François David Quantum Field Theory II 4 φ 4 and perturbation theory We now consider interacting theories. A simple case is the interacting scalar field, with a φ 4 interaction. This corresponds to a 2-body contact repulsive interaction between scalar bosons (the quanta of the field). The classical theory is given by the (Euclidean) action. S E [φ]= Z d d x 1 2 (r x φ) 2 + m 2 2 φ 2 + g 4! φ 4 (4.1) g> 0 is the coupling constant. The 1/4! factor is for convenience (but will be explained later). The real time action is S [φ]= Z d d x - 1 2 (@ μ φ@ μ φ) - m 2 2 φ 2 - g 4! φ 4 (4.2) Classical field configurations are solution of the non linear equation of motion - Δφ + m 2 φ + g 6 φ 3 =0 , Δ = -@ 2 t + @ 2 ~ x (4.3) Quantization can be performed in the canonical framework (see QFT I course). Here we look at the functional integral quantization scheme. Naively we would expect that the theory can be quantized by functional integration like for the free field, the free field action S 0 [φ]= Z d d x 1 2 (r x φ) 2 + m 2 2 φ 2 (4.4) being replaced by the action S [φ. For instance, the correlation functions of the interacting (Euclidean) theory should be given by (I omit the Euclidean suffix “E” when there is no ambiguity) hφ(z 1 ) ··· φ(z N )i g = R D 0 [φ] exp ⇥ - 1 ~ S [φ] ⇤ φ(z 1 ) ··· φ(z N ) R D 0 [φ] exp ⇥ - 1 ~ S [φ; g] ⇤ (4.5) The functional measure being the same as that of the free field D 0 [φ]= Y x dφ(x)= Y n2Z d dφ n 2⇡~ ✏ d-2 -1/2 (4.6) In particular the same lattice regularization scheme can be applied. If this calculation makes sense, on should recover via Wick rotation the real time Green functions, i.e. the vacuum expectation values of time ordered product of the field operators hφ(z 1 ) ··· φ(z N )i g ! Wick rotation h⌦|T [φ(z 1 ) ··· φ(z N )] |⌦i (4.7) |⌦i is the vacuum state of the interacting φ 4 theory, different from the vacuum |0i of the free theory. Perimeter Scholars International 39 Nov. 2012
Transcript
François David Quantum Field Theory II
4 �4 and perturbation theory
We now consider interacting theories. A simple case is the interacting scalar field, with a�4 interaction. This corresponds to a 2-body contact repulsive interaction between scalarbosons (the quanta of the field).
The classical theory is given by the (Euclidean) action.
SE[�] =
Zddx
1
2
(rx �)2
+
m2
2
�2
+
g
4!
�4
�(4.1)
g > 0 is the coupling constant. The 1/4! factor is for convenience (but will be explainedlater). The real time action is
S[�] =
Zddx
�1
2
(@µ�@µ�) � m2
2
�2 � g
4!
�4
�(4.2)
Classical field configurations are solution of the non linear equation of motion
���+m2�+
g
6
�3
= 0 , � = �@2t + @2~x (4.3)
Quantization can be performed in the canonical framework (see QFT I course). Here welook at the functional integral quantization scheme. Naively we would expect that the theorycan be quantized by functional integration like for the free field, the free field action
S0
[�] =
Zddx
1
2
(rx �)2
+
m2
2
�2
�(4.4)
being replaced by the action S[�. For instance, the correlation functions of the interacting(Euclidean) theory should be given by (I omit the Euclidean suffix “E” when there is noambiguity)
h�(z1
) · · ·�(zN)ig =RD
0
[�] exp⇥� 1
~ S[�]⇤�(z
1
) · · ·�(zN)RD
0
[�] exp⇥� 1
~ S[�; g]⇤ (4.5)
The functional measure being the same as that of the free field
D0
[�] =
Y
x
d�(x) =
Y
n2Zd
d�n
2⇡~✏d�2
��1/2
(4.6)
In particular the same lattice regularization scheme can be applied.If this calculation makes sense, on should recover via Wick rotation the real time Green
functions, i.e. the vacuum expectation values of time ordered product of the field operators
h�(z1
) · · ·�(zN)ig !Wick rotation
h⌦|T [�(z1
) · · ·�(zN)] |⌦i (4.7)
|⌦i is the vacuum state of the interacting �4 theory, different from the vacuum |0i of the freetheory.
