[PPT] - From Lattice Strong Dynamics to Phenomenology Ethan T. Neil PowerPoint Presentation

SLIDE 1

From Lattice Strong Dynamics to Phenomenology

Ethan T. Neil (Fermilab) for the LSD Collaboration SCGT12 Workshop, KMI December 4, 2012

Tuesday, December 4, 12

SLIDE 2

Motivation

We have a Higgs! Or is it a Higgs

impostor? A composite?

If the new Higgs-like particle is

composite, presence of a new strongly- coupled sector should reveal itself dramatically with many new resonances.

However, scale where the resonances

appear may high and difficult to reach

directly. First signs of such a sector

may appear in low-energy EW physics!

UV-complete theory determines low-

energy effective description, and fixes all low-energy constants (non- perturbative -> lattice!)

Tuesday, December 4, 12

SLIDE 3

Exploring the space

There exists a large parameter space of theories beyond QCD - cartoon above

shows plane for Nf fundamental fermions only

Many theories in this space can reduce to similar low-energy effective theories of
EWSB. How do the coupling constants change in this space? (Lattice!)
(Not mentioned in my talk, but interesting: bounding the edge of the window,

study of IR-conformal theories. See 1204.6000, G. Voronov - PoS Lattice11)

CBZ

Tuesday, December 4, 12

SLIDE 4

Composite Higgs and setting the scale

Decay constant F gives the EW gauge boson masses, and thus EWSB scale.

For simplest case (one EW doublet), identify v=F=246 GeV.

W 1,2

µ

≡ A1,2

µ − 4

fg∂µφ1,2 Zµ ≡ g p g2 + g⌅2 ✓ A3

µ − g⌅

g Bµ − 4 fg∂µφ3 ◆ ✓

⌅

◆

For QCD, the higher resonances

(ρ,N,...) start around 2πF - separation of scales!

Integrate out --> chiral Lagrangian:

Lχ,LO = F 2 4 Tr ⇥ DµU †DµU ⇤ + F 2B 2 Tr ⇥ m(U + U †) ⇤ U = exp(2iT aπa/F)

where .

B is related to mass generation and the chiral condensate:

h ¯ TTi / F 2B

Caveat: chiral Lagrangian only has “pion” states - if Higgs is a dilaton, then it

needs to be accounted for as well...

Tuesday, December 4, 12

SLIDE 5

The chiral Lagrangian at higher order

At next order in momentum expansion, many new terms appear. Three- and

four-point pion interactions, and interactions with external left/right currents. Once again all LECs fixed by underlying strong dynamics.

Looking on the electroweak side makes connection to experiment clearer...

Z Gasser, 14. Leutwyler/ ('hiral perturbation theory

481

The efIective lagrangian of order p2 then simplifies to ~, = ~ F~{tr (V,, U+V ~' U) + tr (X* U + xU + )}, (6.14) and the constraint which eliminates the U(l) field associated with the z/' becomes det U = 1 .

(6.15)

Since we need the lagrangian ~-2 only at tree graph level, we may use the classical field equations (5.9) obeyed by U to simplify the general expression of order p4. Using the procedure outlined in sect. 3 to impose gauge invariance, Lorentz invariance, P and C, one finds the following expression for the general lagrangian of

rder p4:

~2 = L,(V~'U'V,U) 2 + L2(V,U~V,,U)(V"U~V'U)

+ L3(V~'U+V~,UV'~U "V,,U)+ L4(W' U+VuU)(x +

U +xU +) + L~(V"UW,,U(x + U + U+x))

+ L6( X' U+xU+)2+ L7( X' U-xU+) 2

+ Ls(x~Ux ~ U+xU*xU')

R

p. v +

F,,~V U V U)

tLo(FuvV UV U + t. ~, +

FL Fv-vL'~

+ L,o(U*F~UFC~'~)+ ~,,,\-,`~-I'~R "c~'vR+ _,,,~_ , + H2(x~X) ,

(6.16)

R L

where CA) stands for the trace of the matrix A. The field strength tensors F,~, F~,~ are defined in (3.8). At leading order two constants Fo, Bo suffice to determine the low-energy behaviour

fthe Green functions (recall that we disregard the singlet vector and axial currents) -

at first nonleading order we need l0 additional low-energy coupling constants Lj,..., L,o. (Although the contact terms Hi, /42 are of no physical significance, they are needed as counterterms in the renormalization of the one-loop graphs.)

