Functional-Integral Calculation of Effective Action. Loop Expansion

It is simplicity that makes the uneducated more eﬀective than the educated. Aristoteles (384 BC–322 BC)

22 Functional-Integral Calculation of Eﬀective Action. 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 loop expansion organized by powers of the Planck quantumh ¯ [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 vacuum expectation of the ﬁeld, the average

Φ(x) φ(x) , (22.2) ≡h i 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) ≡ − Z 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] = j(x). (22.6) δΦ(x) −

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 physical field 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 generating functional W [j] from the effective action:

W [j] = Γ[Φ] + d4xj(x)Φ(x). (22.7) Z Let us compare this with the functional-integral formula for the generating functional Z[j]:

4 φ(x)e(i/¯h) A[φ]+ d xj(x)φ(x) Z[j]= D { } . (22.8) φ(x)e(i/¯hR)A0[φ] R D Using (22.1) and (22.5), this impliesR a functional-integral formula for the eﬀective action Γ[Φ]:

i Γ[Φ]+ d4xj(x)Φ(x) (i/¯h) A[φ]+ d4xj(x)φ(x) e h¯ = φ(x)e , (22.9) { } N D { } R Z R where is a normalization factor: N −1 = φ(x)e(i/¯h)A0[φ] . (22.10) N Z D We have explicitly displayed the fundamental quantum of actionh ¯, which is a mea- sure for the size of quantum fluctuations. There are many physical systems for which quantum fluctuations are rather small, 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 expansion in powers ofh ¯, taking into account successively increasing quantum fluctuations. In the limith ¯ 0, the path integral over the field φ(x) in (22.8) is dominated → by the classical solution φcl(x) which extremizes the exponent δ A = j(x). (22.11) ∂φ − φ=φcl(x)

At this level we therefore identify

4 4 W [j] = Γ[Φ] + d xj(x)Φ(x) = [φcl]+ d xj(x)φcl(x). (22.12) Z A Z

Of course, φcl(x) is a functional of j(x), so that we may write it more explicitly as φcl[j](x). By diﬀerentiating W [j] with respect to j, we have from Eqs. (22.3) and (22.7): δW δΓ δΦ δΦ Φ(x)= = +Φ+ j . (22.13) δj δΦ δj δj 22.1 General Formalism 1255

Inserting the classical ﬁeld equation (22.11), this becomes

δ δφcl δφcl Φ(x)= A + φcl + j = φcl. (22.14) δφcl δj δj

Thus, to this approximation, Φ(x) coincides with the classical ﬁeld φcl(x). For the φ4-theory with O(N)-symmetry, the action is

2 4 1 2 m 2 g 2 2 [φ]= d x (∂φa) φa φa . (22.15) A " 2 − 2 − 4! # Z Replacing the φ φ + δφ = Φ + δφ on the right-hand side of Eq. (22.12), we a → cl a s a therefore obtain the lowest-order result for the eﬀective action (which is of zeroth order inh ¯):

2 4 1 2 m 2 g 2 2 Γ[Φ] [Φ] = d x (∂Φa) Φa Φa . (22.16) ≈A "2 − 2 − 4! # Z It equals the fundamental action, and is also referred to as the mean ﬁeld approximation to the eﬀective action. We have shown in Chapter 13 that all vertex functions can be obtained from the derivatives of Γ[Φ] with respect to Φ at the equilibrium value of Φ for j = 0. We 2 1 begin by assuming that m > 0. Then Γ[Φ] has an extremum at Φa 0, and there (n) ≡ are only two types of non-vanishing vertex functions Γ (x1,...,xn): For n = 2, we obtain the lowest-order two-point function

2 2 (2) δ Γ δ Γ (x1, x2)ab = A ≡ δΦa(x1)δΦb(x2) φa(x1)φb(x2) Φa=0 φa=Φa=0

= ( ∂2 m2)δ δ(4) (x x ). (22.17) − − ab 1 − 2 This determines the inverse of the propagator:

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

Thus we ﬁnd in this zeroth-order approximation that Gab(x1, x2) is equal to the free propagator: Gab(x1, x2)= G0ab(x1, x2). (22.19) For n = 4, we ﬁnd the lowest-order four-point function

4 (4) δ Γ Γabcd(x1, x2, x3, x4) = gTabcd, (22.20) ≡ δΦa(x1)δΦb(x2)δΦc(x3)δΦd(x4) where 1 T = (δ δ + δ δ + δ δ ) (22.21) abcd 3 ab cd ac bd ad bc 1Also the case m2 < 0 corresponds to a physical state in a different phase and will be discussed separately. 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. 1256 22 Functional-Integral Calculation of Effective Action. Loop Expansion 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 obtained from tree diagrams built from propagators Gab(x1, x2) and vertex functions. This was explained in Chapter 13. In lowest approximation, we recover precisely the subset of all original Feynman diagrams with a tree-like topology. These are the diagrams which do not involve any loop integration. Since the limith ¯ 0 corresponds to the classical equations of motion with no quantum fluctuations,→ we conclude: Classical field theory corresponds to tree diagrams. The use of the initial action as an approximation to the effective action neglecting fluctuations is often referred to as mean-field theory.

22.2 Quadratic Fluctuations

In order to find theh ¯-correction to this approximation we expand the action in powers of the fluctuations of the field around the classical solution δφ(x) φ(x) φ (x), (22.22) ≡ − cl and perform a perturbation expansion. The quadratic term in δφ(x) is taken to be the free-field action, the higher powers in δφ(x) are the interactions. Up to second order in the fluctuations δφ(x), the action is expanded as follows:

4 [φcl + δφ]+ d xj(x)[φcl(x)+ δφ(x)] A Z 4 4 δ = [φcl]+ d xj(x)φcl(x)+ d x j(x)+ A δφ(x) A ( δφ(x)  Z Z φ=φcl 

δ2 + d4xd4y δφ(x) A δφ(y)+ (δφ) 3 .  (22.23) δφ(x)δφ(y) O Z φ=φcl

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

2 4 i δ Z[j] e(i/¯h) A[φcl]+ d xjφcl δφ exp d4xd4y δφ(x) A δφ(y) . ≈N { } D h¯ δφ(x)δφ(y)  R Z  Z φ=φcl 

