The 4PI Effective Action for φ^4 Theory

0.19 MB·English

by  M. E. Carrington

The 4PI effective action for φ4 theory
M.E. Carrington∗
Department of Physics, Brandon University,
Brandon, Manitoba, R7A 6A9 Canada
and
Winnipeg Institute for Theoretical Physics,
Winnipeg, Manitoba
We work with φ4 theory and study the 4PI effective action at 3-loop order. Wediscuss the rela-
tionship between the equations of motion obtained by taking functional derivatives of the effective
action with respect to the variational parameters, and the Schwinger-Dyson (SD) equations. We
show that the equation obtained by differentiating with respect to the 2-point function is identical
4 to the SD equation for the 2-point function; differentiating with respect to the 3-point function
0 reproduces the SD equation for the 3-point function, up to 1-loop order; and differentiating with
0 respect to the 4-point function reproduces the SD equation for the 4-point function, at the tree
2 level. These results establish a connection between two techniques for performing non-equilibrium
calculations, andprovideastartingplace forastudyof thegaugedependenceof quantitiesderived
n
from an nPI effectiveaction.
a
J
PACSnumbers: 11.15.Tk,11.15.Bt
6
1

1
v I. INTRODUCTION
3
2
1 The study of particles in a hot and dense medium is of current interest in the study of plasma physics, condensed
1 matter physics, nuclear physics, and particle physics. Perturbative calculations of observables in hot dense media
0 are often plagued with infrared divergences and other problems. These problems are connected to the fact that
4 calculations in terms of the bare quantities of the underlying field theory fail to accurately describe the collective
0
behaviourofthehotdensemedium,whichleadstoinconsistencies. Oneattemptstoremedythissituationbyworking
/
h with dressed quantities which take into account the most important collective effects. An arbitrary resummation
p procedurehowever,will produce dressedquantities that violatethe conservationlaws ofthe originaltheory. To avoid
- this problem, one starts with an action functional that respects the original symmetries. Dressed quantities are
p
e obtained from the effective action using a variational procedure which guarentees thermodynamic consistency. Early
h contributions in this field were made by severalauthors (see for example [1, 2, 3, 4]). The formalism was generalized
: to relativistic field theories by Cornwall, Jackiw and Tomboulis [5].
v
The simplestchoiceofsuchaneffective actionisthe 2PIeffectiveaction. Thefull 2PIeffective actioniscompletely
i
X gauge invarient. In order to do practical calculations however, one must truncate the effective action at some order
r in an expansion parameter. This truncated 2PI effective action is also gauge invarient, to the same order in the
a
expansion parameter [6, 7], but dressed quantities which are obtained from functional differentiation of a truncated
2PI effective action are not generally gauge invarient. This problem has been discussed by many authors (see for
example [8, 9]). The source of the problem can be understood diagramatically as follows. The 2PI effective theory
resultsinaperturbationtheorythatusesfullresummedpropagatorsandbarevertices. Equivalently,ateachorderof
theperturbationtheory,thetruncatedtheoryresumscertaintopologiesinpreferencetoothers. Sincegaugeinvarience
results fromthe cancellationofgaugedependent termsthat occurwithin differenttopologies,itis notsurprisingthat
the procedure described above produces gauge dependent results.
Effective theories that involve dressed vertices, as well as dressed propagators, have been proposed [10]. An nPI
effective theory produces a perturbation theory that involves mean fields and dressed 2-point, 3-point n-point
···
functions. A perturbative expansion of this form simultanously resums different classes of topologies. As a result,
one expects that the problems associated with the gauge dependence of the truncated theory will be reduced: at a
given order in the perturbative expansion, gauge dependent contributions should be smaller in a higher order (larger
n) effective theory.
Problems involving the physics of collective phenomena are also studied within a green’s function formalism, in
∗Electronicaddress: [email protected]
2
which dressed propagators and vertices follow from a coupled set of Schwinger-Dyson (SD) equations. As is the
case with effective actions, the full hierarchy of SD equations is gauge invarient, but practical calculations involve
truncation, and the truncated equations are not gauge invarient.
