 2 2 2 2 , (8.63) Z (2π) p + m (p + k) + m where the external momentum k is the sum of the incoming momenta: k = k1 + k2. Using formula (8.16), we can rewrite the Feynman integral (8.63) with Feynman parameters as 1 D ...... d p 1 ...... 2 q q❧ = λ dx 2 (8.64) 0 (2π)D (p2 + m2)(1 x) + [(p + k)2 + m2] x Z Z { − } 1 dDp 1 = λ2 dx (8.65) D 2 2 2 2 Z0 Z (2π) (p +2pkx + k x + m ) λ2 Γ(2 D/2) 1 1 = − dx . (8.66) (4π)D/2 Γ(2) 0 [k2x(1 x)+ m2]2−D/2 Z − The divergence for D = 4 is contained in the Gamma function which possesses poles at D = 4, 6,.... The remaining parameter integral is finite for any D as long as m2 = 0. In terms of ε 6 and g, the expression for the simple loops reads 1 2 ε/2 ...... g 4πµ ...... ε . q q❧. = gµ Γ(ε/2) dx . (8.67) (4π)2 0 "k2x(1 x)+ m2 # Z − 114 8 Regularization of Feynman Integrals In order to separate the pole terms, we expand each term in powers of ε. The Gamma function is expanded with (8.61) to yield 1 2 ...... g 2 ε 4πµ ...... ε 2 . q q❧. = gµ + ψ(1) + (ε) 1+ dx log + (ε ) (4π)2 ε O ( 2 0 "k2x(1 x)+ m2 # O ) Z − g 2 1 4πµ2 = gµε + ψ(1) + dx log + (ε) . (8.68) (4π)2 ( ε 0 "k2x(1 x)+ m2 # O ) Z − The prefactor gµε is, in fact, the coupling constant λ which will be renormalized, and is not expanded in powers of ε. Only the expression multiplying it contributes to the renormalization constant Zg with a pole term that is independent of the arbitrary mass parameter µ. This mass parameter appears only in the finite part, where the freedom of its choice exhibits a degree of freedom in the renormalization procedure. The result (8.66) can be generalized to Feynman integrals of the type (8.63), in which the denominators appear with an arbitrary power. Using Eqs. (8.15) and (8.26), we find dDp 1 1 (8.69) (2π)D (p2 + m2)a [(p k)2 + m2]b Z − Γ(a + b D/2) 1 D/2−a−b = − dx xb−1(1 x)a−1 k2x(1 x)+ m2 . (4π)D/2Γ(a)Γ(b) 0 − − Z h i The parameter integral on the right-hand side develops IR-divergences for m2 = 0 and D =4 if a or b 2. ≥ 8.3.2 Two-Loop Self-Energy Diagrams On the two-loop level, two diagrams contribute to the self-energy: ❡ and . The Feynman ❡q q ❤q integral associated with the first factorizes into two independent momentumq integrals: dDp dDp 1 1 ❤ = λ2 1 2 . (8.70) ❤q D D 2 2 2 2 2 q Z (2π) (2π) p1 + m (p2 + m ) The integral over p1 coincides with the previous integral (8.56), which is expanded in powers of ε in Eq. (8.62). The integral over p2 is calculated using formula (8.26): D ε d p2 1 gµ Γ(2 D/2) 1 λ D 2 2 2 = D/2 − 2 2−D/2 Z (2π) (p2 + m ) (4π) Γ(2) (m ) g 4πµ2 ε/2 = Γ(ε/2) (4π)2 m2 ! g 2 4πµ2 = + ψ(1) + log + (ε) , (8.71) (4π)2 " ε m2 O # after expanding the Gamma function according to formula (8.61). This result can also be 2 found directly in (8.68) for k = 0. The result of the integration over p1 is given in (8.62). The product of the integrations over p1 and p2 gives m2g2 4 ψ(1) + ψ(2) 4 m2 ❤ 0 q = 4 2 +2 log 2 + (ε ) . (8.72) q❤ −(4π) "ε ε − ε 4πµ ! O # 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 8.3 Calculation of One-Particle-Irreducible Diagrams 115 We come now to the calculation of the so-called sunset diagram: D D 2 d p1 d p2 1 1 1 q q♠ = λ D D 2 2 2 2 2 2 , (8.73) Z (2π) (2π) p1 + m p2 + m (q + p1 + p2) + m in which q is the incoming momentum. The calculation of q ❤q is rather difficult, because a naive introduction of the parameter integrals results in divergences of the parameter integral. The problem is solved by lowering the degree of divergence via partial integration. Using the trivial identity ∂pµ/∂pν = δµν and the trace property (8.37), we see that µ µ 1 ∂p1 ∂p2 1= µ + µ . (8.74) 2D ∂p1 ∂p2 ! Inserting this identity into (8.73), and performing a partial integration in which the surface term is discarded, we obtain a sum of two integrals: λ2 dDp dDp 3m2 + q(q + p + p ) = 1 2 1 2 q q♠ −D 3 (2π)D (2π)D (p2 + m2)(p2 + m2)[(q + p + p )2 + m2]2 − Z 1 2 1 2 λ2 = 3m2A(q)+ B(q) . (8.75) −D 