Functional-Integral Calculation of Eﬀective Action. Loop ...users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-21-direffac.pdf · which may be used to form the Legendre transform

It is simplicity that makes the uneducated

more effective than the educated.

Aristoteles (384BC–322BC)

22

Functional-Integral Calculation of EffectiveAction. Loop Expansion

The functional-integral formula for Z[j] can be employed for a rather direct evalua-tion of the effective action of a theory. Diagrammatically, the result will be the loopexpansion organized by powers of the Planck quantum h [1].

22.1 General Formalism

Consider the generating functional of all Green functions

Z[j] = eiW [j]/h, (22.1)

where W [j] is the generating functional of all connected Green functions. The vac-uum expectation of the field, the average

Φ(x) ≡ 〈φ(x)〉, (22.2)

is given by the first functional derivative

Φ(x) = δW [j]/δj(x). (22.3)

This can be inverted to yield j(x) as an x-dependent functional of Φ(x):

j(x) = j[Φ](x), (22.4)

which may be used to form the Legendre transform of W [j]:

Γ[Φ] ≡ W [j]−∫

d4xj(x)Φ(x), (22.5)

where j(x) on the right-hand side is replaced by (22.4). The result is the effective

action of the theory.The first functional derivative of the effective action gives back the current

δΓ[Φ]

δΦ(x)= −j(x). (22.6)

1253

1254 22 Functional-Integral Calculation of Effective Action. Loop Expansion

This equation justifies the name effective action for the functional Γ[Φ]. In the ab-sence of an external current j(x), the effective action Γ[Φ] is extremal on the physicalfield expectation value (22.2). Thus it plays exactly the same role for the fully in-teracting quantum theory as the action does for the classical field configurations.

The Legendre transformation (22.5) can be inverted to recover the generatingfunctional W [j] from the effective action:

W [j] = Γ[Φ] +∫

d4xj(x)Φ(x). (22.7)

Let us compare this with the functional-integral formula for the generating functionalZ[j]:

Z[j] =

∫ Dφ(x)e(i/h){A[φ]+∫

d4xj(x)φ(x)}∫ Dφ(x)e(i/h)A0[φ]

. (22.8)

Using (22.1) and (22.5), this implies a functional-integral formula for the effectiveaction Γ[Φ]:

eih{Γ[Φ]+

∫

d4xj(x)Φ(x)} = N∫

Dφ(x)e(i/h){A[φ]+∫

d4xj(x)φ(x)}, (22.9)

where N is a normalization factor:

N ={∫

Dφ(x)e(i/h)A0[φ]}−1

. (22.10)

We have explicitly displayed the fundamental quantum of action h, which is a mea-sure for the size of quantum fluctuations.

There are many physical systems for which quantum fluctuations are rathersmall, except in the immediate vicinity of certain critical points. For these systems,it is desirable to develop a method of solving (22.9) for Γ[Φ] as a series expansionin powers of h, taking into account successively increasing quantum fluctuations.

In the limit h → 0, the path integral over the field φ(x) in (22.8) is dominatedby the classical solution φcl(x) which extremizes the exponent

δA∂φ

∣

∣

∣

∣

∣

φ=φcl(x)

= −j(x). (22.11)

At this level we therefore identify

W [j] = Γ[Φ] +∫

d4xj(x)Φ(x) = A[φcl] +∫

d4xj(x)φcl(x). (22.12)

Of course, φcl(x) is a functional of j(x), so that we may write it more explicitly asφcl[j](x). By differentiating W [j] with respect to j, we have from Eqs. (22.3) and(22.7):

Φ(x) =δW

δj=

δΓ

δΦ

δΦ

δj+ Φ + j

δΦ

δj. (22.13)

22.1 General Formalism 1255

Inserting the classical field equation (22.11), this becomes

Φ(x) =δAδφcl

δφcl

δj+ φcl + j

δφcl

δj= φcl. (22.14)

Thus, to this approximation, Φ(x) coincides with the classical field φcl(x). For theφ4-theory with O(N)-symmetry, the action is

A[φ] =∫

d4x

[

1

2(∂φa)

2 − m2

2φ2a −

g

4!

(

φ2a

)2]

. (22.15)

Replacing the φa → φcl a + δφs = Φa + δφ on the right-hand side of Eq. (22.12), wetherefore obtain the lowest-order result for the effective action (which is of zerothorder in h):

Γ[Φ] ≈ A[Φ] =∫

d4x

[

1

2(∂Φa)

2 − m2

2Φ2

a −g

4!

(

Φ2a

)2]

. (22.16)

It equals the fundamental action, and is also referred to as the mean field approxi-mation to the effective action.

We have shown in Chapter 13 that all vertex functions can be obtained from thederivatives of Γ[Φ] with respect to Φ at the equilibrium value of Φ for j = 0. Webegin by assuming that m2 > 0.1 Then Γ[Φ] has an extremum at Φa ≡ 0, and thereare only two types of non-vanishing vertex functions Γ(n)(x1, . . . , xn): For n = 2, weobtain the lowest-order two-point function

Γ(2)(x1, x2)ab ≡ δ2Γ

δΦa(x1)δΦb(x2)

∣

∣

∣

∣

∣

Φa=0

=δ2A

φa(x1)φb(x2)

∣

∣

∣

∣

∣

φa=Φa=0

= (−∂2 −m2)δabδ(4)(x1 − x2). (22.17)

This determines the inverse of the propagator:

Γ(2)ab (x1, x2) = [ihG−1]ab(x1, x2). (22.18)

Thus we find in this zeroth-order approximation that Gab(x1, x2) is equal to the freepropagator:

Gab(x1, x2) = G0ab(x1, x2). (22.19)

For n = 4, we find the lowest-order four-point function

Γ(4)abcd(x1, x2, x3, x4) ≡

δ4Γ

δΦa(x1)δΦb(x2)δΦc(x3)δΦd(x4)= gTabcd, (22.20)

where

Tabcd =1

3(δabδcd + δacδbd + δadδbc) (22.21)

1Also the case m2 < 0 corresponds to a physical state in a different phase and will be discussedseparately. In fact, the phase transition is the reason for the nonexistence of tachyon particles, i.e.,of particles that move faster than the speed of light. See the pages 1123, 1294 and 1311.


is the fundamental vertex [compare (22.16)] for the local interaction of strength g.The effective action has the virtue that all diagrams of the theory can be ob-

tained from tree diagrams built from propagators Gab(x1, x2) and vertex functions.This was explained in Chapter 13. In lowest approximation, we recover preciselythe subset of all original Feynman diagrams with a tree-like topology. These arethe diagrams which do not involve any loop integration. Since the limit h → 0corresponds to the classical equations of motion with no quantum fluctuations, weconclude: Classical field theory corresponds to tree diagrams.

The use of the initial action as an approximation to the effective action neglectingfluctuations is often referred to as mean-field theory.