(22.24) We now observe that the fluctuations in δφ will be of average size √h¯ due to theh ¯-denominator in the Fresnel integrals over δφ in (22.24). Thus the fluctuations (δφ)n are on the average of relative orderh ¯n/2. If we ignore corrections of the order h¯3/2, the fluctuations remain quadratic in δφ and we may calculate the right-hand side of (22.24) as

2 4 δ e(i/¯h) A[φcl]+ d xj(x)φcl(x) det A N { } " δφ(x)δφ(y)# R φ=φcl 4 2 1/2 (i/¯h) A[φd]+ d xj(x)φcl(x)+i(¯h/2)Tr log[δ A/δφ(x)δφ(y)|φ=φcl = (det iG0) e { }. (22.25) R 22.2 Quadratic Fluctuations 1257

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

2 4 4 ih¯ δ Γ[Φ] + d xj(x)Φ(x)= [φcl]+ d xjφcl + Tr log A (φcl) . (22.26) Z A Z 2 δφ(x)δφ(y) In the limith ¯ 0, the trace log term disappears and (22.26) reduces to the classical action. → To include theh ¯-correction into Γ[Φ], we expand W [j] as

W [j]= W [j]+¯hW [j]+ (¯h2). (22.27) 0 1 O 2 Correspondingly, the field Φ differs from Φcl by a correction of the orderh ¯ : Φ= φ +¯hφ + (¯h2). (22.28) cl 1 O Inserting this into (22.26), we find

2 4 4 4 i δ Γ[Φ] + d xjΦ = [Φ hφ¯ 1]+ d xjΦ h¯ d xjφ1 + h¯Tr log A A − − 2 δφδφ Z Z Z φ=Φ−¯hφ1

+ h¯2 . O Expanding the action up to the same order inh ¯ gives

2 δ [Φ] 4 i δ Γ[Φ] = [Φ] +h ¯ A j φ1 + d x jΦ+ h¯Tr log A A ( δΦ − ) 2 δφδφ Z φ=Φ

+ h¯2 . (22.29) O But because of (22.11), the curly-bracket term is only of the order (¯h2), so that O we find the one-loop form of the effective action 1 m2 g Γ[Φ] = d4x (∂Φ)2 Φ2 Φ4 Z "2 − 2 − 4! # i g + h¯Tr log ∂2 m2 Φ2 , (22.30) 2 − − − 2 where we dropped the infinite additive constant i h¯Tr log [ ∂2 m2] . 2 − − If we generalize this to an O(N)-invariant theory with N components φa (a = 1,...,N), this expression becomes

2 4 1 2 m 2 g 2 2 Γ[Φ] = Γ0[Φ] + Γ1[Φ] = d x (∂Φa) Φa Φa "2 − 2 − 4! # Z i g + h¯Tr log ∂2 m2 δ Φ2 + 2Φ Φ . (22.31) 2 − − − 6 ab c a b What is the graphical content of the set of all Green functions at this level? In this discussion we shall assume that N = 1. For j = 0, we ﬁnd that the minimum 1258 22 Functional-Integral Calculation of Eﬀective Action. Loop Expansion

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

i g i i i Φ2 h¯Trlog ∂2 m2 Φ2 = h¯Trlog ∂2 m2 + h¯Trlog 1+ ig . 2 − − − 2 2 − − 2 ∂2 m2 2 ! − − (22.32)

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

h¯ ∞ g n 1 i n i i Tr Φ2 . − 2 − 2 n ∂2 m2 nX=1 − − If we insert i G = , (22.33) 0 ∂2 m2 − − then (22.32) can be written as

∞ h¯ h¯ g n 1 n i Tr log ∂2 m2 i i Tr G Φ2 . (22.34) 2 − − − 2 − 2 n 0 nX=1 More explicitly, the terms with n = 1 and n = 2 read:

h¯ g d4xd4yδ(4)(x y)G (x, y)Φ2(y) − 2 − 0 Z2 g 4 4 4 4 2 2 +ih¯ d xd yd zδ (x z)G0(x, y)Φ (y)G0(y, z)Φ (z)+ .... (22.35) 16 Z − 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 of fundamental Φ4-vertices. To systemize the entire expansion (22.34), the leading trace log expression may be pictured by a single-loop diagram

h¯ i Tr log ∂2 m2 = . (22.37) 2 − − 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 functions and lead to more involved tree diagrams. Note that only the ﬁrst two corrections 22.2 Quadratic Fluctuations 1259 are formally divergent, all other Feynman integrals converge. In momentum space, the corrections are, from (22.35),

4 (2) 2 2 g dk i Γ (q) = q m h¯ 4 2 2 (22.38) − − 2 Z (2π) k m + iη 2 4 − (4) g d k i i Γ (qi) = g i 2 + 2 perm . − 2 " (2π)4 k2 m2 + iǫ (q + q k) m2 + iη # Z − 1 2 − − (22.39) The convergence of all higher diagrams in the expansion (22.36) is ensured by the renormalizability of the theory. Indeed, up to n = 4, a counter term may be written down 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

(2) 2 2 g Γ (q) = q + m +¯h D1 , (22.40) − 2 g2 Γ(4)(q ) = g h¯ [I (q + q ) + 2 perm] , (22.41) i − 2 1 2 where D1 and I(q) are the Feynman integrals [compare (11.26) and (11.29)]

-4 1 D1 = d kE 2 2 (22.42) Z kE + m and

-4 1 1 I(q)= d kE 2 2 2 2 . (22.43) Z (kE + m ) [(k + q)E + m ] The integrals can be calculated in D dimensions by separating them into a directional and a size integral as

