functional renormalization group in fermionic...
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Functional Renormalization Groupin fermionic systems
A. Jakovac
Dept. of Atomic PhysicsEotvos Lorand University Budapest
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 1 / 36
Outlines
1 Introduction: FRG and fermions
2 LPA approximation for fermions
3 Fermion systems at finite chemical potential
4 Higgs stability bound and irrelevant operators
5 Conclusions
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 2 / 36
Outlines
1 Introduction: FRG and fermions
2 LPA approximation for fermions
3 Fermion systems at finite chemical potential
4 Higgs stability bound and irrelevant operators
5 Conclusions
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 3 / 36
FRG: a generic view
The behaviour of a subset of the world is governed by somerelevant concepts
eg. for a free falling body, the concepts are the time, positionof the body, velocity, acceleration; eventually air resistance
laws determine connections of the relevant concepts, eg.equation of motion
experience: to different segments of the world belongsdifferent set of relevant concepts ⇒ scientific disciplineseg. quantum mechanics, chemistry, biology, society, astronomy are
all use different concepts.
Their elevant concepts are mutually irrelevant for others.
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 4 / 36
Quantum Field Theory representation
in QFT we represent the concepts by fields,the laws by effective action coming from path integral
eW [J] =
∫Dφ e iS[φ]+Jφ, iΓ[Φ] = W [J]− JΦ, Φ =
δW [J]
δJ.
advantage: constructive definition
for S → S + 12
∫ΦRΦ the change in the effective action Γ
δΓ =i
2STr δR(Γ(1,1) + R)−1 =
i
2δRG
Wetterich equation ( J. Polchinski (1984), C. Wetterich (1993), T. Morris (1994) )
⇒ formally a one-loop equation
alternative definition of the QFT
effect of fermions:STr: trace with fermion loops −1 factorwe need left-right derivative in case of fermionic modes.
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 5 / 36
Regulator
In one-parameter scaling require
if Rk→∞ →∞, no fluctuations are allowed: Γk→∞ = S
if Rk→0 → 0, no correction: Γk→0 = Γ.
⇒ Γk alternates between the classical and effective action.
Some regulator types:
sharp cutoff: Rk(p) =∞Θ(p − k)
⇒ Γk is the perfect actioneW [J] =
∫p<kDΦp e
iΓk [φ]+JΦ
R (p)k
p
8
sharp cutoff
Litim’s regulator
k
k2 2
2
(F.J. Wegner, A. Houghton PRA8 (1973) 401; P. Hasenfratz and F. Niedermayer, NPB414 (1994) 785)
optimized regulator ⇒ momentum regularizationpreg = p [ pΘ(p − k) + kΘ(k − p)] (D. F. Litim, Nucl.Phys. B 631, 128 (2002))
Γk not exactly the perfect action, but similar concept.
CSS regulator: multi-parameter form interpolating betweenchoices (I. Nandori, JHEP 1304, 150 (2013)).
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 6 / 36
Evaluation of the STr in presence of fermionic background
Computation of the STr:
background Φ bosonic as well as ψ fermionic: Γ[Φ, ψ, ψ]
kernel of quadratic fluctuations ϕ, ξ around them
fermionic background can link ξ to ξ and ϕ!
in Nambu representation Ψ =
(ψψT
)Fermion matrix:
(Γ(1,1)k,ΨΨ)ij =
∂
∂ΨiΓk [Ψ]
∂
∂Ψj=
(Γψψ ΓψψΓψψ Γψψ
)Boson-fermion mixing:
(Γ(1,1)k )ij =
(ΓΨΨ ΓΨΦ
ΓΦΨ ΓΦΦ
)These are hypermatrices
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 7 / 36
Evaluation of the tracelog
(A.J., I. Kaposvari, A. Patkos, arXiv:1510.05782)
Computation: STr ∂kRk(Γ(2) + Rk)−1 = ∂k STr ln(Γ(2) + Rk)
Representation(ΓΨΨ ΓΨΦ
ΓΦΨ ΓΦΦ
)=
(1 C0 1
)(0 BA 0
)(1 C0 1
)where A, B, C , C comes consistency. The Tr log of the RHS is easyto evaluate (α = ±1 for bosons/fermions)
STr log Γ(1,1)k = αΨ Tr log ΓΨΨ + αΦ Tr log ΓΦΦ + αΦ Tr log(1− Γ−1
ΦΦΓΦΨΓ−1ΨΨΓΨΦ)
Diagrammatically (for 3-point fermion-boson couplings):
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 8 / 36
Outlines
1 Introduction: FRG and fermions
2 LPA approximation for fermions
3 Fermion systems at finite chemical potential
4 Higgs stability bound and irrelevant operators
5 Conclusions
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 9 / 36
Treatment of the Wetterich equation
With On[ϕ] operator basis: Γk [ϕ] =∑n
gn(k)On[ϕ]
Expanding RHS of FRG eq.