3 − h i In this way, the original quadratically divergent integral is decomposed into a logarithmically divergent integral A(q) and a linearly divergent integral B(q). Let us first evaluate the integral for A(q): dDp dDp 1 q 1 2 A( )= D D 2 2 2 2 2 2 2 . (8.76) Z (2π) (2π) (p1 + m )(p2 + m )[(q + p1 + p2) + m ] Replacing the momentum p2 by p q p1, the integral over p1 coincides with that in the ...... − − − diagram . q q❧. in Eq. (8.63). It can be performed in the same way, and we obtain 1 Γ(2 D/2) 1 dDp 1 A(q)= − dx . (8.77) (4π)D/2 Γ(2) 0 (2π)D [(q + p)2x(1 x)+ m2]2−D/2(p2 + m2)2 Z Z − Applying the Feynman formula (8.15), this becomes 1 1 1 dDp y(1 y)1−D/2 q D/2−2 A( )= D/2 Γ(4 D/2) dx[x(1 x)] dy D − 4−D/2 , (8.78) (4π) − Z0 − Z0 Z (2π) [f(q, p,x,y)] with m2 f(q, p,x,y) = (p2 + m2)y + (p + q)2 + (1 y) " x(1 x)# − − 1 y = p2 +2pq(1 y)+ q2(1 y)+ m2 y + − . (8.79) − − " x(1 x)# − The momentum integral in (8.78) is now carried out with the help of formula (8.26), leading to Γ(4 D) 1 1 y(1 y)1−D/2 q D/2−2 A( )= −D dx[x(1 x)] dy − 4−D . (8.80) (4π) Z0 − Z0 q2y(1 y)+ m2 y + 1−y − x(1−x) h i 116 8 Regularization of Feynman Integrals Inserting D =4 ε and expanding the denominator in ε, we find − Γ(ε) 4π ε 1 1 q −ε/2 ε/2−1 A( ) = 4 2 dx[x(1 x)] dy y(1 y) (4π) m Z0 − Z0 − q2 1 y 1 ε log y(1 y)+ y + − + (ε2) . (8.81) × ( − "m2 − x(1 x)!# O ) − The parameter integrals without the curly brackets can be evaluated with the help of the integral formula [note the difference with respect to (8.8)]: Γ(α)Γ(β) 1 B(α, β) = dy yα−1(1 y)β−1, (8.82) ≡ Γ(α + β) Z0 − and the expansion derived in Appendix 8D: ε2 Γ(n +1+ ε)= n! 1+ εψ(n +1)+ ψ′(n +1)+ ψ(n + 1)2 + (ε3) , (8.83) ( 2 O ) h i yielding 1 Γ(1 ε/2)Γ(1 ε/2) dx x−ε/2(1 x)−ε/2 = − − =1+ ε + (ε2), (8.84) 0 − (1 ε)Γ(1 ε) O Z − − 1 Γ(2)Γ(ε/2) 2 2 dy y(1 y)ε/2−1 = = = 1+ (ε). (8.85) Z0 − Γ(2 + ε/2) ε(1 + ε/2) ε − O Only the latter integral is singular for ε 0. The singularity comes from the endpoint at y = 1. Using this we find for A(q) from the→ first term of the curly bracket of (8.81): 1 4π ε Γ(1 + ε) 2 A(q)= 1+ (ε) 1+ ε + (ε2) . (8.86) (4π)4 m2 ε ε − O O h i The second term in the curly brackets of (8.81) is of order ε. This ε is canceled against a 1/ε coming from the factor Γ(ε). Since the logarithm itself is convergent, a pole term can only appear for y 1 where the y-integral (8.85) diverges. But for y 1, the logarithm goes to zero as log y.→ Therefore, the second term in the bracket yields no pole→ term for A(q), which is therefore entirely contained in (8.86). Expanding Γ(1 + ε) = Γ(1)+ ψ(1)ε + (ε2) according O to formula (8.83), and (4π/m2)ε =1+ ε log(4π/m2)+ (ε2), we obtain O 1 2 1 4π q 0 A( )= 4 2 + 1+2 ψ(1) + 2 log 2 + (ε ) . (8.87) (4π) ε ε m O There are many ways of calculating B(q), most of them involving cumbersome expressions. The easiest way uses the fact that the integrand of B(q) can be rewritten as: µ µ q (q + p1 + p2) 2 2 2 2 2 2 2 (8.88) (p1 + m )(p2 + m )[(q + p1 + p2) + m ] qµ ∂ 1 = µ 2 2 2 2 2 2 2 . − 2 ∂q (p1 + m )(p2 + m )[(q + p1 + p2) + m ] Then the integral for B(q) becomes qµ ∂ dDp dDp 1 q 1 2 B( )= µ D D 2 2 2 2 2 2 . (8.89) − 2 ∂q Z (2π) (2π) (p1 + m )(p2 + m )[(q + p1 + p2) + m ] 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 8.3 Calculation of One-Particle-Irreducible Diagrams 117 Introducing Feynman parameters as before, we find qµ Γ(3 D) ∂ 1 q D/2−2 B( ) = −D µ dx[x(1 x)] − 2 (4π) ∂q Z0 − 1 y D−3 dy y1−D/2 q2y(1 y)+ m2 1 y + × 0 ( − " − x(1 x)#) Z − (3 D)Γ(3 D) 1 q2 D/2−2 = − D − dx[x(1 x)] (4π) Z0 − 1 y D−4 dy y2−D/2(1 y) q2y(1 y)+ m2 1 y + . (8.90) × 0 − ( − " − x(1 x)#) Z − In terms of D =4 ε, this reads − Γ(ε) 4π ε 1 q q2 −ε/2 B( ) = 4 2 dx[x(1 x)] (4π) m Z0 − 1 q2 y dy yε/2(1 y) 1 ε log y(1 y)+ 1 y + + (ε2) . (8.91) × 0 − ( − "m2 − − x(1 x)!