D -D d k SD D−1 d k = D = D dkk Z Z (2π) (2π) Z S¯ D dk2(k2)D/2−1. (22.44) ≡ 2 Z We further simplify all calculations by performing a Wick rotation of all energy ∞ 4 integrals dk0 into i −∞ dk4. Then the integrals d k become what are called 4 µ 2 2 2 2 euclideanR integrals i Rd kE where kE = (k,k4), and Rk = k0 k becomes kE = 2 2 − − (k + k4). By the sameR 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 φ (x) Z−1/2φ(x), (22.45) R ≡ φ 1/2 where Zφ 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 − 1260 22 Functional-Integral Calculation of Effective Action. Loop Expansion constants order by order in perturbation theory. Alternatively, we can add to the bare action suitable counter terms. In either way, we arrive at finite expressions. If we regularize the integrals dimensionally in 4 ǫ dimensions, the counterterms take the forms calculated in Section 11.5, and the− expression (22.40) becomes

(2) 2 2 g −ǫ 2 ΓR (q) = q + m +¯h S¯Dµ m − ( 2 × 1 m2 −ǫ/c 1 Γ(2 ǫ/2)Γ( 1+ ǫ/2) + , (22.46) × 2 − − µ2 ! ǫ  )   while (22.41) is renormalized to g2 Γ(4)(q ) = g h¯ [I (q + q ) + 2 perm] 3I(0) . (22.47) R i − 2 { 1 2 − } The Feynman integral I(q) has, after subtracting the 1/ǫ-pole in Eq. (22.43), the small-ǫ behavior [compare (11.176)]

2 1 −ǫ m q IR(q)= S¯Dµ 1+ L (q) + log + (ǫ), (22.48) −2 " µ2 # O with

1 2 m m L (q) = dx log x(1 x)+ 2 Z0 " − q # m2 √q2 +4m2 √q2 +4m2 + √q2 = 2 + log + log . (22.49) − q2 √q2 √q2 +4m2 √q2 − As far as the original eﬀective action (22.30) to orderh ¯ is concerned, we may add the counter terms as 1 m2 g h¯ g Γ[Φ] = d4x (∂Φ)2 Φ2 Φ4 i Tr log ∂2 m2 Φ2 Z (2 − 2 − 4! ) − 2 − − − 2 4 2 4 g d q 1 4 2 g d q 1 4 4 + 4 2 2 d x Φ (x) 4 2 2 d x Φ . (22.50) 4 Z (2π) q + m Z − 16 Z (2π) (q + m ) Z The divergent integrals in the last two terms can be evaluated with any regularization method, and (22.50) becomes 1 m2 g h¯ g Γ[Φ] = d4x (∂Φ)2 Φ2 Φ4 + i Tr log ∂2 m2 Φ2 Z (2 − 2 − 4! ) 2 − − − 2 2 2 2 m g cm Φ2 cg Φ4. (22.51) − 2 − 4! The third line may be written as hg¯ hg¯ 2 (m2)1−ǫ/2c Φ2 + (m2)−ǫ/2c Φ4, (22.52) − 4 1 16 2 22.2 Quadratic Fluctuations 1261 with the constants

ǫ−2 -D 1 c1 = m d pE 2 2 , (22.53) Z pE + m ǫ -D 1 c2 = m d pE 2 2 2 . (22.54) Z (pE + m ) We evaluate these integrals using the formulas [2]

-D 1 Γ(D/2)Γ(1 D/2) 1 d kE =S ¯D − , (22.55) 2 2 2 1−D/2 Z kE +m 2Γ(1) (m ) and

-D 1 Γ(D/2)Γ(2 D/2) 1 d kE =S ¯D − . (22.56) 2 2 2 2 2−D/2 Z (kE +m ) 2Γ(1) (m )

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

Γ(D/2)Γ(1 D/2) D/2 d-Dk log k2 + m2 =S ¯ − m2 . (22.57) E E D DΓ(1) Z In the so-called minimal subtraction scheme in 4 ǫ dimensions, only the singular − 1/ǫ pole parts of the two integrals are selected for the subtraction in (22.50). In the neighborhood of ǫ = 0, (22.53) and (22.54) become

2 c = S¯ + (ǫ), (22.58) 1 − D ǫ O 1 ǫ c2 =S ¯D 1 + (ǫ). (22.59) ǫ − 2 O Hence we can choose as counter terms the singular terms in (22.52):

−ǫ 2 −ǫ hgµ¯ 2 1 2 hg¯ µ 1 4 m S¯D Φ , S¯D Φ . (22.60) − 4 − ǫ 16 ǫ 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 equal to m. But for m = 0, we must use an arbitrary nonzero auxiliary mass µ as the renormalization scale. Observe that up to the orderh ¯, there is no divergence that needs to be absorbed in the gradient term dDx (∂Φ)2 of the effective action (22.51). These come in as soon as we carry the sameR analysis to one more loop order. Let us calculate the effective potential in the critical regime for a constant field Φ at the one-loop level. It is defined by v(Φ) = Γ [Φ]/V T . The argument in the − E 1262 22 Functional-Integral Calculation of Effective Action. Loop Expansion trace log term is now diagonal in momentum space and the calculation reduces to a simple momentum integral. It follows directly from (22.51) and reads

2 D 2 m 2 g 4 h¯ d qE g Φ v(Φ) = Φ + Φ + D log 1+ 2 2 2 4! 2 Z (2π) 2 qE +m ! hg¯ hg¯ 2 (m2)1−ǫ/2c Φ2 + (m2)−ǫ/2c Φ4. (22.61) − 4 1 16 2 The expansions in powers of ǫ have an important property which has direct consequences for the strong-coupling limit that is measured in critical phenomena. For small ǫ, Eq. (22.47) can be rewritten as

2 (2) 2 2 h¯ −ǫ mE ΓR = qE + m + µ gS¯D log , (22.62) 4 µ2 ! which is, to the same order in g, equal to

h¯ −ǫ 2 1+ 4 µ g S¯D (2) 2 m 2 ΓR (q)= qE + µ . (22.63)  µ2 !    This means that the vertex function at q = 0 has a mass term that depends on the mass m of the φ-ﬁeld via a power law:

2 γ (2) m 2 ΓR (0) = µ . (22.64) µ2 ! The power γ depends on the coupling strength g like h¯ γ =1+ µ−ǫgS¯ . (22.65) 4 D The important point is that this power γ is measurable as an experimental quantity called the susceptibility. It is called the critical exponent of the susceptibility. If the effective action is calculated to orderh ¯2, then the gradient term in the effective action is modified and becomes

d Γ[Φ] = d x ΦR(x)ΓR(ˆq)ΦR(x), (22.66) Z −1/2 where ΦR(x)= Zφ 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 limit m2 0, the renormalization is power-like→ Φ (x−) (µ/µ )−η/2Φ(x), and Eq. (22.63) → R → 0 becomes

2 γ (2) 2 2−η η m 2 ΓR (q)= (q ) µ + µ . (22.67) − " µ2 ! # The power η is called the anomalous dimension of the ﬁeld Φ. 22.2 Quadratic Fluctuations 1263

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. (4) Another power behavior is found for ΓR (0):

2 (4) 3 −ǫ m 2 ΓR (0) g 1+ hgµ¯ S¯D log + (m ) m→→0 4 µ2 ! O 3 −ǫ m2 ¯h 4 gµ S¯D = g . (22.70) µ2 !

Also this power behavior is measurable, and defines the critical index β via the so-called scaling relation 3 γ 2β hgµ¯ −ǫS¯ , (22.71) − ≡ 4 D so that 1 1 β = hgµ¯ −ǫS¯ . (22.72) 2 − 4 D The higher powers of Φ are accompanied by terms which are more and more singular in the limit m 0. From the Feynman integrals in (22.34) we see that the diagrams → in (22.36) behave like m4−ǫ−n for m 0, and so do the associated effective action terms Φn. → The coefficients of the dimensionless quantities (Φ2/µ2−ǫ)≡(Φ˜ 2)n or (gΦ2/µ2)n ≡ (Φˆ 2)n have the general form (m2/µ2)γ−2nβ, so that the effective potential can be written as m2 γ Φ2 v(Φ) = µ4−ǫ f(x), (22.73) µ2 ! µ2−ǫ with

m2 −2β Φ2 Φ2 x t−2β = t−2βΦ˜ 2, (22.74) ≡ µ2 ! µ2−ǫ ≡ µ2−ǫ where we have abbreviated m2 t . (22.75) ≡ µ2 1264 22 Functional-Integral Calculation of Eﬀective Action. Loop Expansion

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

∞ ∞ n −2nβ 2 n f(x)=1+ fnx =1+ fn t Φ˜ . (22.76) nX=1 nX=1 If we start the sum at n = 1, we also get the general form of the vacuum energy − m2 γ+2β v(0) = µ4−ǫ . (22.77) µ2 !

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 temperature behavior of the specific heat

m2 Dν−2 m2 −α C = , (22.78) ∝ µ2 ! µ2 ! which yields the important critical exponent α =2 Dν that governs the singularity − of 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 accuracy was achieved by performing the measurement in a micro-gravity environment on a satellite [3]. We can also rewrite (22.73) in the form

S¯ m2 γ gΦ2 v(Φ) = µ4−ǫ D f¯(y), (22.79) λ µ2 ! µ2 where

∞ ∞ 2 n n −2β gΦ f¯(y)=1+ f¯ny =1+ f¯n t , (22.80) µ2 !! nX=1 nX=1 and

2 −2β 2 λ −ǫ m gΦ −2β ˆ 2 y x = gµ x = 2 2 = t Φ . (22.81) ≡ S¯D µ ! µ !

In the limit m2 0, the expansions (22.76) and (22.80) are divergent since the → coefficients grow like n!. A sum can nevertheless be calculated with the technique of Variational Perturbation Theory (VPT). This was used before in Chapter 3, whose first 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 the coupling strength in the denominator, the limit m2 0 is equivalent to the limit → g , i.e., to the strong-coupling limit. →∞ The strong-coupling limit deserves special attention. The theory in this limit is referred to as the critical theory. This name reflects the relevance of this limit for the behavior of physical systems at a critical temperature where fluctuations are of infinite 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 → 1 S¯ gΦ2 (δ+1)/2 v(Φ) µ4−ǫ D . (22.82) → 4! λ µ2 !

Without ﬂuctuation corrections, δ has the value 3, and (22.82) reduces properly to the mean-ﬁeld potential gΦ4/4!. With the general leading large-Φ-behavior, the potential in the form (22.79) is the so-called Widom scaling expression that depends on y−1/2β m2/Φ1/β [5] in the following general way: ∝

v(Φ) Φδ+1w(m2/Φ1/β). (22.83) ∝ From this eﬀective potential we may derive the general Widom form of the equation of state. After adding a source term HΦ and going to the extremum, we obtain H(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

1 S¯ gΦ2 (δ+1)/2 v(Φ) µ4−ǫ D wˆ (τ) , (22.85) → 4! λ µ2 ! where τ (m2/µ2) /(Φ/µ1−ǫ/2)1/β t/(Φ˜ 2)1/β. ≡ ≡ For small m2,w ˆ(τ) has a series expansion in powers of τ ων :

ων 2ων wˆ(τ)=1+c1τ +c2τ + ...), (22.86) or, since t = µξ−1/ν,

−ω −ων/β −2ω −2ων/β wˆ(τ)=1+¯c1ξ Φ +¯c2ξ Φ + ...). (22.87)

Here ω is the Wegner exponent [6] that governs the approach to scaling. Its numerical value is close to 0.8 [7]. 1266 22 Functional-Integral Calculation of Eﬀective Action. Loop Expansion

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

2 (δ−1)/2 δ +1 2 gΦ H = ∂Φv(Φ) = µ 4! µ2 ! (δ−1)/2 δ +1 λΦ˜ 2 = µ2 , (22.88) 4! S¯D ! where δ 1= γ/β. Note− that the eﬀective potential remains ﬁnite for m = 0. Then, v(Φ) becomes

