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¯[ ∂µϕ∂ ϕ+ m ϕ +V (ϕ)−Jϕ] WE[J] = e = N Dϕ 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! ϕ , 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 concept 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) = { − 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)4} 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. In order to get the contributions to G˜ (p1, ··· , pN ), draw all the possible arrangements which are topologically inequivalent after having identiﬁed the external legs. The number of ways a given diagram can be drawn is the topological weight of the diagram.

4. After having conserved momentum at every vertex, integrate over the R d4q internal loop momenta with (2π)4 .

The result gives the desired Green’s function. Perhaps a more systematic λ 4 way to describe these rules is to attach to each vertex the factor − 4! δ(Σp)(2π) , where Σp is the net incoming momentum at that vertex. The one integrates over all internal momenta. In this manner one obtains an overall (2π)4δ(Σp) where Σp is the net momentum ﬂow into the Green’s function. For instance, the expression (4.1.24) is diagrammatically rewritten as

.6.7.6.7.6.7.6.7.6.7.6.7 (4.28) From the rules we can easily obtain the analytical expressions corresponding to these diagrams:

a) We need one vertex and three propagators. There are four ways to attach the ﬁrst leg of the vertex to 1, three 1 ways to attach the second leg to 2. Hence the weight 4! 4 · 1 3 = 2 . The vertex counts for −λ. Let q be the momentum circulating around the loop. The rules then give

Z 4 λ d q 4 1 1 − (2π) δ(p1 + p2 + q − q) . (4.29) 2 (2π)4 q2 + m2 (p2 + m2)2 b) We need two vertices. There are four ways to attach the first leg of the first vertex to 1, four ways to attach the first leg of the second vertex to 2, three ways to sew the second leg of the first vertex to the second, and two ways to sew the third leg of the first vertex to the second. Hence the weight 1 1 1 4! · 4! 4 · 4 · 3 · 2 = 6 . Note that we did not count that we Perturbative Evaluation of the Path Integral: λϕ4 Theory 7 initially had two vertices to play with. This is because the diagrams are the same irrespective of which vertex was used. The strength of this diagram is (−λ)2 = λ2. If we call the momenta flowing in the internal legs q1, q2, q3, the Feynman rules give

λ2 1 Z d4q d4q d4q 1 2 3 (2π)8 (4.30) 6 (p2 + m2)2 (2π)4 (2π)4 (2π)4

δ(p1 − q1 − q2 − q3)δ(p2 + q1 + q2 + q3) × 2 2 2 2 2 2 .(4.31) (q1 + m )(q2 + m )(q3 + m )

c) We need two vertices: four ways to attach the ﬁrst leg to 1, four ways to attach the third leg of the tied vertex to the other, three ways to tie the fourth leg of the ﬁrst vertex 1 1 1 2 to the second one. Hence 4! 4! 4·3·4·3 = 4 . Strength (−λ) .

d) We need two vertices; four ways to attach one vertex to 1, four ways to attach the other vertex to 2. This leaves three legs from each vertex free to be tied together in one way. For each vertex there are three ways to close the buckle. Hence 1 1 1 2 4! 4! 4 · 4 · 3 · 3 = 4 . Strength (−λ) .

Thus we see that the Feynman rules have reduced the problem to that faced by a child assembling a “Leggo” set. The basic tools are the propagator (line and the vertex. With a bit of skill one can read those directly from the Lagrangian. The same applies for the four-point function:

.7.7 .6.7.6.7.6.7.6.7.6.7 (4.32) .7.7.7.7.7.7 (4.33) corresponding to the analytical expression (4.1.25). We leave it to the reader to verify the correctness of the numerical factors in (4.1.25). Let us remark (N) that in the expression for G˜ (p1, ··· , pN ), we will have the multiplicative QN 2 2−1 factor i=1 p1 + m corresponding to the propagation of the external legs. It is much simpler to deal with the Green’s functions generated by the eﬀective Action. Their relation to G˜(n) is very simple: while G˜(n) are connected, the Γ˜(n) are one particle irreducible. In particular Γ˜(2)(p) is minus 8 Perturbative Evaluation of the Path Integral: λϕ4 Theory the inverse propagator. We have already come in contact with this result in conjunction with the tree diagram, but the result holds true in all orders of perturbation theory. As a result Γ˜(4) contains only the diagrams

.7.9.7.7.7.7.7.7 (4.34) with no propagators for the external legs. In order to show that the Γ˜(n) are one particle irreducible, it is convenient to use as a starting point the deﬁning equations:

δΓ[ϕ] δZ[J] = −J ; = ϕ , (4.35) δϕ δJ and keep on differentiating them. After all this work, let us go back to the Lagrangian where all the ingre- dients of the Feynman rules are easily identified: the four-particle vertex coming from the λϕ4 term, and the propagator coming from the kinetic and mass terms. Hence, after a little bit of practice, one can just read off the Feynman rules from L. The difficult part is to get the right signs and weight factors in front of the diagram.

4.0.1 PROBLEMS

A. Draw the contributions in the two point function G˜(2) that are of order λ3.

B. Draw the contributions to G˜(4) of order λ3.

C. Write the analytical expressions for the diagrams of problem A, including the weights.

∗ λ 4 µ 3 D. Derive the Feynman rules for V = 4! ϕ + 3! ϕ .

∗∗E. Show that Z[J] generates only connected Feynman diagrams.

∗∗∗ eﬀ 2 F. Derive the form of the eﬀective Action S [ϕcl] to order λ , and show that the one particle reducible diagrams do not appear in the Green’s functions it generates. 4.1 Divergences of Feynman Diagrams 9 4.1 Divergences of Feynman Diagrams No sooner has the beauty of the Feynman rules sunk in that one realizes that most of the loop integrations diverge! For instance the O(λ) contribution to G˜(2) involves the integral

Z d4q 1 ; (4.36) (2π)4 q2 + m2 it clearly diverges when q → ∞ since the integrand does not have enough juice to make up for the measure. Such a divergence is called an ultraviolet divergence. It occurs for large momenta, or equivalently, small distances [in x-space it comes from ∆(0)], and clearly has to do with the fact that one is taking too many derivatives with respect to J at the same point. Another example occurs in the “ﬁsh” diagram

2 Z 4 4 λ d q1 d q2 δ(q1 + q2 − p1 − p2) .6.6 ∼ 4 4 2 2 2 2 (4.37) 2 (2π) (2π) (q1 + m )(q2 + m ) 1 λ2 Z d4q 1 = 4 4 2 2 2 2 . (4.38) (2π) 2 (2π) (q + m ) ((q − p1 − p2) + m )

d4q Here as q → ∞, the integral behaves as q4 , which is a logarithm: log q. 2 2 It also diverges! In passing, we note that when m = 0, (p1 + p2) = 0 it also diverges for low q. Such a divergence is called an infrared divergence. Typically such divergences occur only in the massless case and for special value of the external momenta [in this case p1 + p2 = 0 because we are in Euclidean space]. For the moment we take m2 6= 0 and concentrate on the ultraviolet divergences. On the face of it, it is catastrophic to our program to find that our carefully constructed Green’s functions diverge. Still, with the hindsight of History we do not get discouraged, but try to learn more about these divergences. We will find that they appear in a very traceable way, and that by a suitable redefinition of the fields and coupling constants, they will disappear! This is the miracle of renormalization which, as we shall see, occurs only for certain theories. By a mixture of topology and power counting we now show how to tell where the divergences are. Consider a Feynman diagram with V vertices, E external lines and I internal lines. For a start we assume only scalar particles are involved. The number of independent internal momenta is the number of loops, L, in the diagram. The I internal momenta satisfy V −1 relations among them- 10 Perturbative Evaluation of the Path Integral: λϕ4 Theory selves (the −1 appears here because of overall momentum conservation), so that

L = I − V + 1 . (4.39) This relation enables us to compute the naive count of powers of momenta for the diagram. This yields the superﬁcial (apparent) degree of divergence of the diagram D. To compute it, we note that there are — L independent loop integrations, each providing, in d dimensions, d powers of momenta. — I internal momenta, such providing a propagator with two inverse powers of momenta. Hence

D = dL − 2I. (4.40)

We need one more relation among V , E and I. Suppose VN stands for the number of vertices with N legs. In a diagram with VN such vertices, we have NVN lines which are either external or internal. An internal line counts twice because it originates and terminates at a vertex, so that

NVN = E + 2I. (4.41) These relations allow us to express in D terms of the number of external lines and vertices

1 N − 2 D = d − (d − 2) E + V d − N . (4.42) 2 N 2 In four dimensions, this reduces to

D = 4 − E + (N − 4) VN [four dimensions] . (4.43) Furthermore, in the theory of interest to us, N = 4. Hence

D = 4 − E [for λϕ4 theory in four dimensions] . (4.44) The important result here is that the superficial degree of divergence does not depend on the number of vertices, but only on the number of external legs! Thus we have only two candidates with D ≥ 0 — G˜(2) with D = 2 superficial quadratic divergence. — G˜(4) with D = 0 superficial logarithmic divergence. Note that these two- and four-point interactions are already 4.1 Divergences of Feynman Diagrams 11 present in the Lagrangian, a fact which will prove crucial for renormalization. Also, D = 0 does not necessarily mean a logarithmic divergence: the fundamental vertex has D = 0. This analysis does not prove that G˜(6), G˜(8), ··· which have a negative D, converge. This is why D is called superficial. Consider a “n-particle reducible” diagram with E external lines; it is a diagram which can be disconnected by cutting at least n internal lines. In general, if D1 and D2 are the superficial degrees of divergence of the two blobs shown below, then the whole n-particle reducible diagram has

D = D1 + D2 + 4(n − 1) − 2n , (4.45) since the two blobs are connected by n propagators and n − 1 loops:

1.51 (4.46) Note that by deﬁnition, I and II are at least n-particle reducible themselves. In our case, when n = 1, we could have D1 = D2 = 0 and yet we would obtain D = −2, making the diagram apparently convergent. An example of this situation is the “dinosaur” diagram

1.51 (4.47) which clearly diverges because of the two divergent loop integrations. An- other example of a one particle reducible graph is

1.51 (4.48)

In this case D1 = 2, D2 = −2; it is apparently convergent, but is not because of the “wart” on one of the legs. This should also serve to show that a four point function may be more divergent than it would naively seem to be: witness this quadratically divergent four-point function

.8.8 (4.49) Let us remark that by considering Γ˜(n) we avoid such one particle reducible diagrams. When n = 2, D = D1+D2 and we can have D1 (or D2) suﬃciently negative to oﬀset D2 (or D1) and yield a negative D. An example is pictured by the “lobster” diagram

1.51 (4.50) 12 Perturbative Evaluation of the Path Integral: λϕ4 Theory Similarly, when n = 3, we can have diagrams like

1.51 (4.51)

[in this complicated diagram, each vertex has at most one external line em- anating from it]. In λϕ4 theory, diagrams can be at most three-particle reducible because any vertex attached to an external line can be disconnected from the diagram by cutting its remaining three legs. The procedure of hunting for diagrams which have a negative D and yet are divergent is now clear. Take any Feynman diagram and catalog it in terms of its reducibility: in our case it is either one-, two-, or three-particle reducible. It if is three-particle reducible, decompose it into units, and look for breakups of the form

1.51 N ≥ 5 (4.52)

Then in this type of diagram, we have a primitively divergent four-point function, where the second blob can be decomposed in the same way. Similarly, hidden divergences in two-particle reducible diagrams will arise in the diagrams of the form

1.51 N > 4 1.51 N > 2 (4.53) with the same breakup to be repeated in the second blob. Finally, one- particle reducible diagrams which can be decomposed in the form

1.51 N > 5 1.51 N > 3 (4.54) will have hidden divergences. The same decomposition can be carried out for the second blobs until all such structures have been uncovered. This exhaustive catalog shows that truly convergent diagrams do not contain hidden two- and four-point functions. One can understand the origin of hidden ultraviolet divergences in any diagram in a more pedestrian fashion. Consider any loop residing inside a diagram. Integration over the loop momentum in four dimensions will lead to a UV divergence if the loop is bounded by one or two propagators (internal lines) at most. Any more will give UV convergence. A loop bounded by one propagator involves only one vertex,

.8.8 (4.55) 4.1 Divergences of Feynman Diagrams 13 leaving two free legs, which in turn may be attached to the rest of the diagram (or one external, one attached). In this case one can isolate this two-point function from the insides of the diagram. A loop bounded by two propagators involves two vertices and therefore four free legs

