The Effective Action and Its Applications

The Eﬀective Action and its Applications

Tom Charnock based on the lectures of Pete Millington Contents

1 Introduction and Motivation 2

2 Prerequisites 7 2.1 Convex Sets and Convex Functions and Functionals ...... 9 2.2 Legendre and Fenchel Transforms ...... 10 2.3 Schwinger Function ...... 10

3 n-Particle Irreducible (nPI) Effective Action 12 3.1 1PI Effective Action ...... 12 3.2 Two-point Effective Action ...... 14 3.2.1 2PI Effective Action ...... 14

4 (Non)-Equilibrium Field Theory 17 4.1 Imaginary-time Formalism ...... 17 4.2 Schwinger-Keldysh Closed Time Path Formalism ...... 17 4.3 Transport Equation ...... 19

5 Symmetries 21 5.1 Thermal Corrections ...... 21 5.2 Ward-Takahashi Identities ...... 23 5.3 Symmetry Preservation ...... 24 5.4 First-order Phase-transition ...... 24

6 Functional Renormalisation Group 25 6.1 Wetterich Equation ...... 25 6.2 First-order Phase Transitions ...... 27

1 Chapter 1

Introduction and Motivation

ϑ + ∆ϑ

L ϑ x x + ∆x L x − 2 2

Figure 1.1: Classically vibrating string which is transversly displaced.

Start with a classically vibrating string with length L, tension T , and mass per unit length µ. If a segment of the string ∆x, between x and x + ∆x is displaced transversly from equilibrium then the net force of the displacement (for small displacements) is

Fx = T cos ϑ + ∆ϑ − T cos ϑ ≈ 0, (1.1) Fy = T sin ϑ + ∆ϑ − T sin ϑ ≈ T ∆ϑ. ∂2y From the variation in the angle ∆ϑ, it can be seen that F ≈ T ∆x. y ∂x2 As the mass of this segment is µ∆x and the acceleration in the y direction is the second derivative of y with respect to time t, then it can be seen that ∂2y ∂2y 0 = µ − T ∆x. (1.2) ∂t2 ∂x2 This is the wave equation 1 ∂2y ∂2y 0 = − , (1.3) v2 ∂t2 ∂x2 where v = pT/µ. This equation is not only invariant under translations in time and space 0 t → t = t + a, a ∈ R 0 (1.4) x → x = x + b, b ∈ R, but also under boosts (Lorentz transformations) ux t → t0 = γ t − v2 (1.5) x → x0 = γ x − ut ,

2 1 where γ = . p1 − u2/v2 ↑ 1,1 These two sets of symmetries are the isometries of the 1+1 dimensional Poincar´egroup P+(1, 1) = R × SO(1, 1). µ A metric ηµν = diag(1, −1), can be introduced, along with contravariant and covariant vectors, x ≡ (ut, x) and ν xµ ≡ ηµν x = (ut, −x). As well as Poincar´einvariance, this classical wave equation is also invariant under discrete spacetime symmetries

T : t → −t (1.6) P : x → −x, and charge conjugation C. Thus the equation is also invariant under CPT transformation. Due to the linearity and homogeneity of the wave equation it is further invariant under rescalings of y which can be seen by √ φ(t, x) ≡ µy(t, x), (1.7) where the dimensionality of φ(t, x) is [φ] = M 1/2L1/2. This allows the wave equation to be written as

φ(t, x) = 0, (1.8)

µ where ≡ ∂µ∂ is the 1+1 d’Alembertian.

The Hamiltonian for this system is

Z L/2 H(t) = dxH −L/2 (1.9) Z L/2 1 = dx Π2(t, x) + v2∇φ(t, x)∇φ(t, x), −L/2 2 where the Lagrangian L(t) and the Lagrangian density L (t, x) are

Z L/2 1 µ L(t) = dx ∂µφ(t, x)∂ φ(t, x) −L/2 2 Z L/2 (1.10) = dxL (t, x). −L/2

The action is

Z tf S[φ] = dtL(t) ti (1.11) Z tf Z L/2 = dt L (t, x), ti −L/2 with the ﬁrst functional variation

Z tf Z L/2 ∂L ∂L δS[φ] = dt dx δφ + δ∂µφ . (1.12) ti −L/2 ∂φ ∂∂µφ It is assumed that the function and partial variations commute and that the surface terms can be neglected after integrating by parts. Setting this functional variation to zero gives the Euler-Lagrange equations

Z tf Z L/2 ∂L ∂L 0 = dt dx − ∂µ δφ. (1.13) ti −L/2 ∂φ ∂∂µφ Boundary conditions can be chosen to preserve as many symmetries as possible. Such conditions include • φ(t, x) is periodic on [−L/2, L/2]. • φ(t, x) is constant on the line x = vt.

3 The wave equation is a separable second-order partial diﬀerential equation

2 1 ∂ τ(t) 2 2 2 = k v τ(t) ∂t (1.14) 1 ∂2χ(x) = k2, χ(x) ∂x2 with φ(t, x) = τ(t)χ(x). Here k is discrete, k ∈ {kn = 2πn/L|n ∈ Z}. φ(t, x) can be a function of only ωn − knx, with ωn = knv. It is written as

+∞ 1 X 1 φ(t, x) = A e−iωnteiknx + A∗ eiωnte−iknx, (1.15) L |2ω | n n n=−∞ n

∗ 1/2 3/2 −1 where An = A−n which have dimension [An] = M L T . Unlike in quantum ﬁeld theory, the normalisation can be chosen such that +∞ Z L/2 1 X dxφ2(t, x) = |A |2 L n (1.16) −L/2 n=−∞ < ∞,

2 for any ωn|An| which is a decreasing function of n. Substituting this into the Hamiltonian gives

+∞ 1 X E = |A |2. (1.17) L n n=−∞ φ(t, x) can be split into components

+∞ 1 X φ(t, x) = φ (t, x), (1.18) L n n=−∞ where the coeﬃcients are 1 i ˙ An = φn(0, 0) + φn(0, 0) . (1.19) 2 ωn The bandwidth theorem states that ∆t∆ω ≥ 2π as seen with substitution using Fourier transforms. Supposing that 2 2 An have strong support in some region (|An| ≈ |A¯| for n ∈ [nmin, nmax]) then 1 ∆E ≈ |A¯|2∆n L (1.20) |A¯|2 = ∆ω. 2πv

Since ωn = 2πnv/L then ∆ω = 2π∆nv/L. From the ∆E relation it is seen that |A¯|2 ∆E∆t ≥ , (1.21) v which is some number which we can name ~/2. Here ~ does not need to be related to the reduced Planck constant.

The coeﬃcients An, can be promoted to operatorsa ˆn, which act on a Hilbert space spanned by some complete set † of states {|ni, n ∈ N0}.a ˆn and its Hermitian conjugatea ˆn, satisfy † [ˆan, aˆm] = 2~ωnLδnm, (1.22)

(†) where n, m > 0. This commutator means that acting on states with thea ˆn operators give

4 √ √ This meansa ˆ†|ni = n + 1|n + 1i anda ˆ|ni = n|n − 1i. A conjugate operator T , can be introduced with the property that

(†) (†) −1 aˆ−n ≡ T aˆn T (†) (1.24) =a ˆn .

