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

Asymptotically Safe Gauge- Yukawa Theories and fRG based on paper in progress with D. Litim Tugba Buyukbese - University of Sussex 25 July

Asymptotic Safety β i = k ∂ k α i = A α i + B α 2 i + · · · α i ∗ = − A α i ∗ = 0 ✤ Asymptotic = in the UV B ✤ Safe = no poles α � * ✤ Existence of an interacting α � (non-zero) UV fixed point. ✤ Conjectured by Weinberg in 79’ � for QG, and tested many times for various theories and truncations.

Ref: Litim, Sannino JHEP 1412 (2014) 178 Asymptotically Safe Gauge-Yukawa L = 1 2Tr F µ ν F µ ν + Tr( ¯ Qi γ µ D µ Q ) + y Tr( ¯ QHQ ) + Tr( ∂ µ H † ∂ µ H ) − V ◆ 2 ✓ H † H − 1 V = ¯ + ¯ Tr H † H λ 2 (Tr H † H ) 2 λ 1 Tr N f ✤ H is an N f x N f complex scalar matrix , where N f is the number of fermion flavours. (not charged) ✤ SU(N c ) gauge fields. ✤ Q’s are fermions in the fundamental representation. ✏ = N f − 11 ✤ in Veneziano limit 0 ≤ ✏ << 1 N c 2

α y = y 2 N c ◆ 2 ! ✓ ◆ ✓ 11 4 25 + 26 NLO � g = ↵ g 3 ✏ + ↵ g − 2 2 + ✏ 3 ✏ ↵ y (4 π ) 2 � y = ↵ y ((13 + 2 ✏ ) ↵ y − 6 ↵ g ) ✓✓ 701 ◆ ◆ + 53 3 ✏ − 112 g − 27 8 (11 + 2 ✏ ) 2 ↵ g ↵ y + 1 ∆ � (3) = ↵ 2 27 ✏ 2 ↵ 2 4(11 + 2 ✏ ) 2 (20 + 3 ✏ ) ↵ 2 g g 6 y 2 ✏ + ✏ 2 ✓ 20 ✏ − 93 ✓ 385 + 23 ◆ ◆ ∆ � (2) ↵ 2 ↵ 2 y − (44 + 8 ✏ ) ↵ y � 1 + 4 � 2 = ↵ y g + (49 + 8 ✏ ) ↵ g ↵ y − 1 y 6 8 2 NNLO � 1 = − ↵ 2 y (2 ✏ + 11) + 4 ↵ y � 1 + 8 � 2 1 � 2 = − ↵ 2 y (2 ✏ + 11) + 4 ↵ y � 2 + 8 � 2 1 + 8 � 1 � 2 + 4 � 2 2 ✤ 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. Ref: Litim, Sannino JHEP 1412 (2014) 178

↵ g ∗ = 0 . 456140 ✏ + O ( ✏ 2 ) ↵ y ∗ = 0 . 210526 ✏ + O ( ✏ 2 ) � 1 ∗ = 0 . 199781 ✏ + O ( ✏ 2 ) ✏ = N f − 11 � 2 ∗ = 0 . 0625304 ✏ + O ( ✏ 2 ) N c 2 ✤ Fixed points are found by solving the beta functions for zero. ✤ Leading order terms are order epsilon. Ref: Litim, Sannino JHEP 1412 (2014) 178

✤ 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 ) M ij = ∂β i eig ( M ij ) = θ ✓ 2 = 2 . 7368 ✏ + O ( ✏ 2 ) ∂ g j ✓ 3 = 4 . 03859 ✏ + O ( ✏ 2 ) ✓ 4 = 2 . 94059 ✏ + O ( ✏ 2 )

Higher Dimension Operators v k ( i 1 , i 2 ) = u k ( i 1 ) + i 2 c k ( i 1 ) N i (4 π ) 2 j − 2 i j 1 λ 2 j − 2 X u k ( i 1 ) = ✤ We define a N 2 j − 2 f j =2 dimensionless, scale N i (4 π ) 2 j +2 i j 1 λ 2 j + 1 dependent potential X c k ( i 1 ) = N 2 j +1 that describe the f i =0 scalar self i 1 = Tr( h † h ) interactions. ◆ 2 ✓ h † h − 1 (Tr h † h ) i 2 = Tr N f