.8.8 (4.56) which can be attached to the rest of the diagram, or else up to three can serve as external lines. In all these cases, one is led to isolate from the diagram a four-point function, or a two-point function if two of the four legs are attached together (in this case the divergence becomes quadratic). In this way, one sees that UV divergence inside a diagram originate from such loops and that such loops appear in two- and four-point functions nested inside the diagram. Thus, for the λϕ4 theory in four dimensions, a Feynman diagram is truly convergent if its superficial degree of convergence D is positive and if it cannot be split up into three-, two- or one-particle reducible parts of the kind just described which can contain isolated two- and four-point function blobs. Stated more elegantly, a Feynman diagram is convergent if its superficial degree of convergence and that of all its subgraphs are positive. This is known as Weinberg’s theorem, and it holds irrespective of the field theory. This means that the generic sources of the divergences are the two- and four-point functions and nothing else. They are the culprits! So, if we control them in G˜(2) and G˜(4), we have the possibility of controlling the divergences of all the other G˜(N)’s! The graphs which contain the generic divergences are said to be primitively divergent. The fact in λϕ4 theory that the primitively divergent interactions are finite in number (two- and four-point interactions) and are of the type that appears in the Lagrangian, is a necessary ingredient for the successful removal of the ultraviolet divergence by clever redefinitions. A theory for which this is possible is said to be renormalizable. We can see from (4.2.4) that very few theories of interacting scalars satisfy these requirements (see problem). — When d = 4, we see that D grows with the number of vertices for which N > 4. Hence ϕ5, ϕ6, ··· theories in four-dimensions, although perfectly reasonable classically, lead to an infinite number of primitively divergent diagrams (the more vertices the more divergent!). In this case the situation quickly gets out of hand and the hope of tagging the divergences disappears, and hence the renormalizability. — When d = 2 (one space and one time dimension) the situation is reversed. There 14 Perturbative Evaluation of the Path Integral: λϕ4 Theory

D = 2 − 2VN (two-dimensions) , (4.57) and D does not depend on N, which labels the type of interaction! It depends only on the number of vertices, and the more vertices, the more convergent the Feynman diagram! Also D does not depend on the number of external legs! So the only primitively divergent diagrams have one or no vertex. Since divergences occur because of loop integrations, this means that divergences occur only when a leg from one vertex is connected to the same vertex, and not from the interaction between two or more vertices. Such self-inﬂicted divergences are called “normal ordering” divergences. In two dimensions, the only ultraviolet divergences come from “normal ordering,” and not from the type of interaction. Finally, we note that when d ≥ 7, there are no theories with a ﬁnite number of primitively divergent graphs. The last theory in higher dimensions is λϕ3 in six dimensions, where λ is dimensionless, since ϕ now has dimension −2. There the primitively divergent diagrams are few since V does not appear in the expression for D

D = 6 − 2E (ϕ3 in six dimensions) , (4.58) so that the one-, two- and three-point functions are primitively divergent (quartic, quadratic and logarithmic, respectively). This theory, although having an unsatisfactory potential unbounded from below, is interesting in that it shares with the more complicated gauge theories the property of asymptotic freedom. To summarize this section, we have noticed the appearance of ultraviolet divergences in Feynman diagrams with loops (the bad news), but we have seen that we can, at least in our theory, narrow them down as coming only from two primitively divergent Green’s functions (the good news). Hence, if we can arrange to stop divergences from appearing in G˜(2) and G˜(4), we have a hope of stemming the ﬂood and obtaining convergent answers!

4.1.1 PROBLEMS

A. In four dimensions, find all primitively divergent diagrams for the ϕ3 theory. For each, give examples in lowest order of perturbation theory. vskip .5cmB. In three dimensions (d = 3), list all theories of interacting scalars with a finite number of primitively divergent graphs. Give graphical examples for each. 4.2 Dimensional Regularization of Feynman Integrals 15 C. Repeat B, when d = 5, and show that when d ≥ 7 there are no theories with a finite number of primitively divergent graphs.

D. For d = 2 3, 5, 6, ﬁnd the dimensions of the various coupling constants in the theories where there is a ﬁnite number of primitively divergent graphs.

4.2 Dimensional Regularization of Feynman Integrals In the following, we proceed to evaluate the Feynman diagrams. The procedure is straightforward for the UV convergent ones, while special measures have to be taken to evaluate the divergent ones. In those we are confronted with integrals of the form

Z +∞ 4 I4(k) = d `F (`, k) , (4.59) −∞ where for large `, F behaves either as `−2 or `−4. The basic idea behind the technique of dimensional regularization is that by lowering the number of dimensions over which one integrates, the divergences trivially disappear. For instance, if F → `−4, then in two dimensions the integral (4.3.1) converges at the UV end. Mathematically, we can introduce the function

Z I(ω, k) = d2ω`F (`, k) , (4.60) as a function of the (complex) variable ω. Evaluate it in a domain here I has no singularities in the ω plane. Then invent a function I0(ω, k) which has well-deﬁned singularities outside of the domain of convergence. We say by analytic continuation that I and I0 are the same function. A nice example, which is the basis for the method of analytic continuation, is the diﬀerence between the Euler and Weierstrass representations of the Γ-function. For 0, the Euler representation is

Z ∞ Γ(z) = dt e−ttz−1 . (4.61) 0 As such, it diverges when

∞ X (−1)n Z α Z ∞ Γ(z) = dt tn+z−1 + dt e−ttz−1 , (4.62) n! n=0 0 α where α is totally arbitrary. The second integral is well-defined even when 0. The first integral has simple poles whenever z is a negative integer or zero. We find

∞ X (−1)n αn+z Z ∞ Γ(z) = + dt e−ttz−1 . (4.63) n! (z + n) n=0 α This form is valid everywhere in the z-plane. Furthermore, it should not dΓ depend on the arbitrary coefficient α (you can check that dα = 0). When α = 1, it is the Weierstrass representation of the Γ-function. Still, to isolate the singularities we did introduce an arbitrary scale in the process, although the end result is independent of it. Our problem is that integral expressions like (4.3.2) are like Euler’s. We want to find the equivalent of Weierstrass’ representation. Our procedure will be as follows: 1) establish a finite domain of convergence for the loop integral in the ω-plane. For divergent integrals, it will typically lie to the left of the ω = 2 line; 2) construct a new function which overlaps with the loop integral in its domain of convergence, but is defined in a larger domain which encloses the point ω = 2; 3) take the limit ω → 2. We now show how this is done in the case of one-loop diagrams, following the procedures of ‘t Hooft and Veltman, Nucl. Phys. 44B, 189 (1972). Let us split up the domain of integration as

d2ω` → d4` d2ω−4` . (4.64) Next in the 2ω − 4 space, introduce polar coordinates, and call L the length of the 2ω − 4 dimensional `-vector. The integral now reads

Z Z Z ∞ 4 2ω−5 1 I = d ` dΩ2ω−4 dL L 2 2 2 . (4.65) 0 (L + ` + m ) Integration over the angles can be performed (see Appendix B), with the result

ω−2 Z Z ∞ 2π 4 2ω−5 1 I = d ` dL L 2 2 2 . (4.66) Γ(ω − 2) 0 (L + ` + m ) This expression is not well-deﬁned because it is UV divergent for ω ≥ 1, 4.2 Dimensional Regularization of Feynman Integrals 17 and the integration over L diverges at the lower end (“infrared”) whenever ω ≤ 2. Thus, there is no overlapping region in the ω plane where I is well- deﬁned. The IR divergence is, however, an artifact of the break up of the measure. Observe that by writing

1 d ω−2 L2ω−6 = L2 , (4.67) ω − 2 dL2 and integrating by parts over L2, and throwing away the surface term, we obtain

ω−2 Z Z ∞ π 4 2 2ω−2 d 1 I = d ` dL L − 2 2 2 2 , (4.68) Γ(ω − 1) 0 dL L + ` + m where we have used Γ(ω − 1) = (ω − 2)Γ(ω − 2). Now the representation (4.3.9) has an infrared divergence for ω ≤ 1 and the same UV divergence for ω ≥ 1, that is, still no overlapping region of convergence. So we do it again, and obtain

ω−2 Z Z ∞ 2 π 4 2 2ω−1 d 1 I = d ` dL L − 2 2 2 2 , (4.69) Γ(ω) 0 dL L + ` + m an expression which is well-defined for 0 < ω < 1. Note that we had to move the IR convergence region two units to obtain a nonzero region of convergence. Had the loop integral been logarithmically divergent, one such step would have sufficed. Having obtain an expression for I convergent in a finite domain (in this case 0 < ω < 1), we want to continue I of (4.3.10) to the physical point ω = 2. It goes as follows: insert in the integrand the clever expression

1 ∂L ∂` 1 = + µ , (4.70) 5 ∂L ∂`µ and integrate by parts in the region of convergence. We obtain

ω−2 Z Z ∞ 2ω−1 1 2π 4 2 ∂ 2 ∂ L I = − d ` dL `µ + L 2 3 . (4.71) 5 Γ(ω) 0 ∂`µ ∂L (L2 + `2 + m2) A little bit of algebra gives, after reexpressing the right-hand side in terms of I, 18 Perturbative Evaluation of the Path Integral: λϕ4 Theory

2 ω−2 Z Z ∞ 2ω−1 3m 2π 4 2 L I = − d ` dL 4 . (4.72) ω − 1 Γ(ω) 0 (L2 + `2 + m2) This expression displays explicitly a pole at ω = 1. The integral now diverges at the upper end when ω ≥ 2. So we repeat the process and reinsert (4.3.11) in (4.3.13). The result is predictable:

4 ω−2 Z Z ∞ 2ω−1 2 · 3 · 4m π 4 2 L I = d ` dL 5 . (4.73) (ω − 1)(ω − 2) Γ(ω) 0 (L2 + `2 + m2) This is the desired result. The only hint of a divergence comes from the simple pole at ω = 2, since the integral itself now converges. Let us summarize. We first define a finite integral in the ω-plane to be what we mean by R d2ω`F (`, k): in this case it is the expression given by (4.3.10), and constitutes our starting point. Then if the region of convergence does not include ω = 2, we continue analytically by means of the trick (4.3.11). It would be nice to show that for a convergent integral, the procedure that leads to (4.3.10) indeed gives the right answer. Take as an example the convergent integral

Z 1 I = d2ω` . (4.74) (`2 + m2)6

It is easy to see that the same procedure leads to the expression

ω−2 Z Z ∞ π 4 2 2ω−2 d 1 I(ω) = d ` dL L − 2 6 , Γ(ω − 1) 0 dL (L2 + `2 + m2) (4.75) which is perfectly ﬁnite at ω = 2. We ﬁnd

Z (−1) ∞ Z 1 I(2) = d4` = d4` , (4.76) 2 2 2 6 2 2 6 (L + ` + m ) 0 (` + m ) as desired. Therefore, the procedure is entirely consistent. After all this ponderous work, let us turn to a more cavalier interpretation of integrals in 2ω-dimensions, as derived in Appendix B. If we were to blindly plug in the formulae of Appendix B, we would obtain 4.2 Dimensional Regularization of Feynman Integrals 19

Z d2ω` Γ(1 − ω) 1 = πω . (4.77) (`2 + m2) Γ(1) (m2)1−ω We expand around ω = 2, using for n = 0, 1, 2, ··· and → 0 the formula

(−1)n 1 1 π2 Γ(−n+) = + ψ(n + 1) + + ψ2(n + 1) − ψ0(n + 1) + O(2) , n! 2 3 (4.78) where 1 1 d ln Γ(s) ψ(n + 1) = 1 + + ··· + − γ , ψ(s) = , (4.79) 2 n ds n π2 X 1 π2 ψ0(n + 1) = − , ψ0(1) = (4.80) 6 k2 6 k=1 γ being the Euler-Mascheroni constant

ψ(1) = −γ = −0.5772 ··· . (4.81)

The result is

Z d2ω` 1 lim = −π2m2 + ψ(2) . (4.82) ω→2 (`2 + m2) 2 − ω It can be shown that this is the same result as that obtained by integrating (4.3.14). From now on, we are going to use the naive formulae of Appendix B, and not concern ourselves with their justiﬁcation since we know they are legitimate, thanks to ‘t Hooft and Veltman.