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4.1 Perturbation theory, Feynman diagrams
As as been presented for QED, a natural scheme is to assume that g is small and performa series expansion in powers of g. This amounts to consider that the interaction terms aresmall, and represents a small perturbation of the free theory. Thus we expand the interactionterm in the functional integral
exp
�1
~g
4!
Zddx�4
(x)
�=
1X
K=0
1
K!
⇣� g
4! ~
⌘K x
ddx1
. . . dDxK �4
(x1
) · · ·�4
(xK) (4.8)
Then we invert the perturbative seriesP
K and the functional integralRD
0
[�].Z
D0
[�]X
K
· · · =
X
K
ZD
0
[�] · · · (4.9)
CAUTION! This inversion is highly non trivial. In particular it will transform a convergentseries (the Taylor series of a exponential) into a divergent series, possibly asymptotic wheng ! 0, but with zero radius of convergence. Let us close our eyes, cross our fingers and carryon the calculation.
The numerator in 4.5 is
Z(z1
, · · · , zN) =Z
D0
[�] exp
� 1
~ S[�]
��(z
1
) · · ·�(zN) (4.10)
and given by the series
Z(z1
, · · · , zN) =X
K
⇣�g
~
⌘K1
K!(4!)
K
x
ddx1
. . . dDxK h�(z1
) · · ·�(zN)�4
(x1
) · · ·�4
(xK)i0
(4.11)h i
0
denoting the expectation value in the free field theory. Now these e.v. of productsof fields in the free theory can be computed, using Wick theorem. They are given by a sumover all pairings between the M = N + 4K fields.
⌦�(z
1
) · · ·�(zN)�4
(x1
) · · ·�4
(xK)↵0
=
X
pairings
h�(y1
)�(y2
)i0
· · · h�(yM�1
)�(yM)i0
(4.12)
Each term given by a product of L = M/2 free field propagators
h�(y1
)�(y2
)i0
= ~G0
(y1
, y2
) (4.13)
Applying the Feynman diagrammatic rules (represent each propagator by a line), we endup with a representation of 4.12 by a sum of Feynman integrals (Feynman amplitudes) IGassociated with Feynman diagrams (or graphs) G, of the form
Z(N)
(z1
, · · · , zN ; g) =X
GwithN legs
(�g)K ~L�K c(G) IG(z1, · · · , zN) (4.14)
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The amplitude IG is given by the integral over the positions of the K internal vertices (with4 legs) of the graph, of the integrand (the product of the propagators)
IG(zi) =
x
dx1
· · · dxK
Y
lines `2G
G0
(y`, y0`) (4.15)
We do not discuss if theses integrals make senses at this stage. The combinatorial coefficientc(G) comes from the factors K! and 4!, and from the fact that different pairings can give thesame Feynman diagram G, with the same amplitude. These factors are called the symmetryfactor of the diagram, since they are in fact given by the order of a symmetry group associatedto each diagram.
