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 rôle 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 S[φ]

[ﬁg. nach H. Gies] Γ[φ]

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

1 • . . . a “coarse-grained” effective action, averaged over volumes ∼ 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 rôle) • . . . 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] + Jχ Z Zx • deﬁne a scale-dependent generating functional by inserting a cutoﬀ term

Zk[J] = Dχ exp −S[χ] + Jχ − ∆Sk[χ] Z Zx • deﬁne scale-dependent generating functional Wk[J] for the connected Greens functions by

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

• cutoﬀ term for a scalar theory:

1 ∗ ∆S [χ] = χ (q)R (q)χ(q) k 2 k Zq [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])

2 2 2 3. Rk(q) > 0 for q → 0 (e.g. Rk(q) → k for q → 0) (must be an IR regulator, after all!) • examples for popular cutoff functions: 1. without finite UV cutoff Λ → ∞

2 1 Rk(q) = q q2 exp k2 − 1 2. with a ﬁnite UV cutoﬀ Λ

2 1 Rk(q) = q q2 q2 exp k2 − exp Λ2

7 Derivation of the scale-dependent eﬀective action [2]

• exchange dependence on the source J for dependence on expectation value δW [J] φ(x) = k → φ(x) = φa[J(x)] δJ(x) k • employ a modified Legendre transformation and define the scale dependent effective action as

Γk[φ] = −Wk[J] + J(x)φ(x) − ∆Sk[φ] x Z (∗) (∗): cutoﬀ term depends on expectation value φ: crucial for connection| {z } 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) δΓ [φ] δW [J] δJ(y) δJ(y) δ∆S [φ] k = − k + φ(y) +J(x) − k δφ(x) δJ(y) δφ(x) δφ(x) δφ(x) Zy Zy =0 δ = J| (x) − ∆S{z[φ] = J(x) − (R}φ)(x) δφ(x) k k

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): δW [J] ∂ Γ [φ] = −∂ W [J] − k ∂ J + φ(x)(∂ J) −∂ ∆S [φ] = −∂ W [J] − ∂ ∆S [φ] t k t k δJ(x) t t t k t k t k Zx Zx =φ(x)

| {z } =0

• derivative of the cutoﬀ term| (remember{zthat φ is the indep} endent variable in Γk[φ])

1 ∗ 1 ∗ ∂ ∆S [φ] = ∂ φ (q)R (q)φ(q) = φ (q)(∂ R (q))φ(q) t k t 2 k 2 t k Zq Zq

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

=1 = exp| (−Wk[J{z]) (∂t exp(W}k[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] + Jχ − ∆Sk[χ] Z Zx

= exp(−Wk[J]) Dχ(−∂t∆Sk[χ]) exp −S[χ] + Jχ − ∆Sk[χ] Z Zx 1 ∗ = exp(−W [J]) Dχ − χ (q)(∂ R (q))χ(q) exp −S[χ] + Jχ − ∆S [χ] k 2 t k k Z Zq Zx 1 ∗ = − (∂ R (q)) exp(−W [J]) Dχ χ (q)χ(q) exp −S[χ] + Jχ − ∆S [χ] 2 t k k k Zq Z Zx

| {z } 10 Derivation of Flow equation [3]

Express this in terms of the connected Greens functions: δ2 δ2W [J] δW [J] δW [J] exp(−W [J]) exp(W [J]) = k + k k k δJ(q)δJ ∗(q) k δJ(q)δJ ∗(q) δJ ∗(q) δJ(q) ∗ ∗ = hχ (q)χ(q)ik,connected + φ (q)φ(q) ∗ = Gk(q, q) + φ (q)φ(q)

⇒ we ﬁnd for the ﬂow of Wk[J]

1 ∗ ∂ W [J] = − (∂ R (q)) (G (q, q) + φ (q)φ(q)) t k 2 t k k Zq 1 1 ∗ = − (∂ R (q))G (q, q) − φ (q)(∂ R (q))φ(q) 2 t k k 2 t k Zq Zq 1 = − (∂ R (q))G (q, q) − ∂ ∆S [φ] 2 t k k t k Zq • 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 = (∂ R (q))G (q, q) + ∂ ∆S [φ] − ∂ ∆S [φ] 2 t k k t k t k Zq The result for the ﬂow equation for the eﬀective action is