22.2 Quadratic Fluctuations

In order to find the h-correction to this approximation we expand the action inpowers of the fluctuations of the field around the classical solution

δφ(x) ≡ φ(x)− φcl(x), (22.22)

and perform a perturbation expansion. The quadratic term in δφ(x) is taken to bethe free-field action, the higher powers in δφ(x) are the interactions. Up to secondorder in the fluctuations δφ(x), the action is expanded as follows:

A[φcl + δφ] +∫

d4xj(x)[φcl(x) + δφ(x)]

= A[φcl] +∫

d4xj(x)φcl(x) +∫

d4x

{

j(x) +δA

δφ(x)

∣

∣

∣

∣

∣

φ=φcl

δφ(x)

+∫

d4xd4y δφ(x)δ2A

δφ(x)δφ(y)

∣

∣

∣

∣

∣

φ=φcl

δφ(y) +O(

(δφ)3)

. (22.23)

The curly bracket term linear in the variation δφ vanishes due to the extremalityproperty of the classical field φcl expressed by the field equation (22.11). Insertingthis expansion into (22.9), we obtain the approximate expression

Z[j] ≈ N e(i/h){A[φcl]+∫

d4xjφcl}∫

Dδφ exp

i

h

∫

d4xd4y δφ(x)δ2A

δφ(x)δφ(y)

∣

∣

∣

∣

∣

φ=φcl

δφ(y)

.

(22.24)We now observe that the fluctuations in δφ will be of average size

√h due to

the h-denominator in the Fresnel integrals over δφ in (22.24). Thus the fluctuations(δφ)n are on the average of relative order hn/2. If we ignore corrections of the orderh3/2, the fluctuations remain quadratic in δφ and we may calculate the right-handside of (22.24) as

N e(i/h){A[φcl]+∫

d4xj(x)φcl(x)}[

detδ2A

δφ(x)δφ(y)

]

φ=φcl

= (det iG0)1/2 e(i/h){A[φd]+

∫

d4xj(x)φcl(x)+i(h/2)Tr log[δ2A/δφ(x)δφ(y)|φ=φcl}. (22.25)

22.2 Quadratic Fluctuations 1257

Comparing this with the left-hand side of (22.9), we find that to first order in h theeffective action may be recovered by equating

Γ[Φ] +∫

d4xj(x)Φ(x) = A[φcl] +∫

d4xjφcl +ih

2Tr log

δ2Aδφ(x)δφ(y)

(φcl) . (22.26)

In the limit h → 0, the trace log term disappears and (22.26) reduces to the classicalaction.

To include the h-correction into Γ[Φ], we expand W [j] as

W [j] = W0[j] + hW1[j] +O(h2). (22.27)

Correspondingly, the field Φ differs from Φcl by a correction of the order h2:

Φ = φcl + hφ1 +O(h2). (22.28)

Inserting this into (22.26), we find

Γ[Φ] +∫

d4xjΦ = A [Φ− hφ1] +∫

d4xjΦ− h∫

d4xjφ1 +i

2hTr log

δ2Aδφ δφ

∣

∣

∣

∣

∣

φ=Φ−hφ1

+O(

h2)

.

Expanding the action up to the same order in h gives

Γ[Φ] = A[Φ] + h

{

δA[Φ]

δΦ− j

}

φ1 +∫

d4x jΦ +i

2hTr log

δ2Aδφ δφ

∣

∣

∣

∣

∣

φ=Φ

+O(

h2)

. (22.29)

But because of (22.11), the curly-bracket term is only of the order O(h2), so thatwe find the one-loop form of the effective action

Γ[Φ] =∫

d4x

[

1

2(∂Φ)2 − m2

2Φ2 − g

4!Φ4

]

+i

2hTr log

[

−∂2 −m2 − g

2Φ2]

, (22.30)

where we dropped the infinite additive constant i2hTr log [−∂2 −m2] .

If we generalize this to an O(N)-invariant theory with N components φa (a =1, . . . , N), this expression becomes

Γ[Φ] = Γ0[Φ] + Γ1[Φ] =∫

d4x

[

1

2(∂Φa)

2 − m2

2Φ2

a −g

4!

(

Φ2a

)2]

+i

2hTr log

[

−∂2 −m2 − g

6

(

δabΦ2c + 2ΦaΦb

)

]

. (22.31)

What is the graphical content of the set of all Green functions at this level? Inthis discussion we shall assume that N = 1. For j = 0, we find that the minimum


of Γ[Φ] lies at Φ = Φ0 ≡ Φj = 0, just as in the mean-field approximation. Aroundthis minimum, we may expand the trace log in powers of Φ, and obtain

i

2hTrlog

(

−∂2−m2− g

2Φ2)

=i

2hTrlog

(

−∂2−m2)

+i

2hTrlog

(

1+i

−∂2−m2igΦ2

2

)

.

(22.32)

The second term can be expanded in powers of Φ2 as follows:

−ih

2

∞∑

n=1

(

−ig

2

)n 1

nTr(

i

−∂2 −m2Φ2)n

.

If we insert

G0 =i

−∂2 −m2, (22.33)

then (22.32) can be written as

ih

2Tr log

(

−∂2 −m2)

− ih

2

∞∑

n=1

(

−ig

2

)n 1

nTr(

G0Φ2)n

. (22.34)

More explicitly, the terms with n = 1 and n = 2 read:

− h

2g∫

d4xd4yδ(4)(x− y)G0(x, y)Φ2(y)

+ihg2

16

∫

d4xd4yd4zδ4(x− z)G0(x, y)Φ2(y)G0(y, z)Φ

2(z) + . . . . (22.35)

The expansion terms correspond obviously to the Feynman diagrams

+ . . . . (22.36)

Thus, the series (22.34) is a sum of all diagrams with one loop and any number offundamental Φ4-vertices.

To systemize the entire expansion (22.34), the leading trace log expression maybe pictured by a single-loop diagram

ih

2Tr log

(

−∂2 −m2)

= . (22.37)

The subsequent two diagrams in (22.36) contribute corrections to the vertices Γ(2)

and Γ(4) in (22.17) and (22.20). The remaining ones produce higher vertex functionsand lead to more involved tree diagrams. Note that only the first two corrections


are formally divergent, all other Feynman integrals converge. In momentum space,the corrections are, from (22.35),

Γ(2)(q) = q2 −m2 − hg

2

∫ dk4

(2π)4i

k2 −m2 + iη(22.38)

Γ(4)(qi) = g − ig2

2

[

∫

d4k

(2π)4i

k2 −m2 + iǫ

i

(q1 + q2 − k)2 −m2 + iη+ 2 perm

]

.

(22.39)

The convergence of all higher diagrams in the expansion (22.36) is ensured by therenormalizability of the theory. Indeed, up to n = 4, a counter term may be writtendown for each infinity that has the same form as those in the original Lagrangian(22.15). In euclidean form, we may write (22.38) and (22.39) as

Γ(2)(q) = −(

q2 +m2 + hg

2D1

)

, (22.40)

Γ(4)(qi) = g − hg2

2[I (q1 + q2) + 2 perm] , (22.41)

where D1 and I(q) are the Feynman integrals [compare (11.26) and (11.29)]

D1 =∫

-d4kE1

k2E +m2

(22.42)

and

I(q) =∫

-d4kE1

(k2E +m2)

1

[(k + q)2E +m2]. (22.43)

The integrals can be calculated inD dimensions by separating them into a directionaland a size integral as

∫

-dDk =∫ dDk

(2π)D=

SD

(2π)D

∫

dkkD−1

≡ SD

2

∫

dk2(k2)D/2−1. (22.44)

We further simplify all calculations by performing a Wick rotation of all energyintegrals

∫

dk0 into i∫∞−∞ dk4. Then the integrals

∫

d4k become what are calledeuclidean integrals i

∫

d4kE where kµE = (k, k4), and k2 = k2

0 − k2 becomes −k2E =

−(k2 + k24). By the same token we introduce the euclidean version ΓE [Φ] = −iΓ[Φ]

of the effective action (22.30) whose functional derivatives are vertex functions Γ(n).We further introduce renormalized fields

φR(x) ≡ Z−1/2φ φ(x), (22.45)

where Z1/2φ is a field renormalization constant. It serves to absorb infinities arising

in the momentum integrals. The renormalized vertex functions are obtained by cal-culating all vertex functions in D = 4− ǫ dimensions and fixing the renormalization


constants order by order in perturbation theory. Alternatively, we can add to thebare action suitable counter terms. In either way, we arrive at finite expressions.

