Strongly coupled gauge theories:
In and outof the conformal window
Anna Hasenfratz
University of Colorado Boulder
SCGT2014, Nagoya, Mar 7, 2014

In collaboration with A. Cheng,Y. Liu, G. Petropoulos and D. SchaichThursday, March 6, 14

Strongly coupled gauge-fermion systems

Thursday, March 6, 14

Strongly coupled gauge-fermion systems- Attractive candidates for BSM phenomenology

Thursday, March 6, 14

Strongly coupled gauge-fermion systems- Attractive candidates for BSM phenomenology- Interesting non-perturbative QFT’s on their own rightThursday, March 6, 14

Strongly coupled gauge-fermion systems- Attractive candidates for BSM phenomenology- Interesting non-perturbative QFT’s on their own right- Strongly coupled - need non-perturbative investigation

Thursday, March 6, 14

Strongly coupled gauge-fermion systems- Attractive candidates for BSM phenomenology- Interesting non-perturbative QFT’s on their own right- Strongly coupled - need non-perturbative investigation- Nearly conformal models are very different from QCD
yet difficult to distinguish from chirally broken systems

Thursday, March 6, 14

Strongly coupled gauge-fermion systems- Attractive candidates for BSM phenomenology- Interesting non-perturbative QFT’s on their own right- Strongly coupled - need non-perturbative investigation- Nearly conformal models are very different from QCD
yet difficult to distinguish from chirally broken systems
Compare the phase diagram on the m=0 chiral surface:chirally broken
conformal
g2

g2

Pert

Pert

g2

g2
g1

IR

g1

g2 represents all irrelevant operators
Thursday, March 6, 14

Universality
Systems– with identical field content– identical symmetries– at criticality (basin of attraction of the FP)

are expected to show universal critical behavior.
Lattice symmetries:– SU(Nc) gauge preserved ✔
– SU(Nf) x SU(Nf) chiral symmetry is not:• staggered fermions : only U(Nf/4) x U(Nf/4) flavor symm.
• Wilson fermions : no chiral symmetry• Domain Wall fermions : approximate chiral symm.

At the g2 = 0 UVFP all formulations approach continuum fermions
At the g2 ≠ 0 conformal IRFP that is not the case
Universality should be investigated more carefully(but only staggered fermions in this talk)Thursday, March 6, 14

Strongly coupled gauge-fermion systems
Nearly conformal models are very different from QCD,
yet difficult to distinguish from chirally broken systems→ numerical methods from QCD are not always effective
Combine– standard QCD methods– modified methods– new approaches

Thursday, March 6, 14

Strongly coupled gauge-fermion systems
Modified methods– Finite size scaling for Nf=12 (poster):• FSS is inconsistent if only the relevant exponent/
operator is considered• becomes consistent across different observables, gauge
couplings, even actions if the leading irrelevantcorrection is included and predicts
𝜸*m = 0.235(15)– Running gauge coupling with Nf=12 (poster):• Investigated both MCRG and gradient flow matching• After careful (a/L)2 extrapolation gradient flow predicts
an IRFP at g2c = 6.21(25) (Scheme dependent!)

Thursday, March 6, 14

Strongly coupled gauge-fermion systems
New approach:– Running anomalous mass dimension from the spectral
density of the Dirac operator (Nf=4, 8, 12, 16)• Nf=4 : chirally broken; test case
• Nf=8 : near the conformal boundary at 2-loop;• Nf=12 : has been rather controversial• Nf= 16 : weakly coupled conformal

These methods probe the systems very differently:
Finite size scaling: L,m finite ;Running coupling calculations: L finite, m=0;
Spectral density: L➝∞, m=0

Thursday, March 6, 14

Spectral density of the Dirac operator
Spectral density: ρ(λ )λν(λ)=VMode number :∫−λ ρ(λ′)dλ′Chirally broken systems: ρ(0)= Σ / π
Conformal systems are chirally symmetric: ρ(0)= 0(Banks-Casher)

critical behavior suggests, λ≈0.
ν (λ )∝V λα +1
The mode number is RG invariant,(Giusti,Luscher)unchanged under scale change s: V→s4V, λ→λ/s1+γ , ν →ν' ' '→ α is related to the anomalous dimension

Thursday, March 6, 14(Zwicky,DelDebbio;Patella)Conformal system
Eigenvalue density ρ(0)= 0 , scales as ρ (λ ) ∝ λ α ( λ )RG invariance implies
λ provides an energy scale

