Functional Renormalization Group and someApplications
A. Wipf
Theoretisch-Physikalisches Institut, FSU Jena
Humboldt Universität Berlin - NIC,DESY ZeuthenJoint Lattice Seminar
Andreas Wipf Functional Renormalization Group and some Applications
1 Introduction
2 Scale-dependent Functionals
3 Derivation of Flow Equations
4 Functional Renormalization in QM
5 Scalar Field Theories
6 Some applications
Andreas Wipf Functional Renormalization Group and some Applications
Introduction 3 / 45
particular implementation of the renormalization groupfor continuum field theory (symmetries ,)functional methods + renormalization group ideaconceptionally simple, technically demandingscale-dependent Schwinger functional or effective actionlarge values of scale parameter k : high resolution→ UVlowering k : including more and more fluctuations→ IRknown microscopic laws −→ macroscopic phenomena
Andreas Wipf Functional Renormalization Group and some Applications
flow of Schwinger functional Wk [j]: Polchinski equationflow of effective action Γk [ϕ]: Wetterich equationflow from classical action S[ϕ] to effective action Γ[ϕ]
applied to variety of physical systemsstrong interactionelectroweak phase transitionasymptotic safety scenariocondensed matter systene.g. Hubbard model, liquid He4, frustrated magnets, superconductivity . . .effective models in nuclear physicsultra-cold atoms
Andreas Wipf Functional Renormalization Group and some Applications
Γk=Λ = S
Γk=0 = Γ
Theory space
Andreas Wipf Functional Renormalization Group and some Applications
1 K. Aoki, Introduction to the nonperturbative renormalization group and its recentapplications,Int. J. Mod. Phys. B14 (2000) 1249.
2 J. Berges, N. Tetradis and C. Wetterich, Nonperturbative renormalization flow inquantum field theory and statistical physics,Phys. Rept. 363 (2002) 223.
3 H. Gies, Introduction to the functional RG and applications to gauge theories,in Lect. Notes Phys. Volume 852, Springer, Berlin (2012)
4 P. Kopietz, L. Bartosch and F. Schütz, Introduction to the functionalrenormalization group,Lect. Notes in Phys., Volume 798, Springer, Berlin (2010).
5 A. Wipf, Statistical Approach to Quantum Field Theory,Lect. Notes Phys., Volume 864, Springer, Berlin (2013).
Andreas Wipf Functional Renormalization Group and some Applications
Scale-dependent functionals 7 / 45
generating functional of (Euclidean) correlation functions
Z [j] =
∫Dφ e−S[φ]+(j,φ), (j , φ) =
∫ddx j(x)φ(x)
Schwinger functional W [j] = log Z [j]effective action = Legendre transform of W [j]
Γ[ϕ] = (j , ϕ)−W [j] with j(x) from ϕ(x) =δW [j]δj(x)
encodes properties of QFT in most economic way
Andreas Wipf Functional Renormalization Group and some Applications
add scale-dependent IR-cutoff term ∆Sk to S
Zk [j] =
∫Dφ e−S[φ]+(j,φ)−∆Sk [φ]
