Introduction to the Exact Renormalization Group Informal Seminar - - PowerPoint PPT Presentation

Introduction to the Exact Renormalization Group

Informal Seminar Bertram Klein, GSI literature:

• J. Berges, N. Tetradis, and C. Wetterich [hep-ph/0005122].
• lectures H. Gies, UB Heidelberg.
• D. F. Litim, J. M. Pawlowski [hep-th/0202188].

[Wetterich (1993), Wegner/Houghton (1973), Polchinski (1984)].

1

Outline

• motivation
• exact RG
• scale-dependent effective action
• one-loop flow equations for effective action
• hierarchy of flow equations for n-point functions
• truncations
• connection to perturbative loop expansion
• O(N)-model in a derivative expansion: flow equations

2

Motivation

• need to cover physics across different scales
• microscopic theory → macroscopic (effective) theory
• “bridge the gap” between microscopic theory and effective macroscopic description (in

terms of effective/thermodynamic potentials, . . . )

• loose the irrelevant details of the microscopic theory

⇒ How do we decide what is relevant and what is not?

• important rˆ
• le of fluctuations: long-range in the vicinity of a critical point

⇒ How do we treat long-range flucutations?

• universality: certain behavior in the vicinity of a critical point independent from the

details of the theory (e.g. critical exponents)

• often additional complications: need to go from one set of degrees of freedom (at the

microscopic level) to a different set (at the macroscopic level) here: we want to use an average effective action in the macroscopic description

3

Exact RG Flows

What do we mean by “exact” renormalization group flows?

• derived from first principles
• connects (any given) initial action (classical action) with full quantum effective action

⇒ exact flow reproduces standard perturbation theory

• flow in “theory space”: trajectory is scheme-dependent, but end point is not
• truncations project “true” flow onto truncated action

Γ[φ] S[φ]

[fig. nach H. Gies]

4

Goal: A scale-dependent effective action

Our goal is an “averaged effective action” Γk[φ] which is . . .

• . . . a generalization of the effective action which includes only fluctuations with q2 k2
• . . . a “coarse-grained” effective action, averaged over volumes ∼

1 kd (i.e. quantum

flucutations on smaller scales are integrated out!)

• . . . for large k (→ small length scales) very similar to the microscopic action S[φ] (since

long-range correlations do not yet play a rˆ

• le)
• . . . for small k (→ large length scales) includes long-range effects (long-range correlations,

critical behavior, . . . )

• . . . and which can be derived from the generating functional.

How does this look in practice? ⇒ we look at derivation of such an effective action starting from the generating functional for n-point correlation functions

5

Derivation of the scale-dependent effective action [1]

• scalar theory, fields χa, a = 1, . . . , N, d Euclidean dimensions
• start from the generating functional of the n-point correlation functions (path integral rep.)

Z[J] =

• Dχ exp
• −S[χ] +
• x

Jχ

• define a scale-dependent generating functional by inserting a cutoff term

Zk[J] =

• Dχ exp
• −S[χ] +
• x

Jχ − ∆Sk[χ]

• define scale-dependent generating functional Wk[J] for the connected Greens functions by

Zk[J] = exp [Wk[J]]

• cutoff term for a scalar theory:

∆Sk[χ] = 1 2

• q

χ∗(q)Rk(q)χ(q) [cutoff term quadratic in the fields ensures that a one-loop equation can be exact (Litim)]

6

Intermezzo: Properties of the cutoff function

• required properties for Rk(q):
• 1. Rk(q) → 0 for k → 0 at fixed q (so that Wk→0[J] = W[J] and thus Γk→0[φ] = Γ[φ])
• 2. Rk(q) → ∞ (divergent) for k → Λ (k → ∞ for Λ → ∞) (so that Γk→Λ[φ] = ΓΛ[φ] = S[φ])
• 3. Rk(q) > 0 for q2 → 0 (e.g. Rk(q) → k2 for q2 → 0) (must be an IR regulator, after all!)
• examples for popular cutoff functions:
• 1. without finite UV cutoff Λ → ∞

Rk(q) = q2 1 exp

• q2

k2

• − 1
• 2. with a finite UV cutoff Λ

Rk(q) = q2 1 exp

• q2

k2

• − exp
• q2

Λ2

• 7

Derivation of the scale-dependent effective action [2]

• exchange dependence on the source J for dependence on expectation value

φ(x) = δWk[J] δJ(x) → φ(x) = φa

k[J(x)]

• employ a modified Legendre transformation and define the scale dependent effective action as

Γk[φ] = −Wk[J] +

• x

J(x)φ(x) − ∆Sk[φ]

(∗)

(∗): cutoff term depends on expectation value φ: crucial for connection to the “bare” (classical) action S[φ] at the UV scale, and to quench only fluctuations around the expectation value!