7. Loops

To evaluate the one-loop graphs generated by the iagrangian .~.~ we consider the neighbourhood of the solution O(x) to the classical equations of motion. Denoting the square root of this solution by u(x) 13 = u 2 (7.1) we write the expansion around tJ in the form

U = u(! + i~- ~2+.. ")u, (7.2)

where ~(x) is a traceless hermitian matrix. The number of flavours does not play a crucial role in the following analysis. We perform the one-loop calculations for the

[Gasser and Leutwyler, NPB 250 (1985) 465]

(χ = 2Bm)

Tuesday, December 4, 12

SLIDE 6

Next-to-leading order on the EW side

[Appelquist and Wu, Phys.Rev. D48 (1993) 3235] L1 ≡ 1 2α1gg′BµνTr(TW µν) L2 ≡ 1 2iα2g′BµνTr(T[V µ, V ν]) L3 ≡ iα3gTr(Wµν[V µ, V ν]) L4 ≡ α4[Tr(VµVν)]2 L5 ≡ α5[Tr(VµV µ)]2 L6 ≡ α6Tr(VµVν)Tr(TV µ)Tr(TV ν) L7 ≡ α7 Tr(VµV µ)Tr(TVν)Tr(TV ν) L8 ≡ 1 4α8 g2 [Tr(TWµν)]2 L9 ≡ 1 2iα9gTr(TWµν)Tr(T[V µ, V ν]) L10 ≡ 1 2α10[Tr(TVµ)Tr(TVν)]2 L11 ≡ α11g µνρλ Tr(TVµ)Tr(VνWρλ) (6)

Corrections to two-point functions (oblique corrections) should appear first in

low-energy experiments.

L

′

1 ≡ 1

4β1g2f 2[Tr(TVµ)]2.

S ∝ α1 T ∝ β1 U ∝ α8

Dominant contributions to W-W scattering at NLO from α4, α5

Tuesday, December 4, 12

SLIDE 7

A tale of two effective theories

In lattice simulations, no EW charges - work in terms of hadronic chiral
Lagrangian. Zero g,g’, massive pseudo-Goldstones.
On the other side, we can write down an electroweak chiral Lagrangian to

describe gauge-boson interactions; non-zero g,g’, massless Goldstones.

Hadronic EFT EW EFT

g, g0 → 0 p2 ⌧ M 2

ds, M 2 ss

p2 ⌧ M 2

ds, M 2 ss

md → 0

restored symmetry

With no Higgs, massless hadronic Goldstones eaten by W/Z, rest taken
heavy. With a pion “Higgs impostor”, more complicated matching...

Tuesday, December 4, 12

SLIDE 8

A tale of two effective theories

In lattice simulations, no EW charges - work in terms of hadronic chiral
Lagrangian. Zero g,g’, massive pseudo-Goldstones.
On the other side, we can write down an electroweak chiral Lagrangian to

describe gauge-boson interactions; non-zero g,g’, massless Goldstones.

Hadronic EFT EW EFT

g, g0 → 0 p2 ⌧ M 2

ds, M 2 ss

p2 ⌧ M 2

ds, M 2 ss

md → 0

restored symmetry (+ dilaton?) (+ Higgs boson)

With no Higgs, massless hadronic Goldstones eaten by W/Z, rest taken
heavy. With a pion “Higgs impostor”, more complicated matching...

