IR fixed points in gauge theories from lattice simulations Luigi - PowerPoint PPT Presentation

IR fixed points in gauge theories from lattice simulations Luigi Del Debbio Higgs Centre for Theoretical Physics - University of Edinburgh Nagoya, March 2014 � 1

Plan � • RG flows, fixed points, anomalous dimensions • Anomalous dimension of four fermi operators • Schrödinger functional step scaling functions • Results • work in collaboration with L Keegan, C Pena � 2

RG flows Consider a theory with a UV cutoff - integrate out UV modes: µ 0 = µ/b < µ g 0 ; µ 0 ) , O (ˆ g ; µ ) = O (ˆ Dependence of the dimensionless couplings on the cut-off � The effect of integrating out the high-energy modes is compensated by the change in the couplings (and rescaling of the fields). � 1.Scheme-dependence - caveats in RB’s talk � 2. Integrating out the UV degrees of freedom generates all the interactions that are compatible with the symmetries of the system. " # 1 Z 2 ( ∂ µ φ ) 2 + X d D x µ d i ˆ S [ φ ; g, µ ] = g k O k ( x ) k D − 2 O k = ∂ p k φ n k , D − d k = n k + p k � 3 2

RG flows An RG transformation can be seen as a flow in the space of couplings. µ d g k = ˆ c-,*E&'6 e-; q*-'-fu.,( dµ ˆ β k (ˆ g ) � � h rL. R6 {L*, g 0 ) = ∂ ˆ g 0 ˆ ˆ k β 0 k (ˆ β j (ˆ g ) ∂ ˆ g j $z- � 4

IR fixed points Long-distance dynamics is described by µ → 0 ˆ lim g k ( µ ) Jr?lv '- o|'-^v The couplings may have a finite limit in the IR, and flow towards fixed point values: ˆ β k (ˆ g ∗ ) = 0 Scale invariant theory at the fixed point � 5

Hierarchies and scaling dimensions Linearized RG flow in a neighbourhood of a fixed point ✓ µ ◆ − y k µ d dµ ˆ g k = − y k ˆ g k ( µ ) = ˆ ˆ g k , g 0 Λ UV Associated IR scale: g 1 /y k Λ IR ⇠ ˆ Λ UV , y k ⌧ 1 = natural hierarchy ) 0 Global-singlet relevant operators (GSRO) require fine-tuning. Stable hierarchy generated by weakly relevant operators. [Strassler 03, Sannino 04, Luty&Okui 04] YM theory at the GFP is a limiting case: 1 Λ IR ∼ Λ UV exp { − β 0 g 2 } � 6

Hierarchies and the flavor sector GSRO In the SM + elementary Higgs: dim( H † H ) ' 2 L Y = Y u H ¯ Lu R + Y d H † ¯ dimension = 1+3 = 4 Ld R In DEWSB: scalar is composite [Dimopoulos et al 79, Eichten et al 1980] L Y = Y q ¯ dimension = 3+3 = 6 QQ ¯ qq Λ 2 UV Tension with suppressing FCNC f dimension = 6 qq ¯ qq ¯ Λ 2 UV � 7

Fermionic operators Alleviate the problem due to the large dimension of the composite scalar Theory at the EW scale is near a non-trivial fixed point Scaling dimension of the fermion bilinear is smaller dim H = dim ¯ [Holdom,Yamawaki, Appelquist, Eichten, Lane] QQ = 3 − γ m ➡ Find numerical evidence for the existence of a fixed point - scheme independent? � ➡ Characterize the fixed point by computing the mass anomalous dimension � ➡ and the anomalous dimension of 4 fermi operators dim( H ) small , but dim( H † H ) > 2 dim( H ) [Sannino 04, Luty 04, Rattazzi et al 08] � 8

Anomalous dimension of the four-fermi operator Under axial transformations in flavor space: ( ¯ ψψ )( ¯ = 4 i ( ¯ ψψ ) ( ¯ δ a ⇥ ψτ a γ 5 ψ ) ⇤ ψψ ) ( ¯ ψψ ) ( ¯ = 2 i ( ¯ ψτ b γ 5 ψ ) ( ¯ δ b ⇥ ψτ a γ 5 ψ ) ψτ a γ 5 ψ ) + . . . ⇤ Flavor non-singlet lies in the same multiplet Construct a basis of operators that is closed under renormalization: parity-even parity-odd γ µ ⊗ γ µ γ µ ⊗ γ µ γ 5 ( ¯ ψ 1 Γ 1 ψ 2 )( ¯ ψ 3 Γ 2 ψ 4 ) γ µ γ 5 ⊗ γ µ γ 5 γ µ γ 5 ⊗ γ µ ( ¯ ψ 1 Γ 1 T A ψ 2 )( ¯ ψ 3 Γ 2 T A ψ 4 ) 1 ⊗ 1 1 ⊗ γ 5 γ 5 ⊗ 1 γ 5 ⊗ γ 5 σ µ ν ⊗ ˜ σ µ ν ⊗ σ µ ν σ µ ν � 9