# O ) Z − The parameter integrals without the brackets give no pole in ε: 1 Γ(1 ε/2)Γ(1 ε/2) dx [x(1 x)]−ε/2 = − − =1+ ε + (ε2), (8.92) 0 − (1 ε)Γ(1 ε) O Z − − 1 Γ(2)Γ(1 + ε/2) Γ(1 + ε/2) dy (1 y) yε/2 = = Z0 − Γ(3 + ε/2) (2 + ε/2)(1 + ε/2)Γ(1 + ε/2) 1 3 = 1 ε + (ε2) . (8.93) 2 − 4 O The only pole in ε comes from the prefactor Γ(ε) in (8.91). Since the second term in the brackets of (8.91) carries a factor ε, it does not contribute to the pole term of B(q), and we have q2 4π ε Γ(1 + ε) 1 3 B(q) = 1+ ε + (ε2) 1 ε + (ε2) (4π)4 m2 ε O 2 − 4 O h i 2 ε q 4π 1 0 = 4 2 + (ε ). (8.94) (4π) m 2 ε O Expanding (4π/m2)ε as before, the final expression for the pole term of B(q) is q2 1 B(q)= + (ε0). (8.95) (4π)4 2 ε O Together with the result for A(q) in (8.87), we find for the sunset diagram in (8.75) with λ = gµε: 2 2 2 2 m 6 6 3 4πµ q 0 q q♠ = g 4 2 + + ψ(1) + log 2 + 2 + (ε ) . (8.96) − (4π) (ε ε "2 m # 2m ε O ) 118 8 Regularization of Feynman Integrals 8.3.3 Two-Loop Diagram of Four-Point Function ...... ❣...... q There are three two-loop diagrams contributing to the four-point function: , ..... q ..... , . q❤ q q❤ q♠ q q q♠ The first is a product of two independent integrals: D D ...... 3 d p 1 d q 1 ...... = λ , (8.97) q❤ q q❤ − (2π)D [(p k)2 + m2](p2 + m2) (2π)D [(q k)2 + m2](q2 + m2) Z − Z − where k denotes either of the three different momentum combinations k1 + k2, k1 + k3, and ε k1 + k4. The pole term in (8.97) is easily calculated using Eq. (8.68). Setting λ = gµ , we find 2 1 2 2 ...... g 2 4πµ ...... ε . q❤ q q❤. = gµ + ψ(1) + dx log + (ε) (8.98) − (4π)4 ( ε 0 "k2x(1 x)+ m2 # O ) Z − g2 4 4 4 1 4πµ2 = gµε + ψ(1) + dx log + (ε0) . (8.99) − (4π)4 (ε2 ε ε 0 "k2x(1 x)+ m2 # O ) Z − The second diagram is associated with the following integral D D . . 3 d p 1 d q 1 ...... ❣...... q ..... = λ . (8.100) q♠ q − (2π)D [(p k)2 + m2](p2 + m2)2 (2π)D (q2 + m2) Z − Z It is calculated using Eqs. (8.69) and (8.62), replacing again λ by gµε: 2 . . 2 g 2 4πµ ...... ❣...... q ..... = m + ψ(2) + log + (ε) (8.101) q♠ q − (4π)2 " ε m2 ! O # g ε 1 1 x gµε Γ(1 + ε/2) 1+ log 4πµ2 + (ε2) dx − (ε) (8.102) × (4π)2 2 O " 0 k2x(1 x)+ m2 −O # Z − g2 2 1 m2(1 x) = gµε dx − + (ε) . (8.103) − (4π)4 ε " 0 k2x(1 x)+ m2 O # Z − The third Feynman integral, .. .. D D ...... d p d q 1 1 q = λ3 , q q♠ − (2π)D (2π)D (p2 + m2) (k + k p)2 + m2 (q2 + m2) (p q + k )2 + m2 Z 1 2 − − 3 h i h (8.104)i can be written as ...... D 3 d p 1 1 q = λ I (p + k3) , (8.105) q q♠ − (2π)D p2 + m2 (k + k p)2 + m2 Z 1 2 − ...... where I(k) is the same integral (8.63) which occurred in the one-loop diagram .. q q❧.. . Using the result of that integral in (8.67), we obtain ...... g 1 q = (gµε)2 Γ(ε/2) dx (8.106) q q♠ 2 (4π) Z0 dDp 1 1 4πµ2 ε/2 . D 2 2 2 2 2 2 × (2π) p + m (k + k p) + m "(k3 + p) x(1 x)+ m # Z 1 2 − − 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories 8.3 Calculation of One-Particle-Irreducible Diagrams 119 The denominators are combined via (8.17), and we get 2 1 1 ε g 2 ε Γ(ε) −ε/2 ε/2−1 gµ 4 (4πµ ) Γ(2 + ε/2) dx [x(1 x)] dy(1 y) y (4π) Γ(2 + ε/2) Z0 − Z0 − 1 2 2 dz yz(1 yz)(k1 + k2) + y(1 y)k3 × Z0 ( − − −ε 2 1 y 2yz(1 y)k3(k1 + k2)+ m y + − . (8.107) − − " x(1 x)#) − The only pole term comes from the end-point singularity in the integral at y = 1. For y = 1, the curly bracket can be expanded as 2 2 −ε m 2 ... = (k1 + k2) 1 ε log z(1 z)+ 2 + ε . (8.108) { } ( − " − (k + k ) # O ) h i 1 2 Using formula (8D.24), the prefactor of (8.107) is seen to have an expansion 1 [1 εψ(1)] + (ε) , ε − O so that we have ...... 2 1 1 g 1 −ε/2 ε q ε 2 ε 2 −1 q q♠ = gµ 4 (4πµ ) [1 + εψ(1)] dx [x(1 x)] dy(1 y) y (4π) ε Z0 − Z0 − 1 2 2 −ε m (k1 + k2) dz 1 ε log z(1 z)+ 2 + (ε). (8.109) × 0 − " − (k + k ) # O h i Z 1 2 The first two terms in the integral are independent of x and y, and give Γ2 (1 ε/2) Γ(ε/2) Γ(2) 1 m2 − 1 ε dx log x(1 x)+ . (8.110) Γ(2 ε) Γ(2+ ε/2) ( − 0 " − (k + k )2 #) − Z 1 2 The Gamma functions are combined using the following formula: Γ(1+ a ε) n n =1+ ε2 , (8.111) Γ(1+ a′ ε) O Qm m which holds if Q ′ an am =0. (8.112) n − m X X We then find for the prefactor in (8.110): Γ2 (1 ε/2) Γ(ε/2) Γ(2) 2 1 − = [1 + (ε2)]. (8.113) Γ(2 ε) Γ(2+ ε/2) ε (1 ε)(1 + ε/2) O − − Up to this point, the singular part of (8.106) is therefore ...... 