• Variation condition on the action/equation of motion for φ(x)

δΓk[φ] δφ(x) = −

• y

δWk[J] δJ(y) δJ(y) δφ(x) +

• y

δJ(y) δφ(x)φ(y)

• =0

+J(x) − δ∆Sk[φ] δφ(x) = J(x) − δ δφ(x)∆Sk[φ] = J(x) − (Rkφ)(x)

8

Derivation of Flow equation [1]

Flow equation: It describes the change of the scale-dependent effective action at scale k with a change of the RG scale, and thus how the effective actions on different scales are connected. to derive the flow equation we need

• modified Legendre transform
• scale-dependent generating functional of the connected Greens functions
• take the derivative with regard to the scale of the modified Legendre transformation

(introduce t = log (k/Λ) ⇒ ∂t = k∂k): ∂tΓk[φ] = −∂tWk[J] −

• x

δWk[J] δJ(x)

=φ(x)

∂tJ +

• x

φ(x)(∂tJ)

• =0

−∂t∆Sk[φ] = −∂tWk[J] − ∂t∆Sk[φ]

• derivative of the cutoff term (remember that φ is the independent variable in Γk[φ])

∂t∆Sk[φ] = ∂t 1 2

• q

φ∗(q)Rk(q)φ(q) = 1 2

• q

φ∗(q)(∂tRk(q))φ(q)

9

Derivation of Flow equation [2]

• we need the scale derivative of Wk[J]
• first express the derivative in terms of exp(Wk[J])

∂tWk[J] = exp(−Wk[J]) exp(Wk[J])

• =1

∂tWk[J] = exp(−Wk[J]) (∂tWk[J]) exp(Wk[J]) = exp(−Wk[J]) (∂t exp(Wk[J]))

• now go back to the path integral representation: scale dependence appears only in cutoff term

∂tWk[J] = exp(−Wk[J]) ∂t

• Dχ exp
• −S[χ] +
• x

Jχ − ∆Sk[χ]

• =

exp(−Wk[J])

• Dχ(−∂t∆Sk[χ]) exp
• −S[χ] +
• x

Jχ − ∆Sk[χ]

• =

exp(−Wk[J])

• Dχ
• −1

2

• q

χ∗(q)(∂tRk(q))χ(q)

• exp
• −S[χ] +
• x

Jχ − ∆Sk[χ]

• =

−1 2

• q

(∂tRk(q)) exp(−Wk[J])

• Dχ χ∗(q)χ(q)
• exp
• −S[χ] +
• x

Jχ − ∆Sk[χ]

• 10

Derivation of Flow equation [3]

Express this in terms of the connected Greens functions: exp(−Wk[J]) δ2 δJ(q)δJ∗(q) exp(Wk[J]) = δ2Wk[J] δJ(q)δJ∗(q) + δWk[J] δJ∗(q) δWk[J] δJ(q) = χ∗(q)χ(q)k,connected + φ∗(q)φ(q) = Gk(q, q) + φ∗(q)φ(q) ⇒ we find for the flow of Wk[J] ∂tWk[J] = −1 2

• q

(∂tRk(q)) (Gk(q, q) + φ∗(q)φ(q)) = −1 2

• q

(∂tRk(q))Gk(q, q) − 1 2

• q

φ∗(q)(∂tRk(q))φ(q) = −1 2

• q

(∂tRk(q))Gk(q, q) − ∂t∆Sk[φ]

• insert this into the flow equation for Γk . . .

11

Derivation of Flow equation [4]

. . . result for ∂tWk[J] into flow equation: ∂tΓk[φ] = −∂tWk[J] − ∂tSk[φ] = 1 2

• q

(∂tRk(q))Gk(q, q) + ∂t∆Sk[φ] − ∂t∆Sk[φ] The result for the flow equation for the effective action is ∂tΓk[φ] = 1 2

• q

(∂tRk(q)) Gk(q, q)

• This should now be expressed as a functional differential equation for the effective action.

12

Inversion of scale-dependent propagator [1]

What remains to do in order to obtain a (functional) differential equation for the scale-dependent effective action is to express G(p, q) in terms of this effective action G(p, q) = δ2Wk[J] δJ∗(p)δJ(q), φ(q) = δWk[J] δJ∗(q) use variation condition on effective action (from modified Legendre transformation) δΓk[φ] δφ(q) = J∗(q) − φ∗(q)Rk(q) second variation with respect to φ∗(q′) δ2Γk[φ] δφ∗(q′)δφ(q) = δJ∗(q) δφ(q′) − Rk(q) δ(q − q′) ⇒ δJ∗(q) δφ(q′) = δ2Γk[φ] δφ∗(q′)δφ(q) + Rk(q) δ(q − q′). Now start from an identity to show that this is the inverse of G(q′, q)

