Cosmology in Colombia (CoCo) 2020 Carlos Nieto September 2020 1 - PowerPoint PPT Presentation

Ultraviolet completion and predictivity from minimal parameterizations of Beyond-Standard-Model physics Carlos Mauricio Nieto Guerrero 1 1 Universidad Industrial de Santander Cosmology in Colombia (CoCo) 2020 Carlos Nieto September 2020

1 Introduction to Asymptotic Safety 2 AS extensions of the SM 3 AS Gravity Carlos Nieto September 2020

Asymptotic Safety - The concept Definition: j ) = kdg i � ⇒ β i ( g ∗ L = g i ( k ) O i ( ϕ ) = dk = 0 . i Linearized analysis for y i = g i − g ∗ i dy i M ij = ∂β i dt = M ij y j , (1) ∂g j ( S − 1 ) ij M jl S ln = δ in λ n , z i = S − 1 ij y j (2) � k � λ i dz i z i ( t ) = c i e λ i t = c i dt = λ i z i and . (3) k 0 If ℜ ( λ i ) > 0, irrelevant direction. If ℜ ( λ i ) < 0, relevant direction. If ℜ ( λ i ) = 0, marginal direction. Carlos Nieto September 2020

Theory space z 3 g 3 z 2 F P A z 1 g 2 S UV g 1 Q B Figure : Theory space and fixed-point properties. Carlos Nieto September 2020

Running of the SM gauge couplings 3.5 g 1 g 2 g 3 3.0 2.5 2.0 1.5 1.0 0.5 10 12 10 22 10 32 100 k [ GeV ] Figure : Running of the gauge couplings g 1 , g 2 and g 3 . At large values of k , g 1 begins its ascent towards the Landau pole. Carlos Nieto September 2020

Gauge theories in d = 4: one loop SU ( N c ) theory with N f fermions in the fundamental representation L = − 1 4tr F µν F µν + ¯ ψiD / ψ (4) The beta function of α g = g 2 N c (4 π ) 2 is β g = − Bα 2 (5) g B = − 4 ǫ = N F − 11 3 ǫ ; (6) N c 2 N F < 11 2 N c = ⇒ ǫ < 0 = ⇒ B > 0 = ⇒ AF Carlos Nieto September 2020

Gauge theories in d = 4: two loops β g = − Bα 2 g + Cα 3 g C > 0 C < 0 β g β g α g α g B > 0 β g β g α g α g B < 0 Carlos Nieto September 2020

Yukawa couplings Adding scalar fields H ∆ L = tr( ∂ µ H ) † ( ∂ µ H ) + y tr( ¯ ψ L Hψ R + ¯ ψ R Hψ L ) (7) β g = dα g dt = ( − B + Cα g − Dα y ) α 2 g β y = dα y dt = ( Eα y − Fα g ) α y Beta functions for α g and α y = y 2 N c (4 π ) 2 � 4 � 25 + 26 � � 11 � � α 2 β g = 3 ǫ + 3 ǫ α g − 2 3 + ǫ α y g β y = α y [(13 + 2 ǫ ) α y − 6 α g ] (8) Carlos Nieto September 2020

Fixed-point solutions For ǫ < 0, Banks-Zaks � � 4 ǫ ( α g ∗ , α y ∗ ) = − 75 + 26 ǫ, 0 (9) For ǫ > 0, Litim-Sannino � � 13 ǫ + 2 ǫ 2 � � 2 12 ǫ ( α g ∗ , α y ∗ ) = 57 − 46 ǫ − 8 ǫ 2 , (10) 57 − 46 ǫ − 8 ǫ 2 (0 . 456 ǫ + O ( ǫ 2 ) , 0 . 211 ǫ + O ( ǫ 2 )) ≈ [D.F. Litim and F. Sannino, JHEP 1412 (2014) 178 ] Carlos Nieto September 2020

Phase diagram Carlos Nieto September 2020

Fixed-point regimes k ∗ < M pl g i g ∗ i k M pl TeV k ∗ k ∗ > M pl g i g ∗ i k M pl k ∗ TeV Carlos Nieto September 2020