Scale-dependent Schwinger functional
Wk [j] = log Zk [j]
regulator: quadratic functional
∆Sk [φ] =12
∫ddxddy φ(x)Rk (x − y)φ(y)
→ one-loop structure of flow equation
Andreas Wipf Functional Renormalization Group and some Applications
Conditions on cutoff function Rk (p) 9 / 45
cutoff-function ∼ momentum-dependent mass
∆Sk [φ] =12
∫ddp
(2π)d φ∗(p)Rk (p)φ(p)
recover effective action for k → 0:
Rk (p)k→0−→ 0 for fixed p
recover classical action at UV-scale Λ:
Rkk→Λ−→ ∞
regularization in the IR:
Rk (p) > 0 for p → 0
Andreas Wipf Functional Renormalization Group and some Applications
Cut-offs in use 10 / 45
exponential regulator: Rk (p) =p2
ep2/k2 − 1,
optimized regulator: Rk (p) = (k2 − p2) θ(k2 − p2) ,
quartic regulator: Rk (p) = k4/p2 ,
sharp regulator: Rk (p) =p2
θ (k2 − p2)− p2 ,
Callan-Symanzik regulator: Rk (p) = k2
Andreas Wipf Functional Renormalization Group and some Applications
exponential cutoff function and its derivative
p2
Rk
k∂kRk
k2 2k2
k2
2k2
Andreas Wipf Functional Renormalization Group and some Applications
Polchinski equation 12 / 45
scale dependence of Wk from
∂k Wk [j] = −12
∫ddxddy 〈φ(x)∂k Rk (x , y)φ(y)〉k
relates to connected two-point function
W (2)k (x , y) ≡ δ2Wk [j]
δj(x)δj(y)= 〈φ(x)φ(y)〉k − ϕ(x)ϕ(y)
Polchinski equation
∂k Wk [j] =− 12
∫ddxddy ∂k Rk (x , y) W (2)
k (x , y)
−∫
ddxddy W (1)k ∂k Rk (x , y)W (1)
k (y)
Andreas Wipf Functional Renormalization Group and some Applications
Scale dependent effective action 13 / 45
average field of cutoff theory with j
ϕ(x) =δWk [j]δj(x)
≡W (1)k (x)
fixed j → ϕ depends on kfixed ϕ→ j depends on kmodified Legendre transformation:
Γk [ϕ] =(LWk
)[ϕ]−∆Sk [ϕ]
Legendre transform(LWk
)(ϕ) = (j , ϕ)−Wk [j]
Γk not L-transform of Wk [j] for k > 0!need not be convex, but Γk→0 = Γ convex
Andreas Wipf Functional Renormalization Group and some Applications
Derivation of Wetterich equation 14 / 45
vary effective average action (ϕ fixed)
δΓk [ϕ]
δϕ(x)=
∫δj(y)
δϕ(x)ϕ(y) + j(x)−
∫δWk [j]δj(y)
δj(y)
δϕ(x)− δ∆Sk [ϕ]
δϕ(x)
red terms chancel→ effective equation of motion
δΓk [ϕ]
δϕ(x)= j(x)− δ
δϕ(x)∆Sk [ϕ] = j(x)− (Rkϕ)(x)
flow equation: ϕ fixed, j depends on scale, differentiate Γk
∂k Γk [ϕ] =
∫ddx ∂k j(x)ϕ(x)− ∂k Wk [j]−
∫∂Wk [j]∂j(x)
∂k j(x)− ∂k ∆Sk [ϕ]
PE=
12
∫ddxddy ∂k Rk (x , y) W (2)
k (x , y)
Andreas Wipf Functional Renormalization Group and some Applications
∂k Wk [j] only from k -dependence of the parameters
∂k Γk [ϕ] = −∂k Wk [j]− ∂k ∆Sk [ϕ]
= −∂k Wk [j]− 12
∫ddxddy ϕ(x)∂k Rk (x , y)ϕ(y)
use Polchinski equation→
∂k Γk [ϕ] =12
∫ddxddy ∂k Rk (x , y) W (2)
k (x , y)
second derivative of Wk vs. second derivative of Γk :
ϕ(x) = W (1)k (x) and j(x) = Γ
(1)k (x) +
∫ddy Rk (x , y)ϕ(y)
Andreas Wipf Functional Renormalization Group and some Applications
chain rule→
δ(x − y) =
∫ddz
δϕ(x)
δj(z)
δj(z)
δϕ(y)
=
∫ddz W (2)
