Lost in an EFT? You are here! The path to You are here! the - PowerPoint PPT Presentation

Lost in an EFT?

You are here!

The path to You are here! the pagoda

Current LHC …no obvious exploration.. signposts so far

What can we understand about the maze in general?

A first glance at some deep underlying structure of EFTs Ongoing work to understand mathematical structure of quantum EFTs on general grounds with Brian Henning, Xiaochuan Lu, and Hitoshi Murayama The importance and ubiquity of EFTs has been understood for decades — remarkable that very basic questions about their structure are (were) unknown

EFTs invariant under M(spacetime)xG(gauge,global) IR free M=SO(d) M=SO(4) inv. G linear G non-linear

EFTs invariant under M(spacetime)xG(gauge,global) IR free M=SO(d) M=SO(4) inv. G linear G non-linear Brian Henning, Xiaochuan Lu, TM, and Hitoshi Murayama, to appear

EFTs invariant under M(spacetime)xG(gauge,global) IR free M=SO(d) M=SO(4) inv. G linear G non-linear Mostly touch on this for this talk

Practical problem X L = c i O i i Redundancies between operators from equations of motion and integration by parts O 1 ∼ O 2 + δ S O 3 EOM δφ i O 1 ∼ O 2 + d O 3 IBP

10 5 98613 55133 SO H 4 L 30973 10 4 17456 No. of Operators SO H 4 L + EOM 10213 9966 6109 5691 3624 3304 SO H 4 L + EOM + IBP 2196 10 3 1933 1316 1137 810 687 643 495 410 412 312 254 256 10 2 194 169 155 127 106 98 80 74 61 55 47 39 35 34 25 25 22 10 17 16 16 12 11 10 9 8 7 6 6 5 4 4 3 3 3 1 2 2 2 1 1 1 5 10 15 20 25 Mass Dimension EFT of a single real scalar field

10 5 ? ‘naive’=SO( ∞ ) 98613 55133 SO H 4 L is interesting 30973 10 4 (but hard) 17456 No. of Operators SO H 4 L + EOM 10213 9966 6109 5691 3624 3304 SO H 4 L + EOM + IBP 2196 10 3 1933 1316 1137 810 687 643 495 410 412 312 254 256 10 2 194 169 155 127 106 98 80 74 61 55 47 39 35 34 25 25 22 10 17 16 16 12 11 10 9 8 7 6 6 5 4 4 3 3 3 1 2 2 2 1 1 1 5 10 15 20 25 Mass Dimension ∂ µ ∂ ν φ ∂ µ ∂ ν φ φ 2 ∂ µ φ ∂ µ ∂ ν φ ∂ ν φ φ Graph theory: ∂ µ φ ∂ µ φ ∂ ν φ ∂ ν φ

It turns out the structure of an operator basis is controlled by the conformal algebra Organize into irreps. of the conformal group — the basis is spanned by primary operators

Why the conformal algebra? SU(2) toy: build ‘operators’ out of two spin 1/2 states Spin A or Spin B or + + + The ‘Lagrangian’ =

Why the conformal algebra? SU(2) toy: build ‘operators’ out of two spin 1/2 states x = + - + 2 x 2 = 3 + 1

Why the conformal algebra? = R [ ∆ , ( j 1 ,j 2 )] repr. φ ψ α F µ ν O [ ∆ , ( j 1 ,j 2)] ∂ µ φ ∂ β ψ α ∂ ρ F µ ν ∂ µ O [ ∆ , ( j 1 ,j 2)] X ∂ { µ ∂ ν } φ x.. x.. ∂ δ ∂ β ψ α ∂ σ ∂ ρ F µ ν c ∆ ,j 1 ,j 2 = ∆ ,j 1 ,j 2 . ∂ { µ ∂ ν ∂ ρ } φ . . . . . . . . . . . R [ 3 R [1 , (0 , 0)] R [2 , (1 , 0)] 2 , ( 1 2 , 0)]

