Asymptotically Safe Gauge- Yukawa Theories and fRG based on paper - - PowerPoint PPT Presentation

25 July

Asymptotically Safe Gauge- Yukawa Theories and fRG

Tugba Buyukbese - University of Sussex based on paper in progress with D. Litim

Asymptotic Safety

✤ Asymptotic = in the UV ✤ Safe = no poles ✤ Existence of an interacting

(non-zero) UV fixed point.

✤ Conjectured by Weinberg in 79’

for QG, and tested many times for various theories and truncations.

βi = k∂kαi = Aαi + Bα2

i + · · ·

αi∗ = 0 αi∗ = − A B

α*

• α

Asymptotically Safe Gauge-Yukawa

✤ H is an Nf x Nf complex scalar matrix , where Nf is the number of

fermion flavours. (not charged)

✤ SU(Nc) gauge fields. ✤ Q’s are fermions in the fundamental representation. ✤ in Veneziano limit

L = 1 2TrF µνFµν + Tr( ¯ QiγµDµQ) + yTr( ¯ QHQ) + Tr(∂µH†∂µH) − V

✏ = Nf Nc − 11 2

0 ≤ ✏ << 1

V = ¯ λ1Tr ✓ H†H − 1 Nf TrH†H ◆2 + ¯ λ2(TrH†H)2

Ref: Litim, Sannino JHEP 1412 (2014) 178

✤ Computed in perturbation theory. ✤ Important note: Yukawa couples with scalars in 2-loop Yukawa beta

• function. And gauge couples indirectly via Yukawa in 3-loop level.

g = ↵g 4 3✏ + ✓ 25 + 26 3 ✏ ◆ ↵g − 2 ✓11 2 + ✏ ◆2 ↵y ! y = ↵y ((13 + 2✏)↵y − 6↵g) ∆(3)

g

= ↵2

g

✓✓701 6 + 53 3 ✏ − 112 27 ✏2 ◆ ↵2

g − 27

8 (11 + 2✏)2↵g↵y + 1 4(11 + 2✏)2(20 + 3✏)↵2

y

◆ ∆(2)

y

= ↵y ✓20✏ − 93 6 ↵2

g + (49 + 8✏)↵g↵y −

✓385 8 + 23 2 ✏ + ✏2 2 ◆ ↵2

y − (44 + 8✏)↵y1 + 42 1

◆

Ref: Litim, Sannino JHEP 1412 (2014) 178 NLO NNLO

αy = y2Nc (4π)2

1 = −↵2

y(2✏ + 11) + 4↵y1 + 82 1

2 = −↵2

y(2✏ + 11) + 4↵y2 + 82 1 + 812 + 42 2

✤ Fixed points are found by solving the beta functions

for zero.

✤ Leading order terms are order epsilon.

1∗ = 0.199781 ✏ + O(✏2) 2∗ = 0.0625304 ✏ + O(✏2) ↵g∗ = 0.456140 ✏ + O(✏2) ↵y∗ = 0.210526 ✏ + O(✏2)

Ref: Litim, Sannino JHEP 1412 (2014) 178

✏ = Nf Nc − 11 2

✤ Scaling exponents = - (eigenvalues of the stability matrix) ✤ They are the universal quantities that determine the phase structure. i.e. how

does the RG flow approach the fixed point.

✤ Operators can have relevant, irrelevant or marginal directions depending on the

sign of the scaling exponents.

✤ A given theory is predictive as long as it has a finite number of relevant

directions.

✤ Negative eigenvalues: relevant directions , positive eigenvalues: irrelevant

directions.

✓1 = −0.590643✏2 + O(✏3) ✓2 = 2.7368✏ + O(✏2) ✓3 = 4.03859✏ + O(✏2) ✓4 = 2.94059✏ + O(✏2)

Mij = ∂βi ∂gj

eig(Mij) = θ

Higher Dimension Operators

✤ We define a

dimensionless, scale dependent potential that describe the scalar self interactions.

vk(i1, i2) = uk(i1) + i2ck(i1) uk(i1) =

Ni

X

j=2

(4π)2j−2ij

1λ2j − 2

N 2j−2

f

ck(i1) =

Ni

X

i=0

(4π)2j+2ij

1λ2j + 1

N 2j+1

f

i1 = Tr(h†h) i2 = Tr ✓ h†h − 1 Nf (Trh†h) ◆2

Motivations

✤ How do the higher dimensional operators in the scalar

potential affect the fixed point structure?

✤ How many relevant/irrelevant operators do we have? ✤ What does the shape of the potential look like with the

contribution of the higher dimensional operators?

✤ How does the functional methods/inclusion of the

threshold effects due to massive modes affect the fixed point structure?

Functional Renormalisation Group

✤ Since all the higher order scalar self couplings have coupling constants

with negative canonical mass dimensions, these are not perturbatively renormalisable => non-perturbative => fRG

✤ We use Wetterich equation which is an exact RG equation in the form

• f a 1-loop propagator, includes the contribution of all loop orders.

k∂kΓk = 1 2 Tr k∂kRk Γ(2)

k

+ Rk !