Rather than making a general theory, let me give a few explicit examples. remember therules for the Euclidean theory:
1. one line: ~ and one propagator G0
2. one internal vertex •: a factor (�g) ~�1, integrate over its position xi
3. one external vertex #: position zj fixed
For the real-time theory they become
1. one line: ~ and one propagator GFeynman
= iG0
2. one internal vertex •: a factor (�i g) ~�1, integrate over its position xi
3. one external vertex #: position zj fixeds
“Vacuum diagrams” (N = 0 )
• N = 0, K = 1: one diagram
1
8
• N = 0, K = 2: three diagrams
1
128
+
1
16
+
1
48
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Two point function ( N = 2 )
• N = 2, K = 0: one diagram
21
• N = 2, K = 1: two diagrams
1
2 21+
1
8 21
• N = 2, K = 2: seven diagrams
1
4 21+
1
4 21+
1
6 21
+
1
16 21+
1
128 21+
1
16 21
+
1
48 1 2
Four point function ( N = 4 )
• N = 4, K = 0: three diagrams
41
2 3+
41
2 3+
41
2 3
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• N = 4, K = 1: Ten diagrams
3
1 4
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
8
3
1 4�
2
+
1
8
3
1 4�
2
+
1
8
3
1 4�
2
These diagrammatic rules to construct and organize the perturbative expansion in termsof Feynman diagrams are the same than those obtained in the canonical formalism. This isnot surprising since the basic rules (Wick theorem) are the same.
The Feynam amplitudes are more easily represented and calculated in momentum space(taking the Fourier transform) than in position space. We shall discuss that later.
4.2 Connected functions
One sees already that diagrams can be decomposed in connected components, and thatthe amplitudes of disconnected components of a diagram factorize. Similarily, a connected
=
Figure 32: Decomposition of a graph into its connected components
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diagram can be decomposed into a tree structure, whose lines are the propagators, and thevertices the so called irreducible parts. These “one particle irreducible diagrams” (1PI) arethe connected diagrams which stay connected if any of its lines is removed (but the end-pointvertices of the diagrams are not removed). First let us state (without demonstration) some
=
Figure 33: Decomposition of a connected graph C into a tree T whose vertices are itsirreducible parts �i
basic results.
Correlation functions: The correlation functions
G(z1
, · · · , zN) = h�(z1
) · · ·�(zN)ig =Z(z
1
, · · · , zN)Z (4.16)
obtained from the functional integral 4.5, are given by the sum of diagrams which do notcontain any connected vacuum diagram with no external legs. Indeed the contribution ofthese vacuum connected components in Z(z
1
, · · · , zN) is cancelled by the denominator Z,which is precisely the sum over all vacuum diagrams.
For instance, the two point function G(z1
, z2
) is (I omit the ~)
21� g
1
2
21(4.17)
The four point function G(z1
, z2
, z3
, z4
) is0
@41
2 3+
41
2 3+
41
2 3
1
A � g
✓
3
1 4
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
+
1
2
3
1 4�
2
◆+ · · · (4.18)
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Connected functions: We notice that the two point function is connected, but the fourpoint function is not connected. More precisely, it is given by the products of the (connected)two points functions G(z
1
, z2
) = W(z1
, z2
), plus the sum over all connected four point graphs,which constitute the connected four point function W(z
1
, · · · , z4
), so that we can write thecorrelation functions G in terms of the connected functions W
G(z1
, z2
) = W(z1
, z2
) (4.19)
G(z1
, z2
, z3
, z4
) = W(z1
, z2
)W(z3
, z4
)+W(z1
, z3
)W(z2
, z4
)+W(z1
, z4
)W(z2
, z3
)+W(z1
, z2
, z3
, z4
)
(4.20)The connected functions in QFT correspond to cumulants of a probability distribution instatistics, and to connected correlations in statistical mechanics.
The zero point connected function W is defined as the sum of connected vacuum diagrams.It is given at order g2 by
W =
1
2
� g1
8
+ g2
0
BBBB@1
16
+
1
48
1
CCCCA+ · · · (4.21)
with the “closed loop diagram” representing the logarithm of the functional determinant�tr log[��+m2
], with its symmetry factor c(�) = 1/2
= tr log
1
��+m2
�(4.22)
This term represents the partition function of the free field, and is taken into account forconsistency.