If we regularize the integrals dimensionally in 4−ǫ dimensions, the countertermstake the forms calculated in Section 11.5, and the expression (22.40) becomes

Γ(2)R (q) = −

{

q2 +m2 + hg

2SDµ

−ǫm2×

×

1

2Γ(2− ǫ/2)Γ (−1 + ǫ/2)

(

m2

µ2

)−ǫ/c

+1

ǫ

}

, (22.46)

while (22.41) is renormalized to

Γ(4)R (qi) = g − h

g2

2{[I (q1 + q2) + 2 perm]− 3I(0)} . (22.47)

The Feynman integral I(q) has, after subtracting the 1/ǫ-pole in Eq. (22.43), thesmall-ǫ behavior [compare (11.176)]

IR(q) = −1

2SDµ

−ǫ

[

1 + Lm(q) + logq2

µ2

]

+O(ǫ), (22.48)

with

Lm(q) =∫ 1

0dx log

[

x(1 − x) +m2

q2

]

= −2 + logm2

q2+

√q2 + 4m2

√q2

log

√q2 + 4m2 +

√q2√

q2 + 4m2 −√q2. (22.49)

As far as the original effective action (22.30) to order h is concerned, we may addthe counter terms as

Γ[Φ] =∫

d4x

{

1

2(∂Φ)2 − m2

2Φ2 − g

4!Φ4

}

− ih

2Tr log

(

−∂2 −m2 − g

2Φ2)

+g

4

∫

d4q

(2π)41

q2 +m2

∫

d4xΦ2(x)− g2

16

∫

d4q

(2π)41

(q2 +m2)

∫

d4xΦ4. (22.50)

The divergent integrals in the last two terms can be evaluated with any regularizationmethod, and (22.50) becomes

Γ[Φ] =∫

d4x

{

1

2(∂Φ)2 − m2

2Φ2 − g

4!Φ4

}

+ ih

2Tr log

(

−∂2 −m2 − g

2Φ2)

− cm2m2

2Φ2 − cg

g2

4!Φ4. (22.51)

The third line may be written as

− hg

4(m2)1−ǫ/2c1Φ

2 +hg2

16(m2)−ǫ/2c2Φ

4, (22.52)


with the constants

c1 = mǫ−2∫

-dDpE1

p2E +m2, (22.53)

c2 = mǫ∫

-dDpE1

(p2E +m2)2. (22.54)

We evaluate these integrals using the formulas [2]

∫

-dDkE1

k2E+m2

= SDΓ(D/2)Γ(1−D/2)

2Γ(1)

1

(m2)1−D/2, (22.55)

and

∫

-dDkE1

(k2E+m2)2

= SDΓ(D/2)Γ(2−D/2)

2Γ(1)

1

(m2)2−D/2. (22.56)

Further, by integrating (22.55) over m2, we find

∫

-dDkE log(

k2E +m2

)

= SDΓ(D/2)Γ(1−D/2)

DΓ(1)

(

m2)D/2

. (22.57)

In the so-called minimal subtraction scheme in 4 − ǫ dimensions, only the singular1/ǫ pole parts of the two integrals are selected for the subtraction in (22.50). In theneighborhood of ǫ = 0, (22.53) and (22.54) become

c1 = −SD2

ǫ+O(ǫ), (22.58)

c2 = SD1

ǫ

(

1− ǫ

2

)

+O(ǫ). (22.59)

Hence we can choose as counter terms the singular terms in (22.52):

− hgµ−ǫ

4m2SD

(

−1

ǫ

)

Φ2,hg2µ−ǫ

16SD

1

ǫΦ4. (22.60)

These make the effective action (22.51) finite for any mass m.Note, however, that an auxiliary mass parameter µ must be introduced to define

these expressions. If the physical mass m is nonzero, µ can be chosen to be equalto m. But for m = 0, we must use an arbitrary nonzero auxiliary mass µ as therenormalization scale.

Observe that up to the order h, there is no divergence that needs to be absorbedin the gradient term

∫

dDx (∂Φ)2 of the effective action (22.51). These come in assoon as we carry the same analysis to one more loop order.

Let us calculate the effective potential in the critical regime for a constant fieldΦ at the one-loop level. It is defined by v(Φ) = −ΓE [Φ]/V T . The argument in the


trace log term is now diagonal in momentum space and the calculation reduces to asimple momentum integral. It follows directly from (22.51) and reads

v(Φ) =m2

2Φ2+

g

4!Φ4+

h

2

∫

dDqE(2π)D

log

(

1 +g

2

Φ2

q2E+m2

)

− hg

4(m2)1−ǫ/2c1Φ

2 +hg2

16(m2)−ǫ/2c2Φ

4. (22.61)

The expansions in powers of ǫ have an important property which has directconsequences for the strong-coupling limit that is measured in critical phenomena.For small ǫ, Eq. (22.47) can be rewritten as

Γ(2)R =

(

q2E +m2 +h

4µ−ǫgSD log

m2E

µ2

)

, (22.62)

which is, to the same order in g, equal to

Γ(2)R (q) =

q2E +

(

m2

µ2

)1+ h4µ−ǫg SD

µ2

. (22.63)

This means that the vertex function at q = 0 has a mass term that depends on themass m of the φ-field via a power law:

Γ(2)R (0) =

(

m2

µ2

)γ

µ2. (22.64)

The power γ depends on the coupling strength g like

γ = 1 +h

4µ−ǫgSD. (22.65)

The important point is that this power γ is measurable as an experimental quantitycalled the susceptibility . It is called the critical exponent of the susceptibility.

If the effective action is calculated to order h2, then the gradient term in theeffective action is modified and becomes

Γ[Φ] =∫

ddxΦR(x)ΓR(q)ΦR(x), (22.66)

where ΦR(x) = Z−1/2φ Φ(x), and Zφ is the field renormalization constant introduced

in (22.45). It is divergent for ǫ → 0 in D = 4 − ǫ dimensions. In the critical limitm2 → 0, the renormalization is power-like ΦR(x) → (µ/µ0)

−η/2Φ(x), and Eq. (22.63)becomes

Γ(2)R (q) = −

[

(q2)2−ηµη +

(

m2

µ2

)γ

µ2

]

. (22.67)

The power η is called the anomalous dimension of the field Φ.


From (22.67) we extract that the coherence length of the system ξ behaves like

ξ = µ−1(m2)−ν , (22.68)

where

ν ≡ γ/(2− η) (22.69)

is the critical exponent of the coherence length.Another power behavior is found for Γ

(4)R (0):

Γ(4)R (0) →

m→0g

(

1 +3

4hgµ−ǫSD log

m2

µ2

)

+O(m2)

= g

(

m2

µ2

)h 3

4gµ−ǫ SD

. (22.70)

Also this power behavior is measurable, and defines the critical index β via theso-called scaling relation

γ − 2β ≡ 3

4hgµ−ǫSD, (22.71)

so that

β =1

2− 1

4hgµ−ǫSD. (22.72)

The higher powers of Φ are accompanied by terms which are more and more singularin the limit m → 0. From the Feynman integrals in (22.34) we see that the diagramsin (22.36) behave like m4−ǫ−n for m → 0, and so do the associated effective actionterms Φn.

