Towards Partial Compositeness on the Lattice: Baryons with Fermions - PowerPoint PPT Presentation

Towards Partial Compositeness on the Lattice: 
 Baryons with Fermions in Multiple Representations William I. Jay, University of Colorado Boulder � With Tom DeGrand, Ethan Neil, Daniel Hackett (Boulder); � Yigal Shamir, Ben Svetitsky (Tel Aviv); and � Maarten Golterman (San Francisco) � 1. 1. Introduction Introduction � o Lightning review of partial compositeness � o Our lattice model � o Technical specifications � 2. 2. Lattice research program Lattice research program � o Baryons in SU(3) and SU(4) � o Non-relativistic quark models � o Lattice results � 3. 3. Summary and Outlook Summary and Outlook �

� � How does mass generation occur in strongly coupled BSM models? � • Classic “extended technicolor” � – Chiral condensate breaks SU(2) breaks SU(2) L � – Higgs emerges from dynamics: dilaton (?) � • Composite Higgs -- Limited lattice investigation to date (!) � – Chiral condensate preserves SU(2) preserves SU(2) L � – Higgs from SSB: exact Goldstone boson � – SM loops generate potential for Higgs � • Fermion masses from 4-fermion interactions in both cases: � – Partial compositeness means linear couplings Partial compositeness means linear couplings to baryon operators � Trouble with FCNC constraints � qq ¯ qq O ETC ψψ ∼ ¯ ¯ q O PC q ψψψ ∼ ¯ ¯ Ø Better FCNC bounds � Ø Mass mixing � Ø Top quark partner(s) � 2

Ferretti’s Model (1404.7137) � A specific continuum UV theory for partial compositeness � ² SU(4) SU(4) gauge theory � 5 × Q ² Fermions: Fermions: � • 5 sextet 5 sextet Majorana fermions � 6 × q • 6 fundamental 6 fundamental Majorana fermions � • Equivalent Dirac DOF: 2.5 sextet, 3 fundamental � Equivalent Dirac DOF: 2.5 sextet, 3 fundamental ² Symmetry breaking: SU(5)/SO(5) SU(5)/SO(5) in the IR � • Sextet SU(4) is a real representation real representation � • Symmetry breaking pattern is different from QCD different from QCD � Ø Tough theory for lattice simulation � 3

Our Lattice Deformation � (The model we actually simulate) � • Still SU(4) SU(4) Gauge theory � • Modified matter content � – 2 . 5 7! 2 sextet sextet Dirac SU(4) fermions � – fundamental fundamental Dirac SU(4) fermions � 3 7! 2 • Symmetry breaking: SU(4)/SO(4) SU(4)/SO(4) in the IR � • Disclaimer 1: The deformation to SU(4)/SO(4) is not directly relevant for phenomenology. � • Disclaimer 2: Results today come from exploratory runs with partial quenching. Fully dynamical simulations are underway. � 4

Technical Specifications • “Multirep Multirep Milc Milc” with “NDS action NDS action” � – (DeGrand, Shamir, Svetitsky: 1407.4201) � • Wilson-Clover fermions � • SU(4) theory space parameterized by ( β , , κ 4 , , κ 6 ) � • Today � o Exploratory study: partially quenched partially quenched � o Ensemble from DeGrand, Liu:1606.01277 � o V=16 3 x 32 � o 2 x 2 x dynamical fundamental dynamical fundamental fermions fermions � ( β = 10.2, κ 4 = 0.1265, κ 4;critical 4;critical = 0.1284) � m PS PS /m /m V = 0.385(1)/0.560(3) = 0.688 � o Quenched sextet Quenched sextet propagators � 5

Warm-up for baryons in SU(4): � Hyperons in SU(3) � Baryons with (S=-1): uus, uds, dds � ∆ 0 ∆ + ∆ ++ ∆ − n p r r r r T  r r  T T   T T  q T   T Σ ∗ 0 Σ ∗ + Σ ∗− Σ 0 T  r r r  T  T T  Σ + Σ − q r T  r r T  T r  T  Λ Ξ ∗ 0 T  T Ξ ∗−  r r q T  T  T  T  T  T  r r Ω − r Ξ 0 Ξ − Σ ∗ (1390) : I ( J P ) = 1(3 / 2 + ) Σ (1190) : I ( J P ) = 1(1 / 2 + ) Λ (isosinglet) = � lightest QCD hyperon � Λ (1120) : I ( J P ) = 0(1 / 2 + ) 6

� Baryons in SU(4) � Building blocks � q v Fundamental SU(4) fermion: q a � v Sextet SU(4) fermion: Q ab ab with two indices with two indices � Q Quarks in a single representation q v Cousins of QCD nucleons q v Typical baryons: ( qqqq ) SU(4) v 4 fermions: bosons q v Also appearing: ( QQQQQQ ) SO(6) q Quarks in both representations Q v Cousins of QCD hyperons v Chimera baryons ( Qqq ) SU(4) q v 3 fermions: fermions v My code constructs these states (!) q 7