At order g2 in the perturbative expansion, the connected two point functions and fourpoint functions are:
W(z1
, z2
) =
21� g
1
2
21+ g2
0
BBBB@1
4
21+
1
4
21+
1
6
21
1
CCCCA(4.23)
W(z1
, z2
, z3
, z4
) = � g3
1 4
2
+ g2
0
@1
2
3
1 4�
2
+
1
2
3
1 4�
2+
1
2
3
1 4�
2+
+
1
2
3
1 4�
2
+
1
2
3
1
2
4
+
1
2
41
2 3
+
1
2
3
1
2
4
1
CA + · · ·
(4.24)
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And for completeness the connected six point function (you start to see the tree structure ofthe connected functions)
W(z1
, · · · z6
) = g2
0
B@
61
2
3 4
5+ · · · (20 terms)
1
CA
� g3
0
B@1
2
0
B@ 2
1
3 4
5
6
+ · · · (120 terms)
1
CA +
1
2
0
BB@ 2
4
5
61
3
+ · · · (20 terms)
1
CCA
+
1
2
0
BB@2
4
5
6
3
1
+ · · · (60 terms)
1
CCA+
1
2
0
BBB@2
1
3 4
5
6
+ · · · (15 terms)
1
CCCA
1
CCCA
+ · · · (4.25)
4.3 Generating functionals
These diagrammatic manipulations are a bit tedious, but it is a good training to computethese functions at first order. However the functional integral formalism provides a verypowerful tool to manipulate the correlation functions through generating functionals.
Generating functions have been invented long ago by L. Euler. They allow to translatecombinatorial manipulations of objects into algebraic calculations on functions (often easier).
The generating functional for correlation functions is defined (Euclidean theory) as
Z[j] =
ZD
0
[�] exp
� 1
~ (S[�]� j�)
�(4.26)
j = {j(x); x 2 Md} is a classical source term (a real function over space-time), and Z[j] is afunctional of this function. � = {�(x); x 2 Md} is the random field variable integrated overin the functional integral, and represents the quantum field. j� is a compact notation for thescalar product of the functions j and � (considered as vectors in L2
(Md)
j� =
Zddx j(x)�(x) (4.27)
This definition for Z[j] is a compact way to manipulate all the N point functions Z(z1
, · · · zN)defined in 4.10, since expanding in j
Z[j] =X
N
~�N
N !
x
ddz1
· · · ddzN j(z1
) · · · j(zN) Z(z1
, · · · zN) (4.28)
Equivalently, the Z(z1
, · · · zN) are the functional derivatives of the functional Z[j]
Z(z1
, · · · zN) = ~N �
�j(z1
)
· · · �
�j(zN)Z[j]
����j=0
(4.29)
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For the real time theory we define the generating function as
Z[j] =
ZD
0
[�] exp
i
~ (S[�] + j�)
�(4.30)
The generating functional for the correlation functions (the Green functions) G(z1
, · · · , zN)4.16 is obviously
G[j] = Z[j]
Z[0]
(4.31)
Now comes the power of the formalism. The generating functional for the connectedfunctions W(z
1
, · · · , zN), defined as
W [j] =X
N
~�N+1
N !
x
ddz1
· · · ddzN j(z1
) · · · j(zN) W(z1
, · · · zN) (4.32)
(note the additional ~ factor) is simply
W [j] = ~ log (Z[j]) (4.33)
In other words
Z[j] = exp
✓1
~ W [j]
◆= 1 + ~�1 W [j] +
1
2
~�2 W [j]2 +1
6
~�3 W [j]3 + · · · (4.34)
The term W [j]P constructs all the diagrams with P connected components.We shall see later how to obtain all the irreducible amplitudes.
4.4 Perturbation theory and semiclassical expansion
A simple but important observation. With our normalization for the definition of the con-nected functions, let us look at the ~ factor for a single diagram. Each propagator is propor-tional to ~, each interaction vertex gives a factor ~�1. Finally each external line (attachedto an external vertex) gives also a factor ~�1, and there is a final ~1 factor per connectedcomponent, hence the final factor is
~�V+L+1 V = number of vertices, L = number of lines (4.35)
Now a celebrated formula of Euler states that for any connected graph
� V + L+ 1 = B number of internal independent loops of the graph (4.36)
For instance the graph of Fig. 34 has V = 4 and L = 7, hence B = 4. This number B is thenumber of independent internal momenta k for a Feynman amplitude in the momentum rep-resentation, hence the number on momentum integrations to be performed in the evaluationof the amplitude.