4.2.1 PROBLEMS

A. Show that the naive procedure of Appendix B and the more careful one outlines in this section lead to the same results for the integrals Z 1 Z 1 I = d2ω` ,I = d2ω` . (4.83) (`2 + m2) (`2 + m2)2

B. Prove the last four formulae of Appendix B. 20 Perturbative Evaluation of the Path Integral: λϕ4 Theory 4.3 Evaluation of Feynman Integrals Using the techniques of the previous section and the formulae of Appendix B, we proceed to evaluate the low order Feynman diagrams for λϕ4 theory. We have seen that the technique of dimensional regularization is predi- cated on the evaluation of Feynman integrals in 2ω dimensions, where the coupling constant λ is no longer dimensionless. We ﬁnd it convenient to redeﬁne it in terms of a dimensionless coupling constant by the artifact

22−ω λold = λnew µ , (4.84)

2 where λnew is dimensionless and µ is an arbitrary constant with the dimension of mass. Alternatively, we can say that we evaluate the Green’s functions for the theory deﬁned by the action

Z 1 1 λ 2−ω S [ϕ] = d2ωx ∂ ϕ∂ ϕ + m2ϕ2 + µ2 ϕ4 . (4.85) ω 2 µ µ 2 4! The Feynman rules for this theory are the same as the one for the theory in four-dimensions with three exceptions: 1) the scalar product between vectors is summed over their 2ω components, 2) the loop integrals appear 2ω with R d ` and 3) the vertex strength −λ is replaced by (−λ) µ22−ω. (2π)2ω Let us evaluate the lowest order diagrams for this theory. We start with the “tadpole” diagram (in conventional terms such a diagram comes from not normal ordering the interaction term). It is given by

2ω 1 2−ω Z d ` 1 .7.7 ≡ T = (−λ) µ2 (4.86) 2 (2π)2ω (`2 + m2) λm2 4πµ2 2−ω = − Γ(1 − ω) , (4.87) 2(4π)2 m2 where we have used (B–16), and have kept m2 in front because the diagram has dimension of mass squared. By expanding around ω = 2, we obtain

λm2 4πµ2 1 T = − 1 + (2 − ω) ln + ··· − − ψ(2) + ··· (4.88) 32π2 m2 2 − ω λm2 1 m2 = + ψ(2) − ln + O(2 − ω) . (4.89) 32π2 2 − ω 4πµ2 Observe that in this formula, the judicious introduction of the arbitrary scale 4.3 Evaluation of Feynman Integrals 21 µ2 allows us to keep track of dimensions, and that the pole from the expan- 2−ω 4πµ2 sion of Γ cancels against the first order term in the expansion of m2 , leaving us with a finite contribution as ω → 2. This feature will be present in the evaluation of all divergent graphs. We conclude that the divergence in T appears as a simple pole, and that the finite part of T , which in this case does not depend on the external momenta, is totally arbitrary because a change in µ2 affects it. The next diagram is the “fish”

11 p1 + p2 + p3 + p4 = 0 . (4.90)

The Feynman rules give [p = p1 + p2]

2ω 1 4−2ω Z d ` 1 1 11 = (−λ)2 µ2 . (4.91) 2 (2π)2ω (`2 + m2) [(` − p)2 + m2] When there are more than one propagator taking part in a loop integration, it is convenient to introduce the Feynman parametrization based on the formula Z 1 1 Γ(a1 + a2 + ··· ak) a1 a2 ak = (4.92) D1 D2 ··· Dk Γ(a1)Γ(a2) ··· Γ(ak) 0 Z 1 a1−1 ak−1 δ(1 − x1 − · · · − xk)x1 ··· xk ··· dx1 ··· dx (4.93). k a1+···+ak 0 (D1x1 + ··· Dkxk) It allows for a convenient rearrangement of the loop momenta. In this case we use it in the form

1 Z 1 dx 2 2 2 2 = 2 . [` + m ] [(` − p) + m ] 0 [`2 + m2 − 2` · p(1 − x) + p2(1 − x)] (4.94) The denominator can be rewritten in the form

`02 + m2 + p2x(1 − x) , (4.95) where

`0 = ` − p(1 − x) . (4.96)

Since we are dealing with convergent integrals, we use d2ω`0 = d2ω` and 22 Perturbative Evaluation of the Path Integral: λϕ4 Theory relabel the loop integral from ` to `0. The result of these manipulations yields 2 Z 1 Z 2ω λ 24−2ω d ` 1 .7.7 = µ dx 2ω 2 . (4.97) 2 0 (2π) [`2 + m2 + p2x(1 − x)] Thanks to this trick, we can now use (B–16) and integrate, obtaining(4.98) 2 Z 1 λ 24−2ω Γ(2 − ω) 1 .7.7 = µ dx ω 2−ω . (4.99) 2 0 (4π) [m2 + p2x(1 − x)]

Before expanding, remember that this diagram has dimensions µ22−ω, which we show explicitly. After expansion we obtain to O(2 − ω)

.6.6 (4.100)

2 Z 1 2 2 22−ω λ 1 m + p x(1 − x) = µ 2 dx + ψ(1) − ln 2 (4.101) 32π 0 2 − ω 4πµ 2 Z 1 2 2 22−ω λ 1 m + p x(1 − x) = µ 2 + ψ(1) − dx ln 2 (4.102). 32π 2 − ω 0 4πµ Again, observe that this time the finite part depends not only on µ2, which is arbitrary, but also on the external momenta. Let us emphasize that this arbitrariness in the finite part is generic to the method because of the separation of a divergent expression into a divergence plus a finite part [after all ∞ + 5 = ∞ + 6!]. There remains to integrate over the Feynman parameter x. Since x(1−x) is always positive [0 < x(1 − x) < 1/4] over the range of integration, the argument of the logarithm is always positive, making the integral easy to evaluate. We use the formula

√ Z 1 4 √ 1 + a + 1 dx ln 1 + x(1 − x) = −2 + 1 + a ln √ , a > 0 . 0 a 1 + a − 1 (4.103) The result is then

2 " 2 2−ω λ 1 4πµ .7.7 = µ2 + ψ(1) + 2 + ln (4.104) 32π2 2 − ω m2

s q 4m2  # 4m2  1 + p2 + 1 − 1 − ln + O(2 − ω) . (4.105) p2 q 4m2  1 + p2 − 1 4.3 Evaluation of Feynman Integrals 23 In the evaluation of the four-point function, there will be three such contributions with p = p1 + p2, p = p1 + p3, and p = p1 + p4, corresponding to the s-, t- and u-channel contributions [caution: here all momenta are incoming]. This diagram is computed in the Euclidean domain; continuation to Minkowski space will entail changing the sign of p2 and carefully inter- preting the result. As it stands, however, the ﬁnite part has no interesting analytical structure as long as p2 > 0. We will return to this point when the interpretation of the result is discussed in Minkowski space. Using the same techniques, we compute the “double scoop” diagram:

2ω 2ω 1 4−2ω Z d ` 1 Z d q 1 .7.7 = λ2 µ2 (4.106) 4 (2π)2ω `2 + m2 (2π)2ω (q2 + m2)2 (4.107) λ2m2 1 1 4πµ2 = + 2 ln + ψ(2) + ψ(1) (4.108) 1024π4 (2 − ω)2 (2 − ω) m2 4πµ2 4πµ2 1 +2 ln2 + 2 ln [ψ(2) + ψ(1)] + [ψ(2) + ψ(1)](4.109)2 m2 m2 2 2π2 − ψ0(2) − ψ0(1) + O(ω − 2) . (4.110) 3

Note the appearance of a double pole and the arbitrariness of the residue of the simple pole and of the ﬁnite part. Finally, we calculate in detail the “setting sun” diagram

1.51.5 (4.111)

It is worth doing in some detail as it is a two-loop diagram. Call the diagram Σ(p). The Feynman rules give

4−2ω λ2 µ2 Z d2ω` Z d2ωq 1 Σ(p) = 6 (2π)2ω (2π)2ω (`2 + m2)(q2 + m2) ([q + p − `]2 + m2) (4.112) In diagrams involving several loops, the ultraviolet divergences will ﬁnd their way into parametric integrals. For convenience one wants to have as few divergences as possible in these integrals. This means that special techniques have to be applied on (4.4.19) before introducing the Feynman parameters, unless the integral is at worse logarithmically divergent. Using the trick 24 Perturbative Evaluation of the Path Integral: λϕ4 Theory

1 ∂` ∂q 1 = µ + µ (4.113) 4ω ∂`µ ∂qµ in the integrand of (4.4.19) and integrating by parts, we ﬁnd

2 Z 2ω Z 2ω 1 λ 24−2ω d ` d q ∂ ∂ Σ(p) = − µ 2ω 2ω `µ + qµ (4.114) 4ω 6 (2π) (2π) ∂`µ ∂qµ 1 · , (4.115) (`2 + m2)(q2 + m2) ([q + p − `]2 + m2) where we have discarded the surface terms, in keeping with the philosophy of analytic continuation we discussed in conjunction with one-loop diagrams [see ‘t Hooft and Veltman, Nucl. Phys. B44, 189 (1972), and also Curtright and Ghandour, Annals of Physics, 106, 209 (1977)]. Explicit diﬀerentiation gives

2 2ω 2ω 1 λ 4−2ω Z d ` Z d q Σ(p) = − µ2 (4.116) (2ω − 3) 6 (2π)2ω (2π)2ω 3m2 + p · (p + q − `) × (4.117) (q2 + m2)(`2 + m2) ([q − ` + p]2 + m2)2 2 1 λ 4−2ω = − µ2 3m2K(p) + pµK (p) , (4.118) 2ω − 3 6 µ where

Z d2ω` Z d2ωq 1 K(p) = (2π)2ω (2π)2ω (q2 + m2)2(`2 + m2) [(q − ` + p)2 + m2] (4.119) and

Z 2ω Z 2ω d ` d q (p + q − `)µ Kµ(p) = , (2π)2ω (2π)2ω (q2 + m2)(`2 + m2) [(q − ` + p)2 + m2]2 (4.120) where we have freely made several linear changes of variables over loop momenta. Now K(p) is log divergent and Kµ(p) is linearly divergent. We ﬁrst evaluate K(p). It is prudent to introduce Feynman parameters one loop at a time, starting from the most divergent one. So doing we obtain 4.3 Evaluation of Feynman Integrals 25

Z d2ω` Z d2ωq 1 Z 1 1 K(p) = 2ω 2ω 2 dx 2 . (2π) (2π) (q2 + m2) 0 [`2 + m2 + (p + q)2x(1 − x)] (4.121) Integrate over `, using (B–16), to obtain

Γ(2 − ω) Z 1 Z d2ωq 1 1 K(p) = ω dx 2ω 2 2−ω . (4π) 0 (2π) (q2 + m2) [m2 + (p + q)2x(1 − x)] (4.122) Proceed to use (4.4.8) to rewrite

Z 1 Z 1 Z 2ω Γ(4 − ω) ω−2 1−ω d q K(p) = ω dx [x(1 − x)] dy y (1 − y) (4.123)2ω (4π) 0 0 (2π) y ω−4 × q2 + p2y(1 − y) + m2 1 − y + . (4.124) x(1 − x) Integration over q ﬁnally gives

Z 1 Z 1 Γ(4 − 2ω) ω−2 1−ω K(p) = 2ω dx [x(1 − x)] dy y (1 − y) (4.125) (4π) 0 0 y 2ω−4 × p2y (1 − y) + m2 1 − y + .(4.126) x(1 − x) It is convenient to introduce

2 − ω ≡ (> 0 because of the analytic continuation) (4.127) and expand (4.4.29) around = 0. The parametric integral has a pole at = 0 coming from y = 0. Then set

Z 1 Z 1 Γ(2) − −1+ K(p) = 4−2 dx [x(1 − x)] dy y (1 − y) (4.128) (4π) 0 0 y −2 quad × p2y(1 − y) + m2 1 − y + .(4.129) x(1 − x) Using

1 d y−1+ = y , (4.130) dy 26 Perturbative Evaluation of the Path Integral: λϕ4 Theory and integrating by parts, we ﬁnd

Z 1 Z 1 Γ(2) 1 − K(p) = 4−2 dx [x(1 − x)] dy y (4π) 0 0 d y × 1 + 2(1 − y) ln p2y(1 − y) + m2 1 − y + dy x(1 − x) y −2 × p2y(1 − y) + m2 1 − y + . (4.131) x(1 − x) Now the parametric integral can be done by expanding around = 0. The evaluation of Kµ proceeds in a similar way, giving

Z 1 Z 1 2 Γ(2) − pµKµ = p 4−2 dx [x(1 − x)] dy y (1 − y) (4.132) (4π) 0 0 y −2 × p2y(1 − y) + m2 1 − y + . (4.133) x(1 − x)

Here the parametric integral converges so that the p2 singularity is only at the simple pole. Expanding around = 0 gives

Γ(2) 1 K(p) = 1 + − 2 ln m2 + O(2) (4.134) (4π)4−2 Γ(2) 1 p K (p) = p2 + O() . (4.135) µ µ (4π)4−2 2

2 The O( )[O()] terms in the parametric integral for K(p)[pµKµ(p)] are very complicated, and give contributions to the ﬁnite part of Σ(p). Putting it all together we ﬁnd

λ2 3m2 3m2 3 4πµ2 1 Σ(p) = + + ψ(1) + ln + p2 + finite . 6 (16π2)2 22 2 m2 4 (4.136) Observe that we now have arbitrariness at the level of the simple pole (as well as at the level of the finite part). The finite part of Σ(p) is very difficult to obtain. It cannot be done in closed form and necessitates the introduction of the “dilogarithm” (or Spence) function defined by 4.4 Renormalization 27

Z 1 dt Li2(x) ≡ − log(1 − xt) . (4.137) 0 t Experience shows that when masses are present, the ﬁnite parts of two-loop diagrams are quite lengthy to evaluate.