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. ! Tr k ∂ k R k k ∂ k Γ k = 1 Γ k : E ff ective Average Action Γ (2) 2 Γ (2) + R k : The Hessian k k R k = ( k 2 − q 2 ) Θ ( k 2 − q 2 ) : The Regulator Refs: C.Wetterich - Phys.Lett. B301 (1993) 90-94 D.Litim - Nucl.Phys.B631:128-158,2002

✤ 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 ✏ 0 . 442635 ✏ 3 � 3 2 + 3 . 14773 ✏ 0 . 197829 ✏ 3 � 4 2 + 4 . 24573 ✏ − 0 . 42182 ✏ 4 � 5 4 + 4 . 19698 ✏ − 0 . 0912196 ✏ 4 � 6 4 + 5 . 29498 ✏ 0 . 442354 ✏ 5 � 7 6 + 5 . 24622 ✏ 0 . 0561861 ✏ 5 � 8 6 + 6 . 34422 ✏ − 0 . 466105 ✏ 6 � 9 8 + 6 . 29546 ✏ − 0 . 0389432 ✏ 6 � 10 8 + 7 . 39347 ✏ 0 . 486798 ✏ 7 � 11 10 + 7 . 34471 ✏ 0 . 0287923 ✏ 7 � 12 10 + 8 . 44271 ✏ − 0 . 503072 ✏ 8 � 13 12 + 8 . 39395 ✏ − 0 . 0221745 ✏ 8 � 14 12 + 9 . 49195 ✏

λ i = ¯ λ i k − d i β i = − d i λ i + # λ 2 i + · · · Leading order epsilon β i = 0 & d i = 0 = ⇒ power in the fixed point will drop by half

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.

✏ = N f − 11 N c 2

✏ = 0 . 01 1000 100 log u k 10 1 5 10 50 100 500 1000 log i 1

✤ We can compute the power log u k = A log i 1 + B A ≡ 1 law by using: i 1 ∂ i 1 u k ( i 1 ) u k i 1 = Tr( h † h ) 2.3 2.2 ϵ = 0.001 2.1 ϵ = 0.01 A 2.0 ϵ = 0.05 ϵ = 0.1 1.9 ϵ = 1. 1.8 200 400 600 800 1000 i 1

Eigenvalues of the Stability Matrix ✤ are the universal quantities. ✤ Bootstrap hypothesis: α i k − d i α i = ¯ β i = − d i α i + quantum fluctuations � M ij = ∂β i � � ∂α j � α ∗ i θ i = − d i + correction from non-diagonal Ref: Falls, Litim, Nikolakopoulos, Rahmede - arXiv:1301.4191

Coupling Fixed Point Eigenvalues of the Stability Matrix � 1 0 . 199781 ✏ 4 . 03859 ✏ � 2 0 . 0625304 ✏ 2 . 94059 ✏ 0 . 442635 ✏ 3 � 3 2 + 3 . 14773 ✏ 0 . 197829 ✏ 3 � 4 2 + 4 . 24573 ✏ − 0 . 42182 ✏ 4 � 5 4 + 4 . 19698 ✏ − 0 . 0912196 ✏ 4 � 6 4 + 5 . 29498 ✏ 0 . 442354 ✏ 5 � 7 6 + 5 . 24622 ✏ 0 . 0561861 ✏ 5 � 8 6 + 6 . 34422 ✏ − 0 . 466105 ✏ 6 � 9 8 + 6 . 29546 ✏ − 0 . 0389432 ✏ 6 � 10 8 + 7 . 39347 ✏ 0 . 486798 ✏ 7 � 11 10 + 7 . 34471 ✏ 0 . 0287923 ✏ 7 � 12 10 + 8 . 44271 ✏ − 0 . 503072 ✏ 8 � 13 12 + 8 . 39395 ✏ − 0 . 0221745 ✏ 8 � 14 12 + 9 . 49195 ✏

✏ = 0 . 01 35 30 25 20 θ n 15 10 5 0 0 5 10 15 20 25 30 35 n ( in λ n )

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 − d i + O ( ✏ ) 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.