The coefficients of the dimensionless quantities (Φ2/µ2−ǫ)≡(Φ2)n or (gΦ2/µ2)n ≡(Φ2)n have the general form (m2/µ2)γ−2nβ, so that the effective potential can bewritten as

v(Φ) = µ4−ǫ

(

m2

µ2

)γΦ2

µ2−ǫf(x), (22.73)

with

x ≡(

m2

µ2

)−2βΦ2

µ2−ǫ≡ t−2β Φ2

µ2−ǫ= t−2βΦ2, (22.74)

where we have abbreviated

t ≡ m2

µ2. (22.75)


For small Φ, the function f(x) has a Taylor expansion in even powers of Φ, corre-sponding to the diagrams in Eq. (22.36):

f(x) = 1 +∞∑

n=1

fnxn = 1 +

∞∑

n=1

fn(

t−2nβΦ2)n

. (22.76)

If we start the sum at n = −1, we also get the general form of the vacuum energy

v(0) = µ4−ǫ

(

m2

µ2

)γ+2β

. (22.77)

The exponent γ + 2β is equal to Dν, where ν is the exponent defined in (22.69).

By differentiating this energy twice with respect to m2, we obtain the tempera-ture behavior of the specific heat

C ∝(

m2

µ2

)Dν−2

=

(

m2

µ2

)−α

, (22.78)

which yields the important critical exponent α = 2−Dν that governs the singularityof the famous λ-peak in superfluid helium at Tc ≈ 2.7 Kelvin. This is the most ac-curately determined critical exponent of any many-body systems. Such an accuracywas achieved by performing the measurement in a micro-gravity environment on asatellite [3].

We can also rewrite (22.73) in the form

v(Φ) = µ4−ǫ SD

λ

(

m2

µ2

)γgΦ2

µ2f(y), (22.79)

where

f(y) = 1 +∞∑

n=1

fnyn = 1 +

∞∑

n=1

fn

(

t−2β

(

gΦ2

µ2

))n

, (22.80)

and

y ≡ λ

SDx = gµ−ǫx =

(

m2

µ2

)−2β (gΦ2

µ2

)

= t−2βΦ2. (22.81)

In the limit m2 → 0, the expansions (22.76) and (22.80) are divergent since thecoefficients grow like n!. A sum can nevertheless be calculated with the technique ofVariational Perturbation Theory (VPT). This was used before in Chapter 3, whosefirst citation reviews briefly its development.

22.3 Massless Theory and Widom Scaling 1265

22.3 Massless Theory and Widom Scaling

Let us evaluate the zero-mass limit of v(Φ). Since m2 always accompanies thecoupling strength in the denominator, the limit m2 → 0 is equivalent to the limitg → ∞, i.e., to the strong-coupling limit.

The strong-coupling limit deserves special attention. The theory in this limit isreferred to as the critical theory. This name reflects the relevance of this limit forthe behavior of physical systems at a critical temperature where fluctuations are ofinfinite range.

We shall see immediately that for large y, f(y) behaves like a pure power of y:f(y) → y(δ−1)/2, so that

v(Φ) → µ4−ǫ 1

4!

SD

λ

(

gΦ2

µ2

)(δ+1)/2

. (22.82)

Without fluctuation corrections, δ has the value 3, and (22.82) reduces properly tothe mean-field potential gΦ4/4!.

With the general leading large-Φ-behavior, the potential in the form (22.79) isthe so-called Widom scaling expression that depends on y−1/2β ∝ m2/Φ1/β [5] in thefollowing general way:

v(Φ) ∝ Φδ+1w(m2/Φ1/β). (22.83)

From this effective potential we may derive the general Widom form of the equationof state. After adding a source term HΦ and going to the extremum, we obtainH(Φ) = ∂v(Φ)/∂Φ with the general behavior

H(Φ) ∝ Φδh(m2/Φ1/β). (22.84)

Recalling (22.83), we identify the general form of the potential (22.73) as being

v(Φ) → µ4−ǫ 1

4!

SD

λ

(

gΦ2

µ2

)(δ+1)/2

w (τ) , (22.85)

where τ ≡ (m2/µ2) /(Φ/µ1−ǫ/2)1/β ≡ t/(Φ2)1/β .For small m2, w(τ) has a series expansion in powers of των :

w(τ)=1+c1των +c2τ

2ων + . . .), (22.86)

or, since t = µξ−1/ν,

w(τ)=1+c1ξ−ωΦ−ων/β+c2ξ

−2ωΦ−2ων/β + . . .). (22.87)

Here ω is the Wegner exponent [6] that governs the approach to scaling. Its numericalvalue is close to 0.8 [7].


Differentiating (22.85) with respect to Φ yields the following leading contributionto H :

H = ∂Φv(Φ) =δ + 1

4!µ2

(

gΦ2

µ2

)(δ−1)/2

=δ + 1

4!µ2

(

λΦ2

SD

)(δ−1)/2

, (22.88)

where δ − 1 = γ/β.Note that the effective potential remains finite for m = 0. Then, v(Φ) becomes

v(Φ)=g

4!Φ4+

h

2

∫

dDq

(2π)Dlog

(

1 +g

2

Φ2

q2

)

+hg2µ−ǫ

16

SD

ǫΦ4. (22.89)

This displays an important feature: When expanding the logarithm in powers of Φ,the expansion terms correspond to increasingly divergent Feynman integrals

∫

dDq

(2π)D1

(q2)n.

Contrary to the previously regularized divergencies coming from the large-q2 regime,these divergencies are due to the q = 0 -singularity of the massless propagatorsG0(q) = i/q2. This means that they are IR-singularities.

Let us verify that the effective potential remains indeed finite for m = 0. Per-forming the momentum integral in (22.89), the potential becomes

v(Φ) =g

4!Φ4+

h

4SD

(

−1

ǫ

)(

g

2Φ2)2− ǫ

2

+hg2

16SDµ

−ǫ1

ǫΦ4. (22.90)

With the goal of expanding this for a small-ǫ expansion, we divide the couplingconstant and field by a scale parameter involving µ. Then gµ−ǫ and Φ/µ1−ǫ/2 aredimensionless quantities, on which the effective potential depends as follows:

v(Φ)=µ4−ǫ

gµ−ǫ

4!

(

Φ

µ1−ǫ/2

)4

+h

4SD

(

−1

ǫ

)

×(

gµ−ǫ

2

Φ2

µ2−ǫ

)2− ǫ2

+hg2

16

µ−2ǫ

ǫSD

(

Φ

µ1−ǫ/2

)4

.

(22.91)

If we use the dimensionless coupling constant

λ ≡ SDhgµ−ǫ, (22.92)

and the reduced field Φ ≡ Φ/µ1−ǫ/2 as new variables, then

v(Φ) =µ4−ǫ

hSD

λ

4!Φ4 +

1

4

(

−1

ǫ

)

(

λΦ2

2

)2

×(

1− ǫ

2log

λΦ2

2

)

+λ2

16ǫΦ4

]

. (22.93)

22.4 Critical Coupling Strength 1267

To zeroth order in ǫ, the prefactor is equal to µ4 8π2. Thus the massless limit of theeffective potential is well defined in D = 4 dimensions. There is, however, a specialfeature: The finiteness is achieved at the expense of an extra parameter µ.