D 2 2 −ǫ g 4 h¯ d q g Φ hg¯ µ S¯D 4 v(Φ) = Φ + D log 1+ 2 + Φ . (22.89) 4! 2Z (2π) 2 q ! 16 ǫ This displays an important feature: When expanding the logarithm in powers of Φ, the expansion terms correspond to increasingly divergent Feynman integrals

dDq 1 D 2 n . Z (2π) (q ) Contrary to the previously regularized divergencies coming from the large-q2 regime, these divergencies are due to the q = 0 -singularity of the massless propagators 2 G0(q)= i/q . 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

ǫ 2− 2 2 g 4 h¯ 1 g 2 hg¯ −ǫ 1 4 v(Φ) = Φ + S¯D Φ + S¯Dµ Φ . (22.90) 4! 4 − ǫ 2 16 ǫ With the goal of expanding this for a small-ǫ expansion, we divide the coupling constant and ﬁeld by a scale parameter involving µ. Then gµ−ǫ and Φ/µ1−ǫ/2 are dimensionless quantities, on which the eﬀective potential depends as follows:

ǫ gµ−ǫ Φ 4 h¯ 1 gµ−ǫ Φ2 2− 2 hg¯ 2 µ−2ǫ Φ 4 v(Φ)=µ4−ǫ + S¯ + S¯ .  1−ǫ/2 D 2−ǫ  D 1−ǫ/2 4! µ ! 4 − ǫ× 2 µ ! 16 ǫ µ !   (22.91)

If we use the dimensionless coupling constant

λ S¯ hgµ¯ −ǫ, (22.92) ≡ D and the reduced ﬁeld Φ˜ Φ/µ1−ǫ/2 as new variables, then ≡ 2 µ4−ǫ λ 1 1 λΦ˜ 2 ǫ λΦ˜ 2 λ2 v(Φ) = Φ˜ 4 + 1 log + Φ˜ 4 . (22.93)  h¯S¯D 4! 4 − ǫ 2 ! × − 2 2 ! 16ǫ #  22.4 Critical Coupling Strength 1267

To zeroth order in ǫ, the prefactor is equal to µ4 8π2. Thus the massless limit of the effective potential is well defined in D = 4 dimensions. There is, however, a special feature: 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 expanded in integer powers of Φ2. Instead, the expression (22.93) can be rewritten, correctly up to order λ2, as

3 µ4−ǫ 1 λ 2 3 λ 2 λ µ4−ǫ 1 λ 2+ 4 λ v(Φ)= Φ˜ 2 + Φ˜ 2 log Φ˜ 2 = Φ˜ 2 + (λ3). S¯D 6  2 ! 8 2 ! 2  S¯D 6λ 2 ! O     (22.94) This is the typical power behavior of a critical interaction. The power of Φ˜ defines the critical exponent of the interaction, in general denoted by 1 + δ, so that up to the first order in the coupling strength, we identify 3 δ =3+ λ. (22.95) 2 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 all subtracted expressions. The relation between the bare coupling constant gB and the renormalized one g is, to this one-loop order,

3 S¯ g = g + hg¯ 2µ−ǫ D + .... (22.96) B 2 ǫ

For a given bare interaction strength gB, the renormalized coupling depends on the parameter µ chosen for the renormalization procedure. Equivalently we may imagine having defined the field theory on a fine spatial lattice with a specific small lattice spacing a 1/µ. The renormalized coupling constant will then depend on the choice of a. ∝ If we sum an infinite chain of such corrections, we obtain a geometric sum that is an expansion of the equation g g = , (22.97) B 3 S¯ 1 hgµ¯ −ǫ D − 2 ǫ which has (22.96) as its first expansion term. Equivalently we may write

1 1 3 S¯D −ǫ = −ǫ + .... (22.98) hg¯ Bµ hgµ¯ − 2 ǫ 1268 22 Functional-Integral Calculation of Eﬀective Action. Loop Expansion

In this equation, we can go to the strong-coupling limit gB by taking µ to the critical limit µ 0, where we ﬁnd that the renormalized coupling→∞ has a ﬁnite value → 1 3 S¯ = D + .... (22.99) hgµ¯ −ǫ 2 ǫ

For the dimensionless coupling constant (22.92), this amounts to the strong-coupling limit 2 λ hgµ¯ −ǫS¯ ǫ + ..., (22.100) ≡ D → 3 with the omitted terms being of higher order in ǫ. The approach to this limit starting from small coupling is obtained from (22.97) to be

1 hgµ¯ −ǫ = . (22.101) 1 3 S¯D −ǫ + hg¯ Bµ 2 ǫ

At a small bare coupling constant gB, this starts out with the renormalized expression which is determined by the s-wave scattering lengthhgµ ¯ −ǫ =6 4πh¯2a /M × s [recall Eq. (9.266)]. In the strong-coupling limit of gB, which is equal to the critical limit 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-coupling limits are 1 1 1 γ =1+ ǫ, β = ǫ, δ =3+ ǫ. (22.102) 6 2 − 6

They are approached for ﬁnite gB like 1 1 1 γ =1+ hgµ¯ −ǫ, β = hgµ¯ −ǫ, (22.103) 4 2 − 4 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 series in Eq. (15.18) of the textbook [2]. The other critical exponents may→∞be obtained from theg ¯B -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− from − − − Chapter 15 in Ref. [2]. All these series are divergent. But they can be resummed for ǫ = 1 in the strong- coupling limitg ¯B . As an example, the series forg ¯ at N = 2 yields the limiting valueg ¯ g∗ 0→∞.503 (see Fig. 17.1 in [2]). The typical dependence ofg ¯ ong ¯ → ≈ B is displayed in Fig. 20.8 of [2]. For ν and η the corresponding plots are shown in Figs. 20.9 and 20.10 of [2]. We leave it as an exercise to compose from these the dependence of α, β, and δ as functions ofg ¯B. 22.4 Critical Coupling Strength 1269

