Dynamics in QCD like theories with infrared fixedpoint
Volodya Miransky
Western University

March 19, 2012

IR fixed point in two-loop β function of QCD like theories:
W. Caswell, PRL 33, 244 (1974);
D. Jones, Nucl. Phys. B75, 531 (1974);
A. Belavin and A. Migdal, Pisma Zh. Eksp. Teor. Fiz. [JETPLetters] 19, 317 (1974).

First discussion of consequences of existence of this IR fixed point:
T. Banks and A. Zaks, Nucl. Phys. B196, 189 (1982).

The two-loop solution:dα= −bα2 − cα3 .
d ln µIf b > 0 (Nf < Nf∗∗ ≡ 11Nc /2) and c < 0, the β function has azero corresponding to a infrared-stable fixed point atb
α = α∗ = − .
cFor Nc = 3, the condition b > 0 and c < 0 is valid for
Nf > Nf∗ = 8 and Nf < Nf∗∗ = 16.5.
α∗ ' 0.04, 0.14, 0.28, 0.47 for Nf = 16, 15, 14, and 13,
respectively (Nc = 3).

There exist two solutions!

Nonperturbative effects and the phase transition withrespect to Nf
T. Appelquist, J. Terning, and L. C. R. Wijweardhana, PRL 77,1214 (1996);
V. M. and K. Yamawaki, PRD 55, 5051 (1997)
The IR fixed point α∗ (Nf ) formally exists for all Nf > Nf∗ whenb(Nf ) > 0 and c(Nf ) < 0. However, whenα∗ (Nf ) > αcr '2Nc π,
Nc2 − 1 3

quarks get a dynamical mass and decouple:
the IR fixed point is a fake in this case.
Nf∗ < Nf < Nfcr : α∗ (Nf ) > αcr .
11Nc: α∗ (Nf ) < αcr .
2For Nc = 3, Nf∗ = 8 and Nfcr = 12.

Nfcr < Nf < Nf∗∗ =Phase transition in QCD at Nf = Nfcr2mdyn∼ Λ2cr exp − q ∗Cα (Nf )
−1αcr

αcr '

2Nc π,
Nc2 − 1 3

Nfcr ∼ 4Nc > Nf∗ .

Λcr : α(µ) µ=Λcr = αcr ,!
:The Phase Diagram in SU(Nc ) gauge model
V. M. and K. Yamawaki (1997)
Nf

**

NfS2
S1cr

NfA
*

Nf

A

2g(0)
crg(0)
Figure 3The phase√ diagram in an SU(Nc ) gauge model. The coupling constantg (0) = 4πα(0) and S and A denote symmetric and asymmetric phases,
respectively.
A. Deuzeman, M. Lombardo and E. Pallante, PRD 82, 074503 (2010)

Dynamics in the conformal window in QCD-like theories

The two loop β function leads to the following equation for therunning coupling: 
111α(q)(α(µ) − α∗ )q=−−log.
b logµα(q) α(µ) α∗α(µ)(α(q) − α∗ )The case α(µ) < α∗ (T. Appelquist, J. Terning andR. Wijewardhana (1996)):1α(q) ',qΛ(α(Λ) ' 0.8α∗ )b log(q/Λ)α∗α(q) ',qΛ
1 + e −1 (q/Λ)bα∗
Asymptotically free theory with no chiral symmetry breaking.

The case with α(µ) > α∗ (V. M. and K. Yamawaki (1997))

1q1α(q)= b log + ∗ log,α(q)α(q) − α∗Λ̃ αwhere Λ̃ is a Landau pole at which α(q) q=Λ̃ = ∞Does a meaningful continuum limit exist in this case?
The answer is “yes”.
Introduce UV cutoff M with a finite bare couplingα(0) ≡ α(q) q=M > α∗ .
Then, M < Λ̃ with α(Λ̃) = ∞. The Landau pole is unreachable inthe theory with cutoff M and α(0) < ∞.
As M → ∞ with a fixed α(0) = α(q) q=M ,α(q) 'α∗→ α∗
1 − e −1 (q/Λ̃)bα∗Nontrivial conformal field theory

Deformation of theory: Introducing bare fermion mass

The absence of a mass for fermions and gluons is a key point
for not creating bound states in S1 and S2 phases.
The situation changes dramatically if a bare fermion mass isintroduced: even weak gauge, Coulomb-like, interactions easily
produce bound states composed of massive constituents, as it
happens, for example, in QED (positronium like bound states)

The main consequences of the presence of the bare mass:a) Taking (α(0) − α∗ )  α∗ (walking regime), the pole fermion
mass m is expressed through m(0) as m(0) 1/(1+γm )m'MMb) The mass of n-body bound state composed of fermions is(0)M (n) ∼ nm ∼ nM mM