The most important property of the critical potential is that it cannot be ex-panded in integer powers of Φ2. Instead, the expression (22.93) can be rewritten,correctly up to order λ2, as

v(Φ)=µ4−ǫ

SD

1

6

(

λ

2Φ2

)2

+3

8

(

λ

2Φ2

)2

logλ

2Φ2

=µ4−ǫ

SD

1

6λ

(

λ

2Φ2

)2+ 3

4λ

+O(λ3).

(22.94)

This is the typical power behavior of a critical interaction. The power of Φ definesthe critical exponent of the interaction, in general denoted by 1 + δ, so that up tothe first order in the coupling strength, we identify

δ = 3 +3

2λ. (22.95)

At the mean-field level, δ is equal to 3.

22.4 Critical Coupling Strength

What is the coupling strength λ in the critical regime? The counter term propor-tional to Φ4 in Eq. (22.89) implies that we are using renormalized quantities in allsubtracted expressions. The relation between the bare coupling constant gB and therenormalized one g is, to this one-loop order,

gB = g +3

2hg2µ−ǫ SD

ǫ+ . . . . (22.96)

For a given bare interaction strength gB, the renormalized coupling depends onthe parameter µ chosen for the renormalization procedure. Equivalently we mayimagine having defined the field theory on a fine spatial lattice with a specific smalllattice spacing a ∝ 1/µ. The renormalized coupling constant will then depend onthe choice of a.

If we sum an infinite chain of such corrections, we obtain a geometric sum thatis an expansion of the equation

gB =g

1− 3

2hgµ−ǫ SD

ǫ

, (22.97)

which has (22.96) as its first expansion term. Equivalently we may write

1

hgBµ−ǫ=

1

hgµ−ǫ− 3

2

SD

ǫ+ . . . . (22.98)


In this equation, we can go to the strong-coupling limit gB → ∞ by taking µ to thecritical limit µ → 0, where we find that the renormalized coupling has a finite value

1

hgµ−ǫ=

3

2

SD

ǫ+ . . . . (22.99)

For the dimensionless coupling constant (22.92), this amounts to the strong-couplinglimit

λ ≡ hgµ−ǫSD → 2

3ǫ+ . . . , (22.100)

with the omitted terms being of higher order in ǫ. The approach to this limit startingfrom small coupling is obtained from (22.97) to be

hgµ−ǫ =1

1

hgBµ−ǫ+

3

2

SD

ǫ

. (22.101)

At a small bare coupling constant gB, this starts out with the renormalized ex-pression which is determined by the s-wave scattering length hgµ−ǫ = 6×4πh2as/M[recall Eq. (9.266)]. In the strong-coupling limit of gB, which is equal to the criticallimit µ → 0, the value (22.99) is reached.

For the exponents γ, β, and δ in (22.65), (22.72), and (22.95), the strong-couplinglimits are

γ = 1 +1

6ǫ, β =

1

2− 1

6ǫ, δ = 3 + ǫ. (22.102)

They are approached for finite gB like

γ = 1 +1

4hgµ−ǫ, β =

1

2− 1

4hgµ−ǫ, (22.103)

with the just discussed gB-behavior of g.If the omitted terms in (22.100) are calculated for all N and up to higher loop

orders, one finds λ = 2g∗, where g∗ is the strong-coupling limit gB → ∞ of the seriesin Eq. (15.18) of the textbook [2]. The other critical exponents may be obtained fromthe gB → ∞ -limit of similar expansions for ν, η into α = 2− νD, β = ν(D− 2+ η),γ = ν(2 − η), and δ = (D + 2 − η)/(D − 2 + η). These can be extracted fromChapter 15 in Ref. [2].

All these series are divergent. But they can be resummed for ǫ = 1 in the strong-coupling limit gB → ∞. As an example, the series for g at N = 2 yields the limitingvalue g → g∗ ≈ 0.503 (see Fig. 17.1 in [2]). The typical dependence of g on gBis displayed in Fig. 20.8 of [2]. For ν and η the corresponding plots are shown inFigs. 20.9 and 20.10 of [2]. We leave it as an exercise to compose from these thedependence of α, β, and δ as functions of gB.

22.4 Critical Coupling Strength 1269

It should be pointed out that the potential in the first line of (22.94) has anotherminimum, away from the origin, solving

λ

3!Φ3 +

λ2

8Φ3

(

logλ

2Φ2 − 1

2

)

+λ2

16Φ3 = 0. (22.104)

The other minimum solves

λ logλ

2Φ2 = −4

3, (22.105)

which lies at

Φ2 =2

λe−4/3λ. (22.106)

However such a solution found for small λ is not reliable. The higher loops tobe discussed below and neglected up to this point will produce more powers ofλ log(λΦ2/2), and the series cannot be expected to converge at such a large λ. As amatter of fact, the approximate exponentiation performed in (22.94) does not showa minimum and will be seen, via the methods to be described later, to be the correctanalytic form of the potential to all orders in λ for small enough ǫ and λ.

If we want to apply the formalism to a Bose-Einstein-condensate, we must discussthe case of a general O(N)-symmetric version of the effective potential based on theaction (22.31), into which we must insert the number N = 2. The equation for thebare coupling constant is then

gB = g +N + 8

6hg2µ−ǫ SD

ǫ+ . . . , (22.107)

rather than (22.96), so that the strong-coupling limit (22.100) becomes

λ ≡ SDhgµ−ǫ → 6

N + 8ǫ+ . . . , (22.108)

with the other critical exponents (22.102):

γ=1+N + 2

2(N+8)ǫ, β=

1

2− 3

2(N+8)ǫ, δ=3+

9

N+8ǫ. (22.109)

If we carry the loop expansion to higher order in h, we find the perturbationexpansion for the renormalized g = hgBµ

−ǫ/2SD as function of the bare couplinggB = hgBµ

−ǫ/2 SD [taken from Eq. (15.18) in Ref. [2], in particular the higherexpansion terms]:

g

gB= 1− gB

8+N3

ǫ−1 + g2B{

(8+N)2

91ǫ2+ 14+3N

61ǫ

}

+ g3B{

− (8+N)3

271ǫ3− 4 (8+N)(14+3N)

271ǫ2

(22.110)

− [2960+922N+33N2+(2112+480N)ζ(3)]648

1ǫ

}

+ . . . .


Given this gB-dependence of g, we find for γ the expansion

γ= g6 (N+2)− 5 g2

36 (N+2) + g3

72(N+2)(5N + 37)

− g4

15552 (N+2) [−N2 + 7578N + 31060 (22.111)

+ 48ζ(3)(3N2+10N+68) + 288ζ(4)(5N+22)]+. . . ,

in which g has the limiting value (22.110).The expression (m2/µ2)γ in the two-point function can be replaced by an expan-

sion ofm2/m2B in powers of gB that can be taken from Eq. (15.15) in [2], again with g

replaced by (22.110). This expansion can be resummed by Variational PerturbationTheory to obtain a curve of the type shown in Fig. 20.8–20.10 of [2]. This permitsus to relate t = m2/µ2 directly to µ−ǫgB.

22.5 Resumming the Effective Potential

According to Eq. (22.79), the effective action in the condensed phase with negativem2 has an expansion in powers of y:

v(Φ)

µ4−ǫ= tγ

Φ2

µ2−ǫ(f0 + f1y + . . . ), (22.112)

where

y = t−2βgΦ2/µ2 = t−2βΦ2. (22.113)

From Eq. (22.80), we determine f0 = f1 = 1.In the critical regime where t is small, y becomes large, and we must convert

the small-y expansion into a large-y expansion. Near the strong-coupling limit, theWidom function (22.83) has an expansion in powers of (m2)ω/ν ∝ ξ−ω which containsthe Wegner critical exponent ω ≈ 0.8 governing the approach to scaling [7]. Thegeneral expansion for strong couplings is

v(Φ)

µ4−ǫ= tγ

Φ2

µ2−ǫ(t−2βΦ2)(δ−1)/2

(

b0+∞∑

m=1

bm

(t−2βΦ2)mων/2β

)

.

