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(✏)