Fierzing Color trace identity: ( T A ) αβ ( T A ) γδ = 1 2 δ αδ δ βγ − 1 2 N δ αβ δ γδ Fierz transformation: ( Γ ( r ) 1 ) ij ( Γ ( r ) f rs ( Γ ( s ) 1 ) il ( Γ ( s ) X 2 ) kl = 2 ) kj s ψ 1 Γ ( r ) ψ 3 Γ ( r ) ( ¯ 1 T A ψ 2 )( ¯ 2 T A ψ 4 ) 7! � 1 ψ 1 Γ ( r ) ψ 3 Γ ( r ) ψ 1 Γ ( s ) ψ 3 Γ ( s ) 2 N ( ¯ 1 ψ 2 )( ¯ f rs ( ¯ 1 ψ 4 )( ¯ X 2 ψ 4 ) + 2 ψ 2 ) s � 10

Parity-even sector & discrete symmetries Γ 1 Γ 2 = 1 ( ¯ ψ 1 Γ 1 ψ 2 )( ¯ ψ 3 Γ 2 ψ 4 ) ± ( ¯ ψ 1 Γ 1 ψ 4 )( ¯ O ± ⇥ ⇤ ψ 3 Γ 2 ψ 2 ) 2 Q ± 1 = O ± V V + AA ±   ∆ 12 ∆ 13 ∆ 14 ∆ 15 0 Q ± 2 = O ± V V − AA ∆ 21 ∆ 24 ∆ 25 0 0   Q ± 3 = O ± ∆ ± =   ∆ 31 ∆ 34 ∆ 35 0 0 SS − P P     Q ± 4 = O ± ∆ 41 ∆ 42 ∆ 43 0 0   SS + P P ∆ 51 ∆ 52 ∆ 53 0 0 Q ± 5 = O ± T T h i Q ± i,R = Z ± δ jk + ∆ ± Q ± ij jk k scale-independent [Donini et al 99, Guagnelli et al 05] � 11

Parity-odd sector & discrete symmetries Γ 1 Γ 2 = 1 ( ¯ ψ 1 Γ 1 ψ 2 )( ¯ ψ 3 Γ 2 ψ 4 ) ± ( ¯ ψ 1 Γ 1 ψ 4 )( ¯ O ± ⇥ ⇤ ψ 3 Γ 2 ψ 2 ) 2 Q ± 1 = O ± V A + AV Q ± 2 = O ± V A − AV Q ± 3 = − O ± SP − P S Q ± 4 = O ± SP + P S Q ± 5 = O ± T ˜ T ± ±  ±      Q 1 Z 11 Q 1 0 0 0 0 Q 2 Z 22 Z 23 Q 2 0 0 0             Q 3 Z 32 Z 33 Q 3 = 0 0 0             Q 4 Z 44 Z 45 Q 4 0 0 0       Q 5 Z 54 Z 55 Q 5 0 0 0 R � 12

Schrödinger functional Finite-volume renormalization scheme: � • size of the system defines the renormalization scale, • Dirichlet boundary conditions � − 1 ⌧ ∂ S r g 2 ( L ) = k ∂η The renormalized charge is an observable, i.e. it can be measured by numerical simulations. � The definition above extends outside the perturbative regime and yields a nonperturbative coupling. � The coupling depends on one scale only, the finite size of the system. [M Luscher et al] � 13

SF - running of the coupling The running of the coupling as the scale is varied by a factor s is encoded in the step scaling function: Σ ( u, s, a/L ) = g 2 ( g 0 , sL/a ) � � g 2 ( g 0 ,L/a )= u Lattice step scaling is affected by lattice artefacts, i.e. depends on the details of the UV regulator. We can compute the continuous step scaling: σ ( u, s ) = a/L → 0 Σ ( u, s, a/L ) lim L/a: resolution of the simulation Z σ ( u,s ) dx − 2 log s = √ x β ( √ x ) u At the fixed point: σ ( u, s ) = u σ ( u, s ) /u = 1 ⇐ ⇒ � 14