2 1 k k 2 2 q ε g 2 ε ( 1 + 2) x(1 x)+ m 0 q q♠ = gµ 4 2 1+ + εψ(1) ε dx log 2− + (ε ), (8.114) (4π) ε ( 2 − Z0 " 4πµ #) O where we have used the mass parameter µ to make the logarithm dimensionless. We now look at the effect of the last logarithm in (8.108). It is accompanied by a factor ε and vanishes at y = 1, such that it fails to lead to an end-point singularity at y = 1, ε = 0. Therefore, it contributes to order (ε0) and can be neglected as far as the singular parts are concerned, O leaving Eq. (8.110) unchanged. 120 8 Regularization of Feynman Integrals 8.3.4 Two-Loop Vacuum Diagrams For the discussion of the effective potential in Section 10.6 we shall also need the vacuum diagrams in dimensional regularization. Up to two loops, they are (0) 1 ✎☞ 1 ✎☞✎☞ Γ = + r . (8.115) 2 ✍✌ 8 ✍✌✍✌ The analytic expression for the one-loop diagram is, according to Eq. (5.45), 1 1 dDp ✎☞ p2 2 = D log( + m ). (8.116) 2 ✍✌ 2 Z (2π) Recalling formula Eq. (8.27), and expanding the result of the integration in powers of ε, we obtain 2 D/2 1 ✎☞ 1 2 (m ) = D/2 Γ(1 D/2) 2 ✍✌ 2 D (4π) − 1 m4 1 1 1 m2 1 1 = + log ψ(2) + (ε), (8.117) 2 µε (4π)2 "−ε 2 4πµ2 − 4 − 2 # O where the value of the Digamma function ψ(2) can be taken from Eq. (8D.12). The one- loop vacuum diagram is the only diagram with no vertex. To go to the second line with a dimensionless argument of the logarithm, we have inserted the mass parameter µ by inserting the identity 1 = µε/µε and expanded (m2/µ2)ε/2 in powers of ε. Note that the parameter µ in the prefactor 1/µε cannot be absorbed into the coupling constant. The two-loop diagram corresponds to the Feynman integral D 2 1 ✎☞✎☞ λ d p 1 r = D 2 2 . (8.118) 8 ✍✌✍✌ − 8 Z (2π) p + m ! Comparison of (8.118) with (8.56) and (8.57) yields 2 D−2 1 ✎☞✎☞ λ (m ) 2 r = D Γ (1 D/2). (8.119) 8 ✍✌✍✌ − 8 (4π) − This has the ε-expansion in terms of g = λµ−ε: 4 2 1 ✎☞✎☞ 1 m g 1 1 m r = ε 4 2 log 2 ψ(2) 8 ✍✌✍✌ − 2 µ (4π) (ε − ε 4πµ − ! m2 1 m2 2 1 π2 ψ(2) log + log + +2ψ2(2) ψ′(2) + (ε) . − 4πµ2 2 4πµ2 ! 4 " 3 − # O Appendix 8A Polar Coordinates and Surface of a Sphere in D Dimensions The polar coordinates in D dimensions are: (p ....,p ) (p,ϕ,ϑ , . . . .ϑ ) , p2 = p p , (8A.1) 1 D → 1 D−2 µ µ D D−1 2 D−2 d p = p dp dϕ sin ϑ1 dϑ1 sin ϑ2 dϑ2 ... sin ϑD−2dϑD−2 , (8A.2) 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories Appendix 8B More on Dimensional Regularizationof ’t Hooft and Veltman 121 where 0 SD = dpˆ. (8A.4) Z The D 1 angular integrations can be carried out using the integral formula − π/2 1 Γ(x)Γ(y) (sin t)2x−1(cos t)2y−1 dt = , Re x, Re y > 0 . (8A.5) Z0 2 Γ(x + y) 1 k+1 Inserting here y = 2 and x = 2 , we obtain π π/2 k+1 1 k+1 k k Γ( 2 )Γ( 2 ) Γ( 2 ) dt (sin t) =2 dt (sin t) = k+2 = √π k+2 , (8A.6) 0 0 Z Z Γ( 2 ) Γ( 2 ) and thus: D−2 D−2 k+1 π D Γ(1) Γ( ) D Γ(1) k 2 k=2 2 2 SD =2π sin ϑk dϑk =2π D D−3 k+2 =2π D . (8A.7) 0 Γ( )Q Γ( ) Γ( ) kY=1 Z 2 k=1 2 2 The surface of the unit sphere in D dimensions is thereforeQ D 2 π 2 SD = D . (8A.8) Γ( 2 ) Integration of a rotationally invariant integrand gives D−2 π ∞ D 2 k D−1 2 I(D) = d p f(p )=2π sin ϑk dϑk dp p f(p ) Z k=1 Z0 Z0 ∞ Y D−1 2 = SD dp p f(p ) . (8A.9) Z0 These calculations hold initially for integer values of the dimension D. By an analytic continu- ation of SD in D, the final results make sense also for continuous values of D. As an example, we extrapolate analytically a D-dimensional Gaussian integral to any complex D. Using Eq. (8A.9) we find D ∞ ∞ d p −p2 SD D−1 −p2 SD D/2−1 −x D e = D dp p e = D dx x e (8A.10) Z (2π) (2π) Z0 2(2π) Z0 D/2 SD 1 = D Γ(D/2) = . (8A.11) 2(2π) 4π Appendix 8B More on Dimensional Regularization of ’t Hooft and Veltman Dimensional