Itisofinteresttounderstandtheconnectionbetweenthesetwoformalisms. Inthispaperwetakeafirststepinthis
direction by looking at the 4PI effective action for φ4 theory. We work to 3-loop order and discuss the relationship
between the equations of motion obtained by taking functional derivatives of the effective action with respect to the
variational parameters, and the SD equations. We show that the equation obtained by differentiation with respect
to the 2-point function is identical to the SD equation for the 2-point function. Taking the functional derivative
with respect to the 3-point function reproduces the SD equation for the 3-point function, up to 1-loop order. The
functional derivative with respect to the 4-pointfunction reproduces the SD equation for the 4-point function, at the
tree level. These results establish a connection between two techniques for performing non-equilibrium calculations,
and provide a starting place for a study of the gauge dependence of quantities derived from an nPI effective action.
This paper is organized as follows. In section II we review the 2PI effective action. We show explicitly that 1PR
diagrams cancel at 2-loop order, and that 2PR diagrams cancel at 3-loop order. Of course, these results are well
known and are presented only to illustrate the method that we will use in the next section. In section III we study
the 4PIeffective theory. We workat3-looporderandobtainanexpressionfor the 4PIeffective actionas afunctional
of four variational parameters: the mean field ϕ, the 2-point function , the 3-point function V, and the 4-point
G
function U. We take functional derivatives of this expression with respect to the variational parameters and discuss
the connection between these results and the SD equations. Some conclusions are presented in section IV.
II. THE 2PI EFFECTIVE ACTION
The generating functional for connected Green functions is given by,
i 1
W[J,B]= i¯hln φExp S(φ)+J φ + φ φ B (1)
x x x x xy
− Z D (cid:20)¯h(cid:18) 2 (cid:19)(cid:21)
Throughout this paper we use DeWitt notation in which it is understood that repeated continuous indices are inte-
grated over:
J φ := d4xJ[x]φ[x]
x x
Z
We expand around the minimum of the exponent which is obtained from the equation
δS(φ) 1
+J + B φ =0 (2)
x xy y
δφx 2 (cid:12)φ=ϕc
(cid:12)
(cid:12)
We define:
G0 = S(2)+B −1 (3)
xy c xy
(cid:0) (cid:1)
and use the notation
δnS
S(n) = (4)
c δφn(cid:12)φc
(cid:12)
(cid:12)
We calculate perturbatively by shifting the integration variable in (1):
φ=ϕ +√¯hχ (5)
c
and expanding around the classical trajectory.
A. 2-loop order
At 2-loops we obtain the familiar result:
W(2)[J,B]=W [J,B]+W [J,B]+W [J,B] (6)
0 1 2
3
1
W [J,B]=S[ϕ ]+J ϕc + ϕcϕcB
0 c x x 2 x y xy
¯h
W [J,B]=i Trln(G−1)
1 2 0
W [J,B]=h¯2 1 + 1 '$+1
2 h− 8 (cid:19)(cid:19)(cid:16)(cid:16)12 8 (cid:19)(cid:16)(cid:19)(cid:16)iG0,Sc(3),Sc(4)
(cid:18)(cid:18)(cid:17)(cid:17) (cid:18)(cid:17)(cid:18)(cid:17)
wherethesubscriptonthesquarebracketindicatesthatthe&linesand%verticesinthediagramsrepresentthepropagator
G and the vertices S(3) and S(4).
0 c c
We want to perform the Legendre transform to obtain the effective action Γ[ϕ, ] where the variables ϕ and are
G G
defined below. Taking functional derivatives of (1) with respect to J and B we obtain,
δW
= φ =:ϕ (7)
x x
δJ h i
x
δ2W i
= φ φ ϕ ϕ
x y x y
δJxδJy ¯h(cid:16)h i− (cid:17)
δW 1
= φ φ
x y
δB 2h i
xy
The mean field is defined by the first line in (7). To define the 2-point function we start from the definition of the
n-th Green function:
n−1
i
G(n)(x ,x , x )= φ(x )φ(x ) φ(x ) (8)
1 2 n 1 2 n
··· (cid:18)−¯h(cid:19) h ··· i
Subtracting disconnected pieces to obtain the connected 2-point function we have:
¯h (2)(x ,x )= i φ(x )φ(x ) ϕ(x )ϕ(x ) (9)
1 2 1 2 1 2
G − h i−
(cid:0) (cid:1)
and using (7) gives:
δW δW
i¯h (2) = i¯h =2 ϕ ϕ (10)
Gxy − δJ δJ δB − x y
x y xy
From now on we drop the superscript (2) on the two point function. The Legendre transform has the form:
1
Γ[ϕ, ]=W[J,B] J ϕ B (i¯h +ϕ ϕ ) (11)
x x xy xy x y
G − − 2 G
This expressionis constructedso thatthe partialfunctionalderivatives with respectto J andB arezero. We obtain:
δΓ
= J B ϕ (12)
x xy y
δϕ − −
x
δΓ i¯h
= B
xy
δ − 2
xy
G
We note that W[J,B] is an explicit function of ϕ and G and only dependent on the sources J and B because of
c 0
the fact that ϕ and G depend on J and B, as given in (2) and (3). We want to remove the dependence on the
c 0
variables ϕ and G and obtain a function that depends only on ϕ and . The procedure is as follows.