13

Inversion of scale-dependent propagator [2]

• start from the identity

δ δφ∗(q′)φ∗(q) = δ(q′ − q) = δ δφ∗(q′) δWk[J] δJ(q) =

• q′′

δ2Wk[J] δJ∗(q′′) δJ(q) δJ∗(q′′) δφ∗(q′)

• use expression for δJ ∗/δφ∗ established above

= ⇒ δ(q − q′) =

• q′′

δ2Wk[J] δJ∗(q′′) δJ(q)

• δ2Γk[φ]

δφ∗(q′) δφ(q′′) + Rk(q) δ(q′ − q′′)

• ⇒ the scale dependent inverse propagator is given by

G(q, q′) =

• δ2Γk[φ]

δφ∗(q) δφ(q′) + Rk(q) δ(q − q′) −1 (result is as expected, but it is necessary to establish the particular form of the scale dependence of the propagator)

14

Result for the Flow equation

∂tΓk[φ] = 1 2

• q

(∂tRk(q))

• δ2Γk[φ]

δφ∗(q) δφ(q) + Rk(q) −1 Graphical representation (insertion stands for the derivative of the cutoff function ∂tRk)

Γ

k

1 2

The line represents the full propagator (which includes the complete field dependence). In a more abstract representation (where in general the trace also involves any internal indices) ∂tΓk[φ] = 1 2Tr

• (∂tRk)

Γ(2)

k [φ] + Rk

• = ∂t

1 2Tr log(Γ(2)

k [φ] + Rk)

Note that this is not equal to the total derivative of a one-loop effective action (since the terms ∂tΓ(2)

k [φ] are missing), although it is a one-loop flow equation! 15

Flow Equation: Exact?

In principle, there are many different ways to introduce some sort of cutoff function into the path integral, and to obtain in this way a flow equation of type ∂tΓk[φ] = Fk[Γ(2)

k ]

with some functional Fk[γ(p, q)]. We needed to show two properties in order to estalish that the flow is “exact”:

• 1. Is the action at scale k related to the full effective quantum action for k → 0? Are they in

fact connected as Γk→0[φ] = Γ[φ]?

• 2. Is the action at scale k related to the classical/initial action for k → Λ? Are they in fact

connected as Γk→Λ[φ] = S[φ]? [as an analogy, one can think of a proof by induction: one needs to prove the induction step from n to n + 1 (here: flow equation), but also the induction premise, the validity of the statement for n = 1 (here: connection to classical action)]

16

Exactness [1]

We begin by answering question 1 (which is the easier one): How is the scale-dependent action related to the quantum effective action? From the properties of the cutoff function we have lim

k→0 Rk(q) = 0 ⇒ lim k→0 ∆Sk[φ] = 0

and therefore for the scale-dependent generating functional lim

k→0 Zk[J] = Z[J]

thus, by the properties we require from the cutoff, it is trivially true that lim

k→0 Γk[φ]

= −Wk→0[J] +

• x

Jφ − ∆Sk→0[φ] = −W[J] +

• x

Jφ = Γ[φ] is the complete quantum effective action!

17

Exactness [2]

We now answer the second question: How is the effective scale dependent action related to the trival (classical) action? This turns on the properties required of cutoff function and modified Legendre transform.

• start from the modified Legendre transform (this is where the necessity of the modification

really comes in) and exponentiate: exp(−Γk[φ]) = exp

• −
• x

Jφ + ∆Sk[φ]

• exp(Wk[J])

= exp

• −
• x

Jφ + ∆Sk[φ] Dχ exp

• −S[χ] +
• x

Jχ − ∆Sk[χ]

• =
• Dχ exp
• −S[χ] +
• x

J(χ − φ) + ∆Sk[φ] − ∆Sk[χ]

• exp(Wk[J]) has been replaced by the path integral representation of Zk[J].
• now use φ as a “background field”: χ = φ + χ′,
• Dχ ≡
• Dχ′

(no assumptions necessary regarding its relation to the minimum of the classical action S[φ])

18

Exactness [3]

• multiply out the square in the cutoff term (no approximation)

∆Sk[φ + χ′] = ∆Sk[φ] +

• χ′(Rkφ) + 1

2

• χ′(Rkχ′)
• introduce this into expression

exp(−Γk[φ]) =

• Dχ′ exp
• −S[φ + χ′] +
• Jχ′ − ∆Sk[φ + χ′] + ∆Sk[φ]
• =
• Dχ′ exp
• − S[φ + χ′] +
• Jχ′ −
• χ′(Rkφ)
• (∗)

−∆Sk[χ′] − ∆Sk[φ] + ∆Sk[φ]

