Lattice simulations and BSM phenomenologyjulian.tau.ac.il/CECAM/DelDebbio.pdf · 2015-06-22 ·...

Lattice simulations and BSM phenomenology

L Del Debbio

Higgs Centre for Theoretical PhysicsUniversity of Edinburgh

L Del Debbio Lattice BSM Tel Aviv, June 2015 1 / 47

motivations

LHC Run-I −→ discovery of the Higgs boson, mh = 125 GeV

V(φ) = λ

(φ†φ− v2

W2

)2

,GF√

2=

12v2

W, m2

h = 2λv2W

new Physics: strongly-coupled dynamics?

vW = 246 GeV, ΛNP ≈ TeV?

weakly coupled effective field theories for pheno↪→ spectrum, LEC & anomalous dimensions

lattice: developed a set of tools for nonperturbative studiespreliminary studies of vector-like theories −→ landscape of theories

↪→ a change of gear is required


strongly coupled EWSB

Standard Model

gauge sector

flavor sector

BSM - stronglycoupled

low-energyEFT

CFT


physics beyond the SM - new interactions

L = LSM +1

ΛNP

∑i

C(5)i O

(5)i +

1Λ2

NP

∑i

C(6)i O

(6)i

= LSM + ∆LSILH + ∆LF1 + ∆LF2

[Buchmuller & Wyler 86, Grzadkowski et al 10, Contino et al 13]

[Contino et al 13]


physics beyond the SM - new particles

• Higgs is a light scalar in the spectrum – pNG boson/dilaton?[Yamawaki, Appelquist 10, Sannino 11, Grinstein 11]

strongly interacting sector −→ include resonances in the EFT

• e.g. vector particles coupled to scalar: hidden local symmetry [Yamawaki 85]

↪→ new LECs and renormalization of previous LECs

mρ, gρL, αi determine the dynamics of the vector resonance


theoretical assumptions

• power-counting for strongly-interacting theories:

ΛNP = g∗f � f , ∂ ∼ 1/ΛNP

↪→ leading NP effects parametrized by OH,T,6,ψ

• if Higgs is a composite pNGB of a spont broken symmetry G → H

f decay constantOγ ,Og,O6,Ou,d,l violate shift symmetry =⇒ h→ γγ suppression

• flavor alignmentcu,d,l is diagonal in flavor space


current boundsEW precision measurements + LHC data

• custodial symmetry

∆ε1 = ∆ρ = cT , −1.5× 10−3 < cT < 2.2× 10−3 .

• Barbieri-Altarelli parameters:

∆ε3 = cW + cB , −1.4× 10−3 < cW + cB < 1.9× 10−3 .

cW,B ∼ O(

m2W

Λ2NP

)=⇒ ΛNP ∼ TeV

• coupling to the SM fermions is small - except for the top[Contino et al 13]


current bounds


challenges for the latticegiven a microscopic theory −→ numerical evaluation of correlators

ESUVTeVmZ

SUV often called a UV-completion −→ microscopic theory

choice of SUV dictated by the theoretical assumptions→ identify the accessible LECs [Ferretti, Golterman & Shamir]

LEC of the effective lagrangian −→ microscopic correlatorsmake contact with LHC

NB: SUV defines a strongly-interacting theory with broken conformalsymmetry


hierarchies and flavor

linearized RG flow

µd

dµgk = −ykgk =⇒ gk =

(µ

ΛUV

)−yk

gk,0

ΛIR ∼ gk,0 ΛUV, yk � 1 =⇒ natural hierarchy

fine-tuning is associated with the existence of global-singlet relevantoperators

stable hierarchies from weakly relevant operators [Strassler 03, Sannino 04, Luty 04]

YM theory in the UV is a limiting case:

ΛIR ∼ ΛUV exp[− 1

2β0g2

]


hierarchies and flavor

SM composite scalar composite fermionHiggs mass [H†H] = 2 [OSOS] ≈ 6− 2γ –Yukawas [HLuR] = 4 [qqOS] = 6− γ [qLλLOL] = 4 + γFCNC [qqqq] = 6 [qqqq] = 6 [qqqq] = 6

suppression of FCNC, large mass, no new relevant interaction↪→ large mass anomalous dimension γ, small values of γR,L