1 ∂ Γ [φ] = (∂ R (q)) G (q, q) t k 2 t k k Zq

• 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 δ2W [J] δW [J] G(p, q) = k , φ(q) = k δJ ∗(p)δJ(q) δJ ∗(q) use variation condition on effective action (from modified Legendre transformation)

δΓ [φ] ∗ ∗ k = J (q) − φ (q)R (q) δφ(q) k second variation with respect to φ∗(q0)

2 ∗ ∗ 2 δ Γ [φ] δJ (q) 0 δJ (q) δ Γ [φ] 0 k = − R (q) δ(q − q ) ⇒ = k + R (q) δ(q − q ). δφ∗(q0)δφ(q) δφ(q0) k δφ(q0) δφ∗(q0)δφ(q) k Now start from an identity to show that this is the inverse of G(q0, q)

13 Inversion of scale-dependent propagator [2]

• start from the identity

δ ∗ 0 φ (q) = δ(q − q) δφ∗(q0) 2 ∗ 00 δ δWk[J] δ Wk[J] δJ (q ) = ∗ 0 = ∗ 00 ∗ 0 δφ (q ) δJ(q) 00 δJ (q ) δJ(q) δφ (q ) Zq • use expression for δJ ∗/δφ∗ established above

2 2 0 δ Wk[J] δ Γk[φ] 0 00 =⇒ δ(q − q ) = ∗ 00 ∗ 0 00 + Rk(q) δ(q − q ) 00 δJ (q ) δJ(q) δφ (q ) δφ(q ) Zq ⇒ the scale dependent inverse propagator is given by − 2 1 0 δ Γ [φ] 0 G(q, q ) = k + R (q) δ(q − q ) δφ∗(q) δφ(q0) k (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

− 1 δ2Γ [φ] 1 ∂ Γ [φ] = (∂ R (q)) k + R (q) t k 2 t k δφ∗(q) δφ(q) k Zq Graphical representation (insertion stands for the derivative of the cutoﬀ function ∂tRk)

Γ 1 k 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)

1 (∂ R ) 1 ∂ Γ [φ] = Tr t k 6= ∂ Tr log(Γ(2)[φ] + R ) t k 2 (2) t 2 k k Γk [φ] + Rk ! Note that this is not equal to the total derivative of a one-loop eﬀective action (since the terms (2) ∂tΓk [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

(2) ∂tΓk[φ] = Fk[Γk ] 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 Rk(q) = 0 ⇒ lim ∆Sk[φ] = 0 k→0 k→0 and therefore for the scale-dependent generating functional

lim Zk[J] = Z[J] k→0 thus, by the properties we require from the cutoﬀ, it is trivially true that

lim Γk[φ] = −Wk→0[J] + Jφ − ∆Sk→0[φ] k→0 Zx = −W [J] + Jφ = Γ[φ] Zx 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 − Jφ + ∆Sk[φ] exp(Wk[J]) Zx

= exp − Jφ + ∆Sk[φ] Dχ exp −S[χ] + Jχ − ∆Sk[χ] Zx Z Zx

= Dχ exp −S[χ] + J(χ − φ) + ∆Sk[φ] − ∆Sk[χ] Z Zx • exp(Wk[J]) has been replaced by the path integral representation of Zk[J]. • now use φ as a “background ﬁeld”: χ = φ + χ0, Dχ ≡ Dχ0 (no assumptions necessary regarding its relation toR the minimR um of the classical action S[φ])

18 Exactness [3]

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

0 0 1 0 0 ∆S [φ + χ ] = ∆S [φ] + χ (R φ) + χ (R χ ) k k k 2 k Z Z • introduce this into expression