• use equation for φ to simplify term linear in χ′ (φ remains unconstrained)

δΓk[φ] δφ = J − (Rkφ)

• find finally

exp(−Γk[φ]) =

• Dχ′ exp
• −S[φ + χ′] +

δΓk[φ] δφ χ′ − ∆Sk[χ′]

• 19

Exactness [4]

• now, in the limit k → Λ, the cutoff function diverges by requirement
• cutoff term diverges as

lim

ǫ→0 exp

• − 1

2ǫ

• (χ′)2
• ⇒

exp(−∆Sk→Λ[χ′]) → δ[χ′] ⇒ exponential becomes a δ-functional (w/ appropriate normalization)! In the path integral lim

k→Λ exp(−Γk[φ])

= lim

k→Λ

• Dχ′ exp(−S[φ + χ′] +

δΓk[φ] δφ χ′ − ∆Sk[χ′]) =

• Dχ′ exp(−S[φ + χ′] +

δΓk[φ] δφ χ′) δ[χ′] = exp(−S[φ]) ⇒ ΓΛ[φ] = S[φ]. This proves that the scale-dependent effective action coincides with the classical action at the UV scale and that the RG flow actualy connects the action at any scale k to the classical action, and thus concludes the proof of the exactness of the ERG flow.

20

Properties of the flow equation

What are the properties of this flow equation? By definition, it describes the change of the effective action Γk with a change of the RG scale k. Now, at this point, what have we obtained?

• An exact (no approximations so far!) renormalization group flow equation for the effective

action . . .

• . . . which is a nonlinear functional differential equation (since it involves the functional

derivatives Γ(2)

k [φ] of Γk[φ]!) . . .

• . . . and which is of course in its most general form completely unsolvable!

So how do we solve this? There are two questions that need to be asked:

• 1. How do we obtain correlation functions for a larger number of fields from this? [How does

it sprout legs?] (→ hierarchy question important for truncations)

• 2. This does look like a one-loop equation: are higher loop orders indeed contained in this?

Can we recover ordinary perturbation theory?

21

Question 1: Higher n-point functions

How do we obtain flow equations for the higher n-point functions? Simply take the appropriate number of derivatives of the flow equation for the effective action (φ-dependence in Γ(2)

k [φ]):

∂tΓk[φ] = 1 2Tr

• (∂tRk)[Γ(2)

k [φ] + Rk]−1

• take derivatives

δ δφ∂tΓk[φ] = 1 2Tr

• (∂tRk)(−1)[Γ(2)

k

+ Rk]−1 δΓ(2)

k

δφ [Γ(2)

k

+ Rk]−1

• =

−1 2Tr

• (∂tRk)[Γ(2)

k

+ Rk]−1 Γ(3)

k

[Γ(2)

k

+ Rk]−1 Graphical representation: ∂tΓ(1)

k =− 1 2

∂tRk Γ(3)

k

• ✁

22

Higher n-point functions [2]

• one more derivative to get the flow equation for the two-point function:

δ2 δφδφ∂tΓk[φ] = 2 × 1 2Tr

• (∂tRk)[Γ(2)

k

+ Rk]−1 Γ(3)

k

[Γ(2)

k

+ Rk]−1 Γ(3)

k

[Γ(2)

k

+ Rk]−1 −1 2Tr

• (∂tRk)[Γ(2)

k

+ Rk]−1 Γ(4)

k

[Γ(2)

k

+ Rk]−1 Graphical representation [first graph implies a factor 2]: ∂tΓ(2)

k = 1 2

∂tRk Γ(3)

k

Γ(3)

k

− 1

2

Γ(4)

k

∂tRk

What does this imply?

• To find flow equation for Γ(2)

k , we need Γ(3) k

and Γ(4)

k !

• In general, for flow of Γ(n)

k , need Γ(n+1) k

, Γ(n+2)

k

: ⇒ hierarchy of flow equations! How can we do meaningful calculations?

23

Truncations

Problem: In order to calculate the flow of Γ(n)

k , we need Γ(n+1) k

and Γ(n+2)

k

. Solution: We need to truncate the effective action and restrict it to correlators of nmax fields. But: then this is no longer a closed system of equations!

• in principle, need to write down most general ansatz for the effective action
• this ansatz will contain all invariants that are compatible with the symmetries of the theory
• then one truncates by reducing higher n-point functions to contact terms, or to a simplified

momentum dependence

• one neglects even higher correlations outright

This is not an expansion in some small parameter (although of course the assumption is that higher order operators will be irrelevant and suppressed due to the existence of a large scale) For practical applications, this is obviously the most problematic part, and it requires a lot of physical insight to make the correct physical choices.