If the vacuum is T -symmetric (|0i = |0i∗ = |0iT ) then | − ni = |ni∗. A ﬁeld operator can also be introduced

+∞ 1 X 1 φˆ(t, x) = aˆ e−i|ωn|teiknx +a ˆ† ei|ωn|te−iknx, (1.25) L |2ω | n n n=−∞ n along with the conjugate momentum operator

+∞ 1 X 1 Π(ˆ t, x) = aˆ e−i|ωn|teiknx − aˆ† ei|ωn|te−iknx, (1.26) iL 2 n n n=−∞ where it can be seen that

[φˆ(t, x), φˆ(t, x0)] = 0 i (1.27) [φˆ(t, x), Π(ˆ t, x0)] = ~δ(x − x0). 2 The Hamiltonian operator for this system is

+∞ 1 X Hˆ = aˆ† aˆ (1.28) L n n n=−∞ which is normal ordered. The state vector is

+∞ 1 X 1 An |ψ(t)i = e−i|ωn|t|ni. (1.29) L |2ω | n=−∞ n ~

The expectation value for the energy is

E = hψ(t)|Hˆ |ψ(t)i +∞ 1 X (1.30) = |A |2. L n n=−∞

The state evolves with the Heisenberg equation of motion d i |ψ(t)i = − Hˆ |ψ(t)i (1.31) dt ~

0 0 0 −iHˆ (t0−t)/ so there is an evolution operator |ψ(t )i = U(t , t)|ψ(t)i where U(t, t ) = e ~. In the limit that L → ∞, ~kn becomes a momentum p, and ∞ Z ~ X dp → . (1.32) L 2π n=−∞

(†) (†) The operatorsa ˆn become true creation and annihilation operators a (p) following

[ˆa , aˆ† ] → [a(p), a†(p)] n n (1.33) = 2π2Eδ(p − p0) where E = |p|v and

Z dp 1 φˆ(t, x) → a(p)e−iE(p)taipx + a†(p)eiE(p)te−ipx. (1.34) 2π 2E(p)

5 This is the description of a 1+1 dimensional ﬁeld theory.

Small anharmonic terms, such as

λφ4 V (φ) = , (1.35) 4! can be added. These terms will begin to dominate the evolution operator U(t0, t) and loop integrals need to be calculated. These depend on the two-point correlation function of the ﬂuctuations

0 ˆ ˆ 0 0 i∆F (x, x ) = h0|T (φ(x0, x)φ(x0, x )|0i +∞ ~ X 1 0 −i|ω |(x −x0 ) ik (x−x0) 0 (1.36) = Θ(x − x )e n 0 0 e n + (x → x ). L |2ω | 0 0 n=−∞ n

Whenever there is a system with ﬂuctuations the “classical” equations of motion (those in the absence of ﬂuctuations) are not enough to understand the full dynamics.

A technology is needed to take into account the backreation from the fluctuations. This technology is the effective action which systematically organises pertubation theory in the scale of the fluctuations. As an example, for −βH stochastic fluctuations of scale β ≡ kBT , the Gibbs distribution is e which is expanded in orders of β.A well-defined variational procedure can be derived from which the full evolution equations from the system can be obtained. These are an infinite set of couple differential equations for all n-point Green’s functions.

6 Chapter 2

Prerequisites

To account for ﬂuctuations in the action, a source can be added to the Lagrangian density

L(t, x) ⊃ J(t, x)φ(t, x). (2.1)

For the wave equation of Chapter 1 this is equivalent to J(t, x) = φ(t, x) which is proportional to ~. The form of this source can be found in quantum ﬁeld theory using the eﬀective action.

The standard massive λφ4 theory in a 4-dimensional Minkowski space has the Lagrangian density 1 L(x) = ∂ Φ(x)∂µΦ(x) − V (Φ(x)) + J(x)Φ(x), (2.2) 2 µ with 1 λ2 V (Φ(x)) = w2Φ2(x) + Φ4(x), (2.3) 2 4! where m2, λ > 0. Correlations functions can be obtained from the generating functional. This generating functional is the vacuum persistence amplitude

Z[j] = hO(+∞)|O(−∞)ij, (2.4) where j(x) is a test source, diﬀerent to the physical external source J(x). The physical limit does not correspond to J(x) → 0, only to j(x) → 0. Working in the Heisenberg picture a state |Φt(¯x), ti, can be introduced. It is an eigenstate of the ﬁeld operator Φ(ˆ t, x) at the time t with eigenvalue Φt(¯x)

Φ(ˆ t, x¯)|Φt(¯x), ti = Φt(¯x)|Φt(¯x), ti. (2.5)

Dividing [−∞, +∞] into N subintervals of size ∆t the generating functional becomes

N Y Z Z[j] = hO(+∞)| [dΦt` (¯x)]|Φt` (¯x), t`ihΦt` (¯x), t`||O(−∞)i (2.6) `=0 using the identity Z I = [dφt(¯x)]|Φt(¯x), tihΦt(¯x, t|. (2.7)

7 The ﬁrst term is the functional δ function Z Z i 3 δ(Φt`+1 (¯x) − Φt` (¯x)) = N [dΠt` (¯x)] exp d x¯Πt` (Φt`+1 (¯x) − Φt` (¯x)) (2.10) ~ where N is an inﬁnite normalisation. The Hamiltonian operator for this theory is

Z 1 1 Hˆ (t ) = d3x¯ Πˆ 2(t ) + (∇¯ Φ(ˆ t ))2 + V (Φ(t )) − [(J(t ) − j(t ))]Φ(ˆ t ) (2.11) ` 2 ` 2 ` ` ` ` ` wherex ¯ has been dropped for convenience. By inserting a complete set of eigenstates of Πˆ we find ˆ Z i Z −iH(t`)∆t/~ 3 hΦt` , 0|e |Φt` , 0i = N [dΠt` ] exp d x¯Πt` (Φt`+1 − Φt` ) ~ i∆t 1 ¯ 2 1 2 × − V (Φt` ) − [J(t`) + j(t`)]Φt` + (∇Φt` ) + Πt + ··· . ~ 2 2 ` Re-exponentiating gives Z Z i 3 hΦt`+1 , t`+1|Φt` , tì = N [dΠt` ] exp d x¯ Πt` (Φt`+1 − Φt` ) − H(t`)∆t . (2.12) ~ This integral is ill-defined, so Wick rotating to Euclidean space and using analytic continuation, t = iτ gives