1/(1+γm ).

c) At momenta q < m, fermions and their bound states decouple.
There is a pure Yang-Mills theory with confinement. Its spectrum
contains light glueballs, Mgl  m, if α∗ is not too large: 1 11Mgl ∼ ΛYM ∼ m exp − ∗ ,b̄ =Nc .
6πb̄αFor Nc = 3, exp(−1/b̄α∗ ) is 6 × 10−7 , 2 × 10−2 , 10−1 , and3 × 10−1 for Nf = 16, 15, 14, and 13, respectively

The signature of the conformal window for latticecomputer simulations12

1/(1+γm )The universal scaling: M (n) ∼ m(0)
1/(1+γm )and Mgl  M (n)Light glueballs: Mgl ∼ m(0)Z. Fodor, K. Holland, J. Kuti, D. Noǵrádi, C. Schroeder,Phys. Lett. B 681, 353 (2009)
T. DeGrand and A. Hasenfratz , PRD 80, 034506 (2009)
L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago,PRD 80, 074507 (2009)
A. Deuzeman, M. Lombardo and E. Pallante, PRD 82, 074503 (2010)
T. Appelquist, G. Fleming, M. Lin, E. Neil and D. Schaich,
PRD 84, 054501 (2011)
Kenji Ogawa et al., arXiv:1111.1575 [hep-lat]
Y. Aoki, T. Aoyama, M. Kurachi, T. Maskawa, K. Nagai, H. Ohki, A. Shibata,K. Yamawaki, T. Yamazaki, arXiv:1201.4157 [hep-lat]; arXiv:1202.4916[hep-lat].

The structure of the β-function in QCD in the MS scheme

F. Chishtie, V. Elias, V. Miransky, and T. Steele,
Prog. Theor. Phys. 104, 603 (2000)
The questions:1

2

3

What is the Nfcr in the MS scheme?
What is the form of the β-function for Nf > Nfcr in the MSscheme?
What is the form of the β-function for Nf < Nfcr in the MSscheme?

Padé-improvement for β-function
J. Ellis, I. Jack, D. Jones, M. Karliner, M. Samuel,Phys. Rev. D 57, 2665 (1998)
A k-loop truncation:β (k) (x) = −β0 x 2 (1 + R1 x + · · · + Rk−1 x k−1 ),11 − 32 Nfα, β0 =.
π4Then, the [N/M] Padé-approximant (N + M = k − 1) is
1 + a1 x + · · · + aN x N
β [N/M] = −β0 x 2
1 + b1 x + · · · + bM x Mx=

The N + M coefficients a1 , . . . , aN , b1 , . . . , bM are determined by
the requirement that the first k terms in the Taylor expansion ofβ [N/M] replicate β (k) (x).

SU(Nc ) SUSY gluodynamics
I. Kogan and M. Shifman, PRL 75, 2085 (1995)

3Nc x 21αsβ(x) = −; x≡41 − Nc x/2π

There is an UV stable fixed point at x = ∞ in strong couplingphase. The denominator zero xd = N2c corresponds to an infrared
attractor.

Gluodynamics
Four-loop MS β-function:(4)(x) = −i
11 2 h2
3x 1 + 2.31818x + 8.11648x + 41.5383x ,
4β

[2|1]

(x) = −


11 2 1 − 2.7996x − 3.7475x 2x,
41 − 5.1178x

β

[1|2]

(x) = −

1 − 5.9672x
11 2x,
241 − 8.2854x + 11.091x

βThe denominator zero xd (infrared attractor) precedes thenumerator zero (xn ):xd = 0.195 < xn = 0.264 and xd = 0.151 < xn = 0.168 in [2|1]and [1|2] approximant, respectively. If taken seriously, xn zerocorresponds to UV fixed point.
F. Sannino and J. Schechter, Phys. Rev. D 82, 096008 (2010)
Rytov-Sannino β function for Gluodynamics

QCD with fermions

Below an approximant-dependent flavor threshold(6 6 Nfthr 6 8), the approximants [2|1] and [1|2] alwaysexhibit a positive pole prior to the occurence of their first
positive zero, precluding any identification of this zero as a IR
fixed point.
Above the threshold Nfthr , those approximants exhibit a IR
fixed point. Its value becomes less than xcr = 14 at Nf = 11.
This value decreases with increasing Nf and is close to the2-loop value for Nf > 12.

β [2|1]

forNf = 3

β [2|1]

UV fixed point in strong coupling gauge theories(recent discussions):
D. Kaplan, J. Lee, D. Son and M. Stephanov,
PRD 80, 125005 (2009)

forNf = 12

Conclusion

Dynamics in the conformal window is a new territoryand one can expect interesting surprises there.