∂kgn(k) = βn(g)
gn(k) curves: running couplings
βn = 0 fixed points
General operator basis is too large. . .
we should include all operators, also the most exotic ones
in practice we need an Ansatz
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 10 / 36
Local Potential Approximation
Local Potential Approximation (LPA) + wave fnct. renorm. (LPA′)
eg . : ΓLPA =
∫ddx
[Zϕ2
(∂µΦ)2 + Uk(Φ)
]Wetterich equations at finite temperature/chemical potential
∂kRk(p) = 2kΘ(p < k) ⇒ no momentum dependence
G ∼ (1∓ n±(ω)) ⇒ distribution functions
Generic structure of the potential evolution:(Ω−1d = 2(4π)d/2Γ( d
2+ 1))
δRG ⇒ ∂kUk = Ωd−1kd∑i
gin±(ωik ± µ)± 1
ωik
gi multiplicities
ω2ik = k2 + m2
i (Φ) with background dependent masses;in 1-component scalar model ω2
ik = k2 + ∂2ΦU.
if m2i =constant we recover the one loop expression.
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 11 / 36
Local fermionic potential approximation
Does LPA′ work in the fermionic case?
Γk [Ψ]→∫ddx
[Zk ψ∂/ψ + Uk(Ψx)
].
Problem with this approach
ψ2i (x) = 0 because of the fermionic nature⇒ (ψCψ)n = 0 for large enough n!⇒ Uk(Ψ) is a finite polynomial?
But. . .
after bosonization Φ = ψMψ, we can have Ub(Φ) arbitrarypotential
perturbation theory: Γ(n)(x1, . . . , xn) =⟨Tψ(x1) . . . ψ(xn)
⟩6= 0
even in the local limit
⇒ we should properly define the fermionic potential(K.-I. Aoki, K. Morikawa, J.-I. Sumi, H. Terao and M. Tomoyose, Phys. Rev. D61 (2000) 045008)
(AJ, A. Patkos, Phys.Rev. D88 (2013) 065008)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 12 / 36
Understanding fermionic LPA
Assumption behind the LPA
propagators vary in spacetime much slower than vertices
evolution from full Γ
n1x = x = ... x2 n1 2y = y = ... y
LPA=⇒
local approximation
yx
numerical values of the diagrams are close
vertex of the 1st diagram: Γ(n)k (x1, . . . , xn)Ψ(x1) . . .Ψ(x2n)
vertex of the 2nd diagram comes from a limiting process:
U(n)k lim
∆V→0
(1
∆V
∫∆V
ψ(x)ψ(x)
)nnotation−→ U
(n)k (ψ(x)ψ(x))n
heuristically: position is not a point, but a patch(k sets the resolution/compositeness scale)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 13 / 36
Gross-Neveu model with fermionic LPA
(AJ., A. Patkos, P. Posfay, Eur.Phys.J. C75 (2015) 2, arXiv:1406.3195 [hep-th])
Gross-Neveu model: matter content ψi , i = 1 . . .Nf
S [Ψ] =∫ddx
[Nf∑i=1
ψi∂/ψi +g
2NfI
]where I =
(Nf∑i=1
ψiψi
)2
.
chiral symmetry: ψ → −γ5ψ, ψ → ψγ5
O(Nf ) flavour symmetry
Γk [Ψ] must depend on invariant fermion bilinears
Ansatz for the effective action: take just the original invariant I
Γk [Ψ] =∫ddx
[Zk
Nf∑i=1
ψi∂/ψi + U(I )
]could depend on other invariants (eg. (ψγµψ)2)
it is self-consistent to assume just I dependence!