24

Question 2: Comparison to perturbation theory

Claim: although the ERG flow equation is a one-loop RG flow equation, ∂tΓk[φ] = 1 2(∂tRk)pq[Γ(2)

k [φ] + Rk]−1 qp ,

shorthand: ApqBqp′ =

• q

A(p, q)B(q, p′) it contains effects to arbitrary high loop order: expect reproduction of higher loop order perturbation theory!

[in the arguments here, we follow Litim/Pawlowski]

Do a loop expansion of the effective action and compare to perturbation theory: Γ = S +

∞

• n=1

∆Γn In terms of the flow equation, contributions of different loop orders can be identified ∂tΓk =

∞

• n=1

∂t∆Γn,k notation: m-point correlation function to n-loop order at scale k Γ(m)

n,k 25

Comparison to PT: scheme of calculation

Our goal: two-loop result for the effective action (first non-trivial loop order) As a roadmap for the expansion in loop order, here is an outline of the calculation:

• 1. start from effective action at n-loop level and calculate the two-point function
• 2. insert the n-loop two-point function into the (one-loop) flow equation
• 3. isolate the (n + 1)-loop correction to effective action
• 4. integrate flow equation to obtain n + 1-loop correction to effective action

Γn,k

δ2 δφδφ

− → Γ(2)

n,k = Γ(2) n−1,k + ∆Γ(2) n,k

into flow eq. − → (∂tRk) Γ(2)

n,k + Rk

= (∂tRk) 1 Γ(2)

n−1,k + ∆Γ(2) n,k + Rk

isolate − → ∂t∆Γn+1,k = (∂tRk) 1 Γ(2)

n−1,k + Rk

∆Γ(2)

n,k

1 Γ(2)

n−1,k + Rk

integrate w.r.t. k: − → Γn+1,k = Γn,k + ∆Γn+1,k

26

Comparison to PT: effective action at one loop

Effective action at one loop is calculated using the tree-level two-point function Γk,1 = S + ∆Γk,1 Γ(2)

k,0

= S(2) Flow equation for the one-loop correction to the effective action: ∂t∆Γ1,k = 1 2(∂tRk)pq

• S(2) + Rk

−1

qp

Integrate this to get the one-loop correction: ∆Γ1,k = k

Λ

dk′ 1 k′ ∂t′ 1 2[log(S(2) + Rk)]pp One-loop result corresponds with ordinary result (RΛ similar to Pauli-Villars regulator): Γ1,k = S + 1 2

• log(S(2) + Rk)
• pp − 1

2

• log(S(2) + RΛ)
• pp = S + 1

2

• log(S(2) + Rk′)
• pp
• k

Λ 27

Comparison to PT: two-point function at one loop

• find the one-loop contribution to the two-point function
• it is obtained by taking the variation of the one-loop effective action
• ∆Γ(2)

1,k

• qq′

= δ2 δφ(q)δφ(q′)∆Γ1,k = 1 2 δ2 δφ(q)δφ(q′)

• log(S(2) + Rk′)
• pp
• k

Λ

= 1 2

• Gpp′S(4)

pp′qq′ − Gpq′′′S(3) q′′′q′′qGq′′p′S(3) p′pq′

• k

Λ

where one uses δ δφ(q)Gpp′ = δ δφ(q)

• S(2) + Rk

−1

pp′ = (−1)Gpq′S(3) q′q′′qGq′′p′

Graphical representation:

[ ]

1 2

double line: UV regularization from the cutoff

28

Comparison to PT: notation for regularization

• the double line represent the regularization through the presence of the UV cutoff Λ

= = k Λ k k Λ Λ

• RΛ has a similar effect as a Pauli-Villars regulator

29

Comparison to PT: flow of two-loop correction to action

• Now we need to find the two-loop correction to the flow equation for the effective action.
• do this by first inserting the correction to the propagator into the flow equation . . .
• . . . and then isolating the two-loop part

∂tΓ2,k = 1 2(∂tRk)qp

• Γ(2)

1,k + Rk

−1

pq = 1

2(∂tRk)qp

• S(2) + ∆Γ(2)

1,k + Rk

−1

pq

= 1 2(∂tRk)qp

• S(2) + Rk

−1

pq +

+1 2(∂tRk)qp(−1)

• S(2) + Rk

−1

pq′′

• ∆Γ(2)

1,k

• q′′q′
• S(2) + Rk

−1

q′q + . . .

= ∂t∆Γ1,k + ∂t∆Γ2,k + . . . Therefore we find for the correction ∂t∆Γ2,k = −1 2(∂tRk)qpGpq′

• ∆Γ(2)

1,k

• q′′q′ Gq′′q

This needs to be integrated over all scales k. In order to do the k-integration, we need particular properties of Gpq.

