Strongly coupled near- or conformalsystems with fundamental fermions
SCGT12 Workshop

Anna Hasenfratz
University of Colorado

Outline & Summary• Why are these calculations difficult?
– Conformal systems are conformal in the chiral limit only Simulations at finite mass entangle conformal & chirally brokenbehavior– Lattice simulations are often forced to strong coupling Spurious fixed points can change the scaling behavior!

• SU(3) gauge system with 12 flavors– MCRG with improved gauge action
 Emergence of an IRFP
(A.H. 1106.5293)– The phase structure at zero and finite temperature
Novel phase with new symmetry breaking pattern(A. Cheng, A.H., D. Schaich)• SU(3) gauge system with 8 flavors– Novel phase is present – implications for simulations & IR behavior(A. Cheng, A.H., D. Schaich, G. Petropoulos, in preparation)

How can we distinguish QCD-like and conformal systems?
QCD like

Conformal
confining

confining

mmbulk¯=6/g2(arrows: UV to IR)

IRFP¯=6/g2

Lattice simulations can connect the perturbative FP and strong coupling• Found IRFP ? Done ✔
• No IRFP? Show that it is confining before a bulk transition is reachedFinite temperature and bulk phase transitions
QCD likeconfining

NT 4Conformal
8 16 32 ..

confining

NT 4

8 16 32 ..

mdeconfined¯c ∞	 as	 NT  ∞mbulkIRFP¯c ¯bulk as	 NT  ∞In a conformal system••finite temperature transitions run into a bulk (T=0) transition
βbulk separates strong coupling (confining) and weak coupling(conformal) phases

Fixed points and their basin of attraction
In QCD like systems continuum limit is defined at the Gaussian UVFPContinuum scaling is expected in the basin of attraction of G-FP

GaussianFPβ2

β1

Fixed points and their basin of attraction
In conformal systems there is a new IRFP
• asymptotically free around G-FP,• the conformal behavior in the infrared around the IRFP

Gaussian
FPβ2
IRFP

β1

Fixed points and their basin of attraction
If there are two UV fixed points, continuum limit can be defined at both.
The basin of attractions are exclusive, stay in one or the other to get
desired continuum scaling!

UVFP-2β2β1

Gaussian
FPFixed points and their basin of attraction
Pure gauge SU(2), SU(3) has this structure in the fundamental-adjoint
plaquette plane: 1st order transitions ending in a 2nd order endpointβA
UVFP-2GaussianFP0Between the basins of attraction
is a no-man’s landβF

RG flow in the fundamental-adjoint plane
RG flow in pure gauge SU(2)Tomboulis,Velitski (hep-lat/0702015)
The flow runs away from the first order line/end point:

Extension of thefirst order line

βA

UVFP-20βF

GFP

Scaling in the fundamental-adjoint gauge actionSU(3) pure gauge theory
Hasenbusch,Necco JHEP08(2004)005:
Scaling test of the glueball, Tc and r0 at βA=0, - 2.0, - 4.0 shows scaling
breaks down on the strong coupling side

Filled squares: βA=0Open squares: βA=-2.0
Triangles: βA=-4.0

Is UVFP-2 a problem?
••

Not for QCD simulations, those are on the weak coupling side.
BSM models are strongly coupled and simulations can end up in thewrong FP basin

βA
GaussianThis is a problem for spectral studies
as well,FPUVFP-20

not only MCRG!

βFOutline & Summary• Why are these calculations difficult?
– Conformal systems are conformal in the chiral limit only Simulations at finite mass entangle conformal & chirally brokenbehavior– Lattice simulations are often forced to strong coupling Spurious fixed points can change the scaling behavior!

• SU(3) gauge system with 12 flavors– MCRG with improved gauge action
 Emergence of an IRFP
(A.H. 1106.5293)– The phase structure at zero and finite temperature
Novel phase with new symmetry breaking pattern(A. Cheng, A.H., D. Schaich)• SU(3) gauge system with 8 flavors– Novel phase is present – implication to IR behavior(A. Cheng, A.H., D. Schaich, G. Petropoulos, in preparation)

SU(3) gauge with Nf=12 fundamental flavors•Controversial system, likely very close to the conformal window.