4.3.1 PROBLEMS

A. Show that for a > 0

√ Z 1 4 √ 1 + a + 1 I(a) = dx ln 1 + x(1 − x) = −2 + 1 + a ln √ . (4.138) 0 a 1 + a − 1

∗ 4 B. Now let z = a . Find the singularity structure of I(z) in the complex z-plane and specify I(z) when z is real.

C. Derive the form of the ﬁnite part of Σ(p) when m2 = 0.

D. Derive the expression for pµKµ [Eq. (4.4.34)].

∗∗E. Express K(p) to order 2 and write the resulting integrals in terms of dilogarithms.

4.4 Renormalization In the previous sections we have shown how to evaluate Feynman diagrams. In λϕ4 theory we found that some diagrams are ultraviolet divergent, with the divergences showing up only in two- and four-point Green functions (the primitively divergent diagrams). When dimensional regularization was used to regularize these diagrams, the infinities resulted in poles in the dimension plane in = 2 − n/2 > 0, n being the number of space-time dimensions; in addition, the finite part of these diagrams was arbitrary, depending in our scheme on a mysterious mass parameter µ. We are now going to show how to eliminate these poles order by order in λ. The technique is very simple: alter the Feynman rules at each order so as to obtain a finite result as → 0. As a first example, consider the tadpole diagram 28 Perturbative Evaluation of the Path Integral: λϕ4 Theory

λˆ 1 1.5 = m2 + ψ(2) − lnm ˆ 2 + O() , (4.139) 2 where

λ m2 λˆ ≡ , mˆ 2 ≡ . (4.140) 16π2 4πµ2 This inﬁnity can be hidden away by adding to L the additional term

m2 1 λˆ + F (, mˆ 2) ϕ2 , (4.141) 4 1 which we consider to be an extra interaction term. Here F1 is an arbitrary dimensionless function, analytic as → 0; its presence reﬂects the arbitrariness of the procedure. This extra term results in a new Feynman rule indicated by

1 1 1.5 = − m2λˆ + F . (4.142) 2 1 Thus if we compute the inverse propagator to (λ), we find (remember that the correction to the inverse propagator picks up a minus sign for inverting) .7.7 = .7.7 + .7.7 + .7.7 + O(λ2) (4.143) 1 Γ˜(2) (p) = p2 + m2 1 − λˆ ψ(2) − lnm ˆ 2 − F + O(λ2) .(4.144) new 2 1 This very naive procedure makes the theory finite to order λ. The extra term (4.5.3) is called a counterterm. It is crucial to note that its dependence on the field (and its derivatives) is the same as that of a term already appearing in L (in this case the mass term). Now we proceed to order λ2. The one particle irreducible (1PI) four-point function is given by .7.7 .7.7.7.7 (4.145) .7.9.7.9 O(λ2) (4.146) " 3 1 1 Γ˜(4)(p , p , p , p ) = −µ2λ 1 − λˆ + ψ(1) + 2 − lnm ˆ 2 − A(s, t, u) 1 2 2 4 2 3 # + O() + O(λ3) , (4.147) 4.4 Renormalization 29 where

q 2 2 1/2 4m X 4m 1 + z + 1 A(s, t, u) = 1 + ln , (4.148) z q 4m2 z=s,t,u 1 + z − 1 and where s, t and u are Mandelstam variables

2 2 2 s = (p1 + p2) t = (p1 + p3) u = (p1 + p4) . (4.149) As it stands Γ˜(4) is divergent as → 0. To remedy this situation, we add yet another term to L

1 3λˆ 1 µ2λ · + G (, mˆ 2) ϕ4 , (4.150) 4! 2 1 where G1 is an arbitrary dimensionless function of , analytic as → 0. This new counterterm results in an additional Feynman rule denoted by

3 1 .7.7 = − µ2λ λˆ + G . (4.151) 2 1 It is added in the calculation of the new Γ˜(4), yielding a ﬁnite result as → 0:

˜(4) Γnew = 2.61.5 (4.152) 3 1 = −µ2λ 1 − λˆ −G + ψ(1) + 2 − lnm ˆ 2 − A(s, t, u) + O(4.153)(λ3) . 2 1 3 The contribution to Γ˜(2) in O(λ2) can be handled in a similar way, but there the additional Feynman rules (4.5.4) and (4.5.10) must be used. Hence we have diagrammatically for the inverse propagator

.7.7 .7.7.7.7.7.7 .7.7.7.7.7.7.7.7 (4.154) where the extra Feynman rules have induced to this order two new diagrams. These are easily calculated to be (see problem)

m2 1 1 .7.7 λˆ2 + ψ(1) + F − lnm ˆ 2 + ··· , (4.155) 4 2 1 were we show only the poles in . We also have 30 Perturbative Evaluation of the Path Integral: λϕ4 Theory

3m2 1 1 .7.7 λˆ2 + ψ(2) + G − lnm ˆ 2 + ··· . (4.156) 4 2 1 Comparing with the “double scoop” diagram of the previous section

m2 1 1 .7.7 − λˆ2 + ψ(2) + ψ(1) − 2 lnm ˆ 2 + ··· , (4.157) 4 2 we observe that its double pole is exactly canceled by the counterterm diagram (4.5.13) although the simple pole remains. This is obvious from the point of view of diagrams since the combination .7.1 + .7.1 is by definition finite. Adding all the diagrams, we find

λˆ2 m2 1 1 1.1 = − p2 + λ2 + (F + 3G − 1) + ··· + O(λ3) , (4.158) 24 2 2 2 1 1 where, again, we have not shown the ﬁnite part. We are faced again with a divergent expression as → 0. Lo and behold! The lnm ˆ 2 term present in the simple poles of individual diagrams have disappeared as well as the Euler constant present in each ψ(n) but absent in the diﬀerence

1 ψ(n + 1) − ψ(n) = . (4.159) n In order to cancel the poles in (4.5.16) we introduce a new mass counterterm so that the mass counterterm Feynman rule becomes

" ! # m2 λˆ2 1 λˆ2 1.25 = − + λˆ + (F + 3G − 1) + λˆ2F + λFˆ (4.160) 2 2 4 1 1 2 1

2 where F2 is an arbitrary function of andm ˆ , but ﬁnite as → 0. This term is generated by the additional counterterm

" ! # m2 λˆ2 1 λˆ2 ϕ2 + λˆ + (F + 3G − 1) + λˆ2F + λFˆ . (4.161) 4 2 4 1 1 2 1

This takes care of only one type of inﬁnity in Γ˜(2). The other one is canceled by adding yet another term to our ever expanding Lagrangian of the form

" # 1 λˆ2 ∂ ϕ∂µϕ − − λˆ2H , m2 , (4.162) 2 µ 24 2 4.4 Renormalization 31

H2 being arbitrary and analytic as → 0. Thus with all this patching up, we have been able to eliminate the ultraviolet divergences to order λ2. It is clear that we can continue this little game ad nauseam: calculate diagrams to order λ3, with the original L and the counterterms (4.5.19) and (4.5.20); then invent new counterterms which to O(λ3) are chosen to cancel the new divergences, etc. So far in this process, the remarkable thing has been that the counterterms needed to remove the divergences all generate new interactions of the same type as those that were present in the original Lagrangian; we have not been forced (to O(λ2)) to introduce counterterms that correspond to terms of a type absent in L. If it can be shown that this noteworthy matching feature continues to all orders in λ, we will say that the theory is renormalizable. Here we do not attempt a proof for λϕ4 theory, but rather point out where the procedure might fail. Consider a generic two loop diagram

1.1. (4.163) where all external legs have been removed. The diagram is divergent due to the various loop integrations. In accordance with our new rules for the counterterms, it must be added to the three counterterm diagrams, (presented here schematically)

.7.7 .7.7 .7.7 (4.164) Here each • represents the lowest order counterterm vertex needed to cancel the inﬁnity coming from one loop diagrams. These counterterm diagrams 1 2 will therefore contain a coming from • which multiplies a ln p where p is some momentum coming from the loop integration. It would therefore seem that this would generate counterterms of the form

ln p2 , (4.165) which do not correspond to any term in L because of the ln p2 residue, which is highly nonlocal in position space. Such terms would clearly throw a monkey-wrench in the works. This is the famous problem of overlapping divergences. A close study of these diagrams shows that the sum of all the diagrams shown above does not contain any poles with logarithmic residue: they cancel against similar poles contained in the two loop diagram. We have in fact seen as example of this miracle when the lnm ˆ 2 terms canceled from the residue of the simple pole in (4.5.16). [For more details, the reader 32 Perturbative Evaluation of the Path Integral: λϕ4 Theory is referred to the paper of ‘t Hooft and Veltman, Nucl. Phys. 448, 189 (1972).] In the proof of renormalization, it is crucial to be able to prove that these overlapping divergences do, in fact, cancel. Let us assume that this is so that to an arbitrarily high order in λ only counterterms which match the original L are needed to render the theory ﬁnite. This means that the Lagrangian which gives ﬁnite answers has the form

ren L = L + Lc.t. , (4.166) where L is our original Lagrangian

1 1 λ L = ∂ ϕ∂µϕ + m2ϕ2 + µ2ϕ4 (4.167) 2 µ 2 4! and Lc.t. is the counterterm Lagrangian

1 1 λ L = A∂ ϕ∂µϕ + m2Bϕ2 + µ2Cϕ4 . (4.168) c.t. 2 µ 2 4! It is (by assumption, but verified to O(λ2)) exactly of the same form as L, but with specially designed A, B and C so that the Green’s functions generated by Lren are finite as → 0. By defining new fields and parameters, we can rewrite Lren in the form

1 1 λ Lren = ∂ ϕ ∂µϕ + m2ϕ2 + 0 ϕ4 , (4.169) 2 µ 0 0 2 0 0 4! 0 where 1/2 1/2 ϕ0 ≡ (1 + A) ϕ ≡ Zϕ ϕ , (4.170) 1 + B m2 = m2 = m2 (1 + B) Z−1 , (4.171) 0 1 + A ϕ 1 + C λ = λµ2 = λµ2(1 + C)Z−2 , (4.172) 0 (1 + A)2 ϕ are called the bare field, mass and coupling constant, respectively. Note that Lren looks exactly the same as L except for the parameters and the field. Yet Lren leads to a finite theory while L does not. This shows that by cleverly putting all the infinities in ϕ0, m0 and λ0, we can make the theory finite. The infinities are then absorbed by renormalization. The bare quantities all diverge as → 0 while the (renormalized) quantities m, λ all give finite (but so far arbitrary) values as → 0. The latter are to be identified with the physical parameters of the theory. In the path integral 4.4 Renormalization 33 formalism, one integrates over the fields; thus their rescaling by Zϕ can be absorbed provided one rescales the source accordingly, defining a bare source

−1/2 J0 = Zϕ J, (4.173) or a bare classical ﬁeld 1 2 ϕclo = Zϕ ϕcl . (4.174) Starting from the new Lagrangian (4.5.21) we obtain the Green’s functions of the previous section with m and λ replaced by m0 and λ0. However, by expressing the bare parameters in terms of the physical parameters m and λ and by suitable renormalizing J, we obtain ﬁnite Green’s functions. For the 1PI Green’s functions, this equality reads

˜(n) −n/2 ˜(n) Γ0 (p1, ··· , pn; λ0, m0, ) = Zϕ Γ (p1, ··· , pn; λ, m, µ, ) , (4.175) where the Γ˜(n) are finite as → 0. In this equation we can either regard the bare parameters as functions of the renormalized ones or take the bare parameters to be the independent variables; in the latter case the dressed parameters are functions of the bare parameters. In this form we note that the left hand side of (4.5.30) does not depend on µ while the right hand side is explicitly as well as implicitly (through λ and m) dependent on µ. Therefore, by differentiating (4.5.30) with respect to µ, we obtain a differential equation that summarizes the magic of renormalization