30

Comparison to PT: k-derivative of two-point function

The tree-level propagator (with cutoff at scale k) is from here on abbreviated as Gpq =

• S(2) + Rk

−1

pq

Look at derivatives of the propagator w.r.t. the RG scale k: ∂tGpq = ∂t

• S(2) + Rk

−1

pq = (−1)

• S(2) + Rk

−1

pq′ (∂tRk)q′q′′

• S(2) + Rk

−1

q′′q

= (−1)Gpq′(∂tRk)q′q′′Gq′′q Graphically: ∂t

=

• This seems trivial, but is a very important result (and why we recover perturbation theory)
• makes it possible to re-write terms in the flow equations as total derivatives with the correct

combinatorial factors that come from inserting (∂tRk) in all possible propagators!

• with appropriate renaming of indices:

Gpp′S(4)

pp′q′q′′(∂t′G)q′′q′

= 1 2∂t′

• Gpp′S(4)

pp′q′q′′Gq′′q′

• Gpp′S(3)

pp′′qGp′′q′′S(3) q′′p′q′(∂t′G)q′q

= 1 3∂t′

• Gpp′S(3)

pp′′qGp′′q′′S(3) q′′p′q′Gq′q

• 31

Comparison to PT: integrand as total derivative

• using the result for the scale derivative of Gpq, we can write for the two-loop flow correction

∂t∆Γ2,k = −1 2(∂tRk)qpGpq′

• ∆Γ(2)

1,k

• q′′q′ Gq′′q

= 1 2

• ∆Γ(2)

1,k

• q′′q′ (∂tG)q′q′′
• now insert the expression for the one-loop propagator correction
• use the results for ∂tGpq to write this as a total derivative (note combinatorial factors!)

1 2 1 2

• Gpp′S(4)

pp′qq′ − Gpp′S(3) pp′′qGp′′q′′S(3) q′′p′q′

k

Λ (∂t′G)qq′

= 1 2 1 2∂t′ 1 2Gpp′S(4)

pp′qq′Gqq′ − 1

3Gpp′S(3)

pp′′qGp′′q′′S(3) q′′p′q′Gqq′

• now perform the scale integration (and look at the graphical representation, to make this a

bit more transparent) . . .

32

Comparison to PT: effective action at two loops

• integrate with regard to the renormalization scale k

∆Γ2,k = k

Λ

dk′ 1 k′ 1 2 1 2

• Gpp′S(4)

pp′qq′ − Gpp′S(3) pp′′qGp′′q′′S(3) q′′p′q′

k

Λ (∂t′G)qq′

= k

Λ

dk′ 1 k′ 1 2 1 2∂t′ 1 2Gpp′S(4)

pp′qq′Gqq′ − 1

3Gpp′S(3)

pp′′qGp′′q′′S(3) q′′p′q′Gqq′

• 1

4

[ ]

• result of the integration (up to regularization terms)

∆Γ2,k = 1 8Gpp′S(4)

pp′qq′Gqq′ − 1

12Gpp′S(3)

pp′′qGp′′q′′S(3) q′′p′q′Gq′q

k

Λ

]

1 12 1

[8

ren.

⇒ this is indeed the correct perturbative two-loop result! [Figures are taken from Litim/Pawlowski (2002)]

33

O(N)-model: simple example

Action for O(N)-symmetric scalar theory: S[φ] =

• x

1 2(∂µφa)(∂µφa) + 1 2m2φ2 + 1 4λ(φ2)2

• given at some scale Λ

φ = (φa), a = 1, . . . , N, d Euclidean dimensions

• given in terms of couplings at scale Λ
• allows for spontaneous symmetry breaking and light modes

34

O(N)-model: derivative expansion

Effective action for O(N)-symmetric scalar theory:

• can depend only on invariants,
• a priori all invariants are possible
• expand around constant expectation value
• do expansion in terms of number of derivatives

Ansatz for effective action in the derivative expansion Γk[φ] =

• x
• Uk(ρ) + 1

2∂µφaZk(ρ, −∂2)∂µφa +1 4∂µρYk(ρ, −∂2)∂µρ + . . .

• where ρ = 1

2φaφa

(all derivatives in Zk, Yk act only to the right)

35

O(N)-model: flow equation for effective potential

The flow equation for the effective potential is the lowest order term in the derivative expansion ∂ ∂tUk(ρ) = 1 2

• q

∂ ∂tRk(q) 1 M1 + N − 1 M0

• M0(ρ, q2)

= Zk(ρ, q2)q2 + U ′

k(ρ)

+ Rk(q) M1(ρ, q2) = Zk(ρ, q2)q2 + ρYk(ρ, q2)q2 + U ′

k(ρ) + 2ρU ′′ k (ρ) + Rk(q)

where ρ = 1

2φaφa

most interesting: regions with light degrees of freedom → spontaneous symmetry breaking. Note that in order to keep the rescaling invariance, we need the wave function renormalization in the cutoff function (scale argument as the other momenta) Rk(q) = Zkq2 exp(q2/k2) − exp(q2/Λ2) Observe: as they stand, this set of equations is not closed! Missing: flow equation for the wave function renormalizations!