c 0
G
[1]: Calculate a series expansion of the variables ϕ and as functions of ϕ and G by using (7) and (10) and taking
c 0
G
functional derivatives of the perturbative expansion of W[J,B] (Eqn. (6)). We obtain series of the form:
ϕ=ϕ(0)[ϕ ,G ]+ϕ(1)[ϕ ,G ]+ϕ(2)[ϕ ,G ]+ (13)
c 0 c 0 c 0
···
= (0)[ϕ ,G ]+ (1)[ϕ ,G ]+ (2)[ϕ ,G ]+
c 0 c 0 c 0
G G G G ···
4
To lowest order
ϕ(0) =ϕ ; (0) =G (14)
c 0
G
The first order corrections are:
ϕ(1)[ϕ ,G ]= ih¯ (cid:27) (15)
c 0 −2h (cid:19)(cid:16)iG0,Sc(3),Sc(4)
(cid:19)(cid:16)
(cid:18)(cid:17)
(1)[ϕ ,G ]= ih¯ (cid:27)(cid:19)(cid:16)- 1 (cid:27) - 1 (cid:27)(cid:18)(cid:17)- (16)
G c 0 −2h −2 (cid:19)(cid:16) −2 iG0,Sc(3),Sc(4)
(cid:18)(cid:17)
where the arrow on the end of a line indicates explicitly that the line is not truncated.
(cid:18)(cid:17)
[2]: Invert (13) by expanding iteratively in powers of h¯ to obtain expressions of the form:
ϕ =ϕ(0)[ϕ, ]+ϕ(1)[ϕ, ]+ϕ(2)[ϕ, ]+ (17)
c c G c G c G ···
G =G(0)[ϕ, ])+G(1)[ϕ, ]+G(2)[ϕ, ]+
0 0 G 0 G 0 G ···
Substituting (14) and (15) into (13) and keeping terms of order ¯h gives,
ϕ =ϕ ϕ(1)[ϕ, ]+ (h¯2) (18)
c
− G O
G = (1)[ϕ, ]+ (h¯2) (19)
0
G−G G O
and
(G )−1 = −1 −1 (1)[ϕ, ] −1+ (h¯2) (20)
0 xy Gxy −GxzG G zwGwy O
Below we show explicitly that all 1PR diagrams cancel at 2-loop order. Of course, this result is well known. Our
intention is only to illustrate the method. We substitute (6) into (11) to obtain:
1 1 ¯h
Γ(2)[ϕ, ]=S[ϕ ]+J ϕc + ϕcϕcB J ϕ B (i¯h +ϕ ϕ )+i Trln(G−1) (21)
G c x x 2 x y xy− x x−2 xy Gxy x y 2 0
α β β β γ β γ
|{z} |{z} | {z } |{z} | {z } |{z} | {z }
+h¯2 1 + 1 '$+1
h− 8 (cid:19)(cid:19)(cid:16)(cid:16)12 8 (cid:19)(cid:16)(cid:19)(cid:16)iG0;Sc(3);Sc(4)
(cid:18)(cid:18)(cid:17)(cid:17) (cid:18)(cid:17)(cid:18)(cid:17)
&%
Note that since we are working to 2-loop order in this section, we can make the replacements G ; S(3) S(3)
0 c
→ G →
and S(4) S(4) inside of all terms in the square bracket.
c
→
We group the terms as indicated, and consider each set separately.