It should be pointed out that the potential in the first line of (22.94) has another minimum, away from the origin, solving λ λ2 λ 1 λ2 Φ˜ 3 + Φ˜ 3 log Φ˜ 2 + Φ˜ 3 =0. (22.104) 3! 8 2 − 2! 16 The other minimum solves λ 4 λ log Φ˜ 2 = , (22.105) 2 −3 which lies at 2 Φ˜ 2 = e−4/3λ. (22.106) λ However such a solution found for small λ is not reliable. The higher loops to be discussed below and neglected up to this point will produce more powers of λ log(λΦ˜ 2/2), and the series cannot be expected to converge at such a large λ. Asa matter of fact, the approximate exponentiation performed in (22.94) does not show a minimum and will be seen, via the methods to be described later, to be the correct analytic 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 discuss the case of a general O(N)-symmetric version of the effective potential based on the action (22.31), into which we must insert the number N = 2. The equation for the bare coupling constant is then N +8 S¯ g = g + hg¯ 2µ−ǫ D + ..., (22.107) B 6 ǫ rather than (22.96), so that the strong-coupling limit (22.100) becomes 6 λ S¯ hgµ¯ −ǫ ǫ + ..., (22.108) ≡ D → N +8 with the other critical exponents (22.102): N +2 1 3 9 γ =1+ ǫ, β = ǫ, δ =3+ ǫ. (22.109) 2(N +8) 2 − 2(N +8) N +8 If we carry the loop expansion to higher order inh ¯, we find the perturbation −ǫ expansion for the renormalizedg ¯ =hg ¯ Bµ /2¯SD as function of the bare coupling −ǫ g¯B =hg ¯ Bµ /2¯SD [taken from Eq. (15.18) in Ref. [2], in particular the higher expansion terms]:

2 g¯ 8+N −1 2 (8+N) 1 14+3 N 1 = 1 g¯B 3 ǫ +g ¯B 9 ǫ2 + 6 ǫ g¯B − 3 n o 3 (8+N) 1 (8+N)(14+3N) 1 +g ¯ 3 4 2 (22.110) B − 27 ǫ − 27 ǫ n 2 [2960+922 N+33 N +(2112+480 N)ζ(3)] 1 648 ǫ + .... − 1270 22 Functional-Integral Calculation of Eﬀective Action. Loop Expansion

Given thisg ¯B-dependence ofg ¯, we ﬁnd for γ the expansion g¯ g¯2 g¯3 γ = 6 (N +2) 5 36 (N +2) + 72 (N +2)(5N + 37) 4− g¯ (N +2) [ N 2 + 7578N + 31060 (22.111) − 15552 − + 48ζ(3)(3N 2 +10N +68) + 288ζ(4)(5N +22)]+..., in whichg ¯ has the limiting value (22.110). The expression (m2/µ2)γ in the two-point function can be replaced by an expan- 2 2 sion of m /mB in powers ofg ¯B that can be taken from Eq. (15.15) in [2], again withg ¯ replaced by (22.110). This expansion can be resummed by Variational Perturbation Theory to obtain a curve of the type shown in Fig. 20.8–20.10 of [2]. This permits 2 2 −ǫ us to relate t = m /µ directly to µ gB.

22.5 Resumming the Eﬀective Potential

According to Eq. (22.79), the eﬀective action in the condensed phase with negative m2 has an expansion in powers of y: v(Φ) Φ2 = tγ (f¯ + f¯ y + ... ), (22.112) µ4−ǫ µ2−ǫ 0 1 where y = t−2βgΦ2/µ2 = t−2βΦˆ 2. (22.113)

From Eq. (22.80), we determine f¯0 = f¯1 = 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, the Widom function (22.83) has an expansion in powers of (m2)ω/ν ξ−ω which contains ∝ the Wegner critical exponent ω 0.8 governing the approach to scaling [7]. The general expansion for strong couplings≈ is 2 ∞ v(Φ) γ Φ −2β 2 (δ−1)/2 bm = t (t Φˆ ) b0 + . µ4−ǫ µ2−ǫ (t−2βΦˆ 2)mων/2β ! mX=1 (22.114) We may derive this from the rules of VPT in [2, 4], reviewed in some detail in Section 21.7. 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

2 γ 2 p ¯ ¯ y wN = µ t Φ κ f0 + f1 q + ... . (22.115) κ Next we exchange κp by the identical expression √K2 + grp, where r (k2 2 2 ≡ − K )/K . After this we form w1 by expanding wN up to order g, and setting κ = 1. This leads to 2 γ 2 p ¯ p p 1 ¯ y w1 = µ t Φ K f0 1 + 2 + f1 q − 2 2 K K 2 γ 2 = µ t Φ W1(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 derivative dw1/dK vanishes. This determines K from the equation

γ 2 p−1 p(2 p) ¯ 1 ¯ y t Φ K − f0 1 2 f1c q =0, (22.118) 2 − K − K where 2(p q) c − 0.32. (22.119) ≡ p(p 2) ≈ − In the free-particle limit y 0, the solution is K(0) = 1, and → 2 γ 2 w1 = µ t Φ f¯0. (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-y behavior

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

3 W1 =1+ y +0.154436y + .... (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

2 γ 2 p p y w1(y)= µ t Φ K 1 + q , (22.126) − 2 K 1272 22 Functional-Integral Calculation of Eﬀective Action. Loop Expansion

12 W (y) K(y) 1 10 60 8 40 6 4 20 2

2 4 6 8 10 12 14 2 4 6 8 10 y y

Figure 22.1 Solution of the variational equation (22.118) for f¯1 = 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

p p 1 W1(y) Was(y)= K 1 + → − 2 c 0.197 y1.87. (22.127) ≈ Near the limit, the corrections to (22.121) are

∞ hm K(y) = Kas(y) 1+ ymων/2β ! mX=1 ∞ h 0.648 y0.381 1+ m , (22.128) ≈ y0.761 m ! mX=1 with h 0.909, h 0.155,.... Inserting this into (22.116), we ﬁnd 1 ≈ 2 ≈ − ∞ bm W1(y) = Was(y) 1+ ymων/2β ! mX=1 ∞ b 0.197 y1.87 1+ m , (22.129) ≈ y0.761 m ! mX=1 with b 3.510, b 4.65248,.... 1 ≈ 2 ≈