Why the conformal algebra? Keep only scalar primary = R [ ∆ , ( j 1 ,j 2 )] repr. operators φ ψ α F µ ν O [ ∆ , ( j 1 ,j 2)] ∂ µ φ ∂ β ψ α ∂ ρ F µ ν ∂ µ O [ ∆ , ( j 1 ,j 2)] X ∂ { µ ∂ ν } φ x.. x.. ∂ δ ∂ β ψ α ∂ σ ∂ ρ F µ ν c ∆ ,j 1 ,j 2 = ∆ ,j 1 ,j 2 . ∂ { µ ∂ ν ∂ ρ } φ . . . . . . . . . . . R [ 3 R [1 , (0 , 0)] R [2 , (1 , 0)] 2 , ( 1 2 , 0)] All total derivatives

Why the conformal algebra? We have an alternate picture in terms of scattering amplitudes EOM: on-shell particles IBP: momentum conservation

A recipe for constructing a Hilbert series for 4d phenomenological theories

Application to the SM On this operator basis we defined a generating function — Hilbert series Evaluate to count the number of independent operators at a given mass dimension in the SM Buchmuller, Wyler 1986 dim 6, 1 gen. Grzadkowski et. al. 2010 dim 6, 1 gen, corrected Manohar et. al. 2013 dim 6, Nf gen. Lehman, Martin 2014 dim 7, Nf gen. dim 8, 1 gen.

What a Hilbert series looks like Hilbert series H ( D , Q, u, d, L, e, H, F L , .. ) Z Z = PE ( χ Q , χ u , χ d , χ L , χ e , χ H , χ F L , .. ) SO (6) SU (3) × SU (2) × U (1)

What a Hilbert series looks like Characters χ R ( g ) = tr R ( g ) hgh − 1 = e i θ a H a e.g. = y + y − 1 diag( e i θ , e − i θ ) � � χ 2 ,SU (2) ( g ) = tr H ( D , Q, u, d, L, e, H, F L , .. ) Z Z = PE ( χ Q , χ u , χ d , χ L , χ e , χ H , χ F L , .. ) SO (6) SU (3) × SU (2) × U (1) Plethystic exponential: generate all possible operators e.g. ∞ ! L r χ [ 3 X 2 , 0)] ( α r , β r , q r ) χ 2 ,SU (2) ( y r ) PE ( χ L ) = exp 2 , ( 1 r =1

What a Hilbert series looks like H ( D , Q, u, d, L, e, H, F L , .. ) Z Z = PE ( χ Q , χ u , χ d , χ L , χ e , χ H , χ F L , .. ) SO (6) SU (3) × SU (2) × U (1) Haar measures: project out scalar, gauge inv operators e.g. d y 1 Z I y (1 − y 2 )(1 − y − 2 ) dµ SU (2) ( y ) = 2 π i

What a Hilbert series looks like � � HS ( D , Q, u, d, L, e, H, F L , .. ) � � dim 5 � 2 (1 − z 1 z 2 ) 2 � I dz 1 I dz 2 1 − α 2 � 2 � 1 − β 2 � 2 1 − y 2 � 2 � 2 � � � z 2 z 1 − z 2 I I I 1 I 1 − z 2 d α d β dx dy 2 = 6 z 5 1 z 5 2 π i 2 π i 4 α 3 β 3 2 π i 2 π i 2 y 3 2 π i 2 π i x 2 � 2 y 2 ⌘ � 2 y 2 ⌘ y 4 + y 2 + 1 � ⇣ α 2 + α 2 y 4 + α 2 + 1 y 4 + y 2 + 1 � ⇣ β 2 + β 2 y 4 + β 2 + 1 � � � � �  H 2 L 2 + ( H † ) 2 ( L † ) 2 + . . . α 2 y 4 β 2 y 4 = H 2 L 2 + H † 2 L † 2 Neutrino mass

