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

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

Dec 11, 2023 310 likes •560 views

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

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SLIDE 1

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

  • Lightning review of partial

compositeness

  • Our lattice model
  • Technical specifications

2. 2. Lattice research program Lattice research program

  • Baryons in SU(3) and SU(4)
  • Non-relativistic quark models
  • Lattice results

3. 3. Summary and Outlook Summary and Outlook

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SLIDE 2
  • 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

  • How does mass generation occur

in strongly coupled BSM models?

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Ø Better FCNC bounds Ø Mass mixing Ø Top quark partner(s)

Trouble with FCNC constraints

¯ qq ¯ ψψ ∼ ¯ qqOETC ¯ qψψψ ∼ ¯ qOPC

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SLIDE 3

A specific continuum UV theory for partial compositeness

² SU(4) SU(4) gauge theory ² Fermions: Fermions:

  • 5 sextet

5 sextet Majorana fermions

  • 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

Ferretti’s Model (1404.7137)

Ø Tough theory for lattice simulation

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6× q 5× Q

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SLIDE 4

Our Lattice Deformation

  • Still SU(4)

SU(4) Gauge theory

  • Modified matter content

– sextet sextet Dirac SU(4) fermions – fundamental fundamental Dirac SU(4) fermions

  • 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.

(The model we actually simulate)

2.5 7! 2 3 7! 2

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SLIDE 5

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,

, κ4 , ,κ6)

  • Today
  • Exploratory study: partially quenched

partially quenched

  • Ensemble from DeGrand, Liu:1606.01277
  • V=163 x 32
  • 2 x

2 x dynamical fundamental dynamical fundamental fermions fermions (β = 10.2, κ4 = 0.1265, κ4;critical

4;critical = 0.1284)

mPS

PS/m

/mV = 0.385(1)/0.560(3) = 0.688

  • Quenched sextet

Quenched sextet propagators

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SLIDE 6

Warm-up for baryons in SU(4):

Hyperons in SU(3)

T T T T T T T T T T T T T              r r r r r r r r r r ∆− ∆0 ∆+ ∆++ Ξ∗− Ξ∗0 Σ∗+ Σ∗− Σ∗0 Ω−

     T T T T T T T T T T      r r r r r r r r n p Ξ− Ξ0 Σ+ Σ− Σ0 Λ

Σ∗(1390) : I(JP ) = 1(3/2+) Σ(1190) : I(JP ) = 1(1/2+) Λ(1120) : I(JP ) = 0(1/2+) q q q

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Baryons with (S=-1): uus, uds, dds

Λ (isosinglet) = lightest QCD hyperon

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SLIDE 7

Baryons in SU(4)

Building blocks

  • v Fundamental SU(4) fermion: qa

v Sextet SU(4) fermion: Qab

ab with two indices

with two indices

Quarks in both representations

v Cousins of QCD hyperons v Chimera baryons (Qqq)SU(4) v 3 fermions: fermions v My code constructs these states (!)

Quarks in a single representation

v Cousins of QCD nucleons v Typical baryons: (qqqq)SU(4) v 4 fermions: bosons v Also appearing: (QQQQQQ)SO(6)

q q q q q

Q

q q

Q

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SLIDE 8

Baryon Masses in SU(4)

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 entire spectrum of SU(4) baryons entire spectrum of SU(4) baryons

Goal: qualitative understanding

qualitative understanding of baryon spectrum

Gell-Mann 1969

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mqqqq = 4mq + C m2

q

X

i<j

~ Si · ~ Sj = 4mq + C 2m2

q

⇣ ~ S2

tot − 3

⌘ mQqq = mQ + 2mq + C m2

q

✓ ~ S1 · ~ S2 + 2 mq mQ ~ SQ · (~ S1 + ~ S2) ◆

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SLIDE 9

Qqq Lattice Interpolating Fields

  • Color Structure

– Baryons are SU(4) color SU(4) color singlets singlets – Code simulates six degrees of freedom for sextets – 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 – Projection with P± = ½(1±γ4) onto two-component NR basis – Clebsches Cαβγδ enact spin contraction

           1 7! 12 2 7! 13 . . . 6 7! 34

Color singlets Spin contraction Sextet quark: two color indices Fundamental quarks: single color index Minus sign from Wick’s theorem

⌦ ¯ OB(m)OB(n) ↵ =✏abcd✏efghC↵C✏⌘⇣D−1

Q (m|n)ab,ef ↵,✏

× h D−1

q (m|n)c,g ,D−1 q (m|n)d,h ,h − D−1 q (m|n)c,h ,⌘D−1 q (m|n)d,g ,

i

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SLIDE 10

“Chimera” 2-point correlators

  • Strong signals with 50 - 60 configurations
  • Asymmetric correlators, as in QCD (cf. Leinweber 2005, nucl-th/0406032)

Qqq

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SLIDE 11

Chimera Spectrum vs κ6 (fixed κ4)

Qqq

Isotriplet “Σ-like” state lighter than isosinglet “Λ-like” state at small sextet quark mass

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SLIDE 12

SU(4) baryon spectrum vs κ6

Ø Chimera Qqq qq baryons can be light particles in the heavy spectrum Ø Will these features persist with both representations in the sea?

