Walking technicolor on the lattice

Kieran Holland University of the Pacific

Lattice Higgs CollaborationZoltan Fodor (Wuppertal), Julius Kuti (UCSD),

Daniel Nogradi (UCSD), Chris Schroeder (UCSD)

where is UoP?

7/21/08 11:59 AMuniversity of the pacific loc: Stockton, CA - Google Maps

Page 1 of 1http://maps.google.com/maps?um=1&ie=UTF-8&q=university+of+the+p…,0,17511088924321459949&sa=X&oi=local_result&resnum=1&ct=image

Results 1-1 of about 1 foruniversity of the pacific nearStockton, CA

©2008 Google - Map data ©2008 NAVTEQ™ - Terms of Use

A. University of the Pacific3601 Pacific Ave, Stockton, CA - (209) 946-2011!

Stockton “foreclosure capital” of the USA

3 lattice faculty: Jim Hetrick, Jimmy Juge, KHoldest university in California (1851)

outline

• motivation for technicolor

• walking technicolor

• strategy

• something unexpected for fundamental?

• outlook

technicolor

replace Higgs with strong gauge theory

good:avoid triviality, fine-tuningQCD disguised as Electroweak

bad:quark masses - extended technicolorflavor-changing neutral currentselectroweak precision datalight composite Higgs?

what’s new?

walking technicolorwalking technicolor

gauge theory asymptotically free - what if coupling runs slowly?Strong Electroweak Gauge Sector: running couplings

Julius Kuti, University of California at San Diego USQCD Collaboration Meeting, Jefferson Laboratory, April 4 - 5, 2008, 17/22

if beta function near zero, almost fixed point: near-conformal

can generate separation of scalesameliorate FCNC’squark masses natural (Top?)

conformal models (“unparticles”) also getting popular

ΛTC,ΛETC

if coupling walks, separate scalesfix FCNC’s?light composite Higgs?

W,Z masses quark masses

techniquark fundamental rep.need large NF

bad for EW precision

extended technicolor difficult

possible theories(Walking) technicolor

(Sannino)

Fundamental: gray2 antisym: blue2 sym: redadjoint: green

gray: fundamentalblue: 2-index antisymmetricred: 2-index symmetricgreen: adjoint

NF

conformal windowupper curve: AF lostlower curve: chiral SB

perturbative

want to be below windowNc (except Georgi,Luty)

candidates

EW precision prefers small NF

2-index Symmetric NF (χSB) = 2.5

S parameter

NF = 2 3 Goldstone bosons for W’s, Z

fundamental

Nc = 3

NF (χSB) = 11.91

NF = 12 borderline, test caseless likely for new physics

2-index Symm.

fundamental

best candidate?

lattice problems

• large bare coupling: QCD-like for all

• small bare coupling: femto-world, free theory

• Wilson: explicit chiral SB

• Staggered: taste-breaking, what ?

• Overlap: expensive

NF

NF

strategy

examine eigenvalues of the Dirac operator

if chiral SB and1

Fπ

! L !1

mπ

-regime

chiral Lagrangian dominated by zero modes

eigenvalue distributions known Random Matrix Theory

pk(z, µ) z = λΣV, µ = mΣV

λ

quark condensate

if conformal ρ(λ) ∼ λ3+γ anomalous dimension

ε

Σ

staggered simulations

2 and 3 flavors staggered fermions, fundamental rep.

NF = 8, 12

Asqtad action, RHMC algorithmvolume

no rooting i.e. continuum

104

ma = 0.01quark mass

look at 1st eigenvalue distribution

integrated distribution

RMT: fit Σ〈λ1〉

m=

〈z1〉

µpredict distribution

m < λ

p1(λ)∫ λ

0

p1(λ′)dλ′

2 staggered flavors

chiral SB with continuum value ?

consistent with Fleming & co., Pallante & co.

NF = 8 QCD-like

0 0.01 0.02 0.03 0.04

lambda

0

0.2

0.4

0.6

0.8

1

integrated distribution

RMT NF=2

RMT NF=8

simulations

3 staggered flavors

surprise: is outside conformal window?NF = 12

not consistent with Fleming & co.

0.02 0.03 0.04

lambda

0

0.2

0.4

0.6

0.8

1

integrated distribution

RMT NF=3

RMT NF=12

simulations

taste-breaking & effective

0 10 20

!