[α]. The term marked α can be expanded:
δS 1 δ2S
(α)=S[ϕ ]=S[ϕ+(ϕ ϕ)]=S[ϕ]+(ϕ ϕ) + (ϕ ϕ) (ϕ ϕ) (22)
c c c x c x c y
− − δϕ 2 − − δϕ δϕ
x x y
[β]. The terms marked β give:
1
(β)=(J +B ϕc)(ϕ ϕ) B (ϕ ϕ) (ϕ ϕ) (23)
x xy y c− x− 2 xy c− x c− y
δS 1
= (ϕ ϕ) B (ϕ ϕ) (ϕ ϕ)
−δϕc c− x− 2 xy c− x c− y
x
δS δ2S 1
= +(ϕ ϕ) (ϕ ϕ) B (ϕ ϕ) (ϕ ϕ)
c y c x xy c x c y
−(cid:18)δϕ − δϕ δϕ (cid:19) − − 2 − −
x x y
5
where we have used (2) to go from the first to the second line and made the expansion
δS δS δS
= +(ϕ ϕ) (24)
δϕc δϕ c− yδϕ δϕ
x x x y
in going from the second to third lines. The second term in (22) and the first term in (23) cancel exactly. Note that
it is because of this cancellation that we do not need the term (h¯2) which is dropped in (18).
O
[γ]. We use (3) to eliminate the source B. The factors G−1 are expanded using (20) and S(2) is expanded as above.
0 c
We obtain,
¯h ¯h δ2S δ3S
(γ)=i Tr ln −1 i −1 +(ϕ ϕ) (25)
2 G − 2 Gxy(cid:16)Gxy −hδϕxδϕy c− zδϕxδϕyδϕzi(cid:17)
Contributionsto 1PRdiagramscome fromthe lasttermin (22), the lasttwo terms in(23), andthe lasttermin(25).
Combining we have:
1  δ2S  ¯h δ3S
(ϕ ϕ) (ϕ ϕ) +B +i (ϕ ϕ) (26)
c x c y xy xy c z
− 2 − − δϕxδϕy  2G − δϕxδϕyδϕz
−ϕ(1)[ϕ,G] −ϕ(1)[ϕ,G]   −ϕ(1)[ϕ,G]
| {z }| {z } G  | {z }
| {z }
where the substitutions indicated by the underbraces are valid up to corrections of order h¯3 (using (3), (14), (15)
and (18)). Using (15) it is easy to see that both terms correspond to the dumbell graph. The first carries a factor
of h¯2/8 and the second a factor of ¯h2/4. Since symmetry factor for the dumbell graph in (21) is 1/8, we find that,
−
after summing all contributions, the dumbell graphs cancel. This is merely the well known statement that the 2PI
effective action is one-particle-irreducible. Since there are no 2PR diagramsat 2-looporder we have to go to next or-
derintheloopexpansion(orderh¯3)inordertoseecancellationsof2PRgraphs. Thisisthesubjectofthenextsection.
B. 3-loop order
In this section we will calculate the 2PI effective action to 3-looporder and explicitly demonstrate the cancellation
of the 2PR diagrams. Of course, it is well known that both the 1PR and 2PR diagrams cancel in the 2PI effective
action [5]. We remind the reader that our purpose is to demonstrate the technique that will be used in the next
section to study the 4PI effective action. Throughout this section we will ignore all 1PR diagrams. Equivalently, we
assumethatweareallowedtomakethe substitutionϕ ϕeverywhere. Ourgoalistodemonstratethecancellation
c
→
ofthe2PRdiagrams,withoutbotheringwiththe1PRdiagramswhichcanbeshowntocancelusingthesamemethod
as in the previous section.
In order to simplify the equations, we introduce some shorthand notation. We define:
δ2S δ3S
=S(2); =S(3) ; etc. (27)
δϕ δϕ xy δϕ δϕ δϕ xyz
x y x y z
In addition, whenever possible to do so without causing confusion, we will use a type of matrix notation. We give
some examples below:
J ϕ Jϕ (28)
x x
→
(ϕ ϕ) (ϕ ϕ) B (ϕ ϕ)2B
c x c y xy c
− − → −
Note that in principle an expression like (ϕ ϕ)2B is ambiguous since it could mean either
c
−
(ϕ ϕ) (ϕ ϕ) B or (ϕ ϕ) (ϕ ϕ) B (29)
c x c y xy c x c x yy
− − − −
Since we are only dealing with connected diagrams it is clear that the first expression is to be used. The only place
where this shorthand notation can cause confusion is when a given combination of factors could correspond to either
6
of two diagrams with different topologies. In these cases, we will write the indices explicitly.