Tuesday, December 4, 12

SLIDE 9

Lattice Strong Dynamics Collaboration

James Osborn Heechang Na Rich Brower Michael Cheng Claudio Rebbi Oliver Witzel David Schaich Ethan Neil Sergey Syritsyn Mike Buchoff Chris Schroeder Pavlos Vranas Joe Wasem Joe Kiskis Tom Appelquist George Fleming Meifeng Lin Gennady Voronov Saul Cohen

Tuesday, December 4, 12

SLIDE 10

(IBM Blue Gene/L supercomputer at LLNL) (Cray XT5 “Kraken” at Oak Ridge) (Computing cluster “7N” at JLab)

Results to be shown are state-of-the-art for lattice simulation - O(100 million) core-hours for full program Many thanks to the computing centers and funding agencies (DOE through USQCD and LLNL, NSF through XSEDE)

Tuesday, December 4, 12

SLIDE 11

Simulation details

Iwasaki gauge action +

domain-wall fermions, fermion masses from mf=0.005 to mf=0.03, one volume (323x64).

Residual chiral symmetry

breaking reasonably small, mres~0.002. All chiral extrapolations in m=mf+mres.

Results also exist for Nf=8 (five

ensembles, in progress) and Nf=10 (six ensembles, spectrum may indicate IR- conformality, see 1204.6000)

Nf = 2 Nf = 6 amf “Mπ” L Ncfg “Mπ” L Ncfg 0.005 3.5 1430 4.7 1350 0.010 4.4 2750 5.4 1250 0.015 5.3 1060 6.6 550 0.020 6.5 720 7.8 400 0.025 7.0 600 8.8 420 0.030 7.8 400 9.8 360

a ∼ 5mρ.

Runs tuned to

Tuesday, December 4, 12

SLIDE 12

Scale setting Lattice scale fr

Tuesday, December 4, 12

SLIDE 13

Chiral condensate

Condensate fixes other leading-order low-energy constant, B. Once overall

scale is set by F, the ratio B/F is meaningful.

In a composite Higgs theory, mass terms arise from four-fermion operators

and the condensate:

Generically, standard model four-fermi operators also generated are a

problem (FCNC!) Viable models tend to require small coupling and large B/F.

yfH ¯ ff → cf Λ2 ¯ ff ¯ ψψ

Tuesday, December 4, 12

SLIDE 14

Condensate enhancement results

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.2 0.3 0.4 0.5 0.6 mf MB

(B/F)6 (B/F)2 = 1.9 ± 0.1 hψψim F 3

m

(M 2

m/2m)3/2

hψψi1/2

m

M 2

m

2mFm B F ←m→0

LSD preliminary

Nf = 2 Nf = 6 σf ⌘ hB| ¯ ff|Bi|q2→0 = mf ∂MB ∂mf σ6 σ2 = 1.71(4)

Tuesday, December 4, 12

SLIDE 15

Overview: The S-parameter

As stated previously, S measures

corrections from new physics to gauge boson 2-pt functions

Z

Ø

z

ú-

h

Q-

ûn

/,2

@

7

+

7 f

@

{ ú-

w-,

{{

S = 16π(Π0

33(0) − Π0 3Q(0))

We measure the current correlators

at fixed m and q2, and fit. Operator product expansion constrains the form at large momentum:

ΠV −A(q2)

q2→∞

⇥ NT C

8π2 m2 + m⇤ψψ⌅ q2 + O(α) + O(q−4)

[M. A. Shifman, A. I. Vainshtein,

V. I. Zakharov, Nucl. Phys. B 147 (1979)]

= −4π(Π0

V V (0) − Π0 AA(0))

Fit using Pade approximants:

ΠV −A(q2) = P

m amq2m

P

n bnq2n

(Pade (1,2) gives best fit.)

(note: model assumption!)