0 0 0 0 exp(−Γk[φ]) = Dχ exp −S[φ + χ ] + Jχ − ∆Sk[φ + χ ] + ∆Sk[φ] Z Z 0 0 0 0 0 = Dχ exp − S[φ + χ ] + Jχ − χ (Rkφ) −∆Sk[χ ] − ∆Sk[φ] + ∆Sk[φ] Z Z Z (∗) • use equation for φ to simplify term linear in|χ0 (φ remains{z unconstrained)} δΓ [φ] k = J − (R φ) δφ k • ﬁnd ﬁnally

0 0 δΓ [φ] 0 0 exp(−Γ [φ]) = Dχ exp −S[φ + χ ] + k χ − ∆S [χ ] k δφ k Z Z

19 Exactness [4]

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

1 0 2 0 0 lim exp − (χ ) ⇒ exp(−∆Sk→Λ[χ ]) → δ[χ ] →0 2 Z ⇒ exponential becomes a δ-functional (w/ appropriate normalization)! In the path integral

0 0 δΓk[φ] 0 0 lim exp(−Γk[φ]) = lim Dχ exp(−S[φ + χ ] + χ − ∆Sk[χ ]) k→Λ k→Λ δφ Z Z 0 0 δΓ [φ] 0 0 = Dχ exp(−S[φ + χ ] + k χ ) δ[χ ] δφ Z Z = 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 (2) derivatives Γk [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 (2) flow equation for the effective action (φ-dependence in Γk [φ]):

1 − ∂ Γ [φ] = Tr (∂ R )[Γ(2)[φ] + R ] 1 t k 2 t k k k n o • take derivatives (2) δ 1 (2) −1 δΓk (2) −1 ∂tΓk[φ] = Tr (∂tRk)(−1)[Γk + Rk] [Γk + Rk] δφ 2 ( δφ ) 1 − − = − Tr (∂ R )[Γ(2) + R ] 1 Γ(3) [Γ(2) + R ] 1 2 t k k k k k k n o Graphical representation:

(3) (1) ¡ ¡ ¡ Γk

1 ¡ ¡ ¡ ∂tΓ =− ∂tRk k 2 ¡ ¡ ¡ ¡ ¡ ¡

22 Higher n-point functions [2]

• one more derivative to get the ﬂow equation for the two-point function: 2 δ 1 − − − ∂ Γ [φ] = 2 × Tr (∂ R )[Γ(2) + R ] 1 Γ(3) [Γ(2) + R ] 1 Γ(3) [Γ(2) + R ] 1 δφδφ t k 2 t k k k k k k k k k n o 1 − − − Tr (∂ R )[Γ(2) + R ] 1 Γ(4) [Γ(2) + R ] 1 2 t k k k k k k n o Graphical representation [ﬁrst graph implies a factor 2]:

∂ R (3) ∂ R (3) t k

¢ £ ¢ £ ¢ £ t k ¤ ¥ ¤ ¥ ¤ ¥ (2) Γk Γk

1 ¢ £ ¢ £ ¢ £ ¤ ¥ ¤ ¥ ¤ ¥ 1 ∂tΓ = − (4) k 2 ¢ £ ¢ £ ¢ £ ¤ ¥ ¤ ¥ ¤ ¥ 2 Γ

¢ £ ¢ £ ¢ £ ¤ ¥ ¤ ¥ ¤ ¥ k ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § ¦ § What does this imply? (2) (3) (4) • To find flow equation for Γk , we need Γk and Γk ! (n) (n+1) (n+2) • In general, for flow of Γk , need Γk , Γk : ⇒ hierarchy of flow equations! How can we do meaningful calculations?

23 Truncations

(n) (n+1) (n+2) Problem: In order to calculate the ﬂow of Γk , we need Γk and Γk .

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,

1 (2) −1 0 ∂ Γ [φ] = (∂ R ) [Γ [φ] + R ] , shorthand: A B 0 = A(p, q)B(q, p ) t k 2 t k pq k k qp pq qp Zq 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 + ∆Γn n=1 X In terms of the flow equation, contributions of different loop orders can be identified ∞

∂tΓk = ∂t∆Γn,k n=1 X notation: m-point correlation function to n-loop order at scale k (m) Γn,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