In order to obtain an expression for Γ(3)[ϕ, ] we use (11) and expand W[J,B] to three loop order:
G
W[J,B] = W [J,B] + W [J,B] + W [J,B] + W [J,B]. We group terms as in (21) and use expressions of the
0 1 2 3
form(17) to removethe dependence onthe variablesϕ andG andobtaina function thatdepends onlyonϕ and .
c 0
G
There are three different places where 3-loop graphs will appear:
[1] Contributions from W [J,B].
3
[2] When working at order h¯3 it is not sufficient to replace G with in the contributions from W [J,B], as in the
0 2
previous section. Insteadwe must use (19) and write G = (1)[ϕG, ]+ (h¯2). Note that in principle the vertices
0
G−G G O
should also be expanded by writing S(n) = S(n)[ϕ+(ϕ ϕ)] and Taylor expanding as in (24). However, since we
c c
−
are ignoring 1PR diagrams, we make the replacement S(n) S(n).
c
→
[3] In the terms marked (γ), we use (3) to remove the source B, make the replacement ϕ ϕ, and expand G to
c 0
second order. As will be seen below, this expansion involves terms of the form ( (1))2 but te→rms proportionalto (2)
G G
cancel, and thus we will not need anything more than (1) as given in (16) and (19).
G
We calculate below each of these three contributions.
[1] The contribution from W [J,B] (neglecting 1PR graphs) is:
3
W [J,B]=i¯h3 1 1 + 1 '$ (30)
3 h16 (cid:19)(cid:16)(cid:19)(cid:19)(cid:16)(cid:16)−8 (cid:19)(cid:19)(cid:16)(cid:16) 16
(cid:18)(cid:17)(cid:18)(cid:18)(cid:17)(cid:17) (cid:18)(cid:18)(cid:17)(cid:17)
&%
+ 1 '$1 '$+ 1 ''$$
24 −8 (cid:19)(cid:16) 48 iG0;Sc(3);Sc(4)
(cid:18)(cid:17)
&% &% &&%%
Note that the first three diagrams are 2PR and the second three are 2PI. Since we are working to 3-loop order in
this section, we can make the replacements G ; S(3) S(3) and S(4) S(4) inside of all terms in the square
0 c c
→ G → →
bracket.
[2] Consider the graphs that we obtain by substituting (16) and (19) into W [J,B] (ignoring 1PR terms). The result
2
can be represented graphically:
ih¯3 2 '$ (31)
8 h (cid:19)(cid:19)(cid:16)(cid:16)− − (cid:19)(cid:19)(cid:16)(cid:16)(cid:19)(cid:16)iG;S(3);S(4)
(cid:18)(cid:18)(cid:17)(cid:17) (cid:18)(cid:18)(cid:17)(cid:17)(cid:18)(cid:17)
&%
[3] Finally, we consider the expansion of the terms marked (γ) in (21). We use (3) to eliminate the source B and
make the replacement ϕ ϕ since doing so means dropping 1PR graphs. We want to expand the propagator using
c
→
an expression like (20), but working to one higher power in h¯.
