Non-perturbative beta functions and scaling laws
in QCD with many quark flavors
Kyoto RG Workshop @YITP
Aug. 25, 2011
Yuki Kusafuka, Eri Ueno and H. T.
(Nara Women’s Univ. Japan)

Contents:
1. Introduction
2. Non-perturbative beta function
3. RG flow equations for SU(N) gauge theories
4. Aspects of the RG flows
5. “Non-perturbative” gauge beta functions
6. Anomalous dimensions in many flavor QCD
7. Scaling laws in nearly conformal theories
8. Summary and discussions

Based on Y.Kusafuka, H.T., arXiv:11043606
Y.Kusafuka, E.Ueno, H.T., in preparation
1

Introduction
Conformal window of the many flavor QCD
SU(Nc) gauge theory with Nf massless flavors
2-loop gauge beta function

IR (Banks-Zaks) fixed point

T.Banks, A.Zaks, NP B 196 (1982)
W.E.Caswell, PRL 33 (1974)

goes towards strong coupling region as Nf decreases.
Spontaneous breaking of the chiral symmetry for Nf < Ncr
Scale invariance is lost there. ⇒ Fixed point cannot exist !
Conformal window:
2

Introduction
Boundary of the conformal window Nf crAnalysis of the Dyson-Schwinger equations
V.A.Miransky, K.Yamawaki, MPL A4 (1989); PRD 55 (1997)
T.Appelquist et.al. PRL 77 (1996); PRD 58 (1998)

DS equations with the fixed point gauge coupling are examined.
The ladder approximation is used mostly.
Motivation of study: application to New Technicolor models
Lattice simulations of the effective gauge coupling
Y.Iwasaki et. al. PRL 69 (1992)
T. Appelquist et. al. PRL 100 (2008); PRL1 02 (2009);PRD 79 (2009)
A.Hasenfratz, PRD 80 (2009); PRD 82 (2010)
T. DeGrand, et.al. PRD 80 (2009); PRD 82 (2010); PRD 83 (2011)
L. Del Debbio et.al. PRD 81 (2010); PRD 82 (2010)
Z. Fodor et.al.PL B 681 (2009); arXiv:0904.4662
E.Itou et.al arXiv:1011.1964
M.Hyakawa et.al. PRD 83 (2011)3

Introduction
Question: How does the beta function change vs. Nf ?
A fixed point is not allowed at the strong coupling region. Therefore
the perturbative beta functions cannot transform into theasymptotically free one smoothly, then what happens?
Decoupling of fermions?
Chiral symmetry breaking reduces the flavor number effectively. So
the beta function may be modified at the strong coupling region.
But a UV fixed point seems to appear in the strong coupling region
for Nf ≧ Ncr , which contradicts with chiral symmetry breaking.
No fixed point?

4

Introduction⇒ We need some non-perturbative analyses of thebeta functions in the conformal window.
Use of the Wilson (Exact) Renormalization Group
Scale invariance:
Dyson-Schwinger equations treating the chiral order parameters
are useless in the conformal window. Also it would be difficult tostudy almost scale invariant theories by the Lattice MC simulation.
Non-perturbative calculation:
In the Wilson RG, renormalized theories can be defined by therenormalized trajectories (RTs), which are given as the continuum
limit of the Wilson RG flows.
J.Polchinski, N.P. B231 (1984)
Beta function:
Then non-perturbative beta functions can be given by scaletransformation on the RTs.
⇒ So the ERG is a quite suitable framework.

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Non-perturbative beta function
Wilson RG
Wilsonian effective action
Integrating out higher momentum modes

K.G.Wilson, I.G.Kogut (1974): Wilsonian effective action contains infinitely many operators
Wilson RG

Legendre flow equation (Wetterich eq.) for the cutoff effective action

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Non-perturbative beta function
Scalar field theory as a toy model
RG flows inspace
Operator truncation

J.Polchinski, N.P. B231 (1984)

Wetterich eqn (sharp cutoff limit)

Renormalized trajectory⇔ renormalized theory7

Non-perturbative beta function
Renormalized trajectory
Perturbative analysis

Non-perturbative beta function
Renormalized trajectory
We can find the RT numericallywithout perturbative expansion.
“Non-perturbative” beta function
The beta function of a renormalized
parameter is given by the scaletransformation on the RT.