Tuesday, December 4, 12

SLIDE 16

Momentum/mass fits

ΠV −A(q2) = P

m amq2m

P

n bnq2n

0.05 0.10 0.15 0.20 0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 Q2 ⇤VAQ2⇥ Nf ⇥6⇥ 0.05 0.10 0.15 0.20 0.0035 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 0.0000 Q2 2⇥ ⇤VAQ2⇥ Nf ⇥2⇥

0.0

0.5 1.0 1.5 2.0 0.0 0.1 0.2 0.3 0.4 MP

2⇤MV0 2

4⇤ ⇥'VA0⇥

Nf = 2 Nf = 6 (m=1, n=2) (above quantity gives LEC L10.)

Tuesday, December 4, 12

SLIDE 17

S-parameter results

0.0 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0 MP

2⇤MV0 2

S⇥4⇧Nf ⇤2⇥⌅'VA0⇥⇤SSM

S2f (m=0) = 0.35(6) - agrees with other determinations

S(x) = A + Bx + 1 12π N 2

f

4 − 1 ! log(1/x)

S6f drops far below naive scaling estimate at light masses! Still above conjectured bound: Nf = 6 Nf = 2 For 6f, divergence due to PNGBs: S ≥ ND 2π

(F. Sannino, arXiv:1006.0207)

Tuesday, December 4, 12

SLIDE 18

Preview: S-parameter at Nf=8

LSD preliminary

Tuesday, December 4, 12

SLIDE 19

Overview: WW scattering

Direct probe of EW symmetry

breaking physics. Unitarized by the Higgs boson in SM.

Experimental process as shown

(VBF). Relatively clean signal, especially with Z’s, but low rates for large momentum transfer!

At low energy, corrections

appear through LECs α4, α5:

Estimates for 99% CL bounds for 100 inverse fb: µ ∼ 2 TeV

−7.7 × 10−3 < α4 < 15 × 10−3 −12 × 10−3 < α5 < 10 × 10−3

Eboli et. al. 2006

Tuesday, December 4, 12

SLIDE 20

On the lattice: pi-pi scattering

We measure I=2 (“maximal isospin”) pion scattering - identified with WW

scattering on the electroweak side.

Finite-volume scaling of two-particle energy used to extract scattering phase

shift (Luscher method.) Then, fit mass dependence to get LECs:

10 20 30 40 ( MP / FP )

2

1.2
1
0.8
0.6
0.4
0.2

MP/ |

→

k| cot δ LO Nf=2 Nf=6

MP aI=2

P P = − M 2 P

8⇡F 2

P

⇢ 1 + M 2

P

16⇡2F 2

P

 3 log ✓M 2

P

µ2 ◆ − 1 − `I=2

P P (µ)

Plotted on right: MPaPP vs. mass for

Nf=2,6. Good agreement in both cases with zero-parameter LO prediction - triumph of Weinberg.

Tuesday, December 4, 12

SLIDE 21

Getting the LECs

0.005 0.01 0.015 0.02 0.025 0.03

m

10
8
6
4
2

b!

r PP(µ = 0.0229 a

1)

Nf=2 Nf=6

Can’t isolate α4 and α5 for Nf=6

yet - with only I=2 scattering, entangled with other LECs.

α4 + α5 = ( (3.43 ± 0.31) × 10−3 µ ∼ 246 GeV (0.15 ± 0.31) × 10−3 µ ∼ 2 TeV For 2 flavors:

f

b0r

P P(µ) = 256⇡2⇥Lr 0(µ) + 2Lr 1(µ) + 2Lr 2(µ)

+Lr

3(µ) 2Lr 4(µ) Lr 5(µ) + 2Lr 6(µ)

+Lr

8(µ)

⇤ . (24)

Still, comparison with Nf=2

shows a hopeful trend...

At Nf=2 some of the extra LECs don’t exist, and we can get the

linear combination α4+α5 by itself.

Tuesday, December 4, 12

SLIDE 22

Switching gears: composite dark matter

Composite Higgs models tend to have a natural dark matter candidate -

lightest “baryon” can be stable and electroweak neutral.