∂ ∂λ ∂ ∂m ∂ n ∂ ln Z µ + µ + µ − µ ϕ Γ˜(n) = 0 . (4.176) ∂µ ∂µ ∂λ ∂µ ∂m 2 ∂µ The beauty of this equation is that it only involves the renormalized Green’s function Γ˜(n) which is finite as → 0. The various derivatives come from the implicit dependence of Γ˜(n) on µ via λ and m. Define the coefficients

m ∂λ β λ, , ≡ µ (4.177) µ ∂µ m 1 ∂ ln Z γ λ, , ≡ µ ϕ (4.178) d µ 2 ∂µ m 1 ∂ ln m2 γ λ, , = µ . (4.179) m µ 2 ∂µ They are analytic as → 0 and dimensionless — they depend only on λ and m µ . 34 Perturbative Evaluation of the Path Integral: λϕ4 Theory On the other hand, Γ˜(n) has an engineering dimension equal to 4 − n + (n − 2), which can be read oﬀ as the sum of its degree of homogeneity in its dimensionful parameter, i.e.,

∂ ∂ ∂ µ + s + m − [4 − n + (n − 2)] Γ˜(n)(sp; m, λ, µ, ) = 0 (4.180) ∂µ ∂s ∂m where we have introduced a scale s for the momenta. This equation, in conjunction with (4.5.31) can be turned into a scaling equation for Γ˜(n), by ∂ eliminating µ ∂µ . Taking the limit → 0, we obtain

∂ m ∂ m ∂ −s + β λ, + γ λ, − 1 m (4.181) ∂s µ ∂λ m µ ∂m m −nγ λ, + 4 − n Γ˜(n)(sp; m, λ, µ) = 0 . (4.182) d µ This equation summarizes the behavior of Γ˜(n) as one scales it momenta. (An equation of this type for QED was first obtained by Gell-Mann and Low, Phys. Rev. 95, 1300 (1954); see also Peterman and Stueckelberg ,Helv. Phys. Acta 26,499 (1953).) If we could solve it, we would know how the Green’s functions at momenta sp are related to the same functions at a reference p. The difficulty in solving (4.5.36) lies in the fact that the m coefficients β, γ and γm depend on two variables λ and µ . They can be computed explicitly order by order in perturbation theory, but they are at the moment quite arbitrary because we have not stated what the finite part of the counterterms is. This will be done in the next section where various “renormalization prescriptions” will be investigated. Suffice it to say that these coefficients depend on the way we choose the finite part of the counterterms. We can express the bare parameters as a Laurent series in the renormalized parameters

" # a λ, m 2 m P∞ k µ = µ a0 λ, µ , + k=1 k (4.183)   ∞ b λ, m m X k µ m2 = m2 b λ, , + (4.184) 0  0 µ k  k=1 ∞ c λ, m m X k µ Z = c λ, , + (4.185), ϕ 0 µ k k=1 4.4 Renormalization 35 where a0, b0 and c0 are analytic as → 0. Comparing with the counterterms already obtained up to O(λ2), we ﬁnd

m 3 a λ, , = λ 1 + λˆ G + O(λ3) (4.186) 0 µ 2 1 1 m 1 b λ, , = 1 + λFˆ + λˆ2F + λˆ2H + O(λ3) (4.187) 0 µ 2 1 2 2 m m c λ, , = 1 − λˆ2H , + O(λ3) (4.188) 0 µ 2 µ m 3 λ2 a λ, = + O(λ3) (4.189) 1 µ 2 16π2 ! m 1 λˆ2 λˆ2 b λ, = λˆ + (F + 3G − 1) + + O(λ3)(4.190) 1 µ 2 4 1 1 24 m 1 b λ, = λˆ2 + O(λ3) (4.191) 2 µ 2 m λˆ2 c λ, = − + O(λ3) (4.192) 1 µ 24

m In these formulas, we remark that the a, b and c coefficients depend on µ only through the unknown functions F1, F2, G1, H2. This can be understood heuristically by noting that the counterterms are used to eliminate the divergences that occur at very large momentum (∼ mass) scales. There any fixed mass parameter is not expected to play any role. Thus, as long as the counterterms have order by order no finite part, we do not expect the residues of their poles to depend on m. This is exactly what the formula (4.5.40) – (4.5.46) reflect. This remark is at the heart of the mass-independent renormalization prescription we discuss in the next section. The dependence of these coefficients on the arbitrary finite parts of the counterterms reflects the (a priori) prescription dependence of the β- and γ-functions. It follows that the solution of the renormalization group equation (4.5.31) is not be to attempted before a prescription has been chosen. The technical difficulty in m finding solutions lies in the dependence of the coefficients on both λ and µ . However, as we shall see, there is a prescription where the coefficients become mass independent, greatly simplifying the solutions of (4.5.31). Otherwise, one has to solve (4.5.31) in a region where the masses can be neglected, i.e., where momenta are large compared with input mass parameters. Finally, we mention that one can derive another type of renormalization group equation, first obtained by C. Callan [Phys. Rev. D5, 3202 (1972)] 36 Perturbative Evaluation of the Path Integral: λϕ4 Theory and K. Symanzik [Comm. Math. Phys. 23, 49 (1971)]. This kind of equation studies the variation of the Γ’s˜ with respect to the physical mass. ∂ The β and γ coefficients depend only on λ; the γm and µ ∂µ terms do not appear but are replaced by an inhomogeneous term, which can be neglected in the limit of small masses, or equivalently large momenta.

4.4.1 PROBLEMS

A. Compute the value of the extra counterterm diagrams (4.5.12) and (4.5.13) including the ﬁnite part.

B. Verify that the propagator itself is ﬁnite to O(λ2) (hint — the propagator contains one particle reducible graph).

∗∗C. In λϕ3 theory, verify that the overlapping divergences from the diagram .7.7 (4.193) are, in fact, canceled by the counterterm diagram that regulates the one loop diagram .7.7 (4.194)

4.5 Renormalization Prescriptions In the previous section, we carried out in detail the renormalization procedure in λϕ4 theory. Besides the arbitrary scale µ, the elimination of divergences brought with it an extra arbitrariness reflected by the functions F1, F2, G1, H2, ··· which constitute the finite part of the counterterms. From the structure of the Lagrangian that gives finite answers

ren L = L + Lc.t. , (4.195) it is clear that the finite part of Lc.t. can be absorbed in a redefinition (or finite renormalization) of the initial parameters appearing in L since both L and Lc.t. have the same structure. It follows that the finite part of the counterterms can be fixed only by defining the parameters that enter in L. There is, however, much arbitrariness in the method used to define m, λ and ϕ, and the choice of method is dictated by convenience, or by the convergence properties of the perturbation theory. In some cases it is possible to directly relate the renormalized parameters 4.5 Renormalization Prescriptions 37 to physically measured quantities. QED is a case in point where the physical electric charge is equated to the vertex function in the Thomson limit. Specification of the scale at which the renormalized parameters are equated to the relevant Green’s functions is largely arbitrary (in Euclidean space) with one important restriction for theories that involve massless particles. These theories give infrared divergent Green’s function for zero value of input momenta. It would not be wise to choose the subtraction point at the scale where the Green’s function diverges. Such points are to be avoided. Later when the amplitude is continued to Minkowski space, the subtraction scale will appear at a space-like value of input momenta and will not interfere with the singularities that Green’s functions must and do have to quality as transition amplitudes. These appear in the physical region where at least some of the momenta are always timelike. Let us give several examples of subtractions, also called prescriptions: A. This first way of fixing parameters is the most common. We define the input parameters by

˜(2) 2 2 2 Γ (p , mA) ≡ p + mA at p = 0 (4.196) (4) 2 Γ˜ (p1, p2, p3, p4) ≡ −µ λA at pi = 0 . (4.197)

In the absence of infrared divergences which occur when m2 = 0, this prescription is well-defined. We have substituted the input parameters to indicate how they were defined. Note that (4.6.2) embodies two conditions since it fixes the mass as well as the field normalization. These fix the finite parts of the counterterms. In particular, we find that

A 2 A 2 A F1 = ψ(2)−lnm ˆ A ; G1 = ψ(1)−lnm ˆ A ; H2 = 0 , etc. . (4.198)

Ideally, one would prefer to identify the coupling constant with Γ˜(4) at a physical point where the particles are in Minkowski space and on their mass- 2 2 shell (pM = m ). B. One can change the subtraction point at will provided it does not interfere with the continuation to Minkowski space or with infrared singularities. We should add that as long as the subtraction procedure is performed on Euclidean Green functions, it will result in a spacelike subtraction in Minkowski space. Thus our second prescription [H. Georgi and H. D. Politzer, Phys. Rev. D14, 1829 (1976)] is the same as A but carried out in an arbitrary value of p: 38 Perturbative Evaluation of the Path Integral: λϕ4 Theory

˜(2) 2 2 2 2 Γ (p, mB) = p + mB at p = M , (4.199) 1 Γ˜(4)(p , p , p , p ) = −µ2λ at p p = M 2 δ − , (4.200) 1 2 3 4 B i j ij 4 the later point chosen to that s = t = u = M 2. One can, of course, choose any value for s, t and u, and the p2 at which Γ˜(2) is normalized. In this prescription, the unknown functions are now ﬁxed to be

B 2 B F1 = ψ(2) − lnm ˆ B ; H2 = 0 (4.201) Z 1 2 B 2 M G1 = ψ(1) − lnm ˆ B − dx ln 1 + 2 x(1 − x) , etc.(4.202) 0 mB In this case the scale µ has been totally absorbed and replaced by the scale 2 2 2 M , which is equally arbitrary. The numerical value of M /mB now clearly becomes relevant in choosing M. The trouble with the type of prescription outlined above is that the renormalization group equation of the last section is not easily solved except in the deep Euclidean region where all masses can presumably be neglected. Yet equating a coupling constant with the value of an amplitude at some scale has some physical appeal even though the identification may take place at an unphysical point. The reason is that it brings in the masses explicitly in the calculation and allows for a ready identification of various physical thresholds. C. We now present a very beautiful prescription invented by ‘t Hooft and Weinberg [Nucl. Phys. B61, 455 (1973) and Phys. Rev. D8, 3497 (1973), respectively]. It is very simple to state and allows for a simple solution of the “renormalization group equation” (4.5.36) of the last section. In it one simply sets all the finite parts of the counterterms equal to zero, order by order in the coupling, i.e. C C C C F1 = H2 = G1 = F2 = 0 , etc. . (4.203) Then, by comparing with Eqs. (4.5.40) – (4.5.46) of the last section, we notice that all the a, b and c coefficients are independent of m. This prescription is aptly called “mass-independent” renormalization. This mass independence survives to arbitrarily high order; that this is, in fact, true is not hard to understand by means of the following heuristic argument: when the counterterms have no finite part, they just have the “bare bones” structure needed to cancel the infinite behavior at very short distances and no more, but in this regime, i.e. at infinite momenta, all masses can presumably 4.5 Renormalization Prescriptions 39 be neglected, provided the amplitudes are well-behaved as p → ∞ — hence the mass independence. It is true that in prescriptions A and B the mass dependence appeared only through the finite part of the counterterms. This enormous simplification enables us to compute the β, γ and γm coefficients appearing in (4.5.36) in a straightforward way. For instance, we now write

" ∞ # X ak(λ) λ = µ2 λ + . (4.204) 0 k k=1

Diﬀerentiate with respect to µ at ﬁxed λ0 to obtain

∞ ! ∞ 0 ! X ak(λ) ∂λ X a (λ) 0 = 2 λ + + µ 1 + k , (4.205) k ∂µ k k=1 k=1 with the prime denoting diﬀerentiation with respect to λ. In this formula λ ∂λ and ∂µ are analytic at = 0. It follows that

∂λ µ = −2λ − 2a (λ) + 2λa0 (λ) , (4.206) ∂µ 1 1 or taking the limit → 0

∂λ d β(λ) = lim µ = −2 1 − λ a1(λ) , (4.207) →0 ∂µ dλ showing that the β-function appearing in the “renormalization group equation” depends only on λ and is determined by the residue of the simple pole on . Using (4.6.12) and the fact that the residues of the various poles in must vanish in (4.6.11), we ﬁnd that

d d 1 − λ a (λ) = a0 (λ) 1 − λ a (λ) . (4.208) dλ k+1 k dλ 1 The meaning of (4.6.12) is clear; a successful renormalization means that the bare coupling does not depend on what µ is — a change in µ is accompa- nied by a change in λ to as to leave (4.6.10) invariant. Let us now evaluate β in perturbation theory. Using (4.5.43) we ﬁnd

∂λ 3λ2 µ = + O(λ3) . (4.209) ∂µ 16π2 Neglecting the terms of O(λ3), we can easily integrate (4.6.15), obtaining 40 Perturbative Evaluation of the Path Integral: λϕ4 Theory