What a Hilbert series looks like � � (one generation) HS ( D , Q, u, d, L, e, H, F L , .. ) � � dim 6 = H 3 H † 3 + u † Q † HH † 2 + 2 Q 2 Q † 2 + Q † 3 L † + Q 3 L + 2 QQ † LL † + L 2 L † 2 + uQH 2 H † b 6 + 2 uu † QQ † + uu † LL † + u 2 u † 2 + e † u † Q 2 + e † L † H 2 H † + 2 e † u † Q † L † + eLHH † 2 + euQ † 2 + 2 euQL + ee † QQ † + ee † LL † + ee † uu † + e 2 e † 2 + d † Q † H 2 H † + 2 d † u † Q † 2 + d † u † QL + d † e † u † 2 + d † eQ † L + dQHH † 2 + 2 duQ 2 + duQ † L † + de † QL † + deu 2 + 2 dd † QQ † + dd † LL † + 2 dd † uu † + dd † ee † + d 2 d † 2 + u † Q † H † G R + d † Q † HG R + HH † G 2 R + G 3 R + uQHG L + dQH † G L + HH † G 2 L + G 3 L + u † Q † H † W R + e † L † HW R + d † Q † HW R + HH † W 2 R + W 3 R + uQHW L + eLH † W L + dQH † W L + HH † W 2 L + W 3 L + u † Q † H † B R + e † L † HB R + d † Q † HB R + HH † B R W R + HH † B 2 R + uQHB L + eLH † B L + dQH † B L + HH † B L W L + HH † B 2 L + 2 QQ † HH † D + 2 LL † HH † D + uu † HH † D + ee † HH † D + d † uH 2 D + du † H † 2 D + dd † HH † D + 2 H 2 H † 2 D 2 . (3.16) First systematic procedure

Counting in the SM EFT 10000000000 7557369962 2795173575 1000000000 175373592 100000000 75577476 10000000 4614554 5474170 No. of independent ops 3472266 2092441 1000000 Nf=3 261485 257378 100000 90456 44807 15456 10000 11962 3045 Nf=1 1542 993 1000 560 84 100 30 12 10 2 1 5 6 7 8 9 10 11 12 13 14 15 Mass dimension

Never have to count ops again…. Hilbert series for your theory Characters for the repr. of H ( D , ψ 1 , ψ 2 , . . . ) each field Z Z = PE ( χ ψ 1 , χ ψ 2 , . . . ) SO (6) gauge, global Gauge and global symmetries of the EFT

Explicit example of a particular maze…

Explicit example of a particular maze… N ∞ ! 1 Z Y X φ i (1 − t 2 ) P ( t ; x ) H ( φ 1 , .., φ N , t ) = d µ SO ( d ) exp P ( t ; x ) i =1 r =1 in four dimensions 1 H ( φ , t ) | φ 4 = φ 4 (1 − t 4 )(1 − t 6 )

Explicit example of a particular maze… N ∞ ! 1 Z Y X φ i (1 − t 2 ) P ( t ; x ) H ( φ 1 , .., φ N , t ) = d µ SO ( d ) exp P ( t ; x ) i =1 r =1 in four dimensions 1 H ( φ , t ) | φ 4 = φ 4 (1 − t 4 )(1 − t 6 ) n=4

Explicit example of a particular maze… N ∞ ! 1 Z Y X φ i (1 − t 2 ) P ( t ; x ) H ( φ 1 , .., φ N , t ) = d µ SO ( d ) exp P ( t ; x ) i =1 r =1 in four dimensions H ( φ , t ) | φ 5 = φ 5 t 28 + t 22 + t 16 + t 14 + t 12 + t 8 − t 6 + t 4 − t 2 + 1 (1 − t 2 )(1 − t 6 )(1 − t 8 )(1 − t 10 )(1 − t 12 ) n=4 n=5 …

Explicit example of a particular maze… N ∞ ! 1 Z Y X φ i (1 − t 2 ) P ( t ; x ) H ( φ 1 , .., φ N , t ) = d µ SO ( d ) exp P ( t ; x ) i =1 r =1 in four dimensions fixed powers of t … 1 H ( φ , t ) | φ 4 = φ 4 (1 − t 4 )(1 − t 6 ) n=4 n=5 …

Explicit example of a particular maze… in one dimension 1 H ( φ 1 , φ 2 , t ) = (1 − φ 1 )(1 − φ 2 )(1 − t φ 1 φ 2 )

Explicit example of a particular maze… in one dimension 1 H ( φ 1 , φ 2 , t ) = (1 − φ 1 )(1 − φ 2 )(1 − t φ 1 φ 2 ) 1 − t φ 1 φ 2 φ 3 H ( φ 1 , φ 2 , φ 3 , t ) = (1 − φ 1 )(1 − φ 2 )(1 − φ 3 )(1 − t φ 1 φ 2 )(1 − t φ 1 φ 3 )(1 − t φ 2 φ 3 )