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4 Perturbative Evaluation of the Path Integral: Λϕ4 Theory

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4 Perturbative Evaluation of the Path Integral: Λϕ4 Theory 4 Perturbative Evaluation of the Path Integral: λϕ4 Theory In the following, we proceed with the conventional (perturbative) evaluation of the Green's functions in Euclidean space. We start from Z R 4 1 ¯ ¯µ 1 2 2 −ZE [J] − d x¯[ @µ'@ '+ m ' +V (')−J'] WE[J] = e = N D' e 2 2 ; (4.1) where N is an arbitrary (infinite) normalization constant. The connected Green's functions are given by N (N) δ ZE[J] GE (¯x1; ··· ; x¯N ) = − : (4.2) δJ1 ··· δJN J=0 They will be calculated by perturbing in the potential V . For simplicity in the following, we neglect the subscript E and the bar over x, which indicate Euclidean space. Later when confusion with Minkowski space might occur, they will be reinstated. Using the trick of Section 3.3, we obtain − V δ −Z [J] W [J] = N e h ( δJ )i e 0 ; (4.3) where 1 Z [J] = − hJ(x)∆ (x − y)J(y)i ; (4.4) 0 2 F xy and Z d4p eip(x−y) ∆ (x − y) = : (4.5) F (2π)4 p2 + m2 A little algebraic rearrangement yields 1 2 Perturbative Evaluation of the Path Integral: λϕ4 Theory δ Z0 −hV ( )i −Z0 Z[J] = − ln N + Z0[J] − ln 1 + e e δJ − 1 e ; (4.6) which is ready for a perturbative expansion in the potential V . If we let Z − V δ −Z δ ≡ e 0 e h ( δJ )i − 1 e 0 ; (4.7) we arrive at 1 1 Z[J] = − ln N + Z [J] − δ[J] + δ2[J] − δ3[J] + ··· : (4.8) 0 2 3 λ 4 In particular, for V = 4! ' , we can expand in powers of the dimensionless (in four dimensions) coupling constant λ. Setting 2 δ = λδ1 + λ δ2 + ··· (4.9) we find 1 Z[J] = − ln N + Z [J] − λδ [J] − λ2 δ [J] − δ2[J] (4.10) 0 1 2 2 1 1 −λ3 δ [J] − δ [J]δ [J] + δ3[J] + ··· : (4.11) 3 1 2 3 1 From expanding the exponential in (4.1.7) we find 1 δ4 δ [J] = − eZ0[J]h i e−Z0[J] (4.12) 1 4! δJ 4 1 δ4 δ4 Z0[J] −Z0[J] δ2[J] = 2 e h 4 i1h 4 i2 e ; etc. ··· : (4.13) 2(4!) δJ1 δJ2 Using the explicit form (4.1.4) for Z0, we arrive at 1 δ [J] = − h∆ ∆ ∆ ∆ J J J J i + 6 h∆ ∆ ∆ J J i + 3 ∆2 1 4! xa xb xc xd a b c d xx xa xb a b xx (4.14) where all variables x, a, b, c, d, are integrated over in the relevant h· · ·i. Similarly, we evaluate δ2 in a slightly trickier fashion: we note that 1 δ4 Z0[J] −Z0[J] δ2[J] = − 2 e h 4 i e δ1[J] ; (4.15) 2(4!) δJ1 Perturbative Evaluation of the Path Integral: λϕ4 Theory 3 by inserting e−Z0[J]eZ0[J] in the middle of (4.1.12). Next the expansion δ4 δ4e−Z0[J] δ3e−Z0[J] δ δ2e−Z0[J] δ2 e−Z0[J] = + 4 + 6 (4.16) δJ 4 δJ 4 δJ 3 δJ δJ 2 δJ 2 δe−Z0[J] δ3 δ4 +4 + e−Z0[J] (4.17) δJ δJ 3 δJ 4 allows us to write * 1 1 δ3e−Z0[J] δ δ2e−Z0[J] δ2 2 Z0[J] δ2 = δ1 − 2 e 4 3 + 6 2 2 (4.18) 2 2(4!) δJ1 δJ1 δJ1 δJ1 + δe−Z0[J] δ3 δ4 −Z0[J] +4 3 + e 4 δ1[J] : (4.19) δJ1 δJ δJ 1 1 1 Comparison with the expansion (4.1.10) for Z[J] shows that the \discon- 1 2 nected" part 2 δ1 drops out. By disconnected we mean a contribution which can be written as the product of two or more functions of J. This con- cept will become obvious in the diagrammatic representation. The fact that Z generates only connected pieces is true to all orders (see problem). For example, the order λ3 contribution in (4.1.10) is connected: write 1 δ = − eZ0 V V V e−Z0 3 3! x y z xyz 1 = − eZ0 V e−Z0 eZ0 V e−Z0 eZ0 V e−Z0 3! x y z xyz 1 − eZ0 V e−Z0 eZ0 V V e−Z0 + δc 2 x y z xyz 3 1 = δ3[J] + δ [J]δc[J] + δc[J] : 3! 