36

O(N)-model: anomalous dimension

• the anomalous dimension η is given in terms of the wave function renormalization Zk:

η = − d dt log Zk(ρ0(k), q2 = 0)

• the wave function renormalization can be calculated from the two-point function

Γ(2)

k (ρ; p, q) = [Zk(ρ, q2)q2 + M 2]δ(p + q) (neglect explicit q-dependence of Zk(ρ)):

Zk(ρ) = lim

q2→0

∂ ∂q2 δ2 δφ(q)δφ(−q)Γk[φ2] = lim

q2→0

∂ ∂q2 Γ(2)

k (φ2; q, −q)

⇒ we need a flow equation for the two-point function as well . . . . . . which in turn depends on the three- and four-point functions! (Hierarchy of flow equations!) [why anomalous dimension? It’s the anomalous dimension η of the propagator, as in Γ(2)

k

∼ q2( q2

k2 + c)−η/2

→ presence of relevant scale k allows for scaling different from canonical dimension!]

37

O(N)-model: results

• need to close the equations
• possible approach: obtain higher n-point functions from RG-improved flow equations (they

represent total derivative terms which can be integrated w.r.t. k)

• reproduce perturbative β function for four-point coupling to two loops

[Papenbrock/Wetterich hep-th/9403164]

• result with uniform (no momentum, field dependence) wave function renormalization:

βλ = N + 8 16π2 λ2 − 17.26N + 75.95 (16π2)2 λ3

• need momentum and field dependence of wave function renormalization Zk(ρ, q2)
• result there in d = 4 coincides with the perturbative result:

βλ = N + 8 16π2 λ2 − 9N + 42 (16π2)2 λ3

38

O(N)-model: more details on β-functions

• to find critical behavior, re-write flow equations for couplings in scale-invariant form

starting from ∂ ∂tUk(ρ) = 1 2

• q

∂ ∂tRk(q) N − 1 M0 + 1 M1

• (neglect ∂tZk from Rk/Zk → justified for small anomalous dimension)

1 2

• q

∂ ∂tRk(q) N − 1 M0

• −

→ 1 2

• ddq

(2π)d ∂t Rk Zk

• (N − 1)
• q2 + Rk

Zk + k2 U ′

k(ρ)

Zkk2

• ld

0(w)

= 1 4 1 vdkd

• q

∂t Rk Zk

• 1

q2 + Rk

Zk + k2w,

1 vd = 2d+1πd/2Γ(d/2), ld

n(w) = − ∂

∂wld

n−1(w)

• w is a dimensionless variable
• functions ld

n(w) are “threshold functions”, since they cut off modes with masses m2 ≫ k2

⇒ theory becomes theory of effective “light modes”!

39

O(N)-model: β-functions [2]

• in terms of the threshold functions flow equation

∂tUk(ρ) = 2vdkd

• (N − 1)ld

U ′

k(ρ)

Zkk2

• + ld

U ′

k(ρ) + 2ρU ′′ k (ρ

Zkk2

• Now take lowest possible approximation (for symmetry breaking): quartic potential

Uk(ρ) = λk 2 (ρ − ρ0(k))2

• flow equation for minimum ρ0(k) from minimum condition:

d dtU ′

k(ρ0(k))

= ∂tU ′

k(ρ0(k)) + U ′′ k (ρ0(k))∂tρ0(k) ≡ 0

⇒ ∂tρ0(k) = 2vdkd−2Z−1

k

• 3 ld

1

2ρ0(k)λk Zkk2

• + (N − 1)ld

1(0)

• flow equation for coupling λk from second derivative

∂2 ∂ρ2 ∂tUk(ρ) (note U ′′′ k ≡ 0 )

∂tλk = 2vdkd−4Z−2

k λ2 k

• 9 ld

2

2ρ0(k)λk Zkk2

• + (N − 1)ld

2(0)

• 40

O(N)-model: β-functions [3]

• introduce couplings rescaled according to canonical dimensions:

¯ ρ0 = Zkk2−dρ0(k) ¯ λ = Z−2

k kd−4λk

• β-functions: flow equations of the dimensionless couplings (note where η = −∂t log Zk

appears) ∂t¯ ρ0 = βρ = (2 − d − η)¯ ρ0 + 2vd

• 3ld

1(2¯

λ¯ ρ) + (N − 1)ld

1(0)

• ∂t¯

λ = βλ = (d − 4 + 2η)¯ λ + 2vd¯ λ2 9ld

2(2¯

λ¯ ρ) + (N − 1)ld

2(0)

• need in principle anomalous dimension to solve this! As expected, related to long-range

correlations, so it has to be obtained from two-point correlator.