k (x , z){
Γ(2)k + Rk
}(z, y)
relation between curvatures
W (2)k =
1
Γ(2)k + Rk
insert into ∂k Γk ⇒
Wetterich equation
∂k Γk [ϕ] =12
tr
(∂k Rk
Γ(2)k [ϕ] + Rk
)
Andreas Wipf Functional Renormalization Group and some Applications
non-linear functional integro-differential equation
full propagator Γ(2)k [ϕ] enters flow equation
Polchinski and Wetterich equations = exact FRG equationsPolchinski: simple polynomial structure (structural investigations)Wetterich: rational structure stabilizes flow (explicit calculations)UV and IR regulatedin practice: truncation = projection onto finite-dim. spaceproblem: reliable error estimate for flow→ improve truncation, vary regulator, check stability
Andreas Wipf Functional Renormalization Group and some Applications
Functional renormalization in QM 18 / 45
anharmonic oscillator
S[q] =
∫dτ(
12
q2 + V (q)
),
here LPA (local potential approximation)
Γk [q] =
∫dτ(
12
q2 + uk (q)
)leading order in gradient expansionscale-dependent effective potential uk
neglected: higher derivative terms, mixed terms qnqm
Andreas Wipf Functional Renormalization Group and some Applications
in denominator of flow equation Γ(2)k = −∂2
τ + u′′k (q)
LPA: may consider constant q → momentum space
(∂k Γk )[q] =∂
∂k
∫dτ uk (q) =
12
∫dτ
∞∫−∞
dp2π
∂k Rk (p)
p2 + u′′k (q) + Rk (p)
optimal regulator
Rk (p) = (k2 − p2) θ(k2 − p2) =⇒ ∂k Rk (p) = 2kθ(k2 − p2)
⇒ non-linear partial differential equation for uk :
∂k uk (q) =1π
k2
k2 + u′′k (q)
Andreas Wipf Functional Renormalization Group and some Applications
free particle limit fixes subtraction in flow equation (Λ→∞)
∂k uk (q) =1π
(k2
k2 + u′′k (q)− 1)
= −1π
u′′k (q)
k2 + u′′k (q)
assume uΛ(q) even→ uk (q) evenpolynomial ansatz
uk (q) = Ek +ω2
k2
q2 +λk
4!q4 +
∑n=3,4,...
1(2n)!
a2n,k q2n ,
Andreas Wipf Functional Renormalization Group and some Applications
insert and compare coefficients→ infinite set of coupled ode’s
dEk
dk= −1
πω2
k ∆0, ∆0 =1
k2 + ω2k,
dω2k
dk= −k2
πλk ∆2
0 ,
dλk
dk= −
k2∆20
π
(a6,k − 6λ2
k ∆0),
da6,k
dk= −
k2∆20
π
(a8,k − 30λk a6,k ∆0 + 90λ3
k ∆20),
......
initial condition: classical potential at cutoffprojection onto space of polynomials up to given degree n
Andreas Wipf Functional Renormalization Group and some Applications
e.g. crude truncation a6 = a8 = · · · = 0truncated finite system of flow equations
dEk
dk= −
ω2kπ
∆0,dω2
kdk
= −k2λk
π∆2
0,dλk
dk=
6k2λ2k
π∆3
0
at cutoff k = Λ: EΛ = 0, ωΛ = 1, varying λΛ at the cutoff scale→ scale-dependent couplingsdominant contribution from near typical scale k ≈ ω
Andreas Wipf Functional Renormalization Group and some Applications
k
Ek
.1
.2
.3
.4
.5
1 2 3 4
from above:
λ = 2.0λ = 1.0λ = 0.5λ = 0.0
k
ω2k
1.0
1.1
1.2
1.3
1.4
1 2 3 4
from above:
λ = 0.0λ = 0.5λ = 1.0λ = 2.0
The flow of the couplings Ek and ω2k (EΛ = 0, ω2
Λ = 1).