δ2 δφδφ (2) (2) (2) Γn,k −→ Γn,k = Γn−1,k + ∆Γn,k

into ﬂow eq. (∂tRk) 1 −→ (2) = (∂tRk) (2) (2) Γn,k + Rk Γn−1,k + ∆Γn,k + Rk isolate 1 (2) 1 −→ ∂t∆Γn+1,k = (∂tRk) (2) ∆Γn,k (2) Γn−1,k + Rk Γn−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 (2) (2) Γk,0 = S Flow equation for the one-loop correction to the eﬀective action:

−1 1 (2) ∂t∆Γ1,k = (∂tRk)pq S + Rk 2 qp h i Integrate this to get the one-loop correction:

k 0 1 1 (2) ∆Γ = dk ∂ 0 [log(S + R )] 1,k k0 t 2 k pp ZΛ One-loop result corresponds with ordinary result (RΛ similar to Pauli-Villars regulator):

k 1 (2) 1 (2) 1 (2) Γ1,k = S + log(S + Rk) − log(S + RΛ) = S + log(S + Rk0 ) 2 pp 2 pp 2 pp Λ h i h i h i

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

2 2 k (2) δ 1 δ (2) 0 ∆Γ1,k = 0 ∆Γ1,k = 0 log(S + Rk ) qq0 δφ(q)δφ(q ) 2 δφ(q)δφ(q ) pp Λ h k i 1 (4) (3) (3) 0 000 00 0 = Gpp Spp0qq0 − Gpq Sq000q00qGq p Sp0pq0 2 Λ h i where one uses

−1 δ δ (3) 0 (2) 0 00 0 Gpp = S + Rk = (−1)Gpq Sq0q00qGq p δφ(q) δφ(q) pp0 h i 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 Λ = k 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

−1 −1 1 (2) 1 (2) (2) ∂tΓ2,k = (∂tRk)qp Γ1,k + Rk = (∂tRk)qp S + ∆Γ1,k + Rk 2 pq 2 pq h i−1 h i 1 (2) = (∂tRk)qp S + Rk + 2 pq h i −1 −1 1 (2) (2) (2) + (∂tRk)qp(−1) S + Rk ∆Γ1,k S + Rk + . . . 2 pq00 q00q0 q0q h i h i = ∂t∆Γ1,k + ∂t∆Γ2,k + . . .

Therefore we ﬁnd for the correction

1 (2) 0 00 ∂t∆Γ2,k = − (∂tRk)qpGpq ∆Γ1,k Gq q 2 q00q0 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 −1 (2) Gpq = S + Rk pq h i Look at derivatives of the propagator w.r.t. the RG scale k: −1 −1 −1 (2) (2) (2) ∂tGpq = ∂t S + Rk = (−1) S + Rk (∂tRk)q0q00 S + Rk pq pq0 q00q h i h i h i = (−1)Gpq0 (∂tRk)q0q00 Gq00q

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:

(4) 1 (4) G 0 S 0 0 00 (∂ 0 G) 00 0 = ∂ 0 G 0 S 0 0 00 G 00 0 pp pp q q t q q 2 t pp pp q q q q (3) (3) 1 (3) (3) G 0 S 00 G 00 00 S 00 0 0 (∂ 0 G) 0 = ∂ 0 G 0 S 00 G 00 00 S 00 0 0 G 0 pp pp q p q q p q t q q 3 t pp pp q p q q p q q 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

1 (2) 0 00 ∂t∆Γ2,k = − (∂tRk)qpGpq ∆Γ1,k Gq q 2 q00q0 1 (2) 0 00 = ∆Γ1,k (∂tG)q q 2 q00q0 • 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!)