1 1 2 3 c c In the above δ2, δ3 stand for the connected pieces. To arrive at this form, we have used the fact that there are only two types of \disconnectedness": all three x, y, z disconnected, and only one disconnected from the other two; and there are three ways to obtain the latter possibility. The parentheses in (4.1.17) serve to shield other terms from the action of the derivative operators within them. It follows that the term appearing in the expansion of Z can be rewritten, using (4.1.18): 1 1 1 1 δ − δ δ + δ3 = δc + δ3 + δ δc − δ δc + δ2 + δ3 = δc : (4.20) 3 1 2 3 1 3 3! 1 1 2 1 2 2 1 3 1 3 Now, the explicit evaluation of the connected part of δ2 yields, save for the J-independent part, 4 Perturbative Evaluation of the Path Integral: λϕ4 Theory c 1 1 3 1 δ2[J] = + Ja∆ax ∆xy + ∆xx∆yy∆xy ∆ybJb (4.21) 2 6 4 xyab 1 + J ∆ ∆ ∆2 ∆ J (4.22) 8 a ax yy xy xb b xyab 2 + hJ ∆ ∆ ∆ ∆ ∆ ∆ J J J i (4.23) 4! a ax xx xy yb yc yd b c d xyabcd 3 + J J ∆ ∆ ∆2 ∆ ∆ J J (4.24) 2(4!) a b ax bx xy yc yd c d xyabcd 1 + hJ J J ∆ ∆ ∆ ∆ ∆ ∆ ∆ J J J i (4.25): 2(3!)2 a b c ax bx cx xy yd ye yf d e f xyabcdef The resulting connected Green's functions follow from (4.1.2): λ Z G(2)(x ; x ) = ∆(x − x ) − d4y∆ (x − y) ∆(y − y)∆(y − x ) 1 2 1 2 2 1 2 λ2 Z + d4xd4y∆(x − x)∆3(x − y)∆(y − x ) 6 1 2 λ2 Z + d4xd4y∆(x − x)∆2(x − y)∆(y − y)∆(x − x ) 4 1 2 λ2 Z + d4xd4y∆(x − x)∆(x − x)∆(x − y)∆(y − y)∆(y − x ) 4 1 2 +O(λ3) ; Z (4) 4 G (x1; x2; x3; x4) = −λ d x∆(x1 − x)∆(x2 − x)∆(x3 − x)∆(x4 − x) λ2 Z + d4xd4y∆2(x − y)[∆(x − x)∆(x − x)∆(x − y)∆(x − y) 2 1 2 3 4 +∆(x1 − x)∆(x3 − x)∆(x2 − y)∆(x4 − y) +∆(x1 − x)∆(x4 − x)∆(x2 − y)∆(x3 − y)] λ2 Z + d4xd4y∆(y − y)∆(x − y)[∆(x − x)∆(x − x)∆(x − x) 2 1 2 3 3 ∆(x4 − y) + cyclic permutations] + O(λ) ) ; and finally Z (6) 2 4 4 X G (x1; ··· ; x6) = λ d xd y∆(x − y) ∆(x1 − x)∆(xj − x)∆(xk − x) (ijk) 3 ∆(x` − y)∆(xm − y)∆(xn − y) + O(λ ) : where the sum in the last expression runs over the triples (ijk) = (123), Perturbative Evaluation of the Path Integral: λϕ4 Theory 5 (124), (125), (126), (134), (135), (136), (145), (146), (156), with (`mn) assuming the complementary value, i.e.,(`mn) = (456) when (ijk) = (123), etc. The remaining Green's functions get no contribution to this order in λ. Note that the λ0 contribution to G(2), the λ contribution to G(4) and the λ2 contribution to G(6) were all previously obtained in the classical approximation of the last chapter. It is straightforward to derive the p-space Green's functions, using (3.4.26). We find 1 λ 1 Z d4q 1 G~(2)(p; −p) = − p2 + m2 2 (p2 + m2)2 (2π)4 q2 + m2 λ2 1 Z d4q d4q d4q δ(p − q − q − q )(2π)4 + 1 2 3 1 2 3 2 2 2 4 4 4 2 2 2 2 2 2 6 (p + m ) (2π) (2π) (2π) (q1 + m )(q2 + m )(q3 + m ) λ2 1 Z d4q 1 Z d4` d4` δ(` − ` )(2π)4 + 1 2 1 2 2 2 2 4 2 2 4 4 2 2 2 2 4 (p + m ) (2π) q + m (2π) (2π) (`1 + m )(`2 + m ) λ2 1 Z d4q 1 1 Z d4` 1 + 4 (p2 + m2)2 (2π)4 q2 + m2 p2 + m2 (2π)4 `2 + m2 +O(λ3) 4 Z 4 4 (4) Y 1 1 2 d q 1 X 1 G~ (p1; p2; p3; p4) = f − λ + λ p2 + m2 2 (2π)4 q2 + m2 p2 + m2 i=1 i i=1 i 2 Z 4 4 λ d q1 d q2 1 X + δ(q + q − p − p )(2π)4g 2 (2π)4 (2π)4 (q2 + m2)(q2 + m2) 1 2 i j 1 2 (ij) +O(λ3) : In the last expression, the sum ij runs over (ij) = (12), (13), (14) only. Finally G~(6) is given by (3.4.29). These expressions are clearly unwieldy. One needs to devise a clever way of remembering how to generate them. This is exactly what the Feynman rules achieve. We now proceed to state them: 1 1. For each factor p2+m2 draw a line with momentum p flowing through it: 1 .5:2 : : (4.26) p2 + m2 2. For each factor of −λ/4! draw a four-point vertex with the understanding 6 Perturbative Evaluation of the Path Integral: λϕ4 Theory that the net momentum flowing into the vertex is zero: λ − 11 (p + p + p + p = 0) : (4.27) 4! 1 2 3 4 (N) 3.
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