PDF Composite Higgs Dynamics on the Lattice

Composite Higgs Dynamics on the Lattice

Claudio Pica

Claudio Pica

Standard Model Higgs

m_H = 125.09 ± 0.24 GeV

L = v²

4 TrD_µß^†D^µß µ

1+2a H

v +...

∂

°√¯ⁱ_Lß√_R µ

1+c H

v +...

∂

∏_f = p 2

µm_f M

∂_1+≤

, g_V = 2

√m_V^2(1+≤) M^1+2≤

!

J. Ellis and T. You, JHEP 1306 (2013) 103

Claudio Pica

• If EWSB is due to a condensate of strongly-interacting fermions, one would expect, in general, composite scalar particles

• To not be excluded by experiments, this scalar states should mimic a SM-like Higgs boson.

• This could happen if the composite scalar is a light pseudo-

Goldstone boson of some (higher scale) broken symmetry, e.g:

- approximate scale invariance symmetry (dilaton)

- larger chiral symmetry (composite Goldstone boson Higgs)

Can the Higgs be composite?

Claudio Pica

Plus

• Break EW

• UV complete theories available to explore

• Natural

Minus

• Fermion masses vs FCNC

• Electroweak precision data

• Light Higgs

• Higgs couplings

Technicolor

Higgs is the lightest scalar excitation of the condensate

S. Weinberg & L. Susskind, ‘79

Claudio Pica

• Gauge Group: SU, SO, SP, Exceptional

• Matter Representation

• # of Flavors per Representation

Parameters

QCD-like IR fixed point No AF

N

α

Energy Λ_TC

�

Energy

�_U

�^* α

Energy Λ_ETC

Λ_TC

Claudio Pica

Walking

�

Energy

�_U

�^*

α

Energy Λ_ETC

Λ_TC α

Energy Λ_TC

Running IR fixed point

Walking

Holdom, Appelquist, Miranski, Yamawaki, Wijewardhana

Claudio Pica

SU(N) phase diagram

SU(N)

2 3 4 5 6 7 8

2 4 6 8 10 12 14 16 18

N

n f

Fund

A-Sym

Sym

Adj

Ryttov & Sannino 07 Dietrich & Sannino 07 Sannino & Tuominen 04 Ryttov & Shrock 10

Poppitz & Unsal 9, 10 Pica & Sannino 10

Other groups: SO(n), SP(n), exceptional

Walking region? Large

γ

Lattice

Claudio Pica

• Use the Lattice to investigate quantitatively models which can feature the right dynamics:

• SU(2) + 2 Dirac Adjoint SU(2)a - MWT

• SU(3) + 2 Dirac Symmetric SU(3)s - MWT

• SO(4) + 2 Dirac Vector SO(4)v - MWT

• SU(3) + 2 Dirac Fund. + Ungauged SU(3)f - pMWT

• SU(2) +2 Dirac Fund. + … (U - MWT) SU(2)f - MWT

Minimal (Walking) Models

Only one doublet gauged ➙ small S parameter

Claudio Pica

• TC Higgs is the lightest isospin-0 scalar made of TC-fermions

• will contain also a TC-glue component

• analogue to QCD lightest scalar:


f0(500) with mass ~ 400-550 MeV


Sannino & Schechter 95 PRD; Harada, Sannino & Schechter 95 PRD, 96PRL; 


see also: Pelaez - Confinement X

TC Higgs

H ⇠ c₁QQ¯ + c₂QQ¯ QQ¯ + · · ·

Claudio Pica

• Can it be light?

• SM radiative corrections shift the TC Higss mass

TC Higgs mass

t

W Z

R. Foadi, M. Frandsen, F. Sannino, Phys.Rev. D 87, 095001

M_H² = (M_H^TC)² + 3(4⇡F_⇧)² 16⇡²v²



4r_t²m²_t + 2s_⇡

✓

m²_W + m²_Z 2

◆

+ _M²

H (4⇡F_⇧)

Claudio Pica

• Can it be light?

• Narrow due to kinematics [Similar to f0(980) in QCD]

TC Higgs mass

0.0 0.5 1.0 1.5 2.0

200 400 600 800 1000 1200

k r_t M HTC HGeVL

(M_H^TC)² ' M_H² + 12 ²r_t²m²_t

R. Foadi, M. Frandsen, F. Sannino, Phys.Rev. D 87, 095001

Claudio Pica

• The leading order interaction of the TC Higgs with EW goldstone bosons can be described by:


with

• One can estimate the analogue coupling in QCD


from elastic pi-pi scattering data

TC Higgs couplings?