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Figure 34: A famous graph with V = 4 vertices or “islands”, L = 7 links or “bridges” andB = 4 loops or “cycles”
This formula is valid for any kind of graph, not only the diagrams of the �4 theory. Thisjustifies the fact that the perturbative expansions in QFT are also semiclassical expansions(expansions in ~, and are often denoted the “ loop expansion”.
For the �4 theory, one has the specific relation (a link has two end-points, an internalvertex has 4 legs, an external vertex only 1)
2L = N + 4K (4.37)
where N = number of external vertices and K = number of interaction vertices, so that
B = 1�N/2 +K (4.38)
So the ~ or loop expansion is also the interaction coupling expansion in g. But note that forinstance in QED, the loop expansion is an expansion in e2 (i.e. the fine structure constant↵), not in e.
4.5 The effective action at 1 loop
The effective action is an important concept of QFT. Firstly it is the generating functionalfor the “one particle irreducible” amplitudes. But the effective action plays also an importantrole when discussing renormalisation and renormalisation group in QFT, gauge symmetriesand anomalies, QFT in non trivial gravitational backgrounds, etc.
The idea is to look at the properties and observables of the theory, expressed as a func-tion of the v.e.v. of the field h�i. Let us start from the generation functional W [j]. The“background field” '(x) is the e.v. of the field �, for general values of the source term (orexternal classical field) j(x).
'(x) =
�W [j]
�j(x)= h�(x)ij (4.39)
The background field ' is a classical field, which is a functional of the classical source termj. It is a non-local functional, since a local change in j at a given x
0
leads to changes of '(x)for x far away from x
0
. The background field' is the “response” to the external source j.
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If one is interested in the properties of the theory as a function of ' one must considerthe Legendre transform of W , instead of W . One makes the change of variables
'(x)|j ! j(x)|' (4.40)
The effective action � is the functional of ' given by the Legendre transform of the functionalW [j]
�['] = j · ' � W [j] , with j · ' =
Zddx j(x)'(x) (4.41)
This effective action contains all the information of the quantum theory. In fact one canconsider the effective action as the “classical action” of a “non-local theory”, whose equationsof motion give the correlation functions of the quantum theory. For instance, finding theminimum of the effective action �, '
0
such that
��[']
�'(x)
����'0
= 0 (4.42)
amounts to find the v.e.v. of the quantum field � for the quantum theory
h�(x)i = '0
(x) locus of the minimum of �['] (4.43)
Similarity, the two point (connected) function is the inverse of the Hessian of the effectiveaction (we shall use this later)
W(x, y) = h�(x)�(y)ic = (�
00['
0
])
�1
x,y , �
00[']x,y =
�2�[']
�'(x)�'(y)(4.44)
To see the advantage of the functional integral formalism, let us compute explicitly atone loop the quantum effective action for a general theory of a scalar field � with action S[�](not necessarily �4). We stay in the Euclidean case. There is a simple beautiful formula
�['] = S['] +~2
tr [log (S 00['])] +O(~2) (4.45)
where S 00['] is the operator (acting of functions (x)) given by the functional derivative of
order 2 of the classical action S['] (the Hessian) , whose integral kernel is
S 00['](x
1
, x2
) =
�
�'(x1
)
�
�'(x2
)
S['] (4.46)
In other words: at tree level, the effective action equals the classical action; the one-loopterm in the effective action is given by the quantum fluctuations around the background fieldconfiguration ', like for the path integral, but here ' is a general configuration, it does notneed to be a classical solution of the equations of motion.