0

0.1

0.2

0.3

0.4

Pk(!

),

k=

1..

.6

!=6.41

NF

Damgaard et al.PLB 495, 263 (2000)

staggered 1 flavor

eigenvalue distributions

k=1,...,6

superb agreementwith RMT

taste-breaking reduces effective

NOT

NF = 1

NF = 4

NF

crucial when hunting conformal window

criticism

have not measured Fπ, mπ

do not know if -regime conditions metε

have not measured taste-breakingwhat is the effective # of light pions?

can conformal theory with finite quark mass fake RMT with chiral SB?

improve taste-breaking

1 staggered flavor

eigenvalue quartetstaste-symmetry restored

improved Dirac operator essential

running coupling

3

Nf 8 12 8 12

β00 3.889(31) 2.78(12) c20 -3.793(77) -11.9(1.7)

β01 0.80(12) 8.0(1.3) c21 — 85(17)

β02 — -13.6(3.4) c22 — -210(41)

c10 6.402(54) 8.69(48) c30 6.64(47) 13.9(2.1)

c11 -6.74(54) -33.3(4.9) c31 -43.6(3.8) -122(20)

c12 11.0(1.4) 77(12) c32 102.1(8.7) 301(48)

χ2/d.o.f. 1.63 2.39 d.o.f. 106 69

TABLE I: Best fit parameters for equation 9. Omitted parameters

are constrained to zero without significantly increasing χ2/d.o.f..

The extrapolation to the continuum is implemented us-

ing the step scaling procedure [20, 21], a systematic

method which captures the RG evolution of the coupling

in the continuum limit. The basic idea is to match lattice

calculations at different values of a/L, by tuning the latticecoupling β ≡ 6/g2

0 so that the coupling strength g2(L) isequal on each lattice. Keeping g2(L) fixed while chang-ing a/L allows one to extract the lattice artifacts. Previouswork by the ALPHA collaboration has shown that pertur-

bative counterterms greatly reduce O(a/L) artifacts fromthe running coupling. So, our preferred method of extrap-

olating to the continuum limit at each step assumes that

O(a2/L2) errors dominate.In practice, one calculates a discretized version of the

running coupling, known as the step-scaling function. In

the continuum, it is designated

σ(s, g2(L)) = g2(sL), (11)

where s is the step size. On the lattice, a/L terms are alsopresent; one defines Σ(s, g2(L), a/L) similarly, so that itreduces to σ(s, g2(L)) in the continuum limit:

σ(s, g2(L)) = lima→0

Σ(s, g2(L), a/L) (12)

First, a value u = g2(L) is chosen. Several ensembles withdifferent values of a/L are then generated, with β ≡ 6/g2

0tuned using our interpolating function (Eq. 9) so that the

measured value of g2(L) = u on each. Then for each β,a second ensemble is generated with larger spatial extent,

L → sL. The measured value of g2(sL) on the largerlattice is exactly Σ(s, u, a/L). Extrapolation in a/L to

the continuum then gives us the value of σ(s, u). Usingthe value of σ(s, u) as the new starting value, this processmay then be repeated indefinitely, until we have a series of

continuum running couplings with their scale ranging from

L to snL. In this paper, we take s = 2.Our results for Nf = 12 continuum running are pre-

sented in Fig. 1. We take L0 to be the scale at which

g2(L) # 1.6, a relatively weak coupling. The points

shown are for values of L/L0 increasing by factors of

two. The step scaling procedure leading to these points

involves stepping L/a from 4 → 8, 6 → 12, 8 → 16,

! " #! #" $! $" %! %"

$

&

'

(

#!

#$

#&

#'

)*+!)"!!#

"$$)%

!"#$$%&'()*+

,"#$$%&-.