Andreas Wipf Functional Renormalization Group and some Applications
ω = ωk=0 > 0 ⇒ minimum of u0 at origin⇒ E0 = u0(0)
energy of first excited state
E1 = E0 +√
u′′0 (0) = E0 + ω0
reasonably good results with 3-parameter truncation
units of ~ω: energies for different λ; different regulators
ground state energy energy of first excited statecutoff optimal optimal Callan exact optimal optimal Callan exact
order 4 order 12 order 4 result order 4 order 12 order 4 resultλ = 0 0.5000 0.5000 0.5000 0.5000 1.5000 1.5000 1.5000 1.5000λ = 1 0.5277 0.5277 0.5276 0.5277 1.6311 1.6315 1.6307 1.6313λ =2 0.5506 0.5507 0.5504 0.5508 1.7324 1.7341 1.7314 1.7335λ = 3 0.5706 0.5708 0.5703 0.5710 1.8177 1.8207 1.8159 1.8197λ = 4 0.5885 0.5889 0.5882 0.5891 1.8923 1.8968 1.8898 1.8955λ = 5 0.6049 0.6054 0.6045 0.6056 1.9593 1.9652 1.9562 1.9637λ = 6 0.6201 0.6207 0.6196 0.6209 2.0205 2.0278 2.0168 2.0260λ = 7 0.6343 0.6350 0.6336 0.6352 2.0771 2.0857 2.0728 2.0836λ = 8 0.6476 0.6484 0.6469 0.6487 2.1299 2.1397 2.1250 2.1374λ = 9 0.6602 0.6611 0.6594 0.6614 2.1794 2.1905 2.1741 2.1879λ = 10 0.6721 0.6732 0.6713 0.6735 2.2263 2.2385 2.2205 2.2357λ = 20 0.7694 0.7714 0.7679 0.7719 2.5994 2.6209 2.5898 2.6166
Andreas Wipf Functional Renormalization Group and some Applications
negative ω2: mexican hat
∂k uk (q) = −1π
u′′k (q)
k2 + u′′k (q)
denominator positive for large scales⇒ remains positive during the flowflow equation⇒uk (q) increases toward infrared if u′′k (q) is positiveuk (q) decreases toward infrared if u′′k (q) is negativek2 + u′′k minimal at q = 0⇒ double-well potential flattens during flow, becomes convexexpected on general grounds
Andreas Wipf Functional Renormalization Group and some Applications
solution of partial differential equation, ω2 = −1, λ = 1
q
uk
v
.5
V
uk=0
Andreas Wipf Functional Renormalization Group and some Applications
increasing barrier (weak coupling)→ increasingly difficultnumerical solution does bettersplitting induced by instanton effects: beyond leading order LPA
units of ~ω: varying λ; optimized regulator,ground state energy energy of first excited state
optimal optimal pde exact optimal optimal pde exactorder 4 order 12 order 4 order 12
λ = 1 -0.8732 -0.8556 -0.7887 -0.8299λ = 2 -0.2474 -0.2479 -0.2422 0.0049 0.0063 -0.0216λ = 3 0.2473 -0.0681 -0.0679 -0.0652 -0.2241 0.3514 0.3500 0.3307λ = 4 -0.0186 0.0286 0.0290 0.0308 0.3511 0.5753 0.5755 0.5598λ = 5 0.0654 0.0949 0.0953 0.0967 0.5835 0.7455 0.7462 0.7324λ = 6 0.1234 0.1457 0.1461 0.1472 0.7509 0.8842 0.8851 0.8723λ = 7 0.1688 0.1871 0.1876 0.1885 0.8851 1.0021 1.0030 0.9909λ = 8 0.2063 0.2223 0.2228 0.2236 0.9987 1.1052 1.1061 1.0944λ = 9 0.2671 0.2530 0.2535 0.2543 1.1863 1.1972 1.1981 1.1866λ = 10 0.2386 0.2803 0.2808 0.2816 1.0978 1.2805 1.2814 1.2701λ = 20 0.4536 0.4632 0.4639 0.4643 1.7866 1.8638 1.8648 1.8538
Andreas Wipf Functional Renormalization Group and some Applications
Scalar Field Theory 28 / 45
LP approximation
Γk [ϕ] =
∫ddx
(12
(∂µϕ)2 + uk (ϕ)
)scale-dependent effective potential
∂k uk (q) =12
∫ddp
(2π)d∂k Rk (p)
p2 + u′′k (q) + Rk (p)
optimized regulator: p-integration doable→ flow equation
∂k uk (ϕ) = µdkd+1
k2 + u′′k (ϕ), µd =
(4π)−d/2
Γ( d2 + 1)
space-time dimensions in kd+1 and µd
nonlinear PDE
Andreas Wipf Functional Renormalization Group and some Applications
series expansion (even)→ flow equations for infinite set of couplings
kda0
dk= +µdkd+2∆0, ∆0 =
1k2 + a2
,
kda2
dk= −µdkd+2∆2
0a4 ,
kda4
dk= −µdkd+2∆2
0(a6 − 6a2
4∆0),
kda6
dk= −µdkd+2∆2
0(a8 − 30a4a6∆0 + 90a3
4∆20),
......