“Non-perturbative”
Beta function

Perturbative
Beta function

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RG flow equations for SU(N) gauge theories
Wilsonian effective action
Four-fermi operators
Important to describe the chiral symmetry breaking
Symmetries
Gauge symmetry :Chiral flavor symmetry :Parity4 invariant four-fermi operators

9

RG flow equations for SU(N) gauge theories
Invariant four-fermi operators
Apparent invariants

10

RG flow equations for SU(N) gauge theories
Fiertz identities

Current-current interactions11

RG flow equations for SU(N) gauge theories
Spontaneous breaking of the chiral symmetry
K.-I.Aoki, K.Morikawa, W.Souma, J.-I.Sumi, H.T.,M.Tomoyose,

PTP97 (1997), PTP102 (1999), PRD61 (2000): Chiral symmetry breaking

Approximation scheme
Operator truncation

We discard all gauge non-invariant corrections.
Note: Cutoff breaks gauge invariance. Gauge non-invariant
corrections may be controlled by the modified WT identities.

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RG flow equations for SU(N) gauge theories
RG flow equations （sharp cutoff limit）
Four-fermi couplings
H.Gies, J.Jackel, C.Wetterich, PRD 69 (2004)
H.Gies, J.Jackel, EPJC 46 (2006)

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RG flow equations for SU(N) gauge theories
Loop corrections for the four-fermi operators

Large Nc, Nf limit
rescale asNote: Four-fermi couplings gV1, gV2 do not involve in the large Nc

and Nf limit.
Note: The large Nc corrections contain only the ladder diagrams.
But the non-ladder ones come through the large Nf part.

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RG flow equations for SU(N) gauge theories
Gauge coupling
We use the perturbative beta functions in the large Nc, Nf limit
and add a part of higher order corrections via the four-fermi
effective couplings.
1. Vertex correction :H.Gies, J.Jackel, C.Wetterich, PRD 69 (2004)
We discard all vertex corrections with the four-fermi couplings,
since the gauge symmetry should forbid them.
2. Vacuum polarization :The higher order corrections via four-fermi
effective operators should be incorporated
into the vacuum polarization.

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Aspect of RG flows
Numerical analysis of the flow equations
RG flows in large Nc and Nf
RG flows are given in 3 dimensional
coupling space of.
Fixed points in the conformal windowA UV fixed point exists as well as the IRfixed point.
The UV fixed point and the IR fixed point
merge with each other at r = 4.05.
RG flows in (gS, gV) space
One linear combination of gS and gV gives
the relevant operator, which induces thechiral phase transition.

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Aspect of RG flows
RG flows in the 3D space
There is the phase boundary of chiral symmetry and the UV fixedpoint lies on the boundary.
Flows in the unbroken phase approach towards the IR fixed point.
The phase boundary disappears for r < 4.05 and the entire regionbecomes the broken phase.

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“Non-perturbative” gauge beta functions
RT in the conformal window
Perturbative RT
We may extract the RT by solving the truncated RG flow equations
in perturbative expansion.

Note: These equations give continuum
limit of the truncated ERG equation,
not the full QCD.

This “RT” does not seem to give acontinuum limit.
However this “RT” seems to survive
for Nf > (11/2)Nc.
Note: How about the RT of QED?
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“Non-perturbative” gauge beta functions
RTs near boundary of the conformal window
The perturbative continuum limit lines approaches towards thenon-perturbative RT as the flavor number is lowered.
Out of the window the fixed points disappears. However the nonperturbative RT survives as the RT of the asymptotically free QCD.
The perturbative continuum limit seems to converge towards theRT.

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“Non-perturbative” gauge beta functions
Non-perturbative gauge beta functions
We define the non-perturbative gauge beta function by scaletransformation of the gauge coupling on the RTs.
A UV fixed point appears in the gauge beta function due to higher
order corrections generated through the four-fermi operators.
This behavior is not due tochiral symmetry breaking.
The UV fixed point does not
Perturbative
beta function
appear in the strong couplingregion.
“Conformality lost”
The IR fixed point merges
Non-perturbative
Beta function
with the UV fixed point at theedge of conformal window.
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Anomalous dimensions in many flavor QCD
Critical flavor number
Analysis for the many flavor QCD for comparison with the lattice MC.
Solve the RG equations with a finite Nf, (Nc=3).
Use 2-, 3,- and 4-loop perturbative beta functionsNon-perturbative gauge beta functions