A class of BSM models Are there phenomenologically viable AS gauge-Yukawa theories? Finite N f and N c [A. Bond, D. Litim, Eur.Phys.J. C77 (2017) no.6, 429, arXiv:1608.00519 [hep-th]] [A. Bond, G. Hiller, K. Kowalska, D. Litim, JHEP 1708 (2017) 004, arXiv:1702.01727 [hep-ph]] [G.M. Pelaggi, A.D. Plascencia, A. Salvio, F. Sannino, Y. Smirnov, A. Strumia, Phys.Rev. D97 (2018) no.9, 095013 arXiv:1708.00437 [hep-th]] [R.B. Mann, J.R. Meffe, F. Sannino, T.G. Steele, Z.W. Wang and C. Zhang, Phys. Rev. Lett. 119, 261802 (2017), arXiv:1707.02942 [hep-th]] Simplest models use additional vector-like fermions. Carlos Nieto September 2020

A class of BSM models Group: SU c (3) × SU L (2) × U Y (1). Add N f families of vector-like fermions ψ i minimally coupled to SM and Yukawa interactions new scalars S . L = L SM + Tr( ¯ Dψ ) + Tr( ∂ µ S † ∂ µ S ) − y Tr( ¯ ψ L Sψ R + ¯ ψi / ψ R S † ψ L ) . Representation labels ( p, q ), ℓ , Y and N f . � g i � 2 and Couplings: ( α 1 , α 2 , α 3 , α t , α y , α λ ), where α i ≡ 4 π λ α λ ≡ (4 π ) 2 . Carlos Nieto September 2020

A promising model N f = 3, ℓ = 1 / 2, Y = 3 / 2, p = q = 0 α ∗ 1 = 0 . 188 , α ∗ 2 = 0 , α ∗ 3 = 0 , α ∗ t = 0 , α ∗ y = 0 . 778 λ 1 = 33 . 2 , λ 2 = − 3 . 36 , λ 3 = − 0 . 817 , λ 4 = 0 , λ 5 = 0. 1 α 1 α 2 α 3 α t α y 0.100 0.010 0.001 10 - 4 5 10 15 20 25 30 t Carlos Nieto September 2020

Requirements Stability under loop expansions SM matching at Fermi scale Carlos Nieto September 2020

Criteria for stability Small couplings � 2 � g ∗ α ∗ i ≡ � O (1) . i 4 π Small critical exponents i ( g j ), M ij = − d i δ ij + ∂β q β i = − d i g i + β q ∂g j , demand | λ i | � O (1). i Hierarchy in the loop contributions At the FP 0 = β i = A ( i ) ∗ + B ( i ) + C ( i ) ∗ , ∗ demand ρ i < σ i < 1 where ρ i = | C ( i ) ∗ /A ( i ) ∗ | and σ i = | B ( i ) ∗ /A ( i ) ∗ | . Carlos Nieto September 2020

The search Parameter space N f = 1 , 2 , ..., 300 ℓ = 1 / 2 , 1 , ..., 10 Y = 0 , 1 / 2 , 1 , ..., 10 Singlet SU (3) = ⇒ ( N, ℓ ) : (1 , 1) , (2 , 1 / 2) , (3 , 1 / 2) , (4 , 1 / 2) Fundamental SU (3) = ⇒ ( N, ℓ ) : (1 , 1 / 2) , (1 , 5 / 2) , (1 , 3) , (1 , 7 / 2) , (1 , 4) , (1 , 9 / 2) , (2 , 1) , (2 , 3 / 2) , (2 , 2) , (3 , 1 / 2) , (3 , 1) , (4 , 1 / 2) , (5 , 1 / 2) . Adjoint SU (3) = ⇒ ( N, ℓ ) : None Stable models have the U (1) triviality problem. = ⇒ no satisfactory UV fixed points. Carlos Nieto September 2020

Results Instability of fixed points with large scaling exponents. Low-dimensional representations are preferable Stable solutions show that there are no UV perturbative FPs that can be connected to the TeV physics. Stability condition rules out other models appearing in the literature. [A. Bond, G. Hiller, K. Kowalska, D. Litim, Directions for model building from asymptotic safety, JHEP 1708 (2017) 004] Explore non-abelian embeddings. [B. Bajc, F. Sannino, Asymptotically safe grand unification, JHEP 1612 (2016) 141] [A. Eichhorn, A. Held, C. Wetterich, Quantum-gravity predictions for the fine-structure constant, Phys.Rev. D99 (2019) no.3, 035030] Consider gravity+matter systems. Carlos Nieto September 2020