Andreas Wipf Functional Renormalization Group and some Applications
Fixed points 30 / 45
point in space of dimensionless couplings where all β-functions vanish
K2
K1
line of critical points
T < Tc
T > Tc
(K∗1,K∗
2): fixed point
(K1c,K2c): critical point
Andreas Wipf Functional Renormalization Group and some Applications
critical hyper-surface on which ξ =∞RG trajectory moves away from critical surfaceIf flow begins on critical surface→ stays on surfacegenerically: critical points are not fixed pointd ≥ 3 : expect finite set of isolated fixed points K ∗ = (K ∗1 ,K
∗2 , . . . )
RG flow in the vicinity of fixed point K = K ∗ + δKlinearize flow around fixed point
K ′i = K ∗i + δK ′i = Ri(K ∗j + δKj
)= K ∗i +
∂Ri
δKj
∣∣K∗δKj + O(δK 2)
Andreas Wipf Functional Renormalization Group and some Applications
linearized RG transformation,
δK ′i =∑
j
M ji δKj , M j
i =∂Ri
∂Kj
∣∣∣K∗
left-eigenvectors Φαand eigenvalues λα = eyα of Myα > 0: flow repelled by K ∗ → relevant perturbationyα < 0: flow attracted to K ∗ → irrelevant perturbationrel. couplings/exponents→ IR-physics (scaling laws, TD criticalexponents)
Andreas Wipf Functional Renormalization Group and some Applications
Fixed point analysis for scalar models 33 / 45
introduce dimensionless field and potential,
ϕ = k (d−2)/2√µd χ and uk (ϕ) = kdµdvk (χ)
flow equation in terms of dimensionless quantities (rescaled by k#)
k∂k vk + dvk −d − 2
2χv ′k =
11 + v ′′k
, v ′k =∂vk
∂χ. . .
fixed point equation for effective potential (ode)
dv∗ −d − 2
2χv ′∗ =
11 + v ′′∗
constant solution dv∗ = 1→ trivial Gaussian fixed point
Andreas Wipf Functional Renormalization Group and some Applications
scale invariance at fixed point⇒ fixed point ∼ CFT
are there non-Gaussian fixed point solutions?answer depends on spacetime dimension d2d theories:∞ many fixed-point scalar models [min. models; Morris 1994]∞ many fixed-point Yukawa models [Synatschke, AW, ...]3d theories:small number of fixed-pointsfrom various truncations: series expansions and numerical solutionsexample: LPA and polynomial truncation for scalar field theory
w =m∑
n=0
cnρn, ρ =
12χ2
Andreas Wipf Functional Renormalization Group and some Applications
at fixed point: βn(c∗0 , c∗1 , . . . ) = 0 for n = 0,1,2, . . .
always Gaussian fixed point: c∗0 = 13 , 0 = c∗2 = c∗3 = c∗4 = . . .