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Proof: To show 4.45, let us start from the generating functional
Z[j] =
ZD
0
[�] exp
�1
~ (S[�]� j ·�)�
(4.47)
and use the saddle point method. If j 6= 0 the saddle point �c[j] is solution of
S 0x[�c]� j = 0 (4.48)
i.e. explicitely
S 0x[�c] =
�S[�]
��(x)
�����c
= j(x) (4.49)
We rewrite � in the functional integral as
�(x) = �c(x) + ~1/2 ˜�(x) (4.50)
˜� represents the quantum fluctuations, and the normalization factor ~1/2 is such that in thefunctional integral the typical ˜� are of order 1
˜� ' O(1) (4.51)
Now we expand the action S � j · � around �c. We obtain at order ~
S[�]� j ·� = S[�c]� j ·�c + ~ 1
2
˜�·S 00[�c]· ˜�+ · · · (4.52)
Note that the linear term in ˜� disappears, thank to 4.48. Again we use the compact notation
˜�·S 00[�c]· ˜� =
x
ddx ddy ˜�(x)S 00xy[�c]
˜�(y)
withS 00xy[�] =
�2 S[�]
��(x)��(y)the Hessian of S[�]
We now integrate over ˜� to compute Z[j]. At one loop level it is just a Gaussian integralaround the saddle point �c, and we obtain
Z[j] = (det [S 00[�c]])
�1/2exp
✓1
~(j ·�c � S[�c])
◆(1 +O(~)) (4.53)
The generating functional for the connected functions is therefore
W [j] = ~ log(Z[j]) = j ·�c � S[�c] +~2
tr log [S 00[�c]] + O(~2) (4.54)
We now compute the background field '
'(x) =�W [j]
�j(x)= �c(x) +
Z
y
��c(y)
�j(x)
�
��(y)
j ·�� S[�]� ~
2
tr log [S 00[�]] +O(~2)
�
j fixed,�=�c
(4.55)
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But�
��(y)[j�� S[�]]
����j fixed,�=�c
= j(y)� S 0y[�c] = 0
so that 4.55 becomes
'(x) = �c(x)�~2
Z
y
��c(y)
�j(x)
�
��(y)
htr log [S 00�]]
i
j fixed,�=�c
+ O(~2) (4.56)
We need only to keep the O(~0) order, which is simply
'(x) = �c(x) + O(~) (4.57)
Thus, at tree level, we can identify the background field ' with the saddle point �c. This ofcourse was to be expected.
Now we can compute the effective action �['] at one loop. It is the Legendre transformof W [j], defined as
�['] = j ·'�W [j] (4.58)
and using 4.54 it is at one loop
�['] =
⇣S[�c] + j ·('� �c)
⌘+
~2
tr log [S 00[�c]] +O(~2) (4.59)
The first term can be rewritten as S[']. Indeed, since ' = �c +O(~) we can expand S['] atfirst order around �c
S['] = S[�c] + ('� �c)·S 0[�c] +O(~2) (4.60)
so that, using again the saddle point equation 4.48, we obtain
So we obtain the announced result 4.45 for the one loop effective action
�['] = S['] +~2
tr
⇥log
�S 00
[']�⇤
+ O(~2) (4.63)
This is a very simple and important result. It is independent of the precise form of the actionS[�], and extends to any theory with bosonic fields. Now two important remarks:
Perimeter Scholars International 51 Nov. 2012
François David Quantum Field Theory II
Real time: We defined the effective action for an Euclidean QFT. In Minkowski space(real-time) W [j] and �['] are defined as
W [j] =~i
log
✓ZD
0
[�] exp
i
~ (S[�] + j�)
�◆(4.64)
�['] = �j ·'+W [j] , '(x) =�W [j]
�j(x)(4.65)
At one loop we get
�['] = S['] + i
~2
tr
⇥log
�S 00
[']�⇤
+ O(~2) (4.66)
Theories with fermions and gauge fields: This one loop result is modified when thereare fermionic fields, since fermionic fields anticommute. We shall see later how. When gaugefields are present, gauge invariance require some care, we shall see later what has to be done.