Πt` = −iΠτ` and Z Z 1 3 hΦτ`+1 , τ`+1|Φτ` , τ`i = NE [dΠτ` ] exp d x¯ Πτ` (Φτ`+1 − Φτ` ) − H(τ`)∆τ . (2.13) ~ Performing this Gaussian integral gives Z 2 ∆τ 3 1 (Φτ`+1 − Φτ` ) hΦτ`+1 , τ`+1|Φτ` , τ`i = NE exp − d x¯ ~ 2 ∆τ (2.14) 1 + (∇¯ Φ )2 + V (Φ ) − J(τ ) + j(τ ) Φ 2 τ` τ` ` ` τ` so the generating functional is

Z ∆τ Z 1(Φ − Φ )2 Z[j] = DΦ exp − d3x¯ τ`+1 τ` ~ 2 ∆τ (2.15) 1 + (∇¯ Φ )2 + V (Φ ) − J(τ ) + j(τ ) Φ , 2 τ` τ` ` ` τ` where Z N 0 Y DΦ = NE [dΦτ` ] (2.16) `=0

is subject to the boundary condition Φ0 and ΦτN corresponding to the vacuum states. In the limit N → ∞, ∆τ → 0 2 2 then (Φτ`+1 − Φτ` )/∆τ → ∂τ Φ(τ`) giving the generating functional Z Z 1 4 Z[j] = DΦ exp − SE[Φ] − d x J(x) + j(x) Φ(x) , (2.17) ~ where x ≡ xµ = (¯x, x4) and SE is the Euclidean action

Z 1 S [Φ] = d4x (∂ Φ(x)∂µΦ(x) + V Φ(x) . (2.18) E 2 µ

8 f(x) −f(x)

x x

convex concave

Figure 2.1: Convex and concave functions

From this, all n-point disconnected Green’s functions can be generated

( ) ( ) ( ) (n) ˆ ˆ ˆ Gdisc(x1, x2, ··· , xn) = h Φ(x1) Φ(x2) ··· Φ(xn)i (2.19) n Y δ = ~ Z[j] (2.20) δj(x`) `=1 j=0 Z Z 1 4 = DΦ(x1)Φ(x2) ··· Φ(xn) exp − SE[Φ] − d x J(x) + j(x) Φ(x) . (2.21) ~ j=0 The functional derivative is deﬁned such that δ Z Z d4yf(y) = d4yδ4(x − y) δf(x) (2.22) = 1.

The Schwinger function W [j] = −~ ln Z[j], generates the connected Green’s functions n (n) 1 Y δ G (x1, x2, ··· , xn) = − ~ W [j] . (2.23) δj(x`) ~ `=1 j=0 For n = 2 we ﬁnd that the Green’s function is equivalent to the variance

2 (2) δ W [j] G (x1, x2) = −~ δj(x1)δj(x2) j=0 (2.24) 1 = − hΦ(x1)Φ(x2)i − hΦ(x1)ihΦ(x2)i . ~ 2.1 Convex Sets and Convex Functions and Functionals

Let I = (a, b) = {x ∈ R|a < x < b} be an open interval on the real line and f : I → R be a real-valued function. If the line segment passing between any two points on the graph of f lies above or on the graph, then f is said to be convex. The additive inverse of f (−f) is said to be concave. These can be seen in Fig. 2.1. If f is twice differentiable then f is convex iff ∀x ∈ I : f 00(x) > 0. If f is a function of more than one variable and twice differentiable in all variables then f is convex iff Hessf ≡ ∇∇f(¯x) > 0. Since the second derivative of convex or concave functions do not change sign, their first derivatives are monotonic, single-valued and invertible. The function can be specified by (a) (x, f(x)) or (b) the envelope of its tangents. This can be seen in Fig. 2.2

9 f(x)

(a)

(b)

Figure 2.2: Function specﬁed by (x, f(x)) and by the envelope of its tangents.

2.2 Legendre and Fenchel Transforms

Let ω(x) = x∗x where x∗ ∈ R. ∗ is the convex conjugate. If f(x) is convex then ω(x) − f(x) will have a maximum, or conversely f(x) − ω(x) will have a minimum. A family of functions can be deﬁned

Max x ∈ I {x∗x − f(x)}, f is convex ∗f(x∗) ≡ Min x ∈ I {x∗x − f(x)}, f is concave (2.25) Min x ∈ I {x∗x − f(x)}, f is convex f ∗(x∗) ≡ Max x ∈ I {x∗x − f(x)}, f is concave

Note that f ∗(x∗) = −∗f(x∗). This means that x∗ = f 0(x). ω(x) can be shifted to t(x) = ω(x) − f ∗(x∗) or equivalently t(x) = ω(x) +∗ f(x∗). The function t(x) will intersect f at the point x and since t0(x) = f 0(x) then t is the tangent to f at the point x with intercept ∗f(x∗). This is the Legendre transform.

2.3 Schwinger Function

Returning to the Schwinger function, W [j] ≡ W [J]. This means

2 δ W [J] 1 = − hΦ(x1)Φ(x2)i − hΦ(x1)ihΦ(x2)i . (2.26) δJ(x1)δJ(x2) ~ Since the right-hand side is a variance, and thus positive, then the left-hand side must be negative. This means that W [J] is a concave functional J(x) ≡ Jx. The eﬀective action can now be deﬁned as

Γ[φ] ≡ Max Γj[φ] J (2.27) = Max W [J] + Jxφx J Z 4 where the DeWitt notation Jxφx ≡ d J(x)φ(x) can be used. An external source J is deﬁned as

δΓJ [φ] = 0 (2.28) δJx J=J leading to Γ[φ] = W [J ] − Jxφx. Jx ≡ Jx[φ] is a functional of φ and

φx = hΦxi δW [J ] (2.29) = − . δJx It follows that δΓ[φ] = Jx[φ]. (2.30) δφx

10 This gives the quantum equation of motion. In the usual approach to quantum mechanics, the quantum equation of motion is obtained by setting Jx[φ] = 0. There is no reason for this to be true, it amounts to setting the momenta to zero.

11 Chapter 3 n-Particle Irreducible (nPI) Eﬀective Action

3.1 1PI Eﬀective Action

The one-particle irreducible (1PI) eﬀective action as deﬁned in Sec. 2.3 is

Γ[φ] = W [J ] + Jx[φ]φx, (3.1) where W [J ] = −~ ln Z[J ] is the Schwinger function, Z 1 Z[J ] = DΦ exp − S[Φ] − J [φ]Φx (3.2) ~ is the generating functional with

φx = hΦxi δW [J ] (3.3) = − δJx and δΓ[φ] = Jx[φ]. (3.4) δφx

1/2 ˜ If the ﬂuctuations are pulled out of the ﬁeld, for power counting, then it can be written and Φx = ϕx + ~ Φ and the action expanded as

2 1/2 δS[Φ] ~ δ S[Φ] ˜ 3/2 S[Φ] = S[ϕ] + ~ + Φy + O(~ ). (3.5) δΦx Φ=ϕ 2! δΦxδΦy Φ=ϕ In order to eliminate the linear term a stationarity condition is imposed,

δS[Φ] 0 = − Jx[φ]. (3.6) δΦ Φ=ϕ The generating functional becomes Z 1 ˜ 1 ˜ −1 ˜ 1/2 Z[J ] = exp − S[ϕ] − Jx[φ]ϕx DΦ exp − ΦxGxy (ϕ)Φy + O(~ ) (3.7) ~ 2 where 2 −1 δ S[Φ] Gxy (ϕ) = δΦxδΦy Φ=ϕ (3.8) 4 2 00 = δxy − ∂x + V (ϕx) .

