SU(2) with six flavors
A new kind of gauge theory
George T. Fleming (Yale)for the LSD Collaboration

Two-Color Gauge Theories•

Perturbatively, SU(2) gauge theories behave like any other SU(Nc)gauge theory.

•Non-perturbatively, SU(2) could be quite different:•

No complex representations (pseudo-real, real)•Enlarged global symmetry: SUL(Nf)×SUR(Nf)→SU(2Nf).

•

Spontaneous symmetry breaking produces more NG bosons:
SU(2Nf)→Sp(2Nf) gives Nf(2Nf-1) - 1 vs. Nf2 - 1.

•Can we establish the range of Nf over which spontaneous
symmetry breaking occurs, i.e. the conformal window?

Two Colors and BSM Physics•

The special features of two-color gauge theories can lead to newmodels of BSM physics.

•The five NG bosons of the Nf=2 theory can yield a composite
Higgs boson as pseudo-NG boson.

•

Enlarged global symmetry suppresses charge radius andmagnetic moment interactions in composite dark matter models.

•

Enlarged NG boson sector could lead new kind of finite
temperature phase transition for a confining gauge theory.

•If a confining, two-color gauge theory is realized in nature, what
are the implications of this phase transition on cosmology?

Two-Color Conformal Window
Perturbative Estimates•

Caswell-Banks-Zaks established that SU(Nc) gauge
theories with Nf flavors of Dirac fermions in thefundamental representation have IR conformal fixed
points if Nf<11Nc/2.

•

This IR conformal behavior ends for Nf < Nf*(Nc) when
the theory confines.

•Higher-loop calculations (Refs) can be used to test thereliability of perturbative estimates of Nf*.

•A reasonable estimate is Nf* ≲ 4 Nc.

Two-Color Conformal Window
Ladder-Gap Equations•

The rainbow diagram approximation of theSchwinger-Dyson equation gives an estimate of thecritical coupling gc2 of chiral symmetry breaking.

•For SU(2), gc2 ≈ 17.5.

•Comparing this estimate to the IRFP coupling of thetwo-loop beta function gives an estimate of Nfc.

•For SU(2), Nfc ≈ 8, consistent with pert. theory.

Two-Color Conformal Window
Cardy’s a-theorem•

Much has been made of late of the proposed proofof Cardy’s a-theorem. Can it constrain Nfc?

•aUV = 62(Nc2-1) + 11 Nc Nf•

For broken SU(2): aIR = Nf (2Nf-1) - 1•

Given massless gauge dofs count 62 times
massless scalars means the a-theorem, even if
true, provides no useful constraint.

Two-Color Conformal Window
ACS Thermal Inequality Conjecture•

Another way to count massless dofs is via the thermodynamic
2 4free energy: f(T) = 90 F(T) / π T .

•In T→0 limit, massive contributions are suppressed.

•SU(Nc): fUV(0) = 2 (Nc2 - 1) + 3.5 Nc Nf•

SU(2): fIR(0) = Nf (2Nf-1) - 1.

•

ACS conjecture: fUV(0) ≥ fIR(0). If true, this leads to asignificant bound for SU(2): Nfc ≲ 4.7.

•

Further, it is significantly different from perturbative estimates.

Two-Color Conformal Window
Previous Lattice Results•

Numerous lattice results that demonstrate that the SU(2) Nf=2 theory is
confining and chirally broken.

•Iwasaki et al (2004) infinite coupling confinement studies: Nf=3 inside theconformal window.

•Karavirta et al (2011) SF running coupling studies: Nf=4 outside conformalwindow.

•Other running coupling studies suggest Nf=8 (Ohki et al) and Nf=10 (Karavirtaet al) are inside conformal window.

•Nf=6 is a difficult but very interesting case. Several early attempts wereinconclusive (Bursa 2010, Karavirta 2011, Voronov 2011-2).

•

There will be a presentation by N. Yamada about the calculation of the KEKgroup.