search for non-trivial fixed pointsonly one for scalar field in d = 3 (even u):m = 20⇒ c∗1 = −.186066m = 42⇒ c∗1 = −.186041⇒ c∗0 , c
∗2 , . . . , c
∗m−1 ⇒ polynomial approximation to fixed point solution
fixed-point coefficients n!c∗n
c∗0 c∗1 c∗2 c∗3 c∗4 c∗5 c∗6m = 20 0.409534 -0.186066 0.082178 0.018981 0.005253 0.001104 -0.000255m = 42 0.409533 -0.186064 0.082177 0.018980 0.005252 0.001104 -0.000256
c∗7 c∗8 c∗9 c∗10 c∗11 c∗12 c∗13m = 20 -0.000526 -0.000263 0.000237 0.000632 0.000438 -0.000779 -0.002583m = 42 -0.000526 -0.000263 0.000236 0.000629 0.000431 -0.000799 -0.002643
c∗14 c∗15 c∗16 c∗17 c∗18 c∗19 c∗20m = 20 -0.002029 0.007305 0.028778 0.034696 -0.077525 -0.381385 0.000000m = 42 -0.002216 0.006677 0.026544 0.026320 -0.110498 -0.517445 -0.587152
Andreas Wipf Functional Renormalization Group and some Applications
Polynomial approximations vs. numerical solution 36 / 45
numerics: shooting method with seventh-order Runge-Kuttaw∗
0 1 2 3 4 5 6 7 8 9 10 11
1
2
3
4
5
6
m = 10
m = 20
m = 30
m = 40
numerical
solution
Andreas Wipf Functional Renormalization Group and some Applications
Critical exponents
m ν = −1/ω1 ω2 ω3 ω4 ω510 0.648617 0.658053 2.985880 7.502130 17.91349414 0.649655 0.652391 3.232549 5.733445 9.32485818 0.649572 0.656475 3.186784 5.853987 9.14109322 0.649554 0.655804 3.170538 5.977066 8.52281126 0.649564 0.655629 3.182910 5.897290 8.84463230 0.649562 0.655791 3.180847 5.903039 8.90760734 0.649561 0.655749 3.178636 5.922910 8.70258338 0.649562 0.655731 3.180577 5.908885 8.81422542 0.649562 0.655755 3.180216 5.909910 8.84738646 0.649562 0.655746 3.179541 5.915754 8.738608
convergencetwo relevant operators: ω0 = −3 and ω1 = −1/νLPA-prediction: ν = 0.649562 (high-T expansion: ν = 0.630)
Andreas Wipf Functional Renormalization Group and some Applications
Wave function renormalization 38 / 45
next-to-leading in derivative expansion→ Zk (p, ϕ)
simple LPA’ truncation (no p and ϕ-dependence)
Γk [ϕ] =
∫ddx
(12
Zk (∂µϕ)2 + uk (ϕ)
).
flow of Zk : project flow on operator (∂φ)2
must admit inhomogeneous ϕ in flow equation⇒ anomalous dimension
η = −k∂k log Zk
Andreas Wipf Functional Renormalization Group and some Applications
Application: linear O(N) models susy: Litim, Synatschke, Heilmann, AW 39 / 45
scalar field φ ∈ RN , O(N) invariant potentialfixed-point analysis: dimensionless field χ and composite field
% =12χ · χ
dimensionless potential νk (%).N − 1 Goldstone modes, one massive radial modeGoldstone modes drive flow for large Nfinds one Wilson-Fisher fixed point w∗ for all Nfixed point analysis: wk = w∗ + δk
fluctuation δk obeys the linear differential equation
Andreas Wipf Functional Renormalization Group and some Applications
polynomial truncation to high order (40)
N 1 2 3 100 1000−w ′∗(0) 0.186064 0.230186 0.263517 0.384172 0.387935ν = −1/ω1 0.64956 0.70821 0.76113 0.99187 0.99923ω2 0.6556 0.6713 0.6990 0.97218 0.99844ω3 3.1798 3.0710 3.0039 2.98292 2.99554
asymptotic formulas, e.g. ν ≈ 0.9998− 0.9616N
Large-N limit
flow equation can be solved (characteristics), exact relation w∗ ⇔ δ in
w(t , ρ) ≈ w∗(ρ) + eωtδ(ρ), δ = w ′ (ω+d)/2∗ t = log
kΛ
perturbation non-singular→ all critical exponents
ωn ∈ {2n − d |n = 0,1,2, . . . }
Andreas Wipf Functional Renormalization Group and some Applications
Special features of susy theories 41 / 45
on-shell supersymmetry: susy transformations non-linear⇒ no supersymmetric quadratic regulator existsoff-shell supersymmetry: susy transformation linearsupersymmetric quadratic regulator constructed Synatschke, Gies, AW
cutoff function bosons⇔ cutoff function fermions
constructed manifest supersymmetric FRG
N = 1 theory in d = 4: non-renormalization of superpotentialN = 1 theories in d < 4: running superpotential, running couplingsN = 2 theories in d < 4: non-renormalization of superpotentialinstead: flow of Kähler-potential
Andreas Wipf Functional Renormalization Group and some Applications
N = 2 Wess-Zumino model in 3 dimensions 42 / 45
Lagrangian density: complex φ,F , Dirac ψ
L = −14
∫d2θd2θΦΦ† −
{ 12i
∫d2θW + h.c.