4.6 Application to �4
What does this generic compact formula means for the �4 theory? We consider the Euclideantheory for simplicity. At tree level the classical action S['] can be represented graphically asthe “inverted propagator” plus the interaction vertex
S['] =
Zdx
1
2
�(rx '(x))
2
+m2'2
(x)�+
g
4!
Zdx'4
(x)
�
tree
['] =
1
2
+
1
4!
(4.67)
With the convention: barring with a | a propagator (truncation) means applying the operator�� +m2 to it, so that the truncated propagator ���|��� is just (�� +m2
)G0
= �d(x� y),while the twice-truncated propagator is just the inverse propagator, i.e. the operator ��+m2
��|�� = (��+m2
) ���� = 1 , = (��+m2
)
2 ���� = (��+m2
) (4.68)
At one loop (order ~) we have
�
1 loop
['] =
1
2
tr log [S 00[']] =
1
2
tr log
h��x +m2
+
g
2
'2
i(4.69)
The Hessian: The Hessian operator S 00['] = ��x + m2
+
g2
'2 is an elliptic differentialoperator acting on the functions (x) as
S 00[']· (x) = ��x (x) +m2 (x) +
g
2
'2
(x) (x) (4.70)
Perimeter Scholars International 52 Nov. 2012
François David Quantum Field Theory II
We expand in g the tr log[ ] of this operator, using the factorization of determinants (itsa bit formal since we are dealing with infinite dimensional determinants, which are in factinfinite...)
The first term of order g0 gives the total vacuum energy for the free field. We represent itgraphically (the factor �1
2
is chosen for consistency) as a closed loop
1
2
tr log
⇥��x +m2
⇤= �1
2
(4.71)
Since the integral kernels of (��x + m2
)
�1 and '2 are respectively the propagator G0
and'2⇥ a Dirac distribution,
(��x +m2
)
�1
xy = G0
(x� y) , ('2
)xy = '(x)2�(x� y) (4.72)
the trace in the term of order gK is
tr
h�(��x +m2
)
�1'2
�Ki=
x
dx1
· · · dxK '2
(x1
)G0
(x1
� x2
)'2
(x2
) · · ·'2
(xK)G0
(xK � x1
)
=
x
dx1
· · · dxK '2
(x1
)· · ·'2
(xK) G0
(x1
� x2
)· · ·G0
(xK � x1
)
(4.73)
Graphically this product of K free field propagators in the integral is descriped by a closedK-sided polygon
5 2
1
K
34�
6
7
(4.74)
It gives the contribution of the one-loop irreducible diagram with K vertices and 2 external“amputated” lines attached to eack vertex. One can check that the factor (�1)
K�1/(2K+1K)
is precisely the combinatoric (or symmetry) factor attached to each irreducible part, comingfrom the application of the Feynman diagrammatic rules and of the Wick theorem.
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So at one loop, the effective action is indeed given by the sum of all the 1PI (one particleirreducible) diagrams, as depicted here
�
1 loop
= � 1
2
+
1
4
� 1
16
+
1
48
+ · · · (4.75)
Each external � represents a ' field. The truncated propagator ���|��� represents simply a�(x� y) function, enforcing two ' to sit at each internal vertex •.
This result extends to general theories and to higher loop diagrams. The effective actionis the generating functional of the irreducible amplitudes at all order. This means that
�['] =1X
N=0
1
N !
x
ddz1
· · · ddzN '(z1
) · · ·'(zN)�(z1, · · · zN) (4.76)
The N point function �(z1
, · · · zN) is given by the sum of irreducible diagrams amplitudewith N truncated external legs
�(z1
, · · · zN) = �X
1PIG
~B(�g)Kc(G)IG(z1, · · · , zN) (4.77)
B being the number of loops of the diagram and K the number of vertices (for the �4 theory).