•I use nHYP staggered fermions (very good taste restoration) with
fundamental+adjoint plaquette gauge action•

Fermion masses are tiny
• In MCRG studies the simulations can be considered to be in thechiral limit• In phase structure studies we investigate finite volume/massscalingSU(3) gauge with Nf=12 fundamental flavors

MCRG at βA=0 (1004.1004)
remained inconclusive(too close to crossover)This work: βA/βF=-0.25
βF = 12/g2!
2N c!A $= ! F #1+ 2 & = 02g0!F %"The step scaling function around a UVFPThe bare differential step scaling function sb (¯)sb(¯) = ¯ - ¯ʼ where »(¯) = »(¯ʼ)/2(¯=2Nc/g02 )» is the correlation length defined by some physical masssb(¯) is the projection of the RG flow to a lower dimensional coupling spacesb(¯) has the opposite sign of the RG ¯ function

The step scaling function & MCRG
If blocked actions match:S( β(n) )= S( βʼ(n-1) )  ξ(β)=2ξ(βʼ)  sb(β) = β-β’
MCRG finds (β,β’) pairs by matching blocked lattice actions
action space

Two actions are identical if all
operator expectations values agree

Match operators (local expectationvalues) after several blocking steps

Where all the bodies and bombs areburied (and what to do with them)- The RG flow might not reach the renormalized trajectory- Improved blocking is essential- Compare different blocking levels- The blocked lattices are small, finite volume effects are significant- Careful matching on identical volumes helps;- Compare different volumes- Spurious lattice fixed points can effect the results- Check the phase structure

The step scaling function Nf = 12Results from many blocking levels,many volumes are all consistent.
••

At βF=∞ the step scaling functionsb>0
In the investigated β range it isnegative There has to be an IRFP
(around/above β=11.0 ) Indicates a conformal system

The step scaling function – a different action

With βA/βF = -0.15 the IRFP is closer
And MCRG can find it(168 4 matching)Summary of MCRG matching
MCRG:– Optimized, volume-matched MCRG gives consistent
results for Δβ (the step scaling function)– For Nf=12 fermions, SU(3) gauge the step scaling
function is consistently negative, indicating an IRFP andconformal dynamics

Phase diagram studies
Nf=12 (and 8) flavors, SU(3) gauge + nHYP’ fermions(arXiv:1111.2317, A. Cheng, A.H., D. Schaich)

The action:– Fundamental-adjoint gauge : βA/βF = -0.25– nHYP projection has numerical problems when the smeared linkdevelops near-zero eigenvalues• small tweak of the HYP parameters can fix that!
(α1,α2,α3)=(0.40,0.50,0.50) will do the trick• This action is 10-15 times faster than HISQ and can be pushed
to stronger couplings

Previous results on the phase stucture Nf=12
Groningen-INFN group found 2 first order transitions(2010,2011)
m=0.025(tree level imp. staggered)condensate

NT=6,8, 10 and T=0βBoth sets of transitions are converging to bulk transitions

Phase diagram β-m plane Nf=12
Our results are consistent with the phase diagram of Deuzeman et al
We use different action at several (small) mass values, volumes
Finite temperature phase transitions
converge to zero temperature“bulk” transitions

This is robust property ofNf = 12 staggered fermionsFirst order transitions at small mass
turning into crossoverPhase diagram β-m plane Nf=12NT 4

8 16 32 ..
confiningmbulkIRFP¯c ¯bulk as	 NT  ∞It is as if the finite T transitions fissioned into twotransitions before converging to bulk

Or maybe not….

Phase diagram Nf=12
What are the three phases?

<ψψ>~ 10 m

<ψψ>~ 5m

Chiral condensate extrapolates to zero in the chiral limit on theweak coupling side of the “big” jump
 Chiral restoring transition (more on that later!)Is it deconfining?

Phase diagram
Is it deconfining? Polyakov line is very noisy but the blocked Poly line issensitive:
Blocked Poly line is measured on RG
blocked lattices:• improved Poly lineor• Poly line on renormalized
trajectory, after removing UVfluctuations

The blocked Polyakov line sees the “weak” transition strongly
but hardly changes at the “strong” transition

The intermediate phasepossibly only a lattice artifact (bordered by 1st order phasetransitions)- it is not partial restoration of taste symmetry- it does not go away with increasing volume- It cannot extend to g2=0
- Staggered version of Aoki phase?