12 The Gaussian integral yields −1 −1/2 1 Z[J ] = det G ∗ G0 exp − S[ϕ] − Jx[φ]ϕx (3.9) ~

−1 with G0 ≡ G (ϕ = v) where v is the vacuum expectation value. The eﬀective action is

Γ[φ] = S[ϕ] + ~ ln det G−1(ϕ) ∗ G + J [φ](φ − ϕ ) + O( 2). (3.10) 2 0 x x x ~

Supposing that ϕx is both the saddle point of the generating functional and the extremum of the eﬀective action

δΓ[φ] 0 = , (3.11) δφx φ=ϕ then the stationarity condition means that

δΓ[φ] δS[φ] = − Jx[φ] δφx φ=ϕ δφx φ=ϕ (3.12) = 0.

This deﬁnes the source and is called the consistency relation. Since φx = hΦxi, a relation to ϕx can be found

Z 3/2 1 ˜ 1 1 ˜ −1 ˜ ~ λ ˜ 3 2 φz = DΦΦz exp − S[ϕ] − Jx[φ]ϕx + ΦxGxy (ϕ)Φy + ϕxΦx + O(~ ) (3.13) Z[J ] ~ 2 3! Z 1/2λ 1 = ϕ + 1/2 DΦ˜Φ˜ 1 − ~ ϕ Φ3 exp − Φ˜ G−1(ϕ)Φ˜ + O( 2) (3.14) z ~ z 3! w w 2 x xy y ~

= ϕz + ~δϕz (3.15) with λ δϕ = −~ ϕ G (ϕ)G (ϕ) ~ z 2 w zw ww −λ (3.16) = ϕ G

~ It follows that 2 δΓ[φ] δS[ϕ] δϕy ~ δ −1 δϕy = + ln det G (ϕ) ∗ G0 + ~Jx[φ]δϕx δφx δϕy δφx 2 δϕy δφx (3.17) δS[ϕ] ~ δ −1 2 = + ln det G (ϕ) ∗ G0 + O(~ ). δϕy 2 δϕy The consistency relations gives

2 ~ δ −1 Jx[φ] ≡ ln det G (ϕ) ∗ G0 2 δϕy φ=ϕ λ = −~ ϕ G (ϕ) (3.18) 2 x xx =

This leaves the quantum equation of motion λ 0 = −∂2ϕ + V 0(ϕ ) + ~ ϕ G (ϕ). (3.19) x x x 2 x xx

13 Self consistency of this entire procedure requires

δΓ[φ] = Jx[φ] δφx φ=ϕ φ=ϕ (3.20) = Jx[ϕ] =! 0.

It can be shown that this is indeed the case by expanding J [φ] around ϕx,

δJ [φ] J [φ] = Jx[ϕ] + ~ δϕy δφy φ=ϕ (3.21) λ = −~ ϕ G (ϕ), 2 x xx since

2 δJx[φ] δ Γ[φ] = δφy φ=ϕ δφxδφy φ=ϕ 2 δ S[φ] (3.22) = + O(~) δφxδφy φ=ϕ −1 = Gxy (ϕ). This means

−1 Jx[φ] = Jx[ϕ] + ~Gxy (ϕ)δϕy λ (3.23) = J [ϕ] − ~ ϕ G (ϕ) + O( 2). x 2 x xx ~

! Thus Jx[ϕ] = 0..

3.2 Two-point Eﬀective Action

The two point function is deﬁned via

−1 4 2 00 Gxy (ϕ) = δxy − ∂x + V (ϕx) . (3.24)

This does not have any quantum corrections, i.e. it doesn’t see quantum ﬂuctuations. In order to capture these a bi-local source Kxy ≡ Kxy[φ, ∆], can be included. Performing a double Legendre transform gives the two particle irreducible (2PI) eﬀective action.

3.2.1 2PI Eﬀective Action The 2PI eﬀective action is

Γ[φ, ∆] ≡ Max ΓJ,K [φ, ∆] J,K (3.25) = Max W [J, K] + Jxφx + Kxy[φxφy + ~∆xy] . J,K

The extremisation procedure deﬁnes the extremal sources

δΓJ,K [φ, ∆] 0 = (3.26) δJx J=J ,K=K and

δΓJ,K [φ, ∆] 0 = . (3.27) δKx J=J ,K=K

14 The unphysical one- and two-point functions φx and ∆xy are given by δW [J , K] φx = − , (3.28) δJx and 2 δW [J , K] 1 ∆xy = − − φxφy. (3.29) ~ δKxy ~ Finally δΓ[φ, ∆] = Jx[φ, ∆] + Kxy[φ, ∆]φy (3.30) δφx gives the equations of motions for ϕx and δΓ[φ, ∆] ~ = Kxy[φ, ∆] (3.31) δ∆xy 2 gives the equations of motions for some two-point function Gxy. In this case, the stationarity condition becomes

δS[Φ] 0 = − Jx[φ, ∆] − Kxy[φ, ∆]ϕy (3.32) δΦx Φ=ϕ because 1 1 Z[J , K] = exp − S[ϕ] − J [φ, ∆]ϕ − ϕ K [φ, ∆]ϕ x x 2 x xy y ~ (3.33) Z 1 1/2λ λ × DΦ˜ exp − Φ˜G−1[φ, ∆]Φ˜ − ~ ϕ Φ˜ 3 − ~ Φ˜ 4 2 xy y 3! x x 4! x

−1 −1 where Gxy [φ, ∆] = Gxy (ϕ) − Kxy[φ, ∆]. The eﬀective action can be expanded as

2 2 Γ[φ, ∆] = Γ0[ϕ] + ~Γ1[ϕ, G] + ~ Γ2[ϕ, G] + ~ Γ1PR[ϕ, G] 1 (3.34) + J [φ, ∆](φ − ϕ) + K [φ, ∆][φ φ − ϕ ϕ + (∆ − G) ] x x 2 xy x y x y ~ xy where Γ0[ϕ] = S[ϕ] and 1 Γ [ϕ, G] = Tr[ln G−1 ∗ G + G−1 ∗ G − 1] 1 2 0

= (3.35)

= + + + ··· which is the 1PI action when K = 0. The other terms are

1 1 2Γ [ϕ, G] = − − ~ 2 8 12 (3.36)

2 ~ Γ1PR[ϕ, G] = .