Regularization of ’t Hooft and Veltman by Splitting the Integration Measure The splitting of the integration measure may be exemplified using the integral ∞ D 1 D−1 1 I(D)= d p 2 2 = SD dp p 2 2 , (8B.1) Z p + m Z0 p + m 122 8 Regularization of Feynman Integrals where SD is the surface of a unit sphere in D dimensions calculated in Eq. (8A.8). The integral in (8B.1) is obviously UV-divergent for D 2 and IR-divergent for D 0. ≥ ≤ The idea of ’t Hooft and Veltman is to define Feynman integrals in noninteger dimensions by splitting the measure of integration into a physical four- and an additional artificial D 4- − dimensional one. Let the four-dimensional momentum variable part be pk, the remaining D 4 -dimensional one p , so that − ⊥ 4 D−4 1 I(D) = d pk d p⊥ 2 2 2 . (8B.2) Z Z pk + p⊥ + m The D 4 -dimensional integrals can be reduced to a one-dimensional integral by observing − that the integrand depends only on the length of the D 4 -dimensional integration variable. Proceeding as in Appendix 8A, we perform the angular− integration in noninteger dimension which yields the surface of a unit sphere in D 4 dimensions: − 2π(D−4)/2 S = . (8B.3) D−4 Γ((D 4)/2) − The resulting expression for I(D) contains only ordinary integrals: ∞ D−4−1 4 p⊥ I(D)= SD−4 d pk dp⊥ 2 2 2 . (8B.4) Z Z0 pk + p⊥ + m This integral does not converge for any D. For D 2 it is UV-divergent, whereas for D 4 it ≥ ≤ is IR-divergent. The reason is the artificial IR-divergence generated by splitting the measure of integration. An integral with the original region of convergence is constructed by partial integration. For this we rewrite 1 1 d dp pD−4−1 = dp2 pD−6 = dp2 (p2 )D/2−2, (8B.5) ⊥ ⊥ 2 ⊥ ⊥ ⊥ D 4 dp2 ⊥ − ⊥ 2 and integrate (8B.4) partially over p⊥, yielding 2 D/2−2 ∞ ∞ 2 D/2−2 SD−4 4 (p⊥) 2 (p⊥) I(D)= d pk 2 2 + dp⊥ 2 2 . (8B.6) D 4 p + p + m2 0 (p + p + m2)2 Z k ⊥ 0 Z k ⊥ − The surface term is UV-divergent for D 2, and IR-divergent for D 4, and is discarded. The ≥ ≤ remaining integral is UV-divergent for D 2 and IR-divergent for D 2. Hence there is still ≥ ≤ no region of convergence. We must repeat the procedure, and arrive with Γ(D/2 2)(D/2 − − 2)(D/2 1) = Γ(D/2) at the expression − D/2−2 ∞ 2 D/2−1 2π 4 2 (p⊥) I(D) = d pk dp⊥ 2 2 2 3 . (8B.7) Γ(D/2) Z Z0 (pk + p⊥ + m ) This integral is now convergent for 0 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories Appendix 8C Parametric Representation of Feynman Integrals 123 and carrying out a partial integration in the region 0 D/2−2 ∞ D−1 2 4 π 4 p⊥ I(D)= 3m d pk dp⊥ 2 2 . (8B.9) − (D/2 1)Γ(D/2) 0 (p + p + m2)4 − Z Z k ⊥ This expression provides us with the desired analytic extrapolation of (8B.7). The procedure can be repeated to yield a convergent integral in the region 0 D/2−2 ∞ D−1 4 4 π 4 p⊥ I(D)=3 4 m d pk dp⊥ 2 2 . (8B.10) · (D/2 2)(D/2 1)Γ(D/2) 0 (p + p + m2)5 − − Z Z k ⊥ The only traces of the original UV-divergence are the poles for D = 2 and D = 4. The integrations can be performed and the result expressed in terms of Beta functions (8.8). The four-dimensional momentum integration in (8B.10) yields ∞ D−1 ∞ D−1 4 p⊥ 2 Γ(3) p⊥ d pk dp⊥ 2 2 2 5 = π dp⊥ 2 2 3 , (8B.11) Z Z0 (pk + p⊥ + m ) Γ(5) Z0 (p⊥ + m ) and the remaining one-dimensional integration gives (m2)D/2−3 Γ(D/2)Γ(3 D/2) π2 − . (8B.12) 2 Γ(5) The UV-divergence of the integrations is reflected by the pole of the Gamma function at D = 6. After including the prefactor in (8B.10), we obtain D 1 D/2 2 D/2−1 I(D)= d p 2 2 = π (m ) Γ(1 D/2). (8B.13) Z p + m − This is the same result as in (8.43) which was reached without splitting the integration measure. The splitting is useful for integrals that depend on external momentum. Appendix 8C Parametric Representation of Feynman Integrals We follow Itzykson and Zuber [15] in the derivation of a proper-time representation of a Feyn- man integral with L = I V + 1 loops, I internal lines, and V vertices. It is