Lattice field theory and physics beyond the Standard Model C.-J. - - PowerPoint PPT Presentation

Lattice field theory and physics beyond the Standard Model

01/12/2020 National Taiwan University

1

C.-J. David Lin (林及仁) National Chiao Tung University, Taiwan

2

High-energy physics frontiers

Energy Frontier Intensity Frontier Cosmic Frontier Origin of Mass Dark Energy Proton Decay Origin

• f

Universe Dark Matter CP Violation

3

Paths to the origin of mass

Lattice Phenomenology Field Theory Experiments Origin

• f

Mass

4

Outline

Introduction: the standard-model Higgs and why it is not enough Lattice Field Theory and strong dynamics beyond the SM A new approach inspired by BSM physics and condensed matter theory The Higgs-Yukawa model Higgs as a bound state: Composite Higgs Tensor networks

5

The standard model (SM) Higgs

6

For the standard model: complex doublet (4 real scalars)

ϕ →

The standard model Higgs

JCMB 4417

V(ϕ)

Re( )

ϕ

Im( )

ϕ

7

V(ϕ)

Re( )

ϕ

Im( )

ϕ

μ2 ≥ 0 μ2 < 0 Spontaneous symmetry breaking

|⟨0|ϕ|0⟩| = v

for illustration

V(ϕ) = μ2ϕ*ϕ + λ(ϕ*ϕ)2

ϕ(x) = [h(x) + v] eiθ(x) h(x)

Goldstone boson

θ(x)

Choose ⟨0|ϕ|0⟩ = v =

−μ2 2λ

mh = 4vλ

VEV

The standard model Higgs

8

The standard model Higgs

The 44 corridor @ JCMB

The weak gauge boson masses MW,Z ∝ gv The fermion masses mψ ∝ yv Coupled to weak gauge bosons via (coupling )

∂μϕ → Dμϕ g

Coupled to fermions via the Yukawa coupling + h.c.

y ¯ ψLϕψR

for the origin of masses

9

The good, the bad and the ugly

• f the standard model

10

What the LHC revealed to us hitherto

11

Totally compatible with the SM

12

Higgs boson ~125 GeV Searched up here ~2 TeV The Higgs boson is light

What the LHC revealed to us hitherto

13

14

Running coupling in QFT

Figure From Roberto Soldati

Charge screening in Quantum Electrodynamics Interaction/coupling strength changes with distance (energy scale)

15

Possible new physics appears at this scale with the Higgs quartic self coupling

¯ M

¯ λ

The scalar (Higgs) sector is trivial!

125 GeV with the Higgs quartic self coupling

MHiggs ∼

λ

16

125 GeV with the Higgs quartic self coupling

MHiggs ∼

λ

Possible new physics appears at this scale with the Higgs quartic self coupling

¯ M

¯ λ

The scalar (Higgs) sector is trivial!

17

125 GeV with the Higgs quartic self coupling

MHiggs ∼

λ

Possible new physics appears at this scale with the Higgs quartic self coupling

¯ M

¯ λ

The scalar (Higgs) sector is trivial!

18

125 GeV with the Higgs quartic self coupling

MHiggs ∼

λ

Possible new physics appears at this scale with the Higgs quartic self coupling

¯ M

¯ λ

The scalar (Higgs) sector is trivial!

Non-perturbative

19

20

MHiggs ∼ − μ2 − ¯ M2 ∼ (125 GeV)2

Possible new physics appears at this scale , which could be as high as GeV

¯ M 1019

The Higgs is “unnaturally” light!

for illustration

V(ϕ) = μ2ϕ*ϕ + λ(ϕ*ϕ)2

Huge cancellation! Non-perturbative

21

Lattice field theory

22

Basic ingredients

Monte-Carlo simulations with importance sampling c.f. Partition function in statistical physics Matter fields (fermions and scalars) Gauge fields

a

Z = Z DUDφD ¯ ψDψ e−SU e−Sφe−a4 P

x ¯

ψxM[U,φ]ψx

= Z DUDφD ¯ ψDψ det(M[U, φ]) e−SU e−Sφ

In finite 4-dimensional volume

23

The continuum limit

Matter fields (fermions and scalars) Gauge fields

a

means

• r

a → 0 am → 0 ξ/a → ∞

: typical low-energy mass scale

m

: typical long-distance length scale (correlation length)

ξ

The continuum limits are at 2nd-order phase transition points Critical phenomena are crucial for lattice field theory

24

New physics in the Higgs-Yukawa theory?