Eliminating B we have:
i¯h
(γ)= S(2)+ TrlnG−1 G−1 (32)
2 (cid:16)G (cid:2) 0 −G 0 i(cid:17)
Expanding the term in square brackets gives
1
[ ] Trln −1 −1 (1) −1 (1) −1 (33)
→ G −GxyGyx − 2 GxyGyz GzwGwx
(cid:0) (cid:1)
7
where we have used
(G )−1 = −1 −1 (1) −1 −1 (2) −1 (34)
0 xy Gxy −Gxz Gzw Gwy −GxzGzwGwy
Note that the terms containing the factor (2) cancel exactly. Substituting we obtain,
G
¯h 1
(γ)=i S(2) −1 +Trln −1 (1) −1 (1) −1 (35)
2(cid:16)G(cid:2) −G (cid:3) G − 2(cid:0)GxyGyz GzwGwx(cid:1)(cid:17)
Consider the last term in this expression. This term is already explicitly of order h¯3 and therefore we can make the
replacements ϕ ϕ and G in the arguments of (1). Using the 1PI part of (16) we obtain a result that can
c 0
→ → G G
be represented graphically:
ih¯3 2 + '$+ (36)
16h− (cid:19)(cid:19)(cid:16)(cid:16) (cid:19)(cid:16)(cid:19)(cid:19)(cid:16)(cid:16)iG;S(3);S(4)
(cid:18)(cid:17)(cid:18)(cid:17) (cid:18)(cid:17)(cid:18)(cid:18)(cid:17)(cid:17)
&%
Combining (30), (31) and (36) we see that the 2PR terms cancel. We are left with:
Γ(3)[ϕ, ]=S[ϕ]+ih¯ S(2) −1 +Trln −1 (37)
G 2(cid:16)G(cid:2) −G (cid:3) G (cid:17)
+h¯2 1 + 1 '$
h− 8 (cid:19)(cid:19)(cid:16)(cid:16)12 iG,S(3),S(4)
(cid:18)(cid:18)(cid:17)(cid:17)
&%
i¯h3 1 '$1 '$+ 1 ''$$
h 24 −8 (cid:19)(cid:16) 48 iG;S(3);S(4)
(cid:18)(cid:17)
&% &% &&%%
III. 4PI EFFECTIVE THEORY
A. Generating Functional
The generating functional for connected diagrams is given by:
i 1 1 1
W[J,B,K,H]= i¯hln φExp S(φ)+J φ + B φ φ + K φ φ φ + H φ φ φ φ (38)
x x xy x y xyz x y z xyzw x y z w
− Z D h¯h(cid:16) 2 3! 4! (cid:17)i
where K and H are sources. We expand around the solutions to the classical equations of motion by shifting
xyz xyzw
φ=ϕ +√¯hχ (39)
c
where φ is defined by the equation:
c
δS 1 1
+J +Bϕ + Kϕ2+ Hϕ3 =0 (40)
δφ(cid:12)ϕc c 2 c 3! c
(cid:12)
Inthissection,manyofthevariables,lik(cid:12)eϕ asdefinedabove,havecounterpartsforthe2PIeffectiveactiondiscussed
c
intheprevioussection. Toavoidaproliferationofindiceswewillusethesamevariable,andnotdefineanewvariable
with an extra index (like ϕc ). We assume always that variables used in this section correspond to the definitions
4PI
made within this section (unless otherwise indicated). Shifting around this classical solution it is easy to show that
the perturbative expansion of W[J,B,K,H] involves a propagator and vertices given by:
1
G−1[ϕ ]=S(2)+B+Kϕ + Hϕ2 (41)
0 c c c 2 c
V˜ [ϕ ]=S(3)+K+Hϕ
0 c c c
U˜ [ϕ ]=S(4)+H
0 c c
8
where the tilde’s indicate truncated vertices.
Taking functional derivatives of W[J,B,K,H] we obtain,
δW
ϕ:= φ = (42)
h i δJ
δ2W δW
i¯h := i¯h =2 ϕ2
G − δJ2 δB −
δ3W δW
¯h2V := ¯h2 =3! 3i¯h ϕ ϕ3
− − δJ3 δK − G −
δ4W δW
i¯h3U :=i¯h3 =4! +4h¯2Vϕ+3h¯2 2 6i¯h ϕ2 ϕ4
− δJ4 δH G − G −
where ϕ, , V, and U are the connected 1-point function, 2-point function, 3-point function and 4-point function,
G
respectively, and we have used (8).
B. The Effective Action
The effective action is obtained by Legendre transforming the generating functional. The Legendre transformation
isconstructedsothatpartialderivativeswithrespecttothesourcesJ,B,K andH arezero. Wemakethe definition:
Γ[ϕ, ,V,U] (43)
G
=W[J,B,K,H]
1 1 1
Jϕ B(i¯h +ϕ2) K( ¯h2V +3i¯h ϕ+ϕ3) H( i¯h3U 4h¯2Vϕ 3h¯2 2+6i¯h ϕ2+ϕ4)
− − 2 G − 3! − G − 4! − − − G G
Taking partial derivatives we obtain the equations of motion:
δΓ 1 1
= J Bϕ Kϕ2 Hϕ3 (44)
δϕ − − − 2 − 3!