Composite dark matter is interesting even without a direct EWSB connection!

Allows balance between EW interactions (relic density) and lack thereof (direct detection.)

Lattice can contribute in several ways: spectrum, pion-nucleon interactions,
etc. A major application is baryon form factors, which determine recoil rates

in direct-detection experiments.

No longer working with a chiral Lagrangian - baryons will be heavy*. But now

connection to experiment is more obvious: compute baryon form factors, take appropriate combination for EM current.

*(exception: PNGB dark matter - see 1209.6054 and references therein)

Tuesday, December 4, 12

SLIDE 23

Simulation results: form factors

1.0 1.2 1.4 1.6 1.8 2.0 2.2 3.0 2.5 2.0 1.5 1.0 0.5 0.0 mBmB0 ⇥neut 1.0 1.2 1.4 1.6 1.8 2.0 2.2 5 10 15 20 25 30 mBmB0 rneut

2⇥

LSD preliminary LSD preliminary

Results shown for Nf=2,6 theories. “Neutron” charges assumed (+2/3, -1/3),

with hypercharge only (no net weak charge allowed.) hN(p0)|qγµq|N(p)i = up0

"

F q

1 (Q2)γµ + F q 2 (Q2)iσµνqν

2MB

#

up

Form factors Fi(Q2) computed

from three-point function (right). Fit and extract κ, <r2>.

Dirac charge radius

hr2

1i ⌘ 6dF1(Q2)

dQ2

Q2=0

related to the corresponding isoscalar

moment

κ ⌘ F2(0)

Form factors independent of Nf at this precision (for these masses)!

Tuesday, December 4, 12

SLIDE 24

Connecting to experiment

Computed event rate for XENON100 latest results. Dominated by magnetic

moment interaction κ, exclusion for DM up to 5-10 TeV in this model.

Dashed lines show bound from charge radius operator only (e.g. even Nc?)

10−2 10−1 100 101 102 mDM [TeV] 10−15 10−13 10−11 10−9 10−7 10−5 10−3 10−1 101 103 105 Rate, event / (kg·day)

Nf = 2 dis Nf = 2 ord Nf = 6 dis Nf = 6 ord XENON100 [1207.5988], expect ≈ 1 event XENON100 [1207.5988], ≥ 1 event with 95%

LSD preliminary

Tuesday, December 4, 12

SLIDE 25

Conclusion

A composite Higgs sector may reveal itself first through low-energy effects -

deviations in EW precision, WW scattering, etc.

Many UV theories can reduce to one effective theory, but low-energy

constants determined by strong dynamics. Lattice lets us explore these constants and how they evolve in the large parameter space.

LSD program focused on SU(3) thus far, Nf=2 to Nf=6. Hints of interesting

trends for chiral condensate, S-parameter, WW scattering length. Nf=8,10 in progress - stay tuned

Part of getting the low-energy theory right is getting the states right, so

priority focus now on other light states, in particular light scalar! Scalar meson and glueball calculations in progress on all our lattices.

Tuesday, December 4, 12

SLIDE 26

Backup slides

Tuesday, December 4, 12

SLIDE 27

Finite-volume issues and S?

Tuesday, December 4, 12

SLIDE 28

Comparing with the spectrum

0.2 0.3 0.4 0.5 0.6 MV, MA

Nf=2, Axial-Vector Nf=6, Axial-Vector Nf=2, Vector Nf=6, Vector

0.5 1 1.5 2 2.5 MP

2/MV 2

1 1.2 1.4 1.6 1.8 MA / MV

Nf=2 Nf=6

Tuesday, December 4, 12

SLIDE 29

OPE and extrapolation to large q2

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.0012 0.0010 0.0008 0.0006 0.0004 0.0002 0.0000 0.0002 m lim

q2⌥⌃

q2⌅⇤q2⌅⇥

Leading q2⌥⌃ coeff. and ⇥⇧⇧⇤, Pade⇤1,2⌅

Correlator fit Direct meas.