FIG. 1: Continuum running coupling from step scaling for Nf =12. The statistical error on each point is smaller than the size ofthe symbol. Systematic error is shown in the shaded band.

and 10 → 20, and then extrapolating Σ(2, u, a/L) to thecontinuum limit assuming that O(a2/L2) terms dominate.Our analysis contains various sources of systematic er-

ror, which must be accounted for. The interpolating func-

tion (Eq. 9) may not contain enough terms to capture the

true form of g2(L) at large L, where there is sparse data,and although the O(a/L) terms are expected to be small,ignoring them completely in the continuum extrapolation

may introduce a small systematic effect. Here we provide

an estimate of our systematic error by varying our con-

tinuum extrapolation method between extremes. Inspec-

tion of Σ(2, u, a/L) as a function of a/L indicates that

dropping the step from 4 → 8 and performing a constantextrapolation underestimates the true continuum running,

while performing a linear fit to all four steps gives an over-

estimate. We use these extrapolations to define the upper

and lower bounds of the shaded region in Fig. 1, which we

take to be a conservative estimate of the overall systematic

error.

The observed IRFP for Nf = 12 agrees within theestimated systematic error band with three-loop perturba-

tion theory in the SF scheme. An important feature of

the Nf = 12 simulations is that the interpolating curvesare anchored by values of g2(L) that are also above theIRFP. For β ≤ 4.4, g2(L) is large, decreasing as L/a in-creases. This behavior is similar to that found in Ref. [22]

for Nf = 16, and consistent with approaching the IRFPfrom above in the continuum limit. In a future paper, we

will exhibit the continuum evolution from step scaling in

this region.

Our results forNf = 8 continuum running are presentedin Fig. 2. We again start at a scale L0 where g2(L) # 1.6,and exhibit points with statistical error bars for values of

L/L0 increasing by factors of two. The three step-scaling

procedures are the same as in the Nf = 12 case. The pro-cedure stepping L/a from 4 → 8, 6 → 12, 8 → 16 and10 → 20 with quadratic extrapolation again provides thepoints with statistical error bars, and the other two define

4

! " # $ % &!

"

#

$

%

&!

&"

&#

&$

'()!'"!!#

""$'%

!"#$$%&'()*+

,"#$$%&-.

FIG. 2: Continuum running coupling from step scaling for Nf =8. Errors are shown as in Fig. 1. Perturbation theory is shown upto only g2(L) ∼ 10.

the upper and lower bounds of the shaded systematic-error

band. For comparison, we have also shown the results of

two- and three-loop perturbation theory up to g2 ! 10,beyond which there is no reason to trust the perturbative

expansion.

The Nf = 8 running coupling shows no evidence foran IRFP, or even an inflection point, up through values ex-

ceeding 14. The exhibited points with their statistical er-rors begin to increase above three-loop perturbation the-

ory well before this value. This behavior is similar to that

found for the quenched theory [23] and for Nf = 2 [19],although, as expected, the rate of increase is slower than

in either of these cases. The coupling strength reached for

Nf = 8 exceeds rough estimates of the strength requiredto trigger dynamical chiral symmetry breaking [24, 25, 26],

and therefore also confinement. This conclusion must be

confirmed by simulations of physical quantities such as the

quark-antiquark potential and the chiral condensate at zero

temperature.

To conclude, we have provided evidence from lattice

simulations that for an SU(3) gauge theory with Nf

Dirac fermions in the fundamental representation, the value

Nf=8 lies outside the conformal window, and thereforeleads to confinement and chiral symmetry breaking; while

Nf=12 lies within the conformal window, governed by anIRFP. Thus the lower end of the conformal window, N c

f ,

lies in the range 8 < N cf < 12.

This conclusion, in disagreement with Ref. [5], is

reached employing the Schrodinger functional (SF) run-

ning coupling, g2(L). This coupling is defined at the boxboundaryL with a set of special boundary conditions, runsin accordance with perturbation theory at short enough dis-

tances, and is a gauge invariant quantity that can be used to

search for conformal behavior, either perturbative or non-

perturbative, in the large L limit.For Nf=8, we have simulated g2(L) up to values that

exceed rough estimates of the coupling strength required to

trigger dynamical chiral symmetry breaking [24, 25, 26],

with no evidence for an IRFP. For Nf=12, our observedIRFP is rather weak, agreeing within the estimated errors

with three-loop perturbation theory in the SF scheme.

The simulations of g2(L) at Nf=8 and 12 should becontinued to achieve more precision. It is also important

to supplement the study of g2(L) by examining physicalquantities such as the static potential, and demonstrating

directly that chiral symmetry is spontaneously broken for

Nf=8 through a zero-temperature lattice simulation. Sim-ulations of g2(L) for other values of Nf , in particular

Nf=10, will be crucial to determine more accurately thelower end of the conformal window and to study the phase

transition as a function ofNf . All of these analyses should

be extended to other gauge groups and other representation

assignments for the fermions [27].