Thursday, March 6, 14

Conformal system
Eigenvalue density ρ(0)= 0 , scales as ρ (λ ) ∝ λ α ( λ )RG invariance implies
λ provides an energy scale

Thursday, March 6, 14

Conformal system
Eigenvalue density ρ(0)= 0 , scales as ρ (λ ) ∝ λ α ( λ )RG invariance implies
λ provides an energy scale

IR – small λ region:ρ (λ )γ m (λ → 0) = γ m*
γ m ≤1α ≥1predicts the universal anomalousdimension at the IRFP

γm→0
α →3

λIR

Thursday, March 6, 14UVUV – large λ =O(1) region:
if governed by the asymptotically freeperturbative FPγ m (λ =O(1))= γ 0 g2 +...
In between:
scale dependent effective 𝜸mConformal system
Eigenvalue density ρ(0)= 0 , scales as ρ (λ ) ∝ λ α ( λ )RG invariance implies
λ provides an energy scaleρ (λ )γ (λ )
γ m ≤1α ≥1Luscher

γ*γm→0α →3λλIR

Thursday, March 6, 14UVIRUVConformal system
Eigenvalue density ρ(0)= 0 , scales as ρ (λ ) ∝ λ α ( λ )RG invariance implies
λ provides an energy scaleγ (λ )

g2

Luscher
Perturb

γ*IRF

g2g1

Thursday, March 6, 14λIRUVConformal system
Eigenvalue density ρ(0)= 0 , scales as ρ (λ ) ∝ λ α ( λ )RG invariance implies
λ provides an energy scaleγ (λ )

g2

Strong coupling side???

LuscherPertu
rbγ*IRF

g2
g1

Thursday, March 6, 14

λIRUVChirally broken system
Chirally broken systems show only the asymptotically free regionρ (λ )

O(1)γm→0α →3Unphysicalγ (λ )λλIRThursday, March 6, 14

UVIRUVDirac operator eigenvalue spectrum and spectral density-calculate ν (λ ) stochastically-fit ν (λ )∝V λα (λ )+1logν =(α (λ )+1)log(λ )+c-

extract the scale dependent γ m (λ )This should be done in thechiral m=0 and infinite volume L→∞ limit :finite mass, volume introduces only small λ transient effects

Important: fit ν (λ ) , not ρ(λ ) ; The two are not the same!

Thursday, March 6, 14

Results: Nf =4Broken chiral symmetry in IR, asymptotic freedom in UV

2

Thursday, March 6, 14

Results: Nf =4Broken chiral symmetry in IR, asymptotic freedom in UV

UnphysicalIR

2Thursday, March 6, 14

Results: Nf =4
Broken chiral symmetry in IR, asymptotic freedom in UV
UV

2Thursday, March 6, 14

Results: Nf =4Broken chiral symmetry in IR, asymptotic freedom in UV
Can the different
couplings be rescaled?
Lattice spacing from Wilsonflow:

a6.4 / a 7.4 = 2.84(3)a6.6 / a 7.4 = 2.20(5)a7.0 / a 7.4 = 1.45(3)a8.0 / a 7.4 = 0.60(4)2Thursday, March 6, 14

Rescaling: Nf =4
The dimension of λ is carried by the lattice spacing: λlat = λpa
1+γ m ( λβ )Rescale to a common physical scale:⎛ a7.4 ⎞λβ → λβ ⎜⎟⎝ aβ ⎠

Universal curve covering
almost 2 orders of magnitude
in energy!

Perturbative: functional form
from 1-loop PT, relative scale isfitted

2
Most of these data were obtained on deconfined (small) volumes with m=0!
Thursday, March 6, 14

Spectral density results: Nf =12

All simulations are in the m=0 chiral limit,
324 and 364 volumes to take V→∞

Thursday, March 6, 14

Spectral density results: Nf =12
UV:• There is no sign ofasymptotic freedom behavior
for β<6.0,𝜸m grows towards UV• Not in the basin of attractionof the UVFP

All simulations are in the m=0 chiral limit,
324 and 364 volumes to take V→∞

Thursday, March 6, 14

IR:• Extrapolation to λ=0(quadratic with commonintercept for 4 couplings)
predicts 𝜸*m = 0.26(3)• Consistent with conformalbehavior