22.6 Fractional Gross-Pitaevskii Equation

We now extremize the eﬀective action (22.51) with the two-loop corrected quadratic term (22.63). We consider the case of N = 2 where Φ2 = Ψ∗Ψ. Then we take the 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 the time-independent fractional Gross-Pitaevskii equation:

∂w (y) (pˆ2)1−η/2Ψ+ 1 =0. (22.130) ∂Ψ∗ 22.7 Summary 1273

If we use the weak-coupling limit of w1(y) and the gradient term, this reduces to the ordinary time-independent Gross-Pitaevskii equation 2 h¯ 2 † ∇ µ + gSΨ Ψ Ψ(x)=0. (22.131) "−2M − #

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 given particle number N. In a harmonic trap, µ is replaced by µ + Mω2x2/2. Recalling 2 2 2 2 2 2 the relation m = 2Mµ one has m = 2M Mω Rc (1 R /Rc ). The oscillator 2 2 − − − 2 2 2 2 energy Mω Rc corresponds to a length scale ℓO by the relation Mω Rc =¯h /MℓO, 2 2 2 2 2 so that m may be written as m = µ (R /Rc 1) with µ =1/ℓO. In the strong-coupling limit, however, we arrive− at the time-independent fractional Gross-Pitaevskii equation:

2 1−η/2 δ+1 δ−1 (pˆ ) + gc Ψ(x) Ψ(x)=0. (22.133) " 4µη | | # By using the full effective action for all coupling strengths and masses m2 we can derive the properties of the condensate at any coupling strength. Before reaching the strong-coupling limit, we may use Eq. (22.130) to calculate the field strength Ψ as a function of gS. In this regime, the critical exponents η,α,β,δ have not yet reached their strong-coupling values but must be replaced by theg ¯B-dependent precritical values 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-independent 2 | | 2 2 Gross-Pitaevskii equation has to be solved with m (R) 1 R /Rc . More specifi- cally, the bare coupling constant on the right-hand side has∝ to− be determined in such 2 2 a way that m /mB has the experimental size. If the experiments are performed in an external magnetic field B, the s-wave scattering length as has an enhancement −1 factor (B/Bc 1) and Eq. (22.110) can again be used. We can then calculate the density 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 profiles of vortices for various coupling strengths which can be varied from weak to strong by subjecting the BEC to different magnetic fields, raising it from zero up to the Feshbach resonance. The profiles are shown in Fig. 22.2. Note that the central region 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 even powers of the field strength Φ = φ can be resummed to obtain an expression that h i is 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 calculate the shape of a BEC up to the Feshbach resonance, with and without rotation. 1274 22 Functional-Integral Calculation of Effective Action. Loop Expansion

1.0 1.0 GP 0.8 GP 0.8 FGP 0.6 FGP 0.6 0.4 0.4 ∗ ∗ ρ =Ψ Ψ 0.2 ρ =Ψ Ψ 0.2

0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 r r 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 approximation 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 h- 2

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

i i i h¯ i 2 {Γ[Φ]+jΦ} W [j] (A[φcl]+jφcl)− Tr log A [φcl] h¯ W2[φcl] e h¯ = e h¯ = e h¯ 2 φφ e h¯ , (22A.1) where W2[φcl] is deﬁned by

i 1 ¯ 2 φAφφ[φcl]φ+I[φcl,φ] i 2 φe h h¯ h¯ W2[φcl] { } e = D i 1 , (22A.2) φA [φcl]φ φe h¯ 2 φφ R D { } and is the remainder of the fluctuating actionR R 1 [φ , φ]= [φ + φ] [φ ] φ [φ ]φ. (22A.3) R cl A cl − A cl − 2 Aφφ cl The subscript φ denotes functional differentiation and integration symbols. Adjacent space-time variables have been omitted, for brevity. We have renamed δφ(x) by φ(x). Note that by construction, is at least cubic in φ. Thus the path integral (22A.3) may be considered as the generating functionalR Z of a theory of a φ field, with a propagator

[Φ ]= i¯h [φ ] −1, G cl {Aφφ cl } and an interaction that depends in a complicated manner on j via φcl. We know from previous R 2 2 sections, and will immediately verify this explicitly, that ¯h W2 is indeed of order ¯h . Let us again set

φ = φcl +¯hφ1. (22A.4)

We shall express W [j] in the form

W [j]= [φ ]+ φ j +¯h∆ [φ ] (22A.5) A cl cl 1 cl and collect one- and two-loop corrections in the term i ∆ [φ ]= Tr log i¯h −1 +¯hW [φ ]. (22A.6) 1 cl 2 G 2 cl 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 dependence is due to φcl[j]. We may use this fact by expressing j as a function of φcl and by writing

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

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

1 Γ[Φ] = [φ] ¯h [Φ]φ ¯hφ j[Φ] + ¯h2φ j [Φ]φ + ¯h2φ i −1[Φ]φ A − AΦ 1 − 1 1 Φ 1 2 1 G 1 +¯h∆ [Φ] ¯h2∆ [Φ]φ + (¯h3) (22A.8) 1 − 1Φ 1 O 1 = [Φ] + ¯h∆ [Φ] + ¯h2 φ i −1[φ]φ + φ j [Φ]φ ∆ φ . A 1 2 1 G 1 1 Φ 1 − 1Φ 1

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/δφ = [φ ]= i¯h −1[φ ], (22A.11) cl −Aφφ cl − G cl so that

φ = ∆ [φ ]i [φ ]. (22A.12) 1 1φcl cl G cl Inserting this into (22A.8), the ¯h2-term becomes

¯h2 φ i −1[Φ]φ + (¯h3). (22A.13) − 2 1 G 1 O

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

i δ [X]= Tr −1 . (22A.14) D 2 D δX D Hence the eﬀective action becomes, to this order in ¯h,

Γ[Φ] = [Φ] + ¯hΓ [Φ] + ¯h2Γ [Φ] A 1 2 i = [Φ] + Tr log i −1[Φ] + ¯h2W [Φ] A 2 G 2 ¯h2 i δ i δ Tr −1 i Tr −1 . (22A.15) − 2 2 G δΦG G 2 G δΦG We now calculate W [Φ] to lowest order in ¯h. The remainder in (22A.3) has the expansion 2 R 1 1 [Φ; φ]= φ3 [Φ] + φ4 [Φ], R 3! AΦΦΦ 4! AΦΦΦΦ where we have replaced φcl by Φ+ (¯h). In order to obtain W2, we have to calculate all connected vacuum diagrams for the interactionsO r with a φ-particle propagator ¯h [Φ](x, x′). Since every G 1276 22 Functional-Integral Calculation of Eﬀective Action. Loop Expansion contraction brings in a factor ¯h, we can truncate the expansion (22A.16) after Φ4. Thus the only contribution to W2[Φ] are the connected vacuum diagrams

(22A.16) .