A. Belyaev, M.S. Brown, R. Foadi, M. Frandsen, PhysRevD.90.035012

L_H¶¶ = c_¶

v H @_µ¶^a@^µ¶^a

L_溺 = c_º

f_º æ@_µº^a@^µº^a

c_¶ = 1

6

m (MeV) g _⇡⇡ (GeV) c^QCD_⇡ 457⁺¹⁴₁₃ i(279⁺¹¹₇ ) , [35] 3.59^+0.11_0.13 1.0169 ± 0.06 445± 25 i(278⁺²²₁₈) , [35] 3.4 ± 0.5 1.0013 ± 0.17 441⁺¹⁶₈ i(272⁺⁹_12.5) [36] 3.31^+0.35_0.15 1.0035 ± 0.12 474 ± 6 i(254 ± 4) , [37] 3.58± 0.03 1.0264 ± 0.024

443 ± 2 i(216 ± 4) [38] 2.97± 0.04 1.0479 ± 0.020 452 ± 12 i(260 ± 15) [39] 2.65± 0.01 0.8026 ± 0.053

453 i271 , [40] 3.5 1.0255

TABLE I: Fits to m and g _⇡⇡ extrapolated from the elastic ⇡⇡ scattering(see [35] for a summary of recent results). The last column gives c^QCD_⇡ according to Eq. (17).

of QCD, we expect c⇧ ' 1 and, thus, SM-like HWW and HZZ couplings. This need not be the case for walking dynamics, although departures from QCD-like dynamics by no means imply non-standard couplings to massive weak bosons. Finally we note that in TC the HWW and HZZ couplings receive small corrections from tree-level mixing of the electroweak bosons with the TC spin-one resonances⁵ [26].

B. TC Higgs couplings to f f

The interactions of the TC Higgs or the technipions with two SM fermions are due to four- fermion operators of the form ¯f f⁰FF⁰. Here f and f⁰ are SM flavors, whereas F and F⁰ are techniflavors, with TC gauge indices contracted to form a TC singlet. At low energy these operators generate vertices such as Hf f¯ and ⇧^a f f¯ ⁰. Unlike the coupling to the weak bosons, there are no corresponding quantities in QCD which allow us to estimate the value of these couplings. However, Ward identities guarantee that the couplings of two SM fermions with the EW Goldstone bosons, at zero external momenta, are identical to their SM values. Of course these Ward identities tell us nothing about the Hf f¯ coupling. On the other hand, we have just seen that the interactions of the meson and the pions in QCD are well approximated by a linear sigma model. If this holds in TC as well, then a ⇧^a f f¯ ⁰ coupling close to its SM value implies that the TC Higgs couplings to SM flavors are close to their SM values, i.e c_f ' 1. In the case of a true techni-dilaton, Ward-identities related to the spontaneously-broken scale symmetry can be used to determine c_f [42, 43]. Finally, if SM fermion masses are generated by an ETC sector, we expect

5 Similarly when vector mesons are introduced into the linear sigma model of QCD the fits are altered.

Claudio Pica

Plus

• Higgs is massless

• Gauge boson couplings

Minus

• Underlying UV complete theory

• Higgs mass

• Fermion masses/couplings

• EW vacuum alignment

Composite Goldstone Higgs

Higgs is a pseudo Goldstone boson

D.B. Kaplan & H. Georgi, ‘84

Claudio Pica

• SU(2)TC with Nf=2 fund

• SO(6)/SO(5) chiral symmetry breaking

• Common framework for TC and CH

• Fundamental 4D underlying theory, 


Spectrum can be obtained via lattice simulations

SU(2) Composite Higgs

G. Cacciapaglia & F. Sannino, JHEP04(2014)111

Appelquist, Sannino, 98, 99 Ryttov, Sannino, 2008

Katz, Nelson Walker, 2005

Gripaios, Pomarol, Riva, Serra, 2009 Galloway, Evans, Luty, Tacchi, 2010

Claudio Pica

The SU(2) model

L = ° 1

4 F_µ∫^a F ^aµ∫ +U (i∞^µD_µ ° m)U + D(i ∞^µD_µ ° m)D

Q = 0 BB

@

U_L D_L Ue_L De_L

1 CC A

L = ° 1

4 F_µ∫^a F ^aµ∫ + iU ∞^µD_µU + i D∞^µD_µD + m

2 Q^T (°iæ²)C EQ + m 2

°Q^T (°iæ²)C EQ¢†

E = 0 BB

@

0 0 1 0 0 0 0 1

°1 0 0 0 0 °1 0 0

1 CC A

Claudio Pica

• QL = (UL, DL): SU(2)L doublet with zero hypercharge

• other two fields: SU(2)L singlets with hypercharge ±1/2

• two interesting alignments of the condensate:

1. does not break EW

2. does break EW symmetry

• general superposition:

The model

hQQi /

µ iæ₂ 0 0 °iæ₂

∂

¥ ß_B hQQi /= °i E =

µ 0 °i

i 0

∂

¥ ß_H

ß₀ = cosµ ß_B + sinµ ß_H

Claudio Pica

• Fundamental 4d underlying dynamics

• 5 Goldstone bosons

• 3 eaten by W^±,Z

SO(6)~SU(4) ➙ Sp(4)~SO(5)

TC CH

✓

θ ~0

• 1 GB is Higgs-like

• other GB is SM neutral

θ ~ π /2

• GB complex doublet

• SM neutral

• Natural DM candidate

Claudio Pica

• TC-Higgs and PGB Higgs mix

• Top reduces TC-Higgs mass

• Top contributes to PGB Higgs mass

Generic properties

TC CH

✓

Couplings: for θ near zero:

g_{W W h1}

g_{W W h}^SM = 1 +Cµ + O (µ²) g_{t th1}

g_{t th}^SM = 1 + Dµ + O (µ²)

• Modified Higgs phenomenology

• a new TeV scalar lighter than 


vectors?

Claudio Pica

• Rescale TC limit:

Spin one resonances

m_Ω = m_Ω^TC

sinµ m_A = m^TC_A sinµ f_PS sinµ = 246GeV

Claudio Pica

Lattice results

Claudio Pica

SU(2) f N f =2

Ref:R. Lewis, C. Pica, F. Sannino, Phys.Rev. D85 (2012) 014504 [arXiv:1109.3513]

A. Hietanen, R. Lewis, C. Pica, F. Sannino, [arXiv:1308.4130 [hep-ph]]

A. Hietanen, R. Lewis, C. Pica, F. Sannino, JHEP 1407 (2014) 116 [arXiv:1404.2794 [hep-lat]]

R. Arthur, A. Hietanen, V. Drach, M. Hansen, C.P., F. Sannino, arXiv:1602.06559 R. Arthur, A. Hietanen, V. Drach, M. Hansen, C.P., F. Sannino, in preparation

Claudio Pica

Lattice conformal window

SU(N)