Tuesday, December 4, 12

SLIDE 30

From slope to S

S = 1 3π Z ∞ ds s {(Nf/2) [RV (s) − RA(s)] −1 4 " 1 − ✓ 1 − m2

h

s ◆3 Θ(s − m2

h)

#)

Standard model subtraction:

ref. Higgs mass;

we take mh ≡ MV 0 (=1 TeV, roughly)

∼ 4πΠ0

V A(0)

assumes all technifermions carry EW charge!

∆SSM = 1 12π 11 6 + log ✓ M 2

V 0

4M 2

P

◆

✓M 2

V 0

M 2

P

< 1/4 ◆

* *

SM subtraction

removes Higgs scalar contribution to S, cancels IR divergence

Tuesday, December 4, 12

SLIDE 31

Finite-volume again from the spectrum

m → 0

Nf = 2 Nf = 6 Nf = 10

5 6 7 8 9 10 11 12 14 16 18 20 22 MPêFP MNêFP

Tuesday, December 4, 12

SLIDE 32

Scale setting: rho vs. F

0.000 0.005 0.010 0.015 0.020 0.025 0.030 8 9 10 11 12 m MΡ Fm Nf = 2 Nf = 6

QCD

Tuesday, December 4, 12

SLIDE 33

S-parameter SM subtraction

0.0 0.5 1.0 1.5 2.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 MP

2êMV0 2

DSSM

As a function of x ≡ M 2

P/M 2 V 0, then, the SM subtraction is

∆SSM = 8 > > > > < > > > > : 1 12π 11 6 + log ✓ 1 4x ◆ , x < 1/4, 1 12π ✓ 3 4x − 3 32x2 + 1 192x3 ◆ , x ≥ 1/4.

Tuesday, December 4, 12

SLIDE 34

Ordered vs. disordered at 10 flavors

Internal analysis has revealed that frozen topological charge can explain the

discrepancy between our two starts:

100 200 300 400 Q

2

0.24 0.25 0.26 0.27 0.28 MP Q != 0 (calculated) Q = 0 (calculated) M0+B Q

2

extrapolated

Current plan is to measure topological susceptibility (slope of the Q-

dependence) and correct our results

(plot from Meifeng Lin)

Tuesday, December 4, 12

SLIDE 35

Scaling fit results, Nf=10

S V N F C

total

Nf = 10

0.0 0.5 1.0 1.5 2.0 0.1 0.2 0.5 1 2 5 10 20 50 100 g* c2

Obs.

mf ≥ 0.010 mf ≥ 0.015 mf ≥ 0.020 γ?

1.69(16) 1.10(17) 1.35(47)

[68% CI] [1.54,1.86]

[0.95,1.27] [1.06,1.73]

[95% CI] [1.40,2.06]

[0.82,1.46] [0.83,2.27]

CP

0.98(9) 1.44(21) 1.21(37)

CV

1.17(10) 1.70(25) 1.42(44)

CA

1.43(13) 2.14(32) 1.79(56)

CN

1.75(16) 2.53(37) 2.10(65)

CN ?

2.23(25) 3.35(55) 2.87(92)

CF P

0.121(12) 0.190(28) 0.164(51)

CF V

0.165(15) 0.238(35) 0.195(60)

CF A

0.136(13) 0.192(28) 0.154(48)

χ2/d.o.f.

69/31 14/23 3.1/15 (shown for mf>=0.015

Tuesday, December 4, 12

SLIDE 36

Mass deformation

α(µ) µ α? Λ = a−1 M 0

m(µ) µ Λ = a−1 M 0 M 0

m0(Λ)

4) Bound-state masses are set by M, as in QCD-like theory. Three major differences here:

No Goldstones - PS state scales like everything else.
M is controlled by m:
Expansion in am, as opposed to for χPT

M ∼ m1/(1+γ?) aM 2

π/(4πFπ)2

Tuesday, December 4, 12