superficially − UV-convergent for ω(G) = Re(D)L 2I < 0. − An oriented diagram is defined with the help of a so-called incidence matrix ǫv l with v = 1,...,V and l =1,...,I: +1 if v is starting vertex of line l ǫ = 1 if v is ending vertex of line l . (8C.1) v l − 0 if line l is not incident on vertex v ′ The integral of an n-point diagram G containing no tadpole part is called IG(k1,..., kn). The sum of the external momenta entering the diagram at a vertex v is denoted by k¯v. Momen- tum conservation at each vertex is expressed by δ(D)(k¯ ǫ p ), where p are the internal v − l v l l l momenta. The integral has the following general form: P I dDp 1 V ′ k k V D (D) k¯ p IG( 1,..., n)=( λ) WG D 2 2 (2π) δ v ǫv l l , (8C.2) − (2π) pl + m − ! Z lY=1 vY=1 Xl 124 8 Regularization of Feynman Integrals where WG is the weight factor. Each propagator and the δ-distributions at each vertex are expressed by an integral representation: ∞ 1 −τ(p2+m2) 2 2 = dτ e , (8C.3) p + m Z0 D (D) D −iyv· k¯v− ǫ p (2π) δ (k¯ ǫ p ) = d y e ( l vl l). (8C.4) v − v l l v Xl Z P Insertion of these two formulas and interchange of the order of integration in Eq. (8C.2) leads to p-integrations of the following form: 2 D 2 i D i 1 2 d p −τl p − yvǫvlpl d p −τl pl− yvǫvl − yvǫvl l l τl v l 2τl v 4τl ( v ) D e = D e Z (2π) P Z (2π) P P 1 2 1 − yv ǫvl 4τl ( v ) = D/2 e . (8C.5) (4πτl) P The interchange is justified if Re(D)L< 2I for the entire diagram as well as for all subdiagrams, ′ or if the integrals are regularized. The complete expression for IG becomes 2 2 y V V ∞ I −τlm −( v vǫvl) /4τl ( λ) WG e V y k¯ ′ D −i v=1 v· v IG(ki)= − d yv dτl D/2 e . (8C.6) (4π)ID/2 0 τP Z vY=1 Z lY=1 l P After a variable transformation y1= z1+ zV , y2= z2+ zV ,..., yV = zV , and using the fact that V zV ǫv l =0, vX=1 the integration over zV yields the factor V n D D (D) D (D) d zV exp( izV k¯v)=(2π) δ k¯v = (2π) δ ki . − v ! ! Z vX=1 X Xi It is useful to change the notation and remove this factor, since the integrals in Γ(n) are defined without it. Thus we introduce n ′ D (D) IG = (2π) δ ki IG, (8C.7) ! Xi and the remaining integral is rewritten as − 2 V 1 z 1 z −τlm − v ,v v1 ǫv1l τ ǫv2l v2 /4 V −1 ∞ I 1 2 l V −1 V D e −i z k¯ k v=1 v v IG( i)=( λ) WG d zv dτl P D/2 e . − Z v=1 Z0 l=1 (4πτl) Y Y P The matrix 1 [dG]v1v2 = ǫv1l ǫv2l τl Xl 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories Appendix 8C Parametric Representation of Feynman Integrals 125 is nonsingular and its determinant is found to be [16] 1 ∆G det[dG]= , (8C.8) ≡ τl treesXT lY∈T where trees T denotes a sum over all tree diagrams contained in G. Recall the definition of tree diagramsP on page 73 and in Fig. 5.5 on page 73. They consist of a sum over all maximally connected subsets of lines in G which contain no loop but all vertices of G. As each tree has V 1 lines, ∆ is a homogeneous polynomial in τ −1 of order V 1. Integrating over z gives − G l − i V −1 −τ m2− k¯ [d−1] k¯ V ∞ I l v ,v v1 v1v2 v2 ( λ) WG e 1 2 IG(ki)= − dτl D/2 . (4π)LD/2 0 I Pτ ∆ (τ)D/2 Z lY=1 l=1 l G Q D/2 The denominator in the integrand has the final form [MG(τ)] , where 1 MG(τ)= τ1,...,τI = τl, (8C.9) × τl treesXT lY∈T treesXT l/Y∈T which is a homogeneous polynomial of degree I (V 1) = L. − − Let us denote the quadratic form in the exponent of the integrand by QG(k¯,τ). It is given by a ratio of two homogeneous polynomials in τ of degree L + 1 and L respectively: V −1 ¯ −1 ¯ 1 ¯ 2 ¯ kv1 [d ]v1v2 kv2 = kv τl QG(k,τ). (8C.10) v ,v MG ≡ X1 2 cutsXC v∈XG1(C) lY∈C A cut is a subset of lines of G which, if removed, divides G into two parts, say G1(C) and G2(C). A tree becomes a cut if one of its lines is taken away. Therefore a cut has I (V 1)+1 = L+1 lines. With these notations the final form reads − − 2 ¯ ( λ)V W ∞ I e−τlm −QG(k,τ) k G IG( i)= − LD/2 dτl D/2 . (8C.11) (4π) 0 MG(τ) Z lY=1 The dependence on D emerges explicitly from the one-dimensional