2 3 4 5 6 7 8

2 4 6 8 10 12 14 16 18

N

n f

Fund

Claudio Pica

Scale Setting

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS² w0a

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS² w0a

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS²

w0a cut LO cut NNLO

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS² w0a

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS² w0a

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS²

w0a cut LO cut NNLO

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS² w0a

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS² w0a

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS²

w0a cut LO cut NNLO

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS² w0a

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS² w0a

0 1 2 3 4 5

0123456

y² = w₀² (m) m_PS²

w0a cut NNLO

β=1.8 β=2.0 β=2.2 β=2.3

0 1 2 3 4 5

0.40.50.60.70.80.91.01.1

y² = w₀² (m) m_PS² w0w0χ

0 1 2 3 4 5

0.40.50.60.70.80.91.01.1

y² = w₀² (m) m_PS² w0w0χ

0 1 2 3 4 5

0.40.50.60.70.80.91.01.1

y² = w₀² (m) m_PS² w0w0χ

0 1 2 3 4 5

0.40.50.60.70.80.91.01.1

y² = w₀² (m) m_PS² w0w0χ

β=1.8 β=2.0 β=2.2 β=2.3

w₀(m_ps² ) = w₀^¬ °

1 + Ay² +B y⁴ log y²¢ W (t) = t d

d t [t²E(t)], W (w₀²) ¥ 1

Claudio Pica

Goldstone bosons

0 5 10 15

0.00.10.20.30.4

( w₀^χ mPS )²

w 0χ FPS

0 5 10 15

0.00.10.20.30.4

( w₀^χ mPS )²

w 0χ FPS

0 5 10 15

0.00.10.20.30.4

( w₀^χ mPS )²

w 0χ FPS

0 5 10 15

0.00.10.20.30.4

( w₀^χ mPS )²

w 0χ FPS

0 5 10 15

0.00.10.20.30.4

( w₀^χ mPS )²

w 0χ FPS

0 5 10 15

0.00.10.20.30.4

( w₀^χ mPS )²

w 0χ FPS

0 5 10 15

0.00.10.20.30.4

( w₀^χ mPS )²

w 0χ FPS

0 5 10 15

0.00.10.20.30.4

( w₀^χ mPS )²

w 0χ FPS

β=1.8 β=2.0 β=2.2 β=2.3

0 5 10 15

024681012

( w₀^χ mPS )²

w 0χ m PS2 mf

0 5 10 15

024681012

( w₀^χ mPS )²

w 0χ m PS2 mf

0 5 10 15

024681012

( w₀^χ mPS )²

w 0χ m PS2 mf

0 5 10 15

024681012

( w₀^χ mPS )²

w 0χ m PS2 mf

0 5 10 15

024681012

( w₀^χ mPS )²

w 0χ m PS2 mf

0 5 10 15

024681012

( w₀^χ mPS )²

w 0χ m PS2 mf

0 5 10 15

024681012

( w₀^χ mPS )²

w 0χ m PS2 mf

0 5 10 15

024681012

( w₀^χ mPS )²

w 0χ m PS2 mf

β=1.8 β=2.0 β=2.2 β=2.3

m_PS²

m_f =2B

"

1°a_Mx˜log m²_ps

µ² +b_Mx˜+±_M a

w₀^¬ +∞_Mm_ps² a w₀^¬

# f_PS =F

"

1°a_Fx˜log m_PS²

µ² +b_Fx˜ +±_F a

w₀^¬ +∞_Fm_PS² a w₀^¬

#

w₀^¬B = 2.88(15)(17)

w₀^¬F = 0.078(4)(12) ß^1/3/F = 4.19(26)

Claudio Pica

Spin-1 Resonances

0 2 4 6 8

01234

( w₀^χ mPS )²

w 0χ mV

0 2 4 6 8

01234

( w₀^χ mPS )²

w 0χ mV

0 2 4 6 8

01234

( w₀^χ mPS )²

w 0χ mV

0 2 4 6 8

01234

( w₀^χ mPS )²

w 0χ mV

β=1.8 β=2.0 β=2.2 β=2.3

0 2 4 6 8

01234

( w₀^χ mPS )²

w 0χ mV

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

β=1.8 β=2.0 β=2.2 β=2.3

0 2 4 6 8

012345

( w₀^χ mPS )²

w 0χ mA

w₀^¬m_X = w₀^¬m^¬_X + A(w₀^¬m_PS)²+B(w₀^¬m_PS)⁴+C a w₀

m_V / f_PS = 13.1(2.2) m_A/f_PS = 14.5(3.6)

Claudio Pica

Spin-0 Resonances

0 5 10 15

012345

( w₀^χ mPS )²

w 0χ mσ

0 5 10 15

012345

( w₀^χ mPS )²

w 0χ mσ

0 5 10 15

012345

( w₀^χ mPS )²

w 0χ mσ

0 5 10 15

012345

( w₀^χ mPS )²

w 0χ mσ

β=2.0 β=2.2

0 5 10 15

012345

( w₀^χ mPS )²

w 0χ mσ

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

β=1.8 β=2.0 β=2.2 β=2.3

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ ma0

w₀^¬m_X = w₀^¬m^¬_X + A(w₀^¬m_PS)²+B(w₀^¬m_PS)⁴+C a w₀

Preliminar y Preliminar y

m_a0/f_PS = 16.7(4.9) m_æ/f_PS = 19.2(10.8)

Claudio Pica

Spin-0 Resonances

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ mη'

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ mη'

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ mη'

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ mη'

β=2.0 β=2.2

0 5 10 15

0123456

( w₀^χ mPS )²

w 0χ mη'

w₀^¬m_X = w₀^¬m^¬_X + A(w₀^¬m_PS)²+B(w₀^¬m_PS)⁴+C a w₀

Preliminar y

m_¥⁰/f_PS = 12.8(4.7)

Claudio Pica

Resonances - Summary

0510152025

m XF PS

QCD N_f = 2

σ : 0(0⁺ ) η₂ : 0(0^- ) ρ : 1(1^- ) a₀ : 1(0⁺ ) a₁ : 1(1⁺ )

0(0⁺ )

0(0^- ) 1(1^- )

1(0⁺ )

1(1⁺ )

SU(2) N_f = 2

Preliminar y

Claudio Pica

• Lattice simulations yield first principle results for the NP

dynamics and can provide trustable quantitative results on the spectrum are possible and available

• Several interesting models are being investigated, which

include examples of strongly coupled models with light scalars

• The study of scalar sector of strongly coupled model is challenging, but seems within reach

• SU(2) model viable might be viable as composite goldstone Higgs model, but spin 1 and 0 resonances in the 30-50 TeV region for

Conclusions

Thank you!

sinµ ' 0.1