δΓ i¯h ¯h2
= B+Kϕ+Hϕ2 + H
δ − 2 3 G
G (cid:0) (cid:1)
δΓ ¯h2
= (K+Hϕ)
δV 3!
δΓ
=i¯h3H
δU
As in the previous section, the goal is to obtain perturbative expansions for the variables ϕ ,G ,V˜ ,U˜ in terms
c 0 0 0
{ }
ofthe variables ϕ, ,V,U . We proceedasbefore. The firststepis to deriveexpressionsanalogousto(13)by taking
{ G }
functional derivatives of the perturbative expression for W[J,B,K,H]. This procedure leads to expansions of the
form:
ϕ=ϕ(0)[ϕ ,G ,V˜ ,U˜ ]+ϕ(1)[ϕ ,G ,V˜ ,U˜ ]+ϕ(2)[ϕ ,G ,V˜ ,U˜ ]+ (45)
c 0 0 0 c 0 0 0 c 0 0 0
···
= (0)[ϕ ,G ,V˜ ,U˜ ]+ (1)[ϕ ,G ,V˜ ,U˜ ]+ (2)[ϕ ,G ,V˜ ,U˜ ]+
c 0 0 0 c 0 0 0 c 0 0 0
G G G G ···
V =V(0)[ϕ ,G ,V˜ ,U˜ ]+V(1)[ϕ ,G ,V˜ ,U˜ ]+V(2)[ϕ ,G ,V˜ ,U˜ ]+
c 0 0 0 c 0 0 0 c 0 0 0
···
U =U(0)[ϕ ,G ,V˜ ,U˜ ]+U(1)[ϕ ,G ,V˜ ,U˜ ]+U(2)[ϕ ,G ,V˜ ,U˜ ]+
c 0 0 0 c 0 0 0 c 0 0 0
···
The second step is to invert (45) by expanding iteratively in powers of h¯ and obtain the equations of the form:
ϕ =ϕ(0)[ϕ, ,V,U]+ϕ(1)[ϕ, ,V,U]+ϕ(2)[ϕ, ,V,U]+ (46)
c c G c G c G ···
(0) (1) (2)
G =G [ϕ, ,V,U])+G [ϕ, ,V,U]+G [ϕ, ,V,U]+
0 0 G 0 G 0 G ···
V˜ =V˜(0)[ϕ, ,V,U])+V˜(1)[ϕ, ,V,U]+V˜(2)[ϕ, ,V,U]+
0 0 G G G ···
U˜ =U˜(0)[ϕ, ,V,U])+U˜(1)[ϕ, ,V,U]+U˜(2)[ϕ, ,V,U]+
0 0 G 0 G 0 G ···
9
The cancellation of the 1PR and 2PR diagrams works essentially the same as before. The interesting question is
what happens to the 3PRand 4PRdiagrams. To study this point, we will work at orderh¯3 and assume that all 1PR
and 2PR diagrams cancel. Effectively, we make the replacements ϕ ϕ and G everywhere. The 4PI effective
c 0
action is given by (38) and (43). Dropping 1PR and 2PR diagrams w→e obtain (to→ordGer h¯3):
1 1 1 ¯h
Γ(3)[ϕ, ,V,U]=S[ϕ ]+Jϕ + ϕ ϕ B+ Kϕ3+ Hϕ4+i Trln(G−1)
G c c 2 c c 3! c 4! c 2 0
α β β β β γ
|{z} |{z} | {z } |{z} |{z} | {z }
+h¯2 1 + 1 '$
h− 8 (cid:19)(cid:19)(cid:16)(cid:16)12 iG0;Sc(3);Sc(4)
(cid:18)(cid:18)(cid:17)(cid:17) (47)
&%
i¯h3 1 '$1 '$+ 1 ''$$
h24 −8 (cid:19)(cid:16) 48 iG0;Sc(3);Sc(4)
(cid:18)(cid:17)
&% &% &&%%
1 1
Jϕ B(i¯h + ϕ2 ) K( ¯h2V +3i¯h ϕ+ ϕ3 )
− −2 G − 3! − G
β γ β ǫ γ β
|{z} |{z} |{z} | {z } | {z } |{z}
1
H( i¯h3U 4h¯2Vϕ 3h¯2 2+6i¯h ϕ2+ ϕ4 )
− 4! − − − G G
τ ǫ τ γ β
| {z } | {z } | {z } | {z } |{z}
We start by considering the terms marked (α), (β) and (γ). Using (41) and making the replacements ϕ ϕ and
c
→
G we have:
0
→G
(α)=S[ϕ] (48)
(β)=0
¯h
(γ)=i Trln −1+ S(2) −1
2(cid:16) G G(cid:2) −G (cid:3)(cid:17)
Combining these results gives:
¯h
Γ[ϕ, ,V,U] =S[ϕ]+i Trln −1+ S(2) −1 (49)
Part1
G 2(cid:16) G G(cid:2) −G (cid:3)(cid:17)
The interesting contributions come from:
[1] The 3-loop diagrams in (47).