1 λ = λs , (4.210) 1 − 3 λ ln µ 16π2 s µs where λs is the value of λ at some scale µs. It is clear from (4.6.15) that λ increases with µ. Thus, is we start with a small λs (<< 1) at a given scale µs, the effective coupling constant will increase with µ. In so doing, we will have to deal with larger and larger λ and will therefore leave the domain of validity of perturbation theory: λ << 1, or more exactly 3 λ ln µ << 1. Thus at shorter distances, we 16π2 s µs have to add more and more contributions to the right hand side of (4.6.15). Thus, for λϕ4 theory, perturbation theory becomes more reliable at larger distances, that is, in the long range properties of the interaction: the perturbative approach to defining asymptotic states is to be trusted. Should the sign of the right hand side of (4.6.15) ever be found to be negative in a field theory, then perturbation arguments would be misleading for the definition of the asymptotic states, but great for the short distance behavior. (We will see that this situation occurs in Quantum Chromodynamics (QCD) which describes the interaction among quarks — quarks are useful to describe the short range interaction of two protons but they are not asymptotic states, just the constituents of asymptotic states such as the proton.) Note that the expression for β(λ) obtained here is the same as that obtained via the ζ-function evaluation of determinants. This should not be too surprising since the former method was O(~) and here we have only shown the one loop result which is also O(~). Let us now speculate on the possible behaviors of λ as a function of µ, outside of perturbation theory. First of all we note that if β(λ) is given by (4.6.15) even for large λ, then it will blow up at a scale

16π2 µ = µs exp , (4.211) 3λs a very large scale if λs was small to start with. This is called the Landau point after Landau who recognized the same behavior in QED. However, there is no reason to believe that the one-loop contribution to β is valid for large λ. We do not know how to calculate β for large λ, but let us present some possible scenarios, starting from β = 0 at λ = 0, the no interaction point: 1. β(λ) stays positive for large λ; then λ keeps increasing with the scale in a concave or convex curve depending on the sign of β0(λ). If β(λ) blows up for some value of λ, λ itself becomes inﬁnite there (Landau point). 4.5 Renormalization Prescriptions 41 2. β(λ) starts out positive for low λ then turns over and becomes negative, crossing the axis at λF :

beta(λF ) = 0, (4.212) like

1.22 (4.213)

λF is called a ﬁxed point because if for some reason the coupling was origi- nally at λF , it would remain there. We can analyze the behavior of λ near λF by expanding β about λF , yielding the equation

∂λ µ = (λ − λ ) β0(λ ) + ··· . (4.214) ∂µ F F 0 0 We see that the sign of β (λF ) is crucial. If β (λF ) < 0, as in the figure, ∂λ µ ∂µ is positive for λ just below λF , which drives λ to a larger value, i.e., ∂λ towards the fixed point λF , while µ ∂µ is negative when λ is above the fixed point, therefore driving λ towards λF . We then see that λ will be driven towards λF as µ increases: such a fixed point is called ultraviolet stable, because λ will approach the value λF asymptotically as µ → ∞, from above or from below depending on the starting point λs which can be either above or below λF . If there were a field theory for which β behaved as in the curve, at very short distances λ would be more and more like λF . If λF were small, it would mean that if we start from a small λ < λF we never leave perturbation theory! Alternatively, if we started with λ > λF then as the distance decreases, λ would be driven to a perturbative region. These two situations are depicted below.

32.5 (4.215) None of the field theories in four dimensions exhibit this behavior perturba- tively (λF << 1). The point λ = 0 is a fixed point for which β0(0) > 0, which means that ∂λ above it µ ∂µ is positive driving λ away from it as the distance decreases. Such a fixed point is called infrared stable. Finally, we note that for small λ most field theories behave this way, with β(λ) starting out positive. 3. β(λ) starts out negative for low λ, decreasing in value monotonically. This means that λ decreases monotonically with ln µ. In this case the perturbation approximation becomes better at shorter distances, and λ is driven to 42 Perturbative Evaluation of the Path Integral: λϕ4 Theory zero which in this instance becomes an ultraviolet stable fixed point. Such coupling constant behavior for low λ is exhibited by gauge theories in four dimensions — a phenomenon known as asymptotic freedom. This will be analyzed in detail when we study Yang-Mills gauge theories. 4. β(λ) starts out negative, turns over and becomes positive, crossing the axis at λF . 32 (4.216)

0 In this case, β (λF ) > 0 and λF is an infrared point. This means that if at µs, λs < λF , λ will be driven towards 0, but if λs > λF it will be driven away from λF to larger values of λ:

32 (4.217)

We can also derive the variation of the mass with µ, starting from (4.5.41) m with b0 = 1 and bk independent of µ ; it is not hard to ﬁnd that

∂m2 db µ = 2λm2 1 (4.218) ∂µ dλ so that

db (λ) γ (λ) = λ 1 (4.219) m dλ leading to the value in λϕ4 theory

λ 7 λ 2 γ (λ) = + + O(λ3) . (4.220) m 16π2 12 16π2 We also get a recursion formula for the residues of the higher poles:

db db db d λ k+1 = b λ 1 − k 1 − λ a (λ) k = 1, 2, ··· . (4.221) dλ k dλ dλ dλ 1

Finally, we can also derive from the deﬁnition of Zϕ with c0 = 1 and ck m independent of µ the equation

∂ dc µ ln Z = −2λ 1 , (4.222) ∂µ ϕ dλ leading to 4.5 Renormalization Prescriptions 43

dc γ (λ) = −λ 1 , (4.223) d dλ so that in λϕ4 theory

1 λ 2 γ (λ) = + O(λ3) . (4.224) d 12 16π2

The ck coeﬃcients in turn obey

dc dc da dc λ k+1 = c λ 1 − a − λ 1 k , k = 1, 2, ··· . (4.225) dλ k dλ 1 dλ dλ These recursion relations are useful in computing the residues of the higher order poles in terms of those of the simple pole. This is where the power of the procedure resides: if the theory is renormalizable, then many coeﬃcients can be calculated indirectly without the aid of Feynman diagrams. In this ‘t Hooft–Weinberg mass independent renormalization prescription, the renormalization group equation is easy to integrate because the m dependence has dropped out of the coeﬃcients. It now reads

∂ ∂ ∂ −s + β(λ) + (γ (λ) − 1) m + d − nγ (λ) Γ˜(n)(sp; m, λ, µ) = 0 . ∂s ∂λ m ∂m n d (4.226) Introduce scale dependent variables λ¯(s) andm ¯ (s) such that

∂λ¯(s) s = β λ¯(s) λ¯(s = 1) = λ (4.227) ∂s ∂m¯ (s) s =m ¯ (s) γ λ¯(s) − 1 m¯ (s = 1) = m . (4.228) ∂s m Thus (4.6.28) is easy to integrate for it becomes a ﬁrst order diﬀerential equation in s; the result is

Z s ds0 ˜(n) dn ˜(n) ¯ ¯ 0 Γ (sp; m, λ, µ) = s Γ (p;m ¯ (s), λ(s), µ) exp −n 0 γd(λ(s )) . 1 s (4.229) The interpretation of this equation is that under a change of scale of the external momenta the Green’s functions scale in an unexpected way: the coupling constant and mass scale in a nontrivial fashion and the Green’s 44 Perturbative Evaluation of the Path Integral: λϕ4 Theory functions develop beside the engineering dimension dn and anomalous dimension γd for each outside leg. Suppose m had been set equal to zero initially. Then the classical theory would have been invariant under dilatations (and conformal transformations — see Chapter 1). This is no longer true in quantum theory where the regularization technique introduces a scale, either via a high momentum cutoﬀ or via the µ-parameter of dimensional regularization, which breaks dilata- tion invariance [see S. Coleman’s lecture in Aspects of Symmetry, Cambridge University Press, Cambridge, 1985]. Also this shows that the Green’s functions at scaled momenta give a behavior ruled by λ¯(s) andm ¯ (s) which therefore govern the physics at large scales. It is interesting to investigate the behavior of the Green’s functions at large s. Suppose the theory has an ultraviolet stable ﬁxed point at λ = λF . At large scales λ will be driven to λF . Hence λm and γd will be driven to γm(λF ) and γd(λF ), respectively, so that the solution of (4.6.21)

R s γ λ¯(s) dlns sm¯ (s) = m e 1 m( ) (4.230) can be integrated to

m¯ (s) → ms[−1+γm(λF )] (4.231) if we assume that the integral is dominated by the large s range. A similar assumption about the integration over λd leads to

(n) (dn−nγd(λF )) (n) γm(λF )−1 Γ˜ (sp; m, λ, µ) → s Γ˜ (p; ms , λF , µ) . (4.232)

Thus, if 1 − γm(λF ) is positive the mass can be neglected altogether at large scales and also γd(λF ) appears indeed as an anomalous dimension. We see from (4.6.23) that the naive expectation that the masses decouple from the theory at large momenta depends essentially on the integral over the anomalous dimensions which we write as

Z λ¯(s) 0 0 γm(λ ) dλ 0 . (4.233) λ β(λ )

This form suggests that unless γm(λ) and β(λ) have simultaneous zeros, the greater contributions to the integral will come from the fixed points, which “kind of” justifies our earlier assumption. If the ultraviolet stable fixed point is at λF = 0 (as in Yang-Mills theories), 4.5 Renormalization Prescriptions 45 then all is well because the large momentum behavior of the theory is given by perturbation theory, that is γm(0) = 0. In λϕ4 theory we can integrate the various equations by using the lowest perturbative results found for γm and γd. The results are

1/3 m λ¯(s) ¯ 2 m¯ (s) = e7(λ(s)−λ)/(576π ) (4.234) s λ for the scale dependent mass and

n λ¯(s)−λ (n) dn− (n) Γ˜ (sp) ∼ s 36 16π2 Γ˜ (p) (4.235) for the anomalous dimension. These results are only valid for small mass scales because as we have seen, the perturbative β function is positive and, after a certain scale is reached perturbation theory loses its validity.

4.5.1 PROBLEMS

∂λ A. Given β(λ) = µ ∂µ , describe the behavior of the hypothetical ﬁeld theories for which

0 1. β(λF ) = β (λF ) = 0 (4.236) a 2. β(λ ) = 0 λ = λ + k = 1, 2, · · · ∞ (4.237) k k F k

B. Given

∞ ! X bn(λ) m2 = m2 1 + ≡ m2Z , (4.238) 0 n m n=1 show that

1 ∂ ln Z db (λ) γ = − µ m = λ 1 (4.239) m 2 ∂µ dλ and that

db db 1 db λ n+1 = b λ 1 + n β(λ) n = 1, 2, ··· . (4.240) dλ n dλ 2 dλ n+1 2 If bn(λ) = bn, n+1λ + O(λ ), derive a formula relating bn, n+1 to b1,2. 46 Perturbative Evaluation of the Path Integral: λϕ4 Theory C. Given

∞ !1/2 X cn(λ) ϕ = ϕ 1 + ≡ ϕZ1/2 , (4.241) 0 n ϕ n=1 show that

1 ∂ ln Z dc γ (λ) = µ ϕ = −λ 1 (4.242) d 2 ∂µ dλ and that

dc dc 1 dc λ n+1 = c λ 1 + n β(λ) . (4.243) dλ n dλ 2 dλ

D. Verify the renormalization group equation in λϕ4 theory using the results from perturbation theory for β, γm and γd.

4.6 Prescription Dependence of the Renormalization Group Coefficients In the previous section we computed the value of the β and γ coefficients using the ‘t Hooft–Weinberg mass independent renormalization prescriptions to lowest nontrivial order in λ. We have drawn conclusions of physical import from their behavior — yet the alert reader might wonder to what extent these coefficients depend on the prescription for renormalization: for a general prescription of type A or B these coefficients will, in general, depend on masses through the finite parts of the counterterms. Starting from Eqs. (4.5.37) – (4.5.39) we find by differentiating with respect to µ at fixed bare parameters that

h a i ∂λ a0 ∂m m a˙ 0 = 2 a + 1 + µ a0 + 1 + − a˙ + 1 + higher poles , 0 ∂µ 0 ∂µ µ 0 (4.244) from which we deduce that

∂λ 3λ2 3λ2 3 m λ2 µ = −2λ + G + + G˙ + O(λ3) . (4.245) ∂µ 16π2 1 16π2 2 µ 16π2 1 In these formulae, the prime and dot denote diﬀerentiation with respect m to λ and µ , respectively. Comparing with (4.6.15), we see the explicit 4.6 Prescription Dependence of the Renormalization Group Coeﬃcients 47 prescription dependence coming through G1. In particular, we note that the prescription dependence enters in the lowest order because of the mass. In deriving Eq. (4.7.2) it becomes evident that the order λ3 in (4.7.2) depends directly on G1 without multiplicative mass factors. In a similar way we can show that