(22.114)

We may derive this from the rules of VPT in [2, 4], reviewed in some detail in Section21.7. First we rewrite a variational ansatz for the right-hand side of Eq. (22.112)with the help of a dummy parameter κ = 1 as

wN = µ2tγΦ2κp(

f0 + f1y

κq+ . . .

)

. (22.115)

Next we exchange κp by the identical expression√K2 + gr

p, where r ≡ (k2 −

K2)/K2. After this we form w1 by expanding wN up to order g, and setting κ = 1.This leads to

w1 = µ2tγΦ2Kp[

f0

(

1− p

2+

p

2

1

K2

)

+ f1y

Kq

]

= µ2tγΦ2W1(y). (22.116)

22.5 Resumming the Effective Potential 1271

The last term in the first line shows that, for large y, K has to grow like

K ∝ y1/q. (22.117)

We now extremize w1 with respect to K by finding the place where the derivativedw1/dK vanishes. This determines K from the equation

tγΦ2Kp−1p(2− p)

2

[

f0

(

1− 1

K2

)

− f1cy

Kq

]

= 0, (22.118)

where

c ≡ 2(p− q)

p(p− 2)≈ 0.32. (22.119)

In the free-particle limit y → 0, the solution is K(0) = 1, and

w1 = µ2tγΦ2f0. (22.120)

The last term in (22.118) shows once more that, for large y, K will be pro-portional to y1/q. Moreover, it allows to sharpen relation (22.117) to the large-ybehavior

K → Kas(y) = (cy)1/q ≈ 0.648 y0.381. (22.121)

Then the leading large-y behavior of (22.97) is Φ2(cy)p/q ∝ (Φ2)p+1.The first correction to the large-y behavior comes from the second term in the

brackets of (22.118) which, by (22.113), should behave like

K2 → (c t−2βgΦ2/µ2)ων/2β ≈ (c t−2βgΦ2/µ2)0.76. (22.122)

Comparing this with (22.99) and (22.117), we find p = 2(2− η)/ω and q = 4β/ων.For the N = 2 universality class these have the numerical values p ≈ 4.92 andq ≈ 4 × 0.32/(0.8 × 0.66) ≈ 2.63. Solving (22.118), we see that, for small y, K(y)has the diverging expansion

K(y)=1+0.32 y−0.165967 y2+0.155806 y3+. . . , (22.123)

so that W1 has the diverging expansion

W1 = 1 + y + 0.154436y3 + . . . . (22.124)

From Eq. (22.82) we know that the power p+ 1 must be equal to (δ + 1)/2, so that

yp/q = y(δ−1)/2 = y(2−η)ν/2β ≈ y1.87. (22.125)

If we insert (22.117) into (22.97), the extremal variational energy is

w1(y) = µ2tγΦ2Kp(

1− p

2+

y

Kq

)

, (22.126)


2 4 6 8 10

20

40

60

2 4 6 8 10 12 14

2

4

6

8

10

12

y y

K(y)W1(y)

Figure 22.1 Solution of the variational equation (22.118) for f1 = 1. The dotted curves

show the pure large-y behavior.

where K = K(y) is the function of y plotted on the left in Fig. 22.1.

For large y, where K(y) has the limiting behavior (22.121), W1 becomes

W1(y) → Was(y) = Kp(

1− p

2+

1

c

)

≈ 0.197 y1.87. (22.127)

Near the limit, the corrections to (22.121) are

K(y) = Kas(y)

(

1 +∞∑

m=1

hm

ymων/2β

)

≈ 0.648 y0.381(

1 +∞∑

m=1

hm

y0.761m

)

, (22.128)

with h1 ≈ 0.909, h2 ≈ −0.155, . . . . Inserting this into (22.116), we find

W1(y) = Was(y)

(

1 +∞∑

m=1

bmymων/2β

)

≈ 0.197 y1.87(

1 +∞∑

m=1

bmy0.761m

)

, (22.129)

with b1 ≈ 3.510, b2 ≈ 4.65248, . . . .

22.6 Fractional Gross-Pitaevskii Equation

We now extremize the effective action (22.51) with the two-loop corrected quadraticterm (22.63). We consider the case of N = 2 where Φ2 = Ψ∗Ψ. Then we takethe effective potential (22.126) with the extremal K = K(y) as a function of y(plotted in Fig. 22.1). From this we form the derivative ∂w1(y)/∂Ψ

∗ and obtain thetime-independent fractional Gross-Pitaevskii equation:

(p2)1−η/2Ψ+∂w1(y)

∂Ψ∗=0. (22.130)

22.7 Summary 1273

If we use the weak-coupling limit of w1(y) and the gradient term, this reduces tothe ordinary time-independent Gross-Pitaevskii equation

[

− h2

2M∇

2 − µ+ gSΨ†Ψ

]

Ψ(x) = 0. (22.131)

The relation of gS with the previous coupling constant is

gS/2 = g/4!. (22.132)

Here µ is the chemical potential of the particles which is fixed by ensuring a givenparticle number N . In a harmonic trap, µ is replaced by µ + Mω2x2/2. Recallingthe relation m2 = −2Mµ one has m2 = −2M Mω2R2

c(1 − R2/R2c). The oscillator

energy Mω2R2c corresponds to a length scale ℓO by the relation Mω2R2

c = h2/Mℓ2O,so that m2 may be written as m2 = µ2(R2/R2

c − 1) with µ = 1/ℓO.In the strong-coupling limit, however, we arrive at the time-independent frac-

tional Gross-Pitaevskii equation:[

(p2)1−η/2+δ+1

4µηgc|Ψ(x)|δ−1

]

Ψ(x)=0. (22.133)

By using the full effective action for all coupling strengths and masses m2 we canderive the properties of the condensate at any coupling strength. Before reaching thestrong-coupling limit, we may use Eq. (22.130) to calculate the field strength Ψ as afunction of gS. In this regime, the critical exponents η, α, β, δ have not yet reachedtheir strong-coupling values but must be replaced by the gB-dependent precriticalvalues calculated from (22.110) and the corresponding equations for η, α, β, δ.

In a trap, the mass term becomes weakly space-dependent. If the trap is ro-tationally symmetric, then m2 will depend on R = |x| and the time-independentGross-Pitaevskii equation has to be solved with m2(R) ∝ 1− R2/R2

c . More specifi-cally, the bare coupling constant on the right-hand side has to be determined in sucha way that m2/m2

B has the experimental size. If the experiments are performed inan external magnetic field B, the s-wave scattering length as has an enhancementfactor (B/Bc − 1)−1 and Eq. (22.110) can again be used. We can then calculate thedensity profile quite easily in the Thomas-Fermi approximation as done in Ref. [8].

In a rotating BEC we can calculate the different forms of the density profilesof vortices for various coupling strengths which can be varied from weak to strongby subjecting the BEC to different magnetic fields, raising it from zero up to theFeshbach resonance. The profiles are shown in Fig. 22.2. Note that the centralregion becomes more and more depleted at stronger couplings.