25

The Higgs-Yukawa sector of the SM…

at fixed

V(ϕ) = μ2ϕ†ϕ + λ(ϕ†ϕ)2 with y ( ¯ ψLϕψR + h . c.) λ

“Let me describe a typical computer simulation: … the first thing to do is to look for phase transition.”

• G. Parisi in Field Theory, Disorder and Simulations

y

1 − 2λ 8 + μ2

v ≠ 0

v = 1 V4 ∑

X

⟨0|ϕ(x)|0⟩

vs ≠ 0 vs ≠ 0

vs ≠ 0 v ≠ 0

v = vs = 0 v = vs = 0

SYM SYM FM AFM AFM

vs = 1 V4 ∑

X

η(x) ⟨0|ϕ(x)|0⟩

FM

v ≠ 0

y

1 − 2λ 8 + μ2

26

at fixed

V(ϕ) = μ2ϕ†ϕ + λ(ϕ†ϕ)2 with y ( ¯ ψLϕψR + h . c.) λ

is a small corner on the phase diagram

27

It is a challenging yet important task

at fixed

V(ϕ) = μ2ϕ†ϕ + λ(ϕ†ϕ)2 with y ( ¯ ψLϕψR + h . c.) λ

Possible new fixed point? Possible new four-fermion condensate? Possible first-order phase transition?

S.Catterall and D.Schaich, PRD96 (2017) D.Y.-J.Chu, K.Jansen, B.Knippschild, CJDL, JHEP01 (2019) 110 J.Bulava et al., AHEP 2013 (2013) 875612 A.Hasenfratz, K.Jansen, Y.Shen, NPB394 (1993)

Other directions (e.g. 2HDM)?

28

New physics as new interactions?

V(ϕ) = μ2ϕ†ϕ + λ(ϕ†ϕ)2 + λ6(ϕ†ϕ)3 with y ( ¯ ψLϕψR + h . c.)

λ κ

1 − 2λ 8 + μ2

λ6 = 0.001

D.Y.-J.Chu, K.Jansen, B.Knippschild, CJDL, A.Nagy, PLB 744 (2015) 146

29

The Higgs boson as a bound state

V(ϕ)

Re( )

ϕ

Im( )

ϕ

30

V(ϕ)

Re( )

ϕ

Im( )

ϕ

μ2 ≥ 0 μ2 < 0

SSB

|ϕ0| = v

for illustration

V(ϕ) = μ2ϕ*ϕ + λ(ϕ*ϕ)2

ϕ(x) = [h(x) + v] eiθ(x) h(x)

Goldstone boson

θ(x)

Choose ⟨0|ϕ|0⟩ = v =

−μ2 2λ

mh = 4vλ

The standard model Higgs

VEV

31

Different paths to electroweak symmetry breaking

Self-interacting scalars replaced by strongly-interacting fermions and gauge bosons

r0 = 1/Λb

Coupling Bound state formed at low energy as energy Typical bound-sate size r0 = 1/Λb Typical bound-sate mass Λb GeV GeV

2 × 103 ≲ Λb < 104

Not yet seen experimentally

O p p

• s

i t e t

• i

n !

λ V(ϕ)

32

Two issues: I. The light Higgs

Q: Why is the Higgs so light, ?

MHiggs ≪ Λb

A: Resort to spontaneous symmetry breaking

Global symmetry breaking like QCD (Composite Higgs) Scale-invariance breaking unlike QCD (Dilaton Higgs)