∂ ln m 1 λ 1 m λ λ γ = µ = + F˙ + F + O(λ3) , (4.246) m ∂µ 2 16π2 4 µ 16π2 1 2 16π2 1 as well as

∂ 1 λ 2 m λ 2 γ = µ ln Z1/2 = − H˙ (4.247) d ∂µ ϕ 48 16π2 2µ 16π2 2 λ 2 + H + O(λ3) . (4.248) 2 16π2 2 We let → 0 and ﬁnally obtain

m 3λ2 3 m λ2 β λ, = + G˙ + O(λ3) , (4.249) µ 16π2 2 µ 16π2 1 m 1 λ 1 m λ γ λ, = + F˙ + O(λ2) , (4.250) m µ 2 16π2 4 µ 16π2 1 m 1 λ 2 m λ 2 γ λ, = − H˙ + O(λ3) . (4.251) d µ 48 16π2 2µ 16π2 2 These formulae indicate that when masses are present, the prescription dependence enters in the lowest order. Then it is only when the mass can be neglected that we can say that the lowest order renormalization equation coefficients are independent of the prescription. Typically, when prescriptions of type A or B are used, one solves the renormalization group equation in a regime where the mass can be neglected. If µ is the renormalization point, m we can take µ to be exceedingly small by choosing a very large µ. Suppose we have two renormalization prescriptions. They must be related to one another by a finite renormalization since they differ only in the definition of the renormalized parameters. Hence the parameters in one prescription will be related to those in the other as follows

m λ0 = T λ, = λ + O(λ2) , (4.252) µ 48 Perturbative Evaluation of the Path Integral: λϕ4 Theory m0 m m Z0 λ0, = Z λ, U λ, ; U = 1 + O(λ) ,(4.253) m µ m µ µ m0 m m Z0 λ0, = Z λ, V λ, ; V = 1 + O(λ3)(4.254). ϕ µ ϕ µ µ

m In particular, it follows that, where A is some function of λ and µ ,

m0 m m β0 λ0, = β λ, A0 + (γ − 1) A.˙ (4.255) µ µ µ m

In the deep Euclidean region this reduces to an equation of the form

β0(λ0) = β(λ)A0(λ) , (4.256) which shows that a fixed point at λ = λF results in a fixed point at 0 λF = A(λF ) 6= λF . Note that this result is not perturbative. Hence, as can be expected, the presence (or absence) of a fixed point is prescription independent. It can also be shown that the sign of the first derivative of β at a fixed point is prescription independent (see problem). The other renormalization group parameters γm and γ also have some features which are prescription independent, notably their numerical value at a fixed point in the deep Euclidean region.

4.6.1 PROBLEMS

A. Verify Eqs. (4.7.2), (4.7.3) and (4.7.4).

B. Verify to lowest order the renormalization group equation for Γ˜(2) and (4) Γ˜ , keeping the ﬁnite parts F1, F2, H2 and G1.

dβ C. Show that the sign of dλ at a ﬁxed point λF is prescription independent, neglecting masses.

∗ 0 0 0 0 D. Relate γ (λ ) and γm(λ ) to γ(λ), γm(λ) and β(λ) where the prime and unprimed system refers to two mass independent renormalization schemes m [F , G and H functions are first taken independent of µ , but can have numerical values]. Show that the value of γ and γm at λF is prescription independent. 4.7 Continuation to Minkowski Space: Analyticity 49 4.7 Continuation to Minkowski Space: Analyticity At this stage we have obtained the finite Green’s functions at the price of introducing an arbitrary scale. However, we know from the renormalization group that if we alter this scale, nothing happens to the Green’s functions because the change is compensated by a concomitant change in the renormalized parameters and field. There now remains to express the Green’s functions in Minkowski space in order to make contact with reality. This is achieved by means of analytic continuation. Consider an Euclidean Green’s function which depends on momentap ¯1, ··· , p¯N . We first change all the time components of thep ¯ by making them imaginary

p¯ = (¯p0, p¯i) → p = (p0 = ip¯0, pi =p ¯i) . (4.257) As an example let us examine what happens to the propagator. According to the above

1 1 → , (4.258) p¯2 + m2 −p2 + m2 since we are using the Minkowski metric g00 = −gii = +1. This replacement is not entirely satisfactory because the Minkowski space expression develops a pole at p2 = m2, i.e., when

p 2 p0 = ± ~p · ~p + m . (4.259) The continuation process can be regarded as proceeding from the imaginary to the real axis in the p0 plane in a clockwise fashion: indeed the ability to perform this continuation rests on the avoidance of any pole. It follows that the poles in p0 must be taken to be just below (above) the positive (negative) real axis, that is the continuation is from

1 −1 to , (4.260) p2 + m2 p2 − m2 + i where > 0, and the limit → 0+ is to be taken at the end of all calculations. In more complicated cases the poles will be chosen so that their locations do not interfere with the clockwise rotation of the imaginary time axis into the real time axis. In this −i prescription we recognize the familiar device used to make the path integral convergent which we discussed earlier. Another way to look at this is to compare the Feynman rules in Minkowski and Euclidean spaces, say for ϕ4 theory: BIG MESS 50 Perturbative Evaluation of the Path Integral: λϕ4 Theory

 1 2 2 (Euclidean)  p¯ +m 1.02 (4.261)  i p2−m2+i (Minkowski)  −λ (Euclidean)  11 (4.262)  −iλ (Minkowski)  d4k¯ 4 (Euclidean) loop  (2π) (4.263) integration:  d4k (2π)4 (Minkowski) . Consider a Feynman diagram with L loops, V vertices and I internal lines. The diﬀerence between its computation in Minkowski and Euclidean space will be: a factor of (−i) for each vertex and propagator, and a factor of i for each loop because d4k = id4k¯. In addition, m2 in the Euclidean Green’s function is replaced by m2 − i in the corresponding Minkowski space expression. Thus, we arrive at

(n) 2 L+V +I I (n) 2 GM (p1, ··· , pn; m ) = (i) (−1) GE (¯p1 = p1, ··· p¯n = pn; m − i) . (4.264) where we replace in the Euclidean function the momenta by their Minkowski space continuation, i.e.,p ¯ = (¯p0, p¯i) is replaced by p = (ip¯0, p¯i). Using the topological relation L = I − V + 1, (V 6= 0), we obtain

(n) 2 (n) 2 GM (p.m ) = (i)GE (p =p ¯; m − i) , (4.265) valid for any Feynman diagram with V 6= 0. The exception to this rule is the original propagator expression (V = 0) for which i is replaced by −i, as can be seen from (4.8.4) by putting L = V = 0, I = 1. As an example of the procedure, let us consider the four-point function in Minkowski space. We have

" 2 Z 1 2 ˜(4) λ m − i − sx(1 − x) ΓM (p1, p2, p3, p4) = i −λ − 2 dx ln 2 2 (4.266) 2 · 16π 0 m − i + M x(1 − x) # +t- and u-channels . (4.267) 4.7 Continuation to Minkowski Space: Analyticity 51 In the above we have multiplied the Euclidean expression by i, replacedp ¯2 by −p2, m2 by m2 − i. The subtraction was performed on the Euclidean 2 ˜(4) Green’s function atp ¯ip¯j = M (δij − 1/4) so that ΓE → −λ at this symmetric point (prescription B). Note that the subtraction point appears as a parameter and it does not get changed in the continuation process. The −i in the denominator is not necessary since it can never vanish. However, 2 the numerator does suﬀer a change of sign for some value of s = (p1 + p2) . When this occurs the logarithm develops a cut in the complex s-plane. Con- sider the argument of the log

F (s, x) ≡ m2 − i − sx(1 − x) . (4.268) Now x(1 − x) is positive deﬁnite and varies between 0 and 1/4. Thus the least value of s for which F vanishes is

2 s0 = 4m (4.269) where x(1 − x) assumes its largest value. At this point Γ˜(4) develops a branch point. Traditionally, one attaches to it a cut in the complex s-plane extending from 4m2 to +∞ along the positive real s-axis. In a similar way 2 the u- and t-channel contributions give cuts starting at u0 and t0 = 4m . In view of the relation s + t + u = 4m2 , (4.270) there cuts are not all independent. These branch points can be physically understood if we interpret Γ˜(4) as the scattering amplitude for two particles with momenta p1 and p2 to scatter into two particles of momenta p3 and p4:

p1 + p2 → p3 + p4 . (4.271) Assuming that incoming and outgoing particles are on their “mass-shells” 2 2 pa = m a = 1, 2, 3, 4 , (4.272) 2 it is easy to see that s0 = 4m corresponds to the two particles having their minimum energies Ea = m. It is only beyond s0 that the two particles have enough energy to scatter nontrivially into two others. Consequently, s0 is called a physical two-particle threshold. Below it Γ˜(4) is a real function of its argument, but it develops an imaginary part for s > s0. To recapitulate: by continuing Γ˜(4) in Minkowski space, we see a nontrivial analytical structure emerging; this is, or course, the structure demanded by unitarity and causality which will enable us to regard the Minkowski space Green’s functions as transition amplitudes. 52 Perturbative Evaluation of the Path Integral: λϕ4 Theory Another example of a nontrivial analytic structure emerging as a result of continuation in Minkowski space is given by the “setting sun” diagram. There it is easy to see that the best way to ﬁnd the branch points is to look at the argument of the logarithm in the parametric integral. In this case the argument of interest is

y A = −y(1 − y)p2 + m2 1 − y + . (4.273) x(1 − x) It will vanish when

1 1 p2 = m2 + , (4.274) y x(1 − x)(1 − y) and the location of the branch point will be given by the least such value of p2. In order to ﬁnd its value we have to extremize the parametric expression that multiplies m2. In general for the two-point function, branch points will appear at the minimum value of p2 for which

2 2 p = m f(x1, x2, ··· , xn) (4.275) where x1 ··· xN are the Feynman parameters needed for an N-loop diagram. The branch point is then located at

2 2 0 0 p = m f x1, ··· , xN , (4.276) 0 where the points xi are determined by

∂f 0 = 0 at xi = xi , (4.277) ∂xi 0 [it must be checked that xi = xi is, in fact, a minimum]. Such equations are called the Landau equations after Landau who introduced a systematic procedure to hunt for the branch points of Feynman diagrams. Applying this procedure to our case, we ﬁnd from (4.8.14) that the minimum occurs at

1 1 x = y = , (4.278) 2 3 so that the branch point is located at

p2 = 9m2 . (4.279) 4.8 Cross-Sections and Unitarity 53 If you recall that the form of the “setting sun”, 1.4 we see that it corresponds to the minimum energy needed to excite three particles so it is called the three-particle threshold. Hence the propagator has in Minkowski space the following singularity structure: — a pole at p2 = m2 (appropriately displaced by the i prescription, — a branch point at p2 = 9m2 with a cut taken traditionally along the real positive p2 axis extending to p2 = +∞. When higher order of λ are included, it is expected that branch points at higher values of p2 will be encountered. This singularity structure is (of course) consistent with the interpretation of G(2) as a propagator.

4.7.1 PROBLEMS

∗A. Using diagram and physical arguments, ﬁnd the location of branch points in Γ˜(4) including O(λ4).

∗B. Repeat problem A for the propagator.

∗C. Show that Γ˜(4) satisﬁes a dispersion relation that expresses its real part in terms of its imaginary part. [Use only its perturbative value up to O(λ2).]