Baryon Masses in SU(4) � Goal: qualitative understanding qualitative understanding of baryon spectrum � The tool: A non-relativistic quark model � • “Constituent” quark masses with “color hyperfine” interactions � • A NR quark model also makes quantitative predictions for the quantitative predictions for the Gell-Mann 1969 � entire spectrum of SU(4) baryons entire spectrum of SU(4) baryons � m qqqq = 4 m q + C C ⇣ ⌘ X S i · ~ ~ ~ S 2 S j = 4 m q + tot − 3 m 2 2 m 2 q q i<j ✓ ◆ m Qqq = m Q + 2 m q + C S 2 + 2 m q S 1 · ~ ~ S Q · ( ~ ~ S 1 + ~ S 2 ) m 2 m Q q 8

Qqq Lattice Interpolating Fields � • Color Structure �  1 7! 12 – Baryons are SU(4) color SU(4) color singlets singlets �   – Code simulates six degrees of freedom for sextets �  2 7! 13   – Must map indices SO(6) Must map indices SO(6) à SU(4) SU(4) for correlation functions � . . • Spin Structure � .   – Intuition from quark model as guide �   6 7! 34  – Projection with P ± = ½ (1± γ 4 ) onto two-component NR basis � – Clebsches C αβγδ enact spin contraction � Sextet quark: two color indices � Spin contraction � Color singlets � ⌦ ¯ = ✏ abcd ✏ efgh C ↵��� C ✏�⌘⇣ D − 1 Q ( m | n ) ab,ef ↵ O B ( m ) O B ( n ) ↵ , ✏ h i q ( m | n ) c,g q ( m | n ) d,h q ( m | n ) d,g D − 1 � , � D − 1 � ,h − D − 1 q ( m | n ) c,h � , ⌘ D − 1 × � , � Minus sign from Wick’s theorem � Fundamental quarks: single color index � 9

“Chimera” 2-point correlators � Qqq • Strong signals with 50 - 60 configurations � • Asymmetric correlators, as in QCD (cf. Leinweber 2005, nucl-th/0406032) � 10

Chimera Spectrum vs κ 6 (fixed κ 4 ) � Qqq Isotriplet “ Σ -like” state lighter than isosinglet “ Λ -like” state at small sextet quark mass � 11

SU(4) baryon spectrum vs κ 6 � QQQQQQ [J(J+1) Rotor] [J(J+1) Rotor] � qqqq [J(J+1) Rotor] [J(J+1) Rotor] � Constant Constant vs vs κ 6 � Qqq Ø Chimera Qqq qq baryons can be light particles in the heavy spectrum � Ø Will these features persist with both representations in the sea? � 12

Success with the Quark Model � (pending confirmation with both representations in the sea) � • This SU(4) system is not QCD � • But the quark model successfully predicts all the qualitative features of the low-lying hadron spectrum � – Rotor splittings: δ m~J(J+1) � – Relative sizes of QQQQQ QQQQQ, qqqq qqqq, Qqq qq � – Presence of Σ - Λ inversion � • The chimera baryons are comparatively light à good for phenomenology � 13

Summary and Outlook � • We saw preliminary results for SU(4) gauge theory with fermions in mixed representations � – A quark model plays a key role in our understanding the spectrum of this theory. � • Interesting related questions remain (in progress) � – Pheno implications for the Σ - Λ inversion inversion? � – Calculation of the non- non-perturbative perturbative mixing mixing of elementary fermions with composite operators � – Calculation of anomalous dimensions anomalous dimensions for the four-fermion interactions � – Extending Large-N Extending Large-N results to mixed representations � – … � • Other interesting questions we’re actively pursuing � – What does the thermodynamic phase diagram look like? � – Do dynamically separated phases exist? � – Do hierarchies of scales exist? � 14

Thank you for your attention. � 15

Back-up slides � 16

The NDS Action 
 (Slide credit: E. Neil) • HYP smearing: staple sum over “fat links” added to original. nHYP normalizes the V = Ω ( Ω † Ω ) − 1 / 2 smeared link W. � • Q - ½ appears in the fermion force, and small eigenvalues can cause spikes. Q − 1 / 2 = ( Ω † Ω ) − 1 / 2 “nHYP dislocation suppressing” action cancels these with additional marginal gauge terms S NDS : � • Bare gauge coupling S NDS = 1 X X Q � 1 ˜ Tr γ 1 depends on β and γ . x,µ 2 N c x µ We fix the ratio and 1 adjust β to move lattice X X Q � 1 ˜ Q � 1 + γ 2 x,µ ; ν + γ 3 spacing � A x, ρ ; ξ µ 6 = ν ρ 6 = ξ 17

More technical details 1/2 � The “Multirep Multirep MILC MILC” code… � o Runs SU(N c ) gauge theory with simultaneous dynamical fermions in multiple representations dynamical fermions in multiple representations � o Is branched from the MILCv7 code, focusing on Wilson fermions � o Builds with dynamical code generation dynamical code generation using Perl so that N c and representation(s) are fixed during code generation, allowing the C compiler to produce optimized matrix operations optimized matrix operations � o Includes all the modern bells and whistles: Clover term, nHYP smearing, Hasenbusch preconditioning, multi-level integrators, dislocation-suppressing NDS action (DeGrand, Shamir, Svetitsky: 1407.4201) � 18