Rk = (k2 − q2)Θ(k2 − q2)

Refs: C.Wetterich - Phys.Lett. B301 (1993) 90-94 D.Litim - Nucl.Phys.B631:128-158,2002

Γk : Effective Average Action

Γ(2)

k

: The Hessian

: The Regulator

✤ We first take the wave function renormalisation Z=1,

therefore we ignore the effects from the anomalous dimension.

✤ All the beta functions are computed from Wetterich equation

by taking the appropriate derivatives.

✤ We compute the flow only for the scalar fields, where we add

Yukawa contribution from perturbation theory.

✤ We confirm that the beta functions from the non-perturbative

computation match the perturbative computation perfectly. And go on to calculate beyond quartic terms.

✤ Then we compute the fixed points by systematically solving

beta functions one by one up to leading order in epsilon.

Coupling Fixed Point Eigenvalues of the Stab 1 0.199781 ✏ 4.03859 ✏ 2 0.0625304 ✏ 2.94059 ✏ 3 0.442635 ✏3 2 + 3.14773 ✏ 4 0.197829 ✏3 2 + 4.24573 ✏ 5 −0.42182 ✏4 4 + 4.19698 ✏ 6 −0.0912196 ✏4 4 + 5.29498 ✏ 7 0.442354 ✏5 6 + 5.24622 ✏ 8 0.0561861 ✏5 6 + 6.34422 ✏ 9 −0.466105 ✏6 8 + 6.29546 ✏ 10 −0.0389432 ✏6 8 + 7.39347 ✏ 11 0.486798 ✏7 10 + 7.34471 ✏ 12 0.0287923 ✏7 10 + 8.44271 ✏ 13 −0.503072 ✏8 12 + 8.39395 ✏ 14 −0.0221745 ✏8 12 + 9.49195 ✏

Leading order epsilon power in the fixed point will drop by half

λi = ¯ λik−di βi = −diλi + #λ2

i + · · ·

βi = 0 & di = 0 = ⇒

The Global Effective Potential

✤ We compute the GEP by solving the Wetterich equation

for a random potential without assuming an ansatz.

✤ LHS = 0 at the fixed point. ✤ 0 = RHS is a differential equation as a function of the

potential and its derivatives.

✤ We plot the numerical result and compare it to the

quartic truncation of the potential.

✏ = Nf Nc − 11 2

5 10 50 100 500 1000 1 10 100 1000 log i1 log uk

✏ = 0.01

✤ We can compute the power

law by using:

200 400 600 800 1000 1.8 1.9 2.0 2.1 2.2 2.3 i1 A ϵ=0.001 ϵ=0.01 ϵ=0.05 ϵ=0.1 ϵ=1.

log uk = A log i1 + B A ≡ 1 uk i1∂i1uk(i1)

i1 = Tr(h†h)

Eigenvalues of the Stability Matrix

✤ are the universal quantities. ✤ Bootstrap hypothesis:

Ref: Falls, Litim, Nikolakopoulos, Rahmede - arXiv:1301.4191

αi = ¯ αik−di βi = −diαi + quantum fluctuations θi = −di + correction from non-diagonal Mij = ∂βi ∂αj

• α∗

i

Coupling Fixed Point Eigenvalues of the Stability Matrix 1 0.199781 ✏ 4.03859 ✏ 2 0.0625304 ✏ 2.94059 ✏ 3 0.442635 ✏3 2 + 3.14773 ✏ 4 0.197829 ✏3 2 + 4.24573 ✏ 5 −0.42182 ✏4 4 + 4.19698 ✏ 6 −0.0912196 ✏4 4 + 5.29498 ✏ 7 0.442354 ✏5 6 + 5.24622 ✏ 8 0.0561861 ✏5 6 + 6.34422 ✏ 9 −0.466105 ✏6 8 + 6.29546 ✏ 10 −0.0389432 ✏6 8 + 7.39347 ✏ 11 0.486798 ✏7 10 + 7.34471 ✏ 12 0.0287923 ✏7 10 + 8.44271 ✏ 13 −0.503072 ✏8 12 + 8.39395 ✏ 14 −0.0221745 ✏8 12 + 9.49195 ✏

5 10 15 20 25 30 35 5 10 15 20 25 30 35 n (in λn) θn

✏ = 0.01

Conclusions - Outlook

✤ Fixed points with irrelevant directions exist with the inclusion of

the higher dimensional terms.

✤ Higher dimensional couplings are higher leading order in epsilon. ✤ The eigenvalues of the stability matrix are as

• expected. This satisfies the bootstrap hypothesis.

✤ Potential is stable at large field values. Next: Cosmological

implications are to be checked.

✤ Potential’s asymptotic behaviour is very close to a quartic potential.

−di + O(✏)