Here a line stands for [Φ], a four-vertex for G [Φ] = i −1[Φ] , (22A.17) AΦΦΦΦ G ΦΦ and a three-vertex for

[Φ] = i −1[Φ] . (22A.18) AΦΦΦ G Φ Only the ﬁrst two graphs are one-particle irreducible. It is now a pleasant feature to realize that the third graph cancels with the last term in (22A.15). In order to verify this, we write the diagram more explicitly as

¯h2 ′ ′ ′ ′ ′ ′ (22A.19) − 8 GΦ1Φ2 AΦ1Φ2Φ3 GΦ3Φ3 AΦ3 Φ1 Φ2 GΦ1 Φ2 which is now part of iW2. Thus, only the one-particle irreducible vacuum graphs produce the ¯h2-correction to Γ[Φ]:

2 2 3 2 1 i¯h Γ [Φ] = i¯h + i¯h ′ ′ ′ , (22A.20) 2 4!G12AΦ1Φ2Φ3Φ4 G34 4!2 AΦ1Φ2Φ3 GΦ1Φ1 GΦ2 Φ2 GΦ3Φ3 AΦ1Φ2Φ3 whose graphical representation is

2 i¯h Γ2[Φ] = . (22A.21)

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

[Φ], AΦ1...Φn and sum up all connected one particle irreducible vacuum graphs 2

n i¯h Γn[Φ] = nX≥2

. (22A.22)

2Note that each line carries a factor ¯h from the propagator, the n-vertex contributes a factor ¯h1−n. Appendix 22B Eﬀective Action to All Orders in h¯ 1277

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

(22A.23) of lines which emit 1, 2, 3 ... lines via vertices φn in arbitrary combinations. The expansion to orderh ¯n is an expansion according to the number of loops. The general structure of the series (22A.22) becomes rapidly involved and must be organized via 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 the generating functional is

i i W [j] {A[φ0]+φ0j+¯hΓ1[φ0[j]]} Z[j]= e h¯ = e h¯ , (22B.1) N where

i ′ ′ iΓ1[φ0] ′ A φ0+φ −A[φ0]+j[φ0]φ e φ e h¯ [ ] . (22B.2) ≡ G { } Z

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

δW [j] δ [φ ] δΓ [φ ] δφ Φ= = φ + A 0 + j +¯h 1 0 0 . (22B.3) δj 0 δφ δφ δj 0 0

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

δ [φ0] δΓ1[φ0] A + j[φ0]+¯h =0. (22B.4) δφ0 δφ0

This requirement makes φ0 directly equal to Φ and

Γ[Φ] = [Φ] + ¯hΓ [Φ]. (22B.5) A 1 1278 22 Functional-Integral Calculation of Eﬀective Action. Loop Expansion

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

i ′ ′ ′ ′ iΓ1[φ0] ′ A[φ0+φ ]−A[φ0]−φ A [φ0]−h¯ φΓ1 [φ0]φ e = φ e h¯ φ0 φ0 , (22B.6) G { } Z which is a functional integro-diﬀerential equation. It is possible to extract from (22B.6) the desired result. Let us consider an auxiliary generating functional

i ′ 4 Aaux φ0,φ + d xKφ eiWaux[φ0,K] φ′e h¯ [ ] (22B.7) ≡ G Z R with an action

[φ , φ′]= [φ + φ′] [φ ] φ′ [φ ]. (22B.8) Aaux 0 A 0 − A 0 − Aφ0 0

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

= [φ ] (22B.9) G 0 and vertices

6 n

n δ [φ0] (22B.10) 1 5 = A . δφ0 ...δφ0

2 4 3

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 we show this? By comparing (22B.6) with (22B.2) we see that, for the special choice of the current

K K ¯hΓ [φ ], (22B.11) ≡ 0 − 1φ0 0 the auxiliary functional Waux[φ0,K0] becomes identical to the generating functional, that we are trying to calculate. The auxiliary functional also has an auxiliary Legendre transformed field δW [φ ,K] Φ = aux 0 . (22B.12) aux δK With the help of δW [φ ,K] Φ = aux 0 =Φ [φ ,K], (22B.13) aux δK aux 0 we now define an auxiliary effective action

Γ [Φ ] W [φ ,K] d4xKΦ , (22B.14) aux aux ≡ aux 0 − aux Z where Γaux [Φaux] is also a functional of φ0. There is one important property of Γaux [Φaux]. If evaluated at Φ = 0, it collects all one-particle irreducible vacuum graphs of [see (11.45), aux Aaux Notes and References 1279

(11.46)]. But these are exactly the graphs we want. Hence the proof can be completed by showing that the condition K = K = Γ [φ ] leads to 0 − 1φ0 0

Φaux =0. (22B.15)

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

φ′ = φ φ = φ Φ. (22B.16) − 0 − Now Φ φ implies φ′ = 0. But in the auxiliary theory we have ≡ h i h i Φ φ′ . (22B.17) aux ≡ h i

Thus Φaux = 0, and Waux[φ0,K] collects the one-particle irreducible vacuum graphs with propagators 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 to C. De Dominicis, Jour. Math. Phys. 3, 983 (1962). His treatment of collective phenomena was developed further by J.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 see picture 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 Rencontres Mathématiques 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 that book, 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/M 1/β) f˜(ξΦ2/(D−2+η)) like the function f˜(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). 1280 22 Functional-Integral Calculation of Effective Action. Loop Expansion

[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).