22.7 Summary

We have shown that the expansion of the effective action of a φ4-theory in evenpowers of the field strength Φ = 〈φ〉 can be resummed to obtain an expression thatis valid for any field strength, even in the strong-coupling limit. It has the phe-nomenological scaling form once proposed by Widom, and can be used to calculatethe shape of a BEC up to the Feshbach resonance, with and without rotation.


0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

0.2

0.4

0.6

0.8

1.0

r r

GP

FGP

GP

FGP

ρ = Ψ∗Ψρ = Ψ∗Ψ

Figure 22.2 Condensate density from the Gross-Pitaevskii equation (22.131) (GP,

dashed) and its fractional version (22.133) (FGP), both in the Thomas-Fermi approxi-

mation where the gradients are ignored. The FGP-curve shows a marked depletion of the

condensate. On the right hand, a vortex is included. The zeros at r ≈ 1 will be smoothened

by the gradient terms in (22.133) and (22.133), as shown on the left-hand plots without a

vortex. The curves can be compared with those in Ref. [10, 11, 12, 13, 14, 15].

Appendix 22A Effective Action to Order -h2

Let us now find the next correction to the effective action [16]. Instead of truncating the expansion(22.23), we keep all terms, reorganizing only the linear and quadratic terms as in the passage to(22.23). This yields the exact expression

eih{Γ[Φ]+jΦ} = e

ihW [j] = e

ih(A[φcl]+jφcl)−

h2Tr logAφφ[φcl]e

ihh2W2[φcl], (22A.1)

where W2[φcl] is defined by

eihh2W2[φcl] =

∫

Dφeih{ 1

2φAφφ[φcl]φ+I[φcl,φ]}

∫

Dφeih{ 1

2φAφφ[φcl]φ} , (22A.2)

and R is the remainder of the fluctuating action

R [φcl, φ] = A [φcl + φ]−A[φcl]−1

2φAφφ[φcl]φ. (22A.3)

The subscript φ denotes functional differentiation and integration symbols. Adjacent space-timevariables have been omitted, for brevity. We have renamed δφ(x) by φ(x).

Note that by construction, R is at least cubic in φ. Thus the path integral (22A.3) may beconsidered as the generating functional Z of a theory of a φ field, with a propagator

G[Φcl] = ih{Aφφ[φcl]}−1,

and an interaction R that depends in a complicated manner on j via φcl. We know from previoussections, and will immediately verify this explicitly, that h2W2 is indeed of order h2. Let us againset

φ = φcl + hφ1. (22A.4)

We shall express W [j] in the form

W [j] = A[φcl] + φclj + h∆1[φcl] (22A.5)

and collect one- and two-loop corrections in the term

∆1[φcl] =i

2Tr log ihG−1 + hW2[φcl]. (22A.6)

Appendix 22A Effective Action to Second Order in h 1275

The functional W [j] depends explicitly on j only via the second term, in all others the j dependenceis due to φcl[j]. We may use this fact by expressing j as a function of φcl and by writing

W [φcl] = A[φcl] + φcl j[φcl] + ∆1. (22A.7)

We now insert (22.28) and re-expand everything around X rather than xcl to find

Γ[Φ] = A[φ]− hAΦ[Φ]φ1 − hφ1j[Φ] + h2φ1jΦ[Φ]φ1 +1

2h2φ1iG−1[Φ]φ1

+ h∆1[Φ]− h2∆1Φ[Φ]φ1 +O(h3) (22A.8)

= A[Φ] + h∆1[Φ] + h2

{

1

2φ1iG−1[φ]φ1 + φ1jΦ[Φ]φ1 −∆1Φφ1

}

.

From this we get the correction φ1 to the classical vacuum expectation value φcl. Considering W [j]in (22A.5) again as a functional in j, we see that

Φ = δW [j]/δj = φcl + h∆1φcl[φcl]δφcl/δj, (22A.9)

implying that

φ1 = ∆1φcl[φcl]δφcl/δj. (22A.10)

But the derivative δφcl/δj is known from

δj/δφcl = −Aφφ[φcl] = −ihG−1[φcl], (22A.11)

so that

φ1 = ∆1φcl[φcl]iG[φcl]. (22A.12)

Inserting this into (22A.8), the h2-term becomes

− h2

2φ1iG−1[Φ]φ1 +O(h3). (22A.13)

In the derivative ∆1φ, only the trace log term in (22A.6) has to be included, i.e.,

D[X ] =i

2Tr

(

D−1 δ

δXD)

. (22A.14)

Hence the effective action becomes, to this order in h,

Γ[Φ] = A[Φ] + hΓ1[Φ] + h2Γ2[Φ]

= A[Φ] +i

2Tr log iG−1[Φ] + h2W2[Φ]

− h2

2

i

2Tr

(

G δ

δΦG−1

)

iG i

2Tr

(

G δ

δΦG−1

)

. (22A.15)

We now calculate W2[Φ] to lowest order in h. The remainder R in (22A.3) has the expansion

R[Φ;φ] =1

3!φ3AΦΦΦ[Φ] +

1

4!φ4AΦΦΦΦ[Φ],

where we have replaced φcl by Φ+O(h). In order to obtain W2, we have to calculate all connectedvacuum diagrams for the interactions r with a φ-particle propagator hG[Φ](x, x′). Since every


contraction brings in a factor h, we can truncate the expansion (22A.16) after Φ4. Thus the onlycontribution to W2[Φ] are the connected vacuum diagrams

.(22A.16)

Here a line stands for G[Φ], a four-vertex for

AΦΦΦΦ[Φ] = iG−1[Φ]ΦΦ, (22A.17)

and a three-vertex for

AΦΦΦ[Φ] = iG−1[Φ]Φ. (22A.18)

Only the first two graphs are one-particle irreducible. It is now a pleasant feature to realize thatthe third graph cancels with the last term in (22A.15). In order to verify this, we write the diagrammore explicitly as

− h2

8GΦ1Φ2

AΦ1Φ2Φ3GΦ3Φ3′

AΦ3′Φ

1′Φ

2′GΦ

1′Φ

2′(22A.19)

which is now part of iW2. Thus, only the one-particle irreducible vacuum graphs produce theh2-correction to Γ[Φ]:

ih2Γ2[Φ] = ih2 3

4!G12AΦ1Φ2Φ3Φ4

G34 + ih2 1

4!2AΦ1Φ2Φ3

GΦ1Φ1′GΦ2Φ2′

GΦ3Φ3′AΦ1Φ2Φ3

, (22A.20)

whose graphical representation is

ih2Γ2[Φ] = . (22A.21)

This topological property of the diagram is true for arbitrary orders in h. To calculate Γ, wedefine fundamental vertices of nth order

AΦ1...Φn[Φ],

and sum up all connected one particle irreducible vacuum graphs 2

∑

n≥2

ihnΓn[Φ] =

. (22A.22)

2Note that each line carries a factor h from the propagator, the n-vertex contributes a factorh1−n.

Appendix 22B Effective Action to All Orders in h 1277

Note the content of the lines G in terms of fundamental Feynman diagrams. The lines in thesediagrams contain infinite sums of diagrams in the original perturbation expansion. If propagatorsof the original φ-theory are indicated by a dashed line, an expansion of G in powers of Φ revealsthat every line corresponds to a sum

(22A.23)

of lines which emit 1, 2, 3 . . . lines via vertices φn in arbitrary combinations. The expansion toorder hn is an expansion according to the number of loops.

The general structure of the series (22A.22) becomes rapidly involved and must be organizedvia a functional formalism. It is treated in the next section.