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 14 / 36
Computation overview
General structure of the expressions: Γ(2) ∼ G−10 −#Ψ⊗ΨT
⇒ inverse, tracelog can be computed
no flavour mixing ⇒ use background ψ = (ζ, . . . , ζ)
use static background
considerable cancellations occur
Evaluating the integrals using Litim’s regulator we find
∂kUk = kd+1Qd
[4Nf + 1
Z 2k2 + 4IU ′2− 1
Z 2k2 + 4IU ′(U ′ + U)
]where Q−1
d = (4π)d/2Γ( d2 + 1), U = 2U ′ + 4IU ′′.
Fermionic & bosonic type terms! (without explicit bosons)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 15 / 36
Flow and fixed points
For 2 < d < 4: one nontrivial fixed point for any Nf :Flow pattern d = 3
critical exponents (agree with bosonized version for Nf =∞)
Θn = d − 2n
(1 +
(n − 1)(n − 2)
2Nf − 1
)for d = 3 only Θn=1 = 1 is relevant.
d = 2 only Gaussian fixed pointd = 4 non-Gaussian FP→∞ for Nf =∞.
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 16 / 36
Fixed point potential
nth order potential: noconvergence for larger x values.
exact asymptotics can beobtained: y∗as ∼ x
d2(d−1)
⇒ Pade resummation
Physical regime x > 0
⇒ U < 0 as we expected.
physical point is at Ψ = 0.Nf = 2 potentials
Resummation:
y∗(x) = (1 + x2)d
4(d−1) limN→∞
PadeNN
[∑2Nn=1
1n `∗nx
n
(1 + x2)d
4(d−1)
],
PadeNN : resum polynomials with degree 2N to ratio of polynomials with
degree N.
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 17 / 36
Outlines
1 Introduction: FRG and fermions
2 LPA approximation for fermions
3 Fermion systems at finite chemical potential
4 Higgs stability bound and irrelevant operators
5 Conclusions
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 18 / 36
Finite chemical potential
( G.G. Barnafoldi, A. Jakovac, P. Posfay, arXiv:1604.01717 [hep-th] )
Consider a Yukawa-type fermionic model, with the Ansatz
Γ[ϕ,ψ] =
∫d4x
[ψ(i /∂ − gϕ)ψ +
1
2(∂µϕ)2 − U(ϕ)
].
The FRG equations (ω2B = k2 + ∂2
ϕU, ω2F = k2 + g2ϕ2.)
∂kUk =k4
12π2
[1 + 2nB(ωB)
ωB+ 4−1 + nF (ωF − µ) + nF (ωF + µ)
ωF
]At finite T , µ it is a regular differential equation(T. K. Herbst, J. M. Pawlowski and B. J. Schaefer, PRD88 014007 (2013))
( M. Drews, T. Hell, B. Klein and W. Weise, PRD88 096011 (2013))
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 19 / 36
T=0 equation
At zero temperature
∂kUk =k4
12π2
[1
ωB− 4
Θ(ωF − µ)
ωF
]
gφ
Λ
k
µ
µ
D
D
<
>SF
Fermi surface is atωF = µ ⇒ k2 + g2ϕ2 = µ2
an ellipse.
Direct polynomial expansion is impossible because of the jump innF (ωF − µ) for small T .Instead: Solve the equations separately in D> and D<, requirecontinuous solution at the Fermi surface.
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 20 / 36
Solution strategy: the UV domain
D> domain
∂kUk =k4
12π2
[1
ωB− 4
ωF
]the same as for µ = 0 ⇒ find theU> solution using standard methods
gφ
Λ
k
µ
µ
D
D
<
>SF
simplest Ansatz: polynomial expansion to quartic order
U>(k , ϕ) =1
2m2
k +λk24ϕ4,
initial conditions m20, λ0 can be determined from computing
physical observables.
value of the potential at the Fermi surface ϕ = ϕF (k)
U<(k , ϕF (k)) = U>(k , ϕF (k)) ≡ V (k).
⇒ provides boundary conditions for D< domain
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 21 / 36
Solution strategy: the IR domain
D< domain
∂kUk =k4
12π2
1
ωB
& boundary conditions on Fermi surface
transform boundary to a rectangle bychanging k → x(k), ϕ→ y(k , ϕ)
(evolution should contain first order x derivatives
⇒ x(k) must not depend on ϕ).
choice x = ϕF (k), y =ϕ
x
D>
D<
SF
gφ
k
µ
µ
The new differential equation (u = u − V0)
x∂x u = −xV ′0 + y∂y u −g2(kx)3
12π2
1√(kx)2 + ∂2
y u,
with the boundary conditions u(x = 0, y) = u(x , y = ±1) = 0.