[2] Terms obtained from the 2-loop diagrams in (47) with the vertices expanded to order h¯.
[3]Contributions to the terms in (47) markedǫ andτ with the sourcesremovedusing (41) andthe vertices expanded
to order h¯.
Using (41) to remove the sources we have:
¯h2
Γ[ϕ, ,V,U] = V( V˜ S(3)) (50)
G Part2 3! 0 − c
b e
|{z} |{z}
¯h3 ¯h2
i U( U˜ S(4))+ 2( U˜ S(4))
4! 0 − c 8 G 0 − c
c e a e
|{z} |{z} |{z} |{z}
10
+h¯2 1 + 1 '$
h− 8 (cid:19)(cid:16)(cid:19)(cid:16)12 iG0,V˜0,U˜0
(cid:18)(cid:17)(cid:18)(cid:17)+
&%
a b
| {z } | {z }
+i¯h3 1 '$1 '$+ 1 ''$$
h24 −8 (cid:19)(cid:16) 48 iG0,V˜0,U˜0
(cid:18)(cid:17)
&%+ &%+ &&%%
c c c
| {z } | {z } | {z }
Note that the terms marked (a) cancel (up to 2PR terms).
The terms marked (e) are already functions of , V and U.
G
We need to rewrite the remaining terms as functions of ϕ, ,V,U instead of ϕ ,G ,V˜ ,U˜ . We take functional
c 0 0 0
{ G } { }
derivativesofthe perturbativeexpansionofW[J,B,K,H]andfindequationsof the form(45), whichgiveexpansions
of V and U in terms of ϕ ,G ,V˜ ,U˜ . Recall that since we are ignoring 1PR and 2PR terms we can make the
c 0 0 0
{ }
replacements ϕ ϕ and G . We obtain:
c 0
→ →G
@I
@I @ @I
V = @ - i¯h '$- 3 @'$- (51)
h − (cid:0) − (cid:16) −2 (cid:0) (cid:17)iG0,V˜0,U˜0
(cid:0)(cid:9) (cid:0) (cid:0)(cid:9)
(cid:0)(cid:9)
&% &%
@I (cid:0)(cid:18) @I (cid:0)(cid:18)
U = @(cid:0) +3 @ (cid:0) (52)
h − (cid:0)@ (cid:0) @ iG0,V˜0,U˜0
(cid:0)(cid:9) @R (cid:0)(cid:9) @R
Note that in the equations above the two factors of ‘3’ both come from the fact that there are three different
permutationsoftheexternallegsforthecorrespondingdiagrams. Wemustbe carefulwhenusingthesevertexcorrec-
tionstoproducediagrams,sincedifferentpermutationsofexternallegswillproducediagramswithdifferenttopologies.
We define truncated vertices as follows:
V¯ = −3V , U¯ = −4U (53)
G G
In order to rewrite the terms marked (b) we invert (51) to obtain:
V˜ = V¯ + −3V(1) = V¯ +V¯(1) (54)
0
− G −
with
@
@
V¯(1) = i¯h '$3 @@'$ (55)
− h −2 (cid:0) iG0,V˜0,U˜0
(cid:0) (cid:0)
(cid:0)
Substituting (54) and (55) into the ter&ms marke%d (b) we have: &%
¯h2 ¯h2
(b)= V¯V = V −3V (56)
−12 −12 G
The terms marked (c) are explicitly of order h¯3 which means we can use (51) and (52) to write:
(V˜ ) = V¯ (57)
0 xyz xyz
−
(U˜ ) = U¯ +V¯ V¯ +V¯ V¯ +V¯ V¯
0 xyzw xyzw xya ab bzw xza ab byw xwa ab bzy
− G G G