[OSqq] = 6− γ =⇒ mq ∼ ΛIR

(ΛIR

ΛUV

)2−γ

[OSOS] < 4 =⇒ GSRO

[OL,R] = 5/2 + γL,R =⇒ mq ∼ ΛIR

(ΛIR

ΛUV

)γR+γL

theory must be in a neighbourhood of a fixed point:SUV is a deformed CFT.


challenges for the lattice - 2

• theory defined by the action at the cutoff scale SUV

• study the correlators at large distances as a→ 0

• old tools & new tools: experiment with different optionsscaling of the spectrumdensity of eigenvalues

• large volume simulations: finite-size effectsinvestigate carefully the finte-volume effects (glueballs)

• finite-volume techniquesNP running of the couplingcomputation of anomalous dimensions


one problem at a time

• lattice simulations can provide first principle results

• a microscopic description of the strong sector/CFT at the cutoff scale isneeded

• Monte Carlo study of the dynamics of the strong sectorfirst: study the strong sector in isolation

existence of an IRFPscale invariance at large distances, scaling laws, running couplings↪→ anomalous dimensions

computation of LECsrelate the LEC in the effective description to correlators in the UVcomplete theory↪→ estimates of the LECs [Goltermann & Shamir]


RG fixed pointsWilsonian formalism:

S[φ] =

∫dDx gk µ

D−dk Ok(x)

RG flow and fixed points

βk(g) = µd

dµgk(µ)

βk(g∗) = 0

linearized evolution

δgi = gi − g∗i

µd

dµδgi = Lij δgj

µd

dµui = −yi ui


scheme dependence

couplings in different schemes:

gi(µ) = gi (ga(µ), µ/µ)

0 = µd

dµgi(µ) =⇒ βi =

∑a

∂gi

∂gaβa

existence of fixed points is scheme-independent

critical exponents are also scheme-independent


RG fixed points

need to tune m→ 0: simulations are expensive, finite volume effects


dilatation ward identities

〈∂µDµ(x)P〉 = 〈Tµµ(x)P〉 − 〈δxP〉

pure gauge theory, e.g. using dim reg:

S = − 12g2

0

∫dDx Tr FστFστ

TWI:

δρAµ(x)def= ερ(x)Fρµ(x) , δρS =

∫dDx ∂µερ(x) Tµρ(x)

δx,ρPdef=

δPδAA

µ(x)FAρµ(x)

Tµρ = − 2g2

0Tr[

FσµFσρ −δµρ4

FστFστ

]finiteness of Tµν :

〈∂µTµρ(x)P〉 = −〈δx,ρP〉


lattice ward identities

discretized transformations:

δρUµ(x) = ερ(x)Fρµ(x)Uµ(x)

δx,ρP =1a3 FA

ρµ(x)∂AUµ(x)P

variation of the action:

δρS = −a4∑

x

ερ(x){∇µT(1)

µρ (x) + Xρ(x)}

Xρ(x) = aOρ(x) irrelevant operators

Xρ = a[

1Zδ

OR,ρ +1a

(c1

Zδ− 1)∇µT(1)

µρ +1a

c2

Zδ∇µT(2)

µρ +1a

c3

Zδ∇µT(3)

µρ

][Caracciolo et al 90]


lattice ward identities

FAµν(x) = FA

µν(x) + O(a2)

T(1)µρ =

∑σ

FµσFρσ −14δµρ∑στ

TrFστ Fστ

T(2)µρ = δµρ

∑στ

TrFστ Fστ

T(3)µρ = δµρ

∑σ

TrFµσFµσ


lattice Ward identities

Xρ = a[

1Zδ

OR,ρ +1a

(c1

Zδ− 1)∇µT(1)

µρ +1a

c2

Zδ∇µT(2)

µρ +1a

c3

Zδ∇µT(3)

µρ

]

terms appearing in the renormalization of Xρ renormalize Tµρ:

(Tµρ)R =3∑

i=1

ciT(i)µρ

〈∇µTµρ(x)RP〉 = −〈Zδ δx,ρP + OR,ρ(x)P〉


probe in the bulk

(Tµρ)R =∑

i

ciT(i)µρ

〈∇µTµρ(x)RPT〉 = −〈Zδ δx,ρPT + aOR,ρ(x)PT〉

for a probe defined in the bulk:

〈δx,ρPT〉 = Zδ〈δx,ρPT〉 = Zδ(−2〈Tr Lµ(0, x)Fρµ(x)PT〉

)lima→0

a〈OR,ρ(x)PT〉 = 0

[ldd, patella & rago 13]


NP renormalizationDetermination of the ratios ci/Zδ similar to Caracciolo et al:

ci〈∇µT(i)µρ(x)φT,ρ(0)〉 = −Zδ〈δx,ρφT,ρ(0)〉

determination of Zδ using two-point functions:

Φt(x4) =a3

L3

∑xφt(x, x4)

fΦ(d, t, z4) = 〈Φt(z4) a4+d∑

y4=−d

∑yδy,4Φt(0)〉

Zδ =〈Φt(z4)∇4Φt(0)〉

fΦ(d, t, z4)+ O(e−r2/16t) , r = min(d, |z4 − d|)

see also work by Giusti and Pepe for an alternative prescription


scaling laws

mass is the only relevant coupling in the IR - anomalous dimension:

γ =1mµ

ddµ

m

IR dynamics is determined by the value at the IRFP:

ym = 1 + γ∗

scaling of the spectrum:

M ∝ m1/ym

finite size scaling:

LMH = f (x) , x = Lymm , f (x)x→∞∼ x1/ym


mesonic mass ratio at lighter mass


anomalous dimension from eigenvalues

ν(M,m) = C +(M2 − m2)2/(1+γ∗)


summary for the spectrum

��

��

��

��

(��)

(��)�/(�+γ)

spectrum is consistent with scaling hypothesis - using γ∗ = 0.37


finite size scaling

LMH = f (x) , x = Lymm , f (x)x→∞∼ x1/ym

0 5 10 15 20 25 30

L1+γ*m

0

10

20

30

40

MP

SL

L = 8L = 12L = 16L = 24L = 32L = 48

γ* = 0.371

consistent with the existence of IRFP: large volumes are needed!!L Del Debbio Lattice BSM Tel Aviv, June 2015 27 / 47

towards phenomodels of strongly-interacting Higgs sector

light scalar close to the conformal window? SU(3) with 2 sextet (2S)[Kuti et al]

SU(3) with 8, 10, 12 fundamental [Appelquist et al, Kuti et al, Lombardo et al], Itou et al

SU(4) with 2 sextet (2AS) [DeGrand et al]

SU(2) with 2 fundamental [Hietanen et al]

models of composite Higgs: SU(4) with 5 Majorana 2AS [Ferretti, Golterman +

Shamir]

models of dark matter [Pica, Rinaldi, McCullough + Detmold et al]

2 3 4 5 60

5

10

15

Nc

nf


schrodinger functional

finite volume renormalization scheme:

1g(L)2 = k 〈∂S

∂η〉

N

g(L) is an observable, can be measured by numerical simulationsnonperturbative definition of the coupling, MC study of its running

Step scaling function:

Σ(s, u, a/L) = g(sL)2∣∣g(L)2=u , σ(s, u) = lim

a/L→0Σ(s, u, a/L)

log s =

∫ √u

√σ(s,u)

dgβ(g)


results for σ(s, u)


running of the mass

m(µ) =ZA

ZP(µ)m

ZP(L) =√

3f1/fP(L/2) r

f1 = − 112

∫d3u d3v d3y d3z 〈ζ ′(u)γ5τ

aζ ′(v)ζ(y)γ5τaζ(z)〉

fP(x0) = − 112

∫d3y d3z 〈ψ(x0)γ5τ

aψ(x0)ζ(y)γ5τaζ(z)〉

step scaling

ΣP(s, u, a/L) =ZP(g0, sL/a)

ZP(g0,L/a)

∣∣∣∣g(L)2=u

, σP(s, u) = lima→0

ΣP(s, u, a/L)


results for σP(s, u)


results for γ

γ(u) = − logσP(s, u)

log s

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

u

gamma L=8/12

L=10/16

L=8/16


four-fermi operators


four-fermi operatorsStep scaling functions:

Σ±4f (s, u,L/a) = Z±(g0, sL/a) Z±(g0,L/a)∣∣g(L)2=u

σ±4f (s, u) = lima→0

Σ±4f (s, u,L/a) = T exp

{∫ √σ(s,u)

√u

dgγ±(g)

β(g)

}


pheno?More ambitious: models of strongly-interacting Higgs sector

light scalar close to the conformal window? SU(3) with 2 sextet (2S)[Kuti et al]

SU(3) with 8, 10, 12 fundamental [Appelquist et al, Kuti et al, Lombardo et al]

SU(4) with 2 sextet (2AS) [DeGrand et al]

SU(2) with 2 fundamental [Hietanen et al]

models of composite Higgs: SU(4) with 5 Majorana 2AS [Ferretti, Golterman +

Shamir]

models of dark matter [Rinaldi, McCullough + Detmold et al]

2 3 4 5 60

5

10

15

Nc

nf

Information on the spectrum, computation of LECL Del Debbio Lattice BSM Tel Aviv, June 2015 38 / 47

scaling laws: examplesMore generally:

O ∝ mηO

O def 〈0|O|JP〉 JP ∆O η

S qq GS 0+ 3− γ∗ (2− γ∗)/ym

Sa qλaq GSa 0+ 3− γ∗ (2− γ∗)/ym

Pa q(iγ5)q GPa 0− 3− γ∗ (2− γ∗)/ym

V qγµq εµMVFV 1− 3 1/ym

Va qγµλaq εµMVFVa 1− 3 1/ym

Aa qγµγ5λaq εµMAFAa 1+ 3 1/ym

Eigenvalue density of the massless Dirac operator (Banks-Casher):

ρ(λ) ∝ ληqq


overall picture: SU(2) w adjoint fermions


qualitative look


results on larger lattices: heavier mass

infty 1632 24 1248

L

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

1.80M

TM 1TM γ

0γ

5γ

k

TM γ5γ

k

TM γ0γ

k

TM γk

TM γ0γ

5

TM γ5

GB T++

GB E++

GB A++

σt

1/2

σs

1/2

σ1/2


results on large lattices: lighter mass

infty 48 32 2480

L

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90M

TM 1TM γ

0γ

5γ

k

TM γ5γ

k

TM γ0γ

k

TM γk

TM γ0γ

5

TM γ5

GB T++

GB E++

GB A++


results on large lattices: lighter mass


Probe at the boundary

For a probe defined at the boundary

P = φ(x1)R . . . φ(xn)R

lima→0〈Tµρ(x)Rφ1(x1)R · · · φk(xk)R〉 = 〈Tµρ(x)φ1(x1)R · · ·φk(xk)R〉

lima→0〈{

Zδ δx,ρ + aOR,ρ(x)}φ1(x1)R · · · φk(xk)R〉 =

=∑

j

δ(x− xj)∂

∂xj〈φ1(x1)R · · ·φk(xk)R〉


Probe at the boundary

Fixing the renormalization coefficients:

〈∇µTµρ(0)Rφ(x1)R . . . φ(xn)R〉 = 0 , for xi 6= 0

Set c1 = 1, choose several probes P j:

Mij = 〈∇µT(i)µρ(0)R P j〉

ci Mij = 0

Overall normalization:

〈H|∫

dD−1x T00(0, x)|H〉 = MH

[Caracciolo et al 90]


NP renormalization

Determination of Zδ using one-point functions:

Φt(x4) =a3

L3

∑xφt(x, x4)

hΦ(d, t) = 〈a4+d∑

y4=−d

∑yδy,4Φt(0)〉BC

Zδ =〈∇4Φt(0)〉BC

hΦ(d, t)+ O(e−r2/16t) , r = min(d, |z4 − d|)

Several possible choices for the operator φ.


Lattice simulations and BSM phenomenologyjulian.tau.ac.il/CECAM/DelDebbio.pdf · 2015-06-22 ·...

Documents

Transcript of Lattice simulations and BSM phenomenologyjulian.tau.ac.il/CECAM/DelDebbio.pdf · 2015-06-22 ·...