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 22 / 36
Solution strategy: function basis
To respect boundary conditions expand in a basis hn(y):
u(x , y) =N→∞∑n=1
cn(x)hn(y)
boundary conditions hn(±1) = 0
orthonormal1∫
0
dy hn(y)hm(y) = δnm
possible choice:
hn(y) =√
2 cos((2n + 1)πy2 )
We transform the partial differential equation into an ordinaryintegro-differential equation
∂x u = F(x , ∂py u) ⇒ ∂xcn =1∫
0
dy F(x ,∑
cm∂py hm)hn(y)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 23 / 36
Solution strategy: function basis
To perform the integrals analytically
1√(kx)2 + ∂2
y u=
P→∞∑p=0
(−1/2
p
)(∂2
y u −M2)p
((kx)2 + M2)p+1/2
with an appropriate M2 ⇒ optimization.the value of P controls the “loop order”:
P = −1 ⇒ mean field, no bosonic fluctuations
P = 0 ⇒ one-loop with mass M2
P > 0 ⇒ running couplings
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 24 / 36
Parameters
for the coupling of the Lagrangian we choose “nuclear-type”parameters with large scalar mass
v = fπ = 93 MeV , mN = gv = 938 MeV , m2σ = λv2
3 = mN .
Experience: N ∼ 10− 12 is enough
we varied P = −1, . . . 3
M2 was chosen to have fastest (apparent) convergence
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 25 / 36
Results: effective potential
convergence as increasing loop (P) parameter
0.2 0.4 0.6 0.8 1.0 1.2Φ fΠ
0.5
1.0
1.5
2.0
2.5
3.0
U
0.2 0.4 0.6 0.8 1.0 1.2Φ fΠ
-1.0
-0.5
0.5
1.0
1.5
2.0U
orange dashed: mean field, up to down: P = 0 . . . 3
left panel: first order transition point for mean field µMF
right panel: first order transition point for FRG µc = 1.053µMF
MF, one-loop are different
for P > 0: good convergence where the curvature is positive
physical U is convex, Maxwell construction between theminima ⇒ for P →∞ a straight line
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 26 / 36
Results: phase diagram
(λ, g) space: either first order or second order phase transition;border line tricritical points.
0 100 200 300 400 5000
2
4
6
8
10
12
Λ
gcHΛL
2nd order
1st order
HHHHHY
mean field
HHHHHY
one loop
XXXy FRG
mean field predicts strongest phase transition∃ (λ, g) where mean field predicts 1st order, FRG 2nd order PhT.
bosonic fluctuations soften the transition
one-loop is quite good
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 27 / 36
Results: Equation of State
(G.G.Barnafoldi, AJ., P. Posfay, in preparation)
We can calculate the p(µ) curves in different approximations
from here thermodynamics tell us the p(ε) relation (EoS)
bosonic fluctutions make the EoS (somewhat) stiffer.10% stiffer than mean field, 25% stiffer than 1-loop.
pressure vs µ
EoS (p vs ε)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 28 / 36
Results: Equation of State
(G.G.Barnafoldi, AJ., P. Posfay, in preparation)
We can calculate the p(µ) curves in different approximations
from here thermodynamics tell us the p(ε) relation (EoS)
bosonic fluctutions make the EoS (somewhat) stiffer.10% stiffer than mean field, 25% stiffer than 1-loop.
pressure vs µ
EoS (p vs ε)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 28 / 36
Results: stability of compact stars
EoS can be used to calculate the compact star stability curvesusing TOV equation. The result is the M(R) relation.
mass-radius diagram
realistic M(R) curves from this simple model, similar to SQMcurves ⇒ improvement is needed
quantum fluctuations tend to stabilize compact starsca. 5-10% effect in this model.
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 29 / 36
Outlines
1 Introduction: FRG and fermions
2 LPA approximation for fermions
3 Fermion systems at finite chemical potential
4 Higgs stability bound and irrelevant operators
5 Conclusions
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 30 / 36
The Higgs-top system
(AJ., I. Kaposvari, A. Patkos, arXiv:1510.05782 [hep-th]) ⇒ cf. poster of I. KaposvariStandard Model Higgs-sector
important: scalar self-coupling and top-Higgs Yukawa coupling
Fields: σ Higgs field, ψ top field
Action is function of invariants: ρ = 12σ
2, I = σψψ.