τ-integrations. For this reason, formula (8C.11) has been used to define Feynman integrals in D dimension with complex D [12]. The UV-divergences are found in the τ 0 limit. The superficial divergence of the parametric→ integral is seen explicitly if the τ-integration is rescaled by a homogeneity parameter σ: τ στ , such that l → l QG(k¯,στ)= σQG(k¯,τ), (8C.12) L MG(στ)= σ MG(τ). (8C.13) Inserting the trivial identity ∞ I 1= dσ δ σ τl (8C.14) 0 − ! Z Xl into (8C.11) gives then for IG: I −σ[Q (k¯,τ)+m2 τ ] ∞ dσ 1 e G l l I (k )=( λ)V W σI−DL/2 dτ δ(1 τ ) G i G l L PD/2 l − Z0 σ Z0 l=1 [(4π) MG(τ)] − l Y DL X 1 I ¯ 2 2 −I V [QG(k,τ)+ m l τl] = ( λ) WGΓ(I DL/2) dτl L D/2 δ(1 τl). (8C.15) − − 0 [(4π) MG(τP)] − Z lY=1 Xl 126 8 Regularization of Feynman Integrals The integration over σ produces the superficial divergence of G. It is convergent at σ = 0 only if 2I Re(D)L = ω(G) > 0. The integral is analytically continued with the Gamma function − − Γ((2I DL)/2) = Γ( ω(G)/2), which gives rise to simple poles for ω(G)=0, 2,..., where the − − divergences are logarithmic or quadratic. All possible subdivergences come from the remaining τ-integrations, which produce further Gamma functions [13]. We shall exemplify the above formula for a one-loop diagram which has V = 2, I = 2, L =1 and therefore I DL/2=2 D/2. The momentum conservation is taken care of by a factor D (D) n − − (2π) δ ( i ki) to be omitted. k¯ P 1 k¯ D ...... 3 d p 1 1.....✓✏..... 2 2 ...... = λ (8C.16) r r 2 (2π)D (p2 + m2)[(p k¯)2 + m2] ✒✑2 Z − ¯ 2 1 −σ[QG(k,τ1,τ2)+m (τ1+τ2)] 2 3 dσ 2−D/2 e = λ σ dτ1dτ2 D/2 δ(1 τ1 τ2) 2 Z σ Z0 [4πMG(τ1,τ2)] − − D 1 ¯ 2 2 −2 2 3 [QG(k,τ1,τ2)+ m (τ1 + τ2)] = λ Γ(2 D/2) dτ1dτ2 D/2 δ(1 τ1 τ2). 2 − Z0 [4πMG(τ1,τ2)] − − This simple diagram contains only two trees, and . The first consists of line 1, the second T1 T2 of line 2. There is also one cut , which consists of line 1 and 2, such that G1( ) contains vertex 1 and G ( ) contains vertexC 2. The relations for the number of lines of a treeC and a cut 2 C are fulfilled: I( )= V 1 = 1 and I( )= L +1 = 2. We then find for M and Q : T − C G G MG = τl + τl = τ2 + τ1, (8C.17) l/Y∈T1 l/Y∈T2 1 2 QG = k¯ τ1τ2, (8C.18) τ1 + τ2 and the integral becomes D/2−2 k¯ 1 k¯ τ τ 1 ¯2 1 2 2 .. .. k + m (τ1 + τ2) ...... Γ(2 D/2) τ1+τ2 1.....✓✏..... 2 2 ...... = λ dτ dτ δ(1 τ τ ) r r − D/2 1 2 h D/2 i 1 2 (4π) 0 (τ1 + τ2) − − ✒✑2 Z Γ(2 D/2) 1 D/2−2 = λ2 − dτ k¯2τ(1 τ)+ m2 . (8C.19) (4π)D/2 0 − Z h i We have reproduced the result in (8.66), which we obtained with Feynman’s parameter integral formula. Appendix 8D Expansion of Gamma Function The Gamma function is defined by the integral (8.9). This integral has poles for z = 0 and negative integers. Partial integration applied to the integral representation of Γ(z + 1) leads to the identity ∞ Γ(z +1) = z dt tz−1e−t = z Γ(z), (8D.1) Z0 which shows that the Gamma function is the generalization of the factorial to arbitrary complex variables z. In the context of dimensional regularization, we need a series expansion of the Gamma function near zero or negative integer values of z. Using (8D.1), we see that Γ(2 + ε)=(1+ ε)Γ(1 + ε)=(ε + 1) ε (ε 1)Γ( 1+ ε). (8D.2) − − 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories Appendix8D ExpansionofGammaFunction 127 This relates the expansion of Γ( 1+ ε) to the expansion of Γ(2+ ε) in powers of ε, which reads − 1 Γ(2 + ε) = Γ(2)+Γ′(2) ε + Γ′′(2) ε2 + (ε3) (8D.3) 2 O 1 Γ′′(2) = 1+ ψ(2) ε + ε2 + (ε3). (8D.4) 2 Γ(2) O In the second line, we have inserted Γ(2) = 1 as well as the definition of the Euler Digamma function ψ(z): ψ(z) Γ′(z)/Γ(z). (8D.5) ≡ From this follows the recurrence relation: d d log(z 1) log Γ(z 1) 1 ψ(z)= log Γ(z)= − + − = + ψ(z 1), (8D.6) dz dz dz z 1 − − which for integer values of z implies n−1 1 ψ(n)= ψ(1) + . (8D.7) l Xl=1 Analogously, we find for ψ′(z): 1 ψ′(z)= − + ψ′(z 1), (8D.8) (z 1)2 − − and n−1 1 ψ′(n)= ψ′(1) . (8D.9) − l2 