Writing everything in terms of the physical one- and two-point functions ϕ and G, eliminates φ and ∆ so that the action is

2 Γ[φ, G] = Γ0[ϕ] + ~Γ1[ϕ, G] + ~ Γ2[ϕ, G]. (3.37)

15 To recover the standard eﬀective action ϕ and G become the extremum of the eﬀective action

δΓ[φ, ∆] 0 = (3.38) δφx ϕ,G and

δΓ[φ, ∆] 0 = . (3.39) δ∆xy ϕ,G Following the same procedure of matching these to the consistency relation and the Schwinger-Dyson equation it can be shown that

δΓ2[φ, ∆] Kxy[φ, ∆] = −2~ δ∆xy ϕ,G

(3.40)

= − − .

Moreover it remains the case that

δΓ[φ, ∆] = Jx[φ, ∆] + Kxy[φ, ∆]φy δφx ϕ,G ϕ,G (3.41) = 0. and

δΓ[φ, ∆] ~ = Kxy[φ, ∆] δ∆xy ϕ,G 2 ϕ,G (3.42) = 0 and the procedure is self consistent.

The Heisenberg equation of motion is

dΦ(ˆ t, x¯) i = Hˆ (t), Φ(ˆ t, x¯). (3.43) dt ~ This has the same dynamic content as a classical equation of motion derived from a source dependent Hamiltonian Z 1 Z Z H (t) = H(t)− d3x¯J (t, x¯)Φ(t, x¯)− dt0 d2x¯0Φ(t, x¯)K(t, x,¯ t0, x¯0)Φ(t0, x¯0). (3.44) J ,K 2 The Heisenberg equation of motion can be transformed into an inegral equation

i Z t Φ(ˆ t, x¯) = Φ(0ˆ , x¯) + dt0Hˆ (t), Φ(ˆ t0, x¯) (3.45) ~ 0 Iterating once gives

i Z t Φ(ˆ t, x¯) = Φ(0ˆ , x¯) + dx0Hˆ (t0), Φ(0ˆ , xˆ) − ~ 0 0 (3.46) Z t Z t 1 0 00 ˆ 0 ˆ 00 ˆ 00 2 dt dt H(t ), H(t ), Φ(t , x¯) ~ 0 0 and taking the derivative shows

dΦ(ˆ t, x¯) i 1 Z t ˆ ˆ 0 ˆ ˆ 0 ˆ 0 (3.47) = H(t), Φ(0, x¯) − 2 dt H(t), H(t ), Φ(t , x¯ . dt ~ ~ 0

16 Chapter 4

(Non)-Equilibrium Field Theory

4.1 Imaginary-time Formalism

Starting from vacuum expectation values in particular in-out overlaps of states, hO(+∞)|O(−∞)i, ensemble ex- 1 pectation values hOiˆ = Trˆ%Oˆ are wanted, where Z = Trˆ% is the partition function. Supposing that% ˆ describes a Z pure state in the Schroödingerpicture %ˆ(t) = |ψ(t)ihψ(t)| (4.1) which gives 1 Oˆ = hψ(t)|O|ˆ ψ(t)i. (4.2) Z Proceeding to insert complete eigensets of the eigenstates of the Heisenberg-picture field operator from Chapter 2, a closed-time path C = C+ ∪C−, can be built. Analytic continuation allows for the fields to be defined on the positive and negative branches, which are related by T -conjugation, Φ+(x) ≡ Φ+(x0 + iε) and Φ−(x) ≡− (x) ≡ (x0 − iε). In Minkowski spacetime, the “in-in” or Schwinger-Keldysh closed-time path formalism is found.

4.2 Schwinger-Keldysh Closed Time Path Formalism

The partition function in this formalism can be written Z + − i + ˜ ˜ i − ˜ ˜ − ˜ ˜ + Z[%] = DΦ DΦ exp S[Φ , t, ti] exp − S[Φ , t, ii] hΦx |%ˆ(t, ti|Φx i. (4.3) ~ ~ 0 0 A convenient notation to use is a + − S[Φ , t,˜ t˜i] = S[Φ , t,˜ t˜i] − S[Φ , t,˜ t˜i] Z 4 1 a µ b 1 2 a b λ a b c d (4.4) = d x ηab∂µΦx∂ Φx − ηabm ΦxΦx − ηabcdΦxΦxΦxΦx Ωt 2 2 4! where ηab = diag(+1, −1) is a SO(1,1) “metric” and ηabcd = 1 if a = b = c = d = + ≡ 1 and ηabcd = −1 if a = b = c = d = − ≡ 2 and zero otherwise. The ﬁeld can now be written as a doublet Φ+ Φa = (4.5) Φ− where Φ = η Φb = (Φ+, −Φ−). To hide any ignorance in the form of hΦ− |%ˆ|Φ+ i, it is expanded as a ab x0 x0 hΦ− |%ˆ |Φ+ i = exp iK[Φa] x0 x0 x0 1 1 (4.6) = KaΦ + Kab Φ Φ + Kabc Φ Φ Φ . x a,x 2 xy a,x b,y 3! xyz a,x b,y c,z

Figure 4.1:

17 These are poly-local sources. The Schwinger-Dyson equation contains both the quantum and thermal ﬂuctuations

−1 −1 [Gxy ]ab = [Gxy ]ab − [Kxy]ab (4.7)

−1 4 2 2 where [Gxy ]ab = δ (−x − m )ηab. The free propagators are

(0) −1 −1 thermal [Gxy ]ab = [Gxy ]ab − [Kxy ]ab (4.8) which contains only the thermal ﬂuctuations. It is useful to work in momentum space and look for solutions of the form p2 − m2 + iε 0 1 = G(0)(p). (4.9) 0 −p2 + m2 + iε

Inverting na¨ıvely we ﬁnd  1  0 (0) p2 − m2 + iε  G (p) =  −1  . (4.10) 0 p2 − m2 − iε The ﬁrst element refers to the Feynman propagator and the last to the Dyson propagator. Terms proportional to an on-shell δ function δ(p2 − m2), can be added   1 2 2 2 2 +c ˜1(p)δ(p − m )c ˜3(p)δ(p − m ) (0) p2 − m2 + iε  G (p) =  −1  . (4.11) c˜ (p)δ(p2 − m2) +c ˜ (p)δ(p2 − m2) 2 p2 − m2 − iε 4

The unknown functionsc ˜i(p) ≡ Θ(p0)˜ci(p)+Θ(−p0)˜ci(p) are constrained by ﬁeld theoretic requirements, including:

• CPT invariance:c ˜1(4)(p) =c ˜1(4)(−p) andc ˜2(p) =c ˜3(−p).