FRG Ansatz for this system:
Γk =
∫x
[Zψk ψγm∂mψ +
1
2Zσk(∂mσ)2 + hk I + Uk(ρ)
].
FRG equation for the potential:
∂kU = −1
2Tr ln Γ
(2)ΨΨ +
1
2Tr ln Γ(2)
σσ +1
2Tr ln
(1− Γ(2)−1
σσ Γ(2)σΨΓ
(2)−1ΨΨ Γ
(2)Ψσ
)anomalous dimensions −∂k lnZ = η functional of U
η → 0 in IR limit (irrelevant)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 31 / 36
FRG equations, and the running of the Yukawa coupling
∂kU = −1
2Tr ln Γ
(2)ΨΨ +
1
2Tr ln Γ(2)
σσ +1
2Tr ln
(1− Γ(2)−1
σσ Γ(2)σΨΓ
(2)−1ΨΨ Γ
(2)Ψσ
)calculate the RHS (cf. poster of I. Kaposvari)
power expand wrt. the invariants
running of hk Yukawa coupling: expand Γk [I ] to linear order
RHS [I ] = RHS [Is ] + (I − Is)RHS ′[Is ] + . . .
expansion point?
first guess: expand around I = 0(J. Braun, H. Gies and D.D. Scherer, Phys. Rev. D83:085012 (2011))
version A: around the Higgs EoM Is(σ) to zeroth order(ie. the Yukawa coupling does not run)
version B: around the Higgs EoM Is(σ) to first order
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 32 / 36
FRG running, range of stability
Two effects restrict the FRG evolution:
scalar self-coupling drifts running toward Landau pole:λ(ΛLP)→∞ ⇒ upper bound for mH .
top-Higgs Yukawa coupling drifts running towards instability:λ(Λinst)→ 0 ⇒ lower bound for mH .
for a given mH we have maximal cutoff
104 105 106 107 108
100
200
300
400
500
600
700
800
Λ
Gev
mσ
Gev version A
version B
Ref.[7]
104 105 106 107 10860
80
100
120
140
160
Λ
Gev
mσ
Gev
version A
version B
Ref.[7]
Experimental Higgs mass
small difference between results for different expansion⇒ choice of the expansion point is irrelevant
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 33 / 36
Background dependent Yukawa coupling
choose h(%)I = (h0 + h1%+ . . . )I
we expect h1 irrelevant: start evolution with λ = 0 at k = Λ
h1 does not contribute to the running soon
-4 -3 -2 -1 0 10.00
0.05
0.10
0.15
0.20
t
λ2
ln k/Λ
λ
h1 = 0
h1 6= 0
104 105 106 107 108 10960
80
100
120
140
Λ
Gev
mH
Gev
Gies, h(ϱ)=h0
Is from EoM, h(ϱ)=h0
Gies, h(ϱ)=h0+h1ϱ
Is from EoM, h(ϱ)=h0+h1ϱ
Gies, h(ϱ)=h0+h1ϱ, for positive t
Is from EoM, h(ϱ)=h0+h1ϱ, for positive t
Λ(GeV)
mH (GeV)
HHj
variation around the original cutoff
HHYeffect of back bending running
But this operator can have important effect towards UV:instead of λ < 0 (unstable system), λ turns back.
Stability bound changes significantly(AJ., I. Kaposvari, A. Patkos, arXiv:1510.05782 [hep-th])
Other irrelevant operators (eg. gravity) can do similar job( M. Shaposhnikov, Ch. Wetterich, Phys.Lett. B683 (2010) 196-200)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 34 / 36
Background dependent Yukawa coupling
choose h(%)I = (h0 + h1%+ . . . )I
we expect h1 irrelevant: start evolution with λ = 0 at k = Λ
h1 does not contribute to the running soon
-4 -3 -2 -1 0 10.00
0.05
0.10
0.15
0.20
t
λ2
ln k/Λ
λ
h1 = 0
h1 6= 0
104 105 106 107 108 10960
80
100
120
140
Λ
Gev
mH
Gev
Gies, h(ϱ)=h0
Is from EoM, h(ϱ)=h0
Gies, h(ϱ)=h0+h1ϱ
Is from EoM, h(ϱ)=h0+h1ϱ
Gies, h(ϱ)=h0+h1ϱ, for positive t
Is from EoM, h(ϱ)=h0+h1ϱ, for positive t
Λ(GeV)
mH (GeV)
HHj
variation around the original cutoff
HHYeffect of back bending running
But this operator can have important effect towards UV:instead of λ < 0 (unstable system), λ turns back.