Gravitational running 10 - 1 10 - 11 g N 10 - 21 10 - 31 10 12 10 22 10 32 10 42 100 k [ GeV ] Figure : RG flow of the dimensionless Newton coupling g N in the EH truncation. Carlos Nieto September 2020

Gravity plus matter Gravity modifies the beta functions of matter couplings by universal (flavor-independent) terms. β g i = β Matter + f g g i (11) g i β Y i = β Matter + f y Y i (12) Y i β λ i = β Matter + f λ λ i (13) λ i These can generate nontrivial fixed points for matter couplings [S. P. Robinson, F. Wilczek, Complete asymptotically safe embedding of the standard model, Phys. Rev. Lett. 96 (2006) 231601] [A. Salvio, A. Strumia, Agravity, JHEP 1406 (2014) 080] [O. Zanusso, L. Zambelli, G. P. Vacca, R. Percacci, Gravitational corrections to Yukawa systems, Phys. Lett. B689 (2010) 90] Carlos Nieto September 2020

Calculation of top mass 1.2 195 181 UV unsafe trajectories 1 165 150 0.8 120 M t / GeV y t ( k ) 0.6 predictive trajectory 0.4 free trajectories 0.2 0 10 26 10 51 10 76 10 RG scale k in GeV 5(1 − 4Λ) Λ(235 − Λ(103 + 56Λ)) − 96 f g = G N 18 π (1 − 2Λ) 2 , f y = G N 12 π (3 + 2Λ( − 5 + 4Λ)) 2 [A. Eichhorn and A. Held, Phys. Lett. B777, (2018) 217] Carlos Nieto September 2020

Gauge-Quark-Yukawa system Gauge sector dg i 1 16 π 2 b i g 3 dt = i − f g g i . Up-type Yukawas � 3 1 U Y U − 3 2 Y U Y † 2 Y D Y † � Y U Y † U + Y D Y † � β Y U = D Y U + 3 Tr Y U 16 π 2 D � 17 1 + 9 � � 12 g 2 4 g 2 2 + 8 g 2 − Y U − f y Y U . 3 Down-type Yukawas � 3 1 D Y D − 3 � � 2 Y D Y † 2 Y U Y † Y U Y † U + Y D Y † β Y D = U Y D + 3 Tr Y D D 16 π 2 � 5 1 + 9 � � 12 g 2 4 g 2 2 + 8 g 2 − Y D − f y Y D . 3 Carlos Nieto September 2020

Mass basis We introduce the unitary matrices V U L and V D L such that L M U V U † V U = D 2 U = diag [ y 2 u , y 2 c , y 2 t ] , L L M D V D † V D = D 2 D = diag [ y 2 d , y 2 s , y 2 b ] . L Thus, we get the CKM-matrix  V ud V us V ub  L V D †  . V = V U = V cd V cs V cb  L V td V ts V tb Set of variables: y i = ( y u , y c , y t ), y ρ = ( y d , y s , y b ) and 4 CKM elements | V iρ | 2 Carlos Nieto September 2020

Diagonalized basis Up-type Yukawas   dy i y i � 17 � y 2 � y 2 12 g 2 1 + 9 4 g 2 2 + 8 g 2 + 3 2 y 2 i − 3 � y 2 ρ | V iρ | 2 � dt =  3 j + 3 ρ −  − f y y i . 3 2 16 π 2 j ρ ρ Down-type Yukawas   dy ρ y ρ � 5 � y 2 � y 2 12 g 2 Y + 9 4 g 2 2 + 8 g 2 + 3 2 y 2 ρ − 3 � y 2 i | V iρ | 2 � dt =  3 j + 3 α −  − f y y ρ . 3 16 π 2 2 α j i CKM elements  y 2 i + y 2 β | V iρ | 2 = − 3  � j y 2 � V iσ V ∗ jσ V jρ V ∗ iρ + V ∗ iσ V jσ V ∗ � jρ V iρ σ y 2 i − y 2 2 j σ,j � = i  y 2 ρ + y 2 σ � y 2 V ∗ jσ V jρ V iσ V ∗ iρ + V jσ V ∗ jρ V ∗ � �  . + iσ V iρ j y 2 ρ − y 2 σ j,σ � = ρ Carlos Nieto September 2020