Xl=1 The value of ψ(z) at z = 1 is equal to the negative of Euler’s constant γ: ψ(1) = γ = 0.5772156649 .... (8D.10) − − The derivative of ψ(z) has at z = 1 the value ψ′(1) = π2/6. (8D.11) Thus we find: n−1 1 ψ(n) = γ + , (8D.12) − l Xl=1 π2 n−1 1 ψ′(n) = . (8D.13) 6 − l2 Xl=1 Differentiating Eq. (8D.5), we obtain ψ′(z)=Γ′′(z)/Γ(z) [Γ′(z)]2/Γ(z)2, (8D.14) − and thus an expression for Γ′′(z)/Γ(z) which may be inserted into the expansion Eq. (8D.4) to find 1 Γ(2 + ε)=1+ ψ(2)ε + ψ′(2) + ψ(2)2 ε2 + (ε3). (8D.15) 2 O h i 128 8 Regularization of Feynman Integrals For Γ( 1+ ε), we derive the expansion − Γ(2 + ε) 1 Γ( 1+ ε) = = Γ(2 + ε) 1+ ε2 + (ε3) (8D.16) − −ε(1 + ε)(1 ε) −ε O − h i 1 1 1 = + ψ(2) + ε 1+ ψ′(2) + ψ(2)2 + (ε2). (8D.17) − ε 2 2 O A generalization of this formula to any integer n is obtained using repeatedly the identity (8D.1): Γ(n +1+ ε) Γ( n + ε) = (8D.18) − (n + ε)(n + ε 1) ... (ε + 1) ε (ε 1) ... (ε n) − − − Γ(n +1+ ε) = (8D.19) ( 1)nε (1 + ε)(1 ε)(2 + ε)(2 ε) ... (n + ε)(n ε) − − − − ( 1)n Γ(n +1+ ε) = − 2 2 2 (8D.20) n!2 ε (1 ε2) 1 ε 1 ε ... 1 ε − − 2 − 3 − n ( 1)n n 1 = − Γ(n +1+ ε) 1+ ε2 + (ε4) . (8D.21) n!2 ε j2 O jX=1 Together with Eq. (8D.13), this becomes ( 1)n π2 Γ( n + ε)= − Γ(n +1+ ε) 1+ ε2 ψ′(n + 1) + (ε4) . (8D.22) − n!2 ε ( " 6 − # O ) We further need the Taylor expansion of Γ(n +1+ ε) for integer n: 1 Γ′′(n + 1) Γ(n +1+ ε) = Γ(n + 1) 1+ εψ(n +1)+ ε2 + (ε3) " 2 Γ(n + 1) O # ε2 = n! 1+ εψ(n +1)+ ψ′(n +1)+ ψ(n + 1)2 + (ε3) . (8D.23) ( 2 O ) h i Using (8D.21) and (8D.23), the expansion for Γ( n + ε) takes the form: − ( 1)n 1 ε π2 Γ( n + ε)= − + ψ(n +1)+ + ψ(n + 1)2 ψ′(n + 1) + (ε2) . (8D.24) − n! ( ε 2 " 3 − # O ) Notes and References For the dimensional regularization see also M.E. Fisher and K.G. Wilson, Phys. Rev. Lett. 28, 240 (1972); K.G. Wilson, Phys. Rev. D 7, 2911 (1973); K.G. Wilson and J. Kogut, Phys. Rep. C 12, 77 (1974). Many textbooks on quantum field theory treat dimensional regularization. An explicit analytic interpolation is constructed by J.C. Collins, Renormalization, Cambridge University Press, Cambridge, 1984. Explicit calculations for diagrams of the φ4- theory, for instance the sunset diagram (8.73), are found in Ref. [14]. The development of parametric representations of Feynman integrals in the textbook [15] goes 4 H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ -Theories Notes and References 129 back to R.J. Eden, P.V. Landshoff, D.I. Olive, and J.C. Polkinghorne, The Analytic S-Matrix, Cam- bridge University Press, 1966. See also J.D. Bjorken and S.D. Drell, Relativistic Quantum Fields, McGraw-Hill, New York, 1965. The individual citations in the text refer to: [1] W. Pauli and F. Villars, Rev. Mod. Phys. 21, 434 (1949). [2] E.R. Speer, Generalized Feynman Amplitudes, Princeton University, Princeton, 1969. [3] G.’t Hooft and M. Veltman, Nucl. Phys. B 44, 189 (1972). [4] C.G. Bollini, J.J. Giambiagi, Nuovo Cimento B 12, 20 (1972). [5] G.’t Hooft, Nucl. Phys. B 61, 455 (1973). [6] M.E. Fisher and K.G. Wilson, Phys. Rev. Lett. 28, 240 (1972); K.G. Wilson, Phys. Rev. D 7, 2911 (1973). [7] K.G. Wilson and J. Kogut, Phys. Rep. C 12, 77 (1974). [8] J.C. Collins, Renormalization, Cambridge University Press, Cambridge, 1984. [9] G. Leibbrandt, Rev. Mod. Phys. 74, 849 (1975). [10] See Vol. I of E. Goursat and E.R. Hedrick, A Course in Mathematical Analysis, Ginn and Co., Boston, 1904. [11] G. Cicuta and E. Montaldi, Nuovo Cimento Lett. 4, 329 (1972); P. Butera, G. Cicuta, and E. Montaldi, Nuovo Cimento A 19, 513 (1974). [12] J.F. Ashmore, Nuovo Cimento Lett. 4, 289 (1972); Commun. Math. Phys. 29, 177 (1973); [13] E.R. Speer, J. Math. Phys. 15, 1 (1974). [14] P. Ramond, Field Theory, A Modern Primer, Benjamin/Cummings Pub. Comp., 1981. [15] C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York, 1980. [16] N. Nakanishi, Graph Theory and Feynman Integrals, Gordon and Breach, New York, 1971.