Stability bound changes significantly(AJ., I. Kaposvari, A. Patkos, arXiv:1510.05782 [hep-th])
Other irrelevant operators (eg. gravity) can do similar job( M. Shaposhnikov, Ch. Wetterich, Phys.Lett. B683 (2010) 196-200)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 34 / 36
Background dependent Yukawa coupling
choose h(%)I = (h0 + h1%+ . . . )I
we expect h1 irrelevant: start evolution with λ = 0 at k = Λ
h1 does not contribute to the running soon
-4 -3 -2 -1 0 10.00
0.05
0.10
0.15
0.20
t
λ2
ln k/Λ
λ
h1 = 0
h1 6= 0
104 105 106 107 108 10960
80
100
120
140
Λ
Gev
mH
Gev
Gies, h(ϱ)=h0
Is from EoM, h(ϱ)=h0
Gies, h(ϱ)=h0+h1ϱ
Is from EoM, h(ϱ)=h0+h1ϱ
Gies, h(ϱ)=h0+h1ϱ, for positive t
Is from EoM, h(ϱ)=h0+h1ϱ, for positive t
Λ(GeV)
mH (GeV)
HHj
variation around the original cutoff
HHYeffect of back bending running
But this operator can have important effect towards UV:instead of λ < 0 (unstable system), λ turns back.
Stability bound changes significantly(AJ., I. Kaposvari, A. Patkos, arXiv:1510.05782 [hep-th])
Other irrelevant operators (eg. gravity) can do similar job( M. Shaposhnikov, Ch. Wetterich, Phys.Lett. B683 (2010) 196-200)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 34 / 36
Background dependent Yukawa coupling
choose h(%)I = (h0 + h1%+ . . . )I
we expect h1 irrelevant: start evolution with λ = 0 at k = Λ
h1 does not contribute to the running soon
-4 -3 -2 -1 0 10.00
0.05
0.10
0.15
0.20
t
λ2
ln k/Λ
λ
h1 = 0
h1 6= 0
104 105 106 107 108 10960
80
100
120
140
Λ
Gev
mH
Gev
Gies, h(ϱ)=h0
Is from EoM, h(ϱ)=h0
Gies, h(ϱ)=h0+h1ϱ
Is from EoM, h(ϱ)=h0+h1ϱ
Gies, h(ϱ)=h0+h1ϱ, for positive t
Is from EoM, h(ϱ)=h0+h1ϱ, for positive t
Λ(GeV)
mH (GeV)
HHj
variation around the original cutoff
HHYeffect of back bending running
But this operator can have important effect towards UV:instead of λ < 0 (unstable system), λ turns back.
Stability bound changes significantly(AJ., I. Kaposvari, A. Patkos, arXiv:1510.05782 [hep-th])
Other irrelevant operators (eg. gravity) can do similar job( M. Shaposhnikov, Ch. Wetterich, Phys.Lett. B683 (2010) 196-200)
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 34 / 36
Outlines
1 Introduction: FRG and fermions
2 LPA approximation for fermions
3 Fermion systems at finite chemical potential
4 Higgs stability bound and irrelevant operators
5 Conclusions
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 35 / 36
Conclusions
LPA for fermionic systems can be defined ⇒ potentialdepends on (invariant) fermion bilinears to any power
problem at T = 0, µ > 0: presence of a sharp Fermi-surfacesolution: solve FRG equations separately in the high and lowenergy domains, match the solutions on the boundarybosonic fluctuations weaken the phase transitionbut make EoS stiffer, stabilize neutron stars
Higgs stability: position of the maximal cutoff may depend onIR irrelevant operators, like background dependent Yukawacoupling
8th International Conference on the Exact Renormalization Group - ERG2016 ICTP, Trieste, 19 - 23 September 2016 36 / 36