4.8 Cross-Sections and Unitarity We are now almost at the end of the road. We are about to identify the Minkowski space Green’s functions with transition amplitudes. However, not all functions can be transition amplitudes for they must satisfy certain requirements, notably those of unitarity and causality. As you might expect, the Green’s functions of the previous sections are acceptable candidates. In order to state precisely the requirements, let us review the S-matrix formalism and apply it to λϕ4 theory. Suppose it were possible to deﬁne states far away from the region of interaction; in particular in the very distant past or future. Such a concept clearly makes sense in the case of short range forces, as for instance in weak and strong interactions. When long range forces are involved, the concept is trickier and special care must be exercised in the deﬁnition of such states. Schematically let the states be described by kets |α; ±T i where T is a very large time and α represents a complete set of observables. These states obey completeness and orthogonality 54 Perturbative Evaluation of the Path Integral: λϕ4 Theory

X |a, ±T ih±T, α| = 1 (4.280) α hα, ±T |β, ±T i = δαβ . (4.281)

If the system were a harmonic oscillator, α would denote the occupation number, etc. It is crucial to note that these relations hold only for a given time and therefore involve no dynamics, only kinematics. If, when T is large, the interaction can be turned off (short range forces) the states should be easy to recognize because they diagonalize the unperturbed Hamiltonian. In λϕ4 theory, when m2 6= 0, there is no difficulty in recognizing these states. They are made out of the one-particle Wigner states labeled by m and ~p with the energy identified with +p~p2 + m2. If we adopt a more relativistic-looking notation and denote those states by |pi, where p stands for the momentum four vector, they are required to satisfy

Z d4p |piθ(p )δ(p2 − m2)hp| = 1 , (4.282) (2π)3 0

p hp|p0i = 2(2π)3 ~p2 + m2 δ(~p − ~p0) . (4.283)

Then any multiparticle state will be a superposition of noninteracting one- particle states:

|α, ±∞i ∼ |p1, p2, ··· , pni = |p1i ⊗ |p2i · · · ⊗ |pni . (4.284)

In λϕ4 theory there is some justification for believing these states describe the asymptotic states because the large scale behavior of the coupling is such that the free Feynman propagator accurately describes signal propagation, and we know that ∆F does propagate one particle states of the type described above. [Here we are a bit cavalier since strictly speaking ∆F propagates both positive and negative energy states.] We note in passing that if the large distance behavior of the coupling constant were such that it grew with distance, the identification of asymptotic states would have had to be made only for constructs that can escape this formidable force. This is supposedly the case for QCD where quarks are subject to such a force. Hence, they cannot serve as asymptotic states. However, this force only attacks objects with color and allows for the definition of asymptotic states that have no color (hadrons). 4.8 Cross-Sections and Unitarity 55 A question of physical interest is the computation of the transition amplitude

Tαβ = hα, +∞|β, −∞i . (4.285) With Heisenberg we deﬁne an S-matrix with the property

|β, +∞i = Sˆ|β, −∞i . (4.286) Its job is to contain all the dynamical information about the evolution of the physical states in time. Completeness of the states at +∞ and −∞

X X 1 = |β, +∞ih+∞, β| = Sˆ|β, −∞ih−∞, β|Sˆ† (4.287) β β = SˆSˆ† , (4.288) implies that Sˆ is unitary. [You can show that Sˆ†Sˆ = 1 as well.] In physical terms the unitarity of S means that the system cannot disappear into nothing [black holes?]. Most of the time nothing will happen when states are scattered — they are much more likely to miss each other than to interact. For this reason we set

Sˆ = 1 + iR,ˆ (4.289) with R containing the interesting information. Hence, it follows that

† Tαβ = hα, −∞|Sˆ |β, −∞i (4.290) † = δαβ − ihα, −∞|Rˆ |β, −∞i . (4.291) Since the interaction is Lorentz invariant, it is convenient to take this into account and write

4 (4) † Tαβ = δαβ − i(2π) δ (pα − pβ)hα, −∞|Tˆ |β, −∞i , (4.292) where pα(pβ) is the sum of the momenta in the ﬁnal (initial) state. The transition probability over all of space-time is then given by

2 h 4 (4) i † ωαβ = (2π) δ (pα − pβ) hα, −∞|Tˆ |β, −∞ihβ, −∞|Tˆ|α, −∞i . (4.293) 56 Perturbative Evaluation of the Path Integral: λϕ4 Theory The square of the δ-function is quickly understood since (2π)4δ(4)(0) is the volume of space-time [as can be seen by putting the system in a box]. It follows that the transition probability per element of space-time is

4 (4) 2 Ωαβ = (2π) δ (pα − pβ)|hα|Tˆ|βi| . (4.294)

This form is valid for states that satisfy (4.9.2). In our case the momentum states are not normalized to 1 but according to (4.9.4). Hence we ﬁnd, after dividing by the normalization, that

4 (4) (2π) δ (pα − pβ) 2 Ω(pα|pβ) ≡ Ωpαpβ = |hα|Tˆ|βi| , (4.295) (2Eα)(2Eβ) where Eα(Eβ) stands for the product of the energies in the α-(β-) state, each energy being given by

q 2 2 Ei = ~pi + m . (4.296) In scattering experiments one is usually interested in the scattering cross- section of two particles (target and projectile) into many. It is given by

3 3 3 1 d p1d p2 ··· d pN dσ(a + b → 1 + 2 + ··· + N) = Ω(pa, pb|p1, p2, ··· , pN ) 3N , vab (2π) (4.297) where vab is the relative velocity of particles a and b. For equal mass particles it is conveniently expressed as

p 2 4 (pa · pb) − m vab = . (4.298) EaEb Putting it all together we obtain

4 (4) (2π) δ (pa + pb − p1 − · · · − pn) dσ(a + b → 1 + 2 + ··· N) = q (4.299) 2 4 4 (pa · pb) − m

N 3 D E 2 Y d pi × p p |Tˆ|p ··· p (4.300). a b 1 N 2(2π)3E i=1 i

d3p Note that the measure 2E is relativistic since 4.8 Cross-Sections and Unitarity 57

d3p = d4pθ(p )δ p2 − m2 . (4.301) 2E 0 A special case of interest is elastic scattering where N = 2. Deﬁne the center of mass frame where

~pa + ~pb = ~p1 + ~p2 = 0 . (4.302) Then simple kinematics yields

|T |2 dσ(a + b → 1 + 2) = dΩ , (4.303) 64π2s where dΩ = dϕd(cos θ), θ being the angle between the ingoing and outgoing directions:

1.51.5 (4.304) and s is Mandelstam’s variable

2 s = (pa + pb) . (4.305) As you might have expected, the renormalized Green’s functions will be identiﬁed with the matrix elements of Tˆ. Hence it is important to translate the requirements of unitarity into conditions on T and verify they they are met by our identiﬁcation. From the unitarity condition on Sˆ, it follows that

Rˆ − Rˆ† = iRˆ†Rˆ = iRˆRˆ† . (4.306)

This operator equation summarizes the restrictions on R due to unitarity. For instance, let us take its matrix element between two particle states, labeled |1, 2i and h3, 4|:

h3, 4|Rˆ|1, 2i − h3, 4|Rˆ†|1, 2i = ih3, 4|RˆRˆ†|1, 2i . (4.307)

It is not hard to see that when the external particles are spinless

langle3, 4|Rˆ†|1, 2i = h1, 2|Rˆ†|3, 4i . (4.308)

Using 58 Perturbative Evaluation of the Path Integral: λϕ4 Theory

h1, 2|Rˆ†|3, 4i = h3, 4|Rˆ|1, 2i∗ , (4.309) we arrive at

2=h3, 4|Rˆ|1, 2i = h3, 4|RˆRˆ†|1, 2i . (4.310) Now the right hand side of this equation can be rewritten by introducing a set of intermediate states. Since we want to limit ourselves to interactions that involve an even number of states (invariance under ϕ → −ϕ), the lowest energy intermediate state is the two particle state |a, bi = |ai|bi. Hence, using (4.9.3), we arrive at

Z d4a d4b 2=h3, 4|Rˆ|1, 2i = θ(a )θ(b )δ(a2 − m2)δ(b2 − m2)(4.311) (2π)6 0 0 ×h3, 4|Rˆ|a, biha, b|Rˆ†|1, 2i · · · (4.312) where ··· denotes the sum over 4-, 6-, ··· particle intermediate states. In terms of the T -matrix of (4.9.12), this equation becomes

Z d4a d4b 2=h3, 4|Tˆ|1, 2i = θ(a )θ(b )δ(a2 − m2)δ(b2 − m2)δ(a + b − 1 − 2)(4.313) (2π)2 0 0 ×h3, 4|Tˆ†|a, bi a, b|T |1, 2i + ··· . (4.314) Since both a and b are on their mass-shells, the right hand side will be non-zero only when the |1, 2i initial state has enough energy to produce the 2 2 |a, bi intermediate state, that is, when s = (p1 + p2) ≥ 4m . Thus, we see that as a consequence of unitarity and completeness the T matrix elements are real for s < 4m2 and acquire an imaginary part after the two-particle threshold is crossed, and for all other higher thresholds there is a further contribution to the imaginary part of the matrix element of T . Let us now compare this with the four-point function obtained from perturbation theory.

" λ2 Z m2 − i − sx(1 − x) Γ˜(4) = −i λ + dx ln (4.315) 2 · 16π2 m2 + M 2x(1 − x) # +(s → t) + (s → u) + O(λ3) . (4.316)

We have seen that the parametric integral does develop an imaginary part 4.8 Cross-Sections and Unitarity 59 and has a branch point at s = 4m2, all of it consistent with the unitarity equation (4.9.29). So we are led to identify Green’s functions with (−i) times the T matrix. In this case

Γ˜(4)(1, 2, 3, 4) = −ih3, 4|Tˆ|1, 2i . (4.317) We can easily check that this is true: we have already calculated the imaginary part of Γ˜(4); it is or order λ2 and appears only for s ≥ 4m2. On the other hand, we can calculate it from the unitarity equation (4.9.29), by putting in its right hand side the lowest perturbation vertex; this gives to O(λ4)

λ2 Z 2= iΓ˜(4) = d4a d4bθ(a )θ(b )δ(a2 − m2)δ(b2 − m2)δ(1 + 2 − a − b). (2π)2 0 0 (4.318) We leave its veriﬁcation to the reader. This yields only the lowest order contribution to the imaginary part — the 4-, 6-, ··· particle thresholds will add to the imaginary part but in higher orders of λ. With this identiﬁcation between Green’s functions and scattering amplitudes there emerges a new way of computing the imaginary part of diagrams using the unitarity equation. It is useful in perturbation theory because of the quadratic nature of the right hand side of (4.9.29); for if one has computed a, b|T |1, 2i to order λk, it will give the imaginary part to order λk+1. This remark is important because of the optical theorem which re- lates the imaginary part of the forward scattering amplitude to the total cross section. This theorem is simply obtained by putting |3, 4i = |1, 2i in (4.9.29) and comparing the right hand side with the integrated form of (4.9.22). We can arrive at a diagrammatic representation of the unitarity con- straint, if we recall the meaning of the Feynman propagator. We have

i Z eikx ∆ (x) = d4k (4.319) F (2π)4 k2 − m2 + i Z d4k = θ(x ) θ(−k )δ(k2 − m2) (4.320) 0 (2π)3 0 Z d4k +θ(−x ) θ(+k )δ(k2 − m2) , (4.321) 0 (2π)3 0 which expresses the fact that ∆F includes propagation of both positive and negative energy states depending on the sign of x0. Now if we invent a new 60 Perturbative Evaluation of the Path Integral: λϕ4 Theory

2 2 set of rules where the full propagator ∆F is replaced by θ(k0)δ(k − m ), we can arrive at a pictographic way to compute imaginary parts and therefore total cross-sections. This new rule would apply only in Minkowski space, of course. While we have the old rule

i ⇒ , (4.322) p2 − m2 + i invent a new one for the cut propagator

2 2 .6.6 → (2π)θ(k0)δ(p − m ) . (4.323) Note that the cut propagator is not symmetric, as the shading shows. The reason is that since we have to compute TT † on the right hand side of (4.9.29), portions of the diagram to the left of the cut must correspond to the conjugate of diagrams to the right of it although they may be diﬀerent. The interested reader is referred to “Diagrammar” by ‘t Hooft and Veltman, in Particle Interactions at Very High Energies, part B, D. Speiser et al., eds. (Plenum Press, New York, 1974), for more details. Thus our equation (4.9.32) would read diagrammatically

11 = 11 (4.324) The end result is that one can derive general cutting equations that express the imaginary part of diagrams in terms of the sum of all the possible cuts. [This is not as bad as it sounds since many cut diagrams vanish by energy conservation for cutting a Feynman propagator restricts the energy ﬂow to one direction.] This concludes our study of perturbative λϕ4 theory.

4.8.1 PROBLEMS

dσ 1 |T |2 A. Show that dΩ = s 64π2 starting from Eq. (4.9.19). vskip .5cmB. Show that for elastic scattering of spinless particles

h3, 4|Sˆ|1, 2i = h1, 2|Sˆ|3, 4i . (4.325)

C. Compute =iΓ˜(4) by means of the unitarity equation and compare with the result previously obtained from perturbation theory. 4.8 Cross-Sections and Unitarity 61 D. Show that in general for > 0

1 1 = −iπδ(x) + P , (4.326) x + i x 1 1 where P x is the Cauchy principal part of x deﬁned by

1 Z +∞ P = −i dy eixy [θ(y) − θ(−y)] . (4.327) x −∞

∗E. Compute the imaginary part of the setting sun by using the unitarity equation

= [i.4.4 ] = 11 (4.328)

∗∗ 1 1 2 2 h 3 λ 4 F. Given LE = 2 ∂µϕ∂µϕ + 2 m ϕ + 3! ϕ + 4! ϕ , a) derive the Feynman rules b) ﬁnd the change of m, h, and λ with scale of O(~) c) solve the

equations of b) and interpret the result physically. You may use any renormalization prescription, but the mass independent prescription is strongly advised.