∗ • Hermiticity:c ˜4(p) = −c˜1(p) andc ˜2(p) = −c˜3(−p). ˜ ˜ ˜ • Causality:c ˜1(4) ∈ Im. c˜i(p) ≡ (p) ≡ −2πfi(p) can be defined, where f2(p) − f3(p) = sgn(p0). ˜ ˜ ˜ ˜ • Unitarity: f2(p) + f3(p) = 1 + f1(p) + f4(p). These seven constraints can be applied to the eight functions so that (0) 1 1 G(0)(p) = G¯ (p) − 2πif(p) δ(p2 − m2) 1 1 " (0) (0) # (4.12) GF (p) G< (p) ≡ (0) (0) G> (p) GD (p) where   1 2 2 −πiΘ(−p0)δ(p − m ) ¯(0)  p2 − m2 + iε  G (p) =  −1  (4.13) −πiΘ(+p )δ(p2 − m2) 0 p2 − m2 − iε f(p) corresponds to the normal ordered product of fields 1 haˆ†(¯p, 0, t˜ )â(¯p0, 0, t˜ )i ≡ Trˆ%(t,˜ t˜ )â†(¯p, 0, t,˜ t˜ )â(¯p0, 0, t˜ ) i i t Z i i i = (2π)32E(¯p)f(|p¯|, t)δ3(¯p0 − p¯) and so f(|p¯|, t) can be understood as the statistical distribution function. It can be shown that

1 −1 K (p, t,˜ t˜ ) = 2iεf(|p¯|, t) . (4.14) ab i −1 1

18 The solution to the Schwinger-Dyson equation is

(0)ab (0)ab (0)ac (0)db 2 G (p) = G¯ (p) + G¯ (p)Kcd(p)G¯ (p) + O(ε ) 1 1 ⊃ 2iεf p2 − m2 − iε p2 − m2 − iε ε (4.15) ∝ (p2 − m2)2 + ε2 = πδ(p2 − m2)

For the interacting case the Schwinger-Dyson equation is

−1 −1 Gab (x, y) = [G (x, y)]ab − Kab(x, y) + Πab(x, y) (4.16)

2 where Kab is the quantum part. The self-energy Πab(x, y) goes as T so the corrections are large at ﬁnite tempera- tures. It has a matrix structure of the form Π(x, y) −Π<(x, y) Πab(x, y) ≡ ∗ . (4.17) −Π>(x, y) −Π (x, y)

Partially inverting the Schwinger-Dyson equation by convolution from the right by the ﬁll propagator G is

−1 cb b 4 ab Gac (x, z) ∗ G (z, y) = δa δ (x − y) − Πac(x, y) ∗ G (z, y),

−1 4 2 2 where Gac (x, z) = δxy[−x − m ]ηac so that

2 2 b b 4 cb [−x − m ]Ga (x, y) = δa δ (x − y) − Πac(x, y) ∗ G (z, y) which is self-consistent for G. For simplicity the dispersile and oﬀ-shell corrections can be written

2 2 1 [−x − m ]G<(x, y) = − Π>(x, z) ∗ G<(z, y) − Π<(x, z) ∗ G>(z, y) . > 2 The left-hand side describes thet drift terms whilst the ﬁrst and last terms of the right-hand side describe the loss and gain respectively.

4.3 Transport Equation

In order to extract information about the evolution of f(|P¯|, t) the Wigner representation can be used

µ µ µ Rxy = x − y (4.18) and xµ + yµ Xµ = . (4.19) xy 2 The Wigner transform is Z G(q, X) = d4Reiq·RG(x, y). (4.20)

1 Using this 2 = 2 + ∂ · ∂ + 2 so that x R R X 4X 1 1 Z 2 2 2 4 iq·Rxy [q + iq · ∂X + X − m ]iG<(q, X) = − d Re Π>(x, y) ∗ iG<(z, y) − Π<(x, z) ∗ iG(z, y) . 4 > 2

∗ Taking advantage of the fact that G>(q, X) = −G (q, X) and extracting the anti-Hermitian part gives Z 4 iq·Rxy q · ∂X0 iGz(q, X0) = Re d Re iΠ>(x, z) ∗ iG<(z, y) − iΠ < (x, z) ∗ iG>(z, y) . (4.21)

19 The distribution function is therefore given by Z dq0 f(|q¯|, t) = lim 2 Θ(q0)q0iG<(q, X0), (4.22) X0→t 2π so Z dq Z 0 4 iq·Rxy ∂tf(|q¯|, t) = Re lim Θ(q0) d Re iΠ>(x, z) ∗ iG<(z, y) − iΠ < X0→t 2π (4.23) (x, z) ∗ iG>(z, y) .

Working in the interaction picture, the free propagators found earlier can be used showing

Z dq ∂ f(|q¯|, t) = 0 Θ(q )iΠ (q) ∗ iG(0)(q, t) − iΠ < (q) ∗ iG(0)(q, t) . (4.24) t 2π 0 > < >

(0) 2 2 where iG< (q, t) = 2πδ(q − m ) Θ(−q0) + f(|q,¯ t) so that

∂tf(|q¯|, t) = −Γ>f(|q¯|, t) + Γ(1 + f(|q¯|, t), (4.25) with

iΠ< > Γ< = − . (4.26) > 2E(p)

2 If a coupling of gφχ is adden then Γ>,< ∼ g and expanding around equilibrium gives 1 f(|q¯|, t) = − δf(|q¯|, t) (4.27) eβE(¯q) − 1 so that

∂tδf(|q¯|, t) = −(Γ> − Γ<)δf(|q¯|, t). (4.28)

This means

δf(|q¯|, t) = δf(|q¯|, 0)e−(Γ>−Γ<)t. (4.29)

This is just the Boltzmann equation.

20 Chapter 5

Symmetries

5.1 Thermal Corrections

−βHˆ −1 For a canonical ensemble in thermodynamic equilibrium% ˆ = e where β = (kBT ) . Treating% ˆ as an equilibrium evolution operator shows

0 −1 0 G>(t, t ) = Trˆ%Φ(ˆ t)(ˆ% %ˆ)Φ(ˆ t ) 0 = TrΦ(ˆ t + iβ~)ˆ%Φ(t ) 0 (5.1) = Trˆ%Φ(ˆ t )Φ(ˆ t + iβ~) = G<(t + iβ~, t). The partition function can be recast in the path-integral language as

Z Z ~β Z 1 3 1 2 Z(β) = DΦ exp − dτ d x¯ ∂µΦ(x) + V (Φ) . (5.2) ~ 0 2 The imaginary-time correlation function is

+∞ 1 X Z d3p¯ i i ( c)2 hΦ(τ)Φ(τ 0)i = exp − ω (τ − τ 0) exp p¯· (¯x − x¯0) ~ . (5.3) β (2π )3 n ω2 + E2(¯p) n=−∞ ~ ~ ~ n

The ωn = 2πn/β are the “Matsubara” frequencies. The Helmholtz free energy is 1 F = − ln Z(β). (5.4) β The 2PI “thermodynamic” eﬀective action is just 1 F = Tr ln G−1 + G−1 ∗ G − 1 − F (5.5) 2 0 where F0 is the zero-temperature free energy. The Schwinger-Dyson equation is

−1 −1 G (ωn, p¯) = G (ωn, p¯) + Π 2 2 (5.6) = ωn + E + Π.