In the following I explore the properties of theIM phase (at β=2.6 m=0.005)
and contrast it with the weak coupling phase(β=2.7, m=0.005)

New symmetry breaking patternsingle-site shift symmetry (S4) is brokenχn  ξµ(n)χn+µ , Un,µ  Un+µ,µPlaquette expectation value is “striped”xtNew symmetry breaking pattern
Order parameter I: ΔPμ	 =	 ☐n - ☐n+µβ = 2.6 IM phase

β=2.7 weak coupling phase

New symmetry breaking pattern
Order parameter II:
S4 breaking occurs at the fermionic level: staggered link!Lµ = " nU µ (n)" n+ µ # " n+ µU µ (n + µ )" n+2 µlink in µ

shifted in µβ=2.6 IM phase

β=2.7 weak phaseNew symmetry breaking pattern– Single-site shift symmetry is exact in the action.
– When S4 is broken, the phase has to be separated by a “real”
phase transition– The S4 broken phase cannot go away with the volume– S4 is related to taste so this could be a special taste breaking– Both ΔP and ΔL are order parametersConfinement in SSTB phase- Polyakov line is small- Static potential on 123,163 volumes (no volume dependence!)shows a linear term: r0=2.1 – 2.7, √σ =0.40 -- 0.48

β=2.6 – IM phase

β=2.7 – weak couplingDirac eigenvalue spectrum in SSTB phase
Chiral properties: Dirac eigenvalue spectrum on different
volumes
β=2.6 – IM phaseDirac eigenvalue spectrum• RMT predictions require knowledge of dynamics (Nf, chiralbreaking, etc)
• Simple volume scaling is more general. In the chiral limit
implies! (" ) ~ (" # "0 )$n#m
2!
(")
d"
=+&(1/V)%mVn' n # x0 *"n # "0 ~ )( V ,+1/($ +1)

,

D= ym
$ +1- Chirally broken systems: α = 0, λ0 = 0- Free field α = 3- λ0 ≠ 0 : soft edge (very chiral symmetric)Dirac eigenvalue spectrum
Fit: in S4B phase (β=2.6):
soft edge with α=0.6(1), λ0=0.0175 (RMT prediction: α=1/2)
Very chiral symmetric!

log(λn –λ0) ~ log(n/V) /(α+1)

Weak coupling phase for contrast:
Fit in weak coupling (β=2.7) phase
α=1.5, λ0=0  γm=0.61(5) anomalous dim.
β=2.7 – weak phaselog(λn ) ~ log((n-x0)/V) /(α+1)
Consistent results at smaller masses, larger βIntermediate phase:
The meson spectrum is parity degenerate (chiral symm.)shows no finite volume effectOn-site pions + scalar
1-link pions

Rho + axial vector

Intermediate phase:
The meson spectrum is parity degenerate (chiral symm.)
Shows no finite volume effects in S4b phase

On-site pions + scalarThis is not a Goldstone!

Rho + axial vector

Chirally symmetric,
confined

Chirally broken,
confining

Phase diagram?

Intermediate S4b phase
It is a strange phase.
- It is not consistent with the perturbative g2 = 0 fixed point- Chirally symmetric as shown by- Dirac spectrum- Meson spectrum- Confining with a small correlation length
Chirally symmetric & confining phase is nor supposed to exist at all- There is no continuum limit here (1st order transitions)- Lattice might generate new relevant interactions
Does it exist in any other system than Nf = 12?
No for Nf = 4 ✓Yes for Nf = 8,12,16 ✓Intermediate phase with Nf = 8
The phase diagram is eerily similar to Nf = 12 :2 transitions, converging (likely) to bulk ones, intermediate phasem=0.005intermediate phase with the same properties as Nf = 12(S4 breaking, soft edge spectrum, confinement)

Intermediate phase with Nf = 8
In the chiral limit there is a phase with S4 breaking in the infinite volume
limit

Speculations•This phase cannot persist to g2=0
 it has to be bulk transition•

If Nf=8 is below the conformal
window, the SSTB phase is a
staggered lattice artifact•

Simulations near 6/g2 = 2.3 canpick up scaling of this phase

Conclusion: more questions than answers••SU(3) gauge with Nf =12 fundamental flavors is the test case of BSM
calculations:– MCRG indicates an IRFP at relatively weak coupling– The phase structure in the strong coupling is complicated• There are two sets of phase transitions• The intermediate phase– exists with many actions– is chirally symmetric but confining– breaks single-site translational symmetry (taste?)Nf = 8, 16 systems show the same S4b phase; Even if it’s a staggeredlattice artifact, it still has to be noticed (and avoided)