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.

�i = k@k↵i = A↵i +B↵2i + · · ·

↵i⇤ = 0 ↵i⇤ = �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

2TrFµ⌫Fµ⌫ +Tr(Q̄i�µDµQ) + yTr(Q̄HQ) + Tr(@µH

†@µH)� V

✏ =Nf

Nc� 11

20 ✏ << 1

V = �̄1Tr

✓H†H � 1

NfTrH†H

◆2

+ �̄2(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◆↵2g �

27

8(11 + 2✏)2↵g↵y +

1

4(11 + 2✏)2(20 + 3✏)↵2

y

◆

��(2)y = ↵y

✓20✏� 93

6↵2g + (49 + 8✏)↵g↵y �

✓385

8+

23

2✏+

✏2

2

◆↵2y � (44 + 8✏)↵y�1 + 4�2

1

◆

Ref: Litim, Sannino JHEP 1412 (2014) 178

NLO

NNLO

↵y =y2Nc

(4⇡)2

�1 = �↵2y(2✏+ 11) + 4↵y�1 + 8�2

1

�2 = �↵2y(2✏+ 11) + 4↵y�2 + 8�2

1 + 8�1�2 + 4�22

✤ 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 =@�i

@gjeig(Mij) = ✓

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) =NiX

j=2

(4⇡)2j�2ij1�2j � 2

N2j�2f

ck(i1) =NiX

i=0

(4⇡)2j+2ij1�2j + 1

N2j+1f

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

k@k�k =1

2

Tr

k@kRk

�(2)k +Rk

!

Rk = (k2 � q2)⇥(k2 � q2)

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

�k : E↵ective Average Action

�(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 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 ✏

Leading order epsilon power in the fixed point

will drop by half

�i = �̄ik�di

�i = �di�i +#�2i + · · ·

�i = 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

logu k

✏ = 0.01

✤ We can compute the power law by using:

200 400 600 800 10001.8

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

uki1@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

↵i = ↵̄ik�di

�i = �di↵i + quantum fluctuations

✓i = �di + correction from non-diagonal

Mij =@�i

@↵j

����↵⇤

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 ✏

0 5 10 15 20 25 30 350

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(✏)