k 1 1 (4) (3) (3) 0 0 00 00 0 0 Gpp Spp0qq0 − Gpp Spp00qGp q Sq00p0q0 (∂t G)qq 2 2 Λ h i 1 1 1 (4) 1 (3) (3) = ∂ 0 G 0 S 0 0 G 0 − G 0 S 00 G 00 00 S 00 0 0 G 0 2 2 t 2 pp pp qq qq 3 pp pp q p q q p q qq • 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

k k 0 1 1 1 (4) (3) (3) 0 0 00 00 0 0 ∆Γ2,k = dk 0 Gpp Spp0qq0 − Gpp Spp00qGp q Sq00p0q0 (∂t G)qq k 2 2 Λ ZΛ k h i 0 1 1 1 1 (4) 1 (3) (3) = dk ∂ 0 G 0 S 0 0 G 0 − G 0 S 00 G 00 00 S 00 0 0 G 0 k0 2 2 t 2 pp pp qq qq 3 pp pp q p q q p q qq ZΛ 1 4[ ] • result of the integration (up to regularization terms) k 1 (4) 1 (3) (3) ∆Γ = G 0 S 0 0 G 0 − G 0 S 00 G 00 00 S 00 0 0 G 0 2,k 8 pp pp qq qq 12 pp pp q p q q p q q q Λ 1 1 [8 12 ] 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: 1 1 1 S[φ] = (∂ φa)(∂µφa) + m2φ2 + λ(φ2)2 2 µ 2 4 Zx 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 1 1 Γ [φ] = U (ρ) + ∂ φaZ (ρ, −∂2)∂µφa + ∂ ρY (ρ, −∂2)∂µρ + . . . k k 2 µ k 4 µ k Zx 1 a a where ρ = 2 φ φ (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 ∂ 1 ∂ 1 N − 1 U (ρ) = R (q) + ∂t k 2 ∂t k M M Zq 1 0 2 2 2 0 M0(ρ, q ) = Zk(ρ, q )q + Uk(ρ) + Rk(q) 2 2 2 2 2 0 00 M1(ρ, q ) = Zk(ρ, q )q + ρYk(ρ, q )q + Uk(ρ) + 2ρUk (ρ) + Rk(q)

1 a a where ρ = 2 φ φ 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) Z q2 R (q) = k k 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: d η = − log Z (ρ (k), q2 = 0) dt k 0 • the wave function renormalization can be calculated from the two-point function (2) 2 2 2 Γk (ρ; p, q) = [Zk(ρ, q )q + M ]δ(p + q) (neglect explicit q-dependence of Zk(ρ)): 2 ∂ δ 2 Zk(ρ) = lim Γk[φ ] q2→0 ∂q2 δφ(q)δφ(−q)

∂ (2) 2 = lim Γk (φ ; q, −q) q2→0 ∂q2

⇒ 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? (2) 2 q2 −η/2 It’s the anomalous dimension η of the propagator, as in Γk ∼ q ( k2 + c) → 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: N + 8 17.26N + 75.95 β = λ2 − λ3 λ 16π2 (16π2)2

2 • need momentum and ﬁeld dependence of wave function renormalization Zk(ρ, q ) • result there in d = 4 coincides with the perturbative result: N + 8 9N + 42 β = λ2 − λ3 λ 16π2 (16π2)2

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 ∂ 1 ∂ N − 1 1 U (ρ) = R (q) + ∂t k 2 ∂t k M M Zq 0 1 (neglect ∂tZk from Rk/Zk → justified for small anomalous dimension)

1 ∂ N − 1 1 ddq R (N − 1) R (q) −→ ∂ k k d t U 0 (ρ) 2 ∂t M0 2 (2π) Zk 2 Rk 2 k q q + + k 2 Z Z Zk Zkk

d 1 1 Rk 1 1 d+1 d/2 d ∂ d l (w) = ∂t , = 2 π Γ(d/2), l (w) = − l − (w) 0 d 2 Rk 2 n n 1 4 vdk q Zk q + + k w vd ∂w Z Zk • w is a dimensionless variable d 2 2 • functions ln(w) are “threshold functions”, since they cut oﬀ modes with masses m k ⇒ theory becomes theory of eﬀective “light modes”!