SU(2) Nf=6 Thermodynamics
100

90

In QCD, the equation of stateoutside the transition region isdominated by the StefanBoltzmann term.

7060f

•

80

50

403020•The ACS thermal inequality
would mean that all confining
asymptotically-free gauge
theories have QCD-likethermodynamics.

100T
0.4

0.6

0.8

1

¡SB/T4

Tr0161.2

1412•

If SU(2) Nf=6 violates the ACS
thermal inequality, the equationof state should be very different
from QCD-like theories.

10

¡/T4

3p/T4

8
6
4

p4

asqtadlqcd
T-10 MeV
HRG+lqcd

2T [MeV]

0100 150 200 250 300 350 400 450 500 550

LSD Publication
arXiv:1311.4889, accepted to Phys. Rev. Lett.

SU(2) Nf=6 Calculational Details•We use the standard Schrödinger functionalrunning coupling formulation.

•We use step-scaling to compute the lattice
step scaling function:
Σ(u,s,a/L) ≡ g2(g02,sL/a) if u=g2(g02,L/a).

•We compute the continuum step scaling
function by taking the limit:σ(u,s) = Σ(u,s,a/L) as a/L → 0.

•

The quantity [σ(u,s)-u]/u is analogous to thecontinuum beta function.

•We use the Wilson fermion action with one
level of stout smearing, tuned to masslesspoint.

Stout Wilson Parameter Space•

We determined the massless pointvs. coupling in infinite volume limit.

•We also located bulk phasetransition/crossover line.

•Transition crosses massless curvearound g02 = 2.2.

Interpolating the data•

Gennady generated a huge amount of data using many different computersover two years.

•For a slowly running theory, it is impossible to do step scaling tuning the latticespacing by had at each and every step.

•

We compute the SF coupling over a range of g02 < 2.2 and 4 ≤ L/a ≤ 24.

•

We fit (g02)-1 - (gSF2)-1 topolynomial in g02 for each L/a.

0.48
0.46

The functional form is inspiredby perturbation theory but thecoefficients are not constrained
to p.t. values.

0.44
0.42

11−22g0ḡSF

•

0.4
0.38
0.36

L/a=4
L/a=5
L/a=6
L/a=7
L/a=8
L/a=9
L/a=10
L/a=11
L/a=12
L/a=14
L/a=16
L/a=18
L/a=20
L/a=240.34•

We don’t worry about wiggles atvery weak coupling. They don’t
affect the result, as I will
explain.

0.32
0.3
0.28
0.2

0.4

0.6

0.8

1

1.2
g02

1.4

1.6

1.8

2

Extrapolating the step scaling function•Extrapolate Σ(u,s,a/L) to polynomial in a/L to extract σ(u,s).

•At weak coupling (u<6), a constant extrapolation is fine. At stronger coupling(u>6), a higher order continuum extrapolation is required.

•The quadratic term is as important at the linear term unless L/a is very large.
Perhaps linear would be OK with 16→32 and larger volumes.

Discrete beta function•

In the discrete beta function, we don’t see any evidence for a fixed point.

•

We don’t expect that a fixed point will appear as the beta function dipping downto zero. It should cross zero and run backward all the way to strong coupling.

•You might recall for SU(3), Nf=12 the Yale group (pre-LSD) saw clear evidence
for backward running and for SU(3), Nf=8 there was no such evidence.

Comments•

I look forward to hearing about the latest KEK results forSU(2) Nf=6 from Yamada on Friday.

•

I want to strongly emphasize that very slowly runningtheories are very hard to study on the lattice so it may takesome time to get consistent results from all groups.

•Confining two-color theories always have composite Higgs
candidates as pseudo-NG bosons.

•Studying the thermodynamics of the SU(2) Nf=6 theory could
be very interesting.

•In the future, lattice radial quantization might be a betterway to study (nearly-)conformal theories. See my poster.