Critical flavor numbers
Note: Lattice analyses indicate that QCD with 12 flavors is conformal.
e.g. E.Itou et.al. arXiv:10110516
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Anomalous dimensions in many flavor QCD
Anomalous dimensions of fermion mass
Anomalous dimension ofin the ERG approach

RG scheme and gauge independent at the fixed points
Results by the RG equations
Note: the anomalous dimension
is fairly suppressed comparedwith the conventional value inthe large N and ladder approx.
Lattice MC results
T.Appelquist et.al. (2011)

The 3- and 4-loop results are close to the Lattice estimations.
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Scaling laws in nearly conformal theories
Scaling of the dynamical scale in the broken phase“Conformality lost” and the Miransky scaling
V.A.Miransky, K.Yamawaki MPL A4 (1989); PRD 55 (1997)
D.B.Kaplan, J-W.Lee, D.T.Son, M.A.Stephanov, PRD 80 (2009)

Suppose that a UV fixed point and an IR fixed point merge.
The beta function with an external parameter c is given as

Then the fixed point couplings are
BKT type phase transition forFermion mass generation

Miransky scaling23

Scaling laws in nearly conformal theories
Approximation by a parabolic function
We may approximate the RT as a parabolic function as follows;
1. Expand the RG flow equations around the critical fixed point.
: effective couplings near a fixed point

Note: An exactly marginal operator appears at fixed point merger.
The RT passes along the exactly marginal direction.
2. Extract the beta function along
the exactly marginal direction.

3. Find the (imaginary) fixed points
for a off-critical flavor number Nf.
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Scaling laws in nearly conformal theories
Dynamical scale of the chiral symmetry breakingRunning effect must be taken into account in the broken phase.
J.Braun,C.S.Fischer,H.Gies arXiv: 10124279

The four-fermi coupling diverges at the dynamical scale.
Note: Breakdown of the description in terms of the local fermi fields
indicates the spontaneous chiral symmetry breaking.
K-I.Aoki et al.PTP 97 (1997); PTP 102 (1999); PRD 61(2000)

Take difference withobtained by the 1-loop beta function.
Approximation for the scaling law
Large deviation from the Miranskyscaling.
Fit with perturbative beta functions+ the parabolic function is good.

25Scaling laws in nearly conformal theories
Scaling of the explicit fermion mass
Recent lattice analyses
MC simulations of mass deformed QCD (adding a bare fermion mass)
L.Del Debbio, R.Zwicky, PRD82 014502 (2010); arXiv:1009.2894
Z.Foder et. al. arXiv:1104.3124
T.Applequist et.al. arXiv:1106.2148

Scaling law in the conformal window
The RG eqn for a fermion mass and IR enhancement

Decoupling at the scale of the fermion mass:
The scaling law of the dimensionless mass parameter

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Scaling laws in nearly conformal theories
Scaling of the explicit fermion mass
Scaling laws in the slightly broken caseWe may solve the RG equations for the effective couplings including
the fermion mass m on the RT numerically.
It seems to be difficult to distinguish whether the theory is conformal
or chirally broken.
It is necessary to see dynamical mass generation to future work .

?

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Scaling laws in nearly conformal theories
Scaling of the chiral condensate
Hyperscaling relation in the conformal window
We can also deduce the hyper scaling relation by the RG flow equations
asThe linear term (contact term) is dominant for.
⇒ Therefore, it seems to be difficult to see the hyperscaling relation.

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Summary and discussions
We extended the RG flow equations for the gauge couplings so as toinclude the “non-perturbative” corrections through the effective
four-fermi operators.
We gave the non-perturbative gauge beta functions by scaletransformation on the RT, which shows merge of the UV and theIR fixed points. ⇒ manifestation of the “Conformality Lost” picture.
The anomalous dimension of the fermion mass and the critical flavor
number were evaluated for QCD by using the 3-, 4-loop beta functions.
Scaling of the dynamical scale was evaluated by using the beta functions.
The scaling relations of the fermion mass for the mass deformed QCD
were also examined near the boundary of the conformal window.

Future issues
Derivation of the RG flow equation for the gauge coupling
including the four-fermi couplings by the ERG formalism.
Evaluation of the chiral order parameters near the conformal boundary.
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Many Thanks

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