If the potential is V (Φ) = m2Φ2 + λΦ4/4! the self energy (ignoring factors of ~ and c) is

Π = G (5.7) +∞ λ X Z d3k¯ 1 = . 2β (2π)3 ω2 + E2(k¯) + P i n=−∞ n

21 Using the Schwinger-Dyson equation, the free energy becomes 1 F = Tr ln G−1 − Π ∗ G − F . (5.8) 2 0 The leading correction to the mass is

+∞ λ X Z d2k¯ 1 Π = 2β (2π)3 ω2 + E2 n=−∞ n +∞ (5.9) λ β X Z |k¯|2 = d|k¯| . 8π3 2π n2 + (βE/(2π))2 n=−∞ The Matsubara sum gives

+∞ X 1 2π2 βE = coth n2 + (βE/(2π))2 βE 2 n=−∞ 2π2 2 = 1 + . βE eβE − 1 The last term isthe Boltzmann distribution. The thermal part of the self energy is λ Z |k¯|2 1 Π = d|k¯| 4π2 E eβE − 1 λT 2 Z ∞ x (5.10) ' 2 dx x 4π 0 e − 1 λT 2 = . 24 The leading (in λT 2) contribution to the partition function comes from approximating the free energy as 1 F (1) = Tr ln G−1 2 +∞ V X Z d3k¯ (5.11) = ln(ω2 + E2) 2β (2π )3 n n=−∞ ~ which has units of energy [F (1)] = EL3E3L−3T 3E−3T −3 = E. The Matsubara sum can be done by ﬁrst diﬀeren- tiating with respect to E to get

+∞ +∞ X X 2E ∂ ln(ω2 + E2) = E n ω2 + E2 n=−∞ n=−∞ n βE = β coth , 2 and then integrating with respect to E, giving

+∞ X 2 2 −βE ln(ωn + E ) = 2 ln(1 − e ) + βE, (5.12) n=−∞ where the last term can be dropped since the zero temperature part is being subtracted. The leading correction to the free energy is therefore Z 3¯ (1) V d k −βE F = 3 ln(1 − e ) β (2π~) (5.13) π2 (k T )4 ' − V B . 90 (~c)3 The pressure can then be obtained. π2 (k T )4 P = B . (5.14) 90 (~c)3

22 The − correction can now be calculated. This is simple since this diagram is two free propagators and one factor of −λ. +∞ 2 λ V X Z d3k¯ ( c)2 F (1) ' ~ 8 c β (2π )3 ω2 + E2 ~ n=−∞ ~ n (5.15) λ (k T )2 2 = V B . 2(~c)3 24 The second correction to the pressure can now be included. π2 (k T )4 15 λ P = B 1 − + ··· . (5.16) 90 (~c)3 8 24π2 The ﬁnal correction comes from ln G−1 = ln G−1 + Π (5.17) = ln G−1 + ln(1 + P iG). The free energy for this correction is +∞ V X Z d3k¯ πT 2/24 λT 2/24 F (3/2) = ln 1 + − 2β (2π )3 ω2 + k¯2 ω2 + k¯2 n=−∞ ~ n n

(5.18) = + + + ··· V (k T )4 λ 3/2 ' − B 12π (~c)3 24 This ﬁnal correction to the pressure is π2 (k T )4 15 15 λ 3/2 P = B 1 − + + ··· . (5.19) 90 (~c)3 24π2 2 24π2 5.2 Ward-Takahashi Identities

Consider an O(2) model with spontaneous symmetry breaking: 1 1 λ L = (∂ Φ)2 + m2(Φi)2 + (Φi)2(Φj)2 (5.20) 2 µ 2! 4! where i, j = 1, 2 and m2 < 0. In this case the 2PI eﬀective action is 1 Γ[φ, ∆] = − ln Z[J , K] + J i[φ, ∆]φi + Kij [φ, ∆](φi φj + ∆ij ). (5.21) ~ x x 2 xy x y ~ xy i 0i i i i i j i Γ[φ, ∆] is O(2) invariant, so a transformation of φ → φ = φ + δφ where δφ = T jφ can be made, with T j = σ2 the second Pauli matrix. The 2PI ward identity is

δΓ[φ, ∆] i j δΓ[φ, ∆] i kj j l 0 = i T jφx + ij (T k∆xy + T l∆xy). (5.22) δφx δ∆xy ij The second term is (~/2)Kxp[φ, ∆]. The self-consistency relation can be dropped in favour of the Ward identity. The Ward identity gives information on the physical sources. The Goldstone ﬁeld ϕG, the Higgs ﬁeld ϕH, and the Higgs propagator GHH of this theory are all imposed to be extremal, although the Goldstone propagator is not. This means i kj j il 0 = Kij,xy[φ, ∆](T k∆xy + T l∆xy.) (5.23) The Higgs self energy is HH HH −Πxy = Kxy [φ, ∆] (5.24) = + + + √ but ϕH = v = ±|m|/ λ.

23 5.3 Symmetry Preservation

In the Hartree-Fock approximation the last two diagrams are ignored, creating an inconsistency in the truncation procedure. This can be corrected with the Ward identies. The tree-level inverse propagators are G−1,HH(k) = k2 + m2 + 3λ(ϕH)2 (5.25) G−1,GG(k) = k2 + m2 + λ(ϕH)2.

GG HH In the Hartree-Fock approximation, it is necessary that Kxy [φ, ∆] = Kxy [φ, ∆] and so self-consistency means that GG H H Kxy [ϕ, G] 6= 0. Just as before, the corrections can be found. Firstly ϕ = v + ~δϕ where δϕH = −λvGHH(3GHH + GGG) xy yy yy (5.26) = .

−1,GG H 4 HH The ~ corrections to the Goldstone propagator are Gxy ⊃ 2~λvδϕ δxy + ~Πxy , but the Higgs self energy is HH HH GG Πxy = λ(3Gxx + Gxy )δxy. Algebraically

−1,GG 4 Gxy = −δxy (5.27) GG GG HH GG 4 and so the Goldstone boson is massless. If the extremal K was used instead then Π = λ(Gxy + 3Gxy )δxy −1,GG 4 HH GG so Gxy = δxy[− − 2~λ(Gxy − Gxy )], which is the usual problem with the Hartree-Fock approximation. The Goldstone boson mass comes from

(5.28)

local The ~ corrections to the Higgs propagators are −1,HH H HH Gxy ⊃ 6~λvδϕ + Π HH GG (5.29) = −2~λ(3Gxy + Gxy ). The thermal contributions are HH GG Gxy therm ≈ Gxy therm Z d3k¯ 1 1 ≈ (5.30) (2π)3 |k¯| eβ|k¯| − 1 T 2 = . 24 5.4 First-order Phase-transition

In the Hartree-Fock approximation, the Higgs mass is 8λT 2 m2 = −2m2 − . (5.31) H 12 √ 2 2 Since m < 0, there is a critical temperature where mH = 0 when Tc = 3|v|. The mass gap equations are 2 2 2 HH GG mH = 3λ(vFH) + m + λ(3Gxx + Gxx ) 2 2 HH (5.32) = 3λ(vFH) + m + λΠ and 2 2 2 HH GG mG = λ(vFH) + m + λ(3Gxx + Gxx ) 2 2 HH (5.33) = λ(vFH) + m + λΠ . Taking the diﬀerence shows that m2 − m2 v2 = H G . (5.34) HF 2λ 2 2 2 Since, at Tc, mH = 0 by deﬁnition and mG = 0 then vHF = 0 so there is a second order phase transition.