We acknowledge helpful discussions with Urs Heller,

Walter Goldberger, Rich Brower, Chris Sachrajda and

Aneesh Manohar. This work was partially supported by

DOE grant DE-FG02-92ER-40704 (T.A. and E.N.). Nu-

merical computations were based in part on the MILC col-

laboration’s public lattice gauge theory code (version 6)

[28] and were performed on a variety of resources, includ-

ing clusters at Fermilab and Jefferson Lab provided by the

DOE’s U.S. Lattice QCD (USQCD) program, the Yale Life

Sciences Computing Center supported under NIH grant RR

19895, the Yale High Performance Computing Center, and

on a SiCortex SC648 on loan to Yale from SiCortex, Inc.

[1] W. E. Caswell, Phys. Rev. Lett. 33, 244 (1974).

[2] T. Banks and A. Zaks, Nucl. Phys. B196, 189 (1982).

[3] N. Seiberg, Nucl. Phys. B435, 129 (1995).

[4] T. Appelquist, A. G. Cohen, and M. Schmaltz, Phys. Rev.

D60, 045003 (1999).

[5] Y. Iwasaki, K. Kanaya, S. Kaya, S. Sakai, and T. Yoshie,

Phys. Rev. D69, 014507 (2004).

[6] M. Luscher, R. Sommer, P. Weisz, and U.Wolff, Nucl. Phys.

B413, 481 (1994).

[7] A. Bode et al. (ALPHA), Phys. Lett. B515, 49 (2001).

[8] A. Bode, P. Weisz, and U. Wolff (ALPHA), Nucl. Phys.

B576, 517 (2000), erratum-ibid.B608:481,2001.

[9] U. M. Heller, Nucl. Phys. B504, 435 (1997).

[10] C. Sui, Ph.D. thesis, Columbia University, New York, NY

(2001), UMI-99-98219.

[11] K. G. Wilson, Phys. Rev. D10, 2445 (1974).

[12] J. B. Kogut and L. Susskind, Phys. Rev. D11, 395 (1975).

[13] T. Banks, L. Susskind, and J. B. Kogut, Phys. Rev. D13,

1043 (1976).

[14] L. Susskind, Phys. Rev. D16, 3031 (1977).

[15] G. ’t Hooft, Nucl. Phys. B153, 141 (1979).

[16] M. Luscher, Nucl. Phys. B219, 233 (1983).

[17] S. Necco and R. Sommer, Nucl. Phys. B622, 328 (2002).

[18] S. A. Gottlieb, W. Liu, D. Toussaint, R. L. Renken, and R. L.

Sugar, Phys. Rev. D35, 2531 (1987).

[19] M. Della Morte et al. (ALPHA), Nucl. Phys. B713, 378

(2005).

Appelquist, Fleming, Neil, PRL 100 (2008),171607

2,3 staggered flavors, fundamentalNF = 8

NF = 12

Schrodinger functional, box-size L

large L correspond to low energy

8 flavors: coupling large at low energy - - QCD-like

12 flavors: coupling freezes at lowenergy - - conformal

overlap simulations

overlap fermions, 2 flavors, 2-index symmetric rep.

topology-conserving algorithm, volume 64, ma = 0.05

fit condensate, predict 0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7

!

" = 4.850

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7

!

" = 4.975

Microscopicspectral density

Σ(4.850) = 0.083(4)

Σ(4.975) = 0.084(4)

Σ(5.100) = 0.080(4)

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 1 2 3 4 5 6 7

!

" = 5.100

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10 12 14 16

!

" = 4.85

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10 12 14 16

!

" = 4.975

Microscopicspectral density

from RMT

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 2 4 6 8 10 12 14 16

!

" = 5.10

ρ(z) =∞∑

k=1

pk(z)

simulations

theory not QCD-like? consistent with DeGrand, Svetitsky, Shamir

outlook

• fundamental might not be settled

• 2-index symmetric maybe conformal

• first runs - only beginning project

• taste-breaking crucial in RMT

• Asqtad, stout staggered, HISQ, HYP, ... ?

• 2-index symmetric theory more attractive, fundamental theory is the testing ground

NF = 12