33

Two issues: II. The SM fermion mass

1 Λ2

f

¯ ψSMψSM ¯ ff 1 Λ2

f

¯ ψSMψSM ¯ ψSMψSM

mass FCNC The need to suppress flavour-changing neutral-current processes… GeV

Λf ∼ 107

~ 10 - 100 GeV

High to suppress FCNC

34

Two issues: II. The SM fermion mass

1 Λ2

f

¯ ψSMψSM ¯ f f 1 Λ2

f

¯ ψSMψSM ¯ ψSMψSM

mass FCNC The need to suppress flavour-changing neutral-current processes… GeV

Λf ∼ 107

~ 10 - 100 GeV

35

Two issues: II. The SM fermion mass

1 Λ2

f

¯ ψSMψSM ¯ ff 1 Λ2

f

¯ ψSMψSM ¯ ψSMψSM

mass FCNC The need to suppress flavour-changing neutral-current processes… GeV

Λf ∼ 107

~ 10 - 100 GeV

High to suppress FCNC

36

Two issues: II. The SM fermion mass

1 Λf ¯ ψSMψSM ¯ ff 1 Λ2

f

¯ ψSMψSM ¯ ψSMψSM

mass FCNC The need to suppress flavour-changing neutral-current processes… GeV

Λf ∼ 107

~ 10 - 100 GeV

(1/Λ2

f ) → 1/Λf

37

Two issues: II. The SM fermion mass

1 Λf ¯ ψSMψSM ¯ ff

1 Λ2

f

¯ ψSMψSM ¯ ψSMψSM

mass FCNC The need to suppress flavour-changing neutral-current processes… GeV

Λf ∼ 107

~ 10 - 100 GeV

38

Two issues: II. The SM fermion mass

1 Λ2

f

¯ ψSMψSM ¯ ff

1 Λ2

f

¯ ψSMψSM ¯ ψSMψSM

mass FCNC The need to suppress flavour-changing neutral-current processes… GeV

Λf ∼ 107

~ 10 - 100 GeV

1/Λf → (1/Λf)2

39

Two issues: II. The SM fermion mass

1 Λ2

f

¯ ψSMψSM ¯ ψSMψSM

mass FCNC The need to suppress flavour-changing neutral-current processes…

1 Λ2

f

¯ ψSMψSM ¯ ff

GeV

Λf ∼ 107

~ 10 - 100 GeV

40

Two issues: II. The SM fermion masses

Need power-law scaling behaviour in ¯

ff

Dramatically alter the suppression of from to

¯ ψSMψSM ¯ ff 1/Λ2

f

1/Λf

The system is at criticality for a large range of interaction strength c.f. Berezinskii-Kosterlitz-Thouless phase transition

41

The Higgs boson as a bound state: Composite Higgs models

42

Lesson from Quantum Chromodynamics

GeV

Mπ = 0

Mρ , MN , . . . ∼ Λ(QCD)

b

∼ 1 GeV SSB via ⟨0| ¯

qq|0⟩

43

Lesson from Quantum Chromodynamics

GeV

Mπ = 0.132

Mρ , MN , . . . ∼ Λ(QCD)

b

∼ 1 GeV Couple Electroweak ~ 1000 GeV SSB via ⟨0| ¯

qq|0⟩

44

Lesson from Quantum Chromodynamics

GeV

Mπ = 0.132

Mρ , MN , . . . ∼ Λ(QCD)

b

∼ 1 GeV

Couple Electroweak ~ 1000 GeV

Planck scale GeV

1019

Higgs mass 125 GeV

SSB via ⟨0| ¯

qq|0⟩

45

Composite Higgs models

Composite Higgs scale Λ(CH)

b

∼ 103 GeV

Couple

Flavour scale ~ GeV

Λf 107

Planck scale GeV

1019

Higgs mass and VEV ~ GeV

102

Lattice studies

L a t t i c e s t u d i e s

(Many bound states here)

Lattice studies

System at criticality (for SM fermion mass)

SSB via ⟨0| ¯

ff |0⟩

46

Composite Higgs models

Higgs is light because of the SSB for the global symmetry SM fermion masses: via mixing with the hybrid baryons ffF The theory is at/close to criticality to enhance effects of (ψSM ffF)

Fermions in different representations of the gauge group, different from QCD

Recent new direction in the community: spectrum studies hitherto

Different from QCD Similar to QCD

• Sp(4) gauge theory spectrum

Hsinchu-Pusan-Swansea collaboration, PRD101 (2020)

48

Technique for the future: Tensor networks and Hamiltonian formalism

(Matrix Product States)

Logic flow

Hamiltonian formalism for LFT Quantum spin model MPS & variational method for obtaining the ground state Compute correlators and other quantities

No more path integrals as we go back to the canonical formalism

50

Why?