24 Chapter 6

Functional Renormalisation Group

6.1 Wetterich Equation

Begin with 1 1 λ L = ∂ Φ(x)∂µΦ(x) − µ2Φ2(x) + Φ4(x), (6.1) 2 µ 2 4! where µ2 > 0 and λ > 0 so there is a tachyonic bare mass.

Veﬀ (ϕ)

−v +v ϕ

Figure 6.1: Potential with minima at ±v = p6µ2/λ.

The generating functional is Z 1 Z[J ] = DΦ exp − S[Φ] − JxΦx (6.2) ~ and the 1PI effective action Γ[Φ] = W [J ] + Jx[Φ]Φx where W [J ] = −~ ln Z[J ], recalling that δΓ[Φ] = Jx[Φ]. (6.3) δΦx Assuming a constant field configuration ϕ, an effective potential can be defined as 1 V (ϕ) = Γ[ϕ] eff Ω 1 (6.4) = V (ϕ) + ~ Tr ln det G−1 2 Ω where Ω is a spacetime volume. The effective potential contains the original potential and the ~ corrections. The second functional derivative of the Schwinger function is

2 δ W [J ] = − hΦxΦxi − hΦxihΦyi δJxδJy (6.5) > 0

25 so it is a concave functional of J . The second functional derivative of the eﬀective action is

δ2Γ[Φ] δJ = x (6.6) δΦδΦ δΦy but since it is known that δ2W [J ] δΦ = − y (6.7) δJxJy δJx then Γ[Φ] must be a convex functional of Φ. This means that Veff [ϕ] is a convex function of the constant field configuration.

To go from a non-convex bare potential to a convex effective potential functional renormalisation can be performed. This is done by tracking contributions from fluctuations of differing momenta. The Wetterich equation is built from averaging effective actions. Quantum fluctuations of a given momentum p, can be “re-weighted” such that only modes p > k are integrated out and modes with p < k are effectively “frozen”. This can be done by introducing a term 1 S(k)[Φ] = Φ R(k)Φ (6.8) 2 x xy y so that the effective action becomes

(k) (k) (k) Γ [Φ] = W [J ] − S [Φ] + JxΦx. (6.9)

(k) The Fourier transform of Rxy is known as the regulator, which satisﬁes

(k) • limk→0 Rp = 0 so the 1PI eﬀective action is recovered in the infra-red.

(k) • Rp → 0 for p > k so the ultra-violet modes are unaﬀected.

(k) 2 • Rp → k for p < k so the infra-red modes are frozen. An optimal choice fro this is the Litim regulator

(k) 2 2 2 2 Rp = Θ(k − p )(k − p ). (6.10)

Taking the total derivative of Γ(k)[Φ] with respect to the scale k gives

(k) (k) (k) δW [J ] (k) ∂kΓ [Φ] = ∂kW [J ] + ∂kJx[Φ] − ∂kS [Φ] + ∂k Jx[Φ]Φx δJx (k) (k) = ∂kW [J ] − ∂kS [Φ] 1 (6.11) = ∂ R(k)hΦ Φ i − hΦ ihΦ i 2 k xy x y x y 2 (k) −1 1 (k) δ Γ [Φ] (k) = ∂kRxy + Rxy . 2 δΦxδΦy The Wetterich equation is then obtained by Fourier transforming

2 (k) −1 (k) 1 (p) δ Γ [ϕ] (k) 4 ∂kΓ [ϕ] = Tr ∂kRp + Rp δ (p + q) (6.12) 2 δϕpδϕq

(k) (k) where ϕ is now the background field configuration. Writing Γ [ϕ] as Veff (ϕ)/Ω, the effective potential flattens as more modes are integrated out. There are no first-order phase transitions since there is no barrier between the two minima due to the flattening from summing over all paths.

26 (k) Veﬀ (ϕ)

ϕ Figure 6.2: Flattening of the eﬀective potential due to integrating out higher momentum modes.

6.2 First-order Phase Transitions

Breaking the Z2 symmetry by hand removes the degeneracy between the minima. For example there is a ﬁnite tunnelling probability when the Lagrangian becomes 1 1 λ g L = ∂ Φ(x)∂µΦ(x) − µ2Φ2(x) + Φ4(x) + Φ3. (6.13) 2 µ 2 4! 6

V [Φ]

−v +v Φ

Figure 6.3: Breaking the degeneracy between the minima of the potential.

The classical equations of motion in Euclidean space is

0 = −∂2ϕ + V 0(ϕ). (6.14)

A bubble can be described by starting from the false vacuum at τ → −∞ so ϕ → +v and also ending there at τ → +∞ so ϕ → +v. If the non-degeneracy between the minima is extremely small thenϕ ˙ = 0 at τ = 0. The radius 2 2 2 of the bubble is given by r = τ + x where x is arbitrary for the ﬁrst bubble then ϕ|r→∞ = +v and ∂rϕ|r→0 = 0.

−V [ϕ]

−v +v ϕ

Figure 6.4: Bubble formation due to tunnelling between the false and true minima (maxima).

This has a kink solution

ϕ(r) = v tanh γ(r − R) (6.15)

27 √ where γ = µ/ 2. This bubble has the true minimum inside and the false minimum inside. The critical bubble comes from balancing the surface tension to the latent heat of the bubble

B ≡ SE[ϕ] Z ∞ (6.16) = 2π2 drr3L. 0 The radius of the critical bubble, when the bubble wall is thin is

12γ2 R = . (6.17) gv Because B is a maximum with respect to R, there is a negative eigenvalue of the Klein-Gordon operator

1 δ2B λ = 0 B δR2 (6.18) 3 = − R2 and four vanishing eigenvalues due to spacetime invariance. This is bad.

The transition rate can be obtained from the path integral

Z 1 Z = DΦ exp − S[Φ] . (6.19) ~

1/2 P Expanding the ﬁeld about Φ = ϕ + ~ n anφn then

X δ2S[Φ] S[Φ] = S[ϕ] + ~ a a φ φ . 2 n m n δΦδΦ m (6.20) n,m

If the second term has negative eigenvalues then the exponent becomes positive and the path integral blows up.

The zero eigenmodes can be traded for collective coordinates, VT . The negative eigenmode can be dealt with by the method of steepest descent. This picks out the lowest saddle point of the potential. The tunnelling rate per unit value is then given by

Γ 2|ImZ| rate = V VT 2 (6.21) B 5 −1/2 (5) −1 −1/2 −B/ = (2γ )|λ0| det G (ϕ)G(v) e ~. 2π~

The negative eigenvalue suggests that there is an instability arising from ~ corrections to the potential which cannot be seen at tree level. Adding a source term −KxyΦxΦy into the path integral takes this into account so that it can be dealt with.