Qqq QQQQQQ

[J(J+1) Rotor] [J(J+1) Rotor]

qqqq

[J(J+1) Rotor] [J(J+1) Rotor] Constant Constant vs vs κ6

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SLIDE 13

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: δm~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

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SLIDE 14

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?

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SLIDE 15

Thank you for your attention.

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SLIDE 16

Back-up slides

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SLIDE 17

The NDS Action


(Slide credit: E. Neil)

  • HYP smearing: staple sum over “fat links”

added to original. nHYP normalizes the smeared link W.

  • Q-½ appears in the fermion force, and

small eigenvalues can cause spikes. “nHYP dislocation suppressing” action cancels these with additional marginal gauge terms SNDS:

V = Ω(Ω†Ω)−1/2 Q−1/2 = (Ω†Ω)−1/2

SNDS = 1 2Nc X

x

Tr γ1 X

µ

˜ Q1

x,µ

+ γ2 X

µ6=ν

˜ Q1

x,µ;ν + γ3

X

ρ6=ξ

Q1

x,ρ;ξ

1 A

  • Bare gauge coupling

depends on β and γ. We fix the ratio and adjust β to move lattice spacing

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SLIDE 18

More technical details 1/2

The “Multirep Multirep MILC MILC” code…

  • Runs SU(Nc) gauge theory with simultaneous

dynamical fermions in multiple representations dynamical fermions in multiple representations

  • Is branched from the MILCv7 code, focusing on

Wilson fermions

  • Builds with dynamical code generation

dynamical code generation using Perl so that Nc and representation(s) are fixed during code generation, allowing the C compiler to produce

  • ptimized matrix operations
  • ptimized matrix operations
  • 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)

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SLIDE 19

More technical details 2/2

Running parameters and results:

  • 2 x Dynamical fundamental fermions

2 x Dynamical fundamental fermions

  • (β = 10.2, κ4 = 0.1265, κ4;critical

4;critical = 0.1284)

  • mPS

PS/m

/mV = 0.385(1) / 0.560(3) = 0.688

  • Quenched sextet

Quenched sextet propagators

  • Range of kappa values: κ6 = 0.1170 up to

0.1290, κ6;critical

6;critical = 0.1295

  • mPS

PS/m

/mV ranging from 1.15 / 1.23 = 0.93 down to 0.19/0.52 = 0.36

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SLIDE 20

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SLIDE 21

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SLIDE 22

Baryons and the quark model

mB = m1 + m2 + m3 + X

i<j

a cij mimj ~ Si · ~ Sj

mB = 3m + a0 + a1J(J + 1)

mQqq = mQ + 2mq + a m2

q

✓ ~ S1 · ~ S2 + 2 mq mQ ~ SQ · (~ S1 + ~ S2) ◆ mQCD hyperon = ms + 2mu + a m2

u

✓ ~ S1 · ~ S2 + mu ms ~ SQ · (~ S1 + ~ S2) ◆

Two distinct gluon exchanges: sextet quark feels twice as much color force. Formally, this difference is a statement about relative sizes of Casimirs.

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SLIDE 23

References

(A short and scandalously incomplete list)

  • Composite Higgs

– Contino, The Higgs as a Composite Nambu-Goldstone Boson, arXiv:1005.4269 – Contino et al., On the effect of resonances in composite Higgs phenomenology, arXiv: 1109.1570 – Contino and Salvarezza, One-loop effects from spin-1 resonances in Composite Higgs models, arXiv:1504.02750

  • SU(4) models

– Ferretti and Karateev, Fermionic UV completions of Composite Higgs Models, arXiv: 1312.5330 – Ferretti, UV Completions of Partial Compositeness: The Case for a SU(4) Gauge Group, arXiv:1404.7137 – Ferretti, Gauge theories of Partial Compositeness: Scenarios for Run-II of the LHChttp, arXiv:1604.06467

  • Alternative perspectives

– Luty and Okui, Confromal Technicolor, arXiv: hep-ph/0409274 – Vecchi, A dangerous irrelevant UV-completion of the composite Higgs, arXiv: 1506.00623 – Ma and Cacciapaglia, Fundamental Composite 2HDM: SU(N) with 4 flavours, arXiv: 1508.07014

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SLIDE 24

Baryons and Large-N

  • Dashen, Jenkins, and Manohar derived formulae

for strange baryons in the large-N limit

– Depends only on the spin-flavor structure of the baryons, in the QCD case of SU(2) x U(1)

  • Gives a more general / less restrictive prediction

for the spectrum than the quark model.

  • Do these results remain valid with fermions in

mixed representations?

M = a0Nc + a1Ns + a21 J2 Nc + a22 I2 Nc + a23 N 2

s

Nc + O ✓ 1 N 3

c

◆

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