}= |∇φ|2 + iψσ∇ψ − |F |2 −
{∂W∂φ
F − 12∂2W∂φ2 ψ
Tσ2ψ + h.c.}
(nonlocal) susy regulator with cutoff functions r1(k ,p) and r2(k ,p)
flow of Kähler metric
Γk =
∫d3x(
Z 2k (φ)
{|∇φ|2 + iψσ∇ψ − |F |2
}−{∂Wk
∂φF − 1
2∂2Wk
∂φ2 ψTσ2ψ + h.c.})
⇒ Kähler metric Z 2∗ at fixed point P. Feldmann, L. Zambelli, AW, 2018
Andreas Wipf Functional Renormalization Group and some Applications
Some results on supersymmetric systems 43 / 45
FP-structure in 2 ≤ d ≤ 4non-renormalization theorems with F. Synatschke and M. Heilmann
flow with higher derivative operators with R. Flore, T. Hellwig, B. Knorr and O. Zanusso
exact solution for Wk for susy O(N →∞) model with M. Heilmann and D. Litim
fixed point of nonlinear susy O(N) model with R. Flore and O. Zanusso
spike plots for susy theories with T. Hellwig and O. Zanusso
emergence of supersymmetry in Yukawa models with T. Hellwig and L. Zambelli
flow of Kähler metric in N = 2 in d = 3 with P. Feldmann and L. Zambelli
spontaneouly breaking of susy with H.Gies, F. Synatschke
particle massesfinite-temperature phase transition with J. Braun, F. Synatschke
Andreas Wipf Functional Renormalization Group and some Applications
many FRG studies of Gauge Theories and Gravityproblem with local gauge invariancebackground-field methodbimetric approach, .....
missing: convincing FRG studies for SYMsusy cutoffs exist (beyond WZ gauge)mixing of susy transformations with gauge transformationsneeded for comparison with lattice studiesextract effective theories
supersymmetric lattice gauge theoriesmany lattice-results in d = 1,2 many; D. August, B. Wellegehausen, AW
view lattice simulations in d = 4 Münster-DESY collaboration
M. Steinhauser, A. Sternbeck, B. Wellegehausen, AW
Andreas Wipf Functional Renormalization Group and some Applications
thanks to ... 45 / 45
Coworkers:Jens Braun,Holger Gies, Ilya Shapiro; Boris Merzlikin, Luca Zambelli, Omar Zanusso; Polina Feldmann, Raphael
Flore, Marianne Heilmann, Tobias Hellwig, Franziska Synatschke
JHEP 1712 (2017) 132 JHEP 1502 (2015) 109 JHEP 0903 (2009) 028
Phys. Rev. D96 (2017) 125007; Phys. Rev. D92 (2015) 085027; Phys. Rev. D87 (2013) 06501
Phys. Rev. D86 (2012) 105006; Phys. Rev. D84 (2011) 125009; Phys. Rev. D81 (2010) 125001
Phys. Rev. D80 (2009) 085007; Phys. Rev. D80 (2009) 101701
Andreas Wipf Functional Renormalization Group and some Applications