Appendix 22B Effective Action to All Orders in -h

In order to find the general loop expansion, let us expand Φ not around the classical solution φcl

but around some other functional φ0 = φ0[j] whose properties will be specified later. Then thegenerating functional is

Z[j] = eihW [j] = N e

ih{A[φ0]+φ0j+hΓ1[φ0[j]]}, (22B.1)

where

eiΓ1[φ0] ≡∫

Gφ′eih{A[φ0+φ′]−A[φ0]+j[φ0]φ

′}. (22B.2)

On the right-hand side we have inverted the relation φ0[j] and expressed j as a functional of Φ0.The field Φ may be calculated as

Φ =δW [j]

δj= φ0 +

{

δA[φ0]

δφ0+ j + h

δΓ1[φ0]

δφ0

}

δφ0

δj. (22B.3)

Since φ0 is not equal to φcl, the first two terms in the bracket do not cancel. Instead, we shalldetermine φ0 by the requirement that the whole bracket vanishes for j = j[φ0]:

δA[φ0]

δφ0+ j[φ0] + h

δΓ1[φ0]

δφ0= 0. (22B.4)

This requirement makes φ0 directly equal to Φ and

Γ[Φ] = A[Φ] + hΓ1[Φ]. (22B.5)


Since Γ1[φ0] depends on φi also via j[φ0], it looks impossible to solve equation (22B.4) for φ0. Infact, reinserting (22B.4) into (22B.2) we can find a pure equation for φ0:

eiΓ1[φ0] =

∫

Gφ′eih{A[φ0+φ′]−A[φ0]−φ′Aφ0

[φ0]−h′φΓ1φ0[φ0]φ

′}, (22B.6)

which is a functional integro-differential equation. It is possible to extract from (22B.6) the desiredresult. Let us consider an auxiliary generating functional

eiWaux[φ0,K] ≡∫

Gφ′eih

{

Aaux[φ0,φ′]+∫

d4xKφ}

(22B.7)

with an action

Aaux[φ0, φ′] = A[φ0 + φ′]−A[φ0]− φ′Aφ0

[φ0]. (22B.8)

For an arbitrary fixed function φ0, the exponent in Eq. (22B.7) defines an auxiliary field theorywith an external source K. The expansion of Waux[φ0,K] in powers of K collects all connecteddiagrams for the action (22B.8). They are built from propagators

= G[φ0] (22B.9)

and vertices

1 5

2

n

4

6

3

=δnA[φ0]

δφ0 . . . δφ0.

(22B.10)

Thus, if we succeed in enforcing φ0 = Φ, we are dealing exactly with the field theory whose one-particle irreducible vacuum graphs are supposed to make up the corrections Γ1[Φ]. How can weshow this? By comparing (22B.6) with (22B.2) we see that, for the special choice of the current

K ≡ K0 − hΓ1φ0[φ0], (22B.11)

the auxiliary functional Waux[φ0,K0] becomes identical to the generating functional, that we aretrying to calculate. The auxiliary functional also has an auxiliary Legendre transformed field

Φaux =δWaux[φ0,K]

δK. (22B.12)

With the help of

Φaux =δWaux[φ0,K]

δK= Φaux[φ0,K], (22B.13)

we now define an auxiliary effective action

Γaux [Φaux] ≡ Waux[φ0,K]−∫

d4xKΦaux, (22B.14)

where Γaux [Φaux] is also a functional of φ0. There is one important property of Γaux [Φaux]. Ifevaluated at Φaux = 0, it collects all one-particle irreducible vacuum graphs of Aaux [see (11.45),

Notes and References 1279

(11.46)]. But these are exactly the graphs we want. Hence the proof can be completed by showingthat the condition K = K0 = −Γ1φ0

[φ0] leads to

Φaux = 0. (22B.15)

But this is trivial to see: The condition K = K0 was just made up to enforce Φ = φ0. Thefluctuating fields φ′ are the difference between φ and φ0:

φ′ = φ− φ0 = φ− Φ. (22B.16)

Now Φ ≡ 〈φ〉 implies 〈φ′〉 = 0. But in the auxiliary theory we have

Φaux ≡ 〈φ′〉. (22B.17)

Thus Φaux = 0, and Waux[φ0,K] collects the one-particle irreducible vacuum graphs with propa-gators and vertices (22B.11) and (22B.13). This completes the proof.

Notes and References

The technique of treating quantum field theory with the help of an effective action goes back toC. De Dominicis, Jour. Math. Phys. 3, 983 (1962).His treatment of collective phenomena was developed further byJ.M. Cornwall, R. Jackiw, and E.T. Tomboulis, Phys. Rev. D 10, 2428 (1974);H. Kleinert, 30, 187 (1982) (http://klnrt.de/82); Fortschr. Phys. 30, 351 (1982)(http://klnrt.de/84).

The particular citations in this chapter refer to:

[1] C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill (1985);H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Finan-

cial Markets , World Scientific, Singapore 2009 (http:/klnrt.de/b5).

[2] H. Kleinert and V. Schulte-Frohlinde, Critical Properties of φ4-Theories, World Scientific,Singapore 2001 (http://klnrt.de/b8).

[3] J.A. Lipa, J.A. Nissen, D. A. Stricker, D.R. Swanson, and T.C.P. Chui, Phys. Rev. B 68,174518 (2003);M. Barmatz, I. Hahn, J.A. Lipa, R.V. Duncan, Rev. Mod. Phys. 79, 1 (2007). Also seepicture on the title-page of the textbook [2].

[4] H. Kleinert, Converting Divergent Weak-Coupling into Exponentially Fast Convergent

Strong-Coupling Expansions, Lecture delivered at the Centre International de RencontresMathematiques in Luminy, EJTP 8, 15 (2011) (http://klnrt.de/387).

[5] B. Widom, J. Chem. Phys. 43, 3892 and 3896 (1965).

[6] F.J. Wegner, Z. Physik B 78, 33 (1990).

[7] See Section 10.8 in the textbook [2], in particular Eq. (10.151). Also take Eq. (1.28) in thatbook, expand there f(r/ξ) = 1 + c(r/ξ)ω + . . . , and compare with (22.85).

[8] H. Kleinert, EPL 100, 10001 (2012) (http://klnrt.de/399).

[9] See Eq. (10.191) in Ref. [2] and expand f(t/M1/β) ∼ f(ξΦ2/(D−2+η)) like the functionf(x) = 1 + cx−ω + . . . .

[10] S. Giorgini, L.P. Pitaevskii, and S. Stringari, Phys. Rev. A 54, R4633 (1996).

[11] F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463512 (1999)(see their Fig. 3).


[12] L.V. Hau, B.D. Busch, C. Liu, Z. Dutton, M.M. Burns, and J.A. Golovchenko, Phys. Rev.A 58, R54 (1998) (Fig. 2).

[13] F. Dalfovo and S. Stringari, Phys. Rev. A 53, 2477 (1996).

[14] J. Tempere, F. Brosens, L.F. Lemmens, and J. T. Devreese, Phys. Rev. A 61, 043605 (2000).

[15] H. Kleinert, Effective action and field equation for BEC from weak to strong couplings, J.Phys. B 46, 175401 (2013) (http://klnrt.de/403).See also the textbook [2].

[16] J.M. Cornwall, R. Jackiw, E.T. Tomboulis, Phys. Rev. D 10, 2428 (1974);R. Jackiw, Phys. Rev. D 9, 1687 (1976).

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Functional-Integral Calculation of Eﬀective Action. Loop ...users.physik.fu-berlin.de/~kleinert/b6/psfiles/Chapter-21-direffac.pdf · which may be used to form the Legendre transform