Universality
Is this result universal?
– We use nHYP staggered action. What is the prediction fromother actions ?
(In the FSS analysis we showed that stout and HISQ staggered
actions from LH and LatKMI collaborations are consistent withnHYP)– Look at 2x nHYP smeared and naive staggered actions forcomparison

Thursday, March 6, 14

Spectral density results: nHYP and 2nHYP, Nf =12

Combine nHYP and2nHYP staggered actions• λ=0 extrapolation predicts
𝜸*m = 0.27(3)• 2nHYP has smaller UVeffects

Spectral density results are consistent with FSS( 𝜸*m = 0.235(15) )Thursday, March 6, 14

Spectral density results: nHYP and 2nHYP, Nf =12Combine nHYP and2nHYP staggered actions• λ=0 extrapolation predicts
𝜸*m = 0.27(3)• 2nHYP has smaller UVeffects

Spectral density results are consistent with FSS( 𝜸*m = 0.235(15) )Thursday, March 6, 14

Spectral density results: nHYP and unimproved, Nf =12

The unimproved action hasvery strong lattice artifacts• λ=0 extrapolation isdifficult even on 484
but it could be consistentThursday, March 6, 14

Spectral density results: nHYP and unimproved, Nf =12

The unimproved action hasvery strong lattice artifacts• λ=0 extrapolation isdifficult even on 484
but it could be consistentThursday, March 6, 14

Spectral density results: nHYP and unimproved, Nf =12

The unimproved action hasvery strong lattice artifacts• λ=0 extrapolation isdifficult even on 484
but it could be consistent

Beware of unimproved actions - they can biteThursday, March 6, 14

Nf=8 flavors
Expected to be chirally broken– 2-loop PT : close to conformal– Numerical studies: (newer)• Boulder : finite temperature phase diagram cannot distinguishbetween conformal & chiral broken• LatKMI : walking with mixed ChPT/hyperscaling• USQCD : cannot distinguish between conformal & chiral broken

Phase diagram with nHYP action:
black lines: bulk S4b → deconfinedcolored lines: confined → deconfined
No clear confining phase in the chirallimit up to NT=20 !

Thursday, March 6, 14

Anomalous dimension, nHYP, Nf =8
Expected to be chirally broken - looks like walking!

- No asymptotic free scaling atstronger coupling but- Walking across orders ofmagnitude of energy- At stronger coupling the S4bphase develops- Would 2HYP allow stronger

coupling/ explicit chiral breaking?

All simulations are in the m=0 chiral limit,
243x48 and 323x64 volumes

Thursday, March 6, 14

Anomalous dimension, nHYP, Nf =8
Expected to be chirally broken - looks like walking!

- No asymptotic free scaling atstronger coupling but- Walking across orders ofmagnitude of energy- At stronger coupling the S4bphase develops- Would 2HYP allow stronger

coupling/ explicit chiral breaking?

All simulations are in the m=0 chiral limit,
243x48 and 323x64 volumes

Thursday, March 6, 14

Anomalous dimension, nHYP, Nf =8
Expected to be chirally broken - looks like walking!

- No asymptotic free scaling atstronger coupling but- Walking across orders ofmagnitude of energy- At stronger coupling the S4bphase develops- Would 2HYP allow stronger

coupling/ explicit chiral breaking?

All simulations are in the m=0 chiral limit,
243x48 and 323x64 volumes

Thursday, March 6, 14

Anomalous dimension, 2nHYP vs nHYP, Nf =82nHYP breaks chiral symmetry in thechiral limit on 324 before S4b phase- 2nHYP at β=3.8, m=0.0025-Nearly constant 𝜸m ≈ 1 until it
breaks chiral symmetry-2HYP shows less UV andallows stronger coupling

Nf=8 shows walking and chiral symmetry breaking
What else to ask for ? (0++ mass)
Thursday, March 6, 14

Dirac operator eigenvalue spectrum and spectral density
Unique & promising method !
Can distinguish strong and weak coupling region of conformal /chirally broken systems
It is important to look at- the scale dependence of 𝜸eﬀ- several gauge couplings, even actions
Predictions:
Nf=4 : scaling & anomalous dimension
Nf=12 : looks conformal with 𝜸*= 0.26(3)Nf=8 : could be walking with large anomalous dimension!

Thursday, March 6, 14