Mapping field theories onto quantum spin models offers new insights Possible formulation for quantum computers

Matrix product states in a nutshell

| i 2 |ψi =

d

X

j1,...,jn=1

cj1,...,jn|j1, . . . , jni =

d

X

j1,...,jn=1

cj1,...,jn|j1i ⌦ · · · ⌦ |jni .

• bers. While

O(dn) man many real by O(ndD2) xponential in : so

Entanglement-based argument for choosing D

Bond dim

cj1,...,jn =

D

X

α,...,ω

A(1)

α;j1A(2) α,β;j2 . . . A(n) ω;jn = A(1) j1 A(2) j2 . . . A(n) jn

d d d d d d d D D

• S. White, 1992; M.B. Hasting, 2004; F. Verstraeten and I. Cirac, 2006; …

52

The Hamiltonian and matrix elements

σ σ´ σ1 σL σ´1 σ´L

:

ℋ

:

⟨Ψ|ℋ|Ψ⟩

Variational search for the ground state

⟨Ψ| ℋ |Ψ⟩

53

Application to the Thirring model

1+1 dimensional massive Thirring model

STh

• ψ, ¯

ψ

• =
• d2x
• ¯

ψiγµ∂µψ − m0 ¯ ψψ − g 2 ¯ ψγµψ 2

Gapped Critical A critical phase Expect BKT transitions. Perturbation theory

54

Phase structure of the Thirring model

1+1 dimensional QFT 1 dimensional XXZ quantum spin chain with constant and staggered B-field Commonly-studied correlators in QFT can be turned into spin correlates Look at : exponential/power law in the gapped/critical phase

C(r)

Well… but this is challenging….

55

A string correlator in the spin model helps…

Cstring(r) = ⟨0|σz

0σz 1σz 2⋯σz r |0⟩

r → ∞ Cstring(r) → k ≠ 0

Gapped

Cstring(r) → 0

Critical This corresponds to at each lattice site

¯ ψ(x)ψ(x)

NOT a commonly studied quantity in QFT

56

Probing phase structure with Cstring(r)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

• 0.9
• 0.8
• 0.7
• 0.6
• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

m0a ∆(g) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

= cos ( π − g 2 )

Value of at

Cstring(r) r → ∞

M.C.Banuls, K.Cichy, Y.-J.Kao, CJDL, Y.-P.Lin, D.T.-L.Tan, PRD100 (2019)

57

Probing phase structure with Cstring(r)

M.C.Banuls, K.Cichy, Y.-J.Kao, CJDL, Y.-P.Lin, D.T.-L.Tan, PRD100 (2019)

Perturbation theory (qualitative) Simulation with MPS

58

Conclusion and outlook

Strong dynamics can play an important role in BSM physics Lattice Field Theory is a powerful method in this subject New approaches inspired by BSM physics and condensed matter theory

Thank you all for your attention!

59

Backup slides

60

The Higgs boson as a bound state: Dilaton Higgs

61

Lesson from Quantum Chromodynamics

GeV

Mπ = 0

1 GeV

Mρ , MN , . . . ∼ Λ(QCD)

b

∼

SSB via ⟨0| ¯

qq|0⟩

62

Introducing (approximate) scale invariance

GeV

MGB = 0

Bound states at GeV

∼ Λ(DH)

b

∼ 103

SSB via ⟨0| ¯

qq|0⟩

Flavour scale ~ GeV

Λf 107

System at criticality (For Higgs and SM fermion masses) SSB of approximate scale invariance The Higgs as the pseudo GB (dilaton) 125 GeV

63

Looking for viable candidate theories

Z.Fodor, J.Kuti, K.Holland, S.Mondal, D.Nogradi, PRD94 (2016) CJDL, K.Ogawa, A.Ramos, JHEP12 (2015) T.-W.Chiu, PRD99 (2019) T.DeGrand, Y.Shamir, B.Svetitsky, PRD82 (2010) A.Hasenfratz, C.Rebbi, O.Witzel, PRD101 (2020) A.Deuzeman, M.P.Lombardo, T.N.Da Silva, E.Pallante, PLB720 (2013)

SU(3) gauge theories with various fermion contents

T.Appelquist, G.Fleming, E.Neil, PRL100 (2008)

… SU(2) gauge theories with various fermion contents …

64

What does a viable spectrum look like

LSD Collaboration, PRD 99, 2019