SF - running of the mass The renormalized mass is defined as: Z A m ( µ ) = ¯ Z P ( µ ) m In order to study its running we need to compute nonperturbatively: p Z P ( L ) = 3 f 1 /f P ( L/ 2) Z r 0 ( u ) γ 5 τ a ζ 0 ( v ) ζ ( y ) γ 5 τ a ζ ( z ) i , f 1 = � 1 / 12 L 6 d 3 u d 3 v d 3 y d 3 z h ζ Z d 3 y d 3 z h ψ ( x 0 ) γ 5 τ a ψ ( x 0 ) ζ ( y ) γ 5 τ a ζ ( z ) i . f P ( x 0 ) = � 1 / 12 � 15

SF - running of the mass Step scaling functions for the mass: � Σ P ( u, s, a/L ) = Z P ( g 0 , sL/a ) � � Z P ( g 0 , L/a ) � g 2 ( L )= u σ P ( u, s ) = lim a → 0 Σ P ( u, s, a/L ) Relation to the anomalous dimension: "Z √ ◆# ◆ ( d 0 / (2 β 0 )) σ ( u ) ✓ ✓ γ ( x ) u β ( x ) − d 0 σ P ( u ) = exp dx σ ( u ) β 0 x √ u � 16

SF - running of the mass In the neighbourhood of the fixed point: Z m ( µ/s ) Z µ/s dm dq m = − γ ∗ q m ( µ ) µ log | σ P ( s, u ) | = − γ ∗ log s Hence we can define an estimator for the anomalous dimension: γ ( u ) = − log | σ P ( u, s ) | ˆ log | s | Scheme-independent in a neighbourhood of a fixed point. � 17

� 18 y , z O f 1 f 2 [ Γ ] = a 6 X ζ f 1 ( y ) Γ ζ f 2 ( z ) ¯ y , z f 1 f 2 [ Γ ] = a 6 X f 2 ( z ) f 1 ( y ) Γ ζ 0 ζ 0 O 0 ¯ L 3 h O 0 i ; A,B,C = 1 i O 21 [ Γ A ] O 45 [ Γ B ] i 53 [ Γ C ] Q ± F ± SF - four-fermi anomalous dimension

SF - four-fermi anomalous dimension f 1 = � 1 h ± 1 = h 1; γ 5 γ 5 γ 5 2 L 6 h O 0 12 [ γ 5 ] O 21 [ γ 5 ] i 2 = 1 h ± X k 1 = � 1 ✏ jkl h 1; γ j γ k γ l X h O 0 12 [ γ k ] O 21 [ γ k ] i 6 6 L 6 j,k,l k =1 , 2 , 3 3 = 1 h ± X h 1; γ 5 γ k γ k 3 k 4 = 1 F ± i ; A,B,C ( x 0 ) X h ± h 1; γ k γ 5 γ k h ± i ; A,B,C ( x 0 ) = 3 1 k 3 / 2 − η k f η 1 5 = 1 X h ± h 1; γ k γ k γ 5 3 k � Z ± 1 ( g 0 , aµ ) h ± 1; A,B,C ( L/ 2) = h ± 1; A,B,C ( L/ 2) � � g 0 =0 � 19

SF - four-fermi anomalous dimension Step-scaling functions: Σ ± ( s ; u, L/a ) = Z ± ( g 0 , sL/a ) Z ± ( g 0 , L/a ) − 1 � � g ( L ) 2 = u ¯ (Z √ ) σ ( u ) dg γ ± ( g ) σ ± ( s ; u ) = lim a → 0 Σ ± ( s ; u, L/a ) = T exp β ( g ) √ u In a neighbourhood of a fixed point: γ ± ( u ) = log σ ± ( s ; u ) log s � 20

Phase diagram of SU(N) gauge theories Use lattice tools to search for IRFPs in 4D SU(N) gauge theories Non-SUSY Phase Diagram Bound γ = 2 γ = 1 Fund Ladder Iwasaki et al. Catterall, Sannino Del Debbio, Patella,Pica Catterall, Giedt, Sannino, Schneible Appelquist, Fleming, Neil Deuzeman, Lombardo, Pallante Shamir, Svetitsky, DeGrand MWT 2A × 2S Adj × All Orders Beta Function Ryttov and F.S. 07 � 21 [Yamawaki, Appelquist, Miransky, Schrock, Nunez, Piai, Hong, Braun, Gies]

Running coupling 1.10 1-loop 1.08 2-loop Statistical 1.06 1.04 σ (u)/u 1.02 1.00 0.98 0.0 1.0 2.0 3.0 u [Bursa et al 09] � 22

Running of the mass 1.04 1-loop 1.02 Statistical Error 1.00 0.98 0.96 σ P (4/3,u) 0.94 0.92 0.90 � γ ( u ) = − log | σ P ( u, s ) | � ˆ 0.88 log | s | 0.86 0.84 0 0.5 1 1.5 2 2.5 3 3.5 u γ < 0 . 6 [Bursa et al 09] � 23