39 O(N)-model: β-functions [2]

• in terms of the threshold functions ﬂow equation U 0 (ρ) U 0 (ρ) + 2ρU 00(ρ ∂ U (ρ) = 2v kd (N − 1)ld k + ld k k t k d 0 Z k2 0 Z k2 k k Now take lowest possible approximation (for symmetry breaking): quartic potential λ U (ρ) = k (ρ − ρ (k))2 k 2 0

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

d 0 0 00 U (ρ (k)) = ∂ U (ρ (k)) + U (ρ (k))∂ ρ (k) ≡ 0 dt k 0 t k 0 k 0 t 0 − − 2ρ (k)λ ⇒ ∂ ρ (k) = 2v kd 2Z 1 3 ld 0 k + (N − 1)ld(0) t 0 d k 1 Z k2 1 k ∂2 000 • ﬂow equation for coupling λk from second derivative ∂ρ2 ∂tUk(ρ) (note Uk ≡ 0 )

− − 2ρ (k)λ ∂ λ = 2v kd 4Z 2λ2 9 ld 0 k + (N − 1)ld(0) t k d k k 2 Z k2 2 k

40 O(N)-model: β-functions [3]

• introduce couplings rescaled according to canonical dimensions: 2−d ¯ −2 d−4 ρ¯0 = Zkk ρ0(k) λ = Zk k λk

• β-functions: flow equations of the dimensionless couplings (note where η = −∂t log Zk appears) d ¯ d ∂tρ¯0 = βρ = (2 − d − η)ρ¯0 + 2vd 3l1 (2λρ¯) + (N − 1)l1(0) ¯ ¯ ¯2 d ¯ d ∂tλ = βλ = (d − 4 + 2η)λ + 2vdλ 9l2 (2λρ¯) + (N − 1)l2(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! N+8 ¯2 ¯ d = 4: reproduce one-loop βλ = 16π2 λ , 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

−1 2 2 2 0 2 Gk (ρ, q ) = Zk(ρ, q )q + Uk(ρ) = M0(ρ, q ) − Rk(q) ansatz for the propagator (ﬁelds φa are constant expectation values):

1 0 1 00 Γ = (U (ρ) + Z (ρ, q2)q2)φa(q)φa(−q) + φaφb(2U (ρ) + Y (ρ, q2)q2)φa(q)φb(−q) (2)k 2 k k 2 k k Zq 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): 1 Γ = φaλ(1)(ρ; q , q )φa(q )φb(q )φb(−q − q ) + . . . (3)k 2 k 1 2 1 2 1 2 Zq1 Zq2 1 Γ = λ(2)(ρ; q , q , q )φa(q )φa(q )φb(q )φb(−q − q − q ) + . . . (4)k 8 k 1 2 3 1 2 3 1 2 3 Zq1 Zq2

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)

00 0 1 λ(1)(ρ; q , q ) = U (ρ) + q · (q + q )Z (ρ, q · (q + q )) + q2Y (ρ, q2) + . . . 1 2 k 2 1 2 k 2 1 2 2 1 k 1 (2) 00 0 0 λ (ρ; q1, q2, q3) = Uk (ρ) − q2 · q1Zk(ρ, −q2 · q1) − q4 · q3Zk(ρ, −q4 · q3) 1 + (q + q )2Y (ρ, (q + q )2) + . . . 2 1 2 k 1 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 2 (ρ-dependence of Mi(p ) suppressed): 1 ∂ G (ρ, q2) = ∂ R (p)× t k 2 t k Zp −2 2 −1 2 (1) 2 −2 2 −1 2 (1) 2 {4ρM1 (p )M0 ((p + q) )(λk (p, q)) + 4ρM0 (p )M1 ((p + q) )(λk (−p − q, q)) −2 2 (2) (2) −2 2 (2) −M0 (p )[(N − 1)λk (q, −q, p) + 2λk (q, p, −q)] − M1 (p )λk (q, −q, p)}

(1,2) • use continuity conditions from above to replace λk ! • get ﬂow equation for wave function renormalization

1 0 ∂ Z (ρ, q2) = (∂ G (ρ, q2) − ∂ U (ρ)) = −ξ (ρ, q2)Z (ρ, q2) t k q2 t k t k k k • actual anomalous dimension η from this

2 0 2 d 2 2 k ∂ 2 Zk(ρ0, k ) η = − log Zk(ρ0(k), k ) = ξk(ρ0, k ) − 2 2 2 Zk(ρ0, q ) − 2 ∂tρ0 dt Z (ρ , k ) ∂q 2 2 Z (ρ , k ) k 0 q =k k 0

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