• can already analyze fixed point structure (as a function of the dimension d) in this

approximation! d = 4: reproduce one-loop βλ = N+8

16π2 ¯

λ2, find “trivial” fixed point (¯ λ → 0 for k → 0) d = 3: scaling solution, critical point, phase transition. d = 2: N = 1 phase transition/critical point, N ≥ 3 no fixed point/phase transition, N = 2 special: Kosterlitz-Thouless-transition!

41

O(N)-model: more details on 2-point function

• We want to derive the anomalous dimension: need to get it from the two-point function
• We need to derive the flow equation of the two-point function.

Use that in our ansatz the inverse propagator is G−1

k (ρ, q2)

= Zk(ρ, q2)q2 + U ′

k(ρ) = M0(ρ, q2) − Rk(q)

ansatz for the propagator (fields φa are constant expectation values): Γ(2)k = 1 2

• q
• (U ′

k(ρ) + Zk(ρ, q2)q2)φa(q)φa(−q) + 1

2φaφb(2U ′′

k (ρ) + Yk(ρ, q2)q2)φa(q)φb(−q)

• ansatz for the higher couplings [simplified] that takes momentum dependence of couplings into

account in the form in which it appears in the two point function (momentum conserved): Γ(3)k = 1 2

• q1
• q2

φaλ(1)

k (ρ; q1, q2)φa(q1)φb(q2)φb(−q1 − q2) + . . .

Γ(4)k = 1 8

• q1
• q2

λ(2)

k (ρ; q1, q2, q3)φa(q1)φa(q2)φb(q3)φb(−q1 − q2 − q3) + . . . 42

O(N)-model: 2-point function [2]

• need to express couplings in terms of two-point function couplings
• require a continuity condition between the couplings in Γ(3)k, Γ(4)k and in Γ(2)k (roughly:

they have to coincide if one (3-point) or two (4-point) momenta vanish) λ(1)(ρ; q1, q2) = U ′′

k (ρ) + q2 · (q1 + q2)Z′ k(ρ, q2 · (q1 + q2)) + 1

2q2

1Yk(ρ, q2 1) + . . .

λ(2)(ρ; q1, q2, q3) = U ′′

k (ρ) − q2 · q1Z′ k(ρ, −q2 · q1) − q4 · q3Z′ k(ρ, −q4 · q3)

+1 2(q1 + q2)2Yk(ρ, (q1 + q2)2) + . . .

• additional approximation: neglect the extra terms (to close equations)!

Now need to insert this into the general flow equation for the two-point function!

43

O(N)-model: 2-point function [3]

Flow equation for the two-point function in the couplings introduced above (ρ-dependence of Mi(p2) suppressed): ∂tGk(ρ, q2) = 1 2

• p

∂tRk(p)× {4ρM −2

1 (p2)M −1 0 ((p + q)2)(λ(1) k (p, q))2 + 4ρM −2 0 (p2)M −1 1 ((p + q)2)(λ(1) k (−p − q, q))2

−M −2

0 (p2)[(N − 1)λ(2) k (q, −q, p) + 2λ(2) k (q, p, −q)] − M −2 1 (p2)λ(2) k (q, −q, p)}

• use continuity conditions from above to replace λ(1,2)

k

!

• get flow equation for wave function renormalization

∂tZk(ρ, q2) = 1 q2 (∂tGk(ρ, q2) − ∂tU ′

k(ρ)) = −ξk(ρ, q2)Zk(ρ, q2)

• actual anomalous dimension η from this

η = − d dt log Zk(ρ0(k), k2) = ξk(ρ0, k2) − 2 k2 Zk(ρ0, k2) ∂ ∂q2 Zk(ρ0, q2)

• q2=k2

− Z′

k(ρ0, k2)

Zk(ρ0, k2)∂tρ0

44

Summary

What I hope you will take away from today’s talk

• to cover physics across different scales, it is important to have a systematic scheme of

integrating out quantum fluctuations

• the so-called ERG is an “exact” RG scheme in the following sense: its RG flow equation

connects a classical action at some UV scale to the full quantum effective action

• however, a solution relies on some truncation of the effective action ⇒ result of a

calculation is not exact!

• the one-loop RG equation reproduces ordinary perturbation theory to arbitrary order (we

have shown this up to second order/the first nontrivial order)

• example: works for scalar O(N)-model
• different truncations possible: to get (important) anomalous dimension, flow equation for

two-point function is necessary

45