Post on 17-Jul-2020
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 2 / 47
strongly coupled EWSB
Standard Model
gauge sector
flavor sector
BSM - stronglycoupled
low-energyEFT
CFT
L Del Debbio Lattice BSM Tel Aviv, June 2015 3 / 47
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]
L Del Debbio Lattice BSM Tel Aviv, June 2015 4 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 5 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 6 / 47
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]
L Del Debbio Lattice BSM Tel Aviv, June 2015 7 / 47
current bounds
L Del Debbio Lattice BSM Tel Aviv, June 2015 8 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 9 / 47
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
]
L Del Debbio Lattice BSM Tel Aviv, June 2015 10 / 47
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.
L Del Debbio Lattice BSM Tel Aviv, June 2015 11 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 12 / 47
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]
L Del Debbio Lattice BSM Tel Aviv, June 2015 13 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 14 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 15 / 47
RG fixed points
need to tune m→ 0: simulations are expensive, finite volume effects
L Del Debbio Lattice BSM Tel Aviv, June 2015 16 / 47
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〉
L Del Debbio Lattice BSM Tel Aviv, June 2015 17 / 47
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]
L Del Debbio Lattice BSM Tel Aviv, June 2015 18 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 19 / 47
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〉
L Del Debbio Lattice BSM Tel Aviv, June 2015 20 / 47
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]
L Del Debbio Lattice BSM Tel Aviv, June 2015 21 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 22 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 23 / 47
mesonic mass ratio at lighter mass
L Del Debbio Lattice BSM Tel Aviv, June 2015 24 / 47
anomalous dimension from eigenvalues
ν(M,m) = C +(M2 − m2)2/(1+γ∗)
L Del Debbio Lattice BSM Tel Aviv, June 2015 25 / 47
summary for the spectrum
���� ���� ���� ���� ���� ����� �
���
���
���
(��)
(��)�/(�+γ)
spectrum is consistent with scaling hypothesis - using γ∗ = 0.37
L Del Debbio Lattice BSM Tel Aviv, June 2015 26 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 28 / 47
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)
L Del Debbio Lattice BSM Tel Aviv, June 2015 29 / 47
results for σ(s, u)
L Del Debbio Lattice BSM Tel Aviv, June 2015 30 / 47
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)
L Del Debbio Lattice BSM Tel Aviv, June 2015 31 / 47
results for σP(s, u)
L Del Debbio Lattice BSM Tel Aviv, June 2015 32 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 33 / 47
four-fermi operators
L Del Debbio Lattice BSM Tel Aviv, June 2015 34 / 47
four-fermi operators
L Del Debbio Lattice BSM Tel Aviv, June 2015 35 / 47
four-fermi operators
L Del Debbio Lattice BSM Tel Aviv, June 2015 36 / 47
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)
}
L Del Debbio Lattice BSM Tel Aviv, June 2015 37 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 39 / 47
overall picture: SU(2) w adjoint fermions
L Del Debbio Lattice BSM Tel Aviv, June 2015 40 / 47
qualitative look
L Del Debbio Lattice BSM Tel Aviv, June 2015 41 / 47
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
L Del Debbio Lattice BSM Tel Aviv, June 2015 42 / 47
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++
L Del Debbio Lattice BSM Tel Aviv, June 2015 43 / 47
results on large lattices: lighter mass
L Del Debbio Lattice BSM Tel Aviv, June 2015 44 / 47
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〉
L Del Debbio Lattice BSM Tel Aviv, June 2015 45 / 47
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]
L Del Debbio Lattice BSM Tel Aviv, June 2015 46 / 47
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.
L Del Debbio Lattice BSM Tel Aviv, June 2015 47 / 47